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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3answers
129 views

a good modern topology book

I want to study an advanced modern book on topology, but I couldn't find any. I've already studied the first chapters of Munkres' book, but it is not as advanced as books such as Engelking's topology, ...
0
votes
0answers
70 views

Reference request for well known theorem in combinatorics

From where, I can find the proof of the following theorem. I have to to cite it, in my research article. Theorem: The combination $ {n} C {r}$ is the number of possibilities for ...
0
votes
0answers
25 views

Monotonic convergence of Newton's method for boundary value problems

I’m interested in solving nonlinear elliptic boundary value problems of the type $$ -a\Delta u + f(u) = 0, $$ $$ u|_\Gamma = u_0 $$ by Newton’s method when its convergence is global and monotonic. ...
3
votes
1answer
27 views

Statistics books with motivation and historical tidbits about the development of the concepts?

I've seen some questions asking books on statistics. I'm looking something a little different, I'm specifically looking for a book in statistics that teaches the important concepts well and also ...
0
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2answers
29 views

If $u$ is a Sobolev function then $\nabla u = 0$ on $\{ u = c\}$.

There is a result of the form: If $u$ is a Sobolev function on some domain then $\nabla u = 0$ on $\{ x \mid u(x) = c\}$ where $c$ is constant. Can someone point me to a specific reference? I ...
0
votes
1answer
42 views

Examples of how abstract algebra is used to find concrete solutions to a mathematical model?

All references I've seen to abstract algebra show how it helps in the representation of mathematical models...are there any examples of using abstract algebra to calculate actual solutions to a ...
0
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0answers
22 views

Looking for resources on these topics from Linear Algebra

I am looking for Characteristic roots and characteristic vectors of a linear transformation or of a matrix, Algebraic and Geometric multiplicity of a characteristic value, Cayley-Hamilton theorem, ...
0
votes
1answer
79 views

Questions on CW-complexes

I am trying to proof the following two statements. If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a ...
0
votes
1answer
11 views

Differentiability of Regulated Functions

In general, what can be said about differentiability of (real-valued) regulated functions, i.e. such for which the left and right limit exist at every point? Such functions are necessarily continuous ...
0
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0answers
28 views

Need recommendation for following topics in combinatorics

I have to do following topics for my exam .I have 2 months time .However i have never done any combinatorics except that of high school (Permutations ,Combinations etc ) .I want a book which covers ...
9
votes
1answer
162 views

Ramanujan's transformation formula connected with $r_{2}(n)$

Let $r_{2}(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here the sign as well as order of summands matters. Also by convention ...
2
votes
2answers
65 views

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: "In the limit, a sequence of regular polygons with an increasing number of sides becomes ...
0
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0answers
20 views

References for microlocal versions of some theorems

I am trying to introduce myself into Microlocal Analysis. In particular, motivated by some results in Inverse Problems, I would like to find good references for the microlocal versions of Helgason and ...
4
votes
2answers
59 views

Order Properties on Open Sets

Considering the subset order on the open sets of a topological space, it seems natural to ask what kind of total orders exist as suborders of the subset order. One possibility is that each total order ...
0
votes
0answers
12 views

Sub gaussian concentration for Lipschitz functions

It is well know that: if $f:\mathbb{R}^m\to\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$, and $X_1,\dots X_m$ are i.i.d random variables s.t. $X_i\sim N(0,1)$, then for any $t>0$ ...
3
votes
2answers
125 views

Functional Analysis Question?

