All Questions
23,892 questions
4
votes
1
answer
175
views
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
8
votes
1
answer
465
views
Smooth numbers in short intervals
Let $\psi(x,y)$ be the number of positive integers up to $x$ which are $y$-smooth, that is, integers whose prime factors are at most of size $y$. There has been, for a few decades now, a lot of ...
- 14.6k
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
7
votes
0
answers
269
views
Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
- 460
3
votes
1
answer
287
views
Expectation comparison inequality for concave function of symmetric random variables
Suppose that $X_i$, $i\in[n]$ are
independent symmetric
random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
- 6,541
1
vote
1
answer
56
views
How to study the convergence of the sample mode for arbitrary probability spaces
(This is not the problem I actually care about, but an analogy with similar issues to the problem I'm actually considering.)
Consider a probability space with i.i.d. random variables $X_i$ producing ...
- 128k
1
vote
0
answers
25
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Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
- 111
2
votes
1
answer
99
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
3
votes
0
answers
89
views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
10
votes
1
answer
518
views
Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions
Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
CommunityBot
- 1
2
votes
0
answers
205
views
When should the empirical measure of an infinite sequence be defined?
Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a ...
- 6,406
3
votes
1
answer
187
views
Is this property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
11
votes
3
answers
779
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.
- 47.3k
4
votes
1
answer
193
views
Canonical decomposition as wedge sum up to homotopy equivalence
I am curious: is there a canonical way to decompose a finite simplicial complex into a wedge sum up to homotopy equivalence? More formally:
Let $X$ be a finite simplicial complex. Is $X$ homotopy ...
- 16.2k
16
votes
0
answers
188
views
Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
1
vote
0
answers
95
views
References on the claims of moduli spaces of additive compactifications by Hassett-Tschinkel
I am considering the classification problem of Artinian local $\mathbb{C}$-algebras, and notice the paper
Hassett, Brendan; Tschinkel, Yuri, Geometry of equivariant compactifications of (\mathbb{G}^...
- 930
27
votes
4
answers
4k
views
Who introduced the terms "equivalence relation" and "equivalence class"?
Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns the origin of the terms "equivalence relation" and "equivalence ...
- 61
0
votes
1
answer
73
views
Computing spectrum of very simple Schrödinger operator
I asked this question recently on a thread in math stack exchange, but with no real answers suggested. I think this is a relatively simple variation on the classical free Laplacian spectrum, so I ...
- 17.2k
2
votes
0
answers
103
views
Variant on ring objects in the category of complete lattices
Let $L$ be a complete lattice and denote its top and bottom elements by $0$ and $\infty$ respectively. Consider two binary operations $+$ and $\times$ defined on $L$ such that $(L^{op},+,0)$ is a ...
- 4,219
3
votes
1
answer
205
views
Inertia Action on Kummer Sheaves
In 7.0.2 of Katz's book "Gauss Sums, Kloosterman Sums, and Monodromy Groups", Katz states the following (when $x=0$).
Let $\chi:\mathbb{F}_q^\times\to\mathbb{Q}_\ell^\times$ be a ...
- 148k
11
votes
2
answers
432
views
On the convex cone of convex functions
$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of ...
- 91.8k
7
votes
0
answers
250
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
2
votes
1
answer
315
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 ...
- 12.9k
0
votes
0
answers
20
views
Decomposition of measures orthogonal to the algebra $R(K_1 \times \ldots \times K_n)$ - Can it be done via projection-preserving products of bands?
See "Measures orthogonal to tensor products of function algebras" by Marek Kosiek. Here, it is described for the two-dimensional case. It uses another, more general, approach to OB Bekken's ...
- 63
4
votes
0
answers
87
views
Statistics of random Voronoi S-tessellations
Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
2
votes
0
answers
125
views
Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
- 3,738
2
votes
0
answers
120
views
On mollifiers acting between $L^2$ and Sobolev spaces
(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.)
Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by
$$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
- 6,367
0
votes
0
answers
112
views
Vector field connecting two points
I'm now working on somehow an inverse problem of an ODE:
Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t).
Now there is a ...
- 1
2
votes
1
answer
136
views
Eigenvectors of the dual of positive irreducible operators
This question was previously posted on MSE.
Let $E$ be a Banach lattice such that $E$ is an $M$-space. Assume that $T\colon E\to E$ is a positive bounded non-compact irreducible linear operator with ...
