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https://mathoverflow.net/questions/347285 | 6 | Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{n \times n})$ denote the subalgebra of all $\phi \in \mathcal{L}(\mathbb{C}^{n \times n})$ which satisfy
$$
\phi(U^\*AU)... | https://mathoverflow.net/users/102946 | Commutant of the conjugations by unitary matrices | Building up on my comment, I can now give the complete answer. The space of matrices can be decomposed as follows:
$$
\mathbb M\_n(\mathbb C) = \mathbb C\cdot\mathrm{id}\oplus \mathfrak{sl}(n),
$$
where
$$
\mathfrak{sl}(n) = \{X\in\mathbb M\_n(\mathbb C)\mid \mathrm{Tr}(X) = 0\}.
$$
Thus, the conjugation representati... | 6 | https://mathoverflow.net/users/1275 | 347288 | 147,064 |
https://mathoverflow.net/questions/347134 | 5 | From the literature, showed below, I know two gadgets that provide a way to know if a positive integer (a positive quantity of units) is composite or a prime number. I would like to know if in the literature or from your invention it is possible to show other different gadgets that provide us primality tests.
>
> *... | https://mathoverflow.net/users/142929 | Gadgets as primality tests | I interpret a "gadget" as a physical device that operates in an analog, rather than a digital way (to exclude a computer). The OP asks for "primality tests", but if I may broaden the question to include "prime number generators", there is a variety of such gadgets, collected at [unusual and physical methods for finding... | 5 | https://mathoverflow.net/users/11260 | 347295 | 147,066 |
https://mathoverflow.net/questions/347289 | 1 | I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers.
**Conjecture.** *There are infinitely many prime numbers of the form* $$\frac{3a^2-a}{2}+b^4\tag{1}$$
*where* $a$ *and* $b$ *run over positive integers.*
>
> **Question.** Is there any reaso... | https://mathoverflow.net/users/142929 | Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$? | What can be done, which is already remarked in the cited paper by Friedlander and Iwaniec, is that for any positive definite binary quadratic form $f$ which has no local obstructions, there ought to exist infinitely many pairs $(a,b) \in \mathbb{Z}^2$ such that $f(a,b^2)$ is prime. Even though this is widely believed, ... | 7 | https://mathoverflow.net/users/10898 | 347301 | 147,070 |
https://mathoverflow.net/questions/347311 | 5 | André Weil sometimes glosses his Theorem of Decomposition in a simplified polynomial form:
>
> If $P(x,y)$ and $Q(x,y)$ are homogeneous polynomials algebraically
> prime to each other, with integer coefficients, and $x,y$ are integers
> prime to each other, then $P(x,y)$ and $Q(x,y)$ are ``almost" prime to
> eac... | https://mathoverflow.net/users/38783 | What is the elementary proof of Weil's polynomial theorem of decomposition? | If $P(x,y),Q(x,y)$ are relatively prime, then so are the one-variable polynomials $p(x)=P(x,1),q(x)=Q(x,1)$ (since we can homogenize any common factor of $p,q$ to a common factor of $P,Q$). It follows that in $\mathbb Q[x]$ there are two polynomials $a(x),b(x)$ such that $a(x)p(x)+b(x)q(x)=1$. Dehomogenizing and multip... | 11 | https://mathoverflow.net/users/30186 | 347313 | 147,074 |
https://mathoverflow.net/questions/347157 | 4 | I read in [nLab](https://ncatlab.org/nlab/show/algebraic+model+category) : *Every [cofibrantly generated model category](https://ncatlab.org/nlab/show/cofibrantly+generated+model+category) structure can be lifted to that of an [algebraic model category](https://ncatlab.org/nlab/show/algebraic+model+category). It is not... | https://mathoverflow.net/users/24563 | Does any accessible model category come from an algebraic model category? | For reference, here is a more detailed version of Riehl's argument.
**Definition 1** (Garner, [*Understanding the small object argument*](https://link.springer.com/article/10.1007/s10485-008-9137-4), Proposition 3.8) Let $J$ be a category and $D \colon J \to \mathcal{M}^\to$ a functor. Let $f \colon X \to Y$ be a map... | 5 | https://mathoverflow.net/users/30790 | 347324 | 147,078 |
https://mathoverflow.net/questions/347317 | 0 | Suppose that $\theta\_1$ and $\theta\_2$ are independent and identically distributed (i.i.d.) random variables and that $\theta\_j$ has probability density function (PDF) $f\_j = \frac{1}{2\pi}$ ($i.e.$, the uniform distribution) for $j = 1$ and $2$. Next, we define the following random variables $C = \cos \theta\_1 + ... | https://mathoverflow.net/users/103291 | Is the random point $(C,S)$ the same as $(1,0)+(\cos U,\sin U)=(1 + \cos U,\sin U)$, with $U$ a uniform r.v.? | These two scatter plots illustrate the difference, the first is for the points $(C,S)=(\cos\theta\_1+\cos\theta\_2,\sin\theta\_1+\sin\theta\_2)$, the second for the points $(1+\cos U,\sin U)=(1+\cos\theta\_3,\sin\theta\_3)$, where all angles $\theta\_i$ are uniformly distributed in $(0,2\pi)$.
^2}{k\_{i+1}-k\_i}.
$$
Of course, it is not hard to prove this formula, but it still looks a bit mysterious to me.
**Question** Has... | https://mathoverflow.net/users/21620 | Strange formula for area of a convex polygon | Denote $A\_i=(0,b\_i)$. It is a point on $\ell\_i=\{(x,y):y=k\_ix+b\_i\}$, and let $$P\_i=\ell\_i\cap \ell\_{i+1}=\left(\frac{b\_{i+1}-b\_i}{k\_i-k\_{i+1}},\frac{k\_ib\_{i+1}-k\_{i+1}b\_i}{k\_i-k\_{i+1}}\right)$$ be the vertex of the polygon. The (properly oriented) area of $\triangle A\_iP\_iA\_{i+1}$ equals $\frac{(b... | 10 | https://mathoverflow.net/users/4312 | 347331 | 147,082 |
https://mathoverflow.net/questions/347299 | 14 | Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows:
$$S:=\{t\in [0,\infty):nt\in A\text{ for infinitely many }n\in\Bbb N\}$$
What can we conclude about the measure of $S$?
