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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When does a sparse failing subsequence of $(a_n)_n\subset\mathbb{R}_{>0}$ exist when $\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^na_j = c$ exists

My question is a bit longer than the actual title, so here is the full question: Let $(a_n)_n\subset\mathbb{R}_{>0}$ be a sequence of positive real numbers such that the limit $$\lim_{n\to\infty}\...
Cartesian Bear's user avatar
2 votes
0 answers
23 views

What is the name of $\prod_p \mathbb{Z} / p $?

Let $p_1=2,p_2=3,p_3=5, p_4=7, p_5=11,...$ the primes. I recently had Q/A here and here about the ring$$\boxed{(\prod_{n=1}^{+\infty} \mathbb{Z} / p_n\mathbb Z,+,\times)} $$ where $\prod_p \mathbb{Z} /...
Stéphane Jaouen's user avatar
1 vote
0 answers
20 views

Feynman-Kac theorem of the weak solution of parabolic PDEs

Is there any reference on the Feynman-Kac theorem of the weak solution of parabolic PDEs? So far I can only find the one for classical solution.
mnmn1993's user avatar
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3 votes
0 answers
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Classification of Model Structures on Set

Has anyone published classifications of model structures and/or weak factorization systems on the category $\mathcal{Set}$? If so, where? Context Arising from the question whether the model structures ...
Hermann Döppes's user avatar
1 vote
0 answers
39 views

Reference Request: In Zariski &Samuel's Commutative Algebra (volume I), should the definition of length of an ideal be modified?

This is more a reference request question than a technique one. I hope someone who has read Zariski, Samuel's Commutative Algebra (volume I) could answer it. (It seems few people use it nowadays). In ...
Asigan's user avatar
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-1 votes
0 answers
43 views

Grothendieck group of the Category of Finitely Generated R-Modules is Isomorphic to $\mathbb{Z}$

I was reading the top answer to this post: Torsion Grothendieck group. In the top answer, it states, "This is roughly because the Grothendieck group of the category of finitely generated modules ...
Rocky.Racoon.'s user avatar
2 votes
1 answer
40 views

Proof of $K_0(z)=-\left(\log\frac{z}{2}+\gamma\right)I_0(z)+\sum_{n=1}^\infty \frac{(z/2)^{2n}}{n!^2}H_n$

Let $I_{\nu}$ be the Bessel I function of order $\nu$ defined by $$I_{\nu}(z)=\sum_{n=0}^\infty \frac{(z/2)^{2n+\nu}}{n!\Gamma (n+\nu+1)}$$ and let $K_{0}$ be the Bessel K function of order $0$ ...
Nomas2's user avatar
  • 605
4 votes
1 answer
77 views

Asymptotic Behaviour of the Remainder of Certain Alternating Series

Let $a,b >0$ be real constants. Empirical observation (as in: asking WolframAlpha) suggests $$ \lim_{n\to \infty} n \cdot \sum_{k=0}^\infty (\frac{1}{n+ak} - \frac{1}{n+b+ak}) = \frac{b}{a} \tag{$...
Torsten Schoeneberg's user avatar
1 vote
0 answers
22 views

Generalized Hamming weights for binary BCH codes

Given a linear binary code $C$, the $r$-th generalized Hamming weight $d_{r}(C)$ is the minimal support size of an $r$-dimensional subcode of $C$ (so $d_{1}(C)$ is simply the code's distance). For a ...
DanDan's user avatar
  • 359
3 votes
1 answer
101 views

The category of abelian groups with quasi-homomorphisms

Let $A$ and $B$ be abelian groups. Say that a map $f: A \to B$ is a quasi-homomorphism if there exists a finite $D \subseteq B$ such that $$\forall a_1, a_2 \in A: f(a_1 + a_2) - f(a_1) - f(a_2) \in D$...
Smiley1000's user avatar
0 votes
0 answers
18 views

Is there a closed form for $\int _0 ^1 t^{a-1}(1+t)^{b-1}dt$? [duplicate]

$$B(a,b):=\int _0 ^1 t^{a-1}(1-t)^{b-1}dt$$ What happens If we change the negative sign to positive ? $$F(a,b):=\int _0 ^1 t^{a-1}(1+t)^{b-1}dt$$ This question came to me while solving this limit $$...
pie's user avatar
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2 votes
0 answers
56 views

Historically, when have the the real numbers been constructed via the "positive" (non-negative) reals first, and then usual real numbers second?

