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https://mathoverflow.net/questions/341528 | 4 | Let $\Omega : \mathbf{Top} \to \mathbf{Loc}$ the functor that takes a topological space to its locale of opens.
For a locale morphism $f$, write $f^\*$ for the correspoding morphism in $\mathbf{Frm}$, the category of frames and frame homomorphisms.
Let $\times\_t$ denote the product in $\mathbf{Top}$ and $\times\_{\ell... | https://mathoverflow.net/users/125560 | Product of topological spaces and product of corresponding locales | Your map $f$ is known to be an injective dense localic map. See, for example, Proposition 4.2.2 in [1]. In general, it isn't an isomorphism. The reason for this is that $\Omega(X \times\_t Y)$ is quotiented by more equations than $\Omega(X) \times\_\ell \Omega(Y)$ is.
We can think of $\Omega(X \times\_t Y)$ as if it ... | 6 | https://mathoverflow.net/users/46003 | 342413 | 145,406 |
https://mathoverflow.net/questions/342299 | 3 | Let $J\_{k,m}(N)$ be the space of Jacobi forms of weight $k$, index $m$, and congruence subgroup $\Gamma\_{0}(N) \rtimes \mathbb{Z}^{2}$. I do not believe it is relevant here to specify what type of Jacobi form (weak, holomorphic, etc). I can't find hardly any references on how the index, and the congruence subgroup ch... | https://mathoverflow.net/users/105661 | Index and congruence subgroup from scaling variables of Jacobi form | $\varphi(d\_1\tau,d\_2 z)$ will generally be a Jacobi form for a congruence subgroup of this kind only when $d\_1 | d\_2$. Otherwise it will not transform correctly under $(\tau,z) \mapsto (\tau,z+\tau)$.
In this case its congruence group is $\Gamma\_0(d\_1) \rtimes \mathbb{Z}^2$ and its index is $m\frac{d\_2^2}{d\_1... | 2 | https://mathoverflow.net/users/146384 | 342414 | 145,407 |
https://mathoverflow.net/questions/334032 | 10 | I've tried to become familiar with the so-called "internal language of a category" for the last months. However, I'm still not confident enough when, for instance, I find a subobject (of a given object) which is defined through a formula of the internal laguage of the category that I'm considering.
In detail, if $C$ ... | https://mathoverflow.net/users/125249 | Reference request about “internal language of categories” | Here are some resources:
1. Carsten Butz.[*Regular categories and regular logic*](https://www.brics.dk/LS/98/2/), notes, 1998.
2. P. Freyd and A. Scedrov. *Categories, Allegories.* North–Holland, Amsterdam 1990.
3. S. Mac Lane and I. Moerdijk. *Sheaves in Geometry and Logic.* Springer–Verlag, New York 1992.
4. B. Jac... | 2 | https://mathoverflow.net/users/1176 | 342425 | 145,408 |
https://mathoverflow.net/questions/342423 | 0 | Let $H$ be a separable, complex Hilbert space and let $\mathcal{B}(H)$ denote the algebra of bounded linear operators on $H$. Let $T \in \mathcal{B}(H)$. We define $$ A = \{ p(T,T^\*) : p \in \mathbb{C}[z\_1,z\_2] \}.$$ $A$ is the subalgebra generated by $T,T^\*$ - or the $\*$-algebra generated by $T$. If we denote by ... | https://mathoverflow.net/users/145367 | Weak closure of subalgebra generated by an operator and its adjoint | Yes, the unilateral shift $S$ on $l^2(\mathbb{N})$ generates $B(l^2(\mathbb{N}))$ as a von Neumann algebra. This is a consequence of the double commutant theorem and the fact that the only bounded operators which commute with both $S$ and $S^\*$ are scalars.
(To see this, suppose $T$ commutes with both $S$ and $S^\*$... | 3 | https://mathoverflow.net/users/23141 | 342427 | 145,409 |
https://mathoverflow.net/questions/342419 | 2 | It is well-known that $\{e^{i n t}\}\_{n\in\mathbb Z}$ is an orthonormal basis for $L^2(-\pi,\pi)$. A theorem by Kadec (Kadec $1/4$ theorem) studies the perturbed exponential system:
>
> If $\{\lambda\_n\}$ is a sequence of real numbers for which
> $$|\lambda\_n-n|\leqq L<\frac{1}{4}, \ \ n=0, \pm 1, \pm 2, \dots$... | https://mathoverflow.net/users/69931 | Kadec $1/4$ theorem and some examples with $l=\sup_{n\in\mathbb Z} |n - \lambda_n | \geq \frac{1}{4}$ | There is a complete characterization of sequences $\lambda\_n$ for which which
$e^{i\lambda\_n t}$ is a Riesz basis:
G. Semmler,
[Complete interpolating sequences, discrete Muckenhoupt condition, and conformal mapping](http://www.acadsci.fi/mathematica/Vol35/Semmler.html), which permits to construct many examples.
| 3 | https://mathoverflow.net/users/25510 | 342432 | 145,412 |
https://mathoverflow.net/questions/342416 | 1 | Does there exist a smooth projective variety $X$ such that $\operatorname{NEF}(X)-\{0\}$ is strictly contained in $\operatorname{Big}(X)$, where $\operatorname{Big}(X)$ is the interior of $\overline{\operatorname{Eff}(X)}$?
| https://mathoverflow.net/users/nan | $\operatorname{NEF}(X)\subset\operatorname{Big}(X)$? | Probably someone will come up with a simpler example, but here's a Calabi-Yau threefold that fits the bill: look at <https://arxiv.org/pdf/1206.1649.pdf> . Use the example given in section 6. The Picard rank is 2, the nef cone is given in Prop 6.1, and the rays on the boundary of the movable cone are computed in Lemma ... | 1 | https://mathoverflow.net/users/142054 | 342433 | 145,413 |
https://mathoverflow.net/questions/342415 | 2 | Suppose $X\_1, X\_2,\dots, X\_n$ are iid random variable,$P(X\_i=-\infty)$ is allowed,$P(X\_i>v)< e^{-v}\forall v>0$, $X$ is distributed as $X\_i$, if $c$ is a finite real number such that $E(X)<c$, then show that there is $A>0, r<1$ such that $P(X\_1+\dots+X\_n>nc)< Ar^n\forall n$. Can apply Chernoff or Hoefding bound... | https://mathoverflow.net/users/93713 | Sum of independent random variables having exponential tails | By the Markov--Bernstein inequality (incorrectly referred to as Chernoff's)
\begin{equation}
P(X\_1+\dots+X\_n\ge nc)\le e^{-tnc}Ee^{t(X\_1+\dots+X\_n)}
=e^{-tnc}(Ee^{tX})^n=e^{ng(t)}
\end{equation}
for real $t\ge0$, where
\begin{equation}
g(t):=-tc+\ln Ee^{tX}.
\end{equation}
The condition $P(X\_i>v)< e^{-v}\ \f... | 4 | https://mathoverflow.net/users/36721 | 342434 | 145,414 |
https://mathoverflow.net/questions/342440 | 2 | Let $(\Omega,\Sigma,P)$ be a probability space and $A\in \Sigma$ be such that
$P(A) = 1$. Let $X:\Omega \to \mathbb{R}^n$ be a $\Sigma$-measurable map and $\Lambda\_X (B) = P(X \in B)~\forall B \in \mathcal{B}(\mathbb{R^n})$. Now, $X(A)$ need not be measurable but does there exist a measurable set $B \subset X(A)$ such... | https://mathoverflow.net/users/75483 | Unit probability subset of image of a measurable set | It is not true in general. Let $\Omega = V \subset [0,1]$ be a set of outer Lebesgue measure 1 and inner measure $c < 1$ (you may take the complement of a familiar Vitali set of inner measure 0), and let $\Sigma = \{B \cap V : B \in \mathcal{B}([0,1])\}$. Define the measure $P$ on $\Sigma$ by $P(B \cap V) = m(B)$ where... | 4 | https://mathoverflow.net/users/4832 | 342448 | 145,419 |
https://mathoverflow.net/questions/342443 | 6 | I'm currently studying Ando-Hopkins-Rezk's work [Multiplicative orientations of KO-theory and of the spectrum of TMFs](https://faculty.math.illinois.edu/~mando/papers/koandtmf.pdf). At a point a presumably obvious isomorphism is mentioned, which I'm however not able to immediately see (I guess I'm missing something obv... | https://mathoverflow.net/users/8320 | Morphisms from $bstring$ to $X\otimes \mathbb{Q}$ and sequences $s_n\in\pi_n(X)\otimes \mathbb{Q}$ | This is an assemblage of known results, I'll try to put a reference for all of them.
* By a classical theorem of Serre all stable homotopy groups are finite in positive degree. In particular we have $\mathbb{S}\_\mathbb{Q}=H\mathbb{Q}$.
* Rationalization is a smashing localization, so that $\pi\_\*(\mathbb{S}\_\mathb... | 7 | https://mathoverflow.net/users/43054 | 342450 | 145,421 |
https://mathoverflow.net/questions/342183 | 5 | When tensoring finite dimensional representations of the Lie algebra ${\frak sl}\_n$, we have an explicit algorithm given in terms of Young diagrams. See Section 4 of this [paper](http://www.phys.nthu.edu.tw/~class/group_theory2012fall/doc/tensor.pdf).
Do there exist similar pictures for the $B$ and $D$ series? I am ... | https://mathoverflow.net/users/126606 | Tensoring irreducible $B$-series representations/ Type B Littlewood-Richardson | Littelmann used standard monomial theory to give a unified Littlewood–Richardson rule for the simple reductive algebraic groups of types $A$, $B$, $C$ and $D$ (and some others) in which the coefficients enumerate certain generalized standard tableaux. See (a) in the theorem on page 346 of [Littlemann's paper](https://c... | 4 | https://mathoverflow.net/users/7709 | 342454 | 145,422 |
https://mathoverflow.net/questions/342453 | 21 | Let $\mathcal{M}\_{g,n}$ be the moduli space (stack) of stable smooth curves of genus $g$ with $n$ marked points over $\mathbb{C}. $ It's known that by adding stable nodal curves to $\mathcal{M}\_{g,n}$, the resulting space $\overline{\mathcal{M}}\_{g,n}$ is compact. But why is it so? For example, consider the followin... | https://mathoverflow.net/users/146366 | Why not add cuspidal curves in the moduli space of stable curves? | If you add cuspidal curves, then $\overline{\mathcal{M}}\_{1,1}$ will no longer be separated, which is the scheme/stack analogue of Hausdorff. Specifically, consider the families
$$y\_1^2 = x\_1^3 + t^6 \ \mbox{and}\ y\_2^2 = x\_2^3 + 1$$
(so the second family is a constant family with no $t$-dependence). For all nonze... | 41 | https://mathoverflow.net/users/297 | 342459 | 145,424 |
https://mathoverflow.net/questions/342451 | 1 | [See also StackExchange](https://math.stackexchange.com/q/3368742/631742).
