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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4
votes
1answer
34 views

Gauss' original proof of quadratic reciprocity

Is the original proof of quadratic reciprocity due to Gauss available anywhere online? I've been looking for quite a while now, but with no results. Most papers seem not to include it because of it ...
0
votes
0answers
15 views

Why distribution of multiple recursive random number generators is uniform?

I was reading the article of L'Ecuyer on random number generation. The title of this article is "Uniform Random Number Generation". One of the proposed PRNGs there, is multiple recursive random ...
2
votes
0answers
23 views

Reference request: Tubular neighborhood theorem for non-closed manifolds via the exponential map.

Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and ...
-4
votes
1answer
55 views

I need this book by Michael Weinstein, Between nilpotent and solvable [on hold]

I need this book by Michael Weinstein, Between nilpotent and solvable. I live in iran and i can't find this book. Please help me.
1
vote
0answers
17 views

Introductory Reference for Mathematical Physics

I'm a senior undergraduate student studying differential geometry. I have experience with smooth manifolds, some elementary theory of Lie groups, and a little multi-linear algebra. I understand this ...
0
votes
1answer
11 views

Proof for the fact that the method of characteristic equations is a valid method to solve recurrence relations

Could someone kindly point me to a proof of the fact that the method is characteristic equations is a valid way of solving recurrence relations? It seems fairly arbitrary to me. I would be grateful ...
0
votes
0answers
6 views

Property of locally cyclic groups

I am having difficulty proving that: A group $G$ is locally cyclic if and only if $G$ is isomorphic to a subgroup of $\mathbb{Q}$ or $\mathbb{Q/Z}$. Is there any easy way to prove it? Thanks.
0
votes
0answers
12 views

Automorphism group of a locally cyclic group

I am having difficulty proving that: The automorphism group of a locally cyclic group is commutative. Is there any easy way to prove it? Thanks.
0
votes
0answers
27 views

Differentiability of parameter-dependent integrals when derivative exists only almost everywhere

This unanswered question asked in 2013 Differentiation under the Integral Sign (let's call this Q-zero) seems to be taken from this (or pdf ver.). The result on differentiation under the integral ...
1
vote
1answer
44 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
0
votes
0answers
8 views

Is there a solid reference work that covers optimization for discrete and for continuous domains?

I am looking for a good, comprehensive reference on optimization. Currently, I have Lundberg's "Linear and Nonlinear Programming, 3rd Ed", but this completely omits integer programming, except in the ...
0
votes
0answers
15 views

Study materials for Differential equations and Fourier analysis

In two days, on Monday, a new course called Introduction to differential equations starts, and when that ends in one month another called Fourier analysis and its application starts (Both are actually ...
0
votes
1answer
34 views

What's a somewhat fast introduction to (differential) geometry and algebraic topology for someone who knows a lot of analysis but little else?

I never got to learn much about geometry beyond curves and surfaces in Calculus III, and point set topology. So what is a fast introduction to differential geometry (specifically, differential ...
1
vote
3answers
90 views

Good algebra book to cover these topics?

I will be studying two algebra modules next year and I am looking for a comprehensive book that will cover both of them, however due to having very minmal exprience with algebra I am looking for your ...
3
votes
1answer
29 views

Double integral definition of periodic Sobolev spaces

Preliminaries: For $s \geq 0$ define the periodic Sobolev spaces $H^s(\mathbb T)$ with norm $$ \| f \|_{H^s(\mathbb T)}^2 = \sum_{k \in \mathbb Z} \big(1 + |k|^2 \big)^s \, \big|\widehat{f}_k \big|^2, ...
0
votes
1answer
42 views

What are some easier papers/books I can read? [on hold]

I'm trying to improve my ability in reading mathematics papers. My field is more related to biological sciences, but there are a lot of interesting papers I'd like to read that use more mathematical ...
0
votes
0answers
16 views

Combinatorial Convex Optimization: Russian paper

I'm looking for an electronic version of the paper: David Yudin and Arkadi Nemirovski. Informational complexity and effective methods of solution of convex extremal problems. Economics and ...
0
votes
0answers
13 views

Application of Multivariate Analysis on Straight Lines [on hold]

Can anyone suggest to me a reference on the Application of Multivariate Analysis on Straight Lines? The reference should contain the part where we use the concept of orthogonality to formulate ...
2
votes
1answer
28 views

What is a good book for learning Stochastic Calculus?

