Questions tagged [reference-request]

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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What is Fredholm psuedo-inverse?

Background In 1903, Fredholm invented the concept of pseudo inverse for integral operators, then around half a decade later mathematicians came up with the idea of pseudo inverse for matrix. On the ...
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Book on sufficient conditions for central limit theorem?

A few years ago, I took an advanced statics course where the Professor explained the the independent and identically distributed conditions are not necessairy for the Central limit theorem to hold. He ...
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Is $H_m - H_n$ a surjection onto $\mathbb{Q}^+$?

I was wondering whether, for each rational $q$, we may always write $$q = \sum_{k=a}^b \frac 1k$$ For some positive integers $a \leq b$. I get the feeling that this is not true (although an ...
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Reference for the constant term of Eisenstein series on congruence/discrete subgroups

I'm looking for an exposition of the computation of the constant term of a non-holomorphic Eisenstein series associated to a discrete subgroup $\Gamma$ (or at least a congruence subgroup) of $SL(2,\...
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1answer
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Which Smirnov is behind the Smirnov topology?

Good old Steen and Seebach discuss the Smirnov deleted sequence topology in their Counterexamples in Topology (2nd ed. 1978). This is also reported as the $K$-topology, in e.g. Wikipedia etc. ...
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Need reference of books which deals with ideal theory of tensor product

Is there any book which deals with Ideal theory of tensor product of $C^{\ast}-$ algebras?
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62 views

Which measure theory book should I read after studying Terence Tao's Analysis I and Analysis II?

I am a master's statistics student who is just about to finish studying Terence Tao's Analysis I and Analysis II books and I really liked his way of teaching things. Besides its content, I also need ...
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What is the best option of a graduate multivariate statistics book?

Right now I am in the middle of a graduate multivariate statistics but I am feeling to easy at the moment. As a mathematician I study univariate statistics from "The theory of statistical inference : ...
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1answer
38 views

Extrema of $\xi(1/2+it)$ for the Landau Riemann xi function

"Relations and positivity results for derivatives of the Riemann xi function" by Coffey characterizes the Riemann xi function and its relation to the Riemann zeta function, and there has been much ado ...
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Information theory reference: comparison between Mackay to Thomas and Cover

I'm a computational neuroscience student with a background in mathematics. I want to learn information theory over the summer. I am interested in its applications to neuroscience, machine learning, ...
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Haar measure from the Stiefel Manifold

I am reading paper Finite free convolutions of polynomials, which uses Haar meausre from the Stiefel manifold. All I know about measure theory is the Lebesgue measure on the Euclidean space. I want ...
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1answer
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Can someone recommend for me a text-book about applications of Taylor series?

I have text-books about Taylor series but they do not mention Taylor series applications. Can anyone please suggest a few references to learn the same?
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Category of 3-term sequences

Is there a name for the following categorical concept? If so, where can I read about it? Given some category $\mathcal{C}$ we build a new category $\mathcal{C}^3$ whose objects are $3$-term ...
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Boundary Value ODEs Reference Book

Can someone suggest me a book on Boundary Value Problems in ODEs, which start from the general theory, and then go on to specialize for self-adjoint values? All the books I have found discuss the self-...
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1answer
38 views

Coproducts of join-semilattices

I am trying to describe what is the coproduct of two objects in the category of join-semilattices (with a top element). Does anyone have an idea or a reference to read ?
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1answer
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Cantor–Schröder–Bernstein for Lipschitz maps?

Let $X,Y$ be metric spaces and suppose there exist bijective Lipschitz functions $f : X \rightarrow Y$ and $g : Y \rightarrow X$. Does there necessarily exist a bijective bi-Lipschitz function $h : X \...
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1answer
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sufficient amount of computer science a pure mathematics student would need to know

I am a master’s pure mathematics student. Recently I have been studying just a bit of discrete geometry (e.g. configurations of points) and I seem to be enjoying it in that its ideas are simply ...
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62 views

Textbooks in which determinant is defined as an alternating multilinear map

I'm interested in this abstract definition of determinant, i.e. determinant is defined as an alternating multilinear map. Could you please suggest me some Linear Algebra textbooks that define ...
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Reference request for ISI Mtech QROR entrance exam

I like to ask you for any references, books, pdf etc. that comprises of a lot problems with the level of this exam. Entire syllabus for MMA(objective one): Analytical Reasoning: Algebra: Arithmetic,...
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How to utilize the special feature of this recursive problem to reduce computational complexity?

