Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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Extension of the Jacobi triple product identity

The Jacobi triple product and the mathematical identity of it is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would ...
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2 answers
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Reference request: Oldest complex analysis books with (unsolved) exercises?

Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...
Squid with Black Bean Sauce's user avatar
4 votes
1 answer
382 views

Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?

Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.
Squid with Black Bean Sauce's user avatar
4 votes
1 answer
591 views

*Full proof* references for Markov generators with various boundary conditions

(Note: I've migrated this question from math.stackexchange, as the lack of answers there made me believe it was perhaps too advanced for that forum.) Consider the one-dimensional heat equation $$\...
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2 answers
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System of linear first order PDE with constant coefficients

recently in my researches I've come across the following operator $$L\left(\begin{array}{c} a_1\\ \vdots\\ a_n \end{array}\right)=M_1\left(\begin{array}{c} ...
guido giuliani's user avatar
4 votes
2 answers
669 views

Earliest use of deconvolution by Fourier transforms

From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...
AChem's user avatar
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1 answer
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Counting number of points on a lattice in a hypercube

Suppose I have a lattice $\Lambda \in \mathbb{R}^n$. Let $X_i >0$ for $i=1,..,n$. I am interested in some references regarding counting number of points of $\Lambda$ inside $[-X_1, X_1] \times \...
Johnny T.'s user avatar
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1 answer
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Fourier coefficients of real analytic functions on an n-dimension torus

Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $...
Hugo Chapdelaine's user avatar
4 votes
1 answer
569 views

Total sum of characters of the symmetric group $\frak{S}_n$

Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that $$\sum_{\lambda\vdash n}\...
T. Amdeberhan's user avatar
4 votes
2 answers
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Canonical reference for Chern characteristic classes

I'm a little uncertain about the definitions for Chern roots Chern classes Chern characters From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
Tom Copeland's user avatar
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category of simplicial filters

I am looking for references discussing the category $Filt$ of filters (in the sense of set theory, details below), its simplicial category $sFilt$ and its full subcategory $Top\hookrightarrow sFilt$ ...
user117509's user avatar
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1 answer
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Is every invertible-free cancellative monoid action represented by "shifting" certain maps?

[Note: This question is closed. It's current content reflects a draft of a potential new question, modified from the original by adding conditions to the premises; see comments] Let $W,X$ be ...
David Pokorny's user avatar
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2 answers
675 views

Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More precisely,...
asv's user avatar
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Derivatives of Riemann $\xi$ and traces of zeros

Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
Tom Copeland's user avatar
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0 answers
265 views

Primitive Closure Arithmetic

I am researching a system that was designed by myself and what I call PCA (Primitive Closure Arithmetic), because it looks like PRA. The differences are: - PRA uses recursive definition with a ...
Lucas K.'s user avatar
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Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?

If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...
Giovanni Moreno's user avatar
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1 answer
2k views

Singular value decomposition of random rectangular matrices

Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance). What is the ...
valle's user avatar
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1 answer
390 views

Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots, especially minimum bend drawings. An orthogonal drawing employs segments parallel to the axes of a Cartesian coordinate system. A bend is a ...
Joseph O'Rourke's user avatar
3 votes
0 answers
199 views

On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial

Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
thomashennecke's user avatar
3 votes
1 answer
682 views

Deformations of holomorphic/algebraic vector bundles over $\mathbb{P}^3$

I would like to know what can be said about (global) deformations of holomorphic/algebraic rank two vector bundles on $\mathbb{P}^3$. I am particularly interested in the case of topologically trivial ...
Matthias Wendt's user avatar
3 votes
0 answers
77 views

Nowhere dense covering number of a connected $T_2$ space

This is a generalization of an older question. If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a ...
Dominic van der Zypen's user avatar
3 votes
4 answers
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History of the Sampling Theorem

In January, 1949, Shannon publishes the paper Communication in the Presence of Noise, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available here, which establishes the Information Theory. In this paper, the ...
Papiro's user avatar
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4 answers
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Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior?

Q1. Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $...
Mirko's user avatar
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3 answers
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Estimate for products of integers that are relatively prime with $N$

Let $N$ be a positive integer. Are there known estimates for the product of all numbers that are smaller than $N$ and relatively prime with $N$? One can assume that $N$ is free of squares, if this ...
vlk's user avatar
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1 answer
369 views

Compactness in Bishop's constructive mathematics

In Bishop's constructive mathematics, is there any literature on a possible version of the weak König's lemma, or of the compactness theorem for countable models? There is some related information ...
Mikhail Katz's user avatar
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About eigen-functions of the Gaussian kernel

If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
gradstudent's user avatar
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3 votes
0 answers
152 views

Connecting Quillen functors between motivic homotopy categories (of different "types"): references?

