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3
votes
1answer
27 views

Spaces vs. Structures

Examples of spaces I've come across include vector spaces, inner-product spaces, and metric spaces. Examples of structures I've met include rings, fields, and groups. I have always understood spaces ...
6
votes
3answers
231 views

Can Number Theory be visualized?

So I was thinking about a hard euclidean geometry problem, when it hit me just how much more difficult it would become without the aid of a diagram. This got me thinking: Wouldn't it be great if we ...
2
votes
0answers
28 views

(Concrete) mathematical aspects of programming

It is often said that progamming is mathematics as it "makes use" of "discrete mathematics". However, I would like to ask a more concrete question: what are the concepts of a programming ...
1
vote
0answers
24 views

Benefits of combinatorial reasoning?

What I usually do instead of counting something, I form a polynomial whose coefficients count it and go from there. If you had to convince someone why they should learn combinatorial reasoning what ...
0
votes
0answers
12 views

Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have $$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ ...
2
votes
3answers
74 views

What is the smallest positive integer that has never been mentioned?

The set of positive integers is infinite. The set of explicitly mentioned positive integers is finite. Therefore, there is a non-empty set of positive integers that have never been mentioned, and ...
1
vote
1answer
19 views

Projective Geometry with Clifford Algebra - lost Inner Product

Projective geometry may be studied with the tools of Clifford Algebras by adding a new direction (see for example this article). But as far as I understand it, only blades and null spaces are used for ...
0
votes
1answer
21 views

What are equivalent parametric equations?

What are equivalent parametric equations? Is there a fast method to prove that 2 parametric equations are non-equivalent?
3
votes
1answer
24 views

Find a set of minimal natural axioms, from which we construct $\mathbb{Z}$.

I am interested in this question for teaching two very different kind of students. The first (less important to me) is students in their first year in the university. I wish to construct ...
2
votes
1answer
38 views

Is there a type of number sequence that has a nth number actually have multiple answers?

I am just looking for what this type of number sequence this is called? Example: The logic of the sequence is, take the previous numbers in the sequence and add them together in every possible way to ...
7
votes
2answers
180 views

Am I wrong for wanting to give up studying analysis?

I am writing this because I'm having a hard time studying for analysis. This is my second time taking a real-analysis course and from the beginning, somewhere deep down I thought that I might be ...
3
votes
2answers
85 views

How To Deal With Rude Math Prof? [on hold]

I am having difficulties in math, and every time I try to go to my math professor's office hours she always seems annoyed. When I ask her a question she always says something to the effect "I don't ...
2
votes
0answers
33 views

Important question: Can do a very hard problem, 2 months after have no idea how to approach it. Is it Normal?

I have this problem like sometimes I can do a very hard problem (I usually pass hours thinking and then there's this insight and everything opens up and the solution comes up eventually after some ...
-4
votes
0answers
74 views

Open problems of Clay Mathematics Institute. [on hold]

Mathematics research at the end of the $20$th century assumed unfathomable proportion as compared to the situation at the end of the $19$th century. The Clay role is to further the legacy of Hilbert's ...
0
votes
1answer
60 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
votes
2answers
33 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
votes
1answer
9 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
vote
2answers
32 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
274 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 ...
3
votes
0answers
39 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 ...
6
votes
1answer
95 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
vote
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
43 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 ...
0
votes
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 ...
0
votes
0answers
27 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
votes
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
votes
1answer
61 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
vote
0answers
19 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
votes
10answers
924 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
votes
0answers
56 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
votes
2answers
75 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
votes
0answers
50 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
votes
0answers
44 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
107 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
votes
1answer
276 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 ...
0
votes
0answers
23 views

which number satisfy the given relation in graph [closed]

I am unable to find the relation in the graph?
1
vote
0answers
21 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
vote
0answers
32 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
votes
1answer
68 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 ...
1
vote
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
votes
0answers
17 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
votes
0answers
53 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
votes
0answers
34 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
votes
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
19 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
votes
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
vote
0answers
29 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
votes
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 ...