Columbia
geometry
trigonometry
geometric transformation
dilation
calculus
reflection

It depends on what college you attend.  Some schools like Columbia and Chicago have rigorous core curriculums while others like Hopkins and Brown have none at all.

Maybe subject irrelevant to the major are crammed into college curricula because colleges do not believe that they were learned to any depth in high school. Maybe it is necessary to do that in college because K-12 schools cannot be trusted to give students a well-rounded education. Maybe requiring freshmen to take a broad range of courses keeps a number of professors employed who otherwise would not have a job. It seems to be the usual situation in the USA, but it does not need to be so. Since we have always done it that way, we will probably continue to do it that way. Because of our insular mentality, we do not care to find out if it is done differently anywhere else. It probably is not done the same way in many other countries.

Long time ago, in a planet far, far away, I finished 10th grade with a lot of education under my belt, including geometry, algebra, trigonometry, physics, chemistry (general, inorganic, and organic), biology, music, art, Spanish, English, French, world history, world literature, philosophy, civics, geography, space science. At that point I was asked to choose a major and I chose one of the sciences, resultinh in my being required to take a couple of more years of world literature (why?), philosphy (under which they included psychology and logic too), chemistry, physics, biology, and math (including analytical geometry and calculus). After that I was free to start college as a science major and was never again required to study any humanities again. Do I remember Napoleon, Descartes, Sartre, Baudelaire, Smetana? Yes, if dimly. Do I think I would remember any more of Dante's description of the circles of hell if I had had to study it again in college? Not really. Getting rid of the humanities gave me a lot more room in my college schedule for math, physics and chemistry courses.

In my opinion, I think it is great that people who major in sciences have to take world literature and philosophy. Science may be your future profession, but in a sense, world literature and philosophy are life. They raise thought provoking questions about the nature of life and the universe. They raise complicated ethical and moral questions (and the really good literature leaves it to you to ponder and doesn't preach a certain set of morals).

I happen to get rather annoyed at my fellow grad students when I try to talk about literature or make a literature reference that no one gets. Being well-versed in the arts makes communication easier. That's just my opinion, though.

The way my undergrad worked was for there to be certain "categories" and then we could pick electives that fit those categories. That way they didn't force any "core" courses we didn't want to do. We got a broader range of classes that way and we could pick what seemed interesting or useful to us at the same time.

My undergraduate school works similar to what HilbertThm90 described in that there are 5 or 6 categories and we have to take a certain number of classes in each category. In general, I think this is a bad idea. I think college students should be mature and responsible enough to select the courses that will prepare them most for the challenges and experiences that they will face. Forcing college students to take a pre-decided number of course in various areas outside of their concentration just treats them like naïve children who need administrators to tell them what they should learn.

I have no idea what the college administrators are thinking when they sit down and decide that every student who graduates from here should take this many history courses, this many math courses, this many science courses, this many foreign language courses, etc as if anyone who takes less than this will have some sort of a gap in their education. I think many of the administrators are using this archaic notion of a “Renaissance man” who has a complete and well-rounded education as a standard in the guidelines they set for current university students. Maybe if I lived in 16th century academia in Italy, this standard would be okay. But here in 21st century, there is a completely different set of skills needed to succeed.

With that said, I do think being able to discuss a variety ideas from different academic disciplines and being reasonably fluent at this is good. But, I think the best place to achieve this is outside of the classroom. Many colleges are moving to a residential college system where students, at least for their first one or two years, live together with a variety of faculty members and interact with each other at meals, social events, guest lectures, etc. This fosters intelligent discussion and learning in many different areas. I think this is a much more effective way to make students well-rounded than to force them to sit through many hours of listening to a professor go into incredible depth on a single topic.

Four years of college is certainly more than enough to allow a student to specialize and so there is some room to accommodate these extra liberal arts classes. But for students who are very focused and want to graduate in two or three years, I think it is quite unreasonable to tack on an extra semester or an extra year to their college education, possibly costing them tens of thousands of dollars, just to try to make them "well-rounded".

So, I guess for students who attend college for all four years, it might make sense for them to take at least some courses outside of their concentration. I think college students should basically know for example what an English course or a math course is like, just so that it is not a mystery to them. But in general, I think that universities are way too draconian about this. I doubt that graduation requirements are the best way to achieve this. I think it would be enough to give students a general recommendation about what classes to take and let them read it and discuss it with their advisers and decide whether some of the recommendations don’t apply to them for special reasons.

I've often thought about this. I've sat in English class learning compound-complex sentences and wondering WHY I MUST DO THIS! Although I'm not sure if it is true, I have heard of some places testing children from early ages and determining what they would be most successful at, and placing them in a program to help them do that. I think that could work in some cases, although  some students may have a change of heart on what they desire to become.

My sister attends a university known for it's science and medical programs (she's planning to be a doctor). The university offers an Honors program for advanced students in which they learn all of their "core" classes. This only takes up one day a week of their schedules, and the rest of the week can be spent with classes for their major. My sister finds the Honors program to be a great thing. She roomed in her freshman year with a non-honor student, who found it excrutiating to study for finals in subjects she would never use.

My guess is that the administrators of schools want students to be "well-rounded".

How do you feel when so much of the world has a "Math? Who needs that?" reaction?

If my university did not have a requirement that every student take at least some mathematics, then my department would be a whole lot smaller - so much smaller that I might never have been hired in the first place.

No one's given a good reason why students should have to be "well-rounded" and meet distribution requirements and whatnot.

[quote="HilbertThm90"]In my opinion, I think it is great that people who major in sciences have to take world literature and philosophy. Science may be your future profession, but in a sense, world literature and philosophy are life. They raise thought provoking questions about the nature of life and the universe. They raise complicated ethical and moral questions (and the really good literature leaves it to you to ponder and doesn't preach a certain set of morals).[/quote]

Yes, almost all subjects are interesting (to me at least; that's why I'm not good at anything). And it is wonderful that science majors have the opportunity to study a variety of subjects in the humanities. But to be forced against their will to? That's a waste of time - not good. They'll probably forget everything they learn if they learn it to meet some requirement...

Some argue that student's eyes need to be opened, that subjects really start to get fascinating when one delves deeper. But if a student shows no interest in a subject by the time they get to [i]college[/i], I don't think it's worth it to keep pushing them. Students should have the choice to specialize much earlier.

[quote="Kent Merryfield"]How do you feel when so much of the world has a "Math? Who needs that?" reaction? [/quote]

I feel sorry for people like that; they're missing out on something awesome. But I don't feel that they should be forced to learn math.

[quote]If my university did not have a requirement that every student take at least some mathematics, then my department would be a whole lot smaller - so much smaller that I might never have been hired in the first place.[/quote]

I don't want you to lose your job or anything, but that isn't a reason to require students to take mathematics courses.

I'm sure you would agree that trying to teach math to people who simply aren't interested is not nearly as satisfying as teaching it to those who [i]chose[/i] the course.

[quote="Kent Merryfield"]How do you feel when so much of the world has a "Math? Who needs that?" reaction?[/quote]

Well, however much I think they're wrong, people have a right to believe what they think is valuable. I see math as something essential to everyday life (especially living in a NASA and extremely-high-on-engineers town), who else would program our microwaves and high-def TV's, and what else would I do on Tuesdays after school? While others see it as something petty, I don't. You have to respect others opinions.

[quote]If my university did not have a requirement that every student take at least some mathematics, then my department would be a whole lot smaller - so much smaller that I might never have been hired in the first place.[/quote]

And you wouldn't be able to find a job anywhere else? Like, leoxlin, I don't want you to lose your job, but....

