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https://mathoverflow.net/questions/341576 | 5 | Let $M$ be a module over the polynomial ring $\mathbb C[x]$ such that $x+n$ is invertible in $M$ for every integer $n$. Let
$N=\oplus \_{n>0} \mathbb C\_n$ where $\mathbb{C}\_n := \mathbb C[x]/x+n$.
$\mathbf{Question:}$ Is it true that $Ext^1(M,N)=0$?
It is clear that $Ext^1(M,\mathbb C\_n)=0$ but this is not eno... | https://mathoverflow.net/users/3891 | $Ext^1$ for some modules over the polynomial ring in one variable | $\newcommand{\bC}{\mathbb{C}}\newcommand{\bZ}{\mathbb{Z}}$The resolution constructed by Dylan Wilson seems to show that $Ext^1(M,N)$ is non-zero already for $M=\bC[x][\frac{1}{x+n},n\in\bZ]$.
>
> There is an identification $$Hom(\bC[x][\frac{1}{x+n},n\in\bZ],\bigoplus\limits\_{n\in\bZ}\bC[x,\frac{1}{x+n}]/\bC[x])=... | 5 | https://mathoverflow.net/users/39304 | 341691 | 145,147 |
https://mathoverflow.net/questions/341698 | 5 | Let $(\Phi, V)$ be a reduced root system with base $\Delta$ and Weyl group $W$. Let $\ell$ be the length function of $W$ with respect to the set of simple reflections $S = \{s\_{\alpha} : \alpha \in \Delta\}$.
If $\Phi^+$ is the set of positive roots in $\Phi$ with respect to $\Delta$, and for each $w \in W$ we set $... | https://mathoverflow.net/users/38145 | Bruhat order and positive roots made negative | There is a counterexample in $\mathfrak{sl}\_3$. Denote by $\alpha, \beta$ the simple roots, and $s,t$ the corresponding simple reflections. Then $\Phi\_s^- = \{\alpha\}$ and $\Phi\_{st}^- = \{\beta, \alpha+\beta\}$.
It could be that the condition $\Phi\_{w\_1}^- \subseteq \Phi\_{w\_2}^-$ implies $w\_1 \leq w\_2$, I ... | 4 | https://mathoverflow.net/users/15292 | 341702 | 145,152 |
https://mathoverflow.net/questions/339416 | 4 | My question is closely related to [this one](https://mathoverflow.net/questions/66194/idelic-closures-of-units-of-number-fields), but not clearly the same in my opinion.
Let $L$ be a number field, with ring of integers $\mathcal{O}\_L$, and set $L^{\times}\_+\subset L^{\times}$ to be the subgroup of totally positive ... | https://mathoverflow.net/users/106906 | Are totally positive units of $L$ closed (in $L$) with respect to the finite-idelic topology? | I believe this is difficult in general, and I don't think it has been studied much yet.
A partial answer, showing where *not* to look for simple counterexamples:
Unwinding the definitions, everything is taking place inside a compact subquotient of the idèles where all components at finite places have absolute value... | 3 | https://mathoverflow.net/users/49003 | 341705 | 145,154 |
https://mathoverflow.net/questions/341695 | 30 | What are some examples of **successful** mathematical attempts in clinical setting, specifically at the patient-disease-drug level?
To clarify, by patient-disease-drug level, I mean the mathematical work is approved to be used as part of a decision making process to prescribe a specific treatment for a specific patie... | https://mathoverflow.net/users/109419 | Applications of mathematics in clinical setting | An example of a simple mathematical/evolutionary game theory model used to determine treatment scheduling in clinical treatment of metastic and castrate resistant prostate cancer can be found at <https://www.nature.com/articles/s41467-017-01968-5>. While the clinical trial is on-going, initial results show that the mod... | 17 | https://mathoverflow.net/users/134555 | 341727 | 145,160 |
https://mathoverflow.net/questions/341725 | -5 | Can we have a consistent theory whose signature is $(=,\in, S, +, \times)$ standing for identity and membership binary relations and the successor total unary function, addition and multiplication total binary functions with the followings?
Define: $x=\emptyset \equiv\_{df} \not \exists y (y \in x)$
Add all axioms... | https://mathoverflow.net/users/95347 | Can we blend ZFC with true arithmetic? | What you've written is a bit unclear.
If you *do not* extend the ZFC schemes to formulas involving the new signature, then the answer is **yes**: letting $N$ be the standard model of arithmetic, $M$ be some countable model of ZFC, and $f:N\rightarrow M$ bijective, we can "port over" the structure on $N$ to $M$ and ge... | 5 | https://mathoverflow.net/users/8133 | 341729 | 145,162 |
https://mathoverflow.net/questions/341726 | 2 | Let $W$ be the Weyl group of a root system $\Phi$ with base $\Delta$ and system of positive roots $\Phi^+$. Let $S = \{ w\_{\alpha} : \alpha \in \Delta \}$ be the set of simple reflections corresponding to elements of $\Delta$. Let $\theta \subset \Delta$, and let $w\_0 = w\_l w\_{l,\theta}$, where $w\_l$ and $w\_{l,\t... | https://mathoverflow.net/users/38145 | Characterization of all $w$ in the Weyl group satisfying $w \geq w_l w_{l, \theta}$ | Yes, this follows from the fact that $x \mapsto w\_l x$ is an antiautomorphism of the Bruhat order on a finite Coxeter group. (See Björner and Brenti, Proposition 2.3.4, for example, but their $w\_0$ is your $w\_l$) You also need the fact that $w\_l$ is an involution. (For example, Björner and Brenti, Proposition 2.3.2... | 5 | https://mathoverflow.net/users/5519 | 341753 | 145,166 |
https://mathoverflow.net/questions/341679 | 10 | **Question:** What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which *THH* (Topological Hochschild Homology) satisfies descent?
Adaptations of the arguments appearing in Section 3 of [BMS2](https://arxiv.org/abs/1802.03261) show that *THH* has flat descent for simplicial commutative rings... | https://mathoverflow.net/users/113828 | Descent properties of topological Hochschild homology | In Theorem 1.2 of B. I. Dundas and J. Rognes: "Cubical and cosimplicial descent", Journal of the London Mathematical Society (2) 98 (2018) 439-460, DOI 10.1112/jlms.12141, we showed that for each $1$-connected map $\phi : A \to B$, of connective commutative $S$-algebras, the map from $THH(A)$ to the homotopy limit of t... | 5 | https://mathoverflow.net/users/9684 | 341771 | 145,173 |
https://mathoverflow.net/questions/341755 | 8 | I know $\sum\_{k=0}^{n} \sin(k)$ is bounded by a constant
and $\sum\_{k=0}^{n} \sin(k^2)$ is [not bounded](https://mathoverflow.net/questions/201250/is-sum-k-1n-sink2-bounded-by-a-constant-m) by a constant.
Then, what about $\sum\_{k=0}^{n} (|\sin(k)|-2/\pi)$?
From numerical calculation, $\max\_{n=0...10^8}(\sum\_{... | https://mathoverflow.net/users/142913 | Is $\sum_{k=0}^n (|\sin(k)|-2/\pi) $ bounded by a constant $M$? | As a partial answer, we can show that if $S\_N=\sum\_{k=0}^N (|\sin(k)|-2/\pi)$, then:
$S\_N=O(1)-\frac{4}{\pi}\sum\_{k=2}^{N}\sum\_{m=1}^{[k \log^2 k]}\frac{\cos 2mk}{4m^2-1}=O(\sum\_{m=1}^{[N\log^2 N]}{\frac{\min(N\log^2N, ||\frac{m}{\pi}||^{-1})}{m^2}})$,
where as usual $||x||$ represents the distance to the cl... | 2 | https://mathoverflow.net/users/133811 | 341777 | 145,174 |
https://mathoverflow.net/questions/341690 | 3 | In these days, I'm studying Cohen-Lenstra heuristics to understand the paper of Rene Schoof "Class Numbers of Real Cyclotomic Fields of Prime Conductor".
On page 932 of Schoof's paper, there is a sentence "According to Cohen-Lenstra, the probability that M does not occur in a "random $\mathbb{Z}[\zeta\_{d}]$-module m... | https://mathoverflow.net/users/123226 | How can we justify the use of Example 5,4 (of Cohen, Lenstra) assuming their heuristics | $\DeclareMathOperator\Aut{Aut}$I infer from the context that the precise meaning of "a random $\mathbb{Z}[\zeta\_d]$-module modulo a random principal ideal" means that you start by producing a random $\mathbb{Z}[\zeta\_d]$-module with respect to the Cohen--Lenstra probability distribution (i.e. by definition such a "ra... | 3 | https://mathoverflow.net/users/35416 | 341778 | 145,175 |
https://mathoverflow.net/questions/341784 | 2 | Let $\mathbf{A}$ be any $n\times n$ symmetric positive matrix ($A\_{ij}>0$). It is easy to show that the solution to the following optimization problem
\begin{align}
\max\_{\mathbf{x}}~~\mathbf{x^TAx}\,\,;s.t.~~\mathbf{x}\geq \mathbf{0},~~\|\mathbf{x}\|\_2=1
\end{align}
is given by the so-called Perron vector of $... | https://mathoverflow.net/users/27249 | Does the Perron vector maximize $x^TAx$ in the simplex? | No. The Perron vector is in general very far from optimizing the quantity you're looking at.
Here is an example:
Let $A$ be the $n\times n$ tridiagonal matrix with $\frac 13$ on the diagonal and the off-diagonals as well as in the $(1,n)$ and $(n,1)$ entries. (I think of this as a Markov transition matrix on a ring o... | 7 | https://mathoverflow.net/users/11054 | 341785 | 145,176 |
https://mathoverflow.net/questions/341780 | 5 | In Serre's *Local Fields*, at the beginning of the chapter III section 2, he has wrote "it is known that $T$ extends to a non-degenerate bilinear form on the exterior algebra of $V$", where $T$ is a non-degenerated bilinear form over a vector space $V$.
I get confused about this well-know extension. Is there any expl... | https://mathoverflow.net/users/143426 | Extension of a bilinear form to the exterior algebra | Let $k$ be a nonnegative integer. Let $K$ be a commutative ring, and let $V$ and $W$ be two $K$-modules. Let $\alpha : V \times W \to K$ be a $K$-bilinear form. Then, there is a $K$-bilinear form
\begin{align}
\alpha\_k : \wedge^k V \times \wedge^k W &\to K; \\
\left(v\_1 \wedge v\_2 \wedge \cdots \wedge v\_k , w\_1 \w... | 8 | https://mathoverflow.net/users/2530 | 341786 | 145,177 |
https://mathoverflow.net/questions/341773 | 2 | If X is a normal projective variety with an ample line bundle $L$, and $\pi:Y\to X$ a resolution of $X$ and $E$ be the exceptional divisor, then is it true that $A\pi^{\star}L-[E]$ is always ample for $A$ sufficiently large?
| https://mathoverflow.net/users/104334 | Is the pullback of an ample bundle minus the exceptional divisor ample? | You should probably state what you want a little more precisely. As it is currently stated, it allows for the possibility that $\pi$ is not projective in which case there is no chance. The statement also allows a small resolution in which case $E$ is empty and $a\pi^\*L$ is not ample for any $a$.
