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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3
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1answer
31 views

number of loops of length $n$ without crossings in random walk on $\mathbb{Z}^2$

Consider the symmetric random walk on $\mathbb{Z}^2$, where you go in one of the four directions with probability 1/4. We start in 0. My question is whether there are results on counting how many ways ...
1
vote
1answer
44 views

Are there any consistency proofs for propositional or first-order logic?

Take for example the Hilbert-style axiomatizations of the propositional and first-order calculus. Since a crucial point when operating with a proof system is that no contradictions must be found in ...
1
vote
0answers
24 views

Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ being solved?

Let A be a positive semidefinite matrix of order $n$. Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ is being solved? Where $\tilde{A}$ is the matrix of order $n!\times ...
0
votes
0answers
13 views

Reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem.

I want a reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem. I understand some intuition behind taking independent copies of the random ...
2
votes
1answer
61 views

Generalizing Hall's marriage theorem to arbitrary graphs

Given a finite graph G = (V, E) in which each vertex is a finite set, I call system of local representatives the choice for each vertex of one of its elements (the local representative), so that no ...
5
votes
1answer
52 views

Book recommendations for a large number of topics

I have quite a few topics I'm interested in, and with many of these it may be a while before I cover them at university - I'll not be there until October, so I'd like to just get on and learn about ...
0
votes
0answers
17 views

Intuition on primal convergence in dual subgradient method

It is well known that the subgradient method applied to the Lagrange dual of a convex problem may produce a sequence converging to the dual optimum, but the primal iterates produced by this sequence ...
4
votes
0answers
79 views

Is Bourbaki unique?

So my understanding is that a while back a group of mostly French mathematicians, under the pseudonym Bourbaki, wrote a somewhat austerely written series titled "Elements of Mathematic(s)" covering a ...
8
votes
1answer
161 views

Reference Request: what are some books on mathematics you can read without pencil and paper

I am going overseas for the summer, I need a book or two so I can learn about mathematics (overviews, engineering applications, history, connection with other branches of science) without actually ...
0
votes
0answers
11 views

Binomial representation of stochastic process

It is common knowledge that a random walk can be represented in the form of a binomial process. Is it possible to represent any generic stochastic process (including non-linear) of the form ...
1
vote
0answers
33 views

Relation between Extensions and self-Extensions!

This should be considered as very general question regarding the extension group $Ext^i _A (R,S)$, in particular where $i=1$, for $R$ and $S$, a pair of given objects in an abelian category $A$. For ...
1
vote
1answer
18 views

any interpretation for $\left[ \sum_{n\in \mathbb{Z}} q^{n^2} w^{2n} \right]^{-1}$?

One very simple version of the theta function is as a generating function over the perfect squares: $$ \theta(\tau; z) = \sum_{n\in \mathbb{Z}} q^{n^2} w^{2n} $$ Where $q = e^{2\pi i \tau}$ and $w = ...
2
votes
0answers
23 views

Monomorphisms into direct products

Let $G$ be a group. I am interested in the following property: For any groups $A,B$ and monomorphism $G \hookrightarrow A \times B$, either $G \hookrightarrow A$ or $G \hookrightarrow B$. For ...
2
votes
1answer
28 views

Formula for $q$-expansion of weight 2 modular forms

Is there a general formula for finding the $q$-expansion of weight 2 modular forms?
1
vote
0answers
17 views

Exponentially fast decay of alpha-mixing rates for irreducible, aperiodic finite, Markov chains

Let $(X_n)_{n \in \mathbb N}$ be a stationary, aperiodic, irreducible, finite state space Markov chain. Define the $\alpha$-mixing coefficient as: $$\alpha(n) = \sup \{\vert \Pr(A \cap B) - ...
5
votes
5answers
306 views

Advanced / In-Depth Calculus Book for Self-Edification

I am a pre-engineering student currently taking a Single Variable Calculus course at a community college. I recognize that my future success (or not so much) as an engineer will be based, in large ...
2
votes
0answers
29 views

Grothendieck groups

I have started doing a paper Applications of a New $K$-Theoretic Theorem to Soluble Group Rings by Kropholler which proves Kaplansky conjecture for soluble groups. Now looking ahead in paper, I saw ...
1
vote
0answers
180 views

Can you identify this book?

