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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1answer
36 views

Reference Request: Subgroup of free abelian group is free abelian

I have the following reference Question, meaning that I search for a reference for the following statement: Let $F$ be a free abelian group of finite rank and $U$ be a subgroup. Then there is a basis ...
2
votes
1answer
53 views

Reference on Malcev completion

I need a reference for learning Malcev completion, its associated group scheme, and Lie algebra. Thanks!
0
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2answers
38 views

Prove (or derive) the de Polignac formula for the prime decomposition of $n!$

I can't seem to find any papers published dedicated to show that the de Polignac formula has a rigorous derivation. From Wikipedia's entry for the formula: Let $n \geq 1$ be an integer. The prime ...
0
votes
0answers
15 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
4
votes
1answer
55 views

Jacobi Elliptic Functions built from Jacobi theta functions

I believe I understand the general theory of elliptic functions to an extent. What I can't seem to find is the distinct method which was used to show that a particular combination of Jacobi Theta ...
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0answers
32 views

Partial Differential Equations Solver

Anyone knows a good website/program in which you enter your partial differential equation with the initial conditions and it simply solve it?
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5answers
129 views

Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”

If $C$ and $A$ are the circumference and area of a circle, then $$ \frac{C^2}{A}=4 \pi\; . $$ I'm looking for a reference to an elementary but rigorous proof of it. I'm particularly interested in ...
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0answers
35 views

Books for Real analysis similar to or having same essence like Charles Pinter Abstract algebra

I am looking for book which is similar to spirit of pinter's book mentioned in question that is which is suitable for self study and beginners. Can anyone recommend? Thanks
5
votes
2answers
131 views

Two questions on the Grothendieck ring of varieties

1) In the definition of the Grothendieck ring of varieties over a field $k$, which definition of the various notions of "variety" is chosen? Finite type and separated, or maybe more? 2) If ...
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0answers
14 views

An example of frame operator.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in ...
0
votes
0answers
35 views

Finding references about modular forms and symmetric power $L$-functions

I want to write an introduction to my thesis which is about modular forms and symmetric power $L$-functions. Could you give me good references to these two topics as simple as possible to get an ...
2
votes
1answer
41 views

Class field theory of imaginary quadratic fields

Can someone give me a good source that deals in detail with the Class Field Theory of imaginary quadratic fields? The sources on CFT that I have at hand only deal with CFT in general and then proceed ...
2
votes
2answers
36 views

Positiveness of partial sums of type $ \psi * D_N $

In his paper about Extremal Functions for the Fourier Transform (see, for example, here? https://projecteuclid.org/download/pdf_1/euclid.bams/1183552525), Jeffrey Vaaler, while trying to build ...
1
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2answers
107 views

Hardcore Abstract Algebra Book Request

I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind ...
0
votes
4answers
94 views

In need of a group theory textbook.

I am in need of a group theory textbook for a good summer review. I have already studied from various books (mostly "group theory" part from basic algebra books) and the lecture notes of my teacher, ...
8
votes
1answer
54 views

Mathematically rigorous books on programming and computer science

I would like a mathematical approach to programming languages and computer science, not just the theoretical aspects of computer science. Is there any such text out there? After all, the world has a ...
4
votes
2answers
54 views

About Mordell's Theorem (Elliptic Curves)

I've just finished the proof of Mordell's Theorem given in the book "Rational Points on Elliptic Curves " by Silverman. One of the key lemmas used in the proof of the theorem is: Let ...
1
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2answers
92 views

Reference request for this topics

I'm a second year undergraduate statistic student, I need a good reference to learn these topics Markov Chains in discrete time.    1.1. Classification of states, recurrence notions of transience. ...
3
votes
1answer
97 views

Difference between proof-based calculus and analysis?

When I asked a teacher at school what the difference between calculus and analysis he said that calculus is essentially analysis without proofs. So where does proof-based calculus lie? Is it somewhere ...
5
votes
1answer
140 views

What proof uses both the Riemann Hypothesis and its negation?

Some time ago I happened to see a proof that was remarkable in that it used both the Riemann Hypothesis and its negation. That is, it considered the two cases: RH is true, and RH is false, obtaining, ...
1
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0answers
55 views

Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
3
votes
2answers
45 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
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0answers
25 views

Looking for reference to solve a problem (representation theory, I think?)

this question appeared on an algebra qualifying exam: Let $R$ be the group algebra $\mathbf{C}[S_3]$. How many nonisomorphic, irreducible, left modules does $R$ have and why? Would I be able to ...
0
votes
0answers
49 views

Does the restriction of the Thom class of a submanifold to the cohomology of the submanifold give the Chern class of the normal bundle?

Let $X$ be a compact complex manifold of (complex) dimension $d$, let $i\colon Y\hookrightarrow X$ be a regular complex submanifold of (complex) codimension $k$, let $\mathcal{N}_{Y/X}$ be the normal ...
0
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0answers
66 views

How to prove it and how to solve it

Tomorrow I will begin my studies, real analysis, however I have some difficulties in making statements so I thought before starting the study in real analysis, learn how to do demonstrations properly. ...
2
votes
1answer
33 views

Lie algebra and Lie group cohomology, reference request

Who can give me a good reference (better if introductory/motivated) about Lie group cohomology, Lie algebra cohomology, and the link between the two? Thanks.
5
votes
1answer
115 views

How can I calculate the formula of this fractal-like structure?