Unfortunately, even ... | 11 | https://mathoverflow.net/users/10076 | 341787 | 145,178 |
https://mathoverflow.net/questions/341742 | 3 | What are examples for *convex* polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
* $P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
* all 2-faces of $P$ are 4-gons (not necessarily squares, or rectangles).
I know the $d$-cubes, rhombic dodecahedron and rhombic triaco... | https://mathoverflow.net/users/108884 | Are there any more polytopes whose 2-faces are identical 4-gons? | In fact, there are many to be found on Wikipedia under [isogonal figures](https://en.wikipedia.org/wiki/Isohedral_figure), even in three dimensions.
Examples in dimension *four* are obtained as dual polytope of [runcinated 4-simplex](https://en.wikipedia.org/wiki/Runcinated_5-cell#Runcinated_5-cell) or [runcinated 24... | 4 | https://mathoverflow.net/users/108884 | 341802 | 145,183 |
https://mathoverflow.net/questions/341810 | 2 | If $H=(V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) and $\kappa$ is a cardinal, then $c:V\to\kappa$ is a *coloring* if for every $e\in E$ with $|e|>1$, the restriction $c|\_e:e \to \kappa$ is non-constant. By $\chi(H)$ we denote the smallest cardinal $\kappa$ such that there is a coloring $c:V\to \... | https://mathoverflow.net/users/8628 | Chromatically rigid hypergraphs | Let $H$ be a complete graph on $V=\omega$. Suppose $E\_0$ existed. Then removing a single edge would give a finitely colorable graph. Adding this edge back, we can color the graph with at most one more color. Same should work for any infinite $V$.
| 4 | https://mathoverflow.net/users/30186 | 341811 | 145,186 |
https://mathoverflow.net/questions/323207 | 5 | It may be a naive question, but:
>
> If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups?
>
>
>
But "many", typically I have in mind "infinitely many" or even "uncountably many".
I am not familiar with bounded cohom... | https://mathoverflow.net/users/122026 | Second bounded cohomology and normal subgroups | I finally found the answer to my question.
>
> **Proposition:** There exists a finitely presented simple group whose second bounded cohomology group is infinite-dimensional.
>
>
>
Such an example comes from the study of Kac-Moody groups. The simplicity of such groups is studied in Caprace and Rémy's article [S... | 2 | https://mathoverflow.net/users/122026 | 341828 | 145,192 |
https://mathoverflow.net/questions/339686 | 20 | Suppose $G$ and $H$ are groups. $C \subseteq H$ is called a *skew copy* of $G$ in $H$ if $C = hK$ for some $h \in H$ and some subgroup $K$ of $H$ with $K \cong G$.
>
> **Question 1:** Suppose the infinite symmetric group $S\_\mathbb N$ is partitioned into finitely many pieces. Must one of these pieces contain a ske... | https://mathoverflow.net/users/70618 | If $S_\mathbb N$ is partitioned into finitely many pieces, must one piece contain a "skew copy" of every countable group? | The answer to both questions is no.
For every group $G$, by induction on $|G| = \kappa$, we'll construct a partition $G = A \sqcup B$ such that for every $g, h \in G$, if $g$ has infinite order, then $h\langle g\rangle$ meets both $A, B$ on an infinite set -- Call such a partition of $G$ good. If $\kappa = \aleph\_0$... | 6 | https://mathoverflow.net/users/2689 | 341831 | 145,194 |
https://mathoverflow.net/questions/341823 | 6 | Let $E\subset B\_1(0)\subset \mathbb{R}^n$ be a compact set s.t. $\lambda(E)=0$, where $\lambda$ is the Lebesgue measure, and $B\_1(0)$ is the Euclidean unit ball centered at the origin. Is the following integral finite:
$$\int\_{B\_1(0)}-\log d(x,E)d\lambda(x)<\infty?$$
Although this question seems trivial, I have... | https://mathoverflow.net/users/70853 | Integrability of log of distance function | The integral in question is finite for most sets of measure zero, but can diverge to $\infty$ for some sets. An example in one dimension is obtained by constructing a Cantor set where at stage $k$ the middle $1/(k+1)$ proportion is removed from each of the $2^{k-1}$ intervals obtained at stage $k-1$. Thus the $2^k$ int... | 7 | https://mathoverflow.net/users/7691 | 341834 | 145,195 |
https://mathoverflow.net/questions/341824 | 6 | $\require{AMScd}$
I'm considering minimum values (at non-zero integer points) of real, positive-definite, quadratic forms of determinant $1$. These are functions $f:\mathbb{R}^n\to \mathbb{R}$ which can be represented as $x\mapsto x^\intercal Ax$ for some symmetric, positive-definite matrix $A$ with determinant $1$. Ca... | https://mathoverflow.net/users/105628 | The number of quadratic forms attaining Hermite's constant | There are only finitely many inequivalent forms which may be local maxima for the Hermite invariant. Voronoi showed (1908) that the lattices attaining a local maxima are *extreme*, i.e. perfect and eutactic. Voronoi also showed there are only finitely many perfect lattices in each dimension, and these must all be integ... | 8 | https://mathoverflow.net/users/118731 | 341837 | 145,196 |
https://mathoverflow.net/questions/341734 | 3 | For functions spaces we have ratio of two functions namley $\frac{f}{g}$, Can ratio of two operators in von Neumann algebra make sense? If at all it makes sense, what will be the proper definition of it?
| https://mathoverflow.net/users/145907 | On analogue of ratio in operator algebras | Let $a$ and $b$ be elements of a von Neumann algebra $\mathscr{A}$. As Jochen Glueck notes, one needs to differentiate between $a/b$ and $a\backslash b$. I would define $a/b$ to be (provided it exists) the unique element $c$ of $\mathscr{A}$ with $a=cb$ and $\mathop{Ran}(b)^\perp \subseteq \mathop{Ker}(c)$. Such a $c$ ... | 5 | https://mathoverflow.net/users/145927 | 341849 | 145,199 |
https://mathoverflow.net/questions/341838 | 5 | Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}\_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^i(\mathrm{Sp}\_{2n}(\mathbb{Z}),\mathbb{Z}^{2n})$ for $i=1,2$?
(2) $H^i(\mathrm{Sp}\_{2n}(\mathbb{C}),\mathbb{C}^{2... | https://mathoverflow.net/users/nan | Cohomology of linear algebraic groups | Let me focus on (1). It is answered in Lemma A.3 of [this paper](https://arxiv.org/abs/1902.10097) by Krannich. However, let me explain why his answers are 2-torsion. This argument also works for (2) in either interpretation.
That $H^i(Sp\_{2n}(\mathbb{Z});\mathbb{Z}^{2n})$ is 2-torsion is equivalent to it being zero... | 4 | https://mathoverflow.net/users/798 | 341850 | 145,200 |
https://mathoverflow.net/questions/341492 | 4 | Consider the Hall-Littlewood polynomial
$$
P\_\lambda(x\_1,\ldots,x\_n;t)=\sum\_{\sigma\in S\_n/S\_n^\lambda}\sigma\left(x\_1^{\lambda\_1}\cdots x\_n^{\lambda\_n}\prod\limits\_{\lambda\_i>\lambda\_j}\dfrac{x\_i-tx\_j}{x\_i-x\_j}\right),
$$
where $\lambda=(\lambda\_1,\ldots,\lambda\_n)$ is a partition and $S\_n^\lambda... | https://mathoverflow.net/users/62154 | Applying a simple involution to Hall-Littlewood polynomials | The transition matrix from the Schur functions to the HL symmetric functions is $K(t)$, the matrix of Kostka polynomials. This means that the transition matrix from $P(x;t)$ to $P(x;-t)$ is $K(t)^{-1}K(-t)$. This is upper-triangular with respect to the dominance partial order on partitions (or lower-triangular, dependi... | 2 | https://mathoverflow.net/users/61372 | 341857 | 145,202 |
https://mathoverflow.net/questions/341845 | 27 | For which $a,b,c$ does $axy+byz+czx$ represent all integers?
In a recent answer, I [conjectured](https://mathoverflow.net/a/339014/44143) that this holds whenever $\gcd(a,b,c)=1$, and I hope someone will know. I also conjectured that $axy+byz+czx+dx+ey+fz$ represents all integers when $\gcd(a,b,c,d,e,f)=1$ and each v... | https://mathoverflow.net/users/nan | When does $axy+byz+czx$ represent all integers? | Here is a proof of the conjecture. I will refer several times to the book Cassels: Rational quadratic forms (Academic Press, 1978).
**1.** Let $p$ be a prime such that $p\nmid a$. Using the invertible linear change of variables over $\mathbb{Z}\_p$
$$x'=ax+bz,\qquad y'=y+(c/a)z,\qquad z'=(1/a)z,$$
we have
$$x'y'-(abc... | 26 | https://mathoverflow.net/users/11919 | 341858 | 145,203 |
https://mathoverflow.net/questions/341686 | 5 | I apologise for the long-windedness of this question.
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S\_a(x)=\sum\_{n\leq x} d(n)^a.
$$
For $a=1,$ the following is well known
$$
S\_1(x)=\sum\_{n\leq x} d(n)=x \log x + (2 \gamma -1) x +{\cal O}(\sqrt{x})
$$
while fo... | https://mathoverflow.net/users/17773 | Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots,x\}.$ | Even if you take $Z(x)=(1-\varepsilon)x$ for some fixed $0<\varepsilon<1$, you are going to get
$$
M=x(\ln x)^{a\ln 2+o(1)}.
$$
To prove this, observe that both $\omega(n)$ and $\Omega(n)$ have normal order $\ln\ln n$ (here $\Omega$ and $\omega$ are numbers of prime factors with and without multiplicity respectivel... | 4 | https://mathoverflow.net/users/101078 | 341862 | 145,205 |
https://mathoverflow.net/questions/341868 | 4 | This question is about von Neumann's informal definition of ordinals as "sets of all smaller ordinals" and was discussed [in this math.stackexchange question](https://math.stackexchange.com/questions/3189295/motivation-of-the-von-neumann-definition-of-ordinals/3357377#3357377).
When trying to formalize this definitio... | https://mathoverflow.net/users/48826 | Compact definition of ordinals | (Alexander: In light of our exchange of comments, I have slightly revised the first sentence of my response to better reflect my intention.)
Perhaps you will find the following alternative formalization of the familiar informal characterization of a von Neumann Ordinal to be of interest.
A set $\alpha$ is a *von Ne... | 6 | https://mathoverflow.net/users/18939 | 341872 | 145,210 |
https://mathoverflow.net/questions/339999 | 5 | In the article cited below, I. Castellano gives a proof for the following result (Proposition 4.1).
>
> Let $G$ be a compactly generated totally disconnected locally compact group. Suppose that $G$ acts discretely on a tree $\mathcal T$ such that
>
>
> 1. the group $G$ is acting without edge inversions;
> 2. the ... | https://mathoverflow.net/users/57533 | Compactly generated vertex stabilisers in compactly generated t.d.l.c. groups acting on trees | As YCor suggests, proceed by Bass–Serre theory. We can write $G$ in the form
$\frac{G\_{v\_1} \ast \dots \ast G\_{v\_m} \ast F(E)}{\langle \langle \overline{e}\alpha\_e(g)e\alpha\_{\overline{e}}(g)^{-1} \; (g \in G\_e), \; e\overline{e}, \; e \; (e \in E') \rangle \rangle}$
where $E$ is a set of representatives for... | 3 | https://mathoverflow.net/users/4053 | 341874 | 145,211 |
https://mathoverflow.net/questions/341873 | 1 | Erdős' similarity conjecture states that for each infinite set $A\subset \mathbb R$ there is a set $P\subset [0,1]$ of *positive measure* such that for all $t\in \mathbb R$, $\delta\neq 0$ there is some $a\in A$ with $t+\delta a\notin P$. In particular, it is unknown if the sequence $A=\{2^{-n}:n\geq 1\}$ has the prope... | https://mathoverflow.net/users/111012 | A Related Problem to Erdős' similarity conjecture | I want to take this opportunity to give an application of algorithmic randomness theory to this area. Accidentally I am working on the similarity problem recently and found some interesting applications of algorithmic randomness theory to the area.
