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https://mathoverflow.net/questions/404907 | 1 | Let $\boldsymbol{X} = (X\_1,X\_2)^{\rm T}\sim \mathcal{N}\_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where
\begin{eqnarray\*}
\boldsymbol{\mu} = (\mu\_1, \mu\_2)^{\rm T}& = &(\sqrt{\xi\_1\xi\_2/(\xi\_1+\xi\_2)}, 0)^{\rm T}\\
\mathrm{\Sigma} & = &\begin{pmatrix} 1 & -\rho\\
-\rho & 1\end{pmatrix}\\
\rho & = & \sqrt{\xi\_1\... | https://mathoverflow.net/users/120111 | A problem related to stochastic ordering | $\newcommand{\der}{\mathrm{der}}\newcommand{\tdert}{\mathrm{dert}}\newcommand{\erf}{\operatorname{erf}}\newcommand{\eqs}{\overset{\text{sign}}=}\newcommand{\tder}{\widetilde\der}$The answer is no.
Indeed, let $x:=\xi\_1$ and $y:=\xi\_2$, so that $\xi\_3=1-x-y$, $0\le x\le y\le1-x-y$, whence $x\in[0,1/3]$.
Consider ... | 1 | https://mathoverflow.net/users/36721 | 404955 | 166,070 |
https://mathoverflow.net/questions/404948 | 1 | Toom's rule is a 2-dimensional cellular automaton which is known to have two distinct stationary measures in the thermodynamic limit, even after small perturbations to a probabilistic cellular automaton by introducing bit-flip or biased noise. What about more general local, but non-on-site noise. E.g., imagine after ea... | https://mathoverflow.net/users/115363 | Is Toom's rule robust under local but non-on-site noise? | Toom's CA is robust under such noise. In fact, the class of noise distributions he considers is quite general. For a fixed CA $f$ on $\{0,1\}^{\mathbb{Z}^2}$ and a parameter $0 \leq \epsilon \leq 1$, let $M\_\epsilon$ be the set of Borel probability measures $\mu$ on $(\{0,1\}^{\mathbb{Z}^2})^ {\mathbb{N}}$ such that e... | 4 | https://mathoverflow.net/users/66104 | 404973 | 166,074 |
https://mathoverflow.net/questions/404986 | 2 | For any point $x \in \mathbb R^n$, denote by $\delta\_x$ the Dirac Delta measure centered at $x$.
Let $a\_n$ be an sequence of positive numbers with $\lim\_{n \to \infty} a\_n = 0$, and let $d\_i$ be a countable dense subset of $\mathbb R^n$.
Given a rearrangement $a\_{n\_k}$ of $a\_n$, consider the space $\mathbb ... | https://mathoverflow.net/users/173490 | Does $\mathbb R^n$ equipped with a sum of Dirac delta measures admit nowhere locally constant continuous integrable functions? | Partition $(a\_n)$ into two subsequences $(a\_n')$ and $(a\_n'')$ with $\sum a\_n' < \infty$, and partition $(d\_i)$ into two subsequences $(d\_i')$ and $(d\_i'')$ such that $d\_i'' \to \infty$. Pair the $a\_i'$s with the $d\_i'$s and the $a\_i''$s with the $d\_i''$s. Then any bounded continuous function that vanishes ... | 3 | https://mathoverflow.net/users/23141 | 404987 | 166,079 |
https://mathoverflow.net/questions/404990 | 8 | Two morphisms of category $\ \mathbf C\ $ are isomorphic to one
another $\ \Leftarrow:\Rightarrow\ $ they are the opposite edges that are drawn horizontally (aimed East) of a commutative square that has the vertical edges (aimed North) being isomorphisms of $\ \mathbf C$.
**Problem** What is the minimum total number ... | https://mathoverflow.net/users/110389 | Isomorphic morphisms. A 27-morphism category | I see one example with 7 morphisms. It is a subcategory of the category of groups. The only object is the the Klein 4-group $(\mathbb{Z}/2)^2$, and the morphisms are generated by the two projections and the flip. That monoid has 7 elements, and the two projections are conjugate, and hence they are isomorphic. However $... | 11 | https://mathoverflow.net/users/3969 | 404994 | 166,081 |
https://mathoverflow.net/questions/404953 | 3 | The $p$-adic field $\mathbb{Q}\_p$ has topological basis of open sets of the form $a+p^N \mathbb{Z}\_p$ for $0 \leq a \leq p^N-1$ and $N \in \mathbb{Z}$. These are indeed compact open sets. One can define Bernoulli distributions by $$\mu\_{B,k}(a+p^N \mathbb{Z}\_p)=p^{N(k-1)}B\_k \left(\frac{a}{p^N}\right), $$ where $B... | https://mathoverflow.net/users/122445 | Bernoulli distributions and $p$-adic measure on $K$ | Here is why I doubt that the question has been looked at and why it is not well-formulated, but I am really not confident in saying this. If it has been considered, I'd hope someone corrects me here.
$\newcommand{\ZZ}{\mathbb{Z}}\newcommand{\QQ}{\mathbb{Q}}$ Let $\mu$ be a measure taking open subsets of a group $\Gam... | 2 | https://mathoverflow.net/users/5015 | 405005 | 166,084 |
https://mathoverflow.net/questions/405011 | 8 | In a [recent answer to an old MO question](https://mathoverflow.net/a/404872), I made a distinction between a "definition" of a mathematical object in the sense of axioms that characterize it, and a "definition" that explicitly constructs the object in question. For a concrete example, consider the "definition" of the ... | https://mathoverflow.net/users/3106 | Analytic/synthetic distinction in mathematics besides geometry? | The answer is here:
<https://ncatlab.org/nlab/show/synthetic+mathematics>
As you can see, there are several flavors available, such as synthetic topology, probability, domain theory, etc. This effort is by no mean new, but it is true that the categorical approach has once again emphasized the synthetic over the ana... | 7 | https://mathoverflow.net/users/15293 | 405024 | 166,089 |
https://mathoverflow.net/questions/404993 | 0 | The following is a recursion for one point monotone Hurwitz numbers
$$
d \, m\_g(d) = 2(2d-3) \, m\_g(d-1) + d(d-1)^2 \, m\_{g-1}(d)\label{1}\tag{$\*$}
$$
with initial condition $m\_0 (1) =1$ and some of the other numbers are $ m\_0 (2) = 1, m\_1 (3) =10$.
Let denote the generating function by
$$F\_{g}(x) := \sum\_{g\g... | https://mathoverflow.net/users/45170 | Rational solution of differential equation | The recurrence $(\*)$ gives the correct form of $(\*\*)$ as $$(1 - 4x) F'\_g(x) + 2 F\_g(x) = x^2 F'''\_{g-1}(x) + x F''\_{g-1}(x) \tag{\*\*}$$
The boundary condition which gives the the correct form of $F\_0$ is $$(1 - 4x) F'\_0(x) + 2 F\_0(x) = 1 \tag{\*\*\*}$$ yielding $F\_0 = \frac{1 + 2C\_0 \sqrt{1 - 4x}}2$ and ... | 1 | https://mathoverflow.net/users/46140 | 405033 | 166,090 |
https://mathoverflow.net/questions/375409 | 2 | I believe someone working in an alternative set theory called $$\{x|a\in x\}$$ the *essence* of *a*. Does someone recall a reference?
| https://mathoverflow.net/users/37385 | The essence of type free set theory? | Here:
"In [1944] Hailperin gave the first of a number of finite axiomatisations of NF now known. Many of them exploit the function $x\mapsto\{y:x∈y\}$ which is injective and total and is an $\in$-isomorphism. This function was known to Whitehead, who suggested to Quine that $\{y:x∈y\}$ should be called the “essence” ... | 3 | https://mathoverflow.net/users/37385 | 405051 | 166,095 |
https://mathoverflow.net/questions/405057 | 0 | Let the pdf of a multivariate normal distribution be
\begin{equation}
p\_{Z}(\mathbf{z})=\frac{1}{\left(2\pi \sigma^2 \right)^{k/2}}\exp(-{\mathbf{z}}^{\text{T}}\mathbf{z}/2\sigma^2).
\end{equation}
Consider a continuous function $f(\mathbf{s})$ supported on a compact set $\mathcal{S}$. Then let
\begin{equation}
I\_{... | https://mathoverflow.net/users/149696 | Asymptotic moment of a multivariate normal distribution | $\newcommand\si\sigma\newcommand\om\omega\newcommand\z{\mathbf z}\newcommand\R{\mathbb R}$Let us assume that your $d\Omega$ means $d\z$, so that
$$I\_\si(s):=\int\_{\R^k} p\_{Z}(z-s) f(z)\, dz,$$
and we shall assume that this definition holds for all $s\in\R^k$.
So,
$$I\_\si(s)=Ef(s+\si Z),$$
where $Z$ is a standard ... | 2 | https://mathoverflow.net/users/36721 | 405062 | 166,099 |
https://mathoverflow.net/questions/405060 | 0 | Is there a term for the operation $A B A^T$? In colloquial terms, I might call this a "sandwich" of a matrix between another matrix and the transpose of that other matrix.
How about for the special case where $B$ (as well as $A B A^T$) is symmetric? This shows up often in Kalman filtering.
I would like to look up s... | https://mathoverflow.net/users/391616 | Is there a term for the operation of multiplying the product of two matrices by the transpose of the first matrix? | If $B=XAX^T$ classically one says that A and B are congruent.
| 2 | https://mathoverflow.net/users/56010 | 405073 | 166,100 |
https://mathoverflow.net/questions/405064 | 2 | Is there a name for the following property of a commutative ring $R$:
its Jacobson radical $J$ is nilpotent, and $R/J$ is semi-simple?
(It is easily equivalent to: $R$ is a finite product of commutative local rings with nilpotent radical.)
| https://mathoverflow.net/users/76506 | Terminology for commutative ring whose Jacobson radical $J$ is nilpotent with semisimple quotient $R/J$ | The word for a ring $R$ whose Jacobson radical $J$ is nilpotent and such that $R/J$ is semisimple is **semiprimary**. I don’t know if there is a more special word for the commutative case.
| 5 | https://mathoverflow.net/users/19965 | 405076 | 166,101 |
https://mathoverflow.net/questions/405077 | 2 | I am confused by the two conclusions in this paper ([DOI link behind paywall at Springerlink](https://link.springer.com/content/pdf/10.1007/s10878-019-00461-7.pdf)).
It shows that the equitable tree-coloring problem is $W[1]$-hard when parameterized
by treewidth.
However, it also shows that the equitable tree-color... | https://mathoverflow.net/users/178444 | $W[1]$-hard and FPT about the equitable tree-coloring problem | No. Fix $k \in \mathbb{N}$ and let $G$ be an $n$-vertex graph of treewidth at most $k$. If you analyze their polynomial-time algorithm that decides if $G$ has an equitable tree-colouring, you'll notice that the degree of the polynomial depends on $k$. For example, in the proof of Theorem 11, there is a bound of the for... | 2 | https://mathoverflow.net/users/2233 | 405082 | 166,102 |
https://mathoverflow.net/questions/404965 | 10 | $\DeclareMathOperator\Hom{Hom}$It is well-known (see [Breen, Mikhailov, Touzé - Derived functors of the divided power functors](https://arxiv.org/abs/1312.5676) for example) that for $A$ a free abelian group we have
$$ H\_i(K(A,1); \mathbb{Z}) \cong {\bigwedge}^i A$$
the exterior powers of $A$
and that for $B$ an abe... | https://mathoverflow.net/users/184 | Induced map on $H_4$ of Eilenberg–MacLane spaces | Let me summarize comments above. Functoriality of the resulting map $f\_\*:H\_4(K(A,1),\mathbb{Z})\to H\_4(K(B,2),\mathbb{Z})$ translates into a request for the universal map $\wedge^4 A\to \Gamma(A)$ which in turn, being functorial in $A$, allows to guess the answer assuming $A$ is free and
$rk\ A=4$.
