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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2
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0answers
9 views

Cohomology with coefficient $\mathbb{Q}(n)$

What is the definition of $$\mathrm{H}^i(X,\mathbb{Q}(n))$$ for a variety $X$? and What is its relation with $\mathrm{Ext}^*(\mathbb{Q}(0),\mathbb{Q}(n))$? A good reference is also very helpful. ...
2
votes
1answer
14 views

Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
0
votes
1answer
9 views

Relation between length of arc of horocycle and length of chord?

In Hyperbolic geometry: What is the relation between the length of the arc of a horocycle between two points and the length of the chord (segment) between the two points? Also what is the relation ...
0
votes
0answers
38 views

looking for English version of this article

one of my friend request me to post this question here.now he is writing his master thesis and he exposed with this problem, He exposed with an article with France language but unfortunately he ...
3
votes
0answers
58 views

Where can I find linear algebra described in a pointfree manner?

Clearly, some of linear algebra can be described in a pointfree fashion. For example, if $X$ is an $R$-module and $A : X \leftarrow X$ is an endomorphism of $X$, then we can define that the ...
0
votes
1answer
19 views

How to calculate the sum of a general series

In class we learned how to test the convergence of series and how to calculate the sums of arithmetic and geometric series (if they exist) but are there methods to actually calculate the values ...
0
votes
0answers
17 views

On a congruence for the number of finite topologies

I am making search about "On a congruence for the number of finite topologies". I have found a paper. I guess it is written in Russian. How can I find English version of this paper ? I am also ...
2
votes
1answer
24 views

Origin of period function model of primes

There is a web page attributed to Omar Pol, "Sobre el patrón de los números primos: Determinación geométrica de los números primos y perfectos." ("On the pattern of primes: Geometric Determination of ...
0
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0answers
37 views

Taking Putnam as a freshman.

Currently, in 11th grade, I've always thong about participating exams like the Putnam. I have however, sent the problems, and they seem, to be grueling hard!! I have access to problem solving ...
0
votes
0answers
11 views

Find a cyclic rational function such that…

I'm looking for a function of the form $\frac{f(a,b,c)}{f(b,c,a)}$ (or close to this form, e.g. $\frac{(a+b)^2}{b^2+bc+c^2}$) which is roughly equal to $\frac{b^3-a^2-b^2-a^3-ab^2}{b^2c+a^2b+b^3}$ (I ...
-1
votes
3answers
23 views

How can we make this expression small? [on hold]

How can we make the following expression small: $$(bx-ay)^2+(cx-az)^2+(cy-bz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2$$, where $a,b,c,x,y,z$ are nonnegative reals? Note: I'm not looking for an exact answer, ...
1
vote
1answer
41 views

suggesting books on Field/galois theory

I have finished courses on group theory and ring theory and linear algebra and abstract algebra(inrroducton with fraleigh's book) im about to take field/galois theory this semester any good book not ...
1
vote
2answers
44 views

Measure Theory Book for My Background / Need

My current Math background is as follows: 1) Read first 7 chapters of Rudin "Principles of Mathematical Analysis" and solved a lot of the given problems. 2) Completed Munkres "Analysis on Manifolds" ...
2
votes
0answers
26 views

$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated iff there is a random variable $X$ such that $\mathcal{G} = \sigma(X)$.

Where can I find a reference to the proof of the fact that a $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is countably generated if and only if there is a random variable $X$ such that ...
2
votes
3answers
57 views

What is a conventional name for a set of values having no properties except that values are distinct?

I know essentially nothing about math but I'm interested in very low-level concepts. I'm thinking of something like a finite or infinite set (although I'm not married to consider sets per se, maybe ...
0
votes
0answers
22 views

Variants of the change-of-variables formula

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$ $$ \int_B g(x)\ {\rm ...
0
votes
0answers
19 views

Definition of a vector field

Reading Wikipedia, I see that a vector field is defined as a mapping $X: S \rightarrow \mathbb R^n$ where $S \subseteq R^n$. However, I sometimes see mappings $X: S \subseteq R^m \rightarrow R^n$ ...
0
votes
1answer
67 views

Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF

How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous ...
1
vote
0answers
18 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 ...
3
votes
1answer
42 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 ...
3
votes
0answers
28 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 ...
3
votes
1answer
33 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
70 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
vote
0answers
34 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
36 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 ...
0
votes
0answers
15 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
votes
0answers
50 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 ...
1
vote
0answers
22 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
votes
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
vote
0answers
40 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
27 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
votes
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
vote
1answer
34 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 ...
3
votes
3answers
45 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
41 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
votes
0answers
35 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
37 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?
4
votes
0answers
34 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
58 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
46 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
27 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
votes
0answers
12 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
49 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
votes
0answers
26 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?
1
vote
0answers
55 views

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

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
22 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
18 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 ...