For the notation of randomness you may refer to the book by Downey-Hi... | 2 | https://mathoverflow.net/users/14340 | 341876 | 145,213 |
https://mathoverflow.net/questions/341626 | 2 | I have the following set of equation systems, and I would like to find a short, formal way to write it down. My main difficulty is that I cannot find a good way to write the indices of the variables $\omega$. Any suggestion is highly apprechiated.
**n=4:**
$$\omega\_{A,B,a,b}\cdot\omega\_{C,D,c,d}+\omega\_{A,C,a,c}\... | https://mathoverflow.net/users/63938 | Concise formulation of set of equation systems | Let's use an alphabet $X\_i $ indexed by an integer $1\leq i\leq n$, and also accompanying variables $x\_i \in \{ 0,1,\ldots ,c-1 \} $. Up to now, it seems you've only defined the variables $\omega\_{X\_i, X\_j, x\_i, x\_j} $ for $i<j$. For convenience, define also the auxiliary variables $\omega\_{X\_j, X\_i, x\_j, x\... | 3 | https://mathoverflow.net/users/134299 | 341885 | 145,219 |
https://mathoverflow.net/questions/341840 | 1 | Let $\bf A$ be an $n \times n$ non-singular matrix over $\mathbb{F}$.
Let $x$ be a non-zero element of $\mathbb{F}$.
Suppose that ${\bf 1}\_{n}$ is a symbol for the all-one vector of length $n$ over $\mathbb{F}$.
Now consider the following $(n+1) \times (n+1)$ matrix and assume that $\bf B$ is a invertible matrix ove... | https://mathoverflow.net/users/124008 | A closed-form expression for the inverse of a block-matrix | Say that
$$B^{-1}=:\begin{pmatrix} b & X^T \\ Y & M \end{pmatrix}.$$
Then using *Schur's complement formula* (thanks to Nathaniel), $b=(x-{\bf1}^TA^{-1}{\bf1})^{-1}$ and $M=(A-x^{-1}{\bf11}^T)^{-1}$. From this, you can compute the vectors
$$Y=-x^{-1}M{\bf1},\qquad X^T=-b{\bf1}^TA^{-1}.$$
| 4 | https://mathoverflow.net/users/8799 | 341893 | 145,221 |
https://mathoverflow.net/questions/341870 | 7 | Consider a vector $x$ with $0 < x\_1 < \cdots < x\_n < \infty$, and let $0 < \gamma\_1 < \cdots < \gamma\_n < \infty$.
>
> I would like to show that $x^{\gamma\_1}, \ldots, x^{\gamma\_n}$ are
> linearly independent, where $x^{\gamma\_i}$ is defined as the vector
> $(x\_1^{\gamma\_i}, \ldots, x\_n^{\gamma\_i})$.
>... | https://mathoverflow.net/users/145983 | Linear independence of element-wise powers of positive vectors | The following proposition shows that $x^{\gamma\_1}, \dots x^{\gamma\_n}$ are indeed always linearly independent if $x \in \mathbb{R}^n$ has $n$ mutually distinct strictly positive entries.
**Proposition.** For all real numbers $\gamma\_1 < \dots < \gamma\_n$ (be they positive or not) and each tuple $0 \not= (\alpha\... | 10 | https://mathoverflow.net/users/102946 | 341898 | 145,223 |
https://mathoverflow.net/questions/341894 | 3 | We say that a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ is *connected* if the following condition holds:
>
> for all $S\subseteq V$ with $\emptyset\neq S \neq V$ there is $e\in E$ that meets both $S$ and $V\setminus S$, i.e. $$S\cap e \neq \emptyset \neq (V\setminus S)\cap e.$$
>
>
>
Giv... | https://mathoverflow.net/users/8628 | Connected hypergraphs | Let $V=E=\omega$. Connected subgraphs are precisely those which contain as edges $n$ for arbitrarily large $n$. Clearly there is no minimal such subgraph.
| 4 | https://mathoverflow.net/users/30186 | 341904 | 145,226 |
https://mathoverflow.net/questions/341896 | -3 | I would like to know what articles are in the literature about the known as Firoozbakht's conjecture, see the Wikipedia [*Firoozbakht's conjecture*](https://en.wikipedia.org/wiki/Firoozbakht%27s_conjecture).
>
> **Question.** What articles have been published in journals
> whose main goal is the study of Firoozb... | https://mathoverflow.net/users/142929 | Published articles in journals about the Firoozbakht's conjecture, whose main goal or focus is the study of this conjecture | [Verification of the Firoozbakht conjecture for primes up to four quintillion](https://arxiv.org/abs/1503.01744)
[On the Firoozbakht's conjecture](https://arxiv.org/abs/1603.08917)
[Some consequences of the Firoozbakht's conjecture](https://arxiv.org/abs/1604.03496)
[Prime gaps and the Firoozbakht Conjecture](htt... | 10 | https://mathoverflow.net/users/11260 | 341909 | 145,229 |
https://mathoverflow.net/questions/341852 | 5 | Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all its Morse indices are even?
| https://mathoverflow.net/users/114985 | Kähler manifold with even-only singular cohomology | Assuming that by `cohomology' you mean integer coefficients, there is a general result saying what you want for simply connected manifolds of dimension $5$ or more, without the Kähler condition. According to Smale (Generalized Poincare’s conjecture in dimensions greater than four, Ann. Math. 74, No, 2, 391-406 (1961)) ... | 9 | https://mathoverflow.net/users/3460 | 341914 | 145,231 |
https://mathoverflow.net/questions/341749 | 14 | Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable categories. Assume that $f$ induces an equivalence between the groupoids underlying $\mathcal{C}$ and $\mathcal{D}$. Is $f$ necessarily an equivalence? What if $\mathcal{C}$ and $\mathcal{D}$ are assumed to be linea... | https://mathoverflow.net/users/145919 | Are locally presentable categories determined by their objects? | **The answer in general is *no*.**
Let $\mathcal C$ be the category of sets, let $\mathcal D$ be the category of pointed sets (with basepoint-preserving maps), and let $f: \mathcal C \to \mathcal D$ be the functor which adds a disjoint basepoint. Then $f$ is an equivalence on underlying groupoids, but not an equivale... | 21 | https://mathoverflow.net/users/2362 | 341917 | 145,232 |
https://mathoverflow.net/questions/341913 | 2 | Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a solution to the Cauchy functional equation
$$f(a+b)=f(a)+f(b),\quad\forall a,b\in\mathbb{R}.$$
Observe that
$$A:=\{a\in\mathbb{R}:f(a)\geq 0\},\quad B:=\{b\in\mathbb{R}:f(b)< 0\}$$
provide a partition of $\mathbb{R}$ into two closed by sum subsets.
I would like to know i... | https://mathoverflow.net/users/54552 | Sets closed by sum and solutions to the Cauchy functional equation | Suppose we have a decomposition $\mathbb R=A\cup B$ as described in the question and that $0\in A$. (Clearly, $0\in B$ is impossible, so we have to denote by $A$ the set which contains zero.)
Now, let us denote
$$A\_0=\{x\in A; -x\in A\} \qquad\text{and}\qquad A\_1=\{x\in A; -x\in B\}.$$
It's not difficult to see th... | 2 | https://mathoverflow.net/users/8250 | 341918 | 145,233 |
https://mathoverflow.net/questions/341912 | 1 | Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$
>
> How can we calculate the [generalized gradient](https://pdf.sciencedirectassets.com/272585/1-s2.0-S0001870800X02502/1-s2.0-0001870881900323/main.pdf?X-Amz-Security-Token=AgoJb3JpZ2... | https://mathoverflow.net/users/91890 | How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$? | $\newcommand{\de}{\delta}$
The notion of the generalized gradient, as defined in Clarke's paper linked in your question, is applicable only to Lipschitz functions. In general, depending on your measure space, your function $f$ will not be Lipschitz, because the evaluation functional $L^2(\tau)\ni x\mapsto\de\_s(x):=x(s... | 2 | https://mathoverflow.net/users/36721 | 341920 | 145,234 |
https://mathoverflow.net/questions/341932 | 1 | $X\_i$ iid with $P(X\_i=j)=p\_j$, $j=1, \dots, m$.
$\sum\_{j=1}^m p\_j = 1$.
Define $N = \min\{n>0:X\_n = X\_0\}$, compute $E(N)$.
I have two solutions, but different answers:
**Solution 1**
$E(N) = E(N\mid X\_1=X\_0)P(X\_1=X\_0) + E(N\mid X\_1\neq X\_0)P(X\_1\neq X\_0)$
So $x = 1\cdot y + (1+x)(1-y)$, where $x... | https://mathoverflow.net/users/9260 | Expected minimum number | Solution 2 is correct.
In Solution 1, you are assuming that $E(N|X\_1\ne X\_0)=1+EN$, which is not true in general. Indeed, $N$ is the time needed to return to $i$ from a state $i$ -- whereas, on any event of the form $\{X\_0=i\ne j=X\_1\}$, $N-1$ is the time needed to get to state $i$ from $j\ne i$.
| 2 | https://mathoverflow.net/users/36721 | 341935 | 145,236 |
https://mathoverflow.net/questions/341555 | 3 | Let $D$ be a bounded domain of $\mathbb{R}^n$ with smooth boundary $\partial D$. Is the following subspace dense in space $L^2(D)\times L^2(\partial D)$:
$$\{(f,f\rvert\_{\partial D}) : f\in C^\infty(\overline{D}), (\Delta f)\rvert\_{\partial D}=0\}.$$
I tried to use density of $C\_c^\infty(D)$ in $L^2(D)$, but I didn'... | https://mathoverflow.net/users/nan | Density of a functional space | I believe that Daniele Tampieri's idea of using completeness Fichera's Theorem is the right one. However I think there is a simpler proof.
In view of the Hahn-Banach Theorem, proving the density of the space
\begin{equation}
\left\{(f,f|\_{\partial D})\ |\ f\in S \right\}, \quad \text{where }
S=\left\{ f\in C^{\infty}... | 4 | https://mathoverflow.net/users/146025 | 341940 | 145,238 |
https://mathoverflow.net/questions/341713 | 2 | Let $\alpha$ to be the [Kuratowski measure of non-compactness](https://en.wikipedia.org/wiki/Measure_of_non-compactness), in a Banach space $E$.
It's very easy to prove that $\alpha (D\_1\times D\_2)\leq \alpha (D\_1)+\alpha (D\_2)$, where $D\_1$ and $D\_2$ are bounded subsets in $E$.