**Setup.** Let $n\in\Bbb N$. Let $a\_{1,1}, a\_{1,2},\dots, a\_{1,n}\in\Bbb R$ be a given sequence of real numbers that sum to $0$, i.e. $a\_{1,n}=-(a\_{1,1}+a\_{1,2}+\dots+a\_{1,n-1})$. For $i=2,\dots,n$ define
$$a\_{i,j}=a\_{1,j}+a\_{1,j+1}+... | https://mathoverflow.net/users/129831 | Number of zeroes and sign switches in a constructed zero-sum double sequence | The cyclic sequence $a\_1, a\_2, \dots, a\_n, a\_1$ contains two occurrences of (zero or sign change), as it sums up to $0$. Each zero corresponds to a zero in either $1$st or $(n-1)$th row, or in both. Each sign change corresponds to a sign change in exactly one of the two rows (note that a sign change $a\_n, a\_1$ ap... | 3 | https://mathoverflow.net/users/17581 | 342465 | 145,425 |
https://mathoverflow.net/questions/342467 | -2 | Now to specifics:
Let $V \subset \mathbb{A}^3$ be a reducible affine algebraic set defined by two irreducible polynomials $f,g \in K[X,Y, Z]$ of degree $d$ ($K$ algebraically closed). So, if $V$ is regular, is then its projective closure in $\mathbb{P}^3$ also regular?
I tend to think that this is not necessarily t... | https://mathoverflow.net/users/146404 | Is projective closure of a regular affine algebraic set also regular? | This can easily fail. Start backwards: take a projective variety which is not regular, say $X\subseteq \mathbb P^n$ and let $S\subseteq X$ be its singular set. Choose a hypersurface $H$ that contains $S$. Then $\mathbb P^n\setminus H$ is an affine variety and $X\setminus H$ is regular.
In your specific situation, cho... | 3 | https://mathoverflow.net/users/10076 | 342470 | 145,427 |
https://mathoverflow.net/questions/342462 | 1 | Let $\mathcal Q$ be some qualification on formulas in the first order language of set theory (FOL($\in$)), *that is met by at least one formula*; Let $T$ be the first order set theory whose extra-logical axioms are the following *sole* axiom schema:
$\mathcal Q$-Comprehension schema: if $\phi(y)$ is a formula that me... | https://mathoverflow.net/users/95347 | What are the known conditions for a restriction on naive comprehension that enables a generalization of a property all so constructed sets meet? | The answer to the first question is no. Suppose no formula meets qualification Q. Let () be x≠x.
The answer when there is at least one formula that meets qualification Q and the language does not have = as a primitive symbol, is still no. Suppose that the only formulas which meet qualification Q are
(y∈y or not(y∈y)... | 2 | https://mathoverflow.net/users/133981 | 342490 | 145,432 |
https://mathoverflow.net/questions/342485 | 2 | The problem I'm dealing with has the following form. Let $X$ be some uncountable set, and $Y$ be some finite set. Suppose $f: X \times Y \to [0,1]$, and given $\mathcal{H} \subseteq Y^X$, I'm looking for some $P$, a distribution with support on $\mathcal{H}$, such that
$$
\forall x, \ f(x, y) = \int\_{h\in \mathcal{H... | https://mathoverflow.net/users/146408 | Finding a distribution satisfying uncountably many constraints. Any relevant references? | It seems that in general this is an almost arbitrarily hard problem.
Consider the simpler countably infinite case $X=\mathbb N$, $Y=\{0,1\}$.
Thus $H=\{x:h(x)=1\}$ is a "random set".
Fix $g:X\to\{0,1\}$ and let $f(x,g(x))=1$ (which forces $f(x,1-g(x))=0$). Thus $G=\{x:g(x)=1\}$ is an arbitrary subset of $\mathbb N$... | 1 | https://mathoverflow.net/users/4600 | 342491 | 145,433 |
https://mathoverflow.net/questions/342488 | 6 | I have asked this question in [math.se](https://math.stackexchange.com/questions/3365350/decomposing-a-entry-wise-positive-positive-semidefinite-matrix) without any success.
Let $\mathbf{A}$ be a symmetric $n\times n$ positive semi-definite matrix and also such that each of its entries is positive. Does $\mathbf{A}$ ... | https://mathoverflow.net/users/27249 | Rank-one positive decomposition for a entry-wise positive positive definite matrix | No, there exist doubly nonnegative matrices which are not completely positive, see for example [The difference between 5 x 5 doubly nonnegative and completely positive matrices](https://www.sciencedirect.com/science/article/pii/S002437950900281X) (2009).
A graph-based characterization of doubly nonnegative matrices ... | 5 | https://mathoverflow.net/users/11260 | 342495 | 145,435 |
https://mathoverflow.net/questions/342492 | 0 | Thank you for reading.
My question was raised up when I tried to prove an example in the book of Liggett(1985), which is in P13 Example 2.3(a).
Here is a link of the page:
<https://books.google.com/books?id=7JbqBwAAQBAJ&lpg=PR3&dq=liggett%201985&hl=zh-CN&pg=PA13#v=onepage&q=liggett%201985&f=false>
My questions ... | https://mathoverflow.net/users/83917 | A question about positive operator pregenerator | This question is not research level and would be better suited on MathStackExchange.
Positivity here just means $f\ge 0$ $\Rightarrow$ $Tf\ge 0$ (where $f\ge 0$ is defined as $f(x)\ge 0$ for each $x\in X$).
The example is verified using Proposition 2.2: If $f(\eta)=\min f(X)$ then $g=f-f(\eta)1\ge 0$ and since $T... | 0 | https://mathoverflow.net/users/21051 | 342499 | 145,436 |
https://mathoverflow.net/questions/331336 | 3 | I have a question on an argument appearing in this paper [P](http://www.ams.org/journals/tran/2000-352-06/S0002-9947-00-02594-0/S0002-9947-00-02594-0.pdf).
**Setting**
Let $S=(1,\infty) \times (-1,1) \subset \mathbb{R}^2$ be a split, and let $X=(\{X\_t\},\{P\_x\}\_{x \in S})$ be a Brownian motion in $S$ conditioned... | https://mathoverflow.net/users/68463 | Strong Markov property, independence, regular conditional probability | Let $\tau$ be a Markov time, and define the usual $\sigma$-algebras: $$\mathcal F^{<\tau} = \sigma\{X\_t^{-1}(E) \cap \{t < \tau\} : t \geq 0, \, E \text{ — Borel}\}$$ and $$\begin{aligned} \mathcal F\_{\geqslant\tau} & = \sigma\{X\_{\tau + t}^{-1}(E) : t \geq 0, \, E \text{ — Borel}\} \\ & = \sigma\{X\_t^{-1}(E) \cap ... | 1 | https://mathoverflow.net/users/108637 | 342502 | 145,437 |
https://mathoverflow.net/questions/342503 | 8 | (Sorry for what is probably a rather foundational PL-topology question.) By a triangulation of a manifold $M$, I mean a homeomorphism with the geometric realization of a simplicial complex, $h: |K| \to M$. I am wondering if I am given two triangulations $h\_0 : |K\_0| \to M \times \{0\}$ and $h\_1 : |K\_1| \to M \times... | https://mathoverflow.net/users/99414 | Extending a triangulation of the boundary of $M \times I$ | I think, one has to assume that the triangulations are smooth (i.e. restrictions of $h\_i$ to every simplex are smooth). Then the answer is yes, this is a special case of a theorem by Munkres: a $C^r$-triangulation of the boundary of a manifold extends to a $C^r$-triangulation of the manifold, see Theorem 10.6 in
*Mu... | 8 | https://mathoverflow.net/users/98590 | 342512 | 145,440 |
https://mathoverflow.net/questions/339795 | 0 | A complex manifold $M$ is said to be **Fano** if the Chern curvature $2$-form is a positive definite $(1,1)$-form. What happens if the Chern curvature $2$-form is a negative definite $(1,1)$-form? What are such manifolds called, and do they differ in any meaningful way from Fano manifolds?
| https://mathoverflow.net/users/143172 | Negative Definite Fano Manifolds | Such manifold are called of general type (at least in the projective case). If you look at *compact* Riemann surfaces, then the only Fano variety is $\mathbb{P}^1$ while curves of general type have bigger moduli spaces. Their geometry, and even topology, is really different.
| 2 | https://mathoverflow.net/users/142625 | 342520 | 145,441 |
https://mathoverflow.net/questions/342522 | 3 | **The question:**
Let $\pi$ be a Radon probability measure on $[0,1]^d$, $2\leq d < \omega$, that is singular (w.r.t. to the $d$-dimensional Lebesgue measure).
Suppose that for $i\in \{1,\dots,d\}$ and $\alpha \in [0,1]$, the set $S = \{ x \in [0,1]^2 \mid x\_i = \alpha \}$ intersects the support of $\pi$ (but might ha... | https://mathoverflow.net/users/146430 | Singular Radon probabilities on $[0,1]^d$. Is conditioning on $x_i = \alpha$ well-defined? | $\newcommand{\B}{\mathcal B}$
First here, $[0,1]^d$ is a Polish space (i.e., a separable complete metric space). So, $[0,1]^d$ is a [Radon space](https://en.m.wikipedia.org/wiki/Polish_space#Radon_spaces), and hence any (Borel) probability measure is Radon. So, you did not have to say that the probability measure $\pi$... | 1 | https://mathoverflow.net/users/36721 | 342540 | 145,445 |
https://mathoverflow.net/questions/342532 | 3 | Let $X$ be a Banach space where the closed unit ball equals the convex hull of its extreme points. Is it true that this implies $X$ is reflexive?
| https://mathoverflow.net/users/58366 | A Banach space where the closed unit ball is the convex hull of its extreme points | The answer is "No" because there exist nonreflexive Banach spaces in which every point on the surface of the unit ball is extreme. See, for example, Diestel, Geometry of Banach spaces, Chapter 4, Section 2, Theorem 1 and apply it to the natural embedding of $\ell\_1$ into $\ell\_2$.
| 4 | https://mathoverflow.net/users/85406 | 342545 | 145,448 |
https://mathoverflow.net/questions/341941 | 1 | Let $X$ and $Y$ be bounded complete separable metric spaces. Let $C = 2^\omega$ be Cantor space with its standard metric. All product spaces are taken to have the max metric.
Let $F, G \subseteq X\times Y$ be closed sets such that $\inf \{d(a,b) : a \in F, b \in G \} > 0$. For each $y\in Y$, let $F\_y = \{ x\in X:(x... | https://mathoverflow.net/users/83901 | A Uniform Metric Selection Theorem | For $C=\omega^\omega$ the answer is affirmative:
>
> **Theorem 1.** Let $X$ be a metric space, $Y$ be separable metric space, and $F,G\subset X\times Y$ be closed sets such that $\inf\{d(x,y):x\in F,\;y\in G\}>0$. Then there exists a subset $Q\subset Y\times \omega$ and a function $f:Q\to 2^X$ such that
>
>
> $\b... | 2 | https://mathoverflow.net/users/61536 | 342549 | 145,449 |
https://mathoverflow.net/questions/342551 | 0 | Let $(a,b,c,d)$ be a 4-tuple of real numbers such that $a \leq b \leq c \leq d$, but is otherwise arbitrary. Are there real sequences $x\_n$ and $y\_n$, such that lim inf $x\_n$ $+$ lim inf $y\_n$ $= a$, lim inf $(x\_n + y\_n) = b$, lim sup $(x\_n + y\_n) = c$, and lim sup $x\_n$ $+$ lim sup $y\_n$ $= d$?
| https://mathoverflow.net/users/43439 | Can this string of inequalities take on arbitrary values? | No. If $a=0$, then we may add a constant to $x\_i$ and subtract it from $y\_i$ to achieve $\liminf x\_n=\liminf y\_n=0$; this does not affect $b$, $c$, and $d$. Without loss of generality, assume that $\limsup x\_n\geq d/2$; then $c\geq d/2$ as well.
