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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On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR

Convention : All rings are commutative with unity unless stated otherwise . By ideals we will mean to include $\{0\}$ and $R$ also . Let us call an integer $n>1$ a "principal number " if any ring ...
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2answers
15 views

On those integers $n>1$ such that there exist a coommutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer ; we call $n$ a " ring number " if there exist a commutative ring $R$ , with identity , having exactly $n$ ideals ( including $\{0\}$ and $R$ ) ; now since for every ...
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0answers
11 views

A book about multivariate normal distributions?

I'm looking for a book that explains multivariate normal distributions (definition, properties, examples, etc.) in a simple and didactic way (I'm not a mathematician, so nothing too complex) but ...
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1answer
24 views

Literature on generating functions for networks

Are you aware of any material the presents all (or most, or many) the properties and applications of generating functions in the context of graphs? For example I am aware of 'Generating ...
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0answers
14 views

Can you identify this stochastic process?

So I run into this problem the other day and I cannot even think of the keywords I need to use to look it up. For the discrete random variable $X$ we have: $P_{\Delta X(t)} = F\big(X(t-1), ...
0
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1answer
23 views

How to use monotone convergence theorem to show that $\int \sum |f_n| = \sum \int |f_n|$

In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state $$\int \sum |f_n| = \sum \int |f_n|$$ without proof. Can ...
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0answers
6 views

Condition for global cascade

Assume a unidirectional, unweighted network generated according to a degree distribution. Each node is given a value between 0 and 1 called threshold $\phi$. We topple some nodes, the neighbours will ...
5
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2answers
169 views

Is there any wide sense but introductory Book series/websites for mathematics literature? (just to be familiar with)

Is there any Book/Book Series/Website which illustrate advanced mathematics but in concise and basic form. I just want to be familiar with the literature and I don't want a deep exploration of the ...
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1answer
68 views

A kind of planar figures

Studying issues related to the planar shapes I've found some attribute, useful for my investigations: Any segment with origin in mass center and end point on figure's boundary is contained within ...
2
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1answer
55 views

Hodge numbers of a cartesian product of copies of $\mathbb{C}P^1$

I wonder if some works have been done in the context of cohomology space of projective complex manifolds. Specifically I want to study the Hodge diagrams of $\mathbb{C}P^1\times\mathbb{C}P^1$ and ...
3
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1answer
109 views

Hausdorff measure, volume form, reference

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia? Corollary $4$: Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ ...
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2answers
127 views
+400

How many ways are there to prove Cayley-Hamilton Theorem?

I see many proofs for Cayley-Hamilton Theorem in textbooks and net, so I want to know how many proofs are there for this important and applicable theorem.
4
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1answer
131 views

Fractional powers of positive self-adjoint operators

Consider two positive unbounded operators $A$ and $B$ densely defined on a Hilbert space $H$ self-adjoint on a domain $\mathcal{D}(A) = \mathcal{D}(B) = H_1$. By the spectral theorem, we can define ...
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3answers
72 views

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such ...
3
votes
1answer
101 views

Is there a garden of derivatives?

I've found a book called A Garden of Integrals, in which the author shows the evolution of the concept of Integral. I follow AnalysisFact on Twitter, some days ago, they posted the following: The ...
0
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1answer
61 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
0
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1answer
78 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
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0answers
27 views

Does this theorem hold for real varieties? + Reference request - Real varieties.

Let $X$ be a complex variety and let $X_\mathbb R$ be its real part, that is the $X_\mathbb R$ consists of all the real valued points of $X$. I would like to learn a little more about the real ...
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1answer
92 views

General solution of a finite-difference equation with real non-commensurable differences.

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
3
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0answers
34 views

How many ternary functionally complete connectives are there?

Today I was reading up once more on some of the nice results regarding functional completeness, notably Post's celebrated classification theorem with the 5 classes that need to be avoided. (See this ...
2
votes
2answers
65 views

Measure-theoretic analog of homeomorphism and isometry

If $(X,\tau_X)$ and $(Y,\tau_Y)$ are topological spaces and $f:X\to Y$ is a continuous bijective function between them such that $f^{-1}$ is also continuous, then the two topological spaces are said ...
3
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3answers
245 views

Online reference about Geometric Measure Theory.

I would like to find an online reference about the basics of Geometric Measure Theory. The reference should treat such things as regions and isoperimetric surfaces. Can you tell me, where I can find ...
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3answers
259 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
4
votes
1answer
564 views

Continuity of parameter dependent integral (source needed)

I am looking for a reference from a book for the result of continuity of an integral (found in https://www.encyclopediaofmath.org/index.php/Parameter-dependent_integral): Let $D \subset ...
4
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2answers
259 views

Surveys of Current (last 50 years) Mathematics at Graduate / Research level?

