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Questions tagged [reference-request]

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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Smallest element of cycle of length $k$ in Collatz 3x+1 map?

In studies of the Collatz conjecture, what research has asserted the existence of a $k$-length cycle and drawn conclusions about its smallest element $m$? In particular, about the behavior of $m$ as $...
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Laplace operator on certain groups

Q1. What are some canonical ways to define Laplace operator on a discrete group? in particular if $f\in \mathcal L^2(\mathbb Z)$ how is $\Delta f$ defined? Aside considering Laplacian on graphs, are ...
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0answers
29 views

Can a finite set of bases guarantee that a number is Carmichael or prime?

If a positive integer $N>1$ is a Carmichael number, it passes the weak Fermat-pseudoprime-test for every base coprime to it. I wonder whether the converse is true in the following sense : Is ...
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0answers
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How does one call/characterise sewn-together Riemannian manifolds

Suppose $(M_1,g_1)$ and $(M_2,g_2)$ are (semi)Riemannian manifolds with boundaries (resp.) $\Sigma_1$ and $\Sigma_2$. The metrics $g_{1,2}$ yield induced metrics $\gamma_{1,2}$ on those boundaries $\...
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focal point (definition)

I am a begginer in the differential geometry and I need the definition of focal points. All the books I see is defined in riemannian submanifolds with jacobi fields. I know just the notions of ...
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0answers
18 views

Constructing Green's function for PDE from homogeneous solution [on hold]

I have some complicated PDE and I know its homogeneous solution. Is there a way to construct Green's function from this homogeneous solution, akin to the Sturm-Liouville case for ODE (e.g. http://www....
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4answers
85 views

What is the group-like structure on $x^2+y^2+z^2-2xyz=1$?

(Background: this is inspired by Chebyshev polynomials and expanding a function as a Chebyshev series.) Solving for $ z $ gives $$ z=xy \pm \sqrt{(1-x^2)(1-y^2)}, $$ where $-1\leq x,y \leq 1$. Now ...
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1answer
33 views

Reference: $(X,X/G,\pi,G)$ is principal $G$-bundle?

The accepted answer in Projection map between the Stiefel manifold and the Grassmanian says If $G$ is a Lie group and $X$ is a manifold on which $G$ acts freely and properly, then $X/G$ has a ...
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0answers
17 views

vanishing 2-cohomology for $G$-module $\mathbb{C}^G$?

Let $G$ be a countably infinite group. Is it true that $H^2(G, \mathbb{C}^G)=0$ ? Here, $\mathbb{C}^G$ denotes all functions from $G$ to $\mathbb{C}$ and is treated as a $G$-module, i.e. $(gf)(g'):...
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0answers
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What is known about these “transparent” polytopes?

I am looking for the name (if there is one), simple properties and possible literature for the following class of polytopes (by polytope I mean the convex hull of finitely man points in $\Bbb R^n,n\ge ...
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0answers
15 views

How far was base $47$ checked for a generalized Wieferich-prime?

This question is closely related to : Wieferich primes in base $47$ but I would like to know the current search limit for this base. Upto which prime $p$ was $$47^{p-1}\equiv 1\mod p^2$$ verified ...
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0answers
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Minimal surface contained in a hemisphere

I am looking for examples of closed, orientable minimal surfaces of the sphere $\mathbb{S}^3$ that are contained in a hemisphere. Do you know any? Thanks!
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0answers
20 views

Fibonacci sieve and factoring

I want to know different sieve techniques for Fibonacci numbers and how they works. In wikipedia it is written only that the Cassinie's identity are useful in setting up the special number field sieve ...
1
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1answer
21 views

Infinite speed of propagation of the heat equation

Consider the heat equation $u_t = \Delta u$ on $\mathbb{R}^n$ with initial data $u(0, x) = f(x)$. Suppose $f$ is smooth and compactly supported. Do we necessarily have that $u$ has non-compact support ...
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1answer
38 views

Isoperimetric inequality for non-spherical multivariate Gaussian

Disclaimer: Sorry in advance, if the question is not very reasonable. Recently (like a few days ago...), I've started studying isoperimetric inequalities, and my thoughts on the subject are rather ...
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0answers
23 views

Good high school geometry textbook for self study

I am entering my sophomore year of high school, because of how often I changed schools I never took geometry, this summer I took Pre Calc and am taking calc BC in sophomore year. This means I will ...
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1answer
50 views

Expected normalized maximum: expectation of a ratio $\ell_\infty/\ell_1$.