[quote="leoxnlin"]I feel sorry for people like that; they're missing out on something awesome. But I don't feel that they should be forced to learn math. [/quote]
I'm not a big advocate of the whole "core" thing and if my university had a core, I wouldn't be attending.  Sure people should be exposed to new things but if they don't like it, don't push them-as it's just a waste of the student and instructor's time.

I completely disagree with the quoted statement.  Math, along with the ability to read and write, is essentially for functioning in the world.  The world is driven by math in some form, whether it be the stock market or prices at the supermarket.  You need to have some level of skill at math or you won't survive.  I'm not saying that a history major should be exposed to real analysis, but at least calculus.

I agree that people will probably forget the "details," but I think the main concepts will stay. I also don't think very many schools force students to take extremely specific classes with no choice. There is usually a choice, and the choices should be broad enough that you can find one that will be useful to you.

This is especially true of say "English" or "writing" classes. Can you think of a single profession that requires a college education, but doesn't involve a great deal of writing? Learning to be clear communicators is arguably part of what the university has to teach students. People will stop hiring people from your university if it is clear that students consistently graduate and don't acquire this skill, which in turn will lead to less enrollment.

The same is true of many other subjects. It isn't necessarily to be "well-rounded." You just never know when you might need to know about those things. If the time comes and you need to and don't, then that will look bad for your university. We then are back to the lower enrollment if people are walking around saying they graduated from University X, but are unable to do basic things outside of their major (technology breaks and you can't do some math by hand, you write a business letter full of errors, you respond poorly in an ethical dilemma, etc).

Just because you don't like or enjoy something doesn't mean you will never have to do it. I don't like combinatorics (some may say that I hate it, but my harsh feelings have weakened in the past year), but I still learned the basics since I have no idea if it will ever come up in what I do. It would probably look bad on my university if a basic combinatorial problem came up, and I said, oh, this isn't my area I don't know how to do it. That excuse only goes so far.

(Note that two posts went up in the time it took me to write this).

[quote="HilbertThm90"]I also don't think very many schools force students to take extremely specific classes with no choice. There is usually a choice, and the choices should be broad enough that you can find one that will be useful to you.[/quote]
I think for the most part this is true, but the notable exception is Columbia, which does require that all students take the same classes as part of the core.

[quote="math_Llama"]I've taken core classes such as History, English, Math, and Foreign language in school because it was mandatory. It's also occured to me many of these core classes are mandatory in college as well.

Out of pure curiosity, why is it necessary for everyone to take these classes?
I suppose I take English because it is important to learn to communicate with others, and I take math cause it's fun, but it seems so weird to me that future writers, history professors, lawyers, etc. has to learn DeMoivre's theorem (this is included in Alg 2 which is mandatory to graduate from highschool), and that future doctors and engineers have to learn what year Jackson was president and the major philosophers of the traditionalist and revisionist views in America.

I realize that the world is rapidly changing and that lawyers and customer service secretaries may indeed need to learn math etc far beyond what might be necessary today. But if schools were trying to teach us every possible thing that we might ever need in our career and lifetime, all mathletes might as well learn calligraphy as well, because it might help to have nice handwriting when they write a letter to trying to impress their boss, business partner, etc.[/quote]

lucky...the only thing our school does with complex #s is like $ i^ 2\equal{}\minus{}1$, and addition, subtraction, and stuff like that.

Some people learn what they need without any teacher. Others sit through classes and in the long run end up being incompetent anyway. Basic math, science, reading and writing should have been learned long before high school graduation. In an ideal world, a certain competency should be demanded from the earliest grades; those who do not develop it should be told to try again. Knowledge should be retained for the long term rather than learned for a unit test and then forgotten.
A high school diploma should mean that a "well-rounded education" has been already attained. Colleges could assign core classes based on results of placement tests, or they could allow students to "challenge" core courses and test out of them. That would convince me that they care about the educational attainment of the students more than they care for the tuition dollars and the continued employment of old tenured professors. In its turn, a college degree should guarantee competence in the major listed. Reality seems to be otherwise. Maybe it is because grade points are given piecemeal for sundry assignments rather than on the basis of cumulative exams. Maybe too many grade points are awarded for effort and participation. Maybe grading is too often done "on a curve." Maybe "learners" are not motivated to learn for the long term.[quote="HilbertThm90"]I agree that people will probably forget the "details," but I think the main concepts will stay.[/quote]I am not so sure. I work with people who seem to have forgotten most of what they should have learned in their major. For example, a PhD pharmacologist and pharmacist, with degrees from prestigious US universities and teaching pharmacology and toxicology in a US medical school could not tell apart (the names or chemical formulas of) sodium cyanate and sodium cyanide; confused gamma rays with beta radiation; needed help spelling simple words or interpreting simple chemical formulas. A pharmacist working on a doctoral research project argued about how to make a dilution with an undergraduate student. The undergraduate was right. I could list examples for hours.
[quote="HilbertThm90"]This is especially true of say "English" or "writing" classes. Can you think of a single profession that requires a college education, but doesn't involve a great deal of writing? Learning to be clear communicators is arguably part of what the university has to teach students. People will stop hiring people from your university if it is clear that students consistently graduate and don't acquire this skill, which in turn will lead to less enrollment.[/quote]Regarding writing and communicating, I have worked with US-born and educated people with advance degrees who cannot spell, but they rely on their administrative assistants who are very good writers. I know that some people do not learn to write in high school, but many of those do not learn in college either. I was once shown samples of work from advanced undergraduate students in a creative writing class. Not only were the plots and timelines full of holes, but also there were atrocious "spellchecked" misspellings in many of the stories. One writer even used "I" and "eye" interchangeably.[quote="HilbertThm90"]The same is true of many other subjects. It isn't necessarily to be "well-rounded." You just never know when you might need to know about those things. If the time comes and you need to and don't, then that will look bad for your university. We then are back to the lower enrollment if people are walking around saying they graduated from University X, but are unable to do basic things outside of their major ...[/quote]Again with the enrollment! If I were to enroll in a college, I would look at the professors, their work, their attitudes, and references from former students. A school may have fame for something unrelated, and be really lousy for the major you have in mind. If people choose to pay big money for big college names, I say let them. Sadly, you cannot judge the quality of the education by the reputation of the school that issued the diploma. After working with some utterly incompetent graduates from big name universities, and seeing other people do excellent work out of their field or with a degree from a little known small college, or without any degree at all, I do not judge people based on the prestige of their alma mater. I listen to what they (and their references) can tell me.[quote="HilbertThm90"]You just never know when you might need to know about those things.  ... if people are walking around saying they graduated from University X, but are unable to do basic things outside of their major (technology breaks and you can't do some math by hand, you write a business letter full of errors, you respond poorly in an ethical dilemma, etc).[/quote]So, does anyone know what degree Bernard Madoff had and from what university? I wonder if he took an Ethics course in college.

[b]Cores in College:[/b]

"Cores" can exist in colleges, programs within colleges (ie, liberal arts, engineering, etc.) and within a major. A core is just a set of classes that you have to take. There is nothing wrong with that in principle -- in order to get a particular degree, there are certain courses you must have taken. Similarly, in order to graduate with a certain major, the major requirements must be met.

What you probably are objecting to are the liberal arts distribution requirements that exist in almost every college and university. The strictness of the requirements depend on the institution and the program within the institution. Though, keep in mind they're there for a good reason. See the below section on high schools for a list of what I think are some of the important skills developed through humanities courses.

[b]Cores in High School:[/b] [i]The below is largely adapted from a post I made in another Round Table thread.[/i]

I think the following skills should be...

