Questions tagged [existence-theorems]

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45 views

The existence of a copy of a random variable with conditional expectation constraint

Let there be two random variables ­ŁĹő and ­ŁĹî with a certain joint copula. Is it always true that there is another random variable ­ŁĹŹ independent from ­ŁĹî such as the vectors $(X,Y)$ and $(X,Z)$ have the ...
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0answers
107 views

continuity with respect to the diffusion coefficient of the solution of a semilinear parabolic equation

Let $\Omega \subset %TCIMACRO{\U{211d} } %BeginExpansion \mathbb{R} %EndExpansion ^{n},n\geq 1$ be an open bounded subset has a boundary $\Gamma $ of class $% C^{2}$, $Q=\Omega \times \left( 0,T\...
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3answers
1k views

Simplest diophantine equation with open solvability

What is the simplest diophantine equation for which we (collectively) don't know whether it has any solutions? I'm aware of many simple ones where we don't know (whether we know) all the solutions, ...
-1
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1answer
89 views

(maximal) antichains with respect to two different partial orders on the same set

In my recent work I stumbled across a problem of this type: G with two partial oders $\leq$ and $\preceq$ on every set, i.e. for every $n \in \mathbb{N}$ $A_n \subset A_{n+1}$ and $(A_n, \leq) $ and $...
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0answers
165 views

Do functions $f: \mathbb R \to \mathbb R$ with these properties exist?

Basically, I am trying to determine how exactly and in which ways everywhere discontinuous bijections $f: \mathbb R \to \mathbb R$ "behave when they are unusual". More precisely, I am trying to ...
4
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0answers
117 views

Maximum Principles in Parabolic PDE with Neumann Condition

I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
4
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1answer
223 views

Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$

Let $f: \mathbb R \rightarrow (-\infty,+\infty]$ be a lower-semicontinuous convex function. Question Under what futher conditions does there exists a convex decreasing function $\phi: \mathbb R \...
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0answers
98 views

Why should we give special attention to at most polynomially growing solutions of PDEs?

The equation \begin{gather} \frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\ u(0,x) = \...
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0answers
132 views

Interesting geometric flow of space curves with non-vanishing torsion

Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $\mathbb{R}^3$ given by \begin{equation} \partial_t \gamma = \tau^{-\frac{1}{2}} n, \end{...
1
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1answer
98 views

Lotka Volterra existence of Caratheodory solution

I strive to prove that the following system of differential equations: $$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$ has a unique Caratheodory solution ...
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0answers
84 views

Reference for Existence and uniqueness of an Integro-Differential Equation

I have an Integro-Differential Equation (IDE) of the following form: $$ x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds, $$ I have found this classical reference, but the IDEs considered therein ...
4
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1answer
502 views

Existence of solutions to first-order PDE involving convolution

Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation, \begin{align} \frac{\partial}{\partial \alpha} ...
17
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2answers
2k views

Is there a relationship between a quotient group of the fundamental group of X and the fundamental group of a quotient topology of X?

Let ($X$, $x_0$) be a topological space with a base point, and denote the fundamental group of $X$ as $\pi_1(X)$. Let $N$ be a normal subgroup of $\pi_1(X)$. Does there necessarily exist an ...
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0answers
136 views

Theory on interior Helmholtz Equation with mixed Neumann and Robin BC

Let $\Omega \subset \mathbb{R}^n$ bounded with smooth boundary $\partial \Omega$. Let k>0 and $\partial \Omega = \Gamma_1 \cup \Gamma_2$. Consider the problem $\Delta u + k^2 u = 0$ in $\Omega$ $\...
5
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1answer
267 views

Finding a semi-sparse vertex in a grid

Let $H$ be a $r \times r$ grid. Suppose that at most $r/10^5$ vertices of this grid are colored red. For every vertex $v \in V(H)$, let $B_i(v)$ be the ball of radius $i$ centered at $v$. (Or for ...
4
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0answers
80 views

Existence and uniqueness of an elliptic equation coupled with a parabolic equation (mean curvature flow)

Given a parabolic equation on a simply connected smooth domain $\Omega(t)$ with boundary $\Gamma(t)$ $$ \Delta u = 1 \quad on \quad \Omega(t) \\ \nabla u \cdot n + u = g \quad on \quad \Gamma(t) $$ (...
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0answers
246 views

