All Questions
152,887
questions
4
votes
0
answers
89
views
Regularity of weak solutions of an equation related to conformal maps
Let $\Omega\subset R^d$ be a nice bounded domain (say, the unit ball). Consider the following equation for $f:\Omega\to R^d$,
$$
\operatorname{div}((\det \nabla f)^{1-2/d} \nabla f) = 0.
$$
Suppose ...
0
votes
1
answer
111
views
Adic filtration and integral closure
Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.
Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. ...
3
votes
0
answers
313
views
Nearby Cycle Functor and the Limit of a Variation of Hodge Structures
I am reading Ayoub's paper, The Motivic Nearby Cycles and the Conservation Conjecture,
http://user.math.uzh.ch/ayoub/PDF-Files/Leiden.pdf
In section 2.3, he talks a little about the limit of a ...
6
votes
0
answers
216
views
Reference request: Complex geodesic flow
Can someone suggest a book on complex geodesic flow? I am interested in it mainly because I was told these form a very useful class of Riemann surface laminations. Of special interest to me is the ...
5
votes
0
answers
80
views
One-sided version of the random oracle hypothesis
The random oracle hypothesis states that relationships between complexity classes hold iff they hold relativized to a random oracle with probability 1. This is false, but all of the counterexamples I ...
16
votes
3
answers
977
views
Cohomological dimension of $G \times G$
$\DeclareMathOperator\cd{cd}$A question that I have already posted in the Mathematics section, but which seems to be too delicate for that section (see here and here):
Let $\cd(G)$ denote the ...
1
vote
0
answers
160
views
Rigid analytic reductions of the projective line
I'm reading the book "Rigid Analytic Geometry and its Applications" by Fresnel-van der Put, and I'm confused by their example 4.8.5. In the first two parts of the example, they define the analytic ...
19
votes
1
answer
463
views
Is every transitive ZF-model of inaccessible height a truncation of an inner model?
Let $\kappa$ be an inaccessible cardinal and let $M \subseteq V_{\kappa}$ be an inner model of $V_{\kappa}$, i.e., a transitive model of $\mathsf{ZF}$ containing all the ordinals up to $\kappa$.
My ...
7
votes
1
answer
326
views
2x2-determinantal representations of cubic curves
Let $H_d:=\mathbb{C}_d[x,y,z]$ denote the space of homogeneous degree $d$ polynomials in $x$, $y$, $z$ with complex coefficients. I'd like to show that every $f\in H_3$ can be represented as
$$
f=\det\...
1
vote
0
answers
75
views
Stochastic Control with Stochastic Cost-functional
Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also?
That is, let $X_t^u$ is the solution to a controlled SDE
$$
dX_t = \mu(t,u_t,X_t^u)dt ...
3
votes
0
answers
90
views
Decompositions from torus actions and compactness of (sub-)level sets
Let $T=\mathbb{C}^{*}$ act on a smooth complex quasi-projective variety $X$. Assume that the limit point $\lim_{t\to 0}t\cdot x$ exists for every $x\in X$.
From the induced $U(1)$-action and its (...
0
votes
1
answer
366
views
Chromatic polynomial for hyper cube [closed]
Does anyone know the chromatic polynomial of the hyper cube graph Q4?
I need this to verify that my listing of a subset of all DAG's on the 4-cube is correct.
Any help greatly appreciated,
JC
4
votes
0
answers
329
views
Is complete intersection a open or closed property in Hilbert schemes
Fix an integer $N$, $X$ a (smooth) complete intersection subvariety in $\mathbb{P}^N$. Denote by $P$ the Hilbert polynomial of $X$ (as a subvariety in $\mathbb{P}^N$). Consider the Hilbert scheme $\...
26
votes
1
answer
798
views
What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?
The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
13
votes
2
answers
781
views
Category theorists stance on deductive systems
Lambek & Scott: Introduction to higher order categorical logic
says that
"For example, categorists may be unhappy when we treat categories as special kinds of deductive systems and logicians ...
14
votes
3
answers
1k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
3
votes
1
answer
563
views
Jacobi and Weierstrass elliptic function
Jacobi elliptic function $\mathrm{sn}$ is defined as
$$\operatorname{sn}(u,k)=x\Leftrightarrow u=\int_0^x \frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.$$
and Weierstrass sigma function $\sigma$ is defined as
...
0
votes
1
answer
148
views
Does there exist a comparison principle for vector differential equations?