I'm wondering if there are any books or broad survey articles that survey current mathematical areas, i.e. mathematics of the last 50 years. Ideally, these would continue in the style of the various ...
6
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0answers
301 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
7
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3answers
282 views

Encyclopedic dictionary of Mathematics

I'm looking for a complete dictionary about Mathematics, after searching a lot I found only this one ...
2
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2answers
404 views

Gentle introduction to quasi-geodesics

Compared to the concept of geodesics the concept of quasi-geodesics seems to be substantially harder to grasp and digest. I was given a promising hint to the concept of quasi-geodesics here but the ...
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0answers
272 views

Integration of Binomial Differentials Proof/Reference

In Piskunov's Calulus (P375 & P385) & Hardy's Integration of Functions of a Single Variable (P48) mention is made of Chebyshev's theorem on the integration of binomial differentials however no ...
18
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1answer
775 views

Is there such a thing as a mathematical thesaurus?

I want this for two reasons: When writing proofs, I am constantly in need of synonyms of basic words like thus, there exists, for all, such as, contains, etc. A lot of mathematical concepts have ...
0
votes
1answer
27 views

What book about algebraic combinatorics is it?

Recently I found a fragment of a book about algebraic combinatorics on the internet coincidentally. And I found it's really an excellent resource of learning polynomial method, about Combinatorial ...
1
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2answers
43 views

2nd degree differential equation

Can someone please tell me how to solve this differential equation? $${d^2y\over dx^2} +y=\tan(x)$$ I am a beginner in ODE and have absolutely no idea how to proceed. Can you also site a reference ...
4
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0answers
24 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
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0answers
12 views

Spigot algorithms for transcendental numbers

I'm trying to write a program that will compute digits of transcendental numbers using a spigot algorithm. While researching, I found the BBP Formula, and a Compendium of BBP-Type Formulas, alas, I ...
1
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1answer
40 views

Reference Request: Differential Geometry Book [on hold]

What is a good self study book in Differential Geometry. Keep in mind I won't have the advantage of being able to ask a professor any questions.
1
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4answers
2k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
0
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0answers
5 views

CG as an orthogonal projection

I have heard that the Conjugate Gradient method can be viewed as an orthogonal projection onto the Krylov subspace $K(A,r_0)$, but I can't find a reference that deal with it in this way. Could you ...
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0answers
25 views

How can I correctly catalog this partition problem?

Studying the partition problems, I tried to do an special version to apply it to a kind of model of "orbits and energy levels" (explained below), but I am having problems to properly catalog this. ...
0
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1answer
31 views

Linear algebra for modern differential geometry( and other types of modern geometry, like analytic, complex and algebraic)

I wish to study real and complex analysis(for example, Pugh "Real Mathematical Analysis" and Cartan "Elementary theory of analytic functions of one and several complex variables") and modern ...
1
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1answer
102 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
0
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0answers
18 views

Is this orthogonal distance a common pseudometric?

Define $d: V \times W \to \mathbb{R}$ such that $$d(v,w) = \sup_{z \perp w} \frac{\langle z, v \rangle}{\|v\|\|z\|}.$$ Is this a pseudometric that anyone has utilized in the literature? Does it have a ...
1
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1answer
74 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
13
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3answers
455 views

Has someone seen a discussion of the (divergent) summation of $\sum\limits_{k=0}^\infty (-1)^k (k!)^2 $?

In extending my studies of the Eulerian matrix and its suitability for a matrix-based divergent summation procedure I'm trying to proceed to sums of the form $$ S = \sum_{k=0}^\infty (-1)^k (k!)^2 $$ ...
0
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0answers
13 views

measure-theoretic probability, (sets of) null events and non-zero probability

Assuming a well-defined probability space $ (\Bbb{R},\mathscr{B},\Bbb{P}_X) $, where $\mathscr{B}$ is the Borel $\sigma$-field, and for a random variable $X$ having a continuous probability density ...
2
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1answer
40 views

Help finding paper from the 1920's

I have not been able to find a copy of this paper anywhere! B. Knaster еt C. Kuratowski: Sur quеlquеs propriétés topologiquеs dеs fonctions dеrivéеs. Rеnd. dеl Сirc. Math. di Palеrmo, 59 (1925), ...
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3answers
39 views

ordinary differential equation project suggestion [on hold]

My professor asked to write a project on ODE just to experience on how to write projects. It need not be a research project. Being in second rate school from third world country, we never did those ...
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0answers
9 views

Exponential matrix using Laplace transform - reference request [on hold]

I am looking for a book that covers the concept of exponential matrix in detail using the Laplace transform plz
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3answers
390 views

Extension of real analytic function to a complex analytic function

I just learned that real analytic functions (by real analytic, I mean functions $f: \mathbb{R} \to \mathbb{R}$ which admit a local Taylor series expansion around any point $p \in \mathbb{R}$) cannot ...
2
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1answer
75 views

Linear ordinary differential equations and their evolution operators for measurable operators

Consider the following homogeneous IVP: $$\begin{cases} \dot{u}(t)+A(t)u(t)=0 \\ u(0)=u_0 \end{cases}$$ for $u:[0,1]\to \mathbb{R}^n$ (some interval to some finite dimensional Hilbert space, let's ...
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4answers
1k views

n-th roots of unity form a cyclic group in a field of characteristic p if gcd(n,p) = 1

Let $n$ be a positive integer, and let $\mathbb F$ be a field of positive characteristic $p$ with $\gcd(n,p) = 1$. Where can I find some proofs that the group of all $n$-th roots of unity (in an ...