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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7 views

Different notation for Jacobi symbol

Is there a different, sort of established, notation for the Legendre / Jacobi / Kronecker symbol $\left(\frac{a}{b}\right)$? If yes, where is it used (in which texts)? I'm asking, because I ...
2
votes
1answer
30 views

Wide equalizers

If $(f_i : A \to B)_{i \in I}$ is a family of morphisms in a category, we may declare their wide equalizer as a universal morphism $\iota : E \to A$ which satisfies $f_i \iota = f_j \iota$ for all ...
2
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0answers
15 views

Trigonometric polynomials on non-compact and non-abelian groups

Hewitt and Ross define trigonometric polynomial on a locally compact group $G$ as a linear combination of matrix elements of continuous unitary irreducible representations of $G$: $$ f(t)=\sum_{i=1}^n ...
2
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0answers
15 views

Exercises with solutions for probability theory?

I'm reading the book Probability Theory: A Comprehensive Course by Achim Klenke. There are no solutions for the exercises in this book, so I constantly have to annoy people here (but nobody wants to ...
2
votes
1answer
60 views

Where can I find a text with this result?

I ran into $\int\limits_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int\limits_{f(a)}^{f(b)} f^{-1}(x) dx$ proof today and was surprised to find that it has come up on the Math GRE. I consider ...
1
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0answers
23 views

Convergence of infinite products

I wonder, parallel to the theory of summability of infinite series is there a theory for infinite products? Is there any generalized convergence method (such as Cesaro and Abel summability) for the ...
0
votes
1answer
31 views

Can somebody explain with one example the concepts: Lemma-Hypothesis-Theorem-Assumption-Proof-Axiom-Thesis-Determination-Definition-Proof [on hold]

It would be great if someone can give me for each concept a simple explanatory example ! What is the difference between: Lemma Hypothesis (Hypothese) Theorem (Satz) Assumption (Annahme) Proof ...
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0answers
13 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
2
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0answers
37 views

What computations would advance math knowledge a lot?

Suppose we where given a super computer that would be capable of computing anything, but only for one day. We could for instance compute many of the Ramsey numbers. What would be some computations ...
0
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0answers
11 views

Reference for exercises with solutions for affine Lie algebras.

I am doing self-study of affine Lie algebras, and while I have all recommended reference material, I find it hard to prepare for the exam. It was quite easy to study finite-dimensional simple Lie ...
1
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0answers
18 views

Local existence for semilinear wave equations

After having done some research, I could not find a reference for the following. Suppose I have a problem of the following type, on $(t,r) \in \mathbb{R} \times \mathbb{R}^2$: $$ \begin{array}{ll} ...
0
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0answers
19 views

Modeling predator/prey pursuit behavior?

I want to program a model of predator/prey pursuit behavior on a 2D topographical map (not predator/prey diff. eq. model!), primarily for fun, but I'd love to know if there's any literature or ...
1
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0answers
38 views

Origin of $\sigma$-algebra

In what paper, article or book was the notion of an $\sigma$-algebra first defined or mentioned? Or at least how far could this concept traced back?
0
votes
1answer
19 views

one end group with positve first Betti number$\beta^{(2)}_1(G)>0$

Could anyone give me an example of a countable finitely generated (f.g.) discrete group $G$ with one end but have non-trivial $H^1(G,\ell^2G)$? To be precise, consider the following two cases. (1) ...
1
vote
1answer
24 views

Significance and physical meaning of diagonalization of linear maps and bilinear forms, eigenvalues and eigenvectors

In linear algebra, I have studied the diagonalization of a linear map and of a bilinear form; and also the concepts of eigenvalues and eigenvectors. However, the importance of diagonalizing a linear ...
-5
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0answers
48 views

Is maths very very difficult? [on hold]

my teacher always tells me 'the math I have learned is very difficult to proceed for math' . I am presently in 12 class. Can someone explain on this
1
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1answer
33 views

A theory of radicals of integers?

It seems to me that radicals, natural numbers without power factors, generalize the concept of primes. You could ask after the nth radical and the number of radicals less than a specified number. But ...
2
votes
3answers
42 views

Embeddings are precisely proper injective immersions.