Let $A:E\times E\rightarrow E$... | https://mathoverflow.net/users/102228 | A non-condensing operator with a power condensing | **Example 1.** The following kind of nilpotent construction satisfies the properties require in the question:
Let $F$ be an infinite dimensional Banach space and let $E = F \times F$ (say, with the maximum norm). Let $I\_F: F \to F$ denote the identity operator and define $T: E \to E$ as the operator matrix
$$
\begin... | 1 | https://mathoverflow.net/users/102946 | 341944 | 145,240 |
https://mathoverflow.net/questions/341743 | 1 | Let $f:X \to Y$ be a finite morphism, where $X$ is an affine, non-singular curve and $Y$ is an affine, non-singular surface over $\mathbb{C}$. Denote by $\Gamma\_f \subset X \times Y$ the graph of the morphism $f$. When is $\Gamma\_f$ a local complete intersection subscheme of $X \times Y$? Any hint/reference will be m... | https://mathoverflow.net/users/45397 | When is a graph morphism a regular embedding? | All we need here is smoothness of $Y$. In general if $X$ and $Y$ are varieties over a field $k$, with $Y$ smooth, and if $f: X \to Y$ is a morphism, then $ \Gamma\_f \subset X \times Y$ is a local complete intersection, with co-normal bundle isomorphic to $f^\* \Omega\_Y$.
To see this, note that $\pi\_X: X \times Y ... | 2 | https://mathoverflow.net/users/113296 | 341948 | 145,242 |
https://mathoverflow.net/questions/341924 | 5 | I conjecture the following:
>
> Let $U \subset \Bbb C^{n \times n}$ be an affine subspace, and let $S\_U$ denote the "spectrum of $U$", that is
> $$
> S\_U = \{\lambda \in \Bbb C : \det(A - \lambda I) = 0 \text{ for some } A \in U\}.
> $$
> Then either all elements of $U$ have an identical spectrum, or $S\_U = \B... | https://mathoverflow.net/users/34894 | Spectrum of a Subspace of Matrices | To expand on Christian Remling's answer a bit: in his example, setting $det(A(t) - \lambda) = 0$ becomes $(1-\lambda)t+\lambda^3=0$, and solving for $t$ gives $t = x^3/(x-1)$ -- so for any $\lambda \in \mathbb{C}$, we have $x \in S$ by choosing this $t$, with the exception of $\lambda=1$.
In general for an affine sub... | 3 | https://mathoverflow.net/users/97603 | 341949 | 145,243 |
https://mathoverflow.net/questions/341905 | 5 | Tannaka duality for a finite group lets us recover the group algebra $\mathbb{C}[G]$ as the endomorphisms of the forgetful functor $F:RepG\rightarrow Vect$, and taking the monoidal automorphisms recovers the grouplike elements of this hopf algebra, which we can recognise as just our group $G$.
Is there a diagrammatic... | https://mathoverflow.net/users/128502 | What properties of a finite group fibre functor give its endomorphisms a hopf algebra structure? | The tensor product functor $\otimes : \text{Rep}(G) \times \text{Rep}(G) \to \text{Rep}(G)$ is "bilinear": it preserves colimits in both variables. It can therefore be regarded as being a "linear" functor on a "tensor product" category
$$\otimes : \text{Rep}(G) \otimes \text{Rep}(G) \to \text{Rep}(G)$$
which can be... | 4 | https://mathoverflow.net/users/290 | 341976 | 145,256 |
https://mathoverflow.net/questions/341975 | 5 | (Question mildly edited for clarity by request of Matt F.)
If $G$ is a finitely presented group, let $|\cdot|$ denote the word metric with respect to a finite set of generators. Suppose $\nu$ is a finitely supported measure on $G$ with rational weights. A random walk on $G$ is built by at each stage multiplying by an... | https://mathoverflow.net/users/11054 | Is random walk drift rational? | For nearest neighbour random walks on certain free products the rate of escape (or, if you wish, drift) was explicitly calculated by [Mairesse and Matheus](https://arxiv.org/pdf/math/0509211.pdf). In particular, their formula (26) gives an example of a "rational" random walk on the free product of $\mathbb Z\_2$ and $\... | 6 | https://mathoverflow.net/users/8588 | 341977 | 145,257 |
https://mathoverflow.net/questions/341960 | 3 | Let $k$ be an algebraically closed field, $\ell$ is a prime different with characateristic of $k$, and consider the $\ell$-adic etale cohomology. We know the number of connected components of a scheme finite type over $k$ by looking at $H^0$, but how about the number of irreducible components?
Looking at the example... | https://mathoverflow.net/users/102104 | Reference request: number of irreducible components and top dimension etale cohomology | The answer is *yes* (in fact the result also holds over separably closed fields). You can find this statement in Corollary 7.5.21 of:
Poonen - Rational points on varieties.
Poonen gives a sketch of a proof with references to details in SGA4.
| 7 | https://mathoverflow.net/users/5101 | 342006 | 145,269 |
https://mathoverflow.net/questions/342009 | 2 | Let $X$ be a Brauer-Severi variety over a field $k$ of characteristic $0$. In other words, suppose that $X\_{\overline{k}} \cong \mathbb{P}\_{\overline{k}}^n$.
I came across a statement that the map $X \times\_k X \longrightarrow X$ sending an element of $X \times\_k X$ to its first factor having a section (the diago... | https://mathoverflow.net/users/38282 | Maps from products of Brauer-Severi varieties and sections | **This is false.** Take $X$ to be a smooth plane conic without a rational point. Consider the surface
$$S = X \times X.$$
Over the algebraic closure this becomes isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$. Recall that $\mathrm{Pic}(\mathbb{P}^1 \times \mathbb{P}^1) = \mathbb{Z}^2$. Then one easily sees that
$$\m... | 6 | https://mathoverflow.net/users/5101 | 342012 | 145,270 |
https://mathoverflow.net/questions/342011 | 0 | If $(X,d)$ is a metric space, say a function $\tau$ on some class $\mathscr{C}$ of subsets of $X$ is a pre-measure, if $\emptyset \in \mathscr{C}$, $\tau(\emptyset)=0$ and $0\le \tau(C)\le +\infty$ for all $C\in \mathscr{C}.$
If $\tau$ is a pre-measure on some class $\mathscr{C}$ of subsets of $X$, then the set funct... | https://mathoverflow.net/users/129565 | Borel sets and Method I measure | Anything degenerate enough will do.
Suppose $\mathcal C=\{\emptyset,X\}$ and, say, $\tau(X)=1$.
Then $\mu(E)=1$ for every non-empty $E\subset X$.
Therefore any set $E\subset X$ which is not $X$ nor $\emptyset$ will not be measurable.
Indeed take $A=\{x,y\}$ where $x\in E$ and $y\in X\setminus E$.
Then $\mu(A\cap E)... | 1 | https://mathoverflow.net/users/18698 | 342013 | 145,271 |
https://mathoverflow.net/questions/341953 | 9 | I have seen on the [Wikipedia page](https://en.wikipedia.org/wiki/Mary_Cartwright) for the mathematician Mary Cartwright that she achieved many new results in the field of entire functions and the zeroes of entire functions and that many of these were included in her 1956 book on the subject.
I do not have access to ... | https://mathoverflow.net/users/119114 | Contributions of Mary Cartwright to the theory of entire functions | Mary Cartwright proved many important theorems in the theory of entire functions (too many to list them here). For a survey of her contributions I recommend her obituary:
[Zbl 1032.01034](https://zbmath.org/?q=an%3A1032.01034)
Hayman, W. K.
Mary Lucy Cartwright (1900–1998),
Bull. London Math. Soc. 34 (2002), no. 1, 9... | 13 | https://mathoverflow.net/users/25510 | 342014 | 145,272 |
https://mathoverflow.net/questions/342017 | 2 | Let $(\Omega,\leq)$ be a countable linear order. Suppose that for every finite $m \in \mathbb{N}$, and all subsets $S\_1$ and $S\_2$ of $\Omega$ of order $m$, there is an order-automorphism of $(\Omega,\leq)$ that sends $S\_1$ to $S\_2$.
Call the order-automorphism group $A$.
Is there a "fundamental theorem" for ... | https://mathoverflow.net/users/12884 | Fundamental theorem of linear orders | As already observed, you may as well assume $(\Omega,\leq)$ is $(\mathbb{Q},\leq)$. In this case, of course one condition on $\Omega\_1$ and $\Omega\_2$ ensuring your desired property is finiteness.
So the interesting question is about the case when $\Omega\_1$ and $\Omega\_2$ are infinite. We can even just consider... | 5 | https://mathoverflow.net/users/38253 | 342024 | 145,275 |
https://mathoverflow.net/questions/341803 | 4 | Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra, and $V$ an irreducible finite-dimensional $\mathfrak{g}$-module. Then $\mathfrak{g}$ also acts on the symmetric algebra $S(V)$.
>
> Is there a description of the invariants $S(V)^\mathfrak{g}$?
>
>
>
If $V$ is the standard module of th... | https://mathoverflow.net/users/15292 | Invariants in the symmetric algebra of a module | The question subsumes all of 19th Century invariant theory, so I don't think there is much chance of a really explicit answer.
For example, take $\mathfrak{g} = \mathfrak{sl}\_d(\mathbb{C})$ and let $V$ be the $\mathfrak{sl}\_d(\mathbb{C})$-module obtained from the polynomial representation $\nabla^\lambda(\mathbb{C}... | 4 | https://mathoverflow.net/users/7709 | 342033 | 145,279 |
https://mathoverflow.net/questions/299147 | 4 | I am currently looking for a formulation of a Fourier transform on manifolds of constant negative curvature. Specifically, I am looking for generalizations of the two dimensional results on the poincare disk parameterization given in [1].
Using notation $x = (x\_1,x\_2,x\_3,\ldots)$, I have found that the metric of t... | https://mathoverflow.net/users/123944 | Harmonic analysis on constant curvature hyperbolic spaces of arbitrary dimension | This may be explained in Helgason's *Geometric Analysis on Symmetric Spaces*. Theorem 1.3 on Page 201 lists a general Fourier inversion formula.
If I am parsing everything right, the function $g(\lambda)$ you are asking about is given in the formula as $|\mathbf{c}(\lambda)|^{-2}$, and where $\mathbf{c}$ is [Harish-... | 4 | https://mathoverflow.net/users/3948 | 342043 | 145,281 |
https://mathoverflow.net/questions/342053 | 4 | In a [previous question](https://mathoverflow.net/questions/338565/explanation-for-squashing-and-stretching-lorentzian-analogue-of-berger-sphe) by me I asked about Berger spheres and their Lorentzian analogue, *squashed* $AdS\_3$ along Hopf fibres. It was well answered (by <https://mathoverflow.net/users/13268/ben-mcka... | https://mathoverflow.net/users/142501 | Literature Request: Berger Spheres and their Construction | For a different viewpoint, the Berger Spheres or their Lorentzian analogues are well understood using the canonical variation of the metric associated with a Riemannian submersion with totally geodesic fibers, see Section 5 in [this paper](https://projecteuclid.org/download/pdf_1/euclid.ijm/1256046790).
