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https://mathoverflow.net/questions/395658 | 4 | Birkhoff–von Neumann theorem states that a polytope formed by a set of doubly-stochastic matrices has extreme points that are permutation matrices.
I am wondering if there is a similar theorem for a permutation matrix whose elements are replaced by some values of $e^{ix}$.
| https://mathoverflow.net/users/173974 | "Permutation matrix" but non-zero entries are replaced by $e^{ix}$ | A statement similar to the Birkhoff theorem holds: these matrices are the extreme points of the set of matrices for which the $\ell\_1$ norm of each row and column is $\leq 1$. I denote by $K$ this set.
It is clear that every such matrix is an extreme point in $K$. Conversely, let $A$ be an extreme point in $K$. The ... | 7 | https://mathoverflow.net/users/908 | 395673 | 163,441 |
https://mathoverflow.net/questions/395615 | 6 | Can you prove or disprove the following claim:
>
> Let $U(n,P,Q)$ be the nth [generalized Lucas number of the first kind](https://en.wikipedia.org/wiki/Lucas_sequence#Recurrence_relations) and let $m$ be a natural number. Then,
> $$\sqrt{m}=1+\displaystyle\sum\_{n=1}^{\infty} \frac{(-1)^{n+1} \cdot (m-1)^n}{U(n,2,1... | https://mathoverflow.net/users/88804 | The square root of natural number expressed by an infinite series | So, here we have $P=2$ and $Q=1-m$.
Notice that
$$\frac{Q^n}{U\_n(P,Q)U\_{n+1}(P,Q)} = \frac{U\_{n+1}(P,Q)}{U\_n(P,Q)}-\frac{U\_{n+2}(P,Q)}{U\_{n+1}(P,Q)}.$$
By telescoping, it follows that
$$\sum\_{n=1}^k \frac{Q^n}{U\_n(P,Q)U\_{n+1}(P,Q)} = P -
\frac{U\_{k+2}(P,Q)}{U\_{k+1}(P,Q)}.$$
Taking the limit over $k\to\inf... | 10 | https://mathoverflow.net/users/7076 | 395675 | 163,443 |
https://mathoverflow.net/questions/395627 | 5 | The Lambert $W$ Function is defined in [this Wikipedia entry](https://en.wikipedia.org/wiki/Lambert_W_function), while the Hypergeometric Function is defined in [this other Wikipedia entry](https://en.wikipedia.org/wiki/Hypergeometric_function). There exists also a multivariate generalization which solves the following... | https://mathoverflow.net/users/246134 | Relationship between Lambert $W$ function and Hypergeometric function | **Q:** Can the Lambert $W$ function be written as an inverse of a hypergeometric function?
**A:** $x=W(y)$ is the solution of $\_1F\_1(2;1;x-1)=y/e$.
| 3 | https://mathoverflow.net/users/11260 | 395676 | 163,444 |
https://mathoverflow.net/questions/395688 | 2 | For a convex polytope, its face poset is combinatorially determined by vertex-facet incidences. Now suppose we have an arbitrary finite poset that is *ranked*, so I can still speak of vertices and facets. What property should be satisfied for vertex-facet incidences to still store all the information? Is it the propert... | https://mathoverflow.net/users/123731 | What property of ranked poset ensures that it is determined by its vertex-facet incidences? | If $L$ is a finite lattice, then $L$ is determined by its subposet of elements that are join-irreducible or meet-irreducible (or both). In particular, if the only join-irreducibles are atoms (vertices) and only meet-irreducibles are coatoms (facets), then $L$ is determined by the incidences between its vertices and fac... | 5 | https://mathoverflow.net/users/2807 | 395690 | 163,449 |
https://mathoverflow.net/questions/395695 | 2 | Let $f:X\to Y$ be a representable map of finite type (or is finite dimensional enough?) Artin stacks, whose fibres (which are schemes) have dimension at most $n$. Then is it true that $R^qf\_\*\mathbf{Q}\_\ell=0$ for all $q\gg 0$?
Note: by taking atlases, I think it is sufficient to let $X,Y$ be schemes.
---
**... | https://mathoverflow.net/users/119012 | Finiteness result for higher direct image of $\ell$-adic sheaves | $Y$ admits a smooth surjective morphism from a scheme $Z$. Because smooth morphisms are locally of finite type, $Z \to Y$ is locally of finite type, and you can choose an open cover that covers $Y$ and then pass to a finite subcover to make $Z$ of finite type.
Because this morphism is smooth, by smooth base change th... | 4 | https://mathoverflow.net/users/18060 | 395696 | 163,452 |
https://mathoverflow.net/questions/395467 | 1 | Let $X$ be a Banach space such that both $X$ and $X^\*$ have finite cotype. Also assume that $X$ is an inductive limit of finite dimensional Banach spaces $X\_n\subseteq X\_{n+1}.$ Fix $1<p<\infty.$ Is there any known result which can give precise information about finite dimensional subspaces $Y\_n$'s of $X$ with $\su... | https://mathoverflow.net/users/136860 | Banach space containing uniformly complementend $\ell_p^n$s | You can find many relevant results in the book: Pisier, Gilles
Factorization of linear operators and geometry of Banach spaces.
CBMS Regional Conference Series in Mathematics, 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.... | 2 | https://mathoverflow.net/users/37822 | 395701 | 163,454 |
https://mathoverflow.net/questions/337648 | 2 | I asked the same question but get no answer in other place. Here is the following.
For a compact Riemannian surface $\Sigma$. For an initial embedded closed curve $\gamma\_0$ in $\Sigma$, a family $\gamma\_t$ $(0\leq t<T)$ is parametrized by
\begin{equation}
F : S^{1} \times[0, T) \rightarrow \Sigma,
\end{equation}
... | https://mathoverflow.net/users/113274 | Time derivative of area under curve shortening flow | A late answer if the OP is still interested in a solution. In $\mathbb{R}^2$, the area $A$ enclosed by the closed embedded curve $\gamma \colon I \subset \mathbb{R} \to \mathbb{R}^2$ is given by (using the Green's identity) $$A = \frac 12\oint x\,dy - y\,dx = \frac 12\int\_I \left(x\frac{dy}{dz} - y\frac{dy}{dz}\right)... | 2 | https://mathoverflow.net/users/163454 | 395702 | 163,455 |
https://mathoverflow.net/questions/395622 | 4 | What is the number of ways to distribute $m$ indistinguishable balls to $k$ distinguishable boxes given no box can be a unique number of balls?
for example: ($m=19$ and $k=5$)
$$x\_1 + x\_2 + \dots + x\_5 = 19 $$
Some of the accepted ways are:
$2 , 2 , 5 , 5 , 5$
$3 , 3 , 3 , 5 , 5$
$8 , 1 , 1 , 1 , 8$
and so... | https://mathoverflow.net/users/295084 | Number of ways of distributing indistinguishable balls into distinguishable boxes with extra givens | Max is surely correct that there is no simple formula, though a summation or double summation is plausible. Anyway, computation via a recurrence is probably the best. Define $A(m,k)$ to the number of (ordered) compositions $a\_1+\cdots+a\_k=m$ with no unique term. Note that 0 can't be unique either in this definition.
... | 2 | https://mathoverflow.net/users/9025 | 395706 | 163,456 |
https://mathoverflow.net/questions/395692 | 4 | A sequence $a\_n \in \mathbb{C}, \ n = 1, 2, 3, \dots$ is *Abel-summable* if for all $|x| < 1$ the sum
$$g(x) = \sum\_{n = 1}^{\infty} a\_n x^n$$
converges and the limit $\lim\_{x \to 1^{-}} g(x)$ exists. In the case the limit is called the *Abel sum* of the sequence $a\_n$. Notice that for $g(x)$ to converge, $a\_n$ m... | https://mathoverflow.net/users/88679 | Is there a superpolynomial sequence which is Abel-summable? | Let $$f(x):=\sum a\_nx^n=\exp\left(\frac1{1+x}\right), |x|<1.$$
Then $a\_n$ is Abel summable to $\sqrt{e}$, but if we had $a\_n=O(n^c)$, the value of $f(-1+t)$ for small $t$ would be bounded by $O(t^{-c-1})$.
| 4 | https://mathoverflow.net/users/4312 | 395708 | 163,458 |
https://mathoverflow.net/questions/395719 | -3 | What is an example of a topological base ${\cal B}$ for $\mathbb{R}$ with the Euclidean topology such that for every $B\_1\neq B\_2 \in {\cal B}$ we have $B\_1\not\subseteq B\_2$?
| https://mathoverflow.net/users/8628 | Basis of Euclidean topology on $\mathbb{R}$ such that no element is contained in another | A topological space has a basis which is an antichain w.r.t. set inclusion if and only if its Kolmogorov quotient (ie $T\_0$-fication) is discrete.
The reason is that in this case, any two basis elements need to have empty intersection (otherwise their intersection would need to have a basis element as a subset, whic... | 3 | https://mathoverflow.net/users/15002 | 395725 | 163,462 |
https://mathoverflow.net/questions/395713 | 3 | I asked [this](https://math.stackexchange.com/questions/4176670/how-to-check-whether-an-element-in-this-domain-is-irreducible-or-not) question on MSE. Here also I have the same motive in the question.
Let $D= \{\,a\_1x^{r\_1} + \cdots + a\_n x^{r\_n} \, \vert \, a\_i \in \mathbb{C} \text{ for } i= 1,2,\dots,n \text{ ... | https://mathoverflow.net/users/165646 | Do there exist irreducible elements in this domain? | Fix $m\ge 0$. Let $a\_1,\dots,a\_m\in\mathbf{R}$ be linearly independent over $\mathbf{Q}$.
**Claim** The element $1+\sum\_{i=1}^mx^{a\_i}$ is irreducible in $D=\mathbf{C}[\mathbf{R}\_{\ge 0}]$ iff for all integers $n\_1,\dots n\_m\ge 1$, the polynomial $1+\sum\_{i=1}^mz\_i^{n\_i}$ is irreducible in the polynomial al... | 2 | https://mathoverflow.net/users/14094 | 395737 | 163,465 |
https://mathoverflow.net/questions/395683 | 6 | [I fear that I'm missing something obvious here, but I'll dare to ask anyway.]
As we all know, a division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let an *anti-division ring* be a ring where any element other than the identity is a (two-sided) zero divisor.
There ar... | https://mathoverflow.net/users/16537 | Is there any structural characterization of the rings in which every element other than the identity is a (two-sided) zero divisor? | Sorry for answering my own question, but it turned out that what I'm calling "anti-division rings" in the OP were already studied by P.M. Cohn under the name of "$0$-rings" (though Cohn's work on this stuff is seemingly restricted to the commutative setting), see
* P.M. Cohn, *Rings of zero divisors*, Proc. Amer. Mat... | 3 | https://mathoverflow.net/users/16537 | 395739 | 163,466 |
https://mathoverflow.net/questions/395715 | 0 | Let $f(x)$ be a real transcendental function with algebraic coefficients. So $f(x)$ and $x$ are algebraically independent. Let $\alpha$ be a transcendental number, are the numbers $$\alpha+f(\alpha),\ \alpha f(\alpha),\ \alpha/f(\alpha)$$ transcendental?
It is clear these numbers are transcendental if $f(\alpha)$ is al... | https://mathoverflow.net/users/159935 | Transcendence on $ \alpha+f(\alpha), \alpha f(\alpha) $ and $ \alpha/f(\alpha) $ where $ \alpha$ is transcendental | Here are two counterexamples to the specific question at the end.
