All Questions
15,509 questions
1
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0
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46
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Reference on polynomial attached to permutation group
Let $G$ be a permutation group acting on some set. Let $C(g)$ be the set of associated cycles of an element $g\in G$, and define $l(c)$ to be the length of a cycle $c$. Now set
$$T(G) = \sum_{g\in G}\...
0
votes
0
answers
28
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Analyzing simple DDE with simple characteristic test
I'm wondering if anyone can comment on the stability of delay DE given that we can analyze its characteristic equation.
For instance, let's say we have the DDE $\frac{d}{dt}x(t) = x(t-a),$ where $a$ ...
- 11
0
votes
0
answers
86
views
Uniqueness of compatible cycle decomposition for Eulerian trail
Fleischner mentions in his article Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs about the uniqueness of compatible cycle decomposition that ...
3
votes
0
answers
88
views
cubic twists of Mordell curve and their rank
Let $a$ be a non-zero integer. Consider the elliptic curve $E_a/\mathbb{Q}$ given by the equation
$$
E_a: y^2 = x^3 + a.
$$
For a cube-free integer $D$, define the elliptic curves $E_{aD^2}/\mathbb{Q}$...
- 1,283
2
votes
1
answer
191
views
Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
user536681
0
votes
0
answers
73
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Regularity estimates of Double Layer potential
Let $\Omega$ is a bounded open subset of $\mathbb{R}^n,n\ge 2$ with $C^{\infty}$ boundary. Define $$I\left[ \phi \right](x) := -\frac{1}{\omega_n}\int_{\partial \Omega} \frac{(x-y)\cdot \nu_y}{|x-y|^n}...
- 69
4
votes
1
answer
175
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Looking for J.-C. Deville technical report from 2000
Yves Tillé's book Sampling Algorithms mentions several times a technical report by J.-C. Deville:
J.-C. Deville (2000), Note sur l’algorithme de Chen, Dempster et Liu, Tech.
rept. CREST-ENSAI, Rennes....
- 82.7k
1
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0
answers
161
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Where can I find Stefan Wewers' doctoral thesis "Construction of Hurwitz Spaces"?
I am looking for a copy of Stefan Wewers' doctoral thesis titled "Construction of Hurwitz Spaces," which was defended at the University of Essen in 1998. I have tried searching through ...
- 560
1
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0
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66
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Sum of k vectors with largest possible norm
Suppose I have a family of $n$ vectors in $\mathbb{R}^d$:
$v_1,\dots,v_n.$ Is there a poly-time algorithm that computes a subset $S\subset [n]:=\{1,\dots,n\}$ of size $1\leq k\leq n$ for which the (...
5
votes
1
answer
367
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Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.8k
4
votes
1
answer
253
views
Reference to formal approach to homotopy analysis method
I'm currently reading a book about the Homotopy Analysis Method (HAM), but it isn't very rigorous (it explains most things with a single example), which is bothering me.
I'm searching for papers where ...
- 151
2
votes
0
answers
95
views
Kernel of a Mikhlin multiplier is a Calderón–Zygmund kernel (reference request)
Consider any function (convolution kernel) $K(x):\mathbb{R}^d\to\mathbb{R}$. Suppose the Fourier transform of $K(x)$, denoted by $\hat{K}(\xi):\mathbb{R}^d\to\mathbb{R}$ satisfies the standard Mikhlin ...
- 989
3
votes
2
answers
349
views
Reference for proof about a result concerning Sobolev spaces and exponential growth
I'm reading an article and I saw the following affirmation without proof:
Let $u \in H^1(\mathbb{R}^2)$ and $\alpha>0$, then
$$\int_{\mathbb{R}^2}(e^{\alpha u^2}-1)dx<+\infty.$$
Is this claim ...
- 213
7
votes
1
answer
446
views
Road map and references for combinatorial Hodge theory
I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ...
- 839
1
vote
0
answers
75
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Banach lattice hull of a Banach space
I am interested in defining some Banach lattice properties for subsets of arbitrary Banach spaces. So it would be useful to have the notion of Banach lattice hull $E(X)$ of a Banach space $X$.
I ...
- 4,461
3
votes
1
answer
190
views
References for Bernstein-Zelevisnky classification
I am looking for references for the Bernstein-Zelevisnky classification of irreducible representations of GL$(n,F)$ in terms of cuspidal representations. In particular I would like to find something ...
- 367
6
votes
1
answer
241
views
Reference request: acceleration/curvature of curve in metric space
Let $(X,d)$ be a metric space. Given a continuous curve $\gamma_t : [0,1] \rightarrow X$, the metric speed is defined by $$ |\gamma_t^\prime | := \lim_{s\rightarrow t} \frac{d(\gamma_s, \gamma_t)}{|t-...