Assume $A$ is a $n \times n$ matrix of non-negative numbers. $A_i$ is the $i$-th row of $A$. $(a_1, \ldots, a_n)$ and $(b_1, \ldots, b_n)$ belong to $\mathbb R_+^n$. $X= [x_1, \ldots, x_n]^\intercal$ ...
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Linear Algebra Textbook for Reference?

Just want a good linear algebra textbook for reference. I know that there is a lot of good ones, but I am not a mathematician and I don't want anything way too abstract like Axler, Curtis, Hoffman&...
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Introduction to non linear parabolic differential equations.

I would like to know what the basic references to study non-linear parabolic problems, specifically problems of the type \begin{equation*} \frac{\partial u}{\partial t} - \Delta u = f(x,u), \end{...
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1answer
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Reference for $p$-adic Haar integral

I have stumbled upon the notion of a $K$-valued Haar integral on a locally compact group, where $K$ is a non-Archimedean field, as well as the $K$-valued modular function, in an article of Schikhof. ...
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72 views

Integrate $I=\int_0^{\infty} x^n \, e^{ax+\frac{b}{x}} \, \cos(cx) \, dx$?

Is there an expression in terms of some special functions (or a closed form) of the following integral $$I_n(a,b,c)=\int_0^{\infty} x^n \, e^{ax+\frac{b}{x}} \, \cos(cx) \, dx,$$ $n:$ integer, $a\in\...
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How is a function with internal optimization called?

I am looking for references/information about functions like $$ f(x) = \max_{y} g(x,y) $$ $x,y \in R$. I don't know if functions like this have a special name? Are there any good references for ...
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2answers
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Limit superior and inferior of Césaro means are almost surely constant

Reading the book of probability of Achim Klenke I came across the assertion that if $(X_n)$ is a sequence of independent real valued random variables then the limit superior and inferior of the ...
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Reference for eigenvalue perturbation theory

Could someone please share any reference book to start reading eigenvalue perturbation theory. Thanks in advance
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1answer
51 views

Simplicial Homology of Matrix Groups

It is a well known fact that the matrix groups $GL_n(\Bbb R), SL_n(\Bbb R), \dots$ can be considered as submanifolds of $\Bbb R^{n^2}$. I did not yet attend a lecture on Lie groups, so I don’t know ...
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Which families of functions exist which convolve within the same family?

Convolution $(*)$ is an operation defined on two functions as $$(a*b)(t) = \int_{-\infty}^\infty a(\tau)b(t-\tau)d\tau$$ The other day I came to think about an equation regarding it: $$(f_i * f_j) (...
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Solution for the book “Nonlinear programming” [closed]

I am studying the book "Nonlinear Programming: Theory and Algorithms" by M.S. Bazaraa, H. d. Sherali and C. M. Shetty. And I have the official manual solution book, but the there are only selected ...
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9 views

Binary representation unusual relation in theorem about immersed manifolds.

Question I'm reading Schuller's Lectures on the Geometric Anatomy of Theoretical Physics and he states the following theorem. I was surprised by this theorem and would like references for further ...
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1answer
157 views

What book would you recommend to significantly improve my problem solving skills?

I am a straight-A student (going to the ninth grade) and do nearly perfectly in math, the problem is that my school (like many other schools I suppose) makes you memorize the steps, the formulas, etc. ...
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Reference for: a nontrivially graded integral domain is never quasi-local

Let $\Gamma$ be a torsionless grading monoid and $R=\bigoplus_{\alpha\in\Gamma}R_\alpha$ be a $\Gamma$-graded integral domain. I'm interested in the following result: if $R$ is nontrivially graded,...
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What is known about the approximation constant?

A badly approximable irrational is one whose continued fraction denominators are bounded; equivalently, if $\alpha$ is badly approximable then there is a $c(\alpha) > 0$ such that $$c(\alpha) = \...
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1answer
30 views

Good reference that discusses NP hardness in the context of optimization?

