text:
Colorado sees first high school graduation rate decline in a decade
Wed, Jan 12, 2022 9:38 AM
By Robert Davis | The Center Square contributor, The Center Square
(The Center Square) – Colorado's high school graduation rate fell for the first time in a decade last year amidst the ongoing COVID-19 pandemic, according to the Colorado Department of Education (CDE).
The overall graduation rate for the class of 2021 was 81.7%, a drop of 0.2% when compared to the prior year. The state's data also revealed stark divides in graduation and dropout rates among students of different socioeconomic backgrounds.
Even with the drop, Colorado's four-year graduation rate has increased by more than 9% since 2010 when the state changed how it calculates the rate, CDE said.
"We know how tough it was for everyone last school year due to the challenges brought on by the pandemic with schools going to remote learning and others offering hybrid models," Katy Anthes, Colorado's education commissioner, said in a statement.
Colorado's dropout rate was 1.8% in each of the last two years, with more than 8,200 students leaving school in 2020-21, CDE said. It marks the lowest rate the state has recorded over the last decade.
Anthes said it is a "relief" that the dropout rate remained unchanged and also described it as "a concrete display of the dedication and determination of Colorado's students, parents and teachers, especially during these tough times."
But the highest dropout rates were recorded among students of color and students from economically disadvantaged households, both of which recorded rates more than twice as high as their white classmates.
Overall, students of color recorded a graduation rate of 76.1% last year, a drop of 1% from the year before, CDE's figures show. Black, Hispanic, and Native American students all recorded lower graduation rates despite these groups making up less than half of the total graduates from last year.
Colorado schools also graduated fewer students who were either homeless or migrated to the United States.
In the class of 2021, just 53% of students experiencing homelessness graduated, a decline of 3.1%. Migrant students saw an even steeper year-over-year decline at 4.7% even though two-thirds of these students received a diploma.
The only institutional program category that recorded an increase in its graduation rate last year was for students with disabilities, which graduated 4.6% more students than it did in 2020.
The Center Square
Mesa County grand jury to investigate election tampering, misconduct allegations
Pueblo, CO Has One of the Highest Break-In Rates in the Nation
Polis says 'saving Coloradans money' is top priority in state of the state address
Colorado Republicans roll out 'Commitment to Colorado' legislative package

redpajama_set_name: RedPajamaCommonCrawl


text:
Anaheim ID's 1,263 'historic gems' to preserve
A city "Building on Our Past" sign is attached to a fence in front of one of the properties (on Lincoln Avenue) listed on Anaheim's Citywide Preservation Plan.
Anaheim developed an Historic Preservation Plan to identify properties that have historic significance or might in the future. Pictured here is an old citrus packing house on Anaheim Boulevard that may be turned into a restaurant.
Historic property: 116 N. Resh
Historic property: 110 N. Resh St.
Historic property: 902 W. Broadway
Historic property: 1018 W. Lincoln Ave.
Historic property: 300 N. Janss
Historic property: 511 S. Ohio St.
A monument on Lincoln Avenue marks the entrance to one of Anaheim's four historic districts.
The main building at Anaheim High School and the attached Cook Auditorium are among the buildings on in Anaheim's Citywide Historic Preservation Plan.
By Eric Carpenter | Orange County Register
ANAHEIM – A city often criticized for ignoring history when it redeveloped its downtown has approved the first comprehensive plan to help save its remaining "historic gems."
Anaheim's Citywide Historic Preservation Plan took 18 months to compile and lists 1,263 properties – mostly residential and a few commercial ones – with historical significance. (Click here to see the list of properties.)
It includes properties from across 50 square miles, from west Anaheim to Anaheim Hills.
The plan places no restrictions on what owners can do with their properties; it merely lists guidelines for owners to follow if they want to restore their homes or businesses to original standards. In some cases, homes could be eligible for property-tax breaks through the Mills Act.
The one restriction: listed properties cannot be razed for at least 60 days after an owner applies for a demolition permit. That time would allow city staff and preservationists to talk to the owner about buying or moving the property before it's bulldozed.
"This is Anaheim finally realizing the significance of its past," said Gail Eastman, a member of the committee that put together the 74-page report and list.
"It wasn't just Anaheim that in the '70s and '80s got caught up in the exuberance of redevelopment and moving forward. A lot of significant buildings were lost during that time," she said. "You might say this is making up for some of the sins of the past."
The plan, unanimously approved by the City Council on Tuesday night, lists homes deemed to be significant, either because of their architecture or because a person of significance lived there – or both.
It includes an updated list of homes within four previously established historic districts. The committee also identified 109 structures outside of those districts.
The Ramon Peralta Adobe, for instance, is the first home in Anaheim Hills to be included as "historically significant." It was built in 1871 and is the lone adobe left standing in the Santa Ana Canyon.
The list includes homes from the 1950s and '60s that are representative of the architecture of their eras.
About 240 homes throughout the city have already been awarded Mills Act contracts, ranking Anaheim with the third most officially designated historical homes in California – behind Los Angeles and San Diego.
Still, many of the city's downtown historic buildings were demolished in the 1970s and '80s during redevelopment.
Several property owners said they think Anaheim is striking a good balance with the preservation plan.
"It seems like they are fully respecting property rights, while at they same time letting property owners know, 'Hey, you've got something special here. You might want to fix it up and protect it,'" said Mike Garcia, who owns a home listed in the plan on Sycamore Street.
Councilwoman Lorri Galloway said the plan honors "a taste of generations before us" and encourages preservation as a gift to generations still to come.
"Our heritage is of key importance to us," she said. "We are very proud of it."
Contact the writer: 714-704-3769 or ecarpenter@ocregister.com
directory box
Eric Carpenter


text:
Q: Find the limit of $\lim\limits_{(x,y) \rightarrow (0,\pi/2)} \cos x \sin y$ and prove that your result is correct. Find the limit and prove that your result is correct.
$$\lim _{(x,y) \rightarrow (0,\pi/2)} \cos x \sin y$$
My trial:
I have calculated the limit by direct substitution and I found that it is 1. but I have a difficulty in proving this. I know that I should start with the inequality $$|\cos x \sin y -1|< \epsilon$$ and try to isolate the inequalities $|x|$ &$|y-\pi/2|$ from it in order to use that $|x| < \delta$ &$|y-\pi/2| < \delta$ ...... am I correct in my understanding? If so I do not know how to do this ...... could anyone help me please?

A: $| \cos x \sin y -1|= |\cos x \sin y-\cos x \sin( \pi /2)+\cos x \sin( \pi /2) - \sin( \pi/2)| $
$\le| \cos x| \cdot| \sin y - \sin (\pi/2)|+| \cos x-1|.$
Can you proceed ?

A: $f(x):=$
$|\cos x \sin y -\cos x +\cos x -1| \le$
$|\cos x (\sin y -1)| + |\cos x -1| \le$
$ |\sin y-1| +|\cos x-1| .$
$d:= ((x^2+(y-π/2)^2)^{1/2}.$
Since $\cos x$ , $\sin  y$ are continuos:
For $\epsilon/2$ there is a $\delta_1$ s.t.
$|x| \lt \delta_1$  implies
$|\cos x -1| \lt \epsilon/2.$
For $\epsilon/2$ there is a $\delta_2$ s.t.
$|y-π/2| \lt \delta_2$  implies
$|\sin y -1| \lt \epsilon/2;$
Choose $\delta =\min (\delta_1, \delta_2)$ .
Then
$ (x^2+(y-π/2)^2)^{1/2} \lt \delta $
implies
$f(x) \lt |\sin y-1| +|\cos x -1| \lt \epsilon$.

redpajama_set_name: RedPajamaStackExchange


Every day swarms of Londoners take to their daily commute, seemingly leaving the manners and etiquette that our nation was built on behind on the walk to the station.
If you're unlucky enough to spend the rush hour trundling back and forth on the tube, you'll realise pretty quickly that you no longer process humanity as a collection of autonomous individuals, each with their own unique agenda, but only as a sweaty set of wailing walls.
The Underground? More like the underworld! Maybe it is this peculiar effect of being engulfed in these strange canisters speeding through dark tunnels that accounts for the hellish lack of etiquette that marks out Londoners' behaviour on the tube.
Whether it's the rush at the doors, where passengers are not given the consideration or space to get off the train and have to worm themselves out of a muddle of elbows and legs. Mind the Gap? Mind the people! Or it's another Texan stand-off to read the Metro. Or unwelcome leers only pregnant women seem to avoid as no one makes eye contact in case they'll have to give up their seat. Or perhaps it's another uncovered cough and the bout of flu that's bound to follow.
Worst of all, is what an arrangement of shared armrests does to otherwise rational beings. More often than not a Popeye-muscled man, in the same motion as sitting down next to you, will slyly slide and riggle his sinewy, snake-like arm until he has knocked yours off its perch.
I remember an occasion on the Northern Line when two businessmen came on and sat either side of me. The proceeding stops to Waterloo were the Battle of Waterloo relived.
Eating on the tube is another bugbear. Whether it's a MacDonald's burger or a spicy wrap, the odour lingers on long after the wrapping's been discarded on the floor or, worse still, on a seat.
Then there are the people listening to 'music' through their phones, leaving bystanders listening to the tinny sounds of the bass. Why is there this complete lack of grace? Maybe it's the concoction of odours – cheap cologne, bad breath and sweat – that nulls our faculties for empathy and compassion. Or maybe we're too absorbed in our own little worlds, thinking about this or that interview, file or folder.
Perhaps we're just plain rude.
But how is tube etiquette to be revived – if it ever did exist? God only knows, and I've never seen him on the Underground.
Joseph is a creative Individual and proud Londoner. A recent graduate in English Literature from Sussex University, he has returned to his home in the greatest city in the world, and is looking for a postgraduate course right in its heart. London's rich history, the art it has inspired, its people, its customs, its buildings, its character, always nourish and revitalise Joseph's creative energies.


