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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17 views

Upper bound number of self-avoiding walk of length $n$

A self-avoiding walk is a sequence of $n$ neighbor sites on a graph which are all distinct. Let $C_n$ be number of self-avoiding walk of length $n$ in a $d$-dimensional regular lattice. As a ...
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1answer
24 views

path to (co)homology

I want to understand (co)homology, but in the research I've done, it seems as though despite being 'birthed' within algebraic topology, its usefulness has allowed it to permeate throughout all ...
1
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0answers
18 views

Reference for analytic deformation theory

I am reading about deformation theory. I am treating mostly the algebraic case, but I would like to know a bit about all facets of this field of mathematics, so the geometric case is also of great ...
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0answers
7 views

Reference for $q^{\eta\sum_{i}t_{\lambda_{i}}\otimes t_{\mu_{i}}}$ part of an expression for universal $R$-matrix of a quantum group algebra

The Wikipedia entry for quantum groups states that a quantum group $U_{q}\left(\mathfrak{g}\right)$ has an infinite formal sum that plays the role of an $R$ matrix, which is the product of two factors,...
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0answers
28 views

Recommended Textbook for Integral Equation

I am doing a self reading in preparation for the courses I have next semester of which Integral Equation is part of it. I keep on seeing very strange notations in the materials given to me by my ...
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0answers
9 views

Orthogonally block diagonalize a special matrix

Consider a square matrix $A\in\mathbb{R}^{n\times n}$ of the form $$ A = S D $$ where $S\in\mathbb{R}^{n\times n}$ is symmetric and $D\in\mathbb{R}^{n\times n}$ diagonal with elements $\pm 1$ on the ...
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0answers
29 views

Reference Request: Dynkin Basis

I am currently working with the set of commutators of $A$ and $B$. Due to special properties of these, I am interested in working with a basis of these commutators (due to anticommutativity, not all ...
2
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0answers
40 views

Efficient way to rigorously learn AI prerequisites

Question: My formal goal is to be able to rigorously understand the mathematical basis for modern statistical learning methods (ML, deep learning). I am told by math people that this involves: linear ...
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0answers
20 views

Proving the Hilbert space filling curve is nowhere differentiable. [duplicate]

I am trying to understand the proof the Hilbert curve is nowhere differentiable. The Hilbert curve is defined as the mapping $f_h: I \to \mathcal{Q}$ where I in the unit interval in $\mathbb{R}$ and ...
0
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1answer
26 views

Reference request: an analytical proof the Hilbert space filling curve is nowhere differentiable

I am studying space filling curves and I am using the Hans Sagan book. I am trying to understand the nowhere differentiability of the Hilbert curve presented in this book but it does not seem to ...
1
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1answer
24 views

Hausdorff measure vs Lebesgue measure for a hypersurface in $\mathbb{R}^n$

Let $H$ be a compact smooth hypersurface with boundary in $\mathbb{R}^n$. We can compute the Lebesgue measure $\mathcal{L}(H)$ with respect to the induced Lebesgue measure coming from $\mathbb{R}^n$, ...
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0answers
26 views

Deformation complex of Lie algebra structures

I am learning about deformation theory, e.g. through The unbearable lightness of deformation theory by Szendröi. There the standard example of deformations of a structure of associative algebra, ...
2
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0answers
77 views

Even natural numbers are sums of two primes with twins or of two primes without twins

I seems to be very few even numbers that can't be written as a sum of two primes with twins or as a sum of two primes without twins. That is, suppose that $\mathbb P'$ is the set of the primes not ...
4
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1answer
34 views

Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
1
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0answers
11 views

A determinant associated to point sets in the plane

Consider $n$ distinct points in the plane $z_1, \ldots, z_n$. Form the matrix $D$ containing their squared distances as entries: $$ D_{ij} \ = \ |z_i - z_j|^2 \, . $$ Obviously, this matrix is ...
2
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2answers
66 views

Understanding the foundations of Calculus?

