# Questions tagged [uniqueness-theorems]

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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 ...
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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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### 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 ...
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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 ...
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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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### 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 ...
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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 ...
127 views

### 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 $f \equiv 0$. ...
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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 ...
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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 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 & \...
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 ...