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https://mathoverflow.net/questions/427066 | 2 | In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:
Ultravalue $|\bullet|:K \to \mathbb{R}\_{+}$ can be viewed as a valuation on the field which defines an ultrametric $... | https://mathoverflow.net/users/91850 | Are there "pathological convex sets" over ultravalued fields of char 2? | You can see this already with the ultravalued field $\mathbb F\_2$, with $|1|= 1$, $|0|=0$.
In this field, since we must have $\alpha=0$ or $\alpha=1$, the $\alpha x + (1-\alpha) y$ condition is trivial — every set satisfies it.
But not every set is convex. Since every element of the field is at most $1$, a set is ... | 3 | https://mathoverflow.net/users/18060 | 427070 | 173,269 |
https://mathoverflow.net/questions/426996 | 3 | Given a positive integer $n$. For any symmetric $n\times n$ matrix $M$, write $M\_{ik}$ for the $(i,k)$ entry, $M\_k$ for the unordered list (multiset) of entries in row $k$, and $S(M)$ for the unordered list (multiset) of all pairs $(M\_k,M\_{kk})$.
(This is the corrected definition from July 22, 2022, after having re... | https://mathoverflow.net/users/56920 | A combinatorial matrix reconstruction problem | The meaning of the question is unclear due to the word "set". As Carlo proposed in a comment, I'll take it that we have an unordered list of $n$ unordered lists of $n$ elements, and we want to identify the symmetric matrices for which those $n$ lists correspond to the rows up to permutation.
Clearly this is possible ... | 2 | https://mathoverflow.net/users/9025 | 427082 | 173,272 |
https://mathoverflow.net/questions/426459 | 2 | Let $p\in (0,1)$ and $X\_1, X\_2, ...X\_n \sim \text{Bern}(p)$ be $n$ i.i.d. Bernoulli random variables, where the probability that $X\_i$ is $1$ equals $p$.
Fix $a,b>0$ different from $1$ that satisfy $a^p b^{1-p} = 1$, and define $C\_i = X\_i(a-b)+b$. In other words, $C\_i$ is $a$ when $X\_i$ is $1$, and $b$ when $... | https://mathoverflow.net/users/124761 | Self normalized sum of products of i.i.d. random variables | James, using the fact that $X\_i$ only takes on 2 values, write $C\_iX\_i = xC\_i + y$ . Then your numerator is $$\sum^n x \prod^j C\_i + \sum^{n-1} y \prod^j C\_i = \sum^n (x+y) \prod^j C\_i - yC\_n$$. It suffices to show that $$ ~\frac {\prod^n C\_i} {\sum^n \prod^j C\_i } \rightarrow 0$$, where I am going to show th... | 1 | https://mathoverflow.net/users/143907 | 427084 | 173,273 |
https://mathoverflow.net/questions/427043 | 2 | Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \begin{cases} - 1 & \text{ if } X(t,x) >0, \\
1 & \text{ if } X(t,x) < 0 \end{cases}, &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
This is a typical example of ODE that does not have a solution in the classical sense. The right-ha... | https://mathoverflow.net/users/122620 | Regular Lagrangian flow for explicit ODE with discontinuous right-hand side | $\newcommand{\Om}{\Omega}\newcommand{\om}{\omega}\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}$As stated in my previous comment, in Theorem 3.1 of the paper linked by the OP about the existence (and uniqueness) of a Lagrangian flow, there is the condition that $D\cdot b$ be absolutely continuous with respect to t... | 1 | https://mathoverflow.net/users/36721 | 427097 | 173,279 |
https://mathoverflow.net/questions/427092 | 3 | Let $Q$ be a $d$-dimensional Riemannian manifold. A submanifold $M$ of $Q$ is said to be *extrinsically flat* if $R\_{M}(X,Y,Z,W) = R\_{Q}(X,Y,Z,W)$ for all $X,Y,Z,W \in \mathfrak{X}(M)$, where $R\_{M}$ and $R\_{Q}$ are the curvature tensors of $M$ and $Q$, respectively.
It is clear that if the rank of the second fun... | https://mathoverflow.net/users/74033 | Extrinsically flat submanifolds of a Riemannian manifold | If you take the simplest case, in which $Q$ is a $3$-dimensional Riemannian manifold, then there are plenty of extrinsically flat surfaces $M\subset Q$. In fact, if one chooses a 'generic' curve $\gamma$ in $Q$ and a 'generic' normal vector field $\nu$ along $\gamma$, then there will be a unique 'extinsically flat' sur... | 3 | https://mathoverflow.net/users/13972 | 427102 | 173,281 |
https://mathoverflow.net/questions/427103 | 7 | Let $\mathcal{A}$ be the algebra (in the sense of universal algebra) whose underlying set is the four-element set $\{a,b,c,d\}$ and whose structure consists just of the ternary operation $F$ defined as follows:
* $F(x,x,x)=x$.
* If $x\not=y$ then $F(x,x,y)=F(x,y,x)=F(y,x,x)=y$.
* If $x,y,z$ are pairwise distinct, the... | https://mathoverflow.net/users/8133 | Is the equational theory of this "orthocentrish" algebra finitely based? | This algebra is finitely based.
In fact, if you choose any bijection from $\{a,b,c,d\}$ to $\mathbb Z\_2\times \mathbb Z\_2$, then you can transport the operation $F(x,y,z)$ to $\mathbb Z\_2\times \mathbb Z\_2$ it find that it is $x-y+z$. The resulting algebra $\langle \mathbb Z\_2\times \mathbb Z\_2; x-y+z\rangle$ is ... | 14 | https://mathoverflow.net/users/75735 | 427105 | 173,282 |
https://mathoverflow.net/questions/427109 | 1 | Sorry if this is too easy for MO, but I found it in a research paper, so I thought that it was worth posting here.
I was reading a paper by Rabinowitz([this one](https://apps.dtic.mil/sti/pdfs/ADA093444.pdf) to be more precise) and I came across the following theorem:
>
> **Theorem 2.10:** Suppose $H\in C^1(\math... | https://mathoverflow.net/users/486258 | How do I integrate this inequality that appears in a paper of Rabinowitz? | For any unit vector $u$ and real $t>0$, let
\begin{equation}
h(t):=H(tu).
\end{equation}
Then
\begin{equation}
h'(t)=H'(tu)\cdot u=\frac{H'(tu)\cdot(tu)}t\ge\frac{\mu H(tu)}t=\frac{h(t)}t.
\end{equation}
Recall that $H>0$ and hence $h(t)>0$ for all real $t$. So, for all real $t\ge1$
\begin{equation}
(\ln h)'(t)\... | 2 | https://mathoverflow.net/users/36721 | 427113 | 173,283 |
https://mathoverflow.net/questions/427106 | 1 | In "[Sharp threshold phenomena in statistical physics](https://link.springer.com/article/10.1007/s11537-018-1726-x)", *H. Duminil-Copin, Japanese J. of Math. 14, 2019*, a sharp transition of a boolean function is defined as follows:
>
> A sequence of increasing boolean functions $(\mathbf{f\_n})$ undergoes a sharp ... | https://mathoverflow.net/users/90619 | What is the exact definition of a sharp transition? | The parameters $p\_n$ are not arbitrary numerical parameters. They represent the expectation of one binary variable in a product space. Changing them additively is very different from changing the power in the expression $n^{-c}$.
I believe definition 1.1 in the cited paper has a typo, and for $p\_n$ tending to 0, we... | 5 | https://mathoverflow.net/users/7691 | 427116 | 173,284 |
https://mathoverflow.net/questions/427122 | 2 | Let $V\_i$ be a sequence of $k$ dimensional analytic subsets in $\mathbb C^N$. Suppose that the volume of $V\_i$ is uniformly bounded, then Bishop's compactness theorem says that $V\_i$ will convergence by sequence to an analytic subsets $V$.
Q1: What is the precise meaning of "converge" here.
Q2: Is it possible that... | https://mathoverflow.net/users/78863 | Bishop's compactness theorem and convergence of analytic subset | Convergence is taken [in Hausdorff sense](https://en.wikipedia.org/wiki/Hausdorff_distance),
though you can define the structure of a complex variety (the [Barlet space](http://library.msri.org/books/Book37/files/barlet.pdf))
on the set of cycles, taking every irreducible component with positive integer multiplicity. B... | 3 | https://mathoverflow.net/users/3377 | 427126 | 173,286 |
https://mathoverflow.net/questions/427078 | 7 | **Question:**
is there a class of optimization problems for whose solution no efficent algorithm is known, but for which the claimed optimality of a solution can efficiently be verified?
**Edits:**
* There is the publication [Optimality conditions and complete description of polytopes in combinatorial optimizati... | https://mathoverflow.net/users/31310 | Is there an optimization variant of NP completeness | It is unlikely that there is an interesting class of such optimization problems, for the following reason.
Following Chapter 17 of Papadimitriou's book *Computational Complexity*, let EXACT TSP denote the following problem: Given a distance matrix and an integer $B$, is the length of the shortest tour *equal to* $B$?... | 4 | https://mathoverflow.net/users/3106 | 427127 | 173,287 |
https://mathoverflow.net/questions/427014 | 1 | While I understand that doing the above is not possible *in general*, I would like to know more about how to proceed when it *is* possible. That is, what are the common methods people use to analytically characterize basin of attraction boundaries (i.e. via a closed-form expression) or their (Lebesgue) measures? (Regar... | https://mathoverflow.net/users/486157 | Analytically characterizing basins of attraction boundaries and sizes | Despite having searched for such a reference for quite a while before asking my question, I ironically found a reference that seems to fit the bill! [Alberto and Chiang (2015, p. 198)](https://www.cambridge.org/core/books/stability-regions-of-nonlinear-dynamical-systems/522A4FACB373591173A78C710BF004D8) looks like a gr... | 0 | https://mathoverflow.net/users/486157 | 427134 | 173,289 |
https://mathoverflow.net/questions/427129 | 0 | Let $\{X\_i, i \in \mathbb{N}\}$ be a sequence of non-lattice i.i.d. centered random variables, $\mathbb{E} |X\_1| ^3 < 0$. Let $S\_n = \sum\limits \_{i=1} ^n X\_i$ be the corresponding random walk and $W^{(n)} \_t = \frac{S\_{\lfloor nt \rfloor}}{\sqrt{n}}$, $t \in [0,1]$.
I am looking for a reference that conditioned... | https://mathoverflow.net/users/41071 | Invariance principle: Brownian bridge and random walk conditioned on end point | A more general theorem is proved in [1] for the limits of random walks in the domain of attraction of a stable law. In the case described in the problem, one can also use the strong approximation approach in [2], because the probabilities of deviations of order $\sqrt{n}$ are smaller than the probabilities of the event... | 2 | https://mathoverflow.net/users/7691 | 427136 | 173,290 |
https://mathoverflow.net/questions/427132 | 2 | Let $\Omega \subseteq \mathbb{R}^n$ be a bounded domain. Define the set
$$H = \left\{\nabla f : f \in \mathcal{C}^1(\Omega)\right\}.$$
I suspect that $H$ is a Hilbert space (though I am unsure about completeness) with respect to the inner product
$(F,G) \in H^2 \mapsto \int\_\Omega \langle F(x),G(x)\rangle dx$, whe... | https://mathoverflow.net/users/111925 | Is there a name for the space of gradients? | The space in question is not complete: take a function $f$ whose derivative is in $L^2$ but is discontinuous, and consider a sequence of smooth functions converging to it in the Sobolev space $W^{1, 2}$. Then the sequence of gradients is clearly Cauchy in your space $H$ and its limit is $\nabla f \notin H$.