What branch of mathematics should someone study to really understand why calculus works ? I know basic calculus but for me it seems really more like apply the tools to compute stuff. I would like to ...
1
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1answer
73 views

Does anyone know a no-nonsense intro to “logic for mathematics” that I can give to a Year 11 student?

I'm looking for material on propositional and first-order logic to give to a Year 11 student that explains how they're used "in practice." For example, I want to be able to write the null-factor law ...
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0answers
78 views

Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
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2answers
72 views

Alternatives to Chapters 8-10 of Rudin's PMA

S.E advisers, I have been hearing that the chapters on multi-varaible analysis in Rudin's PMA are almost nothing like previous insightful chapters in the single-variable analysis, and I verified ...
2
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0answers
44 views

Homeomorphisms of the circle

I know that there is a vaste litterature about the group of the homeomorphisms of the circle. I would a good reference to start the study of this topic. Thanks in advance.
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1answer
60 views

Olympiad Books for Primary Students

I am a teacher of gifted program in primary school and currently I am developing Olympiad Curriculum (topic-wise) for my students. I have those topics that could need some help in terms of questions: ...
1
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0answers
42 views

Homotopy type of some lattices with top and bottom removed

There was an interesting question on MO which OP removed by some reason. Here is a (more or less) equivalent form. Take a finite cartesian product of finite linear orders, and remove top and bottom. ...
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0answers
42 views

translation of “Der kanonische Modul …”

Do you know a note that is the translation of the following in English? J. HERZOG et al., "Der kanonische Modul eines Cohen-Macaulay-Rings," Lecture Notes in Mathematics No. 238, Springer-...
2
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0answers
32 views

Basic question: Curvature transforms under Complexified Gauge Transformation

Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge ...
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0answers
15 views

Program to deal with Grobner basis for ideals in universal enveloping algebras.

Is there any way to deal with calculations in ideals of the universal enveloping algebra of the current algebra $\mathfrak{sl}_2\otimes \mathbb C[t]$? Particularly, I am interested in Grobner basis ...
2
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1answer
28 views

Characterization of a square matrix.

I would like to see a proof to this fact. For a square matrix the following are equivalent: $A$ has a right inverse. $A$ has rank $n$, where $A$ is $n \times n$. $A$ is invertible.
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0answers
25 views

Russian journals (similar to Monthly) publishing English papers

I am looking for journals comparable to the American Mathematical Monthly that are published in Russia but that they include English articles too. However I guess for each field the same question ...
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0answers
27 views

What is a good introductory book on Rational Choice Theory for a mathematician?

I'm interested in Rational Choice Theory as an approach to political science. Amongst other, related subjects, I'd like to know a thing or two about Arrow's impossibility theorem (and other aspects of ...
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4answers
143 views

Survey articles in Commutative/Homological algebra

I am a graduate student interested in Commutative algebra/Homological algebra. I am comfortable with first eight chapters of Atiyah. I am familiar with some algebraic geometry, first two chapters of ...
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2answers
65 views

Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
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0answers
49 views

(Reference Request) Proofs for basic facts about regular functions on algebraic sets.

I am writing an assignment about algebraic and analytic sets in $\mathbb{C}^n$ and, when searching for references, came across the book Algebraic Geometry III. The book is a bit out of my depth, yet ...
0
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0answers
33 views

Books recommendation

Which books and topics should I study to solve following problem, book should start from beginning. pls help. Let $\mathbb{N}$ denote the set of natural numbers. Define a function $T:\mathbb{N}\...
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1answer
41 views

Formal definition of “proexample”. [closed]

Where in the literature do we find the preferred formal definition of “proexample” as in: the number zero is a proexample for the existential sentence "some integer is neither positive nor negative"? ...
4
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1answer
55 views

What is an injection in a topos?

I am reading the paper of Andrej Bauer that proves there is a realizability topos in which there is an injection from the internal Baire Space $\mathbb{N}^\mathbb{N}$ to the natural numbers $\mathbb{N}...
2
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1answer
69 views

How does a complex algebraic variety know about its analytic topology?

