32 votes

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

This is not always possible. (This answer was worked out in collaboration with Raymond Cheng.) Example. Let $X$ be the pseudocircle: it is the finite quotient of the unit circle $S^1 \subseteq \...
R. van Dobben de Bruyn's user avatar
22 votes
Accepted

Simplest diophantine equation with open solvability

Determining which integers $n$ are a sum of three cubes is a very famous open problem: $$a^3 + b^3 + c^3 = n, \quad a,b,c \in \mathbb{Z}.$$ Conjecturally, $n$ is a sum of three cubes iff $n \not \...
Daniel Loughran's user avatar
13 votes

Simplest diophantine equation with open solvability

It's more complicated than the other answers by MattF and DanielLoughran, but the Erdős–Straus conjecture states that for every integer $n \ge 2$, there exist positive integers $x, y, z$ such that $$\...
Glorfindel's user avatar
  • 2,743
13 votes
Accepted

Is the set of real matrices with at least one real logarithm closed under multiplication?

This is already not true for $2$-by-$2$ matrices: Consider $$ A = \begin{pmatrix}2 & 0 \\0 &\frac12\end{pmatrix}\quad \text{and}\quad B = \begin{pmatrix}-1 & 0 \\0 &-1\end{pmatrix}. $$...
Robert Bryant's user avatar
7 votes

Existence of solutions to first-order PDE involving convolution

Edit: after posting the answer below, I noticed the question requires the study of the operator $$ \int f(y,\alpha)f(y-x,\alpha)\mathrm{d}y\quad\text{ which is not }\quad \int f(y,\alpha)f(x-y,\alpha)\...
Daniele Tampieri's user avatar
4 votes
Accepted

Existence and uniqueness of solutions for continuous and directionally differentiable ODE

No. Consider $x'=g(x,y)$, $y'=h(x,y)$. If we take $g(x,y)=2|x|^{1/2}$ for $y=x^2$ similarly $h=4|x|^{3/2}$ on $y=x^2$, then we can check directly that $x=t^2$, $y=t^4$, $t\ge 0$, and $x=y=0$ are ...
Christian Remling's user avatar
4 votes
Accepted

What is the most general Carathéodory-type global existence theorem?

(N.B. In the below I assume $[a,b] = [0,\infty]$, but the precise values don't matter and appropriate substitutions of $a,b$ into the discussion also gives you the same conclusion.) Once you have a ...
Willie Wong's user avatar
  • 37.4k
3 votes
Accepted

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

No, suppose $X=Y$ a.s. and that they are non-degenerate. If we want $(X, Z)$ to have the same joint distribution as $(X, Y)$, we must also have $X=Z$ a.s. and hence $Y=Z$ a.s. Then $Y$ and $Z$ can ...
Dasherman's user avatar
  • 203
3 votes
Accepted

Lotka Volterra existence of Caratheodory solution

Assuming that $u$ is Lebesgue integrable, $f$ does satisfy a Lipschitz-like condition, so we have (local) existence and uniqueness theorem. Whatever the controls, the sets $\{(0,0)\}$, $\{\, (x, 0): ...
user539887's user avatar
3 votes

Source of equation - theorems about solving quadratic matrix equations

Multiplying by $C$ from the right, the equation is reduced to $$Y^2 + AY - C = O,$$ where $Y=XC$. Solution to such polynomial matrix equations is described in Chapter VIII in F.R.Gantmachers. The ...
Max Alekseyev's user avatar
3 votes
Accepted

Source of equation - theorems about solving quadratic matrix equations

Bini, Iannazzo, Meini, Numerical Solution of algebraic Riccati equations, SIAM books, seems a good starting point to me. It is a monograph that deals both with the symmetric and the non-symmetric case ...
Federico Poloni's user avatar
3 votes

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

Not exactly an answer to your question, but it might be interesting anyway. Suppose your space $X$ is nice enough (path connected, locally path connected, semi-locally simply connected). Then under ...
Najib Idrissi's user avatar
2 votes
Accepted

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

$\newcommand{\R}{\mathbb{R}} \newcommand{\tto}{\underset{\text{onto}}\to}$ Let us answer the reformulated question: given a convex function $g\colon C\to\R$, when is it possible to find a decreasing ...
Iosif Pinelis's user avatar
2 votes
Accepted

Singularity of matrix pencil-like expression

No. The first condition is satisfied if (and only if) there is some vector in the kernel of $A$ that is also in the kernel of $B$. The second condition is satisfied (if and) only if the kernel of $A$ ...
Will Sawin's user avatar
  • 135k
2 votes

Singularity of matrix pencil-like expression

The statement is false. Take $3\times 3$ matrices such that $A_{11}=B_{22}=1$ and all other entries are zero. Then $EA-hB$ has the third column equal to $0$, but the row spaces of $A$ and $B$ are ...
Federico Poloni's user avatar
2 votes

ODE in Banach space

Let $X$ be a Banach space, and let $V:\mathbb{R}\times X\rightarrow X$ be continous in its first argument and at least Lipschitz in its second argument: i.e., that $\|{V(t,x)-V(t,y)}||\leq K||x-y||$ ...
MySheperd's user avatar
  • 866
1 vote
Accepted

Existence of Markov chain on nonnegative integers with specified rates

Define first the modified rates $$ \tilde Q(n,m) = \frac{Q(n,m)}{n + 1} \, . $$ Clearly, $\tilde Q(n, n+k) = \lambda_k$, and $\tilde Q(n, n-k) \leqslant \mu_k$. Assuming that $\lambda_k$ is summable (...
Mateusz Kwaśnicki's user avatar
1 vote

Finding a semi-sparse vertex in a grid

(This is not an answer to the OP's question1, under any of its interpretations, only an extended comment which the comment box is too small to contain, a comment thought to be helpful to the OP. It ...
Peter Heinig's user avatar
  • 6,001
1 vote
Accepted

Existence of analysis regularization solution

Nothing is stated concerning $\lambda$. I will assume that (as always) the regularization parameter $\lambda >0$. Otherwise the whole thing does not really make sense. The expression to be ...
Fabian Wirth's user avatar
  • 1,167

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