Questions tagged [reference-request]
This tag is used if a reference is needed in a paper or textbook on a specific result.
14,543
questions
7
votes
0
answers
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views
Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks
I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...
3
votes
1
answer
186
views
Maximal $\pi/2$-separated subset of the sphere
A subset $A$ of a metric space is called $\varepsilon$-separated if
$$dist(x,y)> \varepsilon \mbox{ for all } x\ne y\in A.$$
(Notice that the inequality in my definition is strict.)
What is the ...
4
votes
1
answer
149
views
Enumerator Polynomials for Linear Anytime Codes
Let $C = \{c \in \mathbb{F}^n_2 : Hc=0\}$ be a binary linear code where $H \in \mathbb{F}^{k \times n}_2$ is a block lower-triangular matrix of full rank called the parity-check matrix of $C$. Clearly ...
2
votes
1
answer
308
views
How to construct the symmetric power function from a modular form?
I want to understand how we construct from a modular form $f$ its symmetric power function $Sym^rf.$ I read that there is a particular representation that does this but I am not familiar with this ...
10
votes
2
answers
481
views
References on quaternionic geometry
Is there any analog, in the quaternionic setting, of Kahler potentials?
In particular I'm interested in a similar construction of the Fubini-Study 2-form on $\mathbb{P}^1(\mathbb{C})$ over the ...
4
votes
1
answer
188
views
A proof of the Ibragimov-Kara-Mahomed commutation relation
Let $u_a(x),\,a=1,2,\ldots n$ be a $n$-component field in Minkowski spacetime
$x^\mu,\,\mu=0,1,2,3$ and let $u_{a,\,\mu}=\frac{du_a}{dx^\mu}$. Let us introduce two operators (we use Einstein ...
16
votes
1
answer
436
views
Types of generating functions (ordinary, exponential, ???) closed under substitution
A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...
8
votes
0
answers
2k
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Possible values of eigenvalues of Hadamard product of Hermitian matrices
One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...
1
vote
2
answers
212
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Completion under weighted limits/colimits
Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance
3
votes
2
answers
1k
views
Propositional logic without negation
As part of a bigger project I am researching a propositional logic, without a negation. And I would like to know, whether this already exists, to avoid double work and have proper references.
In this ...
3
votes
1
answer
150
views
Regularity of maps in algebraic topology for manifolds
Let $M$ be a $n$ manifold such that $\pi_k(M)$ is non trivial. What can we expect about the regularity of a representant $f:S^k\rightarrow M$ of a non-trivial cycle? For example, if $M$ is a manifold ...
4
votes
0
answers
428
views
Is there a well-known tautological bundle over $\mathbb{P}(\Lambda^nV)$?
Let $V$ be a vector space of dimension $>n$, and define the subset $$
K:=\{ ([\omega],v)\mid v\wedge\omega=0 \}\subset\mathbb{P}(\Lambda^nV)\times V\, .
$$
Denote also by $\pi:K\longrightarrow \...
2
votes
2
answers
333
views
Composition operators on fractional-order (periodic) Sobolev spaces
(The question was originally posted on MSE.)
Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication ...
9
votes
1
answer
812
views
Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?
A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already ...
1
vote
0
answers
132
views
Monotonic convergence of Newton's method for boundary value problems
I’m interested in solving nonlinear elliptic boundary value problems of the type
$$
-a\Delta u + f(u) = 0,
$$
$$
u|_\Gamma = u_0
$$
by Newton’s method when its convergence is global and monotonic.
...
5
votes
1
answer
180
views
Resource Constrained Routing with Refueling
What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity?
Especially modeling ...
5
votes
1
answer
635
views
Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems
I would like to find good references for the theorems mentioned above in the title. I am reading chapter VIII of Hörmander's classic, but I wonder whether there is something more up-to-date.
My ...
2
votes
0
answers
245
views
Reference request: proofs of the theorems in the paper "On the representation of the group GL(n, K) where K is a local field"
In the paper On the representation of the group $GL(n, K)$ where $K$ is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers....
3
votes
0
answers
113
views
Reference request concerning PL tangent Stiefel-Whitney classes
I am hoping for a reference for the fact that PL manifolds have tangent Stiefel-Whitney classes.
I understand this as follows: they have tangent microbundles, which in turn lead to spherical ...
2
votes
0
answers
413
views
Newer list of open problems in model theory
In the book Model Theory by C. C. Chang and H. J. Keisler, there is a list of open problems in model theory. More exactly, this list is called "Open problems in classical model theory" (on page 597, ...
1
vote
1
answer
319
views
A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces
(The question was originally posted on Math StackExchange.)
Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-...
6
votes
1
answer
478
views
References for Forcing with Side Conditions
I'm looking for some good references about Forcing with Side Conditions, including expository papers that explain the main ideas with some details in order to give me a fairly clear insight of those ...
1
vote
1
answer
445
views
Does the Laplacian commutes with elements of the basis of the Lie algebra?
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. I know that if $g$ is semi-simple then the Laplace-Beltrami operator on $G$ agrees with the Casimir element and therefore commutes with ...
2
votes
1
answer
567
views
On the upper Banach density of the set of positive integers whose base-$b$ representation misses at least one prescribed digit
Let $b$ be a fixed integer $\ge 2$ and $A$ a proper subset of $\{0, \ldots, b-1\}$. Then define $X$ to be the set of all positive integers whose base-$b$ representation consists only of digits from $A$...