Similarly, since $\limsup y\_n\leq d/2$, we get $b\leq d/2$.
Conv... | 3 | https://mathoverflow.net/users/17581 | 342553 | 145,451 |
https://mathoverflow.net/questions/341722 | 0 | Let $\mathcal X$ be a Polish space, and let $(N\_x)\_{x \in \mathcal X}$ be a system of closed neighborhoods in $\mathcal X$. Define $\Omega := \{(x,x') \in \mathcal X^2 \mid N\_x \cap N\_{x'} = \emptyset\}$, assumed to be open in $\mathcal X^2$. For example in a Banach space $\mathcal X$, an example would be $N\_x = x... | https://mathoverflow.net/users/78539 | Relationship between a certain binary optimal transport and total-variation of modified distributions |
>
> **Claim.** Let us define a relaxation of $d\_N$ as $\tilde d\_N(\mu,\nu)=\inf\_{\gamma\_a,\gamma\_b} TV(\pi^2\_\#\gamma\_a,\pi^1\_\#\gamma\_b)$ subject to $\pi^1\_\# \gamma\_a = \mu$ and $\pi^2\_\# \gamma\_b =\nu$ and $\gamma\_a,\gamma\_b$ are concentrated on $D\_\epsilon = \{(x,x')\in \mathcal{X}^2; \Vert x-x'\V... | 1 | https://mathoverflow.net/users/110925 | 342571 | 145,456 |
https://mathoverflow.net/questions/342476 | 17 | The traditional theory of topological spaces (as formalized by Bourbaki) starts with a set of points, then builds a structure on that. In contrast, the theory of locales starts with a frame of opens (open subspaces), identifying the points (if these are even needed) from that data. I once read something suggesting that... | https://mathoverflow.net/users/8508 | Combination topological space and locale? | The proper term is *topological system* as found in Vickers' book *Topology via Logic*. Vickers actually uses precisely your expanded definition.
What you call "topological" Vickers calls "spatial" and you both use the word "localic" with the same meaning.
| 14 | https://mathoverflow.net/users/46003 | 342586 | 145,461 |
https://mathoverflow.net/questions/342029 | 4 | It is well known that for a matrix $A$ in $\mathfrak{sl}\_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the centralizer of $A$ (elements commuting with $A$).
The minimal dimension is the rank of the Lie algebra, so $n-1$ in this ... | https://mathoverflow.net/users/142627 | Double centralizer in special linear algebra | I just found a counter-example for $\mathfrak{sl}\_3$.
Take $A= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ and
$B = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$.
You can check that the commun centralizer is of dimension 2, but the couple $(A,B)$ does not admit any cycli... | 0 | https://mathoverflow.net/users/142627 | 342587 | 145,462 |
https://mathoverflow.net/questions/341589 | 11 | The extended Lichtenbaum conjecture concerns the relationship of special values of $L$-functions of number fields $K$, to the algebraic $K$-theory and etale cohomology of the ring of integers $O\_K$.
For example, when $K$ is totally real and $n$ is even, it is known that (cf. Section 4.7.4 of [Kahn](https://link.spri... | https://mathoverflow.net/users/798 | Status of the extended form of the Lichtenbaum conjecture | Thanks very much for the question!
Looking it up a bit I found the PhD thesis "[The Lichtenbaum Conjecture at the Prime 2](https://macsphere.mcmaster.ca/handle/11375/14321)
" by Ion Rada (a student of Kolster) which proves that for every abelian number field $K$ and every odd integer $n \geq 3$ one has
$$ \zeta\_K^{\... | 5 | https://mathoverflow.net/users/94140 | 342589 | 145,463 |
https://mathoverflow.net/questions/342590 | 15 | As it is reasonable to think the work of mathematicians will be developed/made in their offices of universities (or in eventual seminars or conferences),
here are the colleagues, books and journals, connection to databases and blackboards.
My belief is that a great part of mathematicians continue, somehow, their wo... | https://mathoverflow.net/users/142929 | The work of mathematicians outside their professional environment | Mathematics differs from most other professions in that the only "resources" which are really needed are paper and pencil. (Even these are not strictly necessary, one can use sand and stick as the ancients did. Some can do even without sand, as the examples of famous blind mathematicians show).
As a result, the worki... | 28 | https://mathoverflow.net/users/25510 | 342597 | 145,467 |
https://mathoverflow.net/questions/342577 | 2 | Let $A(i,j), i,j=0,1,2$ be the covariance matrix of three random variables. If we know all the entries except $A(2,0)$ and $A(0,2)$, how to determine the range of possible values of $A(2,0)$?
| https://mathoverflow.net/users/128758 | How to determine the range of values of A(i,j) in Covariance matrix A? | Let us write
$$A=\left(
\begin{array}{ccc}
a & b & c \\
b & d & e \\
c & e & f \\
\end{array}
\right).
$$
Then $A$ will be a covariance matrix iff it is positive semidefinite ($A\ge0$), that is, iff
$$\text{$a\ge0$, $d\ge0$, $f\ge0$, $ad\ge b^2$, $d f\ge e^2$, }\tag{1}
$$
$a f\ge c^2$, and
$$\det A=-c^2 d + 2 b c ... | 5 | https://mathoverflow.net/users/36721 | 342598 | 145,468 |
https://mathoverflow.net/questions/342567 | 2 | Does the following sum have a closed-form expression? I've tried an Inclusion-Exclusion interpretation, to no avail:
$f(n, p) = \sum\_{i = 1} ^ n \lfloor \frac{n}{i} \rfloor \phi(p i) (-1) ^ i$
($p$ is a prime number.)
Interesting observation: for $p > n$, $f(n, p) / (p - 1)$ is independent of $p$.
| https://mathoverflow.net/users/145778 | Summation involving Euler's totient function | A fast algorithm for calculating the expression:
We first try to remove the $(-1)^i$ part.
Let $g(n, m)$ be the sum $\sum\_{i = 1} ^ n \lfloor \frac{n}{i} \rfloor \phi(m i)$. Then it is clear that $f(n, p) = g(n, p) - 2 g(\lfloor \frac{n}{2} \rfloor, 2p)$.
Therefore we are reduced to calculating $g(n, p)$ and $g(... | 4 | https://mathoverflow.net/users/76332 | 342610 | 145,474 |
https://mathoverflow.net/questions/339332 | 4 | When studying the blow-up for focusing non-linear Schrödinger equation (NLS) one often compares the initial-state to a stationary solution.
In the energy-critical case, this stationary solution is for example
$$ \Delta W+ \vert W\vert^{\frac{4}{N-2}}W=0 $$
see [Kenig-Merle.](https://arxiv.org/pdf/math/0610266.pdf... | https://mathoverflow.net/users/119875 | Ground state for non-linear Schrödinger | A useful perspective on this is given in
*Weinstein, Michael I.*, [**Nonlinear Schrödinger equations and sharp interpolation estimates**](http://dx.doi.org/10.1007/BF01208265), Commun. Math. Phys. 87, 567-576 (1983). [ZBL0527.35023](https://zbmath.org/?q=an:0527.35023).
As observed in that paper, the ground state f... | 5 | https://mathoverflow.net/users/766 | 342615 | 145,476 |
https://mathoverflow.net/questions/342613 | 4 | Is there an algorithm that enumerates all permutations that are "square roots" of derangements, i.e. permutations that, when applied twice, yield a derangement?
Other information about those kind of permutations is also welcome.
| https://mathoverflow.net/users/31310 | Enumerating all permutations that are "square roots" of derangements | Check out "Example 2. Permutations with no small cycles" on pg. 176 of H. Wilf's "generatingfunctionology": <https://www.math.upenn.edu/~wilf/DownldGF.html>. It explains, using generating functions, how the number of permutations in $\mathfrak{S}\_n$ you are looking for is asymptotically $\approx \frac{1}{e^{1+1/2}} n!... | 11 | https://mathoverflow.net/users/25028 | 342616 | 145,477 |
https://mathoverflow.net/questions/339856 | 6 | Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.
For the purposes of the trace formula, one actually needs the fundamental lemma for the group. In his ICM notes, Ngo states that Waldspurger shows how to reduce the group statement to the Lie alg... | https://mathoverflow.net/users/62154 | Reduction to Lie algebra version of fundamental lemma? | For the purpose of this answer let us say that "fundamental lemma" means "fundamental lemma for the unit of the unramified Hecke algebra". I do not think that "FL for Lie algebras => FL for groups" was proved in "Le lemme fondamental implique le transfert". This implication is however proved in greater generality (twis... | 4 | https://mathoverflow.net/users/146489 | 342628 | 145,482 |
https://mathoverflow.net/questions/332460 | 4 | Let $P\_{x,w}$ be the Kazhdan–Lusztig polynomial, $\rho$ be the half sum of positive roots in $\Phi^+$, $M\_x$ be the Verma module with highest weight $x\cdot(-2\rho)$ and $L\_w$ be the simple highest weight module with highest weight $w\cdot(-2\rho)$.
It is well-known that Kazhdan–Lusztig Conjecture is equivalent to... | https://mathoverflow.net/users/110229 | Kazhdan–Lusztig polynomials in terms of Ext groups | The answer is yes, for fairly elementary reasons, though it's not easy to give a reference. The point is partly that the polynomials are undefined for two elements of the Weyl group not related by the Bruhat partial orderijg. More precisely, the "linkage principle"(or "Harish-Chandra principle") ensures that the Hom fu... | 2 | https://mathoverflow.net/users/4231 | 342640 | 145,483 |
https://mathoverflow.net/questions/342510 | 12 | I'm a last year undergraduate student and I have taken a graduate course in geometric group theory.
I'd like to start reading some more advanced stuff in geometric group theory and in particular about automorphisms of free groups. (In specific I'm interested in $\mathrm{Out}(F\_n)$.)
Could you please suggest me mat... | https://mathoverflow.net/users/145318 | Road map to learn about $\mathrm{Out}{F_n}$ | Here are some assorted recommendations.
* Stallings's "Topology of Finite Graphs" and Bestvina's course notes "Folding Graphs and Applications" are a great introduction to a technique that has found wide-ranging applications, both in the study of $\operatorname{Out}(F\_n)$ and more broadly.
* Vogtmann's "What Is Oute... | 15 | https://mathoverflow.net/users/135175 | 342642 | 145,485 |
https://mathoverflow.net/questions/342627 | 10 | The direct sum of real vector bundles endows $BO=\mathrm{colim} BO(n)$ with a natural structure of abelian group up to homotopy. The same applies to the classifying spaces of all groups in the Whitehead tower of $O$, i.e., one has a natural structure of abelian group up to homotopy on $BSO$, $BSpin$, $BString$, etc.
... | https://mathoverflow.net/users/8320 | How are characteristic classes morphisms of infinite loop spaces? (if they are) | Your sequences are all arise in the following standard way. Suppose $x$ is an $(n-1)$--connected spectrum and let $X = \Omega^{\infty} x$. One always has a
fibration sequence
$$y \rightarrow x \rightarrow \Sigma^n H\pi\_n(X)$$
and applying $\Omega^\infty$ to this yields a fibration sequence of spaces
$$Y \rightarrow ... | 11 | https://mathoverflow.net/users/102519 | 342645 | 145,488 |
https://mathoverflow.net/questions/342674 | 2 |
>
>
> >
> > Is there a constant $\alpha$ such that:
> >
> >
> >
>
>
>
$$P\_{n+1} < P\_n \cdot \left(\frac{n+1}{n}\right)^\alpha$$
Or
$$\lim\_{n\to\infty}\frac{\ln\frac{P\_{n+1}}{P\_n}}{\ln\frac{n+1}{n}} < +\infty$$
Where $P\_n$ is $n$-th prime number.