This tell... | 2 | https://mathoverflow.net/users/109533 | 427137 | 173,291 |
https://mathoverflow.net/questions/427147 | 4 | My faculty imposes some numerical "recommendations" for promotions. Instead of arguing this sort of recommendation is ridiculous, I think it is wiser to provide some evidence the recommended numbers are far too high for (pure) mathematics.
Is there data available for the average or mean number of PhD completions when... | https://mathoverflow.net/users/130882 | Supervision numbers in pure mathematics | To establish a base line, you could look into [Some Patterns of PhDs in Mathematics Awarded Annually by Institutions of Higher Education in the United States over the Last Two Decades](https://www.ams.org/journals/notices/202201/rnoti-p96.pdf). This lists for each subfield of mathematics how many Ph.D.'s were awarded o... | 6 | https://mathoverflow.net/users/11260 | 427149 | 173,294 |
https://mathoverflow.net/questions/427158 | 7 | In provability logic, $\square X \rightarrow X$ is not a theorem.
In my head[1] this reads as "if X is provable you don't necessarily have a proof of X".
This has lead to the question, what does provable even mean, if not there exists a proof of X? Is there an example of some proposition that is provable but does n... | https://mathoverflow.net/users/476956 | Difference between provability and the existence of a proof? | First note this isn't a constructive logic, so it's wrong to think of "$X$" as "there is a proof of $X$". (Even in constructive logic I find that dubious.)
Second note that if $X$ is provable then $Y \rightarrow X$ is provable for any $Y$. So it's not correct to say that $\square X \rightarrow X$ is not a theorem. Th... | 12 | https://mathoverflow.net/users/18060 | 427159 | 173,296 |
https://mathoverflow.net/questions/427157 | 3 | The following problem seems easy at a first glance but I can't see the way to prove it. Actually I don't even know if it's true but it is assumed implicitly in a research paper.
Help highly appreciated.
Given two spd matrices $A$, $B$ with
$x^\top Ax \ge x^\top Bx$
then it follows that
$x^\top A^{-1}x \le x^\to... | https://mathoverflow.net/users/486297 | Inverse quadratic norms | The proposed result holds **true**.
I am assuming throughout that $spd$ means [*symmetric positive definite*](https://en.wikipedia.org/wiki/Definite_matrix) and that the matrices $A$ and $B$ are $n$-by-$n$ matrices over $\mathbb{R}$ for some $n > 0$.
Indeed, since $B = P^{\top}P$ for some $P \in GL\_n(\mathbb{R})$ ... | 4 | https://mathoverflow.net/users/84349 | 427166 | 173,297 |
https://mathoverflow.net/questions/401601 | 38 | Let $\mathbb R ^\omega$ be the set of all sequences of real numbers in the product topology.
Let $X$ be the set of all sequences in $\mathbb R ^\omega$ which have at least one 0.
Let $Y$ be the set of all sequences in $\mathbb R ^\omega$ which have at least two 0's.
Are $X$ and $Y$ homeomorphic, and if so, is the... | https://mathoverflow.net/users/95718 | Sequences with 0's in $\mathbb R ^\omega$ | For every $n\in\mathbb N$ consider the space
$$X\_n=\{x\in\mathbb R^\omega:\lvert x^{-1}(0)\rvert\ge n\}.$$
**Theorem.** For any positive integer numbers $n<m$, the spaces $X\_n$ and $X\_m$ are not homeomorphic.
*Proof.* The spaces $X\_n$ and $X\_m$ are not homeomorphic because the space $X\_m$ is a $\sigma Z\_{m-1... | 8 | https://mathoverflow.net/users/61536 | 427167 | 173,298 |
https://mathoverflow.net/questions/427074 | 2 | Let $(X, \Delta)$ be a (projective) klt pair (say over $\mathbb{C}$, but I am also interested in fields of positive characteristic) and $f: X \to Z$ the contraction associated to a $(K\_X + \Delta)$-negative extremal face $F$ of $\overline{NE}(X)$.
>
> Assuming $f$ is a birational morphism, does $Z$ have rational s... | https://mathoverflow.net/users/519 | Singularities of contractions of extremal faces | In characteristic 0, the answer is well known. By assumption there is an ample divisor $A$ such that $K\_X+\Delta+A$ cuts out $F$ and hence by the BPF theorem $K\_X+\Delta+A\sim \_{\mathbb Q,f}0$ and in fact $K\_X+\Delta +A\sim \_{\mathbb Q}f^\*(K\_Z+B\_Z)$ where $(Z,B\_Z)$ is klt; in the birational case $B\_Z=f\_\*(\D... | 5 | https://mathoverflow.net/users/19369 | 427184 | 173,306 |
https://mathoverflow.net/questions/427188 | 2 | Let $J$ denote [Jensen's modification](https://en.wikipedia.org/wiki/Jensen_hierarchy) of the constructible hierarchy. For an ordinal $\alpha$ and an $n\in\mathbb N^+$, let $\rho\_n^{J\_\alpha}$ denote the $\Sigma\_n$-projectum of $J\_\alpha$, the least $\delta\leq\alpha$ such that there is a $\Sigma\_n(J\_\alpha)$-def... | https://mathoverflow.net/users/479330 | Ordering patterns of projecta by least witness | $<\_\rho$ is a wellorder essentially by definition. The ordertype is $\omega^{\omega}$ (ordinal exponentiation of course).
In fact $s<\_\rho t$ iff either $\mathrm{lh}(s)<\mathrm{lh}(t)$,
or $\mathrm{lh}(s)=\mathrm{lh}(t)$ and $s<\_{\mathrm{lex}}t$,
i.e. letting $i$ be least such that $s(i)\neq t(i)$, we have $s(i)<t... | 3 | https://mathoverflow.net/users/160347 | 427192 | 173,311 |
https://mathoverflow.net/questions/427205 | 2 | $\newcommand{\ur}{\mathrm{ur}}\newcommand{\cris}{\mathrm{cris}}$Let $K$ be a finite extension of $\mathbb{Q}\_p$, $G\_K=\operatorname{Gal}(\overline{K}/K)$ and $I\_K \subset G\_K$ its inertial subgroup. Let $V$ be a finite-dimensional representation of $G\_K$. Assume that $V$ is crystalline as $G\_K$-representation. Is... | https://mathoverflow.net/users/146212 | Crystalline when restricted to inertial subgroup | This is purely formal. If $V$ is crystalline, then $V \otimes \mathbf{B}\_{\mathrm{cris}}$ has a basis as a $\mathbf{B}\_{\mathrm{cris}}$-module in which the action of $G\_K$ is trivial. Hence *a fortiori* it has a basis in which the action of $I\_K$ is trivial.
What is much less obvious, but also true, is that the c... | 8 | https://mathoverflow.net/users/2481 | 427207 | 173,314 |
https://mathoverflow.net/questions/427119 | 1 | The theory $\mathsf{TNT}$, introduced by Hao Wang in 1952, adds negative types to simple [Type Set Theory $\mathsf{TST}$](https://en.wikipedia.org/wiki/New_Foundations#The_Type_Theory_TST), so it's written exactly as $\mathsf{TST}$ but with the type indices ranging over $\mathbb Z$ instead of just $\mathbb N$.
Let $\... | https://mathoverflow.net/users/95347 | Can $\mathsf{TNT}$ be modeled in non-well-founded models of $\mathsf{ZF}$? | Per comments, the above conditions might not be enough to ensure the result of interpreting $\sf TNT$, however, the following line would work to answer the first quetion:
Suppose we work in $\sf ZF−Reg.$ on a transitive non-$\omega$-non-well-founded model $M$ of $\sf Finite \ \sf ZF$ (i.e. $\sf ZF -\text{ inf.+ every... | 1 | https://mathoverflow.net/users/95347 | 427213 | 173,315 |
https://mathoverflow.net/questions/427124 | 5 | Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
This is the prototype of non-uniqueness for ODEs in the classical sense. I'm tempted to say, however, that the [Lagrangian flow in the sense of Ambrosio](http://cvgmt.sns.i... | https://mathoverflow.net/users/122620 | Regular Lagrangian flow for "square root example": $\frac{d}{dt} X(t,x) = \sqrt{X(t,x)}$ | Your intuition is right. The key is in the paper you cite, in that they consider uniqueness in the class $L^1\_{\text{loc}}$, which does not allow for concentrations. If you add to this, that the Lagrangian flow conserves mass and that the the classical ODE has uniqueness away from zero, you thus get that the mass tran... | 4 | https://mathoverflow.net/users/51695 | 427214 | 173,316 |
https://mathoverflow.net/questions/426737 | 3 | Following Anton Petrunin’s suggestion, I revise the question to make it less vague.
Let $M^{m}$ be an $m$-dimensional Riemannian manifold, and let $\gamma$ be a unit-speed curve $I \to M^{m}$. We say that $\gamma$ is *planar* if there are a unit vector field $N$ along $\gamma$ and a function $\kappa$ such that
$$\beg... | https://mathoverflow.net/users/74033 | Planar curves in $M^{m}$ vs curves in $M^{2}$ | Here are some comments about the OP's question that don't give a definitive answer to the final question (although the answer may well be 'no', see below), but do provide more information, at least in the simplest possible case, in which $M$ has dimension $3$.
First, by the Gauss equation in dimension $3$, a surface ... | 3 | https://mathoverflow.net/users/13972 | 427222 | 173,321 |
https://mathoverflow.net/questions/427223 | 2 | If I am looking at a collection $\mathcal{C}$ of circles $\{C\_1,...,C\_n\}$ all of which have some radii $\{r\_1,...,r\_n\}$ where $r\_i\in\mathbb{R}^{+}$ for each $i \in[n]$. In $\mathcal{C}$, all the circles are pairwise intersecting i.e. each circle intersects another circle. We do not allow touchings (i.e. the cir... | https://mathoverflow.net/users/485561 | Pairwise intersecting circles in the plane | First, it is sufficient to show that there is always an empty 3-cell inside each circle. This is because we can always do a Möbius transformation to move the outside of a circle inside of it, which will preserve intersections between circles. The same argument can then be used.
Now pick some circle $C$. First, there ... | 2 | https://mathoverflow.net/users/141277 | 427237 | 173,327 |
https://mathoverflow.net/questions/427196 | 1 | I have a curious question I stumbled upon that may be interesting to some.
Consider **real**-valued continuous functions on the circle $f\_1(x),f\_2(x),f\_3(x)$ (so they are periodic in $x \mapsto x+2\pi$).
They have the following properties:
1. $\int\_0^{2\pi} \frac{dx}{2\pi} f\_i(x) = 0$
2. $ \int\_0^{2\pi} \fr... | https://mathoverflow.net/users/317106 | Orthogonal functions on circle with constraints | I think we can do it smoothly. Think of $f=(f\_1,f\_2,f\_3)$ as a map from the circle into $\mathbb R^3$. I need it to take values in the unit sphere. Mark the three obvious great circles on the sphere, and note the twelve quarter-circles in this picture.
Here is a continuous solution. As $x$ goes from $0$ to $\pi/6$... | 3 | https://mathoverflow.net/users/6666 | 427240 | 173,328 |
https://mathoverflow.net/questions/424141 | 11 | Let $G$ be a (connected) reductive group over a perfect field $k$, and let $\xi\in H^1(k,G)$ be a cohomology class.