[b]Mandatory:[/b]

1. Literacy and communication - [i]English, foreign language, general learning environment[/i]
2. Basic interpersonal skills - [i]General learning environment[/i]
3. A (very) basic outline of US history and government, which is necessary to maintain a minimum level of voter competency - [i]History[/i]
4. Knowledge of basic quantitative tools and scientific concepts that will streamline and improve performance in nearly every profession and in daily tasks - [i]Math, science[/i]
5. Awareness of basic principles of health, such as the hazards of smoking and drinking, and how not to get a venereal disease

[b]Encouraged:[/b]

1. An experience in the arts, because the arts provide a medium of communication and propagate culture
2. Leadership through student-led initiative opportunities, such as student government, student clubs and organizations
3. Understanding and interest in mathematics and science to drive academic and technological innovation
4. Athletics, because they promote good health and healthy competition
5. Community service, because if taken seriously, it is an eyeopening experience that exposes students to other environments, and also because it promotes good character

While I don't think someone should be forced to know DeMoivre's Theorem to graduate from high school, I certainly think that four years of English, a few years of history (at least one of which is American), some foreign language, and some art are [b]very [/b]reasonable as graduation requirements. There is [i]no[/i] reason why anyone should be certified as a high school graduate without this basic, important coursework.

@TZF, I strongly agree with the things you identified as important in a high school education and I think you outlined those things quite well (that could easily be the mission statement for a real high school). However, I think you should differentiate your reasons for high school and college.

I think having a comprehensive set of liberal arts requirements for college makes college too similar to high school. I think the education system in the United Stated is in general quite slow in moving students up through the system toward a position where they can solve real-world problems and contribute to some profession or some field of research. Obviously, we live in a very technologically and intellectually advanced era where there is a lot of preparation necessary to achieve mastery of some sort of field. But still, I don't think college should just basically be a repetition of high school with more just more advanced classes.

I think the reasons for going to college should be fundamentally different than the reasons for going to high school and I think the college curriculum should be structured accordingly. Let me try to make an analog of TZF's list for college. This is only a partial list:

[b]Mandatory (for college):[/b]

1) Develop communication skills, both written and verbal, to the point where you feel comfortable writing or talking at length about subject matter at a high but non-technical level.

2) Concentrate in an area that you intend to pursue. Obtain the skills necessary to find work or enter graduate school after college.

3) You should have significant interaction both with other students and adults. You should probably live in dorm and learn how to co-exist with a variety of other people outside your immediate family. Interaction with professors and graduate students and other people outside your age group should be required regularly.

[b]Encouraged (for college):[/b]

1) Maintain a healthy exercise routine, healthy eating habits, healthy relationships. These things should have been learned in high school and they should just be continued.

2) Participate in clubs and other extracurricular activities that involve competitions but which are more oriented to solving real-world problems or making real-world change than at the high school level.

3) Continue volunteering and community service activities, like in high school.

@KMST: I'm pretty certain that every single argument you presented is flawed or changed my argument into something it is not (and hence was a flaw).

For the first thing, I will just repeat my statement "I agree that people will probably forget the "details," but I think the main concepts will stay," that you seem to be refuting, but really are just reinforcing. Every example you have is of someone forgetting details. I agree with this.

Your general argument seems to be that since it doesn't work the way I say it should, then that is reason to get rid of it. This is a poor attitude. The US healthcare system is not working, thus we should get rid of it. No need for it anymore. The arguments are about the necessity of it, not on whether it works or not.

Not to mention that you keep using very specific examples. Just because a few people push their way through college and learn nothing, doesn't mean that everyone does this. It is possible a majority are becoming better communicators, learning how to behave ethically, etc. That is called the fallacy of generalization.

"Sadly, you cannot judge the quality of the education by the reputation of the school that issued the diploma. After working with some utterly incompetent graduates from big name universities, and seeing other people do excellent work out of their field or with a degree from a little known small college, or without any degree at all, I do not judge people based on the prestige of their alma mater."

This is irrelevant to what I was saying. I agree with this. It is good that you don't judge based on degrees, but rather on the person. The sad truth is that a majority of people are not like this (note that I named that fallacy above so that I could mention that this just means a majority of people do not make logical decisions, most people generalize).

I think that yes, people should be reasonably well-versed in all subjects, especially communication in English. In an ideal world, this education should be attained in high school. College should be where you study [i]whatever you want to[/i], whether you choose to be highly specialized or continue studying a variety of things. But this isn't an ideal world. Darn it.
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[hide]Escape: But math is ideal. We can study that instead. YAY![/hide]
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We seem to just be debating back and forth over nothing in this thread. One person says, "we should have to study more stuff" and another says "we shouldn't have to study more stuff", when in reality, everyone agrees quite well on what should and shouldn't be learned.

Or perhaps I like communication skills. I hope not.

[b]KSMT:[/b] Examples of undergraduates being more competent than graduates are really irrelevant here. The point is that there are certain basic skills that should be developed during a college education.

By the way, I am curious to know what perspective you argue from

[b]To Everyone:[/b] Three things:

[b]#1:[/b] Different colleges have different approaches toward trying to make this happen, and [b]it is the responsibility of the student to get into a program that suits his/her particular philosophy of learning[/b]. Examples:

- Students that want a rigorous, complete program in some study that has been laid out by the gods of the field can go to a place with such a program, understanding full well that a lot of specific coursework will be expected of them.
- Students that want to be able to try their hand at many fields can go to a college with fewer major requirements. Students who want the wisdom of established professors and who would benefit from taking the same classes as their peers can go to a college with an established core curriculum.
- Students who love to self-study and don't want to be told what classes they are qualified to take can go to a college that doesn't enforce pre-requisites.
- Students who want to forge their own path through education can go to a place where faculty are encouraging of students designing their own majors.

Again, it is the student's responsibility to do the research, apply to, and get themselves into a program that they want to be in -- if they fail to do so, it is nobody else's fault!

[b]#2:[/b] Having established that it is the student's problem if they are in a program with an unwanted set of mandatory classes, we move to distribution requirements. They're not as bad as you make them out to be.

Distribution requirements are there because it is commonly believed (by people more qualified than you) that everyone can benefit from them.
- There are some places where AP credits or a proficiency exam will get you credit for a particular class, and thus fulfill the requirement. No place is going to say that you [i]have to[/i] take single-variable calculus [i]no matter what[/i].
- There are other classes that you can't jump out of. For example, suppose some schools require a writing seminar for first-year students, and offer a distribution of topics on which they are taught. The supposition is that no matter how good of a writer you are, and no matter what your interests are, you should be able to find a writing seminar that will aid you in your personal growth as a writer. It is highly arrogant for any student to think otherwise.
- Then, in the true spirit of distribution requirements, many places have guidelines like, "you must take classes from at least four of the following eight groups of classes", and then list groups like "literature", "foreign language", "social behavior and analysis", "mathematics", "natural science", ... or whatever. This doesn't force anyone to take, say, history, if they abhor it. I think these are quite reasonable -- obviously nobody goes to college to take five math courses every semester for eight semesters.

[b]#3:[/b] I also want to note that everyone can benefit from some degree of distribution.
- If a student knows the material cold in a class that they have to take, then they can surely find other ways to benefit from the class. For example, someone who knows how to write a good essay might use a writing class as an opportunity to experiment a bit with their writing and try out new styles and techniques.
- Students can meet other students with different backgrounds and interests and gain from their perspectives
- A student might uncover a new interest in a topic they did not know much about or that they thought would be boring as a result of a preconceived notion or a bad previous teacher. This is actually quite big -- universities with many courses of study know that most students don't know enough about the fields of their interest to commit to them from day one, and that most students havent seen enough of other interesting fields. They want students to experiment with their interests a bit before asserting what they want to do. This is an example of the institutional philosophies I was talking about in point #1.

[b]Final thoughts:[/b]

For any situation, it is one's responsibility to do one or more of the following:
1. Avoid an undesirable situation (see point #1)
2. If in the undesirable situation, find something to gain from it (see point #2)
3. If the person can do none of the above, then suck it up, reflect and see what you could've done differently, and don't get into a similar situation in the future.

