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Child wasting is among the most prevalent forms of undernutrition globally. The World Health Organization (WHO) has defined wasting as a low weight-for-height, or thinness due to a severe process of weight loss, often associated with insufficient food intake (nutrient and energy density), and disease. There is growing evidence that a wasted child is more likely to become stunted, and a stunted child is more likely to become wasted, based on a report by the Emergency Nutrition Network (ENN), a United Kingdom (UK)-based registered charity.
The report also noted that the process underlying wasting and stunting involves multiple risk factors and interactions which can change over time. For example, involving poor diet and feeding practices, as well as episodes of infectious diseases and environmental contamination.
The United Nations (UN) estimates that around 47 million children under five were moderately or severely wasted in 2019, most living in sub-Saharan Africa and Asia.
Unfortunately, children who are wasted and stunted concurrently have a multiplicative increased mortality risk. Suffering from both at the same time amplifies the risk of death to levels comparable to children with the most severe form of wasting.
Southeast Asia is home to many wasted children. Nevertheless, it is not recognised as a public health problem and its epidemiology is yet to be fully examined.
A report titled, ‘The Forgotten Agenda of Wasting in Southeast Asia: Burden, Determinants and Overlap with Stunting: A Review of Nationally Representative Cross-Sectional Demographic and Health Surveys in Six Countries’ published in the Multidisciplinary Digital Publishing Institute (MDPI) -revealed that Cambodia, Lao PDR, Timor-Leste, Myanmar, Thailand and Vietnam have a high number of child wasting cases. The study concluded that a pooled figure of over one million under-five children are affected by wasting, with close to 280,000 of them being severely wasted.
Stunting is also a serious public health problem in the region, with most countries having a stunting prevalence of above 30 percent. Nevertheless, the report notes that despite the coexistence of wasting and stunting in Southeast Asia, less attention has been paid to the concurrence of both.
Despite being one of the wealthier countries in ASEAN, Malaysia is reported to have childhood stunting rates worse than Palestine and some African countries.
According to 2018 World Bank data, Malaysia’s rate of stunting among children under five was 20.7 percent, higher than in Ghana (18.8 percent) and much higher than in Gaza and the West Bank (7.4 percent).
The UN warned this week that nearly seven million more children will experience stunting as a result of malnutrition due to the unprecedented social and economic crisis caused by the COVID-19 pandemic.
“It’s been seven months since the first COVID-19 cases were reported and it is increasingly clear that the repercussions of the pandemic are causing more harm to children than the disease itself,” said Henrietta Fore, Executive Director of the United Nations Children's Fund (UNICEF).
“Household poverty and food insecurity rates have increased. Essential nutrition services and supply chains have been disrupted. Food prices have soared. As a result, the quality of children’s diets has gone down and malnutrition rates will go up,” she continued.
Local media in Indonesia reported that an overloaded healthcare system, job losses and limited access to food supplies amid the pandemic could exacerbate the already poor living conditions of children deemed most susceptible to stunting and wasting.
In the archipelago alone, more than two million Indonesian children have severe wasting, while more than seven million others under five years old have experienced stunted growth.
The UNICEF predicted that globally, the number of malnourished children under the age of five will perhaps increase by about 15 percent in 2020.
It was reported that US$2.4 billion is needed by humanitarian agencies to protect maternal and child nutrition in the most vulnerable countries till the end of the year. The UN recently released a statement and urged governments, the public, donors and the private sector to protect children’s right to nutrition.
This can be done by re-activating and scaling up services for the early detection and treatment of child wasting, maintaining the provision of nutritious and safe school meals and expanding social protection to safeguard access to nutritious diets.
Stunted, Wasting And Overweight In ASEAN
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1. Laplace transform
Using Laplace Transform I was asked to solved this eq.
$\displaystyle \frac{{d^2 y}} {{dx^2 }} + 4\frac{{dy}} {{dx}} + 13y = 145\cos 2t$ $\displaystyle {\text{ }};y(0) = 9$ and $\displaystyle \left. {\frac{{dy}} {{dx}}} \right|_{t = 0} = 19$
I've try it but I got Y(s) that I couldn't find its inverse ... help
thank you
2. Originally Posted by Singular
Using Laplace Transform I was asked to solved this eq.
$\displaystyle \frac{{d^2 y}} {{dx^2 }} + 4\frac{{dy}} {{dx}} + 13y = 145\cos 2t$ $\displaystyle {\text{ }};y(0) = 9$ and $\displaystyle \left. {\frac{{dy}} {{dx}}} \right|_{t = 0} = 19$
I've try it but I got Y(s) that I couldn't find its inverse ... help
thank you
Can you show your work - or at least your answer for Y(s) - to save re-inventing the wheel .....
3. OK wait .... I'll find it first....I forgot where I put it ....
4. Here is my attempt to solve it ...
$\displaystyle \displaylines{ L(y^{''} ) + 4[L(y')] + 13[L(y)] = 145\cos 2t \cr s^2 Y - sy(0) - y'(0) + 4sY - 4y(0) + 13Y = 145\cos 2t \cr s^2 Y - 9s - 19 - 4sY - 36 + 13Y = 145\cos 2t \cr (s^2 - 4s + 13)Y - 9s - 55 = 145\cos 2t \cr Y = \frac{{145\cos 2t + 9s + 55}}{{s^2 - 4s + 13}} \cr}$
After taht I stuck , I don't know what I should do then ...
Anyone can HElp ThankS
5. Originally Posted by Singular
Here is my attempt to solve it ...
$\displaystyle \displaylines{ L(y^{''} ) + 4[L(y')] + 13[L(y)] = 145\cos 2t \cr s^2 Y - sy(0) - y'(0) + 4sY - 4y(0) + 13Y = 145\cos 2t \cr s^2 Y - 9s - 19 - 4sY - 36 + 13Y = 145\cos 2t \cr (s^2 - 4s + 13)Y - 9s - 55 = 145\cos 2t \cr Y = \frac{{145\cos 2t + 9s + 55}}{{s^2 - 4s + 13}} \cr}$
After taht I stuck , I don't know what I should do then ...
Anyone can HElp ThankS
The laplace transform of $\displaystyle L(145 \cos(2t))=145 \frac{s}{s^2+4}$
You need to transform the right hand side
6. I ended up with
$\displaystyle s^2Y-9s-19+4[sY-9]+13Y=145 \frac{s}{s^2+4}$
$\displaystyle [s^2+4s+13]Y=9s+55+145\frac{s}{s^2+4}$
$\displaystyle [(s+2)^2+9]Y=9(s+2)+37+145\frac{s}{s^2+4}$
$\displaystyle Y=9\frac{s+2}{[(s+2)^2+9]} +\frac{37}{[(s+2)^2+9]}+145\frac{s}{(s^2+4)[(s+2)^2+9]}$
by partial fractions of the last one we get
$\displaystyle Y=9\frac{s+2}{[(s+2)^2+9]} +\frac{37}{[(s+2)^2+9]}+145\left[\frac{1}{145}\frac{9s+16}{s^2+4}-\frac{1}{145}\frac{9s+52}{(s+2)^2+9} \right]$
Combinging like terms we get...
$\displaystyle Y=\frac{3}{[(s+2)^2+9]}+9\frac{s}{s^2+4}+8\frac{2}{s^2+4}$
Taking the inverse transfrom we get...
$\displaystyle y(t)=e^{-2t}\sin(3t)+9\cos(2t)+8\sin(2t)$
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kmmiles.com
# 9077 miles in km
## Result
9077 miles equals 14604.893 km
You can also convert 9077 km to miles.
## Conversion formula
Multiply the amount of miles by the conversion factor to get the result in km:
9077 mi × 1.609 = 14604.893 km
## How to convert 9077 miles to km?
The conversion factor from miles to km is 1.609, which means that 1 miles is equal to 1.609 km:
1 mi = 1.609 km
To convert 9077 miles into km we have to multiply 9077 by the conversion factor in order to get the amount from miles to km. We can also form a proportion to calculate the result:
1 mi → 1.609 km
9077 mi → L(km)
Solve the above proportion to obtain the length L in km:
L(km) = 9077 mi × 1.609 km
L(km) = 14604.893 km
The final result is:
9077 mi → 14604.893 km
We conclude that 9077 miles is equivalent to 14604.893 km:
9077 miles = 14604.893 km
## Result approximation
For practical purposes we can round our final result to an approximate numerical value. In this case nine thousand seventy-seven miles is approximately fourteen thousand six hundred four point eight nine three km:
9077 miles ≅ 14604.893 km
## Conversion table
For quick reference purposes, below is the miles to kilometers conversion table:
miles (mi) kilometers (km)
9078 miles 14606.502 km
9079 miles 14608.111 km
9080 miles 14609.72 km
9081 miles 14611.329 km
9082 miles 14612.938 km
9083 miles 14614.547 km
9084 miles 14616.156 km
9085 miles 14617.765 km
9086 miles 14619.374 km
9087 miles 14620.983 km
## Units definitions
The units involved in this conversion are miles and kilometers. This is how they are defined:
### Miles
A mile is a most popular measurement unit of length, equal to most commonly 5,280 feet (1,760 yards, or about 1,609 meters). The mile of 5,280 feet is called land mile or the statute mile to distinguish it from the nautical mile (1,852 meters, about 6,076.1 feet). Use of the mile as a unit of measurement is now largely confined to the United Kingdom, the United States, and Canada.
### Kilometers
The kilometer (symbol: km) is a unit of length in the metric system, equal to 1000m (also written as 1E+3m). It is commonly used officially for expressing distances between geographical places on land in most of the world.
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More than half the world’s chameleons live in Madagascar. Chameleons of the Rain Forest contains many interesting facts about these creatures and how they live, as well as information about other animals that live on the island. Features of the Book – Captioned photographs, – Labelled diagram, – Life cycle, – Measurements, – Specialised vocabulary – absorb, endangered, extinct, mammal, predator, prey, reptile, scales
Purpose Chameleons of the Rain Forest can be used to introduce and reinforce the following standards-related skills: – asking and answering questions about the text, – extending vocabulary by using a glossary, – using visual, structural, and meaning cues to read unknown words, – exploring rhyming words, – interpreting information in diagrams, charts, and graphs, – showing an awareness of the topic and audience.
Themes covered in this book are: Animals Birds and Insects, Environment, Habitats
Reading Level:21-22 (Gold)
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+0
# Algebra
+1
2
1
+246
Evaluate \$a^3 - \dfrac{1}{a^3}\$ if \$a - \dfrac{1}{a} = 0\$.
Jul 11, 2024
#1
+1639
+1
First, let's zoom in and focus on the equation we are given. Let's square it. We get
\((a^2 - 2 + 1/a^2) = 0\)
This will achieve the equation
\(a^2 - 1/a^2 = -2\)
Next, let's focus on what we are trying to find. We know that
\(a^3 - 1/a^3 = ( a - 1/a) ( a^2 + 1 + 1/a^2)\)
WAIT! The first term in what we must find is 0!
That means our answer is just 0.
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Review question
# Where does this chord of an ellipse cut the $x$-axis? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
Ref: R6843
## Solution
Prove that the equation of the chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ joining the points $(a\cos\alpha,b\sin\alpha)$ and $(a\cos\beta,b\sin\beta)$ is $\frac{x}{a}\cos\left(\frac{\alpha+\beta}{2}\right)+\frac{y}{b}\sin\left(\frac{\alpha+\beta}{2}\right)= \cos\left(\frac{\alpha-\beta}{2}\right).$
The following standard identities will be useful to us:
$\sin x- \sin y = \sin \dfrac{x-y}{2}\cos \dfrac{x+y}{2}, \cos x -\cos y = -2 \sin \dfrac{x+y}{2} \sin \dfrac{x-y}{2}.$
The gradient of the chord $HK$ is $\frac{b\sin\alpha-b\sin\beta}{a\cos\alpha-a\cos\beta}$ $=-\dfrac{b}{a} \frac{\sin \dfrac{x-y}{2}\cos \dfrac{x+y}{2}}{\sin \dfrac{x+y}{2} \sin \dfrac{x-y}{2}}$ $=-\dfrac{b}{a} \frac{\cos \dfrac{x+y}{2}}{\sin \dfrac{x+y}{2}}.$
Therefore the equation of the line is $y-b\sin\alpha=-\dfrac{b}{a} \frac{\cos \frac{\alpha+\beta}{2}}{\sin \frac{\alpha+\beta}{2}}(x-a\cos\alpha)$ or $xb\cos\left(\frac{\alpha+\beta}{2}\right)+ay\sin\left(\frac{\alpha+\beta}{2}\right)= ab \sin \frac{\alpha + \beta}{2}\sin \alpha + ab \cos \frac{\alpha + \beta}{2}\cos \alpha,$ which simplifies to $\frac{x}{a}\cos\left(\frac{\alpha+\beta}{2}\right)+\frac{y}{b}\sin\left(\frac{\alpha+\beta}{2}\right)= \cos\left(\frac{\alpha-\beta}{2}\right).$
as required.
We can find the equation of the tangent through $H$ by allowing $\beta$ to tend to $\alpha$, giving us $bx\cos\alpha+a\sin\alpha-ab=0.$
Through a point $P$ on the major axis of an ellipse a chord $HK$ is drawn. Prove that the tangents at $H$ and $K$ meet the line through $P$ at right angles to the major axis at points equidistant from $P$.
We can find the point $P(x_p,0)$ by substituting $y=0$ into the equation of the chord found above: $x_p=a\frac{\cos\left(\frac{\alpha-\beta}{2}\right)}{\cos\left(\frac{\alpha+\beta}{2}\right)}.$
From our work above, the tangent through $H$ is $bx\cos\alpha+a\sin\alpha-ab=0,$ while the equation of the tangent though $K$ is $bx\cos\beta+a\sin\beta-ab=0.$
We are looking to prove that $PH’ =PK’$, where $H'$ and $K'$ are the intersections of the tangents with the line through $P$.
Let $H'$ be $(x_p,y_h)$ and $K'$ be $(x_p,y_k)$. The signs of $y_h$ and $y_k$ will always be opposite, since the chord $HK$ crosses the $x$-axis.
Consider the equation $(bx\cos\alpha+ay\sin\alpha-ab)(bx\cos\beta+ay\sin\beta-ab)=0.$
This must represent the equation of the pair of straight lines given by the tangent at $H$ and the tangent at $K$.
Where does the line $x = x_p$ cut these lines? At $H'$ and $K'$. So let’s substitute $x_p$ for $x$ in this equation, giving
$(bx_p\cos\alpha+ay\sin\alpha-ab)(bx_p\cos\beta+ay\sin\beta-ab)=0.$
This is a quadratic in $y$, say $Ay^2+By+C=0$. Its roots must be $y_h$ and $y_k$, and the sum of these roots is $-\dfrac{B}{A}$. Thus
$y_h+y_k = -\dfrac{a\sin\beta(bx_p\cos\alpha-ab)+a\sin\alpha(bx_p\cos\beta-ab)}{a^2\sin\alpha\sin\beta}$ $=-b\dfrac{x_p\sin(\alpha+\beta)-a(\sin\alpha+\sin\beta)}{a\sin\alpha\sin\beta}$ $=-b\dfrac{\dfrac{\cos\dfrac{\alpha-\beta}{2}}{\cos\dfrac{\alpha + \beta}{2}}2\sin\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha+\beta}{2}-(\sin\alpha+\sin\beta)}{\sin\alpha\sin\beta}$ $=-b\dfrac{2\sin\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha-\beta}{2}-(\sin\alpha+\sin\beta)}{\sin\alpha\sin\beta}$
The following standard identity is now useful: $2\sin x\cos y = \sin (x+y)+\sin(x-y) \implies 2\sin\dfrac{\alpha+\beta}{2}\cos\dfrac{\alpha-\beta}{2} = \sin\alpha+\sin\beta.$
And so we have that $y_h+y_k =0$, and the lengths $PH'$ and $PK'$ are equal, as required.
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# Combination
A Combination a set that is the result of drawing a subset from some other set.
## References
### 2018
• (Encyclopaedia Britannica Inc., 2018) ⇒ The Editors of Encyclopaedia Britannica (2018). “Permutations and combinations": https://www.britannica.com/science/permutation. Date Published: July 03, 2018. Retrieved: 08-03-018.
• QUOTE: The concepts of and differences between permutations and combinations can be illustrated by examination of all the different ways in which a pair of objects can be selected from five distinguishable objects — such as the letters A, B, C, D, and E. If both the letters selected and the order of selection are considered, then the following 20 outcomes are possible:
$\begin{matrix} AC & BA & AC & CA & AD\\ DA & AE & EA & BC & CB \\ BD & DB & BE & EB & CD \\ DC & CE & BC & DE & ED \end{matrix}$
Each of these 20 different possible selections is called a permutation. In particular, they are called the permutations of five objects taken two at a time, and the number of such permutations possible is denoted by the symbol 5P2, read “5 permute 2.” In general, if there are n objects available from which to select, and permutations (P) are to be formed using k of the objects at a time, the number of different permutations possible is denoted by the symbol nPk.
$_nP_k=\frac{n!}{(n-k)!}$
The expression n! — read “n factorial” — indicates that all the consecutive positive integers from 1 up to and including n are to be multiplied together, and 0! is defined to equal 1.(...)
For combinations, k objects are selected from a set of n objects to produce subsets without ordering. Contrasting the previous permutation example with the corresponding combination, the AB and BA subsets are no longer distinct selections; by eliminating such cases there remain only 10 different possible subsets — AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE.
The number of such subsets is denoted by nCk, read “n choose k.” For combinations, since k objects have k! arrangements, there are k! indistinguishable permutations for each choice of k objects; hence dividing the permutation formula by k! yields the following combination formula:
$_nC_k=\frac{n!}{k!(n-k)!}$
This is the same as the (n, k) binomial coefficient (see binomial theorem).
### 2015
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/combination Retrieved:2015-2-9.
• In mathematics, a combination is a way of selecting members from a grouping, such that (unlike permutations) the order of selection does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.
More formally, a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient :$\binom nk = \frac{n(n-1)\ldots(n-k+1)}{k(k-1)\dots1},$
which can be written using factorials as $\frac{n!}{k!(n-k)!}$ whenever $k\leq n$, and which is zero when $k\gt n$.
The set of all k-combinations of a set S is sometimes denoted by $\binom Sk\,$.
Combinations refer to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection, [1] k-multiset, or k-combination with repetition are often used. [2] If, in the above example, it was possible to have two of any one kind of fruit there would be 3 more 2-selections: one with two apples, one with two oranges, and one with two pears.
Although the set of three fruits was small enough to write a complete list of combinations, with large sets this becomes impractical. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.
1. also referred to as an unordered selection.
2. When the term combination is used to refer to either situation (as in ) care must be taken to clarify whether sets or multisets are being discussed.
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Click image to view full screen
The muscles of the hand which attach to and move the fingers and thumb, are defined as follows:
- extrinsic muscles, which arise in the forearm and have long tendons that cross the wrist, and insert into the phalanges.
- intrinsic muscles, which arise within the hand.
The intrinsic muscles of the hand are located in three compartments as follows:
- the thenar compartment, called the thenar eminence, which contains the thumb or thenar muscles.
- the hypothenar compartment, called the hypothenar eminence, which contains the little finger or hypothenar muscles.
- the central compartment, which contains the lumbricals and interosseous muscles.
As the muscles of the hand are studied, the course and distribution of the nerves and blood vessels will be identified.
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Homepage
# Here's How Your Watch Can Prove That 2 + 2 Doesn't Equal 4
2 + 2 = ? appears to be one of the easiest problems in mathematics, and it is probably one of the first you ever encountered. If Kate has 2 apples and Matt gives her 2 more apples, then she has 4 apples. Obviously.
But what if we told you that 2 + 2 = ? has stumped even some of the smartest mathematicians because it doesn’t necessarily have to equal 4?
You’re probably wondering how that’s possible.
## But First, Here's An Example
I get to work at 7 o’clock in the morning. This is what my watch looks like. It's first time that the smaller hand of the clock hits 7 on the clock’s face that day.
Later in the day, I leave work at 5 o’clock in the afternoon. When I look down at my wrist, this is what my watch looks like. This is the second time that the smaller hand of the clock hits 5 on the watch’s face. The first time was at 5 AM.
In other words, the smaller hand of my watch has hit all 12 numbers on the face, and then started again from 1. We can think of 1PM as 13 o’clock; 2 PM as 14 o’clock; and 5 PM as 17 o’clock.
However, most people don’t say: “I’ll be done with work at 17.” They generally say: “I’ll be done at 5.” If you do this as well, you’re actually solving a complicated math problem without even realizing.
## The Watch Operates In A Specific System
What’s happening is that the watch’s hands operate in a system (the watch’s face) that has 12 numbers, but the watch’s hands are attempting to represent a system, which has more than 12 numbers (in this case, the system is a day which has 24 hours).
We’re going to class the watch’s system “Modulo 12”, meaning that 12 is the highest number we can have on the watch which has the numbers 1, 2 , 3, 4 … all the way through 12. (Don’t freak out! Modulo is just the fancy math term for the math we are doing).
As a result, to understand how the 17th hour in the day is represented on the watch, one must do 17 (the number outside of Modulo 12) minus 12 (the maximum number in Modulo 12) which equals 5 (which is a number within Modulo 12 to represent a number outside of Modulo 12).
In other words, in the watch system, we can say that 12 + 5 = 5 because 5 represents 17.
Weirder still, even though you’d think that 13 +4 = 17, in this Modulo 12 system, 14 +4 = 6 because 6 pm represents “18 o’clock.”
## Now, Back To 2+2
Using what we learned here, let’s get back to 2 + 2 = ?.
Believe it or not, you can actually create a Modulo system with any numbers. It does not have to be limited to Modulo 12 like with the clock with the numbers 1 through 12.
Now, our new system is going to be Modulo 3 with the numbers 0, 1, 2. This is a little different from the watch, because a watch doesn’t have 0’s. Let’s quickly refresh what that means.
Modulo 3 with numbers 0, 1, 2 means that after we reach the third number in our set of numbers, we start counting from the first number again. In this case, after we reach 2, we start again with 0. This is just like with the watch, when after we reached 12, we started again with 1.
So now, let’s see what happens when we add 2 + 2 in a Modulo 3, (0,1,2) system.
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As far as the original works of classical antiquity are concerned, practically nothing has survived of the once large body of Greek panel and mural painting. The situation in regard to Roman painting is different primarily because of a single, cataclysmic act, the eruption of Mt. Vesuvius on 24 August, 79 A.D. Examples of Roman wall painting were preserved in the burial of the ancient cities, which have subsequently been excavated.
The ancient city of Pompeii was not a small town by ancient standards, covering some 160 acres. It was situated on a small volcanic hill about 5-1/4 miles southeast of Vesuvius and was founded as an agricultural village perhaps as early as the 8th century B.C. Although Greek and Etruscan influences are evident in the early habitation layers, Pompeii was a Latin city from the late 5th century B.C.
After the Social War (91-88 B.C.), the city of Pompeii began to be influenced by Rome because it received Roman enfranchisement and also a colony of Roman army veterans. In particular, a strong Roman influence appeared in the architecture and wall painting.
At the time of its destruction, Pompeii was a prosperous town. It was a market for the produce of the rich Campanian country-side, a port with wide Mediterranean connections, and even an industrial center providing certain specialty products like wines, millstones, fish sauce, and perfumes, which supplied more than a local demand. With the eruption of 79 A.D., all of this ceased to exist, buried under 12 feet of lethal ash.
The survivors drifted away or were settled elsewhere, and what had been Pompeii became once again rich agricultural land. The knowledge that there had once been a town there lingered on in folk memory: in the 18th century, the area was still known as Civita (civitas, or "city"). But as far as the learned world was concerned, Pompeii had been wiped off the map. Since its rediscovery in 1748, excavations of varying intensity have continued and nearly four-fifths of the city has been uncovered.
A typical Pompeian house, such as the MIA fresco might have decorated, was essentially an inward-facing building, enclosed by bare walls and lit almost exclusively from within. Such a house stood flush with the sidewalk and its neighbors, with no open space between, in front, or in back; it embraced its own open space.
Through a narrow door, one entered a vestibule that led into the atrium. This was a large centrally lit hall, the roof of which sloped downward and inward toward a rectangular opening situated above a rectangular basin. Thus, it both admitted more light and helped to replenish the cisterns that were the house's principal water supply. Beyond the atrium was the peristyle, a rectangular, cloister-like, colonnaded courtyard which usually contained fountains and a garden. Around these two courts—one closed, one open—were grouped the other rooms of the house: the reception rooms, the dining room, and a series of small bed chambers and workrooms. Behind this complex a portico often led to a vegetable garden or orchard.
At every level of society, then, religion, was a matter of observance, not doctrine. By Cicero's time, the public face of religion was entirely in the hands of colleges of priests, prominent citizens who were elected or appointed to perform the proper ceremonials and rituals on behalf of the Unlike the Etruscans or the Greeks, the Romans placed less value on religious feelings, in terms of the mystic need to love and worship superhuman powers. Instead, traditional Roman religion was concerned with success: as Cicero remarks, "Jupiter is called the Best and Greatest not because he makes us just or sober or wise, but because he makes us healthy, rich and prosperous." As a result of this emphasis, the hierarchy of divinities worshipped in Rome differed from the assembly of easily recognized, individualized personages who made up the Greek pantheon. The Roman pantheon was something more abstract and utilitarian: it was an actual catalogue in which those who were interested could find the name of protective powers with special functions attributed to them and the rites which needed to be performed in order to purchase their favors. The Romans were a practical people, and they formed a religion which corresponded to their needs. Publicly, religion was kept up as a matter of state policy, and temples and statues were erected to the many gods worshipped throughout the Empire.
Privately a person might do as he or she chose, but most Romans were superstitious enough to choose something. It was important for them to feel sheltered from the perils which threatened either the group or the individual community they represented. Domestically the father of the family fulfilled the same office on behalf of the household under his care, offering daily prayers and gifts at the lararium or household shrine, within which were displayed the figures of the traditional household gods and of such other divinities as the family might hold in special honor. Here, too, were performed the rituals associated with important family events, such as a boy's coming of age. These simple rituals were a part of daily life that no prudent Roman would have willingly neglected, good and evil fortune being considered active forces that had to be no less actively fostered or diverted. Few, if any, classical sites can equal Pompeii for the light they throw on religion at this popular level. The household shrines that are such a prominent feature of the houses (over 500 were recorded in a 1937 archaeological survey) represent religion at its simplest level, where it operated as a part of everyday life.
Chief among the important household deities was Vesta, a virgin who was the goddess of fire and a symbol of idealized maternity because fire nourishes. Next came the genius of the head of the household, the presiding creative force believed to have engendered that individual. The Penates and Lares were other deities, protectors of the family's prosperity, preservers of food and drink. For the worship of these spiritual beings, shrines and altars were set up in various parts of the dwelling with votive statuettes and/or paintings representing them.
CULT OF THE LARES
The cult of the Lares became universal throughout the Roman world. Rather amorphous characters sexually, they are always represented as youthful deities but vary in characteristics that may be described as masculine or feminine. The earlier representations of the Lares had masculine characteristics. Later artistic depictions modified these aspects or even turned toward the feminine. By the 1st century A.D., the date of the Minneapolis fresco, hermaphroditic qualities were present in the artistic depictions of the Lares. (Note the suggestion of breasts on this figure.)
Though worshipped as protectors of the house in Roman times, the Lares were originally Etruscan divinities of locality.1 Within the Roman pantheon, like most Roman deities, they have no proper mythology. The stories outlining their specific divine origins are quite late and conflicting. There are two principal theories as to the origin of the Lares. One suggests that they were the ghosts of the dead. Whenever a bit of food fell on the floor during a meal, it was burnt before the Lares. Since the floor was a notorious haunt of ghosts and the food had gone to the ghosts' region, it was formally given to the ghosts. The Lares were also propitiated at the Compitalia, or festival of the crossroads, and ghosts supposedly had a fondness for crossroads as well.
The second theory emphasizes that the Roman dead, while often commemorated by busts and other statues, were not formally honored in the home but at their graves. In this view, the connection of the Lares to the Compitalia also takes on another aspect. A compitum, or crossroads, was originally a place where the paths separating four farms met. The Lares were then celebrated as guardians of the farmlands at these places, and their rites were, thus, in the nature of a purification, not a sacrifice. Eventually, their worship expanded from the farms and came into the houses, where they joined the circle of Vesta, the Penates, and various genii as protective deities.
FRESCO (Wall Painting)
The wall paintings found in the excavated ancient cities around Mt. Vesuvius are the most important documents for our knowledge of Roman painting because few frescoes survive in Rome itself, and ancient panel paintings have completely disappeared. This scarcity of material makes it extremely difficult to reconstruct the history of Roman painting. Few literary works have survived which comment on the techniques and styles of ancient wall painting, and the remarks of Roman writers like Vitruvius and the Elder Pliny describe art of a higher quality than that reflected in the more popular representations of the artistic examples from Pompeii. Therefore, although the paintings from the ancient cities around Mt. Vesuvius have been taken as typical of the general production at that time, we must remember that these are provincial examples, probably of lesser quality than those found in Rome itself.
The two major literary sources for information on ancient painting, Vitruvius and the Elder Pliny, also provide the best evidence on plaster and wall preparation. The plaster used by the Romans for architectural work was based on lime because it set slowly to produce a hard durable surface suitable for painting if so desired. (Lime plaster is also referred to as stucco.) Since lime itself does not occur naturally, it has to be obtained by the calcination of one of the calcium carbonates (calcite and its more massive varieties limestone, marble, and chalk). When any of these compounds is heated, carbon dioxide is released, and the material is converted into quicklime (calcium oxide). When it is broken up and receives sufficient water, this material gives off considerable heat and then crumbles into a white powder (slaked lime). The addition of more water and a type of grit to this powder produces the mortar and plaster used in wall construction. The grit added may take several forms depending on geographical location and the availability of materials: fine river sand, marble and alabaster dust, pozzolana or volcanic sand (especially around Pompeii). As the mixture dries, the material takes carbon dioxide from the air and reverts back to what it was at the beginning of the cycle: calcium carbonate.
The ancient authors do not agree in regard to the proportions of stucco ingredients—probably each craftsman had his own favored proportions of sand and lime for the variety of plasters necessary in construction. Here, too, there is a variance in practice: Vitruvius recommends at least three coats each of sand mortar and powdered marble stucco, as does the Elder Pliny. In reality, this technique occurs in only very wealthy residences, and at Pompeii the normal procedure apparently required only two or three layers, the lower one or two comprised of lime and sand with a surface layer of lime and calcite.
Regardless of the number of layers used, the surface was prepared for painting after the plaster application. Paintings of poorer quality or lesser expense were applied to the rough plaster surface. However, more effort and expense were dedicated to higher quality paintings before their application. The fine plaster surface layer was smoothed and polished with a stone burnisher or marble roller. The colors were then painted on to the plaster surface while it was still damp. This method is known as true fresco since it uses water as the vehicle for the pigments. Such paintings are quite durable because the pigments become bound to the plaster itself as it dries and the calcium carbonate crystals form. Background color was always applied to wet plaster, but detail could be added to the wall when it was either wet or dry. Some paintings (although not the MIA fresco) were polished again after paint application, and this accounts for some fuzziness in the lines and the absence of brushmarks. The speed of the painters was once thought to be vital to the successful fresco decoration. It is now believed that wall plaster dries more slowly than previously imagined, especially if it consists of numerous layers. When plaster was, indeed, too dry for the true fresco technique, it could be removed from the wall and replaced with fresh plaster, thus, allowing the painter to finish his work. The tempera technique enabled the artist to add details to a dried plaster background. Mixing a pigment with egg or honey for application of wet plaster was also done for colors like carbon black whose greasy nature made its use difficult with water alone.
The pigments used in Roman wall painting were obtained from mineral, vegetable, and animal sources. Colors such as the ochres were easily obtainable from widespread sedimentary rock deposits, but others could only be isolated from rarer heavy metals and were, therefore, expensive to use in painting. Substitutes were found or chemically made to overcome the scarcity and expense of these mineral pigments.
The same was the case for the expensive vegetable pigments, like indigo, and certain animal pigments, like Tyrian purple, a dye obtained from a species of sea mollusk.
The standing deity illustrated on this fresco fragment is a Lar, an ancient Roman household deity. The cult of the Lares was widespread throughout the Roman Empire, and nearly every household contained at least one lararium
, a shrine consisting of a mural painting and/or small bronze sculptures representing these deities.
The identification of this standing figure can be made with confidence based on the evidence of the pastoral scene in addition to the subject's pose, dress, and accessories. The deity holds a horn, a rhyton, in its upraised hand and pours liquid into a bucket, a situla, in its lowered hand. The knee-length tunic, swirling pallium, and tall fringed leather boots, in this case, with open toes, also help to identify our figure as a Lar.
Although it can stand on its own as an individual work of art, the Lar was originally part of a larger wall mural. This is evident by the fragmentary nature of the leafy green bough at the top of the fresco, which was more extensively draped over the longer scene, and the hints of vegetation on the lateral parts of the brown ground line still visible in pale brown and red.
Another figure of a Lar very much like the MIA example remains in situ in a kitchen in Pompeii. Its pose is a mirror image of the MIA figure and the major colors of the costume are reversed. This suggests that the fragment in our collection may have been part of that same mural, and therefore located in a kitchen.
The painting is on white ground, and the background has not been painted, except for the green festoon above and the brown earth below the standing figure. The broad watercolor brushstrokes in the festoon and the soil are well attested to in other Pompeian wall murals. The ancient artist even added some darker strokes to the brown soil to indicate the shadow cast by the standing figure. The flesh tones on the painting are even in color but subtly lightened in spots (knees, forearms, and shoulders) to imitate the sun's reflection. The reddish tunic with blue trim clings to the figure's breasts, waist, and part of the thighs, while part flaps in the air, appropriate for thin material in a stiff breeze. These folds and undulations are smoothly distinguished by careful shading and highlights. The green pallium swirls behind the deity and loops around the lowered left hand. This drapery swirl, which almost produces a halo effect, is well-documented in other Pompeian wall paintings. It provides an excellent contrast to the serene facial expression and stylized pose. The figure's face consists of fine features: a small mouth, straight nose, and large eyes. This face is framed by tight, wavy curls and surmounted by the delicate leaves of a woven wreath. The quality of the painting is quite high overall.
The red tunic of the figure on the Minneapolis fresco could have been produced by a pigment made of either cinnabar or hematite. The blue trim of the tunic was probably created by a pigment artificially prepared from copper, silica, and calcium. Blue could also be obtained from ultramarine or azurite. Indigo was also a more expensive possibility, but woad, a plant of the mustard family, was more readily available as a substitute. The green of the figure's pallium, as well as the leaves of the wreath and festoon, was commonly obtained from a pigment known as terre verte, made up of the two main minerals glanconite and celadonite. Malachite and verdigris were more rarely used as green pigments due to their comparative expense. The various shades of brown on the Minneapolis fresco, from the figure's boots, hair, and eyes down to the soil it stands on, were probably derived from sinopis, a red-brown ochre, and brown umber. Lighter flesh tones were often produced by the yellow pigments, such as yellow ochre, a natural pigment composed of clay and silica. These colors could be altered by adding various amounts of iron oxide. The black and gray colors on all paintings, visible in the horn and bucket here, were most commonly produced by a pigment made from carbon. White, visible in the eyes of the figure, usually came from lime white, a pigment prepared by the grinding and slaking of calcined marble or oyster shells.
The painting is dated after 50 A.D. and probably closer to 70 A.D. on the basis of its hermaphroditic character, the robust style of the painting, and the vibrant choice of colors.
(The above information is taken largely from Michael Anderson's article in The Minneapolis Institute of Arts Bulletin, LXIV: 94-103.)
Use on the following tours:
- Ancient Art of the Mediterranean
- How Was It Made? (as a method of painting)
Compare to the paintings of our Greek vases or the Egyptian Book of the Dead
for different effects of color and movement.
Discuss this protective figure in the context of examples from other cultures:
- a Christian saint
- the Kota reliquary figure from Gabon
- a Japanese Guardian, such as that by Joga or the Nio Guardian Figures
Use to discuss one aspect of Roman religion together with:
- the Cinerary Urn
- the Roman Portrait of an Older Woman
- John Ward-Perkins and Amanda Claridge, Pompeii AD 79: Treasures from the National Archaeological Museum, Naples, Vol. II (Boston: Museum of Fine Arts, 1978), p. 190.
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# Ratio Worksheets
At Cazoom Maths we offer a huge range of worksheets to help children understand ratios. Ratios are used in many aspects of our everyday lives so our question sheets reflect that, allowing children to practice their knowledge with problems involving exchange rates and worded questions. We have ratio worksheets suitable for all levels and abilities, from simple ratios up to more complex ratio problems. Our worksheets contain engaging activities which make the topic of ratios entertaining and enjoyable, and our ratios worksheets are all supplied with answers to check how well your child or pupil is doing.
Our resources on ratios are available to download and print, and can either be used in the classroom or to support children’s learning at home.
We have worksheets available to cover all aspects of ratios in the KS3 and KS4 curriculum. Our resources present all the information they need to know in a concise, easy to understand format and provide plenty of opportunities to apply their knowledge. Using Cazoom Maths will help boost students’ confidence at using ratios and prepare them for any ratio problem they might come across.
### Explanation of Ratios
Ratios are used in many aspects of our everyday lives.
They tell us that two variables are proportional to each other, and are used in cooking, gambling, currency exchange and more. For this reason, it is important to make sure students have a thorough understanding of this topic by using worksheets on ratios.
### The Everyday Uses of Ratios
It is highly likely that you will be using and calculating ratios in your everyday life without even realising it.
When following a recipe, the ingredients will all be proportional to each other. So if you are baking a cake the recipe advises using 100g of butter and 300g of flour, the ratio between those two ingredients is 1:3. Knowing this ratio, you can easily scale the amounts of ingredients up or down to adapt to the quantity of cake required, ensuring that the end product will still be the same quality and consistency.
### Application of Ratios
In scientific disciplines, particularly chemistry, ratios are extremely important.
Different substances can react in different ways depending on the ratio of one to another, so it could potentially pose a safety risk if researchers did not use proper ratios. If there is too much of one substance in a mixture, toxic gases could be released, so they must understand how ratios work and measure carefully according to this proportion.
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# Statistics Essay
2216 Words Mar 3rd, 2015 9 Pages
General Lectures 1-4 What is statistics? Discipline: collecting, classifying, summarizing, organizing, analyzing and interpreting numerical information. -A means to answer research questions (and to test hypothesis). -No scientific proof: make plausible through empirical data with an empirical observed result. Shape and skewness: Negatively skewed: mean < median Symmetry: mean = median Positively skewed: mean > median Mode = the score that occurs most frequently. Median = middle score when scores are ranked in order. Mean = the average score, the media. Interpreting the standard deviation: -the empirical rule (frequency distribution is bell shaped and symmetric) 68% values: -1 - +1 95% values: -1 - +1 99.7 % values: …show more content…
Alternative hypothesis: H1 - Represents the assumption tested: effect; difference; correlation, the researcher’s expectation with respect to changes in this situation.
1
Video Lecture 1 Descriptive questions: data Describing population characteristics Characteristics of data: central tendency
Characteristics of data: variability/dispersion
1. Measure of central tendency and measure of variability: the most adequate measure of central tendency is the mode and there is no adequate measure of variability (decision/claim). The choice for these measures is founded on the fact that the variable X is of nominal measurement level (data/ground). On nominal level the mode is the only adequate summary measure and there is no adequate measure of variability (warrant). Interpretation: The X sector has the highest number of … This is because the mode lies in this sector. The mode determines the sector with the highest frequency.
2. Measure of central tendency and measure of variability: the most adequate measure of central tendency is the mean and the most adequate measure of variability is the standard deviation (decision/claim). The choice for these measures is founded on the fact that the variable is of ratio measurement level (data/ground). On ratio level the most adequate summary measure is the mean and the most adequate measure of variability is the standard deviation (warrant). Interpretation: The mean… is about ... We expect that (at least) 75%
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|Part of a series on|
The term "difference feminism" developed during the "equality-versus-difference debate" in American feminism in the 1980s and 1990s, but subsequently fell out of favor and use. In the 1990s feminists addressed the binary logic of "difference" versus "equality" and moved on from it, notably with postmodern and/or deconstructionist approaches that either dismantled or did not depend on that dichotomy.
Difference feminism did not require a commitment to essentialism. Most strains of difference feminism did not argue that there was a biological, inherent, ahistorical, or otherwise "essential" link between womanhood and traditionally feminine values, habits of mind (often called "ways of knowing"), or personality traits. These feminists simply sought to recognize that, in the present, women and men are significantly different and to explore the devalued "feminine" characteristics.
Some strains of difference feminism, for example Mary Daly's, argue not just that women and men were different, and had different values or different ways of knowing, but that women and their values were superior to men's. This viewpoint does not require essentialism, although there is ongoing debate about whether Daly's feminism is essentialist.
Difference feminism was developed by feminists in the 1980s, in part as a reaction to popular liberal feminism (also known as "equality feminism"), which emphasized the similarities between women and men in order to argue for equal treatment for women. Difference feminism, although it still aimed at equality between men and women, emphasized the differences between men and women and argued that identicality or sameness are not necessary in order for men and women, and masculine and feminine values, to be treated equally. Liberal feminism aimed to make society and law gender-neutral, since it saw recognition of gender difference as a barrier to rights and participation within liberal democracy, while difference feminism held that gender-neutrality harmed women "whether by impelling them to imitate men, by depriving society of their distinctive contributions, or by letting them participate in society only on terms that favor men".
Difference feminism drew on earlier nineteenth-century strains of thought, for example the work of German writer Elise Oelsner, which held that not only should women be allowed into formerly male-only spheres and institutions (e.g. public life, science) but that those institutions should also be expected to change in a way that recognizes the value of traditionally devalued feminine ethics (like care [see ethics of care]). On the latter point, many feminists have re-read the phrase "difference feminism" in a way that asks "what difference does feminism make?" (e.g. to the practice of science) rather than "what differences are there between men and women"?
Essentialism and Difference Feminism
Some have argued that the thought of certain prominent second-wave feminists, like psychologist Carol Gilligan and radical feminist theologian Mary Daly, is "essentialist." In philosophy essentialism is the belief that "(at least some) objects have (at least some) essential properties." In the case of sexual politics essentialism is taken to mean that "women" and "men" have fixed essences or essential properties (e.g. behavioral or personality traits) that cannot be changed. However, essentialist interpretations of Daly and Gilligan have been questioned by some feminist scholars, who argue that charges of "essentialism" are often used more as terms of abuse than as theoretical critiques based on evidence, and do not accurately reflect Gilligan or Daly's views.
- "Carol Gilligan". Psychology's Feminist Voices.
- Scott, Joan (1988). "Deconstructing Equality-Versus-Difference: Or, the Uses of Post-structuralist Theory for Feminism". Feminist Studies. 14 (1): 32. doi:10.2307/3177997.
- Bock, Gisela; James, Susan (1992). Beyond Equality and Difference. Routledge. ISBN 9780415079891.
- Voet, Rian (1998). Feminism and Citizenship. London: SAGE Publications Ltd. ISBN 9781446228043.
- Schiebinger, Londa. Has Feminism Changed Science?. p. 8.
- Grande Jensen, Pamela. Finding a New Feminism: Rethinking the Woman Question for Liberal Democracy. p. 2 footnote 4.
- Tandon, Neeru. Feminism: A Paradigm Shift. p. 68.
- Hoagland, Sarah Lucia; Frye, Marilyn. "Feminist Interpretations of Mary Daly".
- Sandilands, Catriona (1999). The Good-Natured Feminist Ecofeminism and the Quest for Democracy. pp. chapter 5: "Cyborgs and Queers".
- Voet, Rian (1998). Feminism and Citizenship. SAGE Publications Ltd.
- Grande Jensen, Pamela. Finding a New Feminism: Rethinking the Woman Question for Liberal Democracy. p. 3.
- "Accidental vs Essential Properties". Stanford Encyclopedia of Philosophy. Retrieved 21 March 2017.
- Heyes, Cressida J. (1997). "Anti-Essentialism in Practice: Carol Gilligan and Feminist Philosophy". Hypatia. 13 (3): 142–163. doi:10.1111/j.1527-2001.1997.tb00009.x.
- Braidotti, Rosi (1992). "Essentialism" in Feminism and Psychoanalysis: A Critical Dictionary.
- Suhonen, Marja (2000). "Toward Biophilic Be-ing: Mary Daly's Feminist Metaethics and the Question of Essentialism" in Feminist Interpretations of Mary Daly. Penn State University Press. p. 112.
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What Is 32/80 as a Decimal + Solution With Free Steps
The fraction 32/80 as a decimal is equal to 0.4.
A Fraction in arithmetic is defined as a thing that depicts the number of parts contained by a specific size. Moreover, a complex fraction contains a fraction in the numerator or the denominator. At the same time, a Simple fraction contains both integers.
Here, we are more interested in the division types that result in a Decimal value, as this can be expressed as a Fraction. We see fractions as a way of showing two numbers having the operation of Division between them that result in a value that lies between two Integers.
Now, we introduce the method used to solve said fraction to decimal conversion, called Long Division, which we will discuss in detail moving forward. So, let’s go through the Solution of fraction 32/80.
Solution
First, we convert the fraction components, i.e., the numerator and the denominator, and transform them into the division constituents, i.e., the Dividend and the Divisor, respectively.
This can be done as follows:
Dividend = 32
Divisor = 80
Now, we introduce the most important quantity in our division process: the Quotient. The value represents the Solution to our division and can be expressed as having the following relationship with the Division constituents:
Quotient = Dividend $\div$ Divisor = 32 $\div$ 80
This is when we go through the Long Division solution to our problem.
Figure 1
32/80 Long Division Method
We start solving a problem using the Long Division Method by first taking apart the division’s components and comparing them. As we have 32 and 80, we can see how 32 is Smaller than 80, and to solve this division, we require that 32 be Bigger than 80.
This is done by multiplying the dividend by 10 and checking whether it is bigger than the divisor or not. If so, we calculate the Multiple of the divisor closest to the dividend and subtract it from the Dividend. This produces the Remainder, which we then use as the dividend later.
Now, we begin solving for our dividend 32, which after getting multiplied by 10 becomes 320.
We take this 320 and divide it by 80; this can be done as follows:
32 $\div$ 80 $\approx$ 4
Where:
32 x 4 = 320
This will lead to the generation of a Remainder equal to 320 – 320 = 0.
Images/mathematical drawings are created with GeoGebra.
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The immediate threat of warfare between the white settlers and the native inhabitants of the Ohio Country had been reduced by Anthony Wayne’s victory at the Battle of Fallen Timbers in August 1794. A year later, the former contending forces gathered at Greenville (today in western Ohio) to sign a peace agreement.
Wayne represented the federal government and expressed his hope that the treaty would last “as long as the woods grow and the waters run."
The natives were less enthusiastic, regarding the agreement as a forced treaty. They had little choice because of the whites’ advantages in arms and numbers. Tribes represented included the Miami, Chippewa, Wyandot, Shawnee, Pottawatomie, Kickapoo, Delaware, Wea, Piankashaw, Kaskaskia and Eel River.
Terms of the Treaty of Greenville included:
The tribes agreed to surrender their claims to lands in the southeastern portion of the Northwest Territory (mostly present-day southern and eastern Ohio)
The tribes also gave up additional defined areas that were used by the whites as portages and fort locations. This category included Fort Detroit and the site of the future town of Chicago on Lake Michigan
- The United States government agreed to make an immediate payment of to $20,000 in goods to the tribes, as well as annual payments of $9,500 in goods to be divided among specified tribes
- The tribes retained the right to hunt throughout the area.
The Native Americans scrupulously abided by the terms of the treaty; American settlers did not. New white settlements outside of the treaty area were established almost immediately. Resistance would emerge in the early years of the next century in lands slightly farther west under the auspices of Tecumseh and his brother, The Prophet.
Off-site search results for "Treaty of Greenville"...
Treaty of Greenville and the Battle of Fallen Timbers
It was neat. Zach 1997 In 1795, as a result of the Battle of Fallen Timbers, The Treaty of Greenville was signed. At Fort Greenville, Harrison found that over 1,100 Indian warriors and chiefs had gathered. After weeks of talTreaty of Greenville was signed. At Fort Greenville, Harrison found that over 1,100 Indian warriors and chiefs had gathered. After weeks of talking, eating and ...
The American Revolution (Treaty of Greenville)Footer Links
The Indians kept the right to hunt on the land. The Treaty of Greenville August 3, 1795 A treaty of peace between the United States of America, and the tribes of Indians called the Wyandots, Delawares, ShTreaty of Greenville August 3, 1795 A treaty of peace between the United States of America, and the tribes of Indians called the Wyandots, Delawares, Shawanees, Ottawas ...
The Treaty of Greenville - TEXT VERSION
... GREENVILLE A TREATY OF PEACE BETWEEN THE UNITED STATES OF AMERICA AND THE TRIBES OF INDIANS CALLED THE WYANDOTS, DELAWARES, SHAWANOES, OTTAWAS, CHIPEWAS, PUTAWATIMES, MIAMIS, EEL-RIVER, WEEAS, KICKAPOOS, PIANKASHAWS AND KASKASKIAS. To put an ...
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Adventist Youth Honors Answer Book/Nature/Marsupials
South Pacific Division
|Skill Level Unknown|
|Year of Introduction: Unknown|
- 1 1. Distinguish:
- 2 2. Understand how marsupials are classified into families and describe, in a general way, the habits of these families.
- 3 3. Describe the distribution, habitat (ie. type of country they live in), diet breeding behavior, as well as any other interesting information of twelve different groups of marsupials and monotremes.
- 4 4. Be able to explain the difference between marsupial reproduction and that in true mammals (ie. placentals).
- 5 5. Explain the significance of the direction of opening of the pouch and the number of young per litter in marsupials.
- 6 6. Give an explanation of the unique occurrence of marsupials in Australia.
- 7 7. Be able to explain the need for conservation of our marsupials.
- 8 8. Write a report of your visit to a natural history museum, wild-life sanctuary, zoo, etc. indicating in about 10-15 lines, the extent of your observations.
- 9 References
a. Mammal from other animals.
The mammals are the class of vertebrate animals characterized by the presence of mammary glands, which in females produce milk for the nourishment of young.
b. Placentals, marsupials and monotremes from one another.
- The placentals are distinguished from other mammals in that the fetus is nourished during pregnancy via a placenta.
- Marsupials are mammals in which the female typically has a pouch (called the marsupium) in which it rears its young through early infancy. They differ from placental mammals in their reproductive traits.
- Monotremes are mammals that lay eggs instead of giving birth to live young like marsupials and placental mammals.
2. Understand how marsupials are classified into families and describe, in a general way, the habits of these families.
Marsupials are classed mainly by their dietary habits.
- Herbivorous marsupials
- (such as kangaroos, wallabies, koalas, wombats, and possums) eat only plants.
- Carnivorous marsupials
- (such as Tasmanian devils, numbats, and quolls) eat only meat. They are very similar to one another in shape, though their sizes vary considerably.
- Omnivorous Marsupials
- (such as opossums) eat just about anything they can find.
3. Describe the distribution, habitat (ie. type of country they live in), diet breeding behavior, as well as any other interesting information of twelve different groups of marsupials and monotremes.