Sometimes I read a book on optimization and the author states (without proof) that finding a certain solution to the (non-convex) optimization problem is NP hard. I've learned about complexity theory ...
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Poisson process that outputs both events and associated $[0,1]$ weights

I am looking for a stochastic process that generates events according to a rate parameter, as in the standard Poisson process, but also outputs a weight $[0,1]$ associated with each event (as dictated ...
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38 views

Product representation of the exponential series.

Let ${a_n} = \prod\limits_{j = 1}^n {\gcd (j,n)} .$ Comparing OEIS A067911 and A170911 suggests that there are integers $b_n$ such that the $n-$th partial sum of the product $\prod\limits_{k = 0}^\...
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Textbooks on Stochastic Numerical Analysis

I'm trying to find a nice textbook from which to improve my knowledge on numerical analysis for stochastic differential equations, with a particular focus on intuitive derivations of different ...
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1answer
76 views

Any reference(a book) that defines the $n$-dimensional rotation matrix?

I want to refer to a mathematics book that explains the n-dimensional rotation matrix or rotation transformation. Wikipedia concentrates most on 2D or 3D. There are things that one can say definition ...
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2answers
37 views

Motivation For Weight Choice In Pooled Variance

In the formula for pooled variance, the estimated variance of each population of size $n_i$ is weighted by $n_i-1$. Is there a good motivation for this? I would assume the formula is always unbiased,...
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+50

Function field analogy

I am rephrasing the previous question: Can I get good and accessible references to read to understand this particular statement in Wikipedia: "The function field analogy states that almost all ...
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1answer
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References for Learning Inverse Limits (for Profinite Groups)

I'm doing some independent study on Profinite Groups this summer and, as I understand it, it is important to be familiar with the notion of an inverse limit before doing so. The trouble for me is that ...
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1answer
34 views

Why are the $p$-oldforms $f(z)$ and $f(pz)$ linearly independent at level $\Gamma_0(pN)$?

Let $f$ be a newform (normalized eigenform) of weight $k$ and level $\Gamma_0(N)$. Fix $p$ not dividing $N$ and set $f_p(z)=f(pz)$. Viewing $f$ and $f_p$ at level $\Gamma_0(pN)$, why are they ...
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+50

Proof of the classifying space cohomology isomorphism for local coefficients

There are many references which will show that for (say) a finite discrete group $G$, you can construct the classifying space $BG$ which $G$ as its fundamental group and the same cohomology groups, ...
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69 views

Reducing the strength of a category theoretic proof

The motivation for this question is the following: Say we have a formula $\phi$ in peano arithmetic, and we have proof $\pi$ of $\phi$ using possibly higher order arithmetic or category theory (that ...
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44 views

Approximating $t\to f(t) = \sqrt{t}$ by polynomial and bitwise refinement, how fast will it be?

Say that we want to approximate the function $$t\to f(t) = \sqrt{t}$$ on the interval $t\in [0,1]$. We know that polynomial approximation close to $0$ is very bad so we want to avoid that. Instead ...
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2answers
65 views

Taylor Expansion of Logarithm of Determinant near Identity for Non-Diagonalizable Matrix

I have been working on a problem where I need to Taylor expand an expression of the form $\log \det(I-A)$ in terms of traces of the matrices $A^m$ for $m \in \mathbb N$, where $A$ is a general $n \...
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63 views

A good book for conics?

I want to learn about conics, but I don't really like the chapter on our textbook or the calculus & analytic geometry book that I'm following. Conics are so interesting, ellipse and hyperbola ...
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1answer
112 views

English translation of original Fraïssé's paper and texts on Fraïssé theory

I wonder if there is an English translation of Fraïssé's paper "Sur l’extension aux relations de quelques propriet es desordres", appeared in Annales Scientifiques de l'Ecole Normale Superieure....
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Are there references about universal power series and this statement?

The notion of universal power series is linked with this statement : We can find a power series $\sum_{n\ge1} a_nx^n$ such that for all continuous function $f : [0,1] \to \mathbb{R}$ such that $f(0)...

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