This applies to a student's academic career, too. A student should try to choose an academic program that fits their learning needs and philosophy of education. If they can't, they should try to make the best of what they have, knowing that there are likely people that would be more than happy to help them figure out how to do this. And if none of that works, they better give a good thought to where their life is going.

Core classes and distribution requirements are a property of a student's particular academic career. There are good reasons why a diverse education is encouraged in many places. And if that approach doesn't suit a particular student's needs, they are welcome to find a place with a different approach (because there probably is such a place).

[quote="TZF"][b]it is the responsibility of the student to get into a program that suits his/her particular philosophy of learning[/b]. Examples:
[/quote]

I second that and I think it is very awful how many students don't do that at all. In my opinion this should be one of the most important criteria in choosing an undergraduate university. I think every prospective student should thoroughly read through the distribution requirements of their prospective university. And admissions department should make these requirements readily accessible.

 I hate how prospective students focus on things like how beautiful the campus is or other peripheral things and don't pay much attention to what coursework is actually required of them of graduate. Maybe high school students just don't realize there is a such a variety of different requirements at different universities since almost all high schools have the same general requirements.

"I hate how prospective students focus on things like how beautiful the campus is or other peripheral things and don't pay much attention to what coursework is actually required of them of graduate. Maybe high school students just don't realize there is a such a variety of different requirements at different universities since almost all high schools have the same general requirements."

Thanks for the warning. :)

[quote="TZF"][b]KSMT:[/b] Examples of undergraduates being more competent than graduates are really irrelevant here. The point is that there are certain basic skills that should be developed during a college education.

By the way, I am curious to know what perspective you argue from.[/quote]My points obviously did not get across. It must be because I never took a college writing class in any language, and English is not my native language. If I really cannot communicate, maybe I should go and take a college writing course. Maybe those courses are as necessary as advertised.

I fully agree that there are basic skills that should be developed. They should also be maintained. My previous post stated that many of such skills, if ever developed, are not maintained. My use of examples distracted from the fact that I was also making general statements, and led to the idea that the examples I was citing were isolated case of "a few people"

I fully agree with TZF's requirements for high school, as long as the requirements include learning and remembering at least half of the material. However, I do not think that the requirement to take a class, or to get a passing grade in a class means much. If a student takes American Government in 11th grade, and gets an A in that class, my congratulations on knowing how to work the system. If after graduation that student can tell me who represents him in Congress (House and Senate), for how long they were elected, and when their term expires, then we have a competent citizen. Unfortunately, I believe that such competent citizens are only a small minority of the voters. Likewise, the students who did not flunk English, should be competent enough to communicate in writing in an ideal world, and ideally each diploma should guarantee a certain competency. I believe the system is focused on process and paperwork, but does not focus on results and results are never guaranteed.

After the competency in communications and other subjects deemed essential for living in a society but not related to a student's major has been established, there is no need for a college to impose the requirement to pay for and sit in a class in those subjects. That was my main point. Does anyone have a list of the US colleges that do that?

My secondary point is that people forget (maybe as soon as they walk out of each test/exam) what they were supposed to have learned for the long term. Skill/knowledge that is not interesting or continuously used is soon forgotten. That goes to reinforce my argument about the futility of forcing mature students to take classes they do not want to take. I'll agree that you would want to force small children to learn to read and write. College graduates will develop the ability to communicate in written form if and only if they value and need that ability. How much they practiced writing in a college writing class, or how much they forgot after they obtained their passing grade is not that important. If communicating in writing was necessary in college (e.g. essay questions in exams) and is needed in their life as graduates, the skill attained from K-12 school writing will be maintained and further developed.

I was trying to make the point that people seem to forget not only the details, but also much of the essence of what they supposedly learned in college, even knowledge related to their major, which should be most likely to be retained. I gave examples because I do not know of any study that has examined how much of the supposedly learned material is retained. Would the average chemist out of college for over 5 years pass the AP chemistry exam? I am not sure. The chemists around me are competent in the specialized knowledge that they use daily. A few are interested in and know about some other areas. Does the average pharmacist out of college for over 5 years remember the properties of drugs, the appropriate doses, and how to dilute them to get to the desired concentration? I would not bank on it. The examples I gave were not of people forgetting details, or of particular undergraduates being more competent than graduates. I gave examples of people being incompetent in the most essential parts of the knowledge that their various diplomas implied. A pharmacist should know about drugs and how to dilute them, as much as a mathematician should know about the order of operations. A person confusing sodium cyanate with sodium cyanide should not be a high school graduate, much less a professor of toxicology and pharmacology in a medical school. They both should know about chemical formulas taught in high school chemistry, and should be able to spell the words that elementary school students are taught.


How many permutations of 1, 2, 3, 4, 5, 6, 7, 8, 9 have:
1 appearing somewhere to the left of 2,
3 somewhere to the left of 4, and
5 somewhere to the left of 6?
For example, 8 1 5 7 2 3 9 4 6 would be such a permutation.

(a) $9\cdot7!$
(b) $8!$
(c) $5!4!$
(d) $8!4!$
(e) $8!+6!+4!$

You have that many places to put each of those numbers? O_O

I'm really not sure how to solve it, as none of the ideas that I've come up with are answer choices. :?

[quote="mathgeek2006"][hide]I believe that the answer would be[b] [i]E[/i][/b] because you have 8! places to place the 1, 6! places to place the 3, and 4! places to place the 5.[/hide][/quote]

How do you have 8! places to put the 1?  :?

I got

[hide]

 ( 9 choose 2 ) * ( 7 choose 2 ) * ( 5 choose 2) * 3! = 45360

  This happens to be equal to the expression in A. [/hide]


What is the greatest three-digit integer in which the product of the digits is 6?


Hello!

I'd like to know if anyone could help me on this problem.

Let F be a function from Q to Z where Q is the set of rational numbers and Z the set of integers. Show that there are two distinct rational numbers r and s such that [f(r)+f(s)]/2 <= f((r+s)/2).

Thanks!

[hide="Hint"] Assume the opposite; namely, that $ f(r)+f(s) > 2f \left( \frac{r+s}{2}\right)$ for any choice of $ r, s$. [/hide]


geometry
3D geometry
sphere
advanced fields
advanced fields theorems

If a $\Delta$-complex has the same simplicial homology as the 3-sphere, does that imply it is homotopic to the 3-sphere?

In general, can simplicial homology be used to show homotopic equivalence or is singular homology absolutely required?

[url=http://en.wikipedia.org/wiki/Homology_sphere]Homology spheres[/url] do exist... The simplicial and singular homology groups of a $\Delta$-complex are isomorphic: see [url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html]Hatcher's book[/url] for a proof.


Join Ducky Tourney [size=200]NOW![/size] [size=150]NOW![/size] [size=100]NOW! [/size][size=75]NOW![/size] [size=59]NOW![/size] [size=42]NOW![/size]     [size=25]NOW![/size]

Please sign up to the coolest tournament ever!!

(this is a serious tournament btw)

This will be a team tournament, pick your teammates, and if a team is way too strong or too weak, i will have to switch players to make it more even.

the number of people per team will be decided later

Heres how it goes:

Everyone from team $ X$ plays everyone from team $ Y$

Person $ 1$ from Team $ X$ goes against Person $ 1$ from Team $ Y$.

The team with the most wins wins!

The winning team moves on to the next round. I will decide when each game occurs.

Please play responsibly. This may be dangerous for some minds.

[size=50]MYYELLOWDUCKY82 will not be liable for any injury that will most likely result from the tournament. The tournament is not available in all states or minds. Thank you  [/size]

OK!!! THIS IS A COMPLETE COPY OF the happy face tourney by (^_^)!!!! THATS ILLEGAL!!!!