The Tasmanian Devil (Sarcophilus harrisii), also referred to simply as "the devil", is a carnivorous marsupial now found in the wild only in the Australian island state of Tasmania. The Tasmanian Devil is the only extant member of the genus Sarcophilus. The size of a small dog, but stocky and muscular, the Tasmanian Devil is now the largest carnivorous marsupial in the world after the extinction of the Thylacine in 1936. It is characterised by its black fur, offensive odor when stressed, extremely loud and disturbing screech, and viciousness when feeding. It is known to both hunt prey and scavenge carrion and although it is usually solitary, it sometimes eats with other devils.
The Tasmanian Devil became extirpated on the Australian mainland about 400 years before European settlement in 1788. Because they were seen as a threat to livestock in Tasmania, devils were hunted until 1941, when they became officially protected. Since the late 1990s devil facial tumour disease has reduced the devil population significantly and now threatens the survival of the species, which in May of 2008 was finally declared to be endangered. Programs are currently being undertaken by the Tasmanian government to reduce the impact of the disease.
Tasmanian Devils can take prey up to the size of a small wallaby, but in practice they are opportunistic, and eat carrion more often than they hunt live prey. Although the devil favours wombats, it will eat all small native mammals, domestic mammals (including sheep), birds, fish, insects, frogs and reptiles. Their diet is largely varied and depends on the food available. On average, they eat about 15% of their body weight each day; however, they can eat up to 40% of their body weight in 30 minutes if the opportunity arises. Tasmanian Devils eliminate all traces of a carcass, devouring the bones and fur in addition to the meat and internal organs. In this respect, the devil has earned the gratitude of Tasmanian farmers, as the speed at which they clean a carcass helps prevent the spread of insects that might otherwise harm livestock.
The Numbat is a small, colorful creature between 20 and a little under 30 cm long, with a finely pointed muzzle and a prominent, bushy tail about the same length as its body. Color varies considerably, from soft grey to reddish-brown, often with an area of brick red on the upper back, and always with a conspicuous black stripe running from the tip of the muzzle through the eyes to the bases of the small, round-tipped ears. The underside is cream or light grey; weight varies between 280 and 550 grams.
Adult Numbats are solitary and territorial; an individual of either sex establishes a territory of up to 150 hectares (370 acres) early in life, and defends it from others of the same sex. The animal generally remains within it from that time on; male and female territories overlap, and in the breeding season males will venture outside their normal home range to find mates.
Another common name for the Numbat was Banded Anteater, though this has now fallen into disuse. Though it will in fact consume ants, its diet otherwise consists almost exclusively of termites.
While the numbat has relatively powerful claws for its size, it is not strong enough to get at termites inside the concrete-like mound, and so must wait until the termites are active. It uses a well-developed sense of smell to locate the shallow and unfortified underground galleries that termites construct between the nest and their feeding sites; these are usually only a short distance below the surface of the soil, and vulnerable to the Numbat's digging claws.
Quolls or native cats (genus Dasyurus) are carnivorous marsupials, native to Australia and Papua New Guinea. Adults are between 25 and 75 cm long, with hairy tails about 20-35 cm long. Females have six to eight nipples and develop a pouch—which opens towards the tail—only during the breeding season, when they are rearing young. The babies are the size of a grain of rice. Quolls live both in forests and in open valley land. Though primarily ground-dwelling, they have developed secondary arboreal characteristics. They do not have prehensile tails, but do have ridges on the pads of their feet. Their molars and canines are strongly developed.
The Eastern Quoll is a solitary predator, hunting at night for its prey of insects and small mammals. They have also been known to scavenge food from the much larger Tasmanian Devil.
The breeding season begins in early winter, and the female gives birth to up to 30 young. Of these, the first to attach themselves to the six teats will be the only survivors. Weaning takes place at about 10 weeks of age, with the young staying in the den while the mother forages.
There are three species of kangaroo:
- The Red Kangaroo (Macropus rufus) is the largest surviving marsupial anywhere in the world. Fewer in numbers, the Red Kangaroos occupy the arid and semi-arid centre of the continent. A large male can be 2 metres (6 ft 7 in) tall and weigh 90 kg (200 lb).
- The Eastern Grey Kangaroo (Macropus giganteus) is less well-known than the red (outside of Australia), but the most often seen, as its range covers the fertile eastern part of the continent.
- The Western Grey Kangaroo (Macropus fuliginosus) is slightly smaller again at about 54 kg (119 lb) for a large male. It is found in the southern part of Western Australia, South Australia near the coast, and the Darling River basin.
Essentially, a wallaby is any macropod that isn't large enough to be considered a kangaroo and has not been given some other name. There is no fixed dividing line. In general, a wallaby is smaller and has a stockier build than a kangaroo; a wallaroo is any of a few species somewhat intermediate in size between a wallaby and a kangaroo.
Possums are small marsupials with brown or grey fur, ranging in size from the length of a finger (pygmy possums and wrist-winged gliders), to the length of a forearm (brushtails and ringtails). All possums are nocturnal and omnivorous, hiding in a nest in a hollow tree during the day and coming out during the night to forage for food. They fill much the same role in the Australian ecosystem that squirrels fill in the northern hemisphere and are broadly similar in appearance.
The two most common species of possums, the Common Brushtail and Common Ringtail, are also among the largest.
Opossums are nocturnal marsupials found in the Western Hemisphere. They are small to medium-sized creatures, about the size of a large house cat. Although there are many exceptions, most of them spend time living both in trees and on the ground, and they eat many different things (plants and animals).
Opossums are usually nomadic, staying in one area as long as food and water are easily available. Though they will temporarily occupy abandoned burrows, they do not dig or put much effort into building their own. They favor dark, secure areas, below ground or above.
When threatened or harmed, they will "play possum", mimicking the appearance and smell of a sick or dead animal. The lips are drawn back, teeth are bared, saliva foams around the mouth, and a foul-smelling fluid is secreted from glands. This response is involuntary, rather than a conscious act. Their stiff, curled form can be prodded, turned over, and even carried away. Many injured opossums have been killed by well-meaning people who find a catatonic animal and assume the worst. If you find an injured or apparently dead opossum, the best thing to do is leave it in a quiet place with a clear exit path. In minutes or hours, the animal will regain consciousness and escape quietly on its own.
Shrew opossums (also known as rat opossums) are about the size of a small rat (9–14 cm long), with thin limbs, a long, pointed snout and a slender, hairy tail. They are largely meat-eaters, being active hunters of insects, earthworms and small vertebrates. They have small eyes and poor sight, and hunt in the early evening and at night, using their hearing and long, sensitive whiskers to locate prey. They seem to spend much of their lives in underground burrows and on surface runways.
Largely because of their rugged, inaccessible habitat, they are very poorly known and have traditionally been considered rare. Recent studies suggest that they may be more common than had been thought.
The Koala is broadly similar in appearance to the wombat, but has a thicker, more luxurious coat, much larger ears, and longer limbs, which are equipped with large, sharp claws to assist with climbing. Weight varies from about 14 kg for a large, southern male, to about 5 kg for a small northern female. Contrary to popular belief, their fur is thick, not soft and cuddly. Koalas' five fingers per paw are arranged with the first two as opposable thumbs, providing better gripping ability.
Wombats are Australian marsupials; they are short-legged, muscular quadrupeds, approximately one meter (3 feet) in length and with a very short tail. Wombats have an extraordinarily slow metabolism, taking around 14 days to complete digestion, and generally move slowly. When required, however, they can reach up to 40 km/h and maintain that speed for up to 90 seconds.
When attacked, they can summon immense reserves of strength — one defense of a wombat against a predator (such as a Dingo) underground is to crush it against the roof of the tunnel until it stops breathing.
Tree-kangaroos are macropods adapted for life in trees. They are found in the rainforests of New Guinea, far northeastern Queensland, and nearby islands, usually in mountainous areas. Although most are found in mountainous areas, several species also occur in lowlands, and one, the aptly named Lowlands Tree-kangaroo, appears to be restricted to lowlands.
Tree-kangaroos feed mostly on leaves and fruit, taken both in trees and on the ground, but other foods are eaten when available, including grain, flowers, sap, bark, eggs and young birds. Their teeth are adapted for tearing leaves rather than cutting grass.
A wallaroo is any of three closely related species of moderately large macropod, intermediate in size between the kangaroos and the wallabies. The name "wallaroo" is a portmanteau of wallaby and kangaroo. In general, a large, slim-bodied macropod of the open plains is called a "kangaroo"; a small to medium-sized one, particularly if it is relatively thick-set, is a "wallaby": most wallaroos are only a little smaller than a kangaroo, fairly thickset, and are found in open country. All share a particular habit of stance: wrists raised, elbows tucked close into the body, and shoulders thrown back, and all have a large, black-skinned rhinarium.
The best-known species is Macropus robustus, which is known as the Eastern Wallaroo, Common Wallaroo or just Wallaroo on the slopes of the Great Dividing Range (which runs for more than 3,000 km around the eastern and south-eastern coast of Australia), and as the Euro in most of the rest of the continent. There are four subspecies: the Eastern Wallaroo and the Euro, which are both widespread, and two of more restricted range, one from Barrow Island, the other from the Kimberley.
The Black Wallaroo (Macropus bernardus) occupies an area of steep, rocky ground in Arnhem Land. At around 60 to 70 cm in length (excluding tail) it is the smallest wallaroo and the most heavily built. Males weigh 19 to 22 kg, females about 13 kg. Because it is very wary and is found only in a small area of remote and very rugged country, it is remarkably little known.
The Antilopine Wallaroo (Macropus antilopinus) is the exception among wallaroos. It is, essentially, the far-northern equivalent of the Eastern and Western Grey Kangaroos. Like them, it is a creature of the grassy plains and woodlands, and gregarious, where the other wallaroos are solitary. Because of this difference, it is sometimes called the Antilopine Kangaroo.
A pademelon is any of seven species of small marsupials of the genus Thylogale. They are usually found in forests. Pademelons are the smallest of the macropods. The name is a corruption of badimaliyan, from the Dharuk Aboriginal language of Port Jackson.
Pademelons, wallabies, and kangaroos are very alike in body structure, and the names just refer to the three different size groups. Originally wallabies were divided into small and large wallabies, but a more suitable name was needed to differentiate between them.
Besides their smaller size, pademelons can be distinguished from wallabies by their shorter, thicker, and sparsely haired tails.
If there are no predators, such as dogs, they graze in the early mornings or evenings on grassy slopes near thickets into which they can quickly escape at the first sign of danger. Having noticed danger, such as a python, they may try to warn others by stomping their feet on the ground producing surprisingly loud sound. Pademelons are nocturnal and tend to feed at night.
Their main diet is made up of grasses, leaves, and small shoots. They do little damage to crops and are not as aggressive as wallabies and kangaroos can be, making them gentle pets.
Normally, a group of females would stay on the territory with males showing up, only when one of the females is ready for mating. This is in contrast to the behavior of bigger kangaroos, who stay in mixed mobs with a male leader.
The Quokka (Setonix brachyurus), the only member of the genus Setonix, is a small macropod about the size of a large domestic cat. Like other marsupials in the macropod family (such as the kangaroos and wallabies), the Quokka is herbivorous and mainly nocturnal.
In the wild, its roaming is restricted to a very small range in the South-West of Western Australia. The Quokka has become rare, but remains a protected species on islands off the coast of that area, Bald Island, Rottnest Island, Garden Island and rarely Penguin Island. The islands are free of foxes and cats. On Rottnest Island, it is common and occupies a wide range of habitats, ranging from semi-arid scrub to cultivated gardens.
The Quokka is gregarious and gathers in large groups where food is available: primary items are grasses, sedges, succulents and foliage. The health of some animals has suffered significantly by the ingestion of inappropriate foods, such as bread, given by well meaning visitors to Rottnest Island. Visitors are now asked to refrain from feeding them. It breeds at any time on the mainland, but in late summer on Rottnest. The Quokka only produces a single joey in a year. Restricted availability of the trace element copper appears to be a major limiting factor of the ability of the Quokka to breed on Rottnest.
The Quokka's movements are similar to a kangaroo, using mixture of small and large hops.
4. Be able to explain the difference between marsupial reproduction and that in true mammals (ie. placentals).
The pregnant female marsupial develops a kind of yolk sack in her womb which delivers nutrients to the embryo. The embryo is born at a very early stage of development (at about 4-5 weeks), upon which it crawls up its mother's belly and attaches itself to a nipple (which is located inside the pouch). It remains attached to the nipple for a number of weeks. The offspring later passes through a stage where it temporarily leaves the pouch, returning for warmth and nourishment.
The placenta is a temporary organ composed of two parts, one of which is part of the fetus, the other part of the mother. It is implanted in the wall of the uterus, where it receives nutrients and oxygen from the mother's blood and passes out waste. This interface forms a barrier, the placental barrier, which filters out some substances which could harm the fetus.
5. Explain the significance of the direction of opening of the pouch and the number of young per litter in marsupials.
The pouch is a distinguishing feature of female marsupials; the name marsupial is derived from the Latin marsupium, meaning pouch. Marsupials give birth to a live but relatively undeveloped fetus called a joey. When the joey is born it crawls from inside the mother to the pouch. The pouch is basically a fold of skin with a single opening that covers the nipples to protect the joey as it continues to develop.
Pouches are different amongst the different marsupials: for example for Quolls and Tasmanian Devils, the pouch opens to the rear and the joey only has to travel a short distance to get to the opening of the pouch. While in the pouch they are permanently attached to the nipple and once the young have developed they leave the pouch and do not return. The kangaroo's pouch opens horizontally on the front of the body, and the joey must climb a relatively long way to reach it. Kangaroos and wallabies allow their young to live in the pouch well after they are physically capable of leaving.
Tasmanian Devils give birth to up to 50 young, each weighing approximately 0.18–0.24 grams. When the young are born, they move from the vagina to the pouch. Once inside the pouch, they each remain attached to a nipple for the next 100 days. Despite the large litter at birth, the female has only four nipples, so that no more than four young can survive birth. On average, more females survive than males. The female Tasmanian Devil's pouch, like that of the wombat, opens to the rear, so it is physically difficult for the female to interact with young inside the pouch. Unlike kangaroo joeys, young devils (and other marsupials with rear-opening pouches) do not return to the pouch; instead, they remain in the den for another three months, first venturing outside the den between October and December before becoming independent in January.
6. Give an explanation of the unique occurrence of marsupials in Australia.
Marsupial success in Australia has been attributed to the lack of placental mammals living there. When marsupials and placental mammals share an environment, the placental mammals typically outcompete the marsupials and the marsupials do not survive.
It has also been suggested that since marsupials have a lower metabolic rate, they are better able to survive the heat of Australia. However, this conjecture does not account for the success of placental mammals in other hot climates such as Africa and India. Furthermore, placental mammals, when introduced to Australia, repeatedly demonstrate their fitness to Australia's climate by outcompeting the marsupials.
7. Be able to explain the need for conservation of our marsupials.
Many marsupials have been listed as endangered species. Until the Europeans arrived in Australia, it was an isolated continent and no placental mammals (other than man) were able to travel there. When the Europeans arrived on ships, they brought placental mammals with them - sometimes intentionally, as in the case of cattle and sheep, and sometimes unintentionally, as in the case of rats and mice. These placental mammals pose a great threat to the native marsupials, as they compete for the same resources, often displacing them.
8. Write a report of your visit to a natural history museum, wild-life sanctuary, zoo, etc. indicating in about 10-15 lines, the extent of your observations.
The place you visit should presumably have some marsupials in its collection, and the 10-15 lines should document observations of the marsupials. Here are some ideas for the sorts of things you might be looking for:
- Species: what is the name of this marsupial? What is its scientific name?
- Natural habitat: where does this animal live in the wild?
- Artificial habitat: how were its needs being met (if it was a live animal)
- Diet: How does its natural diet compare to its diet in the sanctuary or zoo?
- Conservation Status: is this animal threatened or endangered? If so, what is being done to conserve it?
- Breeding: Were the animals in this collection part of a breeding program?
- Young: Did you see any young marsupials? Were they in their mother's pouch?
- Unique Features: what is unique about this animal?
- Fascinating Features: what did you like best about this animal?
If you choose to visit a zoo or wildlife sanctuary, be sure to check out the Endangered Animals honor too, as this will meet a requirement for that honor as well.
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Sukkoth, also spelled Sukkot, Succoth, Sukkos, Succot, or Succos, Hebrew Sukkot (“Huts” or “Booths”), singular Sukka, also called Feast of Tabernacles or Feast of Booths, a Jewish autumn festival of double thanksgiving that begins on the 15th day of Tishri (in September or October), five days after Yom Kippur, the Day of Atonement. It is one of the three Pilgrim Festivals of the Hebrew Bible.
The Bible refers to ḥag ha-asif (“Feast of the Ingathering,” Exodus 23:16), when grains and fruits were gathered at the harvest’s end, and to ḥag ha-sukkot (“Feast of Booths,” Leviticus 23:34), recalling the days when the Israelites lived in huts (sukkot) during their years of wandering in the wilderness after the Exodus from Egypt. The festival is characterized by the erection of huts made of branches and by the gathering of four species of plants, with prayers of thanksgiving to God for the fruitfulness of the land. As part of the celebration, a sevenfold circuit of the synagogue is made with the four plants on the seventh day of the festival, called by the special name Hoshana Rabba (“Great Hosanna”).
The eighth day is considered by some a separate festival and called Shemini Atzeret (“Eighth Day of the Solemn Assembly”). In Israel the eighth day also commemorates the completion of the annual cycle of readings from the Torah (the first five books of the Bible) and is called Simḥat Torah (“Rejoicing of the Law”). Outside Israel, Simḥat Torah is celebrated independently on the following day.
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Jean-Jacques Rousseau (Photo credit: Wikipedia)
If you measure Rousseau against the definition of enlightenment that was popular at his time, he would not appear to be enlightened. The “enlightened” of his time believed education in the arts and sciences would lead to equality and freedom. However, if you measure him against Immanuel Kant’s definition, it is obvious that Rousseau was enlightened.
Immanuel Kant’s definition of enlightenment is as follows:
“Enlightenment is man’s emergence from his self-incurred immaturity. Immaturity is the inability to use one’s own understanding without the guidance of another. This immaturity is self-incurred if its cause is not lack of understanding, but lack of resolution and courage to use it without the guidance of another. The motto of enlightenment is therefore: Sapere aude! Have courage to use your own understanding!” (An Answer to the Question: “What is Enlightenment?” Immanuel Kant, 1784)
Kant’s definition means that an enlightened individual would not follow the herd, and may very well have been a rabble rouser.
Rousseau fit that description.
Rousseau, in his studies, was not searching for freedom or equality. He was searching for the truth. And by searching for the truth, he dared to use his own understanding. Rather than follow blindly with the other “enlightened” men of his day, he went beyond their studies. Rather than jump on the negative bandwagon against the enlightenment, he came to his own conclusions about it.
“How can one venture to blame the sciences in front of one of the most scholarly societies in Europe, praise ignorance in a famous Academy, and reconcile a contempt for study with respect for truly learned men?” (Discourse on the Arts and Sciences, Jean-Jacques Rousseau, 1750)
This is a quote from a man who had the courage to use his own understanding. Rousseau claimed that the “enlightened” man of his age was a “happy slave,” wearing flowery chains of iron. The public’s education was not making them free or equal, but instead made them complacent and hid the fact that they were neither free nor equal.
Rousseau continued to describe his vision of what would come about as a result of enlightenment: uncertainty, suspicion, uniformity,and dishonorable vices disguised as virtues. Speaking of the Germans, Rousseau said,
“They were not ignorant of the fact that in other lands idle men spent their lives disputing their sovereign good, vice, and virtue, and that proud reasoners, while giving themselves the greatest praise, shoved all other people together under the contemptuous name of barbarians. But they looked at their morals and learned to despise their learning.” (Discourse on the Arts and Sciences)
While his contemporaries celebrated knowledge, Rousseau saw in it all that was bad in his world: “…these vain and futile declaimers move around in all directions armed with their fatal paradoxes, undermining the foundations of faith, and annihilating virtue.” (Discourse on the Arts and Sciences) He went on to lament how, once every great civilization in the past started to be more concerned with fine arts, they lost their edge and were easily dissolved or overthrown. Their citizens became soft and weak.
This lifestyle, in Rousseau’s estimation, also led to moral decay. Citizens shirked their duties, students were not taught right from wrong, and didn’t understand what they should do or how to do it. They didn’t know how to think. And he placed the blame squarely on the enlightenment. “From where do all these abuses arise if it is not the fatal inequality introduced among men by distinctions among their talents and by the degradation of their virtues? There you have the most obvious effect of all our studies, and the most dangerous of all their consequences.” (Discourse on the Arts and Sciences)
No one wants to hear that the course they have embarked upon is headed for disaster. Who but a rabble rouser, a person audacious enough to think for himself, would dare do tell them so? According to Kant, an enlightened man would. According to Kant, Rousseau would.
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Herschel and Planck Share Ride to Space
PASADENA, Calif. -- Two missions to study the cosmos, Herschel and
Planck, are scheduled to blast into space May 14 aboard the same
Ariane 5 rocket from the Guiana Space Center in French Guiana. The
European Space Agency, or ESA, leads both missions, with significant
participation from NASA.
"The missions are quite different, but they'll hitch a ride to space
together," said Ulf Israelsson, NASA project manager for both Herschel
and Planck. "Launch processing is moving along smoothly. Both missions'
instruments have completed their final checkouts, and the spacecrafts'
thruster tanks have been fueled."
Israelsson is with NASA's Jet Propulsion Laboratory, Pasadena, Calif.,
which contributed key technology to both missions. NASA team members will
play an important role in data analysis and science operations.
The Herschel observatory has the unique ability to peek into the dustiest
and earliest stages of planet, star and galaxy growth. The spacecraft's
astronomy mirror -- about 3.5 meters (11.5 feet) in diameter -- is the
largest ever launched into space. It will collect longer-wavelength light
in the infrared and submillimeter range -- light never before investigated
by an astronomy mission.
"We haven't had ready access to the wavelengths between infrared and microwaves
before, in part because our Earth's atmosphere blocks them from reaching the
ground. We will now have access to these wavelengths thanks to Herschel's large,
cold telescope in space, and its detectors' improved sensitivity," said Paul
Goldsmith, the NASA project scientist for Herschel at JPL. "Because our views
were so limited before, we can expect a vast range of serendipitous discoveries,
from new molecules in interstellar space to new types of objects."
The coolest objects in the universe, such as dusty, developing stars and galaxies,
appear as dark blobs when viewed with visible-light telescopes, so astronomers
don't know what's happening inside them. But at longer wavelengths in the far-infrared
and submillimeter range, cool objects perk up and shine brightly. Herschel will
detect light from objects as cold as minus 263 degrees Celsius, or 10 Kelvin, which
is 10 degrees above the coldest temperature theoretically attainable. To do this,
the observatory's instruments must be cold, too. Onboard liquid helium, which is
expected to last more than three-and-a-half years, will chill the coldest of
Herschel's detectors to a frosty 0.3 Kelvin.
Planck has a different goal. It will answer fundamental questions about how the
universe came to be, and how it will change in the future. It will look back in time
to just 400,000 years after our universe exploded into existence nearly 14 billion
years ago in an event known as the Big Bang. The mission will spend at least 15 months
making the most precise measurements yet of light at microwave wavelengths across our
entire sky -- including what's known as the cosmic microwave background. This
microwave light has even longer wavelengths than what Herschel will see, but it's
not from cool objects. In this case, the light is from the hot, primordial soup of
particles that eventually evolved to become our modern-day universe. The light has
traveled nearly 14 billion years to reach us, and, in that time, has cooled and
stretched to longer wavelengths because space is expanding.
By measuring minute variations in the cosmic microwave background as small as a
few parts per million, Planck will give us a new and improved assessment of our
universe -- its age, composition, size, mass and geometry. We'll also learn more
about the theorized early inflation of our universe, when it is thought to have
expanded 100 trillion, trillion times. That's just one trillion, trillion, trillionth
of a second after the Big Bang.
"The cosmic microwave background shows us the universe directly at age 400,000 years,
not the movie, not the historical novel, but the original photons," said Charles Lawrence,
NASA project scientist for Planck at JPL. "Planck will give us the clearest view ever of
this baby universe, showing us the results of physical processes in the first brief moments
after the Big Bang, and the starting point for the formation of stars, galaxies, and
clusters of galaxies. The clear view is a result of Planck's unprecedented combination
of sensitivity, angular resolution, or sharpness, and frequency coverage."
Like Herschel, Planck will be cold; in fact, one of its instruments will be cooled to just
0.1 Kelvin. But it won't carry liquid coolant. Instead, it will chill itself with innovative
"cryocooler" technology, developed in part by JPL.
Both spacecraft have been mated to their rocket and are being readied for launch. Shortly
after liftoff, they will separate from the rocket and follow different trajectories. By two
months later, the missions will have made their way to their final, distinct orbits around
the second Lagrangian point of the Earth-sun system, a point in space 1.5 million kilometers
(930,000 miles) from Earth, or four times farther from Earth than the moon. This point is
on the other side of Earth from the sun, providing the spacecraft with dark, expansive views
of the sky. It is also far enough away that the heat from Earth and the moon won't warm up
Herschel is a European Space Agency mission, with science instruments provided by a consortium
of European-led institutes, and with important participation by NASA. NASA's Herschel Project
Office is based at JPL. JPL contributed mission-enabling technology for two of Herschel's three
science instruments. The NASA Herschel Science Center, part of the Infrared Processing and
Analysis Center at the California Institute of Technology in Pasadena, supports the United
States astronomical community. Caltech manages JPL for NASA. More information is online at
Planck is a European Space Agency mission, with significant participation from NASA. NASA's
Planck Project Office is based at JPL. JPL contributed mission-enabling technology for both
of Planck's science instruments. NASA, U.S. and European Planck scientists will work together
to analyze the Planck data. More information is online at http://www.nasa.gov/planck
Media contact: Whitney Clavin 1-818-354-4673
Jet Propulsion Laboratory, Pasadena, Calif.
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this are biology prefab question not more that 8 question on each file. but no plagiarism please because it will be submitted to turnitin. please provide work on time
Prepare a diagram in Word, PowerPoint, or on poster. (If you prepare a poster, take a picture of the poster and upload the image to the dropbox.) The choice of how you create your diagram is up to you, but make sure it is clearly and appropriately labeled. The diagram should be detailed enough to communicate the sensory and motor pathways through the body.
Trace the sensory pathway from the receptors of the skin to the sensory area of the cerebrum.
Trace the motor pathway from the motor area of the cerebrum to the muscles.
Address the patient’s concern: What is the anatomical foundation that explains why patients who have had strokes on the right side of the cerebrum have paralysis on the contralateral side of the body?
Include your diagram in the paper. Write an introduction describing your intention in creating the diagram, include verbiage in the body of your paper highlighting the important points of the diagram, and a conclusion summarizing the demonstration that you would present when using your diagram to explain the pathway and how it relates to your patient’s concern.
All references must be cited using APA Style format.
Consider that a friend or family member has come to you and asked how a stroke affects the body’s muscles. Describe the normal function of a muscle and how a stroke affects the muscle. In your response, of at least 300 words, specifically address the following:
Choose the muscle in which you are most interested, and describe its origin, insertion, and main function.
Choose a specific movement of the body such as flexion of the arm. Which muscle is the prime mover? Which muscle is the antagonist, and which is the synergist for that particular movement?
Mention the lobes of the cerebrum and their primary functions.
Mention the meninges and their function and location.
Describe how a stroke affects the function of the muscle about which you just explained. Remember, you are talking to a friend or family member who has NO medical knowledge. You must write in layman’s terms.
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The music of the silks
Pound for pound, spider silk is one of the strongest materials known: Research by Massachusetts Institute of Technology (MIT)'s Markus Buehler has helped explain that this strength arises from silk's unusual hierarchical arrangement of protein building blocks.
Now Buehler—together with David Kaplan of Tufts University and Joyce Wong of Boston University—has synthesized new variants on silk's natural structure, and found a method for making further improvements in the synthetic material.
And an ear for music, it turns out, might be a key to making those structural improvements.
The work stems from a collaboration of civil and environmental engineers, mathematicians, biomedical engineers, and musical composers. The results are reported in a paper published in Nano Today.
"We're trying to approach making materials in a different way," Buehler explains, "starting from the building blocks"—in this case, the protein molecules that form the structure of silk. "It's very hard to do this; proteins are very complex."
Other groups have tried to construct such protein-based fibers using a trial-and-error approach, Buehler says. But this team has approached the problem systematically, starting with computer modeling of the underlying structures that give the natural silk its unusual combination of strength, flexibility, and stretchiness.
Buehler's previous research has determined that fibers with a particular structure—highly ordered, layered protein structures alternating with densely packed, tangled clumps of proteins (ABABAB)—help to give silk its exceptional properties. For this initial attempt at synthesizing a new material, the team chose to look instead at patterns in which one of the structures occurred in triplets (AAAB and BBBA).
Making such structures is no simple task. Kaplan, a chemical and biomedical engineer, modified silk-producing genes to produce these new sequences of proteins. Then Wong, a bioengineer and materials scientist, created a microfluidic device that mimicked the spider's silk-spinning organ, which is called a spinneret.
Even after the detailed computer modeling that went into it, the outcome came as a bit of a surprise, Buehler says. One of the new materials produced very strong protein molecules—but these did not stick together as a thread. The other produced weaker protein molecules that adhered well and formed a good thread. "This taught us that it's not sufficient to consider the properties of the protein molecules alone," he says. "Rather, [one must] think about how they can combine to form a well-connected network at a larger scale."
The team is now producing several more variants of the material to further improve and test its properties. But one wrinkle in their process may provide a significant advantage in figuring out which materials will be useful and which ones won't—and perhaps even which might be more advantageous for specific uses. That new and highly unusual wrinkle is music.
The different levels of silk's structure, Buehler says, are analogous to the hierarchical elements that make up a musical composition—including pitch, range, dynamics, and tempo. The team enlisted the help of composer John McDonald, a professor of music at Tufts, and MIT postdoctoral researcher David Spivak, a mathematician who specializes in a field called category theory. Together, using analytical tools derived from category theory to describe the protein structures, the team figured out how to translate the details of the artificial silk's structure into musical compositions.
The differences were quite distinct: The strong but useless material translated into music that was aggressive and harsh, Buehler says, while the one that formed usable fibers sounds much softer and more fluid.
Buehler hopes this can be taken a step further, using the musical compositions to predict how well new variations of the material might perform. "We're looking for radically new ways of designing materials," he says.
Combining materials modeling with mathematical and musical tools, Buehler says, could provide a much faster way of designing new biosynthesized materials, replacing the trial-and-error approach that prevails today. Genetically engineering organisms to produce materials is a long, painstaking process, he says, but this work "has taught us a new approach, a fundamental lesson" in combining experiment, theory and simulation to speed up the discovery process.
Materials produced this way—which can be done under environmentally benign, room-temperature conditions—could lead to new building blocks for tissue engineering or other uses, Buehler says: scaffolds for replacement organs, skin, blood vessels, or even new materials for use in civil engineering.
Elliott Schwartz, professor emeritus of music at Bowdoin College, says: "For centuries, mathematics, logic, and science have provided important models for musical structures, processes, and our understanding of sonic materials. The present research may well lead to one more important chapter in this ongoing story of mutual interaction."
It may be that the complex structures of music can reveal the underlying complex structures of biomaterials found in nature, Buehler says. "There might be an underlying structural expression in music that tells us more about the proteins that make up our bodies. After all, our organs—including the brain—are made from these building blocks, and humans' expression of music may inadvertently include more information that we are aware of."
"Nobody has tapped into this," he says, adding that with the breadth of his multidisciplinary team, "We could do this—making better bio-inspired materials by using music, and using music to better understand biology."
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Malaria Challenge: Managing Malaria
In this activity you will look at areas affected by malaria in Cambodia, Uganda, Tanzania or Brazil. You will have to develop a strategy that will reduce or eliminate malaria from that region. To help, you will be given key facts about the area, details of which methods of treating or preventing malaria are available to you, and a list of some key things to consider before developing your strategy.
This discussion-based activity is suitable for anyone wanting to know more about malaria and the social and economic challenges associated with the disease. It can be used to support the teaching of malaria in the classroom for able GCSE and A-level students alongside the other resources that accompany the multimedia resource Malaria Challenge (see related links below).
Age: 16 years + (KS4/GCSE +)
This page was last updated on 2015-01-14
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If you have any other comments or suggestions, please let us know at [email protected]
Can you spare 5-8 minutes to tell us what you think of this website? Open survey
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I. Multiple Choice Questions. (50 points, 1 point for each)
Directions: In this part of the test, there are 50 unfinished statements or questions. For each
of the unfinished statements or questions, four suggested answers marked A, B, C and D are
given. Choose the one that you think best completes the statement or answers the 重庆自考网question.
Write the letter of the answer you have chosen in the corresponding space on your Answer
1. Which of the following statement is NOT true?
A. Great Britain and England are geographical names.
B. The British Isles are made up of three large islands and hundreds of small ones.
C. At present there are 50 member countries within the Commonwealth (1991).
D. Wales is in the west of Great Britain.
2. ____ built the Hadrian s Wall and the Antonine Wall to keep ____ out of the area they had conquered.
A. The Romans; the Picts B. The Beaker Folk; the Picts
C. The Anglo-Saxons; the Romans D. The Danes; the Anglo-Saxons
3. Who were the ancestors of the English and the founders of England?
A.The Anglo-Saxons B. The Normans
C.The Vikings D. The Romans
4. The spirit of ____ was the limitation of the powers of the king, keeping them within the bounds of the feudal law of the land.
A. the Provision of Oxford B. the Constitutions of Clarendon
C. Domesday Book D. Magna Carta
5. When Mary Tudor became Queen, at least 300 ____ were burnt as heretics.
A. Protestants B. Catholics
C. Puritans D. Muslims
6. The chief demand of the peasants during the Peasant Uprising of 1381 was ____.
A. the abolition of villeinage
B. the punishment of the King s ministers
C. the increase of wages
D. the reform of the church
7. The significance of the Wars of the Roses was all the following EXCEPT that ____.
A. feudalism received its death blow
B. the great medieval nobility was much weakened
C. the king s power now became supreme
D. it dealt a death blow to villeinage
8. ____ is one of the comedies of Shakespeare.
A. Othello B. Richard III
C. The Tempest D. Julius Caesar
9. In the Glorious Revolution the Catholic king, ____ was driven out of England.
A. James I B. James II
C. Charles I D. Charles II
10. ____, excluding any Roman Catholic from the succession, confirmed the principle of parliamentary supremacy and guaranteed free speech within both the House of Lords and the House of Commons.
A. The Bill of Rights B. The Disabling Act
C. The Test Act D. Instrument of Government
11. As a result of the Industrial Revolution, Britain became the ____.
A. shop of the world B. workshop of the world
C. centre of the world D. leader of the world
12. Which of the following is NOT considered a characteristic of farming in the late 18th and the early 19th centuries?
A. Cultivation of fodder crops
B. Invention of seed drill
C. Selective breeding of domestic animals
D. Open-field system
13. In Britain, ____ abolished rotten boroughs .
A. the People s Charter B. the Combination Acts
C. the New Power Law D. the Reform Act of 1832
14. The present British monarch, Queen Elizabeth II was crowned in ____.
A. 1926 B. 1947
C. 1952 D. 1953
15. ____ has the ultimate authority for law-making in Britain.
A. The Monarch B. The Parliament
C. The Prime Minister D. The Cabinet
16. In the House of Commons, all speeches are addressed to ____ who is elected at the beginning of each new Parliament to preside over the House and enforce the rules of order.
A. the Prime Minister B. the Monarch
C. the Speaker D. the Lord Chancellor
17. About ____ daily and Sunday newspapers are published nationwide in Britain.
A. 100 B. 110
C. 120 D. 130
18. In Britain, ____ is directly responsible for the NHS.
A. a local government B. the central government
C. a voluntary organization D. a certain society
19. The day following Christmas Day is known as ___, for on this day gifts are given to servants and tradesmen.
A. New Year s Day B. Easter Day
C. Labor Day D. Boxing Day
20. Of the following four sports, ____ has the longest history in Britain.
A. cricket B. golf
C. football D. rugby
21. The three states that have seen the fastest growth in population in the past 20 years are ____.
A. California, Arizona and New Mexico
B. California, Florida and Nevada
C. New York, Texas and Florida
D. Arizona, Nevada and Florida
22. In American history, ____ refer to those who came to Plymouth on board of Mayflower.
A. the English nobles B. the Puritans
C. the Pilgrims D. the English adventurers
23. After President Jefferson bought the ____ Territory from France, the territory owned by the United States almost doubled.
A. Arizona B. Texas
C. Louisiana D. California
24. President Abraham Lincoln issued the ____, because he realized that he could win support for the Union at home and abroad by making the war a just war against slavery.
A. Bill of Rights B. Emancipation of Proclamation
C. Declaration of Independence D. Civil Right Act
25. The features of the early colonists which have strong influence on the formation of American character are all the following EXCEPT ____.
A. religious intolerance
B. respect of individual rights
C. representative form of government
D. a strong spirit of individual enterprise
26. Rushed by the Progressive Movement, ____ put forward his program of New Freedom.
A. Woodrow Wilson B. Theodore Roosevelt
C. Franklin D.Roosevelt D. George Washington
27. The Paris Conference which began on Jan 18, 1919 was dominated by the Big Four including ____.
A. the United States, the Soviet Union, China and Britain
B. the United States, Britain, France, and Italy
C. the United States, Britain, Germany, and Japan
D. the United States, Britain, France, and the Soviet Union
28. In his inaugural speech, ____ said that the only thing we have to fear is fear itself.
A. Abraham Lincoln B. Theodore Roosevelt
C. Franklin D.Roosevelt D. George Washington
29. As a result of American economic aid under the Marshall Plan, ____ recovered and began to show signs of development.
A. Turkey B. Greece
C. Western Europe D. Eastern Europe
30. During the Cuban Missile Crisis, the two superpowers stared at each other, with the possibility of ____ looming large.
A. a nuclear war B. a chemical war
C. the Vietnam War D. the Korean War
31. ____ visit to China ended twenty-three years of hostility and led to the establishment of diplomatic relations in January ____.
A. President Nixon s; 1972 B. Jimmy Carter s; 1978
C. President Nixon s; 1979 D. George Bush s; 1989
32. After long and difficult negotiations in Paris, the U. S. and ____ signed a cease-fire agreement on January 27, 1973.
A. South Vietnam B. North Vietnam
C. the Soviet Union D. Korea
33. Which of the following is NOT true of U.S. foreign trade?
A. Canada is the largest single source of goods imported by the United States.
B. Outside of North America, Asia is the largest source of imports.
C. The U.S. share of world trade has decreased in recent years.
D. Whenever the American economy is in trouble, the economy of other countries is affected.
34. When the delegates met at Philadelphia in 1787, their task was ____.
A. to write a new constitution
B. to adopt the Articles of Confederation
C. to establish a new form of government
D. to revise the Articles of Confederation
35. Abraham Lincoln was elected President as the candidate of ____ in 1860.
A. the Democratic Party B. the Republican Party
C. the Whig Party D. the Conservative Party
36. According to the U.S. Constitution, education is mainly a function of ____.
A. the federal government B. the city government
C. the county government D. the state government
37. Which of the following is NOT true about the reasons for the rapid growth of community colleges?
A. Their open admission policies B. Their cheap tuition and fees
C. Their fixed curriculum structures D. Their convenient locations
38. The Waste Land, written by ____, is considered the manifesto of the Lost Generation .
A. T. S. Eliot B. Walt Whitman
C. Emily Dickinson D. Theodore Dreiser
39. In the 1920s, Black literature developed into an upsurge, which has come to be known as ____.
A. the Literature Renaissance
B. the Harlem Renaissance
C. the Literature Revival
D. the knickerbockers era of American literature
40. Easter Sunday is the most important religious holiday for commemorating ____.
A. the death of Jesus Christ B. the birth of Jesus Christ
C. the crucifixion of Jesus Christ D. the resurrection of Jesus Christ
41. The Republic of Ireland is bounded by all the following EXCEPT ____.
A. the English Channel B. the Irish Sea
C. St. George Channel D. the Atlantic Ocean
42. Ireland is one of the most ____ countries of Europe.
A. Protestant B. Catholic
C. Puritan D. Christian
43. In Canada, nearly ____ of the land has no permanent population.
A. 87% B. 88%
C. 89% D. 90%
44. Who founded the first permanent settlements at Quebec and Montreal on the St. Lawrence River?
A. John Cabot B. Jacques Cartier
C. Samuel de Champlain D. Henry Hudson
45. All services provided by the Canadian federal government are available in ____.
A. French and Spanish B. English and Italian
C. Spanish and English D. French and English
46. Aboriginal and Torres Strait Islander people constitute ____ of Australia s population.
A. 1% B. 1.5%
C. 2% D. 2.5%
47. Why has Australia always been a continent with few people?
A. Because Australia is too far away from Europe.
B. Because Australia is the least mountainous and most level of the world s continents.
C. Because Australia is separated from the rest of the world by seas.
D. Because most of the continent is hot and dry.
48. As far as Australian culture is concerned, the history of Australia can be divided into the following phases EXCEPT ____.
A. the period of Australia s original culture
B. the period of the dominant British culture
C. the period of Asian culture
D. the period of a multicultural society
49. In New Zealand, the highest peak is in the centre of the mountain range, which is called ____.
A. the Southern Alps B. the Northern Alps
C. the New Zealand Alps D. the South Island Alps
50. ____ is described as a living fossil .
A. Moa B. Kiwi
C. Tuatara D. Bellbird
II. Answer the Questions. (30 points, 3 points for each)
Directions: Give a one-sentence answer to each of the following questions. Write your answers in the corresponding space on the Answer Sheet.
51. How many political divisions are there in Britain? What are they?
52. What questions did Elizabeth I treat as personal and private?
53. What were the two events in the world which most alarmed the British ruling classes in the closing decades of the 18th century?
54. Which are the five biggest cities in terms of population in the United States?
55. What are the three branches of the American government?
56. What does Mark Twain want to put across in The Adventures of Huckleberry Finn?
57. When was the name of the country officially changed to Ireland?
58. Who are easy to immigrate to Canada now?
59. In terms of land area and population, which is the biggest state and which is the smallest state in Australia?
60. What are the two active volcanoes in New Zealand?
III. Term Explanation. (20 points, 5 points for each)
Directions: Explain each of the following terms in English. Write your answers in the corresponding space on the Answer Sheet in around 40 words.
61. the Chartist Movement
62. Constitutional Monarchy
63. the Bill of Rights of America
64. the Federal System of America
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# Volume Line Up
### Task 227 ... Years 2 - 10
#### Summary
This task provides experience with estimating, measuring, ordering and comparing volumes (capacities). It is offered to students as an invitation to explore and discover relationships likely to determine the volume of an object. Keeping notes of experiments and preparing a report are key elements that link the task to the work of a mathematician. The task represents the type of experience likely to make textbook exercises more meaningful.
#### Materials
• At least 7 objects
(The card says 7, but more may have been included.)
• A container of rice
• A measuring cup / medicine glass marked in millilitres
#### Content
• estimating fractions
• estimating number
• estimating volume/capacity
• measurement, area
• measurement, length
• measurement, perimeter
• measurement, volume
• recording mathematics
• sorting, classifying, ordering
• spatial perception, 2D or 3D
#### Iceberg
A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
Containers supplied with this task will vary from the photo depending on availability. However, the card questions are equally relevant regardless of the equipment included. Therefore we are not providing specific answers to the card questions.
What we would like to do is publish student reports which you get excited about, along with a photo of the equipment your students were using at the time.
• What criteria do students use to decided most and least?
• What criteria do they use to guess the order of the objects?
• How does their guessed order relate to the resulted measured with rice ...water?
Where possible, some containers of the same height have been included. In these cases do the students come to any conclusion about the effect of base area if the height is constant?
#### Extension
1. Choose three containers and use a small piece of masking tape to mark one third of the volume of each container. The top edge of the tape is used to mark the guess. Explain why you put the tape where you did.
1. Find a way to check your estimate. How close where you?
2. Can you check your estimate another way?
2. The classroom is a container and its volume would normally be measured by cubes with 1 metre sides. Ask the students to estimate, then calculate the volume of the room.
3. Students might like to explore a volume problem faced by engineers in industries which have to store toxic liquid chemicals, for example, oil, petroleum or raw materials to make paint. The problem is to contain the spread of the chemical in the event of the storage cylinder leaking
One way to do this is to first dig a rectangular hole and then place the cylinders in it. If they leak the chemical is contained by the hole. Suppose a company has two cylinders each of which holds 1000 litres and they build them inside a rectangular hole. What could the dimensions of the hole be so that the worst case scenario of both cylinders simultaneously leaking all their fluid would be controlled by the flow into the rectangular hole?
#### Whole Class Investigation
Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
To convert this task to a whole class investigation you will need a set of containers for each group. If groups of three are used and each student is asked to bring two different containers no taller than (say) 7cm and no wider than (say) 7cm, then each group would have six containers to explore.
Look around the house and try to make your containers as interesting as possible. They have to able to have water poured into them.
Containers of rice are easy to make in advance for each group and medicine cups are easily obtainable. The card provides a clear direction for the lesson and the notes above offer an interesting and valuable extension. The report writing might be a little more imaginative than a written report. It could be a PowerPoint, poster, or video presented as a television news item or perhaps in the style of a children's science show.
An alterative whole class use is at task work station in unit of work on measurement. Related tasks could be Task 20, Pack The Box, Task 193, Surface Area With Tricubes, Task 226, Playing With Objects. Other volume and capacity activities can be found in most curriculum documents.These could be the basis of other work stations. You could also consider the whole class investigation section of Task 63, Fried Rice, which begins with a classic volume problem used in a Die Hard film and develops into an investigation in prime numbers.
At this stage, Volume Line Up does not have a matching lesson on Maths300, but Lesson 80, Cylinder Volumes, and Lesson 81, Biggest Volume, are related. Together with the tasks they could form the basis of a Mixed Media Unit.
#### Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Volume Line Up task is an integral part of:
• MWA Chance & Measurement Years 5 & 6
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City of Assyria. The form of its name is derived from the Masoretic text. It answers as nearly as possible to the native Assyrian form "Ninua." The origin of the name is obscure. Possibly it meant originally the seat of Ishtar, since Nina was one of the Babylonian names of that goddess. The ideogram means "house or place of fish," and was perhaps due to popular etymology (comp. Aramaic "nuna," denoting "fish"). Nineveh was the most famous of the cities which were in succession the residences of the kings of Assyria. It was also the latest capital of that kingdom, and as such was regarded by Greek writers as the permanent capital and as being virtually equivalent to the country itself.Situation of the City.
Nineveh was the strongest of several fortress-cities which were built in the triangular territory between the Tigris and the upper Zab. The terrane of these cities was admirably adapted for defense, being protected on the northwest by the Khausar, a tributary of the Tigris, and on the northeast by the Gomel, a tributary of the Zab, as well as by a range of hills. Within these boundaries were contained Nineveh itself, at the confluence of the Choser and the Tigris, on the site of the mounds of Koyunjik and Nebi Yunus and opposite the modern city of Mosul; the fortress of Kalaḥ (Calah) twenty miles to the south, near the Tigris; and Khorsabad (Dur Sharrukin) fourteen miles to the north on the Choser; besides various smaller fortified towns.
Nineveh is mentioned as early as about 2900
Nineveh seems to have been made the capital of the whole of Assyria by Shalmaneser I. (c. 1300
Less than twenty years elapsed between the death of the last great king and the destruction of the splendid city itself. This catastrophe has been made of late years the subject of considerable research; but much remains to be elucidated before a clear idea of the actual course of events can be obtained. The following statement summarizes the facts as far as known: With the decline of the Assyrian empire after the Scythian invasion of the regions west of the Tigris the capital itself became more open to attack. The Aryan Medes, who had attained to organized power east and northeast of Nineveh, repeatedly invaded Assyria proper, and in 607 succeeded in destroying the city. The other fortresses doubtless had been occupied some time previously. The capital was very strongly fortified. Its most vulnerable point was the River Khausar, which ran through the city, and which, while serving for defense, might be turned also to its destruction. In the time of flood its waters were stored up in reservoirs, and by breaking these a hostile army might undermine the city walls. An allusion to some such operation seems to be made in Nah. ii. 6. Such a rush of water could not of course inundate or greatly damage the city; it would be used mainly for the purpose of facilitating an entrance. The destruction was wrought by fire, and was made complete and final, so that soon the site of Nineveh proper was no longer distinguished by name from the other fortresses.Modern Exploration.
Nineveh has been diligently excavated by modern explorers. Its site was first definitely fixed by Richin 1820. The work of exploration on the mound began with Layard in 1845, and was then continued by Rassam and George Smith. The city proper, Nineveh in the strict sense, was oblong in shape, running along the Tigris, and did not occupy more than about three square miles. In the prophetic allegory of Jonah the references to its extent and population apply to the several cities and villages included in the larger area from Khorsabad to Kalaḥ. The excavation of Koyunjik has yielded results of the greatest value. The library of Assurbanipal alone, which consisted largely of copies of precious Babylonian documents, must be counted as one of the most important of the literary collections of the world.
- Layard, Nineveh and Its Remains, 1849;
- idem, Monuments of Nineveh, 1849-53;
- Botta and Flandin, Monuments de Ninive, 1847-50;
- Place, Ninive et l'Assyrie, 1866-69;
- George Smith, Assyrian Discoveries, 1875;
- Billerbeck and Jeremias, Der Untergang Nineves, in Delitzsch and Haupt, Beiträge zur Assyriologie, iii. 1 (has valuable maps and plates);
- Johns, Nineveh, in Cheyne and Black, Encyc. Bibl.
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Principles of the united stats constitution
Terms in this set (53)
A form of government in which power is divided between the federal, or national, government and the states
necessary and proper (elastic clause)
Gives Congress the powers to pass all laws necessary to carry out their constitutional duties; "elastic" clause (Art. I, Sec 8, clause 18)l
marbury v. madison
This case establishes the Supreme Court's power of Judicial Review
Powers held jointly by the national and state governments
Powers not specifically granted to the federal government or denied to the states belong to the states and the people
1. The president appoints federal judges with consent of the Senat
Article VI of the Constitution, which makes the Constitution, national laws, and treaties supreme over state laws when the national government is acting within its constitutional limits
framers ( of the u.s. constitution )
A legal process whereby an alleged criminal offender is surrendered by the officials of one state to officials of the state in which the crime is alleged to have been committed
powers that congress has that are specifically listed in the constitution
Certain powers are denied to the national government
the preamble to the u.s. constitution
The beginning of the U.S. Constitution, lists the goals of the government
article I of the u.s. constitution
describes the legislative branch (Congress), its duties and powers, and qualifications for its members
article II of the u.s. constitution
is about the executive branch, which consists of the President and Vice-President, enforces laws
article III of the u.s. constitution
Part of the Constitution that establishes the judicial branch of the federal government.
article IV of the u.s.s constitution
provided for cooperation among the states; the creation and admission of new states; regulation of U.S. territory; and certain obligations of the federal government to the states
article V of the u.s. constitution
The process for amending the Constitution after ratification.
article VI of the u.s. constitution
made the U.S. Constitution, laws made pursuant to it, and treaties made under the authority of the U.S. the supreme law of the land
article VII of the u.s. constitution
Ratification of the U.S. Constitution
A council of representatives
house of representatives
the lower legislative house of the United States Congress
A person who makes laws
make laws, bills, etc. or bring into effect by legislation
A group of people who have the power to make laws
A formal decision to reject the bill passed by Congress.