By a theorem of Steinberg (Serre, Galois cohomology, Appendix 1 to Chapter III, Theorem 11.1),
*if $G$ is quasi-split,* then there exists a $k$-torus $i\colon T\hookrightarrow G$ such that
$$\xi\in i\_\* ... | https://mathoverflow.net/users/4149 | Galois cohomology class of a reductive group not coming from a torus | $\newcommand{\la}{\langle}\newcommand{\ra}{\rangle}$The following example is due to Vladimir Chernousov (private communication).
Let $K={\Bbb Q}(x,y,x',y')$, where $x,y,x',y'$ are variables.
Consider the quadratic forms over $K$
$$ f= \la x,y,-xy\ra\qquad\text{and}\qquad f'=\la x',y',-x'y'\ra.$$
Here $ \la x,y,-xy\ra... | 10 | https://mathoverflow.net/users/4149 | 427289 | 173,339 |
https://mathoverflow.net/questions/427272 | 3 | I'm trying to understand variations of Hodge structure. I understand that this is a very broad field, and that many of the concepts have been extended to algebraic geometry over fields other than $\mathbb{C}$, and so forth. In particular, there is a version of the monodromy theorem by Grothendieck, which is rather inco... | https://mathoverflow.net/users/168781 | What does does the monodromy weight filtration represent? | Given a nilpotent endomorphism $N$ of a finite dimension vector space $V$, Jordan canonical form implies that we can decomponse $V$ into a sum of "blocks" on which we can find bases satisfying $Ne\_1 = e\_{2}, Ne\_2=e\_3,\ldots Ne\_k=0$. To make it basis independent, pass to the filtration
$$ \langle e\_k\rangle\subset... | 4 | https://mathoverflow.net/users/4144 | 427294 | 173,342 |
https://mathoverflow.net/questions/427301 | 3 |
>
> In $\mathbb{F\_p}$, $p$ prime what is the larget subset $S$ of quadratic residues with no pair of elements differing by 1?
>
>
>
In [this](https://mathoverflow.net/q/427035/7113) related question Seva gives an example:
"...assuming $p\equiv\pm3\pmod 8$, consider the set of all those quadratic residues $r$ ... | https://mathoverflow.net/users/7113 | Largest subset of quadratic residues with no pair of elements differing by 1 | No.
Let $T$ be any subset of $\mathbb F\_p$. We can consider the general problem of finding a set $S \subseteq T$ with no pair of elements differing by $1$ that maximizes $|S|$.
The same construction works, i.e. $\{x \in T \mid x-y$ is odd for $y$ the largest nonmember of $T$ less than $x \}$ attains the maximal ca... | 2 | https://mathoverflow.net/users/18060 | 427305 | 173,346 |
https://mathoverflow.net/questions/427274 | 5 | Consider a smooth 2d-manifold $M$ and let $g$ be a smooth $(0,2)$-tensor satisfying $rk(g)\geq1$ everywhere. Obviously if $rk(g)=2$ at a point $p\in M$ then $g$ is locally diagonalisable (i.e. there exists a local coordinate system in which $g$ is diagonal). The same conclusion holds if $rk(g)=1$ in a neighbouhood of a... | https://mathoverflow.net/users/153070 | Local diagonalisation of a degenerated 2d metric tensor | The answer is 'yes'. Here is how one can see this: Suppose that $g$ is a $(0,2)$ form on a neighborhood of the origin in the $xy$-plane such that the rank of $g$ is $1$ at the origin and $2$ everywhere else. Let $h = \mathrm{d}x^2 + \mathrm{d}y^2$ and let
$$
g = E(x,y)\,\mathrm{d}x^2 + 2 F(x,y)\,\mathrm{d}x\,\mathrm{d}... | 11 | https://mathoverflow.net/users/13972 | 427307 | 173,347 |
https://mathoverflow.net/questions/427302 | 4 | In category theory, a dagger category is a precategory $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^{\dagger\_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that
* for all objects $A:\mathcal{C}$, $\mathrm{id}\_A^{\dagger\_{A,A}} = \mathrm{id}\_A... | https://mathoverflow.net/users/483446 | $(n,1)$-dagger categories | Well, it is easy to give definitions, the problem is finding the "right" one.
Here "right" can mean that gives the correct notion up to homotopy (many definition will be equivalent) but also it can mean the one that let you the most easily talk about the examples and constructions you care about ( this much more subj... | 12 | https://mathoverflow.net/users/22131 | 427322 | 173,355 |
https://mathoverflow.net/questions/427330 | 1 | Let
* $sd : sSet \to sSet$ denote barycentric subdivsion;
* $cosk\_1 : sSet \to sSet$ denote 1-coskeletalization.
**Question:** Let $X$ be a graph or simplicial set. If the homotopy type of $cosk\_1(X)$ is known, then what can be said about the homotopy type of $cosk\_1(sd(cosk\_1(X)))$?
I'm interested in this qu... | https://mathoverflow.net/users/2362 | Barycentric subdivision and 1-coskeletalization | I don't know anything about simplicial sets, but I think the question for simplicial complexes is easy.
A complex is said to be ["flag"](https://en.wikipedia.org/wiki/Clique_complex) (also known as a clique complex) if every subset $S$ for which every pair $u, v\in S$ form an edge is in fact a face. Evidently a flag ... | 2 | https://mathoverflow.net/users/25028 | 427338 | 173,356 |
https://mathoverflow.net/questions/427313 | 7 | $A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, the second lies in the $y=0$ plane and its center is the point $(1,0,0)$.)
**Is it true that the circles $A$ and $B$ can... | https://mathoverflow.net/users/486410 | Is it possible to separate two linked (geometric) circles in $\Bbb R^3$ by a set homeomorphic to the 2-sphere (with arbitrarily “bad” homeomorphism)? | By Alexander duality, $\mathbb{S}^3\setminus S$ has two connected components with trivial homologies.
On the other hand, $A$ is nontrivial in $H\_1(\mathbb{S}^3\setminus B)$ — a contradiction.
| 9 | https://mathoverflow.net/users/1441 | 427339 | 173,357 |
https://mathoverflow.net/questions/427156 | 2 | I'm reading a proof of **Theorem 1.37** from Santambrogio's *Optimal transport for applied mathematicians: calculus of variations, PDEs, and modeling*. First, I quote related definitions. Let $X,Y$ be Polish spaces and $\overline{\mathbb{R}} := \mathbb R \cup \{\pm \infty\}$.
---
* **Definition 1.10.** Given a fu... | https://mathoverflow.net/users/477203 | Optimal transport for applied mathematicians: how does $\varphi (x) = \inf_{y \in Y} [c(x, y) - \psi (y)] \neq -\infty$ follow in Theorem 1.37? | I hope I did not misunderstand the question, but it seems $\varphi(x) > - \infty$ holds as follows if $(x, y) \in \Gamma$:
For any $(x\_i, y\_i) \in \Gamma$, $i=1, \dots, n$, we see that
\begin{align}
&c(x, y\_{n}) - c(x\_n, y\_n) + \sum\_{i=0}^{n-1} c(x\_{i+1}, y\_i) - c(x\_i, y\_i) \\
&= c(x, y\_{n}) - c(x\_n, y\_n... | 2 | https://mathoverflow.net/users/106046 | 427350 | 173,359 |
https://mathoverflow.net/questions/427351 | 3 | I just want to know if Tangled Type Theory $\mathsf{TTT}$ of Randall Holmes ([see: [Holmes - NF is consistent, p:11](https://arxiv.org/abs/1503.01406), [Holmes - The equivalence of NF-style set theories with “tangled” type theories; the construction of $\omega$-models of predicative NF (and more), p:4-5](https://randal... | https://mathoverflow.net/users/95347 | Can we write Tangled Type Theory without reference to type sequences? | No it is not the same. The type sequences are absolutely needed, in some form.
--Randall Holmes (author of the theory in question)
The theory you describe is provably inconsistent.
The formula has to be the result of replacing the variables in an axiom of ordinary TST (indexed by type in the usual order) with var... | 9 | https://mathoverflow.net/users/485780 | 427355 | 173,360 |
https://mathoverflow.net/questions/403896 | 1 | Let $m = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
It is [known](https://mathoverflow.net/questions/280897) that
$$\gcd(\sigma(q^k),\sigma(n^2)) = \frac{(\gcd(n,\sigma(n^2)))^2}{\gcd(n^2,\sigma(n^2))}$$
and therefore that
$$\gcd(\sigma(q^k),\s... | https://mathoverflow.net/users/10365 | On odd perfect numbers and a GCD - Part III | Let $p^s Q^2$ be an odd perfect number with special prime $p$ satisfying $p \equiv s \equiv 1 \pmod 4$ and $\gcd(p,Q)=1$.
I did some more digging on when the equations
$$\gcd(Q^2, \sigma(Q^2)) = \gcd(\sigma(Q^2), \sigma(p^s))$$
$$\gcd(Q, \sigma(Q^2)) = \gcd(Q^2, \sigma(Q^2))$$
$$\gcd(\sigma(Q^2), \sigma(p^s)) = \gc... | 0 | https://mathoverflow.net/users/10365 | 427357 | 173,361 |
https://mathoverflow.net/questions/427346 | 3 | Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a projective variety of dimension bigger or equal to $2$ (see the edit below for comments on the dimensional hypothesis). Ideal... | https://mathoverflow.net/users/37214 | Existence of rigid objects in the derived category of a smooth projective variety | I am writing up as *one* answer the comments by @Johan, by @Libli, and by myself. If either of them prefers to write an answer, I am happy to delete this answer.
Let $A$ be an Abelian variety. For every scheme $S$ and every $S$-valued point $x\in A(S)$, denote by $\mu\_x$ the associated translation automorphism of th... | 1 | https://mathoverflow.net/users/13265 | 427395 | 173,370 |
https://mathoverflow.net/questions/427372 | 2 | Given $L$ variables $k\_i$ where each $k\_{i} \in \{1, 2, 3, \ldots, N\}$ I want to obtain how many different sums $k\_{1}+k\_{2}+\cdots+k\_{L}$ are generated and the value of these sums.
There are $L^N$ possible sums but many give the same result, e.g. for $L=2$ and $N=3$ we have
* 1 + 1 = 2 (one solution that giv... | https://mathoverflow.net/users/486480 | Different sum combinations of $L$ identical lists of consecutive natural numbers | You are asking for the number of compositions of $s$ into $L$ parts,
with largest part at most $N$. This is a classical problem,
equivalent to finding the coefficient of $x^s$ in the polynomial
$(x+x^2+\cdots+x^N)^L$. Write this polynomial as
$x^L(1-x^N)^L/(1-x)^L$. Expand the numerator and denominator by the
binomial ... | 8 | https://mathoverflow.net/users/2807 | 427396 | 173,371 |
https://mathoverflow.net/questions/427394 | 1 | Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2\_k$ via the family $E\to S$ where $E\subset\mathbb{P}^2\_k\times S$ given is by the equation $y^2=x^3+ax+b$.
1. Is this in some mild sense universal? ... | https://mathoverflow.net/users/36563 | What's a right parameter space of abelian varieties over a non algebraically closed fields? | 1. This is true in a Zariski-local sense. Any elliptic curve together with a nowhere vanishing differential can be put in this form. For an elliptic curve over $S$, the differentials form a line bundle over $\operatorname{Spec} S$, so we can get a nowhere vanishing differential over a Zariski open cover where this line... | 8 | https://mathoverflow.net/users/18060 | 427398 | 173,373 |
https://mathoverflow.net/questions/418931 | 5 | Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as *correlation matrices*. From the positivity of the minors, we know that each matrix element $a\_{ij}$ satisfies $-1 \leq a\_{ij} \leq 1$. The problem is the following: considering t... | https://mathoverflow.net/users/479369 | Maximal eigenvalue of a correlation matrix with some entries fixed as zeros | The question can be solved by considering the Lovász number of a graph whose adjacency matrix has entries $0$ if $a\_{ij} \neq 0$ and $1$ everywhere else. This is proven (up to a typo) in Section 33 [here](https://arxiv.org/pdf/math/9312214.pdf).
| 0 | https://mathoverflow.net/users/479369 | 427402 | 173,375 |
https://mathoverflow.net/questions/427403 | 8 | **Background.** Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H\_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the pushforward $\Delta\_\ast\mu \in H\_d(M\times M)$ is defined.
