All Questions

2
votes
0answers
82 views

Partial well-ordering of formulas

Given a theory $T$, for arbitrary formulas $φ$ and $ψ$ that provably in $T$ denote an ordinal, set $[φ]_T < [ψ]_T$ iff provably in $T$, the ordinal denoted by $φ$ is less than the ordinal denoted ...
5
votes
1answer
282 views

Are framed manifolds cubulatable?

Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} ...
6
votes
0answers
114 views

Motivic cohomology of $n$-sphere

All motivic cohomology groups are taken with $\mu_2$ coefficient and $k$ has characteristic different from $2$. Consider the affine variety $X$ with coordinate ring $k[x_1,\ldots,x_n]/(x_1^2+\ldots + ...
1
vote
0answers
56 views

Equality condition for Araki–Lieb–Thirring inequality

I'd like to have the equality condition in the Araki–Lieb–Thirring inequality $$\operatorname{Tr} [(BAB)^r]\leq \operatorname{Tr} [(B^{r}A^{r}B^{r})],$$ valid for $A,B$ semidefinite positive and $r\...
12
votes
0answers
159 views

What is the centralizer of a Coxeter element?

Let $(W,S)$ be a Coxeter system (of finite rank) and $c \in W$ a Coxeter element. If $W$ is finite, then the centralizer $C_W(c)$ is the cyclic group generated by $c$ (e.g. see the book "Reflection ...
1
vote
0answers
163 views

p-adic Hodge theory for singular projective varieties

In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic ...
1
vote
2answers
299 views

When was the generalization easier to prove than the specific case? [duplicate]

I distinctly remember from my long-ago undergraduate math that there were some interesting cases where a solution (proof) was sought for some specific thing but it wasn't easy to find - and in a few ...
0
votes
0answers
18 views

How to optimize two ranges for the determination of the intersection point between two curves [on hold]

I start this thread asking for your help in Excel. The main goal is to determine the coordinates of the intersection point P=(x,y) between two curves (curve A, curve B) modeled by points. The curves ...
1
vote
0answers
18 views

Bound for the additive period length of certain Sprague-Grundy functions

Let $\left( Y_x \right)_{x=0}^\infty $ be a sequence of finite subsets of $\mathbb{Z}$, and let $G : \mathbb{N}_0 \to \mathbb{N}_0$ be a greedy permutation, defined by $$ G(x) = \operatorname{mex} \...
4
votes
1answer
139 views

Property $\Gamma$ in terms of Correspondences

A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset $\{ x_{1}, x_{2},..., x_{n} \} \subseteq M$ and each $\epsilon >0$, there is a unitary element $u$ in $...
3
votes
0answers
27 views

Subalgebras and ideals of algebra of Vassiliev invariants

Let $\mathcal{K}$ be the commutative monoid whose elements are (isotopy) equivalence classes of knots with composition under the connected knot sum, and $\mathbb{Z}\mathcal{K}$ be the corresponding ...
3
votes
0answers
50 views

The singular cohomology embeds into the symplectic cohomology

Viterbo's theorem on cotangent bundles $M=T^*N$ tells you in particular that singular cohomology $H^*(M)$ gets embedded in $SH^*(M)$ via the $c^*$ map. Having a Weinstein manifold (or more generally ...
2
votes
1answer
43 views

Intensity and compensator for a jump process

Set-up and assumptions. Let $(\mathscr{F}_t, t \geq 0)$ be a right-continuous complete filtration. Let $(X_t, t\geq 0 )$ be a pure jump $\mathbb{R}$-valued process with unit jumps, that is, $$ X_t = \...
2
votes
1answer
105 views

Injective Change of Rings

Sorry if this is too elementary, but when I was going to ask this question on math.stackexchange, I saw the same question with three up-votes and no answer. So I decided to post it here. I am doing ...
1
vote
0answers
39 views

Does the regularity of the initial data have to agree with the solution's spatial regularity in evolutionary PDEs?

