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9
votes
2answers
653 views

Variety vs. Manifold

In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am ...
2
votes
2answers
70 views

Isomorphisms that have names (Or: Base-less isomorphisms)

The answer to this question of mine provided me with the fact that every isomorphism $$ \phi: K^{n} \rightarrow V, $$ where $V$ is an arbitrary vector space of finite dimension $n$ over the field ...
-2
votes
1answer
128 views

Fuzzy logical and integer programming

Is there any way of formulating linear/non-linear programming problems in terms of YES, NO, and MAYBE instead of just $0-1$ programming?
5
votes
1answer
312 views

Highest gain mathematical activity

I have this odd dream that online resources like this can serve as a virtual thesis advisor for future mathematicians who are teaching themselves. Here's another question along these lines. You can do ...
0
votes
2answers
48 views

Averages and Team

I have a question: Suppose $5$ players each score an average of $10$ points per game. Then collectively, do they score on average $50$ points per game? So player 1 scores an average of 10 points ...
3
votes
2answers
179 views

Question on mathematical writing

I am now writing my graduate thesis, it includes some basics mathematical theorems/propositions. I got a trouble in writing, more concretely, I do not know when can I state a mathematical claim as a ...
33
votes
18answers
7k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
12
votes
3answers
3k views

Can the Bourbaki series be used profitably by undergraduates?

Can the Bourbaki series be used profitably by undergraduates and high school students?Are we the target audience? I came across the N.Bourbaki texts while surfing the internet(I have not had the ...
99
votes
7answers
3k views

What remains in a student's mind

I'm a first year graduate student of mathematics and I have an important question. I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks ...
13
votes
8answers
1k views

What would be a good outdoor maths puzzle for children?

I have to find an interesting activity for some 11-year-olds moving to high school this year. It is supposed to take about 30-45 minutes, and I thought of having a mathematical theme. I can make a ...
9
votes
1answer
431 views

Stacks in arithmetic geometry [closed]

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
5
votes
3answers
5k views

Main branches of mathematics [closed]

Can anybody please show me the main branches and subbranches of mathematics and the statistical sciences in a hierarchical form? I am not a mathematician and often in my research I see a lot of new ...
7
votes
4answers
395 views

Extracurricular ideas for UK GCSE level maths student

My daughter is 15 years old and enjoys her maths classes (perhaps only because her maths homework takes her the least amount of time). Until now I have managed to introduce her to subject matter ...
2
votes
1answer
247 views

How much connection is there between Commutative Algebra and Algebraic Topology?

How much connection is there between Commutative Algebra and Algebraic Topology? I am looking for general highlights, not complex details.
9
votes
3answers
627 views

How to deal with the temporary nature of my knowledge?

I'm a self-learner trying to learn Math while enrolled in a wrong major (Humanities). I have gone through the many amazing questions and answers here (& elsewhere, including Prof. Tao's blogs) ...
8
votes
3answers
234 views

How come in statistics there is very little justification for the formulas used and proofs are almost nonexistent [closed]

I don't understand why people accept certain formulas in statistics without a mathematical proof style argument. You see this a lot in statistics textbooks and unfortunately this spills over with the ...
21
votes
3answers
983 views

Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$(\forall a,b\in L) x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$ where $\vee$ is the join operation, and $\wedge$ is the meet ...
5
votes
1answer
123 views

What is good about simple Lie algebras?

Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple. I wonder what beautiful properties does a ...
6
votes
3answers
511 views

The Importance Of Good Teachers and Guidance In the Academics

I'm a first year student for a math degree. I'm very curious on how good students overcome their bad teachers in the journey of learning and grasping the courses material fully, all in the pressure of ...
1
vote
1answer
93 views

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference? And what is other branch of advanced analytic geometry called? in ...
2
votes
1answer
82 views

What is the utility in writing pdfs in terms of their kernel?

Consider the normal distribution. We know that $$p(x| \mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} $$ The kernel is $$ p(x| \mu, \sigma^{2}) \propto ...
24
votes
3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
25
votes
9answers
2k views

Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
7
votes
1answer
754 views

Independent undergraduate research — what to do?

I hope this question is within the scope of this website. I am currently a rising senior, and need to decide on a topic for my independent undergraduate research/thesis. I was hoping to get some ...
14
votes
8answers
2k views

What math should a programmer know?

I am an application programmer focussing on Line Of Business (LOB) applications. I am from non-mathematics and non-CS background. What mathematics should I learn which help me improve my programming ...
5
votes
3answers
318 views

Is there a branch of mathematics that requires the existence of sets that contain themselves?

I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking ...
2
votes
1answer
122 views

Is there a theory that extend real analysis to functions maps into other algebraic structure?

I am studying real analysis now, reading Rudin's book Real and Complex Analysis. One thing confused me is when talking about measurable functions, we assume the function to be, from an abstract space ...
5
votes
1answer
136 views

Status of PL topology

I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological ...
23
votes
5answers
1k views

Importance of rigor

I always have a hard time explaining the importance of rigor to my friends who are not mathematically minded. A lot of past mathematicians develop the foundations of today's mathematics without going ...
17
votes
2answers
460 views

Which results depend on the irrationality of $\pi$?