- 660
1
vote
1
answer
216
views
Flatness of "derived local system sheaves"
Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of $\...
- 2,907
1
vote
0
answers
87
views
Convergence and sequential compactness for nonlinear operators
I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear.
What kind of notions of convergence does one have for such operators? I'm specifically ...
- 251
5
votes
1
answer
1k
views
Is this a new result about hexagon?
Let a hexagon $AB'CA'BC'$ let $AB' \cap A'B=C''$, $BC' \cap B'C = A''$, $CA' \cap C'A = B''$ then three conditions as follows equivalent:
Three lines $AA', BB', CC'$ are concurrent (let the point of ...
- 1,776
3
votes
1
answer
180
views
Analytic continuation to the Mittag-Leffler star using Mittag-Leffler summation
This is a reference request for a theorem I thought I had read in a book by Steven Krantz, but I can no longer find it.
Searching for Mittag-Leffler star, I can find references to the following result....
- 1,134
5
votes
1
answer
340
views
Equations for dual cubic curves
Suppose I have a cubic curve $C$ (over $\mathbb C$) in Weierstrass form $$y^2=x^3+ax+b.$$
I would like to find the degree $6$ equation for protectively dual curve $C^*$. Do you know any place where ...
- 14.3k
2
votes
1
answer
317
views
A reductive group is the complexification of a compact subgroup even if not connected?
The definition of a linear algebraic complex reductive group is sometimes using the connectedness hypothesis for the complex algebraic group sometimes not.
Here I use the following definition : a ...
- 1,128
11
votes
3
answers
783
views
Is every recursively axiomatizable and consistent theory interpretable in the true arithmetic (TA)?
I am looking for a scholarly text that discusses this issue in detail.
- 317
5
votes
2
answers
218
views
Decomposition of symmetric powers of the fundamental representation of $\text{Sp}(2n,\mathbb{C})$
Let $(k,0,...,0)$ denote the highest weights vector of an irreducible representation of $\text{Sp}(2n,\mathbb{C})$. I read in Fulton-Harris, that this representation may be obtained as a direct ...
- 2,907
0
votes
1
answer
151
views
Inequality $(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$
Conjecture: Let $a_1, a_2, \cdots , a_n>0$ and $y \ge x $ then
$$(a_1^x+a_2^x+\cdots+a_n^x)^y \ge (a_1^y+a_2^y+\cdots+a_n^y)^x$$
Equality iff $x=y$
Is the conjecture right? Have you ever seen this ...
- 1,776
3
votes
1
answer
327
views
Derivative norm estimates
Assume $\Phi$ is some diffeomorphism of a certain manifold. Let $\Phi^{-1}$ denote the inverse map and let $(D\Phi)^{-1}$ denote the matrix inverse of $D\Phi$.
QUESTION. Does this norm estimate hold? ...
- 43.2k
3
votes
0
answers
270
views
Categorical General Relativity
What are some good references for GR from a categorical point of view?
This is essentially just a big-list reference request.
I'm aware that the subject exists and can do some basic sleuthing to find ...
7
votes
4
answers
560
views
Reference request: "Higher order eigentuples" as generalized eigenvectors?
I stumbled upon a cute generalization of the eigenvalue problem and would like to know if anybody has seen something like this and can provide references.
The eigenvalue problem for a square matrix $M$...
- 12.7k
4
votes
0
answers
166
views
Szegő's inequality
I know Erdős-Lax's inequality and a couple of proofs. It states that:
If $P(z)=\sum_{v=0}^{n} a_{v} z^{v}$ is a complex polynomial of degree $n$ having no zeros in $|z|<1$, then
$$
\max _{|z|=1}\...
- 2,829
1
vote
1
answer
65
views
Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
1
vote
0
answers
108
views
Primitive element theorem for algebraic functions
Given a function $f(x) : \mathbb{R}^n \to \mathbb{R}$, we call it algebraic if it satisfies a polynomial equality $g(y, x) = 0$.
This is analogous to an algebraic number being the root of a univariate ...
- 326
2
votes
0
answers
210
views
Inverse problems and chaos theory
In the classical theory of inverse problems we want to recover an unknown $u \in U$ from its noisy measurements $y \in L^2$, where $U$ is a Banach space. In particular, we study the following problem:
...