In the table [The 80 known maximal prime gaps](... | https://mathoverflow.net/users/122662 | Is there a constant $\alpha$ such that: $P_{n+1} < P_n.\left(\frac{n+1}{n}\right)^\alpha$? | As $P\_n$ is asymptotically $n\log n$, your question is equivalent to the following. Is it true that
$$P\_{n+1}-P\_n\ll\log n?$$
In other words, is it true that the actual gap between primes is always at most a constant times the average (expected) gap? The answer is "no" by a 1931 result of Westzynthius. For the best ... | 12 | https://mathoverflow.net/users/11919 | 342675 | 145,498 |
https://mathoverflow.net/questions/342678 | -7 | It seems than an analogue of the twin prime conjecture for polynomials in finite fields has been solved: see <https://www.quantamagazine.org/big-question-about-primes-proved-in-small-number-systems-20190926/>
Can one expect the perfectoid spaces introduced by Scholze, which if I understand correctly deal with spaces ... | https://mathoverflow.net/users/13625 | Can Scholze's perfectoid spaces bridge the gap for twin prime conjecture? | As far as I know, perfectoid spaces have not been used in a non-trivial way in analytic number theory. Since all the progress to twin prime conjecture so far has been mostly analytic in nature, an application of perfectoids to twin primes would have to be creative.
The passage from finite characteristic to mixed cha... | 4 | https://mathoverflow.net/users/146523 | 342683 | 145,500 |
https://mathoverflow.net/questions/342684 | 5 | It would be useful to me to have a result of the following kind (which I would need to generalize, but this case is already interesting). Let $r<n$ be positive integers and let $\delta>0$ be a fixed constant such as 1/100. Does there exist a subspace $V$ of $\mathbb F\_2^n$ that is a $\delta r$-separated $r$-net? That ... | https://mathoverflow.net/users/1459 | When can this condition on linear codes be satisfied? | Any maximal $r$-separated set is an $r$-net, otherwise you can augment it with a non-covered point.
But the same holds for linear codes! If a subspace $V$ is $r$-separated but not an $r$-net, you may take a far point $u$: then $\langle V,u\rangle$ is still $r$-separated. So a maximal $V$ fits.
| 12 | https://mathoverflow.net/users/17581 | 342686 | 145,501 |
https://mathoverflow.net/questions/342676 | 4 | Suppose we have $n$ iid Bernoulli's $X\_1,\ldots,X\_n$ with mean $p$, and a family $\mathcal{F}$ of subsets of $[n]$. The question is how to lower bound the probability that there is a set in the family in which all of its $X\_i$'s come up 1 (a "full set"), that is to show
\begin{equation}
\Pr\bigg(\exists I \in \ma... | https://mathoverflow.net/users/143536 | Probability of a subset of Bernoulli's being all 1's | For the concrete question, it’s equivalent to asking for the cdf of the binomial distribution. This is well known.
In general, this is a very hard problem. Janson’s Inequality is not a second moment bound, despite the appearance of the second moment. But it generally is not too helpful in this situation. Maybe have a... | 5 | https://mathoverflow.net/users/36212 | 342688 | 145,502 |
https://mathoverflow.net/questions/342685 | 0 | Let $x\_1, ..., x\_n\in\mathbb{R}^d$. We know that a point $x$ in the convex hull $\text{conv}(x\_1, ..., x\_n)$ may be expressed as convex combinations $x=\sum\_{i=1}^n r\_ix\_i=\sum\_{i=1}^n s\_ix\_i$ with distinct $(r\_1, ..., r\_n)$ and $(s\_1, ..., s\_n)$ with $r\_i, s\_i\geq 0$, for all $i$ and $\sum\_{i=1}^n r\_... | https://mathoverflow.net/users/123506 | Overlap count of convex combination of points | Let $F$ be the smallest face of the polytope $\text{conv}(x\_1, \dots, x\_n)$ containing $x$. If $F$ is a simplex intersecting $\{x\_1, \dots, x\_n\}$ only in its vertices, then there is a unique convex combination giving $x$: the points $x\_i$ that do not belong to $F$ must have the coefficient $r\_i$ equal to $0$. Ot... | 1 | https://mathoverflow.net/users/24076 | 342690 | 145,504 |
https://mathoverflow.net/questions/342680 | 4 | Are there examples of d-regular graphs (i.e. graphs where every node has exactly d adjacent nodes) which are not the 1-skeleton of a simple convex polytope?
UPDATE:
New version of the question: is there an example of a d-dimensional "simple" poset, i.e. a collection of k-dimensional "faces" with $k=0, 1, \dots, d$ ... | https://mathoverflow.net/users/48526 | An example of a "simple poset" which does not belong to a convex polytope | I think a counterexample can be obtained via [simplicial spheres](https://en.m.wikipedia.org/wiki/Simplicial_sphere). It is known that in higher dimensions, most simplicial spheres do not belong to a polytope. The dual of the face lattice of such a $d$-dimensional non-polytopal simplicial sphere is a $d$-regular "simpl... | 5 | https://mathoverflow.net/users/108884 | 342692 | 145,505 |
https://mathoverflow.net/questions/342687 | 3 | Assume that $A$ and $B$ are two dependent random variables. Are there any results on functions $f$ such that
$C =f(A, B)$ and $A$ are independent?
For example, it can easily be shown that $A$ and $C = F\_{B|A}(B, A)$ are independent where $F\_{B|A}(., .)$ is the conditional CDF function of $B$, given $A$. (I am usi... | https://mathoverflow.net/users/nan | Functions $f$ such that $A$ and $f(A,B)$ are independent | I must say that I don't quite understand your notation. There is a complete description of the independent complements in the sense you are asking, but I prefer to formulate it in somewhat different terms, namely in the language of Lebesgue spaces and their measurable partitions due to [Rokhlin](https://mathscinet.ams.... | 5 | https://mathoverflow.net/users/8588 | 342707 | 145,510 |
https://mathoverflow.net/questions/342562 | 3 | Let $\mathcal{M}\_g$ be the moduli stack of smooth genus $g$ curves. Let $F$ be a coherent sheaf on $\mathcal{M}\_g$.
Is $H^i(\mathcal{M}\_g,F)$ always finite dimensional? For example, $F=(f\_\*\Omega^1\_{\mathcal{C}\_{g}/\mathcal{M}\_g})^\vee$, where $f\colon\mathcal{C}\_g\to \mathcal{M}\_g$ is the universal curve?... | https://mathoverflow.net/users/nan | Is $H^i(\mathcal{M}_g,F)$ necessarily finite dimensional for a coherent sheaf $F$? | Let $X$ be a separated Deligne-Mumford stack of finite type over a field of characteristic 0. Let $\pi\colon X \to M$ be the moduli space; assume that $M$ is quasi-projective. I claim that if $\mathrm{H}^i(X, F)$ is finite-dimensional for all locally free sheaves on $X$, then $X$ is proper, or, equivalently, $M$ is pro... | 6 | https://mathoverflow.net/users/4790 | 342731 | 145,517 |
https://mathoverflow.net/questions/342720 | 3 |
>
> **Question**:
>
> is it known, whether there is an upper bound on the exponent of the fastest polynomial-time reduction from $\mathrm{3SAT}$ to $\mathrm{NP}$-$\mathrm{hard}$ problems or, can it be proved that for every problem requiring an $\Theta(n^k)$ time reduction, there is a problem requiring an $\Omega(... | https://mathoverflow.net/users/31310 | Naive question about polynomial time reducibility | It's unlikely there is any upper bound. To see the problem, consider the (artificial) problem
$\text{3SATpad} = \{ \phi \#^{|\phi|^{100}} : \phi \in \text{3SAT}\}$,
where $\#$ is some new symbol. $\text{3SATpad}$ is in $\mathsf{NP}$, and there is an obvious $O(n^{100})$-time reduction from $\text{3SAT}$ to $\text{... | 6 | https://mathoverflow.net/users/658 | 342735 | 145,519 |
https://mathoverflow.net/questions/342725 | 9 | Let $\mathcal{P}\_d\cong\mathbb{A}^d$ denote the set of monic degree $d$ polynomials defined over an algebraically closed field of characteristic $0$, where we identify $f(x)$ with its coefficients. The multiplicities of the roots of $f(x)\in\mathcal{P}\_d$ defines a partition $\pi(f)$ of $d$. For example, if $f(x)=(x-... | https://mathoverflow.net/users/11926 | The locus of polynomials with specified root multiplicities | I may be wrong but I think "coincident root loci" mentioned in Gjergjji's comment was coined by my coauthor Jaydeep Chipalkatti. The ideals of such varieties are in general poorly understood.
For a general study see:
1. J. Chipalkatti, ["On equations defining Coincident Root loci"](https://www.sciencedirect.com/sc... | 6 | https://mathoverflow.net/users/7410 | 342739 | 145,522 |
https://mathoverflow.net/questions/342746 | 1 | Let $H(\mathbb{C})$ be the space of holomorphic functions on the complex plane. Then it is well-known that for $a\neq 0$, the translation operator
$$
t\_a(f)\triangleq f(x)\mapsto f(x+a),
$$
is topologically transitive on $H(\mathbb{C})$. Are there known, sufficient conditions for $f$ to by a cyclic vector of this map;... | https://mathoverflow.net/users/36886 | Cyclic vectors of translation operator | Such functions are called universal entire functions. Actually most entire functions
have this property. For
specific examples, $\zeta$ function has this property.
MR0771576
Duĭos Ruis,
Universal functions and the structure of the space of entire functions. (Russian)
Dokl. Akad. Nauk SSSR 279 (1984), no. 4, 792–795... | 3 | https://mathoverflow.net/users/25510 | 342747 | 145,524 |
https://mathoverflow.net/questions/342566 | 23 | Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ and $H^1(E,L^{-1})$, does this pairing have a nice expression in terms of theta functions?
I'm either looking for som... | https://mathoverflow.net/users/104442 | Theta functions on an elliptic curve and Serre duality | Let $X$ be an elliptic curve $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice of real rank $2$ in $\mathbb{C}$. A theta function is a holomorphic section of a line bundle $L$ on $X$ whose transition from $U$ to $U + \ell$ is given by$$f(z + \ell) = e^{a\_\ell z + b\_\ell} f(z).$$ Here we have $\ell \in \Lambda$, wher... | 13 | https://mathoverflow.net/users/126532 | 342749 | 145,525 |
https://mathoverflow.net/questions/342652 | 24 | I hope it is appropriate to ask this question here:
One formulation of the abc-conjecture is
$$ c < \text{rad}(abc)^2$$
where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being arbitrary natural numbers):
$$ \frac{a+b}{\gcd(a,b)} < \text{rad}(\frac{ab(a+b)}{\gcd(a,b)^3})^2$$
Let $d\_1(a,b) = 1- \f... | https://mathoverflow.net/users/nan | A reinterpretation of the $abc$ - conjecture in terms of metric spaces? | $d\_2$ is indeed a metric. Abbreviating $\gcd(m,n)$ to $(m,n)$, we need to show that
\begin{align\*}
1-\frac{2(a,c)}{a+c} &\le 1-\frac{2(a,b)}{a+b} + 1-\frac{2(b,c)}{b+c}
\end{align\*}
or equivalently
\begin{align\*}
\frac{2(a,b)}{a+b} + \frac{2(b,c)}{b+c} &\le 1 + \frac{2(a,c)}{a+c}.