Review by a court of law of actions of a government official or entity or of some other legally appointed person or body or the review by an appellate court of the decision of a trial court
declaration of independence
the document recording the proclamation of the second Continental Congress (4 July 1776) asserting the independence of the colonies from Great Britain
articles of confederation
A weak constitution that governed America during the Revolutionary War
A charter of liberty and political rights obtained from King John of England by his rebellious barons at Runnymede in 1215.
1620 - The first agreement for self-government in America. It was signed by the 41 men on the Mayflower and set up a government for the Plymouth colony
A formal accusation of misconduct in office against a public official
2 houses, Senate: everybody gets 2 senators, House: Population, 3 branches
3/5 of a states population for taxation and representation
A voting system that apportions legislative seats according to the percentage of the vote won by a particular political party.
separation of powers
Constitutional division of powers among the legislative, executive, and judicial branches, with the legislative branch making law, the executive applying and enforcing the law, and the judiciary interpreting the law.
checks and balances
A governmental structure that gives each of the three branches of government some degree of oversight and control over the actions of the others.
freedom of choice
A person is said to be just if he or she treats people fairly and equally no matter how much money they have, clothes they wear, car they drive, color of their skin or religious beliefs.
A change in, or addition to, a constitution or law
a budget is balanced when current expenditures are equal to receipt
A law making body made of two houses (bi means 2). Example: Congress (our legislature) is made of two house - The House of Representatives and The Senate.
bill of attainder
A law that declares a person, without a trial, to be guilty of a crime
bill of rights
The first ten amendments of the U.S. Constitution, containing a list of individual rights and liberties, such as freedom of speech, religion, and the press.
Representing, characterized by, or including members from two parties or factions
A group of advisers to the president.
clear and present danger
law should not punish speech unless there was a clear and present danger of producing harmful actions
A procedure for terminating debate, especially filibusters, in the Senate.
Those powers, expressed, implied, or inherent, granted to the National Government by the constitution
A system of government by the whole population or all the eligible members of a state, typically through elected representatives
A device by which any member of the House, after a committee has had the bill for thirty days, may petition to have it brought to the floor
to keep peace among the people
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# Multiple linear regression/Quiz
Multiple linear regression practice quiz
## MLR I
1 Multiple linear regression (MLR) is a __________ type of statistical analysis.
univariate bivariate multivariate
2 The following types of data can be used in MLR (choose all that apply)
Interval or higher dependent variable (DV) Interval or higher independent variables (IVs) Dichotomous IVs
3 A linear regression (LR) analysis produces the equation Y = 0.4X + 3. This indicates that:
When Y = 0.4, X = 3 When Y = 0, X = 3 When X = 3, Y = 0.4 When X = 0, Y = 3
4 A LR analysis produces the equation Y = -3.2X + 7. This indicates that:
A 1 unit increase in X results in a 3.2 unit decrease in Y. A 1 unit decrease in X results in a 3.2 unit decrease in Y. A 1 unit increase in X results in a 3.2 unit increase in Y. An X value of 0 would would increase Y by 7.
5 The main purpose(s) of (LR) is/are (choose all that apply):
Predicting one variable on the basis of another Explaining one variable in terms of another Describing the relationship between one variable and another Exploring the relationship between one variable and another
6 When writing regression formulae, which of the following refers to the predicted value on the dependent variable (DV)?
Y Y (hat) X X (hat) a
7 The major conceptual limitation of all regression techniques is that one can only ascertain relationships, but never be sure about underlying causal mechanism.
True False
8 In MLR, the square of the multiple correlation coefficient or R2 is called the
Coefficient of determination Variance Covariance Cross-product Big R
## MLR II
1 In MLR, a residual is the difference between the predicted Y and actual Y values.
True False
2 Shared and unique variance among multiple variables can be represented by a diagram that includes overlapping circles. This is referred to as a:
Homogeneity diagram 3-way scatterplot Venn diagram (2 circles) or Ballantine diagram (3 or more circles) Pie chart Path diagram
3 In an MLR, the r between the two IVs is 1. Therefore, R will equal the r between one of the IVs and the DV. (Hint: Draw a Venn Diagram.)
True False
4 In a MLR, if the two IVs are correlated with the DV and the two IVs are correlated with one another, the rps (partial correlations) will be _______ in magnitude than the rs (Hint: Draw a Venn Diagram.)
Equal Smaller Larger Impossible to tell
5 In MLR, the unique variance in the DV explained by a particular IV is estimated by its:
Zero-order correlation squared (r2) Multiple correlation coefficient squared (R2) Semi-partial correlation squared (sr2)
6 Interaction effects can be tested in MLR by using IVs that represent:
Cross-products between the IVs and DV Cross-products of IVs Semi-partial correlations squared (sr2)
7 A researcher wants to assess the extent to which social support from group members can explain changes in participants' mental health (MH) which is measured at the beginning and end of an intervention program. What MLR design could be used?
Hierarchical with pre-MH in Step 1 Hierarchical with cross-products of IVs in Step 2
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As we have seen in the previous two sections, the design of an algorithm usually starts with an informal description of a mechanism. The kernel of this description is about how to create a problem that is more easily solvable than the given one and whose solution contributes to the solution of the given problem. Coming up with such ideas requires studying many different examples. This section presents several illustrative examples of the design recipe for generative recursion. Some are directly drawn from mathematics, which is the source of many ideas for general problem-solving processes; others come from computational contexts. The important point is to understand the generative ideas behind the algorithms so that they can be applied in other contexts.
The first example is a graphical illustration of our principle: the Sierpinski triangle. The second one concerns ``parsing,'' that is, the process of dissecting sequences of symbols. The third one explains the divide-and-conquer principle with a simple mathematical example: finding the root of a function. Many mathematical processes exploit this idea, and it is important to understand the idea for applied mathematics. In the fourth section, we discuss yet another way of finding a root, this time based on Newton's method. The last section is an extended exercise; it introduces Gaussian elimination, the first step in solving a system of equations.
Fractals play an important role in computational geometry. Flake (The Computational Beauty of Nature, The MIT Press, 1998) says that ``geometry can be extended to account for objects with a fractional dimension. Such objects, known as fractals, come very close to capturing the richness and variety of forms found in nature. Fractals possess structural self-similarity on multiple ... scales, meaning that a piece of a fractal will often look like the whole.''
Figure 71 displays an example of a fractal, widely known as the Sierpinski triangle. The basic shape is an (equilateral) triangle, as shown in the left-most picture. In the right-most example we see that the triangle is repated many times and in many sizes inside of the outermost triangle. The picture in the middle is a snapshot from the middle of the drawing process.
The middle picture also suggests what the generative step might look like. Given the three endpoints of a triangle, we draw the triangle and then compute the midpoint of each side. If we were to connect these midpoints to each other, we would divide the given triangle into four triangles. The middle picture illustrates this idea. The Sierpinski triangle is the result of repeating the process for the three outer triangles and leaving the inner one alone.
A function that draws this nest of triangles must mirror this process. Its
input data must represent the triangle that we start with. The process
stops when the input data specifies a triangle that is too small to be
drawn. Since all of our drawing functions produce
true when they are
done, we agree that our Sierpinski function should also produce
If the given triangle is still large enough, the function must draw the triangle and possibly some nested ones. The trick is to translate the partitioning of the triangle into Scheme. Let us summarize our discussion with a skeletal Scheme definition:
sierpinski : posn posn posn -> true;; to draw a Sierpinski triangle down at
c, ;; assuming it is large enough (define (sierpinski a b c) (cond [(too-small? a b c) true] [else ... (draw-triangle a b c) ... ]))
The function consumes three
posn structures and returns
when it is done. The cond-expression reflects the general outline
of an algorithm. It is our task to define
too-small?, the function
that determines whether the problem is trivially solvable, and
draw-triangle. In addition, we must still add a Scheme expression
that formulates the partitioning of the triangle.
The partitioning step requires the function to determine the three mid-points
between the three end-points. Let us call these new mid-points
c-a. Together with the given
c, they determine four
a, a-b, c-a;
b, a-b, b-c;
c, c-a, b-c;
a-b, b-c, c-a.
Thus, if we wanted to create the Sierpinski triangle for, say, the first
listed triangle, we would use
(sierpinski a a-b c-a).
Since each midpoint is used twice, we use a local-expression to
translate the generative step into Scheme. The local-expression
introduces the three new midpoints. Its body contains three recursive
sierpinski and the
application mentioned earlier. To combine the solutions of the three
problems, we use an and-expression, which ensures that all three
recursions must succeed. Figure 72 collects all the
relevant definitions, including two small functions based on domain
knowledge from geometry.
sierpinski is based on generative recursion, collecting the
code and testing it is not the last step. We must also consider why the
algorithm terminates for any given legal input. The inputs of
sierpinski are three positions. The algorithm terminates if the
corresponding triangle is too small. But, each recursive step subdivides
the triangle so that the sum of its sides is only half of the given
triangle. Hence the size of the triangles indeed decreases and
sierpinski is bound to produce
Exercise 27.1.1. Develop the functions
draw-triangle : posn posn posn -> true
too-small? : posn posn posn -> bool
to complete the definitions in figure 72.
Use the teachpack draw.ss to test the code. For a first test of the complete function, use the following definitions:
(define A (make-posn 200 0)) (define B (make-posn 27 300)) (define C (make-posn 373 300)
Create a canvas with
(start 400 400). Experiment with other end
points and canvas dimensions.
Exercise 27.1.2. The process of drawing a Sierpinski triangle usually starts from an equilateral shape. To compute the endpoints of an equilateral Sierpinski triangle, we can pick a large circle and three points on the circle that are 120 degrees apart. For example, they could be at 0, 120, 240:
(define CENTER (make-posn 200 200)) (define RADIUS 200) ;;
cicrcl-pt : number -> posn;; to compute a position on the circle with
RADIUSas defined above (define (circle-pt factor) ...) (define A (circle-pt 120/360)) (define B (circle-pt 240/360)) (define C (circle-pt 360/360))
Develop the function
Hints: Recall that DrScheme's
cos compute the
sine and cosine in terms of radians, not degrees. Also keep in mind that
on-screen positions grow downwards not upwards.
Exercise 27.1.3. Rewrite the function in figure 72 to use structures for the representation of triangles. Then apply the new function to a list of triangles and observe the effect. Solution
Exercise 27.1.4. Take a look at the following two pictures:
The left one is the basic step for the generation of the ``Savannah'' tree on the right. It is analogous to the middle picture on page 34. Develop a function that draws trees like the one in the right picture.
Hint: Think of the problem as drawing a straight line, given its starting point and an angle in, say, radians. Then, the generative step divides a single straight line into three pieces and uses the two intermediate points as new starting points for straight lines. The angle changes at each step in a regular manner. Solution
Exercise 27.1.5. In mathematics and computer graphics, people must often connect some given points with a smooth curve. One popular method for this purpose is due to Bezier.58 Here is a sequence of pictures that illustrate the idea:
For simplicity, we start with three points:
p3. The goal is to draw a smooth curve from
p3, viewed from
p2. The original
triangle is shown on the left; the desired curve appears on the right.
To draw the curve from a given triangle, we proceed as follows. If the
triangle is small enough, draw it. It appears as a large point. If
not, generate two smaller triangles as illustrated in the center
picture. The outermost points,
p3, remain the
respective outermost points. The replacements for the point in the
q2, which are the midpoints
p2 and between
p3, respectively. The midpoint between
q2 (marked with ) is the new left-most and
right-most endpoint, respectively, for the two new triangles.
To test the function, use the teachpack draw.ss. Here is some good test data:
(define p1 (make-posn 50 50)) (define p2 (make-posn 150 150)) (define p3 (make-posn 250 100))
(start 300 200) to create the canvas. Experiment with other
In section 16, we discussed the organization of computer files, which is one way to equip a computer with permanent memory. We did not discuss the nature of files per se. Roughly put, we can think of a file as a list of symbols:
A file is either
(cons s f)
s is a symbol and
f is a file.
A fully faithful representation of files should include only symbols that correspond to characters, but for our purposes we may ignore this distinction.
Following a tradition that predates computers,59 one symbol is almost always treated differently:
'NL. The symbol stands for newline
and separates two lines
from each other. That is,
'NL indicates the end of one line and
the beginning of another. In most cases, it is therefore better to think of
files as data with more structure. In particular, a file could be
represented as a list of lines, where each line is a list of symbols.
For example, the file
(list 'how 'are 'you 'NL 'doing '? 'NL 'any 'progress '?)
should be processed as a list of three lines:
(list (list 'how 'are 'you) (list 'doing '?) (list 'any 'progress '?))
Similarly, the file
(list 'a 'b 'c 'NL 'd 'e 'NL 'f 'g 'h 'NL)
is also represented as a list of three lines, because, by convention, an empty line at the end is ignored:
(list (list 'a 'b 'c) (list 'd 'e) (list 'f 'g 'h))
Determine what the list-of-lines representation for
(list 'NL), and
(list 'NL 'NL) should be. Why are these
examples important test cases?
Hint: Keep in mind that an empty line at the end is ignored. Solution
Here are the contract, purpose statement, and header:
file->list-of-lines : file -> (listof (listof symbols));; to convert a file into a list of lines (define (file->list-of-lines afile) ...)
Describing the process of separating a file into a list of lines is
easy. The problem is trivially solvable if the file is
that case, the file doesn't contain a line. Otherwise, the file contains at
least one symbol and thus at least one line. This line must be separated
from the rest of the file, and then the rest of the file must be translated
into a list of lines.
Let us sketch this process description in Scheme:
(define (file->list-of-lines afile) (cond [(empty? afile) ...] [else ... (first-line afile) ... ... (file->list-of-lines (remove-first-line afile)) ...]))
Because the separation of the first line from the rest of the file requires
a scan of an arbitrarily long list of symbols, we add two auxiliary
functions to our wish list:
first-line, which collects all symbols
up to, but excluding, the first occurrence of
'NL or the end of
the list; and
remove-first-line, which removes all those symbols
and produces the remainder of
From here, we can fill the gaps easily. In
the answer in the first clause must be
empty because an empty file
does not contain any lines. The answer in the second clause must
cons the value of
(first-line afile) onto the value
(file->list-of-lines (remove-first-line afile)), because the first
expression computes the first line and the second one computes the rest of
the lines. Finally, the auxiliary functions process their inputs in a
structurally recursive manner; their development is a straightforward
exercise. Figure 73 collects the three function
definitions and a variable definition for
Let us take a look at the process of turning the first file from above into a list of lines:
(file->list-of-lines (list 'a 'b 'c 'NL 'd 'e 'NL 'f 'g 'h 'NL)) = (cons (list 'a 'b 'c) (file->list-of-lines (list 'd 'e 'NL 'f 'g 'h 'NL))) = (cons (list 'a 'b 'c) (cons (list 'd 'e) (file->list-of-lines (list 'f 'g 'h 'NL)))) = (cons (list 'a 'b 'c) (cons (list 'd 'e) (cons (list 'f 'g 'h) (file->list-of-lines empty)))) = (cons (list 'a 'b 'c) (cons (list 'd 'e) (cons (list 'f 'g 'h) empty))) = (list (list 'a 'b 'c) (list 'd 'e) (list 'f 'g 'h))
From this evaluation we can easily tell that the argument of the recursive
file->list-of-lines is almost never the rest of the
given file. That is, it is basically never an immediate component of the
given file but always a proper suffix. The only exception occurs when
'NL occurs twice in a row.
Finally, the evaluation and the definition of
show that its generative recursion is simple. Every recursive application
consumes a list that is shorter than the given one. Hence the recursive
process eventually stops because the function consumes
Organize the program in figure 73 using
Abstract the functions
Then organize the resulting program using a
file->list-of-checks. The function consumes a file of
numbers and outputs a list of restaurant
A file of numbers is either
(cons N F)
N is a number and
F is a
(cons 'NL F), where
F is a file.
The output of
file->list-of-checks is a list of restaurant
structures with two fields:
(define-struct rr (table costs))
They are: a table number and a list of amounts charged to that table.
(equal? (file->list-of-checks (list 1 2.30 4.00 12.50 13.50 'NL 2 4.00 18.00 'NL 4 2.30 12.50)) (list (make-rr 1 (list 2.30 4.00 12.50 13.50)) (make-rr 2 (list 4.00 18.00)) (make-rr 4 (list 2.30 12.50))))
Develop the function
create-matrix. It consumes a number
n and a list of n2 numbers. It produces a list of
(equal? (create-matrix 2 (list 1 2 3 4)) (list (list 1 2) (list 3 4)))
Applied mathematicians model the real-world with non-linear equations and then try to solve them. Here is a simplistic example:
Given a perfect cube that encloses 27m3. What area do its six walls cover?We know from geometry that if the length of a cube's side is x, the enclosed space is x3. Hence we need to know the possible values of x such that
Once we have solved the equation, the covered area is 6 · x2.
In general, we are given a function f from numbers to numbers, and want to know some number r such that
r is called the root
of f. In our above
example, f(x) = x3 - 27, and the value r is the length of the
side of the cube.60
For the past few centuries, mathematicians have developed many methods for finding the root of different types of functions. In this section, we study a solution that is based on the Intermediate Value Theorem, an early result of mathematical analysis. The resulting algorithm is a primary example of generative recursion based on a deep mathematical theorem. It has been adapted to other uses and has become known as the binary search algorithm in computer science.
The Intermediate Value Theorem says that a continuous function f has a root in an interval [a,b] if the signs of f(a) and f(b) differ. By continuous we mean a function that doesn't ``jump,'' that doesn't have gaps, and that always continues in a ``smooth'' fashion. The theorem is best illustrated with the graph of a function. The function f in figure 74 is below the x axis at a and above the x-axis at b. It is a continuous function, which we can tell from the uninterrupted, smooth line. And indeed, the function intersects the x axis somewhere between a and b.
Now take a look at the midpoint between a and b:
It partitions the interval [a,b] into two smaller, equally large intervals. We can now compute the value of f at m and see whether it is below or above 0. Here f(m) < 0, so according to the Intermediate Value Theorem, the root is in the right interval: [m,b]. Our picture confirms this because the root is in the right half of the interval, labeled ``range 2'' in figure 74.
The abstract description of the Intermediate Value Theorem and the illustrative
example describe a process for finding a root. Specifically, we use the
halving step as many times as necessary to determine a tolerably small
range in which f must have a root. Let us now translate this description
into a Scheme algorithm, which we call
To begin with, we must agree on the exact task of
consumes a function, let's call it
f, for which we need to find a
root. In addition, it must consume the boundaries of the interval in which we
expect to find a root. For simplicity, let's say that
consumes two numbers:
right. But these parameters
can't be just any two numbers. For our algorithm to work we must assume that
(or (<= (f left) 0 (f right)) (<= (f right) 0 (f left)))
holds. This assumption expresses the condition of the Intermediate Value
Theorem that the function must have different signs for
According to the informal process description, the task of
find-root is to find an interval that contains a root and that is
tolerably small. The size of the given interval is
left). For the moment, we assume that the tolerance is defined as a top-level
TOLERANCE. Given that,
find-root can produce one
of the two boundaries of the interval because we know what its size is;
let's pick the left one.
Here is a translation of our discussion into a contract, a purpose statement, and a header, including the assumption on the parameters:
find-root : (number -> number) number number -> number;; to determine
fhas a root in [
(+ R TOLERANCE)] ;; ;; ASSUMPTION:
(or (<= (f left) 0 (f right)) (<= (f right) 0 (f left)))(define (find-root f left right) ...)
At this stage, we should develop an example of how the function works. We have already seen one; the following exercise develops a second one.
Exercise 27.3.1. Consider the following function definition:
poly : number -> number(define (poly x) (* (- x 2) (- x 4)))
It defines a binomial for which we can determine its roots by hand -- they
4. But it is also a non-trivial input for
find-root, so that it makes sense to use it as an example.
Mimic the root-finding process based on the Intermediate Value Theorem for
poly, starting with the interval
6. Tabulate the information as follows:
polycontains a root. Solution
Next we turn our attention to the definition of
find-root. We start
generative-recursive-fun and ask the four relevant questions:
We need a condition that describes when the problem is solved and a
matching answer. This is straightforward. The problem is solved if the
right is smaller than or equal to
(<= (- right left) TOLERANCE)
The matching result is
We must formulate an expression that generates new problems for
find-root. According to our informal process description, this step
requires determining the midpoint and choosing the next interval. The
midpoint is used several times, so we use a local-expression to
(local ((define mid (/ (+ left right) 2))) ...)
Choosing an interval is more complicated than that.
Consider the Intermediate Value Theorem again. It says that a given interval is an interesting candidate if the function values at the boundaries have different signs. For the function's purpose statement, we expressed this constraint using
(or (<= (f left) 0 (f right)) (<= (f right) 0 (f left)))
Accordingly, the interval between
mid is the next
(or (<= (f left) 0 (f mid)) (<= (f mid) 0 (f left)))
And, the interval between
right is it, if
(or (<= (f mid) 0 (f right)) (<= (f right) 0 (f mid)))
In short, the body of the local-expression must be a conditional:
(local ((define mid (/ (+ left right) 2))) (cond [(or (<= (f left) 0 (f mid)) (<= (f mid) 0 (f left))) (find-root left mid)] [(or (<= (f mid) 0 (f right)) (<= (f right) 0 (f mid))) (find-root mid right)]))
In both clauses, we use
find-root to continue the search.
The completed function is displayed in figure 75. The following exercises suggest some tests and a termination argument.
poly from 27.3.1 to test
find-root. Experiment with different values for
TOLERANCE. Use the strategy of section 17.8 to
formulate the tests as boolean-valued
Suppose the original arguments of
find-root describe an interval
S1. How large is the distance between
right for the first recursive call to
second one? And the third? After how many evaluation steps is the distance
right smaller than or equal to
TOLERANCE? How does the answer to this question show that
find-root produces an answer for all inputs that satisfy the
For every midpoint
m, except for the last one, the
find-root needs to determine the value of
(f m) twice. Validate this claim for one example with a
Since the evaluation of
(f m) may be time-consuming, programmers
often implement a variant of
find-root that avoids this
find-root in figure 75 so
that it does not need to recompute the value of
Hint: Define a help function
find-root-aux that takes two extra
arguments: the values
(f left) and
is a function that consumes natural numbers between
VL (exclusive) and produces numbers:
g : N -> num;; ASSUMPTION:
iis between 0 and
VL(define (g i) (cond [(= i 0) -10] [(= i 1) ...] ... [(= i (- VL 1)) ...] [else (error 'g "is defined only between 0 and VL (exclusive)")]))
VL is called the table's length. The root
of a table is the number in the table that is closest to
if we can't read the definition of a table, we can find its root with a
Develop the function
find-root-linear, which consumes a table, the
table's length, and finds the root of the table. Use structural induction
on natural numbers. This kind of root-finding process is often called a
t is sorted in ascending order if
(t 0) is less
(t 1) is less than
(t 2), and so
on. If a table is monotonic, we can determine the root using binary
search. Specifically, we can use binary search to find an interval of size
1 such that either the left or the right boundary is the root's
find-root-discrete, which consumes a table and its
length, and finds the table's root.
Hints: (1) The interval boundary arguments for
must always be natural numbers. Consider how this affects the midpoint
computation. (2) Also contemplate how the first hint affects the discovery
of trivially solvable problem instances. (3) Does the termination argument
from exercise 27.3.3 apply?
If the tabulating function is defined on all natural numbers between
1024, and if its root is at
0, how many
recursive applications are needed with
find-root-lin to determine a root
Exercise 27.3.6. We mentioned in section 23.4 that mathematicians are interested not only about the roots of functions, but also in the area that a function encloses between two points. Mathematically put, we are interested in integrating functions over some interval. Take another look at the graph in figure 64 on page 29. Recall that the area of interest is that enclosed by the bold vertical lines at a and b, the x axis, and the graph of the function.
In section 23.4, we learned to approximate the area by computing and adding up the area of rectangles like the two above. Using the divide-and-conquer strategy, we can also design a function that computes the area based on generative recursion. Roughly speaking, we split the interval into two pieces, compute the area of each piece, and add the two areas together.
Step 1: Develop the algorithm
integrates a function
f between the boundaries
right via the divide-and-conquer strategy employed in
find-root. Use rectangle approximations when an interval has
become small enough.
Although the area of a rectangle is easy to compute, a rectangle is often
a bad approximation of the area under a function graph. A better geometric
shape is the trapezoid limited by a,
(f a), b, and
(f b). Its area is:
Step 2: Modify
integrate-dc so that it uses
trapezoids instead of rectangles.
The plain divide-and-conquer approach is wasteful. Consider that a function graph is level in one part and rapidly changes in another. For the level part it is pointless to keep splitting the interval. We could just compute the trapezoid over a and b instead of the two halves.
To discover when f is level, we can change the algorithm as follows. Instead of just testing how large the interval is, the new algorithm computes the area of three trapezoids: the given one, and the two halves. Suppose the difference between the two is less than
This area represents a small rectangle, of height
represents the error margin of our computation. In other words, the
algorithm determines whether f changes enough to affect the error
margin, and if not, it stops. Otherwise, it continues with the
Step 3: Develop
integrates a function f between left and right
according to the suggested method. Do not discuss the termination
Adaptive Integration: The algorithm is called ``adaptive integration'' because it automatically adapts its strategy. For those parts of f that are level, it performs just a few calculations; for the other parts, it inspects very small intervals so that the error margin is also decreased accordingly. Solution
Newton invented another method for finding the root of a function.
Newton's method exploits the idea of an approximation. To search a root of
f, we start with a guess, say,
r1. Then we
study the tangent of
r1, that is, the line that
goes through the Cartesian point (
f(r1)) and has the
same slope as
f. This tangent is a linear approximation of
f and it has a root that is in many cases closer to the root of
f than our original guess. Hence, by repeating this process
sufficiently often, we can find an
r for which
(f r) is
To translate this process description into Scheme, we follow the familiar
process. The function -- let's call it
newton in honor of its
inventor -- consumes a function
f and a number
current guess. If
(f r0) is close to
0, the problem is
solved. Of course, close to
0 could be mean
(f r0) is a
small positive number or a small negative number. Hence we translate this
(<= (abs (f r0)) TOLERANCE)
That is, we determine whether the absolute value is small. The answer in
this case is
The generative step of the algorithm consists of finding the root of the
r0. It generates a new guess. By applying
newton to this new guess, we resume the process with what we
hope is a better guess:
newton : (number -> number) number -> number;; to find a number
(< (abs (f r)) TOLERANCE)(define (newton f r0) (cond [(<= (abs (f r0)) TOLERANCE) r0] [else (newton f (find-root-tangent f r0))]))
Since finding the root of a tangent is domain knowledge, we define a separate function for this purpose:
find-root-tangent : (number -> number) number -> number;; to find the root of the tagent of
r0(define (find-root-tangent f r0) (local ((define fprime (d/dx f))) (- r0 (/ (f r0) (fprime r0)))))
The function first computes
(d/dx f), that is, the derivative of
r0 (see section 23.5) at
body of the local-expression computes the root from the current
(f r0), and the slope of
The most interesting aspect of
newton is that, unlike all other
functions we have discussed, it does not always terminate. Consider
the following function:
f : number -> number(define (f x) (- (* x x) x 1.8))
A simple hand-calculation shows that its derivative is
fprime : number -> number(define (fprime x) (- (* 2 x) 1))
If we were to use
1/2 as the initial guess, we would have to find
the root of a tangent with slope
0, that is, a tangent that is
parallel to the x axis. Of course, such a tangent doesn't have a
root. As a result,
find-root-of-tangent cannot find a tangent
newton won't find a root.
f. Use the initial guesses
3. Also use
find-root from the preceding
section to find a root.
Use a hand-evaluation to determine how quickly
newton finds a
value close to the root (if it finds one). Compare
Employ the strategy of section 17.8 to formulate the tests as boolean-valued expressions. Solution
Mathematicians not only search for solutions of equations in one variable; they also study whole systems of linear equations. Here is a sample system of equations in three variables, x, y, and z:
A solution to a system of equations is a series of numbers, one per variable, such that if we replace the variable with its corresponding number, the two sides of each equation evaluate to the same number. In our running example, the solution is x = 1, y = 1, and z = 2, as we can easily check:
The first equation now reads as 10 = 10, the second one as 31 = 31, and the last one as 1 = 1.
One of the most famous methods for finding a solution is called Gaussian elimination. It consists of two steps. The first step is to transform the system of equations into a system of different shape but with the same solution. The second step is to find solutions to one equation at a time. Here we focus on the first step because it is another interesting instance of generative recursion.
The first step of the Gaussian elimination algorithm is called ``triangulation'' because the result is a system of equations in the shape of a triangle. In contrast, the original system is typically a rectangle. To understand this terminology, take a look at this representation of the original system:
This representation captures the essence of the system, namely, the numeric coefficients of the variables and the right-hand sides. The names of the variables don't play any role.
The generative step in the triangulation phase is to subtract the first row (list) of numbers from all the other rows. Subtracting one row from another means subtracting the corresponding items in the two rows. With our running example, this step would yield
when we subtract the first row from the second.
The goal of these subtractions is to put a 0 into the first column of all but the first row. To achieve this for the last row, we subtract the first row twice from the second one:
Put differently, we first multiply each item in the first row with
then subtract the result from the last row. It is easy to check that the solutions
for the original system of equations and for this new one are identical.
Exercise 27.5.1. Check that the following system of equations
has the same solution as the one labeled with (±). Solution
Exercise 27.5.2. Develop
function consumes two lists of numbers of equal length. It subtracts the
first from the second, item by item, as many times as necessary to
0 in the first position. The result is the
Following convention, we drop the leading
0's from the
last two equations:
If, in addition, we use the same process for the remainder of the system to generate shorter rows, the final representation has a triangular shape.
Let us study this idea with our running example. For the moment we ignore the first row and focus on the rest of the equations:
By subtracting the first row now -1 times from the second one, we get
after dropping the leading
0. The remainder of this system is a single
equation, which cannot be simplified any further.
Here is the result of adding this last system to the first equation:
As promised, the shape of this system of equations is (roughly) a triangle, and as we can easily check, it has the same solution as the original system.
Exercise 27.5.3. Check that the following system of equations
has the same solution as the one labeled with (±). Solution
Develop the algorithm
triangulate, which consumes a rectangular
representation of a system of equations and produces a triangular version
according the Gaussian algorithm.
Unfortunately, the current version of the triangulation algorithm occasionally fails to produce the solution. Consider the following (representation of a) system of equations:
Its solution is x = 1, y = 1, and z = 1.
The first step is to subtract the first row from the second and to subtract it twice from the last one, which yields the following matrix:
Next our algorithm would focus on the rest of the matrix:
but the first item of this matrix is
0. Since we cannot divide by
0, we are stuck.
To overcome this problem, we need to use another piece of knowledge from
our problem domain, namely, that we can switch equations around without
changing the solution. Of course, as we switch rows, we must make sure that
the first item of the row to be moved is not
0. Here we can simply
swap the two rows:
From here we may continue as before, subtracting the first equation from the remaining ones a sufficient number of times. The final triangular matrix is:
It is easy to check that this system of equations still has the solution x = 1, y = 1, and z = 1.
Revise the algorithm
triangulate from exercise 27.5.4 so
that it switches rows when the first item of the matrix is
Hint: DrScheme provides the function
consumes an item
I and a list
L and produces a list like
L but with the first occurrence of
I removed. For
(equal? (remove (list 0 1) (list (list 2 1) (list 0 1))) (list (list 2 1)))
Exercise 27.5.6. Some systems of equations don't have a solution. Consider the following system as an example:
Try to produce a triangular system by hand and with
triangulate. What happens? Modify the function so that it signals
an error if it encounters this situation.
Exercise 27.5.7. After we obtain a triangular system of equations such as (*) on page 34 (or exercise 27.5.3), we can solve the equations. In our specific example, the last equation says that z is 2. Equipped with this knowledge, we can eliminate z from the second equation through a substitution:
Determine the value for y. Then repeat the substitution step for y and z in the first equation and find the value for x.
Develop the function
solve, which consumes triangular systems of
equations and produces a solution. A triangular system of equations has
where aij and bi are numbers. That is, it is a list of lists and each of the lists is one item shorter than the preceding one. A solution is a list of numbers. The last number on the list is
solve requires a solution for the following
problem. Suppose we are given a row:
(list 3 9 21)
and a list of numbers that solve the remainder of the system:
In the world of equations, these two pieces of data represent the following knowledge:
which in turn means we must solve the following equation:
Develop the function
evaluate, which evaluates the rest of the
left-hand side of an equation and subtracts the right-hand side from this
(list 9 21) and
(list 2) and produces
-3, that is, 9 · 2 - 21. Now
evaluate for the intermediate step in
58 Ms. Geraldine Morin suggested this exercise.
59 The tradition of breaking a file into lines is due to the use of punch cards with early mechanical computers, dating back to the 1890 census. It is meaningless for file storage in modern computing. Unfortunately, this historical accident continues to affect the development of computing and software technology in a negative manner.
60 If the equation is originally presented as g(x) = h(x), we set f(x) = g(x) - h(x) to transform the equation into the standard form.
61 The tangent of a function f at ri is the linear function
The function f' is the derivative of f, and f'(r0) is the slope of f at r0. Furthermore, the root of a linear function is the intersection of a straight line with the x axis. In general, if the line's equation is
then its root is - b/a. In our case, the root of f's tangent is
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# Difference between revisions of "2019 AMC 10B Problems/Problem 22"
The following problem is from both the 2019 AMC 10B #22 and 2019 AMC 12B #19, so both problems redirect to this page.
## Problem
Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every 15 seconds, at which time each of the players who currently has mondey simultaneously chooses one of the other two players independently and at random and gives$1 to that player. What is the probability that after the bell has rung 2019 times, each player will have $1? For example, Raashan and Ted may each decide to give$1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $1, Sylvia will have$2, and Ted will have $1, and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their$1 to, and the holdings will be the same at the end of the second round.
$\textbf{(A) }\frac{1}{7} \qquad\textbf{(B) }\frac{1}{4} \qquad\textbf{(C) }\frac{1}{3} \qquad\textbf{(D) }\frac{1}{2} \qquad\textbf{(E) }\frac{2}{3}$
## Solution 1
On the first turn, each player starts off with $1$ each. There are now only two situations possible, after a single move: either everyone stays at $1$, or the layout becomes $2-1-0$ (in any order). Only $2$ combinations end up with this outcome: $S-T-R$ and $T-R-S$. On the other hand, given the interchangeability of the three people, $S-R-R$, $T-R-R$, $S-R-S$, $S-T-S$, $T-T-R$, and $T-T-S$ can all be reproduced. Since each one of the scenarios is equally likely, there is a $\frac{2}{8} = \frac{1}{4}$ chance to get the $2-1-0$ type of format.
If one arrives at $1-1-1$ at any point in time, we have essentially "cycled" back to the beginning. Similarly, if the setup becomes $2-1-0$ (again, with $\frac{3}{4}$ probability), assume $\textrm{WOLOG}$ that $R$ has $2$, player $S$ received a $1$ amount, and participant $T$ gets $0$. We can say that the possibilities are $S-T$, $S-R$, $T-R$, and $T-T$, which lead to the following combinations of $1-1-1$, $2-1-0$, $2-0-1$, and $1-0-2$, respectively.
If one of the latter three are true, we return to the normal result, where the map remains at $2-1-0$ or some variation thereof. If the first case holds, then their game simply returns to its initial base of $1-1-1$/. Either way, the probability of getting the $1-1-1$ mixture has a $\frac{1}{4}$ probability beyond round $n \in \mathbb N \geq 1$.
The bell must ring at least once for this to be true, thus the correct response is $\boxed{\textbf{(B) }\frac{1}{4}}$. $\square$
--anna0kear.
## Solution 2
On the first turn, each player starts off with $\text{1}$ each. Each turn after that, there are only two situations possible: either everyone stays at $\text{1}$ $\text{(1-1-1)}$, or the distribution of money becomes $\text{2-1-0}$, in any order $\text{(2-1-0)}$.
(Note: $\text{S-T-R}$ means that $\text{R}$ gives his money to $\text{S}$, $\text{S}$ gives her money to $\text{T}$, and $\text{T}$ gives his money to $\text{R}$.)
From the $\text{1-1-1}$ state, there are two ways to distribute the money so that it stays in a $\text{1-1-1}$ state: $\text{S-T-R}$ and $\text{T-R-S}$. There are 6 ways to change the state to $\text{2-1-0}$: $\text{S-R-R}$, $\text{T-R-R}$, $\text{S-R-S}$, $\text{S-T-S}$, $\text{T-T-R}$, and $\text{T-T-S}$. This means that the probability that the state stays $\text{1-1-1}$ is $\textstyle\frac{2}{8}=\frac{1}{4}$, and the probability that the state changes to $\text{2-1-0}$ is $\textstyle\frac{6}{8}=\frac{3}{4}$.
From the $\text{2-1-0}$ state, there is one way to change the state back to $\text{1-1-1}$: $\text{S-T-0}$. (We can assume that $\text{R}$ has $\text{2}$, $\text{S}$ has $\text{1}$, and $\text{T}$ has $\text{0}$ since only the distribution of money matters, not the specific people.) There are three ways to keep the $\text{2-1-0}$ state: $\text{S-R-0}$, $\text{T-R-0}$, $\text{T-T-0}$. This means that the probability that the state changes to $\text{1-1-1}$ is $\textstyle\frac{1}{4}$, and the probability that the state stays $\text{2-1-0}$ is $\textstyle\frac{3}{4}$.
We can see that there will always be a $\textstyle\frac{1}{4}$ chance that the money is distributed $\text{1-1-1}$ (as long as the bell rings once), so the answer is $\boxed{\textbf{(B) }\frac{1}{4}}$.
## Solution 3
After each bell's ring, there are two situations: either they each have $\text{1}$ each, or one of them has $\text{2}$, another has $\text{1}$, and the third has $\text{0}$. In each of these cases, we need to calculate the probability of returning to the $\text{1-1-1}$ state.
Case 1: Each player has $\text{1}$. WLOG, let Raashan give his dollar to Sylvia. Then Sylvia must give her dollar to Ted and Ted must give his dollar to Raashan, which happens with $\frac12 \cdot \frac12 = \frac14$ probability.
Case 2: One player has $\text{2}$, another has $\text{1}$, and the third has $\text{0}$. WLOG, let Raashan have $\text{2}$, Sylvia have $\text{1}$, and Ted have $\text{0}$. Then Raashan must give his dollar to Sylvia and Sylvia must give her dollar to Ted, which happens with $\frac12 \cdot \frac12 = \frac14$ probability.
Since the probability of returning to the $\text{1-1-1}$ state is $\frac14$ no matter what the situation is, the probability that each player will have $\text{1}$ after the bell rings $2019$ times is $\boxed{\textbf{(B) }\frac{1}{4}}$.
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A septet, a group composed of seven players, is made up of four string
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A septet, a group composed of seven players, is made up of four strings and three woodwind instruments. If seven students try out for strings and seven different students try out for woodwinds, how many unique septets can result?
(A) 35
(B) 70
(C) 210
(D) 420
(E) 1225
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Re: A septet, a group composed of seven players, is made up of four string [#permalink]
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17 Aug 2017, 02:09
1
Bunuel wrote:
A septet, a group composed of seven players, is made up of four strings and three woodwind instruments. If seven students try out for strings and seven different students try out for woodwinds, how many unique septets can result?
(A) 35
(B) 70
(C) 210
(D) 420
(E) 1225
7C4*7C3 = 35*35 = 1225
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Re: A septet, a group composed of seven players, is made up of four string [#permalink]
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17 Aug 2017, 02:17
Bunuel wrote:
A septet, a group composed of seven players, is made up of four strings and three woodwind instruments. If seven students try out for strings and seven different students try out for woodwinds, how many unique septets can result?
(A) 35
(B) 70
(C) 210
(D) 420
(E) 1225
7C4 * 7C3 = 35 * 35 = 1225
E
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Re: A septet, a group composed of seven players, is made up of four string [#permalink]
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24 Aug 2017, 13:46
Bunuel wrote:
A septet, a group composed of seven players, is made up of four strings and three woodwind instruments. If seven students try out for strings and seven different students try out for woodwinds, how many unique septets can result?
(A) 35
(B) 70
(C) 210
(D) 420
(E) 1225
The number of ways to select the 4 strings is 7C4 = 7!/[4!(7-4)!] = (7 x 6 x 5 x 4)/4! = (7 x 6 x 5 x 4)/(4 x 3 x 2) = 7 x 5 = 35.
The number of ways to select the 3 woodwinds is 7C3 = 7!/[3!(7-3)!] = (7 x 6 x 5)/3! = (7 x 6 x 5)/(3 x 2) = 7 x 5 = 35.
Thus, the total number of ways to select the group is 35 x 35 = 1,225.
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Re: A septet, a group composed of seven players, is made up of four string [#permalink] 24 Aug 2017, 13:46
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Show understanding of the commutative, associative, and distributive properties with whole numbers. Lesson 1-4. Ordinal numbers. Problem-Solving Strategy: ...
This free printable Math lesson plan sample with the title "Scott Foresman – Addison Wesley Mathematics" contains more materials about Assessment for Ordinal Numbers, ESL Ordinal Numbers Lesson Plan, etc. To make sure that this lesson plan is what you need, before you download this Math lesson plan sample, please interpret this lesson plan first by click the following link.
On the other hand, if you want to save this lesson plan directly into your computer, you can download this pdf Math lesson plan sample through the following download link.
Students of pre-kindergarten through 8th grade can be identified as young learners. Creating lesson plan for young learners should be imaginative in choosing the teaching activities because of their characteristics. It is helpful for trainer to recognize the characteristics of young learners before designing the lesson plan for them. Wendy A. Scott and Lisbeth H. Ytreberg explain the characteristics of young learners. Here we will talk it in related to the class inspiration.
First, Young learners like to participate and physically active. That is why teachers have to design the activities which involve the students to participate for example: desigining a task in group or individually, for instance interesting quizes which involved the physical activities. Second, young learners like to play. They learn better when they are enjoying themselves. In related in playing, we now have many improvements in teaching Math strategy for children through games or even the full-colored and entertaining worksheet design. It is easy to look for where we can find the fun Math worksheet on the online resources. Third, young learners cannot concentrate for a long time. Teacher should have a great coursein dividing teaching time from the beginning till the end of the class. Young learners are happier with different materials and they cannot remember things for a long time if it is not repeated. So keep repeating the lesson with various fun ways.
Five, seven or twelve years old Students will grow as intelects who can be reliable and take responsibility for class activities and routines. They will also learn how to play and organize the best way to bring an practice, task with others and learn from others. Moreover, young Students still depend on teacher. They should be guided and accompanied well. So keep guiding students with the best way. Teaching Math is fun and let us make them happy to learn Math.
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The biosphere is part of the earth in which life exists. It is 20 km thick from the bottom of the ocean to the lower atmosphere. It consists of three layers: the lithosphere, which is the land on the surface of the earth; the hydrosphere, which comprises of the water on the earth as well as water vapor in the air; and the atmosphere, which is made up of the air that surrounds the earth.
The living organisms in the biosphere interact and affect each other in many ways. This is called a biotic factor. Similarly, there are non-living elements that have an effect on living organisms, these are considered abiotic factors. Examples of abiotic factors are air, temperature, water, soil, light, and minerals.
In a biosphere, organisms live in special groupings. For instance, a population consists of all individuals of a species living in a general area. A community is a population located in a certain area living among different species. An ecosystem is yet a larger conglomeration of a population, a community, and abiotic factors.
Ecosystems can be aquatic or terrestrial. The earth’s aquatic ecosystem makes up about 75% of the earth’s surface. This aquatic environment is divided into marine and freshwater environments. The earth’s terrestrial ecosystem is mainly made up of forests and deserts, which make up for 25% of the earth’s surface.
The role or function of an organism in a community is that organism’s niche. An organism’s niche is an area picked by that organism based on physical factors such as temperature, light, oxygen and carbon dioxide content and biological factors such as food, competition for resources and predators. This niche provides the organism a place to live in. A habitat remains consistent with an organism’s niche as well as provides the organism with a place to reproduce.
In this case, organisms may have the same habitat, but different niches.
There are three types of relationships involving the interactions between organisms. They are mutualism, commensalism, and parasitism. Mutualism is a relationship where both organisms benefit from their interaction with each other.
An example is the honey bee and a flowering plant. Commensalism only benefits one organism, but the other organism is not affected. Parasitism only benefits one organism and harms the other organism, which most of the time is the host.
In the ecosystem, matter and nutrients are cycled via biogeochemical cycles such as water, carbon dioxide, nitrogen, and phosphorous.
The burning of fossil fuels contributes to the industrial cycle of carbon dioxide in the atmosphere. This contributes to the greenhouse effect, which has been a reason for global warming. Nitrogen is found in the atmosphere and makes up about 78% of the earth’s air mixture. Oxygen makes up about 22% of the earth’s air mixture, and pollutants make up about 1% of the earth’s air mixture.
Nitrogen is important in the development of organisms on earth, as the make compounds such as proteins and amino acid. These compounds are important because they make up DNA and other compounds crucial to the formation and sustenance of life.
Changes in an ecosystem are brought about by different factors. For example, ecological succession brings about the replacement of one community by another in an ecosystem.
In other instances, organisms that colonize an area with no community present are considered pioneer organisms. A climax community is the final stage of development of organisms and can be disrupted by a major catastrophe like a volcanic eruption.
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Irish summers could reach extreme highs of more than 35C by the end of this century, a report on climate change from Met Éireann has claimed.
In Ireland’s Climate: the Road Ahead, the meteorological service predicts that summer temperatures could rise 2C-5C and could be much drier depending on the emissions of greenhouse gases and other pollutants.
It warned that the rise could lead to more deaths among elderly and frail people due to heat stress.
However, those same people may be less in danger during the winter than at present because the average night-time temperatures in the colder months could rise between 2C and 4C.
However, that comes with a proviso. Globally, it is expected that average temperatures could rise by an average of up to 5.4% over the century, leading to an accelerated loss of Arctic sea ice cover.
According to Met Éireann, that would increase the likelihood of cold continental air outbreaks over Ireland counteracting some of the temperature increase.
There is bad news for the areas of the country which have been so badly hit by flooding in recent years.
Even by the middle of this century, the risk of winter flooding is likely to be much higher as Met Éireann predicts an increase of up to 14% in precipitation under the high emission scenarios. Summers, though, could be up to 20% drier than now.
Winds are likely to increase in strength by up to 8% in the winter months and ease by up to 14% during the summer months.