1. Let $\alpha$ be the unique real solution to $xe^x=1$. This number is also called the [omega constant](https://en.wikipedia.org/wiki/Omega_constant#Transcendence). It is transcendental by the Lindemann-Weierstrass Theorem which (in particular) says t... | 3 | https://mathoverflow.net/users/38253 | 395743 | 163,467 |
https://mathoverflow.net/questions/395727 | 1 | If $A$ is a commutative ring, $I \subset A$ an ideal and $f:A \rightarrow B$ a ring homomorphism, then the extension of $I$, $I^e = \langle f(a): a \in I \rangle$ does not commute with the radical, I mean, $\sqrt{I^e} \neq (\sqrt{I})^e$ in general.
I'm struggling with the following problem: I have two fields, $K$ and... | https://mathoverflow.net/users/296297 | Extension of the radical and radical of the extension of an ideal | Let $K$ be a field of characteristic $p>0$ and let $a\in K$ an element without $p$th root. Let $L=K(b)$, where $b$ is the $p$th root. Let $I$ be the ideal in $K[x^{\pm 1}]$ generated by $x^p-a$. Then $I$ is reduced, but its extension is not.
| 2 | https://mathoverflow.net/users/9502 | 395744 | 163,468 |
https://mathoverflow.net/questions/259253 | 5 | I am given two metric spaces as two arrays of the same size. Each one is supposed to represent distance between vertices on a mesh in R^3. The meshes are assumed to have the same number of vertices and the correspondence betweeen the vertices is also given. Is there a way to find the a meaningful distance between these... | https://mathoverflow.net/users/103418 | Distance between two metric spaces | Persistent Homology can be used for this purpose. Given two metric spaces $X$ and $Y$, then one can measure the distance between their persistence diagrams $PD\_i(X)$ and $PD\_i(Y)$ where $i \geq 0$. There are many distances that can be defined on the space of persistence diagrams such as the [bottleneck distance](http... | 0 | https://mathoverflow.net/users/103418 | 395747 | 163,469 |
https://mathoverflow.net/questions/395745 | 12 | I wonder is it still an open question that a smooth sphere $\Sigma^{2}\subset S^4$ is unknotted in $S^4$ iff its complement is homotopy equivalent to $S^1$? If it is an open question, how is it related to other known conjectures in 4D?
I know for all the other $n$ this has been settled by Levine 1965 "Unknotting sphe... | https://mathoverflow.net/users/9800 | Unknotted $S^{n-2}$ in $S^n$ | My understanding is this remains an open problem in the smooth category.
I believe there have been a few claims of proofs of this statement in the literature over the years, but as far as I know none of these arguments have been robust. As I believe you are aware, in the topological category this was done by Mike Fre... | 12 | https://mathoverflow.net/users/1465 | 395748 | 163,470 |
https://mathoverflow.net/questions/395749 | 6 | This is my first question in MathOverflow and I will do my best to format it correctly and make it clear.
I am reading a paper on dispersive wave turbulence which introduces the following family of equations:
$$i\psi\_t=|\partial\_x|^{\alpha}\psi+|\partial\_x|^{-\beta/4}\left(\left||\partial\_x|^{-\beta/4}\psi\righ... | https://mathoverflow.net/users/296643 | Fractional derivative notation in wave turbulence | The fractional derivative $|\partial\_x|^\alpha$ is discussed in [One-dimensional wave turbulence](https://maths.ucd.ie/~dias/ZakharovDiasPushkarev.pdf) by Zakharov, Dias, and Pushkarev. (Zakharov introduced the notation.) As they explain below Eq. 2.1, it is indeed defined via the Fourier transform, such that the Four... | 4 | https://mathoverflow.net/users/11260 | 395750 | 163,471 |
https://mathoverflow.net/questions/395755 | 5 | **Definitions:**
A measurable subset $S$ of $\mathbb R$ is said to be *mesoscopic* if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero Lebesgue measure.
>
> **Question:** Is the set of zeroes of a Brownian motion almost surely a mesoscopic set... | https://mathoverflow.net/users/173490 | Largeness of the set of zeroes of a Brownian motion | Yes, the local time (at zero) maps the zero set of Brownian motion to an interval. See e.g. Lemma 6.9 page 159 in [1] for continuity.
[1] Brownian motion, by Peter Mörters and Yuval Peres. Cambridge University Press, 2010 <https://people.bath.ac.uk/maspm/book.pdf>
| 9 | https://mathoverflow.net/users/7691 | 395761 | 163,474 |
https://mathoverflow.net/questions/395141 | 1 | I have posted this problem in MSE long ago:
<https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral>. But it hasn't been solved yet so I post it here. Maybe this problem not so easy. I would like to describe it again.
Consider the polynomial function in $\mathbb{R}^n$: $$f(x)=\sum\... | https://mathoverflow.net/users/145357 | Prove the integral of multi-variable rational fraction is convergent | I'm questioner. Now I have found a reference:
[Volume estimates of sublevel sets of real polynomials](https://arxiv.org/pdf/1711.04544.pdf)
Theorem 4.1 in it can solve this proplem.
This paper has been published at
[Annales Polonici Mathematici](https://www.impan.pl/en/publishing-house/journals-and-series/annales... | 1 | https://mathoverflow.net/users/145357 | 395766 | 163,476 |
https://mathoverflow.net/questions/395603 | 1 | Let $B\_t$ be a standard one dimensional Brownian motion. Is it true that
$$\lim\_{s \to \infty} \frac{\int\_{[0, s]} \mathbf 1\_{ \{|B\_t| \geq \sqrt{2t/\pi} \} } \ dt}{s}$$
exists almost surely?
| https://mathoverflow.net/users/173490 | The long run average amount of time the deviation of Brownian motion spends above its expected value | The limsup is 1 (as noted by Anthony Quas) and the liminf is zero. The first of these follows immediately from Strassen's functional LIL. The liminf can be deduced from the distribution of the running maximum of Brownian motion.
[1] <https://sites.stat.washington.edu/jaw/COURSES/520s/523/HO.523.20/523-Spr2020-L4.pdf>... | 2 | https://mathoverflow.net/users/7691 | 395767 | 163,477 |
https://mathoverflow.net/questions/395448 | 2 | Let $E$ be a separable Banach space and let $T\in L(E,E)$.
Is there a condition on $T$ ensuring that:
$$
\mbox{$\{x\_n\}\_{n=1}^N\subseteq E$ is linearly independent} \Rightarrow
\{T(x\_n)\}\_{n=1}^N\cup \{x\_n\}\_{n=1}^N \mbox{ is a independent in $E$}?
$$
Is $T$ [mixing](https://en.wikipedia.org/wiki/Mixing_(math... | https://mathoverflow.net/users/222170 | Operators "building" linear independant sets | Possibly I misunderstood your question, but it seems to me that an operator satisfying the condition should have $\{x, Tx\}$ linearly independent for nonzero $x$. Then the condition fails to be satisfied for $\{x\_i\}\_{i=1}^N=\{x,Tx\}$, because its image repeats a vector, so there are no operators $T$ satisfying the c... | 3 | https://mathoverflow.net/users/37822 | 395769 | 163,478 |
https://mathoverflow.net/questions/395771 | 3 | Let $(X,d)$ be an arbitrary metric space and let $\Bbb B(x,r)$ denote the closed ball with center $x \in X$ and radius $r>0$. For $p\geq 0$, let $H^p$ denote the $p$- dimensional Hausdorff measure. Under which assumptions on $X$ and $p$ is $H^p(\Bbb B(x,r))< + \infty$? Is this always the case even if the Hausdorff dime... | https://mathoverflow.net/users/114128 | Finiteness of Hausdorff measure of balls | As a counterexample, let $X$ be an infinite-dimensional normed space. For $\varepsilon<r/2$ it follows from Borsuk-Ulam that you need more than $n$ closed sets of diameter $\varepsilon$ to cover the intersection of the boundary $B(0,r)$ with an $n$-dimensional subspace (because these sets are free of antipodal points).... | 8 | https://mathoverflow.net/users/165275 | 395773 | 163,480 |
https://mathoverflow.net/questions/395774 | 1 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). If $\kappa \neq \emptyset$ is a cardinal, we call a map $c:V\to \kappa$ a *coloring* if for each $e\in E$ with $|e|>1$ the restriction $c\restriction\_e$ is non-constant. The smallest cardinal $\kappa > 0$ such that there is a coloring map $c:V\... | https://mathoverflow.net/users/8628 | Large chromatic number in hypergraphs with large edges | For $\kappa=\aleph\_0$ yes: there are (many) models with ultrafilters of character less than $\mathfrak{c}$. Let $E\subseteq[\omega]^\omega$ be a base for an ultrafilter, say $|E|=\aleph\_1<\mathfrak{c}$.
If $f:\omega\to k$ for some $k<\omega$ then $f$ is constant on a member of $E$. The identity map is a colouring of ... | 2 | https://mathoverflow.net/users/5903 | 395784 | 163,482 |
https://mathoverflow.net/questions/395781 | 5 | As the title says, I am interested to know Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$. There is some confusion in the literature.
Let recall that the compactness theorem in $L^p(\Bbb R^d)$ is somewhat a generalization of the Ascoli-Arzelà compactness theorem for space $C(X)$ of continuous functi... | https://mathoverflow.net/users/112207 | Who are the owners of the compactness theorem in $L^p(\Bbb R^d)$? | I consulted the compendum on such topics: Dunford & Schwartz.
*Dunford, Nelson; Schwartz, Jacob T.*, Linear operators. I. General theory. (With the assistence of William G. Bade and Robert G. Bartle), Pure and Applied Mathematics. Vol. 7. New York and London: Interscience Publishers. xiv, 858 p. (1958). [ZBL0084.1040... | 18 | https://mathoverflow.net/users/454 | 395789 | 163,484 |
https://mathoverflow.net/questions/395798 | 1 | Let $G = (V,E)$ be a simple, undirected graph. For $v\in V$ we let $N(v) = \{w \in V: \{v,w\} \in E\}$.
We define the *coloring number* $\text{Col}(G)$ of the graph $G$ to be the smallest cardinal $\kappa$ such that there is a well-ordering $\leq\_{\text{well}}$ on $V$ such that for every vertex $v\in V$ we have $$|N... | https://mathoverflow.net/users/8628 | Graph $G=(V,E)$ with $\chi(G)$ finite and $\text{Col}(G)$ infinite | Take a complete bipartite graph $G=(V\_1,V\_2,E)$ such that $V\_1$ and $V\_2$ are infinite. Then $\chi(G)=2$ and $\text{Col}(G)$ is infinite.
Indeed, consider any well-ordering on $V=V\_1\cup V\_2$. Either there exists a vertex with infinitely many smaller neighbors, or there exists an infinite path $(v\_1,v\_2,\dots... | 5 | https://mathoverflow.net/users/11919 | 395799 | 163,485 |
https://mathoverflow.net/questions/395787 | 21 | A (unital) ring $R$ with the property that every element other than the identity $1\_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring". These rings were first studied by P.M. Cohn (though only in the commutative setting) in
* *Rings of zero divisors*, Proc. Amer. Math. ... | https://mathoverflow.net/users/16537 | Is there any non-commutative ring such that every element other than the identity is a zero divisor? | [Sorry for answering my own question, and the more so because this is happening for the second time in 24 hours.]