\end{align\*}
Furthermore, we may ... | 11 | https://mathoverflow.net/users/5091 | 342750 | 145,526 |
https://mathoverflow.net/questions/342713 | 1 | In my research, I need to find the set of all minimal dependent sets of a given set of vectors. One method is to check every subset of the given set. But this method is very slow when the set of vectors is large. For example, let $S$ be the set of all positive roots of type $B\_5$ root system. Then $S$ consists of the ... | https://mathoverflow.net/users/11877 | How to find all minimal dependent sets of a set of vectors effectively? | Install a recent version of [Macaulay2](http://www2.macaulay2.com/Macaulay2/). Open a Macaulay2 session in a terminal and issue the commands below (the ones starting with "i" for input).
```
i1 : loadPackage "Matroids"
i2 : M = matroid transpose matrix {{0,0,0,0,1}, {0,0,0,1,2}, {0,0,0,1,1}, {0,0,1,1,2}, {0,0,1,2,... | 4 | https://mathoverflow.net/users/94968 | 342757 | 145,527 |
https://mathoverflow.net/questions/342765 | 1 | Let $G$ be a family of functions mapping from $Z$ to $[a, b]$ and $S=\left(z\_{1}, \ldots, z\_{m}\right)$ a fixed sample of size $m$ with elements in $Z$ . Then, the empirical Rademacher complexity of $G$ with respect to the sample $S$ is defined as :
$$
\widehat{\Re}\_{S}(G)=\underset{\boldsymbol{\sigma}}{\mathrm{E}}\... | https://mathoverflow.net/users/125250 | Why we use Rademacher complexity for generalization error when we can have a trained function? |
>
> We will usually get a fixed function by a training algorithm on a training set, and we can give the generalization error regarding this function directly by Hoeffding's inequality.
>
>
>
Nope! The function is special because you used the data to pick it, i.e. it's correlated. By this same reasoning, you coul... | 1 | https://mathoverflow.net/users/29697 | 342781 | 145,532 |
https://mathoverflow.net/questions/342777 | 5 | Let $X$ be a compact pointed metric subspace of the $d$-dimensional Euclidean space $(\mathbb{R}^d,d\_E)$ and let $AE(X)$ denote its Arens-Eells space. Then a result of [Nik Weaver](https://www.researchgate.net/scientific-contributions/9128689_Nik_Weaver) shows that for every Lipschitz map $f:X\rightarrow E$ into a sep... | https://mathoverflow.net/users/36886 | Concrete description of lift in Arens-Eells space | $AE(X)$ is the completion of the space of "molecules", i.e., the finitely supported functions $m: X \to \mathbb{R}$ which satisfy $\sum\_{p \in X}m(p) = 0$. The extension $F$ of $f: X \to E$ satisfies $F(m) = \sum\_{p \in X} m(p)f(p)$. (BTW $E$ need not be separable.)
| 6 | https://mathoverflow.net/users/23141 | 342782 | 145,533 |
https://mathoverflow.net/questions/342752 | 2 | I am trying to determine whether a particular theorem used in the course of Godel’s (1930) proof of the completeness of predicate logic could be proven in an intuitionistic metatheory.
Theorem VI (p. 113 of Godel Collected Works, Vol. I) states that:
For every n,
$P(A) \implies P\_n(A\_n)$
is provable in the ... | https://mathoverflow.net/users/116705 | Possible to prove a lemma from Godel's completeness theorem in intuitionistic logic? | Lemma 1(a) is about derivability of $\newcommand{\x}{\mathbf{x}}\forall \x\, \varphi(\x) \Rightarrow \exists \x\, \varphi(\x)$ in the *object theory*, not the metatheory.
Gödel doesn’t give the proof, but it’s nothing subtle: in the deduction system he considers, and many similar systems, you can just write down the ... | 7 | https://mathoverflow.net/users/2273 | 342789 | 145,539 |
https://mathoverflow.net/questions/342791 | 2 | **Main question**
* In the context of resistor networks, and particularly purely from a graph theory point of view, is there a consistent way of assigning the two electrode nodes in order to compare the conductivity or equivalently the resistance of two networks?
The intuitive choices might be:
* Choosing the fur... | https://mathoverflow.net/users/115841 | Electrode assignment problem in resistive networks | Best practice to determine the resistivity tensor is to use the [van der Pauw method](https://en.wikipedia.org/wiki/Van_der_Pauw_method). For resistor networks this method is used for example in [Nonlinearity of resistive impurity effects on van der Pauw measurements](https://www.researchgate.net/publication/253990320_... | 4 | https://mathoverflow.net/users/11260 | 342795 | 145,540 |
https://mathoverflow.net/questions/342798 | 9 | I'm curious if there is a finite measure on the $\sigma$-algebra of subsets of $[0,1]$ with the Property of Baire, whose null sets are exactly the meagre sets.
I'd also be interested how "nice" such a measure can be like can it be Radon(when restricted to Borel sets) for example.
| https://mathoverflow.net/users/146596 | Is there a measure on $[0,1]$ that is 0 on meagre sets and 1 on co-meagre sets | The answer is no. Assume that such a measure $\mu$ exists.
First, since every singleton in $[0,1]$ is closed with empty interior, $\mu(\{x\}) = 0$ for all $x \in [0,1]$. Write $B\_{x,\epsilon}$ for the open ball around $x$ of radius $\epsilon$ with respect to the standard metric on $[0,1]$. By countable additivity, f... | 13 | https://mathoverflow.net/users/61785 | 342803 | 145,544 |
https://mathoverflow.net/questions/342802 | 6 | I am trying to understand how the Moore spectrum is constructed. And in reading [Foundations of Stable Homotopy Theory](https://www.kent.ac.uk/smsas/personal/csrr/stablemodelcatsCUP.pdf) by David Barnes and Constanze Roitzheim, I see that in example 8.4.7 (pg 340) they show how to construct the Moore spectrum. The cons... | https://mathoverflow.net/users/54401 | How to construct the Moore spectrum? | What you're missing is that $[\mathbb{S},\mathbb{S}]=\mathbb{Z}$. Let now $R$ be the infinite matrix of integers representing $\rho$. Note that since $\rho$ takes value in $\bigoplus\_{I\_1}\mathbb{Z}\subseteq \prod\_{I\_1}\mathbb{Z}$, all of its columns have only finitely many non-zero values. So, for every column of ... | 12 | https://mathoverflow.net/users/43054 | 342807 | 145,545 |
https://mathoverflow.net/questions/342780 | 1 | I'm currently working on understanding certain mean-field limits in kinetic theory, and the equations I'm working with are usually of the form
$$\partial\_t f +v\cdot\nabla\_x f \pm c\nabla(-\Delta)^{-1}\rho\_f\cdot\nabla\_v f=0$$ for $f:\mathbb R\_x^d\times\mathbb R\_v^d\times\mathbb R\_t\to\mathbb R$ and where $\rho\... | https://mathoverflow.net/users/94022 | Reference for singular integral operators such as $(-\Delta)^{-1}$ or $\nabla(-\Delta)^{-1}$ | Make $0$-order operator applying extra $\partial\_j$ and use your reference.
| 1 | https://mathoverflow.net/users/144495 | 342816 | 145,546 |
https://mathoverflow.net/questions/342814 | 3 | Scholze attributes the tilting construction for perfectoid rings to Fontaine, who calls it "a classical construction in $p$-adic Hodge theory".
Would anyone happen to know an early reference where one can see this construction being used to good effect? Are there canonical examples of classic applications?
| https://mathoverflow.net/users/126543 | References for the early history of Fontaine's tilting construction | Jean-Marc Fontaine *Groupes p-divisibles sur les corps locaux.* Astérisque 47-48, Soc. Math. France, Paris (1977), i+262 pp (especially chapter V)
This is probably the canonical answer to your question. Note that the application was found before *tilting* was defined, even for fields, as is usually the case in the hi... | 6 | https://mathoverflow.net/users/2284 | 342838 | 145,550 |
https://mathoverflow.net/questions/342574 | 2 | I've got a very simple question about a homogenous polynomial, for which I cannot see neatly how to proceed (probably due to my limitations in algebraic geometry though). Any help would be greatly appreciated.
So, let $p:\mathbb R^4\to \mathbb R$ be a homogeneous polynomial of degree 4, and assume that $p$ has a non-... | https://mathoverflow.net/users/132086 | Homogeneous polynomial in 4 variable with non degenerate zero | This is indeed true. One can drop the assumption on $p$ to be homogeneous and dimension $4$ is not relevant, though the condition $\deg(p)=4$ is crucial. The statement follows from the following facts.
**Fact 1.** Let $q(x)$ be a polynomial in one variable $x$ such that $\deg(q)=4$ and $q$ changes sign. Then there is... | 3 | https://mathoverflow.net/users/943 | 342842 | 145,551 |
https://mathoverflow.net/questions/342824 | 1 | How would one solve algebraically the following system of algebraic equations?
$$f(a,b):=a(1-b)+ab\frac a{a+b}.$$
$$u = f(a,b),\quad v = f(b,a).$$
Solve algebraically $(a,b)$ in terms of $(u,v)$
---
Multiplying both sides of the equations by $a+b$ would give us a system of cubic equations. But that does not seem... | https://mathoverflow.net/users/32660 | Algebraic solution for a system of algebraic equations? | This is mechanized in current CASes, e.g. the command of Maple 2019.1
```
solve({a*(1 - b) + a*b*a/(a + b) = u, eval(a*(1 - b) + a*b*a/(a + b), {a = b, b = a}) = v}, {a, b}, explicit);
```
performs a long output which can be seen [here](https://www.dropbox.com/s/wd89oddvbjt99bk/solution.pdf?dl=0) exported as a PDF... | 3 | https://mathoverflow.net/users/35959 | 342849 | 145,552 |
https://mathoverflow.net/questions/342852 | 1 | Recently, I have been looking at some articles about bases for cluster algebras and came across the idea of tropical points. I should highlight here that unfortunately I have no background on algebraic geometry (and in what follows, tropical geometry) and so I have been first trying to understand what is the equivalent... | https://mathoverflow.net/users/144655 | Cluster algebras and tropical points | In short, yes. I would strongly recommend the survey by Nakanishi,
*Nakanishi, Tomoki*, [**Tropicalization method in cluster algebras**](http://dx.doi.org/10.1090/conm/580/11486), Athorne, Chris (ed.) et al., Tropical geometry and integrable systems. A conference on tropical geometry and integrable systems, School of... | 3 | https://mathoverflow.net/users/13215 | 342856 | 145,555 |
https://mathoverflow.net/questions/342844 | 1 | By Bishop-Phelps theorem we know that for a real Banach space, the set of all norm attaining bounded linear functionals is norm-dense in $X^\*$, the topological dual of $X$. We also know that in general the class of all norm attaining operators $NA(X,Y)$ from a Banach space $X$ into another Banach space $Y$ need not be... | https://mathoverflow.net/users/41137 | Density of norm-attaining operators | Take a separable $X$ s.t. $B\_X$ has no extreme points (for example, $c\_0$ or $L\_1$), and equivalently renorm it to be strictly convex--call the resulting space $Y$. Show that the identity operator from $X$ to $Y$ cannot be approximated by norm attaining operators.
| 2 | https://mathoverflow.net/users/2554 | 342857 | 145,556 |
https://mathoverflow.net/questions/342855 | 8 | I am learning A-infinity category with Fukaya category in mind, and would like to understand the meaning of A-infinity relations.
