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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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Eigenvalue analysis of $X^T (XX^T + \mathrm{Id})^{-1} X$ for $X$ iid random matrix

Consider the following quantity $$X^T (XX^T + \mathrm{Id})^{-1} X,$$ where $X \in \mathbb{R}^{m\times n}$ is a iid random matrix with 0 mean and finite variance. The empiric covariance matrix ${X^T X}$...
Goulifet's user avatar
  • 2,216
2 votes
1 answer
85 views

Reference Request: Possible generalizations of the stability of $\gamma$-factors

$\DeclareMathOperator\GL{GL}$ Let $F$ be a nonarchimedean local field. Suppose $\pi, \sigma$ are irreducible admissible representations of $\GL_{n}(F)$ and $\GL_{m}(F)$ respectively, with $n \geq m$. ...
Hetong Xu's user avatar
  • 619
4 votes
3 answers
438 views

Generalized Fuchsian-type PDE

Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
Math2024's user avatar
  • 141
1 vote
0 answers
42 views

Frobenius pullback of an integrable connection on a quasi-projective scheme

Let $X_k$ be a smooth quasi-projective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to characteristic $0$, and let $(X_K)^{\text{an}}$ denote the rigid analytic space associated ...
kindasorta's user avatar
  • 1,963
3 votes
1 answer
219 views

Original proof of Lefschetz's theorem on $(1,1)$ classes

Is there a "modern" account of Lefschetz proof of his theorem about $(1,1)$ classes for projective surfaces ? I believe that would be very interesting to understand the original arguments ...
Nicolas Hemelsoet's user avatar
4 votes
1 answer
168 views

Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and ...
strat's user avatar
  • 349
4 votes
0 answers
122 views

Automorphism-invariant positive linear functionals on $C*$-algebras

Let $A$ be a $C^*$-algebra. Does there exist a non-trivial positive linear functional $\nu\in A^*$ which is $\mathrm{Aut}(A)$-invariant? That is, $\nu\circ\alpha=\nu$ for all $\alpha\in\mathrm{Aut}(A)$...
Bedovlat's user avatar
  • 1,939
6 votes
1 answer
101 views

Topological entropy of semi-conjugated dynamical systems

Let $(X,T)$ and $(Y,G)$ be topological dynamical systems. If $(Y,G)$ is a factor of $(X,T)$ it is well known and easy to proof that $h(G)\le h(T)$ , where $h$ denotes the topological entropy. If the ...
Jörg Neunhäuserer's user avatar
2 votes
1 answer
197 views

Decay estimate of moment of an SDE

We consider an SDE $$ d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t, $$ where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...
Akira's user avatar
  • 917
11 votes
1 answer
435 views

Examples of games developed purposely to analyze players' strategies for mathematics research

Background This question is about games that were created, developed, deployed and popularized1 by researchers because they wanted to learn more about some mathematical structure, and did so by ...
Max Muller's user avatar
  • 4,585
1 vote
0 answers
59 views

$F$-structure implies regular singularities + unipotent local monodromy?

Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...
kindasorta's user avatar
  • 1,963
7 votes
4 answers
461 views

A conservative extension of Peano Arithmetic

Ulrich Kohlenbach makes the following intriguing comment here: "In the 70s S. Feferman introduced a mathematically strong system S=restricted(PA^omega)+QF-AC+mu for classical mathematics (and in ...
Mikhail Katz's user avatar
  • 15.6k
4 votes
1 answer
207 views

Double cover the edges of a complete graph by smaller complete graphs

Suppose we have a complete graph $K_n$ on $n$ vertices. Are there any results on the ways to cover $K_n$ with $k$ copies of $K_m$, for $m<n$, such that each edge of $K_n$ is contained in exactly ...
Wallace Rin's user avatar
6 votes
0 answers
245 views

Proof $\pi$ is transcendental without symmetric function theory

This is a crosspost of my question from MSE from 3 weeks ago which was bountied but has received no response. For an algebra assignment, I was asked to do a literature review and write up a proof of ...
Alex Pawelko's user avatar
3 votes
1 answer
188 views

Isocrystal with no $F$-structure

$\DeclareMathOperator\Isoc{Isoc}$Let $X_k$ be a quasiprojective $k$ scheme, with $k$ finite, and let $X_K$ be the rigid analytic space lifting it to the fraction field of its Witt ring, which I denote ...
kindasorta's user avatar
  • 1,963
6 votes
0 answers
80 views