The recently renovated Silken Torre Garden is our 3-star hotel located in calle Hemanos García Noblejas, Ciudad Lineal. This is a magnificent option both for clients travelling to Madrid for a family rest as well as those travelling for business.
In each of the 90 bright and spacious rooms measuring 35m2 in our hotel you will find two integrated spaces, a 32" LCD LED TV, free WiFi connection, climate control, safe and large desk; all designed for enjoying the perfect rest.
Thanks to Hotel Silken Torre Garden's excellent location in Ciudad Lineal you will have access to the city centre, as well as the main streets in Madrid (M30, M40, A3...) for getting around. If, on the other hand, you prefer to leave your car parked in our private garage you will find line 2 of the underground just 100m away, which will take you directly to the city centre in just 10 minutes. Alternatively, you have a number of bus routes with a stop close to the hotel that connect you to any point in Madrid.
In addition, its location makes it the perfect hotel for visiting the IFEMA trade fair, as well as being close to Barajas-Adolfo Suárez airport of Madrid.
Among the most noteworthy services you will find in Silken Torre Garden are our cafeteria, the private car park, the terrace where you can celebrate any kind of event in the open air, or our function room, large and adaptable, where you can hold meetings, give presentations and hold all kinds of business and family events, such as banquets and celebrations; the wedding menu is exquisite and the service, professional like no other.
One of the unique features of the La Torre restaurant is that everything in the kitchen is 100% homemade. Booking in advance is required, and there is a minimum of 10 guests.
In the restaurant itself, you will also be able to enjoy the outstanding breakfast buffet every morning, which includes fruits, cereals, cold meats, pastries, breads, and juices.
In the café, you can also enjoy dinners for a great end to your day after a long and tiring trip for business or tourism.
Silken Torre Garden in pictures.


text:
National Strawberry Day
Even though National Strawberry Month doesn't officially start until May, National Strawberry Day is a good way to get a jump on the festivities. This holiday falls on February 27th every year and gives everyone the opportunity to enjoy a fruit that's been cultivated by humans since the Middle Ages and can be found in dishes all around the world.
The History Of National Strawberry Day
National Strawberry Day was started by the grocery industry in 2013. Why they choose this day in February as the day to host this holiday is beyond anyone's comprehension, however. After all, National Strawberry Month isn't until May, and most of the south's strawberry harvest is in late April or May. Despite these facts, however, people can still celebrate this holiday thanks to strawberries imported from Mexico and South America.
The History Of The Strawberry
We can't discuss National Strawberry Day without first taking the time to acknowledge the rich history of this fruit. Although humans have enjoyed wild strawberries for hundreds, if not thousands, of years, it wasn't until the 14th century that the French started growing them. They would take wild strawberries from the forest and slowly introduce them into their gardens. By the 16th century, the strawberry had traveled around Europe.
Fun Facts About Strawberries
Now that we've uncovered not only the history of this holiday but also the history of the strawberry, it's time to learn some fun facts about these fruits. Below are some things that most people won't know about strawberries—trivia that makes National Strawberry Day a lot more interesting to celebrate.
Strawberries are one of the first fruits to ripen during the Spring.
A strawberries' flavor can be influenced by weather and how ripe it is when it's harvested.
During a typical California strawberry time in California, 10 million baskets of strawberries are shipped each day.
Almost 95% of American households consumer berries in some form.
Although some people think that strawberries are berries, they're actually aggregate fruit.
Americans consume approximately 2-pounds of strawberries per person a year.
An average strawberry has over two hundred seeds.
The perfect growing temperature for strawberries is somewhere between 54 and 80 degrees.
In California, 23,000 acres of strawberries are planted every year.
Almost 3/4th of the entire U.S strawberry crop is produced in California.
Florida is the number two producer of strawberries in the United States.
Florida plants over 5,000 acres of strawberries every year.
Celebrating National Strawberry Day
This holiday is quite easy to celebrate as all it requires is the participant to use strawberries in some way during the course of the day. This can include enjoying a bowl of strawberries with some sugar, having a strawberry pie, or enjoying some strawberry shortcake. And while a person is enjoying strawberries on this day, they can announce their activities to the world using the hashtag #NationalStrawberryDay.
When is National Strawberry Day?
This year (2023) February 27 (Monday) Multiple dates - more
Next year (2024) February 27 (Tuesday) Multiple dates - more
Last year (2022) February 27 (Sunday) Multiple dates - more
Where is National Strawberry Day celebrated?
National Chili Day
Bloomsday
National Plus Size Appreciation Day
National Jukebox Day
National Educational Support Professionals Day
Shrove Tuesday (Pancake Day) February 21, 2023


text:
Q: algebra direct connect pell eqn soln $(p_{nk},q_{nk})$ with $(p_n + q_n\sqrt{D})^k$ Apologies if this question has been asked before - there are many "pell" entries
in this forum.
Given:
(a) $\;D \in \mathbb{Z^+}\;$ is not a perfect square.
(b) The continued fraction expansion of $\sqrt{D}\;$ is
    $\;[a_1, \overline{a_2, \cdots, a_n, 2a_1}].$
(c) Coprime $\;\frac{p_r}{q_r}\;$ denotes the $r^{th}$ convergent of $\sqrt{D}.$
(d) $\;\forall k \in \mathbb{Z^+}, \;(P_k,Q_k)$ is defined by
    $\;(p_n + q_n\sqrt{D})^k = P_k + Q_k\sqrt{D}.$
To prove: (e) $\;\forall k \in \mathbb{Z^+},$
$\;(p_{nk},q_{nk}) = (P_k, Q_k).$
My research: Let $E_1$ denote the pell equation $\;x^2 - Dy^2 = \pm 1.\;$
I reviewed pages 1-7 (through theorem 1.19) of pdf-1,
chapter 1 (only) of pdf-2, chapter 4 (only) of pdf-3, and
html-4.  From these references, I concluded:
(1) $\;\{(p_n,q_n), (p_{2n},q_{2n}), (p_{3n},q_{3n}), \cdots \}\;$ is the
complete set of all positive integer solutions to $E_1.$
(2) $\;\{(P_1,Q_1), (P_2,Q_2), (P_3,Q_3), \cdots \}\;$ is also the
complete set of all positive integer solutions to $E_1.$
(3) The 4 sequences $\;\{p_n, p_{2n}, p_{3n}, \cdots \},
\;\{q_n, q_{2n}, q_{3n}, \cdots \},
\;\{P_1, P_2, P_3, \cdots \},\;$ and $\;\{Q_1, Q_2, Q_3, \cdots \}\;$
are each strictly increasing.
From these conclusions I indirectly deduced result (e) above.  However, as
I have elaborated in my Partial Work section (below), I would like to establish this
result directly, through algebra.
Partial Work: I decided to attempt to algebraically demonstrate that
coprime $\;(p_{2n}, q_{2n}) = ([{p_n}^2 + D{q_n}^2], 2p_nq_n) = (P_2, Q_2).\;$
If successful, I then hoped to find a pattern in the demonstration that would allow
me to show that
$[F_1]\;\; \forall k\in\mathbb{Z^+},
\;(p_{n(k+1)}, q_{n(k+1)}) = (p_{nk}p_n + Dq_{nk}q_n, p_{nk}q_n + q_{nk}p_n).\;$
This would allow me to inductively conclude result (e) above.
$a_1 = $ the floor of $\sqrt{D}.$
Let $A$ denote the continued fraction $\;[a_1, a_2, \cdots, a_n] = \frac{p_n}{q_n}.$
Let $B$ denote the continued fraction $\;[2a_1, a_2, \cdots, a_n].$
Then, $\;B = a_1 + \frac{p_n}{q_n}\;$ and
$\;\frac{p_{2n}}{q_{2n}} = \;$ the continued fraction represented by
$\;[A, B].$
I can now generate 2 non-linear equations ($F_2$ and $F_3,$ below)
for $\;p_{2n}, q_{2n}\;$
that depend on the computation of coprime $p_{n-1}$ and $q_{n-1}.$
I may have to give special consideration for when $\;n=1\;$
(e.g. $\sqrt{2} = [1, \overline{2}]$),
since then $(p_{n-1}, q_{n-1}) = \;$ the artificially contrived $(1, 0).$
$[F_2]\;\; \dfrac{Bp_n + p_{n-1}}{Bq_n + q_{n-1}} = \;$ the ratio $\dfrac{p_{2n}}{q_{2n}}.$
$[F_3]\;\; {p_{2n}}^2 - D{q_{2n}}^2 = (-1)^{2n} = 1.$
Assuming that $p_{2n}$ and $q_{2n}$ are each expressed in terms of
$\;p_n, q_n, p_{n-1}, q_{n-1}, a_1\;$ and $\;D,\;$
I now have to compute $p_{n-1}$ and $q_{n-1}$ in terms of
$\;p_n, q_n, a_1\;$ and $\;D.\;$
Let $\beta $ denote $[\overline{2a_1, a_2, \cdots, a_n}] = a_1 + \sqrt{D}.$
Then $\sqrt{D}$ may be represented by the continued fraction $\;[A,\beta].$
Therefore, (I suspect that) the following two equations will generate unique
values for $p_{n-1}$ and $q_{n-1}:$
$[F_4]\;\; \dfrac{\beta p_n + p_{n-1}}{\beta q_n + q_{n-1}} = \sqrt{D}.$
$[F_5]\;\; p_nq_{n-1} - q_np_{n-1} = (-1)^n\;$
(since the first element of the continued fraction is $a_1,$ rather than $a_0$).
Unfortunately, $F_2, F_3, F_4,\;$ and $F_5\;$ seem very ugly to me, and there is
no way to predict the challenges in $F_1.$  I guess that it is not suprising that
an indirect proof was chosen for the last theorem in html-4.
Anyway, has any professional mathematician wrestled with this?  For example,
would any of the later chapters in Khinchin's pdf-2 be pertinent?  Alternatively, does anyone have any ideas about making the attack less
crude?
My knowledge of field theory is fairly weak, so please be merciful.