- 12.6k
2
votes
1
answer
79
views
There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?
Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
- 452
3
votes
0
answers
117
views
Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?
Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
2
votes
1
answer
272
views
A variant of Hardy's inequality for "convolutions"?
Consider Hardy's inequality on $L^{2}(\mathbb{R}^3)$. This inequality states that:
$$\int_{\mathbb{R}^3} dx \, \frac{|\psi(x)|^2}{|x|^2} \le K \int_{\mathbb{R}^3} dx \, |\nabla \psi(x)|^2.$$
I want to ...
CommunityBot
- 1
2
votes
0
answers
121
views
Singular cohomology as a sheaf of $\infty$-categories
In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
7
votes
2
answers
517
views
Tensor product of a DGA and an $A_\infty$ algebra
In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge ...
9
votes
0
answers
275
views
Has a computer search for inconsistency of large cardinals been carried out before?
In discussion about the consistency of large cardinals, a justification which occasionally appears is that despite years of work, no inconsistency has currently been discovered. For example, here are ...
- 2,031
4
votes
0
answers
167
views
What textbooks/papers should I read to try to make this rigorous?
Consider a surface of revolution $S$ and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with cone points $p,q$ elements of $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm {sup}~ ...
- 383
2
votes
0
answers
80
views
Surjectivity of kernel operator with kernel in $L^1(\nu \times \mu)$
Let $ \mu $ and $ \nu $ be two finite and non discrete measures. Let's begin with a well-known fact. Let $ k \in L^2(\nu \otimes \mu) $, then we can define an operator $ \tilde{T} $ as follows:
$$
\...
4
votes
1
answer
272
views
Is there a non-semistable simple sheaf?
Let $C$ be a smooth projective (irreducible) curve over an algebraically closed field $k$.
A sheaf is said to be simple if its endomorphism algebra is isomorphic to $k$.
It is known that a stable ...
- 5,857
6
votes
1
answer
335
views
Existence of pairwise quasi-complementary but not complementary subspaces
Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
- 986
15
votes
1
answer
601
views
Topological spaces in which countable intersections of dense open sets have dense interior
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.
Now consider the following strengthening of the Baire ...
CommunityBot
- 1
2
votes
1
answer
142
views
Bounded differentiation operator on compact intervals with $L^2$ norm
It is known that the differentiation operator $D$ is not bounded on $C^1([0,1])$ with $L^2$ norm (counterexample: $f(x)=x^n$). Now I am wondering whether there is an infinitely dimensional subspace ...
- 24.2k
2
votes
1
answer
180
views
Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$
Let us consider the heat equation
$$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$
where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if ...
CommunityBot
- 1
2
votes
1
answer
128
views
Density of smooth functions in weighted Sobolev space
Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
- 3,425
4
votes
1
answer
54
views
Krein-Rutman for integral transforms: proof of convergence to leading eigenvector
Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory.
Consider an integral ...
- 630
17
votes
1
answer
3k
views
The homology of the orbit space
Suppose we have an acyclic group $G$ and let $X$ be a contractible CW-complex such that $G$ acts freely on $X$ (we do not suppose that the action is proper).
Is there a way to understand the homology ...
CommunityBot
- 1
6
votes
2
answers
804
views
Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 44.4k
0
votes
0
answers
80
views
Relationship between two minimization problems
Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
0
votes
0
answers
52
views
References on a variant of Geometric Calculus
Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism.
I have read a bit about fractional calculus and time-scale calculus, both very ...
CommunityBot
- 1
4
votes
1
answer
140
views
Reference for Reedy weak factorization systems, not Reedy model structure?
What paper/book is appropriate as the standard reference for Reedy weak factorization systems, not Reedy model structure? Specifically, I would like a reference for the following Propositions 1 and 2:
...
- 366
5
votes
1
answer
490
views
$L^2$ uniform integrability in terms of Fourier coefficients
Given a bounded sequence $(f_n)_n$ in $L^2(\mathbf{T})$ where $\mathbf{T}:=\mathbf{R}/\mathbf{Z}$, the strong compactness of $(f_n)_n$ is equivalent to $$\lim_N \sup_n \sum_{|k|\geq N} |c_k(f_n)|^2=0,$...
CommunityBot
- 1