BY THE WAY I"M JOINING!

sorry guys i guess some other things i put earlier wasn't added.

Join Ducky Tourney [size=200]NOW![/size] [size=150]NOW![/size] [size=100]NOW! [/size][size=75]NOW![/size] [size=59]NOW![/size] [size=42]NOW![/size]     [size=25]NOW![/size]

Please sign up to the coolest tournament ever!!

This will be a team tournament, pick your teammates, and if a team is way too strong or too weak, i will have to switch players to make it more even.

the number of people per team will be decided later

Heres how it goes:

Everyone from team $ X$ plays everyone from team $ Y$ and $ Z$

Person $ 1$ from Team $ X$ goes against Person $ 1$ from Team $ Y$. and person $ 1$ from team $ Z$

The team with the most wins wins!

The winning team moves on to the next round. I will decide when each game occurs.

Please play responsibly. This may be dangerous for some minds.

[size=50]MYYELLOWDUCKY82 will not be liable for any injury that will most likely result from the tournament. The tournament is not available in all states or minds. Thank you  [/size]

[quote="myyellowducky82"][size=59]MYYELLOWDUCKY82 will not be liable for any injury that will most likely result from the tournament. The tournament is not available in all states or minds. Thank you [/size][/quote]

This is really:

MYYELLOWDUCKY82 will not be liable for any injury that will most likely result from the tournament. The tournament is not available in all states or minds. Thank you

[quote="myyellowducky82"]sorry guys i guess some other things i put earlier wasn't added....AGAINNNNNNNN!

Join Ducky Tourney [size=200]NOW![/size] [size=150]NOW![/size] [size=100]NOW! [/size][size=75]NOW![/size] [size=59]NOW![/size] [size=42]NOW![/size]     [size=25]NOW![/size]

Please sign up to the coolest tournament ever!!

This will be a team tournament.

Heres how it goes:

Everyone from team $ X$ plays everyone from team $ Y$ and $ Z$

Person $ 1$ from Team $ X$ goes against Person $ 1$ from Team $ Y$. and person $ 1$ from team $ Z$

The team with the most wins wins!

The winning team moves on to the next round. I will decide when each game occurs.

Please play responsibly. This may be dangerous for some minds.

[size=50]MYYELLOWDUCKY82 will not be liable for any injury that will result from the tournament. The tournament is not available in all states or minds. Thank you  [/size][/quote]

SO if these are the official rules, then does that mean you wont change teans around?

i'll change the teams as a last resort.
sorry guyz for changing the rules a lot.

u change the rules a lot...

just like dinger (neat name, huh?)

wait...r u talking 2 me happy face?
if u r im not mad.
really.

[quote="(^_^)"][quote="ernie"]meh.  maybe dinger's comp was weird too...[/quote]

whatever....

ur just mad that i took u out of my tournament and ur trying to get back at me...[/quote]

dude... im rooting for u, and u put me down like that?

harsh.

[quote="(^_^)"]very harsh...

no i wasn't talking to you ducky, sorry if it seemed like it.[/quote]
oh haha i thought u were yelling at me for no reason.
do u want 2 join the tourney? i need to make a list of the participants.

NOTE: if some more people do not join, the tourney will be CANCELED!

sooo, is anyone actually joining
or is this just a post for chatting?

Okay guys,
the tourney is officially CANCELLED!!!!!! sorry!
I might make a new one soon, so watch for it!
**someone please lock this one, thanks!


calculus
integration
trigonometry
real analysis
real analysis unsolved

can you solve :D

$ \int_{0}^{\infty}\;\frac{\textbf dx}{\sqrt{\;x^{4}\plus{}x^{2}\plus{}1\;}}$

\begin{eqnarray*}\int_{0}^{\infty}\frac{dx}{\sqrt{1+x^{2}+x^{4}}}& = &\int_{0}^{\frac{\pi}{2}}\frac{\sec^{2}\theta\, d\theta}{\sqrt{\sec^{4}\theta-\tan^{2}\theta}}\\ & = &\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-\sin^{2}\theta\cos^{2}\theta}}\\ & = &\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-\tfrac{1}{4}\sin^{2}2\theta}}\\ & = &\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-\tfrac{1}{4}\sin^{2}\phi}}\quad\quad (\phi = 2\theta)\\ & = & K\left(\tfrac{1}{2}\right)\end{eqnarray*}

where $ K(k)\ =\ \int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}$ is the complete elliptic integral of the first kind.


1. Let $a_1,..., a_n,...$ be positive real numbers such that $\frac{a_n+a_{n+2}}{2} < a_{n+1}$ for every $n \geq 1$. Show that $a_n < a_{n+1}$ for every $n \geq 1$.

This is rather evident... :?

We can rewrite the condition as $a_{n+2}-a_{n+1}<a_{n+1}-a_n$. Suppose that for some $n$ we have $a_n \geq a_{n+1}$, then $a_{n+2}-a_{n+1}<0$ by the condition, and by induction the sequence is strictly decreasing from there on. Let $a_{n+2}-a_{n+1}=r$. Then $a_{n+k}=a_{n+k-1}+(a_{n+k}-a_{n+k-1})<a_{n+k-1}+r$, so by induction $a_{n+k}<a_{n+2}+(k-2)r$. Since $r$ is negative, for $k> \frac{a_{n+2}}{r}+2$ we have $a_{n+k}<0$, contradiction.

Yar, I dropped a mark or two because I negated the statement wrongly, I didn't make it greater [b]or equal[/b]  :blush:  :blush:  :blush:

nope, but adrian admonished me for the slip, if that's the right word (=


In this game, we carry on a conversation with various segues.  In other words, we try to create an entire monologue like in this video:

[youtube]OratYKWszU8[/youtube]

I'll start.

Hello.  My name's Francis Pumphandle, but everyone calls me Pip.  I see that there are a lot of people on this forum that enjoy math.  I enjoy math, as well.  There are so many math contests to participate in these days.  One contest in the United States is ARML, which occurs in three places at once - Nevada, Iowa, and Pennsylvania.

[i]And then the next poster could talk about Pennsylvania.[/i]

Now there's a great state, Pennsylvania. Great schools, historical sites, and food. My favorite food is the yam, which is grown in Africa, Asia, Latin America, and Oceania. They also have great music. I love being serenaded with Japanese music. Japan also is the origin of the well known food sushi. It's rice, vegetables, and sometimes seafood wrapped in seaweed.

@Next poster:Seaweed.

I'm not really good at making monologues, though.

seaweed can be found in the sea...
in the sea lives fish...
examples of fish are clown fish..
such as NEMO!!

@next poster: nemo

the name nemo of course brings back memories of the legendary captain of the nautilus.
few characters from the world of literature have been as awe inspiring.
among those that come close one undoubtedly would be albus dumbledore.

(note: the last statement is not my personal view, it's just to keep the game going)

Dumbledore, eh? Have you seen those Harry Potter movies? Great stuff! If you haven't you should make a family night out. It has something for everyone. Though the Harry kid does get on some peoples' nerves of course besides the fact that he is British. Well everyone from the movie is British, and it is kind of filmed in Britian. And to think of it the author of the books is British also. What do people have agianst the Brits? Not like they bother us they live across the Atlantic, well, of course, they do not call it the "Atlantic" they call it the "Pond", so I guess you could say that we live across the "Pond" from them though no one in the U.S. says "Pond" unless they mean a small body of water.  Have you every gone swimming in a pond? What about a lake? It id kind of gross, you know, with the water being all brown and stuff and you can not see where you are swimming.  Not to get off topic though. Back to those Brits. They have weird phrases and words that people from the U.S. do not use even thoguh they are technically the predecssors to Americans. For instance telly, I mean really the is one instance where American English sounds more sophisticated than the British English. Hm. Another! Watercloset, that can mean a bunch of things in the U.S., if an Brit came to the U.S. and asked where the watercloset was in a restuarent the waiter would give them a weird look and, after they continue to insist, they would eventually get shown the plumbing.  The Brits have a bunch of differences with Americans. ...