Reciprocity for algebra objects in two algebraic categories

I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories. So, ...
Nik Bren's user avatar
  • 519
2 votes
0 answers
33 views

0-1 knapsack problem with additional capacity

The 0-1 knapsack problem maximizes the profits of items under a capacity constraint (let's call this capacity $C$). I am interested in an augmented setting where the algorithm is permitted to use a ...
Titan's user avatar
  • 21
1 vote
0 answers
125 views

Can't parse a statement in an article on coalgebras and umbral calculus

This question is cross-posted from MSE. I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", ...
Daigaku no Baku's user avatar
12 votes
1 answer
370 views

Partition into antichains

I've read that the following statement is a result of Balcar, but I am unable to find a reference or a proof: Theorem: If $\kappa\ge \lambda$ are infinite cardinals, then $[\kappa]^{<\lambda}$ can ...
Lajos Soukup's user avatar
  • 1,467
4 votes
1 answer
218 views

Equivalence between vector bundles with integrable connections to isocrystals

Let $k$ be a perfect field, $W(k)$ its Witt ring, and $K$ the fraction field of $W(k)$. Let $X_k$ be a smooth proper curve over $k$, and let $X_K$ be the schematic generic fibre of a smooth proper ...
kindasorta's user avatar
  • 1,963
1 vote
1 answer
104 views

A reliable reference for the statement every $k$-tree is uniquely $(k + 1)$-colorable

I see that every $k$-tree is uniquely $(k + 1)$-colorable in Uniquely_colorable_graph. Wikipedia does not cite any references, even though I know that its proof is not difficult by using mathematical ...
L.C. Zhang's user avatar
  • 1,645
2 votes
2 answers
73 views

Reference request for a subfamily of regular graphs

[Repost of same question math stack exchange which got no answers] I'm looking for literature on the following family of graphs: Call a regular graph $G=(V,E)$ (of regularity degree $d$) nice if there ...
jojo's user avatar
  • 21
9 votes
1 answer
833 views

Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
Song Ye's user avatar
  • 51
2 votes
2 answers
214 views

$L^p$ domination of mixed partial derivatives of the 3rd order by the unmixed ones?

Is it true that for each real $p>1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^3$ one has $$\|D_1D_2D_3u\|_p\le C_p(\|D_1^3u\|_p+\|...
Iosif Pinelis's user avatar
2 votes
1 answer
182 views

Is the category of simplicial $R$-modules closed monoidal?

I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...
SetR's user avatar
  • 91
1 vote
0 answers
132 views

Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments [closed]

I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
total dependent random choice's user avatar
0 votes
0 answers
55 views

Names for product-like algebras involving a "duo of directed pseudoforests"

I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class. In both cases, there is an (infix) binary ...
user1661473's user avatar
5 votes
0 answers
781 views

A fourth-order linear PDE

I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$): $$x^3 f_{xxxt}+ f =0$$ Does anyone know if this type of PDE already appeared in the literature? ...
Math2024's user avatar
  • 141
5 votes
0 answers
97 views

Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$

Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
Tom Copeland's user avatar
  • 9,937
3 votes
1 answer
118 views

Reference request for log-differential forms

I read in a paper of Kato about log-differential forms, that if $X$ is a smooth locally Noetherian log-scheme, and $D$ is a reduced normal crossing divisor, then there is a definition of a sheaf on $X$...
kindasorta's user avatar
  • 1,963
2 votes
2 answers
180 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
1 vote
0 answers
62 views

Etale local systems and proper base change

I am looking for a reference, or a proof, of the following statement: Let $f:Y\longrightarrow X$ be a smooth proper map of quasiprojective $K$ schemes, and let $\overline{f}:\overline{Y}\...
kindasorta's user avatar
  • 1,963
2 votes
0 answers
119 views

Imaginary quadratic fields with prime class number

Let $K$ be an imaginary quadratic field, with class number equal to an odd prime, say $h_K = p$. In the proof Proposition 2.4 of this paper, Fukuda and Komatsu write, "Since $h_K = p$, there ...
matt stokes's user avatar
0 votes
0 answers
56 views

asymptotic expansions for $C^{1+\epsilon}$operators

I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators. More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\...
user avatar
6 votes
0 answers
192 views

Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
red_trumpet's user avatar
  • 1,071
5 votes
1 answer
163 views

Reference request: Fréchet embedding

Given a separable metric space $(X,d)$, we have an isometric embedding $\iota:X\to\ell^\infty$ given by taking $(x_n) _{n \ge 0}$ to be the countable dense subset and sending $\iota(x)_n=(d(x,x_n) - d(...
Gesh's user avatar
  • 115
16 votes
2 answers
2k views

Is Freyd's thesis available online anywhere?