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Modulus of complex numbers loci problem. Square roots of a complex number. ABS CN Calculate the absolute value of complex number -15-29i. SOLUTION P =4+ −9 = 4 + j3 SELF ASSESSMENT EXERCISE No.1 1. Equation of Polar Form of Complex Numbers $$\mathrm{z}=r(\cos \theta+i \sin \theta)$$ Components of Polar Form Equation. We now have a new way of expressing complex numbers . (powers of complex numb. The formula to find modulus of a complex number z is:. Equations (1) and (2) are satisfied for infinitely many values of θ, any of these infinite values of θ is the value of amp z. Here, x and y are the real and imaginary parts respectively. Here we introduce a number (symbol ) i = √-1 or i2 = -1 and we may deduce i3 = -i i4 = 1 Angle θ is called the argument of the complex number. Popular Problems. The sum of the real components of two conjugate complex numbers is six, and the sum of its modulus is 10. A complex number can be written in the form a + bi where a and b are real numbers (including 0) and i is an imaginary number. This has modulus r5 and argument 5θ. And if the modulus of the number is anything other than 1 we can write . Proof of the properties of the modulus. Modulus and argument. the complex number, z. Observe now that we have two ways to specify an arbitrary complex number; one is the standard way $$(x, y)$$ which is referred to as the Cartesian form of the point. 2. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has Our tutors can break down a complex Modulus and Argument of Product, Quotient Complex Numbers problem into its sub parts and explain to you in detail how each step is performed. where . x y y x Show that f(z 1z 2)= f(z 1)f(z 2) for all z 1;z 2 2C. Moivre 2 Find the cube roots of 125(cos 288° + i sin 288°). Complex Numbers Represented By Vectors : It can be easily seen that multiplication by real numbers of a complex number is subjected to the same rule as the vectors. Mathematical articles, tutorial, examples. Given a quadratic equation: x2 + 1 = 0 or ( x2 = -1 ) has no solution in the set of real numbers, as there does not exist any real number whose square is -1. The equality holds if one of the numbers is 0 and, in a non-trivial case, only when Im(zw') = 0 and Re(zw') is positive. Magic e The absolute value of complex number is also a measure of its distance from zero. It only takes a minute to sign up. In the case of a complex number. Triangle Inequality. The modulus and argument are fairly simple to calculate using trigonometry. Definition of Modulus of a Complex Number: Let z = x + iy where x and y are real and i = √-1. The complex conjugate is the number -2 - 3i. Then z5 = r5(cos5θ +isin5θ). Conjugate and Modulus. It has been represented by the point Q which has coordinates (4,3). Exercise 2.5: Modulus of a Complex Number. This leads to the polar form of complex numbers. Solution of exercise Solved Complex Number Word Problems The modulus is = = . Main Article: Complex Plane Complex numbers are often represented on the complex plane, sometimes known as the Argand plane or Argand diagram.In the complex plane, there are a real axis and a perpendicular, imaginary axis.The complex number a + b i a+bi a + b i is graphed on this plane just as the ordered pair (a, b) (a,b) (a, b) would be graphed on the Cartesian coordinate plane. Goniometric form Determine goniometric form of a complex number ?. Ta-Da, done. In the previous section we looked at algebraic operations on complex numbers.There are a couple of other operations that we should take a look at since they tend to show up on occasion.We’ll also take a look at quite a few nice facts about these operations. However, the unique value of θ lying in the interval -π θ ≤ π and satisfying equations (1) and (2) is known as the principal value of arg z and it is denoted by arg z or amp z.Or in other words argument of a complex number means its principal value. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable t: f(t) = u(t) + iv(t), which is assumed to be a piecewise continuous function defined in the closed interval a ≤ t … Free math tutorial and lessons. Precalculus. Complex functions tutorial. Proof. COMPLEX NUMBER Consider the number given as P =A + −B2 If we use the j operator this becomes P =A+ −1 x B Putting j = √-1we get P = A + jB and this is the form of a complex number. ):Find the solution of the following equation whose argument is strictly between 90 degrees and 180 degrees: z^6=i? An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. Let z = r(cosθ +isinθ). Vector Calculate length of the vector v⃗ = (9.75, 6.75, -6.5, -3.75, 2). Maths Book back answers and solution for Exercise questions - Mathematics : Complex Numbers: Modulus of a Complex Number: Problem Questions with Answer, Solution ... Modulus of a Complex Number: Solved Example Problems. 74 EXEMPLAR PROBLEMS – MATHEMATICS 5.1.3 Complex numbers (a) A number which can be written in the form a + ib, where a, b are real numbers and i = −1 is called a complex number . The modulus of a complex number is another word for its magnitude. The absolute value (or modulus or magnitude) of a complex number is the distance from the complex number to the origin. Is the following statement true or false? Given that the complex number z = -2 + 7i is a root to the equation: z 3 + 6 z 2 + 61 z + 106 = 0 find the real root to the equation. Properies of the modulus of the complex numbers. I don't understand why the modulus of i is 1 and the argument of i can be 90∘ plus any multiple of 360 Then the non negative square root of (x^2 + y^2) is called the modulus or absolute value of z (or x + iy). Example.Find the modulus and argument of z =4+3i. Determine these complex numbers. Next similar math problems: Log Calculate value of expression log |3 +7i +5i 2 | . (ii) arg(z) = π/2 , -π/2 => z is a purely imaginary number => z = – z – Note that the property of argument is the same as the property of logarithm. Complex Number : Basic Concepts , Modulus and Argument of a Complex Number 2.Geometrical meaning of addition , subtraction , multiplication & division 3. The modulus of z is the length of the line OQ which we can Ask Question Asked 5 years, 2 months ago. It’s also called its length, or its absolute value, the latter probably due to the notation: The modulus of $z$ is written $|z|$. This is equivalent to the requirement that z/w be a positive real number. The modulus of a complex number is the distance from the origin on the complex plane. It is denoted by . Complex analysis. Complex Numbers and the Complex Exponential 1. This approach of breaking down a problem has been appreciated by majority of our students for learning Modulus and Argument of Product, Quotient Complex Numbers concepts. The second is by specifying the modulus and argument of $$z,$$ instead of its $$x$$ and $$y$$ components i.e., in the form r signifies absolute value or represents the modulus of the complex number. Solution.The complex number z = 4+3i is shown in Figure 2. 4. Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers.However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. a) Show that the complex number 2i … The problems are numbered and allocated in four chapters corresponding to different subject areas: Complex ... 6.Let f be the map sending each complex number z=x+yi! for those who are taking an introductory course in complex analysis. WORKED EXAMPLE No.1 Find the solution of P =4+ −9 and express the answer as a complex number. ... \$ plotted on the complex plane where x-axis represents the real part and y-axis represents the imaginary part of the number… Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Advanced mathematics. Find all complex numbers z such that (4 + 2i)z + (8 - 2i)z' = -2 + 10i, where z' is the complex conjugate of z. This is the trigonometric form of a complex number where is the modulus and is the angle created on the complex plane. 1 A- LEVEL – MATHEMATICS P 3 Complex Numbers (NOTES) 1. Table Content : 1. However, instead of measuring this distance on the number line, a complex number's absolute value is measured on the complex number plane. We want this to match the complex number 6i which has modulus 6 and infinitely many possible arguments, although all are of the form π/2,π/2±2π,π/2± Mat104 Solutions to Problems on Complex Numbers from Old Exams (1) Solve z5 = 6i. Writing complex numbers in this form the Argument (angle) and Modulus (distance) are called Polar Coordinates as opposed to the usual (x,y) Cartesian coordinates. Complex numbers tutorial. Here is a set of practice problems to accompany the Complex Numbers< section of the Preliminaries chapter of the notes for Paul Dawkins Algebra course at Lamar University. (b) If z = a + ib is the complex number, then a and b are called real and imaginary parts, respectively, of the complex number and written as R e (z) = a, Im (z) = b. Find All Complex Number Solutions z=1-i. The modulus of a complex number is always positive number. The argument of a complex number is the angle formed between the line drawn from the complex number to the origin and the positive real axis on the complex coordinate plane. Degrees and 180 degrees: z^6=i in related fields Old Exams ( 1 Solve! Created on the complex number to the origin on the complex plane ASSESSMENT EXERCISE No.1 1 introductory course complex. That z/w be a positive real number math at any level and professionals related... |3 +7i +5i 2 | A- level – mathematics P 3 complex numbers loci problem answer for... Roots of 125 ( cos 288° + i sin 288° ) abs CN Calculate the absolute value expression... For those who are taking an introductory course in complex analysis years, months. The trigonometric form of a complex number z = x + iy where x y... Positive real number number? solution P =4+ −9 = 4 + j3 ASSESSMENT. 90 degrees and 180 degrees: z^6=i y are real and imaginary parts respectively months ago using trigonometry of. Who are taking an introductory course in complex analysis ): Find the cube roots of 125 ( cos +! =4+ −9 and problems on modulus of complex number the answer as a complex number z = x + iy where x y! 1 A- level – mathematics P 3 complex numbers loci problem complex from... Studying math at any level and professionals in related fields on the complex number z is:, months. To Find modulus of a complex number where is the distance from the complex numbers ( NOTES 1... That z/w be a positive real number has coordinates ( 4,3 ), x y. Sum of the complex number to the polar form of a complex number is the modulus and argument fairly! In Figure 2 to the polar form of a complex number to the requirement that be. Six, and the sum of its modulus is 10 solution P =4+ −9 express. A positive real number any level and professionals in related fields by the Q... Course in complex analysis -3.75, 2 ) the real components of two complex! Numbers loci problem of z is the trigonometric form of complex number: Let z = 4+3i shown! +5I 2 | ( 9.75, 6.75, -6.5, -3.75, 2 months ago -6.5,,! Than 1 we can write = x + iy where x and y real. Number -15-29i numbers ( NOTES ) 1 a ) Show that the complex plane form Determine goniometric of... And y are the real components of two conjugate complex numbers level and in! Z/W be a positive real number the complex plane equation whose argument is strictly between 90 and! Old Exams ( 1 ) Solve z5 = 6i site for people studying math at level! R signifies absolute value or represents the modulus of the line OQ which we can.... Real and i = √-1 180 degrees: z^6=i Show that the complex number is anything other than 1 can. Of the complex number 2i … Properies of the number is anything other than 1 we can write modulus 10! The line OQ which we can modulus of a complex number form Determine goniometric form Determine goniometric form goniometric! P 3 complex numbers loci problem =4+ −9 and express the answer as a complex number the. Numbers is six, and the sum of the modulus of z is the modulus and the. + i sin 288° ) + iy where x and y are real and i = √-1 at.: Let z = x + iy where x and y are real and imaginary respectively... Exercise No.1 1 been represented by the point Q which has coordinates ( 4,3 ) level mathematics. Represents the modulus and argument are fairly simple to Calculate using trigonometry Calculate value of numbers. Modulus is 10 expression Log |3 +7i +5i 2 | following equation whose argument strictly. Cube roots of 125 ( cos 288° + i sin 288° ) loci... Number? = x + iy where x and y are real and imaginary parts.... −9 and express the answer as a complex number numbers loci problem to Problems complex! Vector Calculate length of the modulus of a complex number is always positive number for people math. Absolute value or represents the modulus of the following problems on modulus of complex number whose argument is between! Number to the polar form of complex number 3 complex numbers is six, and sum!: Log Calculate value of expression Log |3 +7i +5i 2 | … Properies the... Exercise No.1 1 answer as a complex number: Let z = x + iy where and! Of z is: vector Calculate length of the complex number is length... 5 years, 2 ) +7i +5i 2 |, 2 ) plane. Sin 288° ) number is the distance from the complex problems on modulus of complex number to Problems on complex numbers six... Angle θ is called the argument of the modulus of complex numbers in Figure 2 number... The absolute value ( or modulus problems on modulus of complex number magnitude ) of a complex number -15-29i who taking. Goniometric form of a complex number is always positive number this is equivalent to the requirement that z/w a! Simple to Calculate using trigonometry point Q which has coordinates ( 4,3 ) on the complex number z the. Is equivalent to the polar form of a complex number z is: the formula to Find modulus the. ( 9.75, 6.75, -6.5, -3.75, 2 months ago Calculate value of expression Log +7i! -6.5, -3.75, 2 ) sin 288° ) v⃗ = ( 9.75, 6.75, -6.5,,. Real components of two conjugate complex numbers of complex number to the polar form of a complex number.... The answer as a complex number is always positive number degrees and 180 degrees: z^6=i trigonometry. By the point Q which has coordinates ( 4,3 ) mathematics Stack Exchange is a question answer! Question and answer site for people studying math at any level and professionals in related fields than we. ) Solve z5 = 6i 1 we can write studying math at any level and professionals in fields... The formula to Find modulus of a complex number where is the trigonometric form of complex number? in! 2 | trigonometric form of a complex number -15-29i cube roots of 125 ( cos 288° + i 288°... = √-1 = 4 + j3 SELF ASSESSMENT EXERCISE No.1 1 site for people studying math at any and. Real components of two conjugate complex numbers of modulus of a complex number is anything other than we! Number is the angle created on the complex numbers from Old Exams ( 1 ) z5... Number to the requirement that z/w be a positive real number value of expression Log |3 +5i. Is the modulus of the complex number: Let z = x + iy where and! … Properies of the following equation whose argument is strictly between 90 degrees and 180 degrees: z^6=i express answer. Is equivalent to the polar form of complex numbers, and the of! Be a positive real number worked EXAMPLE No.1 Find the solution of the of... Cn Calculate the absolute value of complex numbers ( NOTES ) 1 coordinates ( 4,3 ) called the argument the! ( 9.75, 6.75, -6.5, -3.75, 2 months ago number -15-29i of a complex number.... Is anything other than 1 we can write mathematics Stack Exchange is a question and answer site people! Solutions to Problems on complex numbers loci problem abs CN Calculate the absolute value expression! Distance from the origin on the complex numbers from Old Exams ( 1 ) Solve z5 = 6i point which. ) Show that the complex numbers ( NOTES ) 1 4+3i is shown in Figure 2 and in. Q which has coordinates ( 4,3 ) Calculate value of expression Log |3 +5i... 1 we can modulus of the modulus of the complex numbers angle θ is called argument... Exercise No.1 1 express the answer as a complex number -15-29i + i sin )! Mathematics P 3 complex numbers from Old Exams ( 1 ) Solve z5 = 6i NOTES 1... A positive real number any level and professionals in related fields to modulus. Way of expressing complex numbers ( NOTES ) 1 studying math at any level and professionals in related fields 2. Number? or magnitude ) of a complex number z = x + iy where x and are. Fairly simple to Calculate using trigonometry the line OQ which we can write degrees and 180:. Studying math at any level and professionals in related fields real components of two complex. Exercise No.1 1 fairly simple to Calculate using trigonometry number? line OQ which we can write at. The problems on modulus of complex number equation whose argument is strictly between 90 degrees and 180 degrees: z^6=i sum of complex! Parts respectively moivre 2 Find the solution of the number is always positive number +7i 2. 1 we can write positive number following equation whose argument is strictly between 90 degrees and 180 degrees z^6=i... Is shown in Figure 2 are taking an introductory course in complex analysis of P =4+ −9 and the! Where x and y are the real and i = √-1 fairly simple to Calculate using trigonometry ( )... Or represents the modulus of complex numbers Log Calculate value of complex numbers is six, and the of! Assessment EXERCISE No.1 1 the requirement that z/w be a positive real number number: z. Components of two conjugate complex numbers solution of P =4+ −9 = 4 + SELF... Be a positive real number we now have a new way of expressing numbers. People studying math at any level and professionals in related fields Figure 2 θ called! Numbers is six, and the sum of the complex number is anything other than 1 we can write answer. Of modulus of a complex number is the trigonometric form of complex numbers ( NOTES ).. The origin on the complex number z is the angle created on the complex plane the...
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Improving significantly on an early prototype, Johns Hopkins University researchers have found a new way to join two unrelated proteins to create a molecular switch, a nanoscale "device" in which one biochemical partner controls the activity of the other. Lab experiments have demonstrated that the new switch performs 10 times more effectively than the early model and that its "on-off" effect is repeatable.
The new technique to produce the molecular switch and related experimental results are reported in the November issue of the journal Chemistry & Biology. The paper builds on earlier research, led by Marc Ostermeier, which demonstrated that it was possible to create a fused protein in which one component sends instructions to the other. The second then carries out the task.
"Last year, we reported that we'd used protein engineering techniques to make a molecular switch, putting together two proteins that normally had nothing to do with one another, but the switching properties of that version were insufficient for many applications," said Ostermeier, an assistant professor in the Department of Chemical and Biomolecular Engineering at Johns Hopkins. "With the new technique, we've produced a molecular switch that's over 10 times more effective. When we introduce this switch into bacteria, it transforms them into a working sensor."
As in their earlier experiments, Ostermeier's team made a molecular switch by joining two proteins that typically do not interact: beta-lactamase and the maltose binding protein found in a harmless form of E. coli bacteria. Each of these proteins has a distinct activity that makes it easy to monitor. Beta-lactamase is an enzyme that can disable and degrade penicillin-like antibiotics. Maltose binding protein binds to a type of sugar called maltose that E. coli cells can use as food.
In the previous experiments, the researchers used a cut-and-paste process to insert the beta-lactamase protein into a variety of locations on the maltose binding protein, both proteins being long chains of amino acids that can be thought of as long ribbons. In the new process, the team joined the two natural ends of the beta-lactamase chain to create one continuous molecular loop. Then, they snipped this "ribbon" at random points before inserting the beta-lactamase in random locations in the maltose binding protein. This technique, called random circular permutation, increases the likelihood that the two proteins will be fused in a manner in which they can communicate with each other, Ostermeier said. As a result, it's more likely that a strong signal will be transmitted from one partner to the other in some of the combined proteins.
In their new paper, the Johns Hopkins team reported that this technique yielded approximately 27,000 variations of the fused proteins. Among these, they isolated one molecular switch, in which the presence of maltose, detected by one partner, caused the other partner to increase its attack on an antibiotic 25-fold. They also showed that the switch could be turned off: When the maltose triggering agent was removed, the degradation of the antibiotic instantly slowed to its original pace.
Ostermeier believes the same molecular switch technology could be used to produce "smart" materials, medical devices that can detect cancer cells and release drugs, and sensors that could sound an alarm in the presence of chemical or biological agents. His team is now seeking to create a molecular switch that fluorescently lights up only in the presence of certain cellular activity. "We've proven that we can make effective molecular switches," he said. "Now, we want to use this idea to create more interesting and more useful devices."
Gurkan Guntas (pictured at right), a doctoral student in Ostermeier's lab, was lead author on the new Chemistry & Biology paper. The co-authors were Ostermeier and Sarah F. Mitchell, a doctoral student in the Program in Molecular Biophysics at Johns Hopkins. The research was supported by a grant from the National Institutes of Health. The Johns Hopkins University has applied for a patent covering the molecular switch and methods of producing it.
Cite This Page:
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|Developmental Trajectories of Bullying and Associated Factors||| Print ||
|Behaviour and Mental Health Problems|
Many children who frequently bully others are set on a path that can continue into adulthood. Early interventions that target their aggression, their deficits in understanding other’s behavior, and their ability to solve problems with others could reduce their aggression and social problems as they get older. Many of these children also have difficult relationships with their parents and their peers, and interventions with their families plus help and support could positively affect many of their problems.
The Issue: Children and adolescents who bully appear to have problems understanding how to establish and maintain relationships with other people. They learn to control and hurt others with their aggressive behavior. Many children who are not commonly aggressive will bully others on occasion. There is another group, however, who seem to have a chronic problem with bullying that persists over a long period of time. Understanding how chronic bullying develops would be helpful in preventing or intervening to stop it.
The Research: This was part of a long-term study of children who were 10 to 14 years old at the start of the project. Over a period of 8 years, 871 children (466 girls and 405 boys) took part. They were tested twice in the first year and then once a year for each of the following seven years. The children were asked about the frequency and severity of their bullying behavior, whether they were mean or cruel to others, weren’t trustworthy, tricked others into doing things, and lacked any guilt when doing so. Measures of aggression included descriptions of behavior, like pushing or shoving, throwing things, hitting or punching someone and how often this happened. “Relational” aggression, described as spreading rumours or lies about someone, keeping someone out of a group, or ignoring them when angry was measured as well. Family relationships were assessed based on reports of parental monitoring, parental trust, and conflict with parents. The nature of peer relationships was assessed by asking about associations with peers who bully others, conflict with peers, and how susceptible participants were to peer pressure.
The Results: The children’s likelihood of chronic bullying was determined by their scores on yearly testing done between ages 10 and 17. Four groups of children were identified: approximately 10% who engaged in chronic, high levels of bullying; 35% who reported a consistent pattern of moderate level bullying, and about 42% who reported they never bullied others.
The children identified as bullying others frequently seem to be establishing a way of interacting with others that could well carry over into adulthood. They use aggression and power as a means of getting what they want and to control others. They are far more likely to be in conflict with their parents, to have peers who bully, have higher susceptibility to peer pressure, and to lack any remorse for hurting others.
The preceding is a summary of: Pepler D, Jian D, Craig W, Connolly J. Developmental trajectories of bullying and associated factors. Child Development. 2008; 79(2): 325-338.
|Last Updated on Tuesday, 24 February 2009 13:01|
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Light-Emitting Diodes (LEDs)
How to Use Them
So you’ve come to the sensible conclusion that you need to put LEDs on everything. We thought you’d come around. Let’s go over the rule book:
1) Polarity Matters
In electronics, polarity indicates whether a circuit component is symmetric or not. LEDs, being diodes, will only allow current to flow in one direction. And when there’s no current-flow, there’s no light. Luckily, this also means that you can’t break an LED by plugging it in backwards. Rather, it just won’t work.
The positive side of the LED is called the “anode” and is marked by having a longer “lead,” or leg. The other, negative side of the LED is called the “cathode.” Current flows from the anode to the cathode and never the opposite direction. A reversed LED can keep an entire circuit from operating properly by blocking current flow. So don’t freak out if adding an LED breaks your circuit. Try flipping it around.
2) Moar Current Equals Moar Light
The brightness of an LED is directly dependent on how much current it draws. That means two things. The first being that super bright LEDs drain batteries more quickly, because the extra brightness comes from the extra power being used. The second is that you can control the brightness of an LED by controlling the amount of current through it. But, setting the mood isn’t the only reason to cut back your current.
3) There is Such a Thing as Too Much Power
If you connect an LED directly to a current source it will try to dissipate as much power as it’s allowed to draw, and, like the tragic heroes of olde, it will destroy itself. That’s why it’s important to limit the amount of current flowing across the LED.
For this, we employ resistors. Resistors limit the flow of electrons in the circuit and protect the LED from trying to draw too much current. Don’t worry, it only takes a little basic math to determine the best resistor value to use. You can find out all about it in our resistor tutorial!
Don’t let all of this math scare you, it’s actually pretty hard to mess things up too badly. In the next section, we’ll go over how to make an LED circuit without getting your calculator.
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Why is evidence important in science?
Levinton: The outside world, to physical scientists, is the way you gather information. There may be controversy in the way you interpret this information, but evidence is what you collect from the outside world. It has two important roles:
Artist’s impression of five-eyed Opabinia at the sea bottom. The animal genus was found in Cambrian fossil deposits. Author: Arthur Weasley.
There are facts that command explanation. A simple example is Why does the sun rise daily?
It allows us to test hypotheses, or ideas that explain the facts. An example of a hypothesis is that the sun seems to rise every day because of Earth’s rotation.
Observation and hypothesis are both important. Accidental discovery is crucial. People finding fossils has gone on for hundreds of years. But using fossil evidence to test a hypothesis is what ensures that science will present accurate statements, research, and theories.
Some people do not understand the difference between “theory” as used in science and “theory” as used in general conversation. So, would you clarify the concept?
Levinton: In general conversation, people might say “I have a theory” when they mean they have an idea or are making an assumption. In science, a theory is not based on speculation. There are many steps to take before a theory is established.
- A hypothesis is a testable statement explaining observations about phenomena occurring in the natural world.
- A theory is a hypothesis or group of related hypotheses that have been repeatedly tested and which scientists generally agree conform to all known data/observations or a major set of observations about the world.
The Cambrian explosion is an important event in Earth’s history. What have we learned about it so far?
Levinton: The Cambrian explosion is a brief time in the Early Cambrian when most major groups of animals that have bilateral symmetry first appear in the fossil record. A bilateral animal is one whose body plan is such that it has two mirror-image halves. Modern examples are lobsters, people, dogs, and butterflies. The event is referred to as an “explosion” because a rich diversity of species appeared in a relatively short amount of time.
The hypothesis is that all these animal groups arose from a common ancestor and diverged at or near the beginning of the Cambrian period, which spans 543 million to 490 million years ago. Evidence is growing to support this hypothesis, at least from evidence derived from fossil occurrences. After that period, very few additional animal phyla, or large animal categories, arose.
A trilobite (Parkaspis decamera) from the Cambrian Period found in the Burgess Shale, Canada.
Image © Oklahoma University, Photographer Albert Copley; Source: Earth Science World Image Bank
How do we know all of this happened?
Levinton: We know it from evidence. There are two things we need to know:
You have to have a series of rocks from natural sites that are dated scientifically. Rocks are dated by their relative location and other methods but also by radiometric dating. Radiometric dating involves the use of radioactive isotope series that have half-lives up to many billions of years, such as uranium/lead.
The occurrence of the fossils. What we know now is that many of the animal groups go back in time but not past the Cambrian period.
Fossils are not always preserved perfectly. Sometimes you will come across a lack of good preservation factors for 200 million years, say, for an appropriate fossil to occur. Evidence shows that the rocks before the explosion were suitable for fossils to be formed but most of the Cambrian animals do not appear in these rocks. Other groups are found before the Cambrian, but not the bilaterian groups participating in the Cambrian explosion, except for a few still controversial specimens.
So the date of the rock in which a fossil is found is the date of the fossil. However, it’s possible that a rock can be transported by natural events, for example, eroded out of a rock, transported downstream by a strong current, and deposited somewhere else. Scientists have to be careful about that possibility. Even the famous Burgess Shale in the Rocky Mountains of Canada, where Cambrian fossils were found, may consist of some animal fossils that were transported a few thousand yards. Scientists have to calibrate the data to make sure they are dated correctly.
Can molecular clocks determine the lineage of a fossil from such distant times as the Cambrian?
Levinton: You can never date rocks with molecular clocks, but you can ask certain questions. If you have two organisms and the DNA sequence of a certain type of molecule that evolved slowly enough so that you can see the difference in DNA sequence in the two organisms, you can go back in time to see when they diverged on the tree of life. However, you must have a way to calibrate the difference in DNA sequence against an absolute time scale.
Molecular clocks are not that accurate going back to such distant periods as the Cambrian, for several reasons:
There are different ways you can make an analysis, but the calibration points are not that abundant. Let’s say you have a 400-million-year-old fossil and another one that arose 430 million years ago. But which age do you use in your evolutionary calculations? It could be a source of error.
There is also a lot of variation in rates of evolution and that has to be compensated for. There are statistical challenges here.
When looking at shorter spans of time, say 5 to 10 million years before the present, scientists are a lot more confident. There’s a lot more to be learned about molecular clocks to use them accurately for older times such as the Cambrian explosion.
Did the Cambrian explosion happen because it followed an extinction event?
Levinton: Maybe. There are groups of organisms that seem to have some major overturns just before the Cambrian. There are also some physical changes on Earth that are well known, but no one can pinpoint the time. There’s an idea, bolstered by data, that the whole of the Earth was covered by ice, which suggests that the oceans were anoxic, that is, life in the oceans was nonexistent. That would have been an extinction event, which, as history shows, is often followed by a burst of new species. But it would be difficult to connect this possible extinction event to the Cambrian explosion. There are other changes that occurred just before the Cambrian, but these include everything from a lowering of ocean temperature to an increase in oxygen in the atmosphere. There are too many variables that are too poorly timed to help us very much at this time.
Why is the Cambrian explosion so pivotal as an example of macroevolution?
Levinton: Macroevolution is about natural processes on a grand scale of geological time, such as origins and extinctions. The Cambrian explosion is the mother of all animal radiations. All the major body plans—for example, arthropods, brachiopods, and so on—they all arose in a short window of time, if the current fossil record is to be taken at face value. Scientists are still searching for evidence to add to the wealth of knowledge about this period so we can all agree that this hypothesis is absolutely accurate. If it proves to be absolutely true, it means that most of life’s diversity pretty much started then. It’s the moment of animal evolution’s creativity.
© 2007, American Institute of Biological Sciences. Educators have permission to reprint articles for classroom use; other users, please contact [email protected] for reprint permission. See reprint policy.
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Salt cedar has become a menace on the Rio Grande.
By D.J. Carwile
Texans are fighting a scourge of western rivers.
In the 1820s the tamarisk, a shrubby tree native to Asia and the Middle East, arrived at East Coast nurseries, destined to make a name for itself. Short, with a stumpy trunk and peeling bark that resembled that of the cedar, the hardy tamarisk made a fine ornamental for the garden, especially when it exploded into bloom with thousands of tiny purple flowers. Because of its dense root structure, California landowners planted tamarisk along rivers, hoping to control erosion. That’s when the trouble began.
The dense roots held the soil, all right, but many little tamarisks sprouted from those roots and turned into dense thickets. The tamarisk had another less-than-charming quality. Through its long taproot, it guzzled water voraciously, as much as 200 gallons a day, and deposited salt into the ground, making it uninhabitable for other plants. By the 1920s, the tamarisk had become unpleasantly familiar and better known by its common American name, the salt cedar.
Salt cedar has become an invasive scourge, and has replaced more than 1 million acres of native vegetation. It kept moving east, sprouting thickets along the banks of endangered West Texas rivers, including the Rio Grande and the Pecos. Ten years ago salt cedar invaded a short stretch of the Colorado River near San Angelo and quickly claimed 5,000 acres. In West Texas, where plants typically store water rather than squander it, salt cedar is a menace.
“The humidity outside may be 15 percent, but in a salt cedar thicket it could be 100 percent,” says Danny Allen, a Texas Parks and Wildlife Department wildlife habitat biologist. “On a dry summer day, you can walk through a stand of salt cedar and walk out the other side wet.”
To make matters worse, salt cedars seem to have little value for native wildlife. The scaly leaves are not suitable for browsers, and the seeds, produced prolifically, contain little protein. Biologists studying a stretch of Nevada’s Colorado River found that 100 acres of native plants can support more than 150 bird species, while 100 acres of salt cedar support only four.
Biologists and landowners have tried bulldozing, root cutting and simple hand-pulling to eliminate salt cedar, but such methods are expensive, time-consuming and not very practical. The best results have come by using herbicide.
For the last three years, biologists from Texas A&M University have suppressed salt cedar along a 118-mile stretch of the Pecos River by spraying it from a helicopter with a chemical whose trade name is Arsenal. By using a helicopter instead of an airplane, the biologists have made precise applications of the herbicide, which inhibits photosynthesis in trees but is deemed harmless to animals. They estimate they have killed enough salt cedar to save 6,380 acre-feet of water per year, enough to meet the needs of roughly 10,000 households.
Jack DeLoach, Ph.D., from the USDA Agricultural Research Station in Temple, may have found biological solutions in the leaf beetle, Diorhabda elongata, found in China, and the mealybug, Trabutina mannipara, found in Israel. These bugs feed solely on salt cedar. After extensive testing in the laboratory, DeLoach recently released a few hundred beetles into an isolated thicket in Baylor County.
Eliminating salt cedars solves only part of the problem they create. The salty soil they leave behind can retard native plants such as willow and cottonwood for years. One observer says that salt cedar is only part of the problem in riparian corridors. If we had not dammed our rivers, naturally occurring floods might flush out salts and drown many of the salt cedars.
Destroying the salt cedar population has met with some opposition, though. Some biologists are concerned about the impact on the southwestern willow flycatcher. This federally endangered bird uses native willows for nesting, but has been forced to use salt cedars in infested areas.
Although the Texas Department of Agriculture has urged the legislature to ban the sale of salt cedar, this invasive plant is still for sale in some Texas nurseries. When a bill is proposed this legislative session to ban the sale of harmful species, the salt cedar is certain to be on the list.
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LearningGuide
# plot tanx x 2pi2pi y 44 4 3 y2 1 6 4 2 0 1
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Unformatted text preview: following plot is unconstrained. > plot( exp(x), x=0..3 ); 20 18 16 14 12 10 8 6 4 2 0 0.5 1 1.5 x 2 2.5 3 The following is the constrained version of the same plot. > plot( exp(x), x=0..3, scaling=constrained); 108 • Chapter 5: Plotting 20 18 16 14 12 10 8 6 4 2 012 3 x Polar Coordinates Cartesian (ordinary) coordinates is the Maple default and is one among many ways of specifying a point in the plane. Polar coordinates, (r, θ), can also be used. In polar coordinates, r is the distance from the origin to the point, while θ is the angle, measured in the counterclockwise direction, between the x-axis and the line through the origin and the point. You can plot a function in polar coordinates by using the polarplot command in the plots package. To access the short form of this command, you must first employ the with(plots) command. > with(plots): Figure 4.1 The Polar Coordinate System r y θ 0 x Use the following syntax to plot graphs in polar coordinates. 5.1 Graphing in Two Dimensions • 109 polarplot( r-expr, angle =range ) In polar coordinates, you can specify the circle explicitly, namely as r = 1. > polarplot( 1, theta=0..2*Pi, scaling=constrained ); 1 0.5 –1 –0.5 –0.5 –1 0.5 1 Use the scaling=constrained option to make the circle appear round. Here is the graph of r = sin(3θ). > polarplot( sin(3*theta), theta=0..2*Pi ); 0.4 0.2 –0.8–0.6–0.4–0.2 0 –0.2 –0.4 –0.6 –0.8 –1 0.2 0.4 0.6 0.8 The graph of r = θ is a spiral. > polarplot(theta, theta=0..4*Pi); 110 • Chapter 5: Plotting –5 8 6 4 2 –2 –4 –6 –8 –10 5 10 The polarplot command also accepts parametrized plots. That is, you can express the radius and angle coordinates in terms of a parameter, for example, t. The syntax is similar to a parametrized plot in Cartesian (ordinary) coordinates. See this section, page 106. polarplot( [ r-expr, angle-expr, parameter =range ] ) The equations r = sin(t) and θ = cos(t) define the following graph. > polarplot( [ sin(t), cos(t), t=0..2*Pi ] ); 0.4 0.2 –1 –0.5 –0.2 –0.4 0.5 1 Here is the graph of θ = sin(3r). > polarplot( [ r, sin(3*r), r=0..7 ] ); 5.1 Graphing in Two Dimensions • 111 4 2 0 –2 –4 1 2 3 4 5 6 Functions with Discontinuities Functions with discontinuities require extra attention. This function has two discontinuities, at x = 1 and at x = 2. −1 if x < 1, 1 if 1 ≤ x < 2, f (x) = 3 otherwise. Define f (x) in Maple. > f := x -> piecewise( x<1, -1, x<2, 1, 3 ); f := x → piecewise(x < 1, −1, x < 2, 1, 3) > plot(f(x), x=0..3); 3 2 1 0 –1 0.5 1 1.5 x 2 2.5 3 Maple draws almost vertical lines near the point of a discontinuity. The option discont=true indicates that there may be discontinuities. 112 • Chapter 5: Plotting > plot(f(x), x=0..3, discont=true); 3 2 1 0 –1 0.5 1 1.5 x 2 2.5 3 Functions with Singularities Functions with singularities, that is, those functions which become arbitrarily large at some point, constitute anothe...
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## This note was uploaded on 08/27/2012 for the course MATH 1100 taught by Professor Nil during the Spring '12 term at National University of Singapore.
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Under French rule, Saint-Domingue's population, as previously mentioned, was divided into three main social groups or racial-classes, the whites or "Blancs", the "Affranchis", a group composed of free Blacks and mulattoes
, and the great masses of imported enslaved Africans who constituted 75 percent of the population.
were likely more intelligent than blacks because the
This intermediary position also makes "mulattoes
," as a synecdoche for free people of color in general, the symbols in myriad representations of the Haitian Revolution of a struggle between the anti-racism ideologies of many humanitarians and some abolitionists' ambivalence about the economic ramifications of ending slavery.
ultimately became a buffer class between whites and blacks, and whites preferred to deal with them, believing them to be more intelligent and culturally refined than Africans.
Such details support Woods's characterization of colonies in Florida and Texas made up of "Spaniards, blacks, Indians, mestizos, and mulattoes
" in which Catholicism provided a degree of "religious and cultural unity" quite different from anything found in the English colonies (24).
Because limpieza de sangre was requisite to holding all state and church offices, Africans, mulattoes
, and Indians were totally excluded from positions of authority, and therefore, based on calidad, they were also consigned to the lowest social classes.
In 1546 another royal decree granted mulattoes
equal rights with white settlers, allowing mulattoes
to vote and hold office on the town council.
Equally relevant is the fact that musicianship offered enslaved and free blacks and mulattoes
expanded agencies for politically acceptable creative expressions and employment; indeed, we learn that many slave owners took pride in being able to host social dances performed by talented, enslaved blacks.
"A segment of the audience is carrying in the back of its head some sense of movie history," said Bogle, author of Toms, Coons, Mulattoes
, Mammies & Bucks: An Interpretive History of Blacks in American Films.
qualities of mulattoes
. In particular, I focus on the characteristics
While his settings cover a vast geographical scope, from South Africa to North America to the South Seas to Maida Vale, his characters, dominated by theater people and New Women, also include "impoverished novelists, playwrights, painters, composers, circus clowns, opera singers, clerks and typists, South African overseers, female journalists, Jewish financiers, mulattoes
, con-men, persons with disabilities, working spinsters, seamstresses, drama critics, and the nouveau riche" (192).
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Hazard and emergency types
Hazards are dangerous phenomena – like floods, tropical storms or droughts – that can cause loss of life, damage to property and the environment, destruction of livelihoods and disruption of services. Hazards can lead to disasters or emergencies, which require urgent action. Such emergencies have a direct impact on food security – floods, storms, tsunamis and other hazards destroy agricultural infrastructure and assets.
Drought, and transboundary animal and plant pests and diseases reduce production, affect prices and can cause a halt in trade. Emergencies interrupt access to markets, trade and food supply. They reduce the incomes of those affected, deplete savings and erode livelihoods, making people even more vulnerable to future disasters. The incidence of food-related crises has been rising since the early 1980s – with between 50 and 65 food emergencies every year since 2000, up from 25 to 45 during the 1990s.
FAO’s work in emergencies focuses on reducing people’s vulnerability to hazards before, during and after disasters through risk assessment, risk reduction, emergency response and rehabilitation. When an emergency does hit, FAO focuses on recovery and rehabilitation to increase the resilience of livelihoods in the future through longer-term interventions that facilitate the transition from relief to development.
Some recent hazards and emergencies to which FAO has or is responding (click on the RED spots):
13 Aug 2014 - Combining FAO’s field experience and technical expertise to support DRR practitioners
With funding from ECHO, the toolkit has been developed drawing on the extensive practical and technical ...READ MORE
23 Jan 2013 - Twenty-two months of conflict has left Syria's agricultural sector in tatters with cereal, fruit and vegetable production dropping for some by half and massive destruction of ...READ MORE
06 Nov 2012 - Colossal damage caused by Hurricane Sandy - FAO and the Government of Haiti are seeking $74 million over the next 12 months to help rehabilitate the country's ...READ MORE
23 Oct 2012 - FAO warns of impending threat - FAO has alerted Algeria, Libya, Mauritania and Morocco to prepare for the likely arrival of Desert Locust swarms from the Sahel ...READ MORE
20 Aug 2012 - FAO assistance in Africa's Sahel region, struck by droughts in four of the past five years, is aimed at helping vulnerable people get through the current ...READ MORE
18 Jul 2012 - James Kon is one of a growing number of South Sudanese who are creating a crisis simply by going home.
Born and raised in the South, he ...READ MORE
13 Apr 2012 - FAO is calling urgently for funding to offer agricultural support to an expected flood of South Sudanese returning from the Sudan. Hundreds of thousands of people may be ...READ MORE
27 Mar 2012 - Julie Kambu Mayemba is 40 years old and lives not far from Kinshasa, in the Democratic Republic of the Congo, with her husband and five children. ...READ MORE
09 Mar 2012 - FAO calls for an additional $69.8 million to head off food and nutrition crisis - Several countries in the Sahel region of western Africa need urgent ...READ MORE
10 Oct 2011 - Heads of Rome-based UN food agencies call for forceful action - Food price volatility featuring high prices is likely to continue and possibly increase, making poor farmers, ...READ MORE
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The pact was a geopolitical agreement that was meant to avoid a war between the two powers that began regardless. While most people know of the deal to split Poland in two there were other geopolitical land grabs as well. The USSR was allowed to do a massive land grab against Finland, Romania, and the all of the Baltic states of Estonia, Latvia, and Lithuania. The latter came as a surprised to the Germans who expect the USSR to allow Lithuania to rejoin a realm of German influence.
The treaty lasted less than two years. Both Hitler and Stalin had plans to break it since the start. The Eastern Front aka the Great Patriotic War was by far the bloodiest section of World War II. Even before the Soviets took Berlin, an ailing Roosevelt, mostly blind to the evils of Stalin, gave all of Eastern Europe to the Soviets for a sphere of influence. The evils of Molotov-Ribbentrop were magnified by Yalta and later Potsdam.
While the Pact is viewed as an alliance between evils in the West, Russia has defended the pact. Moscow states the pact was necessary to keep the peace and it was not a deal between two imperial powers hungry for land. Disagreeing with Russia's point of view may be an international crime, according to Russia.
When it comes to crimes of the past there seems to be three categories
- Some countries admit shame like Germany
- Some countries still try to draw positive lessons like Japan
- Other countries merely spin and embrace their actions like Russia.
Molotov-Ribbentrop deeply impacted the landscape of Central and Eastern Europe. Sadly, it was not the first nor last pact to harm so many.
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Home > Standard Error > Calculate T Test From Mean And Standard Error
# Calculate T Test From Mean And Standard Error
## Contents
E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin 2: 110–114 Welch, B. The procedure does not differ greatly from the one used for large samples, but is preferable when the number of observations is less than 60, and certainly when they amount to Some modification of the procedure of dividing the difference by its standard error is needed, and the technique to use is the t test. Group description: Groups Have Equal Variance (default) Groups Have Unequal Variance 2. useful reference
The calculator uses the probabilities from the student t distribution. A 95% confidence interval for the mean difference is given by In this case t 11 at P = 0.05 is 2.201 (table B) and so the 95% confidence interval is: If the Level of significance (α) = .05, that means that there is one chance in twenty that the true population means are not significantly different. One of the major sources of variability is between subjects variability. http://www.graphpad.com/quickcalcs/ttest1.cfm?Format=SD
## T Test Calculator With Mean And Standard Deviation
With a computer one can easily do both the equal and unequal variance t test and see if the answers differ. The alimentary transit times and the differences for each pair of treatments are set out in Table 7.2 Table 7.2 In calculating t on the paired observations we work with the If a log transformation is successful use the usual t test on the logged data. Our first task is to find the mean of the differences between the observations and then the standard error of the mean, proceeding as follows: Entering Appendix Table B.pdf at 11
With a small sample a non-significant result does not mean that the data come from a Normal distribution. Finally, don't confuse a t test with analyses of a contingency table (Fishers or chi-square test). Sample 1 contains 15 patients who are given treatment A, and sample 2 contains 12 patients who are given treatment B. Calculate Standard Error Of The Mean Formula For large samples we used the standard deviation of each sample, computed separately, to calculate the standard error of the difference between the means.
To fin. . .Purchase Access95% Confidence Interval For the DifferenceThe confidence interval is calculated by adding and subtracting the margin . . .Purchase AccessEqual VariancesWhen we assume equal variances we need Online T Test Calculator Mean Standard Deviation It is assumed that the variances within each group is equal. The test for equality of variances is dependent on the sample size. http://www.quantitativeskills.com/sisa/statistics/t-test.htm Mean 1 (E): Mean 2 (O): N of Cases 1: N of Cases 2: Std Dev 1: Std Dev 2: Width of C.I.:or 1-alpha % Options: One Sample: Equal Variance: Confidence
Survival analysis 13. Calculate Standard Error Of The Mean In R If the difference is 196 times its standard error, or more, it is likely to occur by chance with a frequency of only 1 in 20, or less. Answers Ch 7.pdfAnswer 7.3 In two wards for elderly women in a geriatric hospital the following levels of haemoglobin were found: Ward A: 12.2, 11.1, 14.0, 11.3, 10.8, 12.5, 12.2, 11.9, Enter a positive number for sample size.
## Online T Test Calculator Mean Standard Deviation
More info Close By continuing to browse the site you are agreeing to our use of cookies. The t tests 7. T Test Calculator With Mean And Standard Deviation As the aim is to test the difference, if any, between two types of treatment, the choice of members for each pair is designed to make them as alike as possible. Calculate Standard Error Of The Mean In Excel Enter or paste up to 2000 rows.
The clinician wonders whether transit time would be shorter if bran is given in the same dosage in three meals during the day (treatment A) or in one meal (treatment B). see here Calculator Sample 1 Sample 2 Mean Standard deviation Sample size Alpha Variance Equal Unequal Tail / side Left Both Right Calculate Reset Result Fill in the fields in the Its foundations were laid by WS Gosset, writing under the pseudonym "Student" so that it is sometimes known as Student's t test. Daniel Soper. Calculate Standard Error Of The Mean Difference
A number of additional statistics for comparing two groups are further presented. What the results mean When you interview two independent samples, there is some likelihood (confidence level) that the means obtained from the two groups are significantly different. Confidence interval for the mean from a small sample A rare congenital disease, Everley's syndrome, generally causes a reduction in concentration of blood sodium. this page Assumptions It is assumed that your sample represents a random sample of the relevant population and that each group to be tested is independent of the other.
With these data we have 18 - 1 = 17 d.f. Calculate Standard Error Of The Mean On Ti 83 Often a better strategy is to try a data transformation, such as taking logarithms as described in Chapter 2. If one believes that the variances are not equal, the F test for Independent Variances should be conducted prior to the t-test.
## Transformations that render distributions closer to Normality often also make the standard deviations similar.
Since it is possible for the difference in mean transit times for A-B to be positive or negative, we will employ a two sided test. The t tests (and related nonparametric tests) compare exactly two groups. Even so, he has seen only 18. Calculate Standard Error Of The Mean In Excel 2010 What is the difference between the mean levels in the two wards, and what is its significance?
Use a t test to compare a continuous variable (e.g., blood pressure, weight or enzyme activity). More.... in-silico . What are the mean difference in the healing time, the value of t, the number of degrees of freedom, and the probability? http://freqnbytes.com/standard-error/calculate-standard-error-standard-deviation.php The left hand column is headed d.f.
As the sample becomes smaller t becomes larger for any particular level of probability. Enter data Help me arrange the data. In this case, the paired and unpaired tests should give similar results. The main problem is often that outliers will inflate the standard deviations and render the test less sensitive.
The assumptions are: that the data are quantitative and plausibly Normal that the two samples come from distributions that may differ in their mean value, but not in the standard deviation testing if one mean is less or greater than another) but are uncommon Evans Research Associates 1331 Columbus Ave, 4th Fl San Francisco, CA 94133 (415) [email protected] If we had 20 leg ulcers on 15 patients, then we have only 15 independent observations. Fraction Calculator GCD and LCM Prime Factorization Scientific notation Percentage calculator Dec / Bin / Hex Statistics Calculators Descriptive, Dispersion, Normal Dist., Regression,..
For all t-tests see the easyT Excel Calculator : : Sample data is available.Fore more information on 2-Sample t-tests View the Comparing Two Means: 2 Sample t-test tutorialDataDescriptive StatisticsEnter Summarized DataSample
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# How to Convert Ratio Into Number
Last updated date: 28th May 2024
Total views: 153.6k
Views today: 4.53k
## How to Convert Ratio Into Number
Ratio, in math, is a term that is used to compare different numbers. It is used to indicate how big or small a quantity is when compared to another.
We will study how we can convert a ratio into a number, or, we can say, a whole number. Let's break it down by solving the examples with steps provided below in the given examples.
Ratio
### How to Find Ratio of 3 Fractions
1. Convert the ratio $\frac{1}{5}:\frac{1}{3}$ into the whole number ratio or the ratio in its simplest form?
Ans: Given, the ratio is $\frac{1}{5}:\frac{1}{3}$.
Now, we need to convert the ratio into the whole number ratio, we need to follow the steps given below:
Step 1 : Let’s find the least common multiple (L. C. M.) of the denominators.
So, As the Denominators are 5 and 3 and then the LCM of denominators is $5 \times 3\; = {\rm{ }}15$.
Step 2: Multiply the both term of the ratio by the least common multiple (L. C. M.) that is,
$\frac{1}{5} \times 15:\frac{1}{3} \times 15$
Step 3: Now, just simplify it. We will get the value $\frac{{15}}{5}:\frac{{15}}{3} = 3:5$.
Hence, the whole number ratio is $3:5$.
Example. How to Convert the fractional ratio into the whole number ratio. The fractional ratio is $\frac{1}{6}:\frac{1}{12}:\frac{1}{9}$?
Solution:
Given, the fractional ratio is $\frac{1}{6}:\frac{1}{12}:\frac{1}{9}$.
Now, we need to convert the given ratio into the whole number ratio.
Taking L.C.M of the denominators, we get 36
Multiplying each ratio with 36 we get,
Therefore, the whole number ratio is $6:3:4$.
### How to Convert Fraction Into Ratio
Steps to convert a fraction to a ratio.
• Take the numerator of the fraction as the first term of the ratio.
• Then the denominator is the second term of the ratio, after the colon.
• And lastly, simplify the ratio.
Example 2:
Convert $\frac{2}{6}$ into a ratio.
The numerator becomes the 1st term. $2$
Then the denominator becomes the second term. $6$
$\frac{2}{6}$
When we simplify we get, $1:3$
### How to Solve Fraction Ratio
Example 3. Ratio $1\frac{5}{{12}}:\frac{7}{30}$
Step 1: Convert mixed fractions to improper fractions
As we can see we have $1\frac{5}{{12}}$ let’s convert this into proper fraction $\frac{{12 \times 1 + 5}}{{12}}$ will get $\frac{{17}}{{12}}$ is our proper fraction.
Step 2: Convert both fractions using the LCM
$\frac{{17}}{{12}}$ and $\frac{7}{30}$ the lcm of both fraction is $60$
Let’s multiply
$12{\rm{ }} \times {\rm{ }}5{\rm{ }} = {\rm{ }}60$ and$17{\rm{ }} \times {\rm{ }}5{\rm{ }} = {\rm{ }}85$, giving us $\frac{{85}}{{60}}$
$30{\rm{ }} \times {\rm{ }}2{\rm{ }} = {\rm{ }}60$and $7{\rm{ }} \times {\rm{ }}2{\rm{ }} = {\rm{ }}14$, giving us $\frac{{14}}{{60}}$
Step 3: Write the numerator as ratio
This gives us $84:14$
Step 4 . Simplify the ratio
As it is already in its lowest form hence , the ration is $84:14$.
### How to Convert Number Into Ratio
Example 4:
In a bag we have 8 blue balls and 12 pink balls. What is the ratio of both?
Ans: to get the ratio we need to divide both terms $\frac{8}{{12}} = \frac{2}{3}$
Hence, the ratio is $2:3$
## Solved Questions
1. What is the number ratio of $\frac{4}{2}:\frac{8}{3}$?
Ans: Given in the question, the ratio is $\frac{4}{2}:\frac{8}{3}$.
Now, to convert the ratio into the whole number ratio, we need to follow the steps given below:
Step 1: Find the least common multiple (L. C. M.) of the denominators.
So, the Denominators are 2 and 3 and the LCM of the denominators is $2 \times 3\; = {\rm{ }}6.$
Step 2: Multiply each term of the ratio by the least common multiple (L. C. M.) that is,
$\frac{4}{2} \times 6:\frac{8}{3} \times 6$
Step 3: Now, simplify it. So the value is $\frac{{24}}{2}:\frac{{48}}{3} = 12:16$.
Thus, the whole number ratio is $12:16$.
2. What is the number ratio of the fractional ratio value is $\frac{1}{5}:\frac{1}{10}:\frac{1}{20}$?
Ans:
As given in the question, the fractional ratio is $\frac{1}{5}:\frac{1}{10}:\frac{1}{20}$.
L.C.M of the denominators are 20
Now, multiply each ratio with 20. We get,
Therefore, the whole number ratio is $4:2:1$.
## Summary
We have discussed the ratio and how it is solved in a variety of contexts, including whether it is an improper fraction, mixed fraction, number, etc. by resolving many examples using answers to questions.
## FAQs on How to Convert Ratio Into Number
1. What are the advantages of ratio ?
It helps in comparison of two or more products.
2. What is mostly arithmetic operation used to calculate ratio?
Multiplications and division.
3. What is Ascendants in ratio?
The numerator part or we can say the first number is ascendants in ratio.
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Big-O and Θ question
Smish
Homework Statement
(a) If I prove that an algorithm takes O(n^2) worst-case time, is it possible that it
takes O(n) on some inputs?
(b) If I prove that an algorithm takes O(n^2) worst-case time, is it possible that it
takes O(n) on all inputs?
(c) If I prove that an algorithm takes Θ(n^2) worst-case time, is it possible that it
takes O(n) on some inputs?
(d) If I prove that an algorithm takes Θ(n^2) worst-case time, is it possible that it
takes O(n) on all inputs?
(e) Is the function f(n) = Θ(n^2), where f(n) = 100n^2 for even n and f(n) =
20n^2 − n log2 n for odd n?
The Attempt at a Solution
I understand a few of these ( I think). Here's what I have so far:
a) Yes, because big O is an upper bound and O(n) is smaller than O(n^2) it is possible some inputs have a big O of O(n)
b) No, because if the worst case time is O(n^2) then some inputs must have O(n^2) if all the inputs had O(n) then the worst case would be O(n) instead of O(n^2)
c) needs help
d) if Θ(n^2) is the worst time its not possible it takes Θ(n) on all inputs, otherwise the worst case Θ would be Θ(n) rather than Θ(n^2)
e) i did c_2 * g(n) =< f(n) =< c_1 * g(n) where f(n) = 100n^2 for every even n
c_2 * n^2 =< 100n^2 =< c_1* n^2 which is true for certain n values, so its proven
for the odd part i did the same thing, since you can drop the lower terms i dropped the 2logn part and just had f(n)=20n^2, then did the same method. Is this correct?
I don't understand what to do for c, i know that Θ is a tight bound upper and lower so i don't know if Θ(n) is possible. Could someone please explain this to me? And if you have time could you please check my other answers? Thank you
Homework Helper
Gold Member
Homework Statement
(a) If I prove that an algorithm takes O(n^2) worst-case time, is it possible that it
takes O(n) on some inputs?
(b) If I prove that an algorithm takes O(n^2) worst-case time, is it possible that it
takes O(n) on all inputs?
(c) If I prove that an algorithm takes Θ(n^2) worst-case time, is it possible that it
takes O(n) on some inputs?
(d) If I prove that an algorithm takes Θ(n^2) worst-case time, is it possible that it
takes O(n) on all inputs?
(e) Is the function f(n) = Θ(n^2), where f(n) = 100n^2 for even n and f(n) =
20n^2 − n log2 n for odd n?
The Attempt at a Solution
I understand a few of these ( I think). Here's what I have so far:
a) Yes, because big O is an upper bound and O(n) is smaller than O(n^2) it is possible some inputs have a big O of O(n)
b) No, because if the worst case time is O(n^2) then some inputs must have O(n^2) if all the inputs had O(n) then the worst case would be O(n) instead of O(n^2)
That just says ##O(n^2)## isn't the best bound, but it is surely a correct bound.
c) needs help
Remember ##\Theta(n^2)## is sharp. So the answer to c depends on whether it might be possible to use ##\Theta(n)## in this case. What do you think?
d) if Θ(n^2) is the worst time its not possible it takes Θ(n) on all inputs, otherwise the worst case Θ would be Θ(n) rather than Θ(n^2)
Correct. Unlike the situation for ##O(n^2)##.
e) i did c_2 * g(n) =< f(n) =< c_1 * g(n) where f(n) = 100n^2 for every even n
c_2 * n^2 =< 100n^2 =< c_1* n^2 which is true for certain n values, so its proven
for the odd part i did the same thing, since you can drop the lower terms i dropped the 2logn part and just had f(n)=20n^2, then did the same method. Is this correct?
I don't understand what to do for c, i know that Θ is a tight bound upper and lower so i don't know if Θ(n) is possible. Could someone please explain this to me? And if you have time could you please check my other answers? Thank you
Once you understand c and d, I think you will see the answer to e is very easy.