Note that $\Ga... | 3 | https://mathoverflow.net/users/8906 | 405083 | 166,103 |
https://mathoverflow.net/questions/405023 | 8 | Are there two non-isotopic knots $K,K'$ in $S^3$ with the same $\mathrm{SL}\_2(\mathbb C)$ $A$-polynomials? If it's an open problem, has anyone suggested a method for finding them, or a reason why no such pair should exist (i.e., why the $A$-polynomial should be a complete knot invariant)?
My first guess for how to f... | https://mathoverflow.net/users/113402 | Distinct knots with same $A$-polynomial | The torus knots $T\_{7,15}$ and $T\_{3,35}$ have the same A-polynomials. In general, if $p,q>1$ are coprime and odd then $T\_{p,q}$ has A-polynomial $(L-1)(LM^{pq}+1)(LM^{pq}-1)$, which only depends on the product $pq$. This follows from a few observations:
1. The [original paper](https://link.springer.com/article/10... | 5 | https://mathoverflow.net/users/428 | 405091 | 166,107 |
https://mathoverflow.net/questions/405097 | 2 | Suppose that $(X,\leq )$ is an ordered set, we can define the maximum and the infimum of this set,now let $(X,A)$ be a measurable space and let $M(X,A)$ be the set of all measures on $(X,A)$, we now define an order $\leq$ by declaring that $\mu \_{1}\leqslant \mu \_{2}$ iff $\mu \_{1}\left ( A\_{0} \right )\leq \mu \_{... | https://mathoverflow.net/users/206621 | Supremum with respect to the order of measures on $(X,A)$ | $\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}\newcommand\A{\mathscr A}\newcommand\N{\mathbb N}$Yes, $\mu$ is a measure.
Indeed, let $\A:=A$. Say that a partition of a set $B\in\A$ is measurable if all its pieces are in $\A$.
For any $B\in\A$, let $P(B)$ denote the set of all countable measurable partitions of... | 1 | https://mathoverflow.net/users/36721 | 405103 | 166,109 |
https://mathoverflow.net/questions/405069 | 9 | A [classical result](http://www-users.math.umd.edu/%7Ewmg//fricke.pdf) of Fricke--Klein--Vogt from the late 1800s implies that the character variety $\chi\_\mathbb{C}$ associated to the free group $F\_2$ and the algebraic group $\mathrm{SL}\_2(\mathbb{C})$ is isomorphic to $\mathbb{C}^3$.
Question: What happens if we... | https://mathoverflow.net/users/41301 | Character variety of the free group | Let $\pi$ be a free group of rank 2. The character variety of $SL\_{2,k}$-representations of $\pi$ is always isomorphic to affine 3 space $\mathbb{A}^3\_k$ *for any ring $k$*.
Let $A[\pi] = A[\pi]\_k$ denote the coordinate ring of $SL\_{2,k}\times SL\_{2,k}$, which we identify with $Hom(\pi, SL\_{2,k})$ by picking a ... | 5 | https://mathoverflow.net/users/15242 | 405109 | 166,113 |
https://mathoverflow.net/questions/405099 | 11 | For the statement of Carlson's theorem please see,
<https://en.wikipedia.org/wiki/Carlson%27s_theorem>.
There is an extension of Carlson's theorem that says that the condition that $f$ needs to vanish on integers can be replaced with $f$ vanishing on a subset A of integers provided that $A$ has upper density 1. Thi... | https://mathoverflow.net/users/50438 | An extension of the Carlson's theorem in complex analysis | If $f$ is bounded on the imaginary line, (and has exponential type) then
$f$ has completely regular growth in the sense of Levin-Pfluger, with indicator
$c|\cos\theta|$. This implies that density of zeros on the positive ray must
be zero. Moreover, density of zeros in any angle $|\arg z|<\pi/2-\epsilon$
and in the vert... | 10 | https://mathoverflow.net/users/25510 | 405112 | 166,114 |
https://mathoverflow.net/questions/405105 | 9 | Is it known when
$2^{2n}-2^n+1$
is prime?
It seems to be only when n is 1,2,4 or 32.
| https://mathoverflow.net/users/45242 | When is $2^{2n}-2^n+1$ prime? | Expanding on the remarks by Pace Nielsen (with an additional result that we need only consider $n>2$ a multiple of $4.$):
$x^n-1=\prod\_{d \mid n}\phi\_d(x)$. The factors are irreducible over the rationals.
In particular $$x^6-1=\phi\_1(x)\phi\_2(x)\phi\_3(x)\phi\_6(x)=(x-1)(x+1)(x^2+x+1)(x^2-x+1)$$
And $$\phi\_6... | 13 | https://mathoverflow.net/users/8008 | 405124 | 166,116 |
https://mathoverflow.net/questions/105926 | 7 | I am trying to derive the [classic paper](http://alexandria.tue.nl/repository/freearticles/597601.pdf) in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" want to prove that the average height $h\_n$ of a tree wi... | https://mathoverflow.net/users/26078 | On "The Average Height of Planted Plane Trees" by Knuth, de Bruijn and Rice (1972) | I think the problem may be in the bijection. Consider instead the following bijection between (planted plane) trees with $n$ vertices and Dyck paths from $(0,0)$ to $(n-1,n-1)$: perform a depth-first search on the tree, moving $\uparrow$ in the Dyck path when you follow an edge away from the root, and $\rightarrow$ in ... | 1 | https://mathoverflow.net/users/46140 | 405130 | 166,117 |
https://mathoverflow.net/questions/405117 | 7 | EDIT: After a little prompting by Mark Grant, I answered the first question in the comments. The second question remains open.
---
Let $M$ be a compact $3$-manifold with $\pi\_2(M) \neq 0$. The sphere theorem says that there is an embedded 2-sphere in $M$ that is nontrivial in $\pi\_2$.
In his book "Group theor... | https://mathoverflow.net/users/392655 | Two details from Stallings's proof of the sphere theorem | What you need for your second question is that $H^1\_c(\tilde{M})$ is a free abelian group. As you noted, this is isomorphic to $H\_2(\tilde{M})$. Now, $\tilde{M}$ is a connected non-compact 3-manifold. A basic theorem of Whitehead says that a connected noncompact PL $n$-manifold is homotopy equivalent to an $(n-1)$-di... | 4 | https://mathoverflow.net/users/317 | 405140 | 166,120 |
https://mathoverflow.net/questions/405080 | 3 | Let $C
\_p$ be the Schatten-p-classes on a separable Hilbert spaces, $p\ge 1$.
Let ${\rm Tr}$ be the standard trace.
Let $y\in C\_p$ be a self-adjoint operator (or even a positive operator) and let $x\_n \in C\_{p'}$, $\frac1p+\frac1{p'}=1$, be a sequence of self-adjoint operators with $\|x\_n\|\_{p'}=1$ such that ${\r... | https://mathoverflow.net/users/91769 | norm estimates for Schatten class | I understand that you are happy with the case $y\geq 0$, so let's assume that for simplicity (I expect that small modifications should deal with arbitrary self-adjoint $y$). In that case, **and if** $1<p<\infty$, it is true that $\| u\_n^\* y - y\|\_p \to 0$. I am not sure about the extreme cases $p=1,\infty$, where th... | 2 | https://mathoverflow.net/users/10265 | 405146 | 166,124 |
https://mathoverflow.net/questions/405095 | 3 | Let $R$ be a Noetherian commutative ring. A complex of $R$-modules $P^{\bullet}$ is K-projective if for any acyclic complex $A^{\bullet}$, the complex of abelian groups $ Hom(P^{\bullet}, A^{\bullet})$ is acyclic. K-projective complexes were defined by Spaltenstein:
<http://www.numdam.org/article/CM_1988__65_2_121_0.... | https://mathoverflow.net/users/375103 | K-projectivity for rings of finite homological dimension | There's a nice, short proof in
*Positselski, Leonid; Schnürer, Olaf M.*, [**Unbounded derived categories of small and big modules: is the natural functor fully faithful?**](http://dx.doi.org/10.1016/j.jpaa.2021.106722), J. Pure Appl. Algebra 225, No. 11, Article ID 106722, 23 p. (2021). [ZBL1464.18015](https://zbmath... | 6 | https://mathoverflow.net/users/22989 | 405151 | 166,126 |
https://mathoverflow.net/questions/405150 | 6 | Let $X$ be a complex smooth projective variety of dimension $d$. Let $K(X) := K(\text{Coh}(X))$ denote the Grothendieck group of coherent sheaves on $X$. For two coherent sheaves $E$ and $F$ on $X$, define their Euler pairing by
$$\chi(E,F) = \sum\_{i=0}^{d} (-1)^i \text{dim Ext}^i(E,F).$$
The Euler pairing descends to... | https://mathoverflow.net/users/146366 | Why does the Chern character descend to the numerical Grothendieck group for surfaces? | First, taking $F$ to be the structure sheaf of a point you "kill" $\mathrm{ch}\_0(E)$.
Next, since intersection pairing is non-degenerate on
$$
\mathrm{Im}(\mathrm{Pic}(X) \to H^2(X,\mathbb{Q})) = \mathrm{NS}(X)
$$
you can choose $F\_i$ to be a collection of the structure sheaves of curves on $X$ to "kill" $\mathrm{c... | 5 | https://mathoverflow.net/users/4428 | 405157 | 166,128 |
https://mathoverflow.net/questions/405160 | 6 | Let $p$ be a prime and let $n\geq 2$ be an integer.
The set $\mathbb{A}^n(\mathbb{F}\_p)$ has $p^n$ elements so it has $2^{p^n}$ subsets. How many of those subsets are of the form $V(\mathbb{F}\_p)$ with $V\subset \mathbb{A}^n$ a closed geometrically irreducible subvariety?
| https://mathoverflow.net/users/393396 | Sets of $\mathbb{F}_p$-points of closed subvarieties of $\mathbb{A}^n$ | All of them. One can even take $V$ to be a smooth hypersurface whose extension to projective space is smooth, by Poonen's Bertini theorem. This guarantees irreducibility.
Poonen's Bertini theorem states that there exists a smooth hypersurface in $\mathbb P^n$ of degree $d$ for all $d$ sufficiently large satisfying an... | 13 | https://mathoverflow.net/users/18060 | 405165 | 166,130 |
https://mathoverflow.net/questions/405163 | 3 | What is an example of a rigid object $A$ in an abelian monoidal category $\mathcal{M}$ that is not projective as an object in $\mathcal{M}$? (Since $\mathcal{M}$ is abelian projective just means that all exact sequences of the form $ 0 \to R \to Q \to P \to 0$ split.) What are sufficient conditions for such a rigid obj... | https://mathoverflow.net/users/371382 | A non-projective rigid object in an abelian monoidal category | If by "rigid" you mean the same thing as dualizable, an example is given by $\mathcal M = Mod\_{R[G]}$ for some commutative ring $R$ and group $G$, with monoidal structure given by $\otimes\_R$ and the diagonal action.