If we apply the intersection pairing to $\Delta\_\as... | https://mathoverflow.net/users/8032 | On the Euler characteristic of a Poincaré duality space | It follows from Poincaré duality in $M\times M$. I'll suppress gradings and be vague about signs here.
Let $x\_i$ be a rational homology basis for $M$. Let $a\_i$ be the cohomology basis that is dual to this basis in the linear algebra sense: $\langle a\_i,x\_j\rangle=\delta\_{ij}$. Let $y\_i$ be the homology basis t... | 14 | https://mathoverflow.net/users/6666 | 427404 | 173,376 |
https://mathoverflow.net/questions/419755 | 5 | Let $R$ be a commutative unital ring and let $M$ be a unital $R$-module. A non-Archimedean ring seminorm on $R$ is a map $|\cdot| \colon R \rightarrow \mathbb{R}\_{\geq 0}$ which satisfies
$$ | 0\_R| = 0, \quad |1\_R| \leq 1, \quad |r - s| \leq \max \{ |r|, |s|\}, \quad |r \cdot s|\leq | r|\cdot |s| $$
for all $r,s \in... | https://mathoverflow.net/users/471381 | An example where the non-Archimedean tensor product of normed modules is only seminormed? | Let $k$ be a field. Pick some $r \in (0,1)$ and let $R$ be the ring $k[[t]]$ endowed with the absolute value that is trivial on $k$ and such that $\vert t\vert =r$. Let $M$ be $k((t))$ endowed with the absolute value that extends that on $R$. Pick $s \in (0,r)$ and let $N$ be the ring $k[[t]]$ endowed with the absolute... | 3 | https://mathoverflow.net/users/4069 | 427406 | 173,377 |
https://mathoverflow.net/questions/427447 | 7 | Let $(\mathcal{X} , \|\cdot \|\_\mathcal{X})$ be a Banach space and $\mathcal{X}'$ its topological dual. We denote by $\| \cdot \|\_{\mathcal{X}'}$ the dual norm and define also the topological dual $\mathcal{X}''$ of the Banach space $(\mathcal{X}',\|\cdot\|\_{\mathcal{X}'})$. The unit ball of $\mathcal{X}'$ is denote... | https://mathoverflow.net/users/39261 | Compactness of the unit ball of a Banach space for topologies finer than the weak* topology | As pointed out by Goulifet, my previous answer was wrong. In fact almost the exact opposite is true: on any dual Banach space there is *no* locally convex vector space topology strictly stronger than the weak${}^\*$ topology that makes the unit ball compact. That's because any stronger topology for which the unit ball ... | 10 | https://mathoverflow.net/users/23141 | 427450 | 173,387 |
https://mathoverflow.net/questions/427474 | 3 | Suppose that $D\_{KL}(p\_1\parallel q)<1$ and $D\_{KL}(p\_2\parallel q)<1$. I'm trying to show that either $D\_{KL}(p\_1\parallel p\_2)$ or $D\_{KL}(p\_2\parallel p\_1)$ will have an upper bound close to 1 provided ${q}$ is fixed. It seems intuitive that if $p\_1$ and $q$ are similar enough and if $p\_2$ and $q$ are al... | https://mathoverflow.net/users/486746 | An inequality relating the Kullback-Leibler divergence of two discrete distributions with constant reference distribution | The answer is no. E.g., let $p\_1,p\_2,q$ be pdf's on $[0,1]$ such that $q=1$, $p\_1=\frac1{1-h}\,1\_{[0,1-h]}$, and $p\_2=\frac1{1-h}\,1\_{[h,1]}$, for some $h\in(0,1)$. Then
$$D\_{KL}(p\_1\parallel q)=D\_{KL}(p\_2\parallel q)=\ln\frac1{1-h}\to0$$
as $h\downarrow0$, whereas
$$D\_{KL}(p\_1\parallel p\_2)=D\_{KL}(p\_2\p... | 4 | https://mathoverflow.net/users/36721 | 427476 | 173,390 |
https://mathoverflow.net/questions/427481 | 0 | Given some variables $x\_1, x\_2, \dots, x\_n$, the Vandermonde determinant is given by
$$V\_n(x\_1,\dots,x\_n):=\det(x\_j^{i-1})\_{i,j=1}^n=\prod\_{i<j}(x\_j-x\_i).$$
One can take as special cases: $x\_j=j$ or $x\_j=q^j$.
I was playing around with the following variation $A\_n=((i+j-1)^{i-1})\_{i,j=1}^n$. It turns o... | https://mathoverflow.net/users/66131 | A variant of numeric Vandermonde which failed symbolically? | Matrix $A\_n$ can be generalized to $\big[(j+f(i))^{i-1}\big]\_{i,j=1}^n$ for any function $f(i)$ not just $f(i)=i-1$.
Since
$$(j+f(i))^{i-1} = \sum\_{k=0}^{i-1} c\_{i,k}\cdot j^k,$$
where $c\_{i,k} := \binom{i-1}k f(i)^{i-1-k}$, we have
$$\big[(j+f(i))^{i-1}\big]\_{i,j=1}^n = \begin{bmatrix}
c\_{1,0} & 0 & 0 & \dot... | 7 | https://mathoverflow.net/users/7076 | 427482 | 173,392 |
https://mathoverflow.net/questions/427465 | 12 | *This is related to a couple recent MO/MSE questions of mine, namely [1](https://mathoverflow.net/questions/427135/what-algebraic-properties-are-preserved-by-mathbbn-leadsto-beta-mathbbn),[2](https://math.stackexchange.com/questions/4498457/which-ultrafilters-are-sort-of-commutators). Belatedly, I've tweaked this post ... | https://mathoverflow.net/users/8133 | Ultrafilter subtraction and "zero" | Both your guesses are correct. To see this, it's helpful to reformulate the way you're thinking about the subtraction operator on $\beta \mathbb Z$. Beginning with subtraction on $\mathbb Z$, you can first extend this to an operator $\beta \mathbb Z \times \mathbb Z \rightarrow \beta \mathbb Z$ by setting $\mathcal U -... | 16 | https://mathoverflow.net/users/70618 | 427486 | 173,393 |
https://mathoverflow.net/questions/427479 | 7 | Let $X$ be a normed vector space with its dual $X^\*$. Let $S^\*$ be the unit sphere of $X^\*$. We have known that if $X$ (or $X^\*$ ) is reflexive then the weak-star and weak topology of $X^\*$ coincide and thus the weak-star closure (or weak closure) of $S^\*$ is the unit ball. I have the following questions:
1. If... | https://mathoverflow.net/users/487196 | Weak star closure of unit sphere in dual space | Suppose the weak$^\*$ sequential closure of $S^\*$ contains $0$. So there is a sequence $(f\_n) \subseteq X^\*$ with $\|f\_n\|=1$ for each $n$, and with $f\_n\rightarrow 0$ weak$^\*$. For $f\in B^\*$ the unit ball, we seek a bounded sequence $(t\_n)$ of scalars such that $f+t\_nf\_n \in S^\*$ for each $n$, as then $f+t... | 3 | https://mathoverflow.net/users/406 | 427491 | 173,394 |
https://mathoverflow.net/questions/427399 | 11 | Let us say that three consecutive positive integers $(m-1,m,m+1)$ have a big prime factor if the largest prime factor $p$ of $N:=(m-1)m(m+1)$ satisfies $e^p>N$.
I ckecked that it is true for all $m<10^8$, except for $m=3,9,15,49,55,99,351,441,2431$ where $p=3,5,7,7,11,11,13,17,19$, respectively.
**Question**: Is it... | https://mathoverflow.net/users/34538 | Do consecutive integers have a big prime factor? | Thanks to Szpiro’s/$abc$ conjecture, the answer to your question is true for sufficiently large $N$, up to a factor of $3$ times the primes $p$ (which factor, I believe, can be removed if one is a bit careful in the estimation of the prime factors of $N$ below $3\log m$ in the proof below). Note that the problem you wi... | 6 | https://mathoverflow.net/users/166628 | 427500 | 173,396 |
https://mathoverflow.net/questions/427484 | 2 | I am reading [a paper by Szekeres and Peters](https://www.cambridge.org/core/journals/anziam-journal/article/computer-solution-to-the-17point-erdosszekeres-problem/0EC7876789232266D60439A4C00D86D9) on computing the 17-point case of [the Erdős–Szekeres conjecture](https://en.wikipedia.org/wiki/Happy_ending_problem). The... | https://mathoverflow.net/users/487612 | Determining if a polygon is convex using relations on orientation of each ordered triple of points | It's important to note that when it talks about *ordered points*, this is ordered by $x$-coordinate and not (as one might otherwise suppose) by traversing the edges of the polygon.
Suppose we have $n$ points, $p\_1$ to $p\_n$, ordered by $x$-coordinate. Observe that $p\_1$ and $p\_n$ are on the convex hull. Now draw ... | 3 | https://mathoverflow.net/users/46140 | 427502 | 173,397 |
https://mathoverflow.net/questions/427499 | 3 | It seems to me that a proof of $\alpha\_{n}=o(n)$ where the quantity $\alpha\_{n}$ is defined in [About Goldbach's conjecture](https://mathoverflow.net/questions/61842/about-goldbachs-conjecture) together with the main result of <https://kyushu-u.pure.elsevier.com/en/publications/exceptional-zeros-and-the-goldbach-prob... | https://mathoverflow.net/users/13625 | Does asymptotic Goldbach imply GRH? | Granville (2008) proved that a sufficiently strong average error term for the Goldbach-Hardy-Littlewood conjecture is equivalent to RH, and a refinement of it is equivalent to GRH. See MR2357316 and MR2492859. A closely related result was proved by Bhowmik-Ruzsa (2018) and Bhowmik-Halupczok-Matsumoto-Suzuki (2018). See... | 12 | https://mathoverflow.net/users/11919 | 427505 | 173,399 |
https://mathoverflow.net/questions/427469 | 3 | Let $X$ be a complex manifold endowed with a holomorphic closed 2-form $\omega$ whose associated map $\omega : TX \to T^\*X$ is invertible. Can we always embed $X$ as an open subset of a compact complex manifold $Y$ endowed with a holomorphic Poisson structure $\pi : T^\*Y \to TY$ such that $\pi|\_X = \omega^{-1}$? We ... | https://mathoverflow.net/users/479013 | When does a holomorphic symplectic manifold compactify to a Poisson manifold? | No, not even under the nicest possible algebraicity assumptions like quasiprojective.