A: I just had insights that facilitated the proof.
Given:
(a) $\;D \in \mathbb{Z^+}\;$ is not a perfect square.
(b) The continued fraction expansion of $\;\sqrt{D}\;$ is
    $\;[a_1, \overline{a_2, \cdots, a_n, 2a_1}].$
(c) Coprime $\;\frac{p_r}{q_r}\;$ denotes the $r^{th}$
    convergent of $\sqrt{D}.$
(d) $\;\forall k \in \mathbb{Z^+}, \;(P_k,Q_k)$ is defined by
    $\;(p_n + q_n\sqrt{D})^k = P_k + Q_k\sqrt{D}.$
To prove: $\;\;$ (e) $\;\forall k \in \mathbb{Z^+},$
$\;(p_{nk},q_{nk}) = (P_k, Q_k).$
Proof:
In general, continued fraction results may have to adjust to the first element
in $\;\alpha = [a_1, a_2, \cdots]$ being $a_1,$ rather than $a_0.$
When $\;n=1\;$ (i.e. $\;\sqrt{2} = [1,\overline{2}]),\;$ the artificial nature
        of
$\;(p_{n-1}, q_{n-1}) = (p_0, q_0) = (1,0)\;$ suggests that this case
should be handled separately.
$\underline{\text{case 1:} \;\; n>1}$
In Old-s_pdf, the analysis in pages 114-115 (page 61/85 in the pdf file),
establishes the following:
(1) $\;p_{n-1} = Dq_n - a_1p_n\;$ and $\;q_{n-1} = p_n - a_1q_n.$
(2) $\;{p_n}^2 - D{q_n}^2 = (-1)^n.$
However, examination of this analysis indicates that it can be extended to
$\alpha_{[(nk)+1]} = \sqrt{D} + a_1\; :\;k\in\mathbb{Z^+} \;\Rightarrow\;
\sqrt{D} = \dfrac{(\sqrt{D} + a_1)p_{nk} + p_{[(nk)-1]}}
    {(\sqrt{D} + a_1)q_{nk} + q_{[(nk)-1]}}\;$
This leads to
(3) $\;p_{[(nk)-1]} = Dq_{nk} - a_1p_{nk}\;$ and $\;q_{[(nk)-1]} = p_{nk} - a_1q_{nk}.$
Also, since $\;p_{nk}q_{[(nk)-1]} - q_{nk}p_{[(nk)-1]} = (-1)^{nk},\;$ this leads to
(4)  $\;{p_{nk}}^2 - D{q_{nk}}^2 = (-1)^{nk}.$
(5) $\;{P_k}^2 - D{Q_k}^2 \;=\; (P_k + Q_k\sqrt{D})(P_k - Q_k\sqrt{D})
    \;=\; (P_1 + Q_1\sqrt{D})^k (P_1 - Q_1\sqrt{D})^k\;$
[since conjugation is multiplicitive]
$\;=\; (p_n + q_n\sqrt{D})^k (p_n - q_n\sqrt{D})^k
    \;=\; ({p_n}^2 - D{q_n}^2)^k \;=\; (-1)^{nk}
    \;=\; {p_{nk}}^2 - D{q_{nk}}^2.$
(6) Suppose that $\;\frac{p_{nk}}{q_{nk}} =\;$
the ratio $\;\frac{P_k}{Q_k}.$
Then, since $\;p_{nk}, q_{nk}, P_k,\;$ and $\;Q_k$ are all positive,
$\exists s>0 \ni (p_{nk}, q_{nk}) = (sP_k, sQ_k) \;\Rightarrow\;
{p_{nk}}^2 - D{q_{nk}}^2 = s^2({P_k}^2 - D{Q_k}^2).$
Therefore, by result (5), $\;(p_{nk}, q_{nk}) = (P_k, Q_k).$
..........................
Let $A$ denote the continued fraction
        $\;[a_1, a_2, \cdots, a_n] = \frac{p_n}{q_n}.$
Let $B$ denote the continued fraction $\;[2a_1, a_2, \cdots, a_n].$
Then, $\;B = a_1 + \frac{p_n}{q_n}\;$ and
$\;\frac{p_{2n}}{q_{2n}} = \;$ the continued fraction represented by
        $\;[A, B].$
Therefore,
$\;\dfrac{Bp_n + p_{n-1}}{Bq_n + q_{n-1}} = \;$
        the ratio $\;\dfrac{p_{2n}}{q_{2n}}.$
By the previous results, this leads to
$\dfrac{p_{2n}}{q_{2n}} \;=\; \dfrac{(a_1 + \frac{p_n}{q_n})p_n + (Dq_n - a_1p_n)}
    {{(a_1 + \frac{p_n}{q_n})q_n + (p_n - a_1q_n)}}
\;=\; \dfrac{a_1p_n + \frac{{p_n}^2}{q_n} + Dq_n - a_1p_n}
{a_1q_n + p_n + p_n - a_1q_n} $
$=\; \dfrac{{p_n}^2 + D{q_n}^2}{2p_nq_n} \;=\; \dfrac{P_2}{Q_2}.\;$
Therefore, by result (6), $\;(p_{2n}, q_{2n}) = (P_2, Q_2).$
(7) Thus, when $n>1,$ the conjecture is true for $k=2.$
..........................
Inductively assume that when $n>1,$ the conjecture is true for
        $\;\{1, 2, \cdots, k\}.$
$P_{k+1} + \sqrt{D}Q_{k+1} \;=\; (P_k + \sqrt{D}Q_k)(P_1 + \sqrt{D}Q_1)$
$=\; (P_kP_1 + DQ_kQ_1) + \sqrt{D}(P_kQ_1 + P_1Q_k)$
$=\;$ [by inductive assumption]
$\;(p_{nk}p_n + Dq_{nk}q_n) + \sqrt{D}(p_{nk}q_n + p_nq_{nk}) \;\Rightarrow$
$(P_{k+1},Q_{k+1}) \;=\; (p_{nk}p_n + Dq_{nk}q_n, p_{nk}q_n + p_nq_{nk}).$
Let $C$ denote the continued fraction representation of $\;\dfrac{p_{nk}}{q_{nk}}.$
Then $\;\dfrac{p_{[n(k+1)]}}{q_{[n(k+1)]}} = \;$ the continued fraction represented by
        $\;[C, B].$
Therefore,
$\;\dfrac{Bp_{nk} + p_{[(nk)-1]}}{Bq_{nk} + q_{[(nk)-1]}} = \;$
        the ratio $\;\dfrac{p_{[n(k+1)]}}{q_{[n(k+1)]}}.$
By the previous results, this leads to
$\dfrac{p_{[n(k+1)]}}{q_{[n(k+1)]}} \;=\;
\dfrac{(a_1 + \frac{p_n}{q_n})p_{nk} + (Dq_{nk} - a_1p_{nk})}
    {(a_1 + \frac{p_{n}}{q_n})q_{nk} + (p_{nk} - a_1q_{nk})}
\;=\; \dfrac{a_1p_{nk} + \frac{p_np_{nk}}{q_n} + Dq_{nk} - a_1p_{nk}}
{a_1q_{nk} + \frac{p_nq_{nk}}{q_n} + p_{nk} - a_1q_{nk}}$
$=\; \dfrac{p_np_{nk} + Dq_{nk}q_n}{p_nq_{nk} + p_{nk}{q_n}}
\;=\; \dfrac{P_{k+1}}{Q_{k+1}}.\;$
Therefore, by result (6), $\;(p_{[n(k+1)]}, q_{[n(k+1)]}) = (P_{k+1}, Q_{k+1}).$
(8) Thus, when $n>1,$ the conjecture is true by induction.
..........................
$\underline{\text{case 2:} \;\; n=1}$
When $n=1, \;\sqrt{D} = [a_1, \overline{2a_1}].$
(9) Therefore, $\;(p_1, q_1) = (a_1, 1), \;(p_2, q_2) = (2{a_1}^2 + 1, 2a_1),\;$
and for $\;2\leq k \in\mathbb{Z^+}, \;(p_{k+1}, q_{k+1})
\;=\; (2a_1p_k + p_{k-1}, 2a_1q_k + q_{k-1}).$
$\sqrt{D} + a_1 = [\overline{2a_1}] \;=\; 2a_1 + \dfrac{1}{\sqrt{D} + a_1} \;\Rightarrow$
$D + {a_1}^2 + 2a_1\sqrt{D} = 2a_1\sqrt{D} + 2{a_1}^2 + 1 \;\Rightarrow\;$
(10) $\;D = {a_1}^2 +1.$
(11) From (9) and (10), $\;{p_1}^2 - D{q_1}^2 = -1.$
..........................
Let $R$ represent the continued fraction $\;[a_1, 2a_1, \cdots, 2a_1]\; ([k+1] $ elements).
This implies that $\;R = \frac{p_{k+1}}{q_{k+1}}.$
Let $\;S = \;$ the continued fraction represented by $\;[\overline{2a_1}] \;\Rightarrow S = a_1 + \sqrt{D}.$
Then $\sqrt{D} = \;$ the continued fraction represented by $\;[R, S].$
(12) Therefore, $\;\sqrt{D} \;=\; \dfrac{Sp_{k+1} + p_k}{Sq_{k+1} + q_k}.$
This is very similar to equation (4.39) given in Old-s_pdf, on page 114
(page 61/85 in the pdf file).  That pdf's subsequent analysis will be paralleled
with result (12) above as the starting point and will lead to :
(13) When $n=1,\;$ and $\;k\geq 2:$
$p_k = Dq_{k+1} - a_1p_{k+1} \;$ and $\;q_k = p_{k+1} - a_1q_{k+1}.$
(14) Continuing the parallel analysis, result (13) will lead to
${p_k}^2 - D(q_k)^2 = (-1)^k.$
(15) The analysis and conclusions in results (5) and (6) from case 1 also
apply when $n=1.\;$  Therefore,
$\;{P_k}^2 - D{Q_k}^2 = (-1)^k = {p_k}^2 - D(q_k)^2.$
Therefore, $\;\frac{p_k}{q_k} = \frac{P_k}{Q_k} \;\Rightarrow\;
(p_k, q_k) = (P_k, Q_k).$
..........................
(16) From results (9) and (10) above, it is immediate that
$(p_2, q_2) = (2{a_1}^2 + 1, 2a_1) = ({a_1}^2 + D, 2a_1) = (P_2, Q_2).$
Inductively assume that when $n=1,\;$ the conjecture is true for
$\;\{1, 2, \cdots, (k+1)\}.$
$P_{k+2} + \sqrt{D}Q_{k+2} \;=\; (P_{k+1} + \sqrt{D}Q_{k+1})(P_1 + \sqrt{D}Q_1)$
$=\; (P_{k+1}P_1 + DQ_{k+1}Q_1) + \sqrt{D}(P_{k+1}Q_1 + P_1Q_{k+1})$
$=\;$ [by inductive assumption]
$\;(p_{k+1}a_1 + Dq_{k+1}) + \sqrt{D}(p_{k+1} + a_1q_{k+1}) \;\Rightarrow$
$(P_{k+2},Q_{k+2}) \;=\; (p_{k+1}a_1 + Dq_{k+1}, p_{k+1} + a_1q_{k+1}).$
..........................
Let $T$ denote the continued fraction representation of $\;\dfrac{p_{k+1}}{q_{k+1}}.$
Then $\;\dfrac{p_{k+2}}{q_{k+2}} = \;$ the continued fraction represented by
        $\;[T, 2a_1].$
Therefore,
$\;\dfrac{2a_1p_{k+1} + p_k}{2a_1q_{k+1} + q_k} = \;$
        the ratio $\;\dfrac{p_{k+2}}{q_{k+2}}.$
Using result (13), this leads to
$\dfrac{p_{k+2}}{q_{k+2}} \;=\;
\dfrac{2a_1p_{k+1} + (Dq_{k+1} - a_1p_{k+1})}{2a_1q_{k+1} + (p_{k+1} - a_1q_{k+1})}
\;=\; \dfrac{a_1p_{k+1} + Dq_{k+1}}
{a_1q_{k+1} + p_{k+1}}
\;=\; \dfrac{P_{k+2}}{Q_{k+2}}.\;$
Therefore, by result (15), $\;(p_{k+2}, q_{k+2}) = (P_{k+2}, Q_{k+2}).$
(17) Thus, when $n=1,$ the conjecture is also true by induction.