- 43
6
votes
0
answers
141
views
Historical background of finding the roots of cubic equations using continued fractions
I came across an algebra problem book written in 1899 for students of Dar al-Fonun ([dɒːɾolfʊˈnuːn], meaning, "polytechnic college",) the only modern educational institute in Iran at the ...
- 2,437
3
votes
1
answer
187
views
Reference Request: Preservation of étale maps under rigid analytic GAGA
Let $K$ be a finite extension of $\mathbb{Q}_p$. As the title says, I am looking for a reference in which it is shown that given an étale map $f:X\rightarrow Y$ between smooth algebraic $K$-varieties, ...
- 541
0
votes
2
answers
180
views
Inversion formula for discrete sine and cosine transforms
$\newcommand{\wh}[1]{{\widehat{#1}}}
\newcommand{\R}{{\mathbb{R}}}
$I am looking for a proof of the inversion formulas for the discrete sine and cosine transforms, i.e. a proof of the fact that these ...
- 113
3
votes
0
answers
79
views
Reference request for unitary Shimura varieties
Let $K$ be an imaginary quadratic field and let $X$ be the Shimura variety associated to the unitary group $U(m,n)$ over $K$ (after a suitable choice of PEL datum).
Is there a reference that explains ...
- 1,949
0
votes
0
answers
159
views
Understanding the Hilbert scheme of subvarieties of $\mathbb{CP}^n$
EDIT: migrated to MSE.
I am looking to get a more concrete understanding of the Hilbert scheme of projective subvarieties, specifically over $\mathbb{C}$, and to obtain good references on this subject....
- 1,763
3
votes
0
answers
92
views
Reference for the monoidal category structure $X \otimes Y = X + Y + X \times Y$ on a distributive category
Given a distributive category $\mathscr C$ (more generally a rig category), we can define a (semicocartesian) monoidal category structure on $\mathscr C$ with tensor product given by $X \otimes Y := X ...
- 10.7k
6
votes
1
answer
350
views
Where is the original theorem shooting a club to kill a Mahlo cardinal?
I just want to make sure that I have the correct reference for the original theorem of shooting a club of singular cardinals to make a Mahlo cardinal become a non-Mahlo inaccessible cardinal. I can't ...
- 163
3
votes
1
answer
158
views
How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
- 807
4
votes
0
answers
143
views
Part II to Ketonen's "Set Theory for a Small Universe I. The Paris-Harrington Axiom"
There is an unpublished manuscript "Set Theory for a Small Universe I. The Paris-Harrington Axiom" by Ketonen which appeared early in the study of the Paris-Harrington theorem, around 1979. ...
- 2,031
2
votes
0
answers
80
views
Prove uniqueness of Radon transform without using Fourier transform
The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity):
If a continuous function with compact support has zero ...
- 807
1
vote
0
answers
167
views
Perfect complexes in a family
Consider a simple normal crossings variety $X=\bigcup_{i=1}^k X_i$ over $\mathbb{C}$ where $X_i$ are smooth projectiv and a flat family $\mathcal{X}\xrightarrow{\pi}\mathbb{A}^1_{\mathbb{C}}$ with $\...
- 341
0
votes
0
answers
61
views
A parabolic–hyperbolic in 3d: $\partial_t u(x,y,t)=\frac{1}{2}(\partial_{xx}u(x,y,t)-\partial_{yy}u(x,y,t))$
I was just wondering if somebody can provide some references for the parabolic–hyperbolic pde
$$\partial_t u(x,y,t)=\frac{1}{2}(\partial_{xx}u(x,y,t)-\partial_{yy}u(x,y,t)).$$
Apparently, the IVP ...
- 5,474
4
votes
0
answers
154
views
Is there a notion of "locally flat" for CW complexes?
A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\...
- 13.6k
10
votes
2
answers
598
views
Is there a "simplest" way to embed a graph in 3-space?
I consider embeddings of graphs into 3-space with edges embedded as arbitrary curves. In the simplest (non-trivial) case the graph $G$ is a cycle or union of cycles, in which case the embeddings can ...
- 13.6k
5
votes
0
answers
148
views
Tensor product of a Verma module of the highest weight and a Verma module of the lowest weight, $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$
$\DeclareMathOperator\sl{\mathfrak{sl}}\newcommand\hw{\mathrm{hw}}\newcommand\lw{\mathrm{lw}}$Consider $\mathfrak{g}=\sl_2(\mathbb{C})$. Fix $\lambda,\mu\in\mathbb{C}\setminus \frac{1}{2}\mathbb{Z}_{\...
- 51
1
vote
0
answers
122
views
Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds
Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
- 111
1
vote
0
answers
74
views
Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense:
Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 1,759