Let $Z$ be the product of two curves of genus $\geq 2$. Choose a nonzero $2$-form $\omega$ on $Z$, the wedge of a nonzero one-form on each of the two curves. Let $X$ be obtained from $Z$ by removing the locus where $\omega$ vanishes... | 8 | https://mathoverflow.net/users/18060 | 427509 | 173,402 |
https://mathoverflow.net/questions/427506 | 1 | Suppose we have i.i.d. samples $x\_i\sim N(0,\Sigma)$ and $y\_i\sim x\_i^T\omega^\*+\xi\_i,\xi\_i\sim N(0,1)$ where $\omega^\*$ is the fixed point of:
$$\omega\_{i+1} = \omega\_i − \eta\nabla\_\omega f(\omega\_i, x\_i
, y\_i), \quad \omega\_0 = 0$$
where $f(\omega, x\_i, y\_i) = \frac{1}{2}|y\_i − \omega^T x\_i|^2 $and... | https://mathoverflow.net/users/348579 | Using gradient descent in probability case | Lets first assume $\zeta\_i=0$ and ask the following:
* under which conditions $w^\*$ is a stable fixed point?
If it's not a stable fixed point for noise-free case, then you won't end up with a fixed mean or stationary distribution after adding additive noise. This means $w^\*$ being a fixed point is a necessary co... | 1 | https://mathoverflow.net/users/7655 | 427511 | 173,403 |
https://mathoverflow.net/questions/427510 | 6 | Let $V$ be a finite dimensional vector space. Let $\Lambda$ be a collection of subspaces of $V$ such that, if $X$ and $Y$ are in $\Lambda$, then $X\cap Y$ and $X+Y$ are in $\Lambda$. This makes $\Lambda$ into a lattice; we impose that $\Lambda$ is a distributive lattice, meaning that
$$X \cap (Y+Z) = (X \cap Y) + (X \c... | https://mathoverflow.net/users/297 | Distributive lattice of subspaces | Just to not leave this open, a proof can be found in Proposition 7.1 of Chapter 1 of Quadratic Algebras by Alexander Polishchuk and Leonid Positselski as alluded to by Mariano on MSE <https://math.stackexchange.com/questions/63493/non-distributivity-of-subspaces>.
| 7 | https://mathoverflow.net/users/15934 | 427513 | 173,404 |
https://mathoverflow.net/questions/425082 | 12 | Some of the earliest writings on the [Joyal model structure on simplicial sets](https://ncatlab.org/nlab/show/model+structure+for+quasi-categories) include Jacob Lurie's account in [Higher Topos Theory](https://arxiv.org/abs/math/0608040v1) from 2006,
as well as Joyal's own account in [The Theory of Quasi-Categories an... | https://mathoverflow.net/users/402 | When did the Joyal model structure on simplicial sets originate? | Here is what André Joyal wrote in an email to me:
>
> No, I have not discovered the model structure for quasi-categories in the 1980's.
> I became interested in quasi-categories (without the name) around 1980
> after attending a talk by Jon Beck on the work of Boardman and Vogt.
> I wondered if category theory coul... | 15 | https://mathoverflow.net/users/402 | 427514 | 173,405 |
https://mathoverflow.net/questions/384709 | 6 | The following identity is not hard to prove:
$$
\sum\_{1\leq i\_1<i\_2<\ldots <i\_{2n}\leq N} (-1)^{i\_1+\ldots+i\_{2n}}\frac{(1-x\_{i\_1})(1-x\_{i\_3})\ldots(1-x\_{i\_{2n-1}})}{(1-x\_{i\_2})(1-x\_{i\_4}) \ldots (1-x\_{i\_{2n}})\ \ }=
\frac{(x\_{1}-x\_2)(x\_{2}-x\_{3}) \ldots (x\_{N-1}-x\_{N})}{(1-x\_{2})\ \ldots\ \ \ ... | https://mathoverflow.net/users/21620 | An identity for rational functions leading to equations for multiple polylogarithms | If we denote $1-x\_i=y\_i$, this reads as $$\sum\_{1\leq i\_1<i\_2<\ldots <i\_{2n}\leq N} (-1)^{i\_1+\ldots+i\_{2n}}\frac{y\_{i\_1}y\_{i\_3}\ldots y\_{i\_{2n-1}}}{y\_{i\_2}y\_{i\_4} \ldots y\_{i\_{2n}}}=
\frac{(y\_{2}-y\_1)(y\_{3}-y\_{2}) \ldots (y\_{N}-y\_{N-1})}{y\_{2}\ldots y\_{N-1}y\_{N}}\\
=(1-y\_1/y\_2)(1-y\_2/y\... | 3 | https://mathoverflow.net/users/4312 | 427523 | 173,406 |
https://mathoverflow.net/questions/427462 | 3 | Work in ZFC with no large cardinal assumptions. Say that a (parameter-definable) class $X \subseteq ORD$ is *club* if it is closed and unbounded in the sense that:
1. For each $\beta \in ORD$, there exists $\gamma \geq \beta$ such that $\gamma \in X$, and
2. For each $\delta, \epsilon \in ORD$, and each increasing fu... | https://mathoverflow.net/users/2362 | Is the definable club filter normal? | Work in ZF. As Asaf was probably mentioning in his comment, you have asked two distinct questions, one in the title (referring to a filter), and one in the main body of the question (just referring to clubs). Moreover, the notion asked about in the title hasn't been defined clearly. Given what you wrote in the body of ... | 6 | https://mathoverflow.net/users/160347 | 427524 | 173,407 |
https://mathoverflow.net/questions/427153 | 1 | Let $W$ be a standard one dimensional Brownian motion, and consider the SDE
$$dX\_t = \sigma(X\_t) \, dW\_t, \, \, \, X\_0 = 1 \, \text {a.s.}$$
Assume $\sigma$ is regular enough that the above SDE admits a globally defined solution.
Suppose $|\sigma(x)| \to 0$ as $x \to 0$.
**Question:** Is it true that almost... | https://mathoverflow.net/users/173490 | Is the solution to this SDE always positive? | To complement Nawaf's answer ([1](https://mathoverflow.net/questions/427153/is-the-solution-to-this-sde-always-positive#comment1098413_427153) [2](https://mathoverflow.net/questions/427153/is-the-solution-to-this-sde-always-positive#comment1098415_427153)), I thought I'd present a short argument how with some additiona... | 3 | https://mathoverflow.net/users/80052 | 427532 | 173,410 |
https://mathoverflow.net/questions/427544 | 6 | Let $X$ be an object in one of the well-known symmetric monoidal model categories of spectra. E.g., an $\mathtt S$-module in the sense of EKMM, or an orthogonal spectrum, or a $\Gamma$-space, etc.
One can form the spectrum $X\wedge X$, which is a "naive" spectrum with an action of $\Sigma\_2$, and one can take the ca... | https://mathoverflow.net/users/6668 | What are the naive fixed points of a non-naive smash product of a spectrum with itself? | In both the orthogonal world and the EKMM world one can set things up so that $G$-spectra are just spectra with an action of $G$, and the naive and genuine equivariant stable categories are the homotopy categories for two different model structures on the same underlying geometric category. I will take that point of vi... | 7 | https://mathoverflow.net/users/10366 | 427546 | 173,413 |
https://mathoverflow.net/questions/427517 | 2 | Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
$$
\Omega(x^L, y^L, z^L) = \langle x, [y, z] \rangle,\quad (x,y,z \in \mathrm{Lie}(G))
$$
where $\langle,\rangle$ is th... | https://mathoverflow.net/users/385475 | Is the restriction of the Cartan 3-form on conjugacy classes exact? | Yes, it is exact, and there is in fact a canonical 2-form on each conjugacy class whose derivative is your $\Omega$. This was an important observation when studying D-branes in WZW models, see, e.g. <https://arxiv.org/abs/hep-th/0008038> or <https://arxiv.org/abs/hep-th/0205233>.
| 4 | https://mathoverflow.net/users/3473 | 427570 | 173,417 |
https://mathoverflow.net/questions/427549 | 7 | In the algebraic group $G = \operatorname{PGL}\_4(\mathbb{C})$, let $E$ denote the subgroup of elements of order dividing 2 in the diagonal maximal torus; it is generated by the images of the three matrices
$$
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
\end{bmatrix},\quad
\b... | https://mathoverflow.net/users/488802 | $N_{G}(E)/C_{G}(E)$ is the Weyl group of $G$? | $\newcommand{\ZZ}{{\mathcal Z}\_G}
\newcommand{\NN}{{\mathcal N}\_G}
\newcommand{\zz}{{\mathfrak z}\_G}
\newcommand{\Lie}{{\rm Lie\,}}
\renewcommand{\tt}{{\mathfrak t}}
\renewcommand{\gg}{{\mathfrak g}}
\newcommand{\X}{{\sf X}}
\newcommand{\Z}{{\Bbb Z}}$
**Yes,** this is true for any $n\ge 3$.
Let $G$ be a semisimple... | 9 | https://mathoverflow.net/users/4149 | 427574 | 173,419 |
https://mathoverflow.net/questions/427575 | 15 | **Motivation.** As I was travelling in the UK, I used a physical copy of the "A-Z Road Atlas BRITAIN" for getting around. I was impressed that whenever I wanted to go from the map segment shown on page 23, say (showing a part of the West Midlands), to the segment east of the one I was looking at, I only had to turn une... | https://mathoverflow.net/users/8628 | Page-turning number of a graph | The page-turning number of a graph $G$ is also known as the *bandwidth* of $G$ (<https://en.wikipedia.org/wiki/Graph_bandwidth>).
The Wikipedia page also contains values of the bandwidth for some special graphs, including the $m\times n$ grid graph, in which case it is equal to $\min\{m,n\}$. This was proved by Jarmila... | 17 | https://mathoverflow.net/users/24076 | 427582 | 173,422 |
https://mathoverflow.net/questions/427558 | 1 |
>
> What concepts in the real world can be described by adjunctions?
>
>
>
For example, parents and children are adjoint to one another. Specifically, work in $ZFC$ plus a finite class of atoms $\mathscr{X}$ (one for each person) and let $${\sf Par}:\mathcal{P}(\mathscr{X})\to\mathcal{P}(\mathscr{X})$$ $${\sf Ch... | https://mathoverflow.net/users/92164 | Adjunctions in the real world | If you're willing to admit your example as "describing a concept by an adjunction", then I would argue that *any* binary relation can be "described by an adjunction", namely [the Galois connection that it induces](https://ncatlab.org/nlab/show/Galois+connection#InducedFromARelation).
To be sure, your example isn't qu... | 5 | https://mathoverflow.net/users/49 | 427587 | 173,424 |
https://mathoverflow.net/questions/427551 | 1 | Consider $N \times N$ matrices
$$A = \begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & 0 & & 0 \\
\vdots & 1 & 0 & \ddots & \vdots \\
0 & & \ddots & \ddots & 0 \\
0 & 0 & \cdots &1 & 0 \\
\end{bmatrix}$$
and
$$B=\operatorname{diag}( \cos(2\pi\cdot 0/N),...,\cos(2\pi\cdot (N-1)/N)).$$
Does anybody know why the eig... | https://mathoverflow.net/users/119875 | Eigenvalues invariant under 90° rotation | Assume $N$ is even (this is false when N is odd).
Let $X=2B, Y=A+A^T$.
Let
$$P = \begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & \zeta & \zeta^2 & \cdots & \zeta^{N-1} \\
1 & \zeta^2 & \zeta^4 & \cdots & \zeta^{2(N-1)} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & \zeta^{N-1} & \zeta^{2(N-1)} & \cdots & \zeta^{(N-... | 5 | https://mathoverflow.net/users/160416 | 427608 | 173,431 |
https://mathoverflow.net/questions/425687 | 4 | We call a ring extension (where $R$ and $S$ are commutative) $R \subset S$ essential if for every ideal $I$ of $S$ we have
that $I \cap S \neq 0 \implies I \cap R \neq 0$.