There has been something that has been bugging me for the longest time, at least since grad school. In the teaching of mathematics, during the construction of the "usual" real numbers, why ...
Rex Butler's user avatar
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0 answers
51 views

Solving the quaternion sum of four squares/reduced norm

Let $K$ be an algebraic totally real number field, and let $O_K$ its ring of integers denote by $Q$ the quaternion algebra over $K$ I am looking for references on the resolution of the reduced norm ...
Don Freecs's user avatar
1 vote
0 answers
19 views

Reference for semirings with exponentiation

A commutative semiring $R$ with exponentiation has an additional operation $R \times R \to R, (a, b) \mapsto a^b$ (called exponentiation), satisfying the following six axioms ($\forall a, b, c \in R$):...
Geoffrey Trang's user avatar
0 votes
0 answers
23 views

Why do we take the absolute value in big-O?

The various usual definitions for the statement $f(x) = O(g(x))$ as $x \to \infty$ either stipulate that $f$ and $g$ are (eventually) nonnegative, or restrict their behavior in the definition, e.g. ...
Electro's user avatar
  • 343
1 vote
0 answers
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Reference request: if $f$ is real analytic and $\{x:f(x)=0\}$ has a limit point, then $f=0$

I would like to know a reference for the following theorem. Theorem: If $f$ is a real analytic function on an open set $G$ in $\mathbb{R}^p$ and if $\{x:f(x)=0\}$ has a limit point in $G$, then $f(x)=...
rfloc's user avatar
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2 votes
0 answers
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A common technique in number theory to evaluate integration

Sorry this question is a bit weird ... I saw a technique in some number theory book (that I don't remember the name - I checked Titchmarsh but I couldn't find) that in order to evaluate some integral (...
Ali's user avatar
  • 173
0 votes
0 answers
16 views

Request bibliographic reference(s) for finite alternating sums with Eulerian numbers

I would like to know a bibliographic reference for a math formula. I found this formula on Wikipedia, but no reference is given. It's the 2nd formula ($A(n,k)$ is an Eulerian number): $$\sum_{k=0}^{n-...
Roy's user avatar
  • 1
4 votes
1 answer
60 views

Reference request non existence of minimal resolution.

In this page of Wikipedia(https://en.wikipedia.org/wiki/Resolution_of_singularities), it writes, the hypersurface in $\mathbb{A}_\mathbb{C}^4$ defined by the equation $xy-zw$ has no minimal resolution....
George's user avatar
  • 291
2 votes
0 answers
37 views

Has there been research on definability predicates, just like there has been research on truth predicates?

I know that there has been research on truth predicates, namely, formal theories of arithmetic with a truth predicate. You basically add a predicate $T$ to the language that models truth. I wonder, ...
user107952's user avatar
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11 votes
6 answers
1k views

Book recommendations for Combinatorics for Computer Science Students

I am a computer science student with an interest in competitive programming. I am currently looking to deepen my understanding of combinatorics, as it is a crucial part of algorithm design and ...
Abdelrhman Sersawy's user avatar
0 votes
0 answers
20 views

Hasse principle for quadratic forms vs algebraic groups

Can the Hasse-Minkowski theorem for quadratic forms over a number field be recovered from the Hasse principle for algebraic groups? More specifically, by the former I mean the theorem that two ...
Cyclicduck's user avatar
2 votes
0 answers
54 views

Dedekind eta and Lie algebras

Reference request and a formal question I am currently self-studying modular forms and read the Wikipedia page for Dedekind's eta function. There it reads that: "The theory of the algebraic ...
ProtoZone's user avatar
1 vote
0 answers
22 views

Why the term "geometric rough paths".