The question might be open. In fact, a positive answer would imply an equally positive answer to a question stated in the introduction of Melvin Henriksen's paper
* "Rings with a unique regular element"... | 12 | https://mathoverflow.net/users/16537 | 395800 | 163,486 |
https://mathoverflow.net/questions/395649 | 1 | This question has also been asked on <https://math.stackexchange.com/questions/4174928/bessel-process-conditioned-to-stay-positive>
Suppose the stochastic process $(X\_t)\_{t\ge 0}$ with start in $X\_0:=x>0$ is the solution of the SDE $$ dX\_t = dB\_t + \frac{\rho-1}{2X\_t} \, dt $$ with $B\_t$ denoting Brownian moti... | https://mathoverflow.net/users/101850 | Bessel process conditioned to stay positive | This conditioning of a Bessel process is a Doob transform. For $\rho$ in $(0,2)$ it leads to a Bessel process of dimension $4-\rho$. See
Goeing-Jaeschke, A., Yor, M. (2003) A Survey on some generalizations of Bessel processes. Bernoulli
9, 313–349 and the book by Revuz-Yor, Continuous Martingales and BM.
the LIL for Be... | 2 | https://mathoverflow.net/users/7691 | 395802 | 163,488 |
https://mathoverflow.net/questions/395806 | 0 | Let $(\Omega,\mathcal{F},(\mathcal{F}\_t)\_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t\_n\}\_{n=1}^{N}$ be a partition of $[0,T]$ with $t\_0=0,t\_n<t\_{n+1},t\_N=T$. Fix $k\in \mathbb{Z}^+$, and suppose that for each $n=0,\dots,N$ we are given some
$$
X\_{t\_n}\in L^1\_{\mathbb{... | https://mathoverflow.net/users/298030 | A martingale extension/interpolation problem | Assume the filtration is large enough. Then Kellerer and Strassen both proved indepently the existence of such a martingale is equivalent to
$$\int fd\mu\_{t\_n} \le \int fd\mu\_{t\_{n+1}},\quad \mbox{for all convex functions} f:\mathbb R^k \to\mathbb R \mbox{ with linear growth},~~~~~(\ast)$$
where $\mu\_{t\_n}$ d... | 0 | https://mathoverflow.net/users/261243 | 395807 | 163,491 |
https://mathoverflow.net/questions/395548 | 8 | My naïve cartoon picture of the construction of étale cohomology is this:
1. start with a scheme, associate to it a Grothendieck topology (making a site).
2. A functor from the Grothendieck topology to abelian groups (say) has all the relevant properties of a presheaf (by the definition of a Grothendieck topology) an... | https://mathoverflow.net/users/5339 | Cohomology of Grothendieck topology | Artin, M. Grothendieck topologies. (English) [Zbl 0208.48701](https://www.zbmath.org/?q=an%3A0208.48701) Cambridge, Mass.: Harvard University. 133 p. (1962). ([pdf copy](https://www.math.nagoya-u.ac.jp/%7Elarsh/teaching/S2013_AG/grothendiecktopologies.pdf))
These notes seem to fit your description precisely. They are... | 11 | https://mathoverflow.net/users/1310 | 395809 | 163,492 |
https://mathoverflow.net/questions/395783 | 4 | Given a set $S\subseteq \mathbf{R}^n $ and $ x \in \overline{S} $ we define the tangent cone $ T\_S(x) $ to be the collection of all vectors $ v \in \mathbf{R}^n $ such that
$$ \liminf\_{r \to 0+} r^{-1} \mathrm{dist}(x + r v, S) =0. $$
It seems reasonable that there exists an example of a *closed* set $ S \subseteq ... | https://mathoverflow.net/users/88920 | Tangent cone of null sets | One can use your infinite-density example, but replace the outer lines with very sparse dotted lines:
$$S = (\{0\} \times \mathbb{R}) \cup \bigcup\_{i=1}^\infty \{i^{-1},-i^{-1}\} \times \left[\bigcup\_{j \in \mathbb{Z}} [i^{-2}j,i^{-2}j+i^{-4}]\right] $$
By symmetry it is enough to consider the intersection with $... | 3 | https://mathoverflow.net/users/51695 | 395830 | 163,496 |
https://mathoverflow.net/questions/395729 | 2 | Let $A$ be a Banach algebra and $Bil(A)$ denote the space of bounded bilinear forms on $A$.
$Bil(A)$ is a Banach $A$-bimodule with the module operations
\begin{eqnarray\*}
\beta a(x,y) &:=& \beta(ax,y) \\
a \beta(x,y) &:=& \beta(x,ya)
\end{eqnarray\*}
for each $\beta\in Bil(A)$ and each $a,x,y\in A$.
Further, for each ... | https://mathoverflow.net/users/164350 | Weak sequential continuity of certain bilinear forms on Banach algebras | YES.
Consider the Jolissaint--Lafforgue Sobolev algebra $H\_\ell^s(\Gamma)$.
(I don't know the common name for it.)
Here we take $\Gamma=F\_\infty$ to be the free group of countably infinite rank, $\ell$ the standard word length, and $s>2$.
It is the completion of the complex group algebra ${\mathbb C}\Gamma$ under
the... | 4 | https://mathoverflow.net/users/7591 | 395832 | 163,497 |
https://mathoverflow.net/questions/395819 | 4 | $\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\pd{pd}$Let A be a finite dimensional algebra over some field k and S a nonprojective simple left A-module. Suppose the projective dimension $\pd\_A(S)$ of $S$ is finite. Let $n$ be a nonnegative integer such that $n+1\leq \pd\_A(S)$. Then we can find 2 simple left $A$-... | https://mathoverflow.net/users/134942 | Can we always choose 2 nonisomorphic simple modules to satisfy the following nonvanishing extension conditions? | Let $A$ be the path algebra of the quiver with three vertices $1$, $2$, $3$ with arrows $a:1\to 2$, $b:2\to 3$, $c:3\to 1$, modulo relations $bca=0$, $cabc=0$, so the indecomposable projectives are uniserial modules
$$P\_1=\begin{matrix}S\_1\\S\_2\\S\_3\\S\_1\end{matrix},\quad\quad\quad
P\_2=\begin{matrix}S\_2\\S\_3\\S... | 5 | https://mathoverflow.net/users/22989 | 395838 | 163,501 |
https://mathoverflow.net/questions/395824 | 4 | Let A be a finite dimensional symmetric k-algebra over some field k. The set of units of A is denoted by U(A). Suppose G is a cyclic group of prime order which acts via inner algebra automorphism on A, say, there is a homomorphism $\phi : G \rightarrow U(A)$ such that $a\cdot g:= a ^{\phi(g)}$ for all $a\in A, g\in G$ ... | https://mathoverflow.net/users/134942 | Is the fixed subring a symmetric algebra? | Let $k$ be a field of characteristic $2$, and let $A$ be the path algebra over $k$ of the quiver with two vertices, $v\_1$ and $v\_2$, and arrows $a:v\_1\to v\_2$ and $b:v\_2\to v\_1$, modulo the relations $aba=0$ and $bab=0$.
Then $A$ is a symmetric algebra (with symmetrizing form given by $\varphi(ab)=\varphi(ba)=1... | 7 | https://mathoverflow.net/users/22989 | 395842 | 163,503 |
https://mathoverflow.net/questions/395840 | 3 | I wonder whether the following question have a positive answer within $ZFC$.
>
> **Question** If $\{A\_n\}\_{n\in \omega}$ is a sequence of analytic sets so that $\bigcup\_n A\_n=2^{\omega}$, then there must be some $n$ so that $A\_n$ has a pointed subset.
>
>
>
A pointed set is a perfect set $P$ of reals in w... | https://mathoverflow.net/users/14340 | Analytic sets and Turing determinacy | I believe this fails under $V=L$. If $P$ is a pointed perfect set and $X$ is a real, let $P(X)$ be the element of $P$ where at every split we choose according to the next bit of $X$. So $X\oplus P \equiv\_T P(X)$.
We'll build the $A\_n$ via an $\omega\_1$ length construction as follows. At stage $\alpha$, we consider... | 1 | https://mathoverflow.net/users/32178 | 395850 | 163,505 |
https://mathoverflow.net/questions/395851 | 2 | Assume just for sake of simplicity that $R = k[x\_1 , \dots , x\_n]$ is a standard graded polynomial ring over a field. If one considers the ideal
$$I = \left({x}\_{1}{x}\_{3},{x}\_{2}^{2},{x}\_{2}{x}\_{3},{x}\_{3}^{2}\right)$$
and computes the minimal free resolution, the very last differential takes the form
$$\begin... | https://mathoverflow.net/users/73780 | Can the differentials in a minimal free resolution ever have a "long" row of $0$'s? | Since I can not comment on the other's posts as commenting needs at least 50 reputation, I'm forced to release an answer. It seems that the second power, $I^2$, of your ideal $I$ has a zero row of length 4. Compute it via Macaulay2 and look at the matrix of the very last differential.
| 1 | https://mathoverflow.net/users/127857 | 395863 | 163,508 |
https://mathoverflow.net/questions/395869 | 6 | I'm interested in the following assertion about the Davis-Putnam-Robinson-Matijasevich theorem
>
> Given a recursive function $f:\mathbb{N}\rightarrow\mathbb{N}$, i.e. its index, we can *effectively* get a polynomial $p\in\mathbb{Z}[x,y,z\_1,\dots,z\_n]$ that satisfies
> $$f(x)=y\iff\exists z\_1\dots z\_n \in \math... | https://mathoverflow.net/users/282044 | Given some recursive function, can we effectively associate it a polynomial as in the DPRM theorem? | Davis (also available on the MAA [site](https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/MartinDavis.pdf)) begins section 7 by saying "an explicit enumeration of all the Diophantine sets of positive integers will now be described".
He then gives theorem 7.1, from which one can get a polynomial form ... | 7 | https://mathoverflow.net/users/nan | 395872 | 163,511 |
https://mathoverflow.net/questions/395844 | 1 | Let $\mathbf X := (X, \mathcal F, \mu)$ be a standard probability space.
For an ergodic measure preserving transformation $T$, we define the *ergodic robustness* $\mathcal R(T)$ of $T$ as follows:
For $0 \leq r \leq 1$, let $C\_r \subset \mathbb N^{\mathbb N}$ be the subset of monotonically increasing sequences whose... | https://mathoverflow.net/users/173490 | Robustness of ergodic dynamical systems | Yes. Let $\nu$ be a probability measure on $[0,1]$. Let $T$ be the left shift on a sequence space $X:=[0,1]^{\mathbb N}$ equipped with the product $\sigma$-field and the product measure $\mu:=\nu^{\mathbb N}$. Then for **every** strictly increasing sequence
$\{n\_k\}$ of positive lower density and $f \in L^1 (X)$ we ha... | 1 | https://mathoverflow.net/users/7691 | 395877 | 163,512 |
https://mathoverflow.net/questions/380515 | 4 | Assuming Goldbach's conjecture, denote as usual by $r\_{0}(n)$ for any large enough positive integer $n$ the smallest positive integer $r$ such that both $n-r$ and $n+r$ are prime.