In particular, as $N=1$, it means $dd=0$. As $N=2$, it means that $d$ satisfies Leibniz's rule if we regard the second operation as multiplication. As $N=3$, it means that the multiplicati... | https://mathoverflow.net/users/124549 | Meaning of A-infinity relations | For your first question. Suppose $(A,d,m,m\_3,m\_4\dots)$ is an $A\_\infty$-algebra. The operation $m\_3$ gives a homotopy between $m(-,m(-,-))$ and $m(m(-,-),-)$, which I will abusively denote as $a(bc)$ and $(ab)c$.
Now consider the two operations $A^{\otimes 4} \to A$ given by $a(b(cd))$ and $((ab)c)d$. Using $m\_... | 13 | https://mathoverflow.net/users/36146 | 342866 | 145,559 |
https://mathoverflow.net/questions/342863 | 4 | Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$. Suppose that $(T\_n)$ is a sequence in $\mathcal{E}'(\mathbb{R})$ that converges to $T$ in the weak topology $\sigma(\mathcal{E}',\mathcal{E})$ on $\mathcal{E}'(\mathbb{R})$.
Does this imply that $(T\_n)$ converges to... | https://mathoverflow.net/users/142650 | Convergence in $\sigma(\mathcal{E}',\mathcal{E})$ versus $\beta(\mathcal{E}',\mathcal{E})$ | The answer is yes. First, since $\newcommand{E}{\mathcal{E}}\E$ is a Fréchet space, it is barrelled, and so any $\sigma(\E',\E)$-bounded subset of $\E'$ is equicontinuous, and therefore bounded in any dual topology. Convergent sequences (including their limits) are compact sets, and therefore bounded. So each $\sigma(\... | 3 | https://mathoverflow.net/users/61785 | 342874 | 145,562 |
https://mathoverflow.net/questions/342794 | 0 | We have a process $\{X\_{t}\}\_{t\geq 0}$ ,with fixed parameter $\epsilon>0$, starting from zero that satisfies
* The process is strictly monotone $X\_{t+r}-X\_{t}>0$ with moments existing $p\in(-\infty, \beta)$ for some $\beta>0$. (In the interval [0,1] we also have the lower bound $X\_{t+r}-X\_{t}>cr^{b}$ where $b>... | https://mathoverflow.net/users/99863 | Markov with epsilon memory and Quantitative Strong Markov property | Q1: I have not seen such a process, but I can easily imagine one: Let $Z\_t$ be any non-negative Lévy process with positive drift, and let $X\_t = \int\_{t-\epsilon}^t Z\_s ds$.
Q2: No, unless we know something more about $X\_t$. Imagine the process $X\_t$ as above, with $Z\_t$ having frequent extremely large jumps. ... | 1 | https://mathoverflow.net/users/108637 | 342884 | 145,564 |
https://mathoverflow.net/questions/342891 | 5 | Looking for example at $R$-modules for some commutative $R$, we have the direct sum and the tensor product acting analogously to addition and multiplication.
After studying a little bit about co-algebras and bi-algebras, I wondered if there is any way to define something that would be analogous to comultiplication in ... | https://mathoverflow.net/users/94076 | Comultiplication on objects in an (abelian?) category | Sure, we can define such things. Let's work in the Morita 2-category $\text{Mor}(k)$ over a commutative ring $k$, which has
* objects $k$-algebras $A$,
* morphisms $k$-bimodules (with composition given by tensor product), and
* 2-morphisms bimodule homomorphisms.
Equivalently, applying the forgetful functor $\text{... | 7 | https://mathoverflow.net/users/290 | 342894 | 145,568 |
https://mathoverflow.net/questions/342767 | 3 | Let $\phi$ be a homogeneous symplectomorphism of tangent bundle $\dot{T}^\*M=T^\*M-0\_M$ and let $\alpha\_M$ be the canonical Liouville 1-form of $\dot{T}^\*M$. Then is it true that $\phi^\*\alpha\_M=\alpha\_M$?
| https://mathoverflow.net/users/143888 | Any homogeneous symplectomorphism of cotangent bundle $\dot{T}^*M=T^*M-0_M$ preserves the canonical Liouville form? | Yes this is true. You can prove it in Darboux coordinates $(x,\xi)$.
Let $\phi=(\phi\_1,\phi\_2)$ be the symplectomorphism. Since it is homogeneous, $\phi(x,t\xi) = (\phi\_1(x,\xi), t\phi\_2(x,\xi))$ and differentiating wrt $t$ you get
$$ \partial\_\xi\phi\_1 \cdot \xi = 0$$
and
$$ \partial\_\xi\phi\_2 \cdot \xi = \p... | 4 | https://mathoverflow.net/users/145904 | 342895 | 145,569 |
https://mathoverflow.net/questions/342867 | 4 | Suppose that $c$ is a nonnegative integer and $A\_c = (a\_n)$ and $B\_c = (b\_n)$ are strictly increasing complementary sequences satisfying
$$a\_n = b\_{2n} + b\_{4n} + c,$$
where $b\_0 = 1.$ Can someone prove that the sequence $A\_1-A\_0$ consists entirely of zeros and ones?
Notes:
$$
A\_0 = (2, 10, 17, 23, 3... | https://mathoverflow.net/users/61426 | Difference of two integer sequences: all zeros and ones? | The same method as in [this answer to a previous your question](https://mathoverflow.net/a/342287/17581) works as well.
**As for $(A\_0)$.** Starting with a guess
$$
7n+2\leq a\_n\leq 7n+3,
$$
and trying to prove it inductionally, we arrive at $b\_{6n+2}\geq 7n+4$ and $b\_{6n}\leq 7n+1$, hence
$$
t+\left\lfloor\fra... | 3 | https://mathoverflow.net/users/17581 | 342901 | 145,571 |
https://mathoverflow.net/questions/342904 | 12 | I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $N\times S^1$.
| https://mathoverflow.net/users/40517 | Manifolds with nonwhere vanishing closed one forms | If $f: M \to S^1$ is a submersion (and so a fiber bundle if $M$ is compact) then $f^\*d\theta$ is a nowhere-vanishing closed 1-form. There are many more such manifolds and fibrations than just products, and the manifolds have the name mapping tori.
If $(M,\omega)$ is a manifold equipped with a nowhere vanishing close... | 24 | https://mathoverflow.net/users/40804 | 342906 | 145,572 |
https://mathoverflow.net/questions/342913 | 1 | I am working with GP-UCB and need to calculate RKHS norm as in Theorem 6 of [Srinivas et.al 2012](https://ieeexplore.ieee.org/document/6138914). I found on page 3 column 1 like:
>
> The induced RKHS norm $||{f}||\_k=\sqrt{<f,f>}\_k$ measures smoothness of $f$ w.r.t. $k$.
>
>
>
I am new to this field so canno... | https://mathoverflow.net/users/128364 | How to calculate or estimate RKHS norm? | For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://arxiv.org/abs/1807.02582) by Kanagawa et al. (2018).
In particular, they give a characterisation of the RKHS associated to a shift-invariant ke... | 5 | https://mathoverflow.net/users/140545 | 342916 | 145,573 |
https://mathoverflow.net/questions/342922 | 3 | Let $S\_\omega$ be the group of bijections $f:\omega\to\omega$, and let $F\_\omega = \{\pi\in S\_\omega: \exists N\in \omega(\pi(k) = k \text{ for all } k\geq N)\}$. It is easy to see that $F\_\omega$ is a normal subgroup of $S\_\omega$.
Is $S\_\omega/F\_\omega$ isomorphic to a subgroup of $S\_\omega$?
| https://mathoverflow.net/users/8628 | Is $S_\omega/F_\omega$ embeddable to $S_\omega$? | No, it is not.
McKenzie (1971) observed that the "direct sum" of $\ge\aleph\_1$ non-abelian groups cannot be embedded into $S\_\omega$ (indeed, it yields an ascending chain of centralizers in $S\_\omega$ of type $\omega\_1$, and this is not possible).
On the other hand, the direct sum of $\aleph\_1$ (or $2^{\aleph... | 6 | https://mathoverflow.net/users/14094 | 342923 | 145,576 |
https://mathoverflow.net/questions/333469 | 6 | Let me first make sure I have the correct definitions because my question will be about the difference about the two and there may be some massive confusion on my part.
A topological space $X$ is said to be **completely regular** or **Tychonoff** when it is Hausdorff and satisfies the following equivalent conditions:... | https://mathoverflow.net/users/17064 | Tychonoff-ization and Urysohn (functionally Hausdorff) topological spaces | **0a** Correct, except for one point: is $X$ is not completely regular then there is no $\beta X$, so the third equivalence in `functionally Hausdorff' does not exist.
**0b** Not quite, the paper mentioned in your edit maps $X$ into a Tychonoff cube and lets $X'$ be the image, see 0a: there is no $\beta X$ available.... | 5 | https://mathoverflow.net/users/5903 | 342940 | 145,582 |
https://mathoverflow.net/questions/342418 | 4 | Let $B$ be the open unit ball in $\mathbb C^n$. Consider the space $\mathcal F$ of holomorphic embeddings of $B$ in $\mathbb C^n$ equipped with the compact-open topology. (A holomorphic embedding of $B$ in $\mathbb C^n$ is a holomorphic map $f: B\to \mathbb C^n$ such that $f(B)$ is open and there is a holomorphic inver... | https://mathoverflow.net/users/102829 | Space of holomorphic embeddings of open unit ball in ${\mathbb C}^n$ | The space of holomorphic embeddings with Jacobi matrix = 1 at zero is contractible due to the standard construction $f\_t=(1/t)f(zt)$, and $f\_0$ defined as a limit will be equal to the identity map ($t$ changes from 1 to 0). The similar proof also works in the smooth category.
| 4 | https://mathoverflow.net/users/33286 | 342951 | 145,586 |
https://mathoverflow.net/questions/342920 | 3 | There is a step in the proof of Theorem 4.11 of [this set of notes](https://www.math.u-psud.fr/~crovisie/00-CP-Trieste-Version1.pdf) that I don't quite see.
The set up is that $f$ is a $C^2$-diffeomorphism on some Riemannian manifold $M$, and that $E \oplus F = T M$ is a dominated splitting for $f$, and it is conclu... | https://mathoverflow.net/users/114097 | A question about the proof of Hölder continuity for dominated splittings | thanks for the comments. The notes were at several points written quite in a hurry, we plan to work on them some day (but don't know when).
For this specific proof, notice that $\epsilon\_k$ can be chosen to be of the order $\epsilon\_1 \|Df^k|\_{F}\|^{-1}$ and so the fact that $\lambda \|Df|\_F\|^\theta <1$ allows ... | 1 | https://mathoverflow.net/users/5753 | 342953 | 145,587 |
https://mathoverflow.net/questions/342941 | 3 | The five primes, 131, 157, 211, 349, 739, are neither in arithmetic or geometric progression, but are instead the sum of the five corresponding terms of an arithmetic and geometric progression.