In the case o... | 9 | https://mathoverflow.net/users/48356 | 342056 | 145,284 |
https://mathoverflow.net/questions/342038 | 2 | Let $G$ be a simple graph and $S$ be the set of all sub graphs of $G$. Define two operations on $S$ as: $union$ of two graphs $ G\_1$ and $G\_2$ is,
$G\_1\cup G\_2$
$=\langle V(G\_1)\cup V(G\_2), (E(G\_1)\cup E(G\_2)\rangle$
and the graphs $intersection$ is, $G\_1\cap G\_2$
$=\langle V(G\_1)\cap V(G\_2), (E(G\_1... | https://mathoverflow.net/users/116857 | Define a homomorphism of a set of graphs to its power set | A possible answer is the following, if you are willing to relax the definitions (in a very minor way) of union and intersection of two graphs.
For clarity, let me introduce an additional notation. For any subgraph $H$ of $G$, considered as a graph with vertex set $V(G)$ (this is trivially possible by adding those ver... | 1 | https://mathoverflow.net/users/110337 | 342059 | 145,286 |
https://mathoverflow.net/questions/342042 | 4 | Let $n$ be a large integer, $p$ be a small number (say, $p=C/n$ for some constant $C \ll n$), and consider the tail of the binomial distribution $B(n,p)$, after $T$:
$$
\delta = \sum\_{s=T}^{n} p^s (1-p)^{n-s} \binom{n}{s}
$$
I'd like to find a $T$ large enough that $\delta$ is on the order of $1/n$. [Berry-Esseen]... | https://mathoverflow.net/users/120465 | How far do I have to go for the tail of a binomial distribution with small $p$ to be $O(1/n)$? | $\newcommand{\de}{\delta}$
We have
\begin{equation}
\de=\sum\_{j=k}^n a\_j,
\end{equation}
where $k:=T$,
\begin{equation}
a\_j:=a\_{n,j;p}:=\binom nj p^j q^{n-j},
\end{equation}
$p=c/n$, $c:=C\in(0,\infty)$ (a constant), and $q:=1-p=1-c/n$.
Suppose now that $n\to\infty$, $b\in(0,\infty)$ (a constant), and
\be... | 8 | https://mathoverflow.net/users/36721 | 342064 | 145,289 |
https://mathoverflow.net/questions/342041 | 20 | Is it possible to constructively prove that every $q \in \mathbb H$ has some $r$ such that $r^2 = q$? The difficulty here is that $q$ might be a negative scalar, in which case there might be "too many" values of $r$. Namely, $r$ could then equal any vector quaternion of magnitude $\sqrt{|q|}$. The presence of this seem... | https://mathoverflow.net/users/75761 | Is it possible to constructively prove that every quaternion has a square root? | Reduction to LLPO (Lesser Limited Principle of Omniscience).
The statement LLPO is the following (from Wikipedia): For any sequence *a*0, *a*1, ... such that each *a**i* is either 0 or 1, and such that at most one *a**i* is nonzero, the following holds: either *a*2i = 0 for all *i*, or *a*2i+1 = 0 for all *i*, where ... | 14 | https://mathoverflow.net/users/75761 | 342065 | 145,290 |
https://mathoverflow.net/questions/342057 | 6 | In *Example 1.1.4* of the book [*Grobner Deformations of Hypergeometric Differential Equations*](https://www.springer.com/gp/book/9783540660651), it is stated without proof that
$$\partial^2 \in D\cdot \langle x\partial^4, x^3\partial^2 \rangle \tag{$\star$}$$
where $D$ denotes the Weyl algebra over $\mathbb{k}[x]$... | https://mathoverflow.net/users/123926 | Testing ideal membership in the Weyl algebra: a simple example | Following my nose gave the following argument. Writing $I$ be the left ideal generated by $x\partial^2$ and $x^3$ and using $\cdot$ to stress multiplication we get
$$ x^2 \cdot x\partial^2 - \partial^2\cdot x^3 = [ x^3, \partial^2] = -6x^2\partial - 6x\in I$$
So $$\frac{1}{6}x\partial\cdot (-6x^2\partial - 6x) + x... | 5 | https://mathoverflow.net/users/345 | 342082 | 145,295 |
https://mathoverflow.net/questions/342073 | 3 | Wikipedia states under the entry for the [von Mangoldt function](https://en.m.wikipedia.org/wiki/Von_Mangoldt_function):
*The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann zeta function zeros.*
(I believe "ordinates" should be changed... | https://mathoverflow.net/users/12178 | Fourier transform of the von Mangoldt function? | Start with the explicit formula
$$\sum\_{n \le x}\Lambda(n) =\frac1{2i\pi} \int\_{2-i\infty}^{2+i\infty} \frac{-\zeta'(s)}{\zeta(s)}\frac{x^s}s ds=1\_{x > 1}\sum Res(\frac{-\zeta'(s)}{\zeta(s)}\frac{x^s}s)$$ $$=1\_{x > 1}( x - \sum\_\rho \frac{x^\rho}{\rho} - \frac12 \log 2\pi - \sum\_{k=1}^\infty \frac{x^{-2k}}{-2k}... | 3 | https://mathoverflow.net/users/84768 | 342096 | 145,301 |
https://mathoverflow.net/questions/328741 | 8 | Let $X$ be a proper, geodesic, $\delta$-hyperbolic metric space (e.g. a hyperbolic group), and let $x\_0$ be a basepoint for $X$. There seem to be two different definitions of "horofunction" for $X$, and I'd like to understand the relationship between them.
### First Definition
>
> **Definition 1.** For each $p\i... | https://mathoverflow.net/users/6514 | Two definitions of horofunction for Gromov hyperbolic spaces | I don't know whether this was known before, but Collin Bleak, Francesco Matucci, and I have settled this question in the course of our work on our recent paper [[1]](https://arxiv.org/abs/1711.08369). The answer is that any horofunction satisfying Definition 1 satisfies Definition 2, but there exist hyperbolic groups w... | 3 | https://mathoverflow.net/users/6514 | 342100 | 145,303 |
https://mathoverflow.net/questions/342060 | 0 | Add a primitive a one place function symbol $c$ to represent "True cardinality" of a set, to the first order language of set theory.
Add the following axiom schema:
**1. Cardinal Equality:** If $\phi(x,y)$ is a formula in which *both and only* $x,y$ occur free, and only occur free, then all closures of:
$\forall... | https://mathoverflow.net/users/95347 | Can ZFC commit cardinality errors? | We assume |X| is the least von Neuman ordinal for which there is a bijection from it to X. Then ZFC cannot "commit cardinality error of the second kind". This is true because your axiom scheme and
rule of inference hold in ZFC when c(X) is |X|.
In order to verify the rule of inference holds when c(X) is |X|, suppos... | 2 | https://mathoverflow.net/users/133981 | 342106 | 145,305 |
https://mathoverflow.net/questions/341996 | 2 | Let $M$ be an affine complex manifold, let $A$ be an abelian scheme over $M$. Let $\mathcal{A}$ be the sheaf of local sections of $A$ over $M$. We can equip $M$ with etale topology $M\_{et}$ or complex topology $M\_{an}$. There is a natural comparison map $$\gamma\colon H^1(M\_{et},\mathcal{A})\to H^1(M\_{an},\mathcal{... | https://mathoverflow.net/users/nan | Analytic and algebraic torsor of abelian scheme | Here is an example when $\gamma $ is not injective.
In general, if $A=A\_0\times M$ is a constant abelian scheme, choose a presentation for $(A\_0)\_{an}$ as $\mathbb{C}^g/\mathbb{Z}^{2g}$. This induces a short exact sequence of sheaves on $M\_{an}$: $$0\to\underline{\mathbb{Z}}^{2g}\to\mathcal{O}\_{M\_{an}}^{\oplus ... | 1 | https://mathoverflow.net/users/39304 | 342111 | 145,308 |
https://mathoverflow.net/questions/342114 | 0 | Let $SF$ be the schema of stratified comprehension.
Take the theory $SF + Infinity + Choice + \text {Extensionality fails everywhere}$
Is the following consistent with this theory?
$\exists \iota \forall x (x \in \iota \leftrightarrow \exists a (x = \langle a, b \rangle \land \forall m (m \in b \leftrightarrow m... | https://mathoverflow.net/users/95347 | Can global failure of Extensionality in fragments of NFU permit existence of singleton relation set? | No.
Let s be the relation whose elements are all the pairs (x,y) such that
x is the only element of y [the set the poster postulates].
Now let R be the set of all x such that for all y such that (x,y) E s,
y is not a subset of x. This definition is stratified: if s exists, R must exist. This set R is the Russell cl... | 4 | https://mathoverflow.net/users/130007 | 342123 | 145,312 |
https://mathoverflow.net/questions/342121 | 4 | Let $O$ be a matrix sampled from the Haar measure on $O(n)$. Let $X$ be the upperleft $k\times k$ submatrix of $O$.
In a physics research project I am interested in the distribution of $X$, say $\rho(X)$. What I was able to prove for $k=1$ and $k=2$ is that:
$k=1$: $\rho(X)\propto(1-X^2)^{(n-3)/2}$.
$k=2$: $\rho(... | https://mathoverflow.net/users/146159 | Distribution of Submatrix of Orthogonal Matrix | The probability distribution of $X$ was calculated in [Random-matrix theory of thermal conduction in superconducting quantum dots](https://arxiv.org/abs/1004.2438). In the context of that physics problem, the $k\times k$ upper-left submatrix $X$ of an $n\times n$ orthogonal matrix $O$ is the reflection matrix of a supe... | 4 | https://mathoverflow.net/users/11260 | 342124 | 145,313 |
https://mathoverflow.net/questions/342144 | 3 | Let $F$ be a number field, $p\in\mathbb{Z}$ a prime which is unramified in $F$ and $G$ a connected reductive group over $F$. Moreover $G$ is supposed to be quasi-split over $p$.
Does there exist a finite set of primes $S\subset\mathbb{Z}$ which does not contain $p$ and a split reductive group $\mathcal{G}$ defined o... | https://mathoverflow.net/users/143426 | split integral model of a reductive group | If I understand the question correctly, the answer is no, but I think it can be changed to yes if the word quasi-split is replaced by unramified.
Let E be a quadratic extension of the rational numbers ramified at p. Let G be the special unitary group associated to the quadratic extension E/Q. This is a quasi-split gr... | 2 | https://mathoverflow.net/users/425 | 342151 | 145,322 |
https://mathoverflow.net/questions/342138 | 7 | Let $N$ be a positive integer and $0 \leq s < N$.
