Questions tagged [uniqueness-theorems]
The uniqueness-theorems tag has no usage guidance.
41 questions
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Uniqueness of bubbling points in Struwe's global compactness theorem
I am reading the following paper of Struwe in which he proves the following result:
Proposition 2.1:
Let $n\geq 3$, $\lambda \in \mathbb{R}$ and $\Omega$ be a smoothly bounded domain in $\mathbb{R}^{n}...
- 547
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Reduction of the $0$-handle data in Lurie's classification of TFT
A vital part of Jacob Lurie's classification of fully extended topological
field theories [1], very roughly, says that any representation of the
n-Cobordism category $Z: {\rm Cob}_{{n}} \to C$ depends ...
- 5,230
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Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
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205
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Uniqueness for Volterra equation with initially (linearly) unbounded kernel
Letting $D:=\{(x,y):\ 0\leq x\leq y\leq1 \text{ and } y>0\}$, I have a continuous function $k:D\to[0,\infty)$ that satisfies some properties that I list below. I'm interested in continuous and ...
- 537
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46
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Uniqueness results for linear first order systems of PDEs
Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$):
$$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
- 1,205
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Reference for (general case) of uniqueness of singular value decomposition (SVD)
My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values.
I believe that the statements and proofs on this StackExchange posts are ...
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Uniqueness of the "algebraic closure" of a commutative ring
There are several ways to generalize the notion of "algebraic closure" from fields to arbitrary commutative rings. A good overview is On algebraic closures by R. Raphael. I am more ...
- 63.1k
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116
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Existence and uniqueness of a posterior distribution
I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...
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243
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Existence and uniqueness of solutions for continuous and directionally differentiable ODE
Given $f:\mathbb{R}^n \to \mathbb{R}^n$ continuous and directionally differentiable (i.e., such that the directional derivative of $f$ exists for any direction) at a neighborhood $N$ of $x_0\in\mathbb{...
- 63
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Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
- 1,075
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154
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Uniqueness of the solution to systems of first-order linear PDEs
Context:
Let $\Omega \subset \mathbb{R}^p$ be an domain.
For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
- 51
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Definitions of determinant by unique features
A well-known definition of the determinant is:
The determinant is the only function of a vector space of dimension $n$ to its underlying field which is multilinear, alternating and normalized.
See e....
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1
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314
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Question on possibility of uniquely defining the FRFT via certain properties
I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ...
- 286
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133
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Infinite dimensional matrix solvability
In order to solve a boundary problem for a conducting hemisphere, the following matrix equation arises (derived from the boundary condition on the curved part of the surface) where we must solve for ...
- 573
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Uniqueness of a solution to an equation
Let $t \in [0,1]$ or $t \in (0,1)$ be distributed according to $F(t)$.
Now consider the following equation:
\begin{equation}
\frac{\int_{\underline{t}}^{\overline{t}}(\gamma-t(2\gamma-1))dF(t)}{\int_{...
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2
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136
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Uniqueness of a second order linear ode
I am currently confronted with the following equation $$
0=w''(t)(t^2-t)+w'(t)((2n-1)t^2-n)+w(t)(n-1)^2t
$$ for $t\in(-1,1)$. So $w:(-1,1)\rightarrow\mathbb{R}$. The following assumption is also in ...
- 21
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3
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2k
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Uniqueness of solution to heat equation when initial condition is a generalized function
Let $u(t,x)$ be a solution to the heat equation $$\partial_t = \partial_{xx} \quad (t,x) \in [0,T) \times [-1,1]$$ subject to the initial/boundary conditions
$$u(0,x) = f(x), \quad x \in [-1,1], \\
u(...
- 103
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Uniqueness continuation property for parabolic equation
Consider the following parabolic equation:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial \rho }{\partial t}-\Div\left( a\left( x\right) \nabla
\rho \right) +p(x)\rho = 0 & \...
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Proving the uniqueness of the solution to a functional equation involving integral
Consider the functional equation
$$
g\left(a\right) = \int_0^1 \frac{e^{c(a,h)+f(h)}}{1+e^{c(a,h)+f(h)}}dh
$$
and this holds for all $a$. $g(a)$ and $c(a,h)$ are known functions on a continuous ...
7
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Looking for an electronic copy of Holmgren's old paper
I would like to know if anyone has an electronic copy of the following paper:
"Holmgren, E.: Über Systeme von linearen partiellen Differentialgleichungen. Översigt Vetensk. Akad. Handlingar 58, ...
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Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid?
This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold:
https://arxiv.org/abs/1709.10033
What's the current ...
