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0
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1answer
24 views

Definable real numbers

Reading this Wikipedia page I found this definition: A real number $a$ is first-order definable in the language of set theory, without parameters, if there is a formula $\phi$ in the language ...
2
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2answers
25 views

Book recommendation for a new student on complex analysis

Please consider the following topics 1.Analytic functions 2.Cauchy's theorem and Cauchy Integral formula 3.Maximum Modulus Principle 4.Laurent Series 5.Singularities 6.Theory of residues and ...
0
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1answer
7 views

online notes on symmetric spaces

Can anyone suggest some good online lecture notes on symmetric spaces? I am interested in reading from Helgason, which is a very tough book to read. So I am searching for some places where the ...
1
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2answers
30 views

What does an ideal generated by a subset look like?

I am acquainted with principal ideals relatively well but I am struggling to understand the more general definition of ideals generated by an arbitrary subset of a ring. I suppose my question comes ...
4
votes
3answers
265 views

How to check my answer in combinatorics problems

Combinatorics problems (combinations and permutations) are an absolutely maddening subject for me. I can seem to work my way to the answer, provided I already know the correct answer. However, I can ...
2
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0answers
36 views

Generalizing in Mathematics

I was reading the book "Fermat's Last Theorem" by Simon Singh when it hit me that this theorem is so contrived, andyet it lead to several important breakthroughs in mathematics and especially the ...
5
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1answer
77 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
1
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0answers
18 views

Is it more useful to study lots of methods/theorems or work with details?

Suppose one has to learn a new subject on book that contains hundreds of pages and hundreds of problems. Is it more useful to learn book by reading it and skipping those parts I don't understand and ...
1
vote
1answer
42 views

Why do some authors use terms “non-ascending” and “non-descending” instead of ascending and descending?

In my math book, everywhere the author has used "non-ascending" instead of descending and "non-descending" instead of ascending. I was wondering if there is some special meaning or use associated with ...
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2answers
26 views

Books on Lebesgue Integration

I am having Measure Theory as a subject in my course.It is having these as topics: 1.Lebesgue measure on the line 2.Measurable functions 3.Lebesgue integral 4.Convergence almost everywhere ...
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0answers
26 views

Different methods used to show the existence of integer solutions

Let $A_{n},B_{n},C_{n}$ be three sequences of positive integers. I want to know the different methods used to show the existence of integer solutions $x$ and $y$ for the equation: ...
2
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2answers
64 views

Request for apps for Mathematical Drawing [duplicate]

I have been for long looking for some software apps which can help me draw various mathematical and geometrical figures and drawings.Can someone please tell me something about these which will run on ...
0
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1answer
58 views

Understanding the condition of $\diamondsuit^+$

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which ...
1
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0answers
18 views

Etymology of the term coherent in sheaf theory

Why has been introducted the term coherent in sheaf theory? What's its intuitive significance? In italian we translate it with the term "coerente", in the sense of compactness.
20
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10answers
908 views

Surprising applications of topology [on hold]

Today in class we got to see how to use the Brouwer Fixed Point theorem for $D^2$ to prove that a $3 \times 3$ matrix $M$ with positive real entries has an eigenvector with a positive eigenvalue. The ...
2
votes
0answers
23 views

Red-Black tree - “Insert-Delete” [on hold]

I am looking at red-black trees. Unfortunately in my lecture notes, the operations "Insert" and "Delete" are not well explained. Could you explain to me steps that we have to do for these two ...
3
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0answers
50 views

Combinatorics project ideas for high school students

It's that time again! Last year I asked for high school project ideas in the area of algebraic geometry, this year it's combinatorics (you can include graph theory and combinatorial game theory if you ...
0
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2answers
72 views

Universe as a finite 3-manifold without boundary

My question is soft and imprecise, as I know very little differential topology. Nevertheless, I hope it makes some $\epsilon>0$ of sense. Assume the Universe is a 3-manifold without boundary, ...
0
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0answers
49 views

Taking Putnam as a freshman.

Currently, in 11th grade, I've always thong about participating exams like the Putnam. I have however, sent the problems, and they seem, to be grueling hard!! I have access to problem solving ...
2
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0answers
43 views

What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
3
votes
1answer
100 views

How important is the own talent for research of your PhD supervisor?

Currently I am in the process of finding a PhD. Some potential supervisors are more didactical than others, some are nicer and warmer than others, and some are more famous mathematicians than others. ...
7
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1answer
274 views

The Purpose of Master Thesis

I am posting this question in the aftermath of the earlier posting in this link. Here are what I would like to know more about master and PhD thesis: (1) I understand that schools' math departments ...
-1
votes
1answer
64 views

Best schools for commutative algebra [on hold]

I will be applying to graduate programs this fall and I was curious which schools have the best commutative algebra groups. I know berkeley and michigan are up there, but what are others?
0
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0answers
20 views

which number satisfy the given relation in graph [on hold]

I am unable to find the relation in the graph?
1
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0answers
20 views

Different notation for Jacobi symbol

Is there a different, sort of established, notation for the Legendre / Jacobi / Kronecker symbol $\left(\frac{a}{b}\right)$? If yes, where is it used (in which texts)? I'm asking, because I ...
1
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0answers
31 views

Is Problem Solving Strategies by Engel sufficient?