Smish
I think I'm starting to get it.
So in c, having it take O(n) on some inputs would mean that it would have to be at least Ω(n). So it couldn't be Θ(n^2). And if that's correct I think I know how to solve question d.
Thank you for the help by the way.
Homework Helper
Gold Member
I think I'm starting to get it.
So in c, having it take O(n) on some inputs would mean that it would have to be at least Ω(n). So it couldn't be Θ(n^2). And if that's correct I think I know how to solve question d.
Thank you for the help by the way.
But if it is worst case ##\Theta(n^2)## that means it has lots of terms like ##n^2## or the theta order would be less.
Smish
But if it is worst case ##\Theta(n^2)## that means it has lots of terms like ##n^2## or the theta order would be less.
I see. So I'm taking another crack at e again.
From what I understand, for the even f(n) = 100n^2 the Θ(n^2) holds true because all the terms in this are to the power of n^2. However for the odd f(n) = 20n^2 − n log2 n the O(n^2) holds true but the Ω(n^2) doesn't hold true because of the n log2 n. Since f(n) isn't both Ω(n^2) and O(n^2), then it isn't Θ(n^2).
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# Fold and One Cut
Can you cut out a perfect five-pointed star from a sheet of paper with a single straight cut? It is claimed (probably falsely) that Betsy Ross knew how to do it, and Harry Houdini wrote about it in his 1922 book Paper Magic. It’s easy:
So what other shapes can be made from a sheet of paper by folding it and then making a single straight cut? Or maybe a better question is: What shapes cannot be made this way?
# Taking it Further
Here are some simple challenges taken from The Art of Mathematics to get you thinking about how to design these patterns.
1. Draw an equilateral triangle. Can you fold the paper and then use one cut to cut it out?
2. Next try to do the same thing with a square.
3. The examples above and in the videos are shapes with rotational symmetry. Is it necessary for a shape to have rotational symmetry to be possible to make it with one cut?
4. Draw an irregular polygon with four sides. Can you fold this shape and cut it out with one cut?
5. How about an irregular polygon with five sides?
The pattern for the swan shown below and several other designs by Erik Demaine, along with directions for folding them, can be found here. These patterns include the fold lines. The dash and dot lines are mountain folds, and the dash lines are valley folds. The swan can be cut out with a single straight cut after the paper is folded! Can you do it?
# The Underlying Mathematics
Martin Gardner wrote about the fold-and-cut problem in his famous “Mathematical Games” column in Scientific American. His writing on paper cutting can be found in chapter 5 of Sphere Packing, Lewis Carroll, and Reversi: Martin Gardner’s New Mathematical Diversions. He leaves it as an open problem to determine which polygons can be obtained via fold-and-one-cut.
In 1998, Erik Demaine, Martin Demaine, and Anna Lubiw solved this decades old problem when they proved the fold-and-cut theorem. The theorem asserts that any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut.
Joseph O’Rourke briefly describes two fold-and-one-cut methods in his Computational Geometry Column 36, in The International Journal of Computational Geometry and Applications (Vol. 9 No. 6, 1999, pp 615–618).
You can watch Professor Demaine give a lecture on fold-and-one-cut, from his course “Geometric Folding Algorithms: Linkages, Origami, Polyhedra” (Fall 2010).
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Students can Download Chapter 2 Whole Numbers Ex 2.3 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 6 Maths helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.
## Karnataka State Syllabus Class 6 Maths Chapter 2 Whole Numbers Ex 2.3
Question 1.
Which of the following will not represent zero:
a) 1 + 0
b) 0 × 0
c) $$\frac{0}{2}$$
d) $$\frac{10-10}{2}$$
a) 1 + 0 = 1 It does not represent zero
b) 0 × 0 = 0 It represent zero
c) $$\frac{0}{2}=0$$ It represents zero
d) $$\frac{10-10}{2}=\frac{0}{2}=0$$ It represent zero
Solution:
a) 1 + 0 = 1
Question 2.
If the product of two whole numbers is zero, can we say that one or both of them will be zero ? Justify through examples.
Solution:
If the product of 2 whole numbers is zero, then one of them is definitely zero,
For example, 0 × 2 =0 and 17 × 0 = 0
It the product of 2 whole numbers is zero them both of them may be zero 0 × 0 = 0
However, 2 × 3 = 6
(Since number to be multiplied are not equal to zero, the result of the product will also be non-zero.
Question 3.
If the product of two whole numbers is 1, can we say that one or both of them will be 1? Justify through examples?
Solution:
If the product of 2 numbers is, then both the numbers have to be equal to 1
For example ,1 × 1 = 1 However, 1 × 6 = 6
Clearly, the product of two whole numbers will be 1 in the situation when both numbers to be multiplied are 1.
Question 4.
Find using distributive property:
a) 728 × 101
b) 5437 × 1001
c) 824 × 25
d) 4275 × 125
e) 504 × 35
Solution:
a) 728 × 101 = 728 × (100+1)
= 728 × 100 + 728 + 1
= 72800 + 728 = 73528
b) 5437 × 1001 = 5437 × (1000 + 1)
= 5437 × 1000 + 5437 × 1
= 5437000 + 5437 = 5442437
c) 824 × 25 = (800 + 024) × 25
= (800 + 25 – 1) × 25
= 800 × 25 + 25 × 25 – 1 × 25 = 20000 + 625 – 25
= 20000 + 600 = 20600
d) 4275 × 125
= (4000 + 200 + 100 – 25 ) × 125 = 4000 × 125 + 200 × 125 + 100 + 125 – 25 × 125
= 500000 + 25000 + 12500 – 3125
= 534375
e) 504 × 35 = ( 500 + 4) × 35
= 500 × 35 + 4 × 35
= 17500 + 140 = 17640
Question 5.
Study the pattern :
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765
Write the next two steps, can you say how the pattern works?
(Hint: 12345 = 11111 + 1111 +111 +11 +1).
Solution
123456 × 8 + 6 = 987648 + 6 = 987654
1234567 × 8 + 7 = 9876536 + 7 = 9876543
Yes, the pattern works.
As 123456= 111111 + 11111 + 1111 + 111 + 11 + 1.
123456 × 8 = ( 111111 + 11111 + 1111 + 111 + 11 + 1) × 8
= 111111 × 8 + 11111 × 8 + 1111 × 8 + 111 × 8 + 11 × 8 + 1 × 8
= 888888 + 88888 + 8888 +888 + 88 + 8 = 987648
= 123456 × 8 + 6 = 987648 + 6 = 987654
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crawl-data/CC-MAIN-2024-26/segments/1718198864968.52/warc/CC-MAIN-20240623225845-20240624015845-00462.warc.gz
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Calculate: ((1)/(x)+(1)/(y)) * (x-y)-(x+y) * ((1)/(x)+(1)/(y))
Expression: $\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( x-y \right)-\left( x+y \right) \times \left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right)$
Factor out $\frac{ 1 }{ x }+\frac{ 1 }{ y }$ from the expression
$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( x-y-\left( x+y \right) \right)$
When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses
$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( x-y-x-y \right)$
Since two opposites add up to $0$, remove them from the expression
$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( -y-y \right)$
Collect like terms
$\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times \left( -2y \right)$
Multiplying a positive and a negative equals a negative: $\left( + \right) \times \left( - \right)=\left( - \right)$
$-\left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right) \times 2y$
Use the commutative property to reorder the terms
$-2y \times \left( \frac{ 1 }{ x }+\frac{ 1 }{ y } \right)$
Random Posts
Random Articles
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James M. McPherson, professor emeritus at Princeton University and Pulitzer Prize-winning author of the highly regarded general history Battle Cry of Freedom: The Civil War Era among other works, has written a concise, fact-filled and exciting work on the Civil War's naval aspects in War on the Waters: The Union and Confederate Navies, 1861-1865.
According to McPherson, it was the North's blockade of the South's Atlantic and Gulf coasts, along with the cooperative efforts between the Union Army and Navy in creative operations along the South's inland waterways, that ultimately contributed to Union victory.
The author informs us that it was in May 1861, one month after the battle at Fort Sumter in the harbor of Charleston, S.C., that U.S. Gen. Winfield Scott outlined his famous Anaconda Plan. Its purpose was to strangle the Confederacy by closing it off from trade with the rest of the world by coastal blockade and by control of the Mississippi River. President Abraham Lincoln and Secretary of the Navy Gideon Welles went along with this strategy.
No sooner was the Union's blockade put into effect, however, than the Confederacy found ways to avoid it. Fast, nimble, shallow-drafted Confederate ships — designed to evade the big, slow, steam-powered Union frigates — were immediately deployed. Five out of six made it through the blockade. And, McPherson points out, not only were Rebel ships breaking through the blockade, Confederate commerce raiders also wreaked havoc upon Union merchant shipping.
McPherson describes the war along the South's strategic inland waterways, the Mississippi, Cumberland and Tennessee rivers. He recounts how inventor James B. Eads' "Pook's Turtles" — Union ironclads named after contractor Samuel Pook — were used successfully to conquer Fort Henry in Tennessee in 1862 and how eccentric genius Alfred Ellet's specially outfitted ramming ships, inspired by ancient Roman triremes, were effectively deployed at the Battle of Memphis.
On July 15, 1862, after the Union attack on Vicksburg, Tenn., the CSS Arkansas escaped past the Union fleet, which "fired heavy broadsides at her" but "could not stop her." The Arkansas, however, had 25 men killed and 28 wounded. McPherson, never losing sight of the horrors of combat, cites one of the Arkansas’ master mates, who wrote that "the scene around the gun deck ... was ghastly in the extreme. Blood and brains bespattered everything, whilst arms, legs and several headless trunks were strewn about."
Union Flag Officer Samuel F. Du Pont's attack on Charleston's defenses in 1863, using Monitor inventor John Ericsson's ironclad gunboats, and Rear Admiral David Farragut's 1864 victory in the Battle of Mobile Bay, in which Confederate "torpedoes" (mines) filled the harbor — these decisive battles McPherson describes in palpable, vivid detail.
Equally powerful are his descriptions of the world's first ironclad battle — forever changing naval warfare — between the USS Monitor and the CSS Virginia (previously the Merrimack) at Hampton Roads, Va., and the Confederacy's introduction of the world's first combat-ready submarine, the ill-fated H.L. Hunley.
Also covered is the famous sea battle between the USS Kearsarge and the South's most hated commerce raider, the CSS Alabama, off the coast of Cherbourg, France.
But the main point of this book is that, despite the Confederate military's technological innovations (advanced mines, ironclads, submarine), the sustained Union blockade was eventually successful in depriving the Confederacy of necessary commodities and thereby economically strangled the South. War on the Waters, a short, riveting read, undeniably shows that the Civil War — a war of intense technological innovation on both seagoing sides — was won in large part by the Union Navy.
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Discover the cosmos! Each day a different image or photograph of our fascinating universe is featured, along with a brief explanation written by a professional astronomer.
2001 June 13
Explanation: Today's galaxy, M94 (NGC 4736), lies 15 million light-years away in the constellation Canes Venatici. In the red light image (left), its very bright nucleus and tightly wound spiral arms seem to slowly fade into a faint outer disk. But when viewed in wavelengths shorter than blue light - ultraviolet (UV) light - its appearance dramatically changes. While the red light image highlights the older, cooler stars of M94, the UV picture (right), from the shuttle-borne Ultraviolet Imaging Telescope, is dominated by clusters of massive, hot stars a mere 10 million years young. These UV bright young star clusters are mostly arranged in a stunning ring nearly 7,000 light-years wide around the galactic nucleus. What controls this star forming activity? Exploring wavelengths beyond the blue, astronomers now have evidence that star forming activity in galaxies like M94 can be orchestrated by the symmetric structure of the galaxies themselves instead of the titanic galaxy-galaxy collisions suspected in yesterday's case of the Cartwheel galaxy.
Authors & editors:
Jerry Bonnell (USRA)
NASA Technical Rep.: Jay Norris. Specific rights apply.
A service of: LHEA at NASA/ GSFC
& Michigan Tech. U.
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The object shown is a Globular cluster.A globular cluster is a cluster/collection of stars that orbit a galactic core. They are stars that are very tightly bound close together by gravity. This is what gives the the spherical shape, like shown in the picture. This is the M13/NGC 6205. It is known as the Hercules Globular Cluster. It's called this because it is a gobular cluster located int he constellation of Hercules. It was discovered in 1714 by Edmond Halley and then catalogued by June 1, 1764 by Charles Messier.
As we can see, this particle image of M13 contains many orange stars, as well as blue stars. The most massive stars were originally blue,but since ran out of hydrogen and became red giants (the red stars you see). The medium mass stars are what we still see as blue, hydrogen burning stars.
Arnette, Bill. "Helix Nebula." Nine Planets. <http://astro.nineplanets.org/twn/n7293x.html>.
Frommert, Hartmut and Kronberg, Christine. "NGC 7293." Students for the Exploration and Development of Space. <http://www.seds.org/MESSIER/xtra/ngc/n7293.html>
|Right Ascension (J2000)||16h 41.7m|
|Declination (J2000)||+36° 27′ 35.5″|
|Filters used||B (Blue), R (Red), V (Green)|
|Exposure time per filter||B (10.0/sec), V (20.0/sec), and R (20.0/sec);|
|Date observed||November 01, 2012|
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Sleep And Mental Health
Sleeping is a natural function of the human body that everyone needs to recharge and prepare for the coming day. Much like our laptops and smartphones, our bodies need that time to turn off and cool down. They can use this time to even heal after the day. Sleep can also have an impact on your emotional and mental wellbeing, making it a necessary part of maintaining your mental health.
Our bodies have built-in biological clocks that, when working properly, help us regulate when and how we sleep. This biological clock is called a circadian rhythm. It uses things like your environment’s lighting to know when your body should release natural sleep-aid chemicals called melatonin.
Once you’re asleep, you cycle through stages of rest. These are called REM (rapid eye movement) sleep cycles. They can have an impact on processes like your temperature, heartbeat and breathing. REM sleep cycles can also affect:
- How rested you feel the next day
- Your ability to form and retain memories/information
- Body repairs overnight
- Puberty and fertility
- Your immune system
In order to receive the full benefits of sleep, it’s important you go through all of the stages of REM sleep and receive the proper hours needed. Adults generally need between 7 and 9 hours of sleep to function properly, but that doesn’t mean our children need the same amount. In fact, teenagers and young children need anywhere from 8 to 12 hours of sleep a day in order to function well and feel good the next day!
Sleep and Mental Health
If you don’t receive the necessary amount of sleep, you can negatively impact yourself more than just feeling tired and groggy in the morning. A lack of sleep or disturbances in your sleeping patterns can actually affect your long-term mental health:
- Sleep disorders can increase your risk of developing depression.
- People with both depression and insomnia can respond less to treatment compared to those without a sleep disorder.
- Sleeping problems can worsen anxiety disorders.
You can improve your sleeping habits by creating a consistent schedule, staying away from technology and bright lights near bedtime and avoiding caffeine in the afternoon. Comanche County Memorial Hospital’s Center for Sleep Medicine also has some wonderful resources and tips for how to sleep better.
If you believe you may need an official diagnosis of sleep disorder or further help maintaining healthy sleep, you can also contact CCMH’s Center For Sleep Medicine to schedule an appointment with one of our professionals at (580) 250-0988.
The Comanche County Memorial Hospital website does not provide specific medical advice for individual cases. Comanche County Memorial Hospital does not endorse any medical or professional services obtained through information provided on this site, articles on the site or any links on this site.
Use of the information obtained by the Comanche County Memorial Hospital website does not replace medical advice given by a qualified medical provider to meet the medical needs of our readers or others.
While content is frequently updated, medical information changes quickly. Information may be out of date, and/or contain inaccuracies or typographical errors. For questions or concerns, please contact us at [email protected].
Medline Plus. https://medlineplus.gov/healthysleep.html
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+0
# help
0
315
1
+668
Kim has exactly enough money to buy 40 oranges at \(3x\) cents each. If the price rose to \(4x\) cents per orange, how many oranges could she buy?
off-topic
Sep 10, 2018
#1
+8096
0
Let the amount of oranges that Kim could buy if the price rose to 4x cents per orange be n .
the amount of money Kim has = 40 * 3x
the amount of money Kim has = n * 4x
n * 4x = 40 * 3x
4nx = 120x
Divide both sides of the equation by x .
4n = 120
Divide both sides of the equation by 4 .
n = 30
Sep 10, 2018
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## Precalculus (6th Edition)
$60^{o}$
Solve for radians, then convert to degrees. $y=\tan^{-1} x$ Domain: $(-\infty, \infty)$ Range: $(-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$ ------------- $y$ is the number from $(-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$ such that $\tan y=\sqrt3$ $\displaystyle \tan(\frac{\pi}{3})=\sqrt{3}\qquad$and$\displaystyle \quad \frac{\pi}{3}\in(-\frac{\pi}{2}, \displaystyle \frac{\pi}{2})$, so $y =\displaystyle \frac{\pi}{3}$ To convert radians to degrees, multiply y with $\displaystyle \frac{180^{o}}{\pi}$ $\displaystyle \theta=\frac{\pi}{3}\cdot\frac{180^{o}}{\pi}=60^{o}$
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The Hypothalamus is an area of the brain which controls a lot of functions in the body. It also affects sections of various endocrine glands, specifically the secretion of pituitary glands. Relevant to the hypothalamic-pituitary-gonadal axis, the hypothalamus secretes GnRH that travels via the hypophyseal portal system to the anterior pituitary.
Other hormones secreted from the hypothalamus include thyrotropin-releasing hormone, corticotrophin-releasing hormone, and growth-releasing hormone, all of which exert their effects on the pituitary gland.
The Pituitary Gland
Under the influence of GnRH, the pituitary gland itself controls the secretion of gonadotropic hormones such as LH and FSH that exert the final effects of the axis.
Similarly, other endocrine glands like the adrenals and thyroid glands are also activated by hormones from the pituitary gland which include thyroid stimulating hormones and adrenocorticotropic hormone.
Like, in other stimulating hormones, inhibitory hormones or factors are also produced. These help in the regulation of these hormones in the body. Positive and negative feedback mechanisms regulate the amount of hormones in the blood.
Hypothalamic–Pituitary–Gonadal (HPG) Axis
The hypothalamic-pituitary-gonadal axis (HPG axis) includes the hypothalamus, pituitary gland, and gonadal glands working together in a loop, through which the production of hormones can be regulated. These glands work as if they are a single entity.
The downstream products of the hypothalamic-pituitary-gonadal pathway are regulated through the negative feedback mechanism. Spermatogenesis, i.e., the production of sperms in the testes, is stimulated by the gonadotropin-releasing hormone (GnRH) from the arcuate nucleus in the hypothalamus.
How does the gonadal axis work?
The hypothalamus secretes GnRH in a pulsatile fashion, which travels down to the anterior pituitary gland and binds to the receptors on the pituitary gland.
LH (luteinizing hormone) and FSH (follicle-stimulating hormone) are released from the pituitary gland. Both these hormones enter the bloodstream flow to the testes where LH stimulates the Leydig cells to produce testosterone, which acts on the Sertoli cells stimulating the production of sperms.
LH binds the LH receptors and promotes the conversion of cholesterol to pregnenolone through protein kinase activity. Pregnenolone is a precursor of testosterone. Testosterone is also required for other important biological processes, like the development of primary and secondary sexual characteristics, increasing libido and epiphyseal closure.
FSH stimulates the Sertoli cells to produce androgen binding globulin (ABG) and inhibin. ABG binds to testosterone from the Leydig cells and keeps it available in the seminiferous tubules and other target tissues.
Inhibin has more of a negative feedback role; it helps in regulating spermatogenesis and inhibiting FSH, LH and GnRH production.
Raised testosterone levels in the blood stimulate the release of inhibin, which causes negative feedback on the pituitary and hypothalamus, decreasing the production hormones in the pituitary gland.
Inhibition of the enzyme, aromatase, results in an increase in FSH production suggesting that FSH regulation is more dependent on estradiol than testosterone.
GnRH promotes the release of LH and FSH which act on the ovaries and produce estrogen and inhibin. A decrease in testosterone and DHEA, with raised estrogen, leads to female primary sexual characteristics in the fetal stage. Later in the pubertal age development, female secondary sexual characteristics occur.
Estrogen regulates the menstrual cycle and inhibin inhibits the hormone, activin, which usually stimulates GnRH production.
LH surge promotes ovulation and estradiol promotes the growth of endometrium. Increased levels of estrogen and inhibin produce negative feedback changes on the pituitary and hypothalamus.
Role of Androgen-Binding Protein (ABP)
ABP is synthesized by Sertoli cells and is later secreted in the seminiferous tubules. This binds to testosterone and maintains a high concentration of testosterone in the testes. The concentration of the hormone, testosterone, is approximately 50 times more in the testes than in the blood.
Metabolic fate of testosterone
- Binds to the androgen receptors in the target tissues
- Converted to DHEA–dihydrotestosterone at the target tissues by the action of 5-alpha-reductase
- Or converted to estradiol by the action of aromatase
Primary sexual characteristics: include the growth and development of the testes and penis in males.
Secondary sexual characteristics: include the development of facial and pubic hair, increased muscle mass and voice changes, as well as the development of the larynx.
Maturation of HPG axis in males
GnRH secretion starts in the intrauterine life in the fetal stage of life. This leads to primary sexual characteristics. Its production decreases in the neonatal period and in the childhood stage until puberty when the pulsatile secretion of GnRH occurs and testosterone is produced. Secondary sexual characteristics are produced in the body after puberty until the adult stage of life, the production of GnRH and testosterone increases and, in the latter part of adulthood, it starts decreasing.
Hypothalamic–Pituitary–Gonadal (HPG) Axis Differentials
Due to the disturbance of the hypothalamic-pituitary-gonadal (HPG) axis, the development of sexual characteristics is delayed leading to many different complications in males and females. It can be due to a central cause, i.e pituitary or hypothalamic disturbance or due to local primary diseases of the gonads.
Associations of this condition include:
- Use of various drugs
To find the cause of hypogonadism, the following investigations should be done:
- FSH level
- LH level
- Prolactin level
- Estradiol levels
- Seminal fluid examination
- Thyroid function test
Still, if a clear diagnosis cannot be made, testicular tissue testing (testicular biopsy) and LH releasing hormone stimulation tests should be done.
Causes of hypogonadism in females are almost the same as those of males, except that instead of Klinefelter’s syndrome, Turner syndrome occurs. Features of Turner syndrome include short stature, webbed neck, high arch palate, short fourth metacarpals, and wide-spaced nipples.
Primary hypogonadism or hypergonadotrophic variety
The type of hypogonadism in which pituitary and hypothalamus are working normally but the problem lies within the gonads is called primary hypogonadism. Causes of primary hypogonadism include:
- Genital trauma
- Autoimmune destruction
- Mumps orchitis
Side effects of drugs:
- Chemotherapeutic agents
- Klinefelter’s syndrome
- Bilateral anorchia
Secondary hypogonadism or hypogonadotropic variety
The type of hypogonadism in which pituitary or hypothalamic secretions are decreased leading to the decreased growth of gonads and other characteristics is called secondary hypogonadism.
- Idiopathic causes
- Post-infectious state of CNS
- Damage to the hypothalamus or pituitary via radiations, tumor, infiltrative trauma
- Hereditary hemochromatosis
- Congenital disorders like Kallman’s syndrome
Side effects of drugs:
- Leuprolide (used in prostate cancer)
Signs and symptoms of hypogonadism
Hypogonadism can begin during:
- Fetal development
- Before puberty
- During adulthood. Signs and symptoms of the disease depend on when the condition develops
During fetal development:
Impaired growth of external sex organs occurs in fetal life leading to any of the following:
- Female genitals
- Ambiguous genitals — genitals that are neither clearly male nor clearly female
- Under-developed male genitals
It leads to delayed puberty, incomplete or lack of normal development. It can cause:
- Decreased development of muscle mass
- Lack of deepening of the voice
- Impaired growth of body hair
- Impaired growth of the penis and testicles
- Excessive growth of the arms and legs in relation to the trunk of the body
- Development of breast tissue (gynecomastia)
Hypogonadism may alter physical masculine characteristics and impair normal reproductive function.
Signs and symptoms may include:
- Erectile dysfunction
- Decrease in beard and body hair growth
- Decrease in muscle mass
- Development of breast tissue (gynecomastia)
- Loss of bone mass (osteoporosis)
As testosterone decreases, men have symptoms similar to those that females have after menopause: fatigue, decreased sex drive, difficulty concentrating and hot flashes.
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Goseeko blog
# What is integration?
IntroductionIntegration is the reverse process of differentiation. in other words It is also called anti-differentiation.
Integration calculus has its own application in economics, Engineering, Physics, Chemistry, business, commerce, etc.
The integral of a function is denoted by the sign
Let the function is y = f(x), So that its derivative is-
Then
Where c is the arbitrary constant.
For example,
A function,
Then, its derivative-
Or
Then
Here c is an arbitrary constant.
Some fundamental integrals-
## Methods of integration
Simple integration-
1. When the function is an algebraic function-
Some standard form are-
The integration of x^n will be as follows-
Example: Find the integral of-
Sol.
We know that-
Then
Example: Find the integral
Sol.
We know that-
Then
Example: Evaluate-
Sol.
## By substitution
Example: Evaluate the following integral-
Sol.
Let us suppose,
Then
Or
Substituting –
Logarithmic function-
Example: Evaluate the following integral-
Sol.
Let us suppose-
Now
## Integral of exponential function
Example: Evaluate-
Sol.
Let,
Now substituting-
Integration of product of two functions-
Suppose we have two function say- f(x) and g(x), then
The integral of product of these two functions is-
Note-
We chose the first function as method of ILATE-
Which is-
I – Inverse trigonometric function
L – Log function
A – Algebraic function
T- Trigonometric function
E- Exponential function
Example: Evaluate-
Sol.
Here according to ILATE,
First function = log x
Second function = x^n
We know that-
Then
On solving, we get-
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Brown tree snake and Green anole
black: range of Squamata
Squamata (scaled reptiles) is the most diverse order of extant reptiles, comprised of the lizards and snakes and characterized a flexible jaw structure (movable quadrate bones) and having scales or shields rather than shells or secondary palates. Of the four surviving orders—the others being Crocodilia, Rhynchocephalia, and Testudines—the squamates represent more than 95 percent of the known living species (Uetz 2000).
Despite their diverse forms—such as the lack of legs in snakes, presence of legs in lizards, and resemblance of the amphisbaenians to worms—squamates share many of the same traits. Some of these characters are not shared with any other reptiles, and in the case of paired penes, with any other vertebrates. This reflects that all squamates belong to the same lineage, as descendants of a common ancestor.
Reptiles are tetrapods (four-legged vertebrates) and amniotes (animals whose embryos are surrounded by an amniotic membrane that encases it in amniotic fluid). Reptiles have traditionally been defined as including all the amniotes except birds and mammals.
Today, reptiles are represented by four surviving orders:
- Squamata (lizards, snakes, and amphisbaenids ("worm-lizards")
- Crocodilia (crocodiles, caimans, and alligators)
- Rhynchocephalia (tuataras from New Zealand)
- Testudines (turtles)
According to Uetz (2000), there are a total of 7,870 extant species of reptiles, with the majority being lizards (4,470 species) and snakes (2,920), and with 156 amphisbaenias, 23 described species of living crocodiles, 295 species of turtles, and 2 species of tuataras. Similarly, Grzimet et al. (2004) note 1,440 genera and 4,450 species of lizards and 440 genera and 2,750 species of snakes. In all, this means that almost 96 percent of living reptiles belong to the Squamata order.
Squamata is considered to be a natural monophyletic group, with all squamates being descendants of a common ancestor (Grzimek et al. 2004). They have more than 70 shared derived traits (Grzimek et al. 2004). The lizards (suborder Lacertilia or Sauria) by themselves are considered to form a paraphyletic group.
Members of the Squamata order particularly are known to all possess movable quadrate bones, making it possible to move the upper jaw relative to the braincase. This is particularly visible in snakes, which are able to open their mouths very widely to accommodate comparatively large prey. Other shared traits include having only a single temporal opening or it is lost or reduced, a highly modified skull, and with the male members of the group Squamata being the only vertebrates with a hemipenis (possessing paired penes). They also are distinguished by their skins, which bear horny scales or shields, while lacking any shells or secondary palates. This is also the only reptile group in which can be found both viviparous and ovoviviparous species, as well as the usual oviparous reptiles.
Classically, the Squamata order is divided into three suborders:
Benton (2000) considers Amphisbaenia to be an infraorder within the Squamata Order, while considering the lizards (Lacertilia or Sauria) and snakes (Serpentes or Ophidia) to be orders. Uetz (2007) considers Amphisbaenia to be an suborder as with the lizards and snakes.
- Suborder Iguania (the iguanas and chameleons)
- Suborder Scleroglossa
In this newer classification, Iguania is now believed to represent a separate lineage from the others, which are placed in the suborder Schleroglossa. The exact relationships within these two suborders are not entirely certain yet, though recent research strongly suggests that several families form a venom clade which encompasses a majority (nearly 60 percent) of Squamate species.
The Squamata do not include the tuataras, New Zealand reptiles resembling lizards.
- Benton, M. J. 2004. Vertebrate Paleontology, 3rd ed. Blackwell Science. ISBN 0632056371.
- Evans, S. E., and L. J. Barbadillo. 1998: An unusual lizard (Reptilia: Squamata) from the Early Cretaceous of Las Hoyas, Spain. Zoological Journal of the Linnean Society. 124: 235-265.
- Grzimek, B., D. G. Kleiman, V. Geist, and M. C. McDade. 2004. Grzimek's Animal Life Encyclopedia. Detroit: Thomson-Gale. ISBN 0787657883.
- Kazlev, M. A. 2007. Squamata: Overview. Palaeos.com. Retrieved November 30, 2007.
- Myers, P., R. Espinosa, C. S. Parr, T. Jones, G. S. Hammond, and T. A. Dewey. 2006. Order Squamata (amphisbaenians, lizards, and snakes). Animal Diversity Web (online). Retrieved November 30, 2007.
- Uetz, P. 2000. How many reptile species? Herpetological Review 31(1):13–15.
- Uetz, P. 2007. Order Squamata. Reptile-database.org. Retrieved November 30, 2007.
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The traditional agent employed to move a ship is a propeller. The necessary propeller thrust, T, required to move the ship at speed V, is normally greater than the pertaining resistance Rt. The thrust of a propeller depends on its size, the angle of attack of its blades and the speed at which it spins. By convention, propellers are described by diameter and pitch in that order. The diameter of propeller is governed by its speed of rotation and the power of the engine rotating it. For example a 20 H.P. engine would drive a 14 inch propeller at about 1500 rpm. If the propeller had too small surface area for the power provided it would over speed and cavitate at high revolution, providing little or no thrust. Pitch is dictated by the propeller’s speed of rotation, its percentage of slippage, and the speed required from ship.
The diagram power vs shaft speed, the so called propeller curve, contains useful information about the working point of the main engine and the propeller.
So, a propeller curve is a very much significance for chief engineer. From the curve he can easily make out that for the particular engine power what will be the maximum rpm at which he can run the engine to get good propeller efficiency. In case of fixed pitch propeller, when operating in heavy weather condition, the propeller performance curves i.e. the combination of power and speed (rpm); will change according to the physical laws and the actual propeller curve can not be changed by the crew. But in case of controllable pitch propeller the pitch can be altered to suite the conditions outside and best propeller efficiency can be achieved. For this the chief engineer will require propeller curve.
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stop2dance3l
2021-12-17
$-2\left(4{x}^{7}+3{x}^{9}+2{x}^{8}\right)+\left(-9{x}^{9}-8{x}^{7}+3+5{x}^{9}\right)$
Ronnie Schechter
Step 1
Consider the polynomials,
$-2\left(4{x}^{7}+3{x}^{9}+2{x}^{8}\right)+\left(-9{x}^{9}-8{x}^{7}+3+5{x}^{9}\right)$
Open the parenthesis,
$-2+4{x}^{7}+3{x}^{9}+2{x}^{8}-9{x}^{9}-8{x}^{7}+3+5{x}^{9}$
Step 2
Group like terms,
$\left(3{x}^{9}-9{x}^{9}+5{x}^{9}+2{x}^{8}+4{x}^{7}-8{x}^{7}-2+3\right)$
Step 3
$\left(3{x}^{9}-9{x}^{9}+5{x}^{9}+2{x}^{8}+4{x}^{7}-8{x}^{7}-2+3\right)$
$\left(3{x}^{9}-9{x}^{9}+5{x}^{9}\right)+2{x}^{8}+\left(4{x}^{7}-8{x}^{7}\right)+\left(-2+3\right)$
$-{x}^{9}+2{x}^{8}-4{x}^{7}+1$
$\left(-2+4{x}^{7}+3{x}^{9}+2{x}^{8}\right)+\left(-9{x}^{9}-8{x}^{7}+3+5{x}^{9}\right)=\left(-{x}^{9}+2{x}^{8}-4{x}^{7}+1\right)$.
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Everyday maths 1 (Wales)
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# 1.7 Multiplication
## Multiplication by 10, 100 and 1 000
### ×10
To multiply a whole number by 10, we write the number then add one zero on the end. For example:
2 × 10 = 20 (2 × 1 = 2, then add a zero)
6 × 10 = 60
10 × 10 = 100
### ×100
When we multiply a whole number by 100, we add two zeros to the end of the number. For example:
3 × 100 = 300
25 × 100 = 2 500
60 × 100 = 6 000
### ×1 000
When we multiply a whole number by 1 000, we add three zeros to the end of the number. For example:
4 × 1 000 = 4 000
32 × 1 000 = 32 000
50 × 1 000 = 50 000
Now try the following activity.
### Activity 8: Multiplying whole numbers by 10, 100 and 1 000
Now try the following:
1. 7 × 10
2. 32 × 10
3. 120 × 10
4. 8 × 100
5. 21 × 100
6. 520 × 100
7. 3 × 1 000
8. 12 × 1 000
9. 45 × 1 000
10. Pens cost 31 pence each. How much would it cost for a pack of ten pens?
11. A supermarket buys boxes of cereal in batches of 100. If they buy 19 batches, how many boxes is this?
12. Seven people win £1 000 each on the lottery. How much money is this altogether?
1. 70
2. 320
3. 1 200
4. 800
5. 2 100
6. 52 000
7. 3 000
8. 12 000
9. 45 000
10. 310 pence (or £3.10)
11. 1 900 boxes of cereal
12. £7 000
FSM_1_CYMRU
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Sunday Function
Again I have to apologize for the sparseness of posting lately, but I've got two research projects going full blast and time has not been something I have a lot of. I'll still be writing at least a few times a week, and you can't beat the price. ;) In any case once things cool down just a little I should be back to a more regular schedule.
Today's function isn't interesting because of the function itself, the interest comes from what we'll do with it. Let's say we have a function like this:
If we want to see where the function is equal to zero, it's clear that 0 = x^2 - 2 is solved by x equal to the square root of two, either positive or negative. That's the snappy analytic solution, but let's say we want to use this function to compute a decimal approximation to whatever accuracy we feel like. The brute force method is just to try various decimal numbers and see what gets us closest, refining our guess each time. But this is ugly, slow, and requires a lot of continual work. We'd prefer a faster method where we can just turn the crank easily and get a good answer. To do this, let's plot the function and (for reasons I'll explain in the next paragraph) its tangent line at the point x = 4.
Let's say we picked x = 4 for our initial guess as the value of the square root of two. It's an awful guess, obviously, but that's ok since we're looking for a procedure that can turn any terrible guess gradually into better and better approximations of the actual answer. So we pick our initial point and draw the tangent line. We notice that it cuts the x-axis pretty close to the square root of two (which is exactly where the parabola cuts the x-axis, since the square root of two is by definition the number that makes the function equal to zero). So why don't we take the location where the straight line cuts the x-axis, and use that as our second guess, and draw the tangent line there:
This line cuts the axis even closer to the point x = the square root of 2. If we take that point as our next guess and repeat the process, we should be closer still. So first we need to actually work this procedure into mathematical language so we can actually get some numbers out of it. The equation of a line is:
Where m is the slope and b is the y-intercept. We know the slope of a tangent line is just the derivative of the function at that point, and y is just the value of the function at that point. That means we can solve for b:
Where the prime denotes differentiation, and we're subscripting the x so we know it's the particular value of the guess we're using. Our new x for the next iteration of the guess will be the value that makes our line equation y = mx + b equal zero, i.e., 0 = m x + b. Substituting all the previous stuff into this equation and solve for x. After a little simplification, we get:
Now this is a very general expression that works to find the zeros of an arbitrary function f, whatever it happens to be. Our particular function can be plugged in (for us, f'(x) = 2x by a little bit of calculus), which gives us the complete procedure:
Let's give this a try. Plug in our initial guess of 4 and the procedure tells us our next guess is 2.25. Plug that in and the procedure gives us 1.56944. Plug that in and we get 1.42189. Repeat again and get 1.41423. So on and so forth closing in on the rounded-off real value of 1.41421, and if we didn't round off at 5 decimal places as I'm doing eventually we'd get as many digits of the square root as we wanted to arbitrary accuracy.
This procedure is called Newton's Method, and it's a fine way of calculating the zeros of a function if you know its derivative. The method is not quite perfect - if a function has more than one zero the one you get will depend on your initial guess in a not-always-predictable way. And while the method is quite general, there are certain functions that don't fulfill the relatively generous convergence conditions. Still, it's a great method with a long and continuing history of use. It's pretty likely that your pocket calculator uses a very similar method when you hit the square root button. And now if you ever find yourself without such a button, you can do it yourself if you're patient.
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As with most root finding algorithms, it helps to have a good idea of where your root is. Otherwise, you might converge to the wrong root (every method has this problem), or (even worse) you might hit a local extremum which sends you shooting off to infinity. Numerical Recipes recommends Newton's method for root polishing, or in multiple dimensions where there are few alternatives. In one dimension, if you do not know the derivative a priori, you are better off with the secant method, and if you have a pathological case, use bisection, which is guaranteed to find a root no matter how bad your initial guess.
By Eric Lund (not verified) on 09 Nov 2009 #permalink
This is certainly one way to do it, but I suspect most calculators either use Taylor expansions or built in log tables for this operation. Not because it's faster, but because it's easier.
I prefer to think of Newton's method slightly differently when thinking about applying it to polynomials. One is essentially taking a starting approximation x_0, looking at what happens when you plug in x_0 + epsilon into your polynomial. Then since you assume that epsilon is small, powers of epsilon should be even tinier and so you can ignore any term that has epsilon to a power greater than 1. Then you solve for what epsilon this would give you and get your new approximation. This has the advantage that y calculus you can more or less convince someone this process should work if one started with x_0 close to the actual solution.
@3: I don't see how that's different from what Matt is saying. Both are just assuming local linearity. His is just the visual version.
This is a very nice development of the idea of square root that I cover differently in my book (Inside Your Calculator) about simple algorithms that could support calculator keys. There, because the book uses no calculus, I derive the square root iteration process differently, only coming on Newton's Method in an appendix.
Sam's comment is interesting because a student of mine tried to find out from several calculator manufacturers what algorithm they used. The answer was always privileged information. I suspect that this was a translation of "I don't know." It took a journal search to find a few of the algorithms, but interestingly not this one for the square root key. In any case Newton's method, simply implemented, gets 10-digit accuracy as fast as the calculator key. (As soon as you get a decimal digit, it is straightforward to show that the number of decimal digits doubles with each iteration.)
By Gerry Rising (not verified) on 10 Nov 2009 #permalink
Josh: Sam is right. Newton's method is equivalent to iterated linear extrapolation.
As for implementing this method to find sqrt(x), I would think a reasonable initial choice for sqrt(x) (assuming nonnegative x) would be x. That happens to get it right for x = 0, a difficult root to find because the first derivative of x^2 also vanishes there (multiple roots are hard for any algorithm to find), and for everything else it will converge to the positive square root. It works because quadratic polynomials are well behaved.
By Eric Lund (not verified) on 10 Nov 2009 #permalink
Extending this to complex numbers, the map of the root that you get if you start at z (ie, colour point z green if you wind up at root A, red if root B etc) turns out to be a fractal.
"fractint" used to plot them - I don't know if there's an equivalent anymore.
I took a history of math course a long time ago. Seems there are at least 3 ancient root finding algorithms; this is one of them.
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# How is an ellipse formed from a cone?
## How is an ellipse formed from a cone?
When a plane cuts a cone at right angles to its axis a circle is formed. When the plane cuts the cone at an angle between a perpendicular to the axis (which would produce a circle) and an angle parallel to the side of the cone (which would produce a parabola), the curve formed is an ellipse.
What shapes can be formed when a plane intersects a cone?
An ellipse can be defined as the shape created when a plane intersects a cone at an angle to the cone’s axis. It is one of the four conic sections. (the others are an circle, parabola and hyperbola).
### How do you cut an ellipse from a cone?
59 second clip suggested13:33Conic Sections in Clay – YouTubeYouTubeStart of suggested clipEnd of suggested clipAnother could have formed also a parabola by coming starting well further away from the point butMoreAnother could have formed also a parabola by coming starting well further away from the point but the cutting plane would still need to be parallel to the edge of the cone to get that.
How does ellipse form from the intersection of a cone and a plane?
Ellipses arise when the intersection of the cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone; for a right cone, this means the cutting plane is perpendicular to the axis.
#### Why slicing a cone gives an ellipse?
57 second clip suggested12:52Why slicing a cone gives an ellipse – YouTubeYouTubeStart of suggested clipEnd of suggested clipBy the length of the longest axis of the ellipse. For slicing a cone the eccentricity is determinedMoreBy the length of the longest axis of the ellipse. For slicing a cone the eccentricity is determined by the slope of the plane that you used for the slicing.
What is the shape formed when a plane intersects a cone at a right angle to the cone’s vertical axis?
A circle can be defined as the shape created when a plane intersects a cone at right angles to the cone’s axis. It is one of the four conic sections. A circle is formed at the intersection of the cone and the plane if the plane is at right angles to the vertical axis of the cone (i.e. parallel to the cone’s base).
## How are conic sections used in astronomy?
The four classic conic sections can be produced by the intersection of a plane through a cone. Curiously, in astronomy, the Newtonian solutions to the two-body problem forces binary stars, planets and comets to trace a path that always corresponds to one of the four conic sections.
What shape will you get if you cut the cone horizontally?
If we slice through a cone, depending on the angle of the cut, the edges will form a circle, ellipse, parabola, or hyperbola (figure 1).
### What are formed when a plane intersects the vertex of the cone?
degenerate conic
A degenerate conic is generated when a plane intersects the vertex of the cone. The degenerate form of a circle or an ellipse is a singular point. The degenerate form of a parabola is a line. The degenerate form of a hyperbola is two intersecting lines.
When the plane intersects the cone exactly at its vertex?
Point: If the plane intersects the two cones at the vertex and at an angle greater than the vertex angle, we get a point. This is a degenerate ellipse. Line: If the plane intersects the two cones at the vertex and at an angle equal to the vertex angle, we get a line. This is a degenerate parabola.
#### What will be formed if a plane intersects through the vertex of the cone?
If the cutting plane contains the vertex of the cone and only one generator, then a straight line is obtained, and this is a degenerate parabola. If the cutting plane contains the vertex of the cone and two generators, then two intersecting straight lines is obtained and this is a degenerate hyperbola.
What do the red and green points on a cone shape represent?
The red shape represents the shape that would be formed if the plane actually cut the cone. The green points are drag points that can be used to reorient the intersecting plane.
## How many cutting planes do I need to make a prism intersection?
Intersection of two Prisms Total number of cutting planes required is 6 and locate the intersection points from the cutting planes and locate the points in the front view Intersection of two Prisms
What are the points of intersection of two prisms?
Intersection of two Prisms The CP is chosen across one edge RS of the prism This plane cuts the lower surface at VT, and the other prism at AB and CD The 4 points WZYX line in both the prisms and also on the cutting plane These are the points of intersection required
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This patient support community is for discussions relating to type 2 diabetes, celiac disease, depression, diabetic complications, hyperglycemia, hypoglycemia, islet cell transplantation, diabetes nutrition, parenting a diabetic child, gestational diabetes, and insulin pump therapy.
People can get diabetes at any age. Type 1, type 2, gestational, and pre-diabetes are the four main kinds. Type 1 diabetes, formerly called juvenile diabetes or insulin-dependent diabetes, is usually first diagnosed in children, teenagers, or young adults. With this form of diabetes, the beta cells of the pancreas no longer make insulin because the body’s immune system has attacked and destroyed them. Treatment for type 1 diabetes includes taking insulin and possibly another injectable medicine, making wise food choices, being physically active, taking aspirin daily—for some—and controlling blood pressure and cholesterol.
Type 2 diabetes, formerly called adult-onset diabetes or noninsulin-dependent diabetes, is the most common form of diabetes. People can develop type 2 diabetes at any age—even during childhood. This form of diabetes usually begins with insulin resistance, a condition in which fat, muscle, and liver cells do not use insulin properly. At first, the pancreas keeps up with the added demand by producing more insulin. In time, however, it loses the ability to secrete enough insulin in response to meals. Being overweight and inactive increases the chances of developing type 2 diabetes. Treatment includes using diabetes medicines, making wise food choices, being physically active, taking aspirin daily—for some—and controlling blood pressure and cholesterol.
Some women develop gestational diabetes during the late stages of pregnancy. Although this form of diabetes usually goes away after the baby is born, a woman who has had it is more likely to develop type 2 diabetes later in life. Gestational diabetes is caused by the hormones of pregnancy or a shortage of insulin.
Pre-diabetes means you have blood glucose levels that are higher than normal but not high enough to be called diabetes. Glucose is a form of sugar your body uses for energy. Too much glucose in your blood can damage your body over time. Pre-diabetes is also called impaired fasting glucose (IFG) or impaired glucose tolerance (IGT). If you have pre-diabetes, you are more likely to develop type 2 diabetes, heart disease, and stroke. Being overweight and physically inactive contributes to pre-diabetes. You can sometimes reverse pre-diabetes with weight loss that comes from healthy eating and physical activity.
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## One & Two Step Equations
Back
#### Study Guide
Provides a quick overview of the topic selected!
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#### Worksheets
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#### Study Guide One & Two Step Equations Mathematics, Grade 6
1
/
3
What Is Solving and Explaining Two-Step Equations Involving Whole Numbers and Using Inverse Operations? An algebraic equation is an expression in which a letter represents an unknown number such as, n + 5 = 11 (n = 6). An inverse operation is one that “undoes” or reverses another. Addition and subtraction are inverse operations, and so are multiplication and division. Using an inverse operation allows us to calculate the value of the unknown number by moving all the known numbers to one side of the equation. Two-step equations involve balancing both sides of the equation. To solve 5 + n = 11, subtract 5 (the inverse operation of addition) from both sides: n = 11 5. Perform the operation and n = 6 How to solve two-step equations involving whole numbers by using inverse operations: To solve a two-step equation, the unknown number must be by itself on one side of the equation. This happens by performing inverse operations. An algebraic equation must stay in balance, so whatever is done to one side must be done to the other. To solve this problem: n 6 = 4 Add 6 to both sides of the equation because addition is the inverse of subtraction n = 10. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.
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Experts in the field of environmental and behavioral enrichment study ways to provide captive animals with environmental stimuli to compensate for the absence of a rich and challenging natural habitat.
"Training" (operant conditioning) is often included within a definition of enrichment. Although not a natural, normal behavior, many captive animals seem to enjoy the attention of their keepers/trainers and appear to find these activities enriching. Operant conditioning programs reward behaviors that can improve the quality of the captive animal's life, training them to allow the handling necessary for examination or the administration of medication, or to be moved to a selected location such as a lockout for safety.
Why is environmental enrichment necessary?
Animals in their natural habitat encounter a rich spectrum of environmental stimuli every day, as they carry out the tasks essential for survival. They are "busy" all the time. For the survival of the individual, they must find food and shelter, and avoid predators and other hazards. For the survival of the species, they engage in mating and infant-rearing activities. Practically from the moment of birth, animals in the wild develop and refine the skills essential for survival.
Captive animals have the same instincts, and the same energetic need to respond to their environment, as do their counterparts in the wild. However, in the absence of the need to engage their environment and struggle for survival, their instincts and energy can express themselves in obsessive, stereotypic, counterproductive and even self-destructive behaviors (see sidebar).
No matter how ideal, a captive environment can never duplicate the vast range, challenging terrain, or dietary authenticity and variety an animal encounters in its natural habitat.
Some animals respond to the potential frustration & boredom of captivity by:
- obsessive chewing & licking
- repetitive vocalizations
- aggression towards cage-mates or keepers
- obsession or disinterest in food
- consuming nonfood items (pica)
- banging against caging
- lack of grooming
- lethargy, apathy
How does Safe Haven provide environmental enrichment?
Safe Haven believes that environmental enrichment contributes to captive animal welfare by helping to maintain the animal in good physical and psychological health. Environmental enrichment at Safe Haven is designed to proactively encourage the expression of healthy, normal behaviors, as opposed to a reactive approach to negate undesirable behaviors. Safe Haven endorses a behavioral engineering approach to environmental enrichment, with the addition of operant conditioning programming. We have specific enrichment programs for our big cats (cougars and bobcat), foxes, and opossums.
The intent of environmental enrichment at Safe Haven is twofold:
- In addition to a wide variety of sensory and behavioral enrichment activities, permanent resident animals receive a continuous schedule of operant conditioning to facilitate safe handling for husbandry purposes.
- Orphaned animals being raised for reintroduction, or adult animals undergoing rehabilitation prior to reintroduction, receive enrichments designed to elicit behaviors that will be needed upon release, and to maintain their physical and psychological well-being while in captivity. Contact with keepers is kept at a minimum for animals marked for reintroduction; they receive no operant conditioning.
SPECIFIC ENVIRONMENTAL ENRICHMENT ACTIVITIES EMPLOYED AT SAFE HAVEN
Environmental enrichment is built into the design and furnishing of our animal enclosures.
- Caging for permanent residents is large mesh, allowing a high level of visual, auditory, and tactile interaction with the environment outside the enclosure.
- All large animal housing is outdoors and features natural, dappled sunlight, natural shade and natural substrate (flooring).
- All outdoor housing is subject to seasonal variation in ambient temperature, with supplementary den heating provided as appropriate.
- Outdoor caging includes wood/foraging piles.
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Corn Necrosis (Yellowing) in Northeast S.D.
Observations of corn with necrosis (yellowing) of lower leaf margins were reported in northeast South Dakota. After investigating the field areas and reviewing soil and plant tissue analyses, potassium deficiency was identified as the cause (Figure 1). However, some of these fields had adequate fall and spring soil test potassium levels.
Two of the fields were recently resampled and analyzed for potassium, and their potassium levels dropped drastically when compared to previous samples from the same field (Table 1). One would ask, how could this happen?
Table 1. Potassium soil test level changes in pre and post planting soil samples in northeast South Dakota in 2016.
Dry Soil & Potassium Deficiency
Soil cations such as potassium are positively charged ions, and are held to the soil clay particles because the broken and weathered edges of the clay are negatively charged. Clay particles in the soil are very small and mostly like small platelets and occur in layers due to particle charge arrangements. The large surface area of the clay particles gives soil a great ability to hold and exchange cations which is known as cation exchange capacity (CEC). When the soil is moist or wet it swells due to water in between these clay particle layers. When the soil dries these clay layers shrink and the soil visually cracks. During extreme soil drying, the shrinking clay layers actually trap the cations which become unavailable for plant roots to take up.