We try to divide $s$ into $N$ using the Euclidean algorithm:
$N = q\_1 s + r\_1 $
$r = q\_2 r\_1 + r\_2 $
$\vdots$
$r\_{K-1} = q\_{K-1} r\_K$
If we choose $-r\_{i-1}/2 \leq r\_i < r\_{i-1}/2$, I think this determines the $q\_i$'s uniquely, but I don't think th... | https://mathoverflow.net/users/92401 | Average number of iterations for the Euclidean algorithm to terminate | This algorithm correspons to [nearest integer continued fractions](http://mathworld.wolfram.com/NearestIntegerContinuedFraction.html) or centered continued fraction. The length of such fraction $l(a/b)$ can be expressed in terms of Gauss - Kuz'min statistics for classical continued fraction expansion, see [The mean num... | 11 | https://mathoverflow.net/users/5712 | 342156 | 145,324 |
https://mathoverflow.net/questions/342159 | 3 | I need an example of a periodic function $q:\mathbb{R} \to \mathbb{R}$ with period $\pi$ such that if we consider the differential equation
\begin{equation}\tag{1}
y''(x)+(\lambda -q(x))y(x)=0
\end{equation}
and the boundary conditions
\begin{equation}\tag{2}
y(0)=y(\pi), \quad y'(0)=y'(\pi)
\end{equation}
\begin{eq... | https://mathoverflow.net/users/142048 | Example of differential equation with periodic boundary conditions that has at least two simple eigenvalues | For the choice $q(x)=2q\cos (2x)$ (with $q$ on the right hand side a constant, I'm trying to stick to standard notation), the differential equation is known as Mathieu's equation, with solutions described, e.g., in the [NIST Handbook](https://dlmf.nist.gov/28.2#vi). The solutions for (1),(2) are the ones commonly denot... | 2 | https://mathoverflow.net/users/134299 | 342177 | 145,327 |
https://mathoverflow.net/questions/342185 | 0 | So I am stuck at this situation. Let $\{A\_n\}$ be a weakly convergent sequence in $B\_2(H)$ converging to $0$ in the weak topology on $B\_2(H)$. Which means that $\left<A\_n,D\right>=\operatorname{tr}(D^\*A\_n)\to 0$ for each $D\in B\_2(H)$. I want to prove/disprove that $\|A\_n\|\_2\to 0$,i.e $A\_n\to 0$ . Clearly $$... | https://mathoverflow.net/users/145729 | Weak convergence of Hilbert Schmidt operators | Well, the space of Hilbert-Schmidt operators is a Hilbert space, so you are asking whether weak convergence to zero implies norm convergence to zero in a Hilbert space. The answer is no. For instance, let $A\_n$ be the rank 1 projection onto $e\_n$. This converges weakly but not in norm to zero.
| 7 | https://mathoverflow.net/users/23141 | 342190 | 145,330 |
https://mathoverflow.net/questions/341556 | 4 | I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of **parabolic PDEs** (basically the heat equation). As key words here we can cite for instance:
1) Carleman estimate (to prove uniqueness and stability results).
2) Regularization theory (to r... | https://mathoverflow.net/users/124904 | Interesting questions for inverse parabolic problems | [Inverse Problems for Partial Differential Equations](https://books.google.nl/books?id=K_aNMWE5O38C) (third edition, 2017) by Victor Isakov concludes each chapter with a collection of open research problems. Chapter 9 is specifically devoted to inverse parabolic problems. I reproduce one of the "open problems" from tha... | 3 | https://mathoverflow.net/users/11260 | 342191 | 145,331 |
https://mathoverflow.net/questions/341669 | 5 | Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
* Say that the action is acylindrical if for every $\epsilon$, there exist $R,N$ such that for every two points $x,y\in X$ with $d(x,y)\geq R$, there are at mos... | https://mathoverflow.net/users/111917 | Relations between boundaries of groups acting on hyperbolic spaces with WPD elements | I think the equivariant embedding you ask for is given in Theorem 3.2 of this paper:
<https://arxiv.org/pdf/1601.00101.pdf>.
Actually, there is such an embedding any time you cone off uniformly quasiconvex subspaces of a hyperbolic space.
Added:
Theorem 3.2 states that if $X$ and $Y$ are hyperbolic and $f \colo... | 2 | https://mathoverflow.net/users/27850 | 342192 | 145,332 |
https://mathoverflow.net/questions/342074 | 2 | As described in the title, what is the (topological) Euler characteristic of the homogeneous space $SL\_m(\mathbb{C})/SO\_m(\mathbb{C})$?
| https://mathoverflow.net/users/98788 | Euler Characteristic of $SL_m(\mathbb{C})/SO_m(\mathbb{C})$ | Let $G$ be a connected complex reductive affine algebraic group, and let $H$ be an algebraic subgroup (assume reductive). Then the Euler characteristic of the homogeneous space $G/H$ can be computed in many different ways.
As noted in the comments, $G/H$ is homotopic to $K/J$ where $K\subset G$ and $J\subset H$ are m... | 6 | https://mathoverflow.net/users/12218 | 342202 | 145,336 |
https://mathoverflow.net/questions/342140 | 2 | I'm trying to learn about negative association of random variables. A definition can be found here: <http://www.cs.cmu.edu/~dwajc/notes/Negative%20Association.pdf>.
Now, consider the following question:
Let $X\_1$, $X\_2$ be independent but not necessarily identically distributed
random variables. Let $σ\_1, σ\_2$... | https://mathoverflow.net/users/146166 | Simple question regarding negatively associated random variables | The answer is no. Why?
Suppose $X\_1=2$ a.s. and $X\_2$ takes the values 1 and 3 with Probability $1/2$ each. Let $f$ be the indicator of $[3,\infty)$ and let $g$ be the indicator of $[2,\infty)$. Then the expectation of $f(Y\_1) g(Y\_2)$ is $1/4$. But $f(Y\_1)$ has expectation $1/4$ and $g(Y\_2)$ has expectation $3/4... | 4 | https://mathoverflow.net/users/7691 | 342208 | 145,338 |
https://mathoverflow.net/questions/341333 | 4 | Let $f: \mathbb R \rightarrow (-\infty,+\infty]$ be a lower-semicontinuous convex function.
Question
========
* Under what futher conditions does there exists a convex decreasing function $\phi: \mathbb R \rightarrow \mathbb R$ such that $f(x) = \sup\_y x\phi(y)-\phi(-y)$ for all $x \in \mathbb R$ ?
* Construct suc... | https://mathoverflow.net/users/78539 | Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$ | $\newcommand{\R}{\mathbb{R}}
\newcommand{\tto}{\underset{\text{onto}}\to}$
Let us answer the reformulated question: given a convex function $g\colon C\to\R$, when is it possible to find a decreasing convex function $\phi\colon\R\to\R$ such that
\begin{equation}
\phi\circ(-\phi^{-1})=g? \tag{1}
\end{equation}
Here $C$... | 2 | https://mathoverflow.net/users/36721 | 342212 | 145,341 |
https://mathoverflow.net/questions/342217 | 6 | Given a positive integer $P>1$, let its prime factorization be written $$P=p\_1^{a\_1}p\_2^{a\_2}p\_3^{a\_3}\cdots p\_k^{a\_k}$$
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a\_1, a\_2,\ldots,a\_k)$
>
> Is the follows property true or false?
>
>
> **The property**: *Let $n$ is a positive integer then ... | https://mathoverflow.net/users/122662 | Is min exponents of three positive integers $n$, $n+1$ and $n+2$ $=1$ true or false? | It is actually an old conjecture of **Erdős, Mollin, and Walsh** that the pattern you have noticed does indeed go on forever, i.e., there are no three consecutive [powerful numbers](https://en.wikipedia.org/wiki/Powerful_number).
| 15 | https://mathoverflow.net/users/4600 | 342219 | 145,342 |
https://mathoverflow.net/questions/341830 | 22 | Let $[x]\_q=\frac{1-q^x}{1-q}$, $[n]\_q!=[1]\_q[2]\_q\cdots[n]\_q$ and ${\binom{x}{n}}\_{q}=\frac{[x]\_q[x-1]\_q\cdots[x-n+1]\_q }{[n]\_q!}$.
Computer experiments suggest that
$$\det \left(q^\binom{i-j}{2}\left(\binom{i+r}{j}\_{q}x+\binom{i+r-j}{j}\_{q}\right)\right)\_{i,j = 0}^{n - 1} = (1+x)^n$$
Any idea how to p... | https://mathoverflow.net/users/5585 | A q-rious identity | Let $A\_n$ be the matrix involved in the problem and let $L\_n=\left((-1)^{i-j}\binom{i}{j}\_q\right)\_{i,j=0}^{n-1}$.
Observe that $L\_n$ is lower-triangular with 1's in the diagonal. Multiplying, we have:
$$L\_nA\_n=\left(x\,u\_{i,j}(r)+u\_{i,j}(r-j)\right)\_{i,j=0}^{n-1}$$
where
$$u\_{i,j}(r)=\sum\_{k=0}^i(-1)^{i... | 4 | https://mathoverflow.net/users/146235 | 342221 | 145,343 |
https://mathoverflow.net/questions/342146 | 1 | For a topological space $X$ and one point $x\in X$, we call the cofinal type of neighborhood bases of $x$ are cofinally finer than $\omega^\omega$-base if for any neighborhood base $\mathfrak{N}$ of $x$ there exists a convergent map $f:\mathfrak{N}\rightarrow \omega^\omega$ ,i.e.,for any $\alpha\in \omega^\omega$ there... | https://mathoverflow.net/users/111291 | Is there some characterization of $\omega^\omega$-base related to $S_\omega$? | This question has negative answer.
To construct a counterexample, consider the set $X=(\omega\times\omega)\cup\{\infty\}$ where $\infty\notin(\omega\times\omega)$ is any point. Then set $X$ is endowed with the topology $\tau$ consisting of the sets $U\subset X$ satisfying the following condition:
$\bullet$ if $\inf... | 2 | https://mathoverflow.net/users/61536 | 342235 | 145,348 |
https://mathoverflow.net/questions/342234 | 0 | Suppose $f:\mathbb C \to \mathbb C$ is an entire function on the complex plane of order $1$.
Additionally, suppose that:
$$ \forall\, c \in \mathbb R, \quad \lim\_{t \to \pm \infty} \, f(t+ic) =0.$$
Can one conclude that $f \equiv 0$?
| https://mathoverflow.net/users/50438 | A question about entire functions of order 1 | The answer is no: think of $f(z):=\frac {\sin z} z$. Indeed,
$$\frac {\sin (t+ic)} {t+ic}=\frac {\sin(t)\cos(ic)+\cos(t)\sin(ic)} {t+ic}. $$
| 6 | https://mathoverflow.net/users/35959 | 342237 | 145,349 |
https://mathoverflow.net/questions/342218 | 5 | [This](http://www.ecs.umass.edu/~mduarte/images/IntroCS.pdf) excellent introduction to Compressive Sensing cites a couple of (seemingly) interesting Caratheodory papers from 1907-1911.
These are:
* [46] C. Caratheodory. Uber den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht anneh... | https://mathoverflow.net/users/23206 | Translated version of a Caratheodory article | Google translate output of the first page of [the 1907 paper](https://link.springer.com/content/pdf/10.1007/BF01449883.pdf), no postprocessing (other than LaTeXing the formulas).
>
> **Over the range of variability of the coefficients of power series that
> do not assume given values.**
>
>
> **Introduction.** ... | 3 | https://mathoverflow.net/users/11260 | 342243 | 145,350 |
https://mathoverflow.net/questions/342246 | 2 | Working in an order $\mathcal{O}$ in an imaginary quadratic field $K = \mathbb{Q}(\sqrt{d})$ and given an invertible ideal $\mathfrak{a}\subseteq \mathcal{O}$, I would like to produce another integral ideal $\mathfrak{b}$ in the same equivalence class of the ideal class group, i.e. $\mathfrak{b}=\alpha\mathfrak{a}$ for... | https://mathoverflow.net/users/131900 | Representative in ideal class group coprime to the conductor | There is probably better algorithmically but
* $C = \{ b\in O\_K,bO\_K\subset O\}$ is a $O$ and $O\_K$ ideal.
Choose some $O\_K$-representatives of $O\_K/C$ and let $f(a+C) = a O\_K \cap O$ whose class doesn't depend on the representative $a$.
