All Questions
153,323
questions
4
votes
1
answer
252
views
Interpretation of the integral "with respect to a plane wave" in terms of Radon transform
This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform:
$\newcommand{\R}{\mathbb{R}}$
$$
Rf(\varphi,s)=\int_{-\infty}^\infty f(s\...
- 203
4
votes
1
answer
256
views
Is there a Whitehead-type theorem in T-equivariant cohomology?
Let $T$ be a real torus, and let $X$ and $Y$ be $T$-spaces. Under what conditions (if any) will the existence of graded $H^*_T$-algebra isomorphism between the $T$-equivariant cohomologies of $X$ and $...
- 4,870
4
votes
2
answers
469
views
On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$
Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the ...
- 12.1k
4
votes
1
answer
198
views
When does $\ell_1(\Gamma)$ embed into $L_1(\mu)$?
Suppose we are given an uncountable set $\Gamma$ and a measure space $(\Omega, \mathcal{F}, \mu)$. I would like to know when the Banach space $\ell_1(\Gamma)$ embeds into $L_1(\mu)$. Of course, this ...
4
votes
1
answer
330
views
Extension of characters of abelian locally compact groups
Let $G$ be an abelian locally compact group and $H$ be its closed subgroup. It is known from Pontryagin duality theory that every unitary character of $H$ can be extended to $G$. I think this is true ...
- 83
4
votes
1
answer
388
views
Reference request: harnack inequality for distributional solutions of the heat equation
Dear Math Overflowers,
I'm looking for references on the parabolic Harnack inequality for distributional solutions of the heat equation on the whole space
$$
\partial_t u=\Delta u\quad\text{and}\quad ...
- 5,163
4
votes
1
answer
572
views
To which automorphic forms/rep's over a function field can we associate a Galois representation?
As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic ...
- 43
4
votes
1
answer
489
views
Is "ultracompact" taken?
Almost-huge cardinals are characterizable in terms of coherent towers of supercompactness measures, with a certain property of the direct limit model (see Kanamori's book). A useful large cardinal ...
- 18.1k
4
votes
1
answer
902
views
Interpetation of torsion and curvature in terms of families of nearby geodesics
Let $M$ be a Riemannian manifold with affine connection such that the metric is covariantly constant (so that the connection equals the Levi-Civita connection up to torsion).
I know the ...
- 2,359
4
votes
1
answer
179
views
Progress on group languages characterizations
Def. A group language is a recognizable language whose syntactic monoid is a group.
q1. Is it known a "nice" combinatorial characterization of group languages ?
q1.1. If no, is it well understood ...
- 434
4
votes
1
answer
606
views
Characterization of the Laplace Transform
One of the main properties of the Laplace transform is given by the convolution theorem.
$$\mathcal{L}(f*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$
Question: Is there a full characterization of the ...
4
votes
1
answer
333
views
The space of circular triangles?
A circular triangle is a closed simple curve in the Euclidean plane $\mathbb{R}^2$ that can be expressed as the union of three circular arcs. Data naturally associated with such a figure includes:
...
- 3,049
4
votes
1
answer
110
views
An corona pair (a,b) satisfies (A_2)
Let $b$ be a non-extreme point in the unit ball of $H^\infty$. Let $(a,b)$ be a corona pair, that is $|a|+|b|$ is bounded away from zero in the unit disc. Also let $|a|^2$ satisfy Hunt-Muckenhoupt ...
- 103
4
votes
1
answer
1k
views
Books on the Hardy-Littlewood circle method
Are there any good books providing an introduction to the Hardy-Littlewood method that do not require much of a background in complex analysis?
- 1,831
4
votes
1
answer
436
views
Other elliptic curves for $x^4+y^4+z^4 = 1$
Given,
$$a^4+b^4+c^4 = d^4\tag{0}$$
we have the identity,
$$(-11980 + 1673 u + 54u^2)^4 + (36 - 2321 u + 3u^2)^4 + t^4 = (24677 + 203 u + 71u^2)^4$$
where,
$$591800025 + 20030510 u + 1671327 u^2 +...
- 12.1k
4
votes
1
answer
275
views
Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)
Given correlation matrix $B$ (positive semi-definite with ones in the diagonal).
1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$.
2)Find the correlation matrix $A$ which ...