 :P

American food is kind of gross, and the only real American foods (pizza and pasta: Italian) are mac & cheese (not bad, but unhealthy. Yeah, I guess you could work out and stuff, but that would be too tiring. Best just to not eat it) and hamburgers. Hamburgers are so disgusting nowadays. Just about the only hamburger fastfood restaurant worth eating from is In-n-Out, which assured you eat only the most fresh and delicous burgers. Companies like Burger King on the other hand.... BLEH! Especially McDonalds, the food is tasteless, dry, and... It just tastes so bad you can't possibly describe it. I don't see why so many people actually buy food from McDonalds....

Have fun...

Woohoo! Someone with my viewpoint about McDonalds!

McDonalds food is basically tasteless white filth served with the lowest form of wheat: the bun. I doubt buns are even the real thing; they are too squishy. The drinks they serve there? Just ""pleasure""*does finger thing* drinks that kill brain cells. Brain cells are the most important cells in the body. They don't grow back like your other cells. When they are dead, they are dead. You don't get a second chance. Just like "Curiosity killed the cat.".

the average rate of cats being killed by a car is increasing every year. however, the average rate of squirrels being killed by cars are skyrocketing...
however, the lack of knowledge and eating McDonald's fries have left squirrels hopeless like this:

ahh...roadkill. did you know in West Virginia, you can take roadkill home for dinner? and WV is a great place. a great source of unrenewable resources and home to part of the Appalachian Mountains. Is that how you spell Appalachian? thats a funny word. i like funny words. Have you ever noticed the viewpoint of funny in diferent perspectives? Different things sound different to other people. and sound! what is sound? what is the sound of 1 hand clapping? i think thats part of a deep question used in Buddhism


Well, by applying twice Cauchy- Schwartz, we get
$ \frac{xy}{2}.2xy.(x^2 \plus{} y^2) \leq \left(\frac{x\plus{}y}{2}\right)^2\frac{1}{2}\left(\frac {(x \plus{} y)^2}{2}\right)^2 \equal{} 2$
The equality holds if and only if $ x \equal{} y \equal{} 1$

ineq$ \Leftrightarrow x^2y^2(4 \minus{} 2xy)\leq 2\Leftrightarrow (xy)^3 \minus{} 2(xy)^2 \plus{} 1\geq 0\Leftrightarrow(xy \minus{} 1)(x^2y^2 \minus{} xy \minus{} 1)\geq 0,Right ( xy\leq 1)$

Well, since $ x\plus{}y\equal{}2$ do $ x\equal{}1\plus{}a$ and $ y\equal{}1\minus{}a$, with $ 0 \le a \le 1$.
Then we get: $ ((1\minus{}a)(1\plus{}a))^2((1\plus{}a)^2\plus{}(1\minus{}a)^2) \le 2$.
But this is the same as $ (1\minus{}a^2)^2(1\plus{}a^2) \le 1$, or, $ (1\minus{}a^2)(1\minus{}a^4) \le 1$, which is obviously true.

$t=xy,f(t)=x^2y^2(x^2+y^2)=8t-6t^2(0<t\le 1),f ' (t)>0,f(t)\le f(1)=2,$

$x^2y^2(x^2+y^2)\le 2.$

Irish Mathematical Olympiad 2000
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=196400&hilit=Ireland：
$ \iff$  $ x^2y^2(4\minus{}2xy)\,\leq\,2$
$ \iff$  $ x^3y^3\minus{}2x^2y^2\plus{}1\,\geq\,0$
$\iff (xy\minus{}1)^{2}\plus{}(\frac{1}{xy}\minus{}1)\geq 0$ $( xy\in (0,1))$

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=523349
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=523362&p=2955008#p2955008
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=521368&p=2939566

[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&p=344620#p344620]India 2002:[/url]
If $x$, $y$ are positive reals such that $x + y = 2$ show that $x^3y^3(x^3+ y^3) \leq 2$

[b]Generalization of Ireland 2000[/b] ：
Let $x$, $y$ are positive reals such that $x + y = 2$ , for $n\ge 1$ , we have\[x^ny^n(x^2+ y^2) \leq 2 .\]
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=542249

[quote="stvs_f"]Let $ x; y \geq 0$with x + y = 2. Prove that:
 \[ x^2y^2(x^2 \plus{} y^2) \leq 2\].[/quote]

$ x^2y^2(x^2 \plus{} y^2)\le xy(x^2 \plus{} y^2)= 2xy(2-xy)\le 2(\frac{xy+2-xy}{2})^2= 2.$
[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=360304&hilit=Bulgaria+2010]Bulgaria 2010:[/url]
Let $a,b,c>0$and$a+b+c=3$,Prove that\[abc(a^{2}+b^{2}+c^{2}) \le 3\].
[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&p=3129961#p3129961]Inspired by Bulgaria 2010:[/url]
Let $a,b,c,d>0$ and $a+b+c+d=4$ ,  Prove that\[abcd(a^{2}+b^{2}+c^{2}+d^{2}) \le 4.\]

http://www.artofproblemsolving.com/community/c6h1246458p6396061:
It's obvious that $xy\le 1$, then
$x^2y^2(x^2+y^2)=2x^2y^2(2-xy)\le 2xy(2-xy)\le 2$
by AM-GM.

[quote=stvs_f]Let $ x; y \geq 0$with x + y = 2. Prove that:
 \[ x^2y^2(x^2 \plus{} y^2) \leq 2\].[/quote]
[b][color=#f00]Irish 2000[/color][/b]
[url=https://artofproblemsolving.com/community/c1068820h2028088p14282030] Estonia  2001[/url]
Let $x$, $y$ are positive reals such that $x + y = 2$ , [url=https://artofproblemsolving.com/community/c6h542249p3128662]show that[/url]
$$xy(x^2 \plus{} y^2) \leq 2$$


The given definition of a quotient vector space $V/W$ over a subspace $W$ of vector space $V$ is:
$V/W=\{v_1+w_1|v_1\in V, w_1\in W\}$

Since $w_1\in W\in V$, $w_1+v_1\in V$ by closure of vector spaces. On the other hand, for any $v_1\not\in W$, choose the zero vector for $w_1$ to get $v_1+w_1=v_1$. That said, how is $V/W$ different from $V$?
Also, can someone please give an example of a quotient vector space? Thanks in advance.

I don't think that's the correct definition of a qoutient vector space. The one you proposed describes the [i]sum[/i] V + W of two subspaces of the same vector space. The correct one should be:

$ V / W = \{ W + v | v \in V \} $, where $ W $ is a subspace of $ V $

and where the set $ W + v_0 $ is the set $ \{ w + v_0 | w \in W \} $. Note that $ W / V $ is a set and not a family; the definition may produce many pairs $ v_1, v_2 $ for which $ W + v_1 = W + v_2 $, so we disregard repeated sets. Then $ V / W $ is a partition of $ V $ into sets of the form $ W + v $, called [i]cosets[/i], and in order to turn $ V / W $ into a vector space, addition and scalar multiplication are defined on it by $ (W + v_1 ) + (W + v_2) = W + (v_1 + v_2) $ and $ c(W + v_1) = W + cv_1 $ (you can check that these are well-defined operations that satisfy the vector space axioms).

I think there was an example about a quotient space in $\mathbb{R}^2$.
The subspace was defined as the set of vectors $\{(x,y)^T|y=3x\}$.
Then the quotient space is the set of all lines in $\mathbb{R}^2$ parallel to $y=3x$.
I don't really understand why that is the case.