Then the object $R$ with a trivial $G$-action is the unit, hence it's rigid, but it's rarely projec... | 2 | https://mathoverflow.net/users/102343 | 405166 | 166,131 |
https://mathoverflow.net/questions/405139 | 6 | $\DeclareMathOperator\holim{holim}\DeclareMathOperator\hocolim{hocolim}$Let $\mathcal{F}$ be a simplicial (pre)sheaf on some site $\mathcal{C}$ (assume the site has enough stalks; if you like also assume every representable functor on $\mathcal{C}$ is a sheaf). Suppose $\mathcal{G}$ is a (pre)sheaf of groups acting on ... | https://mathoverflow.net/users/392998 | Homotopy quotients, fixed points and stalks of simplicial (pre)sheaves | Taking stalks always commutes with taking homotopy orbits,
since filtered colimits of simplicial sets are also filtered homotopy colimits, and homotopy colimits commute with homotopy colimits.
Taking stalks commutes with taking homotopy fixed points
with respect to a finite group
only if additional fibrancy condition... | 3 | https://mathoverflow.net/users/402 | 405175 | 166,132 |
https://mathoverflow.net/questions/316614 | 5 | Let $f = (f\_1, \dotso, f\_n):\mathbb{R}^n \to \mathbb{R}^n$ be a smooth map and let $J$ be its Jacobian (determinant of the matrix with $ij$-th entry $\partial\_i f\_j$). We introduce the zero sets of $J$ and its derivatives
$$Z\_0 = J^{-1}(0), \quad Z\_1 = Z\_0 \cap (\nabla J)^{-1}(0), \quad \dotso$$
Here we define $... | https://mathoverflow.net/users/84963 | Generic properties of Jacobians of smooth functions | This is answered in the paper <https://link.springer.com/content/pdf/10.1007/s00526-020-01740-6.pdf> (OA), Lemma 6.13; this Lemma is proved in the Appendix of the paper. The final answer is that one needs to take $k(n) = \left \lceil{n + 1}\right \rceil $ derivatives to obtain that generically $Z\_{k(n) - 1} = \emptyse... | 1 | https://mathoverflow.net/users/84963 | 405204 | 166,141 |
https://mathoverflow.net/questions/405174 | 1 | Let $m\geq 2$ be a fixed integer.
Let
$$f(n):=\begin{cases}
mf\left(\frac{n}{m}\right),&\text{if $n\mod m = 0$;}\\
1,&\text{otherwise}
\end{cases}$$
then if we have
$$a(n):=\begin{cases}
1,&\text{if $n=0$;}\\
a\left(\frac{n}{m}\right)+a\left(n-f\left(\frac{n}{m}\right)\right),&\text{if $n\mod m = 0$;}\\
a\left(\left\... | https://mathoverflow.net/users/231922 | Recurrence for the sum | Let $n=m^tk$ where $m\nmid k$. Then $f(n)=m^t$.
Furthermore, if $t>0$, then $f(n/m)=m^{t-1}$ and $n-f(n/m)=m^{t-1}(mk-1)$. It follows that $a(n)=a(m^{t-1}k)+a(m^{t-1}(mk-1))$ and further by induction on $t$,
$$
(\star)\qquad a(n)=\sum\_{i=0}^t \binom{t}{i} a\big(m^ik-\frac{m^i-1}{m-1}\big).
$$
---
**CASE $m>2$.*... | 4 | https://mathoverflow.net/users/7076 | 405210 | 166,145 |
https://mathoverflow.net/questions/405098 | 1 | Let $\boldsymbol{X} = (X\_1,X\_2)^{\rm T}\sim \mathcal{N}\_2(\boldsymbol{\mu}, \mathrm{\Sigma})$, where
\begin{eqnarray\*}
\boldsymbol{\mu} = (\mu\_1, \mu\_2)^{\rm T}& = &(\sqrt{\xi\_1\xi\_2/(\xi\_1+\xi\_2)}, 0)^{\rm T}\\
\mathrm{\Sigma} & = &\begin{pmatrix} 1 & -\rho\\
-\rho & 1\end{pmatrix}\\
\rho & = & \sqrt{\xi\_1\... | https://mathoverflow.net/users/120111 | A problem related to bivariate normal stochastic order | $\newcommand{\der}{\mathrm{der}}\newcommand{\tdert}{\mathrm{dert}}\newcommand{\erf}{\operatorname{erf}}\newcommand{\eqs}{\overset{\text{sign}}=}\newcommand{\tder}{\widetilde\der}$The statement about the maximum is true, but the proof has hardly anything to do with stochastic ordering arguments. Rather, the problem is r... | 2 | https://mathoverflow.net/users/36721 | 405215 | 166,146 |
https://mathoverflow.net/questions/405212 | 4 | Let $Y$ be a space, and $G$ a group. For simplicity we can take $G$ to be finite, $Y$ to be a point, if this avoids technical issues. Then for $X/Y$ a $G$ torsor, so a sheaf of sets with $G$ action on $X$, locally isomorphic to the sections of $G\times U \xrightarrow{\pi} U$, we can consider the automorphisms of this m... | https://mathoverflow.net/users/128502 | What extra structure does the group of automorphisms of a torsor carry? | I am not sure what kind of criterion you are after. You can give an answer in terms of cocycles or an answer in terms of killing an obstruction. Briefly, if $\mathcal{G}$ is a bundle of groups on $Y$ which is locally isomorphic to the trivial bundle of groups $\underline{G}$ with fiber $G$, then the first obstruction t... | 4 | https://mathoverflow.net/users/439 | 405218 | 166,147 |
https://mathoverflow.net/questions/405227 | 4 | I am from physics background so I apologize in advance if my question is trivial.
Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$ ([here](https://core.ac.uk/download/pdf/82773183.pdf)). I wonder if it is possible to work out an ideal triangulatio... | https://mathoverflow.net/users/394173 | Ideal triangulation of hyperbolic 3-manifold with generic mapping class group | The hyperbolic manifolds considered by Kojima are closed, so do not admit ideal triangulations in the "usual sense". (They admit "partially flat, spun, ideal triangulations" but I am not sure that is what you are interested in.)
It is an open question whether or not every finite volume cusped hyperbolic three-manifol... | 3 | https://mathoverflow.net/users/1650 | 405230 | 166,152 |
https://mathoverflow.net/questions/405196 | 20 | Recall that $A(X)$, the K-theory of a connected, pointed space X, is defined as the K-theory spectrum of the ring spectrum $\Sigma^\infty\_+ \Omega X$ (or via a plethora of alternative definitions). Is it known if the homotopy type of $A(X)$ determines the homotopy type of $X$? If not, what is the best one can hope for... | https://mathoverflow.net/users/134512 | Does Waldhausen K-theory detect homotopy type? | The answer to the question
>
> Does the homotopy type of $()$ determine the homotopy type of $$?
>
>
>
is No in general. As you say, $A(X)$ is determined by the homotopy type of $\Sigma^\infty \Omega X\_+$ as an associative (or $A\_\infty$) ring spectrum, and this ring spectrum does not uniquely determine $X$,... | 23 | https://mathoverflow.net/users/6668 | 405235 | 166,154 |
https://mathoverflow.net/questions/405238 | 0 | Let $E$ be a Banach space, $\mathfrak{M}\_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}\_E$ the family of all relatively compact sets, and $Ker \mu=\{X\in \mathfrak{M}\_E$ such that: $\mu(X)=0\}$.
>
> **"Standard" Definition:**
> A mapping $\mu:\mathfrak{M}\_E\rightarrow \mathbb R^+$ i... | https://mathoverflow.net/users/102228 | A MNC with maximum property but not singular | 1. First, according to [paper you cited](https://dml.cz/bitstream/handle/10338.dmlcz/105982/CommentatMathUnivCarol_021-1980-1_10.pdf), $\mathfrak{M}\_E$ is **the** [not "a"] family of all nonempty bounded subset**s** of $E$.
2. The examples of MNC's given in the [paper you cited](https://dml.cz/bitstream/handle/10338.d... | 1 | https://mathoverflow.net/users/36721 | 405255 | 166,162 |
https://mathoverflow.net/questions/405256 | 22 | I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to give a simple, convincing example where it's useful to use the two-generating set $\{(1,2),(1,2,...,n)\}$. (I always find i... | https://mathoverflow.net/users/17988 | What is the standard 2-generating set of the symmetric group good for? | Let $f\in\mathbb{Q}[x]$ be an irreducible polynomial of prime degree $p$, with exactly $2$ non-real roots.
You can view the Galois group of $f$ (i.e., the Galois group of the splitting $f$) as a subgroup of $S\_p$. Complex conjugation shows that the Galois group contains a transposition. You can use Cauchy's theorem ... | 37 | https://mathoverflow.net/users/95685 | 405257 | 166,163 |
https://mathoverflow.net/questions/405253 | 1 | Can every polynomial $P(x)$ with integer coefficients we represented in the form
$$
P(x) = Q^2(x)R(x),
$$
where $Q(x)$ and $R(x)$ are polynomials with integer coefficients such that $R(x)$ has no repeated (complex) roots?
If this is true it should be well-known, then a reference would be helpful.
Intuition: if $\al... | https://mathoverflow.net/users/89064 | Square-free representation of polynomials with integer coefficients | Yes. Recall that $\mathbb{Z}[x]$ is a UFD and recall that, if $p(x)$ is irreducible in $\mathbb{Z}[x]$, then the roots of $p(x)$ are distinct.
So write $P(x) = \prod p\_i(x)^{a\_i}$, a product of irreducible polynomials in $\mathbb{Z}[x]$, and put $Q(x) = \prod p\_i(x)^{\lfloor a\_i/2 \rfloor}$ and $R(x) = \prod\_{a\... | 3 | https://mathoverflow.net/users/297 | 405258 | 166,164 |
https://mathoverflow.net/questions/405214 | 3 | In the book [Number Theory IV](https://www.springer.com/gp/book/9783540614678) from Parshin, one can find this statement (precisely at p. 215) with the comment "*it is easy to see that...*":
>
> Let $a\_{ij}$ ($1\le i,j\le m$) be complex numbers with $|a\_{i,j}|\le H\_i$ ($1\le j\le m$).
> One assumes that the line... | https://mathoverflow.net/users/33128 | Minoration of linear forms | We claim that
$$
\sum\_{i=1}^m \frac{|L\_i(w)|}{H\_i} \geqslant \frac{|w|\_\infty}{(m-1)!}\frac{|\Delta|}{H\_1 \cdots H\_m},
$$
where $|w|\_\infty = \max\_{1 \leqslant j \leqslant m} |w\_j|$.
Indeed, first assume that $|A|\_\infty = \max\_{1\leqslant i,j \leqslant m} |a\_{ij}|\leqslant 1$, and $H\_i = 1$ for all $i$.... | 2 | https://mathoverflow.net/users/393977 | 405265 | 166,166 |
https://mathoverflow.net/questions/405252 | 4 | I cannot find the exact same question asked anywhere in this site. I know the related Green-Tao theorem but the gaps between consecutive primes can grow unbounded so it does not seem helpful to answer this question.
What I have tried: assume the largest gap is D and without loss of generality it appears infinitely ofte... | https://mathoverflow.net/users/393027 | Are there arbitrarily long arithmetic progressions in every increasing sequence of positive integers with bounded gaps between consecutive terms? | You may of course use Szemeredi theorem, as suggested by Alexander Kalmynin.
If you need a more elementary argument, you may apply Van der Waerden theorem as follows: assuming that the gaps are bounded by $T$, color every positive integer $n$ to the color $i\in \{1,\dots,T\}$ if $nT+i$ belongs to your set (so each la... | 9 | https://mathoverflow.net/users/4312 | 405274 | 166,171 |
https://mathoverflow.net/questions/405270 | 2 | Let $X$ be an random $n \times d$ matrix with entries drawn iid from $N(0,1/d)$ and let $w$ be a unit-vector in $\mathbb R^d$. With $\lambda>0$, and define $G:=X^\top(XX^\top + \lambda I\_n)^{-1}X$. Finally, defined $\alpha := w^\top G^2 w$.