Is a book like, Problem Solving Strategies by Arthur Engel sufficient for the Putnam Exam or should I consult something else? I asked a similar question asking for recommendation, no one discussed ...
0
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1answer
67 views

Book recommendation for Putnam/Olympiads

I have been concentrating on olympiad questions, and PUTNAM exams, Putnam is my main focus. Can you suggest a book from one of these: Problem Solving Strategies By Arthur Engel Putnam and Beyond by ...
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0answers
24 views

Understanding the Jacobian past calculus

What's taught in calculus: In the calculus of multiple variables I learned that the Jacobian $$\textbf ...
0
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0answers
16 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
2
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0answers
52 views

What computations would advance math knowledge a lot?

Suppose we where given a super computer that would be capable of computing anything, but only for one day. We could for instance compute many of the Ramsey numbers. What would be some computations ...
0
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0answers
32 views

Mathematically compare two sets of oscillatory data

I need to compare two sets of oscillatory data from a biological process. The data is quite noisy. Both sets have a similar frequency spectrum. I wonder what is the appropriate mathematical tool to ...
0
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0answers
43 views

What is umbral calculus, really? [duplicate]

I've seen this page on umbral calculus as well as wikipedia and and another question asked on this website (What's umbral calculus about?), but I still cannot realize what really umbral calculus ...
0
votes
1answer
18 views

Motivation for the binary entropy function

What is the motivation for the definition of the binary entropy function $H(x) = -p\log_2(p) - (1-p)\log_2(1-p)$? I understand that we want the entropy to be zero at $p = 0$ and $p = 1$ (no ...
0
votes
1answer
38 views

Study of systems of Linear Differential Equations?

Is there any area of mathematics that deals with and formalizes systems of Linear DEs, akin to how Linear Algebra deals with systems of linear equations? Does it provide any insightful results?
0
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0answers
22 views

Modeling predator/prey pursuit behavior?

I want to program a model of predator/prey pursuit behavior on a 2D topographical map (not predator/prey diff. eq. model!), primarily for fun, but I'd love to know if there's any literature or ...
1
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0answers
28 views

Solutions $n^2 = -1 \mod (p_n-1)$

Consider the equation $n^2 = -1 \mod (p_n-1)(*)$ where $p_n > n$ and $f(n) = p_n$ is the largest prime that satisfies the equation. $f(n)$ gives $p_n$ assuming there is a solution to the equation ...
0
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0answers
26 views

Naive explanation of the key concept of Teichmuller theory

I heard that Teichmuller theory has a great array of applications, is an active field of research, and has great relevance to other branches of mathematics. However, I have not the necessary ...
3
votes
1answer
105 views

New primality test, now what (publishing and proof)? [closed]

Over my research, I found a new relatively simple way to calculate whenever a number is prime or not. What's exciting is that it runs in $O(\log^2 n)$ running time (where $n$ is the number of digits ...
1
vote
0answers
88 views

How do you remember “Mathematics” [closed]

Sorry for a bad title/a silly question. How do you remember various theorems/techniques in mathematics especially advanced topics where there is no exercise or topics on what you have not done formal ...
1
vote
2answers
33 views

Significance and physical meaning of diagonalization of linear maps and bilinear forms, eigenvalues and eigenvectors

In linear algebra, I have studied the diagonalization of a linear map and of a bilinear form; and also the concepts of eigenvalues and eigenvectors. However, the importance of diagonalizing a linear ...
1
vote
1answer
38 views

Completely self-contained (and as elementary as possible) introduction to Teichmuller Theory

Can you recommend a completely self-contained and elementary (as much as it can be) introduction to Teichmuller Theory?
-2
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0answers
47 views

Are these Putnam ranges actually accurate? [closed]

From Putnam 2013 Score cutoffs, statistics You see that fifth-place is $88/120$. Is that actually accurate? many people score from 20-40 points, so then I would assume fifth place at least to be much ...
7
votes
3answers
118 views

Moscow State Oral Exam

I have heard that during the 1960s, prospective students had to take an 'Oral Maths' exam (alongside written maths, physics and Russian literature). I having trouble imagining what type of exam this ...
1
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0answers
49 views

Advance mathematics but not basic mathematics? [closed]

Here is the issue, As a high school student I have explored several area of mathematics, complex analysis, real-analysis, number theory, but now I cant seem to do simple SAT questions!? A simple ...
1
vote
0answers
38 views

For a given integer $n>1$ , for which type of rings $R$ is it true that $(xy-yx)^n=0 , \forall x,y \in R \implies R$ is commutative?

For a given integer $n>1$ , for which type of rings $R$ is it true that $(xy-yx)^n=0 , \forall x,y \in R \implies R$ is commutative ? (It is obvious indeed that if $R$ is an integral domain or a ...
1
vote
1answer
34 views

How should I think when combining multiple inequalities?

When reading/writing papers, I have always find it not obvious when two or more inequalities are combined. For example, taken from my current research $$\text{Pr}(X \le ab) \le -a (1-p)^{-N} (1 - ...
-4
votes
4answers
169 views

How much bigger is 1 than 0? [closed]

Bear with me -- nonsense to ensue. How much bigger is one than zero? The obvious answer is one. One is one bigger than zero. The backstory: Greg Fishel on WRAL said something along the lines of ...
1
vote
1answer
34 views

Physically, what meaning have Taylor series which have their lower order terms equal to zero, but their higher order terms non zero?

Usually, when using a Taylor series to describe a function (which may itself be a model of some physical phenomenon), we often throw out the higher order terms, as they are quite small relative to the ...
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0answers
88 views

useful exact sequences [closed]

There are some exact-sequences or long-exact-sequences that are great help in proving to prove some surprising theorem, or have some interesting applications. What's your favorite exact ...
0
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0answers
24 views

Collection of solved problems in linear algebra [duplicate]

Apart from Schaum's 3000 Solved Problems in Linear Algebra, what are some good collections of worked problems in linear algebra?