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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A Near Closed-Form Expression of Strict Partition Function Inquiry [closed]

I am an independent researcher working in various fields of mathematics and sciences. I am working on a strict partition problem. I believe I have found a very fast exact solution that is a near-...
jables's user avatar
  • 1
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0 answers
26 views

Are the coefficients in the stationary phase approximation computed explicitly somewhere

In Stein's "Harmonic analysis" book, page 334, one can find the asymptotic expansion An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
Medo's user avatar
  • 676
0 votes
0 answers
18 views

Find transseries from difference equation

I want to find a method to solve equations of the form $f(x+1)=f(x)+g(x)$ for a given function $g$ and $f(x)=0$. The paper here has solutions for $f(x+1)=\lambda(x)f(x)+g(x)$, which is more general ...
opfromthestart's user avatar
0 votes
0 answers
45 views

Reference request for equivalent Lipschitz smoothness conditions

For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
William Kong's user avatar
2 votes
0 answers
179 views

What are the Hodge and log Hodge groups of $M_{g,n}$?

I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
Leo Herr's user avatar
  • 1,084
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0 answers
22 views

reference request: mercer expansion and kernel underlying Sobolev spaces?

Let us define the periodic Sobolev spaces, for $s > n/2$ by $$ H_{s}([0, 1]^n) = \{f : [0, 1]^n \to \mathbb{R} :\mbox{for}~j\leq s, f^{(j)} |_{\partial[0, 1]^n} \equiv 0, ~~ \int_{[0, 1]^d} (f^{(s)...
Drew Brady's user avatar
2 votes
0 answers
99 views

Bourgain-Gamburd-like theorems in the non-algebraic case

For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
John Rached's user avatar
3 votes
1 answer
114 views

Integral involving Legendre polynomial

In a physics problem the following integral shows up $$\int\limits_0^{2\pi}P_m(\cos{(\theta-\alpha)})\,\cos^{m+2}{(n\alpha)}\;d\alpha,$$ where $P_m$ is the Legendre polynomial and $n,m$ are integer ...
Zurab Silagadze's user avatar
0 votes
0 answers
126 views

Connectedness of deleted symmetric product

Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
Peluso's user avatar
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7 votes
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Stochastic analysis on nuclear Fréchet spaces

This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise. A lot of the time in infinite-...
J_P's user avatar
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1 vote
0 answers
57 views

Finitely presentable groups are residually finite if and only if they are universally pseudofinite

Suppose $G$ is finitely presentable. Then residual finiteness of $G$ is equivalent to $G$ satisfying the universal theory of finite groups (equivalently, to every existential statement true in $G$ ...
tomasz's user avatar
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5 votes
2 answers
432 views

On the derivative of the Bernstein polynomial

$\newcommand\Z{\Bbb Z}\newcommand\De{\Delta}$For a natural $n$ and a function $g\colon\Z\to\Bbb R$, let $B_n g$ be the corresponding Bernstein polynomial, so that $$(B_n g)(x)=\sum_{k\in\Z} g(k)\binom ...
Iosif Pinelis's user avatar
1 vote
0 answers
96 views

Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module

Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
kindasorta's user avatar
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0 answers
126 views

A question and reference about Bombieri's article continued fraction of algebraic numbers

Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...
XL _At_Here_There's user avatar
4 votes
0 answers
154 views

Friedman's proof of covering lemma for $L$

There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...
Lxm's user avatar
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1 vote
2 answers
166 views

Name for a certain type of cardinal

I'm not a set-theorist, but I hope this question is appropriate. This is just a question about names: Fix a cardinal $\lambda$. I'd like to know if there is a name for regular cardinals $\kappa$ such ...
Maxime Ramzi's user avatar
  • 13.2k
0 votes
1 answer
31 views

Isometric path cover number of the 2 dimensional grid graph

I am looking for a proof of the fact that at least $2n/3$ isometric paths (i.e. shortest paths between the end points) are required to cover the vertices of the $n\times n$ grid graph (i.e. Cartesian ...
Pritam Majumder's user avatar
5 votes
0 answers
110 views

If chromatic polynomials for two graphs agree, can I always find an edge such that the two deletion-contraction minors have same chromatic polynomial?