>
> **Question.** *In the limit $n,d \to \infty$ with $n/d \to \rho \in (0... | https://mathoverflow.net/users/78539 | Asymptotics of $w^\top G^2 w$, where $w$ is a unit-vector, $G:=X^T(XX^T+t I_n)^{-1}X$, $t > 0$, and $X$ is an $n\times d$ gaussian random matrix | Let me calculate the expectation value of $\alpha$. The probability distribution of $X$ is invariant under orthogonal transformations, so without loss of generality I can orient the unit vector $w$ along one of the axes, $w\_i=\delta\_{ip}$, $p\in\{1,2,\ldots d\}$. Then
$$\mathbb{E}[\alpha]=\mathbb{E}\left(X^T(XX^T+\la... | 2 | https://mathoverflow.net/users/11260 | 405278 | 166,173 |
https://mathoverflow.net/questions/405284 | 2 | Prove that there exists five matrices $B\_i \in \mathbb{F}\_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B\_i$'s form an invertible matrix in $\mathbb{F}\_2^{10\times 10}$.
I am interested in a proof of existence that could be generalized to matrices with other dimensions.
Follow-up question when there... | https://mathoverflow.net/users/nan | Existence of matrices with some invertibility properties | If $B$, $B'$ are $5\times 10$ matrices of rank $5$ over $\mathbb{F}\_2$, the condition that the matrix formed by stacking $B$ on top of $B'$ is invertible is equivalent to the condition that the row span of $B$ and the row span of $B'$ have trivial intersection. The row spans are $5$-dimensional subspaces of $\mathbb{F... | 3 | https://mathoverflow.net/users/5263 | 405288 | 166,178 |
https://mathoverflow.net/questions/405295 | 37 | Does there exist a map $f:\Bbb R^n \rightarrow \Bbb R^m$, where $n<m$ and $ n,m \in\Bbb N^+$ such that $f$ is surjective and differentiable?
| https://mathoverflow.net/users/369324 | Is there a differentiable map surjective from low to high dimension？ | $\DeclareMathOperator\R{\mathbf{R}}$It's easy to check that the image of any locally Lipschitz map $f:\R^n\to\R^m$ has measure zero when $n<m$ (this encompasses the case of class-$\text{C}^1$ maps, but not the case of differentiable maps).
Indeed, extend $f$ to $F:\R^m\to\R^m$ by $F(x,y)=f(x)$. This is still locally ... | 25 | https://mathoverflow.net/users/14094 | 405303 | 166,184 |
https://mathoverflow.net/questions/405318 | -1 | Let $M\in \mathbb{R}^{N\times N}$ be a given matrix and $k\ge 2$ be a given integer. Then my question is the following optimization problem:
Is there a polynomial-time solution to the following problem: $$S^\star = \arg\max\_{\substack{S\subset [N]:\\ |S|\le k}} \sum\_{i,j\in S}M\_{ij}?$$
This seems to be hard in g... | https://mathoverflow.net/users/64194 | Finding a $k$-subset which maximizes a matrix sum | This is the 0-1 [quadratic knapsack problem](https://en.wikipedia.org/wiki/Quadratic_knapsack_problem), which is NP-hard. The binary decision variables $x\_i$ indicate whether $i\in S$, and the knapsack capacity is $k$.
You can solve it via integer linear programming as follows. For $i<j$, let binary decision variabl... | 3 | https://mathoverflow.net/users/141766 | 405320 | 166,190 |
https://mathoverflow.net/questions/405316 | 0 | The following is a recursion for one point monotone Hurwitz numbers
$$
d \, m\_g(d) = 2(2d-3) \, m\_g(d-1) + d(d-1)^2 \, m\_{g-1}(d)\label{1}\tag{$\*$}
$$
with initial condition $m\_0 (1) =1$ and some of the other numbers are $ m\_0 (2) = 1, m\_1 (3) =10$.
Let denote the generating function by
$$F\_{g}(x) := \sum\_{d\g... | https://mathoverflow.net/users/45170 | Rational solution for linear differential equation | I must say that I don't much understand the motivation coming from number theory, but your question about rational solutions of ODEs has a definite answer, provided the equation has polynomial or rational coefficients. A standard reference is
>
> *Abramov, S. A.*, [**Rational solutions of linear differential and di... | 2 | https://mathoverflow.net/users/2622 | 405330 | 166,194 |
https://mathoverflow.net/questions/345008 | 2 | Let $P\subset\Bbb R^d$ be a *vertex-transitive polytope* aka. an *orbit polytope*.
Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\Bbb R^d)$?
Probably one approach to the question is as follows: can there be a vertex-transitive polytope $P\subse... | https://mathoverflow.net/users/108884 | A matrix that commutes with all symmetries of a vertex-transitive polytope | There are examples of vertex-transitive polytopes with **irreducible** symmetry groups for which there are still a **non-scalar** transformations $T\in\mathrm O(\Bbb R^d)$ that commutes with all the symmetries of the polytopes.
Fix a group $G$ with the following properties:
* $G$ is neither abelian nor generalized ... | 0 | https://mathoverflow.net/users/108884 | 405332 | 166,195 |
https://mathoverflow.net/questions/405294 | 3 | **Motivation**
The notion of uniform integrability is important for formulating the Vitali convergence theorem. Unfortunately, different authors define uniform integrability differently, which causes quite a lot of confusion, as evident in the number of questions about uniform integrability and Vitali convergence the... | https://mathoverflow.net/users/49284 | Confusion around uniform integrability and Vitali convergence theorem | $\newcommand\R{\mathbb R}\newcommand{\ep}{\epsilon}\newcommand{\de}{\delta}$
Your claim is not quite correct.
E.g., let $\mu$ be the Lebesgue measure over $X:=\R$. Let
$$\Phi:=\{f\_n\colon\, n\in\mathbb N\},$$
where
$$f\_n(x):=\frac1{1+(x-n)^2}$$
for real $x$. Then
$\Phi$ is uniformly bounded in $L^1$, does not escap... | 1 | https://mathoverflow.net/users/36721 | 405336 | 166,197 |
https://mathoverflow.net/questions/405321 | 1 | Let $\Gamma\_0,\Gamma\_1,...$ be regeneration epochs.
If $(X\_n)\_{n \in \mathbb{N}}$ is a $\lambda$ biased random walk on a Galton-Watson tree, than the regeneration epochs are defined as:
$\Gamma\_0:=\inf\{\iota \ | X\_i\neq X\_{\iota} \ \forall i\leq \iota \ \text{and} \ X\_{j}\neq (X\_{\iota})\_\* \ \forall j\g... | https://mathoverflow.net/users/396065 | Random walks on GW-trees (regeneration epochs/survival set) | You need to assume that the bias satisfies $\lambda<m$ where $m$ is the mean offspring. Then you can find the proof in Lemma 3.3 page 253 of [1].
[1] Lyons, Russell, Robin Pemantle, and Yuval Peres. "Biased random walks on Galton–Watson trees." Probability theory and related fields 106, no. 2 (1996): 249-264.
<http... | 1 | https://mathoverflow.net/users/7691 | 405346 | 166,201 |
https://mathoverflow.net/questions/405323 | 9 | The "pants" bordism in dimension n is a bordism which goes from $S^n \sqcup S^n$ to $S^n$ witnessing the connected sum operation - equivalently by attaching 1-handle to the trivial bordism, equivalently doing surgery on a 0-sphere.
The "copants" bordism is the same manifold, but thought of as a bordism from $S^n$ to ... | https://mathoverflow.net/users/184 | Framed version of the "copants bordism"? | The key point is the identification $\tau(S^n)+\mathbb{R}$ with the restriction of tangent bundle of the bordism. I will read bordism from bottom to top.
Even to obtain pants bordism between standard spheres you need to choose the $\textit{inward}$ normal at bottom and $\textit{outward}$ normal at top for the above ide... | 5 | https://mathoverflow.net/users/8906 | 405354 | 166,203 |
https://mathoverflow.net/questions/405358 | 1 | The following theorem is well-known:
>
> **Theorem.** Let $M$ be a smooth manifold. A smooth covector field $\omega \in \Omega^1(M)$ is conservative, that is, $$\int\_{\mathbb{S}^1}f^\*\omega = 0 \qquad \forall f \in C^\infty(\mathbb{S}^1,M),$$ if and only if $\omega$ is exact.
>
>
>
For a proof, see for examp... | https://mathoverflow.net/users/98139 | Generalisation of conservative covector fields | Consider a 2-torus. The second homotopy is trivial, so the spheres can't feel the homology, or cohomology. The same for any manifold obtained by quotienting a simply connected manifold by a cocompact discrete group action, so all surfaces of genus 2 or more.
| 3 | https://mathoverflow.net/users/13268 | 405360 | 166,205 |
https://mathoverflow.net/questions/405349 | 7 | Let $T:H\to H$ be a continuous operator on a Hilbert space.
Assume there exists an orthonormal base $(e\_j)\_{j\in\mathbb N}$, such that the sequence $Te\_j$ tends to zero.
Must $T$ be compact?
| https://mathoverflow.net/users/nan | Criterion for compactness | $T$ is not necessarily compact. Let me produce a counterexample.
Let $H$ be any infinite dimensional real or complex separable Hilbert space. Let $(f\_{j,k})\_{1\leq k\leq j},(e\_{j})\_{j=1}^{\infty}$ be orthonormal bases for $H$.
Then let $T:H\rightarrow H$ be the bounded linear operator defined by letting
$T(f\_{j,... | 12 | https://mathoverflow.net/users/22277 | 405362 | 166,206 |
https://mathoverflow.net/questions/404734 | 4 | I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in [Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses](https://eudml.org/doc/157211) as follows, $(1\leq k\leq n-1)$:
\begin{align\*}
x\_{2k-1}&=\frac{a^2}{z\_n}\cos \frac{z\_k}{a}\\
x\_{2k}... | https://mathoverflow.net/users/171387 | A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$ | The isometric immersion that you describe above is the higher dimensional pseudosphere. Now, concerning your final question, I presume that you need to search about isometric immersions of the hyperbolic space $\mathbb H^n$ by means of a warped product representation (of $\mathbb H^n$) into the Euclidean space.
Now, ... | 4 | https://mathoverflow.net/users/85181 | 405373 | 166,208 |
https://mathoverflow.net/questions/405353 | 1 | I am dealing with a knapsack-like problem with one difference from the conventional problem: the “weights” can be positive or negative and the constraint is $\sum w\_i x\_i \ge 0$ instead of $\sum w\_i x\_i \le W$. The "values" can also be positive or negative.
Can this be transformed to a knapsack problem or is it s... | https://mathoverflow.net/users/396728 | Knapsack like problem with nonnegative weight constraint | Assuming $x\_i$ is binary, perform a change of variables $\bar{x}\_i:=1-x\_i$ where $w\_i>0$:
\begin{align}
\sum\_i w\_i x\_i
&= \sum\_{i:w\_i>0} w\_i x\_i + \sum\_{i:w\_i<0} w\_i x\_i \\
&= \sum\_{i:w\_i>0} w\_i(1-\bar{x}\_i) + \sum\_{i:w\_i<0} w\_i x\_i \\
&= \sum\_{i:w\_i>0} w\_i + \sum\_{i:w\_i>0} (-w\_i)\bar{x}\_... | 1 | https://mathoverflow.net/users/141766 | 405378 | 166,209 |
https://mathoverflow.net/questions/405021 | 2 | This question is inspired by [Characterization of functors whose right adjoint is monadic?](https://mathoverflow.net/questions/385363/characterization-of-functors-whose-right-adjoint-is-monadic).