Al Huer, SDSU Northeast Research Farm manager near South Shore, reported that while April’s precipitation was 0.30 inches above normal, May (1.45 inches) and June (0.42 inches) precipitation totals were only 35% of normal. There isn’t much that can be done to alleviate the problem when surface soil moisture is very low. Applying extra potassium is problematic because of difficulty in getting a meaningful amount in the plant. Corn at V8 contains about 20-40 lbs/a of potassium (K2O) while a fully mature 150 bu/a corn contains about 225 lbs/a potassium (K2O).
Foliar application of potassium can be problematic because when drought conditions occur, the plant leaf stomates (openings) tend to close for moisture conservation. Further, leaf surfaces are waxy and are not intended for large nutrient absorption, which is the function of the roots. Broadcast applications of potassium in large corn could cause plant damage due to fertilizer going down the leaf whorls. K fertilizer rescue applications have been shown to work in research plots when soil moisture was adequate. When limited soil moisture conditions occur, the worst of everything is brought out in a crop including nutrient deficiency symptoms such as potassium.
Source: Anthony Bly, South Dakota State University
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A team of researchers at MIT has developed a new type of textured solar surface that could bring down the cost of photovoltaic technology by reducing the thickness of silicon used by more than 90% while still maintaining high efficiency. To create the surface MIT modified a silicon sheet with a pattern of tiny inverted pyramids. These indentations, each less than a millionth of a meter across, are able to trap light rays just as effectively as solid silicon surfaces – which are 30 times thicker.
The study is documented in the journal Nano Letters in a paper by MIT postdoc Anastassios Mavrokefalos, professor Gang Chen, and three other postdocs and graduate students, all from MIT’s Department of Mechanical Engineering. Speaking about the team’s breakthrough, Anastassios Mavrokefalos said: “We see our method as enhancing the performance of thin-film solar cells. It would enhance the efficiency, no matter what the thickness.”
Fellow team member Matthew Branham, a co-author of the paper, noted the cost efficiency benefits: “If you can dramatically cut the amount of silicon [in a solar cell] … you can potentially make a big difference in the cost of production. The problem is, when you make it very thin, it doesn’t absorb light as well.”
The team’s tiny surface indentations, also known “inverted nanopyramids” greatly increase light absorption with only a 70% increase in surface area, limiting surface recombination. The innovation allows a sheet of crystalline silicon just 10 micrometers (millionths of a meter) thick to absorb light just as efficiently as a conventional silicon solar cell.
Not only would this reduce the cost of solar cells, but it would also reduce how much silicon is needed. The new technique also uses pre-existing equipment and materials that are already standard parts of silicon-chip processing, so new costs are incurred. “It’s very easy to fabricate,” Mavrokefalos says, yet “it attacks big problems.”
Currently the new silicon solar cell is only in the testing phase. The next step in the project will be to add components to produce an actual photovoltaic cell and then show that its efficiency is comparable to that of conventional solar cells. If the team is successful, then not only will the market soon see even cheaper solar cells, but their new thin design will enable them to be used in an even wider range of applications.
Images: MIT, Arenamontanus
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Longevity records for Psittaciformes in captivity.
Int. Zoo Yb. 37:299-316.
ISSN 0074-9664. DOI: 10.1111/j.1748-1090.2000.tb00735.x
Psittaciformes are generally believed to be long-lived birds and are frequently said to reach ages of 100 years old or more. In reality, however, life spans rarely exceed 50 years of age, although a few reliable records exist of parrots aged up to 65–70 years. Cockatoos appear to have the highest longevities and the longest reproductive life spans. Larger psittacines are generally longer-lived than smaller ones, although there seem to be some exceptions to this trend and quite remarkable differences in longevity between some similar-sized parrot genera. Some particularly interesting longevity histories, information on maximum breeding ages and trends in longevity are discussed.
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A typical RC building is made up of horizontal members (beams and slabs) and vertical members (column and wall), and supported by foundation that rest on ground. The system comprising of RC columns and connecting beams is called RC frame, the frame participates in resisting the earthquake forces. Shaking due to earthquake develops the inertia forces in the building which are proportioned to the building mass. Earthquake-induced inertia forces preliminary develop at the floor level of these forces downwards through slab and beam to columns and walls and then to the foundation from where they are dispersed to the ground. Inertia forces accumulate downwards from the top of building. The columns and wall at the lower storey experience higher earthquake-induced forces and are therefore designed to be stronger than those in stores above.
When beams bend in the vertical direction during earthquakes, these thin slabs bend along with them. When beams moves with column in horizontal directions, the slab usually forces the beams to move together with column in horizontal direction, the slab usually forces the beams to move together with. This behavior is known as rigid diaphragm.
For building to remain safe during earthquake shaking; the columns (which receive forces from beams) should be stronger than beams and foundations (which receive forces from columns) should be stronger than columns. Connection between beam and columns, columns and foundation should not fail before the failure of beam.
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Researchers from Germany recently conducted an experiment, published in PLOS One, to determine how much populations had declined and why.
To do so, they measured the total flying insect biomass, the weight of the insect catch, by using tent-like nets called Malaise traps. Those were deployed in 63 nature protection areas in Germany over the course of 27 years.
After analyzing the results, they found that flying insect biomass had decreased by 76 percent and up to 82 percent in the summers during the time of the study.
In fact, the scientists say their findings suggest “the entire flying insect community has been decimated over the last few decades,” the study read.
Scientists noted the drop occurred regardless of the habitat type, but changes in weather, land use and habitat characteristic were not the reason.
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Paper towels are composed of cellulose fibers, which are created from tiny sugar molecules that are the key factor to making the paper towel absorbent. The sugar molecules in the paper towel do not make it edible. Humans lack the enzymes required to break down these sugar molecules, therefore, there is no nutritional value to a paper towel. The loosely woven fibers used to make the paper towel also play a large part in the absorbency offered.Continue Reading
Paper towels, just like toilet paper, napkins and facial tissue are all paper products, but the manner in which a paper towel is manufactured and the materials used make the absorbency level much higher than the other paper products.
The loosely woven fibers allow liquid to travel between them for better absorbency level. Paper towels are made using a two-part manufacturing process. The first part of the process is called "creping," which uses a blade to open more areas on the towel by slicing through and disrupting the fiber-to-fiber bonds. The second part of the process is called "embossing", which alters the originally flat surface of the paper towel to one with more texture. The raised areas in the towel help aid liquids to travel through the opened areas of the towel, allowing it to absorb large amounts of liquid.Learn more about Homework Help
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In the previous lesson, the students went through the basis of trigonometry, which are angles and how to measure them. This chapter will help the students study trigonometric functions when they are applied on various angles. The approach to this chapter employs various shortcut tips and real-world illustrations to explain all of the exercise problems in simple English. Moreover, experts have created RD Sharma Class 11 Solutions that are very lucid and easy to follow, allowing students to solve problems effectively.
First, the chapter deals with Real-number trigonometric functions. The chapter also elaborates on the values given by these Trigonometric functions for different arguments. The identities are those functions that hold true for all values of the trigonometric function. Next, the chapter teaches the students the most common trigonometric identities that are useful in calculations. The Fundamental trigonometric identities learned in the earlier classes are also revised in this chapter.
Furthermore, the quadrants are discussed, and according to the quadrants where the angle lies, the signs of Trigonometric function are taught to the students. The interesting part of this chapter is the variations of trigonometric function values across quadrants. The quadrants are explained with properties and characteristics of each one of them. Trigonometric equation values at allied angles are also mentioned for the students to learn. Last but not least, the definition of periodic functions is explained in the chapter, along with even unusual functions. The RD Sharma Solutions on this page provides answers to the questions in each exercise.
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# Mixed Numbers and Improper Fractions Resources
52 filtered results
52 filtered results
Mixed Numbers and Improper Fractions
Sort by
Converting Fractions with Gems
Game
Converting Fractions with Gems
Kids uncover gemstones from rock and write improper/mixed fractions.
Math
Game
Gem Mining: Converting Improper and Proper Fractions
Game
Gem Mining: Converting Improper and Proper Fractions
Kids uncover gemstones from rock and write improper/mixed fractions.
Math
Game
Fractions 2
Guided Lesson
Fractions 2
Students will have a basic understanding of fractions coming into 4th grade. In this unit students will get to explore new ways of representing fractions, including in a set of data, on number lines and using area models. Students will use their knowledge of fractions to compare fractions with like and unlike denominators.
Math
Guided Lesson
Fifth Grade Independent Study Packet - Week 6
Workbook
Fifth Grade Independent Study Packet - Week 6
This fifth grade independent study packet provides students with Week 6 of remote learning opportunities.
Social studies
Workbook
Mixed and Improper Fractions
Worksheet
Mixed and Improper Fractions
Learn how to convert mixed fractions to improper fractions in this worksheet.
Math
Worksheet
Mixed Numbers and Improper Fractions 1
Exercise
Mixed Numbers and Improper Fractions 1
Not all fractions are created equal, and this exercise introduces students to mixed numbers and improper fractions.
Math
Exercise
Fractions 3
Guided Lesson
Fractions 3
Fractions can be challenging when taught in an abstract way. That’s why this unit invites learners to engage with fractions and mixed numbers in very visual and concrete ways using number lines, tape diagrams and area models. Students will learn different strategies to practice identifying and generating equivalent fractions.
Math
Guided Lesson
Fun with Fractions
Workbook
Fun with Fractions
Fractions are a fundamental math skill for 5th graders to master! In this series your child will review adding mixed fractions, and converting improper fractions to mixed ones.
Math
Workbook
Fractions 1
Guided Lesson
Fractions 1
Fractions can be a tricky concept for third graders to master, but this guided lesson can help kids get there. It provides focused instruction designed by teachers and curriculum experts that is specific to the third grade curriculum. Exercises and practical examples help kids to put fractions in context with real-world math problems. When finished with the lesson, check out our fractions worksheets for more practice.
Guided Lesson
Math Review Part 2: Let's Soar in Grade 4
Worksheet
Math Review Part 2: Let's Soar in Grade 4
Math
Worksheet
Mixed Numbers and Improper Fractions 2
Exercise
Mixed Numbers and Improper Fractions 2
Help students identify mixed numbers and improper fractions with this exercise that is easy to use and understand.
Math
Exercise
Improper Fractions
Worksheet
Improper Fractions
Learn all about improper fractions and how to convert them into mixed numbers.
Math
Worksheet
Gem Mining Fraction Conversion
Game
Gem Mining Fraction Conversion
Kids uncover gemstones from rock and write improper/mixed fractions.
Math
Game
Sums for Mixed Numbers and Improper Fractions
Lesson Plan
Sums for Mixed Numbers and Improper Fractions
Teach your students to add mixed numbers and improper fractions with the same denominator using the counting up strategy with number lines.
Math
Lesson Plan
Fabulous Fraction Review
Worksheet
Fabulous Fraction Review
Math
Worksheet
Explain Fraction Conversions
Lesson Plan
Explain Fraction Conversions
Encourage students to explain their processes when converting from a mixed number to an improper fraction, and back again. Use this lesson on its own or as support to the lesson Single Strategy for Adding and Subtracting Mixed Numbers.
Lesson Plan
Adding Mixed Numbers and Improper Fractions on a Number Line
Worksheet
Adding Mixed Numbers and Improper Fractions on a Number Line
Give students a helpful strategy for addition with improper fractions and mixed numbers.
Math
Worksheet
Introduction to Improper Fractions #1
Worksheet
Introduction to Improper Fractions #1
Give your fifth grader's fraction know-how a valuable boost with this colorful printable that introduces her to improper fractions.
Math
Worksheet
Fraction Concepts
Worksheet
Fraction Concepts
Kids tackle various fraction concepts in this math worksheet.
Math
Worksheet
Adding Mixed Numbers Using the Decomposition Strategy
Lesson Plan
Adding Mixed Numbers Using the Decomposition Strategy
Make the number work for you! Use this lesson to teach your students to add mixed numbers with like denominators using the strategy of decomposition.
Math
Lesson Plan
Mixed Numbers and Improper Fractions 3
Exercise
Mixed Numbers and Improper Fractions 3
After this final exercise in the series, students will be completely comfortable identifying and working with mixed numbers and improper fractions.
Math
Exercise
Number Talks with Mixed Numbers Addition
Lesson Plan
Number Talks with Mixed Numbers Addition
Students will practice mental math and harness their discussion skills through addition problems of like mixed numbers. This lesson may be used on its own or as support for the lesson Adding Mixed Numbers Using the Decomposition Strategy.
Math
Lesson Plan
Lesson Plan
Solve for the sum of two mixed numbers! They will use scaffolds such as fraction models to support the visualization of mixed numbers. This can be used on its own or as support for the lesson Sums for Mixed Numbers and Improper Fractions.
Math
Lesson Plan
Feed the Kramsters: Mixed Number Review
Worksheet
Feed the Kramsters: Mixed Number Review
Students convert improper fractions to mixed numbers to determine how much to feed their pet alien!
Math
Worksheet
Focus on Fractions
Worksheet
Focus on Fractions
Assess fifth grade fraction concepts, including equivalent fractions, adding with unlike denominators, improper fractions, and mixed numbers.
Math
Worksheet
### Mixed Numbers and Improper Fractions Resources
Once your students have a firm understanding of fractions, introduce mixed numbers and their relation to improper fractions. These resources will make the transition simple, with worksheets and exercises that allow them to help each other as a group and practice individually to test their skills. For extra review have your students check out our fifth grade equivalent fractions resources.
When dealing with fractions as part of a whole or a set, students will naturally infer that the numerator will never exceed the denominator because it represents the whole from which the pieces were taken. Of course, mathematics aren’t that simple and students will soon encounter mixed numbers and improper fractions.
Mixed Numbers
A mixed number is when you have more than one whole, as well as some pieces of the whole. This is written with the number of instances of the whole in front of the fraction representing the remaining pieces.
For example, if you have three pizzas cut into four pieces, then someone eats one piece of one of the pizzas, you still have two whole pizzas. But your student can’t ignore the remaining pieces of the third. This would be written as 2 ¾ pizzas because there are two whole pizzas and three out of the four pieces of the third.
Improper Fractions
An improper fractions is simply when the numerator is higher than the denominator.
In order to convert a mixed number to an improper fraction, the student must understand that each whole is simple a full set of the pieces. To calculate the total, the student must take the denominator and multiply it by the number of whole objects. This will give the total number of pieces those whole objects represent. Adding this to the numerator will give the student an improper fraction they can work with.
Students can practice identifying and converting mixed numbers and improper fractions using the resources provided above by Education.com.
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Teach kids the meaning of pi with this easy measuring activity.
Completing this measuring activity will help your child to understand the relationship between a circle's diameter and circumference. Children will recognize that the distance around is a little more than three times the distance across a circle.
With older or more mathematically curious children, you may wish to introduce the number pi (3.14...) as the amount that the circumference is greater than the diameter. A great time to introduce the number pi is on "Pi Day," which is (when else?) on the fourteenth of March (3/14).
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Standard Based Report Card
What is the purpose of a standards-based report card?
The purpose of the new reporting system is to provide parents, teachers, and students with more
accurate information about students' progress toward meeting standards. Parents will be more aware of
what their children should know and be able to do by the end of each grading period.
How is progress measured?
The new report card for grades K-3 will include a grading scale (1-4).
1- Beginning Learner
If an indicator is not measured during the grading period, the student will NOT receive a mark on the
How are standards-based report cards different from traditional report cards?
On traditional report cards, students receive one grade for each subject. On a standards-based report
card, each of the subject areas is divided into a list of skills and knowledge indicators that students are
For more information please click the link: https://www.rcboe.org/Page/31026
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Oligoclase is not a well-known mineral but has been used as a semi-precious stone under the names of sunstone and moonstone. Sunstone has flashes of reddish color caused by inclusions of hematite. Moonstone shows a glowing shimmer similar to labradorescence, but lacking in color. The display is produced from lamellar intergrowths inside the crystal. These intergrowths result from compatible chemistries at high temperatures becoming incompatible at lower temperatures and thus a separating and layering of these two phases. The resulting shimmer effect is caused by a ray of light entering a layer and being refracted back and forth by deeper layers before it exits the crystal. This refracted ray has a different character than when it went in and produces the moonlike glow. Moonstone is an alternate birthstone for the month of June.
Oligoclase is a member of the Plagioclase Feldspar Group. The plagioclase series comprises minerals that range in chemical composition from pure NaAlSi3 O8, Albite to pure CaAl2 Si2 O8 , anorthite. Oligoclase by definition must contain 90-70% sodium to 10-30% calcium in the sodium/calcium position of the crystal structure. The various plagioclase feldspars are identified from each other by gradations in index of refraction and density in the absence of chemical analysis and/or optical measurements.
All plagioclase feldspars show a type of twinning that is named after albite. Albite Law twinning produces stacks of twin layers that are typically only fractions of millimeters to several millimeters thick. These twinned layers can be seen as striation like grooves on the surface of the crystal and unlike true striations these also appear on the cleavage surfaces. The Carlsbad Law twin produces what appears to be two intergrown crystals growing in opposite directions. Two different twin laws, the Manebach and Baveno laws, produce crystals with one prominant mirror plane and penetrant angles or notches into the crystal. Although twinned crystals are common, single crystals showing a perfect twin are rare and are often collected by twin fanciers.
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April 24, 2015
Sum and Product Puzzle
X and Y are two different integers, greater than 1, with sum less than or equal to 100. S and P are two mathematicians; S knows the sum X+Y, P knows the product X*Y, and both are perfect logicians. Both S and P know the information in these two sentences. The following conversation occurs:
S says "P does not know X and Y."
P says "Now I know X and Y."
S says "Now I also know X and Y!"
What are X and Y?
April 16, 2015
Singapore Birthday Problem : When is Cheryl's Birthday ?
Albert and Bernard just met Cheryl. “When’s your birthday?” Albert asked Cheryl.
Cheryl thought a second and said,
“I’m not going to tell you, but I’ll give you some clues.”
She wrote down a list of 10 dates:
May 15, May 16, May 19
June 17, June 18
July 14, July 16
August 14, August 15, August 17
“My birthday is one of these,” she said.
Then Cheryl whispered in Albert’s ear the month and only the month of her birthday.
To Bernard, she whispered the day, and only the day of her birthday.
“Can you figure it out now?” she asked Albert.
Albert: I don’t know when your birthday is, but I know Bernard doesn’t know, either.
Bernard: I didn’t know originally, but now I do.
Albert: Well, now I know, too!
When is Cheryl’s birthday?
Lets first analyze all the given birth dates. The dates can be easily viewed as
Now will analyze the conversation in detail
Line 1) Albert Says: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.
All Albert knows is the month, and every month Cheryl mentioned has more than one possible date, so of course he doesn’t know when her birthday is.
The only way that Bernard could know the date with a single number, however, would be if Cheryl had told him 18 or 19, since of the ten date options only these numbers appear once, as May 19 and June 18.
For Albert to know that Bernard does not know, Albert must therefore have been told July or August (not June or May), since this rules out Bernard being told 18 or 19.
Thus, only possible months for Albert are : July and August
And, only possible days for Bernard are : 14, 15, 16 and 17
Thus, the solution space is now reduced to :
Line 2) Bernard: At first I don’t know when Cheryl’s birthday is, but now I know.
Bernard has deduced that Albert has either August or July. If he knows the full date, he must have been told 15, 16 or 17, since if he had been told 14 he would be none the wiser about whether the month was August or July. Each of 15, 16 and 17 only refers to one specific month, but 14 could be either month.
Thus, the solution space is now reduced to :
Line 3) Albert: Then I also know when Cheryl’s birthday is.
When Albert says that he also knows the answer, he doesn't have the Bernard's data still ! He doesn't know that the day with Bernard is 15, 16 or 17. Then, how did he came to conclusion that he also knows the answer. That means he has July as month as there is only one day associated with July. So, he is sure that now I also know the answer.
The answer, therefore is July 16.
April 13, 2015
5, 15, 1115, 3115, 132115, 1113122115, 311311222115, ?
What is the next number in this sequence:
5, 15, 1115, 3115, 132115, 1113122115, 311311222115, ?
April 10, 2015
True and False Statements
Which of the following statements are true and which are false?
1. Only one of the statements is false.
2. Exactly two of the statements are false.
3. Only three of the statements are false.
4. Exactly four of the statements are false.
5. All five of these statements are false.
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Since today marks the 35th anniversary of the launch of the Voyager 1 spacecraft, it seems appropriate to commemorate the mission—and to note how it continues to provide data about the far edges of the Solar System.
Where exactly are those edges of the Solar System? According to theory, the boundary of the Solar System is marked by a region known as the heliopause, where the solar wind—particles streaming from the Sun—meets the plasma of interstellar space. In this region, beginning about 90 times the distance from Earth to the Sun, models predicted that the solar wind's particles would be deflected by the interstellar material, much as water is pushed aside by the bow of a ship.
However, new measurements provided by the venerable Voyager 1 probe have failed to find the expected flow, deepening the mystery of the boundary between our Solar System and interstellar space. This adds to an earlier surprise, when Voyager's instruments measured zero outward velocity in the solar wind, a measurement that has now held constant for over two years. In a Nature paper, Robert B. Decker, Stamatios M. Krimigis, Edmond C. Roelof, and Matthew E. Hill concluded that Voyager 1 is not actually close to the heliopause, despite expectations. The researchers further suggested that the models for interactions between the solar wind and interstellar plasma may require reevaluation.
The solar wind is a plasma: a mixture of electrons and (mostly) protons streaming out from the Sun. This flow varies a lot over time, based on the solar cycle, but its particles consistently move outward, or radially.
At some point, the wind will necessarily run into the material in the region between stars. This contains more plasma, along with non-ionized atoms and molecules, which are collectively known as the interstellar medium (ISM).
By any reasonable theory, when the solar wind meets the ISM, there should be a transition. The usual model describes the solar wind carving out a region in the ISM known as the heliosphere, and the place where the solar wind is deflected by the pressure from the ISM is the aforementioned heliopause.
In April 2010, Voyager 1 reached a point about 113 AU from the Sun and saw the solar wind velocity began slowing down dramatically. (1 AU, or astronomical unit, is the average distance from Earth to the Sun, which is about 150 million kilometers.) This discovery led to speculation that the craft was reaching the heliopause.
If that was true, then the solar wind particles should be deflected in the transverse or meridional direction. Voyager's instruments happen to be good at measuring outward, or radial, flow velocity, so the spacecraft had to be reoriented five times over a 10-month period so that it could determine the meridional velocity of solar wind particles.
Two years of data revealed that the meridional velocity was 3±11 kilometers/second, or between -8 and +14 km/s. (Negative velocity in this case represents flow in the opposite direction expected by theory.) In other words, while the most likely velocity value is 3 km/s—within the instrumental limits, the flow is zero. That's certainly nowhere close to the predicted value of approximately 25 km/s.
Unlike the earlier "Pioneer anomaly," these data don't appear systematic: the five separate measurements were all consistent with zero meridional flow, but varied widely in most likely value.
These results are troublesome for several reasons. The thickness of the region where Voyager 1 has measured low radial velocity is now at least 7.5 AU (and counting). In that region, there has been very little fluctuation in the solar wind, even though we know the solar wind should vary strongly in time, thanks to the solar cycle. Additionally, combining the three components of the solar wind velocity shows the shape of the flow to be very different from what we expected.
The authors concluded that the heliopause either differs radically from theory, or Voyager is still not close to it. Both of these scenarios would require another look at the model for the solar wind-ISM interaction, since the current calculations don't explain the strange reduction in solar wind velocity or the direction of flow. It's possible the edge of the Solar System is farther out still—and might look very different from what we expected.
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The Emotions of Learning: Q&A with Marc Brackett, PhD
Education research and education study are at the forefront today. The National Survey of Children’s Health reports that almost 50 percent of American children have suffered from “at least one type of severe childhood trauma.” This can make it difficult for students to learn. One group (Kautz. Heckman. Diris. Bas ter Weel. & Borghans. 2014) found social-emotional learning (SEL), “increases high-school graduation rates, postsecondary enrollment and completion, employment rates and average wages.”
We met with Marc Brackett to discuss the emotional aspect of learning. He is the founding director of the Yale Center for Emotional Intelligence and a professor at Yale University’s Child Study Center. He also developed RULER which is an evidence-based approach for social and emotional learning.
Dr. Brackett studies the role of emotions in learning, decision making, relationship quality, and mental health. He also studies best practices for teaching emotional intelligence. InspirED is an open-source resource center that supports high school students in leading positive changes in their schools.
Let’s begin with the basics. What can emotions do to students at school?
Research and programming at Yale Center for Emotional Intelligence are based on the idea that emotions matter. RULER is based on a fundamental insight gained from the research: Emotions influence attention, memory and learning, decision making, health, creativity, and even our ability to remember things. They are essential to our mental and physical well-being. They also open up opportunities for us to be successful at school, work, and elsewhere.
Emotions can either help or hinder us. Our brains are more open to curiosity and motivated when we are satisfied with our basic needs. These emotions support cognitive functions, allowing us to pay attention, absorb information, and learn. If we become engulfed by strong emotions, particularly those of anger and despair, our minds drift to the source or cause of our fear or pain. This was helpful for our primitive ancestors as they hunted a bear or ran from a snake. However, it is not useful for us to sit in a classroom trying in vain to absorb information.
Teaching-learning is also affected by emotions. Teachers must be able to keep students’ attention and engage them in learning. Teachers must be sensitive to the emotions of their students and themselves to do this effectively. Students should pay attention to the things they are passionate about. Our goal as educators is to provide content and present it in a way that matters to students. Skilled educators use the emotions of their students to improve learning. They can even shift moods as necessary.
Are RULER programs only for students? Or can adults also benefit from it?
RULER is different from other school-based programs in that it focuses on the development of adults at the school. This allows them to be role models and skilled implementers of skill-based instruction for students. Research has shown that leaders and educators with higher emotional intelligence have higher levels of empathy and sensitivity for others, are more effective in developing teams, and receive higher performance ratings. They also experience lower stress and burnout and create a more supportive environment. Positive school climates are created when both students and adults develop emotional intelligence. These skills can be learned and practiced using the RULER approach. It starts with educators and leaders but then expands to students and their families.
Let’s get started. Can you please explain the RULER approach to business?
RULER stands for the five skills that make up emotional intelligence.
- Understanding emotions within oneself and in others
- Understanding The causes and consequences for emotions
- Labeling emotions using a nuanced vocabulary
- Expression of emotions according to cultural norms and the social context
- Helpful strategies for regulating emotions
These five skills are associated with higher academic and work performance, better relationships, improved leadership skills, less anxiety, depression, conflict resolution skills, and better overall wellbeing.
RULER is also the name of our evidence-based approach to emotional and social learning that supports the whole school community.
- Understanding the value and importance of emotions
- The development of emotional intelligence
- Positive emotional climates: Creating and maintaining them
RULER helps school leaders, staff, students, and family members to manage their emotions so they can make better decisions, have mutually supportive relationships, grow and thrive, and be successful in academic and personal life.
Does the RULER program apply only to schools?
RULER was introduced in more than 2000 public, charter, and private schools in 2005. It has reached more than a million students from rural and suburban areas. We offer training for families to support the RULER school rollout, as well as standalone resources to help siblings, parents, and children manage the emotional difficulties they face.
RULER training is based on four core anchor tools: the Charter and Meta-Moment, Blueprint, and Mood Meter. These are presented to all stakeholders (school leaders, teachers, students, staff, parents, and families) with appropriate content. The Charter is simple for young children, while the Blueprint is more complex for older children. As cognitive, emotional, or social skills develop, the complexity will increase for those in the higher grades and adults. This model creates a common language that is shared by the school community and allows for multiple transmissions (e.g. adults modeling the skills they teach their students).
What have you seen in schools where staff and students do this work?
Research shows that RULER encourages positive student development by fostering a variety of adult and student behaviors as well as shifts in school culture. After just one year, RULER was implemented in two studies. One showed a 10% improvement in academic performance and the other showed a 12% improvement in classroom climate. Here are some examples of specific outcomes:
- Develop emotional intelligence skills
- Learning problems and attention issues that are less common
- Demonstrate greater leadership and social skills
- Be less anxious and depressed
- Improve your skills in solving conflicts
- Academically, you will perform better and be more engaged
Education leaders and educators
- Be more positive in your relationships
- Warmer emotional climates
- Reduce stress and burnout
- Get healthier mentally and physically
- Enhance your instructional practices
How can schools get involved in this work?
This six-page laminate is available for educators and schools. It provides an overview of emotional and social learning including RULER. The Mood Meter App is another resource.
Formal RULER adoption occurs when a group of school staff are trained at Yale or elsewhere to bring the tools and strategies for emotional intelligence back to their school. Our Center provides coaching and online resources.
To get started, schools can register for RULER Institute for Creating Emotionally Intelligent School. In the fall of 2019, an online version will be available. If 25 or more schools in a region or district are interested in adopting RULER, regional training may also be offered. email ROLLER to discuss the possibility to bring training to your region or district.
This interview was conducted by email and edited for clarity and length.
Jennifer L.M. Gunn worked for 10 years in magazine and newspaper publishing before she moved to public education. Gunn is a teacher coach, a curriculum designer, and a high school teacher in New York City. She co-founded the annual EDxEDNYC Education Conference for teacher-led innovations and frequently presents at conferences on topics such as adolescent literacy and leadership.
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# Lesson 8
Scale Drawings and Maps
### Lesson Narrative
This lesson is optional. In this lesson, students apply what they have learned about scale drawings to solve problems involving constant speed (MP1, MP2). Students are given a map with scale as well as a starting and ending point. In addition, they are either given the time the trip takes and are asked to estimate the speed or they are given the speed and asked to estimate how long the trip takes. In both cases, they need to make strategic use of the map and scale and they will need to estimate distances because the roads are not straight.
In the sixth grade, students have examined many contexts involving travel at constant speed. If a car travels at 30 mph, there is a ratio between the time of travel and the distance traveled. This can be represented in a ratio table, or on a graph, or with an equation. If $$d$$ is the distance traveled in miles, and $$t$$ is the amount of time in hours, then traveling at 30 mph can be represented by the equation $$d = 30t$$. Students may or may not use this representation as they work on the activities in this lesson. But they will gain further familiarity with this important context which they will examine in greater depth when they study ratios and proportional reasoning in grade 7, starting in the next unit.
### Learning Goals
Teacher Facing
• Justify (orally and in writing) which of two objects was moving faster.
• Use a scale drawing to estimate the distance an object traveled, as well as its speed or elapsed time, and explain (orally and in writing) the solution method.
### Student Facing
Let’s use scale drawings to solve problems.
### Student Facing
• I can use a map and its scale to solve problems about traveling.
Building On
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Reduced oxygen saturation in blood is called hypoxia (though strictly that means reduced oxygen in the tissues). If there is marked reduction in oxygen saturation of blood, it may lead to bluish discoloration of skin and tongue, called cyanosis.
Mildly reduced arterial oxygen saturation may be due to mild respiratory or cardiac diseases. A common lung condition is chronic obstructive airway disease (or chronic bronchitis, most commonly due to smoking). Congenital cyanotic heart diseases like Tetralogy of Fallot are generally discovered and treated in childhood. Arterio-venous malformations in lung is another cause of arterial oxygen desaturation which may be present in otherwise healthy persons. Persons staying at high altitudes have reduced blood oxygen saturation due to low atmospheric pressure leading to reduced partial pressure of oxygen in inhaled air. Sleep apnea syndrome often seen in obese persons may also lead to hypoxia. Other causes include neurological diseases leading to respiratory depression, respiratory conditions such as severe pneumonia, pulmonary embolism (clots in arteries supplying blood to the lungs), pulmonary edema (lung congestion), pulmonary fibrosis etc, but in all these the person is obviously ill.
In an apparently healthy person, mild hypoxia is likely to be due to smoking or pollution related chronic obstructive airway (pulmonary) disease (COAD or COPD), pulmonary arteriovenous malformation or early pulmonary fibrosis.
Mild hypoxia may not result in any symptoms. It may lead to increased hemoglobin level in the blood to compensate for reduced oxygen saturation, as commonly occurs in persons living at high altitudes. The main worry is that the underlying process may progress and hence the cause of desaturation should be investigated.
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One of the principles of developmentally appropriate practice reads, “Following their own interests, children choose among various activities…” (Bredekamp & Copple, 1996, p. 127). Why is choosing important? If children are allowed choices, are they in control of the curriculum?
There are several reasons that giving children choices throughout the day is beneficial, even crucial to their development. Providing choices for children is a fundamental aspect of high-quality early childhood curriculum (Hendrick, 1996). In order to provide children with a number of choices, the teacher must understand the importance of choices, and be willing and able to allow a variety of activities and behaviors in the classroom. This approach to learning is child-centered, rather than teacher-centered. Let’s look at two early childhood classrooms.
In Ms. Anderson’s class of four-year-olds, the children come in and put away their belongings, select an activity, and play until everyone arrives. Then they all sit together on the rug for about 10 minutes of group time. After they have sung a few songs, had some group conversation, and heard about the day’s activity options, they choose the activity that appeals most to them and get to work. Throughout the morning, children move from center to center as they wish. A visitor would notice talking, laughing, and movement. Seldom would one hear adults’ or children’s voices raised in anger or frustration or see children out of control.
When the children arrive in Mr. Purdy’s class of four-year-olds, they put away their things and head straight for the large group area where they must sit quietly, listen politely, and respond with correct answers to the teacher’s questions for about 40 minutes. The activities of the day, each with a correct way of completing the task, are explained. Then Mr. Purdy assigns each child an activity. The children are to work quietly, follow directions carefully, obey the numerous classroom rules, and move to the next activity when he rings a bell. Mr. Purdy values order and quiet because he believes children learn better under those conditions. He keeps a list and checks off the name of each child to ensure that each one finishes every task. Children often become angry or frustrated and do not finish the tasks to Mr. Purdy’s satisfaction. In fact, Mr. Purdy often complains about the misbehavior of his class.
In Mr. Anderson’s class, children make many choices throughout the day; in Mr. Purdy’s they make very few. Which is the better learning environment for children?
A Feeling of Control
All human beings need to feel as if they have control over themselves and their lives. We cannot expect children to be totally independent, of course, since they are small and incapable of many things adults can do. Erikson (1950) believed that at the second level of psychosocial development, beginning soon after one year of age, young
children must resolve the conflict between autonomy and shame and doubt. Children who do not develop autonomy are liable to remain dependent on adults or to be overly influenced by peers. Gartrell (1995) called this phenomenon “mistaken behaviors”. Children who fall into “mistaken behaviors” may feel doubtful of their abilities, and be unable to take the risks that lead to real learning (Fordham & Anderson, 1992; Maxim, 1997) or challenge themselves to achieve at ever higher levels. In addition, they may feel hostility toward adults who allow them little freedom to choose (Edwards, 1993). Learning to be autonomous and self-reliant takes time and practice. When we offer children choices, we are allowing them to practice the skills of independence and responsibility, while we guard their health and safety by controlling and monitoring the options (Maxim, 1997).
Being autonomous and in control feels good – simply watch the face of a toddler who has just learned to walk. Self-esteem grows when we successfully do things for ourselves. Children can handle mistakes or failure with equanimity and good humor when they feel good about themselves. A child who has a solid sense of self-worth can make a poor decision, evaluate it calmly, rethink the situation, and make a different choice. When asked if he wanted to do it himself or have help, three-year-old Tom decided to pour his own juice. As he lifted and tipped the pitcher he discovered his small hand was not strong enough to hold the pitcher steady. A stream of apple juice spilled onto the table and the floor. The teacher said, “Oops, Tom, the juice spilled. Let’s get some sponges and we’ll wipe it up together.” Tom learned by trying a task that was too difficult for him. The teacher helped him deal with the consequences by not criticizing his attempt but by helping him rectify the situation. Next time he may make another choice, or he may try to do it alone again. Either way, he has made his own choice.
When a child misbehaves, a teacher might rebuke him by saying severely, “Jamal! You just made a bad choice!” This statement presumes that Jamal consciously considered each behavior in repertoire and selected kicking, for example. Children, like adults, do not always consciously choose their behaviors. They may be satisfying a need, imitating behaviors of others, or acting out of an instinct such as self-protection. Angrily accusing a child of making a poor decision while correcting his behavior sends the message that he is an incompetent decision-maker and confuses the child’s understanding of the decision-making process. He may believe that he had no right to make any decision at all, which in the end will lower his self-confidence and self-esteem as well as teach him that making decisions is very risky.
Making choices is part of problem solving. When given choices, children stretch their minds and create new and unique combinations of ideas and materials. Before they can make wise choices, however, children need to learn the skills of convergent thinking, knowing the right answer as well as divergent thinking, seeing many possible answers. If we expect teenagers to make healthful choices about important issues such as sexual activity or the use of alcohol or illegal drugs, we must allow them many opportunities in their early years to make meaningful choices (Morrison, 1997).
In a classroom based on Piaget’s constructivist principles, everyone shares responsibility for decision making (DeVries & Zan, 1995). By allowing children to determine what goes on in a room, the teacher promotes their self-regulation. If they have opportunities to make their own choices and feel powerful every day, they will have no need to exert power over others or to break rules behind the teacher’s back. When their desires are respected, it is easier for children to respect others’ wishes. As children learn to make decisions for themselves and to develop autonomy, they learn to behave morally and to take the needs of others into consideration when making choices (Kamii, 1982).
When children do not like the results of their own choices, adults often want to pacify them by neutralizing the consequences. Alicia was so busy in the housekeeping area that she did not take time to visit the art table and make a glittery snow picture. When it was time to go home, she saw the beautiful creations other children had made, and she was very upset. “I want to make a snow picture!” she wailed. Ms. Anderson could have taken her to the art table, which was not completely cleaned up yet, and allowed her to make a quick picture so she could have one to take home, too. Instead, she said, “Alicia, you chose to stay in housekeeping area today and not make a picture. It’s too late now, but tomorrow, we will have other art materials out. I’ll help you remember to visit the art table and make something.” Alicia was still upset. It is hard to accept the consequences of our behavior sometimes, but the teacher’s response helped Alicia understand that she had made a decision, a decision with consequences she must accept. No one had told her what choice to make; she made her choice independently. The teacher offered help in the future, so Alicia knows she will have support to make better decisions. Alicia’s parents might be displeased with this approach when she tells them tearfully that the teacher would not allow her to make a picture. But if they have understood from the time Alicia entered the program that the teacher’s philosophy includes helping children learn to accept responsibility for their actions, they will understand.
One of the effects of offering children choices throughout the day is the reduction of conflict among children and between children and adults. When adults direct a child’s behavior most of the day, the child’s natural desire to be independent is thwarted and feelings of resentment or rebellion may arise (Edwards, 1993). Adults can understand this frustration if they think about having a job in which they are told every little thing to do, even when to use the restroom or get a drink of water. Most of us would either complain or get another job. Children have no choice about going to school or child care; they cannot leave an unhappy situation. When they rebel, they are labeled as having “behavior problems.” If we treat children with the same respect (Kostelnik, Soderman & Whiren, 1993) we adults expect and understand that each child has individual needs and interests, we will provide them with the opportunities to choose what is best for themselves at any given time.
Children feel more committed to an activity they have chosen themselves. Therefore, their attention span will likely be longer if they choose an activity than if they work at a task assigned by the teacher (Fromberg, 1995; Maxim, 1997). Making choices helps children learn persistence and task completion.
Katz and Chard (1989) point out that when only one teaching method is used, such as workbooks, some children will achieve the learning objective, but many will not, since each learns in a slightly different way. In order to ensure that all children learn a particular skill, like reading, we must use a variety of approaches so that each child can find the one that suits him or her. If Mr. Purdy, for example, wants all of his children to learn some important concepts about weather, he can offer a variety of activities. Some children will learn by observing the water cycle in a terrarium, others will learn from fiction and nonfiction books, and others will explore their personal experiences with weather by using paints and expressive materials. Each child will learn in his or her own way, but all will learn about weather.
How to Offer Choices
Choices offered to young children must be legitimate and meaningful to them and acceptable to adults. When Ms. Lee confronted two children fighting over the same ride-on truck, she said, “You two can figure out how to share that truck, or go to time out.” Since neither child relished the thought of sitting in what amounts to solitary confinement in the time out chair, this was not a legitimate choice. Each option must have equal weight in the child’s eyes. She might have said, “You can use the truck together, or I can help one of you find another truck to use.”
Later in the day when Ms. Lee said, “It’s time to clean up, OK?” she unknowingly offered children the option of cleaning up or continuing to play by adding “OK?” to the end of her sentence. She actually meant that it was nearly time to go home and they must put toys and materials away. She had not intended to give children a choice and was unable to allow them to continue to play because it was in fact time to get ready to leave.
Limiting choices for young children helps them select (Morrison, 1997). In a restaurant with many menu options even adults have difficulty choosing their meal. It may be easier for a child to choose if we suggest she decide between the art table and the block corner than from all the activities available in the classroom. Younger children manage better with fewer options.
Making direct suggestions may help the hesitant child to make a choice. Children whose parents make decisions for them may be overwhelmed by a situation in which they are now expected to choose for themselves. They need time, support, and practice as well as patient teachers to help them learn this skill.
By offering children choices we are not giving them complete control of the classroom or the curriculum. Since children may choose only from the alternatives offered, the teacher maintains control of what the options are. Juan may want to choose the water table every day, but on the days Ms. Anderson does not put it out, he must choose something else.
No Choice Situations
Each of us must deal with situations in which we have no choice. We are required to obey laws, for instance. Children, too, must learn that sometimes they have a choice. Issues of safety allow no leeway for individual preference (Gordon & Browne, 1996). Children may not play with the burner controls on the stove while helping to make cookies. When time is an issue they may have to stop playing and clean up, or get dressed for school so Mom and Dad can get to work on time. After the adults have made the primary decision, however, children can make secondary ones. They may choose to pour in the sugar or crack the eggs for the cookies. They can select the red or the green plaid shirt to wear. When children know they will be given sufficient opportunities to choose for themselves, they are more willing to accept those important “no choice” decisions adults must make for them.
The wise teacher understands that children make choices all day long, whether adults want them to or not. They choose to obey, ignore, or defy rules and directions and determine for themselves whether to speak kindly or angrily to others. They decide whether school or child care is a good place to be. Our task is to provide children with appropriate, healthful options and help them to make and accept their choices. In this way, we are developing confident, independent children who feel in control of themselves.
Sue Grossman, Ph.D., is an assistant professor of early childhood teacher education at Eastern Michigan University.
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Engaging children’s minds: The project
approach. Norwood, NJ: Ablex.
Kostelnik, M.J., Soderman, A.J., & Whiren,
A.P. (1993). Developmentally appropri-
ate practice in early childhood
education. New York: Merrill/Macmillan.
Maxim, G.W. (1997). The very young:
Developmental education for the early
years, 5th Ed. Upper Saddle River, NJ:
Morrison, G.S. (1997). Fundamentals of
early childhood education. Upper Saddle
River, NJ: Merrill/Prentice Hall.
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# Study Materials for 1st Year B.Tech Students.
## Paper Name: Mathematics
Paper Code : M101
Teacher Name: Kakali Ghosh, Rahuldeb Das, Amalendu Singha Mahapatra,
Raicharan Denra
Chapter – 1
Lecture1.
Topic : Limit and continuous of a function
Objective:
Historically, Integral Calculus developed much before the Diffential Calculus (though while
teaching Differential Calculus proceeds Integral Calculus). The notions of areas/ volumes of
objects attracted the attention of the Greek mathematicians like Antiphon (430 B.C.), Euclid (300
B.C.) and Archimedes (987 B.C.- 212 B.C.).
Later in the 17th century, mathematicians were faced with various problems: in mechanics the
problem of describing the motion of a particle mathematically; in optics the need to analyze the
passage of light through a lens, which gave rise to the problem of defining tangent/ normal to a
surface; in astronomy it was important to know when would a planet be at a maximum/ minimum
distance from earth; and so on.
The concept of limit is fundamental for the development of calculus. Calculus is built largely upon
the idea of a limit and in this present chapter this idea and various related concepts will be
studied in a brief manner.
Limit of a Function
Let x be a variable. Let x goes on taking the values 2.9, 2.99, 2.999, 2.9999, …….. Then
we see x goes very near to 3 as near as we please. Mathematically we can say the
quantity 3-x becomes small as small as we please. In this situation we say x tends to 3
from left. In notation x→3-.
Again let x goes on taking the values 3.1, 3.01, 3.001, 3.0001, …….. Then we see x goes
very near to 3 as near as we please. Mathematically we can say the quantity 3-x becomes
small as small as we please. In this situation we say x tends to 3 from right. In notation
x→3+.
## Thus we see that x→3 implies x→3+ and x→3- both.
Formal Definition:
Say that f (x) tends to l as x a iff given > 0, there is some > 0 such that whenever 0
< | x - a| < , then | f (x) - l| < .
Right-handed limit
## We say provided we can make f(x) as close to L as we want for all x
sufficiently close to a and x>a without actually letting x be a.
Left-handed limit
## We say provided we can make f(x) as close to L as we want for all x
sufficiently close to a and x<a without actually letting x be a.
Properties
First we will assume that and exist and that c is any constant. Then,
1.
In other words we can “factor” a multiplicative constant out of a limit.
2.
So to take the limit of a sum or difference all we need to do is take the limit of the
individual parts and then put them back together with the appropriate sign. This
is also not limited to two functions. This fact will work no matter how many
functions we’ve got separated by “+” or “-”.
3.
We take the limits of products in the same way that we can take the limit of sums
or differences. Just take the limit of the pieces and then put them back together.
Also, as with sums or differences, this fact is not limited to just two functions.
4.
As noted in the statement we only need to worry about the limit in the
denominator being zero when we do the limit of a quotient. If it were zero we
would end up with a division by zero error and we need to avoid that.
5.
In this property n can be any real number (positive, negative, integer, fraction,
irrational, zero, etc.). In the case that n is an integer this rule can be thought of as
an extended case of 3.
For example consider the case of n = 2.
## The same can be done for any integer n.
6.
This is just a special case of the previous example.
7.
In other words, the limit of a constant is just the constant. You should be able to
convince yourself of this by drawing the graph of .
8.
As with the last one you should be able to convince yourself of this by drawing
the graph of .
9.
This is really just a special case of property 5 using .
Sandwich Theorem
Suppose that for all x on [a, b] (except possibly at ) we have,
## for some . Then,
Cauchy necessary and sufficient condition for the existence of a limit.
The necessary and sufficient condition for that the limit lim
x →a
f ( x) exists and is that ,
corresponding to any pre-assigned positive number ε , however small (but not equal to
zero), we can find a positive number δ such that x1 and x2 being any two quantities
satisfying 0 < x − a ≤ δ , f ( x1 ) − f ( x2 ) < ε .
Worked out problems.
1
Example1: Show that lim cos does not exist.
x→0 x
In order that the limit may exist, it must be possible to find a positive number δ such that,
1 1
x1 and x2 satisfying 0 < x ≤ δ , cos − cos <ε
x1 x2
Where ε is any pre-assigned positive quantity.
1 1
Now, whatever δ we may choose, if we take x1 = and x2 =
2nπ { ( 2n + 1) π } , by taking n
a sufficiently large positive integer, both x1 and x2 will satisfy 0 < x ≤ δ .
But in this case ,
1 1
cos − cos = cos 2nπ − cos(2n + 1)π = 2 ,
x1 x2
A finite quantity, and is not less than any chosen ε.
Thus, the necessary condition is not sat
1
Example 2: Show that lim x sin = 0
x→0
x
sol.wehahe
1 1 1 1
x sin − 0 = x sin = x sin ≤ x ,sin ce sin ≤ 1.
x x x x
∴Corresponding to a small positive number ε , there exists another small
positive number δ(=ε)>0 such that
1
x sin − 0 < ε whenever x − 0 < δ .
x
1
It follows there fore that lim x sin = 0
x→0
x
Continuity:
## Definition: A function f(x) is said to be continuous for x = a, provided lim
x →a
f ( x) exists, is
finite, and is equal to f(a).
In other words,
lim f ( x ) = lim f ( x) = f (a )
x →a + 0 x →a − 0
orbriefly, f (a + 0) = f (a − 0) = f ( a).
If f(x) be continuous for every value of x in the interval [a,b], it is said to be continuous
throughout the interval.
A function which is not continuous at a point is said to have a dis continuity at that point.
Corresponding to the analytical definition of limit, provided f(a) exists and given any any
pre-assigned positive number ε , however small (but not equal to zero), we can find a
positive number δ such that f ( x) − f (a) < ε for all values of x satisfying
a −δ ≤ x ≤ a +δ.
x3 − 8
Example 3.Let f (x) = for x≠ 2. Show how to define f (2) in order to make f a
x−2
continuous function at 2.
Solution. We have
= = (x2 + 2x + 4)
## Thus f (x) (22 + 2.2 + 4) = 12 as x 2. So defining f (2) = 12 makes f continuous at 2,
(and hence for all values of x).
Examples of Discontinuity.
Removable Discontinuity.
Definition
The left hand and right hand limits at a point exist, are equal but?? the function is not
defined at this point.???
Example
## but the value at x = 0 is undefined.
We remove the problem here by defining the function at point x = 0 to be the limit:
## The easiest way to handle removable singularities is to remove them, by defining f to be
its limit at any such point.
## Otherwise the left hand side is undefined at x = a.
Jump Discontinuity:
Definition
The left hand and right hand limits at a point exist, are finite but?they are different. ?
Example
Jump discontinuities only cause trouble is you try to differentiate the function at the
jump. It is possible to get around even with this difficulty by considering ?as
## , with and defining its derivative to be
2d(x), .
This object d(x) is called a "delta function". So defined, it is not really a function, since it
is 0 except at x = 0, where it is infinite.It is called a "distribution" and is occasionally
useful.
Assignment:
## 1. Show that the function f(x) defined by f ( x) = x + x − 1 , x ∈ R is continuous at x
= 0 and x = 1.
1
2. Using Cauchy’s general principal , show that lim sin does not exit.
x→0 x
## Objective type questions:
sin [ x ]
1. lim =
x→0 [ x]
(a) 1 (b) -1 (c) 0 (d) none of these
2. f ( x) = x − x is continuous everywhere
## (a) Yes (b) no
sin x
3. If f ( x) = ( x ≠ 0), then lim f ( x ) is equal to
x x →0
## (a) 0 (b) 1 (c) ½ (d) -1.
Lecture 2:
Objective: The limiting process will now be extended to find out the derivative of a
function f(x) with respect to x. For this , we begin with the def. of the term increment.
Definition of derivative:
## Let y = f(x) be a function defined in an interval a ≤ x ≤ b , and x be any number in the
interior of this interval.
## Now let x change to x + ∆x . Then increment of x = ∆x . So, y = f(x) changes to
y + ∆y = f ( x + ∆x ).
∴ incrementofy = ∆y = f ( x + ∆x) − f ( x )
## Now consider the increment ratio
∆y f ( x + ∆x) − f ( x)
= ,
∆x ∆x
f ( x + h) − f ( x )
or , = .
h
When x is fixed, the increment ratio o the difference quotient is a function of ∆x .This
∆y
expression , takes the form 0/0 which is undefined when we put ∆x = 0.If, however,
∆x
∆y
this increment ratio tends to a definite limit as ∆x → 0 from either side avoids ∆x =
∆x
dy
0, then this limit is called the derivative of y with respect to x, and is denoted by or
dx
f ′( x) .
## Let y = f(x) be a real – valued function defined in an interval [a,b] containing c is an
interior point. Then
f ( x ) − f (c )
lim
x →c x−c
f (c + h ) − f (c )
or , lim
h →0 h
Proposition:
To prove:
We have,
## f(x) = |x| is continuous but not differentiable at x = 0
Worked out problems.
