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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8 views

Universal Property of free objects

I am working on free objects, I am restricting myself primarely to groups, rings and modules (with maybe algebras) so in a sense in the concrete category (if I am not mistaken. This is a thesis work I ...
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0answers
32 views

requirement of multiple choice question for linear algebra

I want to prepare myself for a multiple choice examination. Is it ok if someone introduce a good and complete multiple choice question books for Linear Algebra and Calculus to me? Thanks,
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0answers
18 views

Relation between Alexander duality and linking numbers

I just have the feeling that there must be some relation between Alexander duality and linking numbers, but I don't know what is that. Will anyone tell me anything about that? Or could anyone give ...
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2answers
46 views

Applications of Linear Algebra in software engineering.

I'm a software engineering and mathematics student, I was searching for disciplines of mathematics that would go well with my engineering degree, and found a lot of people recommended that software ...
1
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0answers
32 views

what is relation between topology and geometry? [on hold]

what is relation between general topology and geometry ?( and example in this relation .) is there simple book in relation between general topology and geometry ?
2
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2answers
69 views

What's the mathematics behind 3D modelling? [on hold]

I'm highly interested about 3D modelling in software, and I know that it has some deep mathematics behind it too. I would like to learn what specific topics are behind it mathematically. As long as I ...
1
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1answer
11 views

Proof of the PAC generalization error bound using VC dimension

There is a theorem in PAC ("Probably Approximately Correct" model, in computational learning theory) that reads as follows: To guarantee that any hypothesis that perfectly fits the training data ...
0
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1answer
22 views

Does every variety contain a smooth subvariety?

Let $X$ be a projective variety (not necessarily irreducible) in $\mathbb{P}^n_{\mathbb{Q}}$. Given any such $X$, does it always contain a smooth subvariety $Y$ and can we say anything about $\dim Y$? ...
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0answers
10 views

Knotted cross-sections of unknotted spheres

Livingston's notes on concordance mention "embeddings of $S^2$ into $\mathbb{R}^4$ which are unknotted, but have non-trivial knots as cross-sections. There are other such unknotted two spheres with ...
4
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3answers
59 views

What kind of mathematics do I need to know to understand virtual reality very well? [on hold]

In my life I want to develop the virtual reality industry and I wanted to know what kind of mathematics will help me get there, I'm sure that every mathematical subject is helpful, but I mean ...
1
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1answer
36 views

Enough Projectives in Category of Groups

Working on homology and completion a question has arisen in my head. I know that $R$-mod as a category has enough projectives in it, and as such the category of abelian groups has it as they are in ...
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9answers
524 views

A good book for beginning Group theory

I am new to the field of Abstract Algebra and so far it's looking to me quite tough. So far I have encountered the following books in group theory - Contemporary abstract algebra by Joseph Gallian and ...
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0answers
17 views

Relation between error of estimate and rate of convergence

How is bounds on estimated error of an iterative algorithm related to rate of convergence? Referring to references is appreciated.
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0answers
18 views

Hampton Rule Hay Stacks [on hold]

"All hay in stacks is to be measured about October 1st, 1934 by Hampton rule" What is the Hampton rule method or unit?
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0answers
28 views

Are there some new “function or even topic” in lie theory with special functions? [on hold]

Every one: I research in Lie theory with special functions. But I saw a lot of research for most of the special functions and polynomials. I wish you could recommend a specific kind of these special ...
0
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0answers
30 views

How to find quadratic residues in the polynomial ring $k[t]$?

I have a question: Given a field $k$, finite or infinite, and an element $p(t)$ in the polynomial ring $k[t]$. I am searching for results of any kind about how to find quadratic residues in the ...
1
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0answers
13 views

matrix valued integrating factor of one forms (reference request)

I have $N$ 1-forms $\omega_1(x), \ldots, \omega_N(x)$. I want to know if there exists an invertible linear combination of these forms which yields $N$ closed forms. In other words: does an invertible ...
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0answers
13 views

Calculating sheaf of differential operators for smooth scheme

I have heard that if $X$ is a smooth scheme over $k$, then we can calculate the sheaf of differential operators $\mathcal{D}_X$ by considering étale morphisms from an affine open set to ...
3
votes
1answer
30 views

Intuitive understanding into the mean curvature flow

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $$S = ...
1
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1answer
31 views

A sign in a book that I can understand what it mean: $1_{x}$, $1_{y}$.