$f(O\_K/C)$ contains the kernel of $\phi : Cl(O) \to Cl(O\_K), \phi(I... | 4 | https://mathoverflow.net/users/84768 | 342253 | 145,354 |
https://mathoverflow.net/questions/342261 | 1 | While doing a research work, I had to read about the Glauber dynamics for an Ising model. A wonderful account on this is given in the book *Markov Chains and Mixing Times* by Levin, Peres and Wilmer.
In Chapter 15 (proof of Theorem 15.1), the authors seem to show that the Glauber dynamics chain is $(1-\alpha/n)$-con... | https://mathoverflow.net/users/123578 | Coupling argument involved in the contracting and mixing properties of the Glauber dynamics for an Ising model | The Hamming distance is a graph metric and therefore it suffices to check the expected distance between neighboring states contracts. Concretely, given two arbitrary initial states of Hamming distance $k$, find a path of length k connecting them by changing spins one by one. Then use the triangle inequality for the tra... | 2 | https://mathoverflow.net/users/7691 | 342263 | 145,357 |
https://mathoverflow.net/questions/342260 | 6 | I have posed the following question on [math.stackexchange.com](https://math.stackexchange.com/q/3365266/64809) but have not received an answer. So I would like to seek experts' opinion here.
Consider the set of all binary sequence of length $n+1$, $B=\big\{(b\_i)\_{i=0}^n\,\big| b\_i\in\{0,1\}, \forall i\big\}$. Con... | https://mathoverflow.net/users/32660 | Guessing the number of other $1$'s in a binary sequence | Since every person knows his assigned digit, the problem is equivalent to guessing the sum of all digits.
Label the persons with $0,1,...,n$. Let person $0$ guess $0$ when given a $0$, and $n+1$ when given a $1$. Let person $k$ guess $k$ for every $k\in \{1,...,n\}$.
At least one person can correctly guess the sum ... | 6 | https://mathoverflow.net/users/125498 | 342264 | 145,358 |
https://mathoverflow.net/questions/342273 | 7 | Is there a formula for $\int \chi(K \cap gL) \: dg$ (where $\chi$ is Euler characteristic) analogous to the kinematic formula for $\int \mu(K \cap gL) \: dg$ (where $\mu$ is Lebesgue measure)? In both expressions $K$ and $L$ are compact convex bodies, $g$ varies over a group of isometries acting on the ambient space, a... | https://mathoverflow.net/users/3621 | Kinematic formula for Euler characteristic | Yes, this is called the principal kinematic formula:
$$\int \chi(K \cap gL)\, dg = \sum\_{k=0}^n c\_{nk} V\_k(K) V\_{n-k}(L),$$
where $V\_i$ are the intrinsic volumes, and $c\_{nk}$ certain constants. See e.g. Section 4.4 in
*Schneider, Rolf*, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and... | 9 | https://mathoverflow.net/users/98590 | 342278 | 145,361 |
https://mathoverflow.net/questions/203627 | 19 | I initially asked this question on [MSE](https://math.stackexchange.com/q/1219052/39599) but I haven't had any luck.
---
The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the manifolds are real analytic, is every continuous map between them ... | https://mathoverflow.net/users/21564 | Is the analytic version of the Whitney Approximation Theorem true? | The result is not stated in Grauert's paper. On the other hand, Grauert proves that every real analytic manifold $M$ sits as a real analytic totally real submanifold, and analytic deformation retraction, in a Stein manifold $M\_{\mathbb{C}}$. So every continuous map $\phi \colon M \to N$ of real analytic manifolds exte... | 10 | https://mathoverflow.net/users/13268 | 342279 | 145,362 |
https://mathoverflow.net/questions/342142 | 0 | Let $SF$ be the schema of stratified comprehension.
Take the theory $SF + Infinity + Choice + \text {Extensionality fails everywhere}$.
Are the following consistent with this theory?
1. $\forall X (|X| \leq |P\_1(X)|)$
2. $\forall X (Infinite(X) \to |X|=|P\_1(X)|)$
It is known that these two statements fail in... | https://mathoverflow.net/users/95347 | Can removal of extensionality avoid cardinality errors in stratified theories? | This is a partial answer, it only answers the first question.
Working in **NFU + Infinity + Choice + $|U|>|P(V)|$** [which is consistent relative to ZFC]
Since we have choice then there is a function $H$ from a partition $K$ on $U$ that has all of its pieces (i.e. elements of $K$) equinumerous to $|V|$, to $P1(P1(V... | 0 | https://mathoverflow.net/users/95347 | 342280 | 145,363 |
https://mathoverflow.net/questions/336536 | 1 | Is it possible to choose $k$ fixed point free maps $f\_i$ from an arbitrarily large finite set $X$ to itself such that:
$$\max\_{A\subset X} \vert A \setminus \cup\_{i=1..k} f\_i(A)\vert = O(\vert X\vert^{1-\epsilon})$$
for some $\epsilon >0$?
I am mostly interested in the case $k=2$ or $3$.
| https://mathoverflow.net/users/112954 | Almost covering every set with few images | The maximum is attained at a set $A$ disjoint with al its images (just replace $A$ with $A\setminus\bigcup\_i f\_i(A)$). For choosing that, you need to find an independent set in a digraph which is a union of $k$ graphs of out-degrees $1$, i.e., in a digraph of out-degree $k$.
Any such graph contains a vertex of tot... | 2 | https://mathoverflow.net/users/17581 | 342282 | 145,364 |
https://mathoverflow.net/questions/336348 | 8 | Define $A=(a\_n)$ and $B=(b\_n)$ by $b\_0=1$ and
$$a\_n=b\_n+b\_{2n}$$
for $n \geq 0$, where $A$ and $B$ are increasing and every positive integer occurs exactly once in $A$ or $B$. Can someone prove that
$$b\_{3n+2}=4n+4$$
for $n \geq 0$? Initial terms:
$$A=(2,7,10,14,18,23,26,31,34,38,43,46,50,\ldots)$$
$$... | https://mathoverflow.net/users/61426 | Simple-looking sequences $A$ and $B$ defined by a complementary equation | THis can be shown by a bit less concrete estimates that in @Deld's answer.
**[EDIT]** Initially I thought $(b\_i)$ starts with $b\_1$, but it actually starts with $b\_0$. Here is a proof for the actual case; the previous answer is left below.
We show by induction on $n$ that
$$
4n+2\leq a\_n\leq 4n+3; \qquad(\*)
... | 1 | https://mathoverflow.net/users/17581 | 342287 | 145,366 |
https://mathoverflow.net/questions/342283 | 2 | Consider the smooth vector fields $X=(X\_1,X\_2,...,X\_m)$ defined in a open bounded set $\Omega\in R^n$. And the non-isotropic dimension is $Q$ (see <https://arxiv.org/pdf/1502.06332.pdf> page 398)
In the paper above (page 398) the author gave a embedding that if $f\in H\_{X,0}^1$,
$$\|f\|\_{L^{p^{\*}}\left(\Omega^{\p... | https://mathoverflow.net/users/145357 | subelliptic Sobolev compact embedding theorem | The compactness for exponents below the embedding exponent is standard and true in a great generality. In your case compactness follows from results in Section 8 of [1]. Sobolev embeddings for vector fields are discussed in Section 11 of [1]. See also Theorem 4 in [2] for a very general compactness criteria. Both paper... | 3 | https://mathoverflow.net/users/121665 | 342296 | 145,369 |
https://mathoverflow.net/questions/342298 | 1 | Let $M\subset B(\mathcal{H})$ be an infinite dimensional vN algebra in standard form. Fix $\xi\neq 0 \in \mathcal{H}$, does there exist $M\ni x\_{\xi}\neq I$ such that $x\_{\xi}(\xi)=\xi$?
| https://mathoverflow.net/users/145907 | On existence of certain operators in von Neumann algebra | This is false.
Consider, for example, the case of M being the von Neumann algebra
of bounded complex-valued functions on an infinite countable set I.
It acts on the Hilbert space of square-summable functions on I,
which is the standard form of M.
Take any ξ∈H that is everywhere nonvanishing (e.g., n↦1/n).
Then any x∈M ... | 4 | https://mathoverflow.net/users/402 | 342303 | 145,371 |
https://mathoverflow.net/questions/342294 | 6 | Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
The following equality is famous:
$$\cfrac{q^{1/5}}{R(q)} = \prod\_{k>0} \cfrac{(1-q^{5k-2})(1-q^{5k-3})}{(1-q^{5k-1})(1-q^{5k-4})} ( = f(q))$$
The coefficients of $f(q)$ can be positive or negative. In fact,
$$f(q)... | https://mathoverflow.net/users/69834 | Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative? | Notice that we can write
$$f(q)=\prod\_{n\geq 1} (1-q^n)^{-\left(\frac{n}{5}\right)}$$
therefore
$$g(q)=\prod\_{k\geq 1} f(q^k)=\prod\_{n\geq 1} (1-q^n)^{-a(n)}$$
where $a(n)=\sum\_{d|n}\left(\frac{d}{5}\right)$, where $\left(\frac{d}{5}\right)$ is the Legendre symbol. Now, $a(n)$ is easily seen to be multiplicative wi... | 12 | https://mathoverflow.net/users/2384 | 342310 | 145,373 |
https://mathoverflow.net/questions/342293 | 4 | With $f(x\_1,x\_2,x\_3,x\_1+x\_2+x\_3;\,1/3,1/3,1/3):= \frac{(x\_1+x\_2+x\_3)!}{x\_1!\,x\_2!\,x\_3!\, 3^{x\_1+x\_2+x\_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in [wiki/Multinomial\_distribution](https://en.wikipedia.org/wiki/Multinomial_distribution), for any $x\in \math... | https://mathoverflow.net/users/101850 | Understanding equiprobable trinomial identity | $\newcommand{\Pr}{\mathbb{P}}$
I claim that this limit is in fact not zero. For each $N$, write $\Pr\_N$ to be the probability measure on the triple $(X\_1,X\_2,X\_3)$ that is trinomial with $N$ trials and parameters all equal to $1/3$. Then the sum you are interested in is \begin{align}\sum\_{N} \Pr\_N(X\_1 = x-1, 0... | 1 | https://mathoverflow.net/users/69870 | 342313 | 145,375 |
https://mathoverflow.net/questions/342051 | 2 | In the paper [Multiply Twisted Products](https://arxiv.org/abs/1207.0199v1) by Yong Wang, general definitions for so called *warped* and *twisted* products are given:
A **(singly) warped product** $B \times\_b F$ of two pseudo-Riemannian manifolds $\left(B\,,g\_B\right)$ and $\left(F\,,g\_F\right)$ with a smooth func... | https://mathoverflow.net/users/142501 | Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$ | The short answer is "what physicists mean by 'warped' and 'twisted' geometry" is not the same as "what differential geometers mean by 'warped' and 'twisted' geometry". The use is a lot more qualitative and a little loosey-goosey, but on the other hand very easy to visualize.
From the physicists' point of view, a **t... | 3 | https://mathoverflow.net/users/3948 | 342317 | 145,379 |
https://mathoverflow.net/questions/342300 | 6 | Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$.
Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$.
Let $\sigma$ be an *anti-holomorphic involution* of $G({\Bbb C})$, that is, a an automorphism of the real Lie group
$$\sigma\colon G({\Bbb C})\to G({\Bbb... | https://mathoverflow.net/users/4149 | Anti-holomorphic involutions of a complex linear algebraic group | (1): No; (2,3): Yes (and also for unipotent groups).