As is well known, there is a comparison principle for scalar differential equations $\dot x(t)=a(x(t))$ and $\dot y(t)=b(y(t))$ with $x(t_0)=y(t_0)$ and $a(\cdot)\leq b(\cdot)$, with the result being ...
8
votes
1
answer
697
views
A direct proof of a property of symmetric 2x2-determinants
Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix.
Let $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
1
vote
0
answers
182
views
Puiseux's theorem's converse
Puiseux's theorem asserts that
given a polynomial equation $P(x,y)=0$, its solutions in $y$, viewed as functions of $x$, may be expanded as Puiseux series that are convergent in some neighbourhood of ...
3
votes
1
answer
510
views
Weighted sum of standard Brownian bridges
Let $\{B_j\}_{j=1}^k$ be a sequence of Brownian bridges.
Let us consider $$X(t)=\sum_{j=1}^m w_j(t)B_j(t),$$ where $w_j$ are positive weight functions.
Then what can we say about (distribution or may ...
2
votes
1
answer
104
views
Limit of biggest share of the pie
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest ...
6
votes
1
answer
309
views
Creating an additive structure over the set of all finite groups?
I'm trying to form a ring (or ring-like structure) out of the set of all finite groups.
Has anyone created/encountered an operation "+" before with the follow properties.
Let $G_1, G_2$ be finite ...
1
vote
0
answers
83
views
Diffraction across an absorbing wall
I need help finding the procedure for the solution of the following differential equation.
This is equation is: Find $u:\mathbb{R}^2 \to \mathbb{C}$ such that for $C>0$
$ \begin{cases} u_{xx}+ ...
5
votes
1
answer
250
views
curious relation between orders of generators of a finite group
Let $G$ be a finite group, and $g,h\in G$ be such that $G = \langle g,h\rangle = \langle g,hgh^{-1}\rangle$ - that is, $g,h$ generate $G$, and so does $g,hgh^{-1}$.
Let $n := |G|$, $r := |g|$, and $e ...
11
votes
1
answer
290
views
Conjugation action on the classifying space of circle
Let $T$ denote the unit sphere in complex plane. $\mathbb{Z}/2$ acts by complex conjugation on $T$. There is an induced action on $BT$. The cohomology of the homotopy orbit space of this action is
$$
...
4
votes
3
answers
258
views
Iterative matrix inversion with $L^\infty$ norm
The usual conjugate gradient type algorithms for iteratively finding the inverse of a matrix applied to a vector, $x = A^{-1} y$, works by minimizing $\|Ax - y\|^2$ where $\| \cdot \|$ is the $L^2$-...
4
votes
0
answers
199
views
Self-Dual Equations Elliptic Complex
I'm posting this on MathOverflow because I'm not sure if it would get much of a response on Math.SE. Please feel free to remove it if it is not "research-level".
I was studying Donaldson's self-dual ...
5
votes
2
answers
392
views
Reference Request: Derived group of $\mathscr R_u(B)$
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ ...
1
vote
0
answers
220
views
Conditional convergence and rearrangements
Is the concept of conditional convergence of series whose terms are real numbers a topic of research in analysis or merely something to be aware of?
The question is prompted by the conjunction of my ...
4
votes
2
answers
279
views
Cohomology Ring of a small category $\mathsf{C}$
Assume that $\mathcal{C}$ is a small category and that $\mathcal{F} \in \mathsf{ob(Ab^{\mathsf{C}})}$, is a covariant functor. When our category has finitely many objects then a classical theorem from ...
4
votes
1
answer
510
views
compact objects and derived categories
Sorry in advance if my question does not have the required level.
Let $K$ be a commutative ring (let say an integral domain for simplicity) and let $0\neq s\in K$. Let $K[s^{-1}]$ be the localized ...
7
votes
2
answers
507
views
"Reversion" of class $J(\theta)$ interpolation property for Besov spaces
In (function space) interpolation theory, a Banach space $E$ is of class $J(\theta)$ (for $0 < \theta < 1$) if $$X \cap Y \subseteq E \subseteq X+Y,$$ where $(X,Y)$ are Banach spaces and form an ...
4
votes
0
answers
213
views
Does it make sense to regard the graph of any function as being a "sort-of-null set"?
Following the nice answer to Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"?, the particular situation that I am especially interested in (...
1
vote
0
answers
107
views
Characters of a quadratic extension and convergence
Let $F$ be a non-archimedean local field, $\chi$ a quasi-character of $F^\star$ and $\psi$ a positive character of $E^\star$. I would like to understand why the usual Rankin-Selberg zeta integrals ...