In a book that I read about mapping, It said: *Any mapping $f:X\rightarrow Y$ satisfies: $f1_{x}=1_{y}f=f$ *$g:Y\rightarrow X$ is a reverse mapping with $f:X\rightarrow Y$ only when $gf=1_{x}$ and ...
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1answer
30 views

To understand elliptic partial differential equations

I am a graduate university student of mathematics. I would like to study elliptic partial differential equations on my own. I have tried this lecture note though I cannot understand it all as I never ...
0
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1answer
21 views

Reference request: Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of some nonzero functional $f$.

I know the following statement is true, but I am looking to find a good reference that proves this quite nicely Every vector subspace of codimension $1$ of a vector space $A$, is the kernel of ...
0
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0answers
3 views

Fractional Sobolev space on a compact 1-D segment

Fractional Sobolev space $H^s_p(\mathbb R), s>0, 1<p<\infty$ is a space of tempered distributions $f$ that satisfy $F^{-1}((1+|\xi|^2)^{s/2} F(f)) \in L^p(\mathbb R)$. Here, $F$ denotes the ...
2
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1answer
25 views

Number of points satisfying a quadratic equation over $GF(q)$

I am stuck on the following problem. Let $GF(q)$ be the finite field of order $q$, where $q$ is an odd prime power. How can I show by elementary methods that the number of points $(x,y,z)$ satisfying ...
3
votes
2answers
104 views

Is category theory ambiguous? or it just is the case for beginners? [on hold]

First of all, I have to say that I'm not going to offend anyone/anything here; I just need some clarification/studying tips about category theory. I'm totally new in category theory and this happens ...
0
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0answers
45 views

Several options using Black-Scholes equation(s)

Could someone provide me some information about the modelling of several options at the same time by using Black-Scholes (probably coupled) equations? Any reference to papers and/or books shall be ...
2
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0answers
27 views

How to read Spectral Theory of Graphs

My background is a course is Linear Algebra -Hoffman,Kunze Graph Theory-Clark. I am reading Spectral Theory of Graphs. My professor has asked me to start from the book Spectra of Graphs by ...
0
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0answers
18 views

Good introductory book in geometric probability

I recently came across the proof of the Buffon theorem and I was fascinated by geometric probability. Could someone indicate me a good introductory book? Maybe with many exercises?
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0answers
4 views

Counting subgraphs of bounded extremal degrees

Let $m\leq n-1$. Is there a closed expression counting the subgraphs of minimum degree $\geq m$ (resp. maximum degree $\geq m$) on $n$ labelled vertices?
0
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1answer
56 views

Regarding Apostol's theory of integration

I have some questions regarding the theory of integration as discussed in Tom Apostol's Calculus. Integration is defined using step functions. My question is, is this definition he presents equivalent ...
3
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2answers
45 views

Proofs of Liouville Theorem

Are there proofs of Liouville theorem (bounded functions holomorphic in $\mathbb{C}$ are constants) without using the Cauchy theorem?
0
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0answers
6 views

References: Manifold with boundary, corners, and stratified manifolds

I would like to know some references on: Manifolds with boundary and corners Stratified manifolds I am looking for introductory texts, possibly with some physical insights. Thank you
4
votes
1answer
113 views

Why is “$\pi^2= g $” where $g$ is the gravitational constant?

Some months ago a professor of mine showed us a 'proof' of why $g\approx 9.8 ~\text{m}/\text{s}^2$ (the gravitational acceleration at the surface of the Earth) is 'equal' to $\pi^2\approx9.86\dots$ ...
3
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1answer
25 views

How to see that the prime gaps functions isn't monotonic?