Let's say we have some evolutionary PDE and the initial data $u_0$ is in the space $X$. For example $X=H^s(\Omega)$ for some $s$. My question is if the solution has to have the same spatial regularity,...
0
votes
0answers
52 views

On an approach on the Hilbert-Polya Conjecture suggested by Schumayer and Hutchison

In their expository paper, ''Physics of the Riemann Hypothesis arxiv.org/abs/1101.3116v1'', Hutchison and Schumayer suggested the following approach on the Hilbert Polya conjecture, via quantisation ...
4
votes
1answer
83 views

“Twisted” direct sums of Banach spaces

It is hard to provide motivation, so I just want to state this definition "as is". Suppose I have a Banach space $E$ and two commuting, injective operators $R_0, R_1$ on $E$ which satisfy the ...
5
votes
0answers
73 views

Second bounded cohomology and normal subgroups

It may be a naive question, but: If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups? But "many", ...
-1
votes
0answers
45 views

Theorem of Lebesgue point [on hold]

Let $h \in {L^1}\left( {0,T} \right)$, then for almost $t \in \left( {0,T} \right)$ prove that $$n\int\limits_t^{t + \frac{1}{n}} {\left( {1 - n\left( {s - t} \right)} \right)h\left( s \right)ds} \to ...
1
vote
1answer
38 views

Tightly knit graphs on $\omega$

We say that a simple, undirected graph $G = (\omega, E)$ on the vertex set $\omega$ is tightly knit if there is a positive integer $n>2$ such that for all $v,w\in \omega$ there is a cycle $C$ of ...
2
votes
0answers
114 views
+50

Equal principal minors of matrix plus rank-1 and inverse

Given an invertible real matrix $A$ and real column vectors $b$ and $c$. For which $A$,$b$ and $c$ are all corresponding principal minors of $B = A-bc^T$ and $A^{-1}$ equal? According to a ...
1
vote
1answer
97 views

Complete intersection subvariety of projective variety

I am not able to find any literature which studies complete intersection subvarieties of a projective variety, all good references consider CI in projective space. My guess is, a subvariety X of ...
1
vote
0answers
39 views

Pure (ordered) Subgroups

Let $H,G$ be abelian groups with $H \leq G$. We say that $H$ is a pure subgroup of $G$ if for every $n \in \mathbb N$ and $h \in H$ the following holds: If $h$ is $n$-divisible in $G$, then $h$ is $n$-...
11
votes
1answer
411 views
+500

Applications of integral p-adic Hodge theory

What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $\mathbb{Z}_p$-...
3
votes
0answers
106 views

A recursion with a number-theoretic function

For a positive integer $Q$, let $$ s(Q) := Q\,\sum_{p\mid Q} \frac1p, $$ where the sum extends over all prime divisors of $Q$; also, let $s(0)=0$. Thus, we have, for instance, $s(1)=0$, while $s(p^\...
1
vote
0answers
99 views

Quasi homomorphism from integers to reals

Reference : Lemma 5.3 from "Groups acting on circle" by Etiyenne Ghys. Let $f:\mathbb{Z}\rightarrow \mathbb{R}$ be a quasi-homomorphism, i.e $|f(a+b)-f(a)-f(b)|\leq D$ $\forall$ $a$ and $b$ in $\...
1
vote
0answers
26 views

Is every critically subsimple Laver-like algebra a quotient of a critically simple Laver-like algebra on the same number of generators?

A finite reduced Laver-like algebra is a finite algebra $(X,*,1)$ that satisfies the identities $1*x=x,x*1=1,x*(y*z)=(x*y)*(x*z)$ and where there is a natural number $n$ and a function $\mathrm{crit}:...
1
vote
0answers
85 views

Fine tuning the growth rate of the degrees of polynomials

Let $r$ be an integer with $r>1$. Suppose that if $k\geq 0$, then $p_{k}(x)$ is a polynomial with nonnegative integer coefficients with $p_{k}(0)=1$ but where $p_{k}\neq 1$. Suppose that $$\...
15
votes
1answer
268 views

Where to find some subset of Khovanskii's 15 proofs of the BKK theorem?

(I asked this question on MSE, but someone suggested it would be better asked here.) I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^*)^n$ to a ...
1
vote
0answers
66 views

Number of misplaced elements for a partition of a set of coloured items

We are given a set $V$ of $n$ items, where each item is tinted with exactly one color in $C=\{c_1, c_2, \ldots, c_k\}$, in such a way that for each $i\in [k]$ there exists at least one item tinted ...
1
vote
0answers
85 views

Factorizations of etale morphisms

Let $f:X \rightarrow Y$ be a finitely presented separated etale morphism, with $Y$ quasicompact and quasiseparated. By Zariski’s main theorem, we can factor $f$ as $f= g \circ j$ with $j$ an open ...
0
votes
0answers
21 views

Show that the transition semigroup of the strong solution to a Langevin-type SDE is immediately differentiable