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Formula 1 driver Lewis Hamilton developed a love for cars while growing up in the Hertfordshire town of Stevenage.
He began karting when he was eight years old and within four years he had been signed by McLaren.
Now, the 32-year-old has put the first laps on the Mercedes car he hopes will make him world champion for the fourth time in 2017.
Before Hamilton's new Mercedes W08 was unveiled at Silverstone, the BBC retraced the 56-mile (90km) route from his hometown to the home of the British Grand Prix.


This is a big week for me. On Thursday I turn 44 – for my good friends in Louisiana, I'll make 44 this week.
Twice as good as 22, as a friend told me last week. Let's hope she's right about that. I don't want this to be the year that my metabolism, joints and body in general betray me.
This is also a year that I am focusing on INTENTION!
It's not a new idea or concept, just one that has taken a bit of time to come into focus. One of the catalysts for this word was reading the book 7 by Jen Hatmaker. In full disclosure, I did not replicate any of the challenges that Jen set for herself. She didn't intend it to be a "do as I do", but an experiment in dealing with excess. I roll in it and simultaneously hate it and am working to eliminate it.
In many ways, this book totally changed how I look at nearly every aspect of my life. It didn't dictate how many foods I eat or items of clothing I wear, but opened my eyes to choices I was making. It even opened my eyes to how many factors in my life were defaults because I wasn't actually making choices. That's passively existing, not actively living. I'll take the latter, thank you very much.
Dayspring (they are AWESOME, check them out) is graciously helping me celebrate by giving away a copy of 7 by Jen Hatmaker to one of you. Not only this, but 4 books this week; a different book from today until my birthday on Thursday. You'll have to come back to find out which ones. Here's a hint: they are all brand-spankin' new. One is so new it isn't even out yet! Actually, as of today, none have been released yet. You are in for the good stuff here.
So, I'm gonna party like it's my birthday. And I want you to come along for the ride.
Have a cupcake and make a wish. Remember, birthday calories don't count!
Have you ever read a book that totally shifted your everyday life? You know, other than the Bible? Tell me about it.
It's been years since I read it, and I don't remember anything other than the title but "Don't Sweat the Small Stuff" made a big difference. I realized as it said, in the long run, it's all small stuff. So many of the things we stress over are things we have no control of. All we do is make ourselves miserable.
Thanks. If we can't control it or change it, then it isn't worth the stress - yes!! So glad you visited.
You know I am all about celebrating all MONTH long!! I have been wanting to read this book for forever, but keep forgetting to get it.
I think for us, we should go 2 months!!! I haven't read that by Tim Keller, but need to put it on my list!! Thanks for sharing it.
The Freedom of Self-Forgetfulness is one I was going to mention! Tiny little book packed with truth and freedom. LOVE it and it's truly a must read! Seriously would only take about 30 minutes to read; maybe that's another reason I love it?
HAPPY BIRTHDAY WEEK!!!!!!!!!!!!!!!!!!!!!!!! Ok, that is all.
Thank you!! Don't forget to enter if you haven't read this already!
Happy Birthday to you!! That's wonderful. And what fun giveaways! Let's see...Secrets in the Dark by Frederick Buechner is one that has changed me.
Ooh, another one to add to my list! Thanks for sharing!
One book that broke my heart was Scared by Tom Davis. It is fiction, but based on his real life experience working with orphans in Africa. The AIDS pandemic has left many communities there with almost no adults, no elders. He also wrote Priceless about his experiences with human trafficking in Eastern Europe.
Wow, that must've been incredibly difficult for you to read, Amy. I'll have to look up his books, they sound powerful.
"Lioness Arising" just had so many powerful truths illuminated for that particular point in my life.
Another one I've never heard of. You guys are a great resource. Thanks for sharing.
That is a good message and something easy to forget. Thanks for sharing!
"Overcoming Fear, Worry, and Anxiety" by Elyse Fitzpatrick. She is a Christian counselor and this book explains how fear escalates to serious anxiety. I struggled with this and went to a Christian counselor and this is the first book she recommended to me. Understanding what is happening and the connection between our mind and how our body reacts is powerful.
Excellent. What a great resource to have. It's crazy how the mind and body work together.
Oh man. There have been so many books that shifted my life in many little ways. Honestly, I think maybe one of the first things to affect me was the Anne of Green Gables series - I wanted to live with the passion and verve Anne has.
Confession: I have never read those. I should change that soon!
Yes, absolutely. I even wrote a blog post about it this week: http://www.leeanngtaylor.com/seeds-new-life/. Radical by David Platt.
I need to check this out. I've seen the book, but don't know much about it. Thanks!