Suppose now that $R \subset S$ is an extension that is flat. Can we find a ring homomorphism $S \to T$ such that the composite $R \to S \to T$ is ... | https://mathoverflow.net/users/145417 | Flat essential ring extensions | **Edit**: I have added some additional remarks to the answer below.
As already [mentioned](https://mathoverflow.net/questions/425687/flat-essential-ring-extensions#comment1099107_425687), when $R$ is a domain and $S$ is a flat overring, we can find $S \to T$ such that $T$ is an overring of $R$ which is both flat and ... | 2 | https://mathoverflow.net/users/31923 | 427609 | 173,432 |
https://mathoverflow.net/questions/427585 | 6 | The Chebyshev polynomial $U\_n(x)$ of the second kind is characterized by
$$
U\_n(\cos\theta)=\frac{\sin(n+1)\theta}{\sin(\theta)}.
$$
It seems that
$$\operatorname\*{Res}\_x \left( U\_n(x)+tU\_{n-1}(x),\sum\_{k=0}^{n-1}U\_k(x) \right) =(-1)^{\frac{n(n-1)}{2}} t^{\left\lfloor\frac{k}{2} \right\rfloor}2^{n(n-1)},$$
wher... | https://mathoverflow.net/users/120597 | Resultant of linear combinations of Chebyshev polynomials of the second kind | Since $U\_n + t U\_{n-1}$ is of degree $n$ and $\sum\_{k=0}^{n-1} U\_k$ is of degree $n-1$ with leading coefficient $2^{n-1}$, the resultant factors as
$$ 2^{n(n-1)} (-1)^{n(n-1)} \prod\_{j=1}^{n-1} (U\_n(x\_j) + t U\_{n-1}(x\_j))$$
where $x\_1,\dots,x\_{n-1}$ are the zeroes of $\sum\_{k=0}^{n-1} U\_k$.
Fortunately, ... | 12 | https://mathoverflow.net/users/766 | 427618 | 173,433 |
https://mathoverflow.net/questions/427273 | 10 | It is known from Serre's classical result that every p-torsion occurs in the homotopy groups of every sphere. Is it known: do elements of every order occur in homotopy groups of spheres?
| https://mathoverflow.net/users/148161 | Do elements of every order occur in homotopy groups of spheres? | As others suggested, I am posting my earlier [comment](https://mathoverflow.net/questions/427273/do-elements-of-every-order-occur-in-homotopy-groups-of-spheres#comment1098754_427273) as an answer:
---
$\DeclareMathOperator\denom{denom}$Sure. Let $n$ be a positive integer. Let $N$ be the product of $2n$ and Euler ... | 17 | https://mathoverflow.net/users/nan | 427625 | 173,435 |
https://mathoverflow.net/questions/427520 | 3 | Let $0<\theta\_1,\theta\_2<\pi/2$. Suppose $\psi$ is a smooth real-valued function with compact support.
Consider the oscillatory integral
$$I(t):=\int\_{0}^{1}\frac{1}{(y-e^{\dot{\imath}\theta\_1})
(y-e^{\dot{\imath}\theta\_2})}\int\_{\mathbb{R}}\psi(x)e^{\dot{\imath} t x^2y^2}dx dy.$$
I am trying to find the asympt... | https://mathoverflow.net/users/116555 | Asymptotic behavior of a double oscillatory integral | $\newcommand{\R}{\mathbb R}\newcommand\sgn{\operatorname{sgn}}\newcommand{\vpi}{\varphi}$Obviously, for $a:=\sqrt{2\pi}\,\psi(0)$ we have
\begin{equation\*}
\psi(0)=f(0),
\end{equation\*}
where
\begin{equation\*}
f:=a\vpi
\end{equation\*}
and $\vpi$ is the standard normal density. Letting now $g(x):=\frac{\psi(x)-f(... | 6 | https://mathoverflow.net/users/36721 | 427631 | 173,438 |
https://mathoverflow.net/questions/427620 | 13 | Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$.
>
> What can be said about $X$?
>
>
>
Is there a special term for such a property? Any characterizations in terms of the inner struc... | https://mathoverflow.net/users/53155 | When can we divide continuous functions? | The kind of completely regular space you are looking for is an $F$-space.
Suppose that $X$ is a completely regular space. Then we say that a subset $A\subseteq X$ is $C^\*$-embedded if whenever $f:A\rightarrow\mathbb{R}$ is bounded and continuous, there is a continuous $g:X\rightarrow\mathbb{R}$ with $g|\_A=f$. We sa... | 16 | https://mathoverflow.net/users/22277 | 427638 | 173,440 |
https://mathoverflow.net/questions/426630 | 8 | Let $\Omega$ be a bounded domain with a Lipschitz boundary. Consider the Dirichlet-to-Neumann map $\Lambda:H^{\frac{1}{2}}(\partial \Omega)\to H^{-\frac{1}{2}}(\partial \Omega)$ defined via
$$ \langle \Lambda f, h\rangle = \int\_{\Omega} \nabla u\cdot \nabla v \, dx,$$
for any $f,h \in H^{\frac{1}{2}}(\partial \Omega)$... | https://mathoverflow.net/users/50438 | Dirichlet-to-Neumann map on Lipschitz domains | Let $u$ be the solution of the Dirichlet problem for Laplacian in a Lipschitz domain with boundary data $g$. Then, for every $s\in [1/2,3/2]$,
$$
\| u \|\_{H^{s}\,(U)} \leq C \| g \|\_{H^{s-1/2}\,\,\,\,(\partial U)} .
$$
This is a classical result of [Jerison & Kenig](https://www.sciencedirect.com/science/article/pii/S... | 4 | https://mathoverflow.net/users/5678 | 427643 | 173,442 |
https://mathoverflow.net/questions/427661 | 1 | It is stated in [Caruso - An introduction to $p$-adic period rings](https://arxiv.org/abs/1908.08424) (the remarks following equation *(2)*) that the $p$-adic étale cohomology groups of an algebraic variety $X$ over a finite extension $K$ of $\mathbb{Q}\_p$ are not $\mathbb{C}\_p$-admissible. In fact, $\mathbb{C}\_p\ot... | https://mathoverflow.net/users/170999 | $p$-adic étale cohomology groups are not $\mathbb{C}_p$-admissible | The characterization of $\mathbb C\_p$-admissibility in the notes is that the action of the inertia group factors through a finite quotient.
A standard calculation in étale cohomology shows that for $\mathbb P^1$, the action of the Galois group on $H^2( \mathbb P^1\_{\overline{K}}, \mathbb Q\_p)$ is by the inverse of... | 4 | https://mathoverflow.net/users/18060 | 427665 | 173,445 |
https://mathoverflow.net/questions/427343 | 3 | We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \mathbb{P}^2$ be a plane quintic curve describing the locus of degenerate conics of $f$, and let $\pi: \widetilde{C} \to C$ b... | https://mathoverflow.net/users/160814 | Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two? | I'll show the rank is at most 4 in characteristic 2, modulo some claims that I think can be justified.
Your exact sequence
$$0 \to \mathcal{O}\_C \to \pi\_\*\mathcal{O}\_{\widetilde{C}} \to \mathcal{O}\_C
\to 0$$
induces a long exact sequence on cohomology
$$0 \to H^0(C, \mathcal{O}\_C(1) ) \to H^0(C, \pi\_\*\... | 2 | https://mathoverflow.net/users/18060 | 427668 | 173,446 |
https://mathoverflow.net/questions/426076 | 7 | Let $\mathbb P$ be a forcing that does not collapse $\omega\_1$, $\theta$ sufficiently large and regular and $X\prec H\_\theta$ a countable elementary substructure with $\mathbb P\in X$ as well as $p\in \mathbb P\cap X$. This is a typical situation if one deals with (semi)proper forcings. Let $\delta=X\cap\omega\_1$. A... | https://mathoverflow.net/users/125703 | For which class of forcings does the "name dichotomy" hold? | It turns out that it is not provable in $\mathrm{ZFC}$ that even $\sigma$-closed forcings satisfy the name dichotomy. The answer to all three questions is thus no (unfortunately).
I will show that under $V=L$, $\mathbb P:=\mathrm{Add}(\omega\_1, 1)$ does not satisfy the name dichotomy:
Consider the function $f:\ome... | 0 | https://mathoverflow.net/users/125703 | 427669 | 173,447 |
https://mathoverflow.net/questions/427673 | 4 | I am reading the survey paper: "The de-Rham Witt complex and Crystalline cohomology" by Luc Illusie.
In math line (2.1.12), Illusie considers the pairing $\langle-,-\rangle:\Omega\_{X/S}^1\times T\_{X/S}\longrightarrow \mathcal{O}\_X$ of the tangent and cotangent bundles on a scheme $X$ of characteristic $p$, relativ... | https://mathoverflow.net/users/174655 | Pairing of cotangent and tangent bundles | For (1), recall that if $R$ is a ring, then a derivation $D: R \to R$ satisfies the Leibniz rule, which by induction on $n$ implies that if $D^n$ denotes the $n$-fold iterate of $D$, then
$$D^n(fg) = \sum\_{i=0}^n \binom{n}{i} D^i(f) D^{n-i}(g).$$
Since $\binom{p}{i} \equiv 0$ for $0<i<p$, this implies that if $R$ is a... | 8 | https://mathoverflow.net/users/102390 | 427677 | 173,448 |
https://mathoverflow.net/questions/427453 | 7 | Convex polyhedra are rigid by Cauchy’s theorem. Steffen’s polyhedron is an example of a non-convex polyhedron that is flexible (i.e., non-rigid). However, it appears to have edges of different lengths. My question: are there flexible polyhedra with equilateral triangular faces? I am interested in both finite and (non-t... | https://mathoverflow.net/users/486571 | Are polyhedra with equilateral triangular faces rigid? | This depends on how do you define a "polyhedron". If you accept a doubly covered lozenge (two copies of two adjacent equilateral triangles), then no. But under reasonable nondegeneracy conditions the answer is mostly likely yes. We worked on this problem in our recent [Domes over curves](https://arxiv.org/abs/2005.0255... | 8 | https://mathoverflow.net/users/4040 | 427679 | 173,450 |
https://mathoverflow.net/questions/427657 | 2 | Let $M$ be a compact Riemannian manifold. I'll call a set of non-empty subsets $C\_1,\dots,C\_N$ a *geodesic tiling* of $M$ if:
* Each $C\_n$ is closed (geodesically) convex hull of a finite number of $\{p\_{i,n}\}\_{i=1}^{k\_n}$ in $C\_n$; i.e. the smallest geodesically convex subset of $M$ containing the points $\{... | https://mathoverflow.net/users/36886 | Do all compact manifolds admit geodesic tiling | If dimension $=2$, then "yes".
If dimension $\ge 3$, then "no".
If such a tiling would exist, then the common boundary of a pair of $C$'s would contain a geodesic hypersurface, but generic Riemannian manifold does not have such hypersurfaces.
For more on convex hulls in Riemannian world, see [our paper](https://arx... | 3 | https://mathoverflow.net/users/1441 | 427680 | 173,451 |
https://mathoverflow.net/questions/427594 | 7 | Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is *quasi-regular* if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $A\_n\varphi(x)$ of Birkhoff averages converges as $n\to\infty$, where $$A\_n\varphi(x)=\frac1n\sum\_{j=0}^{n-1} \varphi... | https://mathoverflow.net/users/24676 | Are all quasi-regular points on Polish spaces generic points? | $A\_n \varphi (x) = \int\_X \varphi \mathrm{d} \mu\_n$ for a Borel probability measure $\mu\_n$, in fact a measure with finite support
$\{ T^j(x) \mid j=0,1,\ldots,n-1\}$.