I can guess that $\lambda (S)... | https://mathoverflow.net/users/129539 | Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure | Let $f(t) = 1$ if $t \in A$ and $f(t) = 0$ otherwise. Suppose that $a > 0$. Then
$$ \begin{aligned}
\int\_a^{2 a} \operatorname{card} \{n : n t \in A\} dt & = \int\_a^\infty \biggl(\sum\_{n = 1}^\infty f(n t) \biggr) dt \\
& = \sum\_{n = 1}^\infty \int\_a^{2 a} f(n t) dt \\
& = \sum\_{n = 1}^\infty \frac{1}{n} \int\_{n... | 17 | https://mathoverflow.net/users/108637 | 347332 | 147,083 |
https://mathoverflow.net/questions/347316 | 11 | Let $\pi\_p$ be an irreducible representation of $GL\_2(\mathbb{Q}\_p)$. Assume $\pi\_p$ is ramified,hence it will have a positive conductor. Consider $sym^3(\pi\_p)$ which is a representation of $GL\_4(\mathbb{Q}\_p)$. I want to know how to calculate the conductor of $sym^3(\pi\_p)$. What is the relation between condu... | https://mathoverflow.net/users/140336 | conductor formula | The [PhD thesis of Manami Roy](https://hdl.handle.net/11244/321046) (Univ. Oklahoma, 2019) gives a very detailed formula for the conductor of $\operatorname{Sym}^3(\pi\_p)$, where $\pi\_p$ is the representation of $\mathrm{GL}\_2(\mathbf{Q}\_p)$ coming from an elliptic curve. See Chapter 5 in particular.
(More preci... | 10 | https://mathoverflow.net/users/2481 | 347333 | 147,084 |
https://mathoverflow.net/questions/347355 | -1 | For $A\subseteq \mathbb{N}$, let the *upper density* of $A$ be defined by $$\mu^+(A) = \lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
Let $$A = \{n\in\mathbb{N}: 2n \text{ is the sum of } 2 \text{ primes}\}.$$ Can it be shown that $\mu^+(A) > 0$? What about $\mu^+(A) \geq 1/2$?
| https://mathoverflow.net/users/8628 | Statement about upper density of even numbers satisfying the Goldbach condition | The state-of-the art is contained in [this paper of János Pintz](https://arxiv.org/abs/1804.09084). Read the introduction, especially (1.9).
| 6 | https://mathoverflow.net/users/11919 | 347358 | 147,094 |
https://mathoverflow.net/questions/346675 | 0 | For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum\_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia [*Dedekind psi function*](https://en.wikipedia.org/wiki/Dedekind_psi_function)), then there exist integers $n\geq 1$ that satisfy $$\psi(\sigma(n))=2... | https://mathoverflow.net/users/142929 | Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function | There are no other solutions than $n=3$ and those from Claim: $n=2^{p-1}$ such that $2^p-1$ is prime.
Consider several cases.
1. $n=2^ks$ is even (here $k\geqslant 1$ and $s$ is odd). Then $$\frac{\psi(\sigma(n))}{n}=\frac{\psi((2^{k+1}-1)\sigma(s))}{(2^{k+1}-1)\sigma(s)}\cdot \frac{2^{k+1}-1}{2^k}\cdot \frac{\sig... | 4 | https://mathoverflow.net/users/4312 | 347359 | 147,095 |
https://mathoverflow.net/questions/347363 | 3 | I'd like to ask a question on "Asymmetry of Outer Space" by Yael Algom-Kfir & Mladen Bestvina.
In Example $2$, page $4$ it says "Note that in this case the asymmetry can be explained by the fact that the injectivity radius $\text{injrad}(x\_k)$ of $x\_k$ goes to $0$, and in fact $d(x\_k,x\_2)\sim -\log{\text{injrad}(... | https://mathoverflow.net/users/145318 | Asymmetry of outer space - injectivity radius | Given a geodesic metric space $X$ and a point $p\in X$, the injectivity radius $\mathrm{injrad}(p)$ is the maximum value of $r$ such that every point in the open ball $B(p,r)$ is connected to $p$ by a unique geodesic. Injectivity radius is important in the study of Riemannian manifolds (where it is often defined in ter... | 8 | https://mathoverflow.net/users/6514 | 347364 | 147,096 |
https://mathoverflow.net/questions/347318 | 5 | Let $L^1\_{loc}$ denote the set of all functions from $\mathbb{R}$ to itself which are locally integrable. For every infinite compact subset $K\subseteq \mathbb{R}$, let $L^1\_{m\_K}$ denote the space of Lebesgue measurable functions supported on $K$.
Clearly the collection $\mathcal{K}$ of all such compact subsets ... | https://mathoverflow.net/users/36886 | Can $L^1_{loc}$ be represented as colimit? | Interpreting the question as Dmitri Pavlov I assume that $L^1\_K$ is the space of $L^1$-functions with support in $K$ so that we have an inductive spectrum of Banach spaces (identifying almost everywhere equal functions). The colimit (=inductive limit) in the category of locally convex spaces (i.e., the union endowed w... | 6 | https://mathoverflow.net/users/21051 | 347372 | 147,098 |
https://mathoverflow.net/questions/347334 | 23 | Consider the moduli space $M\_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that $M\_g$ has dimension $3g-3$.
A modern proof involves deformation theory and Riemann-Roch theorem. In particular, one n... | https://mathoverflow.net/users/146366 | How did Riemann prove that the moduli space of compact Riemann surfaces of genus $g>1$ has dimension $3g-3$? | Riemann combines what is called Riemann-Roch and Riemann-Hurwitz nowadays.
He considers the dimension of the space of holomorphic maps of degree $d$ from the Riemann surface of genus $g$ to the sphere. He computes this dimension in two ways. By Riemann-Roch this dimension is
$2d-g+1$, for a fixed Riemann surface. (Inde... | 32 | https://mathoverflow.net/users/25510 | 347378 | 147,100 |
https://mathoverflow.net/questions/346431 | 1 | By the Banach-Mazur theorem, every separable Banach space $X$ embeds into $C([0,1])$. When $X$ is reflexive, it is not possible to find a sequence of disjointly supported, non-negative functions in any isometric image of $X$ in because this would generate a copy of $c\_0$. I am intersted in some special families of fun... | https://mathoverflow.net/users/148734 | Strictly increasing functions in reflexive subspaces of $C([0,1])$ | This is impossible. Each such function has norm 1 and only one supporting functional (point value at 1) in $C[0,1]$ so a'fortiori in this Hilbert space. But in Hilbert space the supporting functional uniquely defines the element.
| 7 | https://mathoverflow.net/users/149336 | 347379 | 147,101 |
https://mathoverflow.net/questions/347274 | 8 | Roughly, this question asks how the Bruhat (strong) order in type $D$ can be understood like the Bruhat orders in types A and B=C. I'll review how types A and B work before asking my question. As a side note, I tried to ask this question earlier today, then deleted it with the intention of fixing some errors and reaski... | https://mathoverflow.net/users/297 | Rank matrices for type $D$ Bruhat order | Here are some more thoughts about how the Type D Bruhat order is more complicated than the Type A and Type B/C orders. These ideas might even suggest that giving a "rank matrix"-like description of the partial order is "impossible" in Type D.
There is a certain property of posets called "clivage" (by Lascoux and Schü... | 2 | https://mathoverflow.net/users/25028 | 347381 | 147,103 |
https://mathoverflow.net/questions/347383 | 8 | **Conjecture** If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$.