We call a map $f: X \to Y$ between topological spaces proper if $f^{-1}(K)$ is compact for all compact $K \subset Y$. Where can I find a reference that embeddings are precisely proper injective ...
1
vote
1answer
40 views

Are functions infinite dimensional vectors? [on hold]

Are functions infinite dimensional vectors? There are a few sources on the internet that makes this claim, but they do not cite any sources which makes me feel like they are just using it as an ...
2
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0answers
31 views

Coproduct of Lie algebras

Fix a commutative ring $k$ and look at the category of Lie algebras over $k$. How do coproducts in that category look like? Notice that what is usually called the "direct sum" of Lie algebras is not ...
1
vote
1answer
36 views

Completely self-contained (and as elementary as possible) introduction to Teichmuller Theory

Can you recommend a completely self-contained and elementary (as much as it can be) introduction to Teichmuller Theory?
3
votes
0answers
24 views

Semigroup satisfying the cancellation property which cannot be embedded in a group

There is a basic result about commutative semigroups which says Any commutative semigroup satisfying the cancellation property can be embedded in a group. However, I read that Not every ...
2
votes
0answers
53 views

Multivariable Calculus or Differential Geometry (Analysis on Manifolds) after single variable calculus

Background: Applied Mathematics program, finished with single variable calculus, and in parallel with basic analysis. (Not knowledge of multivariable calculus yet) Please feel free to recommend ...
2
votes
2answers
45 views

Strong solutions to an elliptic PDE

I would like a reference for the following result (you can assume more regularity and replace $C^2(\bar\Omega)$ with $C^2(\mathbb R^n)$ if needed): Let $\Omega\subset\mathbb R^n$ be a bounded ...
0
votes
0answers
45 views

Theorem of Lefschetz

If anyone has the book of James D. Lewis entitled: A survey of Hodge conjecture on page $58$, There are the famous theorem of Lefschetz $(1,1)$ "without proof it seems to me." Is that so? Could you ...
0
votes
1answer
24 views

Reference for unbounded operators

I've run into some unbounded operators in my research and need to learn some of the theory of unbounded operators. Particularly I want a rigorous treatment that discusses symmetric operators, ...
0
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0answers
10 views

Reference request for Hermite Normal Form (HNF)

I need a reference (preferably a basic well-known text-book or paper) that contains Hermite Normal Form (HNF) topic, and specifically includes: definitions proof of existence and uniqueness of the ...
1
vote
1answer
45 views

Families of Elliptic Curves

I am looking to test some properties of elliptic curves and I would like to have a variety of different families to test. I was wondering if there was, say, a catalogue of the different interesting ...
0
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0answers
25 views

Good resource to learn geometric interpretations of matrices

I need a good resource to learn matrices and all its properties through geometry. I feel geometry gives an insight into many matrix operations and a good resource will be useful for many students who ...
0
votes
0answers
24 views

Collection of solved problems in linear algebra [duplicate]

Apart from Schaum's 3000 Solved Problems in Linear Algebra, what are some good collections of worked problems in linear algebra?
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0answers
54 views

Did psychologists really publish experimental support for $R_{3,3}=6$ [on hold]

A friend of mine told me that psychologist or sociologists once published a result in which they noticed that in larger groups of children there always seemed to be three children who where all ...
1
vote
1answer
20 views

Characterizing the Prüfer $p$-group

I've been trying to solve these questions for the past few hours with no luck: If $G$ is an infinite abelian group all of whose proper subgroups are finite, then $G$ is a Prüfer $p$-group for some ...
0
votes
0answers
16 views

Give a suitable way to study Fourier Transforms:

Give a suitable way to study Fourier Transforms. In the website called the fourier transform, gives somewhat good approach to meet it. But, I need to clarify onething. I am doing my pure papers ...
1
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0answers
60 views

Removed due to revision [on hold]

