This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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5
votes
1answer
74 views

Pretty lower bound on the gamma function

According to http://functions.wolfram.com/06.05.29.0006.01, for every $x\geq 2$ it is $$ \left( \frac{x}{e}\right)^{x-1} \leq \Gamma(x) \leq \left( \frac{x}{2}\right)^{x-1}, $$ where $\Gamma$ is the ...
2
votes
0answers
30 views

Topology/Geometry book for self study

I'm looking for a book on topology and geometry that is suited for self study. That means I would like it to include exercises and solutions. Could you kindly recommend some literature?
2
votes
2answers
47 views

Book for Hilbert spaces.

Which book either on functional analysis or specifically for Hilbert spaces has the best way of explaining with most examples and to the point without much applications. I studied Limaye's book and ...
0
votes
1answer
30 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
3
votes
1answer
41 views

Subjects or recent progress in Tropical geometry or similar suitable for undergraduate investigation

I'm worried this might not be fitting for this forum, but it's basically a literature and reference request. I'm looking to do a project in algebra where we are supposed to research some topic ...
8
votes
10answers
350 views

Real analysis book suggestion

I am searching for a real analysis book (instead of Rudin's) which satisfies the following requirements: clear, motivated (but not chatty), clean exposition in definition-theorem-proof style; ...
2
votes
0answers
45 views

Shortlist of problems in linear algebra

A while ago I remember seeing a very nice shortlist of problems in linear algebra. It was a list of about 40-50 problems. The idea was that if you solve them, you learn linear algebra very well and ...
1
vote
1answer
14 views

Behavior of Points on Conformal Mapping Boundary

Carathéodory's theorem states that given a conformal mapping $f: J \to D$ from a Jordan region to the unit disc, we can extend this to a homeomorphism from the Jordan curve bounding $J$ to the unit ...
4
votes
2answers
104 views

Ordinal exponentiation identity with natural sum of exponents

This is related to a previous question on How to think about ordinal exponentiation? One possible definition for the natural product $\alpha\otimes\beta$ of ordinals is based on Cantor Normal Forms ...
0
votes
0answers
44 views

Geometric Meaning of Luna's Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
3
votes
2answers
75 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
3
votes
2answers
90 views

Claim: Mathematical models of the economy have thousands of variables

A quote from the book Linear algebra done right by Axler is as follows: "Mathematical models of the economy have thousands of variables" I find this hard to ...
1
vote
1answer
61 views

Chapter dependency tree for Hartshorne's Algebraic Geometry

I'm self-studying Hartshorne's Algebraic Geometry and I need some guidance. I've studied chapter I (varieties) and sections 1, 2 and 3 of chapter II (schemes). Do I need to study all sections in ...
1
vote
0answers
31 views

When does it become impossible to lose a game of FreeCell?

After spending more time than I should playing FreeCell, I've been wondering about the following: At what point in the game does it become impossible for someone to lose? "losing" probably isn't ...
2
votes
0answers
23 views

counting function for minima of continuous function?

Given an absolutely continuous function $f(x)$ with $x\in[a,b]$ and $f(x)\geq 0$ (e.g. a signal pack), I am trying to deduct analytically another function $g(x)$ which counts (similar a step ...
2
votes
1answer
51 views

“Half-primitive root”?

I've made a topic related to this (but containing a different question) and it got no responses, so I was wondering if I've stumbled on something new or if it's obvious and I'm just not seeing it at ...
2
votes
2answers
41 views

Measure-theoretic analog of homeomorphism and isometry

If $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces and $f:X\to Y$ is a continuous bijective function between them such that $f^{-1}$ is also continuous, then the two topological spaces are said ...
0
votes
0answers
37 views

History of a result from Bézout

BÉZOUT'S THEOREM: Let $F$ and $G$ be projective plane curves of degree $m$ and $n$ respectively. Assume $F$ and $G$ have no common component. Then $\displaystyle\sum_{P}I(P,F\cap G)=mn$ $I(P,F\cap ...
0
votes
1answer
67 views

How to deal with probability problems in proper class sample spaces?