Recently the following uninteresting clock picture was posted by one of my non-mathematically inclined friends to my facebook wall, saying that it was funny and possibly thinking that I would find it ...
3
votes
2answers
462 views

How much algebra is there in Noncommutative Geometry?

My Professor of Homological Algebra got me into some Hochschild (co)homology and then suggested to continue with formally smooth algebras, noncommutative differential forms and so forth. Now, my ...
4
votes
7answers
3k views

Simple applications of complex numbers

I've been helping a high school student with his complex number homework (algebra, de Moivre's formula, etc.), and we came across the question of the "usefulness" of "imaginary" numbers - If there ...
3
votes
1answer
116 views

Varieties given by non-algebraic equations

In algebraic geometry one (mostly) studies varieties given by polynomial equations. Such equations define algebraic varieties and there are many "dictionaries" available. For example, the category ...
24
votes
4answers
4k views

What does it really mean for something to be “trivial”?

I see this word a lot when I read about mathematics. Is this meant to be another way of saying "obvious" or "easy"? What if it's actually wrong? It's like when I see "the rest is left as an exercise ...
1
vote
0answers
84 views

I'm interested in different meanings of “normal”~ [duplicate]

Possible Duplicate: What is it to be normal? I've learned in algebra class that "normal" means a linear operator is commutative with its adjoint; also we say that $H$ is a normal ...
9
votes
5answers
727 views

What are special functions for?

If you read enough mathematics, you eventually come across several so-called "special functions". I'm always left wondering what on Earth these things are actually for. We have the Euler Gamma ...
9
votes
5answers
3k views

Why do we consider prime numbers important, and what are their applications other than number theory in pure math?

Why do we consider prime numbers important, and what are their applications other than number theory in pure math? I know that Number theory is devoted to studying prime numbers, but there must be ...
2
votes
3answers
626 views

notation of distribution

I have a question: Does $$N(0, x)$$ mean a normal distribution with mean $0$ and variance $x$? Or standard deviation $x$? The notation seems ambiguous sometimes.
1
vote
2answers
923 views

Struggling to Bridge the Gap (to Rudin's Principles of Mathematical Analysis)

After taking an introductions to proofs course and abstract algebra, I have been trying to study from Rudin's Principles of Mathematical Analysis. Unfortunately, I still find it very very difficult to ...
10
votes
1answer
570 views

Undergrad Student Trying to Figure Out What to Study

this is my first time on stack exchange and I am seeking advice for my future studies. Some background first; I am a undergraduate student pursuing a degree in mathematics and I hope to pursue ...
4
votes
1answer
573 views

What Are R-Modules Used For?

Kind of a simple question, but what exactly are R-modules used for? Do they have any engineering applications? EDIT: If it helps, I'll give some more context to the question... I am a graduate ...
2
votes
2answers
670 views

Recommendation for a precalculus textbook

I'm a high school senior interested in pursuing a major in quantitative economics, which I understand is heavily math-intensive. However, as it stands, my academic strengths are more verbal (780 on ...
9
votes
4answers
2k views

What do I need to know to understand the Riemann hypothesis

Which kinds of fields of mathematics do I have to know about in order to understand the Riemann hypothesis millenium prize problem?
0
votes
1answer
199 views

Combination Conceptual Understanding

I'm currently studying some combinatorics and I'm trying to understand combinations ${{n}\choose{r}}$ conceptually. I don't have trouble understanding permutations (n!) and r-permutations P(n,r) ...
7
votes
4answers
41k views

How to figure out the log of a number without a calculator?

I have seen people look at log (several digit number) and rattle off the first couple of digits. I can get the value for small values (aka the popular or easy to know roots), but is there a formula. ...
5
votes
6answers
2k views

A book for self-study of matrix decompositions

I am a third year math student and I noticed that there are many uses for decomposing a matrix (I mean decompositions like SVD, LU etc'). Is there a good book for self-study of the subject ? Note ...
5
votes
3answers
198 views

Dealing with many entities that need a symbol

What does one do when one needs a lot of symbols and one has exhausted the useful symbols of the latin and greek alphabets? (I say useful symbols because letters like iota (ι) and upsilon (υ) seem too ...
1
vote
6answers
193 views

Taking the derivative of $y = \dfrac{x}{2} + \dfrac {1}{4} \sin(2x)$

Again a simple problem that I can't seem to get the derivative of I have $\frac{x}{2} + \frac{1}{4}\sin(2x)$ I am getting $\frac{x^2}{4} + \frac{4\sin(2x)}{16}$ This is all very wrong, and I do not ...
6
votes
3answers
355 views

Characterizations of primes

Let $\mathbb{P}$ be the primes set. We know from Wilson's Theorem that $$(p-1)!\equiv-1 \pmod p \iff p \in \mathbb{P}$$ What another formulas we have with an if and only if ($\iff$) statement to ...
0
votes
1answer
215 views

Name of probability distribution

Does this distribution have a name: $f(x) = yx^{y-1}$ for $0 < x<1$ and $y>0$? It looks like an exponential distribution. Or is it a nameless distribution?