For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it: (a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here ...
Mikhail Bondarko's user avatar
3 votes
1 answer
424 views

Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?

I need to use the Lebesgue differentiation theorem for doubling metric measure spaces and was wondering if Carnot groups are doubling. If yes, is there any reference you can point me to? Thank you.
Axiom's user avatar
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0 answers
197 views

Finitistic dimension via tilting modules

is the following true (all algebras and modules are assumed to be finite dimensional): The finitistic dimension of an algebra is equal to the supremum of projective dimensions of tilting modules? It ...
Mare's user avatar
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2 votes
0 answers
206 views

Variational formulation for elliptic interface problem

Where can I find a paper that deals with the following interface problem with variational methods? In particular, what is the correct variational formulation of the problem (that is, a functional ...
user avatar
2 votes
2 answers
343 views

Matching with probabilistic edges

Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the ...
Alexi's user avatar
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2 votes
0 answers
156 views

Are there hereditarily square-boxed plane continua?

A plane continuum is a bounded, closed and connected subset of the plane. A bounding box $B$ for a plane continuum $C$ is a rectangle $B=[a,b]\times[c,d]$ (including sides and interior) such that $C$ ...
Mirko's user avatar
  • 1,345
2 votes
1 answer
241 views

Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's "Classical Groups"

First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but didn'...
InvisiblePanda's user avatar
2 votes
1 answer
3k views

Supplementary notes to Mumford's The Red Book of Varieties and Schemes

I am a graduate student with good mathematical maturity (I took advanced courses like category theory, commutative algebra...). I want to study algebraic geometry from Mumford's red book. I find it ...
Jesse Solomon Scott's user avatar
2 votes
2 answers
673 views

$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear

I know the following is a well-known result. Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$ Furthermore, there is ...
user33122's user avatar
  • 125
2 votes
2 answers
179 views

$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
Iosif Pinelis's user avatar
2 votes
0 answers
700 views

Confusing notation for sets of unordered vs ordered pairs

Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$. One may also consider ...
Matthieu Latapy's user avatar
2 votes
2 answers
3k views

Quotient of an algebraic group by a closed algebraic subgroup

Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed and normal subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" ...
Jesko Hüttenhain's user avatar
2 votes
1 answer
269 views

Mapping between Notations

As in my other question, it is assumed that the (total) function describing a given notation is denoted as $address:p\rightarrow \Bbb{N}$ and assumed to be bijective. Suppose we are given two ...
SSequence's user avatar
  • 861
2 votes
1 answer
315 views

The study of dynamics of a polynomial vector field via Green's function methods

In the litterature, in particular in the papers on dynamical investigation of polynomial vector fields on the plane, are there some research devoting to study the Green's function for the PDE which is ...
Ali Taghavi's user avatar
1 vote
2 answers
549 views

Prove Liouville theorem without using mean value property

How can I prove the following Liouville theorem without using the mean value property? If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$, then $...
Lao's user avatar
  • 217
1 vote
0 answers
70 views

Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \...
Kei's user avatar
  • 267
1 vote
1 answer
231 views

Elliptic interface problem without conditions on the interface

Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$. For a model case, consider a ball split in a smaller ball and an anulus. Consider the following elliptic ...
user avatar
1 vote
1 answer
276 views

SDE with non-degenerate diffusion visits every point

I am asking an extension of the question here for SDEs of the Ito form. Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}...
John's user avatar
  • 483
1 vote
1 answer
658 views

Smooth structures on closed $3$-manifolds are unique up to diffeomorphism?

Hi! I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the ...
Malte Muth's user avatar
38 votes
6 answers
7k views

Companion to theoretical physics for working mathematicians

In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...
15 votes
2 answers
442 views

Generalizations of summation methods of divergence series

If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
Serge the Toaster's user avatar
13 votes
5 answers
3k views

Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...
Miroslav Korbelar's user avatar
92 votes
0 answers
16k views

Hironaka's proof of resolution of singularities in positive characteristics

Recent publication of Hironaka seems to provoke extended discussions, like Atiyah's proof of almost complex structure of $S^6$ earlier... Unlike Atiyah's paper, Hironaka's paper does not have a ...
Henry.L's user avatar
  • 7,951
83 votes
4 answers
13k views

How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...
Hailong Dao's user avatar
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