I have a question about functional analysis. I know in finite dimensional space $\mathbb{C}^n$, all bases have the same cardinality. However let us consider $L^{2}[-\pi,\pi]$ which has TWO bases ...
5
votes
0answers
67 views

Infinite Sums which turn out to be Riemann Integrals

I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be $$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots ...
1
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0answers
55 views

Recommendations for a thorough logic textbook

I'm looking for a (possibly introductory) textbook on logic that covers the motivation behind conventions in logic, like the definition of the implication. Prof. J. Lau has an excellent webpage, ...
0
votes
0answers
9 views

A reference to study Boundary conditions of diffusion processes

I am trying to learn about Wentzell Boundary condition and (A,L) diffusion in the sense of Watanabe's paper (On the existence and uniqueness of diffusion processes with Wentzell's boundary condition ...
0
votes
1answer
19 views

An inductive limit of amenable groups is amenable

It is a Theorem that an inductive Limit of amenable Groups is amenable. Could someone sketch me a proof of this, or give me a reference? I couldn't find one. Thanks in advance. Edit: I wanted it for ...
6
votes
2answers
145 views

Cancellation problem: $R\not\cong S$ but $R[t]\cong S[t]$ (Danielewski surfaces)

I would like to understand why the two rings $$ R={\mathbb{C}[x,y,z]}/{(xy - (1 - z^2))} \\ S=\mathbb{C}[x,y,z]/{(x^2y - (1 - z^2))} $$ are not isomorphic, but $R[t]\cong S[t]$. This example is ...
5
votes
0answers
54 views

a new(?) operation using products of multiplicities

Does the operation $$n \odot m := \prod_{p \text{ prime}} p^{v_p(n) \cdot v_p(m)}$$ on positive integers have a common name? Has this operation been studied somewhere? Notice that $\odot$ is ...
2
votes
0answers
15 views

Understanding $SL_3(D)$ where D is a central division algebra

Suppose that $K$ is a non-archimedean local field of positive characteristic and $D$ is a four-dimensional central division algebra over $K$. The group $SL_{3}(D)$ can be embedded as a $K$-form of ...
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0answers
21 views

Topological entropy and degree of smooth mappings

Where can I find the literature "Topological entropy and degree of smooth mappings" by Misiurewicz. Thanks for any help.
0
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0answers
15 views

Does the Laplacian commutes with elements of the basis of the Lie algebra?

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with ...
3
votes
1answer
73 views

Explanation of a joke on abelian groups (grapes).

Q: What's purple and commutes? A: An Abelian grape. Q: What is lavender and commutes? A: An Abelian semigrape. Q: What's purple, commutes, and is worshipped by a limited number of people? A: A ...
2
votes
1answer
22 views

Convergence of a sequence of subspaces

Let $E_n\subset \mathbb R^n$ be a sequence of subspaces. What does it mean $E_n$ convergence to a subspace $E\subset \mathbb R^n$? I saw this when reading about hyperbolic sets. Where can I read ...
5
votes
1answer
56 views

A product version of Riemann integral

Motivated by Riemann sum in Riemann integral and motivated by relations between infinite series and infinite products we ask: Assume that $f:[0, 1]\to \mathbb{R}$ is a positive function. Assume ...
1
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0answers
27 views

Fourier coefficients of eigenforms

How does one prove that the fourier coefficients of a normalized eigenform for Hecke operators $T_p$ on $S_k(N)$ all lie in a fixed number field? If the proof is lengthy, a reference to a book that ...
2
votes
0answers
22 views

Terminology for the difference of real dimension and scheme-theoretic dimension

Consider the scheme $\mathrm{Spec} \left(\mathbb{R}[x_1,\cdots,x_n]/(x_1^2+\cdots+x_n^2-a)\right)$ where $a$ is a real number. Scheme-theoretically, this has dimension $n-1$. But the dimension of the ...
7
votes
2answers
243 views

Recommendation for books on topology (light reads)

Are there any books on topology which can be read without having to do any exercises and look up definitions every second line? Something to read while relaxing, and not meant to replace a textbook ...
1
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1answer
45 views

Is there any PDE that applies specifically to Number Theory?

Given the advanced results obtained by analytic means in Number Theory, it puzzles me why I don’t recall ever seeing a partial differential equation used to good effect in Number Theory. Is there ...
6
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2answers
537 views

Are Vector Spaces over non-commutative fields ever studied?