Suppose I have non-isomorphic graphs $G$ and $H$ (which have at least one edge), but such that their chromatic polynomials are the same. Can I then always find an edge $e$ in $G$ and $f$ in $H$ such ...
Per Alexandersson's user avatar
0 votes
0 answers
17 views

reference request: product measures defined by a subsequence of measures

Suppose $\{\mu_n\}_{n\in\mathbb{N}}$ is a sequence of pairwise equivalent probability measures, each of which is defined on $\mathbb{R}$. Let $\bigotimes_n\mu_n$ be the product measure defined on $\...
Sanae Kochiya's user avatar
1 vote
0 answers
51 views

A question on generalized bases

I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
Dumbest person on earth's user avatar
2 votes
0 answers
99 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
8 votes
0 answers
164 views

Elkies' family of elliptic curves of rank 19

There is a widely cited fact that Elkies had found that infinitely many curves of rank 19 in 2006, in "Z^28 in E(Q), etc. Email to the number theory mailing list at [email protected]&...
Stepan Nesterov's user avatar
2 votes
1 answer
65 views

Seeking Article "Generating random lattices according to the invariant distribution" by M. Ajtai

I am searching for a specific article titled "Generating random lattices according to the invariant distribution" authored by Ajtai. Despite being widely cited in various papers, I have been ...
LATTICE's user avatar
  • 21
0 votes
0 answers
43 views

Two-parameter “$\varepsilon$-$\delta$ filtration” given a function between metric spaces

Let $X,Y$ be metric space and $f : X \to Y$ a (not necessarily continuous) function. I'm interested in the two-parameter filtration $(X_{\varepsilon, \delta})_{{\varepsilon, \delta} > 0}$ where $X_{...
user1892304's user avatar
7 votes
1 answer
163 views

A reference for forcing projections

The idea of a projection $\pi\colon\mathbb{Q}\to\mathbb{P}$ of forcing notions is something like a combinatorial stand-in for the fact that forcing with $\mathbb{Q}$ produces a generic for $\mathbb{P}$...
Miha Habič's user avatar
  • 2,279
6 votes
0 answers
77 views

The meet of two dominant permutations in weak order of $S_n$

A permutation is called dominant if its Lehmer code is a partition, or equivalently if it avoids the pattern $132$. I can prove that given a permutation $v\in S_n$, there is a unique dominant ...
Matt Samuel's user avatar
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2 votes
0 answers
46 views

Does this filtration have a name?

In the context of Ethier&Kurtz Markov Processes: Characterization and Convergence (Chapter 4, equation (3.2)) as well as the two papers Martingale problems for conditional distributions of Markov ...
Mushu Nrek's user avatar
6 votes
1 answer
294 views

Literature about formalization of "natural reasoning" in mathematical logic

In "Logic of sheaves of structures", X. Caicedo justifies the logic he introduces stating (more or less) that assertions about a point should really be understood as assertions about a ...
user524506's user avatar
1 vote
0 answers
108 views

What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...
HASouza's user avatar
  • 293
0 votes
1 answer
80 views

Properties of limit set for cellular automata

Is anyone familiar with results about properties of the limit set of the local rule for a cellular automaton? I haven't been able to find any good materials on the subject from an initial search, and ...
Keen-ameteur's user avatar
-3 votes
0 answers
208 views

Are there known examples like this almost official exposition of ZFC that is very weak?

Pseudo-ZFC is a theory written in the usual language of set theory, i.e. mono-sorted first order logic with equality and membership. The extra-logical axioms are: Extensionality: $\forall x \forall y:...
Zuhair Al-Johar's user avatar
2 votes
2 answers
157 views

A Inequality in the paper by Kenig, Ponce and Vega

I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle", ...
Sarthak's user avatar
  • 63
2 votes
1 answer
271 views

Counting points on elliptic curves

Consider the Legendre family of elliptic curves $$E_a: y^2=x(x-1)(x-a).$$ Let $p$ be an odd prime. QUESTION. Is the following true? If $p\equiv 3\pmod4$ then number of solutions to $E_2$ over the ...
T. Amdeberhan's user avatar
1 vote
0 answers
126 views

Isomorphic quotients of a countably infinitely-generated free abelian group

Let $F$ denote the free abelian group on countably infinite generators. I am trying to understand the relationship between normal subgroups $A$ and $B$ of $F$ with isomorphic quotients. So is there a ...
medvjed's user avatar
  • 11
6 votes
2 answers
230 views

Reference for Langlands dual homomorphisms

I am looking for a reference that explains in detail the existence of Langlands dual homomorphisms. It seems that in the literature two references are given most often. The first is Borel's article ...
user449595's user avatar
2 votes
1 answer
79 views

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
  • 2,174
2 votes
1 answer
75 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
  • 579
3 votes
3 answers
280 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
1 vote
0 answers
36 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
  • 1,373
3 votes
1 answer
188 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
147 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
  • 291
4 votes
0 answers
92 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
89 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
188 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
  • 749
10 votes
0 answers
330 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
  • 4,435
1 vote
0 answers
58 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
  • 1,373
7 votes
4 answers
414 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
  • 15.1k
4 votes
1 answer
197 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
212 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
179 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
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