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2
votes
0answers
32 views

Get function definition from an equation

My question: I have to find a function $g$ fulfilling the equation $$2\frac{t_k \cdot t_0 - 1}{t_k-t_{-1}} = g(t_k) + g(t_{k+1}) + t_{k+1}\cdot g(t_k)g(t_{k+1})$$ Whereby $t_{n+1}=t_n + h$ with $t_0, ...
10
votes
3answers
735 views

What's so special about the group axioms?

I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly arbitrary. I'm not sure what exactly it is about the ...
0
votes
1answer
64 views

Music of primes

In http://plus.maths.org/content/music-primes DuSatoy describes the relation between the prime number staircase and harmonics from music. So in the article he uses music as an analogy. But I wonder ...
4
votes
1answer
48 views

What are the subjects an analytic number theorist must be well versed with after undergraduate studies?

I am a mathematics major and I aspire to be an analytic number theorist. In general, what are the subjects an analytic number theorist must be well versed with after undergraduate studies (i.e. in ...
1
vote
2answers
151 views

Stochastic Processes Solution manuals.

Does anyone have a link or a pdf stash of solution manuals for stochastic processes ebooks? I am doing a self-study on this course and I can't seem to find any solution manual online to cross-check ...
0
votes
0answers
62 views

Importance of set theory

Currently I write an article about set theory (in German). Therefore I want to answer the question, why students need to learn something about sets. Among other reasons I have currently the following ...
57
votes
11answers
4k views

How do people who study intensely abstract mathematics “imagine” or understand the concepts they are studying or being taught? [closed]

This question is probably to the actual people who study such mathematics, rather than any "third-party". I haven't studied any such mathematics, but I can imagine that some (probably most of it) of ...
3
votes
1answer
61 views

Can differential equations have orders of derivatives that are dependent on the independent variable?

I was wondering if you can construct and/or solve a differential equation of the following form, $$\frac{d^{(f(x))}y}{dx^{(f(x))}}=g(x)$$ where $y$ is dependent only on $x$ and $f(x)$ is a function ...
2
votes
1answer
87 views

I find “closed formula” for divisors of an positive integer, it is useful or not? [closed]

I find this formula, but i don't know this is worth or not. $$d(n)=\sum_{i=1}^{n} \lim_{j\to\infty} (cos\left(\frac{\pi n}{i} \right))^{2j}$$ It is possible to improve it to deduce formula for $P_n$ ...
1
vote
1answer
51 views

How to write corollary of many results?

I have a few results that I want to write a corollary for, however I don't know where to include this. Should I write it under the last result as "Corollary of results A, B and C" (where C is the last ...
1
vote
1answer
49 views

Recommended books to purchase after a two-year break

I am going to begin my master's degree after a two-year break since my bachelor's degree and I am a little bit scared that I forgot a lot of things. Since I would rather purchase books than look back ...
3
votes
2answers
81 views

Understanding how to prove limit theorems for sequences.

How do you know what arbitrary value to choose for epsilon as in this document, and when should the triangle inequality be considered when writing your proof?
1
vote
2answers
39 views

Intuition behind group action on a set

In Algebra Chapter 0 the definition of a group action on a set is given as: An action of a group $G$ on a set $A$ is a set function $P:G\times A\rightarrow{A}$ such that $P(e_G,a)=a$ and ...
6
votes
2answers
164 views

How to intrinsically think about simplicial objects.

It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps ...
2
votes
0answers
21 views

Learning Logic and Metric Spaces

I want to learn how to manipulate metric spaces. I have been reading this book about metric spaces. ...
2
votes
1answer
34 views

Motivation behind Distributive Property

Distributivity is inherent in the definitions of rings, boolean algebras, etc. But why distributivity specifically? Let me qualify that question a bit and try to get to the bottom of what I mean by ...
2
votes
4answers
99 views

Counting numbers vs Natural numbers; Peano Axioms

I can feel that my question is going to be a somewhat lengthy one, but I will try my best to deliver it in as short a form as I can manage. So to begin, I've always thought that the numbers such as ...
0
votes
0answers
21 views

Standard notation for different variables?