Are there arbitrarily long sequences of primes with this property, that is, that they are the sum of the corresponding term... | https://mathoverflow.net/users/60732 | Primes from arithmetic and geometric progressions | Claim: I actually have almost zero knowledge in this domain, and this answer is totally based on the correctness of [this wiki page](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem). Moreover, it could be that I misunderstood the content of that page, so it would be kind of some expert to point out to me if I m... | 4 | https://mathoverflow.net/users/76332 | 342956 | 145,589 |
https://mathoverflow.net/questions/342943 | 0 | Let $(M,\omega)$ be a compact Kähler manifold with Kähler form $\omega$. Furthermore, denote by $c\_{1}$ the first Chern class of $M$. Assume one of the following $c\_{1}>0$, $c\_{1}<0$ or $c\_{1}=0$. My question is the following.
**Question:** How can one show that there exists a real number $\lambda \in \mathbb{R}$... | https://mathoverflow.net/users/146678 | First Chern class with sign | See Demailly, **Complex Algebraic and Differential Geometry**, p.333. By definition, the first Chern class of a vector bundle is positive (negative, zero) if it is positive (negative, zero) as a cohomology class, i.e. representable by some positive (negative, zero) $(1,1)$-form, i.e. a closed $(1,1)$-form which is posi... | 1 | https://mathoverflow.net/users/13268 | 342957 | 145,590 |
https://mathoverflow.net/questions/342954 | 0 | Can we prove that for any simple polygon with more than 3 vertices there always exists a diagonal which:
* is inside the polygon
* doesn't intersect with any edges
* splits the polygon in two polygons in such a way that the difference between their vertex counts is smaller than 2 (e.g. splits a polygon with 29 vertic... | https://mathoverflow.net/users/146660 | Can we prove that simple polygons can always be split in half (vertex-wise) by diagonals? | I think the following is a counterexample on six vertices: Start with an equilateral triangle $ABC$ and a larger equilateral triangle $A'B'C'$ with the same center and parallel edges. Now consider the hexagon $AA'BB'CC'$. The only candidates for bisecting diagonals (since we have an even number of vertices) are $AB'$, ... | 2 | https://mathoverflow.net/users/39747 | 342958 | 145,591 |
https://mathoverflow.net/questions/342973 | 1 | Given an nondegerate ellipsoid $E$ in $\mathbb{R}^d$, described as $E = \{x\in\mathbb{R}^d: (x-x\_0)^TQ\_0(x-x\_0)\leq 1\}$ and let $\chi\_E$ be the characteristic function supported on $E$. I am thinking about how to construct a family $\{f\_n:\mathbb{R}^d\to\mathbb{R}\}\_{n= 1}^\infty$, where each $f\_n$ can be writt... | https://mathoverflow.net/users/123506 | Family of funcitons that approximates uniform density on an ellipsoid | There are many good ways to accomplish this. One is by using the logistic function $L$, defined by the formula
$$L(u):=\frac1{1+e^{-u}}.$$
Then
$$g\_n(u):=L(n(1-u))
\left\{
\begin{aligned}
\uparrow1&\text{ if }0\le u<1,\\
\downarrow0&\text{ if }u>1
\end{aligned}
\right.
$$
as $n\to\infty$.
So, assuming that $Q\_0$ ... | 1 | https://mathoverflow.net/users/36721 | 342981 | 145,599 |
https://mathoverflow.net/questions/342706 | 3 | All known examples for double exponential lower bounds for real quantifier elimination involves polynomial inequalities with degree $>1$.
Is there an example of double exponentiality with polyhedral inequalities where polyhedral inequalities refers degree is at most $1$?
**Conjecture**: There is none and if you hav... | https://mathoverflow.net/users/136553 | Lower bound for polyhedral real quantifier elimination | I believe the following is a counterexample to the stronger $O(\mathrm{poly}(n,t,m))$ conjecture. Start with variables $x\_i$ for $1 \leq i \leq n$ and $t\_{ij}$ for $1 \leq i < j \leq n$. Consider the formula
$$\phi=\exists t\_{12} \ldots \exists t\_{n-1,n}\left(\bigwedge\_i x\_i = \sum\_{j<i} t\_{ji} - \sum\_{j>i} t\... | 4 | https://mathoverflow.net/users/297 | 342991 | 145,601 |
https://mathoverflow.net/questions/341933 | 13 | It is mentioned in the introduction to [1] that (Cartesian) differential categories might be the unifying framework for differentiation in various branches of mathematics including combinatorics. It is also mentioned in [2] and other papers on tangent categories that there are tangent categories of combinatorial specie... | https://mathoverflow.net/users/62782 | Combinatorial species and differential categories | I contacted Geoffrey Cruttwell with regards to this question. Here is his reply:
>
> There hasn’t been any published paper on a tangent category of combinatorial species. However, the ideas can be found in a talk by my co-author, Robin Cockett here: [Can you Differentiate a Polynomial?](https://www.mscs.dal.ca/~sel... | 7 | https://mathoverflow.net/users/62782 | 342996 | 145,603 |
https://mathoverflow.net/questions/342985 | 4 | This is a [cross-post from stats.stackexchange.com](https://stats.stackexchange.com/q/407121/44368). No answer has appeared there. Since this is a theoretical question, mathoverflow.net seems to be a more appropriate venue for it.
---
What is the analog of the central limit theorem or concentration theorem for re... | https://mathoverflow.net/users/32660 | Central limit theorem for resampling | First, we need to fix the notation a bit. Let $X\_1,X\_2,\dots$ be iid zero-mean unit-variance random variables (r.v.'s). For each natural $n$, let the $n$-tuple $(J\_1,\dots,J\_n):=(J\_{n,1},\dots,J\_{n,n})$ of r.v.'s be independent of the $X\_k$'s and have the multinomial distribution with parameters $n,1/n,\dots,1/n... | 6 | https://mathoverflow.net/users/36721 | 342997 | 145,604 |
https://mathoverflow.net/questions/342999 | 9 | Let $\mathbb S[z]$ be the free $E\_\infty$-ring spectrum generated by the commutative monoid $\mathbb N$. That is, $\mathbb S[z] = \Sigma^\infty\_+ \mathbb N$.
In Bhatt-Morrrow-Scholze II (<https://arxiv.org/abs/1802.03261>), they define a map out of this spectrum by choosing an element of the target. In particular,... | https://mathoverflow.net/users/136914 | Universal property of $\mathbb S[z]$ and $E_\infty$-ring maps | $\newcommand{\E}{\mathbf{E}}$Dylan answered question 3 (and hence question 1) in the comments, but here's another equivalent way to see it: $\E\_\infty$-maps $S^0[z]\to R$ with $R$ a discrete ring (i.e., viewed as an Eilenberg-Maclane spectrum) are the same data as $\E\_\infty$-maps $\tau\_{\leq 0} S^0[z] = H\mathbf{Z}... | 6 | https://mathoverflow.net/users/102390 | 343007 | 145,609 |
https://mathoverflow.net/questions/343004 | 2 | I had asked this question in [math.se](https://math.stackexchange.com/questions/3375396/is-this-graph-problem-np-hard?noredirect=1#comment6946013_3375396) without any success
Let $A$ be the symmetric $n\times n$ adjacency matrix for a graph where $A\_{ij}$ is the positive edge value between node $i$ and $j$ (thus ful... | https://mathoverflow.net/users/27249 | Is this graph problem NP-Hard? | The problem is at least as hard as the [clique problem](https://en.wikipedia.org/wiki/Clique_problem): Take a graph $G$. Let $G'$ be a new graph with vertices $V(G)+c$ and edges $E(G)+\{(g,c)|g\in V(G)\}$.
Let $A\_{ij}=1$ if $ij$ is an edge in $G'$, and $0.5$ otherwise. As $A\_{ic}=1$ for all vertices $i$, we have
... | 5 | https://mathoverflow.net/users/125498 | 343010 | 145,610 |
https://mathoverflow.net/questions/342860 | 5 | If the answer to this question is in affirmative, then this would yield a good answer to [this question](https://mathoverflow.net/questions/342763).
Let $f\colon \mathbb C\to\mathbb C$ be an entire function whose values on the real line are real and bounded from below by
$$
f(x)\geq \sqrt{\frac{\sinh x}{x}}, \qquad... | https://mathoverflow.net/users/17581 | Entire function not less than $\sqrt{\sinh x/x}$ on the real line | The answer is no. Let $f$ be the function described under Remark 1 of the OP. By continuity, there exists a $\delta >0$ such that $f(x) - \sqrt{(\sinh x)/x} > 0.05$ on the real axis for $|x|\leq \delta $ (note that, at $x=0$, the aforementioned difference is $2/\sqrt{\pi } -1 = 0.128$).
Consider furthermore the funct... | 5 | https://mathoverflow.net/users/134299 | 343016 | 145,612 |
https://mathoverflow.net/questions/237662 | 16 | It is well-known that if $AC$ holds and if $j: L(V\_{\lambda+1}) \to L(V\_{\lambda+1})$ is a non-trivial elementary embedding with $crit(j) < \lambda,$
then $\lambda$ has countable cofinality (and in fact it is the least fixed point of $j$ above $crit(j)$).
>
> **Question.** Is $AC$ needed to show that $\lambda$ ha... | https://mathoverflow.net/users/11115 | The axiom $I_0$ in the absence of $AC$ | It is consistent that $AC$ fails and there exists a non-trivial elementary embedding $j: L(V\_{\lambda+1}) \to L(V\_{\lambda+1})$ with $crit(j) < \lambda,$
and $\lambda$ has uncountable cofinality. See [BERKELEY CARDINALS AND THE STRUCTURE OF $L(V\_{δ+1})$](https://www.cambridge.org/core/journals/journal-of-symbolic-lo... | 3 | https://mathoverflow.net/users/11115 | 343019 | 145,614 |
https://mathoverflow.net/questions/342965 | 3 | Let $\alpha \in \mathbb{C}$ be a zero of a monic irreducible polynomial $f \in \mathbb{Z}[X]$. Define $K = \mathbb{Q}[\alpha]$ and $A := \mathbb{Z}[\alpha]$, then $A \subseteq O\_K$, where the latter denotes the ring of integers of the number field $K$.
Let $p$ be a prime number, and assume that $f$ factors modulo $p... | https://mathoverflow.net/users/125074 | Generators of prime ideals and factorization of polynomials | $\newcommand{\Z}{\mathbb{Z}}$No.
It is easier to construct a counter-example by starting from the local case.
Let $f\in\Z\_p[X]$ be monic irreducible of degree $e>1$ such that $F = \mathbb{Q}\_p[X]/f$ is totally ramified, let $\alpha = X \bmod f$, and let $v$ be the normalised valuation on $F$.
Assume that $f = X... | 3 | https://mathoverflow.net/users/40821 | 343024 | 145,615 |
https://mathoverflow.net/questions/343021 | 3 | Is there a Hausdorff space $(X,\tau)$ such that $|X|>1$, and whenever $U, V\in \tau$ with $U\cong V$ (with the subspace topologies) we have $U=V$?
**Note.** There is an infinite $T\_0$-space with this property, namely $(\omega, \omega+1)$. I don't know about $T\_1$-spaces, though.
| https://mathoverflow.net/users/8628 | Hausdorff space such that the open sets are pairwise non-isomorphic | There are many such spaces in Continuum Theory.