- 41
4
votes
1
answer
447
views
Off-diagonal holomorphic extension of real analytic functions on $\mathbb{C}^n \times\mathbb{C}^n$
I am struggling trying to understand an statement in a paper I am reading:
Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$ $\mathbb{C}$ whose ...
- 601
4
votes
1
answer
521
views
Total variation distance between diffusion processes with different volatility coefficient
Preamble:
This question is similar to the one in total variation distance between two solutions of SDE . The difference is that in my case the drift is the same but there are different diffusion ...
- 315
4
votes
2
answers
410
views
Regarding Ricci curvature of Markov chains
In Ricci curvature of Markov chains on metric spaces Yann Ollivier, defines a coarse Ricci curvature for a Markov chain with transition kernels $\{m_x\}$ defined on a metric space $(X,d)$ as follows: ...
- 1,639
4
votes
1
answer
364
views
Inducing metric spaces
Let $f\colon \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$ be a function. We say that $f$ has the property of inducing metric spaces, whenever for all metric space $(X,d)$, $(X, f \circ d)$ is also a ...
- 243
4
votes
2
answers
606
views
Uniform distribution in (non-compact) locally compact spaces
I'm trying to understand how much of the theory of uniformly distributed sequences in compact spaces can be extended to locally compact spaces.
Following L. Kuipers and H. Niederreiter - Uniform ...
- 41
4
votes
1
answer
413
views
Schanuel's conjecture and abstract elementary classes
Are there any connections between Schanuel's conjecture and abstract elementary classes. More precisely
Question. Is there any conjecture in abstract elementary classes whose truth implies the ...
user39869
4
votes
1
answer
324
views
"Small" subfields of algebraically closed fields
Sufficient background:
Let $\mathcal{M}=(M,...)$ be an $\mathcal{L}$-structure and $X\subset M$.
Definition. $X$ is large if there exists a function $f:\mathcal{M}^n \overset {\leq k} \rightarrow \...
- 147
4
votes
3
answers
271
views
Invariants in $S^n(S^k(\mathbb{C}^w)$
Are the formulas for the multiplicity of $SL(w)$ invariants in $S^n(S^k(\mathbb{C}^w)$ known? This is a very classical topic. If no, in what ranges one can compute it (for certain paramaters fixed - e....
user27328
4
votes
1
answer
163
views
An algebraically independent set of real as a range of an increasing function
Is there an strictly increasing function $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that its image is algebraically independent (over $\Bbb{Q}$) ?
user39070
4
votes
1
answer
318
views
n-simplex in an intersection of n balls
Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
- 195
4
votes
1
answer
245
views
Are the heredity ideals in an heredity chain always finitely generated?
An ideal $J$ of a ring $A$ is a heredity ideal of $A$ if:
$J^2=J$;
$J$ is a projective $A$-module;
$J (\operatorname{Rad}A)J=0$.
A (unitary) semiprimary ring $A$ is said to be quasihereditary if ...
- 425
4
votes
1
answer
710
views
How should we understand the relative interior in Berkovich spaces
I'm reading Berkovich's book on analytic spaces. The notion of relative interior confuses me. Is there anyway to see how it "looks like"? For instance, if $r <1$, what is the relative interior of
\...
- 125
4
votes
1
answer
241
views
dimension of a union of grassmannians
I'm working on a dynamical systems problem and have arrived at a naive question in differential topology? geometric measure theory?
I have a smooth path $\gamma\colon \mathbb R\to\mathcal G_2(\...
- 22.5k
4
votes
2
answers
813
views
p-group with abelian centralizer
I will be so thankful if someone helps me with the following question. There exists finite non-abelian p-groups G (except non-abelian groups of order $p^3$) with the following properties:
all non-...
4
votes
2
answers
1k
views
Algorithm to decide if ideal is principal
Suppose $R = \mathbb{Q}[x_1, ..., x_n]/I$, and $J \subset R$ is a given height one ideal. Is there a quick algorithm one could write to determine if $J$ is a principal ideal or necessarily not ...
- 3,505
4
votes
1
answer
283
views
Who first proved there's an $\omega$-model of $\mathsf{WKL}_0$ in which all sets are low?
I am trying to pin down: who first proved that $\mathsf{WKL}_0$ has an $\omega$-model in which every set is of low degree? As shown in Simpson's Subsystems of Second Order Arithmetic (Theorem IX.2.17),...