I was wondering if there is a rule for choosing which letters to give to variables when naming or substituting them? In Gauss' Disquisitiones Arithmeticae, he uses many gothic lettering, but I haven't ...
15
votes
4answers
456 views

Books in the spirit of Problems and Theorems in Analysis by George Pólya and Gábor Szegő

In the Preface of the first German Edition of the book Problems and Theorems in Analysis by George Pólya and Gábor Szegő, one can read [emphasis mine] : The chief aim of this book, which we trust ...
1
vote
1answer
46 views

How to get the most of Euclid's Elements?

I have bought a sample of Euclid Elements and I wonder which will be the best way to study it. I know that it's a great classic and I dont want to miss anything. I will appreciate your help.
2
votes
0answers
63 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
1
vote
0answers
47 views

Seminar topic in Combinatorial Group Theory

I am doing a course on "Combinatorial Group Theory" and we have a choice of giving a presentation on topics related to the course, Course outline is as follows- Free groups: groups defined by ...
3
votes
0answers
81 views

A mathematical abstraction of colours?

Does any formal system exist built upon the mechanics of Additive Colour Theory? Additive Colour Theory is the description of the behaviour of the visible spectrum of light as it combines-the ...
0
votes
1answer
37 views

Information about the cyclic group $C_{p^n}$, where $p$ is prime and $n\geq 2$

My question is quite generic and I need it to have an idea of the knowledge of this group. In particular, I am interested in the extension of number fields whose Galois group is $C_{p^n}$ (if there is ...
8
votes
2answers
142 views

Why is it difficult to define n-category?

Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally. If we have a definition for 1-categories (category) and a definition for ...
2
votes
0answers
34 views

Symmetry of two-sided ideals

I was thinking about two-sided ideals and have some intuition-guided, soft questions regarding them. Since I don't have anyone to talk to about such subject matter, I thought I'd ask. Let a be an ...
5
votes
1answer
97 views

Why are fundamental theorems called fundamental?

I am just curious about some theorems that I had been used for some time. We have Fundamental Theorem on Calculus the Fundamental Theorem of Calculus. In number theory we have Fundamental Theorem of ...
3
votes
3answers
72 views

What is the relationship between the Archimedean Property and Calculus?

My textbook begin with a chapter dedicaded to the Real Numbers. It introduces firstly integers and rational numbers, then it introduces the irrational numbers, which appeared in the first place as a ...
2
votes
1answer
69 views

What is the importance of last axiom of vector space [duplicate]

What is the importance of last axiom of vector space. Axiom is $1u=u$.
7
votes
2answers
132 views

Which other fields of mathematics are relevant to modern set theory?

I am currently studying mathematics as an undergrad. Over the last year I discovered mathematical logic and set theory and took some courses in these subjects. Especially set theory is really ...
1
vote
1answer
52 views

Classification of Mathematical Objects

In Mathematics we are always classifying things up to some transformations, for example, we classify Vector Spaces up to isomorphisms (in the finite-dimensional case it's easy, there is a invariant ...
0
votes
1answer
65 views

What is some prerequisite of global differential geometry other than real analysis and advanced calculus?

What are some prerequisites for global differential geometry other than real analysis and advanced calculus? It introduces a lot of new definitions other than the ones from the local differential ...
0
votes
0answers
25 views

What should I know before reading Ordinary Differential Equations of VI Arnold?

I have just adquired some math books and VI Arnold is one of them. But it was more sophisticated that I may think, so I want to know what should I know to be able to read it. Actually I have on my ...
5
votes
1answer
115 views

Journals similar to The Mathematical Intelligencer?