I did the following fractal-like structure manually, and I was trying to convert it to a formula (or an algorithm including formulas) to compute some parts of the drawing, but I get lost due to the ...
1
vote
1answer
75 views

Modeling, Measuring, and Maximizing “Mixedness”

Possible key-terms: combinatorial optimization techniques; simulated annealing; genetic algorithms; Kirkman's schoolgirl problem; Steiner triple systems; orthogonal regrouping. Background: My class ...
2
votes
0answers
34 views

Set of functions from a finite field to the integers

Has the set of functions $\mathbb{F}_q \to \mathbb{Z}$ endowed with pointwise addition and additive/multiplicative convolution been studied? Does anyone know a reference or keywords to search for? ...
0
votes
1answer
30 views

Reference help about fuzzy logic and fuzzy set

Good afternoon, I'm trying to learn about Fuzzy alone, I was using some texts on the Internet about it, but it was very difficult to learn. I want to learn about Fuzzy set as shaper of uncertainty. ...
1
vote
1answer
19 views

Reference for the statement “bilinear form $a$ is symmetric if and only if the operator $S$ is self-adjoint”

Thanks to Riesz representation theorem, a continues bilinear (sesquilinear) form on Hilbert space $$a: \mathcal H\times \mathcal H\rightarrow\mathbb R \ \ (\text{or} \ \ \mathbb C)$$ can be ...
1
vote
1answer
16 views

Is optimal bound for Alcuin's triangular city problem known?

Alcuin's triangular city problem is Problem 28 from Propositiones ad Acuendos Juvenes. There is a triangular city which has one side of 100 feet, another side of 100 feet, and a third of 90 ...
0
votes
1answer
20 views

Intervals of integers modulo n

Do the following related concepts appear anywhere in literature? Denoting an "interval" in the integers modulo $n$ by $[i,j] = \{i, i+1, \dotsc, j\}$. For example, in modulo 6, $[5,3] = ...
1
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1answer
25 views

What is this 2-D array of numbers called?

Define $c(n,r)$ ($n\in\Bbb N;r\in\Bbb Z$) by setting $c(0,-1)=-1$, $c(0,0)=1$, and $c(0,r)=0$ otherwise, with all further $c(n, r)$ given recursively by $$c(n+1,r)=rc(n,r-1)-(r+1)c(n,r)$$(rather in ...
8
votes
2answers
112 views

Good resources for studying independence proofs

I've finished most of Enderton's set theory. And I intend to spend some time studying independence proofs. I'm more interested in independence of axiom of choice not CH. From I know so far, there ...
0
votes
0answers
19 views

Quartic spline explanation

I need to use a quartic spline (natural spline) for a robotics application. However, most of the documents explaining splines focus on quadratic and cubic splines. Could you please give me some ...
7
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0answers
157 views

What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
0
votes
2answers
50 views

On proving $(f^{-1})'(b) = \frac{1}{f'(a)}. $ where $b = f(a)$.

Could somebody kindly provide a proof or a reference to a proof of this fact: Let $ I $ be an open interval, and suppose that $ f: I \to \mathbb{R} $ is one-to-one and continuous on $ I $. If $ f ...
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0answers
25 views

Need ''The diameter of the prime graph of a finite group'' by Lucido

I need a paper by Lucido, I am studying prime graphs to make a presentation in Group theory class, which is here on degruyter. Now, my college doesn't have access to degruyter (although is has to many ...
1
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0answers
17 views

Discrete version of Sylvester's Law of Inertia

Given a matrix $A\in\mathbb{C}^{n\times n}$, we denote by symbol $\mathrm{In}_d (n_<,n_>,n_1)$, the discrete inertia (or inertia w.r.t. the unit circle) of $A$, where $n_<,n_>$ and $n_1$ ...
2
votes
1answer
27 views

The 1-Norm on a Quantum Group as a Supremum

To this MO question, Yemon Choi comments that If $\tau$ is a faithful normal trace on a von Neumann algebra $M$, then IIRC $\tau(|x|)$ is equal to the supremum of $|\tau(xy)|$ as $y$ runs over all ...
0
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0answers
31 views

The equivalent of least squares, but for vectors

Given a set of poins, one can use a fitting method such as least squares to find the straight (or the parabola, or the 3rd grade equivalent) that's closest to all points at the same time (via ...
3
votes
1answer
74 views

Reference Book for Calculus

I've made this post as detailed as I can so as to give you a fair idea of my level. Calculus (particularly Integration) is my passion and frankly I spend all my free time learning as much of it as I ...
0
votes
1answer
45 views

Group Theory: Suggest video lecture (In English)

Please suggest video lecture for following topics in Group Theory. Revision of definition and examples of groups, subgroups. Cyclic Groups, Classification of subgroups of cyclic groups. Permutation ...
1
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1answer
38 views

Transcendence of Values of Beta Function

Wikipedia mentions that the number $$a = \dfrac{\Gamma\left(\dfrac{1}{4}\right)}{\pi^{1/4}}$$ is transcendental. Since $\Gamma(1/2) = \sqrt{\pi}$ the above number $a$ seems to connected to a ...
9
votes
1answer
120 views

What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
0
votes
0answers
57 views

Prerequisites “Homology Theory of Algebraic Varieties” by Wallace

I bought this book because the title was very interesting, the description as well and the price very cheap. You can read it here in PDF. Unfortunately, I realized after reading the first lines I was ...
0
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0answers
20 views

Where can I find ths paper about a diophantine equation?

https://scholar.google.com/scholar?cluster=15345230964505461139 I couldn't access this paper anywhere. It would be very helpful if anybody could explain the way how Nagell solved the diophantine ...
2
votes
0answers
9 views

Special class of Brenke Polynomials

I was wondering if there are any particular papers dealing with a particular class of Brenke Polynomials, defined as $$A(t)B(xt)=\sum_{n\ge 0}P_n(x)t^n$$ where $A=B$ or, where $A(t)=C(B(t),t)$ for a ...
4
votes
6answers
440 views

Elementary number theory - prerequisites

Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really ...