Let's define the notion of "staircase number" as any such integer $n$ such that the elements of the sequence $r\_{0}(n), r\_{0}(n)^2,\cdo... | https://mathoverflow.net/users/13625 | Staircase numbers | Strong staircase conjecture follows from a the following version of the $k$-tuple conjecture: for any admissible tuple $T$ and a finite set $S$ disjoint from it, there are infinitely many integers $n$ such that $n+T$ contains only primes, and $n+S$ contains only composite numbers. This version follows from the Dickson'... | 1 | https://mathoverflow.net/users/30186 | 395878 | 163,513 |
https://mathoverflow.net/questions/395873 | 6 | Q1: Is it true that a knot $S^2\hookrightarrow S^4$ has an inverse iff it is trivial? Or it is also an open question?
See relatedly [Unknotted $S^{n-2}$ in $S^n$](https://mathoverflow.net/questions/395745/unknotted-sn-2-in-sn).
Q2: It is easy to see that if a knot $f\colon S^2\hookrightarrow S^4$ has an inverse tha... | https://mathoverflow.net/users/9800 | Invertible 2-knots in $S^4$ | Q1: This is true in the topological category and unknown in the smooth setting. In the topological setting, the fundamental group of $S^4 - K\_1 \# K\_2$ is $G\_1 \*\_\mathbb{Z} G\_2$ where $G\_i$ are the fundamental groups of $S^4 -K\_i$. If this is $\mathbb{Z}$ then I think the $G\_i$ are both $\mathbb{Z}$. By the ar... | 7 | https://mathoverflow.net/users/3460 | 395884 | 163,515 |
https://mathoverflow.net/questions/395865 | 15 | $\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\diag{diag}$In [SGA3](https://en.wikipedia.org/wiki/S%C3%A9minaire_de_G%C3%A9om%C3%A9trie_Alg%C3%A9brique_du_Bois_Marie), Expose XXIV, Lemme 7.2.2 it says (let's say our base scheme $S$ is an algebraically closed field $k$): if $G$ is reductive ... | https://mathoverflow.net/users/299730 | Pushout of group schemes (question on a lemma in SGA3) | The Lemma is clearly wrong. There is no way to recover $G$ from $G'$, $T$ and $T'$ alone (not even up to isomorphism) since that data do not determine the radical $R:={\rm rad}(G)\subseteq T$. Maybe the authors had in mind the amalgamated product of $T$ and $R\times G'$ over $R\times T'$.
| 9 | https://mathoverflow.net/users/89948 | 395885 | 163,516 |
https://mathoverflow.net/questions/395856 | 8 | Let $G$ be a finite group and $KG$ its group algebra over some field $K$ with $\mathrm{char}\ K$ dividing the order of $G$. It's well-known that the Green correspondence is compatible with the Brauer correspondence. Suppose we are dealing with Green correpondence between indecomposable modules of a block $B$ and its Br... | https://mathoverflow.net/users/134942 | Is there always a simple module whose Green correspondent is a simple module under some conditions? | The answer is "no" in general. I presume you mean that $B$ is a block of $KG$, and $b$ is its local Brauer correspondent.
Consider the case $G = {\rm SL}(2,3)$ with $p = 3.$ Then $G$ has three $3$-blocks.
One is the principal $3$-block, one is $3$-block of defect zero. The third block is a non-principal $3$-block $B$... | 7 | https://mathoverflow.net/users/14450 | 395888 | 163,518 |
https://mathoverflow.net/questions/395758 | 2 | Let $u(k,j) = 1$ if $j=0$, $0$ if $j > k$, or else it is $j\*u(k-1,j-1) +(j+1)\*u(k-1,j) $. Prove that $ \sum\_{i=0}^{2k} {n \choose i+1} u(2k,i) +\sum\_{i=0}^{2k} {-n-1 \choose i+1} u(2k,i)=0. $ This problem is provable using Bernoulli numbers but I'm interested if there's a proof that doesn't require the Bernoulli nu... | https://mathoverflow.net/users/265714 | Show that $\sum_{i=0}^{2k} [ {n\choose i+1} + (-1)^{i+1}{n+i+1\choose i+1} ] \sum_{j=0}^i {i\choose j}(-1)^j (i+1-j)^{2k} =0.$ | As pointed out by Ira Gessel, $u(k,j)=j!S(k+1,j+1)$. Correspondingly, the sum in question reduces to
$$f\_{2k}(n) + f\_{2k}(-n-1),$$
where
$$f\_k(t):=\sum\_{i=1}^{k+1} S(k+1,i)\frac{(t)\_i}i,$$
where $(t)\_i := t(t-1)\cdots(t-i+1)$ is the falling factorial.
Using the recurrence $S(k+1,i)=iS(k,i)+S(k,i-1)$, we get
$$f... | 2 | https://mathoverflow.net/users/7076 | 395897 | 163,521 |
https://mathoverflow.net/questions/395862 | 3 | I'm having some issues with the spectral decomposition of the integral operator
\begin{equation}
(Af)(x)=\int\_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}.
\end{equation}
Since
\begin{equation}
\int\_0^1\int\_0^1|x-y|^2\,dx\,dy<\infty \text{ and } |x-y|=|y-x|,
\end{equation}
this is a self-adjoint Hilbert-Schmidt int... | https://mathoverflow.net/users/299642 | Hilbert-Schmidt integral operator with missing eigenfunctions | I figured a positive eigenvalue. For $\lambda>0$, if $f(x) := \alpha\exp(\tau x)+ \exp(-\tau x)$ with $\alpha$ a constant and $\tau=\sqrt{2/\lambda}$ then $Af-\lambda f=0$ if and only if
$$\frac{e^{-\tau}-\alpha e^{\tau}-(\alpha-1)}{\tau}x+\frac{(-1-\tau) e^{-\tau}-\alpha(1-\tau)e^{\tau}-\alpha-1}{\tau^2}=0,\text{ for ... | 1 | https://mathoverflow.net/users/299788 | 395898 | 163,522 |
https://mathoverflow.net/questions/395883 | 3 | Let $L$ be an $n-$dimensional lattice. The Voronoi region of $L$ is given by
$$
\mathcal{V}(L)=\big\{x\in\mathbb{R}^n~|~ \|x\|\_2\leq \|x-v\|\_2~\forall v\in L\setminus\{0\}\big\}.
$$
Considering the half-spaces
$$
H\_v=\left\{x\in\mathbb{R}^n~\Big|~\langle x,v \rangle\leq \frac{1}{2}\|v\|\_2^2\right\}
$$
the normal ve... | https://mathoverflow.net/users/138478 | Covering radius of a lattice from relevant vectors | A convex polyhedra has two (equivalent) representations:
1. The *H-representation*, as an intersection of finitely many halfspaces.
2. The *V-representation*, as the convex hull of finitely many vertices.
The problem of converting between these is the problem of "Polyhedral Representation Conversion". In this langu... | 1 | https://mathoverflow.net/users/101207 | 395905 | 163,524 |
https://mathoverflow.net/questions/395718 | 2 | Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to satisfy this functor in order to "extend" to a functor $F^\*$ form the category of $\operatorname{Spec}(L)$-Schemes to the c... | https://mathoverflow.net/users/158462 | Extending functors between K-algebras to schemes | You are talking about *descent*. As you suggest in the last paragraph it is very useful in algebraic geometry.
I would prefer to reverse the question: instead of starting with a functor defined on affine schemes and trying to extend it to all schemes, you can consider a functor defined on all schemes and ask whether ... | 4 | https://mathoverflow.net/users/1310 | 395909 | 163,526 |
https://mathoverflow.net/questions/395827 | 1 | I was hoping someone could help me with the understanding of a particular truncated object. Here are some background:
For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an object in the category of complexes $\mathbf{C}(\mathcal{A})$ in $\mathcal{A}$ by setting $A$ as the degree zero object a... | https://mathoverflow.net/users/172132 | A question about a truncated object | One misconception in your post is in the definition of the derived category: we do *not* say that $A$ and $B$ are quasi-isomorphic if $H^i(A) \cong H^i(B)$ for all $i$. Instead we should define a quasi-isomorphism as a chain map inducing isomorphism on cohomology, and invert the quasi-isomorphisms, and this produces a ... | 5 | https://mathoverflow.net/users/1310 | 395911 | 163,528 |
https://mathoverflow.net/questions/395916 | 2 | Let $(X\_t,Z\_t)\_t$ be an $\mathbb{R}^{n}\times \mathbb{R}^m$-valued time-homogeneous Markov process on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}\_t)\_t,\mathbb{P})$ with transition kernel $\kappa$ and where $\mathcal{F}\_t$ is the right-continuous filtration generated by this process. Let $\mathc... | https://mathoverflow.net/users/298030 | Preservation of the Markov Property under Conditioning | No. E.g., let $n=m=1$ and $X\_t=Z\_t=B\_t$, where $B$ is the standard Brownian motion. Take the natural filtrations, so that
$E(f(X\_t)|\mathcal G\_t)=f(B\_t)$. Let $f(x)$ to be something like $\max(0,x)$. It should be easy to show that the process $(f(B\_t))$ is not Markov.
Indeed, to simplify calculations, let $f(x... | 1 | https://mathoverflow.net/users/36721 | 395917 | 163,530 |
https://mathoverflow.net/questions/395921 | 4 | A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $$H^1\_{\text{dR}}(X) \cong H^{0,1}(X) \oplus H^{1,0}(X)$$
where the groups on the right are the Dolbeault cohomology gr... | https://mathoverflow.net/users/175051 | Relationship between Dolbeault and de Rham cohomology on Riemann surface | (This would a comment, but it's hard to squeeze all the notation into the comment box.)
If you are comfortable with sheaf theory, then you can use the exact sequence
$$0\to \mathbb{C}\to \mathcal{O}\_X\to \Omega\_X^1\to 0$$
to get
$$\to H^0(X,\Omega\_X^1)\xrightarrow{\iota} H^1(X,\mathbb{C})\xrightarrow{\pi} H^1(X,\m... | 6 | https://mathoverflow.net/users/4144 | 395925 | 163,532 |
https://mathoverflow.net/questions/395894 | 3 | Let $\mu\_. : \mathbb{R}^+ \rightarrow M\_F(\mathbb{N}) $ a function. We set up :
$$ \mu\_t = \sum a\_i(t) \delta\_i$$
where each $a\_i$ is a positive continuous function from $\mathbb{R}^+$ to $\mathbb{R}^+$. If we have the following hypotheses :
1. Given $\mu\_0$, $<\mu\_0, \chi^2> := \int x^2 \mu\_0(dx) = \sum\_i ... | https://mathoverflow.net/users/164762 | Second moment of a measure with size biaised variation | Yes: If $a\_i(t)>0$ for some real $T>0$, all $t\in[0,T)$, and all $i$, then the $a\_i$'s will be decreasing on $[0,T)$, so that for all $t\in[0,T)$
we will have
$$0<\sum\_i a\_i(t) i^2\le\sum\_i a\_i(0) i^2.$$
---
**Comment:** In fact, if $a\_i(t)>0$ for some real $T>0$, all $t\in[0,T)$, and all $i$, then
$$T\le ... | 2 | https://mathoverflow.net/users/36721 | 395927 | 163,534 |
https://mathoverflow.net/questions/395918 | 2 | We can define a limit of a sequence of points in a locale in the usual way: $x$ is a limit of $\{ x\_i \}\_{i \in \mathbb{N}}$ if, for every open $U$ containing $x$, there exists $N$ such that $x\_n$ belongs to $U$ for every $n > N$. Now, let's say that we have a sequence of localic maps $\{ f\_i : X \to Y \}\_{i \in \... | https://mathoverflow.net/users/62782 | Convergence of localic maps | There is a pretty good notion of convergence of maps of locales, though I have never seen anything in the literature about it (maybe I should write something about it ?).