Are Vector Spaces over non-commutative fields ever studied? If not, why? If they are, I'd like to learn a bit about them, could you guys recommend me a good book on the subject? Cheers.
1
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0answers
25 views

Books that introduce many subjects?

I am looking for a rigorous math textbook that introduces many subjects in university (undergraduate) math. One book that seems to meet this criteria is Alan Beardon's Algebra and Geometry but I ...
2
votes
0answers
29 views

Sources for simple probability brain teasers

I am searching for a book that can supply me with probability brain teasers, that can be solved using little arithmetic/mental math, paired with somewhat detailed solutions. Any suggestions? ...
-2
votes
3answers
47 views

Book to learn Mathematical Probability theory? [closed]

What are some good references to , good book to learn Mathematical Probability theory ? Please help .
1
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0answers
30 views

Book to learn Darboux integral

What are some good references to , good book to learn Darboux integral ( https://en.wikipedia.org/wiki/Darboux_integral ) ? Please help .
1
vote
1answer
50 views

Composition operators on fractional-order Sobolev spaces

Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication provided $s > 1/2$. This is proved for ...
3
votes
1answer
43 views

Multicategories with out-arities

Basically, my question is: Why the emphasis on domains in the notion of multicategory? I will now give the formal framework to state it correctly. Passing from categories to multicategories ...
10
votes
1answer
169 views

Estimating the “size” of the mathematical research literature

The other day I was telling one of my friends that mathematics, as a living science, possesses quite an extensive research literature. How extensive then, she asked. Unfortunately, I didn't have ...
3
votes
1answer
60 views

(Nonunique) Solvability of Sylvester Equation

I am interested in stating existence of solution of a Sylvester equation $$ AX - XB = C, $$ where $A$, $B$, $C$, and $X$ are $(n,n)$ matrices. Existence of a unique solution $X$ is given, if $A$ ...
1
vote
0answers
28 views

largest distance between vertices on a polyhedron

I have a polyhedron defined by m inequations and n unknowns. I am interested in the largest distance between two vertices (the number of edges I have to follow from one vertex to another). I am ...
4
votes
1answer
59 views

Probability theory required for learning statistics rigorously

I would like to learn statistics rigorously. The only book that I can find that seems to do statistics rigorously is this book "Theory of statistics" by Schervish (which seems advanced): ...
1
vote
1answer
52 views

Anyone know a good standard reference for Lie group and Lie algebra facts?

I'm writing something and I need to refer to a mathematical fact; unfortunately I got it from Wikipedia, which does not source the specific piece of info! It relates to a choice of simple roots for ...
2
votes
1answer
52 views

literature on advanced calculus [closed]

I need your opinions on this particular textbook: Advanced Calculus by Robert C. Buck. In my first year in college I finished two semesters of single-variable calculus and now I'm looking for a proper ...
1
vote
2answers
54 views

A road-map through “Combinatorial Set theory: With gentle intro to independence proofs”

I'm going to study independence proofs form Halbeisen's book. It seems that some material is not needed to study independence proofs, so it seems that the book contains more material than my needs. ...
1
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0answers
25 views

Any online videos on a course taught from Munkres?

Are there any vidoes available on the Internet --- for watching online or for download --- of any (general) topology course taught using the book Topology by James R. Munkres, 2nd ed? If so, please ...
0
votes
1answer
57 views

Mathematics in Chemistry [closed]

What are the applications of mathematics in chemistry, biology ? Please give some references including text books if any.
-12
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2answers
118 views

Ayn Rand and athematics [on hold]

I am an honors undergraduate in mathematics. I have taken an interest in objectivism. I came across a discovery of Ms. Ayn Rand's in mathematics: In a triangle the inscribed circle touches the ...
0
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0answers
20 views

Eigenvalues of normalized adjacency matrix

Can anyone introduce some references on the eigenvalue estimation of normalized adjacency matrix, i.e., $W=D^{-1}A$ ($D$ is the degree matrix and $A$ is the adjacency matrix of the corresponding ...