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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5
votes
1answer
98 views
+50

Coordinate-free notation for tensor contraction?

I am not sure if I can prevent this question from being too vague or with too large an overlap with other similar math.SE questions, but I will do my best... A standard linear operation in tensor ...
0
votes
0answers
40 views

Groupoid objects in the category of algebras

Can anyone give me some references where I could read about groupoid objects in the category of algebras? References about groupoid objects in other categories would also be welcome.
0
votes
1answer
38 views

Best source to study partial differential equations (PDE) [duplicate]

Want to understand partial differential equations (linear and non-linear) more deeply. I am not a mathematician and I am more interessted in a more practical source that is teaching this topic from a ...
0
votes
0answers
52 views

Order Magic riddle

I have to solve this problem for some course and no one in my class has an idea how to solve it... There are 4 objects on a table: spoon, fork, knife and a spatula, which are positioned in an order ...
0
votes
0answers
31 views

Lebesgue-Stieltjes integral and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
1
vote
0answers
53 views

$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

Conjecture For $n \ge 1 $ , $m \ge 1$ $\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$ where $\pi\left(n\right)$ is the prime counting function . Does this conjecture ...
1
vote
1answer
45 views

Reference: Mahlo cardinals remain Mahlo in L

The following is stated on Wikipedia for Mahlo cardinals. Unfortunately, it's not sourced. Where can I find details? I wasn't able to google any articles dealing with Mahlo cardinals in L. Since ...
13
votes
2answers
137 views

Egg vs. chicken: trig functions, exponential, real and complex

This is something I was shaky about when I took calculus, real analysis, and then complex analysis. Specifically, is the following chain of definitions circular in any way? Define the set ...
0
votes
0answers
40 views

A request of a journal theorem

I am reading the paper Groups whose proper subgroups are finite-by-nilpotent by Maoqian Xu and he has some references that I can't find. Does somebody have (B. Bruno, On p-groups with ...
1
vote
0answers
9 views

Irrationality of $\min$ and $\arg\,\min$ of $\Gamma|_{[1, 2]}$

The Gamma function achieves a local minimum at $x^* \approx 1.46163$, where $\Gamma(x^*) \approx 0.88560$. Can $x^*$ and $\Gamma(x^*)$ easily be proven irrational? Are they transcendental?
1
vote
1answer
29 views

Degenerate zeroes, fundamental theorem of algebra.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two ...
0
votes
3answers
46 views

Reference request: Analytic study of the trigonometric functions

I'm looking for a source (or sources) which develop a complete theory of the trigonometric functions with no reference to circle geometry. That is, it is purely analytic. The starting point could be ...
0
votes
0answers
11 views

K-wise identical marginal distributions

Suppose I have two joint distributions described by the two sequences of random variables,$X_1, \ldots, X_n$; $Y_1, \ldots, Y_n$. Is there a name/theory/reference for when these two distributions ...
1
vote
0answers
28 views

Continuity of PDE solutions with respect to coefficients

Suppose I have a PDE, for example the Fokker-Planck one: $$ \frac{\partial}{\partial t}u(x,t)=-\frac{\partial}{\partial x}(\mu(x,t)u(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(D(x,t)u(x,t)). $$ ...
-1
votes
0answers
19 views

Getting Started in Bayesian Statistics [duplicate]

I want to start learning about Bayesian statistics. What resources would you recommend? If possible, I think for future readers it would be helpful if answers could be broken up into (1) overview ...
1
vote
1answer
19 views

Where should I begin the study of fixed point theory, especially of multi-valued maps?