A map of locale $f:X \to Y$ can be thought of as a point of $Y$ in the internal logic of the sheaf topos $Sh(X)$. And a sequence/net of such maps $... | 2 | https://mathoverflow.net/users/22131 | 395928 | 163,535 |
https://mathoverflow.net/questions/395920 | 5 | **Motivation**: I need to find a mapping from $n$-dimensional Euclidean space to real numbers such that the distance between each pair of points in the quoted space is relatively-preserved after the application of the mapping.
>
> **Question**: Given $a, b, c \in \mathbb{R}^{n}$ and assuming that $||a-b|| \le ||a-c... | https://mathoverflow.net/users/106458 | Is there a mapping from Euclidean space to real numbers which relatively preserves distance? | If I understood correctly, such a mapping must be constant if $n\geq 2$.
Permuting the names of the variables, the condition implies that $f$ must send every equilateral triangle in $\mathbb{R}^n$ to an "equilateral triangle" in $\mathbb{R}$, which can only be a single point. Since every pair of points in $\mathbb{R}... | 11 | https://mathoverflow.net/users/142382 | 395933 | 163,536 |
https://mathoverflow.net/questions/395912 | 3 | If $f$ is plurisubharmonic (not identically $-\infty$) on a neighbourhood of $0$ then the [Lelong number](https://en.wikipedia.org/wiki/Lelong_number) of $f$ at $0$ is defined by $$\nu\_{f}(0) = \liminf\_{|z|\rightarrow 0}\dfrac{f(z)}{\log|z|}.$$
My question: How about $\displaystyle\limsup\_{|z|\rightarrow 0}\dfrac{... | https://mathoverflow.net/users/300739 | A question about Lelong number | Your first question is unclear. The answer to the second question is positive: you can have a plurisubharmonic function such that $f(x\_k)=-\infty$
on some sequence $x\_k\to 0$. For this function $\limsup\_{x\to 0}f(x)/\log|x|=+\infty$. Then by small modification you can make this $\limsup$ finite but as large as you w... | 3 | https://mathoverflow.net/users/25510 | 395935 | 163,537 |
https://mathoverflow.net/questions/395537 | 3 | Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a probability measure on $\mathcal B(E)$ and $\varphi\_\mu$ denote the characteristic function$^1$ of $\mu$.
Assume $\mu$ is *infinitely divisible*, i.e. there is a sequence $(\mu\_n)\_{n\in\mathbb N}$ of probability measures on $\mathcal B(E)$ such that$^2$ $$\m... | https://mathoverflow.net/users/91890 | Can we show that the characteristic function of an infinitely divisible probability measure has no zeros | $\newcommand\vpi\varphi\newcommand\R{\mathbb R}$
1. The approach involving (4) will not work, because [Lévy's continuity theorem](https://en.wikipedia.org/wiki/L%C3%A9vy%27s_continuity_theorem#:%7E:text=In%20probability%20theory%2C%20L%C3%A9vy%27s%20continuity,convergence%20of%20their%20characteristic%20functions.) w... | 3 | https://mathoverflow.net/users/36721 | 395940 | 163,539 |
https://mathoverflow.net/questions/395931 | 2 | Consider the following self-adjoint matrix
$A\_X = \begin{pmatrix} 0 & -i \\ i & X \end{pmatrix},$ where $i$ is the imaginary unit and $X$ is a uniformly distributed random variable on some interval $[-\varepsilon,\varepsilon].$
Now, take the product
$$M\_n = A\_{X\_n} \cdot...\cdot A\_{X\_1}$$
where $X\_1,...,... | https://mathoverflow.net/users/150549 | Random sequence with positive Lyapunov exponent? | We have the classical result of Furstenberg: consider a random walk $\mu$ on $SL(n,\mathbb{C})$ with finite first moment such that the semigroup generated by the support of the measure is strongly irreducible and unbounded. Then its greater Lyapunov exponent is positive.
(You can find this statement e.g. in [this sur... | 2 | https://mathoverflow.net/users/91134 | 395944 | 163,540 |
https://mathoverflow.net/questions/395941 | 3 | Let $X$ be a Gorenstein curve over a field an consider the compactified Jacobian parametrizing torsion-free, rank-1 sheaves on $X$.
Is there a chance that the dual functor $Hom(\\_, \mathcal O\_X)$ is well defined (geometric, i.e. preserve flatness of families)?
Or maybe is it the Gorenstein-dual functor $Hom(\\_, ... | https://mathoverflow.net/users/91935 | Dual family of torsion-free rank-1 sheaves on Gorenstein curves | Yes. This follows from Theorem 1.10(ii) of the paper of Altman-Kleiman cited below.
More precisely, let $S$ be a scheme and let $\mathcal{F}$ be a locally finitely presented $\mathcal{O}\_{X\_S}$-module on $X\_S$, flat over $S$, with the property that $\mathcal{F}$ is torsion-free rank $1$ in every geometric fibre of... | 4 | https://mathoverflow.net/users/110362 | 395945 | 163,541 |
https://mathoverflow.net/questions/395352 | 3 | Let $f^n$ be a family of $C^1$ functions and $f(x)=-|x|^2+1$ such that
$$f^n\to f$$
in $C^1$ sense as $\varepsilon\to 0$. I want to ask that does the level set $\{f^n=0\}$ converges to $\{f=0\}$ in some sense as $\varepsilon\to 0$? Is $\{f^n=0\}$ still a $C^1$ curve for $\varepsilon $ sufficiently small?
| https://mathoverflow.net/users/176547 | Convergence of a level set when $f^n\to f$ in $C^1$ sense | $\newcommand\de\delta$Apparently, (i) by $\epsilon\to0$ you meant $n\to\infty$ and (ii) by "$f^n\to f$ in $C^1$ sense" you meant that
$$\sup\_x|f^n(x)-f(x)|+\sup\_x|\nabla f^n(x)-\nabla f(x)|\to0. \tag{1}$$
If so, then the answers to both of your questions are positive:
>
> Question 1: "does the level set $\{f^n=0\... | 2 | https://mathoverflow.net/users/36721 | 395946 | 163,542 |
https://mathoverflow.net/questions/395951 | 3 | While trying to understand a proof in a paper, I came upon the following a calculation needing the following identity:
$$\lim\_{t\to 0} \int\_{-\infty}^\infty \left(e^{-\log(4\pi i t)/2} e^{ik^2/4t} -\delta(k)\right)f(k)\,dk=0.$$
for $f\in\mathcal{S}(\mathbb{R})$ and $t>0$. Of course, this means that the exponential ke... | https://mathoverflow.net/users/152473 | How to prove that this one-parameter family of distributions converges to the Dirac measure? | Allow me to replace $k$ by $x$. The kernel
$$G(x,t)=(4\pi it)^{-1/2}e^{ix^2/4t}$$
is the Green function of the Schrödinger equation, which can be written in the integral form
$$G(x,t)=\frac{1}{2\pi}\int\_{-\infty}^\infty e^{-ikx}e^{-ik^2 t}dk.$$
In the limit $t\rightarrow 0$ we then have an integral representation of t... | 5 | https://mathoverflow.net/users/11260 | 395959 | 163,547 |
https://mathoverflow.net/questions/395890 | 5 | Introduce the sequence (this is [A047781 on OEIS](https://oeis.org/A047781))
$$t\_n=\sum\_{k=0}^{n-1}\binom{n-1}k\binom{n+k}k$$
and denote the set $T(ij)=\{n\in\mathbb{N}: \text{the ternary digits of $n$ contain $i$ or $j$ only}\}$.
>
> **QUESTION.** Is this true modulo $3$?
> $$t\_n\equiv\_3\begin{cases} 1 \qquad ... | https://mathoverflow.net/users/66131 | Modulo $3$ calculations for a binomial-sum sequence | The answer is Yes.
The generating function for $t\_n$ is
$$\sum\_{n\geq 0} t\_n x^n = \frac14\big(\frac{1+x}{\sqrt{1-6x+x^2}}-1\big).$$
Correspondingly,
$$\sum\_{n\geq 0} t\_n x^n \equiv \frac{1+x}{\sqrt{1+x^2}}-1 \pmod{3}.$$
It follows that for $n>0$,
$$t\_n \equiv \binom{-1/2}{\lfloor n/2\rfloor}\equiv (-1)^{\lfloo... | 7 | https://mathoverflow.net/users/7076 | 395964 | 163,548 |
https://mathoverflow.net/questions/393283 | 5 |
>
> Is there a category with binary biproducts but no zero morphism?
>
>
>
I'm wondering if the definition of [biproducts](https://ncatlab.org/nlab/show/biproduct) as objects that are simultaneously products and coproducts that obey some identities on the projections/injections is 'different' than the definition... | https://mathoverflow.net/users/92164 | Category with binary biproducts but no zero morphism | Karvonen has an [article on arXiv](https://arxiv.org/pdf/1801.06488.pdf) showing that if a category has all binary biproducts (using the alternative definition Tom mentioned) then it has zero morphisms (Corollary 3.3).
| 2 | https://mathoverflow.net/users/54401 | 395968 | 163,551 |
https://mathoverflow.net/questions/395966 | 6 | Let ${f : E \rightarrow S}$ be a geometric morphism (between toposes).
For $s$ in $S$ and $x$ in $E$ let ${\pi : f^\* s \times x \rightarrow x}$ be the obvious projection in $E$.
Let ${u \rightarrow f^\* s \times x}$ be a complemented subobject of ${f^\* s \times x}$.
Is the image of $u$ along $\pi$ complemented ... | https://mathoverflow.net/users/121350 | Images of complemented subobjects in toposes | No, not even if $E=S$, $f$ is the identity morphism, and $x=1$. In that special case, your question asks whether $\forall z\in s\,\big((z\in u)\lor \neg(z\in u)\big)$ (in the internal language of $S$) implies $(\exists z\in s\,z\in u)\lor\neg(\exists z\in s\,z\in u)$. When $s$ is $\mathbb N$, this is the limited princi... | 11 | https://mathoverflow.net/users/6794 | 395969 | 163,552 |
https://mathoverflow.net/questions/374391 | 2 | **The Problem**
For a given graph $G$, the cubic subgraph problem asks if there is a subgraph where every vertex has degree 3.
The cubic subgraph problem is NP-hard even in bipartite planar graphs with maximum degree at most 4.
Suppose we have an oracle that decides if a bipartite graph contains a "two from cubic s... | https://mathoverflow.net/users/167250 | Two from cubic subgraph hardness | A formal proof has been produced.
See <https://arxiv.org/abs/2105.07161>
| 1 | https://mathoverflow.net/users/167250 | 395974 | 163,553 |
https://mathoverflow.net/questions/395983 | 6 | Let $X$ be a compact complex manifold, and $f: Y\to X$ a proper surjective holomorphic map with fiber $\mathbb{CP}^n$. Is there always a holomorphic vector bundle $E$ of rank $n+1$ such that $Y$ is biholomorphic to $\mathbb{P}(E)$ over $X$?
| https://mathoverflow.net/users/192152 | Does any projective bundle on a compact complex manifold have an associated holomorphic vector bundle? | It is classically known that this is true when $\dim X=1$ ("ruled surfaces" = "geometrically ruled surfaces").
It is also true when $\dim X=2$, provided that $H^2(X, \, \mathcal{O}\_X)=H^3(X, \, \mathbb{Z})=0$.
