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 \...
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 \...
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
$$\...
13
votes
Accepted
Is the set of real matrices with at least one real logarithm closed under multiplication?
This is already not true for $2$-by-$2$ matrices: Consider
$$
A = \begin{pmatrix}2 & 0 \\0 &\frac12\end{pmatrix}\quad
\text{and}\quad
B = \begin{pmatrix}-1 & 0 \\0 &-1\end{pmatrix}.
$$...
7
votes
Existence of solutions to first-order PDE involving convolution
Edit: after posting the answer below, I noticed the question requires the study of the operator
$$
\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)\...
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 ...
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 ...
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 ...
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): ...
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 ...
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 ...
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 ...
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 ...
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 $A$ ...
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 ...
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 $\|{V(t,x)-V(t,y)}||\leq K||x-y||$ ...
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 (...
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 ...
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 ...
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