Here, $A\otimes B$ denotes the [Kronecker Product](https://en.wikipedia.org/wiki/Kronecker_product) of two matrices. ... | https://mathoverflow.net/users/18474 | Eigenvectors of Kronecker Product | Counterexample: the matrix $I \times I$ has eigenvectors that are not in product form, since every vector is an eigenvector of it and not every vector can be written in product form.
| 8 | https://mathoverflow.net/users/1898 | 347385 | 147,105 |
https://mathoverflow.net/questions/347377 | 22 | Which natural number can be represented as a product of a sum of natural numbers and a sum of their inverses? I. e. does there exist for a natural $n$ a set of natural numbers $\{a\_1, a\_2,...a\_m\}$ such that $n = (a\_1 + a\_2 + ...+a\_m)(\frac{1}{a\_1} + \frac{1}{a\_2} + ... +\frac{1}{a\_m})$? Call $n$ good if such ... | https://mathoverflow.net/users/149334 | Harmonic sums and elementary number theory | A note on the observation "$n$ good implies $2n + 2$ good":
First remark is that $n$ is good iff there are positive *rational numbers* $a\_1, \dotsc, a\_m$ such that $n = (a\_1 + \dotsc + a\_m)(1/a\_1 + \dotsc + 1 / a\_m)$. This is because one can multiply all $a\_i$ by the lcm of their denominators.
Second remark ... | 19 | https://mathoverflow.net/users/76332 | 347387 | 147,106 |
https://mathoverflow.net/questions/347394 | 7 | Historically (as I gather from [Learning Class Field Theory: Local or Global First?](https://mathoverflow.net/questions/6932/learning-class-field-theory-local-or-global-first)), global class field theory was proved first, and then used to deduce local class field theory. But nowadays most treatments do the local theory... | https://mathoverflow.net/users/125639 | How to prove local class field theory from global class field theory | That's the approach taken in Lang's *Algebraic Number Theory* Springer GTM 110. Lang develops global class field theory, and then in Chapter XI Section 4 he finishes "the proof of the complete splitting theorem and derives local class field theory, describing the effect of the Artin map on the local component $k\_v^\*$... | 6 | https://mathoverflow.net/users/11926 | 347397 | 147,110 |
https://mathoverflow.net/questions/347406 | 2 | An $\Omega$-algebra over a field $K$ is a $K$-algebra $A$ with a set of multilinear operators $\Omega$, where $\Omega=\bigcup\_{m=1}^{\infty} \Omega\_{m}$ and each $\Omega\_{m}$ is a set of $m$-array multilinear operators on $A$. On the other side, let consider the definition of Hom-Lie algebras as follows:
A Hom-alg... | https://mathoverflow.net/users/40491 | Can Hom-Lie algebras be seen as an $\Omega$-algebras? | Yes, Hom-Lie algebras can be considered in the framework of multiple operated algebras. We have finished this paper and we will pose it on Arxiv in the next time. Please see our recent paper [Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras](https://arxiv.org/abs/1909.10577)
f... | 2 | https://mathoverflow.net/users/83928 | 347413 | 147,116 |
https://mathoverflow.net/questions/347366 | 1 | Consider a general-sum game with $N$ players. Let $u\_i(a\_1, \ldots, a\_N)\colon \prod\_{i=1}^N A\_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes action $a\_i \in A\_i$, where $A\_i $ is the action set of player $i$. Let $\sigma^\*$ be any notion of correlated e... | https://mathoverflow.net/users/81633 | Perturbation of the value of a general-sum game at a equilibirium | Consider the following 2x2 two player game
\begin{array}{c|c}
1,1 & 0,1 \\
\hline
1,0 & 0,0
\end{array}
In this game, all strategy profiles are Nash equilibria, and consequently every point in the unit square is an equilibrium payoff (and a correlated equilibrium payoff).
Take now the following perturbation of this gam... | 2 | https://mathoverflow.net/users/64609 | 347415 | 147,117 |
https://mathoverflow.net/questions/346418 | 3 | **Background**
--------------
Let $(U\_t)\_{t \in \mathbb{R}}$ be the (translation) $C\_0$-group on $L^1(\mathbb{R})$ defined by
$$
U\_t(f)(x) = f(x-t) \quad \text{for almost every } x \in \mathbb{R}
$$
(for $t \in \mathbb{R}$ and $f \in L^1(\mathbb{R})$).
The [Wiener Tauberien theorem](https://en.wikipedia.org/wik... | https://mathoverflow.net/users/36886 | Relaxed/Truncated Version of Wiener's Tauberian Theorem | No, you have the simple counter example: $$g\_{T}(x)=\frac{1}{T}1\_{0\leq x\leq T}. $$
Since $f\in L^1$, there exist $M>0$ with $\|f|\_{[-M,M]}\|\_{L^1}=\|f\|\_{L^1}-\epsilon/N$. Moreover for any $(\beta\_i,t\_i)\_{i\leq N}$ we have $$\|g\_T - \sum \beta\_i U\_{t\_i}f|\_{[-M,M]}\|\_{L^1}\geq \frac{T-2NM}{T} $$ that goe... | 1 | https://mathoverflow.net/users/99045 | 347418 | 147,118 |
https://mathoverflow.net/questions/347430 | 0 | Let $p(z)=\prod\_{k=1}^n(z-z\_k)$ and $p\_k(z)=\prod\_{i=1,i\neq k}^n(z-z\_i). $
Then $p'(z)=\sum\_{k=1}^np\_k(z).$
Let $q(z)=(1/n)p'(z)= (1/n)\sum\_{k=1}^np\_k(z).$ Suppose $p(z)$ has all its zeros in a convex polygon $C.$ Then by Gauss-Lucas Theorem $q(z)$ has all its zeros in $C.$ Now $q(z)$ can be thought of as a p... | https://mathoverflow.net/users/128472 | Convex polygon containing the zeros of a convex linear combination of polynomials | Yes it can. If $z$ is a root of $Q(z)$ outside $C$, then $\sum a\_k/(z-z\_k)=0$, therefore (take the complex conjugate) we get $\sum a\_k(z-z\_k)/|z-z\_k|^2=0$, but all summands in LHS belong to the same half-plane.
| 2 | https://mathoverflow.net/users/4312 | 347435 | 147,124 |
https://mathoverflow.net/questions/346656 | 4 | For a finite set $S$ of $n$ elements, say a weight function is a function $f \colon S \to \mathbb{R}$. For any subset $T \subseteq S$, define $f(T) = \sum \_{x \in T} f(x)$. Define two weight functions $f\_1, f\_2$ on the same set $S$ to be *equivalent* if $f\_1(T\_1) \leq f\_1(T\_2) \Leftrightarrow f\_2(T\_1) \leq f\_... | https://mathoverflow.net/users/90005 | Number of nonequivalent weight functions on a set of $n$ elements | It turns out my question is answered in a recent paper by [Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn](https://arxiv.org/abs/1904.01361) on the complexity of integer programming. Letting $w \in \mathbb{R}^n$ be the vector of weights, the weight of a sub... | 2 | https://mathoverflow.net/users/90005 | 347439 | 147,125 |
https://mathoverflow.net/questions/347425 | 5 | I know that for compact Kähler manifolds $M$ there is an isomorphism:
$$ H^p(M, \Omega\_M^q) = H^q(M, \Omega\_M^p) $$
where $\Omega\_M$ is the sheaf of holomorphic $1$-forms. It is because $H^p(M, \Omega\_M^q) = H^{p,q}\_{\bar{\partial}}(M)=\mathcal{H}^{p,q}(M)$ the set of harmonic forms on $M$. We can then apply conju... | https://mathoverflow.net/users/148020 | $h^{p,q} = h^{q,p}$ on complex smooth projective scheme | I remember seeing such a proof in an article by Messing, who attributed it to Gabber. Let $X$ be a smooth projective variety of dimension $n$ over a field of characteristic $0$.