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Which generation is better at relationships: Millennials or Baby Boomers? | The Tylt
Which generation is better at relationships: Millennials or Baby Boomers?
It's an old adage: 50 percent of marriages end in divorce. But according to a new study, divorce rates in the U.S. dropped 18 percent between 2008 and 2016, and millennials are the driving force behind the change. Despite the increasing public acceptance of divorce, millennials seem to be turning away from the option. Even so, baby boomers, a generation plagued by divorce, continue to chastise younger generations for steering away from tradition. Could millennials actually be better at relationships?
By Jessie Blaeser
#MillennialsKnowLove
#LoveHardAtAnyAge
A new study (yet to be published in a peer-reviewed journal) reports that divorce rates are dropping among millennials. This is the same generation known for being "lazy, narcissistic and prone to jump from job to job," as reported by LifeScience.com. Millennials may be antsy when it comes to jobs, but it seems this quality does not translate to their relationships.
USA Today reports on the new study conducted by University of Maryland professor Philip Cohen, highlighting that:
...newly married women are now 'more likely to be in their first marriages, more likely to have BA degrees or higher education, less likely to be under age 25, and less likely to have own children in the household,' which Cohen writes can all affect the risk of divorce.
Couples are waiting until they're more economically stable to marry.
These factors are likely to result in declining divorce rates across the board.
https://www.usatoday.com/story/news/nation-now/2018/09/26/millennials-blame-lower-us-divorce-rate-study/1429494002/
Baby boomers, on the other hand, have roughly doubled their divorce rate since the 1990s. According to Pew Research's Renee Stepler, the climbing divorce rates among baby boomers is due in part to their aging. She says:
During their young adulthood, Baby Boomers had unprecedented levels of divorce. Their marital instability earlier in life is contributing to the rising divorce rate among adults ages 50 and older today, since remarriages tend to be less stable than first marriages.
On top of this, now that baby boomers are reaching the later stages of life, there are more reasons to divorce even if couples did not have the early-onset marital problems typical of their generation. According to Market Watch's Angela Moore:
Statistically speaking we're healthier and probably going to be living a lot longer — possibly 30 years longer — than average retirees once did. The surge in late-in-life — or "gray" — divorce is one possibly unintended consequence of this so-called longevity bonus.
https://www.marketwatch.com/story/your-failing-marriage-is-about-to-make-the-retirement-crisis-worse-2017-03-13
It's important to keep in mind that the odds may be stacked against baby boomers when it comes comparing divorce rates among generations. As the senior generation, those who have been divorced are more likely to get married again, and the divorce rate for second marriages is even higher than for the first. Pew Research points out:
The risk of divorce for adults ages 50 and older is also higher among those who have been married for a shorter time.
Millennials might face similar increases in divorce rates a few decades down the line. The fact is, baby boomers have a longer record to track, making the comparison between the two generations somewhat unfair.
http://www.pewresearch.org/fact-tank/2017/03/09/led-by-baby-boomers-divorce-rates-climb-for-americas-50-population/
When it comes to marriage, millennials are known for being more open to a number of factors that likely contribute to the health of a relationship. Business Insider's Shana Lebowitz reports that millennials are more likely to have sex and live together with their partner before actually getting married. Millennials are also more open to partners of different religions.
But engaging in long-term, serious relationships without the official marriage title might result in the same fate as those in the "gray divorce" group. This trend among baby boomers indicates that in an age where divorce is socially acceptable, most relationships have an expiration date. Millennials aren't better at relationships; they've just had less time to be in them.
https://www.businessinsider.com/how-millennials-gen-x-and-baby-boomers-approach-marriage-2017-11#millennials-have-complex-attitudes-about-gender-roles-marriage-and-family-9
Burning Questions: A Climate Change Series, Part 1
Ashley Hord | Courte
Q&A: How LeBron Could Change Public Education with I Promise School
Jonathon Musgrave/Th
This is part of the American story
tell me everything
Is it wrong to have children outside of marriage?
Do we really need a men's rights movement?
Should we get rid of marriage?
Did the baby boomers ruin America?
Should Millennials form their own political party?
Do you feel appreciated at work?
Should polygamy be legalized?
Would you trade vacation days to pay off your student loans?


"Peddle Boaters" - Valley Paddler"
Nashawannuck Pond provides countess hours of recreational fun in all seasons. Fishing, ice skating, sailing, kayaking and even green ice sculpturing keep Easthampton residents active year around. We are pleased to feature user generated content on this page and on our Facebook page, and are happy to give credit to videographers and photographers. To submit additional work, please contact a member of the steering committee.


text:
Category "Manufacturing News"
CSL's massive profit increase as vaccines, new products star
Features, Manufacturing NewsBy Peter Roberts February 18, 2021 Leave a comment
Australia's largest company by market capitalisation, biotechnology manufacturer CSL has powered ahead in the first half of the financial year reporting a net profit after tax up 44 per cent on the previous corresponding period. The company, which is in the final stages of manufacturing the AstraZeneca Covid-19 vaccine, reported a NPAT of $1.8 billion…
Manufacturing NewsBy Peter Roberts February 18, 2021 Leave a comment
Noxopharm's loss on R&D activities Clinical stage drug development company Noxopharm has reported a loss of $6.7 million before tax for the first half of the financial year. The company, which is commercialising its drug Veyonda (pictured) for the treatment of advanced prostate cancer, spent $3 million on R&D and received a R&D tax refund…
Goanna Ag and Myriota combine for farm sensor solutions
Farm sensor manufacturer Goanna Ag and IoT nanosatellite operator Myriota have combined to launch two innovative products to help farmers manage water resources, Go Rain and GoTank. GoRain allows farmers to monitor rainfall at remote locations, providing information on rainfall variability that enables data driven decision-making. GoTank updates livestock farmers via the industrial IoT on…
PPK Group looks to divest mining equipment business, focus on technology
Technology commercialisation business and mining equipment manufacturer PPK Group has suffered a drop in profit as the company continues to aggressively expand its portfolio of businesses. The company reported today earnings before interest tax amortisation and depreciation of $924,000 in the first half year, down from a profit of $3.1 million. Total revenues were hit…
Treasury Wine Estates profit slumps as it exits China
Treasury Wine Estates, the maker of Penfolds wines, today reported earnings before interest and taxes (EBITS) down 23 per cent in the first half year to $284 million as it was hit by China's punishing tariffs against Australian products. The slump, which was widely expected, was driven by the company's exit from the premium China…
Dimplex buys majority stake in Thermofilm
Manufacturing NewsBy Editor February 17, 2021 Leave a comment
Glen Dimplex Australia has acquired a majority stake in Dandenong South-based Thermofilm.
New Environex plastics recycling plant at Toowoomba will create 30 new jobs
Features, Manufacturing NewsBy Editor February 16, 2021 Leave a comment
Environex has been awarded a $1.3 million grant under the Queensland government's Resource Recovery Industry Development Program, which will contribute to a new recycling facility at Toowoomba's Wellcamp Airport.
Massive Covid boost for Ansell
Global latex and protective glove manufacturer Ansell has reported a massive sales boost from the Covid-19 pandemic. The Melbourne company reported sales for the first half year up 24.5 per cent to $1.2 billion (US$937.8 million) driven by rising healthcare and industrial sales. Healthcare sales were up 37 per cent on the previous corresponding period…
Lavo to make hydrogen electrolysers in Queensland
Low-price domestic hydrogen electrolyser and storage system manufacturer Lavo has committed to building a new factory in Queensland reported to cost $20 million. The company and its JV partner Nedstack said in a statement they would develop a fuel cell and LAVO unit (pictured) manufacturing facility in Springfield City, a private sector CBD development near…
Fleet Space to launch fifth satellite
IIot communications services provider Fleet Space Technologies will launch its fifth nanosatellite, Centauri 3 in March aboard Rocket Lab's They Go Up So Fast mission. The locally manufactured Commercial Nanosatellite will head for Low Earth Orbit (LEO) at 550 km from Launch Complex 1 on New Zealand's Māhia Peninsula. Centauri 3 is part of the…
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BRAD HORN/Nevada Appeal Joana Camargo, 13, of Yerington, samples salsas from Killer Salsa during the Salsa y Salsas cultural celebration at the Legislative Mall on Saturday.
Teaching tolerance and community spirit might not be the mission of a day spent on the green of the Legislative Mall – but the co-mingling of a diverse community during the Salsa y Salsas festival engendering a little more togetherness, may well be the result.
The 11th annual festival to benefit Nevada Hispanic Services, a referral agency for Hispanics in need of help here, turned out more than 1,000 attendees enjoying dollar tacos, native dance and booths featuring information from how to set up a checking account to how domestic violence victims can receive pro bono legal services.
"It's not that needs in the (Hispanic) community have grown more complex," said Luis Pierrott, who was on hand to spread word of an the upcoming Latino Community Day at Western Nevada College. "It's just there is a more sophisticated outlook.
If the notion of integration went largely unspoken by the elders attending the festival, it was illustrated with cross-cultural classmates playing on the grass between bands and dancing troupes.
Students made impromptu parental introductions while waiting for Yerington's Grupo Tualoc Aztec dance troupe to warm up on the makeshift courtyard stage.
The festival didn't center around food and dance only. Community service groups also drew attention.
Encuentro Matrimonial Mundial, a service group to help provide counseling to local Hispanic couples and families, passed out brochures on an upcoming weekend retreat for married couples.
Outreach was also the goal of the Domestic Violence Victims Assistance Project, said Valerie Clark, an attorney for the group.