Existence and uniqueness of solution for nonlinear system

Under what conditions will a solution $y \in \Re^n$ exist for this system of nonlinear equations? If it exists, will it be unique? $$ \mu_k = \int_0^\infty g_k(x) \, f(x, y) \, dx \qquad \forall \, k ...
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0answers
69 views

Existence of a shift invariant selection map

Some time ago I asked this question, but now realise that it is harder than I anticipated. Therefore I am taking a step back to the following problem. Let $X$ and $Y$ be two sets and $F$ a point to ...
1
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1answer
102 views

Existence of analysis regularization solution

I am interested in the optimization problem known as "analysis regularization": $$ {\rm argmin}_{x \in \mathbb{R}^{p}}\frac{1}{2}\|y - Ax\|_2^2 + \lambda \|D^T x\|_1,$$ where $y \in \mathbb{R}^n$, $...
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2answers
232 views

Source of equation - theorems about solving quadratic matrix equations

I have seen outlined in this comment os mathoverflow how to solve quadratic matrix equations of the form $$ XCX + AX = I $$ where $X \in \mathbb{R}^{n\times n}$, $C = C^T > 0 \in \mathbb{R}^{n\...
2
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0answers
98 views

Positive existential theory of $(\mathbb{Z}; +, |_n)$

I am reading a paper and there is the following theorem: Let $n$ be a fixed integer, and $n >1$. Denote divisibility in $\mathbb{Z}[\frac{1}{n}]$ by $|_n$, thus for all $x, y \in \...
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0answers
128 views

Probabilistic proof for expander existence [closed]

I am new to probabilistic proofs and trying to understand them better. Apparently, a common probabilistic proof focuses on the existence of expanders (eg. vertex expanders). I've been using the search ...
5
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0answers
227 views

When does a “stable” assignment of buyers into goods exist?

Consider a setting of $n$ buyers and $m$ goods. We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...
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0answers
143 views

Does there exist this special kind of homeomorphism?

Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
2
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2answers
178 views

$L^\infty_\mathrm{loc}$ assumption in global existence for Boltzmann equation

In short: In P. G├ęrard's paper on the existence of global solutions to the Boltzmann equation from 1988 (or equivalently Cercignani's book), why are the stated assumptions (especially $A_n \in L^\...
2
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1answer
589 views

Global Solutions of Ordinary Differential Equations

Background Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying, $f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$, for every ...
5
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0answers
82 views

reference on existence result for nonlinear elliptic PDE

During my work, I came to the question of existence of weak solutions to the following elliptic equation $$\triangle u + \partial_{1} u + \partial_2(f(x_1,x_2,u)) = 0 \hbox{ in } \Omega$$ with ...
2
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0answers
256 views

Existence and uniqueness of heteroclinic orbits

I am looking for conditions on a nonlinear dynamical system $\frac{d\vec{x}}{dt} = \vec{F}(\vec{x})$ that guarantee the existence of a unique heteroclinic orbit between a stable attractor of this ...
22
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12answers
2k views

Instances where an existence result precedes the constructive version

The basic motivation here is to encourage and inspire - via examples - the pursuit of alternate proofs of existing results that might be more accessible and intuitive by cataloging success stories. ...
4
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2answers
789 views

Are there cusp forms for the full modular group Sp(2,Z) and representations det^3 \otimes Sym^2j(\rho_standard)

What are modular forms or cusps forms, resp. ? We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts. The ...
9
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3answers
1k views

Existence of Rational Orthogonal Matrices

Question: Let $A\in\mathbb{R}^{n \times n}$ be an orthogonal matrix and let $\varepsilon>0$. Then does there exist a rational orthogonal matrix $B\in\mathbb{R}^{n \times n}$ such that $\|A-B\|<\...
0
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1answer
484 views

Continuous variation from solution of easy problem to solution of hard problem

I asked this question a week ago over on math.stackexchange and got no reply, so I am asking here with slightly different wording. I am trying to prove that there exists a solution to a problem. I ...
21
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3answers
2k views

Is this 1974 claim still valid?

In G. F. Simmons' Differential Equations book (p.141), the following claim is made: ÔÇť... As a matter of fact, there is no known type of second order linear differential equation- apart from those with ...