Suppose that $p+q=i\le n$.
Serre duality gives $h^{pq}= h^{n-p, n-q}$. Now put this together with algebraic proofs of hard Lefschetz for $\el... | 9 | https://mathoverflow.net/users/4144 | 347442 | 147,126 |
https://mathoverflow.net/questions/347433 | 3 | Let $R$ be a Riemann surface and let $\varphi=\varphi(z)dz^2$ be a nonzero holomorphic quadratic differential on $R$. A differentiable curve $\gamma$ on $R$ is called a **horizontal trajectory** if along the curve $\varphi(z)dz^2>0$. My question is, if $\gamma$ is closed, can it be freely homotopic to zero?
My conjec... | https://mathoverflow.net/users/143284 | Can a closed horizontal trajectory on a Riemann surface be freely homotopic to $0$? | Indeed, it can't. I'll give two proofs (the second proof is shorter but needs the first 3 sentences of the first proof)
*Proof 1.* Suppose by contradiction that such a contractible curve exists. Then, since $\gamma$ is a simple loop, it must bound a disk $D$ on $R$. So we have a compact disk $D$ with a flat metric wi... | 2 | https://mathoverflow.net/users/943 | 347443 | 147,127 |
https://mathoverflow.net/questions/346653 | 3 | Let $\overline{\mathcal{M}}\_g$ be the moduli stack of genus $g$ stable (nodal) curves and let $\overline{M}\_g$ denote its coarse moduli space. In 1969, in the paper "The irreducibility of the space of curves of given genus", Deligne and Mumford constructed and showed that $\overline{\mathcal{M}}\_g$ is a smooth Delig... | https://mathoverflow.net/users/146366 | When is the coarse moduli space of genus $g$ stable curves smooth? | A more detailed description of the singular locus of $\mathrm{M}\_g$ is as follows.
**Theorem.** Let $\mathrm{C}$ be a smooth curve of genus $g$.
If $g=2$, then $[\mathrm{C}]$ is a singular point of $\mathrm{M}\_2$ if and only if $\mathrm{C}$ is given by $y^2=x^6-x$.
If $g=3$ and $\mathrm{C}$ is not hyperellipt... | 3 | https://mathoverflow.net/users/104669 | 347460 | 147,133 |
https://mathoverflow.net/questions/347456 | 2 | Are there conditions which guarantee that the heart of a triangulated category is Grothendieck? Is the compatibility between the t-structure with filtered colimits enough?
| https://mathoverflow.net/users/111070 | When is the heart of a triangulated category Grothendieck? | This topic has been studied by Parra & Saorín. For details, see their
Direct limits in the heart of a t-structure: the case of a torsion pair.
*J. Pure Appl. Algebra*, **219** (2015), no. 9, 4117–4143.
The case of the derived category of a commutative Noetherian ring is treated in
Hearts of t-structures in the ... | 3 | https://mathoverflow.net/users/6348 | 347461 | 147,134 |
https://mathoverflow.net/questions/347438 | 10 | Consider the following alternative definition of finite reflection group:
>
> **Definition:** A *finite reflection group* $\Gamma\subset\mathrm O(\Bbb R^d)$ is a finite group generated by orthogonal transformations $T\in\mathrm O(\Bbb R^d)$ with eigenvalues $\{-1^1,1^{d-1}\}$. (the exponents denote multiplicites)
>... | https://mathoverflow.net/users/108884 | Generalized root systems and reflection groups | If we place no restrictions on $k$, then this is precisely the class of finite groups that are generated by involutions.
In particular, if $G$ is any finite group of order $n$, then in the left regular representation of $G$ any involution acts as an $n\times n$ orthogonal matrix of order two and trace zero. Such a ma... | 11 | https://mathoverflow.net/users/6514 | 347480 | 147,139 |
https://mathoverflow.net/questions/347459 | 5 | A result of Higman states that there exists a finitely-presented group $G$ in which all other finitely-presented groups embed - I'll call such a group universal. Every countable group embeds in a 2-generated group, so there are 2-generated universal groups.
I was told that someone somewhere wrote down some explicit ... | https://mathoverflow.net/users/99414 | Explicit short presentation of a 2-generated universal group? | As I wrote in my comment above, the OP is about two different classes of groups: 2-generated universal countable groups (these contain all countable groups and are not finitely presented) and universal finitely presented group (these contain all recursively presented groups and are finitely presented). The question abo... | 3 | https://mathoverflow.net/users/nan | 347486 | 147,142 |
https://mathoverflow.net/questions/347478 | 5 | Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$
$$\int\_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int\_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \vert f(x) \vert^2 \ dx? $$
The question comes from the fact that it is very easy to see that
$$\int\_{\mathbb R} \vert ... | https://mathoverflow.net/users/nan | Elementary calculus estimate or not? | This really belongs to MSE rather than to MO, but I'm too lazy to initiate the moving process, so I'll just answer.
There may be more intelligent ways to do it, but you ***can*** also integrate by parts and get what you want, say, for smooth functions with compact support, after which you should carefully pass to the... | 18 | https://mathoverflow.net/users/1131 | 347492 | 147,145 |
https://mathoverflow.net/questions/347473 | 4 | When is an algebra defined by generators and relations finite-dimensional and satisfies Poincaré duality?
**EDIT:** My question is not very concrete. Rather I am wondering if there is anything known in the following direction.
Assume we are given a commutative algebra $A$ (say over complex numbers) which is graded ... | https://mathoverflow.net/users/16183 | When is an algebra defined by generators and relations finite-dimensional and satisfies Poincaré duality? | Say $I = (f\_1,\dotsc,f\_l)$ is the ideal generated by the $f\_i$. The $f\_i$ are homogeneous; let’s add an assumption that none of the $f\_i$ are constant (degree zero). The following conditions are equivalent:
1. $A$ is finite-dimensional (as a vector space over $\mathbb{C}$).
2. The [radical](https://en.wikipedia.... | 9 | https://mathoverflow.net/users/88133 | 347495 | 147,148 |
https://mathoverflow.net/questions/347491 | 3 | I always encounter two definitions: two-sided and oriented (hypersurface or submanifold). What is the difference of them? Which one is stronger?
| https://mathoverflow.net/users/22815 | Difference of two-sided and oriented | If the submanifold $M$ of a manifold $N$ is co-dimension one, being *two-sided* typically means it has a trivial normal bundle, i.e. $M$ splits the normal bundle into two path components. These are the two sides.