I wanted to understand where the term "geometric rough paths" comes from, and when was it used for the first time. In this MathOverflow post, Martin Hairer suggests that the term geometric ...
Oscar's user avatar
  • 866
2 votes
0 answers
38 views

"Infinite-dimensional Courant–Fischer"

Let $T$ be a positive self-adjoint operator, possibly unbounded, on a Hilbert space with domain $D$ and spectrum $\sigma(T)$. I know that $$\inf\sigma(T)=\inf\limits_{\substack{x\in D\\\|x\|=1}}\...
zjs's user avatar
  • 1,145
2 votes
1 answer
130 views

Is it true but unassertable that there are undefinable real numbers?

I know of Joel David Hamkins's analysis of the so-called "math tea argument", namely that there are undefinable real numbers. Supposedly, he debunked this argument by constructing a ...
user107952's user avatar
  • 20.7k
4 votes
1 answer
52 views

Projective plane over the octonions (Cayley plane)

You can use octonion algebra's $\mathbb{O}$ over a field to coordinatize projective planes. They are called Cayley planes as far as I know. You can't use the usual approach with homogeneous ...
Vincent Batens's user avatar
1 vote
2 answers
43 views

Book recommendation for stochastic integral wrt local martingales

I am looking for a book (or a chapter of a book) for stochastic integral wrt local martingales. The book should contain a rigourous introduction to the definition. It should also contain proofs for ...
Mingzhou Liu's user avatar
1 vote
1 answer
56 views

Are there affine rings that are properly affine?

Let $A$ be an abelian group. If $B$ is a nonempty set equipped with a transitive free action of $A$ (written additively), say that $B$ is an affine abelian group (equivalently, $B$ is an $A$-principal ...
Smiley1000's user avatar
-1 votes
0 answers
35 views

Looking for an optimal control article that was only available as Bell Labs Technical Memo (by Holtzman and Halkin) [closed]

I am looking for the following reference: J. M. HOLTZMAN and H. HALKIN, Directional convexity and the existence of optimal controls, Bell Telephone Laboratories, Technical Memorandum, 1965. There is ...
ExpressionCoder's user avatar
2 votes
1 answer
69 views

Real degrees after forcing a random real

We know a lot about the properties of the real degrees if we assume the existence of an $L$-generic Cohen real (e.g. see Abraham and Shore Degrees of Constructibility of Cohen Reals). Among other ...
Lorenzo's user avatar
  • 2,601
4 votes
0 answers
50 views

Can we efficiently check whether a number is a Zumkeller number?

A positive integer $n$ is a Zumkeller number iff its divisors can be partitioned into two sets with equal sum. If $\sigma(n)$ denotes the divisor-sum-function , this means that there are distinct ...
Peter's user avatar
  • 84.5k
0 votes
0 answers
25 views

Definition Gauge Field / Connection on principal bundle

Reading Division Algebras and Supersymmetry I by John C. Baez and John Huerta, one passage confused me a bit: A connection $A$ on a principal $G$ bundle over $M$. Since the bundle is trivial, we ...
anonymous250's user avatar
-3 votes
0 answers
36 views

Refer me books through which I can learn advanced calculus. [duplicate]

You can also refer websites or books PDF.
Shantanu Binekar's user avatar
3 votes
1 answer
39 views

Origin of a Relation in the Proof of Theorem 10.6.4 in Beardon's Book

I'm studying the proof of the following theorem in Beardon's book. Theorem 10.6.4: A group $ G $ is a $(p, q, r)$-Triangle group if and only if it is a discrete group of the first kind with signature ...
Rowing0914's user avatar
3 votes
1 answer
96 views

What can semigroup theory do better in the study of PDEs compared to alternative methods?