Braxton Miller made a spectacular juggling catch for 45 yards in a 49-7 victory against Rutgers this year. In this photo the ball is headed towards his knees.
Braxton Miller secures the ball without it hitting the ground against Rutgers.
COLUMBUS – Braxton Miller's final season in an Ohio State uniform has been a year later and considerably quieter than anyone would have imagined after his first three seasons in scarlet and gray.
The original timeline most people pictured was that maybe he would lead OSU to a national championship last season after being stopped short in 2013 with the same sort of spectacular plays he had produced in his first three seasons.
In those three seasons, the Wayne High School graduate passed for 5,295 yards and 52 touchdowns and rushed for 3,054 yards and 32 touchdowns. No Ohio State quarterback has ever run for more yards.
But two shoulder surgeries in 2014 to repair a torn labrum changed all that. Now instead of leaving Ohio State as a star quarterback, he is going out as a No. 3 receiver, who occasionally lines up at quarterback to run the ball.
Miller has 22 catches for 321 yards and three touchdowns and has rushed for 218 yards this season.
In case anyone has forgotten how dazzling Miller was and how important he has been to Ohio State's success the last four seasons, OSU coach Urban Meyer has been reminding them all week.
"It's immense the feelings we all have for him because we all know how average we were, especially on offense our first year. If you don't have that kid, you start your coaching career here at Ohio State with a .500 or 8-5 or 8-4 record or whatever it is," Meyer said on Wednesday, referring to OSU's 12-0 record in his first season as coach in 2012.
"If you don't get momentum those other great things don't happen or certainly doesn't happen in that time frame. Braxton is the ultimate competitor. He just means a lot to us," he said.
Thursday on his weekly radio show, Meyer said, "I made the comment about Braxton that it is going to be a hard one when you look out there and don't get to coach him anymore. Braxton is really dear to my heart and our program because of his selflessness.
— His scrambling 40-yard pass to Devin Smith with 20 second to play to beat Wisconsin 33-29 in 2011.
— An 81-yard run through Indiana's defense to score in 2011 and a 67-yard touchdown run against the Hoosiers a year later.
— His 63-yard touchdown pass to Smith to give Ohio State the lead for good in a 17-16 win over Michigan State in 2012.
— A 1-yard touchdown run against Penn State when Miller started forward, stepped back to avoid a tackler, then dived into the end zone. He called this his favorite play earlier this season.
— And, of course, the spin move on a 53-yard touchdown run against Virginia Tech this season.


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Posted inNational
Ivermectin Targeted by Media…but Why?
Friday, October 15, 2021 Saturday, October 16, 2021
by Sharon Rondeau
(Oct. 15, 2021) — "Ivermectin" was trending in second place on Twitter early Friday afternoon just below an announcement of the murder of British parliament member David Amess.
When clicking the "Ivermectin" trend, the reader was brought to a column with an October 1, 2021 headline from Reuters stating, "Ivermectin is not authorized for treatment of COVID-19, according to the FDA."
Below that, whether or not from the article, were three points under the caption, "What you need to know," stated as:
— Ivermectin is an antiparasitic drug that is used to treat several neglected tropical diseases, according to the National Institutes of Health (NIH)
— A study by the US Centers for Disease Control and Prevention (CDC) found a 24-fold increase in the amount of ivermectin distributed from retail pharmacies as compared to the the pre-pandemic baseline
— The FDA warns that taking medication intended for animals can cause toxic reactions in humans because doses are highly concentrated
Just below that is a tweet from the FDA which arguably gives the impression that no "human" form of ivermectin exists. "Some people are taking ivermectin, a drug often prescribed for animals, to try to prevent or treat #COVID19," the tweet reads. "FDA has not approved or authorized ivermectin for this use, and it can be dangerous for people."
The tweet (above) links to an undated FDA-issued page titled, "Why You Should Not Use Ivermectin to Treat or Prevent COVID-19."
"The most effective ways to limit the spread of COVID-19 include getting a COVID-19 vaccine when it is available to you and following current CDC guidance," the article states, in part, in its conclusion.
On August 26, 2021, the CDC issued an orange-letter "Health Advisory" titled, "Rapid Increase in Ivermectin Prescriptions and Reports of Severe Illness Associated with Use of Products Containing Ivermectin to Prevent or Treat COVID-19."
However, according to the documentary, "The Story of Ivermectin and COVID-19," ivermectin was found to be highly effective in treating COVID-19 last year. In early December, New York-based Dr. Pierre Kory testified to the Senate Homeland Security Committee, then chaired by Sen. Ron Johnson (R-WI), about positive outcomes he and his colleagues witnessed with use of the drug in those infected with COVID-19.
Kory told the committee:
…Although we, like many, are extremely encouraged by the apparent successes in developing effective vaccines, we also are dismayed at the near complete absence of guidance and research on effective early, at-home, or preventative treatment options apart from vaccines, a reality we find unconscionable. Our hospitals are overflowing with over 100,000 COVID-19 patients admitted, and new record deaths are reported each passing day. It will take months for the vaccine to be distributed to the general public and further time to have sufficient impact in this crisis, so we are here to stress the need for effective early treatment. My organization of critical care specialists have spent the almost nine months tirelessly reviewing the scientific literature to gain insight into this virus and the disease process and to develop effective treatment protocols. All the while, we were working long hours in Intensive Care Units full of COVID patients. I was proud to testify in front of the committee about our MATH+ Hospital Treatment Protocol in May which I would like to mention has had nearly every single component of its combination therapies validated in clinical studies and our paper detailing and reporting on the impacts of the treatment protocol will be published within days in the Journal of Critical Care Medicine.
And so, it is with great pride as well as significant optimism, that I am here to report that our group, led by Professor Paul E. Marik, has developed a highly effective protocol for preventing and early treatment of COVID-19. In the last 3-4 months, emerging publications provide conclusive data on the profound efficacy of the anti-parasite, anti-viral drug, anti-inflammatory agent called ivermectin in all stages of the disease. Our protocol was created only recently, after we identified these data. Nearly all studies are demonstrating the therapeutic potency and safety of ivermectin in preventing transmission and progression of illness in nearly all who take the drug.
Before proceeding, I want to bring attention to two critical deficits in our national treatment response that has made this hearing necessary in the first place. Besides the early interest and research into hydroxychloroquine, we can find no other significant efforts to research the use of any other already existing, safe, low-cost therapeutic agents. Seemingly the only research and treatment focus that we have observed on a national scale is with novel or high-cost pharmaceutically engineered products such as remdesivir, monoclonal antibodies, tocilizumab, with all such therapies costing thousands of dollars. This is consistent with conclusions drawn by a physician consulting to Congress about Covid-19 when she concluded, "There is a pervasive problem on the Hill with how we prove the value of a low cost treatment." Another barrier has been the censorship of all of our attempts at disseminating critical scientific information on facebook and other social media with our pages repeatedly being blocked.
Finally, we believe the lack of clinical experts on the existing task forces is further hindering progress on identifying effective therapeutics. We can identify almost no members with any similarities to the skill set, clinical knowledge base, and patient care experience to our group of expert clinicians. Existing members all seem to be either physician leaders of large health care organizations or have research backgrounds. Although many must have had some bedside experience in the care of patients in their careers, there seem to be almost none that have been at the bedside of COVID-19 patients in any appreciable fashion during this pandemic. Expert clinician panels such as ours have large amounts of valuable insights and wisdom and we are extremely pleased to share our recent discovery of the immense potency of Ivermectin in COVID-19.
Kory's organization, the Front Line COVID-19 Critical Care Alliance (FLCCC), contains testimonies from citizens of their experiences with ivermectin against COVID-19; pointers on how to "overcome the pharmacy barrier" in obtaining prescriptions for the drug; and articles focusing on issues arising from the growing usage of ivermectin against the virus.
Ivermectin was approved for use by the FDA in 1996 as an anti-parasitic and is currently recommended by the CDC for treating scabies, river blindness and "intestinal strongyloidiasis."