Let $F : \mathbf C \rightleftarrows \mathbf D : U$ be an adjunction, and suppose that we want to establish when the canoni... | https://mathoverflow.net/users/152679 | Characterisation of functors whose left adjoint is Kleisli | Try this. I confess I haven't written out the proofs yet, but will do so if you (and the community) think this is an appropriate answer. Please excuse my renaming your categories according to my personal convention.
First, $U:{\mathcal A}\to{\mathcal S}$ must be faithful and reflect invertibility, cf Beck's theorem.
... | 2 | https://mathoverflow.net/users/2733 | 405379 | 166,210 |
https://mathoverflow.net/questions/405370 | 4 | Let $A$ be a finite-dimensional (not necessarily unital) associative algebra over the field of complex numbers $\mathbb{C}$ (but I'm also interested in more general fields). Assume the multiplication on $A$ is non-degenerate, which means that $A= AA$ and if $a \in A$ satisfies $aA = 0$ or $Aa = 0$, then $a=0$. Is it tr... | https://mathoverflow.net/users/nan | Is a non-degenerate finite-dimensional algebra unital? | There's a four-dimensional counterexample over any field.
$A$ has basis $\{e,a,b,c\}$, with all products of basis elements zero except for
$$e^2=e,\quad ab=c,\quad ea=a,\quad ec=c,\quad be=b,\quad ce=c.$$
(This is a codimension one ideal in the path algebra of the quiver with two vertices, two arrows $a$ and $b$ in... | 5 | https://mathoverflow.net/users/22989 | 405381 | 166,211 |
https://mathoverflow.net/questions/405385 | 10 | Let $P(z) = \prod\_{i = 1}^n (z - z\_i) \in \mathbb{C}[z]$ be a monic polynomial having all roots $z\_1, \dots, z\_n$ on the unit circle $\mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$.
What is known about upper bounds for
$$M(P) :=\max\_{|z| = 1} |P(z)|$$
in terms of the distribution of $z\_1, \dots, z\_n$ in $\mathb... | https://mathoverflow.net/users/388007 | Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots | Let me first start with the other side: does the maximum being small guarantee that the roots are equidistributed? This is indeed the case, and is a beautiful theorem of Erdos and Turan. For a recent exposition see [Equidistribution of zeros of polynomials](https://arxiv.org/abs/1802.06506) (published paper in the Amer... | 11 | https://mathoverflow.net/users/38624 | 405397 | 166,220 |
https://mathoverflow.net/questions/405395 | -1 | If $B\_k(x)$ are the Bernoulli polynomials, then (by definition, if you like) we get that
$$\sum\_{k=0}^{\infty}B\_k(x)\frac{t^k}{k!}=\frac{te^{tx}}{e^t-1}$$
My question is whether or not there is a known formula for
$$\mathcal{G}(x;t):=\sum\_{k=0}^{\infty}B\_{2k+1}(x)\frac{t^{2k+1}}{(2k+1)!}.$$
The motivation ... | https://mathoverflow.net/users/159298 | Closed form for odd part of Bernoulli Polynomial generating function, $\sum_{k=0}^{\infty}B_{2k+1}(x)\frac{t^{2k+1}}{(2k+1)!}$ | Using Pietro Majer's bisection formula we find by a straightforward computation (I did it with Maple, but I'm sure it could be done without too much difficulty by hand) that the OP's formula for
$$\sum\_{k=1}^{\frac{m-1}{2}}k\mathcal{G}\left(\frac{k}{m},t\right)$$ is indeed true.
| 2 | https://mathoverflow.net/users/10744 | 405399 | 166,222 |
https://mathoverflow.net/questions/405048 | 3 | John Isbell defined a notion of a median algebra (although the original idea is due to Birkhoff). A median algebra is a set $S$ with a ternary operation $[x,y,z]$ satisfying a set of [axioms](https://en.wikipedia.org/wiki/Median_algebra). The example to have in mind comes from lattice theory. Suppose $(L, \wedge, \vee,... | https://mathoverflow.net/users/128639 | Medians in lattice theory | Yes. There is a nice notion of an $n$-ary median term in a distributive lattice and in a median algebra as long as $n$ is odd.
Proposition: Suppose that $1\leq k\leq n$.
1. There is a term $t$ in the language of lattices such that if $x\_{1}\leq\dots\leq x\_{n}$, then $t(x\_{1},\dots,x\_{n})=x\_{k}$ and where $t$ s... | 4 | https://mathoverflow.net/users/22277 | 405400 | 166,223 |
https://mathoverflow.net/questions/405361 | 6 | Let $B \subset \mathbb R^n$ be the unit ball.
Consider a Borel measurable set $E \subset B$ with positive Lebesgue measure $|E|>0$ (say $|E| = |B|/2$).
Then, Lebesgue's density theorem, says that for a.e. $x\in E$
$$
\lim\_{r \downarrow 0} \frac{|B(x,r)\backslash E|}{|B(x,r)|} = 0.
$$
We can restate it as follows: ... | https://mathoverflow.net/users/173610 | Set where the speed of convergence is uniform in Lebesgue's density theorem | Let $$f\_n(x) = \sup\_{r \in {\mathbb Q} \cap [\frac{1}{n+1},\frac{1}{n})} \frac{|B(x,r)\setminus E|}{|B(x,r)|}\,,$$ so that $f\_n(x) \to 0$ for a.e. $x \in E$. By Egorov's theorem [1], for every $\epsilon>0$ there is a subset
$\tilde{E} \subset E$ with $|E \setminus \tilde{E}| <\epsilon$, such that
$f\_n(x) \to 0$ uni... | 4 | https://mathoverflow.net/users/7691 | 405405 | 166,224 |
https://mathoverflow.net/questions/405363 | 3 | Let $0\neq \beta\in\overline{\mathbb{Z}}$ and let $n$ be a positive integer coprime to $N\_{\mathbb{Q}(\beta)/\mathbb{Q}}(\beta)$. Say that $n$ is a Fermat pseudoprime to base $\beta$ if
$$\beta^{n^{[\mathbb{Q}(\beta):\mathbb{Q}]}-1}\equiv 1\pmod n.$$
If $p$ is a prime which does not divide $N\_{\mathbb{Q}(\beta)/\math... | https://mathoverflow.net/users/165478 | Can we construct composite Fermat pseudoprimes to integral algebraic bases? | The answer to both questions is yes. Just like Alford, Granville and Pomerance's proof that there are infinitely many Carmichael numbers, my proof that there are infinitely many Carmichael numbers where the minimal polynomial splits completely is essentially a constructive proof.
Both proofs need to jump through some... | 1 | https://mathoverflow.net/users/11722 | 405406 | 166,225 |
https://mathoverflow.net/questions/405434 | 4 | $\DeclareMathOperator\SL{SL}$Let $d$ be a metric on the upper-half plane $\mathbb H = \{(x,y) : y > 0\}$ which is invariant with respect to the action of $\SL(2, \mathbb R)$ to $\mathbb H$ which is defined by
$$A \cdot z = \frac{az+b}{cz+d}, \ \ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \SL(2, \mathbb R), \ ... | https://mathoverflow.net/users/116429 | Gromov hyperbolicity for (non-geodesic) metrics on the upper-half plane invariant with respect to SL(2, R) action | (Recall that an arbitrary metric space is Gromov-hyperbolic if
$$\sup\_{a,b,c,d}\Big((ab+cd)-\max(ac+bd,ad+bc)\Big)<\infty;$$
where $ab$ is the distance between $a$ and $b$. Note that this passes to subspaces.)
The answer is no: a counterexample being the square root of the usual distance on $\mathbf{H}^2$, which is ... | 7 | https://mathoverflow.net/users/14094 | 405441 | 166,234 |
https://mathoverflow.net/questions/405439 | 0 | Let $(X\_n)\_{n\in\mathbb{N}\_0}$ be a biased Random Walk on Galton-Watson tree with $\lambda\in(\lambda\_c,m)$.
How can I obtain the following equation:
$\sum\_{k=0}^{n-1}\mathbb{E}\_{e\_\*}[|X\_{k+1}|-|X\_k| \ | \mathcal{S}]=\sum\_{k=0}^{n-1}\mathbb{E}\_{e\_\*}[\frac{\nu(X\_k)-\lambda}{\nu(X\_k)+\lambda} \ | \mat... | https://mathoverflow.net/users/396065 | Random walks on GW-trees (transformation) | It would be more useful if you specify exactly where you are reading this and what is $e\_\*$. Is the walk on a GW-tree or an augmented GW tree?
The underlying reason for this identity is that if a vertex $v$ has $b$ children with weight 1 each, and one parent with weight $\lambda$, then the net drift (=expected increm... | 1 | https://mathoverflow.net/users/7691 | 405445 | 166,236 |
https://mathoverflow.net/questions/405435 | 4 | Let $Y$ be a Stein manifold and $D\subset\subset Y$ be a Stein domain. I think $\overline D$ has connected boundary, and it should be somewhere, but I cannot find a reference for this. Thanks
| https://mathoverflow.net/users/70148 | Connectedness of boundary of a Stein domain | This is a consequence of the fact that any Stein manifold with complex dimension at least 2 has one end. This follows from Hartogs extension across compact sets in Stein manifolds. You can find a proof of this Hartogs
extension in the paper of Serre on Serre duality Commentarii Mathematici Helvetici volume 29 1955 page... | 7 | https://mathoverflow.net/users/4696 | 405448 | 166,239 |
https://mathoverflow.net/questions/405170 | 3 | Let $(\mathcal{M},\otimes)$ be a symmetric monoidal model category; I'll assume for simplicity that every object is fibrant. Suppose that the unit $I$ is NOT cofibrant. I'm interested in whether the derived tensor product with the unit is oplax/strong monoidal.
On the one hand, since $-\otimes^L I$ is the left derive... | https://mathoverflow.net/users/138396 | On the derived functor of the tensor product in a monoidal category | Yes, the tensor product with a cofibrant replacement can be turned into a lax monoidal functor, where the lax structure maps are weak equivalences.
Consider the model category $\def\Mon{{\rm Mon}} \Mon(M)$ of monoids in $M$.
This model structure exists if $M$ satisfies the monoid axiom,
which is almost always true in... | 2 | https://mathoverflow.net/users/402 | 405453 | 166,240 |
https://mathoverflow.net/questions/405425 | 3 | This question comes from book [Ju-Yi Yen and Marc Yor](https://link.springer.com/book/10.1007/978-3-319-01270-4) [P59](https://i.stack.imgur.com/DJGja.png) and [P60](https://i.stack.imgur.com/3C4qD.png),
On page 59, "Define $\mathcal{Z}\_\omega=\{t:B\_t(\omega)=0\},$ and $\tau\_l$ is the inverse local time. The compl... | https://mathoverflow.net/users/147009 | How to prove excursion process is a Poisson point process? | The standard reference for Brownian excursions is Chapter 12 in the classic book [1]. Other developments of excursion theory can be found in [2]-[4] and many other sources.
To develop the intuition, a good approach is to start with random walks.
(See, e.g., the exposition of local time in [6]).