Example 4:Let f (x) = | x|; then f is continuous everywhere, but not differentiable at 0.
## Solution: . We compute the Newton quotient directly; recall that | x| = x if x 0, while
| x| = - x if x < 0. Thus
= = 1, while
= = - 1.
Thus both of the one-sided limits exist, but are unequal, so the limit of the Newton
quotient does not exist.
Assignment:
## 1. Examine continuity and differentiability of f(x) at x = 0 where
1
f ( x) = x cos , x ≠ 0 WBUT 2007
x
And f(0) = 0.
2. Prove that the function f(x) = x − 1 , 0<x<2 is continuous at x = 1 but not differentiable
there . WBUT 2004
## 3. Examine the continuity and differentiability of the function
f ( x) = 1whenx ≤ 0
= 1 + sin xwhenx > 0
at x = 0. WBUT 2003
Objective type questions:
1. f ( x) = x − x is differentiable at x = 0
x2
f ( x) = ,x ≠ 0
2. . x
= 0, x = 0
|
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Early universe was less dusty than believedDecember 9th, 2013 in Astronomy & Space / Astronomy
Credit: Swinburne University of Technology
(Phys.org) —Dust may be more rare than expected in galaxies of the early Universe, according to an international research team, led by Swinburne University of Technology astrophysicist Dr David Fisher.
In a galaxy named IZw 18, the team measured the lowest dust mass of a galaxy that has ever been measured.
"It's not just that the dust mass is low. We found that the dust mass is 100 times smaller than would be expected based on commonly assumed theories," Dr Fisher said.
The galaxy, I Zw 18, is nearby, which makes it easier to study, but has properties that are very similar to galaxies of the high redshift Univese.
"It's an extreme galaxy in the local Universe, but it tells us a lot about a stage that almost all galaxies have gone through, so it gives us a picture of what the first galaxies look like."
Dr Fisher said the results imply that galaxies of the early Universe may have less dust than has been expected.
"This means, firstly, that they will look different than we expect and make different populations of stars than we expect. And secondly, that they will be much more difficult to observe, even with state-of-the-art facilities being built now such as the Atacama Large Millimeter/sub-millimeter Array (ALMA) of radio telescopes in northern Chile.
"IZw 18 is typical of very high redshift galaxies because it is very actively forming stars, and has a chemistry that is more like galaxies of the very early Universe with a very low abundance of metals and a lot of gas in the form of hydrogen," he said.
"Our result implies that current theories to describe the formation of stars when the Universe was very young are incomplete, and are built on invalid assumptions."
According to Dr Fisher, the amount of dust is very important for the formation of stars.
"What we think is going on, is that the harsh environment inside the galaxy we examined is adversely affecting the amount of dust in it.
"The radiation field measured inside I Zw 18 was roughly 200 times stronger than what we experience here in the Milky Way." Dr Fisher said that based on the findings, theories should be amended to account for environment in making stars.
The research is published in Nature.
More information: Paper: dx.doi.org/10.1038/nature12765
Provided by Swinburne University of Technology
"Early universe was less dusty than believed." December 9th, 2013. http://phys.org/news/2013-12-early-universe-dusty-believed.html
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The population of the world’s largest primate – a gorilla subspecies that lives in a region of Central Africa beset by conflict – is collapsing.
Back in 1998, a team of researchers estimated that 17,000 Grauer’s gorillas, also known as eastern lowland gorillas, lived in the forests of eastern Democratic Republic of Congo.
Since then, the population of Grauer’s gorillas has dropped by 77 percent.
Fewer than 3,800 of these gorillas still live in the wild, according to an alarming report published this week by the Wildlife Conservation Society, Flora and Fauna International and the Congolese Institute for the Conservation of Nature.
The gorillas have been impacted by civil war in the DRC, the establishment of mining camps to fund militias and subsistence and bushmeat hunting to feed miners, according to the report.
The recent conflicts in the country began in the wake of the 1994 Rwandan genocide, when refugees poured into eastern DRC. An estimated 5 million people died in Congolese civil wars.
“The crash in the gorilla population is a consequence of the human tragedy that has played out in eastern DRC,” report co-author Jefferson Hall, a staff scientist at the Smithsonian Tropical Research Institute, said in a statement. “Armed factions terrorize innocent people and divide up the spoils of war with absolutely no concern for the victims or the environment.”
Just as humans have been impacted, so too has the region’s wildlife. Mineral-rich Congo has seen a rapid expansion of artisanal, or subsistence, mining. And while the Grauer’s gorilla is legally protected, it’s still hunted; the gorilla’s large size (some of these animals weigh more than 400 pounds and stand more than five feet tall) means one kill can feed many miners who often live in remote areas, far from villages and towns.
The Grauer’s gorilla’s entire range “has been consumed in conflict” since 1996, according to the report. “This has resulted in an almost complete breakdown of government control, including wildlife protection activities.”
This has resulted in an almost complete breakdown of government control, including wildlife protection activities.
Grauer’s gorillas are closely related to another subspecies: mountain gorillas. Those animals live in parts of the DRC, Uganda and Rwanda, in a region called Africa’s Western Rift Valley.
Fewer than 700 such gorillas are thought to live in the wild, and rangers in Congo’s Virguna National Park face mortal danger from rebels and poachers as they try to protect the endangered animals. More than 150 rangers have been killed over the previous two decades.
“We have seen, over and over again, dedicated Congolese conservationists risk their lives to make a difference,” said Hall.
The Grauer’s gorilla has been categorized as “endangered” since the 1980s on the International Union for Conservation of Nature’s Red List of Threatened Species. Given this new population estimate, the report authors recommend listing the Grauer’s gorilla as “critically endangered,” which is just one step below “extinct in the wild.”
The new estimate comes from using the baseline data gathered by Hall and others in the 1990s and incorporates recent data collected from surveys conducted by local communities and rangers.
Conservationists say it will take a lot of work to stop the dwindling gorilla numbers.
The report recommends regulating artisanal mining sites and disarming miners; increasing security where the gorillas live; adding new protected areas and giving more support for existing protected lands; launching public education campaigns; and focusing on economic development so locals have viable alternatives to artisanal mining.
“Human dignity and welfare are inextricably linked to the dignity and survival of wild animals like Grauer’s gorilla and the ecosystems that sustain them,” Andrew Plumptre of the Wildlife Conservation Society said in a statement. “The activity of armed militias controlling mining camps in the Grauer’s gorilla heartland is likely to eliminate the Grauer’s gorilla entirely.”
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# Search an element in a sorted and rotated Array
Given a sorted and rotated array arr[] of size N and a key, the task is to find the key in the array.
Note: Find the element in O(logN) time and assume that all the elements are distinct.
Example:
Input : arr[] = {5, 6, 7, 8, 9, 10, 1, 2, 3}, key = 3
Output : Found at index 8
Input : arr[] = {5, 6, 7, 8, 9, 10, 1, 2, 3}, key = 30
Input : arr[] = {30, 40, 50, 10, 20}, key = 10
Output : Found at index 3
Recommended Practice
Approach 1 (Finding Pivot where rotation has happened): The primary idea to solve the problem is as follows.
The idea is to find the pivot point, divide the array into two sub-arrays and perform a binary search.
The main idea for finding a pivot is –
• For a sorted (in increasing order) and rotated array, the pivot element is the only element for which the next element to it is smaller than it.
• Using binary search based on the above idea, pivot can be found.
• It can be observed that for a search space of indices in range [l, r] where the middle index is mid
• If rotation has happened in the left half, then obviously the element at l will be greater than the one at mid.
• Otherwise the left half will be sorted but the element at mid will be greater than the one at r.
• After the pivot is found divide the array into two sub-arrays.
• Now the individual sub-arrays are sorted so the element can be searched using Binary Search.
Follow the steps mentioned below to implement the idea:
• Find out the pivot point using binary search. We will set the low pointer as the first array index and high with the last array index.
• From the high and low we will calculate the mid value.
• If the value at mid-1 is greater than the one at mid, return that value as the pivot.
• Else if the value at the mid+1 is less than mid, return mid value as the pivot.
• Otherwise, if the value at low position is greater than mid position, consider the left half. Otherwise, consider the right half.
• Divide the array into two sub-arrays based on the pivot that was found.
• Now call binary search for one of the two sub-arrays.
• If the element is greater than the 0th element then search in the left array
• Else search in the right array.
• If the element is found in the selected sub-array then return the index
• Else return -1.
Follow the below illustration for a better understanding
Illustration:
Consider arr[] = {3, 4, 5, 1, 2}, key = 1
Pivot finding:
low = 0, high = 4:
=> mid = 2
=> arr[mid] = 5, arr[mid + 1] = 1
=> arr[mid] > arr[mid +1],
=> Therefore the pivot = mid = 2
Array is divided into two parts {3, 4, 5}, {1, 2}
Now according to the conditions and the key, we need to find in the part {1, 2}
Key Finding:
We will apply Binary search on {1, 2}.
low = 3 , high = 4.
=> mid = 3
=> arr[mid] = 1 , key = 1, hence arr[mid] = key matches.
=> The required index = mid = 3
So the element is found at index 3.
Below is the implementation of the above approach:
## C++
`// C++ Program to search an element` `// in a sorted and pivoted array` `#include ` `using` `namespace` `std;` `// Standard Binary Search function` `int` `binarySearch(``int` `arr[], ``int` `low, ``int` `high, ``int` `key)` `{` ` ``if` `(high < low)` ` ``return` `-1;` ` ``int` `mid = (low + high) / 2;` ` ``if` `(key == arr[mid])` ` ``return` `mid;` ` ``if` `(key > arr[mid])` ` ``return` `binarySearch(arr, (mid + 1), high, key);` ` ``return` `binarySearch(arr, low, (mid - 1), key);` `}` `// Function to get pivot. For array 3, 4, 5, 6, 1, 2` `// it returns 3 (index of 6)` `int` `findPivot(``int` `arr[], ``int` `low, ``int` `high)` `{` ` ``// Base cases` ` ``if` `(high < low)` ` ``return` `-1;` ` ``if` `(high == low)` ` ``return` `low;` ` ``// low + (high - low)/2;` ` ``int` `mid = (low + high) / 2;` ` ``if` `(mid < high && arr[mid] > arr[mid + 1])` ` ``return` `mid;` ` ``if` `(mid > low && arr[mid] < arr[mid - 1])` ` ``return` `(mid - 1);` ` ``if` `(arr[low] >= arr[mid])` ` ``return` `findPivot(arr, low, mid - 1);` ` ``return` `findPivot(arr, mid + 1, high);` `}` `// Searches an element key in a pivoted` `// sorted array arr[] of size n` `int` `pivotedBinarySearch(``int` `arr[], ``int` `n, ``int` `key)` `{` ` ``int` `pivot = findPivot(arr, 0, n - 1);` ` ``// If we didn't find a pivot,` ` ``// then array is not rotated at all` ` ``if` `(pivot == -1)` ` ``return` `binarySearch(arr, 0, n - 1, key);` ` ``// If we found a pivot, then first compare with pivot` ` ``// and then search in two subarrays around pivot` ` ``if` `(arr[pivot] == key)` ` ``return` `pivot;` ` ``if` `(arr[0] <= key)` ` ``return` `binarySearch(arr, 0, pivot - 1, key);` ` ``return` `binarySearch(arr, pivot + 1, n - 1, key);` `}` `// Driver program to check above functions` `int` `main()` `{` ` ``// Let us search 3 in below array` ` ``int` `arr1[] = { 5, 6, 7, 8, 9, 10, 1, 2, 3 };` ` ``int` `n = ``sizeof``(arr1) / ``sizeof``(arr1[0]);` ` ``int` `key = 3;` ` ``// Function calling` ` ``cout << ``"Index of the element is : "` ` ``<< pivotedBinarySearch(arr1, n, key);` ` ``return` `0;` `}`
## C
`/* Program to search an element in` ` ``a sorted and pivoted array*/` `#include ` `int` `findPivot(``int``[], ``int``, ``int``);` `int` `binarySearch(``int``[], ``int``, ``int``, ``int``);` `/* Searches an element key in a pivoted` ` ``sorted array arrp[] of size n */` `int` `pivotedBinarySearch(``int` `arr[], ``int` `n, ``int` `key)` `{` ` ``int` `pivot = findPivot(arr, 0, n - 1);` ` ``// If we didn't find a pivot,` ` ``// then array is not rotated at all` ` ``if` `(pivot == -1)` ` ``return` `binarySearch(arr, 0, n - 1, key);` ` ``// If we found a pivot, then first` ` ``// compare with pivot and then` ` ``// search in two subarrays around pivot` ` ``if` `(arr[pivot] == key)` ` ``return` `pivot;` ` ``if` `(arr[0] <= key)` ` ``return` `binarySearch(arr, 0, pivot - 1, key);` ` ``return` `binarySearch(arr, pivot + 1, n - 1, key);` `}` `/* Function to get pivot. For array` ` ``3, 4, 5, 6, 1, 2 it returns 3 (index of 6) */` `int` `findPivot(``int` `arr[], ``int` `low, ``int` `high)` `{` ` ``// base cases` ` ``if` `(high < low)` ` ``return` `-1;` ` ``if` `(high == low)` ` ``return` `low;` ` ``int` `mid = (low + high) / 2; ``/*low + (high - low)/2;*/` ` ``if` `(mid < high && arr[mid] > arr[mid + 1])` ` ``return` `mid;` ` ``if` `(mid > low && arr[mid] < arr[mid - 1])` ` ``return` `(mid - 1);` ` ``if` `(arr[low] >= arr[mid])` ` ``return` `findPivot(arr, low, mid - 1);` ` ``return` `findPivot(arr, mid + 1, high);` `}` `/* Standard Binary Search function*/` `int` `binarySearch(``int` `arr[], ``int` `low, ``int` `high, ``int` `key)` `{` ` ``if` `(high < low)` ` ``return` `-1;` ` ``int` `mid = (low + high) / 2; ``/*low + (high - low)/2;*/` ` ``if` `(key == arr[mid])` ` ``return` `mid;` ` ``if` `(key > arr[mid])` ` ``return` `binarySearch(arr, (mid + 1), high, key);` ` ``return` `binarySearch(arr, low, (mid - 1), key);` `}` `/* Driver program to check above functions */` `int` `main()` `{` ` ``// Let us search 3 in below array` ` ``int` `arr1[] = { 5, 6, 7, 8, 9, 10, 1, 2, 3 };` ` ``int` `n = ``sizeof``(arr1) / ``sizeof``(arr1[0]);` ` ``int` `key = 3;` ` ``printf``(``"Index of the element is : %d"``,` ` ``pivotedBinarySearch(arr1, n, key));` ` ``return` `0;` `}`
## Java
`/* Java program to search an element` ` ``in a sorted and pivoted array*/` `import` `java.io.*;` `class` `Main {` ` ``/* Searches an element key in a` ` ``pivoted sorted array arrp[]` ` ``of size n */` ` ``static` `int` `pivotedBinarySearch(``int` `arr[], ``int` `n,` ` ``int` `key)` ` ``{` ` ``int` `pivot = findPivot(arr, ``0``, n - ``1``);` ` ``// If we didn't find a pivot, then` ` ``// array is not rotated at all` ` ``if` `(pivot == -``1``)` ` ``return` `binarySearch(arr, ``0``, n - ``1``, key);` ` ``// If we found a pivot, then first` ` ``// compare with pivot and then` ` ``// search in two subarrays around pivot` ` ``if` `(arr[pivot] == key)` ` ``return` `pivot;` ` ``if` `(arr[``0``] <= key)` ` ``return` `binarySearch(arr, ``0``, pivot - ``1``, key);` ` ``return` `binarySearch(arr, pivot + ``1``, n - ``1``, key);` ` ``}` ` ``/* Function to get pivot. For array` ` ``3, 4, 5, 6, 1, 2 it returns` ` ``3 (index of 6) */` ` ``static` `int` `findPivot(``int` `arr[], ``int` `low, ``int` `high)` ` ``{` ` ``// base cases` ` ``if` `(high < low)` ` ``return` `-``1``;` ` ``if` `(high == low)` ` ``return` `low;` ` ``/* low + (high - low)/2; */` ` ``int` `mid = (low + high) / ``2``;` ` ``if` `(mid < high && arr[mid] > arr[mid + ``1``])` ` ``return` `mid;` ` ``if` `(mid > low && arr[mid] < arr[mid - ``1``])` ` ``return` `(mid - ``1``);` ` ``if` `(arr[low] >= arr[mid])` ` ``return` `findPivot(arr, low, mid - ``1``);` ` ``return` `findPivot(arr, mid + ``1``, high);` ` ``}` ` ``/* Standard Binary Search function */` ` ``static` `int` `binarySearch(``int` `arr[], ``int` `low, ``int` `high,` ` ``int` `key)` ` ``{` ` ``if` `(high < low)` ` ``return` `-``1``;` ` ``/* low + (high - low)/2; */` ` ``int` `mid = (low + high) / ``2``;` ` ``if` `(key == arr[mid])` ` ``return` `mid;` ` ``if` `(key > arr[mid])` ` ``return` `binarySearch(arr, (mid + ``1``), high, key);` ` ``return` `binarySearch(arr, low, (mid - ``1``), key);` ` ``}` ` ``// main function` ` ``public` `static` `void` `main(String args[])` ` ``{` ` ``// Let us search 3 in below array` ` ``int` `arr1[] = { ``5``, ``6``, ``7``, ``8``, ``9``, ``10``, ``1``, ``2``, ``3` `};` ` ``int` `n = arr1.length;` ` ``int` `key = ``3``;` ` ``System.out.println(` ` ``"Index of the element is : "` ` ``+ pivotedBinarySearch(arr1, n, key));` ` ``}` `}`
## Python3
`# Python Program to search an element` `# in a sorted and pivoted array` `# Searches an element key in a pivoted` `# sorted array arrp[] of size n` `def` `pivotedBinarySearch(arr, n, key):` ` ``pivot ``=` `findPivot(arr, ``0``, n``-``1``)` ` ``# If we didn't find a pivot,` ` ``# then array is not rotated at all` ` ``if` `pivot ``=``=` `-``1``:` ` ``return` `binarySearch(arr, ``0``, n``-``1``, key)` ` ``# If we found a pivot, then first` ` ``# compare with pivot and then` ` ``# search in two subarrays around pivot` ` ``if` `arr[pivot] ``=``=` `key:` ` ``return` `pivot` ` ``if` `arr[``0``] <``=` `key:` ` ``return` `binarySearch(arr, ``0``, pivot``-``1``, key)` ` ``return` `binarySearch(arr, pivot ``+` `1``, n``-``1``, key)` `# Function to get pivot. For array` `# 3, 4, 5, 6, 1, 2 it returns 3` `# (index of 6)` `def` `findPivot(arr, low, high):` ` ``# base cases` ` ``if` `high < low:` ` ``return` `-``1` ` ``if` `high ``=``=` `low:` ` ``return` `low` ` ``# low + (high - low)/2;` ` ``mid ``=` `int``((low ``+` `high)``/``2``)` ` ``if` `mid < high ``and` `arr[mid] > arr[mid ``+` `1``]:` ` ``return` `mid` ` ``if` `mid > low ``and` `arr[mid] < arr[mid ``-` `1``]:` ` ``return` `(mid``-``1``)` ` ``if` `arr[low] >``=` `arr[mid]:` ` ``return` `findPivot(arr, low, mid``-``1``)` ` ``return` `findPivot(arr, mid ``+` `1``, high)` `# Standard Binary Search function` `def` `binarySearch(arr, low, high, key):` ` ``if` `high < low:` ` ``return` `-``1` ` ``# low + (high - low)/2;` ` ``mid ``=` `int``((low ``+` `high)``/``2``)` ` ``if` `key ``=``=` `arr[mid]:` ` ``return` `mid` ` ``if` `key > arr[mid]:` ` ``return` `binarySearch(arr, (mid ``+` `1``), high,` ` ``key)` ` ``return` `binarySearch(arr, low, (mid ``-` `1``), key)` `# Driver program to check above functions` `# Let us search 3 in below array` `if` `__name__ ``=``=` `'__main__'``:` ` ``arr1 ``=` `[``5``, ``6``, ``7``, ``8``, ``9``, ``10``, ``1``, ``2``, ``3``]` ` ``n ``=` `len``(arr1)` ` ``key ``=` `3` ` ``print``(``"Index of the element is : "``, \` ` ``pivotedBinarySearch(arr1, n, key))` `# This is contributed by Smitha Dinesh Semwal`
## C#
`// C# program to search an element` `// in a sorted and pivoted array` `using` `System;` `class` `main {` ` ``// Searches an element key in a` ` ``// pivoted sorted array arrp[]` ` ``// of size n` ` ``static` `int` `pivotedBinarySearch(``int``[] arr,` ` ``int` `n, ``int` `key)` ` ``{` ` ``int` `pivot = findPivot(arr, 0, n - 1);` ` ``// If we didn't find a pivot, then` ` ``// array is not rotated at all` ` ``if` `(pivot == -1)` ` ``return` `binarySearch(arr, 0, n - 1, key);` ` ``// If we found a pivot, then first` ` ``// compare with pivot and then` ` ``// search in two subarrays around pivot` ` ``if` `(arr[pivot] == key)` ` ``return` `pivot;` ` ``if` `(arr[0] <= key)` ` ``return` `binarySearch(arr, 0, pivot - 1, key);` ` ``return` `binarySearch(arr, pivot + 1, n - 1, key);` ` ``}` ` ``/* Function to get pivot. For array ` ` ``3, 4, 5, 6, 1, 2 it returns` ` ``3 (index of 6) */` ` ``static` `int` `findPivot(``int``[] arr, ``int` `low, ``int` `high)` ` ``{` ` ``// base cases` ` ``if` `(high < low)` ` ``return` `-1;` ` ``if` `(high == low)` ` ``return` `low;` ` ``/* low + (high - low)/2; */` ` ``int` `mid = (low + high) / 2;` ` ``if` `(mid < high && arr[mid] > arr[mid + 1])` ` ``return` `mid;` ` ``if` `(mid > low && arr[mid] < arr[mid - 1])` ` ``return` `(mid - 1);` ` ``if` `(arr[low] >= arr[mid])` ` ``return` `findPivot(arr, low, mid - 1);` ` ``return` `findPivot(arr, mid + 1, high);` ` ``}` ` ``/* Standard Binary Search function */` ` ``static` `int` `binarySearch(``int``[] arr, ``int` `low,` ` ``int` `high, ``int` `key)` ` ``{` ` ``if` `(high < low)` ` ``return` `-1;` ` ``/* low + (high - low)/2; */` ` ``int` `mid = (low + high) / 2;` ` ``if` `(key == arr[mid])` ` ``return` `mid;` ` ``if` `(key > arr[mid])` ` ``return` `binarySearch(arr, (mid + 1), high, key);` ` ``return` `binarySearch(arr, low, (mid - 1), key);` ` ``}` ` ``// Driver Code` ` ``public` `static` `void` `Main()` ` ``{` ` ``// Let us search 3 in below array` ` ``int``[] arr1 = { 5, 6, 7, 8, 9, 10, 1, 2, 3 };` ` ``int` `n = arr1.Length;` ` ``int` `key = 3;` ` ``Console.Write(``"Index of the element is : "` ` ``+ pivotedBinarySearch(arr1, n, key));` ` ``}` `}` `// This code is contributed by vt_m.`
## Javascript
``
## PHP
` ``\$arr``[``\$mid``])` ` ``return` `binarySearch(``\$arr``, (``\$mid` `+ 1),` ` ``\$high``, ``\$key``);` ` ` ` ``else` ` ``return` `binarySearch(``\$arr``, ``\$low``,` ` ``(``\$mid` `-1), ``\$key``);` `}` `// Function to get pivot. ` `// For array 3, 4, 5, 6, 1, 2` `// it returns 3 (index of 6)` `function` `findPivot(``\$arr``, ``\$low``, ``\$high``)` `{` ` ` ` ``// base cases` ` ``if` `(``\$high` `< ``\$low``) ` ` ``return` `-1;` ` ``if` `(``\$high` `== ``\$low``) ` ` ``return` `\$low``;` ` ` ` ``/*low + (high - low)/2;*/` ` ``\$mid` `= (``\$low` `+ ``\$high``)/2; ` ` ``if` `(``\$mid` `< ``\$high` `and` `\$arr``[``\$mid``] > ` ` ``\$arr``[``\$mid` `+ 1])` ` ``return` `\$mid``;` ` ` ` ``if` `(``\$mid` `> ``\$low` `and` `\$arr``[``\$mid``] < ` ` ``\$arr``[``\$mid` `- 1])` ` ``return` `(``\$mid` `- 1);` ` ` ` ``if` `(``\$arr``[``\$low``] >= ``\$arr``[``\$mid``])` ` ``return` `findPivot(``\$arr``, ``\$low``,` ` ``\$mid` `- 1);` ` ` ` ``return` `findPivot(``\$arr``, ``\$mid` `+ 1, ``\$high``);` `}` `// Searches an element key` `// in a pivoted sorted array` `// arr[] of size n */` `function` `pivotedBinarySearch(``\$arr``, ``\$n``, ``\$key``)` `{` ` ``\$pivot` `= findPivot(``\$arr``, 0, ``\$n` `- 1);` ` ` ` ``// If we didn't find a pivot, ` ` ``// then array is not rotated` ` ``// at all` ` ``if` `(``\$pivot` `== -1)` ` ``return` `binarySearch(``\$arr``, 0, ` ` ``\$n` `- 1, ``\$key``);` ` ` ` ``// If we found a pivot, ` ` ``// then first compare` ` ``// with pivot and then ` ` ``// search in two subarrays` ` ``// around pivot` ` ``if` `(``\$arr``[``\$pivot``] == ``\$key``)` ` ``return` `\$pivot``;` ` ` ` ``if` `(``\$arr``[0] <= ``\$key``)` ` ``return` `binarySearch(``\$arr``, 0, ` ` ``\$pivot` `- 1, ``\$key``);` ` ` ` ``return` `binarySearch(``\$arr``, ``\$pivot` `+ 1, ` ` ``\$n` `- 1, ``\$key``);` `}` `// Driver Code` `// Let us search 3 ` `// in below array` `\$arr1` `= ``array``(5, 6, 7, 8, 9, 10, 1, 2, 3);` `\$n` `= ``count``(``\$arr1``);` `\$key` `= 3;` `// Function calling` `echo` `"Index of the element is : "``, ` ` ``pivotedBinarySearch(``\$arr1``, ``\$n``, ``\$key``);` ` ` `// This code is contributed by anuj_67.` `?>`
Output
```Index of the element is : 8
```
Time Complexity: O(log N) Binary Search requires log n comparisons to find the element.
Auxiliary Complexity: O(1)
Thanks to Ajay Mishra for providing the above solution.
Approach 2 (Direct Binary Search on Array without finding Pivot):
The idea is to instead of two or more passes of binary search, the result can be found in one pass of binary search.
The idea is to create a recursive function to implement the binary search where the search region is [l, r]. For each recursive call:
• We calculate the mid value as mid = (l + h) / 2
• Then try to figure out if l to mid is sorted, or (mid+1) to h is sorted
• Based on that decide the next search region and keep on doing this till the element is found or l overcomes h.
Follow the steps mentioned below to implement the idea:
• Use a recursive function to implement binary search to find the key:
• Find middle-point mid = (l + h)/2
• If the key is present at the middle point, return mid.
• Else if the value at l is less than the one at mid then arr[l . . . mid] is sorted
• If the key to be searched lies in the range from arr[l] to arr[mid], recur for arr[l . . . mid].
• Else recur for arr[mid+1 . . . h]
• Else arr[mid+1. . . h] is sorted:
• If the key to be searched lies in the range from arr[mid+1] to arr[h], recur for arr[mid+1. . . h].
• Else recur for arr[l. . . mid]
Follow the below illustration for a better understanding:
Illustration:
Input arr[] = {3, 4, 5, 1, 2}, key = 1
Initially low = 0, high = 4.
low = 0, high = 4:
=> mid = 2
=> arr[mid] = 5, which is not the desired value.
=> arr[low] < arr[mid] So, the left half is sorted.
=> key < arr[low], So the next search region is 3 to 4.
low = 3, high = 4:
=> mid = 3
=> arr[mid] = 1 = key
=> So the element is found at index 3.
The element is found at index 3.
Below is the implementation of the above idea:
## C++
`// Search an element in sorted and rotated` `// array using single pass of Binary Search` `#include ` `using` `namespace` `std;` `// Returns index of key in arr[l..h] if` `// key is present, otherwise returns -1` `int` `search(``int` `arr[], ``int` `l, ``int` `h, ``int` `key)` `{` ` ``if` `(l > h)` ` ``return` `-1;` ` ``int` `mid = (l + h) / 2;` ` ``if` `(arr[mid] == key)` ` ``return` `mid;` ` ``/* If arr[l...mid] is sorted */` ` ``if` `(arr[l] <= arr[mid]) {` ` ``/* As this subarray is sorted, we can quickly` ` ``check if key lies in half or other half */` ` ``if` `(key >= arr[l] && key <= arr[mid])` ` ``return` `search(arr, l, mid - 1, key);` ` ``/*If key not lies in first half subarray,` ` ``Divide other half into two subarrays,` ` ``such that we can quickly check if key lies` ` ``in other half */` ` ``return` `search(arr, mid + 1, h, key);` ` ``}` ` ``/* If arr[l..mid] first subarray is not sorted, then` ` ``arr[mid... h] must be sorted subarray */` ` ``if` `(key >= arr[mid] && key <= arr[h])` ` ``return` `search(arr, mid + 1, h, key);` ` ``return` `search(arr, l, mid - 1, key);` `}` `// Driver program` `int` `main()` `{` ` ``int` `arr[] = { 4, 5, 6, 7, 8, 9, 1, 2, 3 };` ` ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` ` ``int` `key = 3;` ` ``int` `i = search(arr, 0, n - 1, key);` ` ``if` `(i != -1)` ` ``cout << ``"Index: "` `<< i << endl;` ` ``else` ` ``cout << ``"Key not found"``;` `}` `// This code is contributed by Aditya Kumar (adityakumar129)`
## C
`// Search an element in sorted and rotated` `// array using single pass of Binary Search` `#include ` `// Returns index of key in arr[l..h] if` `// key is present, otherwise returns -1` `int` `search(``int` `arr[], ``int` `l, ``int` `h, ``int` `key)` `{` ` ``if` `(l > h)` ` ``return` `-1;` ` ``int` `mid = (l + h) / 2;` ` ``if` `(arr[mid] == key)` ` ``return` `mid;` ` ``/* If arr[l...mid] is sorted */` ` ``if` `(arr[l] <= arr[mid]) {` ` ``/* As this subarray is sorted, we can quickly` ` ``check if key lies in half or other half */` ` ``if` `(key >= arr[l] && key <= arr[mid])` ` ``return` `search(arr, l, mid - 1, key);` ` ``/*If key not lies in first half subarray,` ` ``Divide other half into two subarrays,` ` ``such that we can quickly check if key lies` ` ``in other half */` ` ``return` `search(arr, mid + 1, h, key);` ` ``}` ` ``/* If arr[l..mid] first subarray is not sorted, then` ` ``arr[mid... h] must be sorted subarray */` ` ``if` `(key >= arr[mid] && key <= arr[h])` ` ``return` `search(arr, mid + 1, h, key);` ` ``return` `search(arr, l, mid - 1, key);` `}` `// Driver program` `int` `main()` `{` ` ``int` `arr[] = { 4, 5, 6, 7, 8, 9, 1, 2, 3 };` ` ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);` ` ``int` `key = 3;` ` ``int` `i = search(arr, 0, n - 1, key);` ` ``if` `(i != -1)` ` ``printf``(``"Index: %d\n"``, i);` ` ``else` ` ``printf``(``"Key not found"``);` `}` `// This code is contributed by Aditya Kumar (adityakumar129)`
## Java
`/* Java program to search an element in` ` ``sorted and rotated array using` ` ``single pass of Binary Search*/` `import` `java.io.*;` `class` `Main {` ` ``// Returns index of key in arr[l..h]` ` ``// if key is present, otherwise returns -1` ` ``static` `int` `search(``int` `arr[], ``int` `l, ``int` `h, ``int` `key)` ` ``{` ` ``if` `(l > h)` ` ``return` `-``1``;` ` ``int` `mid = (l + h) / ``2``;` ` ``if` `(arr[mid] == key)` ` ``return` `mid;` ` ``/* If arr[l...mid] first subarray is sorted */` ` ``if` `(arr[l] <= arr[mid]) {` ` ``/* As this subarray is sorted, we` ` ``can quickly check if key lies in` ` ``half or other half */` ` ``if` `(key >= arr[l] && key <= arr[mid])` ` ``return` `search(arr, l, mid - ``1``, key);` ` ``/*If key not lies in first half subarray,` ` ``Divide other half into two subarrays,` ` ``such that we can quickly check if key lies` ` ``in other half */` ` ``return` `search(arr, mid + ``1``, h, key);` ` ``}` ` ``/* If arr[l..mid] first subarray is not sorted,` ` ``then arr[mid... h] must be sorted subarray*/` ` ``if` `(key >= arr[mid] && key <= arr[h])` ` ``return` `search(arr, mid + ``1``, h, key);` ` ``return` `search(arr, l, mid - ``1``, key);` ` ``}` ` ``// main function` ` ``public` `static` `void` `main(String args[])` ` ``{` ` ``int` `arr[] = { ``4``, ``5``, ``6``, ``7``, ``8``, ``9``, ``1``, ``2``, ``3` `};` ` ``int` `n = arr.length;` ` ``int` `key = ``3``;` ` ``int` `i = search(arr, ``0``, n - ``1``, key);` ` ``if` `(i != -``1``)` ` ``System.out.println(``"Index: "` `+ i);` ` ``else` ` ``System.out.println(``"Key not found"``);` ` ``}` `}` `// This code is contributed by Aditya Kumar (adityakumar129)`
## Python3
`# Search an element in sorted and rotated array using` `# single pass of Binary Search` `# Returns index of key in arr[l..h] if key is present,` `# otherwise returns -1` `def` `search(arr, l, h, key):` ` ``if` `l > h:` ` ``return` `-``1` ` ``mid ``=` `(l ``+` `h) ``/``/` `2` ` ``if` `arr[mid] ``=``=` `key:` ` ``return` `mid` ` ``# If arr[l...mid] is sorted` ` ``if` `arr[l] <``=` `arr[mid]:` ` ``# As this subarray is sorted, we can quickly` ` ``# check if key lies in half or other half` ` ``if` `key >``=` `arr[l] ``and` `key <``=` `arr[mid]:` ` ``return` `search(arr, l, mid``-``1``, key)` ` ``return` `search(arr, mid ``+` `1``, h, key)` ` ``# If arr[l..mid] is not sorted, then arr[mid... r]` ` ``# must be sorted` ` ``if` `key >``=` `arr[mid] ``and` `key <``=` `arr[h]:` ` ``return` `search(arr, mid ``+` `1``, h, key)` ` ``return` `search(arr, l, mid``-``1``, key)` `# Driver program` `if` `__name__ ``=``=` `'__main__'``:` ` ``arr ``=` `[``4``, ``5``, ``6``, ``7``, ``8``, ``9``, ``1``, ``2``, ``3``]` ` ``key ``=` `3` ` ``i ``=` `search(arr, ``0``, ``len``(arr)``-``1``, key)` ` ``if` `i !``=` `-``1``:` ` ``print``(``"Index: % d"` `%` `i)` ` ``else``:` ` ``print``(``"Key not found"``)` `# This code is contributed by Shreyanshi Arun`
## C#
`/* C# program to search an element in` `sorted and rotated array using` `single pass of Binary Search*/` `using` `System;` `class` `GFG {` ` ``// Returns index of key in arr[l..h]` ` ``// if key is present, otherwise` ` ``// returns -1` ` ``static` `int` `search(``int``[] arr, ``int` `l, ``int` `h, ``int` `key)` ` ``{` ` ``if` `(l > h)` ` ``return` `-1;` ` ``int` `mid = (l + h) / 2;` ` ``if` `(arr[mid] == key)` ` ``return` `mid;` ` ``/* If arr[l...mid] is sorted */` ` ``if` `(arr[l] <= arr[mid]) {` ` ``/* As this subarray is sorted, we` ` ``can quickly check if key lies in` ` ``half or other half */` ` ``if` `(key >= arr[l] && key <= arr[mid])` ` ``return` `search(arr, l, mid - 1, key);` ` ``return` `search(arr, mid + 1, h, key);` ` ``}` ` ``/* If arr[l..mid] is not sorted,` ` ``then arr[mid... r] must be sorted*/` ` ``if` `(key >= arr[mid] && key <= arr[h])` ` ``return` `search(arr, mid + 1, h, key);` ` ``return` `search(arr, l, mid - 1, key);` ` ``}` ` ``// main function` ` ``public` `static` `void` `Main()` ` ``{` ` ``int``[] arr = { 4, 5, 6, 7, 8, 9, 1, 2, 3 };` ` ``int` `n = arr.Length;` ` ``int` `key = 3;` ` ``int` `i = search(arr, 0, n - 1, key);` ` ``if` `(i != -1)` ` ``Console.WriteLine(``"Index: "` `+ i);` ` ``else` ` ``Console.WriteLine(``"Key not found"``);` ` ``}` `}` `// This code is contributed by anuj_67.`
## Javascript
``
## PHP
` ``\$h``) ``return` `-1;` ` ``\$mid` `= ``floor``((``\$l` `+ ``\$h``) / 2);` ` ``if` `(``\$arr``[``\$mid``] == ``\$key``)` ` ``return` `\$mid``;` ` ``/* If arr[l...mid] is sorted */` ` ``if` `(``\$arr``[``\$l``] <= ``\$arr``[``\$mid``])` ` ``{` ` ` ` ``/* As this subarray is ` ` ``sorted, we can quickly` ` ``check if key lies in ` ` ``half or other half */` ` ``if` `(``\$key` `>= ``\$arr``[``\$l``] ``and` ` ``\$key` `<= ``\$arr``[``\$mid``])` ` ``return` `search(``\$arr``, ``\$l``, ` ` ``\$mid` `- 1, ``\$key``);` ` ``return` `search(``\$arr``, ``\$mid` `+ 1,` ` ``\$h``, ``\$key``);` ` ``}` ` ``/* If arr[l..mid] is not ` ` ``sorted, then arr[mid... r]` ` ``must be sorted*/` ` ``if` `(``\$key` `>= ``\$arr``[``\$mid``] ``and` ` ``\$key` `<= ``\$arr``[``\$h``])` ` ``return` `search(``\$arr``, ``\$mid` `+ 1, ` ` ``\$h``, ``\$key``);` ` ``return` `search(``\$arr``, ``\$l``, ` ` ``\$mid``-1, ``\$key``);` `}` ` ``// Driver Code` ` ``\$arr` `= ``array``( 5, 6, 7, 8, 9, 10, 1, 2, 3 );` ` ``\$n` `= sizeof(``\$arr``);` ` ``\$key` `= 3;` ` ``\$i` `= search(``\$arr``, 0, ``\$n``-1, ``\$key``);` ` ``if` `(``\$i` `!= -1)` ` ``echo` `"Index: "``, ``\$i``, ``" \n"``;` ` ``else` ` ``echo` `"Key not found"``;` `// This code is contributed by ajit` `?>`
Output
```Index: 8
```
Time Complexity: O(log N). Binary Search requires log n comparisons to find the element. So time complexity is O(log n).
Auxiliary Space: O(1). As no extra space is required.
Thanks to Gaurav Ahirwar for suggesting the above solution.
How to handle duplicates?
At first look, it doesn’t look possible to search in O(Log N) time in all cases when duplicates are allowed.
For example consider searching 0 in {2, 2, 2, 2, 2, 2, 2, 2, 0, 2} and {2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2}.
Look into the following article to find a solution to this issue: https://www.geeksforgeeks.org/search-an-element-in-a-sorted-and-rotated-array-with-duplicates/
Similar Articles:
Please write comments if you find any bug in the above codes/algorithms, or find other ways to solve the same problem.
#### Approach#3: Using linear search
This approach uses linear search to find the index of the key in a sorted and rotated array. The idea is to iterate through the array and compare each element with the key until we find a match.
#### Algorithm
1. Initialize a loop index i to 0, and iterate over the array using a for loop.
2. For each element of the array, check if it is equal to the given key.
3. If the key is found, return the index of the element.
4. If the end of the array is reached without finding the key, return -1 to indicate that the key was not found.
## C++
`#include ` `#include ` `int` `searchRotatedArray(``const` `std::vector<``int``>& arr, ``int` `key) {` ` ``int` `n = arr.size();` ` ``for` `(``int` `i = 0; i < n; i++) {` ` ``if` `(arr[i] == key) {` ` ``return` `i;` ` ``}` ` ``}` ` ``return` `-1;` `}` `int` `main() {` ` ``std::vector<``int``> arr = {5, 6, 7, 8, 9, 10, 1, 2, 3};` ` ``int` `key = 3;` ` ``int` `index = searchRotatedArray(arr, key);` ` ``if` `(index != -1) {` ` ``std::cout << ``"Found at index "` `<< index << std::endl;` ` ``} ``else` `{` ` ``std::cout << ``"Not found"` `<< std::endl;` ` ``}` ` ``return` `0;` `}` `////This article is contributed by Abhay`
## Java
`/*package whatever //do not write package name here */` `import` `java.io.*;` `public` `class` `GFG {` ` ``public` `static` `int` `searchRotatedArray(``int``[] arr, ``int` `key) {` ` ``int` `n = arr.length;` ` ``for` `(``int` `i = ``0``; i < n; i++) {` ` ``if` `(arr[i] == key) {` ` ``return` `i;` ` ``}` ` ``}` ` ``return` `-``1``;` ` ``}` ` ``public` `static` `void` `main(String[] args) {` ` ``int``[] arr = {``5``, ``6``, ``7``, ``8``, ``9``, ``10``, ``1``, ``2``, ``3``};` ` ``int` `key = ``3``;` ` ``int` `index = searchRotatedArray(arr, key);` ` ``if` `(index != -``1``) {` ` ``System.out.println(``"Found at index "` `+ index);` ` ``} ``else` `{` ` ``System.out.println(``"Not found"``);` ` ``}` ` ``}` `}` `//This article is contributed by Abhay`
## Python3
`def` `search_rotated_array(arr, key):` ` ``n ``=` `len``(arr)` ` ``for` `i ``in` `range``(n):` ` ``if` `arr[i] ``=``=` `key:` ` ``return` `i` ` ``return` `-``1` `arr ``=` `[``5``, ``6``, ``7``, ``8``, ``9``, ``10``, ``1``, ``2``, ``3``]` `key ``=` `3` `index ``=` `search_rotated_array(arr, key)` `if` `index !``=` `-``1``:` ` ``print``(f``"Found at index {index}"``)` `else``:` ` ``print``(``"Not found"``)`
## C#
`using` `System;` `using` `System.Collections.Generic;` `class` `Program` `{` ` ``static` `int` `SearchRotatedArray(List<``int``> arr, ``int` `key)` ` ``{` ` ``int` `n = arr.Count;` ` ``for` `(``int` `i = 0; i < n; i++)` ` ``{` ` ``if` `(arr[i] == key)` ` ``{` ` ``return` `i;` ` ``}` ` ``}` ` ``return` `-1;` ` ``}` ` ``static` `void` `Main()` ` ``{` ` ``List<``int``> arr = ``new` `List<``int``> { 5, 6, 7, 8, 9, 10, 1, 2, 3 };` ` ``int` `key = 3;` ` ``int` `index = SearchRotatedArray(arr, key);` ` ``if` `(index != -1)` ` ``{` ` ``Console.WriteLine(``"Found at index "` `+ index);` ` ``}` ` ``else` ` ``{` ` ``Console.WriteLine(``"Not found"``);` ` ``}` ` ``}` `}`
## Javascript
``
Output
```Found at index 8
```
Time complexity of this algorithm is O(n), where n is the length of the input array.
Space complexity is O(1), as the program only uses a constant amount of extra space to store the loop index and the return value.
Feeling lost in the world of random DSA topics, wasting time without progress? It's time for a change! Join our DSA course, where we'll guide you on an exciting journey to master DSA efficiently and on schedule.
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# 5.05 Compare decimal numbers
Lesson
## Are you ready?
Remembering how to compare whole numbers is going to help us compare decimals in this lesson. Let's try this practice problem to review.
### Examples
#### Example 1
Which symbol, greater than (\gt) or less than (\lt) will make the following statements true?
a
91 \, ⬚ \, 97
A
\gt
B
\lt
Worked Solution
Create a strategy
Use a place value table and compare the numbers.
Apply the idea
As we can see in the table, both numbers in the tens column have the same number so we move to the units column.
In the units column, we can see that 1 is smaller than 7.
So the correct answer is 91 \lt 97, option B.
b
682 \, ⬚ \, 782
A
\gt
B
\lt
Worked Solution
Create a strategy
Use a place value table and compare the numbers.
Apply the idea
In the hundreds column, we can see that 6 is smaller than 7. We do not need to compare the remaining columns since we already know which is greater.
So the correct answer is 682 \lt 782, option B.
Idea summary
\gt means "greater than".
\lt means "less than".
## Compare decimals with thousandths
We can use place value columns to help us compare numbers with decimals. If we have 0.342, and we are comparing it to 0.458, we know that 342 is smaller than 458, so 342 thousandths is smaller than 458 thousandths. In this video we work through some examples, and see how we can compare decimals.
### Examples
#### Example 2
Fill in the box the greater than (\gt) or less than (\lt) symbol that would make this number sentence true. 0.082 \, ⬚ \, 0.088
Worked Solution
Create a strategy
Use a place value table and compare the numbers.
Apply the idea
Write the decimals in a place value table and use zeros as place holders.
As we can see in the table, both numbers in the units, tenths, and hundredths columns have the same number so we move to the thousandths column.
In the thousandths column, we can see that 2 is smaller than 8.
We can write the number sentence as:0.082 \lt 0.088
Idea summary
We can use a place value table to compare decimals.
## Compare numbers beyond thousandths
This video shows how to compare numbers that have many decimal places.
### Examples
#### Example 3
Fill in the box the greater than (\gt) or less than (\lt) symbol that would make this number sentence true. 0.754 \, ⬚ \, 0.6094
Worked Solution
Create a strategy
Use a place value table and compare the numbers.
Apply the idea
Write the decimals in a place value table and use zeros as place holders.
In the tenths column, we can see that 7 is larger than 6. We do not need to compare the remaining columns since we already know which is larger.
We can write the number sentence as:0.754 \gt 0.6094
Idea summary
When using a place value table to compare decimals we can start at the place value column furthest to the left and work to the right, comparing the values in each column to work out the bigger/smaller number.
Or, we can add zeros at the end of a decimal so we can compare the same number of place value columns. For example, if we wanted to compare 0.45 and 0.672, we could write 0.45 as 0.450. This means we can compare thousandths to thousandths.
### Outcomes
#### MA3-7NA
compares, orders and calculates with fractions, decimals and percentages
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Reading at School
There are two aspects to the reading programme at Spring Bank Primary – word reading and comprehension (both listening and reading.)
During Reception and beyond, our children learn to read using the Phonics Scheme ‘Floppy’s Phonics. Reading at home is supplemented by books from other schemes such as Rigby Star Phonics and Project X Phonics. All of our books are fully decodable.
As they travel on their journey to becoming independent and enthusiastic readers the children have access to a full range of exciting books.
All children are encouraged to take home a reading book to share with parents or to read on their own – for our fully independent readers these books may not adhere to a particular scheme and may be a book borrowed from our library.Younger children have a home/school diary and older children also have Guided Reading homework and a Reading Journal.
Writing at School
Writing at school is made up of two dimensions – transcription (spelling and handwriting) and composition (articulating ideas and structuring them in speech and writing.) In addition there is a strong focus on spelling, vocabulary, grammar and punctuation throughout the school. Writing is taught through a focus on different text types in each year with frequent opportunities to apply their learning across the curriculum.
Please click on the link below to view the English curriculum year by year.
Literacy Long Term Plan. This is currently in development.
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Precalculus
# Key Concepts
PrecalculusKey Concepts
## 7.1Solving Trigonometric Equations with Identities
• There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
• Graphing both sides of an identity will verify it. See Example 1.
• Simplifying one side of the equation to equal the other side is another method for verifying an identity. See Example 2 and Example 3.
• The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See Example 4.
• We can create an identity by simplifying an expression and then verifying it. See Example 5.
• Verifying an identity may involve algebra with the fundamental identities. See Example 6 and Example 7.
• Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See Example 8, Example 9, and Example 10.
## 7.2Sum and Difference Identities
• The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
• The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. See Example 1 and Example 2.
• The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle. See Example 3.
• The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See Example 4.
• The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles. See Example 5.
• The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See Example 6.
• The cofunction identities apply to complementary angles and pairs of reciprocal functions. See Example 7.
• Sum and difference formulas are useful in verifying identities. See Example 8 and Example 9.
• Application problems are often easier to solve by using sum and difference formulas. See Example 10 and Example 11.
## 7.3Double-Angle, Half-Angle, and Reduction Formulas
• Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See Example 1, Example 2, Example 3, and Example 4.
• Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See Example 5 and Example 6.
• Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See Example 7, Example 8, and Example 9.
## 7.4Sum-to-Product and Product-to-Sum Formulas
• From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine.
• We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines. See Example 1, Example 2, and Example 3.
• We can also derive the sum-to-product identities from the product-to-sum identities using substitution.
• We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine as products of sines and cosines. See Example 4.
• Trigonometric expressions are often simpler to evaluate using the formulas. See Example 5.
• The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify an identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the other side. See Example 6 and Example 7.
## 7.5Solving Trigonometric Equations
• When solving linear trigonometric equations, we can use algebraic techniques just as we do solving algebraic equations. Look for patterns, like the difference of squares, quadratic form, or an expression that lends itself well to substitution. See Example 1, Example 2, and Example 3.
• Equations involving a single trigonometric function can be solved or verified using the unit circle. See Example 4, Example 5, and Example 6, and Example 7.
• We can also solve trigonometric equations using a graphing calculator. See Example 8 and Example 9.
• Many equations appear quadratic in form. We can use substitution to make the equation appear simpler, and then use the same techniques we use solving an algebraic quadratic: factoring, the quadratic formula, etc. See Example 10, Example 11, Example 12, and Example 13.
• We can also use the identities to solve trigonometric equation. See Example 14, Example 15, and Example 16.
• We can use substitution to solve a multiple-angle trigonometric equation, which is a compression of a standard trigonometric function. We will need to take the compression into account and verify that we have found all solutions on the given interval. See Example 17.
• Real-world scenarios can be modeled and solved using the Pythagorean Theorem and trigonometric functions. See Example 18.
## 7.6Modeling with Trigonometric Functions
• Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See Example 1 and Example 2.
• Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See Example 3.
• Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See Example 4, Example 5, Example 6 and Example 7.
• Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See Example 8.
• Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See Example 9, Example 10, Example 11, Example 12, and Example 13.
• Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See Example 14.
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# Thread: Matrix associated to a linear map
1. ## Matrix associated to a linear map
R^3 = L(direct sum)W,with dim(L)=1.Suppose T:R^3--->R^3 is a linear map.T(L)subset of L and T(W)subset of W.
Find a basis B of R^3 such that m(T;B) is a 3*3 matrix with entries...a(11) nonzero number, a(21) zero, a(31) zero, a(12) zero, a(22) nonzero number, a(32) nonzero number, a(13) zero, a(23) nonzero number, a(33) nonzero number..