I confess I am not sure if the following question is appropriate for this S.E.. If not, simply let me know and I will remove it by myself. As The American Mathematical Monthly may be viewed in some ...
1
vote
0answers
28 views

How to filter out non-mathematical results in online searching?

This similar question has been asked; yet I am still looking for an efficient way to filter out non-mathematical materials in an online searching. For example, is there any "mathematical" searching ...
2
votes
3answers
97 views

Good math books to get on with my education? [closed]

im looking for a set of good math books. Since i live in norway im going through the norwegian school system, where i choose to go whats called a electrical line. Im not gonna go to deep into the ...
2
votes
0answers
82 views

Visualising algebraic topology

I'm new to algebraic topology and although I can follow the arguments it would be nice to be able to visualise important concepts like homology and excision. Can anyone recommend a book or other ...
1
vote
1answer
23 views

Some websites for numerical values of orbit of $x$ under $f$ in a dynamical system

I don't know anything about programming. Is there some website to calculate the orbit of $x$ under $f$ in dynamical system? For example, if $x_{n+1}=2x_n(1-x_n)$, how to a have a list of first many ...
0
votes
3answers
81 views

Real-analysis--words of motivation [closed]

I am studying real-analysis (I have an exam tomorrow) and am suffering. Can anyone give me some words of motivation?
0
votes
2answers
58 views

Self-studying differential forms and tensors

I am interested in understanding the generalized Stokes' Theorem. From my understanding, this theorem involves differential forms and exterior algebra, and tensors (to some extent). I'm not ...
1
vote
0answers
56 views

What are some really cool problems that involve “least squares and Eigenvalue problems”?

I am required to find a research topic in this domain, so I'm really interested in finding out what kind of problems are covered in this domain, and how others are using these techniques to solve ...
4
votes
6answers
134 views

Notable examples of “impossible” results ruled out by earlier barrier or no-go theorems or widespread beliefs

there is a certain style of type of proof in mathematics something like a "barrier theorem" but which also relates to widespread mathematical beliefs/ "conventional wisdom". an example would be the ...
3
votes
0answers
354 views

Can I get a PhD in Stochastic Analysis given this limited background?

General advice on PhD apps welcome Given my limited background in stochastic analysis and other information (below), can I apply for a PhD with stochastic analysis for my dissertation topic? 1/4 I ...
1
vote
0answers
32 views

How a proof like this should be written?

I'm studying with Apostol's Calculus I, and it has required me to carry out some proofs, which I'd never done before (which has turned out to be pretty neat, by the way). When proving a statement ...
2
votes
2answers
55 views

Banach Spaces which are not $L^p$

Most of the times, when I think of Banach Spaces I think of $L^p$ spaces. I would like to know if there is any Banach space which can not be written as $L^p$ space. Please also indicate any ...
2
votes
0answers
88 views

What is the legacy of Bourbaki?

As I was preparing a short lecture (for amateurs) on the mathematics of the '900, I realized that this year marks the 70-th anniversary of the founding of the Bourbaki group. I remember that Bourbaki ...
4
votes
0answers
70 views

Asperger's and math [closed]

I am 19, recently diagnosed with Asperger's syndrome. Question to anyone here who has AS: Has your AS helped you or held you back while learning math? If so, in what ways?
4
votes
1answer
76 views

What's With The Diagonal Morphism?

Given a morphism $X \to Y$ of schemes, we can construct a diagonal morphism $\delta: X \to X \times_Y X$ via the universal property of the fiber product applied to the identity map $X \to X$. ...
1
vote
1answer
43 views

Beyond calculating just as an exercise, are there any other reasons to calculate an indefinite integral?

Calculus textbooks often have us calculate indefinite integrals as exercises. However, beyond calculating just as an exercise, are there any other reasons to calculate an indefinite integral?
0
votes
0answers
20 views

Book for computational classes

Recently I started reading about computational classes in quantum mechanics. I just have basic experience in theory of computation. ( I am reading sipsers currently ). The papers I am reading keep on ...