How should one begin one's study of fixed point theory, especially of multi-valued maps? What background --- in topology, analysis, functional analysis, algebra, and set theory --- should one have? ...
9
votes
1answer
66 views

Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows: $p_0 = a_0$ $p_1 = a_0x+a_1$ $p_2 = a_0x^2+a_1x+a_2$ $p_n = ...
0
votes
0answers
16 views

Laplace operator on a compact riemannian manifold $(M^2,g)$

I'm studying some things about conformally covariant operators and I found this equation that there is an extensive literature about it, second the author. Let be $\Delta_{g_w}$ the Laplace operator ...
3
votes
1answer
34 views

Recommend of English video about math

My mother tongue is not English, but sometime there are some math reports or conference using English .Because my English is poor, I can't understand it well and can't suitably describe my question. ...
1
vote
1answer
33 views

Finitely generated projective modules form exact category

I am looking for a proof of the fact that the category of f.g. projective modules over some commutative ring is exact. Actually, I would already be happy to know why this category is closed under ...
3
votes
0answers
15 views

Is there a theory of “sums-of-squares residues”?

The theory of quadratic residues is long- and well-studied. Recall that, [somewhat simplified] if $x,a,b$ are integers, with $0 \le a < b$, such that $$x \equiv a^2\!\!\!\pmod{b},$$ then we say ...
2
votes
0answers
16 views

The simplest reference for Wedderburn decomposition

Assume $k$ is a field and $A$ is a $k$-algebra then the Wedderburn decomposition says $A=A_{sep}\oplus Nil(A)$, $A_{sep}\rightarrow A_{red}$ via $a\mapsto \bar{a}:=a+Nil(A)$. Where $A_{sep}$ means ...
0
votes
0answers
63 views

What are the classic or great books for Algebra, Geometry, and Trigonometry that are similar to what Spivak, Courant, and Apostol are for Calculus?

There are classic textbooks for Calculus like Spivak, Courant, Apostol, etc that do a fantastic job at explaining the fundamental concepts and theory along with great problem sets. My question is the ...
1
vote
0answers
47 views

Slick Definition of the Category of Cartesian Closed Categories

I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ...
1
vote
3answers
42 views

Solving $rX_1^2+sY_1^2+tZ_1^2=rX_2^2+sY_2^2+tZ_2^2$ completely in integers

Given pairwise relatively prime integers $r,s,t$, I’m looking for a complete solution (i.e., integer parameterization or similar) for the Diophantine equation $$ ...
2
votes
1answer
72 views

Does PA prove that Con(PA) implies Con(ZF-I) and Con(NFU)?

I read from many sources that PA and ZF-I (a suitable axiomatization of ZF minus Infinity plus its negation) are bi-interpretable, but is PA enough to prove that they are equiconsistent? Specifically ...
1
vote
1answer
41 views

Generating all prime powers $\leq N$

Some very good algorithms exist to generate all primes $p$ up to some bound $N$, like the sieve of Erastothenes and the sieve of Atkin. However, suppose I want to generate a (sorted) list of all prime ...
1
vote
1answer
71 views

Algebraic geometry reference for people with limited background [closed]

My background in algebra is groups and rings from Herstein's Topics in Algebra and field theory from Gallian's Contemporary Abstract Algebra. With this background, can I read any algebraic geometry? ...
17
votes
3answers
1k views

Why does what I've written fail to define truth?

I stumbled across a set of axioms for first order logic a bit ago. Intrigued, I decided to try to write it all down and organise what I read. After I did that, it seemed to me as though one could ...
1
vote
0answers
36 views

Book 2 of Visual Complex Functions

I am having a lot of fun in reading Visual Complex Functions by prof Wegert. (it is a very interesting read and well-recommended by me). Inside it, he regularly let things be and postpone until part ...
0
votes
0answers
21 views

A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 ...
6
votes
3answers
476 views

What does “the average continuous function is nowhere monotonic” mean?

I plan on asking my professor what he meant by "average continuous function," but as it is possible that this is a concept as vague as the statement, I was hoping to get some interesting ...
9
votes
0answers
99 views

Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
0
votes
0answers
30 views

Is Engelking and Sieklucki's “Topology: A Geometric Approach” a Good Introduction to Algebraic Topology?