Technically, being two-sided is unrelated to being oriented, but there are cases where they are related.... | 9 | https://mathoverflow.net/users/1465 | 347497 | 147,149 |
https://mathoverflow.net/questions/347482 | 7 | Say that a function $f$ defined on $\mathbb{Q}^n$ is *midpoint convex* if $f((x+y)/2) \le (f(x) + f(y))/2$. Say that it is *rationally convex* if, for $\lambda \in \mathbb{Q} \cap [0,1]$ and $\bar \lambda = 1-\lambda$, we have $f(\lambda x + \bar\lambda y) \le \lambda f(x) + \bar\lambda f(y)$.
Clearly every rationall... | https://mathoverflow.net/users/5010 | Does midpoint-convex imply rationally convex? | Assume that $g$ defined on $\mathbb{Q}^n$ is midpoint convex. First we show that we can extend the midpoint inequality to arbitrary means:
$g((x\_1+\dots+x\_m)/m)\leq (g(x\_1)+\dots+g(x\_m))/m$ for any $m\in\mathbb{Z}\_{\geq1}$
We can easily prove this for $m=2^k$ by using midpoint convexity $k$ times.
For gener... | 12 | https://mathoverflow.net/users/7113 | 347499 | 147,151 |
https://mathoverflow.net/questions/347496 | 1 | For this question, all Banach spaces are over the reals.
Let $1\leq p<\infty$. Recall that a sequence $(x\_n)$ in a Banach space $E$ is weakly $p$-summable if
$$ \Vert (x\_n) \Vert\_{p,w} := \sup\_{\gamma\in E^\* \colon \Vert\gamma\Vert\leq 1} \left( \sum\_{n=1}^\infty \vert\gamma(x\_n) |^p \right)^{1/p} < \infty .$$... | https://mathoverflow.net/users/763 | Does taking the modulus preserve weak $p$-summability of sequences in $L_q$? | It seems to me that the answer is no for all $1 < p < 2$: consider $x\_n = \frac{e^{2\pi i nx}}{n^{r}}, n = 1, 2, \ldots$ for $r = \frac{1}{p}$. It is easy to see that the sequence $\{ |x\_n|\}$ is not weakly $p$-summable by testing against $\gamma = 1$. Let us show that $\{ x\_n\}$ is weakly p-summable:
We want to s... | 1 | https://mathoverflow.net/users/104330 | 347500 | 147,152 |
https://mathoverflow.net/questions/342466 | 19 | Topos theory can be seen as a categorification of topology via the following analogies.
\begin{array}{|c|c|}
\hline
\text{locales}&\text{Grothendieck toposes}\\\hline
\text{open sets}&\text{sheaves}\\\hline
\text{continuous maps}&\text{geometric morphisms}\\\hline
\text{bases}&\text{sites}\\\hline
\text{topological s... | https://mathoverflow.net/users/4613 | Has this "backwards" perspective on toposes been studied? | Actually, the closure operator of a topology is a finite *colimit* preserving monad on a powerset.
| 4 | https://mathoverflow.net/users/49 | 347502 | 147,153 |
https://mathoverflow.net/questions/345110 | 4 | The source <https://en.wikipedia.org/wiki/Accuracy_and_precision> says that in statistics "precision" is understood to be a measure of statistical variability within samples. The lower the variability within the sample, the higher the "precision." That's okay. It is a technical term. But I wonder if there a standard te... | https://mathoverflow.net/users/38783 | Looking for a statistical term close to "precision" | I have heard statisticians and data scientists use the word 'granularity' to express the idea you are looking for. Here is a quote from the dedicated Wikipedia article: 'The granularity of data refers to the size in which data fields are sub-divided'. The article also gives an enlightening example of proper usage: 'A k... | 2 | https://mathoverflow.net/users/99279 | 347517 | 147,159 |
https://mathoverflow.net/questions/347516 | 0 | I noticed that for any vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ where $\mathbf{a},\mathbf{b}\in \mathbb{R}^{m\times 1}$ and $\mathbf{c}\in \mathbb{R}^{n\times 1}$, there exists the equality that
$$\mathbf{a}^\top \mathbf{b} \mathbf{c}=\mathbf{c}\mathbf{b}^\top \mathbf{a}$$
I can prove it as follows,
Denote the ... | https://mathoverflow.net/users/121882 | Proving equality of a vector multiplication example | $\mathbf{a}^\top \mathbf{b}$ is a scalar, so $\mathbf{a}^\top \mathbf{b} = (\mathbf{a}^\top \mathbf{b})^\top = \mathbf{b}^\top \mathbf{a}$ and the term can be moved to the other side of $\mathbf{c}$ (again because it's a scalar).
General remark: the second product in your LHS is a scalar-vector multiplication, which... | 1 | https://mathoverflow.net/users/1898 | 347518 | 147,160 |
https://mathoverflow.net/questions/347520 | 2 | I have heard that there are ways to express sums of rational functions in terms of polygamma functions, and I would like to read more about it. However, I don't know the literature about special functions very well. Can anybody suggest possible references? Thank you.
| https://mathoverflow.net/users/127070 | Reference request: sums of rational functions and polygamma functions | a journal publication with many generalisations is
[Infinite sums as linear combinations of polygamma functions](https://www.semanticscholar.org/paper/Infinite-sums-as-linear-combinations-of-polygamma-Pilehrood-Pilehrood/6c3a7246b5cf2478c5714593c7ee8bd1802c8887), Kh. Hessami Pilehrood, T. Hessami Pilehrood, Acta Arit... | 1 | https://mathoverflow.net/users/11260 | 347523 | 147,161 |
https://mathoverflow.net/questions/347513 | 3 | Is the following true? If so, I would be grateful for a reference that contains such a result and its proof.
Let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a real analytic function, and $V:=\{\mathbf{x}\in \mathbb{R}^d: f(\mathbf{x}) =0\}$ be its variety. Then there exists a decomposition $V=V\_{d-1}\cup \dots \cup V... | https://mathoverflow.net/users/142650 | Decomposition of a real analytic variety | You may want to look at Lojasiewicz's structure theorem; for a statement and proof (I won't reproduce it here since the complete theorem statement is over a page long) see Chapter 6, section 3 of [Krantz's *A Primer of Real Analytic Functions*](https://www.springer.com/gp/book/9780817642648).
(The theorem as stated ... | 3 | https://mathoverflow.net/users/3948 | 347537 | 147,167 |
https://mathoverflow.net/questions/347540 | 82 | There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
1. A total characterisation, where somehow "all of the information" about a field has been uncovered.
2. A negative result, rendering the field... | https://mathoverflow.net/users/149435 | What are examples of (collections of) papers which "close" a field? | Let me preface this by saying that this is just my own account, based on various conversations I've had over the years with many mathematicians, of the following example.
In 1976, William Thurston [proved](http://dx.doi.org/10.2307/1971047) that a closed smooth manifold has a codimension one foliation if and only if ... | 59 | https://mathoverflow.net/users/49247 | 347544 | 147,169 |
https://mathoverflow.net/questions/335102 | 2 | Inspired by the two posts which are linked bellow we ask the following question:
**Question:** For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ with $D(f)=(\Delta\circ L\_X-L\_X\circ \Delta)(f)$ where $\Delta$ is the standard Laplacian.