Peter Freyd is a great category theorist. His PhD dissertation, Functor Theory, dates from Princeton in 1960. It's cited as [14] in Mitchell's book Theory of categories. In fact, Google scholar says ...
David White - gone from MO's user avatar
3 votes
0 answers
56 views

Self-enrichment for a closed monoidal bicategory

First, there are two possible generalization of the notion of closed category, vertical and horizontal. I'm interested in the vertical one, something saying, I guess, that a monoidal bicategory $\...
Nikio's user avatar
  • 351
2 votes
2 answers
204 views

Devaney chaos and topological entropy

I am searching for dynamical systems on compact spaces which are Devaney chaotic but have topological entropy zero. On the interval such systems do not exist. I think on the Cantor space and on the ...
Jörg Neunhäuserer's user avatar
1 vote
0 answers
47 views

Enumeration of uniform polyhedra

[I already asked this question on MSE (here) but got no answer so I am trying here] It is known that there are two infinite classes of polyhedra (prisms and antiprisms) together with $75$ uniform ...
Martin's user avatar
  • 1,101
0 votes
0 answers
74 views

Potential typo in "Complete Systems of Two Addition Laws for Elliptic Curves" by Bosma and Lenstra

Here is a link to the article: https://www.sciencedirect.com/science/article/pii/S0022314X85710888?ref=cra_js_challenge&fr=RR-1. Pages 237-238 give polynomial expressions $X_3^{(2)}, Y_3^{(2)}, ...
Vik78's user avatar
  • 527
6 votes
2 answers
732 views

Prove positivity of a binomial sum

Some problems appear easy on the face of it, but perhaps they are not. Here is an instance of a certain calculation which is slightly reformulated from its original encounter in a current work. I have ...
T. Amdeberhan's user avatar
0 votes
0 answers
74 views

Random walks on groups

I recently started reading Wolfgang Woess' book titled "Random Walks on Infinite Groups". In the section where he introduces Markov chains and random walks on a set $X$, he has defined a ...
Dimitri's user avatar
1 vote
0 answers
47 views

Optimal orbital tranfers: reference for statements & proofs

I know that Pontryagin's contribution to optimal control in the 1950'ies was allegedly inspired by the then-nascent rocket industry and the question of how to get a rocket into orbit with minimum fuel....
5th decile's user avatar
  • 1,451
4 votes
1 answer
180 views

Generic absoluteness

In Theorem 14 of "On The Question Of Absolute Undecidability" Peter Koellner describes a generic absoluteness result which could be summed up as $\Sigma^2_1(\Gamma^\infty)$-generic ...
Rupert's user avatar
  • 2,055
1 vote
1 answer
123 views

Reference request regarding faithfully exact functors between abelian categories

I am looking for a reference for the following result (or any subresult) in any book or notes: Lemma. Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor between abelian categories. The following ...
Elías Guisado Villalgordo's user avatar
2 votes
0 answers
132 views

Is there a name for a normal, projective variety where every effective divisor is ample?

Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...
Schemer1's user avatar
  • 789
2 votes
0 answers
52 views

Regularization for Newtonian n-body collisions in $\mathbb{R}^3$

In working with binary collisions in the Hamiltonian formulation of the Newtonian $n$-body problem, two common regularization techniques that deal with binary collisions are the Levi-Civita technique, ...
user12994's user avatar
1 vote
1 answer
86 views

A Kolmogorov inequality for sums of contiguous subsequences

If $X_1, \ldots X_n$ are independent real-valued random variables such that $E[X_k] = 0$ and $E[X_k^2]$ is finite for each $k$, Kolmorogov's inequality gives an upper bound on $P[\max_{1\le k \le n}|...
Rob Arthan's user avatar
4 votes
0 answers
70 views

Coloured Jones polynomial of the mirror image of a multicomponent link

This question has been reposted from MathStackExchange It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/...
hopftype's user avatar

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