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
10 views

Reference request for a special product

I have the product $$\prod_{k=0}^n (1+a_k)$$ Does this product have a special name under which I can find some of its properties? I appreciate any reference for this product. Note: Because of the ...
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0answers
10 views

Reference Request: James reduced product

I would like to quickly learn the basics of James reduced product (also called James construction). Anyone know some suitable material for beginners?
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0answers
21 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
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0answers
11 views

Quotients of finitely generated semigroups are finitely generated

Is it true that the quotient of a finitely generated semigroup by a normal subsemigroup is finitely generated? If so, could you suggest a place where the proof is written down?
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1answer
24 views

Citation for some facts [on hold]

I need citations for the following facts, but really cannot remember from where I studied these knowledge. Two conformal mappings agree everywhere, if they agree on a dense set. Conformal mapping ...
2
votes
1answer
34 views

Is there a type of number sequence that has a nth number actually have multiple answers?

I am just looking for what this type of number sequence this is called? Example: The logic of the sequence is, take the previous numbers in the sequence and add them together in every possible way to ...
1
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0answers
20 views

Books for basic level college student

I have just finished a college class on multivariable calculus and I am currently taking linear algebra. I want to delve more into the topics of math, but the books that I have been looking over seem ...
0
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0answers
5 views

Introduction to the boundary element method, with convergence analysis

I've looked through several textbooks on the BEM, but while they show how to set up the boundary integral formulation and how to discretize it, none give any indication of what the convergence ...
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0answers
2 views

Basis argument in the proof of convergence of Galerkin method

In Evans' book partial differential equations among others, the proof for convergence of the approximation solution $u = \sum_i^N c_i \varphi_i$ is based on the assumption that $\{\varphi_i\}_{i\in ...
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2answers
34 views

hyperbolic geometry and affine transformations

A very famous geometer recently commented to me that "hyperbolic geometries are the only geometries invariant under affine transformations". It is unclear to me what this comment even means. Can ...
2
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1answer
30 views

Reference request: Controllable and Observable form for transform function

I came across some online material a year ago that claimed that a the ABCD matrix of a transfer function $$G(z) = \frac{b_1 z+b_2}{z^2+a_1z + a_2}$$ can be directly computed from the coefficients of ...
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2answers
23 views

Need a Probability Theory book that also focusses on Analysis

I am in search for a Probability Theory book which also contains elements and proofs from Analysis. A non-Measure Theoretic approach is most desirable. I have gone through great books like Ross but I ...
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0answers
16 views

Prove that $\left\{e^{i n t}\right\}_{n\in\mathbb Z}$ is a Riesz basis on $L^2[-\pi,\pi]$.

Prove that $$\left\{e^{i n t}\right\}_{n\in\mathbb Z}$$ is a Riesz basis on $L^2[-\pi,\pi]$. Can I have any reference or any suggepstion please? Thanks.
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0answers
16 views

$\ell_q$ is not finitely representable in $\ell_p$ if $2<q<p$.

This seems to be a well known result in Banach space theory. It is referenced, for example, in Pietsch's book "History of Banach spaces and Linear Operator". Where can I find a proof? Who was the ...
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0answers
12 views

Unit balls and the Schatten norms

I have a very naive question: Let $A$ and $B$ $n \times n$ (complex) matrices with operator norms $\|A\| \leq 1$ and $\|B\| \leq 1.$ Pick a $1 \leq p < \infty.$ Then with a constant $K_p$ ...
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0answers
31 views

Has anyone proposed a “maximum subset” symbol?

From a ZFC perspective, there is a unique set $\emptyset$, which is the empty subset of every set. Further, every set has a maximum subset, namely itself. However from a structural perspective, there ...
1
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1answer
15 views

Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...
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2answers
31 views

Reference Request on a good Linear Algebra book [duplicate]

So I'm looking for a linear algebra book with a strong focus on proofs. It would be great if the book also uses concepts from regular abstract algebra like isomorphisms etc instead of dancing around ...
3
votes
2answers
38 views

Suggestions for calculus review for mGRE

I'm currently studying for the mGRE that will be given in October of this year and I'm having trouble deciding which books to use for the calculus review. I'm currently brushing up in Stewart's ...
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0answers
14 views

Statistics on ordered lists

Participants of a study had to rank several proposition by their importance: 1st, 2nd, 3rd, and so on... Two cases are of interest for me: The "ideal" case: there are $n$ propositions and each ...
1
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2answers
24 views

What is a third proportional?

I searched online, couldn't find anything clear. If I had two numbers, $a,b$, what is their third proportional? Apparently it can be either $c$ such that $a/b=b/c$ or $b/a=a/c$, but obviously these ...
3
votes
2answers
36 views

Suggestion to a book with lots of number theory problems

What I am looking for is a book that contains "infinitely many problems", starts from the easiest to high level(that can be found in national and even international olympiads). Are there such books, ...
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0answers
38 views

What is the proof of this procedure

This wikipedia article describes the general procedure for finding the Asymptote of algebraic curves without mentioning any proof. I tried googling but it produced no relevant results. where can I get ...
2
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0answers
37 views

Generalizing in Mathematics

I was reading the book "Fermat's Last Theorem" by Simon Singh when it hit me that this theorem is so contrived, andyet it lead to several important breakthroughs in mathematics and especially the ...
6
votes
1answer
91 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
0
votes
1answer
21 views

References for mean curvature flow in Riemannian manifolds

I am interested in mean curvature flows (MCFs) in Riemannian manifolds. But most textbooks about MCF seem to treat mainly MCFs of hypersurfaces in $\mathbb{R}^{n+1}$. Should I study them before ...
2
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0answers
25 views

Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
3
votes
3answers
97 views

book recommendation on functional analysis

I recently started studying functional analysis. I have many ebooks loaded on my laptop, but can't figure out which one to start with. I've asked my instructor, and he says there aren't any specific ...
2
votes
1answer
15 views

Homotopically equivalent to Čech nerve?

I see a theorem without proof on Gelfand & Manin: Suppose $\mathfrak U=\{U_\alpha\}_\alpha$ is a locally finite open covering of the topological space $X$ such that each finite intersection ...
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votes
3answers
67 views

Outline for high school combinatorics class?

I am a high school student and I have taken all the math classes that my school provides (through calculus AB). I have been looking at a possible independent study for next year and I have landed on ...
3
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0answers
45 views
+50

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 $\mathrm{Ext}^*(\mathbb{Q}(0),\mathbb{Q}(n))$? Another question: Is this a notion which ...
2
votes
1answer
21 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 ...
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0answers
50 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
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0answers
68 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
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1answer
21 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 ...
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0answers
28 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 ...
3
votes
1answer
31 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 ...
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0answers
50 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
57 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
60 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
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0answers
30 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
60 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
26 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 ...
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0answers
23 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
70 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
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0answers
20 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
46 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 ...