I am looking for a very conceptual book on analysis to recommend it to one of my juniors. I have a book in my mind which I once read during my UG days in my college library on one fine morning, a book ...
0
votes
0answers
20 views

Partial differential equation involving a random process (literature advice)

In articles like this one (end of page one and page two), physicists often tend to treat a random process with discrete time and countable space set as a differentiable function (whose domains are ...
0
votes
1answer
52 views

Self-adjoint extension of the Laplacian

Let $M$ be a complete Riemannian manifold and $-\Delta$ denote the Laplace-Beltrami operator on $M$. We can prove that $(-\Delta f, g) = (\nabla f, \nabla g) = (f, -\Delta g)$, when $f, g \in ...
-2
votes
1answer
45 views

A reference for “every set containing an element $a$, allows a abelian group structure with identity element $a$”

Do you know a reference which contains a proof for this proposition: A set containing an element $a$, allows an Abelian group structure with identity element $a$. ? By reference, I ...
1
vote
1answer
43 views

When do three cubics form an arithmetic progression?

Are there any solutions to the diophantine equation $x^3+y^3=2z^3$ other than the trivial ones? What about $x^4+y^4=2z^4$? I think I remember these equations in one of Euler's work, but having ...
0
votes
0answers
69 views

Linear Algebra Book like Calculus Made Easy

Now, I know that there are a tons of reference requests for Linear Algebra books but mine is very specific: what is a nice, short, concise, simple, to the point book that gets at the heart of Linear ...
2
votes
1answer
26 views

Uniform convergence for functions with jumps

We know that Fourier partial sums (integrals) do not converge uniformly for BV functions with jumps due to Gibb's phenomenon. Is there any other types of sums/procedures that use only Fourier ...
0
votes
0answers
16 views

Legendre–Fenchel transformation $ f^{\ast}(x^\ast)=\sup_{x\in\mathbb{R}^n}\{\langle x,x^\ast\rangle -f(x)\} $

The convex conjugate also known as Legendre–Fenchel transformation of a convex function $f:\mathbb{R}^n\to\mathbb{R}\cup\{+\infty\}$ is definite by $$ ...
2
votes
1answer
35 views

Explicit Calculation of Banded Toeplitz Matrix Eigenvalues

I recently found a paper which detailed a method of finding the eigenvalues of the $n\times n$ banded Toeplitz matrix $$ \left[ \begin{array}{ccccccc} a_0 & a_1 & a_2 & \dots & a_s ...
0
votes
1answer
29 views

Probability theory book for self-sudy [duplicate]

I want to know a book on Probability theory (Measure-theoretic) for self-study, like at the level of Terence Tao's analysis and measure theory book (he does not have any prob. th. book I guess). I ...
0
votes
0answers
28 views

A type of continuous time Markov process

I am looking for a stochastic process model with the following features. It is a continuous time Markov process---modelling, if you like, the evolution of a population. New arrivals are added to ...
0
votes
0answers
16 views

Please recommend a book/article for Newton-Raphson method

There are so many search results I'm a bit lost. I would like to read an article to fully understand it, including the math end, and the appliction side, please recommend one that contains also ...
1
vote
0answers
25 views

Limits of multivariable (two) functions [closed]

Does someone know a thorough book or some pdf where I can learn to solve them ?
2
votes
1answer
79 views

A categorical approach to algebraic geometry

I learned Algebraic Geometry in a geometrical viewpoint, e.g. Hartshorne's book. But I want to learn algebraic geometry categorically, for examples, i) Sheafification $\mathcal{F}^+$ of a presheaf ...
2
votes
1answer
28 views

What was the paper about flower-shaped knots?

I read a article about the possibility to bring knots in a "polar rose" projection, where there is only one crossing of higher multiplicity. The overcrossing/ undercrossing information is thus more ...
6
votes
1answer
73 views

About $\mathbb{R}$ as a vector space over $\mathbb{Q}$

I want to understand better the structure of the vector space $\mathbb{R}$ over $\mathbb{Q}$. I know that it is an infinite dimensional vector space with a non countable Hamel basis, and it is cited ...
0
votes
0answers
31 views

What are some good books to study Non -Commutative Rings?