**Example 1.** Take a dendrite $T$ containing exactly one separating point $x\_n$ of each degree $n\ge 3$ (the degree of a separating point $x$ is the number of connected components of $X\setminus\{x\}$) such that the set $D=\{x\_n\}\_{n\ge 3}$ is dense in $T$.
Then ... | 2 | https://mathoverflow.net/users/61536 | 343028 | 145,616 |
https://mathoverflow.net/questions/343026 | 1 | I wondered, inspired in a result mentioned from [1] (page 45), what should be the asymptotic behaviour of the sequence on assumption of the First Hardy–Littlewood conjecture
$$\sum\_{\substack{\text{primes }p\_n\leq x\\\text{such that }p\_n+2\text{ is prime}}}(p\_{n+1}-p\_n)^2$$
as $x\to\infty$. Thus the summation ... | https://mathoverflow.net/users/142929 | A question about a sum that involves gaps between twin primes, on assumption of the First Hardy–Littlewood conjecture | You want to estimate $x \to +\infty$:
$$\sum\_{\substack{\text{primes }p\_n\leq x\\\text{such that }p\_{n+1}+2\text{ is prime}}}(p\_{n+1}-p\_n)^2$$
Let $n\in 2\mathbb{N}$, and consider the 3 tuple $\mathcal{H}\_3 = (0,n,n+2)$.
The 3-tuple $(0,n,n+2)$ is admissible iff $n = 1 \pmod 3$ or $n = 0 \pmod 3$.
Let $\pi\_{... | 1 | https://mathoverflow.net/users/164630 | 343031 | 145,617 |
https://mathoverflow.net/questions/342947 | 2 | It is well-known that the weighted backshift operator $B\_{\lambda}:\ell^p \rightarrow \ell^p$ is hypercyclic (with $\lambda>1$); that is, there exists a dense set of sequences $X\subseteq \ell^p$ for which
$$
\overline{\left\{B^n(x)\right\}\_{n \in \mathbb{N}}
} = \ell^p \qquad (\forall x \in X).
$$
Is there a know... | https://mathoverflow.net/users/36886 | Hypercyclic vector for backshift operator | Probably not an example you are looking for. Rolewicz's proof is essentially an application of Baire's theorem and you can make this in a certain sense construtive: Choose a dense countable set $\{x^k: k\in \mathbb N\}$ of finite sequences (i.e., terminating with zeros) in $\ell^p$, align $y^k=(x^k\_1,\lambda x^k\_2, \... | 3 | https://mathoverflow.net/users/21051 | 343033 | 145,618 |
https://mathoverflow.net/questions/342926 | 4 | Let $A \subset \mathcal{B}(H)$ a subalgebra, not necessarily a $\*$-algebra. In Murphy's book 'C\*-algebras and Operator Theory', in Remark 4.2.1 you can find a proof of the failure of strong compactness for the ball of $\mathcal{B}(H)$:
If the ball is strongly compact, then the identity map of the ball with the rela... | https://mathoverflow.net/users/145367 | Coinciding weak operator and strong operator topologies | I'm not sure there is a definitive answer here, as it's asked if there are "invariants", not a complete characterisation. Further, it's not clear to me if the question is asking about compactness or just whether the WOT and SOT are different.
Indeed, if all we are interested in is to show that the SOT is not the same... | 3 | https://mathoverflow.net/users/406 | 343062 | 145,620 |
https://mathoverflow.net/questions/343018 | 1 | The Lipschitz-Free space (also known as [Arens-Eells spaces](http://projecteuclid.org/euclid.pjm/1103043959)) $\mathcal{F}(X,d)$ over a pointed metric space $(X,d)$ is a well-studied object. In many instances, we have "concrete" representations of it.
**Example**
---
* Example 3.10 of [Weaver's book](https://r... | https://mathoverflow.net/users/36886 | Known Lipschitz-free spaces | Let me try again, at least for the case of metric spaces. In my opinion, the question has been answered completely but I don‘t think you will like the description. The sloppy way to state it is that these free spaces are always spaces of measures on the underlying space. Thus for your example of a discrete space, the s... | 3 | https://mathoverflow.net/users/131781 | 343066 | 145,622 |
https://mathoverflow.net/questions/343072 | 5 | Specifically, I am interested in the case where one eigenvalue is exactly $0$. Given an $n \times n$ symmetric matrix, I would like to find the closest $n\times n$ symmetric matrix that has one eigenvalue that is equal to $0$.
Although the $n\times n$ matrices form a [Hilbert space](https://en.wikipedia.org/wiki/Hil... | https://mathoverflow.net/users/49465 | Projecting a symmetric matrix onto the space of linear operators with a particular eigenvalue | Any (real) symmetric matrix may be diagonalised by an orthogonal matrix. So let your matrix be $A$ and let $O$ be an orthogonal matrix such that $D=OAO^{-1}$ is diagonal. Note that the transformation $A\mapsto OAO^{-1}$ is distance preserving. Now get $D'$ by replacing whichever diagonal entry of $D$ has least absolute... | 7 | https://mathoverflow.net/users/4613 | 343077 | 145,627 |
https://mathoverflow.net/questions/343065 | 3 | Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ and $f \in W^{1,p} (\partial \Omega)$. Can $f$ be extended to a function $u \in W^{1,p}(\Omega)$ such that $u|\_{\partial \Omega}=f$ and
$$\lVert u\rVert\_{W^{1,p}(\Omega)}\leq C\lVert f\rVert\_{W^{1,p}(\partial \Omega)}?$$
What are the minimal assumptions that g... | https://mathoverflow.net/users/115905 | Continuous extension of functions |
>
> **Theorem.** If $\Omega\subset\mathbb{R}^n$, $n\geq 2$, is a bounded and smooth domain, then there is a bounded extension operator $$
> E:W^{1,p}(\partial\Omega)\to W^{1,q}\cap C^\infty(\Omega), \quad
> \text{where $1<p<\infty$ and $q=\frac{np}{n-1}$.} $$
>
>
>
Note that $q>p$ so $W^{1,q}(\Omega)\subset W... | 4 | https://mathoverflow.net/users/121665 | 343092 | 145,631 |
https://mathoverflow.net/questions/342900 | 5 | Let $Y$ be a nonsingular variety and $X\subset Y$ a closed subscheme. A *linear scheme* over $X$ is a scheme of the form $\textbf{Spec}\, ( Sym \_{\mathcal{O}\_X} F) $, where $F$ is a coherent sheaf on $X$.
If the embedding $X\subset Y$ is regular, i.e. if every point of $Y$ has a neighborhood over which the ideal $... | https://mathoverflow.net/users/145172 | when is the normal cone a linear scheme? | I believe the answer is negative. Take $Y= \mathbb{A}^3= \textbf{Spec}\, k[x,y,z]$ and $X= V(xz,yz)$, so $I=(xz,yz)$. Note that $X$ is the union of the plane $z=0$ and the line $x=y=0$. Then $X\subset Y$ is not regular: any neighborhood of the origin contains a point in the plane and a point in the line, so their local... | 2 | https://mathoverflow.net/users/145172 | 343098 | 145,634 |
https://mathoverflow.net/questions/343097 | 11 | What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H\_1=(1)$ and $$ H\_N=\begin{pmatrix}H\_{N/2} & H\_{N/2} \\ H\_{N/2} & -H\_{N/2}\end{pmatrix}, $$ where $N=2^n$?
Edit: The answer below provides a "literal" answer to the problem. However, is there a deeper meaning to... | https://mathoverflow.net/users/7581 | What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix? | The $2^n\times 2^n$ dimensional Hadamard matrices $H\_{2^n}$ are also called [Sylvester matrices](https://en.wikipedia.org/wiki/Hadamard_matrix#Sylvester%27s_construction) or [Walsh matrices.](https://en.wikipedia.org/wiki/Walsh_matrix) There are only two distinct eigenvalues $\pm 2^{n/2}$, so the eigenvectors are not ... | 12 | https://mathoverflow.net/users/11260 | 343101 | 145,637 |
https://mathoverflow.net/questions/343086 | 6 | It is well-known that there exist Groebner bases for ideals in polynomial ring $\mathbb Q[x]$ which can be found algorithmically Moreover, I don't think it is hard to show that there exist Groebner bases for ideals in $\mathbb Z[x]$. But I am having trouble defining Groebner Bases for submodule of free $\mathbb Z[x]$-m... | https://mathoverflow.net/users/142603 | Groebner Bases for submodule over polynomial ring with integer coefficients | There is a description of the appropriate Groebner basis algorithm in this book:
Franz Pauer, Andreas Unterkircher.
[Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations.](https://doi.org/10.1007/s002000050108)
AAECC 9, 271–291 (1999)
I've implemented it in t... | 6 | https://mathoverflow.net/users/1465 | 343102 | 145,638 |
https://mathoverflow.net/questions/342859 | 3 | For a compact Kaehler manifold $M$, a basic structural result for its de Rham cohomology is the hard Lefschetz theorem. See [here](https://en.wikipedia.org/wiki/Lefschetz_hyperplane_theorem#Hard_Lefschetz_theorem) or [here](https://ncatlab.org/nlab/show/hard+Lefschetz+theorem) for an overview of the result.
What hap... | https://mathoverflow.net/users/126606 | Non-compact hard Lefschetz theorem | Hard Lefschetz implies Poincaré duality with real coefficients, in particular $H^0\cong H^{2n}$ and for a non-compact connected manifold that *never* happens ($H^{2n}=0$).
For example: $M:=\mathbb{C}P^2\setminus\mathbb{C}P^1$ is contractible, but hard Lefschetz theorem (as stated) should have given an isomorphism be... | 4 | https://mathoverflow.net/users/13842 | 343110 | 145,642 |
https://mathoverflow.net/questions/343125 | -1 | Let $G$ be an infinite group, let $S\_0\subseteq G$ be a subgroup and suppose that ${\frak C}$ is a collection of subgroups of $G$ such that
1. $C \cong S\_0$ for all $C\in {\frak C}$, and
2. for all $C, C'\in {\frak C}$ we have $C\subseteq C'$ or $C'\subseteq C$.
It is easy to see that $\bigcup{\frak C}\subseteq G... | https://mathoverflow.net/users/8628 | Union of an ascending chain of subgroups in group $G$ isomorphic to subgroup $S_0\subseteq G$ | This can fail on cardinality grounds. For instance if $G$ is a vector space of dimension $\aleph\_1$ and $\mathfrak{C}$ is a chain of $\aleph\_1$ subspaces each of dimension $\aleph\_0$, then all of the groups in $\mathfrak{C}$ are isomorphic but their union is not isomorphic to any of them.
| 6 | https://mathoverflow.net/users/83901 | 343126 | 145,646 |
https://mathoverflow.net/questions/343124 | 1 | Does anyone have a reference for the definition of a canonically polarized manifold? Typically, at least from what I have seen, a polarized manifold is a compact Kähler manifold $X$ together with an ample line bundle $L \to X$. I cannot seem to find a definition of canonically polarized, however.
| https://mathoverflow.net/users/105103 | Definition of Canonically polarized manifold? | Canonically polarised means that the polarising line bundle L is a power of the canonical bundle (top exterior power of the cotangent bundle). In particular it only makes sense if the canonical bundle is ample.
| 4 | https://mathoverflow.net/users/10839 | 343128 | 145,647 |
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