2
votes
0
answers
176
views
Lattice Sieving in Number Field Sieve
I am currently going through Pollard's article on Lattice Sieving and have a few confusions. Firstly, how to figure that $C$ and $D$ in the two-dimensional array so that every $(c,d)$ pair corresponds ...
1
vote
0
answers
611
views
Bound of the eigenvalues of a matrix product of two diagonal and one symmetric PSD matrices
Let B be a positive diagonal matrix, and C a PSD symmetric matrix with eigenvalues $\Lambda$, with $\lambda_i = {0,1}$. I'm trying to find a reference for the following bounds or similar
$B^{1\over 2}...
0
votes
2
answers
149
views
How to prove that a subset in a vector space is an indicatrix?
Suppose that $\pi:(V_1,F_1)\to V_2$ is a linear surjective map, where $V_1$ and $V_2$ are vector spaces and $F_1$ is a Minkowski norm on $V_1$. Let $B_1$ be the unitary ball on $V_1$. Define $B_2:=\pi(...
8
votes
1
answer
271
views
Can Alexandrov surfaces of CAT(0) type be approximated by CAT(0) polyhedra?
The theory of such surfaces goes back to the book by Alexandrov and Zalgaller (1967 English translation) and from a more analytic viewpoint, work by Reshetnyak where everything is translated into ...
5
votes
0
answers
162
views
Zero Range Process: Construction of Feller process by the generator
A Zero Range Process is the Feller process with generator
$Af(\eta)=\sum_{x,y} p(y-x)f(\eta(x)) (f(\eta^{x,y}-f(\eta))$
where $\eta\in \mathbb{N}_0^S$ denotes a configuration of particles at sites $...
3
votes
1
answer
203
views
When the endomorphism ring of the injective envelope of any simple module is division ring?
I have a question. I will be thank you if you give me some hints.
This is the question:
Let $R$ be a ring and for any simple R-module T, we have End(E(T)) is isomorphic to End(T) which is a division ...
2
votes
1
answer
237
views
How to encode subgraphs as hyperedges
Hi i was reading a paper "Propagating Distributions on a Hypergraph by Dual Information Regularization by Koji Tsuda", and one section stood out to me.
hypergraphs have more flexibility
in ...
9
votes
2
answers
929
views
Average size of extreme points of convex hull of $N$ points
Fix $n$ a (small) integer.
Let $N$ be a (big) integer. Consider $N$ random points in the $n$-dimensional unit cube $[0, 1]^n$. The $N$ points are independently uniformly distributed.
Define $V(N)$ ...
4
votes
0
answers
178
views
Construct Lyapunov-Foster function given invariant distribution
Consider a discrete time Markov chain on a countable state space which is irreducible, aperiodic, and has a given invariant distribution $\pi$. Then the chain is necessarily positive recurrent and ...
2
votes
0
answers
67
views
Question about the mutation of a cluster seed associated to any word of the braid semigroup
Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ ...
2
votes
1
answer
199
views
Alternative notation for Kleene star
I am writing a paper which use two different operations on sets of works $X$, both of which I want to denote by a star, $X^{\ast}$. One of these operations is the Kleene star, and for whatever reason ...
5
votes
1
answer
167
views
Resolving $\mathbb Z_n$ action on $\mathbb C^2$
Consider a diagonal action of $\mathbb Z_n$ on $\mathbb C^2$ generated by $(z_1,z_2)\to (\mu^pz_1,\mu^qz_2)$, with $\mu^n=1$.
Question. Is it always possible to find a smooth blow up $X\to \mathbb ...
4
votes
0
answers
222
views
Computation Hasse unit index for biquadratic fields
For a totally real (resp. imaginary) biquadratic number field $K$ with quadratic subfields $K_1$, $K_2$ and $K_3$, is there an explicit method to determine Hasse unit index $(U_K:U_{K_1}U_{K_2}U_{K_3})...
9
votes
1
answer
849
views
Do the Lebesgue-null sets cover "all the sets can naturally be regarded as sort-of-null sets"?
Let $F$ be the set of bijective Borel-measurable functions $f \colon [0,1] \to [0,1]$ that preserve the Lebesgue measure.
Is it the case that for every non-Lebesgue-measurable set $A \subset [...
4
votes
1
answer
142
views
The order of the solution of Liouville equations at singularity
If I consider a Liouville equations in the unit disk $D \setminus\{0\} \subset \mathbb{R}^2$ with singularity at $x=0$,
$$\Delta u= e^{2u}$$
If I define the order of $u$ at origin is defined to be
$$...