This question has two parts. The first is a reference request regarding a result I assume is standard, and the second is a soft question asking for philosophy and intuition about an issue the first ...
1
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1answer
25 views

What does a function of matrices do to the eigenvalues of matrices in its domain? Two examples and request for generalization if possible

I think, for example, that if $\lambda$ is an eigenvalue of a matrix $A$, then $\lambda^2$ is an eigenvalue for $A^2$ and that $\frac{1}{\lambda}$ is an eigenvalue for $A^{-1}$ provided $A$ is ...
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1answer
30 views

Convergence of finite metric spaces to an infinite one

Let $\{(M_i, d_i)\}$ be an infinite sequence of finite metric spaces, where $|M_i|$ is strictly increasing with $i$. Is there a standard definition of what it means for the sequence $\{(M_i, d_i)\}$ ...
0
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1answer
32 views

How to show this equality for operator norm?

Let $(X,\Sigma_\mu,d\mu)$ and $(Y,\Sigma_\nu,d\nu)$ be two positive $\sigma$-finite measure space and let $M(d\mu)$ and $M(d\nu)$ be spaces of complex-valued $d\mu$-measurable and $d\nu$-measurable ...
1
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1answer
18 views

Connection between focal points and singularities of the normal exponential map

I am looking for nice references on focal points of Riemannian submanifolds. In particular, I would like to see a proof for the connection between focal points and singularities of the normal ...
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2answers
56 views

Cyclically reduced words

This is just a reference request. I'm trying to find out whether there are some well developed notes/theory out there (books and the like) focusing on cyclically reduced words in groups. Quickly ...
0
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0answers
2 views

Reference request for "isomorphism upto compact kernel /cokernel “

Let $A$, $B$ be abelian topological groups with a map $f :A \to B$. Assume also that the kernel and cokernel of this map are compact. Then we call f an isomorphism upto compactness. Now let $A, B, C$...
0
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1answer
15 views

Riemann-Stieltjes integral and unbounded variation

I was studying the Riemann-Stieltjes integral and I was looking for the proof that function with unbounded variation can't be integrated. Thanks
0
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0answers
42 views

what is the asymptotic expansion of this $_2F_2$ function?

We need to expand the function $_2F_2(a+b x,1; 1+a+b x, b x; x)$ near $x=+\infty$. Where $a$ is complex, $b>1$. When $x\to+\infty$, both the parameters and the variable goes to infinity, we can ...
0
votes
0answers
19 views

Original statement of Wiener's $1/f$ theorem

I'm studying Wiener's 1/f theorem, and I got curious about which was its original statement.I've been looking online but found nothing. I want to know if Wiener also proved the $n-$dimensional ...
0
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0answers
29 views

Representation of $A_5$

Can someone give me a proper reference (a book probably)for how a 3 dimensional representation of the Alternating group $A_5$ is related to the reflection group $H_3$ or the Icosahedral group ? Thanks
2
votes
1answer
81 views

Discrete logarithm modulo powers of a small prime

Is there an efficient way to compute $x$ in $2^x \equiv b \pmod {p^m}$, where $p$ is a small odd prime and $m$ could be a large integer? I know the solution is of the form $x=\phi(p^m) k + y$ for ...
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1answer
29 views

What mathematical background is preliminary for reading and understanding books/papers on wavelets?

Please excuse my english. I have had the following math courses for mechatronics engineering education: Calculus (single and multivariable) Linear algebra (introductory) Differential equations (ode'...
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1answer
19 views

Are symmetric and $\Delta$-metric common terminologies?

In these notes on metric spaces, the author also defined something known as "symmetric", and $\Delta$-metric. I have never seen these terminologies before. Are these terms standard usage? Can ...
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votes
5answers
910 views

Formal definition of “counterexample”.

What is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexample”...
5
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0answers
78 views

Is the error I noticed a harmless typo?

Here http://arxiv.org/PS_cache/arxiv/pdf/1002/1002.0442v1.pdf , at page $2$ at the bottom, it is stated that the number of primes not exceeding $x$, denoted by $\pi(x)$, satisfies the double-...