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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 ...
Raz Kupferman's user avatar
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 $ux \in R$. ...
Avi Steiner's user avatar
  • 3,031
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 ...
Wenzhe's user avatar
  • 2,961
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 ...
Divakaran Divakaran's user avatar
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 ...
Alex Mennen's user avatar
  • 2,090
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 ...
Stephan Mescher's user avatar
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 ...
pw1's user avatar
  • 144
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 ...
Alexander Block's user avatar
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\...
Dima Pasechnik's user avatar
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 ...
ABIM's user avatar
  • 5,019
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 (...
Qfwfq's user avatar
  • 22.7k
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
Jacob's user avatar
  • 17
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 $\...
Ron's user avatar
  • 2,116
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 ...
Gro-Tsen's user avatar
  • 29.9k
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 ...
Gergely's user avatar
  • 291
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: $$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$ ...
TOM's user avatar
  • 2,218
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 ...
O-V-E-R-H-E-A-T's user avatar
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 ...
W. Nyway's user avatar
  • 135
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 $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
Dima Pasechnik's user avatar
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 ...
user1337's user avatar
  • 463
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 ...
Janak's user avatar
  • 213
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 ...
Dominic van der Zypen's user avatar
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 ...
Sidharth Ghoshal's user avatar
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}+ ...
Sebastian Naranjo Alvarez's user avatar
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 ...
stupid_question_bot's user avatar
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 $$ ...
user95770's user avatar
  • 143
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 $L^2$-...
Fetchinson0234's user avatar
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 ...
Rohil Prasad's user avatar
  • 1,591
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 $\Delta$ ...
D_S's user avatar
  • 6,100
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 ...
Michael Hardy's user avatar
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 ...
mayer_vietoris's user avatar
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 ...
symmetry 's user avatar
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 ...
Hannes's user avatar
  • 2,175
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 (...
Julian Newman's user avatar
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 ...
Wolker's user avatar
  • 541
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 ...
swati setia's user avatar
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}...
Pedro G.'s user avatar
  • 111
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(...
Majid's user avatar
  • 227
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 ...
Mikhail Katz's user avatar
  • 15.1k
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 $...
Joseph Doob's user avatar
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 ...
Najmeh Dehghani's user avatar
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 ...
stevestark's user avatar
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 $V(N)$ ...
WhatsUp's user avatar
  • 3,232
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 ...
Ian's user avatar
  • 315
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 $\prod$ ...
Daisy's user avatar
  • 338
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 ...
user111368's user avatar
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 ...
aglearner's user avatar
  • 14k
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})...
A. Maarefparvar's user avatar
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 [...
Julian Newman's user avatar
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 $$...
mnmn1993's user avatar

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