Let $\varrho\in C^1(\mathbb R)$ with $\varrho>0$ $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$ $\mu$ denote the measure with density $\varrho$ with respect to $\lambda$ $b:=2^{-...
4
votes
1answer
149 views

rank of Jacobian of Fermat curve and Chabauty-Coleman method

Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. ...
3
votes
0answers
108 views

Quotient of a smooth projective surface by an involution

Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?
2
votes
0answers
131 views

Translation of Soergel's 1990 paper on category O

Is there any English translation for the folowing paper of Soergel? Kategorie $\mathcal{O}$, perverse Garben, und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445,...
1
vote
0answers
41 views

Lower-bound probability of non-centered quadratic form

Let $X\sim N(\mu,\sigma^2I)\in \mathbb{R}^n$ be a non-centered ($\mu\neq 0$) Gaussian vector with independent coordinates. I'm wondering if there is any sharp lower bound of the following probability: ...
1
vote
0answers
110 views

Sources of derived schemes in geometric representation theory

What are some derived schemes naturally arising in geometric representation theory? Some examples include: Steinberg scheme Hilbert scheme Moduli stack of local systems. Now, this looks like a ...
2
votes
1answer
273 views

A clarification of an argument in “Perfectoid spaces”

The page 50 of (the arXiv version of) the above-mentioned paper of P. Scholze says "Now the Poincare duality pairing implies that $H^i(Y_{\mathbb{C}_p, et}, \bar{\mathbb{Q}}_l)$ is a direct summand of ...
0
votes
0answers
29 views

What is the locus defined by those equations?

I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by $\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$. I know that if $\...
0
votes
0answers
17 views

Number of local minima of a particular non-convex composite function

I am interested in proving that the following equation on the interval $x \in [c,1]$ is minimized either at the endpoints or where $x=\sqrt{c}$: $f(x)=\frac{-1}{\log(1-x)}+\frac{-1}{\log(1-\frac{c}{x}...
3
votes
0answers
116 views

Are Chain Complexes Related to the Tangent Bundle Construction?

For a scheme $X$ over $\text{Spec}(K)$, we can consider maps $\text{Sch}(\text{Spec}(K[d] / d^2), X)$, which we can think of as the tangent bundle over $X$. A map $\text{Spec}(K) \rightarrow S$ picks ...
4
votes
0answers
104 views

Absolute oscillator in Langton's Ant

We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...
0
votes
0answers
34 views

Perform a univariate integral, involving a Gauss hypergeometric function

This is a follow-up question to the one posed in Compute the two-fold partial integral, where the three-fold full integral is known . (I hope that doing so is viewed as a legitimate step. If not so, I ...
7
votes
1answer
172 views

Factoring $\frac{1}{1-rx}$ into an infinite products of polynomials

I am looking for examples of sequences of polynomials $(p_{k}(x))_{k=1}^{\infty}$ with positive integer coefficients where $p_{k}(0)=1$ for all $k\geq 1$ and where there is a positive integer $r$ ...
0
votes
1answer
58 views

Finding the minimum sum of a subset of entries of a given matrix with combinatorial constraints

Given a matrix $M\in\mathbb{N}^{n\times n}$, let $Z$ be the set of all the $M$'s entry subsets $S$ such that (i) no two entries of $S$ are on the same row or column of $M$ and (ii) $|S|=n$. Clearly we ...
0
votes
0answers
101 views

Analytical solution of a system of nonlinear PDEs

I am looking for an analytic solution for the equations $$\left\{ \begin{eqnarray} \frac {\partial v} {\partial x} &=& -m \frac {\partial u} {\partial t} \\ \frac {\partial u} {\partial x} &...
5
votes
1answer
123 views

Moore Graphs and Finite Projective Geometry

In a comment on a blog post from 2009 about the hypothetical Moore graph(s) of degree 57 and girth 5, Gordon Royle offered the following observation (reproduced here in full for the sake of ...
3
votes
1answer
78 views

A problem with sequences with composition of $\log$s

If $(a_n)_{n \ge 1}$ is a non-negative sequence s.t., $$\sum\limits_{n = c_k}^\infty \frac{a_n}{\log^{(k)} n} < \infty, \, \forall k \ge 1 \overset{?}{\implies} \sum\limits_{n \ge 1} a_n < \...
1
vote
0answers
49 views

Reference question: $C^1$ estimate for a stable minimal surface

I am looking for an answer/reference to the following question: Let $(M,g)$ be a complete Riemannian manifold and $\Sigma\subset M$ a closed, stable minimal surface. Is it possible to prove $C^1$ ...
4
votes
0answers
98 views

Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post. I had discussed my computation of $$ \Omega_5^{...

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