On the abelian group $\mathbb{G}\_{\mathrm{a}}\times \mathbb{G}\_{\mathrm{m}}=\mathbf{C}\times\mathbf{C}^\*$, consider the anti-holomorphic involution $$(z,w)\mapsto (\bar{z},\exp(i\bar{z})\bar{w}):$$
it is not "anti-regular".
In the semisimple case, it's the sam... | 5 | https://mathoverflow.net/users/14094 | 342328 | 145,382 |
https://mathoverflow.net/questions/342305 | 5 | Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$.
Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix.
Define $S^k=\frac{1}{\sqrt k}\sum\_{i=1}^k X^i$, so that $\mathbb C[S^k] = I$.
**Question**
We would like to prove a bound on ... | https://mathoverflow.net/users/5429 | Expected supremum of normalised random walk | Let $X\_i:=X^i$, $S\_k:=\sum\_1^n X\_i$, $T\_k:=S\_k/\sqrt k$, $|\cdot|:=\|\cdot\|\_2$, $n\in\{1,2,\dots\}$, and $m:=\lceil\log\_2 n\rceil$, so that $2^m\ge n$. Then
\begin{equation}
\max\_{k=1}^n|T\_k|\le\max\_{k=1}^{2^m}|T\_k|
=\max\_{j=0}^m\max\_{2^{j-1}<k\le 2^j}|T\_k|
\le\max\_{j=0}^m 2^{-(j-1)/2}\max\_{2^{j-1... | 3 | https://mathoverflow.net/users/36721 | 342329 | 145,383 |
https://mathoverflow.net/questions/342308 | 11 | Let $K\subset\mathbb{R}^n$ be any compact set. Let $\operatorname{Unp}(K)$ be the set of points in
$$
\operatorname{Unp}(K)=\{x\in\mathbb{R}^n\setminus K:\, \exists ! y\in K \ \ |x-y|=d(x,K)\}.
$$
Here are some properties.
1. The distance function is Lipschitz and hence differnetiable a.e. (Rademacher's theorem). I... | https://mathoverflow.net/users/121665 | Set of points with a unique closest point in a compact set | I *think* the following is a counterexample in $\mathbb{R}^2$. Consider the curve whose polar coordinates expression is $r = \sum\_{n=1}^\infty \frac{1}{a\_n} \sin(2\pi a\_n\theta)$, where $a\_n = 100^n$, say. Let $K$ be this curve together with all points interior to it.
It seems to me that any point $x$ with a uniq... | 7 | https://mathoverflow.net/users/23141 | 342330 | 145,384 |
https://mathoverflow.net/questions/342320 | 0 | Given vector field $f: \mathbb{R}^2 \to \mathbb{R}^2$, with $f(0)=0$
ODE: $\dot{x}=f(x)$ generates a flow $\Phi^{t}$. so $\Phi^{t}(0)=0$ for all $t \in \mathbb{R}$
So time-one map $\Phi^1$ is diffeo. Assume $0$ is hyperbolic fixed point,
let $W^s(0)$ be stable manifold for $\Phi^1$ through $0$. Then for any $x \... | https://mathoverflow.net/users/124254 | flow, stable manifold and tangent | Yes. The stable manifold of $\Phi^1$ is the stable manifold of $(\Phi^t)\_{t\in \mathbb R}$. Since 0 is a fixed point, the stable manifold of 0 is time-invariant. So $\Phi^t(x)$ moves along the stable manifold. In particular, the derivative at $t=0$ is both $f(x)$ and a tangent vector to the stable manifold.
| 3 | https://mathoverflow.net/users/11054 | 342332 | 145,385 |
https://mathoverflow.net/questions/342347 | 3 | **Question:** Is the conjecture as follows true or false?
>
>
> >
> > For any integer $n>1$, there always exists at least one prime number $p$ with
> >
> >
> > $$n < p< n+\left(\ln\Big(\frac{n}{\ln n}\Big)+1\right)^2$$
> >
> >
> >
>
>
>
The conjecture was checked true with $n$ up to $10^8$ and some [The... | https://mathoverflow.net/users/122662 | For any integer $n>1$, there always exists at least one prime number $p$ with $n < p< n+\left(\ln\Big(\frac{n}{\ln n}\Big)+1\right)^2$ | False.
Let $n=1693182318746371$. The next prime after $n$ is $1693182318747503$.
$(\ln(\frac{n}{\ln n})+1)^2 \le1057$, but the prime gap is $1132$.
| 12 | https://mathoverflow.net/users/125498 | 342355 | 145,389 |
https://mathoverflow.net/questions/338941 | 3 | It is well known that there exists a Quillen equivalence,
$$\mathfrak{C}: Set\_{\Delta} \rightleftarrows Cat\_{\Delta}: N, \enspace \mathfrak{C} \dashv N$$
between Joyal's model structure on simplicial sets and Bergner's model structure on simplicially enriched categories. I will call the left adjoint the rigidific... | https://mathoverflow.net/users/141150 | Rigidification of marked simplicial sets | Let $f : X \to Y$ be a map of marked simplicial sets.
Then $f$ is a weak equivalence if and only if the induced functor on localizations of simplicial categories (where marked arrows are localized) is an equivalence.
Indeed, $f$ is a weak equivalence if and only if $R(f)$ is a weak equivalence, where $R$ is a fibrant... | 1 | https://mathoverflow.net/users/62782 | 342357 | 145,390 |
https://mathoverflow.net/questions/338195 | 7 | I ask this question here since I asked it [here](https://math.stackexchange.com/questions/3310262/the-model-category-structure-on-tmon-and-homotopy-colimits) on Math.SE, and got no answers after a week of a bounty offer.
I am trying to understand the homotopy colimit of a diagram of topological monoids, and whether t... | https://mathoverflow.net/users/103150 | The model category structure on $\mathbf{TMon}$ | See this question:
[Model Structure/Homotopy Pushouts in topological monoids?](https://mathoverflow.net/questions/11059/model-structure-homotopy-pushouts-in-topological-monoids)
especially the answer by John Francis.
First of all, monoids have classifying spaces, and so a pushout
diagram $M\_1\gets M\_0 \to M\_... | 1 | https://mathoverflow.net/users/3634 | 342362 | 145,392 |
https://mathoverflow.net/questions/342389 | 4 | It's known that every irreducible projective singular cubic curve in $\mathbb{CP}^2$ is isomorphic (actually projectively equivalent) to either a cuspidal cubic $$y^2z=x^3,$$ or a nodal cubic $$y^2z=x^3+x^2z.$$ If we focus on the affine case, is it still true that every irreducible affine singular cubic curve in $\math... | https://mathoverflow.net/users/146366 | Are there only two irreducible affine singular cubic curves? | If you remove some (non-singular) points from the curve $y^2=x^3$, you get a singular cubic which is still affine (it is an open affine subset), yet it is not isomorphic to the original curve. Indeed, any isomorphism would induce an isomorphism between the regular loci of the curves. These loci are just projective line... | 4 | https://mathoverflow.net/users/6506 | 342390 | 145,399 |
https://mathoverflow.net/questions/342386 | 9 | Solovay proved that if $\kappa$ is inaccessible, then if we adjoin a generic $G \subseteq \mathrm{Col}(\omega,{<}\kappa)$, then in the extension, every set of reals in $L(\mathbb R)$ is Baire- and Lebesgue-measurable and has the perfect subset property. In several publications, such as “Martin’s Maximum Part 1” by Fore... | https://mathoverflow.net/users/11145 | Solovay’s model | The usual proof (at least the proof that I know) uses the fact that in $V[H]$, the forcing $P/H$ (or $P:Q$) is again equivalent to the Levy collapse, and in particular sufficiently homogeneous, so that every formula with parameters in $V[H]$ (in the forcing language of $P/H$) has Boolean value $1$ or $0$.
| 9 | https://mathoverflow.net/users/14915 | 342391 | 145,400 |
https://mathoverflow.net/questions/342387 | 3 | I have read in several texts that in order to show that a given functor from the category of schemes (over, say an algebraically closed field $k$) to $\mathsf{Set}$ is representable, it suffices to check that it is representable in the subcategory of affine schemes. How is this the case? I have not been able to find a ... | https://mathoverflow.net/users/128914 | Criteria for representability of a functor from schemes to sets | The statement
>
> to show that a given functor [$F\colon \mathsf{Sch}^{\mathit{op}}\to\mathsf{Set}$] is representable, it suffices
> to check that it is representable in the subcategory of affine schemes
>
>
>
isn't exactly true: for example, if $X$ is a scheme that's not affine, $\mathrm{Hom}(-, X)$ is repre... | 10 | https://mathoverflow.net/users/97265 | 342392 | 145,401 |
https://mathoverflow.net/questions/342366 | 0 | I'm trying to derive some weights expression for a boosting algorithm on a L2-ISE loss function, and i have trouble with the taylor expension.
Suppose that $f$ and $g$ are two densities from $\mathbb{R}^d$ to $R$. Denote by $L$ the loss from estimating $g$ by $f$ i.e the functional :
$$L(f) = \int \left(f(x) - g(... | https://mathoverflow.net/users/143783 | Taylor expension of a simple integral | If you want to be mathematically precise, you'd have to say a bit more about what space you're doing this on and whether you're thinking of a Frechet derivative or a Gateaux derivative, etc., but for practical purposes, you can think of functions like you would of components of a vector, just labeled by a continuous in... | 1 | https://mathoverflow.net/users/134299 | 342395 | 145,402 |
https://mathoverflow.net/questions/342393 | 13 | The following seems like an extremely basic algebraic topology question, but it's not something I ever learned, nor does it look familiar to the algebraic topologists I've asked.
>
> Let $f:X\to Y$ be a map, inducing $f^\*:H^\*(Y)\to H^\*(X)$. Hence the image $R$ of $f^\*$ is a subring of $H^\*(X)$. Is there a natu... | https://mathoverflow.net/users/391 | Image of a map on cohomology rings | No. Consider the Hopf map $\eta:S^3\to S^2$. If there were such a space $Z$, it would have $\widetilde H^\*(Z)=0$, so at the very least $Z$ would be stably trivial, forcing $\eta$ to be stably trivial; but it’s not.
| 25 | https://mathoverflow.net/users/3634 | 342396 | 145,403 |
https://mathoverflow.net/questions/342401 | 2 | Consider the integers $\alpha,\beta,\gamma,n>0$.
In which cases does the relation
>
> $$
> \gamma^n=\sum\_{k=1}^{n-1}\binom{n}{k}\alpha^{n-k}\beta^k=(\alpha+\beta)^n-\alpha^n-\beta^n
> $$
>
>
>
hold?
The problem rises in the context of [Waring's formula](https://it.wikipedia.org/wiki/Formule_di_Waring) (... | https://mathoverflow.net/users/124302 | Power of an integer as exact sum of mixed terms | Euler in 1769 [conjectured](https://en.m.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture) that for all integers n and k greater than 1, if the sum of k nth powers of positive integers is itself a nth power, then k is greater than or equal to n:
$$a^n\_1 + a^n\_2 + ... + a^n\_k = b^n ⇒ k ≥ n$$
The conjecture h... | 3 | https://mathoverflow.net/users/4600 | 342402 | 145,404 |
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