However, it fails for a general smooth basis. You can see the discussion at p. 190 of
*Barth, Wolf P.; ... | 9 | https://mathoverflow.net/users/7460 | 395986 | 163,555 |
https://mathoverflow.net/questions/394252 | 8 | The notion of a "compact category" was introduced by Isbell$\color{red}{^{1,2}}$. A locally small category $\mathcal{C}$ is called *compact* when every functor $\mathcal{C} \to \mathcal{D}$ into any category $\mathcal{D}$ which preserves all (possibly large!) colimits is a left adjoint. Equivalently, every presheaf $\m... | https://mathoverflow.net/users/2841 | Strongly compact categories (reference request) | In [Adjoints to functors from categories of algebras](https://www.tandfonline.com/doi/abs/10.1080/00927877508822061?journalCode=lagb20) Rattray defines that a category $\mathcal{A}$ has **LAP** (left adjoint property) when every continuous functor on $\mathcal{A}$ has a left adjoint, or equivalently every continuous fu... | 2 | https://mathoverflow.net/users/2841 | 395994 | 163,556 |
https://mathoverflow.net/questions/395997 | 4 | Suppose $G$ is a finite group with a dihedral maximal subgroup. Suppose that $G$ is not isomorphic to $\operatorname{PSL}(2,q)$ for some any prime-power $q$. Is $G$ always solvable?
| https://mathoverflow.net/users/134942 | Finite groups with a dihedral maximal subgroup | Not necessarily . The Suzuki simple groups ${ \rm Sz}(2^{2n+1})$ also have dihedral maximal subgroups. These are the normalizers of the Hall subgroup of order $2^{2n+1}-1.$
| 9 | https://mathoverflow.net/users/14450 | 395999 | 163,557 |
https://mathoverflow.net/questions/395973 | 17 | There is a paper (not accepted for publication yet) that contains several conjectures. Some of these conjectures were proven recently.
The referee of the original paper requires to substitute the proven "Conjectures" with the "Results". However, there are several papers that cite these conjectures, so I feel it would... | https://mathoverflow.net/users/3840 | Conjectures or Results? | The standard way is to leave the conjectures as they are, and add a remark, or a footnote, saying that "after this paper was written (or after it was submitted for publication) this conjecture was proved"
and give a reference.
| 41 | https://mathoverflow.net/users/25510 | 396007 | 163,560 |
https://mathoverflow.net/questions/395990 | 10 | Let $G$ be a compact topological group. Then $G$ is a CQG with function algebra $C(G)$ and the usual comultiplication on $C(G)$. Is there an easy description of the dual discrete quantum group $\widehat{G}$?
In the case of commutative compact groups, I would hope that there is a connection with the usual dual of a co... | https://mathoverflow.net/users/216007 | What is the discrete quantum group associated to a compact group? | I believe that really the question is being asked in the context of [Locally compact quantum groups](https://en.wikipedia.org/wiki/Locally_compact_quantum_group). This is a framework using the machinery of $C^\*$ and von Neumann algebras, with (amoung many aims, and many different motivations) the aim of extend Pontrya... | 8 | https://mathoverflow.net/users/406 | 396014 | 163,562 |
https://mathoverflow.net/questions/395742 | 4 | We say that a simplicial complex $K$ is acyclic if it's integral reduced simplicial homology groups are trivial in all dimensions.
For a vertex ${v} \in K$, we define the link
$$lk(v) :=\{\sigma \in K \; | \; \sigma \cup \{v\} \in K, \sigma \cap \{v\} = \emptyset\}.$$
A simplicial complex is an integral generalis... | https://mathoverflow.net/users/103150 | An acyclic simplicial complex where all links are generalised homology spheres | If all vertex links in a finite simplicial complex $K$ are homology $n-1$-spheres (i.e., homeomorphic to $n-1$-manifolds with the same homology as an $n-1$-sphere), then the simplicial complex $K$ is a closed homology $n$-manifold. As such it has a mod-2 fundamental class: $H\_n(K;\mathbb{F}\_2)\cong \mathbb{F}\_2$. Ap... | 3 | https://mathoverflow.net/users/124004 | 396015 | 163,563 |
https://mathoverflow.net/questions/395957 | 3 | Let $S$ be a scheme and $f : X\to S$ be an $S$-scheme. This question asks for examples of maps of sets $X(S) \to X(S)$ that do not come from an $S$-scheme endomorphism of $X$, but that, roughly, specialize to maps $X\_s(\kappa(s))\to X\_s(\kappa(s))$ that do come from a $\kappa(s)$-scheme endomorphism of the fiber $X\_... | https://mathoverflow.net/users/nan | Interpolation of scheme-theoretic endomorphisms of closed fibers | Choose a smooth projective $X/R$ of positive dimension, and pick a set-theoretic splitting $\varphi:X(\kappa(s))\to X(R)$ of the reduction map $\pi:X(R)\to X(\kappa(s))$. Take $a=\varphi\circ \pi$, and let $a\_0$ and $\alpha\_0$ be the identity. Then $a$ is constant on residue disks, so is locally constant for the anal... | 2 | https://mathoverflow.net/users/5263 | 396030 | 163,566 |
https://mathoverflow.net/questions/396026 | 4 | Let $T$ be an $n$-dimensional area-minimising hypersurface in $\mathbf{R}^{n+1}$. If $T$ has bounded area growth in the sense that there is a constant $C > 0$ so that $\mathcal{H}^n(T \cap B\_R) \leq C R^n$ for all $R > 0$, then there are rigidity theorems for $T$. For example, when $n \leq 6$ then the work of Simons [... | https://mathoverflow.net/users/103792 | Area-minimising hypersurface with unbounded area growth | This is a straightforward comparison argument. Let $\omega\_n$ be the volume of $\partial B\_1\subset \mathbb{R}^{n+1}$.
For generic $R$, one has $\partial B\_R \cap T=\tau$ a smooth submanifold. By Alexander duality, there is a subset, $\Omega$, of $ \partial B\_R\setminus \tau$ so $\partial \Omega=\tau$. Clearly, $... | 4 | https://mathoverflow.net/users/127803 | 396037 | 163,571 |
https://mathoverflow.net/questions/396032 | 1 | Let $A\in \mathcal{M}\_{m\times m}(\mathbb R)$ , $det(A)=1$ , $A$ is positively definite. Which matrices $P$ satisfy the equation
$$P^TAP=A$$
In fact I am interested in sequences of traces $tr P^n$ of the iterations of such solutions.
In dimension $2$ one can show that
$$P^n=\left(
\begin{array}{cc}
\cos n\phi& -\b... | https://mathoverflow.net/users/46230 | Matrix equation $P^TAP=A$ | As $A=L^\top L$, for some $L\in M\_{m\times m}(\mathbb{R})$, $\det L=1$, you can rewrite $P^\top L^\top LP=L^\top L$ and then multiply both sides by $L^{-1}$, etc., obtaining $(L^{\top})^{-1}P^\top L^\top LPL^{-1}=(LPL^{-1})^\top LPL^{-1}=I$, i.e. each $LPL^{-1}$ must be orthogonal.
As traces are preserved under conj... | 3 | https://mathoverflow.net/users/11100 | 396039 | 163,573 |
https://mathoverflow.net/questions/396041 | 3 | Let $S$ be a Boolean topos.
Let ${f : E \rightarrow S}$ be a hyperconnected geometric morphism.
For $s$ in $S$ and $x$ in $E$ let ${\pi : f^\* s \times x \rightarrow x}$ be the obvious projection in $E$.
Let ${u \rightarrow f^\* s \times x}$ be a complemented subobject of ${f^\* s \times x}$.
Is the image of $u... | https://mathoverflow.net/users/121350 | Images of complemented subobjects in hyperconnected toposes over Boolean bases | No.
Take $S$ to be Sets, then for any set $s$, $f^\* s \times x$ is the coproduct of $s$-copies of $X$, and a complemented subobject of $f^\* s \times x$ is the same as an $s$ indexed collection of complemented subobject of $x$. The the image by the projection $f^\*s \times x \to x$ is their union.
So, for $S= Set$... | 6 | https://mathoverflow.net/users/22131 | 396043 | 163,575 |
https://mathoverflow.net/questions/396044 | 8 | Let $T$ be a triangulation of sphere. We say that $T$ is $k$-colorable if the triangles of $T$ can be assigned with $k$ colors such that any two triangles with a common edge have different colors.
I am interested in $2$-colorable triangulations. Easy examples are boundaries of bipyramids over even polygons such as th... | https://mathoverflow.net/users/2083 | When is a triangulation of sphere two-colorable? | As Fedor Petrov mentions in the comments, a necessary and sufficient condition is that each vertex has even degree. Here is a proof.
Let $T^\*$ be the dual graph of the triangulation. That is, the vertices of $T$ are the faces of the triangulation, and two faces are adjacent if they share an edge. Rephrased, your que... | 11 | https://mathoverflow.net/users/2233 | 396046 | 163,577 |
https://mathoverflow.net/questions/395982 | 4 | Usually when one has a short exact sequence of bundles,
\begin{eqnarray}
0\rightarrow A \rightarrow B\rightarrow C\rightarrow 0,
\end{eqnarray}
then there is an associated long exact sequence,
\begin{eqnarray}
0\rightarrow S^2A\rightarrow A\otimes B\rightarrow \Lambda^2B\rightarrow \Lambda^2C\rightarrow 0.
\end{eqn... | https://mathoverflow.net/users/140074 | Antisymmetric product of complexes | You can interpret your identity as saying: if $cone(A \to B) \simeq C$ then $\wedge^2(cone(A \to B)) \simeq \wedge^2 C$, where $\wedge^2$ of complexes is defined using the Koszul symmetric monoidal structure on complexes. As discussed in the comments, this is only true when $2$ is invertible because to prove it we need... | 2 | https://mathoverflow.net/users/131945 | 396057 | 163,579 |
https://mathoverflow.net/questions/396058 | 6 | From what I currently understand, under certain conditions one may turn the usual Kantorovich problem - a minimisation problem in terms of measures into a maximisation problem in terms of functions. By “turn into” I mean that the optimal values for both problems agree.
The Kantorovich potential associated to the prob... | https://mathoverflow.net/users/173490 | What is the intuition behind the Kantorovich potential in optimal transport? | I recommend the interpretation with bakeries and cafes! In Villani's "Optimal Transport Old and New" in Chapter 5 "Cyclic monotonicity and Kantorovich duality you'll find this:
>
> I shall start by explaining the concepts of cyclical monotonicity and Kantorovich duality in an informal way, sticking to the bakery an... | 5 | https://mathoverflow.net/users/9652 | 396063 | 163,580 |
https://mathoverflow.net/questions/396061 | 3 | In the following paper:
<https://perso.crans.org/besson/publis/mva-2016/MVA_2015-16__Kernel_Methods__Homework__Besson_Clement_Zerbib.en.pdf>
problem 2, Kernel 9. it is shown that $K(x,y) = \frac{\min(x,y)}{\max(x,y)}$ is a positive definite kernel.
I am asking myself, if there is a feature mapping $\phi$ in a Hilbe... | https://mathoverflow.net/users/165920 | Is there a feature mapping for this kernel $k(x,y) = (\frac{\min(x,y)}{\max(x,y)})^2$? | The native Hilbert-space of $K^2$ is well known.