The counting measure o... | 3 | https://mathoverflow.net/users/7691 | 405455 | 166,241 |
https://mathoverflow.net/questions/405436 | 2 | Let $D\_1$ be a domain with smooth boundary and assume that $D\_1$ is a proper subset of $D\_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D\_2\setminus D\_1$ is connected. We write $L^2(D\_2\setminus D\_1)$ for the set of functions in the space
$$\{f \in L^2(D\_2)\,:\,\... | https://mathoverflow.net/users/50438 | Density of traces of solutions to an elliptic equation | The answer is yes: take any smooth function $g\_0$ on $\partial D\_1$ and solve the Dirichlet problem
$$
\begin{cases}
\Delta g = 0 & \text{ on } D\_1\\
g = g\_0 & \text{ on } \partial D\_1.
\end{cases}
$$
Now extend $g$ to a smooth function on $\mathbb{R}^n$. Multiply by a smooth cutoff function $\eta$ which is $1$ on... | 1 | https://mathoverflow.net/users/378654 | 405457 | 166,243 |
https://mathoverflow.net/questions/405268 | 5 | Short version: what are some interesting hyperdoctrines for classical (not intuitionistic) first-order logic, that are not models in the traditional sense? (Where the terminal and initial hyperdoctrines are "uninteresting".)
Long version:
The categorical semantics of first-order logic are given by [hyperdoctrines](... | https://mathoverflow.net/users/15201 | What are some interesting hyperdoctrines that are not classical models? | Maybe I am wrong, but it seems to me that the other answers are misunderstanding the question. The emphasis on syntactic hyperdoctrines seems to me beside the point.
A (classical, first-order) hyperdoctrine is a categorical (semantic) structure, consisting of a category $C$ and a functor $P: C \to \rm BoolAlg$ togeth... | 4 | https://mathoverflow.net/users/49 | 405465 | 166,245 |
https://mathoverflow.net/questions/382462 | 0 | Is there a discrete topological dynamical system $(X,f)$, where $X$ is a compact metric space (with distance $d$), which is transitive but not minimal, such that $h(f)>0$ and every point is a full entropy point?
By transitive I mean that there is a point with a dense orbit. By minimal I mean that every orbit is dense... | https://mathoverflow.net/users/167834 | Non-minimal system in which every point is a full entropy point | I guess this is not hard to do with a subshift. Take $X$ to be your favorite minimal positive entropy subshift on $\{0,1\}$ (such examples are constructed in Hahn-Katznelson, among other works).
Now, choose any $x \in X$ and define $y$ on $\{0,1,\*\}$ as
$y = .\* \ x\_1 \* x\_2 x\_3 \* x\_4 x\_5 x\_6 \* x\_7 x\_8 x\_... | 1 | https://mathoverflow.net/users/116357 | 405471 | 166,249 |
https://mathoverflow.net/questions/405470 | 4 | Two players take turns coloring edges on an $n$-by-$n$ grid. Both players use the same color. Every time a player surrounds a square of the grid, they mark that square with their name and go again. A player cannot pass; they must move if it is their turn. The goal is to make as many squares as possible. What is the sco... | https://mathoverflow.net/users/83174 | Who wins this two player game of making squares? | This just the game of [Dots and Boxes](https://en.wikipedia.org/wiki/Dots_and_Boxes). There is a huge literature on this game. In particular, Berlekamp's book referenced in the above link shows how difficult this game is.
| 13 | https://mathoverflow.net/users/2807 | 405472 | 166,250 |
https://mathoverflow.net/questions/405468 | 15 | Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called *dessins d'enfants* which are bipartite graphs embedded on an oriented surface so that each component of their complement is a topological disk. To be more precise, given a smooth projective curve ... | https://mathoverflow.net/users/128556 | Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants? | If $X$ is a (smooth projective) curve over $\overline{\mathbb{Q}}$, we define
>
>
> >
> > The Belyi degree $\deg\_B(X)$ of $X$ to be the minimum degree of a Belyi map $X\to \mathbb{P}^1\_{\overline{\mathbb{Q}}}$.
> >
> >
> >
>
>
>
The Belyi degree is a function on $\mathcal{M}\_g(\overline{\mathbb{Q}})$ w... | 11 | https://mathoverflow.net/users/4333 | 405480 | 166,253 |
https://mathoverflow.net/questions/405262 | 18 | This question was previously posted [here](https://math.stackexchange.com/questions/4185940/is-the-p-adic-density-of-the-image-of-a-polynomial-always-rational) on MSE.
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. For $n\in\mathbb N$, let $I\_n$ be the number of integers $i\in\{... | https://mathoverflow.net/users/394725 | Is the p-adic density of the image of a polynomial always rational? | Using the strategies suggested by @Merosity on MSE and @Gro-Tsen and @RP\_ on MO, I have found a proof that the density is indeed always rational.
Let $P$ be a polynomial with integer coefficients, and let $p$ be a prime number. If $P$ is a constant polynomial, then we obtain $\delta=0$ which is rational. So assume t... | 7 | https://mathoverflow.net/users/394725 | 405487 | 166,257 |
https://mathoverflow.net/questions/405479 | 1 | Poincaré inequality is stated as follows:
>
> Let $ \Omega $ be bounded, connected, open subset pf $ \mathbb{R}^n $, with a $ C^1 $ boundary $ \partial \Omega $. Assume that $ 1\leq p<\infty $ and $ u\in W^{1,p}(\Omega) $. Then there exists a constant $ C $, depending only on $ n, p $ and $ \Omega $, such that
> $$... | https://mathoverflow.net/users/241460 | How to prove Poincaré inequality by using extension theorem? | It seems to me that some useful information you can find in paragraph 1.5 of
"Differentiable Functions on Bad Domains" by V. G. Mazia, S. Pobozchi.
| 1 | https://mathoverflow.net/users/362006 | 405492 | 166,259 |
https://mathoverflow.net/questions/405476 | 5 | I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r\_\sigma$, well-known in the theory of cardinal characteristics of the continuum.
For a compact metrizable space $X$, let $\mathfrak r\_X$ be the smallest cardinality of a family $\mathcal R$ of infinite subsets of $\omega$ s... | https://mathoverflow.net/users/61536 | The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system | Unfortunately, $\mathfrak r\_{(2^\omega,f)}\ge\mathfrak r$. Indeed, let $\mathcal R$ be a family of infinite subsets of $\omega$ such that $|\mathcal R|=\mathfrak r\_{(2^\omega,f)}$ and for any $x=(x\_n)\_{n\in\omega}\in 2^\omega$ there exists $R\in\mathcal R$ such that the sequence $(f^n(x))\_{n\in R}$ converges in $2... | 7 | https://mathoverflow.net/users/43954 | 405503 | 166,262 |
https://mathoverflow.net/questions/405477 | 11 | In a [comment](https://mathoverflow.net/questions/405256/what-is-the-standard-2-generating-set-of-the-symmetric-group-good-for#comment1038737_405273) at the recent question [What is the standard 2-generating set of the symmetric group good for?](https://mathoverflow.net/questions/405256/what-is-the-standard-2-generatin... | https://mathoverflow.net/users/4177 | Low-order symmetric group 2-generation: n=5,6,8 | It is also possible to use the (exceptional) outer automorphism of order $2$ of $S\_{6}$ to give an "explanation" of why $S\_{6}$ is not $\{2,3\}$-generated, along the lines I used in comments for $S\_{5}$ above. Take a $6$-cycle $\sigma \in S\_{6}.$ Then $\sigma^{2}$ is a product of two disjoint three cycles and $\sig... | 12 | https://mathoverflow.net/users/14450 | 405504 | 166,263 |
https://mathoverflow.net/questions/405498 | 2 | Suppose $\Omega\subset\mathbb{R}^n$ is a regular open set, $f\in L^2(\Omega)$ and consider the following elliptic problem.
$$-\Delta u + u=f'(u) , \;\;u\_{|\partial \Omega}=0,$$
where $f'$ is the derivative of a function $f:\mathbb{R}\to\mathbb{R}$ and $f'(u)\in L^2(\Omega)$ for all $u\in H^1\_0(\Omega)$ (choose any as... | https://mathoverflow.net/users/105925 | Semi-linear elliptic problem, energy functionals, Fréchet derivatives and the Newton method in Banach spaces | Let me write out the equation $J''(u)w = g$ for $g\in H^{-1}$ and $w\in H^1\_0(\Omega)$. This is equivalent to
$$
\int\_\Omega \nabla w \cdot \nabla v - f''(u)wv = g(v) \quad \forall v\in H^1\_0(\Omega).
$$
This is the weak formulation of
$$
-\Delta w - f''(u)w = g$$
plus boundary conditions. To show existence of solut... | 2 | https://mathoverflow.net/users/48485 | 405506 | 166,264 |
https://mathoverflow.net/questions/405460 | 5 | Consider a continuous ODE,
$$\dot x = f(x), f \in C^1$$
$\dot x = 0$ for all $x \in K \subset \mathbb{R}^n$, where we assume that $K$ is a closed but unbounded set of non-isolated equilibrium. For example, $K$ could be a line, or a half space, or a ray, etc.
>
> Suppose that this ODE admits an energy function $... | https://mathoverflow.net/users/145401 | Which result guarantees convergence of solution of an ODE to a set of non-compact, non-isolated equilibrium? | The extension to the unbounded setting is due to [Hale 1969: Dynamical systems and stability](https://www.sciencedirect.com/science/article/pii/0022247X69901759), where he states your statement provided that the positive orbit $\mathscr{O}^+(x\_0)$ of the initial point $x\_0$ is contained in a compact set. See [Theorem... | 3 | https://mathoverflow.net/users/13400 | 405515 | 166,267 |
https://mathoverflow.net/questions/404099 | 1 | Let $X$ be an algebraic variety over $\mathbb{C}$ (the ground field is not important but this makes things easier I think) and $G$ an algebraic group acting over it. Let's say we know that there's a normal subgroup $K \cong \mathbb{G}\_m$ such that $K$ acts trivially on $X$ and $G/K$ acts in a free way on $X$.
I thin... | https://mathoverflow.net/users/146464 | Cohomology of quotient stack | Let $G$ be a linear algebraic group, and $K\subset G$ a closed normal subgroup with quotient $L = G/K$. So all three groups are linear algebraic. Further, assume $L$ is connected.
Let $X$ be a variety with $G$-action such that $K$ acts trivially, $L$ acts freely and we have a quotient $X/L$.
**Step 1:** The canoni... | 1 | https://mathoverflow.net/users/392998 | 405516 | 166,268 |
https://mathoverflow.net/questions/405512 | 3 | I am looking for a reference for the **Stallings-Epstein-Waldhausen construction** (constructing an incompressible surface in a 3-manifold from a nontrivial splitting of the fundamental group).
I know of Proposition 2.3.1 in the following article, but was hoping for a more comprehensive / textbook-like treatment.
M... | https://mathoverflow.net/users/129446 | Reference request: Stallings-Epstein-Waldhausen construction | As an answer to your first question: You can find an exposition of the pull-back construction in Scott's notes "[An introduction to 3-manifolds](https://www.math.cuhk.edu.hk/course_builder/1617/math6071b/scott_An%20Introduction%20to%203-manifolds-1974.pdf)".
As an answer to your second question: If the group we are s... | 3 | https://mathoverflow.net/users/1650 | 405532 | 166,270 |
https://mathoverflow.net/questions/405482 | 0 | Any element in a boolean extension field $a\in GF(2^n)$ can be presented by a boolean vector $a\_{(2)} \in GF(2)^n$. For any element $a\in GF(2^n)$, there exists a boolean matrix $M\_a\in GF(2)^{n\times n}$ such that $(ab)\_{(2)} = M\_a \cdot (b)\_{(2)}$.
