Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,545
questions
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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]&...
2
votes
1
answer
65
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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 ...
0
votes
0
answers
43
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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_{...
8
votes
1
answer
186
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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 $\mathbb{P}$...
6
votes
0
answers
80
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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 ...
2
votes
0
answers
47
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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 ...
7
votes
1
answer
312
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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 ...
1
vote
0
answers
108
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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 ...
1
vote
1
answer
119
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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 ...
-3
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0
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216
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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:...
2
votes
2
answers
171
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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",
...
3
votes
1
answer
284
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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 ...
1
vote
0
answers
127
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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 ...
6
votes
2
answers
236
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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 ...
2
votes
1
answer
95
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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 ${X^T X}$...
2
votes
1
answer
78
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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 $n \geq m$. ...
3
votes
3
answers
304
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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 $t$...
1
vote
0
answers
39
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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 ...
3
votes
1
answer
196
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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 ...
4
votes
1
answer
148
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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 ...
4
votes
0
answers
96
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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 $\alpha\in\mathrm{Aut}(A)$...
6
votes
1
answer
93
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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 ...
2
votes
1
answer
190
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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 ...
10
votes
0
answers
337
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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 ...
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-...
7
votes
4
answers
431
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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 ...
4
votes
1
answer
201
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 ...
6
votes
0
answers
216
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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 ...
3
votes
1
answer
183
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 ...
5
votes
0
answers
75
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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, ...
2
votes
0
answers
31
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 ...
1
vote
0
answers
121
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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", ...
12
votes
1
answer
368
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 ...
3
votes
1
answer
154
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 ...
1
vote
1
answer
103
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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 ...
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 ...
9
votes
1
answer
679
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 ...
2
votes
2
answers
192
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+\|...
2
votes
1
answer
171
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 ...
1
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0
answers
126
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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 ...
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 ...
4
votes
0
answers
412
views
A 4th-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? ...
5
votes
0
answers
91
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$ ...
3
votes
1
answer
113
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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$...
2
votes
2
answers
178
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^...
2
votes
0
answers
102
views
soft question, proof sketch: Constructing spaces of analytic sections. silva spaces
This is a follow up on this question Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations to add more details.
In the linked question I ...
1
vote
0
answers
52
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}\...
2
votes
0
answers
116
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 ...
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+\...
6
votes
0
answers
189
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 ...