I need to revise a few things before reuploading.
0
votes
1answer
46 views

Euler's complete works

If Euler's works are still being published then what is this?: http://eulerarchive.maa.org/pages/E786.html Is it only some of his works? I thought "complete works" meant literally all. Thanks
1
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1answer
76 views

About terminology “Orthogonal” and “Orthonormal”

This question may not be of theoretical importance in Linear algebra, but I came to this question, while looking definition of orthogonal transformation in intuitive way. Let $V$ be an inner product ...
1
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0answers
22 views

Proper usage for term, addend, factor, multiplicand, expression, formula

The definitions and usage of the following words seem to vary, depending on the source text: term addend factor multiplicand expression formula The words are being used in the context of ...
0
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1answer
42 views

Choosing a Project Topic.

I am a undergraduate student and recently i have been assigned to project (one of courses). But i have to choose my own topic. I want to work in the field of ...
-1
votes
1answer
87 views

In preparation forcing and large cardinal textbooks

Everybody in set theory refers to texts like Kunen, Jech and possibly Halbeisen's books as elementary references for forcing. Also Drake and Kanamori's books are well-known references for large ...
6
votes
1answer
73 views

Theorems discovered without observation

Can you name me a few theorems that were discovered without first observing some special cases? In other words, by brute logic: Starting from the known and logically deducing the unknown? EDIT: As an ...
6
votes
0answers
135 views

Do hom-sets really live in the category Set?

In familiar introductory books on category theory, one of the first examples of a category given is Set. And what category is that? Typically no explanation is given at this stage. But of course ...
1
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0answers
12 views

Statistics from the Bayesian point of view

I am looking for text book that develops the theory of statistics from assuming nothing more than Bayes Theorem. Could you please refer me to material? I would prefer text books and papers, however ...
2
votes
1answer
38 views

Morphism $f\colon X\to S$ is proper iff $f^{-1}(V_j)\to V_j$ is proper for some open cover $\{V_j\}$ of $S$? (Lemma 28.42.3 of Stacks Project)

I was browsing the Stacks Project, and Lemma 28.42.3 says that a morphism $f\colon X\to S$ is a proper morphism if and only if there exists an open covering $S=\bigcup V_j$ such that $f^{-1}(V_j)\to ...
1
vote
1answer
54 views

Diophantine Equation: $a^3=a(b^2+c^2+d^2)+2bcd$

Let $a,b,c,d\in \mathbb{Z}$. Solve $$a^3=a(b^2+c^2+d^2)+2bcd$$ I've tried everything but I haven't been able to find a general solution. Note: We may assume $\gcd(a,b,c,d)=1$ because of homogeneity. ...
-1
votes
1answer
45 views

Basic elementary number theory [duplicate]

I just enrolled in a class called "Elementary Number Theory" and I am left confused in every class due to the different notations and proofs shown. Is there a really basic book on Number Theory out ...
1
vote
3answers
80 views

Video lectures on Partial Differential Equations

Would anyone happen to know any introductory video lectures / courses on partial differential equations? I have tried to find it without success (I found, however, on ODEs). It does not have to be ...
2
votes
2answers
62 views

The singular homology and cohomology of manifolds vanishes in high dimensions

Let $M$ be an $n$-manifold. It seems that there are two results that the $p$-th singular homology and cohomology of $M$ are zero if $p>n$. But I can not find them in my books of algebraic ...
3
votes
2answers
49 views

Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?

Let $G$ be a nonabelian group with center $Z(G)$. Let $\rho: Z(G) \to \text{GL}_n({\bf C})$ be an irreducible representation. Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not?
3
votes
2answers
38 views

Frobenius reciprocity, proof if $V$ irrep of $G$ then multiplicity of $V$ in regular rep of $G$ is $\dim(V)$

I know the standard proof of the fact that if $V$ is an irreducible representation of $G$ then the multiplicity of $V$ is the regular representation of $G$ is $\dim(V)$. Does there exist a proof using ...
1
vote
1answer
47 views

Is Tetris a packing or covering problem?

I am looking for some information about packing and covering problems. Some texts mention Tetris without further elaboration. Now, I am wondering if Tetris is a kind of packing or covering problem. ...