I am in search of a good book for learning Stochastic Calculus from a purely mathematical/statistical point of view. Almost all the books I see are based on Finance. Also, please specify the ...
2
votes
0answers
25 views

Abelian hypersurfaces

Is there any way to tell from its defining equation when an affine hypersurface has a group law? I am aware that all such groups are matrix groups; in that case I might want to ask when a variety over ...
0
votes
1answer
44 views

Estimate for partial sums of a series equivalent to the Riemann hypothesis

The sums $$S_N=\sum_{n=1}^N\frac{\mu(n)}{n},$$ where $\mu$ is the Moebius function, are known to tend to 0 as $N\to+\infty$. As far as I remember, there was an estimate on $S_N$ equivalent to the ...
2
votes
1answer
28 views

Multiplicative version of convex hull

The convex hull of a finite set of points, $(x_i,y_i) \in \mathbb{R_+}^2$ ($i=1,...,n$), is defined as: $$\left\{(\sum_{i=1}^{n} \alpha_i x_i,\sum_{i=1}^{n} \alpha_i y_i) \mathrel{\Bigg|} (\forall i: ...
6
votes
0answers
44 views
+50

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]\cdot[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
0
votes
0answers
14 views

Geometrical Interpretation of Risk Ratio (RR) [on hold]

Can anyone suggest any book or article on the topic "Geometrical Interpretation of Risk Ratio (RR)" ?
0
votes
0answers
25 views

do Carmo or Pressley as introduction to Differential Geometry?

I'm choosing between these authors for self-studying Differential Geometry. The contents seem pretty similar for both books and I was wondering which book I should choose. The level of these books ...
0
votes
0answers
9 views

Reference needed: Chi-square goodness of fit test, independence test, …

I want to really understand what is a Chi-square test and how it works: when is it needed, what motivates its use, etc. The same thing is needed for "Independence tests" and analysis of variances. Is ...
1
vote
1answer
24 views

What are those variations of norms called?

Let $V$ be a vector space with a function $\|\cdot\|$ on it that satisfies all the axioms of norms except for scalability condition $\|\alpha \mathbf{x}\| = |\alpha| \|\mathbf{x}\|$ replaced with ...
0
votes
1answer
18 views

Reference about conformal map

I am here because I want to know if someone knows of some good e fast books or references about conformal map . More precisely I need of the propeties of the conformal maps on manifolds with boundary. ...
0
votes
0answers
16 views

“Asymptotic” $\mathbb{R}$-algebras

Definition. By an asymptotic $\mathbb{R}$-algebra, I mean an $\mathbb{R}$-algebra $F$ of functions $\mathbb{R} \rightarrow \mathbb{R}$ satisfying: $$\mathop{\forall}_{f:F}\left[\left(\lim_{x ...
2
votes
1answer
46 views

Error in Billingsley?

Problem 8.25 in the third edition of Probability and Measure by Billingsley (1995, p. 142) is as follows: Suppose that an irreducible [Markov] chain of period $t>1$ has a stationary ...
0
votes
0answers
18 views

What do we call the number that measures how good of an asymptote $g$ is to $f$, and what are the basic results about this number?