>
> Is there a vecto... | https://mathoverflow.net/users/36688 | Keeping track of limit cycles via certain second order differential operator | I believe the answer is yes.
Let $X$ be the vector field $2x \partial\_y - y \partial\_x$. The level sets of $y^2 + 2x^2$ are orbits of $X$, they have the shape of ellipses.
It is easy to compute $D(f) = 2 \partial^2\_{xy} f$. So if you just let $f(x,y) = xy$ you in fact have $D(f) \equiv 2 \neq 0$, and in partic... | 1 | https://mathoverflow.net/users/3948 | 347559 | 147,174 |
https://mathoverflow.net/questions/347514 | 0 | Denote $$T(m)=\sum\_{1\leq n\_m\leq n\_{m-1}\leq\dots\leq n\_2\leq n\_1\leq m}\prod\_{i=1}^{m}\binom{n\_i}{n\_{i+1}}.$$
Is there a name for this kind of summation and is there a good estimate for $\ln T(m)$ bounded above and below by constant factors (preferably additively)?
| https://mathoverflow.net/users/136553 | Terminology and approximation to logarithm of a sum of products of binomial coefficients | Notice that the product of binomial coefficients can be expressed as a [multinomial coefficient](https://en.wikipedia.org/wiki/Multinomial_theorem):
$$\prod\_{i=1}^m \binom{n\_i}{n\_{i+1}} = \binom{n\_1}{n\_1-n\_2,n\_2-n\_3,\dots,n\_{m-1}-n\_m,n\_m}.$$
Denoting $d\_i:=n\_i-n\_{i+1}$ for $i<m$ and $d\_m:=n\_m$, and noti... | 3 | https://mathoverflow.net/users/7076 | 347560 | 147,175 |
https://mathoverflow.net/questions/347543 | 0 | I've the following number:
$$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$
Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has a given value) besides that $x\in\mathbb{N}^+$ and $x\ge2$.
I want to check if the number is a perfect square, so I ... | https://mathoverflow.net/users/149418 | Mistake in SageMathCell code, finding integral points on elliptic curves | After about an hour and a half running SageMath 9.0.beta7 on my computer, I see this:
```
sage: E = EllipticCurve([0, 2484, 0, -3122149536, 131871507195024])
sage: P = E.integral_points()
sage: for p in P:
....: if p[0] % 57132 == 0:
....: print(p[0] // 57132, p[1] // 57132)
....:
(-1, 201)
(0, 201)
(1, 21... | 3 | https://mathoverflow.net/users/4194 | 347563 | 147,176 |
https://mathoverflow.net/questions/347511 | 5 | I am looking for a reference proving the following statement:
>
> For every $n,m \geq 2$, the groups $T\_n$ and $T\_m$ are isomorphic if and only if $n=m$.
>
>
>
Here, $T\_k$ denotes the variation of Thompson's group $T$ (compared to $F$, $T$ acts on the circle instead of the interval) where dyadic numbers are... | https://mathoverflow.net/users/122026 | Isomorphism problem among Thompson's groups | As far as I can tell, a solution to this problem has not appeared in the literature. Unless I'm mistaken, the best partial result was obtained by Liousse in [this 2008 paper](https://link.springer.com/article/10.1007/s10711-007-9216-y), where it is proven by examining possible orders of elements that $T\_m$ is not isom... | 11 | https://mathoverflow.net/users/6514 | 347573 | 147,181 |
https://mathoverflow.net/questions/347575 | 4 |
>
> **Question**. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a [Radon measure space](https://en.wikipedia.org/wiki/Radon_measure). Let $$\rho\colon G\to U(L^2(X))$$
> be a unitary, strongly continuous, representation. Is it true that, if $g\_n\to \infty$, then
> $$
> \int\_X \over... | https://mathoverflow.net/users/13042 | Do all unitary representations weakly converge to zero at infinity? | No, there are simple counterexamples. E.g., take $G = \mathbb{R}$ and $X = \mathbb{C}$ with Lebesgue measure, and define $\rho\_t f(z) = f(e^{2\pi i t}z)$ for $t \in \mathbb{R}$ and $f \in L^2(\mathbb{C})$. Then $\rho\_t$ is the identity for any integer $t$, so $\rho\_n \to {\rm id}$ strongly, not to zero.
| 4 | https://mathoverflow.net/users/23141 | 347579 | 147,186 |
https://mathoverflow.net/questions/285173 | 11 | Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $n\geq 3$ and $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X\_0\in \mathbb{Z}^{2\times n}$, set
$$
R'(T,Q,X\_0) = \{ X\in X\_0G(\mathbb{Z}) : |X\_{ij}|\leq T\}.
$$
**Dubious heuristic:** If $Q$ is indefinite th... | https://mathoverflow.net/users/116794 | Upper bounds for lattice points in orbits, and representations of binary quadratic forms | This is not an answer, but a long comment providing related known results.
In the particular case $Q=I\_{n,1}= diag(I\_n,-1)=diag(1,\dots,1,-1)$ (called the Lorentzian case), Ratcliffe and Tschantz gave in [1](https://sites.google.com/prod/view/emiliolauret) an asymptotic formula for the number of $x=(x\_1,\dots,x\_... | 2 | https://mathoverflow.net/users/20052 | 347581 | 147,188 |
https://mathoverflow.net/questions/347565 | 1 | I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros.
But I want know if is possible to generate a two-variable polynomial giving its coordinates where value is zero?
This problem appears on calculus of... | https://mathoverflow.net/users/149442 | Generate a two-variable polynomial from its "roots | There a number of very well researched techniques that can be used to solve your problem in a practical sense. Most of these have come from computer graphics and computer vision where taking a set of points on a surface, possibly with noise, and trying to create a surface that either interpolates or approximates those ... | 1 | https://mathoverflow.net/users/7113 | 347583 | 147,189 |
https://mathoverflow.net/questions/347519 | 3 | Let $X$ be a smooth complex projective variety and $p:Y\to X$ be a locally trivial in analytic topology $\mathbb CP^k$-bundle. Suppose we have a line bundle $L$ on $Y$, restricting to $\mathcal O(1)$ on $\mathbb CP^k$-fibres.
**Question.** Is it true that there is a line bundle $L'$ on $X$ such that $p^\*L'\otimes L$... | https://mathoverflow.net/users/13441 | Constructing a very ample line bundle on a projective bundle | Since $R^ip\_\*L=0$ for $i>0$, by semi-continuity theorem, you see that $p\_\*L$ is a vector bundle of rank $k+1$. You can twist by a sufficiently ample bundle $L'$ on $X$ to make it globally generated. Thus, you have $O\_X^m\to p\_\*L\otimes L'$ surjective and thus you get an embedding $Y\subset X\times \mathbb{P}^{m-... | 3 | https://mathoverflow.net/users/9502 | 347586 | 147,191 |
https://mathoverflow.net/questions/347564 | 1 | Let $Y$ be a connected CW-complex and $F\subset Y\times Y$ be a closed connected subspace such that the composition $F\subset Y\times Y \rightarrow Y$ is a bijective map, where
$Y\times Y\rightarrow Y $ is given by $(y\_1,y\_2)\mapsto y\_{1}$.