What's the advantage of having a quotient vector space then? Also, can I have another elementary example?


[hide="a"]
Let $a_{n}$ be the number of such good sequences. Given an $n+1$-digit sequence, remove the last digit. If the last $n$ digits form a good sequence, then any of the five options for the last digit work. Otherwise, a 0 occurs before any of the 1's in the remaining digits (that can't happen) or there are no more 1's (in this case, the removed digit must have been 1 and there can't be any 0's, but any combination of 2's, 3's, and 4's is fair game). So we get
\[a_{n+1}=5a_{n}+3^{n}\]
[/hide]

[hide="b"]
$a_{1}=1$ because the 1 has to appear as a digit, and then... oh, of course! there's no more room!
[/hide]

[hide="c"]
We find $\boxed{a_{n}=\frac{5^{n}-3^{n}}{2}}$. This may be proven via induction, or it may be [i]counted directly[/i] as follows:

There are $5^{n}$ total strings, and $3^{n}$ do not contain any 0's or 1's. Therefore $5^{n}-3^{n}$ contain 0 or 1 (or both). Now we can split these $5^{n}-3^{n}$ into pairs such that the second string in each pair looks like the first one, except that 0's and 1's have been switched. (i.e. [b]20342113[/b] gets paired with [b]21342003[/b]). In each of these pairs, exactly one string is good (because if a 1 comes before a 0 (if it appears) in one string, the other string will have a 0 before a 1 (if it appears), and vice versa.) Since there are $\frac{5^{n}-3^{n}}{2}$ pairs, there are $\frac{5^{n}-3^{n}}{2}$ good strings.
[/hide]

I haven't found a recurrence relation, but I found an explicit formula:

[hide="blah"]
Consider where the first $0$ is found. It could be found in the $2nd, 3rd,...,nth$ position or it could not exist at all.

Consider the latter case. Since we could use only the numbers $1,2,3,4$ in the $n$ available spaces, there are $4^{n}$ combinations. However, because at least one $1$ must exist, we have to subtract the number of combinations with only $2,3,4$. There are $3^{n}$ combinations, and thus $4^{n}-3^{n}$ cases.

Now the case where the first zero is placed at the $kth$ place. To the right of the zero, any of the five numbers could exist; therefore, there are $5^{n-k}$ choices there. To the left of the zero, we recall the previous information and conclude that there are $4^{k-1}-3^{k-1}$ combinations there. Thus, we have $5^{n-k}(4^{k-1}-3^{k-1})$.

Let $f(n)$ denote the number of valid n-sequences.
Summing, we have
$f(n)=4^{n}-3^{n}+\sum_{k=2}^{n}5^{n-k}(4^{k-1}-3^{k-1})$

We could split up the summation into components of $3^{k-1}$ and $4^{k-1}$.

Let $k=\sum_{k=2}^{n}5^{n-k}3^{k-1}$

We have
$k=5^{n-2}3^{1}+5^{n-3}3^{2}+\cdots+3^{n-1}$
$\frac{3}{5}k=5^{n-3}3^{2}+\cdots+3^{n-1}+\frac{1}{5}3^{n}$.

$(1-\frac{3}{5})k=5^{n-2}3^{1}-\frac{1}{5}3^{n}$.

We can do the same with the $4^{k-1}$ component and then substitute in to the equation to get

$f(n)=4^{n}-3^{n}+(4*5^{n-1}-4^{n})-\frac{1}{2}(3*5^{n-1}-3^{n})$.
$f(n)=\frac{5^{n}-3^{n}}{2}$[/hide]

[quote="scorpius119"][hide="c"]
We find $\boxed{a_{n}=\frac{5^{n}-3^{n}}{2}}$. This may be proven via induction, or it may be [i]counted directly[/i] as follows:

There are $5^{n}$ total strings, and $3^{n}$ do not contain any 0's or 1's. Therefore $5^{n}-3^{n}$ contain 0 or 1 (or both). Now we can split these $5^{n}-3^{n}$ into pairs such that the second string in each pair looks like the first one, except that 0's and 1's have been switched. (i.e. [b]20342113[/b] gets paired with [b]21342003[/b]). In each of these pairs, exactly one string is good (because if a 1 comes before a 0 (if it appears) in one string, the other string will have a 0 before a 1 (if it appears), and vice versa.) Since there are $\frac{5^{n}-3^{n}}{2}$ pairs, there are $\frac{5^{n}-3^{n}}{2}$ good strings.
[/hide][/quote]

Very tricky direct counting  :ninja:

thanx to all.... is there any easy way the above question could be solved...... any how i appreciate everybody help

thanx much


x,y are natural number.find all x,y that:
3^x =2^x *y +1


group theory
abstract algebra
superior algebra
superior algebra theorems

Hello,
Is there an easy way for finding all of the cyclic subgroups of $A_{4}$. It is easy when the group is cyclic, then one can find a generator and then use it to find all the other subgroups, but what do you do in the case when this is not the case.


If I watch the sky for one hour, the probability that I see a shooting star is exactly 1%. Suppose that I watch the sky for [i]n[/i] hours. What is the probability that I see an odd number of shooting stars?

[hide="solution"]The probability that she does not see a shooting star in one hour is $ \frac{99}{100}$. The probability that she does not see a shooting star in 6 hours is $ \frac{99^6}{100^6}$. The probability that she does see a shooting star in 6 hours is therefore $ \frac{100^6\minus{}99^6}{100^6}\equal{}\boxed{\frac{58519850599}{1000000000000}}$[/hide]

@1=2

You didn't read the question too carefully.

This is $ \dbinom n1\cdot0.01^1\cdot0.99^{n \minus{} 1} \plus{} \dbinom n3\cdot0.01^3\cdot0.99^{n \minus{} 3} \plus{} \cdots$...not sure how to simplify this.

@1=2: Read the problem again (if you haven't already). All of it.

I don't think I understand the question.
Is it possible that two shooting stars are seen in one hour?

I assumed that you can only either see one or none within one hour, which gives the partial solution above.

Yes, you can only see one or none shooting stars per hour.


And the solution is unique because $4^{x}+5^{x}$ is strictly increasing and $(4-x)^{4-x}+(5-x)^{5-x}$ is strictly decreasing (at least for $x\le 3$).


Evaluate the following limit:

$\lim_{h \to 0}\frac{\sqrt{h+2}-\sqrt{2}}{\sin(h)}$

[hide="Solution"]\begin{eqnarray*}\lim_{h\to 0}\frac{\sqrt{h+2}-\sqrt{2}}{\sin h}&=& \lim_{h\to 0}\frac{\sqrt{h+2}-\sqrt{2}}{h}\cdot\frac{h}{\sin h}\\ &=& \lim_{h\to 0}\frac{h+2-2}{h(\sqrt{h+2}+\sqrt{2})}\cdot\frac{h}{\sin h}\\ &=& \lim_{h\to 0}\frac{1}{\sqrt{h+2}+\sqrt{2}}\cdot\frac{h}{\sin h}\\ &=& \frac{1}{2\sqrt{2}}\cdot 1\\ &=& \frac{\sqrt{2}}{4}\end{eqnarray*}[/hide]

[hide]or just use L'Hopital's rule;
this is an indeterminate form of 0/0;
so the limit is really:
lim (h-->0) of (1/(2*sqrt(h+2) -0) /cos(h)
 =1/(2*(sqrt(2))/1
=sqrt(2)/4

sorry about the bad syntax.[/hide]

[hide]Also if $f\left(x\right)=\sqrt{x}$, then you have \begin{eqnarray*}\lim_{h\rightarrow 0}\frac{\sqrt{h+2}-\sqrt{2}}{\sin h}&=&f'\left(2\right)\lim_{h\rightarrow 0}\frac{h}{\sin h}\\ &=& \boxed{\frac{1}{2\sqrt{2}}}\end{eqnarray*} [/hide]


By considering each bracket separately, find the value of:

${(2+x^2)^3}-{(2-x^2)^3}$

[hide="hint"]use difference of cubes[/hide]

[hide]
Using difference of cubes, we get $2x^6+24x^2$[/hide]


Show that there are infinitely many $ (x,y) \in \mathbb{N}^2$ satisfying $ x^{x\minus{}y} \equal{} y^{x\plus{}y}$.