**The question:**
For any non-zero $\vec v\in GF(2)^n$,
is the... | https://mathoverflow.net/users/165642 | Does binary extension field multiplication matrix have random rows? | The function $a \mapsto M\_a$ is a linear transformation from $GF(2)^n$ to $GF(2)^{n \times n}$, so $a \mapsto \vec{v}^{T} M\_a$ is a linear transformation from $GF(2)^n$ to the dual vector space $(GF(2)^n)^\*$. Both of these vector spaces are $n$-dimensional. To see that the linear transformation $a \mapsto \vec{v}^{T... | 2 | https://mathoverflow.net/users/8049 | 405540 | 166,272 |
https://mathoverflow.net/questions/405545 | 1 | I need to solve the following equation for $P \in\mathbb{R}^{r\times d}$
$$P - G\_1G\_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0,$$
where the other quantities are known: $A\in\mathbb{R}^{d\times d}$, $G\_1 \in\mathbb{R}^{r\times d}$, $G\_{2} \in\mathbb{R}^{d\times d}$, $\lambda\in\mathbb{R}$.
I have already ... | https://mathoverflow.net/users/156139 | Matrix equation with projection matrix | The solution for $P$ to
$$P - G\_1G\_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0$$
is
$$P=(1 +\lambda )G\_1 G\_2 A^{-1},$$
as one can check by substitution into
$$G\_1G\_2P^\top(PAP^\top)^{-1}P=G\_1 G\_2(G\_1G\_2A^{-1})^\top\bigl(G\_1G\_2(G\_1G\_2A^{-1})^\top\bigr)^{-1}G\_1G\_2A^{-1}=G\_1G\_2A^{-1}.$$
| 5 | https://mathoverflow.net/users/11260 | 405548 | 166,274 |
https://mathoverflow.net/questions/405511 | 4 | Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}\_{\text{qc}}(\mathcal{D}\_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}\_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}\_X$-modules with quasi-coherent cohomology.
If $f:X\to Y$ is a morphism between such schemes, we have natural fun... | https://mathoverflow.net/users/131975 | Six functor formalism for quasi-coherent $D$-modules | They do not exist in general. The simplest example is maybe to take $X\rightarrow Y$ to be the closed embedding of the origin inside $\mathbb{A}^1.$ Then $f\_\*$ sends a vector space $V$ to the $\mathcal{D}\_{\mathbb{A}^1}$-module $V\otimes\delta\_0$, where $\delta\_0$ is the irreducible $\mathcal{D}\_{\mathbb{A}^1}$-m... | 3 | https://mathoverflow.net/users/51424 | 405559 | 166,278 |
https://mathoverflow.net/questions/405566 | 4 | It is known that in a stable $\infty$-category $\mathcal{C}$ and $X \in \mathcal{C}$, the suspension $X[1]$ is defined by the pushout of $0\leftarrow X \rightarrow 0$. However this does not make sense in usual category since the pushout is the zero object $0$.
Now all we can do is following the formal definition of l... | https://mathoverflow.net/users/133871 | How to understand pushout/pullback in a stable $\infty$-category | Let me try to give some intuition by examining two important examples. One should start from the definition: the suspension $ΣX$ is the universal choice of $Y$ filling of a square
$$\require{AMScd}
\begin{CD}
X @>{p}>> \ast\\
@V{p}VV @VVV \\
\ast @>>> Y
\end{CD}\,.$$
To understand the suspension we need to understand ... | 8 | https://mathoverflow.net/users/43054 | 405571 | 166,282 |
https://mathoverflow.net/questions/405573 | 5 | Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c\_1(\mathscr T)$ is of form $r[Z]$. Is there anything we can say about the positivity of $r$?
Any help is appreciated.
| https://mathoverflow.net/users/313627 | First Chern class of torsion sheaves | The coefficient $r$ is equal to the length of $\mathcal{T}$ at the generic point of $Z$, so it is positive.
| 7 | https://mathoverflow.net/users/4428 | 405575 | 166,283 |
https://mathoverflow.net/questions/405580 | 5 | At the end of
<https://encyclopediaofmath.org/index.php?title=Cubic_hypersurface#References>
it is stated that the diagonal cubic hypersurface
$$
\sum\_{i=0}^{2m+1} a\_i x\_i^3 = 0, m\ge 2
$$
(and presumably $a\_i\not=0$) is rational. Is this true over the complex numbers, or any field of characteristic zero? W... | https://mathoverflow.net/users/66397 | diagonal cubic hypersurfaces | Yes, this is true over $\mathbb{C}$, and rather easy. You can assume your equation is $\sum x\_i^3=0$. For convenience, let me call the coordinates $x\_0,\ldots ,x\_m;y\_0,\ldots ,y\_m$. Then your hypersurface $X$ contains the $m$-planes $P\_1: x\_i=-y\_i$ and $P\_2: x\_i=-\rho y\_i$, with $\rho =e^{2\pi i/3}$. Note th... | 12 | https://mathoverflow.net/users/40297 | 405584 | 166,285 |
https://mathoverflow.net/questions/405452 | 2 | Let $G$ be an affine algebraic group (let's say over $\mathbb{C}$). If necessary one can assume $G$ to be reductive. Imagine one has $X$ over which $G$ acts freely: moreover, we have a locally closed subvariety $Y$ such that $X=G \cdot Y$.
Moreover, one has a subgroup $H \subseteq G$ such that $H$ stabilizes $Y$. We ... | https://mathoverflow.net/users/146464 | Quotient variety and subgroups | I don't think that this is true without extra hypotheses like normality of $X$ (if $X$ is normal then it follows as explained by Damian Rossler in the comments). Here is a possible example.
We will take $G = GL\_2$ and $H=1$ (the latter is just the trivial group). Let $C$ be the cuspidal curve $C = Spec(k[x,y]/(y^2-x... | 4 | https://mathoverflow.net/users/339730 | 405588 | 166,286 |
https://mathoverflow.net/questions/405589 | 2 | Let X and Y be separable Banach spaces.
Let $f:X\rightarrow Y$ be a Baire-1 function, which is the pointwise limit of a sequence of continuous functions $f\_n:X\rightarrow Y$.
Define $E$ as the set of $x$ such that $f\_n(x\_n)\rightarrow f(x)$ fails to hold for some sequence $\{x\_n\}$ approaching x.
Is $E$ empty... | https://mathoverflow.net/users/401135 | Convergence of sequences for Baire-1 functions | Recall:
**1.** Given $f\_n\in C(X,Y)$ point-wise convergent to $f$, the above set $E$ always contains the discontinuity set of $f$ (starting from any $x\_n\to x$ with $ f(x\_n)\not\to f(x)$ one has for a subsequence $ f\_{k\_n}(x\_n)-f(x\_n) \to0$, and re-naming the sequences one can also assume $ f\_{n}(x\_n)-f(x\_n... | 2 | https://mathoverflow.net/users/6101 | 405592 | 166,287 |
https://mathoverflow.net/questions/405543 | 7 | $\newcommand{\oUConf}{\widehat{\mathrm{UConf}}}\newcommand{\UConf}{\mathrm{UConf}}\newcommand{\oGr}{\widehat{\mathrm{Gr}}}\newcommand{\Gr}{\mathrm{Gr}}\newcommand{\SO}{\mathrm{SO}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\String}{\mathrm{String}}\newcommand{\U}{\mathrm{U}}\newcommand{\SU}{\mathrm{SU}}\newcommand{\O... | https://mathoverflow.net/users/130058 | Geometric models for the classifying spaces of the spin and string covers of the orthogonal, symplectic, and symmetric groups | For $\def\B{{\rm B}} \def\bB{{\bf B}} \def\Spin{{\rm Spin}} \def\String{{\rm String}} \B\Spin(n)$, simply equip the $n$-planes with a [spin structure](https://mathoverflow.net/questions/122748/what-is-a-spinor-structure/122798#122798), as originally proposed by Stolz and Teichner.
For $\B\String(n)$, equip the $n$-pl... | 5 | https://mathoverflow.net/users/402 | 405603 | 166,289 |
https://mathoverflow.net/questions/405578 | 2 | $\DeclareMathOperator\FSym{FSym}$Let $p$ be a prime and $S$ be a transitive Sylow $p$-subgroup of $\FSym(\mathbb{N})$, the finitary symmetric group of the set of all natural numbers.
Question: Is $S$ totally imprimitive and uniserial (i.e. for every $i\in \mathbb{N}$, there exists a unique system of imprimitivity on ... | https://mathoverflow.net/users/98061 | On Sylow subgroups of finitary symmetric groups | Yes: all transitive Sylow $p$-subgroups of $\mathrm{FSym}(\mathbb{N})$ are permutation isomorphic to the infinite wreath product $C\_p \wr C\_p \wr \ldots $. This permutation group is uniserial and has a unique system of blocks of size $p^i$ for each $i \in \mathbb{N}$; one such block is an orbit of the canonical subgr... | 3 | https://mathoverflow.net/users/7709 | 405611 | 166,290 |
https://mathoverflow.net/questions/405015 | 2 | A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the monograph [1] below.
A positive function $q(x)$ is said to be logarithmically completely monotonic on an interval $I\subset... | https://mathoverflow.net/users/147732 | What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$? | For $k\in\mathbb{N}\_0=\{0,1,2,\dotsc\}$ and $z\ne0$, let
\begin{equation}\label{exp=k=sum-eq-degree=k+1}
H\_k(z)=\textrm{e}^{1/z}-\sum\_{m=0}^k\frac{1}{m!}\frac1{z^m}.
\end{equation}
For $\Re(z)>0$, the function $H\_k(z)$ has the integral representations
\begin{equation}\label{exp=k=degree=k+1-int}
H\_k(z)=\frac1{k!(k... | 1 | https://mathoverflow.net/users/147732 | 405612 | 166,291 |
https://mathoverflow.net/questions/405610 | 5 | Consider two diffusions given by
$$X\_j(t)=\int\_0^t a\_j(s,X\_j(s))\,dW\_s$$
for $j=1,2$ and $t\ge 0$, where $W\_\cdot$ is a standard Wiener process/Brownian motion and the $a\_j$'s are smooth enough functions such that $0\le a\_1\le a\_2$.
Does it then follow that $P(|X\_1(1)|>x)\le P(|X\_2(1)|>x)$ for all real $x$... | https://mathoverflow.net/users/36721 | A comparison of diffusions | The inequality is not true in general — additional assumptions are needed. I think some kind of monotonicity of $a\_1$ and $a\_2$ should help, but this is merely a guess.
Here is a counterexample. Consider $a\_2 = 1$, so that $X\_2(1) = W(1)$. Let $a\_1(s, x) = 1$ when $|x| < 1$ and $a\_1(s, x) = 0$ otherwise. Then $... | 7 | https://mathoverflow.net/users/108637 | 405618 | 166,292 |
https://mathoverflow.net/questions/405632 | 4 | Chapter 2, Exercise 25 of R. Stanley's "Enumerative Combinatorics" Vol. 1 asserts that
$$ \sum\_{m,n \geq 0} \left(\sum\_{t \geq 0} f\_i(m,n)t^i\right)\frac{x^m}{m!}\frac{y^n}{n!} = e^{-x-y}\sum\_{m,n \geq 0} (1+t)^{mn} \frac{x^m}{m!}\frac{x^n}{n!},$$
where $f\_i(m,n)$ is the number of $m\times n$ $0,1$-matrices with a... | https://mathoverflow.net/users/25028 | $0,1$-matrices with $1$ in every row/column vs. all $0,1$-matrices | If you multiply both sides by $e^{x+y}$ there's a simple bijective proof: In a nutshell, every 0-1 matrix consists of a matrix with a 1 in every row and column together with some all-zero rows and columns.