By the assumption about the point $x$, the sequence of $\mu\_n$ is weakly Cauchy, where "weakly" refers to the duality between the space $M\_\sigma(... | 4 | https://mathoverflow.net/users/95282 | 427690 | 173,453 |
https://mathoverflow.net/questions/427686 | 6 | This starts with a vaguely-recalled result (which may be false!): that if $\mathcal{U}$ is a measure on the least measurable cardinal $\kappa$, then every elementary $j: L[\mathcal{U}]\rightarrow M$ such that $(j,M)$ is first-order definable over $L[\mathcal{U}]$ with parameters from $L[\mathcal{U}]$ "comes from" itera... | https://mathoverflow.net/users/8133 | Can there be no complexity bound on the definable elementary $V\rightarrow M$? | Yes, this is possible. If there is a proper class of measurable cardinals and $V = \text{HOD}$, then any class of ordinals $A$ is definably encoded by the iterated ultrapower $j\_A : V\to M$ that hits the $\text{HOD}$-least measure on the $\alpha$-th measurable iff $\alpha\in A$. You can recover $A$ by looking at the g... | 7 | https://mathoverflow.net/users/102684 | 427700 | 173,456 |
https://mathoverflow.net/questions/427708 | 3 | **Starting point.** The struggle for a magic square consisting of distinct square numbers is still ongoing, but it has produced an amusing landmark result called the [Parker square](https://www.bradyharanblog.com/the-parker-square). One of the issues is that square numbers are very scarce - which leads to the following... | https://mathoverflow.net/users/8628 | $3\times 3$ magic squares consisting of entries of a dense set $D\subseteq \mathbb{N}$ | Yes.
By Szemerédi's theorem, your set contains an arithmetic progression of arbitrary length. In particular, it contains a progression of length 9, say it's $d\_1,\ldots,d\_9$. Then
$$
\begin{pmatrix}
d\_2 & d\_7 & d\_6\\
d\_9 & d\_5 & d\_1\\
d\_4 & d\_3 & d\_8
\end{pmatrix}
$$
is a magic square.
| 10 | https://mathoverflow.net/users/101078 | 427709 | 173,458 |
https://mathoverflow.net/questions/280429 | 9 | Given $n\in\mathbb{N}$, consider the numbers $\{1,\ldots,n^2\}$ and a permutation $\pi\in S\_{n^2}$. It induces pairs $(1,\pi(1))$, $\ldots$, $(n^2,\pi(n^2))$.
Consider an $n\times n$ grid. How many possibilities are there to place the pairs in the grid such that each row grows from left to right in the first value o... | https://mathoverflow.net/users/41187 | Bound on number of nxn grids with lexicographical ordering / poset structure | We have indeed found a proof based on an injection of the set of valid placements into the set of valid placements of the identity permutation. Find details in Theorem 9 of Version 4 of this ArXiv paper: <https://arxiv.org/pdf/1710.03435.pdf>.
| 0 | https://mathoverflow.net/users/41187 | 427712 | 173,461 |
https://mathoverflow.net/questions/427728 | 2 | There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is:
Theorem: If polynomial $P(x,y)$ with rational coefficients is irreducible over ${\mathbb Q}$ but not absolutely irreducible, then the equation $P(x,y)=0$ has at most finitely many rational solutio... | https://mathoverflow.net/users/89064 | Rational solutions to $P(x,y)=0$ for $P$ reducible over ${\mathbb C}$ | The main idea of the proof already appears in what you've written, but here are some more details.
Factor $P(x,y) = Q\_1(x,y) \cdots Q\_n(x,y)$ into irreducibles, where the factorization takes place over a number field $K \supsetneq \mathbb Q$. Let $(a,b)$ be a rational solution to $P(x,y) = 0$. Then $Q\_i(a,b) = 0$ ... | 4 | https://mathoverflow.net/users/66491 | 427741 | 173,470 |
https://mathoverflow.net/questions/427704 | 33 | I submitted a short paper and received a positive review and a negative review. The editor (he) briefly wrote the following things:
1. He thinks my original result could be mistaken because of XYZ
2. He presents an alternative theorem (not entirely in mathematical language but with a combination of math and English),... | https://mathoverflow.net/users/122649 | The editor wrote the paper for me | [Comments combined into a community wiki answer.]
*Copyright* is the wrong word in this context; the correct word is *authorship*. A reasonable course of action is to propose to the editor that you and he be coauthors of the paper. If the editor agrees, then the paper would need to be re-submitted to a different jour... | 56 | https://mathoverflow.net/users/3106 | 427744 | 173,472 |
https://mathoverflow.net/questions/427380 | 6 | Let $\Gamma=(V,E)$ be a finite connected graph.
*Pretty standard notation.* Given a set $S\subset V$, write $\Gamma|\_S$ for the restriction of $\Gamma$ to $S$, i.e., the subgraph $(S,\{\{v,w\}\in E: v,w\in S\})$. Write $\partial\_{\textrm{edge}} S$ for the set of edges $\{v,w\}\in E$ with $v\in S$, $w\notin S$. By $... | https://mathoverflow.net/users/398 | Existence of connected set with large edge boundary | If I am not mistaken, then the following recursive construction disproves statement A:
* Let $G\_1$ consist of a $K\_8$ and an additional vertex $r\_1$ connected to half of the vertices of this $K\_8$
* For $k > 1$ take $2^{2^k}$ copies of $G\_{k-1}$, connect each pair of copies of $r\_{k-1}$ by an edge, and add a ve... | 4 | https://mathoverflow.net/users/97426 | 427750 | 173,473 |
https://mathoverflow.net/questions/427739 | 1 | Suppose we are given a reductive group $G$, its closed subgroup $H$ (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that
(1) The $G$ action on $X$ is free and each orbit is closed.
(2) For $g\in G$, $gY=Y$ if and only if $g\in H$ if and only if $gY\cap Y\neq \emptyset$.
(... | https://mathoverflow.net/users/488976 | Free closed group action on varieties | It is true if $X$ is normal and $Y$ is irreducible: Let $S:=X//G$. Since $X$ is affine and $G$ is reductive, the morphism $X\to S$ is a principal $G$-bundle (consequence of Luna's slice theorem, see Luna's original paper). This implies that also the geometric quotient $T:=X/H$ exists and $X\to T$ is a principal $H$-bun... | 2 | https://mathoverflow.net/users/89948 | 427754 | 173,475 |
https://mathoverflow.net/questions/427707 | 2 | Let $\{X\_k\}$ be a sequence of random variables, with $X\_k\in\{+1, -1\}$ for $k>0$, generated as follows.
First, define $S\_n=X\_1+\dots +X\_n$, with $X\_0=S\_0=0$, and let $0<\beta<\frac{1}{2}$.
Then, if $|S\_n|<n^\beta$, let
$P[X\_{n+1}=1]= P[X\_{n+1}=-1]=\frac{1}{2}$. Otherwise:
* If $S\_n> n^\beta$, then $P[X... | https://mathoverflow.net/users/140356 | Bernoulli trials with small dependencies: asymptotics (central limit theorem, law of the iterated logarithm) | For $0<\beta \le 1/2$, any limiting distribution of the rescaled process $S\_n/n^\beta$ will be fully supported in $[-1,1]$, so it will not be normal.
The downward drift will imply (using Hoeffding's inequality and a union bound over the largest $m<n$ such that $S\_m<m^\beta$) that $P(S\_n>n^\beta+k) \le Ce^{-c\_\eps... | 1 | https://mathoverflow.net/users/7691 | 427771 | 173,480 |
https://mathoverflow.net/questions/427769 | 3 | Let $X$ be a compact manifold with boundary $\partial X$ with $
\dim X\setminus \partial X=n$. Moreover, $X$ and $\partial X$ are both aspherical. Then what's the $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))$, where each $\Sigma $ is a boundary component? By Lefschetz duality, $H^n(X,\partial X)=\mathbb Z$. I am t... | https://mathoverflow.net/users/104837 | Cohomology of the coned off space | You're going to need some more assumptions about your manifold if you want $H^n(X,\partial X)=\mathbb{Z}$ (think about connectedness and orientability).
In general (even without those assumptions), $H^n(X\cup\_{\Sigma\subset \partial X} C(\Sigma))\cong H^n(X,\partial X)$ for $n>1$:
Using the long exact sequence of ... | 4 | https://mathoverflow.net/users/6646 | 427778 | 173,482 |
https://mathoverflow.net/questions/427777 | 0 | In a research paper, when one quotes a numbered equation, is it necessary to put the definite article *the* in front of *equation* like "the equation (3.2)", or is it ok to omit *the* and simply write "equation (3.2)"?
| https://mathoverflow.net/users/32746 | How to quote an numbered equation in a paper | In ordinary English it is correct to omit the definite article 'the' because 'equation (3.2)' serves as the proper name for the equation.
| 3 | https://mathoverflow.net/users/124943 | 427779 | 173,483 |
https://mathoverflow.net/questions/421450 | 2 | $\newcommand{\suc}{\operatorname{succ}}\newcommand{\IsPrime}{\operatorname{IsPrime}}$I'm self learning Homotopy type theory reading the HoTT book. I understand that if $A+B$ and $\neg A :\equiv A\rightarrow 0$ are inhabited, then there is an inhabitant of $B$.
**Proof** : Compose $A\rightarrow 0$ with $0\rightarrow B... | https://mathoverflow.net/users/33937 | Homotopy type theory : how to disprove that $0=\operatorname{succ}(0)$ in the type $\mathbb{N}$ | $\def\opn{\operatorname}$
$\def\mb{\mathbb}$
To prove this result, we shall define the observational equality principle for $\mb{N}$.
---
The observational equality $\opn{Eq}\_{\mb{N}} : \mb{N} \rightarrow \mb{N} \rightarrow \mathcal{U}$ satisfies the following equations :
\begin{align}
\opn{Eq}\_{\mb{N}}(0\_\mb{... | 2 | https://mathoverflow.net/users/368143 | 427786 | 173,484 |
https://mathoverflow.net/questions/427675 | 3 | $\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a uniformizer $\pi$. If $Y=\Sp(B)\subset X$ is a connected affinoid subdomain, the conditions above imply that the rings of pow... | https://mathoverflow.net/users/476832 | Bounded torsion of quotients of affine formal models | As it turns out, the quotient $C$ does not necessarily have bounded $\pi$-torsion. For example, let $X=Sp(K\langle t\rangle)$ and $Y=Sp(K\langle \frac{t}{\pi}\rangle)$ be the disc of radius $\pi$. Then all the elements of the form $\frac{t^{n}}{\pi^{n}}$ are power bounded in $K\langle \frac{t}{\pi}\rangle$. Furthermore... | 1 | https://mathoverflow.net/users/476832 | 427789 | 173,486 |
https://mathoverflow.net/questions/427804 | 29 | One has the nice identities
$${xy\choose 1}={x\choose 1}{y\choose 1},$$
$${xy+1\choose 2}={x+1\choose 2}{y+1\choose 2}+{x\choose 2}{y\choose 2}$$
and
$${xy+2\choose 3}={x+2\choose 3}{y+2\choose 3}+4{x+1\choose 3}{y+1\choose 3}+{x\choose 3}{y\choose 3}.$$
(The proof is essentially trivial by interpreting ${z\choose k}... | https://mathoverflow.net/users/4556 | Reason for breakdown of a nice binomial identity | $\def\des{\operatorname{des}}$Let $\des(\pi)$ be the number of descents of the permutation $\pi$. Then for any permutation $\pi$ in $S\_k$, we have
\begin{equation\*}\binom{xy+k-\des(\pi)-1}{k} =\sum\_{\sigma\tau=\pi}\binom{x+k-\des(\tau)-1}{k}
\binom{y+k-\des(\sigma)-1}{k}.\tag{$\*$}\label{star}
\end{equation\*}
If ... | 65 | https://mathoverflow.net/users/10744 | 427813 | 173,492 |
https://mathoverflow.net/questions/427815 | 13 | Let $n,m\in\mathbb{N}$. Is there a formula for the number of subgroups of index $n$ in $\mathbb{Z}^m$? Perhaps in terms of the divisors of $n$?