[b]Extra:[/b] How complete a solution set can you find?

[quote="The Zuton Force"]Show that there are infinitely many $ (x,y) \in \mathbb{N}^2$ satisfying $ x^{x \minus{} y} \equal{} y^{x \plus{} y}$.

[b]Extra:[/b] How complete a solution set can you find?[/quote]

[hide]Set $ x \equal{} ky$ and you get $ (ky)^{y(k \minus{} 1)} \equal{} (y)^{y(k \plus{} 1)}$, or $ k^{k \minus{} 1}\cdot y^{k \minus{} 1} \equal{} y^{k \plus{} 1}$, or $ k^{k \minus{} 1} \equal{} y^2$ so we can take an odd integer $ k$ and get an integer $ y$, and an $ (x,y) \in \mathbb{N}^2$.[/hide]


Why are there two Temperal's?

[url=http://www.artofproblemsolving.com/Forum/usercp.php?mode=viewprofile&u=30944]Temperal[/url]

[url=http://www.artofproblemsolving.com/Forum/usercp.php?u=28419&mode=viewprofile]Temperal[/url]

And they have two different blogs?

The fake Temperal has the "T" in his name different from a usual Latin "T". Maybe it's a cyrillic "T" - this is the way some trolls have created fake accounts on the Wikipedia.

  Darij

Ok. But how did you figure that out?

And, is there a way to stop this? Like check through usernames for these "fake" letters before continuing.

Any good ASCII converter will tell you.

e.g. http://www.mikezilla.com/exp0012.html


induction
combinatorics unsolved
combinatorics

We want to select as many subsets of $ \{1,2,3,...,n\}$ as possible so that any two selected subsets have at least one element in common. What is the largest number of subsets we can select?

$ 2^{n\minus{}1}$.

- If you select $ 2^{n\minus{}1}\plus{}1$ (or more) subsets then, from the pigeon-hole principle, you will have select at least one subset and its complementary, so that the desired condition will not be satisfied.

- By induction, it is easy to prove that we may select at least $ 2^{n\minus{}1}$ subsets as desired : if it ok for $ n$, then select $ 2^{n\minus{}1}$ such subsets $ A$ in $ \{1, \cdots, n \}$ and add the $ 2^{n\minus{}1}$ subsets $ A \cup \{n\plus{}1\}$ to have the result for $ n\plus{}1$.

Pierre.

(or just take those sets that include the element $ 1$.)

Nonmathematically, there is no "s" in Miklós Bóna's family name (and note also that he has two accents).

[quote="JBL"](or just take those sets that include the element $ 1$.)
[/quote]

Well, since the induction initializes with $ \{1\}$ for $ n\equal{}1$, this leads to the same sets. But, my way to say it is clearly more complicated, thus it may be thought that it has more depth  :D

Pierre.


[quote="filomen"]If $ a,b,c > 0$, then

$ \prod[(a^2 \plus{} b^2)^2 \minus{} a^2b^2]\geq(\sum a^3b^3)(\sum a^2b)(\sum ab^2)$[/quote]

$ \Longleftrightarrow \sum_{sym} (2a^8b^4 \plus{} a^8b^2c^2 \minus{} 2a^7b^4c \plus{} 4a^6b^4c^2 \minus{} 2a^6b^3c^3 \minus{} 3a^5b^5c)\geq 0$.

 true due to Muirhead theorem.

 :lol:


advanced fields
advanced fields theorems

i have been trying to generate pythagorean triplets. i could generate it using the recursive way starting from 3,4,5 using ,

  gentrip (a - 2 * b + 2 * c, 2 * a - b + 2 * c, 2 * a - 2 * b + 3 * c);
  gentrip (a + 2 * b + 2 * c, 2 * a + b + 2 * c, 2 * a + 2 * b + 3 * c);
  gentrip (-a + 2 * b + 2 * c, -2 * a + b + 2 * c, -2 * a + 2 * b + 3 * c);

But how do we find the nth one with sorted by c (hypotenuse). I mean scaling it would slow it considerably because we have to sort it. Is there any way to generate the primitive along with non primitive triplets . Can we generate the next term if we know the current term.

We can consider both (a,b,c) and (b,a,c) to be different triplets.

I'm trying to generate these on a computer , the problem is scaling and huge memory and sorting them.
n<10^6

Any suggestions

You haven't defined a total order.  After you sort by hypotenuse, how do you sort different triplets with the same hypotenuse?  (You should be aware that there are different triplets with the same hypotenuse.)

That having been said, the set of all Pythagorean triplets is generated (uniquely) by

$ (2kmn, k(m^2 \minus{} n^2), k(m^2 \plus{} n^2))$

where $ \gcd(m, n) \equal{} 1$ and $ m \not \equiv n \bmod 2$.  Generating them in any kind of order would be much more annoying than generating a bunch of them and then sorting them afterwards (although I'm not sure why you would want to sort them).

[quote="t0rajir0u"]You haven't defined a total order.  After you sort by hypotenuse, how do you sort different triplets with the same hypotenuse?  (You should be aware that there are different triplets with the same hypotenuse.)

That having been said, the set of all Pythagorean triplets is generated (uniquely) by

$ (2kmn, k(m^2 \minus{} n^2), k(m^2 \plus{} n^2)$

where $ \gcd(m, n) \equal{} 1$ and $ m \not \equiv n \bmod 2$.  Generating them in any kind of order would be much more annoying than generating a bunch of them and then sorting them afterwards (although I'm not sure why you would want to sort them).[/quote]

The triplets are sorted in the order (c,a,b) where c is the hypotenuse and  (a < b).
I want sort because if we generate triplets the nth triplet would generated from equation above needn't be the sorted nth triplet i want .  Now generating the primitives (a,b,c), (b,a,c) and scaling them would generate in some order and we to sort to get the right order. We have 9,12,15 (scale of 3,4,5) after 5,12,13 . Now i want nth triplet with this order defined.Now nth triplet is n/2th triplet if we remove all (a,b,c) but that itself is a problem.

> $ (2kmn, k(m^2 \minus{} n^2), k(m^2 \plus{} n^2)$
 Yes this generates all the triplets but it generates duplicates also. Where do we stop k for the nth one,  Say we generate for m < 1000 and n<m , then we have approx n*(n-1)/2 triplets but then scaling each of the triplets would be meaningless because many of scale of (3,4,5) would come before the last 10 triplets.  i think even  m<900 is redundant. For each primitive we have to scale.

example in order

1) 3 4 5
2) 4 3 5
3) 6 8 10
4) 8 6 10
5) 5 12 13
6) 12 5 13
7) 9 12 15
8) 12 9 15
9) 8 15 17
10) 15 8 17

[quote="tryitn1"]Yes this generates all the triplets but it generates duplicates also. [/quote]
Reread the post more closely.  [b]No[/b] duplicates are generated if you specify that $ \gcd(m, n) \equal{} 1$ and $ m, n$ have opposite parities.

[quote="tryitn1"]The triplets are sorted in the order (c,a,b) where c is the hypotenuse and  (a < b). [/quote]
That describes the order the elements of the triplet are in; how are the [i]different triplets[/i] ordered with respect to each other?  Do you want $ (25, 15, 20)$ to come before or after $ (25, 7, 24)$?