I only found this book incidentally while looking at Engelking's more well-known "General Topology". I posted a link here. ...
0
votes
0answers
27 views

Representable bifunctors

Is there a notion of representability for functors in the form $F:C^{op} \times C \to Set$? Can anyone please give me a reference? Thanks.
4
votes
0answers
55 views

Hilbert's Inequality - improved???

Assume for convenience that $a_n\ge0$ (this also clarifies why certain inequalities below are in fact stronger than certain other inequalities below). Of the various inequalities Hilbert proved, I'm ...
0
votes
0answers
40 views

Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
4
votes
1answer
46 views

Using calculus results for functions of operators

I am interested in the conditions required for functions of operators to be manipulated as if it were a real valued function with a real domain. In an applied maths text I am using the following is ...
0
votes
1answer
23 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
1
vote
1answer
30 views

Asymptotic analysis references

I'm self studying asymptotic analysis with Bruijn (1981) - Asymptotic Methods in Analysis Bleistein and Handelsman (1986) - Asymptotic Expansions of Integrals but the texts are terse, without too ...
0
votes
1answer
16 views

Proving This Theorem on Independence

I'm trying to find a good resource for proving the following theorem, stated in Shreve's "Stochastic Calculus for Finance II," p. 73: Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let ...
0
votes
0answers
35 views

Is there a name for these inequalities? Where can I look them up?

Consider the operators $A,B,C$ on Hilbert space $\mathcal H$: Show that: $$ \left \vert \left \vert AB \right \vert \right \vert \le \left \vert \left \vert A\right \vert \right \vert \left \vert ...
1
vote
1answer
17 views

When does $A\mathbf{v} = \lambda B\mathbf{v}$ admit a basis of solutions?

Let $A, B \in \mathbb{C}^{n \times n}$ be Hermitian matrices, and consider the so-called generalized eigenvalue problem $$A\mathbf{v} = \lambda B\mathbf{v}$$ where $\lambda \in \mathbb{C}$ is called a ...
1
vote
0answers
50 views

Self-extensions of a skyscraper sheaf

Let $V$ be a smooth variety over a field $k$. For a point $x \in V$ we denote the skyscraper sheaf of length 1 by $$ k(x) = \mathcal{O}_x/m_x. $$ Then by taking the Koszul resolution of $k(x)$ one ...
3
votes
0answers
39 views

Matrix with roots of unity entries

For a given prime p, i am interested in the norms of matrices which have root of unity entries, i.e., $M_{k,l} \in \{1, \zeta, \dots, \zeta^{p-1}\}$ where $\zeta = \exp{(2\pi I/p)}$. Are there any ...
0
votes
0answers
45 views

reference request for $L^p(\partial\Omega)$ in real analysis textbooks

Let $\Omega$ be a bounded open set in $\mathbb{R}^d$. Would anybody come up with a real analysis textbook which contains detailed introductory treatment of the space $L^p(\partial\Omega)$?
2
votes
1answer
55 views

bounded generation and groups with infinitely many ends

Following section 7.1 in Peterson-Thom's paper here, we say a countable group $G$ is boundedly generated by the subgroups $G_1, \cdots, G_n$, if there exists an integer $k\in\mathbb{N}$, such that ...
1
vote
0answers
33 views

Reference Quest: Measure Theoretic and Functional Analytic Intro to Stochastic Processes

Does anyone have any recommendations for a good book which introduces and cleanly and rigorously explains the measure theory and functional analysis implicit in and relevant to stochastic processes, ...
0
votes
0answers
23 views

Classical Complex Analysis by Mario O. Gonzalez

My friend found a copy of the book Classical Complex Analysis by Mario O. Gonzalez, and we found that this book was being sold online for upwards of $800. I am really curious what would drive the ...
3
votes
2answers
69 views

Self-study mathematics subject sequence and recommended books

I am a Physics student but I finally found that I've entered the wrong department that I am in fact much more interested in mathematics. I want to self-learn mathematics. I am now reading Artin ...