Let $g(n)$ be the distance between the $n$th prime and the next. By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic. Further we know that ...
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1answer
46 views

Math for a computer engineering graduate course [on hold]

I'm going to take a master's course in Computer Engineering. Now I've not taken a CS course in my undergrad, hence my knowledge in some of the core areas of CS like algorithms are limited. I'm quite ...
4
votes
1answer
52 views

How to put my knowledge of probability and statistics to practice

Background: I am a masters student in stochastic analysis. My course is very theoretical, which in general is fine by me, it is what I enjoy the most. From the more data-friendly subjects, I have (or ...
1
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1answer
88 views

What kind of algebraic structure is $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$?

Let $\left( \mathbb{R}_{\geq 0},+,\cdot \right)$ denote the non-negative real numbers with usual addition and usual multiplication. Obviously, this is not a field, because $0$ is the only additively ...
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0answers
18 views

What are some modern books on Markov Chains with plenty of good exercises?

I would like to know what books people currently like in Markov Chains (with syllabus comprising discrete MC, stationary distributions, etc.), that contain many good exercises. Some such book on ...
3
votes
1answer
34 views

Is $\mathsf{nCob}$ bicomplete?

Let $\mathsf{nCob}$ be the category of $n$-cobordisms, whose objects are $(n-1)$-dimensional closed manifolds and morphisms are bordisms. Is this category bicomplete, or even finitely bicomplete? ...
0
votes
1answer
64 views

A weaker than “zero derivative” condition implies that a function is constant?

Let $f:(a,b)\to R$ be a continuous function such that $\limsup\limits_{n\to \infty}\frac{f(x_n)-f(x_0)}{|x_n-x_0|}\leq 0$ for every $x_0\in(a,b)$ and sequence $x_n$ converging to $x_0$ such that ...
2
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1answer
24 views

$S_n$ as a Coxeter group? (with Matsumoto's theorem)

I need to find a book in which basic $S_n$ theory is covered, mainly the part about Bruhat order, length of an element $w \in S_n$ and invariance modulo braid relations of the expressions ...
1
vote
0answers
7 views

Number of translated cubes covering a given hypercube in $\mathbb{R}^n$

Let $\Omega \subset \mathbb{R}^n$ be open and bounded, and let $Q \subset \Omega$ be a hypercube. Furthermore, denote by $D$ the $n$-dimensional unit cube $(0,1)^n$. Let $k \in \mathbb{N}$ be big, ...
1
vote
1answer
52 views

Name of and references for the equivalence relation $x \sim y :\Longleftrightarrow x^2 = y^2$

Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements $x,y$ of a field $\mathbb{F}$: $$ x \sim y :\Longleftrightarrow ...
2
votes
2answers
35 views

Is there an alternative approach to the definition of prime numbers based on the definition of the natural logarithm?

Is there an alternative approach to the definition of prime numbers based on the definition of the natural logarithm? These are my thoughts about it, the questions are at the end: Basically when a ...
1
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0answers
10 views

On the existence of magic squares of every order different from $2$

I was reading about magic squares and suppose that we speak here only of the magic squares that have in itself numbers from $1$ to $n^2$. It is easy to see that we cannot have $2$x$2$ magic square ...
1
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1answer
22 views

Online course for numerical methods/analysis of PDEs

Could anybody recommend an online course for implementing numerical methods to solve PDEs which can supplement reading? This is with a view to writing an implementation to solve the Monge-Ampere ...
3
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0answers
34 views
+100

Linking circles inside an immersed surface

A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the unknot has the disk as a Seifert surface.) ...
1
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1answer
63 views

What kinds of transformations preserve network topology?

I have been reading a number of "network science" papers where the authors perform transformations on networks that seem to preserve the topology of those networks. By "topology", I mean a collection ...
0
votes
1answer
41 views

linear algebra in infinite dimension

I look for an advanced linear algebra (A complete book but wich deals indiferently with infinite/finite vector space). To give an idea i expect a book that (for exemple) would prove the existence of a ...
2
votes
1answer
59 views

Euler and Bernoulli Polynomial Identity Proof

Given that the Euler Polynomials $E_n(z)$ are defined in terms of the generating function $$\frac{2e^{xz}}{e^x+1}=\sum_{n=0}^\infty E_n(z)\frac{x^n}{n!}$$ and that the Bernoulli Polynomials $B_n(z)$ ...