Suppose we have a (potentially very complicated) smooth function $f : \mathbb{R} \rightarrow \mathbb{R},$ and we're trying to approximate it (in the limit as the input value goes to $+\infty$) by a ...
-1
votes
1answer
15 views

Zeitz's ACoPS vs Larson's PSTP

Which of the following books is better to prepare for a mathematical competition at the undergraduate level? The art and craft of problem solving (ACoPS) or Problem solving through problems (PSTP). ...
0
votes
1answer
32 views

Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other

Let's consider following spaces: $L^2(\mathbb{R}) = L^2(\mathbb{R}, \mathbb{C}, \mu_L)$ --- space of $\mathbb{C}$-valued functions defined on $\mathbb{R}$ for which the square of the absolute value ...
1
vote
2answers
187 views

Works of Kurt Gödel

I'd like to know how to get started with Gödel's work and theorems. I have a decent knowledge of tensors, Einstein field equations. When it comes to logic and set theory, I'm a beginner. Can anyone ...
0
votes
1answer
19 views

Conditions for convergence of derivatives from pointwise convergence

Let $\{f_n\}_n$ be a sequence of functions $\mathbb R \rightarrow \mathbb R$ which converges pointwise to $f$, ie: $$f_n(x) \rightarrow f(x) \hspace{10pt}\hbox{for all $x$}.$$ What additional ...
1
vote
0answers
8 views

Geodesics of Sasaki metric

I would like to ask the community for a reference on the following question: Let $(M,g)$ be a Riemannian manifold and $(T^1M,g_S)$ be the unit tangent bundle with the Sasaki metric. Is it true that ...
2
votes
1answer
29 views

Error estimation for the Wallis product

From the Wallis product we know $$\prod_{k=1}^{\infty} \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot ...
0
votes
0answers
28 views

Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
1
vote
1answer
36 views

Gradient flow of Dirichlet energy

I have heard that the gradient flow of the Dirichlet energy gives a solution of the heat equation, i.e. if $u(t,x) \in C^1( [0,\infty) \times \mathbb R^d)$ solves $$ u_t(t,x) = - dE(u(t,x)), $$ where ...
2
votes
0answers
50 views

QFT and topology

I have had a course in topology, I have heard of homotopy quantum field theory and topological field theory, but I dont know anything about QFT, what would be a good starting point to learn about the ...
3
votes
0answers
21 views

Properties of spectra

I've been trying to understand spectra better, and I was wondering what would be a good source to learn from. My issue with the classical treatments (e.g., Adams, Switzer) is that people keep telling ...
0
votes
0answers
7 views

Nakayama automorphism $\sigma$ of Hecke Algebra $^0H^f_n$ is not inner for $n\geq 3$?

With $R=\mathbb{Z}[q_1,q_2]$, the Hecke algebra $H^f_n$ of $S_n$ is defined to be the $R$-algebra generated by $T_1,\dots,T_{n-1}$ satisfying $T_iT_{i+1}T_i=T_{i+1}T_iT_{i+1}$, $T_iT_j=T_jT_i$ if ...
2
votes
0answers
30 views

Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, ...
7
votes
0answers
65 views

Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
0
votes
1answer
27 views

Reference on manifolds with boundary

I am here because I want to know if someone knows of some good e fast books or references about manifolds with boundary. Help me please.
1
vote
0answers
25 views

Reference request about Thm which use Transversality to compute Homotopy Groups [duplicate]

I'm following the following notes, and my attention was caught by Theorem $1.1.4$. I am unable to find any reference of the proof. Could you suggest me some books in which there is a proof of this ...
3
votes
1answer
49 views

graph partition, second smallest eigenvalue.

In spectral graph partition theory, the eigenvector corresponding to the second smallest eigenvalue of the laplacian matrix of a graph, in general, is used to partition the graph. What is the ...
0
votes
1answer
16 views

Lattice points in simplices - reference request

I found this paper http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf which, in formulas (1.2) and (1.3), relates the number of non-negative and positive integer values that are ...
4
votes
1answer
85 views

A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
-1
votes
0answers
19 views

reference for unsolved problems in ETCS [closed]

I am looking for unsolved problems in the theory of "elementary theory of category of sets" are there references for (foundational) problems in the category of sets as foundation?