I assume the domain of $K$ is $\mathbb{R}^{>0}\times\mathbb{R}^{>0}.$ Note that:
$$ K(x,y)=\begin{cases}
\frac{x}{y} \text{ for } x\leq y\\
\frac{y}{x} \text{ for } x\geq y.\\
\end{cases}$$
Set $x = \exp u$ and $y = \exp v$ to obtain
$$ K(x,y)^2=K(\exp 2u, \exp 2v)=\ex... | 7 | https://mathoverflow.net/users/7695 | 396072 | 163,581 |
https://mathoverflow.net/questions/396088 | 6 | Let $K$ be a field and let $\Lambda\_{1}$ and $\Lambda\_{2}$ be two finite-dimensional $K$-algebras with Jacobson radicals $J\_{1}$ and $J\_{2}$ respectively. How to show or where can I find the proof of the following statement?
>
> $\Lambda\_{1} / J\_{1} \otimes\_{K} \Lambda\_{2} / J\_{2}$ is always semisimple if ... | https://mathoverflow.net/users/118028 | Tensor of finite-dimensional algebra over perfect field is semisimple | We have $gldim A \otimes\_K B= gldim A + gldim B$ if A and B are seperable algebras over the field $K$, see <https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/on-the-dimension-of-modules-and-algebras-viii-dimension-of-tensor-products/58116B52E52F0F6165E84AE11284CCF6> corollary 18.
Now being ... | 6 | https://mathoverflow.net/users/61949 | 396089 | 163,584 |
https://mathoverflow.net/questions/396086 | 1 | In the paper *Floer cohomology of Lagrangian intersections
and pseudo-holomorphic disks 2*, in the part of the preliminaries the author considers a Hamiltonian action of the isometry group $G$ of $\mathbb{C}\mathbb{P}^n$ on $\mathbb{C}\mathbb{P}^n$. The action is described in terms of the dual of the lie algebra and th... | https://mathoverflow.net/users/nan | Moment map in Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $2$ | The choice of $\xi$ amounts to choosing a Hamiltonian circle action on $\mathbb{CP}^n$ by isometries. I.e. choosing a suitable 1-parameter subgroup of the $PU(n+1,\mathbb{C})$.
Consider the Hamiltonian circle action $$z.[z\_{0} : z\_{1} : \ldots : z\_{n}] = [z\_{0} : z z\_{1} : \ldots :z^{n} z\_{n}] , $$
Which will... | 1 | https://mathoverflow.net/users/99732 | 396090 | 163,585 |
https://mathoverflow.net/questions/396092 | 6 | Most definitions of the rational numbers as a higher inductive type in univalent homotopy type theory (such as those in the cubical Agda library for example) require either the use of a quotient set or a 0-truncation constructor. Is there a way to define the rational numbers as a higher inductive type without using eit... | https://mathoverflow.net/users/nan | Defining rational numbers without using quotients or 0-truncations | One version of the theory of continued fractions is as follows. We can define operations $S,T,J\colon\mathbb{Q}^+\to\mathbb{Q}^+$ by $S(x)=x+1$ and $J(x)=1/x$ and $T(x)=JSJ(x)=x/(x+1)$, then we can define $M$ to be the free monoid generated by $S$ and $T$. We then have an evaluation map $M\to\mathbb{Q}^+$ given by $m\m... | 12 | https://mathoverflow.net/users/10366 | 396095 | 163,586 |
https://mathoverflow.net/questions/396077 | 2 | This question is based on [http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf](http://www.science.smith.edu/%7Ejorourke/Papers/FoldingPP.pdf).
Therein is stated the theorem: *Every convex polygon folds to an infinite number (a continuum) of noncongruent convex polyhedra.*
Question: What could one say in th... | https://mathoverflow.net/users/142600 | Convex polyhedra that can be folded from convex polygons | This answers the easiest question posed (1), and addresses part of the more general question (2).
>
> (1) "If there are convex polyhedrons that *cannot* be folded from convex polygons,..."
>
>
>
Yes, there is an abundance of such polyhedra. For example, let $P$ be a
cube. To unfold it to the plane, one must fo... | 4 | https://mathoverflow.net/users/6094 | 396098 | 163,589 |
https://mathoverflow.net/questions/376011 | 13 | *This was [asked and bountied](https://math.stackexchange.com/questions/3885238/are-these-finite-ish-sets-closed-under-union) at MSE without success.*
Throughout, we work in $\mathsf{ZF}$.
Say that a set $X$ is $\Pi^1\_1$-pseudofinite if for every first-order sentence $\varphi$, if $\varphi$ has a model with underl... | https://mathoverflow.net/users/8133 | Is this notion of finiteness closed under unions? | No, that class doesn't need to be closed under unions. I’ll describe a permutation model with two $\Pi\_1^1$-pseudofinite sets whose disjoint union is not $\Pi\_1^1$-pseudofinite. You can use Jech-Sochor to get a ZF model.
Fix a finite field $K.$ Consider the class of tuples $M=(X^M,Y^M,e^M)$ such that $X^M$ and $Y^M... | 3 | https://mathoverflow.net/users/164965 | 396101 | 163,591 |
https://mathoverflow.net/questions/396091 | 2 | Is it true that
$$f(x)=\lim\_{n\to\infty} 2 \sum \_{k=0}^n \left((k-1) \text{Li}\_k\left(\frac{f(x)}{n^2}\right)-x \text{Li}\_{k-1}\left(\frac{f(x)}{n^2}\right)\right)?$$
Here, $f(x)$ is an arbitrary function that I tested. I found this by chance, but numerically it looks OK (tried 5000 terms with $\exp$, $\cosh$, ... | https://mathoverflow.net/users/10059 | This equality numerically looks well. Is there any justification? | $\newcommand\Li{\text{Li}}$This follows immediately because uniformly over all $k\ge-1$ we have $\Li\_k(z)\sim z$ as $z\to0$, where $\Li\_k$ is the [polylogarithm function](https://en.wikipedia.org/wiki/Polylogarithm).
---
**Details:** For $k\ge-1$ and $|z|\downarrow0$,
$$\Big|\frac{\Li\_k(z)}z-1\Big|=\Big|\frac ... | 8 | https://mathoverflow.net/users/36721 | 396103 | 163,592 |
https://mathoverflow.net/questions/396052 | 11 | It's well-known that there are no rigorously constructed and physically relevant QFTs. There is, however, a lot of mathematical work on effective field theories and renormalization, such as the books by Costello and by Salmhofer. My question is: does this mathematical work allow one to give mathematically rigorous (alb... | https://mathoverflow.net/users/81654 | State of rigorous effective quantum field theories | I will leave aside what is meant by "effective field theory" in a purely mathematical context and just presume that the question asks whether renormalized interactive perturbative QFT (using formal power series in $\hbar$ and the coupling constants) can be mathematically well-defined. The answer is Yes (in multiple dif... | 8 | https://mathoverflow.net/users/2622 | 396108 | 163,595 |
https://mathoverflow.net/questions/396102 | 4 | Let $M$ be a connected projective complex manifold with a smooth anticanonical divisor $D$ ($D \sim -K\_M$).
In an answer to a [previous question](https://mathoverflow.net/questions/391540/possible-number-of-components-of-anticanonical-sections-of-projective-manifolds),
It is told that $D$ may have at most two componen... | https://mathoverflow.net/users/69559 | Projective manifold whose anticanonical section is composed of two components | If you look at the proof of the theorem referenced in the answer to that linked question, near the bottom of p.801 and top of p.802 it is established that if you run an MMP (the chosen boundary divisors specified in the proof), the first time you encounter a Fano contraction (which you must), then it is a $\mathbb P^1$... | 5 | https://mathoverflow.net/users/10076 | 396122 | 163,603 |
https://mathoverflow.net/questions/396131 | 1 | For any set $X$, let $[X]^2=\{\{a,b\}:a\neq b \in X\}$. If $n\in\mathbb{N}$ is a positive integer, let $S\_n$ denote the collection of bijections $\varphi:\{0,\ldots,n-1\}\to\{0,\ldots,n-1\}$. Let $E\_n\subseteq [S\_n]^2$ be given by $$E\_n = \{\{\varphi, \psi\}: \varphi, \psi \in S\_n\text{ and } (\exists a\neq b\in \... | https://mathoverflow.net/users/8628 | Hamiltonian cycle in $S_n$ with transpositions | Yes, $(S\_n, E\_n)$ contains a Hamiltonian cycle for every $n \geq 3$. This follows by the [Steinhaus–Johnson–Trotter algorithm](https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm). The transpositions can even be chosen to be consecutive elements in the previous permutation.
| 4 | https://mathoverflow.net/users/2233 | 396140 | 163,606 |
https://mathoverflow.net/questions/396137 | 4 | This is a follow-up to this question:
[Reduction to graph subgroups for Bredon homology when the $G\_1\times G\_2$ is $G\_2$-free](https://mathoverflow.net/questions/395183/reduction-to-graph-subgroups-for-bredon-homology-when-the-g-1-times-g-2-is-g/395285#395285)
In his (very nice) answer Gregory Arone stated the fo... | https://mathoverflow.net/users/123432 | Isomorphism of coends | Knowing that $Q\_!F : y\mapsto \int^z \hom(Qz,y)\otimes Fz$ (this is often called "pointwise formula" for Kan extensions) it is easy to derive the isomorphism in question:
$$
\begin{align}
Q\_!F\;\otimes\_{\cal C\_0} G& := \int^y Q\_!F(y)\otimes Gy \\
&\cong \int^y \int^z \hom(Qz,y)\otimes Fz \otimes Gy \\
&\cong \in... | 6 | https://mathoverflow.net/users/7952 | 396144 | 163,608 |
https://mathoverflow.net/questions/396155 | 2 | Let $A$ be a finite dimensional algebra over some field $K$. Denote the finistic dimension of A by fin($A$), that is, the supremum of the projective dimensions of finite generated modules whose projective dimensions are finite. Let $K\subset L$ be a field extension. If we tensor $A$ with $L$ over $K$. Then we get a new... | https://mathoverflow.net/users/134942 | Finistic dimensions under scalar extensions | We have $fin(A)=fin(A')$ at least for finite extensions by theorem 16 of <https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/on-the-dimension-of-modules-and-algebras-viii-dimension-of-tensor-products/58116B52E52F0F6165E84AE11284CCF6> .
Namely for two algebras $A$ and $B$ over a field $K$, we ... | 1 | https://mathoverflow.net/users/61949 | 396158 | 163,612 |
https://mathoverflow.net/questions/392452 | 3 | Zagier lectures on "From 3-manifold invariants to number theory" - do you know about texts of that or on the discussed web of ideas? ([https://www.mpim-bonn.mpg.de/de/node/10791])
| https://mathoverflow.net/users/451 | Zagier's "From 3-manifold invariants to number theory"? | In the following page there is some information on the lecture series. In particular, it is said that some student(?) has been taking notes which were initially private, but he decided to distribute it with kindness. You can find that through a link in that page.
<https://www.math.sissa.it/course/phd-course/3-manifol... | 1 | https://mathoverflow.net/users/303767 | 396160 | 163,613 |
https://mathoverflow.net/questions/395868 | 9 | Suppose that I am given the graph $G = (V,E)$ where $V = \{ 1, 2, \dots 2N+1 \} \times \{ 1, 2, \dots 2N+1 \} $ and there is an edge between two vertice $(n,m)$ and $(n',m')$ if and only if $\vert n-n'\vert + \vert m-m'\vert = 1$.