I've recently come across semigroup theory in my mathematical physics class and while the theory itself feels nice to work with, I have not yet understood what does the theory offer for the study of ...
Cartesian Bear's user avatar
0 votes
0 answers
26 views

Growth of workers over time if cost of adding more is some function f

Motivation I'm curious about some behaviors in those incremental cookie-clicker-like video games. I've got some thoughts, but as I've not done rigorous math in some years, I decided to seek ...
Vepir's user avatar
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1 vote
0 answers
26 views

References on solid sequence spaces

I have been trying to search in different books on functional analysis for examples and properties of solid sequence spaces (https://en.wikipedia.org/wiki/Solid_set), however I cannot find any ...
NoetherNerd's user avatar
0 votes
0 answers
26 views

Reference Request for Proof of Landau' Formula

I tried to find a proof for Landau' Formula (https://mathworld.wolfram.com/LandausFormula.html). Original paper (1911) is in German and I searched classic books (Montgomery, Titchmarsh, Ivic) and none ...
Ali's user avatar
  • 173
1 vote
0 answers
38 views

On proving that an Artin L-function is cusp form

I am reading the article by JEAN-PIERRE SERRE, On a Theorem of Jordan, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 2003. I have problem in the following part: Since $S_{3}$ is a dihedral group, ...
Aster Phoenix's user avatar
7 votes
2 answers
123 views
+100

Using the binomial expansion to find derivative formulas

While differentation the function $f(x) = (1-e^x)^{5}$, I realize one can use the Binomial Theorem to find formulas for nth derivative: The idea is as follows. Let $f(x) = (1-e^x)^n$. We have $$ f(x) =...
ILoveMath's user avatar
  • 10.7k
0 votes
1 answer
40 views

Solving $Ax=b$: Projection onto subspace with a canonical basis of largest error

The goal is to solve the linear system $Ax = b$, where $A$ is symmetric and positive definite (SPD). Consider the one-dimensional projection method given by equation (1): $$x_{k+1} = \operatorname{...
Meow's user avatar
  • 165
2 votes
1 answer
80 views

Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $

Background The Norwegian mathematician and astronomer Carl Størmer did important work on the equation $$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$ ...
Max Muller's user avatar
  • 7,048
2 votes
0 answers
28 views

Book/Notes Recommendation for Quadratic Optimization

While self-studying and planning on doing some small-time research/fun based on Philip Wolf's "The Simplex Method for Quadratic Programming", I got interested in the notion of quadratic ...
Miss Mae's user avatar
  • 1,586
1 vote
1 answer
45 views

References for Weil restriction?

I'm reading some people's notes and some fantastic answers on stackexchange on Weil restriction. I don't have a background in algebraic geometry, so some of the simplified versions, particularly of $\...
Batrachotoxin's user avatar
4 votes
1 answer
43 views

Kahler geometry and topology in modern physics

How are tools and concepts Complex and algebraic geometry (and also algebraic topology) used in modern physics, such as in string theory? Is there any introductory text which deals with this topic (ie ...
user720386's user avatar
5 votes
1 answer
39 views

Proof of a martingale condition regarding martingale transform

I tried to work on my previous question regarding a Converse of a martingale transform theorem, and was told by a stochastics PhD student at my university that indeed if an integrable, adapted process ...
Jodasilva's user avatar
  • 131
0 votes
0 answers
21 views

Family of probability distributions on $\{0, 1, \ldots, n\}$ that can approximate any deterministic distribution

Consider a family $\mathscr{F}$ of probability distributions with the following properties: $\mathscr{F}$ can be parameterized by a discrete parameter $n \in \mathbb{N}$ and, for some fixed $d \in \...
user76284's user avatar
  • 5,977
5 votes
0 answers
24 views

Defining Derivative on Tropical Semiring

I have tried to find references that are related to calculus on tropical semiring, but I was not able to find appropriate references. So, I used to Thompson's approach to define deriviative on ...
30412's user avatar
  • 51
1 vote
2 answers
106 views

On the volume of parallel rectangles contained in oblique rectangle

Consider the following lemma in these notes on measure theory (page 31 in the pdf): Lemma 2.29. If an oblique rectangle $\tilde{R}$ contains a finite almost disjoint collection of parallel rectangles ...
psie's user avatar
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