The NIH contains a number of medical papers relating initial positive findings from ivermectin's use against COVID-19.
The video indicates that the drug Remdesivir, which the National Institutes of Health (NIH)'s National Institute of Allergy & Infectious Disease Director Dr. Anthony Fauci promoted last year for use in COVID-19, is manufactured by Gilead and costs more than $3,100 per treatment course. In contrast, ivermectin, which is no longer patented, costs very little.
As of April 2021, the (NIH) stated:
Remdesivir is an intravenous nucleotide prodrug of an adenosine analog. Remdesivir binds to the viral RNA-dependent RNA polymerase and inhibits viral replication through premature termination of RNA transcription. It has demonstrated in vitro activity against SARS-CoV-2.1 In a rhesus macaque model of SARS-CoV-2 infection, remdesivir treatment was initiated soon after inoculation; the remdesivir-treated animals had lower virus levels in the lungs and less lung damage than the control animals.2
Remdesivir is approved by the Food and Drug Administration (FDA) for the treatment of COVID-19 in hospitalized adult and pediatric patients (aged ≥12 years and weighing ≥40 kg). It is also available through an FDA Emergency Use Authorization (EUA) for the treatment of COVID-19 in hospitalized pediatric patients weighing 3.5 kg to <40 kg or aged <12 years and weighing ≥3.5 kg. Remdesivir should be administered in a hospital or a health care setting that can provide a similar level of care to an inpatient hospital.
Remdesivir has been studied in several clinical trials for the treatment of COVID-19. The recommendations from the COVID-19 Treatment Guidelines Panel (the Panel) are based on the results of these studies. See Table 2a for more information.
"The Story of Ivermectin and COVID-19" illustrates Gilead's contributions to the NIH and that several NIH doctors took part in approving Remdesivir for the virus.
Anecdotal evidence from hospital healthcare workers suggests Remdesivir given during the late stages of COVID-19 can be harmful or even fatal.
As for ivermectin and COVID-19, the NIH reports, in part:
Reports from in vitro studies suggest that ivermectin acts by inhibiting the host importin alpha/beta-1 nuclear transport proteins, which are part of a key intracellular transport process that viruses hijack to enhance infection by suppressing the host's antiviral response.4,5 In addition, ivermectin docking may interfere with the attachment of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) spike protein to the human cell membrane.6 Ivermectin is thought to be a host-directed agent, which may be the basis for its broad-spectrum activity in vitro against the viruses that cause dengue, Zika, HIV, and yellow fever.4,7-9 Despite this in vitro activity, no clinical trials have reported a clinical benefit for ivermectin in patients with these viruses. Some studies of ivermectin have also reported potential anti-inflammatory properties, which have been postulated to be beneficial in people with COVID-19.10-12
Some observational cohorts and clinical trials have evaluated the use of ivermectin for the prevention and treatment of COVID-19. Data from some of these studies can be found in Table 2c.
The FLCCC cites a number of papers written by medical cohorts on the benefits of ivermectin against the coronavirus.
The NIH's National Library of Medicine contains a number of papers and citations with reference to ivermectin and its use against COVID-19. One such paper published at clinicaltrials.gov written by researchers at the Afyonkarahisar Health Sciences University in Turkey relates a trial involving "Patients who were hospitalised with a pre-diagnosis of severe COVID-19 pneumonia and thereafter diagnosis of COVID-19 was also confirmed microbiologically with polymerase chain reaction (PCR) positivity in respiratory tract samples were included into the study. They were randomized to the study and control group, respectively. Single numbered patients were accepted as study group and double numbered patients as control group."
According to the authors, "Hydroxychloroquine, favipiravir and azithromycin (HFA) standard treatment protocol were given to the control group as recommended in the 'COVID-19 (SARS-CoV-2 Infection) Guide' prepared by the Republic of Turkey Ministry of Health."
Former President Donald J. Trump and physicians in the U.S. have been marginalized, censored and deplatformed by "Big Tech" and the mainstream media for promoting such therapeutics as hydroxychloroquine for coronavirus. However, Trump's "Operation Warp Speed" was intended to produce not only vaccines in a shortened time frame, but also "diagnostics and therapeutics." In August last year, House Minority Leader Kevin McCarthy said on FNC's "Fox & Friends" that approximately 600 therapeutics for the disease were under development.
Today, some physicians are condemning what they see as a "therapeutic nihilism," or the absence of outpatient treatment for coronavirus patients. Those doctors are now "hunted" and punished by the medical establishment, Dr. Peter A. McCullough has said on numerous occasions.
In an August 25 opinion column at The Deseret News by Dr. Justus R. Hope, the author describes the 1990s "gaslighting" by tobacco companies as to the addictive qualities of nicotine. "Today, we know they lied to us because it was in their financial interests. They conducted a classic gaslight. Gaslighting is a form of deceit used by habitual or pathological liars. They will often lie to your face and never back down even when confronted with proof of their deception…Americans know that cigarettes cause cancer, heart disease, stroke, emphysema, and COPD. In addition, they cause seven out of every ten cases of lung cancer. And cigarette smoking is highly addictive."
Further, Hope states:
But currently, we are witnessing an even greater gaslight than from the cigarette executives. We see in 2021 gaslighting about vaccines, Ivermectin, and public health. We are lied to daily by corporate interests who seek to protect a much greater industry than tobacco.
Unlike tobacco, which was plastered all over the television with Joe Camel and the Marlboro Man, today we are bombarded with video clips of overflowing hospitals and dying patients. Rather than tobacco's message of glamor, we relentlessly hear Big Pharma's marketing message of doom. First, this was rehearsed with Ebola, later with Zika, and now Big Pharma has perfected it with COVID-19.
The public may be aware that the Pharmaceutical companies profit handsomely from new drugs and that there is a very cozy and cooperative relationship between FDA, NIH, and Big Pharma. Still, almost no one realizes they are gaslit daily with a coordinated marketing campaign.
At 2:02 p.m. EDT, "Ivermectin" was no longer trending on Twitter; however, as of 2:58 p.m., the topic briefly returned in third place.
Update, 9:49 p.m. EDT: Approximately eight minutes ago, Kory tweeted, "PLEASE realize there is literally a global war on Ivermectin. If it didn't work, you wouldn't see this level of heinous ferocity trying to kill it. I worry so few realize what has happened to US society. They are literally trying to kill us for $".
Update, 3:08 p.m., October 16, 2021: A compilation of the results of 63 studies of ivermectin is here.
Tagged: CDC, COVID-19, Dr. Anthony Fauci, FLCCC, ivermectin, National Institutes of Health, Operation Warp Speed, remdesivir, Twitter
Dennis Becker says:
Saturday, October 16, 2021 at 10:41 PM
This might be why:
https://onepagericu.com/blog/debunking-ivermectin-a-complete-guide
Sharon Rondeau says:Article author
Sunday, October 17, 2021 at 8:37 AM
There are so many conflicting reports: https://www.msn.com/en-in/news/other/uttar-pradesh-government-says-early-use-of-ivermectin-helped-to-keep-positivity-deaths-low/ar-BB1gDp5U
https://covid19criticalcare.com/testimonials/
But is it not better for a physician to try something rather than nothing?
Levi Morton says:
Sunday, October 17, 2021 at 7:19 PM
No; if ivermectin provides no benefit and is harmful, then doing nothing is better.
James Carter says:
Monday, October 18, 2021 at 9:35 AM
Yes; since 63 world-wide studies indicate ivermectin provides substantial benefit and is relatively unharmful, then trying something is indeed better than doing nothing except watching people die.
Monday, October 18, 2021 at 1:01 PM
"Relatively unharmful" still violates "First, do no harm."
Regardless, an anonymous website making claims about studies isn't proof of effectiveness.
The overwhelming number of comments below that anti-ivermectin article take issue with and/or debunk it.
Tom Polhaus says:
There are only 20 comments. Some take issue (the author response to several of them) and some don't.
Not particularly overwhelming.
Nikita's_UN_Shoe says:
Saturday, October 16, 2021 at 3:32 PM
Golden Rule of the medical establishment:
Thou shall not prescribe any effective medical treatment for any medical condition that does not result in a substantial financial gain at the U.S. and foreign stock markets.
Byron Bush says:
"The powers that be…" have declared war against anyone… or anything… that does not parrot their agenda. Truth is not a narrative… and the positive results of Ivermectin and other economical drugs speak for themselves. Keep speaking the TRUTH.
KJ Alexander says:
Saturday, October 16, 2021 at 10:20 AM
The propaganda is manipulated daily to fit the agenda of those who profit and those who want to destroy. How does one say the next virus will be far worse than this one unless of course, they are producing it? Or let's keep the fear and chaos alive and well and it will get more people to be given vaccines. And to think taxpayer dollars help fund some of this.