Let $\alpha_1,\dots,\alpha_k > 0$, and $1\leq s\leq k$ be an integer. Suppose $S\subseteq[k]$ is a random subset chosen uniformly among all subsets of $[k]$ of size $s$. Is there anything known ...
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3answers
36 views

Reference Request-Discrete Mathematics

Being a student of Graduate Level, we are learning some good pure mathematics nowdays. Studying Discrete Mathematics is somewhat different I thought. We are said that Finite Set is a set that has a ...
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1answer
37 views

Epsilon delta study material [duplicate]

I want to practice epsilon-delta method for problems in analysis. Does anyone have any suggestions for books with solutions, or solution books available, for undergraduate analysis? Edit: I didn't ...
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1answer
35 views

Is there public access to a paper, in which the first $k$-tuple conjecture was proposed? [on hold]

How did Hardy and Littlewood derive this conjecture and what needs to be done to prove it?
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0answers
33 views

Is there an analogue to “simple extensions” for category theory?

Given a category $\mathcal{C}$ and two objects $A,B \in \mathcal{C}$, we can construct the "polynomial category" $\mathcal{C}[x : A \to B]$ constructed by adjoining an indeterminate arrow $x : A \to B$...
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1answer
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What does the “integral” in Integral domain stand for? [duplicate]

This will be an odd question, but... In the name Integral domain, what is the meaning of word integral? Does it refer to integral such as adjective for noun integer? Or does it instead refer to noun ...
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1answer
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Definitions and properties of limits of stochastic processes in continuous time

Many books on stochastics take ample time to explain what it means for a sequence of random variables to convergence a.s., in $L_p$, in probability, in distribution, what $\limsup$ and $\liminf$ mean, ...
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0answers
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Where I can download “Principe de concentration-compacité en calcul des variations”

Recently, I read Orbital stability of standing waves for some nonlinear Schrödinger equations. I have some question about the concentration compactness method, so I want to read the Lions' Principe ...
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0answers
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Logarithmic height of algebraic numbers

Let $a$ and $b$ algebraic numbers over $\mathbb{Q}$. Do you know (or recall) if there are simple uppper bounds relating the logarithmic height of $ab$ (or $a/b$) with the logarithmic height of $a$ ...
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0answers
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Introduction to analysis: recommended things to understand

I'm beginning my analysis class next semester and was wondering what should I be very familiar with. For example, in the first page of "Principles of mathematical anylysis" by Walter Rudin, he starts ...
3
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1answer
52 views

Are these trees useful and how are they named in the literature

I am working in a project with trees and it turns out that the specific problem I am trying to attack is easy on some types of trees. If we consider the rooted layout of the tree with the root being ...
2
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0answers
33 views

Polynomial approximation in Sobolev spaces

Let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $p>1$, and let $f \in W^{1,p}(U)$. I have recently heard that $f$ can be locally approximated in $W^{1,p}$ by polynomials. That is, ...
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0answers
36 views

Known proofs of isodiametric inequality

Isodiametric inequality is the following statement: Suppose that $A\subset\mathbb R^n$ is a compact set of diameter at most 2. Then the volume of $A$ is less than or equal to the volume of a unit ...
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0answers
29 views

Incircle bisectors and related measures

This question was inspired by in-triangle-abc-d-is-a-point-on-ac..., show-that-am2-pp-a. Cevians $|AD_a|=d_a$, $|BD_b|=d_b$, $|CD_c|=d_c$ divide $\triangle ABC$ into three pairs of triangles, ($\...
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0answers
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Mathematical Information-Action Ratio [on hold]

I have recently listened to the song 'Four Out of Five' from the Arctic Monkeys and somewhere in the lyrics the expression 'Information-Action Ratio' is heard. Then, I found this article from ...
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0answers
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Riemannian Optimization Tutorial?

Looking for a good quick guide about Riemannian optimization, from an applied perspective such as machine learning. Referrals to easy to read references would be really appreciated.
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1answer
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Literature request: Quantifying 'memory' of dynamical systems?