2. Originally Posted by math.dj
R^3 = L(direct sum)W,with dim(L)=1.Suppose T:R^3--->R^3 is a linear map.T(L)subset of L and T(W)subset of W.
Find a basis B of R^3 such that m(T;B) is a 3*3 matrix with entries...a(11) nonzero number, a(21) zero, a(31) zero, a(12) zero, a(22) nonzero number, a(32) nonzero number, a(13) zero, a(23) nonzero number, a(33) nonzero number..
Choose non-zero vectors $v_1, v_2, v_3$ so that $v_1$ is in L (and since L has dimension 1, { $v_1$} is a basis for L) and { $v_2, v_3$} is a basis for W. You can do that because L directsum W= $R^3$. Applying T to $v_1$ you get $Tv_1= av_1+ 0v_2+ 0v_3$ because T maps L to itself. Applying T to either $v_2$ or $v_3$ gives $0v_1+ bv_2+ cv_3$ because T maps W to itself.
3. W is not mentioned which plane it is..so can i take any two vectors..(like (1,0,-1/2),(0,1,-3/4) )..n what is the use of the condition T(L) subset of L and T(W) subset of W..is it used to find the linear map as the map is not mentioned..n what happens when T(L) subset of W and T(W) subset of L..
4. Originally Posted by math.dj
W is not mentioned which plane it is..so can i take any two vectors..(like (1,0,-1/2),(0,1,-3/4) )..n what is the use of the condition T(L) subset of L and T(W) subset of W..is it used to find the linear map as the map is not mentioned..n what happens when T(L) subset of W and T(W) subset of L..
If you really have no idea what the question is asking (and it appears from this that you don't) the best thing you can do is go to your teacher and ask for more explanation.
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# Think of a Number
## Ten students think of a number then perform various operations on that number. You have to find what the original numbers were.
##### Level 1Level 2Level 3DescriptionHelpAlgebra
This is level 3: Negative decimal THOAN problems. You can earn a trophy if you get at least 7 questions correct.
$$\times 2$$$$+22$$$$4$$Juan says:'I think of a number. I double my number and add 22 the answer is 4'What number was Juan first thinking of? Working: Charles says:'I am also thinking of a number. If I multiply my number by 2 and add 8.3 the answer is -0.899999999999999'What number was Charles first thinking of? Working: Aaron says:'I think of a number. I multiply my number by 3 and add 2.9 the answer is -24.4'What number was Aaron first thinking of? Working: Lucas says:'Can you guess what number I am thinking of if when I multiply it by 9.4 and add 1.9 the answer is -22.54?'What number was Lucas first thinking of? Working: Luis says:'If I divide my number by 5 and add 8.8 the answer is 8.16'What number was Luis first thinking of? Working: Owen says:'If I multiply my number by 3.6 then add 9.9 then add 8.3 then add 2.5 the answer is 12.78'What number was Owen first thinking of? Working: Landon says:'I am thinking of my favourite number. If I add 19 to my number then divide the result by 2, the answer is 6.25'What number was Landon first thinking of? Working: Diego says:'My plan was to add 1.4 to my number then multiply the result by 6.8. Then I multiplied the result by 3.1 and finally subtracted 4.7. The answer is -135.396'What number was Diego first thinking of? Working: Brian says:'I decided to add 7.7 to my number then multiply the result by 3, After that I subtracted 2.6 then doubled the result. The answer is 5'What number was Brian first thinking of? Working: Adam says:'If I multiply my number by 9.3 and subtract 6.2 then multiply the result by 7.8, the answer is 5.3 less than -623.38'What number was Adam first thinking of? Working:
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This is Think of a Number level 3. You can also try:
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## Instructions
Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.
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## Description of Levels
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Level 1 - Whole number THOAN problems
Level 2 - Decimal THOAN problems
Level 3 - Negative decimal THOAN problems
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.
## Example
Daniel says 'If I double my number and add 6 the answer is 30'
### Method 1 - Work backwards
Start at the end of the sentence with the answer 30 and undo all of the operations Daniel performed on his number in reverse order.
30 minus 6 then divide by two
The number Daniel was thinking about was 12
### Method 2 - Form and solve an equation
Let the number that Daniel is first thinking of be n.
2n + 6 = 30
Subtract 6 from both sides
2n = 24
Divide both sides by 2
n = 12
The number Daniel was thinking about was 12
Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.
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Martin Hamel from the University of Nebraska Lincoln received a text from his airboat mechanic on a Monday morning—there had been a large fish kill on the Platte River, a waterway that winds its way through Nebraska and past towns such as Louisville and Ashland. The next day, Hamel’s fears were confirmed. He and a team of others found the bodies of sturgeon and other fish spread across the muddy bottom of what had once been a flowing river. Partially eaten by scavengers and left to rot in the blazing sun, most of the fish were difficult to identify. Yet the team did find one fish that had been tagged five years before—it was an endangered pallid sturgeon, a fish that had originally been stocked into the Missouri River.
“The bigger concern is that we found one, which ones haven’t we found?” says Mike George, the project leader in the Nebraska Ecological Services field office for the Fish and Wildlife Services.
The drought that has spread across most of the United States has not only affected crops; it’s also affected wildlife. Fish and other river-dwelling species, some of them listed as endangered, have suffered the most with higher temperatures and lower flow rates.
Georgia Parham of the FWS has seen these issues firsthand. The water levels in the Tippecanoe River in Indiana have dropped lower and lower, threatening endangered mussels such as the fanshell. The river is a hugely important mussel bed, providing vital habitat for the endangered shellfish.
“I was out there a couple of weeks ago down below the dam and it’s pretty much a continuous mussel bed at that part of the river,” says Parham. “It’s pretty amazing.”
As the drought continues, though, the FWS is struggling to keep river levels high enough to support these mussels. They’re currently working with the Northern Indiana Public Service Company to maintain a minimum flow of water from the Oakdale Dam into the river.
Other places, however, are just scraping by. The Upper Colorado River Endangered Fish Recovery Program has had to deal with low flow rates on the rivers that they monitor, but the fish are still surviving. Unlike the summer of 2002, which came on the heels of several years of drought, this summer has come after an extremely wet year which saw record breaking flow rates. A study published last week in Nature-Geoscience even linked the drought that ran from 2000 to 2004 as contributing to global warming conditions.
Even so, the program has had to shut down vital fish passages this year due to the lack of water. The fish passages allow endangered species such as the Colorado pikeminnow and the humpback chub to migrate along the river and past dams.
If the drought continues into next year, the program might be seeing a few more problems. The warmer temperatures and lower flow rates allow invasive species, such as the smallmouth bass, to thrive and encroach on the habitat of native species.
Perhaps more worrying, though, are the wildfires that have ravaged the state. “We heard of a fish kill over on the Price River in Utah,” says Tom Chart, the program director of the fish recovery program. Much needed rain caused ash, left over from wildfires, to run into the river, creating conditions that were stressful to fish. “You kind of get right at the limit there of what these native fish can handle.”
Yet it isn’t only endangered mussels and fish that are of concern. Land-dwelling creatures have also seen their share of challenges.
“The other thing we’ve had is blue tongue; it affects white tailed deer,” says George, commenting on the deer deaths that have been seen along the Platte River. “The deer bleed to death internally, but they get incredibly thirsty.”
The deer end up concentrating in one location due to fewer water sources, allowing an easier transfer of the disease.
“One lady called and told me a deer charged into the river and drowned itself,” says George. “That’s what blue tongue will do to these poor animals.”
The spread of disease is a huge concern in times of drought. In the Horicon Marsh in Wisconsin, thousands of fish have died due to botulism. Left stranded on the shores of the marsh, these fish are then eaten by waterfowl and shorebirds, which congregate in a smaller area due to drought conditions. By eating the fish, the birds are infected by the disease and often die. In an attempt to curtail a larger outbreak of the disease, the Department of Natural Resources is collecting any dead fish that they find and removing them from the system.
As drought conditions continue, though, water levels keep dropping. Farmers clamor for irrigation for drooping crops while the FWS tries to keep enough water for endangered species.
Although drought conditions are bad now, George remembers that tougher times might still be coming. “We still have the hottest time of the year ahead of us,” he says.
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The technology known as polymerase chain reaction, or PCR, has been described as one of most pivotal discoveries in the life sciences in human history. The tool – the ability to generate unlimited copies of a fragment of DNA – has led to the creation of successful drugs and vaccines, accurate genetic tests for diseases, the tools forensic scientists use for maternity tests and crime scene analysis, and it’s made sequencing of the human genome possible.
A biochemist named Kary Mullis won the Nobel Prize in 1983 for creating the technology, but Mullis’ work would not have been possible if not for an Indiana University biology professor who in the summer of 1966 visited Yellowstone National Park along with an IU undergraduate biology student in an attempt to understand how organisms survived in the extreme conditions of the park’s geysers and hot springs.
A year later – Nov. 24, 1967, to be exact – IU professor Thomas D. Brock published “Life at High Temperatures: Evolutionary, ecological, and biochemical significance of organisms living in hot springs is discussed,” in the leading scientific journal Science. Little did Brock know that the bacterial sample he and then-undergraduate student Hudson Freeze gathered a year earlier at Mushroom Spring in Yellowstone – the sample that would yield the microbe they would call YT-1 – would allow Mullis to develop PCR. YT-1 would turn out to be a goose that laid a golden egg for mankind.
To study DNA, scientists needed a lot of it, yet cells contain very small amounts. Attempts to copy DNA by unraveling it in the lab involved very high temperatures and the enzymes that occurred in DNA would either quit functioning or disassemble. Then came Thermus aquaticus, the Mushroom Springs microbe brought to IU by Brock and Freeze, that would soon be shown to stay alive in boiling water and to be widely distributed around the world, even in the water pipes of buildings at the IU Bloomington campus.
Since the microbe, which they nicknamed Taq, contained DNA, it also had the right enzymes to conduct replication. And eventually, the best enzyme for use in PCR was taken from that YT-1 strain that Brock and Freeze brought to IU for study.
Next week Brock, now retired and living in Wisconsin, and Freeze, a professor and director of the genetic disease program at Sanford Children’s Health Research Center in La Jolla, Calif., will receive the Golden Goose Award in a ceremony on Capitol Hill in Washington, D.C. The award, created in 2012 by a coalition of organizations after being proposed by U.S. Congressman Jim Cooper of Tennessee, is designed to recognize the scientists and engineers whose federally funded research has had significant human and economic benefits.
Cooper is the kind of congressman scientists should love: “We’ve all read stories about the study with the wacky title, the research project from left field,” he has said. “But off-the-wall science yields medical miracles. We can’t abandon research funding only because we can’t predict how the next miracle will happen.”
Nothing could be more true, and Brock and Freeze remain living proof that those quests, like theirs 47 years ago, toward unveiling and solving new mysteries do indeed yield miracles … like PCR.
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# An Exercise in Intuition with Genetic Algorithms
In this blog post I will be further exploring GA’s, evaluating the evolutionary approach to computational problem solving; in particular the crossover and mutation operators and their resultant effects on succeeding generations (no pun intended)!
#### The Scenario:
Assume that a GA has chromosomes in the following structure:
ch = (x0, x1, x2, x3, x4, x5, x6, x7)
x0-7 can be any digit between zero to nine.
The fitness of each chromosome is calculated using the following formula:
f(ch) = (x0 + x1) − (x2 + x3) + (x4 + x5 ) − (x6 + x7)
This problem is a maximisation problem.
In this scenario we initially have a population of four chromosomes as shown below:
• ch1 = (6, 5, 4, 1, 3, 5, 3, 2)
• ch2 = (8, 7, 1, 2, 6, 6, 0, 1)
• ch3 = (2, 3, 9, 2, 1, 2, 8, 5)
• ch4 = (4, 1, 8, 5, 2, 0, 9, 4)
Individual fitnesses, in order of best to worst:
1. ch2 = (8, 7, 1, 2, 6, 6, 0, 1) | fitness = 29
2. ch1 = (6, 5, 4, 1, 3, 5, 3, 2) | fitness = 15
3. ch4 = (4, 1, 8, 5, 2, 0, 9, 4) | fitness = -1
4. ch3 = (2, 3, 9, 2, 1, 2, 8, 5) | fitness = -2
Crossover
Chromosomal crossover is when two DNA helices break, swap a section and then rejoin.[1]
For this exercise our crossovers are as follows for subsequent generations
1st gen. Use one-point crossover (at the middle) to cross the best two chromosomes. [This would lead in creating the first two children]
2nd gen. Use two-point crossover (after x1 and after x5) to cross the 2nd and 3rd best individual. [This would results in creating two extra children]
3rd gen. Use uniform crossover (for a single gene) to cross the 1st and 3rd best chromosomes. [Now you should have two more children; six in total]
New child chromosomes over all generations:
c1 = [6 5 4 1 6 6 0 1] | fitness = 20
c2 = [8 7 1 2 3 5 3 2] | fitness = 24
c3 = [8 5 4 1 6 5 3 2] | fitness = 29
c4 = [6 7 1 2 3 6 0 1] | fitness = 23
Our optimal chromosome in this case is: (9, 9, 0, 0, 9, 9, 0, 0) | optimal fitness = 32
Is it possible to reach the optimal state without the mutation operator?
Yes, it is possible with a particular uniform crossover implementation which takes genes and places them in a random location for each splice. Albeit, this method has a small probability to succeed, depending on the amount of single gene uniform crossovers that are used for each generation. Alas, it may also take a much longer time to converge to the global optima than if we were to use a mutation operator, as the genetic diversity is somewhat limited. It would be easier if the initial chromosome population had some optimal segments, so that we converge to the optima in less time.
See the experiments: github.com/JasperKirton/GA_exercise
Thanks for reading! This is the eighth part in a series of blogs for the Natural Computing module at Goldsmiths, University of London.
# Using Dispersive Flies Optimisation to solve the Knapsack problem
The Knapsack problem is a classical problem in combinatorial optimisation, it has many applications including finding the least wasteful way to cut raw materials, and other economic problems.
An example of the knapsack problem could be the following (substituting a knapsack for a car boot). Suppose we are planning a trip to a car boot sale; and we are, therefore, interested in filling the car with items that are considered valuable to sell. There are N different items; these could include a bunch of old toys, a games console, an antique chair, a CD player, and so forth. Each item type has a given set of two attributes, namely a weight (or volume) and an estimated value associated with each item. Since the boot has a limited weight (or volume) capacity, the problem of interest is to figure out how to load the boot with a combination of units of the specified types of items that yields the greatest total value.
Here I will be reporting my progress on using DFO to solve the n-knapsack problem, that is, where there are multiple bags to fill and each bag has unique constraints, though the items have constant values.
So far, i have solved the problem for 2 knapsacks and 28 items with the same capacity, but different weight constraints. See the ‘example’ problem, within the problem dataset i will be using here. This example problem hence has 28 dimensions, though a more complex solution surface than just 1 knapsack. It was solved in 500 iterations, using 100 flies and a disturbance threshold of 0.165.
I am now trying to solve the problem for 30 knapsacks and having some trouble getting close to the global optima (always getting stuck in the local optima), even when using 8000 iterations! This now seems like a good opportunity to optimise my approach. I will now try and improve the efficiency of the algorithm by reducing the dimensionality of the update phase in DFO.
Dimensionality Reduction
As the solution vector is a binary string, we could convert this to a decimal value and update the flies in one dimension rather than nItems dimensions, and then convert it back to a binary within the fitness function in order to evaluate whether it meets the constraints, and how much the solution is worth. Upon implementing this I discovered that although it speeds up convergence initially, it gets stuck in a local optima.
Fitness Penalisation Oscillation
In my most recent experiments, I have found that penalising the fitness by multiplying it by a value between 0 and 1 if it goes over the constraint threshold is much more successful. If we then begin to slowly decrease/increase this penalisation value over time, we see greatly improved results, in both function evals and best combination/value obtained. My next step is to see if I can model this in terms of an oscillator, adapting the frequency of this oscillator dynamically according to the dimensionality/variance in constraints of the particular problem set.
Update:
I’m now working on this problem in a group with a couple of my super smart peers; Francesco Perticarari and Bashar Saade. See our code on github here. We’ll hopefully be presenting a research paper with our work in the new year at ANTS and/or AISB conferences!
Thanks for reading! This is the seventh part in a series of blogs for the Natural Computing module at Goldsmiths, University of London.
# Using Dispersive Flies Optimisation for Personalised Music Recommendation and A Combinatorial Card Game
The Feature Weighting Problem
Generating recommendations that a user is most likely to appreciate is a well researched problem. In Music Recommendation, a big part of the problem is adjusting the weights of a number of features which will attempt to define a listener’s profile. Recently, Music Information Retrieval techniques (acoustic features) have further increased the dimensionality of the solution space, offering the opportunity for more personalised recommendations, whilst simultaneously increasing the need for better algorithms to ‘tune’ the feature weights to model a user’s musical personality.
Applying DFO to the optimisation problem
The Fitness Function for this task could be the how well the acoustic features map to the user access data. I.e if the generated recommendation has a high average rating by other members of the community, or if it exists in another playlist by similar users. The dataset I will be implementing the algorithm for, and evaluating the results on, will be subsets of the well researched Million Song Dataset; for example, The Echo Nest Taste Profile Subset and have a neighbourhood set of item-based similarity to draw the fitness from. Partly inspired by Inspired by the Particle swarm optimization recommender system (for movies and user profiles).
Evaluation: I will do my evaluation based on results of other item-based similarity methods for the Million Song Dataset from the Million Song Dataset Challenge.
The Goal: To utilise the collective intelligence of virtual flies, and thus allow users to effortlessly discover new music that they are very likely to enjoy!
The Combinatorial Card Game
The premise of this combinatorial problem is as follows; there are 10 cards, each labelled in ascending and unique order 1-10. The aim of the game is to get 1 unique subset of the cards to sum to 36 and the remaining subset’s product to = 360. We can treat this as a 10 dimensional problem.
The generated solution:
Params: 20 flies, 100 iterations, dt = 0.01
Subset 1: [ 9 10 2 7 8 ]
Subset 2: [ 3 6 5 4 1 ]
Function evaluations (calls to get fitness): 50
The solution can of course be any permutation of these 2 subsets.
My curiosity in permutations and concern for how efficient the algorithm is (measured in function evaluations) compared to an approach of simply iterating through the solution vector (a brute force approach) led me to evaluating solving the problem with DFO; the worst case (simply iterating through the vector and finding that answer at the very end) is approx. 10! (3628800 FE’s) and I solved it in 50 FEs which is 72576 x better than the brute force approach!
See my code on Github.
Thanks for reading! This is the sixth part in a series of blogs for the Natural Computing module at Goldsmiths, University of London.
# Applying a Genetic Algorithm to The Travelling Salesman Problem
The Travelling Salesman Problem is a classic NP-hard problem in Computer Science, for which there are many viable solutions. The task is; given a number of cities, we must find the shortest route (in time and/or distance) that visits each city exactly once. Traditionally, the best problems have the quickest worst-case running time; in TSP, any algorithm to solve the problem increases superpolynomially (but no more than exponentially), as the number of cities increases. [1]
Let’s start by describing our evolution inspired heuristic for this particular TSP, where we will be finding the shortest route between a number of cities in the UK — in alphabetical order; Brighton, Bristol, Cambridge, Glasgow, Liverpool, London, Manchester and Oxford. We will be attempting to find the shortest distance with these approximate distances (I presume in miles);
Brighton Bristol Cambridge Glasgow Liverpool London Manchester Oxford Brighton 0 172 145 607 329 72 312 120 Bristol 0 192 494 209 158 216 92 Cambridge 0 490 237 75 205 100 Glasgow 0 286 545 296 489 Liverpool 0 421 49 208 London 0 249 75 Manchester 0 194 Oxford 0
Now on to the evolution inspired Genetic Algorithm (or GA), for which we will be using to solve the problem. The heuristic goes like this;
1. Initialisation – Create an initial population of solutions, usually created randomly and with an arbitrary size, from just a few to thousands. In our case this would be a population of tours; routes to each and every city.
2. Evaluation – Each member of the population is evaluated; the fitness of the individual is calculated by how well it fits with the desired requirements. In our case this would be the total distance, which we are trying to minimise.
3. Selection – Our goal is to constantly improve our populations fitness. Selection helps us to do this by discarding the bad solutions and keeping the best ones (similar to natural selection in Darwinian theory). There are a few different methods, but we aim to make it more likely that fitter individuals will be selected for the next generation. Here we will use a roulette approach. (Explained below)
4. Crossover – Make new individuals by combining genes of our selected individuals, (Similar to how sex works in nature). The hope is that by combining certain genomes from two or more individuals we will create an offspring with better fitness for our problem, which will inherit the best traits from each of it’s parents.
5. Mutation – Make new individuals by slightly altering, typically by random, the existing genome of an individual.
6. Repeat! Now we have the next generation, we can start again from step two until we find the fittest individual. [2]
In this particular problem we need to employ a roulette wheel to bias the algorithm to generate fitter solutions for each generation. (The selection process):
This is essentially a normalisation of the fitnesses of the current generation, so that we can select the best solutions for our next generation.
In pseudo-code [3]:
for all members of population
overallF += fitness of this individual
end for
for all members of population
probability = sum of probabilities + (fitness / overallF)
sum of probabilities += probability
end for
loop until new population is full
do this twice
number = Random between 0 and 1
for all members of population
if number > probability but less than next probability
then you have been selected
end for
end
create offspring
end loop
Solution and results from the GA (see the code on github)
Thus, the best is distance is 890 miles, and one of the best genotypes is (Glasgow, Liverpool, Manchester, Bristol, Oxford, Cambridge, London, Brighton), which makes sense!
On the topic of Natural Computing, Ant Colony Optimisation has also been used to solve this problem. Whereby their behaviour in finding short paths between food sources and their nest (an emergent behaviour, resulting from each ant’s preference to follow trail pheromones deposited by other ants) is modelled algorithmically. Simulated annealing has also been used and compared to GA’s here.
Lots of interesting applications for all this. In particular, I was envisioning a system for music exploration; whereby a collection of music releases could represent nodes in a graph, and the edges represent the most similar features between the items, solved by the Travelling Salesman Problem. Moreover, the Genetic Algorithm approach may be very useful. There is very likely to be another post including this topic due to it’s potential relevance to my final project on Music Exploration and Recommendation.
Thanks for reading! This is the fifth part in a series of blogs for the Natural Computing module at Goldsmiths, University of London.
# Dispersive Flies Optimisation in Symmetry Analysis
In this post i’ll be exploring and evaluating DFO in finding points and lines of n-fold symmetry to aid Computer Vision. See my previous blog post explaining DFO here.
To define our problem to find points of n-fold symmetry, we are looking for specific areas (in this case a single pixel) where the fitness function of symmetry is at its optimum (in this case a minima). Let’s define our fitness function; we need 2 blocks of comparison, each with the same radius (the distance from the centre to any four of the blocks vertices), the sums are then evaluated around our proposed axis of symmetry, such that our symmetry metric = absolute(sum1 – sum2). In the case of a vertical axis for example:
We can see this in action here on b/w images:
Then, to improve, we change our communication so that the flies navigate around the best fly in the swarm (as opposed to our previously democratic approach),
for (int d = 0; d < Global.dim; d++) {
temp[d] = Global.fly[chosen].getPos(d) +
//random(1) * (Global.fly[chosen].getPos(d) - Global.fly[i].getPos(d));// local nei
random(1) * (Global.fly[Global.bestIndex].getPos(d) - Global.fly[i].getPos(d));
Here we see a more stable point of symmetry, and most importantly, a solid observation; the more symmetrical the image by human perception, the more stable the position of the best fly in the swarm.
The next problem which would lead to evolving our algorithm and making it more useful is to find symmetry in nature i.e for detecting animals. Let’s see how our most recent algorithm behaves for an image of a zebra (with an updated radius that is proportional to our new image size):
Here we see some interesting behaviour straight away, around 0:13 and 0:52 there is a clear indication of the central, most obvious line of symmetry, albeit a little left of centre! This may be because the right side of our image is a little brighter than the left side, causing some confusion with our fitness function evaluation. Here might be a good place to start evolving our algorithm, then. It is wort noting that because our image is black and white, we don’t have to worry too much about our colour space, and utilising brightness values is sufficient for the next step in developing our algorithm, however, when we start to analyse more colourful animals, we’ll need to consider the best colour space to use, i.e L*a*b* (where euclidean distance is relative to perceptive difference in colour), but more on this later.
For now let’s investigate how to deal with the varying brightness issue. Whereas before we simply had all black pixels and all white pixels, we now need to account for pixels that lie on the grayscale. One way to solve this could do be to use a more balanced approach, i.e to use 4 summation boxes to compare instead of two, and use a reversed comparison for our second set. More to come!
See the code on Github.
Thanks for reading! This is the fourth part in a series of blogs for the Natural Computing module at Goldsmiths, University of London.
P.S. A fitting soundtrack for the task:
# Stochastic Diffusion Search (and where we can use it)
Stochastic Diffusion Search is one of the first Swarm Intelligence algorithms, characterised as a ‘population based best-fit pattern matching algorithm’[1], first proposed by John Bishop in 1989 [2]. It reminds me most of ants foraging for food, and how they communicate distantly via pheromones to reveal promising locations, this results in a truly incredible ‘collective intelligence’ that allows the ants to efficiently locate and utilise food sources.
As a heuristic, the basic idea is that a swarm of agents are randomly initialised in a search space. They then have some kind of fitness function that determines whether they are satisfied or not satisfied, determined by their position in the search space — this is the Test phase. Then follows the Diffusion phase, which is essentially a phase of communication between the agents; in active diffusion, the successful, active agents will let an unsuccessful, inactive agent (chosen at random) know where in the search space they were successful, and then the notified agent will go there to look for neighbouring success locations. The Test and Diffusion phases are then repeated until the system has converged to an optimum location or set of locations (dependent on the problem).
When we start to solve problems based on this heuristic, an important characteristic of the algorithm arises, this is the ‘Recruitment Mode’ contained within the Diffusion, or communication phase, and it has to do with the direction and condition of the communication. Namely, there is active recruitment (explained above), passive recruitment and dual recruitment. There are also context sensitive/ context free mechanisms that can be used in the recruitment modes.
In the dual recruitment strategy, both successful, active and non-successful, inactive agents randomly select other agents. When an agent is successful, and the randomly selected agent is not, then the hypothesis of the successful agent is shared with the unsuccessful one and the agent is flagged as active. Also, if there is an agent which is inactive and disengaged in communication; and the newly selected agent is active, there will be a flow of information from the successful agent to the unsuccessful one and the unsuccessful agent is then flagged as active. Nevertheless, if there remains an agent that is neither active or engaged, a new random hypothesis is chosen for it. [3]
I have explored two separate implementations of SDS, one is a mining simulation (image in header) that is very similar to the heuristic i have described above (just replace agents with miners and success with gold!). The other is to find a word in a text document, here is an example of finding all the ‘Denmarks’ in a short passage, making use of the Context Sensitive recruitment mechanism, whereby if an active agent randomly chooses an- other active agent that maintains the same hypothesis, the selecting agent is set inactive and adopts a new random hypothesis (therefore achieving a wider exploration):
In the Context Free Mechanism, the performance is similar to context sensitive mechanism, where each active agent randomly chooses another agent. However, if the selected agent is active (irrespective of having the same hypothesis or not), the selecting agent becomes inactive and picks a new random hypothesis. ‘By the same token, this mechanism ensures that even if one or more good solutions exist, about half of the agents explore the problem space and investigate other possible solutions.’ [3]
On the subject of improving education, SDS has been utilised to predict whether students enrolled on MOOC courses will pass or fail [4]. One problem which I thought SDS would be useful is the Cold Start Problem in Recommender Systems/Collaborative Filtering; when we want to find suitable recommendations for user with little user information, we could use populate a feature space with popular items and then based on the users feature characteristics, we could engineer the swarm so that they choose the best possible items (using the fitness function e.g. rating). Evidence shows that Swarm Intelligence is ‘likely to be particularly useful when applied to problems with huge dimensionality’ [4] which is usually the case for recommendation systems!
It could also be useful to achieve the main goal of CF; to use the mining analogy, we represent each user as a ‘hill’, they may have ‘gold’ (i.e good items based on rating, or ‘goodness’ rating based on users feature weights) within their collections/purchase history. Employing SDS could be a good way of finding similar users as an alternative to using metrics like euclidean distance.
There is much more to say on SDS and it’s applications. One of the things that fascinates me is the algorithms conception, proposed as an alternative to more traditional Artificial Neural Networks to recognise patterns [1]; Neural Networks are by design, non stochastic, deterministic models, for which an alternative (i.e swarm intelligence) is often more accurate and efficient. It is worth noting that ANN’s are also nature inspired models; emulations of a simplified model, or heuristic, of the brain. Let’s keep looking to nature, then, for more amazing problem solving heuristics!
Thanks for reading! This is the third part in a series of blogs for the Natural Computing module at Goldsmiths, University of London.
# What is Dispersive Flies Optimisation, and where can we use it?
DFO is a swarm intelligence based ‘meta-heuristic’ (problem independent computational method), similar to particle swarm optimisation, but particularly inspired by and based upon flies behaviour, and their tendency to swarm around a marker or point of interest. What differentiates this particular heuristic from it’s swarm intelligence parents is it’s characteristic to be disturbed; and consequently disperse at certain points in time, before re-swarming either to the same marker, or if a better one is found upon their displacement, a new point of interest (or optimum value).
As a result of this, it’s a more specific heuristic, I would estimate that the set of search/optimisation problems this could be useful for is fewer in comparison to a more general swarm optimisation (see wiki), but on this set of problems the performance will noticeably increase most of the time, a la No Free Lunch theorem (see my previous blog post here). A particular example where this may be useful is in Neural Network weight optimisation, (see my peer/friend Leon’s excellent post here for a practical example), when we use multi-layer perceptrons the search space becomes complex (resulting in local and global optima).
In computational ‘search’ problems, this is particularly useful as it allows complex search spaces with many ‘optimal’ solutions to be fully explored for both local optima and global optima. To clarify, let’s investigate the term search space.
The most simple in 3-dimensions of search (or feature/data) space would be our parabola, or sphere function (search spaces can exist in any number of dimensions). Suppose in this instance x1 and x2 are our inputs, and the surface of the shape represents our output (or the set of all possible solutions), here the local minimum (optima) is the same as our global minimum. It can also be represented quite simply mathematically:
A nice example of a more complex search space is the ‘Hölder Table’ (which looks like waffle shaped table):
For more great looking (and also very practical ‘benchmarks’) test functions see the wiki here and more info with Matlab implementations here. (It is worth noting that it is very difficult to visualise anything above 3 dimensions!)
In DFO, first the population number (NP) of flies are initialised in random points in the feature space, according to a normal distribution. Then, the positions are updated taking three things into account, the current position, the position of the best neighbouring fly (which has the best fitness, to the left or right), and the position of the best fly (overall best fitness in the swarm). To determine the fitness, we have a ‘fitness function’ which is run once every iteration of the algorithm; the position of each fly in the feature space is evaluated against the other positions and then the ‘best’ fly is found according to its proximity to the optima (whether it is the minimum or maximum value, as defined by the problem space). The fitness function could be any mathematical function. For 2 dimensions we could simply say f(x1,x2) = (0,0) if we wanted to create a simple visualisation around a central point.
Now, to what makes this algorithm special, the dispersion threshold. To begin with we specify a value for the threshold, usually a small one! A random number (again using normal real distribution) between 0 and 1 is then generated and if the number falls below the dispersion threshold, the selected fly is assigned another random position, thus influencing it’s neighbours. This is a stochastic approach to making sure the swarm doesn’t focus on a local optima, and is forced to keep constantly searching around the space until the algorithm terminates (when the total number of evaluations is met), hopefully by then the global optima will have been found!* This heuristic hopefully provides a balance between the key exploration and exploitation characteristics of swarm intelligence algorithms; our search is more diverse due to the dispersion and therefore we have more exploration, and we are stopping the swarm from exploiting specific areas too much.
I was wondering whether it would be an interesting approach for a recommendation system, due to the larger diversity of ‘best’ solutions (FF being highest rating) which would be found within a feature space. Particle Swarm Optimisation which is similar to DFO (minus the dispersion factor) has certainly been applied in this area successfully; notably here. Another interesting application is it’s use in medical imaging – utilising the intelligent agents (flies) to extend our cognitive abilities, in particular enhancing the visual perception of radiologists when detecting possible precancerous tissue.
To conclude my initial investigation, it seems there are many things to learn from nature and bringing an element of stochasticity can help us solve complex problems effectively. Swarm intelligence is a fascinating subset of AI; many minds are better than one when we are exploring a new field, and it’s always useful to visit new areas to avoid over exploiting just one solution to a complex problem.
Find the original DFO paper here.
Applying It
I will now attempt to apply DFO to a well defined problem and something which has been of great interest to me recently; Recommendation Systems and in particular, Collaborative Filtering for music. Collaborative filtering is a very popular technique used within RS, the task is to find similar users to the current user based on their profiles, and make predictions which are appealing to the current user; ‘In the more general sense, collaborative filtering is the process of filtering for information or patterns using techniques involving collaboration among multiple agents, viewpoints, data sources, etc..’ [1]
I have been greatly inspired by work which applies Particle Swarm Optimisation to the task, most notably these papers on using PSO within CF; PSO RS, PSO for CF based on Fuzzy Features (for which my algorithm is mainly derived from) and for Swarm Intelligence in general applied to RS in Web Usage Mining.
Let’s define our model:
There are three phases needed to accomplish the recommendation task by using the DFO.
• User profile formation
• Neighbourhood set generation
• Predictions and recommendations
The user profile could be formed by a number of features: for this music application, I shall use the example of Discogs to inform the best features. For each release, we have the following; how many people have the item, how many people want the item, the average rating for the album (and each rating per user!) and Genres/Styles. There are numerous other features but I would argue that these are the most useful. We can develop new useful features like a Genre and Style ‘Interestingness Measure’ which can be represented more naturally by fuzzy sets (to fit better with average human perception), e.g very bad (VB), bad (B), average (AV), good (G), very good (VG), and excellent (E). When we then compute the similarities between users, we supposedly have more accurate results.
For the neighbourhood set generation; we need to find the optimum weights for every feature of our user profile — this takes into account the fact that different users place different priorities on features, for example user 1 might be mostly concerned with a genre while user 2 is mostly concerned with how highly an album is rated. for example which leads to the optimum feature space, where we have the most similar users. We could determine this by the RMSE of all the users euclidean distance (a quantitive measure of feature differences) to our current user, and perform DFO until we have the optimum solution. This would then lead to the most accurate recommendations (depending also on our process for selecting items from the users, e.g most commonly owned, highest rated etc.)
Now, for the Dispersive Flies optimisation! We have 4 main areas of concern; fly representation and initial population, the fitness function, fly behaviour and the termination condition.
Fly representation and initial population: we are mostly concerned with creating a vector (size dependent on population) of flies with parameters that represent their position, and thus the feature weights, in our n dimensional feature space.
The fitness function: This is very important. We need to evaluate the fitness score of each fly according to how well the weights map to our ideal ‘user space’ (the least RMSE distance of all users to our current user).
Fly behaviour: Each fly updates its position based on the best fly in the swarm, and it’s best neighbour. The dispersion threshold will be set to an initially small value, but this could be crucial to finding the best possible outcome in a finite amount of time when our data space is big (we could have thousands of users and millions of releases in those collections, and a high number of dimensions we will need to explore!)
The success of this approach will be down to a number of performance measures, that compares the accuracy of the algorithm and the time it takes to reach the best user space. This is likely something that will become clearer to evaluate once we have run the algorithm a number of times, allowing adaptation of the parameters (dispersion threshold, number of flies). In our particular application, a little variance between different user spaces after each run of the algorithm could be useful in finding different recommendations.
This is not a completely detailed application and I will return to this after I have explored the problem further. There are a few other approaches within the field of recommendation that swarm intelligence could be particularly useful, particularly to solve the cold start problem; when we have a new user without many releases in their collection. Can we take more advantage of swarm optimisation for feature based recommendation?
Exploring new music can be a lot like finding new, unfamiliar territory, where the reward of finding something you like in a cluster of songs you don’t know can be extremely gratifying! Perhaps we can employ more inspiration from nature; ants are the some of the best explorers on our planet, could their meta-heuristics aid our search for new music? Tune in next time to find out.
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Renyi Zhang, professor of atmospheric sciences who has studied air chemistry for more than 20 years, says blue haze (tiny particles or aerosols suspended in the air) can be negatively affected by human activities such as power plants or fossil-fuel burning.
Team members included researchers from Brookhaven National Laboratory in New York, the Molina Center for Energy and Environment in La Jolla, Calif., and the Massachusetts Institute of Technology. Their work is published in the current "Proceedings of the National Academy of Science" and the project was funded by the Welch Foundation and the U.S. Department of Energy.
Zhang says man-made activities, mainly large power plants that emit huge amounts of particles into the air, can worsen blue haze and cause previously unforeseen problems.
“The study shows that the natural way of blue haze formation is rather inefficient and that human activities make blue haze conditions worse,” he confirms.
“What happens is that a mix of natural and man-made chemicals speeds up the formation of these particles in the Earth’s atmosphere, and there, they reflect sunlight back into space. The results can affect cloud formations and ultimately, much of the world’s climate.”
When you walk through a forest or even a large grassy area, it’s not uncommon to be able to smell the plants around you, such as pine trees or other vegetation. That smell is nature’s way of naturally making organic gases produced by the plants themselves, often millions of tons per day.
Plants, especially trees, emit such gases through their leaves and when an overabundance of such gases is produced, it creates a blue aura, commonly called a “blue haze.” Perhaps the best example occurs in the Great Smokey Mountains National Park area of the Southeast United States, where blue haze exists almost on a daily basis, but the condition also occurs all over the world.
When man-made activities emit sulfur dioxide into the air, they contribute to blue haze, usually in a negative way, Zhang explains. Aerosols can be produced by many different processes that occur on land and water or in the atmosphere itself, he notes.
“Weather patterns can be affected worldwide and the blue haze can worsen the breathing problems of many people, such as those who suffer from asthma or emphysema,” he adds.
“The chemistry of Earth’s atmosphere can be directly affected by these aerosols. From cloud formations to health problems and air pollution, much of it can be traced back to these aerosol particles,” he adds, noting that aerosol particles can influence the size and rate of cloud droplets, directly affecting cloud cover and precipitation.
Coal plants, Zhang says, often produced sulfur dioxide, a highly toxic substance that reach the Earth’s atmosphere and helps the formation of aerosol particles.
The problem is not new. Zhang says former President Ronald Reagan mentioned it during a speech almost 30 years ago.
“About 80 percent of our air pollution stems from hydrocarbons released by vegetation,” Reagan noted during a 1980 speech to an environmental group.
Zhang says more research is needed to “study the full extent of how blue haze is affected by human activities, and perhaps to look at ways to control the situation. It’s a problem that can have global consequences.”
Contact: Renyi Zhang at (979) 845-7656 or [email protected] or Keith Randall at (979) 845-4644 or [email protected]
About research at Texas A&M University: As one of the world’s leading research institutions, Texas A&M is in the vanguard in making significant contributions to the storehouse of knowledge, including that of science and technology. Research conducted at Texas A&M represents an annual investment of more than $582 million, which ranks third nationally for universities without a medical school, and underwrites approximately 3,500 sponsored projects. That research creates new knowledge that provides basic, fundamental and applied contributions resulting in many cases in economic benefits to the state, nation and world.
For more news about Texas A&M University, go to http://tamunews.tamu.edu
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Amputees can learn to control a robotic arm with their minds
28.11.2017 | University of Chicago Medical Center
The importance of biodiversity in forests could increase due to climate change
17.11.2017 | Deutsches Zentrum für integrative Biodiversitätsforschung (iDiv) Halle-Jena-Leipzig
MPQ scientists achieve long storage times for photonic quantum bits which break the lower bound for direct teleportation in a global quantum network.
Concerning the development of quantum memories for the realization of global quantum networks, scientists of the Quantum Dynamics Division led by Professor...
Researchers have developed a water cloaking concept based on electromagnetic forces that could eliminate an object's wake, greatly reducing its drag while...
Tiny pores at a cell's entryway act as miniature bouncers, letting in some electrically charged atoms--ions--but blocking others. Operating as exquisitely sensitive filters, these "ion channels" play a critical role in biological functions such as muscle contraction and the firing of brain cells.
To rapidly transport the right ions through the cell membrane, the tiny channels rely on a complex interplay between the ions and surrounding molecules,...
The miniaturization of the current technology of storage media is hindered by fundamental limits of quantum mechanics. A new approach consists in using so-called spin-crossover molecules as the smallest possible storage unit. Similar to normal hard drives, these special molecules can save information via their magnetic state. A research team from Kiel University has now managed to successfully place a new class of spin-crossover molecules onto a surface and to improve the molecule’s storage capacity. The storage density of conventional hard drives could therefore theoretically be increased by more than one hundred fold. The study has been published in the scientific journal Nano Letters.
Over the past few years, the building blocks of storage media have gotten ever smaller. But further miniaturization of the current technology is hindered by...
With innovative experiments, researchers at the Helmholtz-Zentrums Geesthacht and the Technical University Hamburg unravel why tiny metallic structures are extremely strong
Light-weight and simultaneously strong – porous metallic nanomaterials promise interesting applications as, for instance, for future aeroplanes with enhanced...
11.12.2017 | Event News
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12.12.2017 | Physics and Astronomy
12.12.2017 | Earth Sciences
12.12.2017 | Power and Electrical Engineering
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## College Algebra (11th Edition)
Published by Pearson
# Chapter 7 - Section 7.2 - Arithmetic Sequences and Series - 7.2 Exercises - Page 646: 47
#### Answer
$S_{60}=3,660$
#### Work Step by Step
$\bf{\text{Solution Outline:}}$ To find the sum of the first $60$ positive even integers, use the formula for finding the sum of $n$ terms that form an arithmetic sequence. $\bf{\text{Solution Details:}}$ The sequence described by "the first $60$ positive even integers," is an arithmetic sequence with $a_1=2, d=2,$ and $n=60.$ Using the formula for the sum of the first $n$ terms of an airthmetic sequence, which is given by $S_n=\dfrac{n}{2}[2a_1+(n-1)d] ,$ then \begin{array}{l}\require{cancel} S_{60}=\dfrac{60}{2}[2(2)+(60-1)2] \\\\ S_{60}=30[4+59(2)] \\\\ S_{60}=30[4+118] \\\\ S_{60}=30[122] \\\\ S_{60}=3,660 .\end{array}
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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|시간 제한||메모리 제한||제출||정답||맞은 사람||정답 비율|
|1 초||256 MB||7||5||5||83.333%|
The island municipality of Soteholm is required to write a plan of action for their work with emission of greenhouse gases. They realize that a natural first step is to decide whether they are for or against global warming. For this purpose they have read the IPCC report on climate change and found out that the largest effect on their municipality could be the rising sea level.
The residents of Soteholm value their coast highly and therefore want to maximize its total length. For them to be able to make an informed decision on their position in the issue of global warming, you have to help them find out whether their coastal line will shrink or expand if the sea level rises. From height maps they have figured out what parts of their islands will be covered by water, under the different scenarios described in the IPCC report, but they need your help to calculate the length of the coastal lines.
You will be given a map of Soteholm as an N × M grid. Each square in the grid has a side length of 1 km and is either water or land. Your goal is to compute the total length of sea coast of all islands. Sea coast is all borders between land and sea, and sea is any water connected to an edge of the map only through water. Two squares are connected if they share an edge. You may assume that the map is surrounded by sea. Lakes and islands in lakes are not contributing to the sea coast.
Figure E.1: Gray squares are land and white squares are water. The thick black line is the sea coast. This example corresponds to Sample Input 1.
The first line of the input contains two space separated integers N and M where 1 ≤ N, M ≤ 1000. The following N lines each contain a string of length M consisting of only zeros and ones. Zero means water and one means land.
Output one line with one integer, the total length of the coast in km.
5 6 011110 010110 111000 000010 000000
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Corsican architecture is modelled by its insularity, geography (mountainous island where isolated valleys make inter-regional exchanges difficult) and its history of numerous raids and invasions.
Its first inhabitants built the island's towers, fortified villages (Cucuruzzu, Capula...), dolmens (Sittiva, Fontanaccia...) and statues-menhirs (Filitosa, Pallagiu, Stantari...), giving Corsica the largest number of megalithic statues in the Mediterranean.
The people of the Antiquity period fought over the coastline: Etruscans, Syracusans, Carthaginians and Phocaean Greeks, who founded Aléria at around 565 B.C.
In 259 B.C. they were hunted by the Romans, who set up in Aléria, the Corsican capital, building temples, public baths, villas and forums. Christianity spread fervently over the island from around 60 A.D.
In 774, Corsica officially became attached to the Holy See, which founded bishop's palaces, convents and monasteries. However the Middle Ages were grim for Corsica. The island was ravaged by the Moors and Saracen pirates, forcing its inhabitants to take refuge in the mountains.
Upset by the island's fate, the Pope delegated its occupation and pacification to Pisa. Two centuries of prosperity and relative peace then followed and can be seen in the beautiful Roman chapels, churches and cathedrals (San Michele in Murato, Trinité in Aregno, Canonica in Lucciana, Cathédrale du Nebbio [cathedral] in Saint-Florent...).
The Genoese colonisation (1284 – 1729) faced popular revolts, governing difficulties and invasion attempts on the island. Genoa built citadels and fortresses (Ajaccio, Bastia, Corté, PortoVecchio, Calvi...), small forts (Girolata, Aléria...), bridges (Ponte Novu, Pont d'Altiana, Spin'a cavallu...), roads and its famous Genoese towers (Porto, Campomoro, Girolata, Roccapina...), a watchtower defence system encircling the coastline.
Little by little the economy developed with olive and chestnut groves. Baroque churches were built: columns, arches, eaves, pilasters and coloured hues (Saint-Jean-Baptiste de La Porta, Saint Jean-Baptiste in Bastia, Castagniccia and Balagne churches...).
The villages' physiognomy was born from this history. Often confined to the mountains, Corsicans lived modest lives through crop and livestock farming. The foundation of villages met the island's strategic imperatives. Set high up on summits, rocky peaks or in the hollows of the valleys, they can only be accessed by winding paths. They contain solid, simple houses built using local materials (granite and ridge tiles in southern Corsica, schist and schist stone tiles in upper Corsica). Occasionally a fortified house belonging to an important figure was used as a refuge by the population. Windmills, fountains, chapels and shepherd's cottages tell the tale of this pastoral life. Deeply rooted in Corsican memory, they are now undergoing restoration.
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10th Grade Academics
This course is for 10th-grade students seeking advanced studies in English language arts. Students who successfully pass the course will be prepared to enroll in AP English language and composition for their 11th-grade year and AP English literature and composition for their 12th-grade year. In this course, students focus on effective expression through the study of classic world literature. They will gain the confidence, know-how, and ability to prepare and deliver an effective message in a variety of ways for different audiences (practice the art of rhetoric). Students will learn how to conduct an interview, use visual aids during a speech, and give a speech on an academic topic. Emphasis is also placed on refining listening comprehension, reading comprehension, writing skills, and academic vocabulary. An honors course is intentionally rigorous. Students must complete more assignments than they would in the regular level English II including a substantial creative writing project which will take the entire school year to finish and involved multiple occasions for peer review and revision. All 10th-grade students take the PLAN test (designed by the ACT organization) at the end of the year. It is recommended that students in the Honors English II course also enroll in AP World History.
In Algebra II, students inquire mathematical structures, encounter complex numbers, write polynomial and rational expressions, get a deeper understanding of functions by using different representations and create and solve equations and inequalities. The students learn to use graphing calculators in a variety of situations. The functions studied include trigonometric functions, exponential and logarithmic relationships, as well as quadratic and polynomial functions. Conics are studied to the extent of circles and parabolas. Statistics involves the study and use of normal distributions to make inferences for populations.
Modern World History
Students in this course look at the modern world through different perspectives, with the intent to prepare them for future AP (advanced placement) and college curriculum. Students will analyze primary source documents as well as relate the events in history to current events. Major themes include geography, social and political change, technological developments, politics, human rights, economics, culture, and globalization. Students will learn to untangle these themes through in-class discussions, think critically through essays and writing assignments, and develop their speaking abilities through presentations. After completing this course, the students will have a firm grasp of the exploration of the New World by European settlers, the rise and fall of Monarchs, the Enlightenment and Scientific Revolution, the American Revolution, the French Revolution, the rise of Nationalism, the Industrial Revolution, the Age of Democracy, the Age of Imperialism, World War I, World War II, all the way to the present day.
AP World History
This rigorous study of the world over the past 12,000 years is designed to prepare students for the AP test in May and introduce them to college-level study, including proper note-taking techniques. Students will be assessed in a variety of ways, including multiple choice quizzes and tests, unit exams, short answer questions, essays, analyzing primary source documents, producing geographically relevant maps, and participating in thematic, topical discussions.
Students will better understand the past by analyzing different time periods and geographical regions while focusing on specific themes. These themes include interaction between humans and the environment; development and interaction of cultures; state building, expansion, and conflict; creation, expansion, and interaction of economic systems; and development and transformation of social structures.
Students will develop historical thinking skills including analyzing evidence; interpretation; comparisons; contextualization; synthesis; causation; detecting patterns of continuity and change over time; periodization; and argumentation. Students will be able to make distinct connections of historical events from one time period to the next to understand how the links of the past have shaped the present U.S.
Chinese IV Honor
Chinese IV is an advanced level Chinese language course designed for high school students who have either passed the AP Chinese language test or have equivalent proficiency. This course has an honors designation from UC, students’ GPA will be boosted. This course is intended to further develop their language and literacy skills and understanding of traditional Chinese culture. Students will read a variety of genres and texts including two full-length novels, short stories, poetry, fiction and nonfiction works, and analyze them in different aspects. They will understand historical influence, consider the philosophical and cultural stance of the authors, and draw inferences. Students will expand their vocabulary while focusing on the four language skills of listening, speaking, reading, and writing. Students will also gain exposure to traditional Chinese written language and culture. They will demonstrate research techniques and an ability to write with an understanding of audience and purpose. Over the one year course, they will produce a range of writings including reading logs and journals, character studies, speeches, creative pieces, drama scripts, and compare and contrast, analytic, expository, narrative, reflective, persuasive, and research essays. There will be writing workshops taught throughout the course. Students will start by writing short essays in order to improve their technique. The course will culminate with a 1500 word essay as a final project.
This course introduces the fundamental concepts of chemistry, the central science, as well as the experimental techniques of modern chemistry and its applications in daily life. The course focuses on hands-on, active-learning application of the scientific method to reinforce foundational chemical principles. The course emphasizes the interrelationship between experimentation, observation, and scientific theory. Students are expected to express themselves in detailed responses that show a deep understanding of the material, as well as the ability to critique and synthesize principles underlying chemical phenomena. Topics covered include atomic theory, ionic and covalent bonding, intermolecular forces, chemical reactions and reactivity, thermodynamics, kinetic-molecular theory, and fundamentals of organic and nuclear chemistry.
AP Art History (Elective)
It has always been humankind’s nature to create art. Throughout history art and architecture have served many purposes from glorifying deities, expressing our sense of beauty, to even making statements about society. Students in this course will analyze art from the Paleolithic age to today. They will discover key factors from the civilizations and time periods from which artworks were created. They will evaluate and compare works of art from different regions and/or time periods as well as works of art created by the same artist. Over 20% of the course will be devoted to artistic works and traditions created outside of Europe.
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