What are some good books to study Non -Commutative Rings? I want to study structure of semisimple rings and Wedderburn -Artin Theorem in particular . The book should provide motivations and have ...
1
vote
2answers
33 views

Graph theory, colourability in 3-space.

Can somebody explain colourability in $3$-space or share really good material that I can read? I understand the rules of $4$-colourability in planar universe: sections sharing the same border cannot ...
1
vote
2answers
33 views

Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
2
votes
1answer
18 views

Reference request- Darboux cubic of a triangle

Hi everyone on Math Stackexchange, I'm recently interested in the geometry of a triangle, and my studies now seems to require some knowledge on cubic curves related to a triangle, in particular the ...
0
votes
1answer
32 views

Drawing a regular pyramid

I've been asked for help in high school mathematics (some basic stereometry) but I'm not sure how to solve this exercise: Draw a regular triangular pyramid given the lengths of edges $3.8\ cm$ ...
-1
votes
0answers
32 views

I need to a book in modern geometry proofs grad level

I need to book in modern geometry proofs grad level Hi , I need books that have the modern geometric proofs or geometry for teacher's class in grad level , That has the postulates, theorem and ...
2
votes
0answers
63 views

Request for a comparison between these 3 (advanced?) functional analysis books?

It would be helpful if I can get some comparison between these three books, T. Tao, An epsilon of room, I, Graduate Series in Mathematics 117, American Mathematical Society (2010). T. Tao Analysis ...
4
votes
1answer
33 views

Definition of a “modular Galois representation”

I am trying to pin down a definition for a $n$-dimensional modular Galois representation $$\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_n(A).$$ I am just looking for ...
1
vote
3answers
42 views

Fourier transform of a radial function

Consider a function $f \in L^2(\mathbb{R}^n)$ such that $f$ is radial. My question is, is the Fourier transform $\hat{f}(\xi)$ automatically radial (I can see it is even in each variable $x_i$), or we ...
0
votes
0answers
25 views

Are the names such as $N_1$-space, $N_2$-space used for various countability axioms

In this question the OP mentioned "$N_i$-hierarchy for various countability axioms and also that the name $N_2$-space or $N_2$-property is used for second countable space. I did not encounter this ...
1
vote
0answers
9 views

Proof remainder of polynomial division GF(2) can be calculated by LSFR

I have been reading that CRC, which is the calculation of the remainder of $x/P(x)$ in GF(2) can be implemented with a Linear Shift Feedback Register. However, I can't find the proof for this, or ...
1
vote
1answer
13 views

representation theory of two-step nilpotent Lie algebras

Does anyone know of any good reference about the representation theory of two-step nilpotent Lie algebras, like whether their irreducible representations can be classified?
-1
votes
0answers
32 views

Norm equivalent to Sobolev norm?

On the hyperbolic space $\mathbb{H}^n$, it is known that the spectrum of the Laplacian satisfies $\text{Spec}(-\Delta) \subset [\frac{(n - 1)^2}{4}, \infty)$. Consider the operator $P = -\Delta + a$, ...
2
votes
0answers
31 views

Heat kernel on hyperbolic space

On $\mathbb{R}^n$, the heat kernel $p(t, x, y)$ (for the Laplacian) satisfies the following pointwise bounds: $p(t, x, y) \leq Ct^{-n/2}, x, y \in \mathbb{R}^n, t > 0$. I was wondering if there are ...
0
votes
1answer
35 views

Continuity of the eigenvalues.

I came across a statement of which I do not understand the meaning. The smallest eigenvalue of a $k \times k$ symmetric matrix $M$, $\inf_{ \{v \in R^k | ||v|| = 1 \} } v'Mv$, is continuous in ...
2
votes
1answer
43 views

Beam equation and method of separation of variables: some references.

Let us consider the beam equation \begin{align} z_{tt}+z_{xxxx}&=0\ \ \ \ \ \ \ \ \ 0<t<T, \ 0<x<\pi \notag \\ z(t,0)=z(t,\pi)=z_{xx}(t,0)=z_{xx}(t,\pi)&=0\ \ \ \ \ \ \ \ \ ...
2
votes
0answers
39 views

Neighbor-full partition of $\{0,1\}^n$

What is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that has the maximum number of elements? (which we call $k(n)$) We say $A$ and $B$ are neighbors if their ...