In advance: Please forgive my lack of adequate vocabulary, I fear my current vague understanding would rather confuse my question than help solve it. Imagine you are standing at the shore of a river. ...
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1answer
34 views

Reading material for Posets and Partitions.

I am currently pursuing a course on Discrete Mathematics(DM). The book that my instructor suggested is K.H.Rosen DM. But in today's lecture, he gave a very descriptive talk on POSETS, Lattices, Chains ...
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Transformation on surface patches that preserves length of curves.

suppose $\mathcal{S} \subset \mathbb{R}^3$ is a surface patch, and let $\mathcal{C}$ a simple curve on $\mathcal{S}$, suppose $f$ is a transformation/function/mapping such that $f(\mathcal{S}) = \...
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Okay, but what if I'm not using bibtex and I still want to cite a math paper in arXiv? [closed]

I don't know why these people sctrictly talked about bibtex when there's clearly other people in the world How to cite preprints from arXiv? because it definitely doesn't answer my question, I'm not ...
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0answers
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Is there a good resource for learning about univariate polynomial equations over the hyperbolic numbers?

Solving equations over the hyperbolic numbers is weird; for example, the equation $x^2 = 1$ has four solutions; not only are $1$ and $-1$ solutions, but so too are $j$ and $-j$. Question. Are there ...
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1answer
30 views

Reference Request - Lagrange Multipliers

What is a good source to learn about Lagrange Multipliers with proofs?
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0answers
27 views

Textbook recommendation for modern algebra [duplicate]

I once tried to study abstract algebra (I don't remember which textbook), and I found the concepts of groups, rings, fields, etc too confusing. I now want to study modern algebra (is it an ...
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0answers
27 views

Reference Request: Bunge's proof that cocomplete regular atomic categories are presheaves

The fact that a category is a presheaves category if and only if it is cocomplete, regular and atomic is due to Bunge but I cannot find the proof anywhere, does anybody know where it appears? Side ...
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33 views

Magnitude of possible $66$-th idoneal number

Here : https://en.wikipedia.org/wiki/Idoneal_number it is mentioned that $65$ idoneal numbers are known and there is at most one additional idoneal number (probably the list is complete) Assuming ...
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0answers
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Subschemes of projective varieties

I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective any zero-locus $X$ of homogeneous polynomials in the ...
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0answers
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Looking for the Deligne's example of a non trivial topos with no points

In the topos entry of wikipedia, says: there is an example due to Pierre Deligne of a nontrivial topos that has no points (see below for the definition of points of a topos). I'm looking for the ...
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0answers
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Request for reference regarding properties of convolution of functions.

This question is in reference to the earlier questions Convolution of compactly supported function with a locally integrable function is continuous? and Convolution of locally integrable and compactly ...
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0answers
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Is there a question that’s similar to the famous 1988 IMO question 6 [closed]

I’m sure all remember the famous question that follows: Let $a$ and $b$ be positive integers such that $ab+1$ devides $a^2+b^2$. Show that $\frac{a^2+b^2}{ab+1}$ is the square of an integer
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Is there a notion “above” that of perfect numbers?

When trying to understand a notion, it often gives great insight to see it as the "shadow" of something bigger, carrying more information. The notion of categorification relies on this idea. A basic ...
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0answers
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theories where angles exist without a metric

(moved from https://mathoverflow.net/questions/307703/theories-where-angles-exist-without-a-metric) The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible ...
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2answers
102 views

Reference for Kaplansky's proof that in $\mathbb{C}[G]$, $ab=1$ implies $ba=1$

Here on the Wikipedia page for Group Rings, talking about group rings over infinite groups, The case [of a group ring $R[G]$ where $G$ is an infinite group] where $R$ is the field of complex ...
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7answers
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What problems have been frequently computationally verified for large values?

Although any theorem (or true conjecture) can be computationally checked, many long-standing open problems have been computational verified for very large values. For example, the Collatz Conjecture ...
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0answers
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Reference request: the space of formalisms for proving function totality

I'd like to develop an intuition about the space of axiomatic systems (formalisms) that can be used to prove totality of Turing machines. To this end, I'm interested in the set of "totality proof ...