Suppose that we remove some arbitrary edges between vertices $(n,m)$ and $(n',m')$ with $... | https://mathoverflow.net/users/143779 | Are there more paths exiting a box in $\mathbb{Z}^2$ to the right if I remove some edges to the left | Expanding on Anthony Quas' comments above, it is indeed possible to show that there are cases when there are more paths to the left than to the right.
Let $H$ be the graph obtained from $G$ by removing all edges from $(N-1,i)$ to $(N,i)$ except one (which then clearly is a bridge in $H$).
Any walk in $H$ that ends ... | 5 | https://mathoverflow.net/users/97426 | 396165 | 163,615 |
https://mathoverflow.net/questions/396148 | 8 | Let $\varphi:\mathbb R^n \to \mathbb R^n$ be just some continuous function.
If the image of $\varphi$ happens to contain $\mathbb Q^n$, does it follow that in fact all of $\mathbb R^n$ is contained in the image as well?
No, it does not. For instance, the map given by
$$(x,y)\overset{\varphi}{\longmapsto} (xy-1+\pi,... | https://mathoverflow.net/users/2502 | Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective? | A counterexample for $n=2$ is the map $\varphi(x,y) = (x,(x^2-2)y)$. Each point $(r,s)\in\mathbb{Q}^2$ is the image of $\left(r,\frac{s}{r^2-2}\right)\in\mathbb{Q}^2$, but e.g. $(\sqrt{2},1)\not\in\varphi(\mathbb{R}^2)$.
| 13 | https://mathoverflow.net/users/5263 | 396170 | 163,618 |
https://mathoverflow.net/questions/396171 | 2 | In my little research project, I faced the following problem: Assume that $\rho$ is a probability density function with support $[0,\infty)$ and mean $\mu >0$. Let $$H[\rho] = \iiint\_{y,v,w\geq 0} \rho(v)\rho(w)\rho(y)\left(\frac{|v-y|+|w-y|}{2} - \left|\frac{v+w}{2} -y\right|\right)\,\mathrm{d}v\,\mathrm{d}w\,\mathrm... | https://mathoverflow.net/users/163454 | Trying to bound one functional by another functional | Such a function $f$ must be identically zero.
Indeed, by approximation, the problem can be restated as follows:
>
> Suppose that $f\colon[0,\infty)\to[0,\infty)$ is a function such that for any nonnegative iid random variables (r.v.'s) $X,Y,Z$ we have
> $$H\ge f(G),\tag{1}$$
> where
> $$G:=E|X-Y|,\quad H:=E|X-Y|-... | 4 | https://mathoverflow.net/users/36721 | 396175 | 163,619 |
https://mathoverflow.net/questions/396176 | 1 | This should probably be not that hard, but I would like to see a nifty way of proving it.
Consider the double-indexed sequence given by
$$f(n,k)=\binom{2n + 2k}{n + k}\binom{n + k}{n - k}3^k.$$
>
> **QUESTION.** For $1\leq k\leq n$, does this hold true for the $3$-adic valuations?
> $$\nu\_3(f(n,k))>\nu\_3(f(n,0)... | https://mathoverflow.net/users/66131 | Inequality for $3$-adic valuation | Notice that
$$
3^{-k-1}f(n,k+1)/3^{-k}f(n,k)=\frac{{2n+2k+2\choose n+k+1}}{{2n+2k\choose n+k}}\frac{{n+k+1\choose n-k-1}}{{n+k \choose n-k}}.
$$
The first factor here equals
$$
\frac{(2n+2k+2)!}{(2n+2k)!}\frac{(n+k)!^2}{(n+k+1)!^2}=\frac{2(2n+2k+1)}{n+k+1}.
$$
The second is
$$
\frac{(n+k+1)!}{(n+k)!}\frac{(n-k)!}{(n-k-... | 5 | https://mathoverflow.net/users/101078 | 396180 | 163,620 |
https://mathoverflow.net/questions/396031 | 0 | By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}\_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries
$$\mathbf{\Gamma}\_{n\times k} \sim \mathbf{Q}\_{n\times k}\mathbf{R}\_{k\times k}$$
for $\mathbf{Q}\_{n\times k}, \: \mathbf{R}\_{k\times k}$ ... | https://mathoverflow.net/users/nan | Factorisation of Gaussian random matrix into random Hermitian and correction factor | Write the SVD of $\Gamma$, say $\Gamma = \sum\_i q\_i s\_i v\_i^T$.
with $s\_1,...,s\_n>0$ the singular values and $q\_i, v\_i$ are the left and right singular vectors.
If $Q=[q\_1|...|q\_n]$, $B=diag(s\_1,...,s\_n)$ and $V=[v\_1|...|v\_n]$ then $\Gamma=QBV^T$. The crux of the matter is that $(Q,B,V)$ are mutually inde... | 1 | https://mathoverflow.net/users/141760 | 396184 | 163,622 |
https://mathoverflow.net/questions/396194 | 9 | Over the years of my study of set theory, I have encountered several sentences of the form *V = X*: *V = L*, *V = HOD*, *V = WF* (the exclusive assertion of the cumulative hierarchy), and (if I understand [this paper](https://arxiv.org/abs/math/0612636)) *V = HW* ("hereditarily winning").
Now, in general epistemology... | https://mathoverflow.net/users/147890 | The justifiable universe | I find the question interesting.
But I believe that your division of set theories into justificatory types is undermined by the fact that we have instances of bi-interpretable theories that cross the type boundaries.
For example, ZFC set theory is bi-interpretable with the antifoundational theory ZFC-foundation+AFA... | 9 | https://mathoverflow.net/users/1946 | 396204 | 163,627 |
https://mathoverflow.net/questions/396192 | 2 | I asked this question on Math Stack Exchange and did not receive any answers or comments.
Suppose $M$ and $N$ are monoidal categories and let $M\times{N}$ denote the associated product category. $M\times{N}$ comes equipped with two natural projection functors $\pi\_{M}:M\times{N}\rightarrow{M}$ and $\pi\_{N}:M\times{... | https://mathoverflow.net/users/135352 | Pushout of the diagram of a monoidal product category along its projection functors | The answer is that it depends (immensely) on what you mean by "category of monoidal categories".
Let me start with a general statement:
Suppose $C$ is a pointed category.
Suppose further that $C$ has finite products, and let $M,N\in C$; and $P$ with maps $p: M\to P, q: N\to P$ such that $p\circ \pi\_M = q\circ \p... | 3 | https://mathoverflow.net/users/102343 | 396210 | 163,629 |
https://mathoverflow.net/questions/396189 | 15 | I would like to prove the following inequality. It arises from my study of random matrices.
I have verified the inequality for $q\in \{0.01,0.02, \ldots, 0.99\}$ and $1\le n\le 100$.
Let $n$ be any positive integer and $0\le q\le 1$. Then the following inequality is true.
$$\sum\_{k=0}^n(-1)^k\binom{n}{k}(q^k-q^n)^n\... | https://mathoverflow.net/users/306951 | Combinatorial inequality involving alternating signs | Mark each box of an $n\times n$ table with probability $q$. By inclusion-exclusion the difference RHS-LHS equals to the probability that there exists a full row (with all boxes marked) but there does not exist a full column: that's because for given $k$ rows the probability that (they are full but no column is full) eq... | 21 | https://mathoverflow.net/users/4312 | 396211 | 163,630 |
https://mathoverflow.net/questions/385579 | 0 | Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem.
[This answer](https://math.stackexchange.com/a/198091/64809) on math.stackexchange.com claims the following proposition regarding a special case.
Using Farkas' lemma to ... | https://mathoverflow.net/users/32660 | Confining a polytope to one side of an affine hyperplane | Here is a proof that uses directly the [separating hyperplane theorem](https://en.wikipedia.org/wiki/Hyperplane_separation_theorem), which states the following.
Given a matrix $A$ and column matrix $b$,
\begin{align}
Ax=b, &\text{ for some column matrix }x\ge0 \\
&\iff \\
y^TA\ge0 &\text{ for some column matrix }y \i... | 0 | https://mathoverflow.net/users/32660 | 396227 | 163,634 |
https://mathoverflow.net/questions/396205 | 5 | Recently, I've encountered the following question:
Assume that $n\_{1}, \ldots, n\_{k}$ are (not necessary distinct) natural numbers. If
$$ (\sum\_{i = 1}^{k}\sqrt{n\_{i}}) \in \mathbb{N},$$ can we conclude that all $n\_{i}$'s are perfect squares? Is there any famous theorem that answer this question? Or, can anyon... | https://mathoverflow.net/users/125843 | Sum of square roots of natural numbers | Let us show a more general statement, and then show how it implies your question: given distinct positive squarefree numbers $n\_1, n\_2, \dots, n\_k$, the numbers $\sqrt{n\_1}, \dots, \sqrt{n\_k}$ are linearly independent over $\mathbb{Q}$.
Proof: Suppose that
$$\sum\_{i = 1}^{k} a\_i \sqrt{n\_i} = 0$$
where without... | 14 | https://mathoverflow.net/users/88679 | 396241 | 163,636 |
https://mathoverflow.net/questions/396231 | 11 | I'm new to asking questions on MathOverflow, so forgive me if this question is not the kind of thing to be asked here.
Let $q$ be a positive integer and let $N$ be an integer with $1 \leq N \leq q$. The estimate
$$
\sum\_{\substack{n= 1\\ (n,q)=1}}^N 1 = N \frac{\phi(q)}{q} + O(2^{\omega(q)})
$$
is a classical and st... | https://mathoverflow.net/users/307675 | Improving the error term in a classic sieving problem | In fact there are moduli $q$ with arbitrarily many prime factors where the error term can be shown to be as large as $2^{\omega(q)-2}$. The following construction is due to D.H. Lehmer, [The distribution of totatives](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/distribution-of-totati... | 14 | https://mathoverflow.net/users/38624 | 396249 | 163,639 |
https://mathoverflow.net/questions/396255 | 7 | Consider the surface group $S\_g=\langle a\_1,b\_1,a\_2,b\_2,\dots,a\_g,b\_g \mid [a\_1,b\_1][a\_2,b\_2]\cdots[a\_{g},b\_{g}]=1\rangle$, which is the fundamental group of the closed orientable genus-$g$ surface.
Suppose $2\leq m<n$ and let $p:S\_n\to S\_m$ be the canonical projection, which is the identity on generat... | https://mathoverflow.net/users/5801 | Factoring a projection of surface groups through a free group | The homomorphism $p$ is non-trivial on second (co)homology, since it is induced by a degree one map of surfaces ("collapse $n-m$ holes to a point"). This can also be seen using the structure of the cohomology rings of orientable surfaces.
Hence no such factorisation exists, as free groups have zero (co)homology above... | 11 | https://mathoverflow.net/users/8103 | 396257 | 163,640 |
https://mathoverflow.net/questions/396230 | 5 | I am looking for examples of projective varieties (over $\mathbb{C}$) of dimension, say $n$ which cannot appear as an exceptional divisor of a blow-up of $\mathbb{P}^{n+1}$ along some closed subscheme. Any idea/reference will be most welcome.
| https://mathoverflow.net/users/58203 | Which projective varieties cannot appear as an exceptional divisor of a blow-up of the projective space | The exceptional divisor is the $\operatorname{Proj}$ of the normal sheaf, hence it is covered by rational curves. This excludes all the non-uniruled varieties.
| 6 | https://mathoverflow.net/users/7460 | 396259 | 163,642 |
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