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In five years people might accuse me of contributing to a bubble related to 3D Printing (Swedish 3D printer manufacturer Arcam's stock is already trading at P/E 63, whereas the American firm 3D Systems' P/E is above 100…). Having spent my first week at the University of Cambridge and the Institute for Manufacturing (IfM) researching the topic, it is difficult not to be excited about its potential. While I will return to this subject with solid data and hopefully a more complete picture, let me share some initial thoughts below.
Google Trends measures the amount of attention a particular topic receives. As indicated in the graph above, interest in the term '3D Printing' has increased dramatically over the past 3 years.
The term 3D Printing is somewhat perplexing as our understanding of printing has been associated with creating a pattern on a sheet of paper. Additive manufacturing is frequently used as a synonym and it is perhaps easier to make sense of the technology by using this notion instead.
Additive manufacturing can be thought of as the opposite of subtractive manufacturing. So what is subtractive manufacturing then? Historically, manufacturing has largely been based upon a set of inputs which undergo a process of reduction – material is removed from the inputs by casting, machining and milling (etc) them into a desired shape.
Additive manufacturing, on the other hand, is not based upon the removal of material. Instead, a three dimensional CAD file (a digital drawing of a component) is used as instruction telling a 3D printer how to manufacture an object. Using a mix of inputs, the printer then builds the object gradually, layer by layer until the it is ready. Material is not removed – material is added. Anyone with access to a CAD file and the right 3D printer can therefore manufacture something.
Before 3D Printing, we never used the term subtractive manufacturing, it wasn't needed as all manufacturing was subtractive. Well it isn't any longer.
Why then is this such a radical innovation and what are the potential implications?
1. The end of economies of scale?
Putting those two curves in the same graph gives us an idea about how they are different.
2. Variation is no longer expensive!
The industrial revolution was highly related to economies of scale. By building huge factories and introducing process innovations such as the assembly line, the cost per unit decreased. What used to be affordable only for wealthy people quickly became within reach for everybody and we saw the emergence of a consumption society of abundance. Cars, food and clothes became available in ever larger quantities and lower prices. Up until the 1970s, this was largely the growth miracle of industrial capitalism. The graph below serves as a neat illustration.
The introduction of industrialized agriculture created huge economies of scale and prices declined sharply.
With economies of scale, however, came a seemingly inevitable tradeoff in all manufacturing between variation and cost. If you wanted to increase variation, output would also by necessity be more expensive as less scale economies would be obtained. Henry Ford summarized this dilemma eloquently when he stated that Ford's customers could have any color they wished as long as the car was black.
Over the years, extensive work has been done in operations management to cope with this tradeoff and doing so has proven to be a key source of competitive advantage in many industries. Working with platforms and modularization, firms such as Volkswagen and the Swedish truck manufacturer Scania have been able to offer a higher degree of variation while still reaping the cost benefits related to economies of scale. Other firms like GM have struggled to position themselves in this tradeoff. As the company sought to lower its cost, the car portfolio became increasingly streamlined, implying that variation declined and that brands lost their unique identities.
While huge improvements have been made since Henry Ford's days, the tradeoff between variation and cost remains largely intact – essentially we've only learnt to cope with it in better ways, but it is in many regards inherent in industrial capitalism itself (figure below).
If we put the two curves above in the same graph, we get a better idea about how 3D printing is currently emerging.
3D printing is being rapidly adopted in those applications where the need for variation is very high, but has been too costly to accomplish through subtractive manufacturing techniques. More than 10 million hearing aids have already been manufactured using 3D printers. Dental and medical implants are other early adopters of the technology as personalization can be accomplished at a low cost.
The argument outlined above is perhaps the main reason why I believe 3D printing is a truly radical innovation. Fundamentally, the technology is a serious attempt to overcome one of the greatest tradeoffs in industrial capitalism. EOS, a German manufacturer states on their website that economies of scale are fading. I put this statement next to a 20£ bill below which features Adam Smith, arguably the founder of modern economics who among other things pointed at economies of scale as a distinct characteristic of the industrial revolution. The point is that 3D printing has the potential to distort a key feature of industrialized economies.
As 3D printing is additive, it becomes much easier and cheaper to manufacture things which are geometrically complex. Subtractive manufacturing is inherently clumsy and some shapes are more or less impossible (or very expensive) to manufacture through techniques of milling, machining or casting. Again, there is no such tradeoff with 3D printing.
Consequently, 3D printing is being adopted rapidly in applications where geometrically complex components are needed. The aerospace industry would be one example of this. Reduced weight creates huge savings in fuel costs over the lifetime of an aircraft, by using 3D printing it becomes possible to create components that are lighter yet solve the same problem. According to Terry Wohlers, an expert on the 3D printing industry, Boeing is currently using 3D printed parts on some of their military aircrafts.
4. Product development becomes continuous and distributed?
Rapid prototyping was one of the earliest applications of 3D printing technology. In development work, engineers could quickly and iteratively obtain physical versions of their ideas at little cost. As the performance of 3D printers has improved, they are increasingly being used not only for prototyping but for real products. Since manufacturing is no longer a difficulty and since it can be done by anyone with access to a 3D printer, development work becomes separated from manufacturing. In most industrial corporations, there's a clash between R&D and operations. Manufacturing has historically imposed constraints on development work, with 3D printing this is no longer the case in certain applications.
One consequence of the above is that product development can become continuous and distributed.
Traditional R&D work comes in a batch. An idea moves from concept phases through product development and is then scaled up, manufactured and launched. The project had a starting time and an end. Eventually the product will become obsolete in the marketplace and the firm needs to have a new batch of development projects resulting in new, better generations of products. Basically, product lifecycles come and go.
With 3D printing, development takes place in the non-physical domain (in CAD software) and manufacturing is not a challenge. Hence, development work doesn't have to be organized as a batch, it has the potential of becoming a continuous process. That is, you don't develop a product and sell one million units of it, then launch a new product. Instead, you develop one product, someone buys it, the same CAD file is improved, another person buys two units, then the next person buys a couple of units which are different from the first version. As the ancient Greek philosopher Heraclitus put it: "no man ever steps in the same river twice, for it's not the same river and he's not the same man".
Secondly, assuming production is the same and that it requires nothing but access to a 3D printer, development work can become distributed in space. I can work on the design of a product, send it over to someone on the other side of the planet who then develops it further, just like a word document. Manufacturing is not a concern as we have access to the same 3D printing technology, which is not constrained by economies of scale or geometrical complexity.
Continuous and Distributed. That's Wikipedia compared to Britannica. In the printed era, encyclopedias came as a batch (an edition) and they were developed by one organization. Being only one organization imposed constraints on the quality of the content and improvements couldn't be made until the next edition (batch) would be released many years later. For these reasons, Wikipedia was bound to outperform printed encyclopedias over time – one organization could not compete with the accumulated knowledge of millions of individuals and it couldn't match the continuous updating Wikipedia offered. Will the same thing happen with product development work thanks to 3D printing?
1. Which manufacturers will end up like Britannica with obsolete products due to the weaknesses of having a centralized, batch style development process?
2. How can companies successfully leverage online communities to make improvements to their products?
3. If manufacturing becomes democratized and available to anyone and it becomes easy to access CAD files online, how do firms ensure any returns from their development efforts?
5. Time and space no longer an issue?
Subtractive manufacturing requires a fair bit of skills in logistics and supply chain management. Industrial firms have developed very precise channels making sure the right inputs are available at the right place, at the right time. Managing inventories, lead times, process times etc is by no means a trivial task.
With 3D printing, it becomes a lot easier to cope with time and space. Files can be sent electronically to one place where an object is printed when it is needed, the only required logistics would be related to the input material and the printer itself in the first place. The right thing, at the right time, without all the complexities related to transportation of components and inputs back and forth across the planet.
When it comes to literature, the only two actors needed are the author and the reader. All other actors (publishers, distributors, bookstores, websites) are intermediaries and several of them are being gradually removed as fewer are required when physical distribution is not necessary.
What will happen to logistics companies and those firms obtaining a competitive advantage from superior supply chain management skills in those applications where 3D Printing prospers?
A sixth advantage with additive manufacturing is related to waste. Subtractive manufacturing removes material, which often becomes useless. This is not only costly for a firm, it is ultimately also a drag on the environment. With additive manufacturing, you only use the resources needed to create a particular product. 3D printing will therefore be competitive in those applications where waste from subtractive manufacturing is highly expensive.
• Affordability: Prices for commercial 3D printers range from $15,000 to more than $300,000.
• Available materials: The amount of materials that can be used is still a limiting factor. Plastics, ceramics and certain metals are among the materials that can be used at present. Also, material costs are quite high and still impose constraints.
• Precision: the robot arm of a 3D printer is still not as precise as an industrial engineering process, especially when it comes to adding several thin layers of different materials.
• Strength: especially in load-bearing applications, strength is still an issue.
• Time: Especially for larger objects, it might take a lot of time to print something.
As we're dealing with three dimensions, all input costs increase exponentially by the power of three when you create a larger object. If you have a 3*3*3cm toy it requires 27 cm^3 of material and time. An object twice as large needs 6*6*6 = 216 cm^3 of material and time. 216/27 = 8, if you double the size, cost increases by a factor 8! And if you triple the size it will increase by a factor 27. This is an inherent weakness in the technology and it helps us explain why it has initially emerged in rather small applications where the buyer is not very price sensitive, and where geometrical complexity and/or customization is important. Hearing aids are indeed good illustrations of this point.
Having mentioned some of the limitations, the crucial question is if those technological obstacles will be overcome and when this would happen. At present, it seems that the technology is improving rapidly. Some of the leading manufacturers are gaining momentum, leaving the red numbers behind. As these companies grow they will invest progressively more in R&D while also attracting new firms to the industry, both factors contributing to increased performance and more widespread adoption.
Improvements have indeed been very rapid in recent years. Nowadays the cheapest personal 3D printer sells for $500, five years ago you couldn't find any 3D printers below $15 000. We can also expect the amount of available materials to increase in the coming years. The market is expected to expand from $1.4 billion in 2011 to about $3 billion in 2016.
Bearing current limitations in mind, 3D printing will probably continue to grow in its current application areas: prototyping, and manufacturing (primarily in aerospace, automotive and medical industries). Whether it will remain in these areas or have a more widespread impact across the economy ultimately depends on the pace of development and the limitations of the technology.
In the short term, we tend to overestimate the potential of a technology. Its long term impact, on the other hand, we often underestimate.
Digital photography was hyped already back in the early 1980s. Nothing happened over the coming twenty years. But once the shift ook place it exceeded our imagination, for all actors involved: film manufacturers went bust, and so did many camera manufacturers, the industry became highly competitive, photography was amateurized putting an entire profession under pressure, camera stores and photographic stores went out of business and images are now captured and shared instantly in ways that would have been hard to even conceive of 20 years ago.
Short term hype, long term impact beyond our expectations – this is usually the pattern for new technology. Why would 3D printing be an exception? It has so many new and radically different properties that may have far reaching consequences, now the questions are how and when.


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Addressing the interplay of identity and community from diverse disciplinary perspectives, this research explores the interface between individual and society, and how sense of belonging is accumulated through culture, media, language, and history, and affected by the changing nature of communities and nations, including structures of governance, finance and administrative justice. We address questions like: what is the experience of using Welsh in daily life and how can we support it with technologies for digital language equality? How can the public best hold secretive state and commercial surveillant entities publicly accountable? How does literary and creative output enhance our sense of place as well as promote it to the world?
Research Our Impact
From the food we eat, the environment we live in, our health and wellbeing to improving the economy and the law and of the land, Bangor's research transforms the lives of millions of people around the world.
Take a look at our Research Strategy to find out how we're consolidating our position amongst the top 40 UK universities for Research Quality.
Research Excellence Framework (REF)
The Research Excellence Framework (REF) is the current system for assessing the quality of research in UK higher education institutions.
Find out about our current and past research activities.
Bangor University has achieved the Athena SWAN Bronze Award which identifies our commitment to gender equality and diversity.
Innovation at Bangor
Providing environments, events and activities to support innovation.
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