If you restate this in terms of bipartite (or more precisely, bicolored—bipartite with a specified bipartition) ... | 4 | https://mathoverflow.net/users/10744 | 405633 | 166,293 |
https://mathoverflow.net/questions/405186 | 6 | Let $p:X\to S$ be the unique map from a (locally compact) topological space $X$ to a point. Since $\underline{\hom}(\underline{\mathbb{Z}},-)$ is the identity functor, we have that $\Gamma(X,-)=\hom(\underline{\mathbb{Z}},-)$ and so
$$H^i(X,-)=\operatorname{Ext}^i(\underline{\mathbb{Z}},-)=\hom\_{\mathsf{D}(X)}(\underl... | https://mathoverflow.net/users/131975 | Why does $p_*p^! A$ deserve to be called homology with coefficients in $A$? | One nice geometric way to view homology is a measurement of your space $X$ given by probing $X$ with other, nicer spaces, for instance, singular homology probes with simplices. One good reason to call this abstract thing homology is that it too admits elements corresponding to nice enough maps from test objects into $X... | 3 | https://mathoverflow.net/users/128502 | 405640 | 166,295 |
https://mathoverflow.net/questions/405066 | 5 | Let $P$ be a parabolic subgroup of $\mathrm{GL}\_n$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by
$$
\mathcal{P}\_u:=\{gP\in G/P \mathrel\vert g^{-1}u g\in P\}\subseteq G/P.
$$
It is known that these varieties admit affine pavings; see for instance, [Fresse - Ex... | https://mathoverflow.net/users/41301 | Number of points of parabolic Springer fibres | The earliest reference I could find that works out the polynomial $f\_{\lambda, \nu}(q)$ is the paper
>
> R. Hotta, N. Shimomura ["The Fixed Point Subvarieties of Unipotent Transformations
> on Generalized Flag Varieties and the Green Functions"](https://eudml.org/doc/163262) Math. Ann. 241, 193-208 (1979)
>
>
> ... | 2 | https://mathoverflow.net/users/2384 | 405647 | 166,298 |
https://mathoverflow.net/questions/405659 | 2 | I want to find a condition on $\delta(G)$ (ex. $\delta(G) \geq an$) that guarantees $\kappa(G)=\delta(G)$ where $\kappa(G)$ is the vertex-connectivity of a bipartite graph $G$, and $\delta(G)$ is the minimum degree of $G$.
In other words, I want to prove that the statement
>
> If $\delta(G) \geq an$, then $\ka... | https://mathoverflow.net/users/384338 | Connectivity and the minimum degree of bipartite graph | This becomes true at $a = \frac{1}{3}$.
**Claim.** If $G$ is an $n$-vertex bipartite graph such that $\delta(G) \geq \frac{1}{3}n$, then $\delta(G)=\kappa(G)$.
*Proof.* Let $(A,B)$ be the bipartition of $G$. Suppose $\kappa(G)<\delta(G)$ and let $X \subseteq V(G)$ be such that $G-X$ is disconnected and $|X|=\kappa(... | 5 | https://mathoverflow.net/users/2233 | 405664 | 166,300 |
https://mathoverflow.net/questions/405663 | 2 | Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K\_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular locus of $Y$ consists of a single point of type $\frac{1}{2}(1,1,1)$, $C'$ passes through the singular point of $Y$, and $-... | https://mathoverflow.net/users/14514 | Existence of terminal $3$-fold flips | Yes - there are very many such examples, and you can cook up examples by a procedure called 'Mori's algorithm'.
A k2A flipping neighbourhood is a 3-fold flipping contraction $f\colon(C\subset X)\to (P\in Y)$, where the exceptional curve $C$ is irreducible and $X$ has two $cA\_{d\_i-1}/\tfrac1{r\_i}(1,-1,a\_i,0)$ hype... | 4 | https://mathoverflow.net/users/104695 | 405675 | 166,302 |
https://mathoverflow.net/questions/405624 | 4 | Consider a diffusion given by
$$X\_t=\int\_0^t a(s,X\_s)\,dW\_s$$
for $t\ge 0$, where $W\_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $0$.
Does then $X\_1$ have a bounded pdf?
---
This [interesting answer by James Martin](https://mathoverflo... | https://mathoverflow.net/users/36721 | Bounded density for diffusions with diffusion coefficients bounded away from $0$ | The "yes" answer follows immediately from [Theorem 2.5 in this paper by Kusuoka](https://reader.elsevier.com/reader/sd/pii/S0304414916300850?token=83AF411073D65047F42CC4F378ABD1D841ADA751E9CEE4DCE66CD933F7D9311AC9D96047079D3456564E461FAA7C60B7&originRegion=us-east-1&originCreation=20211007170037), which implies that $X... | 1 | https://mathoverflow.net/users/36721 | 405686 | 166,307 |
https://mathoverflow.net/questions/405687 | 8 | I want to characterize Hausdorffness of a locally convex space only using categorical terms of the additive category LCS of locally convex spaces and continuous linear maps, i.e., terms like mono- or epimorphisms, categorical limits or colimits, or images and kernels are allowed but the toplological definition *Distinc... | https://mathoverflow.net/users/21051 | Is Hausdorffness a categorical property in the category of locally convex spaces? | For a category $\mathcal{C}$, let $\mathcal{C}'$ denote the full subcategory of $\mathcal{C}$ whose objects are the non-terminal objects of $\mathcal{C}$.
In a category, say that an object $Y$ is final if for every object $X$ there exists an epimorphism $X\to Y$.
In turn, say that an object of $\mathcal{C}$ is pre-... | 6 | https://mathoverflow.net/users/14094 | 405690 | 166,308 |
https://mathoverflow.net/questions/404668 | 12 | Suppose $E$ is a complex-oriented ring spectrum whose formal group law is isomorphic to the additive one. As the title suggests, we might as well change the complex orientation so that the formal group law is literally the additive one. Is $E$ an $H\mathbb{Z}$-algebra?
| https://mathoverflow.net/users/163893 | Is every complex oriented ring spectrum with additive FGL an Eilenberg-Maclane spectrum? | This is an answer to the question in the title, which is what I had meant to ask: is an $E$ as in the question body an $H\mathbb{Z}$-module? (the last sentence of the question body is stronger and likely has a negative answer)
The answer is yes. Here is an outline of the proof. The vast majority of the following was ... | 4 | https://mathoverflow.net/users/163893 | 405695 | 166,310 |
https://mathoverflow.net/questions/405542 | 1 | This question can be seen as a continuation of [Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$](https://mathoverflow.net/questions/396424/lipschitz-continuity-of-mathbb-p-taut-with-respect-to-t)
Consider the martingale given as
$$X\_t=1+\int\_0^t a(s,X\_s)dW\_s,\quad \forall t\ge 0.$$
Denote $\ta... | https://mathoverflow.net/users/261243 | First hitting time for non-homogeneous diffusion martingale | For $h:=\Delta t>0$, you had
$$P(\tau>t)-P(\tau>t+h)=\int\_{(0,\infty)}
P(M\le-x|X\_t=x)\,P(X\_t\in dx),$$
where
$$M:=\inf\_{t\le u\le t+h}J\_u,\quad J\_u:=\int\_t^u a(s,X\_s)\,dW\_s.$$
By [Doob's martingale inequality](https://en.wikipedia.org/wiki/Doob%27s_martingale_inequality#Statement_of_the_inequality),
$$P(M\le-... | 3 | https://mathoverflow.net/users/36721 | 405700 | 166,313 |
https://mathoverflow.net/questions/405706 | 2 | I want to prove the following statement:
>
> Let $u$ be a vertex in a $2$-connected graph $G$. Then $G$ has two spanning trees such that for every vertex $v$, the $u,v$-paths in the trees are independent.
>
>
>
I tried to show this, but surprisingly, I have proved another statement.
>
> A graph with $\vert... | https://mathoverflow.net/users/384338 | Two independent spanning trees of $2$-connected graph | Yes, this is true. We will prove the following stronger lemma.
**Lemma.** Let $G$ be a $2$-connected graph and $u \in V(G)$. Then $G$ contains two spannings trees $T\_1$ and $T\_2$ such that for all $a,b \in V(G) \setminus \{u\}$ (possibly $a=b$), either
* the $ua$-path in $T\_1$ and the $ub$-path in $T\_2$ are int... | 1 | https://mathoverflow.net/users/2233 | 405710 | 166,319 |
https://mathoverflow.net/questions/405707 | 5 | It would lead me too far to explain how I stumbled upon the somewhat obscure identities
$$\sum\_{m=0}^n \binom{n}{m} (1-m)^m m^{n-m}=(-1)^n d\_n, \quad \sum\_{m=0}^n \binom{n}{m} (1-m)^{m-1} m^{n-m}=1,$$
where $d\_n=n!\sum^{n}\_{k=0} (-1)^k /k!$ is the $n$-th de Montmort number, when doing some algebro-geometric consid... | https://mathoverflow.net/users/104669 | A certain type of combinatorial identity, involving de Montmort numbers | In John Riordan's book *Combinatorial Identities*, page 21, is the formula
$$\sum\_{k=0}^n \binom{n}{k}(x+k)^k (y+n-k)^{n-k}=
\sum\_{k=0}^n \binom{n}{k} k!\, (x+y+n)^{n-k}.$$
(There is a typo in the formula given in the book—this is the correct version.) Riordan writes, "This is usually called Cauchy's formula," but I... | 5 | https://mathoverflow.net/users/10744 | 405716 | 166,323 |
https://mathoverflow.net/questions/405717 | 6 | How many sublattices does the powerset lattice $2^n$ contain for $n$ finite? (up to equality, not isomorphism)
I thought for sure this would be easy to find on OEIS, but so far I am coming up empty.
I really am interested in seeing a list of small examples, say up to $n=5$ or $6$ maybe, although perhaps already thi... | https://mathoverflow.net/users/2362 | How many sublattices are contained in the powerset lattice of a finite set? | [OEIS A306445:](https://oeis.org/A306445) 2, 4, 13, 74, 732, 12085, 319988, 13170652, 822378267, 76359798228, 10367879036456, 2029160621690295, 565446501943834078, 221972785233309046708, 121632215040070175606989, 92294021880898055590522262, 96307116899378725213365550192, 137362837456925278519331211455157, 2663792545369... | 14 | https://mathoverflow.net/users/75735 | 405719 | 166,324 |
https://mathoverflow.net/questions/405631 | 11 | It's well known that $1$ is the sum of three cubes infinitely many different ways but is it true for perhaps the tetrahedral numbers as well? Let $T\_n = (1/6)n(n+1)(n+2)$. Then the following are the first solutions less than 6000 of the form $T\_a - T\_b - T\_c = 1$.
$$
\begin{array}{c c c}
a & b & c\\
6 & 5 & 4\\
8 &... | https://mathoverflow.net/users/265714 | Prove that $1$ is the sum of three tetrahedral numbers infinitely many different ways | There are infinitely many solutions. I'll show below that there are infinitely many positive integers $k$ for which $93k^{2} - 288k + 276 = z^{2}$ for some positive integer $z$. From such a $z$, we get a solution by setting
$a = \frac{z+3k}{6} - 1$, $b = 2k-3$, and $c = \frac{z-3k}{6} - 1$. Here's a table of solutions ... | 16 | https://mathoverflow.net/users/48142 | 405721 | 166,325 |
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