I'm looking for a (easy) example of such $Y$ and such $F$ such that th... | https://mathoverflow.net/users/17895 | closed connected subspace of a cartesian product | Start with any example of a continuous bijection $f:A\to B$ between connected CW spaces that is not a homeomorphism. For example, $A$ a closed half-line and $B$ a circle.
Let $Y$ be $A\times B$, choose a point $p\in B$, and let $F\subset Y\times Y$ consist of all points $((a,b),(a',b'))$ such that $b=f(a')$ and $b'=p... | 3 | https://mathoverflow.net/users/6666 | 347588 | 147,193 |
https://mathoverflow.net/questions/347599 | 2 | Let $G\_i = (V\_i, E\_i)$ be simple, undirected graphs for $i=1,2$. A *graph homomorphism* is a map $f:V\_1\to V\_2$ such that $\{f(v), f(w)\}\in E\_2$ whenever $\{v,w\}\in E\_1$.
By $\text{Hom}(G\_1, G\_2)$ we denote the collection of graph homomorphisms from $G\_1$ to $G\_2$. Note that it is possible that $\text{Ho... | https://mathoverflow.net/users/8628 | Exponential object in the category of simple, undirected graphs | The title of your question asks about "exponential object[s] in the category of simple, undirected graphs". I can tell you what they are.
(The body of your question asks about a construction in a specific paper, which I haven't read, so I can't answer *that* question directly. But of course, exponentials are unique w... | 8 | https://mathoverflow.net/users/586 | 347600 | 147,198 |
https://mathoverflow.net/questions/203536 | 1 | I am looking for a specific paper, that I have found very difficult to trace.
*C. De Concini, V. Kac - Quantum Groups at roots of 1*
Specifically, the paper is cited as follows (on De Concini's webpage).
*De Concini, Corrado; Kac, Victor G. Representations of quantum groups at roots of 1.
Operator algebras, unita... | https://mathoverflow.net/users/60707 | Help finding paper: De Concini, Kac - Quantum Groups at roots of 1 | I was facing the same problem recently and I just found a copy of this paper at this link:<http://www.math.harvard.edu/~yfu/Kac-DeConcini.pdf>.
| 3 | https://mathoverflow.net/users/149464 | 347603 | 147,199 |
https://mathoverflow.net/questions/347590 | 23 | There are various differentiations/derivatives.
For example,
* Exterior derivative $df$ of a smooth function $f:M\to \mathbb{R}$
* Differentiation $Tf:TM\to TN$ of a smooth function between manifolds $f:M\to N$
* Radon-Nikodym derivative $\frac{d\nu}{d\mu}$ of a $\sigma$-finite measure $\nu$
* Fréchet derivative $D... | https://mathoverflow.net/users/149275 | Most general definition of differentiation | That's a real can of worms. There are tons of different notions of differentiability for functions lacking classical smoothness: the Gateaux derivative, the weak derivative, the distributional derivative, the directional derivative, the subgradient (for convex functions), Clarke's generalized gradient, Hadamard differe... | 30 | https://mathoverflow.net/users/9652 | 347609 | 147,202 |
https://mathoverflow.net/questions/347553 | 6 | It is known that $NS\_{\omega\_2}$ cannot be saturated (namely there cannot be $\aleph\_3$ many stationary subsets of $\omega\_2$ any two of which have non-stationary intersection). However, it may be the case when it is restricted to a stationary subset. It is also known that the stationary subset cannot be $\omega\_2... | https://mathoverflow.net/users/119731 | Saturation of non-stationary ideal on $\omega_2$? | I believe it is still open whether $\mathrm{NS}\_{\omega\_2} \restriction \mathrm{cof}(\omega\_1)$ can be saturated. But it was known by early unpublished work of Woodin that $\mathrm{NS}\_{\omega\_2} \restriction S$ can be saturated, where $S$ is a stationary-costationary subset of $\mathrm{cof}(\omega\_1)$. The compl... | 3 | https://mathoverflow.net/users/11145 | 347611 | 147,203 |
https://mathoverflow.net/questions/347558 | 0 | Consider a set of functions $\mathcal{F}$ on $E$ where $E \subset\mathbb{R}^k$ - e.g. the class of $L\_1$ functions on $[0,1]$ - and endow it with a suitable metric $d$ that makes it Polish.
1. Consider a partition $\{\mathcal{F}\_i, i\in I\}$ (say at most countable) of $\mathcal{F}$ and denote by $d\_i$ the restrict... | https://mathoverflow.net/users/148849 | Topologies and Borel $\sigma$-fields on disjoint unions | *Remarks and hints, not a solution*
**Question 1** A disjoint union $\mathcal F = \bigcup\_{i \in I} \mathcal F\_i$ in a metric space has the disjoint union topology if and only if all sets $\mathcal F\_i$ are open in $\mathcal F$. Is that question 1? The answer is yes. Why not try to prove it?
In particular, whe... | 2 | https://mathoverflow.net/users/454 | 347613 | 147,204 |
https://mathoverflow.net/questions/347551 | 1 | I would be thankful to anyone who can present an analytical solution to the following inhomogeneous PDE equation:
$$\frac{\partial{u}}{\partial{t}}= \alpha\frac{\partial^2{u}}{\partial{x^2}}-ku$$
$$u(0,t) = 0$$
$$u(1,t) = M\_R$$
$$u(x,0) = x\*f(x)$$
where k, $\alpha$ and $M\_R$ are constants and k>0.
| https://mathoverflow.net/users/149417 | Analytical solution to inhomogeneous parabolic PDE | Set first $u= ve^{-kt}$ so that
$
\partial\_t u+ku=e^{-kt}(\partial\_t v-kv+kv)=e^{-kt}\partial\_t v.
$
The equation becomes
$$
\partial\_t v=\alpha\partial\_x^2 v, \quad v(0,t)=0, \quad v(1,t)=M\_R e^{kt}, \quad v(x,0)=x\ast f(x).
$$
You may assume $\alpha =1$ by writing $v(x,t)=w(x, \alpha t)$ and get then
$$
\partia... | 1 | https://mathoverflow.net/users/21907 | 347615 | 147,205 |
https://mathoverflow.net/questions/346848 | 4 | Let $p>2$ be a prime integer and let $\mathbb{F}\_{p^2}$ be the corresponding finite field. Consider a subgroup $H$ of $SL\_2(\mathbb{F}\_{p^2})$ which satisfies the following conditions:
1. The matrix $\left(\begin{smallmatrix} 0& 1\\ -1 &0 \end{smallmatrix}\right)\in H$.
2. At least half of all elements in $H$ have... | https://mathoverflow.net/users/123491 | $PGL_2$ image of special subgroups in $SL_2(\mathbb{F}_{p^2})$ | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$Let $H<\SL\_2(p^2)$ for $p$ an odd prime. Let $Z=Z(\SL\_2(p^2))$. The only important supposition is that half of all elements of $H$ have trace $0$. Let $h\in H$ be such an element.