| https://mathoverflow.net/users/41644 | Number of finite index subgroups in a free abelian group | Yes. This is given by OEIS sequence [A160870](https://oeis.org/A160870). The number of subgroups of index $n$ in $\mathbf{Z}^m$ is there denoted $T(n,m)$. There is a recursive formula in terms of the divisors of $n$ given at this page. The initial conditions are
$$ T(n,1) = 1 \quad \textrm{ for all } n\in \mathbb{N}$$
... | 19 | https://mathoverflow.net/users/120914 | 427822 | 173,495 |
https://mathoverflow.net/questions/427819 | 4 | In Section 4.2.4 of [1], the authors write
>
> In this section we consider a causal linear process
> $$
> X\_t = \sum\_{j = 0}^\infty a\_j \varepsilon\_{t - j}, \quad t \in \mathbb{N},
> $$
> where, without loss of generality, $\sum\_{j = 0}^\infty a\_j^2 = 1$ and $\varepsilon\_t$ $(t \in \mathbb{Z})$ are i.i.d. ze... | https://mathoverflow.net/users/302666 | Why is every Gaussian process a linear process? | $\newcommand\ep\varepsilon\newcommand\si\sigma\newcommand\N{\mathbb N}$Your counterexample is correct. Indeed, if
$$X\_t=\sum\_{j=0}^\infty a\_j \ep\_{t-j} \tag{1}\label{1}$$
for $t\in\N$, $\sum\_{j=0}^\infty a\_j^2=1$, and the $\ep\_t$'s are iid zero-mean random variables with variance $\si^2\in(0,\infty)$, then for $... | 6 | https://mathoverflow.net/users/36721 | 427824 | 173,497 |
https://mathoverflow.net/questions/427818 | 2 | In ${\mathbb R}^n$, a vector $a=(a\_1,\ldots,a\_n)$ is said to *majorize* another vector $b=(b\_1,\ldots,b\_n)$ if for any convex function $f\colon\mathbb R\to\mathbb R$, we have
$$\sum\_{i=1}^nf(a\_i)\ge \sum\_{i=1}^nf(b\_i).$$
Note that this does not depend on the order of the coordinates. This natural pre-order has ... | https://mathoverflow.net/users/477827 | Spectral majorization for symmetric matrices | Yes. This notion of majorization of Hermitian matrices has been investigated before in the context of quantum information theory, and there are several good characterizations of this notion of majorization. This notion of majorization is quite natural since it gives a notion of whether one density operator is 'more mix... | 1 | https://mathoverflow.net/users/22277 | 427827 | 173,500 |
https://mathoverflow.net/questions/427436 | 8 | There is a well known Morita equivalence between the group C\*-algebra $C^\*(H)$ and $C\_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an incarnation of Mackey's imprimitivity theorem, see Rieffel's `Morita Equivalence for Operator Algebras', Example 1.
In topo... | https://mathoverflow.net/users/151844 | Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras | To answer the question in the final paragraph: yes, there is such a construction. If $H$ is a closed subgroup of $G$, and if $H$ acts on a $C^\*$-algebra $A$, then one defines the induced $C^\*$-algebra $\operatorname{Ind}\_H^G A$ to be the collection of all continuous, bounded functions $f:G\to A$ satisfying:
1. $f(... | 3 | https://mathoverflow.net/users/85913 | 427832 | 173,501 |
https://mathoverflow.net/questions/427842 | 86 | Consider the following Turing machine $M$: it searches over valid ZFC proofs, in lexicographic order, and if it finds a proof that $M$ halts, then it halts.
If we fix a particular model of Turing machine (say single-tape Turing machine), and if we fix an algorithm to verify that a given string is a valid ZFC proof of... | https://mathoverflow.net/users/5534 | Will this Turing machine find a proof of its halting? | It is a very nice question. The answer is yes, the machine will find a proof of its own halting nature, and it will halt when it does so.
I claim this is a consequence of [Löb's theorem](https://en.wikipedia.org/wiki/L%C3%B6b%27s_theorem). Let $M$ be a Turing machine such as you describe. Note that it is not quite co... | 75 | https://mathoverflow.net/users/1946 | 427846 | 173,506 |
https://mathoverflow.net/questions/427857 | 9 | For any simple, undirected graphs $G, H$, let $G\times H$ denote their [category-theoretical product](https://en.wikipedia.org/wiki/Tensor_product_of_graphs).
What is an example of an infinite connected graph $G$ with $G \cong G \times G$?
(Note that the totally disconnected graph $G = (\omega, \emptyset)$ has $G \... | https://mathoverflow.net/users/8628 | Example of a connected graph $G$ with $G \cong G \times G$ | The category of graphs and homomorphisms has countable powers. Given a graph $H = (V(H), E(H))$, its countable power is the graph $H^\omega = (V(H^\omega), E(V^\omega))$ where $V(H^\omega) = V(H)^\omega$ is the countable power of vertices, and $E(H^\omega)$ is defined for $a, b \in V(H)^\omega$ by
$$\{a, b\} \in E(H^\o... | 18 | https://mathoverflow.net/users/1176 | 427861 | 173,509 |
https://mathoverflow.net/questions/427200 | 2 | Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX\_t = \sigma(X\_t) \, dW\_t \;, \quad X\_0 = 1 \;.$$
where $\sigma:\mathbb R \to \mathbb R$ is a Lipschitz continuous function.
For every $M > 0$, let $A\_M$ denote the event
$$\{\underset{0 \leq t \leq 1}{\text{ma... | https://mathoverflow.net/users/173490 | Large noise limit for SDE with general volatility coefficients | The answer is no, as can be seen in the case $\sigma(u) = u$, so that $X\_t = \exp(W\_t - t/2)$. For the result to be true, Markov's inequality implies that the law of $W$, conditional on $A\_M$, would need to give probability $1/2$ to the event $\sup\_{t \le 1}|W\_t - Mt + t/2| < K\exp(-M/2)$ for some fixed $K>0$. By ... | 5 | https://mathoverflow.net/users/38566 | 427865 | 173,511 |
https://mathoverflow.net/questions/317712 | 6 | Let $A,B \in \mathrm{SL}\_3(\mathbb{Z})$. Set
$$S = \langle A \rangle \cdot \langle B \rangle = \{A^mB^n : m,n \in \mathbb{Z}\}.$$
>
> Is $S$ closed in the profinite topology on
> $\mathrm{SL}\_3(\mathbb{Z})$ ?
>
>
>
Equivalently (using the congruence subgroup property), I am asking whether for every $C \in \... | https://mathoverflow.net/users/38889 | Are double cosets of cyclic subgroups separable in a special linear group? | Although this is an older question, I think that the answer may still be of interest.
I realised that a positive answer follows from the work of Grunewald and Segal [*Grunewald, Fritz; Segal, Daniel*, [**Conjugacy in polycyclic groups**](http://dx.doi.org/10.1080/00927877808822268), Commun. Algebra 6, 775-798 (1978).... | 3 | https://mathoverflow.net/users/7644 | 427882 | 173,518 |
https://mathoverflow.net/questions/427889 | 0 | Since the Johnson graph/triangular graph $J(n,2)$ is the complement of the Kneser graph $K(n,2)$, which is also incidentally the line graph of the complete graph $K\_n$, I thought whether the same can be said about the line graphs of the complete hypergraphs $H\_n^k$. That is, can we say that the line graphs of the com... | https://mathoverflow.net/users/100231 | Line graphs of complete hypergraphs as complement of Kneser graphs | Yes, the correspondence goes the following way: The complete k-uniform Hypergraph has n vertices and the edges are given by all k-element subsets of {1,...,n}. Thus, the line graph has those k-element subsets as vertices and they are adjacent if and only if they intersect non-trivially. The complement graph, thus, has ... | 2 | https://mathoverflow.net/users/486126 | 427892 | 173,519 |
https://mathoverflow.net/questions/427874 | 0 | **Note**: This is a simplified version of the following [question](https://math.stackexchange.com/questions/4504786/determine-if-an-integral-expression-is-in-l2-mathbbr). I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a separate question b... | https://mathoverflow.net/users/51290 | Determine if an integral expression is in $L^2(\mathbb{R})$ | From Theorem 0.3.1 of [1], we have that if a kernel operator
$$
T(f)\equiv \int\_0^\infty K(x,x') f(x')dx',\;\;K(x,y)=e^{x-x'}\chi\_{y>x>0}
$$
is such that
$$
\sup\_{x'}\left(\int\_0^\infty K(x,x') dx\right),\;\;
\sup\_{x}\left(\int\_0^\infty K(x,x') dx'\right)\leq C
$$
then
$$
\|T(f)\|\_{L^2(\mathbb{R}^+)}\leq C \|f\|... | 2 | https://mathoverflow.net/users/51290 | 427904 | 173,525 |
https://mathoverflow.net/questions/427900 | 3 | How to prove the following identity?
Let $r = (r\_1, r\_2, \ldots, r\_d)$ and $c = (c\_1, c\_2, \ldots, c\_d)$ be sequences of natural numbers such that $s = r\_1 + r\_2 + \cdots + r\_d = c\_1 + c\_2 + \ldots + c\_d$.
Denote by $\mathcal{M}(r,c)$ the set of matrices whose rows sums and column sums are $r$ and $c$ r... | https://mathoverflow.net/users/44193 | Combinatorial identity concerning integral matrices with prescribed row sums and column sums | Here is a quick way to see this with generating functions. By using the notation $[\mathbb x^r]f(\mathbb x)$ for the coefficient of a monomial $\mathbb x^r$ in a formal power series $f(\mathbb x)$, we can write:
$$LHS=\left[\prod\_{i,j}x\_i^{r\_i}y\_j^{c\_j}\right]\prod\_{i,j}e^{x\_iy\_j}=\left[\prod\_{i,j}x\_i^{r\_i... | 9 | https://mathoverflow.net/users/2384 | 427907 | 173,526 |
https://mathoverflow.net/questions/427879 | 0 | Let $A=\mathbb{R}[x\_1,\dots,x\_n]$ be the algebra of real polynomials in $n$ variables. Fix polynomials $p\_1,\dots,p\_k\in A$.
Consider the subset
$$M:=\{(q\_1,\dots,q\_k)\in A^k|\, p\_1q\_1+\dots+p\_kq\_k=0\}.$$
Clearly $M$ is an $A$-submodule of $A^k$. Necessarily $M$ is finitely generated.
**I am wondering if ... | https://mathoverflow.net/users/16183 | Software to compute generators of a module over polynomial ring | What you compute is the "*syzygy module*" of $p\_1,\ldots,p\_k$.
You can try the following M2 script to check the computation times in Macaulay 2 for different $n$'s and $k$'s.
For me $n=16, k=6$ was a matter of minutes, but $k > 6$ seems to take much longer.
If you decrease the sparsity of your $p\_i$ by setting... | 2 | https://mathoverflow.net/users/21940 | 427908 | 173,527 |
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