- 2,276
4
votes
2
answers
286
views
Problems similar to Borsuk’s Theorem in the plane
Consider a 2-dimensional Borsuk's theorem:
Every bounded set $S$ in the plane can be partitioned into three parts with diameter smaller than the diameter of $S$.
I wonder if there are any results ...
- 181
4
votes
2
answers
314
views
Reference Request: Representing Positive Integers as Differences with Minimal Hamming Weight
Background of my reference request is an observation that I made, while I was still in school: there are two ways to calculate $x*999$: either do it directly, by applying the multiplication algorithm ...
- 12.7k
4
votes
2
answers
2k
views
Least quadratic residue and nonresidue
For a prime $p$ denote by $r(p)$ (resp. $n(p)$) the smallest prime $q$ which is a quadratic residue (resp. nonresidue) modulo $p$.
It was shown by Linnik that for any fixed $\epsilon>0$ the number ...
- 641
4
votes
1
answer
173
views
Number of times a Gaussian process crosses zero in an interval
Using a probabilistic method for number theoretic purposes, I have encountered the following question (it may be very standard):
Let $X_t$ be a Gaussian process $(t>0)$ such that $X_0=0$. What ...
- 2,248
4
votes
1
answer
393
views
Field extension of fields [closed]
Is the field of real numbers $\mathbb{R}$ a finite extension of some subfield $k\subset \mathbb{R}$?
- 169
4
votes
1
answer
609
views
Conductor of abelian varieties
Let $A$ be a non-zero abelian variety defined over a number field $F$. Let $v$ be a finite place of $F$, and let $f_v(A)$ be the usual conductor exponent of $A$ at $v$ (defined e.g. on p.500 of the ...
- 107
4
votes
1
answer
1k
views
An inequality involving operator and trace norms
Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the ...
- 1,639
4
votes
1
answer
456
views
Describing the affine Grassmanian via $G$-bundles on $\mathbb{P}^1$
Let $G$ be a simple algebraic group, $\mathcal{O}=\mathbb{C}[[t]], \mathcal{K}=\mathbb{C}((t))$ and let $\text{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$ be the affine Grassmanian. My main question:
Why ...
- 2,186
4
votes
1
answer
265
views
Replacement axiom and existence of end-extensions (Chang and Keisler's Model Theory)
I am currently reading Chang and Keisler's model theory textbook, and there is something that I don't seem to be able to understand about their proof (through the Omitting Types Theorem) that every ...
- 43
4
votes
1
answer
468
views
Isomorphisms between topological vector spaces [closed]
Let $f : X \to Y$ be a continuous map of complete topological vector spaces. Suppose that $A \subseteq X$ and $B \subseteq Y$ are proper, dense linear subspaces, and that the restriction map $f : A \...
- 8,392
4
votes
2
answers
411
views
Is it impossible for the dimension of a topological space to increase under a smooth map?
First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...
- 3,235
4
votes
1
answer
325
views
To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function
I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...
- 672
4
votes
1
answer
386
views
locally free sheaves and logarithmic connections
I know that, when $\mathcal{F}$ is a coherent sheaf on a smooth algebraic variety $X$ over a field $k$ equipped with a connection
$$
\nabla: \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X,
$$ then $\...
- 41
4
votes
1
answer
233
views
On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument
Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator ...
- 363
4
votes
1
answer
254
views
Manifold of immersions of a manifold
Let M be a Riemannian manifold, S a closed immersed submanifold of M, and consider the space of smooth perturbations, i.e. functions X : S → NS (the normal bundle of S in M), p → (p, Xp) for some Xp ∈...
- 3,512
4
votes
2
answers
453
views
lie algebras, Kac Moody, and quantum mechanics book
Hi all, I've just finished a graduated course on Kac-Moody algebras, and I'm really looking for some reading in regard to their applications to Quantum Mechanics. Can you help?
- 377
4
votes
1
answer
196
views
Covering a convex body with its smaller homothetic copies
Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...
- 285
4
votes
1
answer
287
views
Set of small numbers with distinct $k$-sums
Let $A$ be a set of $n$ positive integers with distinct $k$-sums. In other words, if $a_1\le\cdots\le a_k$ and $b_1\le\cdots\le b_k$ are elements of $A$ such that $a_1+\cdots+a_k=b_1+\cdots+b_k$, then ...
- 41