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Uniqueness of global solution
I am reading Section 3.3 of this paper, and trying to understand the proof of uniqueness of a global solution to the following equation defined on the Torus $\mathbb{T}^3$
\begin{align*}
\mathrm{d} \...
- 101
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494
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Fokker-Planck: uniqueness and convergence to stationary distribution
Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
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2
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376
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Is the converse of Osgood criterion for ODEs also true?
Namely, Assuming that $f$ is a continuous real function and $f(0)=0$ , $f(x)>0 $ when $x\neq 0$,
Consider the differential equation $x'= f(x)$ with the initial value $x(0)=0$ , is it true that if ...
- 141
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736
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Proof: If a reproducing kernel exists for a Hilbert space, then it is unique
I really want to prove the statement in the title but I'm struggling with it. Here my current state:
Proof via contradiction. Let $\mathcal{H}$ be a RKHS with two reproducing kernels $k$ and $\hat{k}$ ...
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3
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160
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Are <sum, product, N> triplets unique and hard to solve? [closed]
This question comes from some reasoning I made myself about a "joke block chain" where every new block is labeled with a triplet <S, P, N> where where S = sum of the N transactions so ...
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3
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1
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309
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A simple question on the Navier-Stokes system
The Navier-Stokes system for incompressible fluids in $\mathbb R^3$
reads as
\begin{align}
&\frac{\partial v}{\partial t}+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0, \quad \text{div} v=...
- 16.2k
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What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?
I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the ...
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313
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Some doubts on proof of pathwise uniqueness of a stochastic differential equation
I quote a paper from Delbaen and Shirakawa (2002). I will write in italics my observations/questions.
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\...
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133
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Implicit function theorem when $dF/dy = 0$ but under monotonicity constraint of the implicit function $y(x)$
I am looking for an extended version of the implicit/inverse function theorem that would show uniqueness of a strictly increasing implicit function, even when the derivative condition is violated (e.g....
- 151
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41
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Existence and Uniqueness of lifting Hele-Shaw problem
I am researching for the existence and uniqueness of solutions for the equation in figure below
enter image description here
$$\nabla\cdot u = \frac{\dot b(t)}{b(t)} \text{ in }\Omega(t) \tag{1}$$
The ...
2
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1
answer
179
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Unique continuation of the Hilbert transform
Let's consider the usual Hilbert transform $H$ defined as
$$Hf = P.V. (\frac{1}{x}*f).$$
A well-known unique continuation principle states that if $Hf = f =0$ on some interval $I$, then $f \equiv 0$. ...
- 903
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1
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937
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Existence and uniqueness of geodesics in low regularity
Consider a Riemannian manifold $(M,g)$.
How much regularity is required of $g$ so that for any $x\in M$ and $v\in T_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to ...
- 8,086
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If a PDE has a unique classical solution, must it have a unique viscosity solution?
If a PDE has a unique classical solution, must it have a unique viscosity solution?
The particular problem I am interested in is parabolic, but I would be interested in the general case.
A short ...
- 383
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Reference request: existence/uniqueness of solutions to convection diffusion equations
I am looking for a reference wherein existence and uniqueness results are proven for a system of PDEs of the form
$$
\frac{\partial Q}{\partial t} + A \frac{\partial Q}{\partial x} = f(Q,x,t) + \frac{...
- 111
3
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Uniqueness of minimizers in the Calculus of Variations
Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \...
- 391
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Uniqueness of function with range $\mathbb{S}^2$ under a constraint
Assume $g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2$ are two bijective functions defined on the set $A$. Now assume a constraint $C$: $\forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\...
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1
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118
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Reference request: uniqueness for a certain PDE systems
I'm working on a system of the following form:
$$(1) \,\,\,\,\,\ \begin{cases} u_{tt} + L_1u + L_2v= f, \\ \nabla u - \nabla v - \nabla v_t=0 \end{cases} $$
where $u(x,t)$ and $v(x,t)$ belong to ...
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Lotka Volterra existence of Caratheodory solution
I strive to prove that the following system of differential equations:
$$\begin{cases} x'=x-u(t)xy\\ y'= -y+u(t)xy \\ x(0)=x_0>0\\ y(0)=y_0>0 \end{cases}$$
has a unique Caratheodory solution ...
- 1,759
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122
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Ricci flow on locally symmetric noncompact manifold
As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
- 123
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2
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119
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Uniqueness problem for an elliptic system
I want to prove the uniqueness of the solution of the following problem:
$$\eqalign{
& - d\,\Delta u + u = {u^p} \text{ in } \Omega \cr
& u > 0 \text{ in } \Omega \cr
& \...
- 617