Here my focus is mainly on $Ord$ and questions such as: An ordinal is chosen by random. What is the probability of the event that it is a cardinal number? A coin is tossed proper class many ...
2
votes
0answers
37 views

Prerequisite to study smooth 4-manifolds

I am quite interested in understanding smooth 4-manifolds. What are the necessary prerequisites in order to start my study? Also can you please suggest me some good books from where I can start? ...
6
votes
3answers
68 views

Video Lessons in Complex Analysis

Does anybody have some link for good video lessons of a complete course in Complex Analysis? Grateful.
1
vote
1answer
46 views

Accounts of the proof of Fermat's Last Theorem

I would like to collect a set of references to pieces of Wiles' 1995 proof of Fermat's Last Theorem. Has anyone recompiled the proof into another paper? Are there any books or articles that describe ...
3
votes
1answer
174 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

Could you point out some references (undergraduate level) that give a geometric understanding of vector spaces, matrices, and linear applications? As far as I know, many textbooks start with an ...
1
vote
1answer
41 views

Definition of unipotent linear algebraic groups over non algebraically closed fields

Suppose we have a field $F$ with $\text{char}\ F=0$ and $F$ is not necessarily algebraically closed. What is the definition of a unipotent linear algebraic group over $F$? I'd really appreciate ...
0
votes
1answer
42 views

Inequalities textbook request [duplicate]

At university I have got a problem set with lots of inequalities. Unfortunately there are no explanations given how to do them. In Highschool we only did very easy inequalities. Therefore I am looking ...
0
votes
1answer
14 views

Relating maximal elements of downsets to minimal elements of the complement

Denote by $\mathcal{P}(S)$ the set of non-empty subsets of a finite $S$. Suppose that $A\subset \mathcal{P}(S)$ is a downset, i.e., every subset $Q$ of any $P\in A$ is also contained in $A$. We can ...
0
votes
1answer
53 views

Any algorithm or theorem to decide whether two functions are equivalent? [duplicate]

Any algorithm or theorem to decide whether two functions that are polynomials,rationals and analytic over $\mathbb{N}$ or $\mathbb{Q}$ or $\mathbb{R}$ or $\mathbb{C}$ are equivalent ?
1
vote
0answers
17 views

Combinatorial designs give triangulations of complete graphs

I recently attended a talk on combinatorial design theory. The speaker mentioned briefly that the Fano plane, and other designs give rise to triangulations of complete graphs (the Fano plane gives a ...
0
votes
2answers
74 views

Adding Substructures by Forcing

Consider a first order language $\mathcal{L}$ and an $\mathcal{L}$-structure $M$. Let $V$ be a model of ZFC (or ZF) the general question is that what would happen to classes, $Sub(M):=\{N~;~N~\text{is ...
0
votes
0answers
11 views

Reference: Differential operators and principal symbols

I am looking for good references about differential-/pseudodifferential operators and principal symbols. thanks
0
votes
0answers
17 views

Good Convex Analysis book

I am an average student and have just a very basic knowledge of this subject. Thank you
0
votes
0answers
26 views

Ref. request for Linear algebra over noncommutative rings

This is not a real question but more a reference question. I am looking for introductory articles/blog entries/books that discuss the obstructions to do linear algebra over a non-commutative ring $R$. ...
2
votes
1answer
54 views

Infinite direct sum of Hilbert spaces

Let $\{H_i\}_{i \in I}$ be an infinite collection of Hilbert spaces. I am trying to understand their "Hilbert space direct sum". $\bigoplus H_i$ (algebraic sum) is an inner product space in a ...
8
votes
0answers
106 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, ...
0
votes
0answers
37 views

Good source for my “language of math” class?