10
votes
1
answer
542
views
Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?
It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...
1
vote
1
answer
718
views
Geometric interpretation of Chern classes over flag manifolds
I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able ...
4
votes
2
answers
811
views
Künneth formula for Bredon cohomology theory
Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...
0
votes
0
answers
54
views
Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$
Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin.
Is it true that
$$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...
4
votes
1
answer
259
views
Injectivity of the Funk transform for nonsmooth functions
Let $S^{n-1}$ be the unit sphere in $\mathbb R^n$ and $\Gamma_n$ the collection of great circles on it.
Assume $n\geq3$.
The Funk transform of a function $f:S^{n-1}\to\mathbb R$ is a map $Ff:\Gamma_n\...
11
votes
2
answers
1k
views
Concrete examples of covering from the 3-torus to the 3-sphere
There is a two-fold branched covering from 2-torus to the 2-sphere, $T^2 \rightarrow S^2$, whose covering transformation group is generated by the map $x \mapsto -x$ (Note that $T^2$ is an abelian ...
16
votes
3
answers
2k
views
Your favorite papers on geometric group theory
I would like to improve my "depth of understanding" in geometric group theory. So I am interested in short and accessible papers on subjects related to this field but which are not always ...
2
votes
0
answers
77
views
Maximum cardinality general factor of a graph
Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...
2
votes
1
answer
247
views
Well-ordered reference
I'm wondering if there is a standard reference for the following straightforward facts about well-orderings. (They are quite easy to prove, I just need a handy/standard reference.)
Let $(S,<)$ be ...
7
votes
1
answer
776
views
Infinite-dimensional admissible representations of SL(2,C)
I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper "Infinite Irreducible ...
1
vote
0
answers
87
views
Reduced products of (abelian and triangulated) categories: references?
For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...
3
votes
0
answers
103
views
Nemytskii/superposition operator without separability of Banach space?
Let $T:[0,1] \times X \to \mathbb{R}$ be a nonlinear map where $X$ is a Banach space. Suppose that $T$ is a Caratheodory map, so that $t \mapsto F(t,x)$ is measurable and $x \mapsto F(t,x)$ is ...
3
votes
0
answers
484
views
Riemann-Roch formula for nodal curves
Let $X$ be an irreducible, reduced, projective curve over an algebraically closed field, with at worst nodes as singularities. Let $\mathcal{F}$ be a trivial vector bundle on $X$ of rank $r$. Consider ...
0
votes
0
answers
117
views
How to solve $\sqrt{-1}\partial\bar{\partial}u=\omega$
I'm looking for references on the study of the equation $\sqrt{-1}\partial\bar{\partial}u=\omega$,especially when $\omega$ is a k\"ahler metric on $\Omega\setminus S$,where $\Omega\subset \mathbb{C}^n$...
2
votes
1
answer
641
views
Elementary bound on operator norm on symmetric tensors: reference request
My education didn't really cover Tensors very well, so I'm getting stumped by a quite elementary question.
Let $T_k$ be a type k symmetric tensor. We can define the "operator norm" (or the induced ...
0
votes
0
answers
121
views
Bound of Chebyshev function and zeros of zeta function
It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
2
votes
1
answer
146
views
Irreducible unitary representations of semidirect groups of a discrete abelian group by a discrete group
Recently in a paper we get the following result:
Let a discrete group $\Gamma$ act on a discrete abelian group $G$ by group automorphisms. Every irreducible unitary representation $\pi$ of $G\rtimes\...
5
votes
4
answers
620
views
Integrals involving the Tricomi hypergeometric function
I am looking for a reference for the two following equalities involving the Tricomi function $U$ and the Meijer function $G$. I have found these formulas on the website http://functions.wolfram.com/, ...
7
votes
1
answer
196
views
How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?
This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...
4
votes
2
answers
323
views
Non-Forking and Related Concepts
Is the importance of developing forking machinery in the way we set it up, or is it in the fact that it allows us to come up with a notion of independence via the properties of non-forking? I'm ...
0
votes
0
answers
406
views
Set as a (strict) infinite-category?
First, let me say that I have no idea if such a post has its place here. However, I believe that the ideas I'm going to present are important. The goal of this thread is three fold:
1) trying to ...
1
vote
2
answers
359
views
binomial/factorial identity mod p
In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result.
Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
4
votes
0
answers
173
views
Do more generalizations of Schur's inequality exist?
I meet this following problem
If $$n\ge 3,\sum_{i=1}^{n}\left(\prod_{j\neq i}(a_{i}-a_{j})\right)\ge 0$$
where $a_{i}$ are real numbers.
when $n=3$, it is Schur's inequality
so which $n$ such ...
2
votes
1
answer
280
views
Convergence of weighted double sum of random variables
I'm looking for convergence results of particular weighted sum:
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation ...
5
votes
0
answers
270
views
Moore spectra are not E-infinity (oldest known proof)
Fix a prime $p$. Let $M_p(i)$, the $i$-th Moore spectrum at the prime $p$, be the cofiber of the map
$$ S^0 \overset{p^i}\longrightarrow S^0 $$
where $S^0$ be the sphere spectrum. In the Mathoverflow ...
1
vote
1
answer
113
views
Differences of consecutive ordered fractional parts
Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a_1 < a_2 < \cdots < a_n.$$ Let $D_n$ be the set of ...