I'm having a hard time in my "language of math" class (proofs, sets, etc). Right now we're doing finite sets. Are there any good online resources for this class? Thanks
1
vote
3answers
82 views

Book recommendation for ordinary differential equations

This question has been posted before, but I need book with specific qualifications. I do not need books for engineers, book that is centered around calculations and stuff. I need to find a book that ...
0
votes
0answers
22 views

Compact hypersurface in $\mathbb{R}^n$

Let $S$ be an $(n-1)$ dimensional hypersurface in $\mathbb{R}^n$. If we say that $S$ is compact, does this necessarily mean that $S$ has no boundary? Eg. $S$ can be a sphere but not a sphere cut in ...
2
votes
1answer
24 views

Reference for this theorem: $a, b$ coprime, $f(k) := ka \bmod b$, then $f$ is bijection on $\lbrace 0, …, b−1 \rbrace$.

I need to use the following theorem in a paper but have to expect that some of the audience (physicists) is not familiar with it, so I would like to reference it: Let $a$ and $b$ be two coprime ...
1
vote
0answers
17 views

Equivalence of condition on function

In a book I am studying it states that a condition on a function $g$ as follows: Given the function $g: \Omega \times \mathbb{R} \mapsto \mathbb{R}$ is a Caratheodory function satisfying $$\sup_{|u| ...
0
votes
0answers
16 views

Standard examples of operator

In a text I am reading it says that we can consider an operator $A: X \rightarrow X^{*}$ (where $X := W^{1,p}(\Omega)$) which is defined as $$Au = -\text{div}(a(x,u,\nabla u))$$ where $a: \Omega ...
5
votes
0answers
47 views

Real affine variety of $d$ orthonormal vectors in $\mathbb R^n$

I'm interested in the affine variety $$ V = \left\{ \, A\in \mathbb R^{d\,\times\, n} \, \middle| \, A\,A^T = I \, \right\} \subseteq \mathbb R^{d\, \times\, n}, $$ where $n\ge d$ and $I$ is the ...
6
votes
2answers
88 views

Proving if A and B are matrices such that A, B, and AB are normal, then BA is also normal.

If A and B are matrices such that A, B, and AB are normal, then BA is also normal. I've seen this statement around, although I've only seen the site/publication/etc... state that it was proven by ...
1
vote
1answer
54 views

Fixed Point Equivalences of Axiom of Choice

Axiom of Choice has many known equivalences. Also there are many known fixed point theorems (unproved statements) which provide useful information about existence of fixed points for particular ...
4
votes
1answer
86 views

Theorems which later turned out to be vacuous

Has it ever happened that a theorem of the form If $P$, then $Q$ was proven and published, perhaps with great difficulty, only for someone to realize later that the condition $P$ of the theorem ...
1
vote
1answer
25 views

Free loop space of classifying space as a disjoint union of classifying spaces of centralizer proof reference request.

I am looking for a reference for the proof or explanation of why for a discrete group $G$ we have that the free loop space of its classifying space is the disjoint union of centralizeers of $g$ where ...
17
votes
6answers
355 views

Open source lecture notes and textbooks

This question is inspired by the popular "Best Sets of Lecture Notes and Articles". Indeed, I would like to collect a "big-list" of open source (that is, with $\LaTeX$ code available) high-quality ...
0
votes
0answers
12 views

anisotropic vector valued function definition

Does anyone have insight into the significance of the following statement which defines an anistropic function $$a:\Omega \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$$ The ...
1
vote
1answer
81 views

series and inequality

I found this homework in an old paper written Let $n\in \mathbb{N}^*$ and $x_1,x_2,\ldots,x_n \in \mathbb{R}$ such that $ \sum_{i=1}^{n}\left|x_{i}\right|=1$ and $\sum_{i=1}^{n}x_{i}=0$ ...
0
votes
1answer
35 views

What is the difference between Difference equations and Recurrence relations?

Is there any difference between Difference equations and Recurrence relations? Some people are use them as difference equations and some are use as recurrence relations. I couldn't find in anywhere. ...
4
votes
0answers
86 views

Flatness of homomorphisms of graded-commutative rings

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...