As Mark Wildon [observes](https://mathoverflow.net/questions/346848/pgl-2-imag... | 5 | https://mathoverflow.net/users/801 | 347618 | 147,207 |
https://mathoverflow.net/questions/347536 | 5 | It is well known that the finite subgroups of $SL(2,\mathbb{C})$ up to conjugacy are the binary polyhedral groups (or Klein groups). There are two infinite families (cyclic groups and binary dihedral groups) and three exceptional groups (binary tetrahedral, binary octahedral, binary icosahedral). These groups and the r... | https://mathoverflow.net/users/106706 | Finite subgroups of $GL(2,K)$ with $K\neq\mathbb{C}$ | An answer to your questions is provided by this article of Beauville:
*Beauville, Arnaud*, [**Finite subgroups of (\mathrm{PGL}\_2(K)).**](http://www.arxiv.org/abs/0909.3942), García-Prada, Oscar (ed.) et al., Vector bundles and complex geometry. Conference on vector bundles in honor of S. Ramanan on the occasion of ... | 7 | https://mathoverflow.net/users/801 | 347619 | 147,208 |
https://mathoverflow.net/questions/347616 | 4 | Let $\mathbb{D}^2$ be the closed two-dimensional unit disk, and let $g:\mathbb{D}^2 \to \mathbb{R}$ be a non-constant **harmonic** function (smooth up to the boundary).
>
> Does there exist a sequence of smooth one-forms $\sigma\_n$ on $\mathbb{D}^2$ such that
>
>
> 1. $\sigma\_n \to dg$ in $L^2$.
> 2. $\sigma\_n... | https://mathoverflow.net/users/46290 | Approximate a one-form on the disk with nowhere vanishing one-forms satisfying an asymptotic vanishing of some derivatives | Your third condition implies
$$ \| (\delta d + d\delta) (\sigma\_n - \sigma) \|\_{L^1} \to 0 $$
Notice that up to constants, $\delta d + d\delta$ is identical (on the flat disk) to the standard Laplacian acting componentwise.
Since the disk is compact, you also have by your first condition and Holder
$$ \| \si... | 3 | https://mathoverflow.net/users/3948 | 347624 | 147,209 |
https://mathoverflow.net/questions/347637 | 3 | $\require{AMScd}$
This is basic level question, but this kind of questions usually find no answer on stackexchange.
I am trying to introduce my self to categories theory and advanced algebraic topology. I have just learnt about group and cogroup structures. My question is about *cogroup* structures.
An object w... | https://mathoverflow.net/users/137622 | Why is $\mathbb{S}^1$ a cogroup object in $\mathbf{Top.}$? | This is a statement about the homotopy category. Consider the following fact:
Let $X$ be an object in a category $C$ such that the corepresentable functor $h^X:C\to \operatorname{Set}$ factors through the forgetful functor $\operatorname{Grp}\to \operatorname{Set}$. Then these data precisely specify the diagrams you ... | 12 | https://mathoverflow.net/users/1353 | 347640 | 147,212 |
https://mathoverflow.net/questions/347644 | 5 | I'm reading Voevodsky and Morel's book '$\mathbb{A}^1$-homotopy theory of schemes'. In Remark 3.1.15, it says that for any simplicial fibrant sheaf $F$ and open sets $U\subseteq V$, $F(V)\to F(U)$ is a fibration.
Prove by definition. We have a bifunctor
$$\begin{array}{ccccc}sSet&\times&Shv(Sm/k)&\to&sShv(Sm/k)\\(S&,... | https://mathoverflow.net/users/149491 | Do infinite products commute with trivial cofibrations, for simplicial sets? | This fact admits a much easier proof.
To show that for any simplicial fibrant sheaf F and open sets U⊆V the map F(V)→F(U) is a fibration
it suffices to show that F(V)→F(U) has a right lifting property with respect to
horn inclusions.
Since F(V)→F(U) can be rewritten as Map(U→V,F),
we can move U→V using the two-variab... | 6 | https://mathoverflow.net/users/402 | 347646 | 147,214 |
https://mathoverflow.net/questions/347622 | 10 | In Serre's book *Trees* [Se, p. 68] it says:
>
> 3) For $SL\_2$ the situation is different. It is clear that
> $SL\_2(\mathbf Z )$ does not have property (FA). It is the same with
> $SL\_2(A)$ when $A$ is the ring of integers of an imaginary quadratic
> field not isomorphic to $\mathbf{Q}(\sqrt{- 1})$ or
> $\ma... | https://mathoverflow.net/users/12604 | Is property FA of Serre known for $SL_2(\mathbb{Z}[i])$ and $SL_2(\mathbb{Z}[\zeta_3])$ | Here is an answer for all Bianchi groups $SL(2, O\_d)$: Such a group admits a nontrivial graph of groups decomposition (equivalently, does not have the property FA) unless $d=3$, in the latter case, it does not split, i.e. has the Property FA. For details, see:
C. Frohman and B. Fine, Some Amalgam Structures for Bia... | 10 | https://mathoverflow.net/users/21684 | 347650 | 147,216 |
https://mathoverflow.net/questions/346916 | 4 | We say the rook graph, $R\_n$, is the cartesian product of $K\_n \times K\_n$. Let $S$ be the set of graphs that are an induced subgraph of $R\_n$ for some $n$.
Does there exist some constant $c$ such that if $G \in S$ is c-connected, it follows that $G$ is Hamiltonian? I know that if $c$ does exist, $c\geq 3$.
If ... | https://mathoverflow.net/users/130484 | Does 2-connectivity imply Hamiltoniancy for subgraphs of the rook graph | It's known that $3\leq c \leq7$.
$S$ is the class of line graphs of bipartite graphs, by [ISGCI](http://graphclasses.org/classes/gc_736.html).
The line graph of [this graph](https://math.stackexchange.com/questions/955285/2-connected-graphs-with-a-line-graph-containing-no-hamilton-cycle/955298#955298) is 2-connect... | 4 | https://mathoverflow.net/users/125498 | 347651 | 147,217 |
https://mathoverflow.net/questions/347654 | 17 | A manifold $X$ has the fixed-point property if for every continuous map $f:X→X$ there is $x∈X$ with $f(x)=x$. Examples of such spaces are disks and the real projective plane $\mathbb{RP}^2$.
**Question:** If a compact manifold $X$ has the fixed-point property, does $X\times X$ necessarily have the fixed-point propert... | https://mathoverflow.net/users/125498 | Compact manifold $X$ having fixed-point property but $X\times X$ does not | Not a full answer, but the answer seems to not be known. The question is open for closed manifolds (see the mathscinet [review](https://mathscinet.ams.org/mathscinet-getitem?mr=3584128) of Kwasik and Sun's [paper](https://arxiv.org/pdf/1609.05802.pdf), MR3584128). The answer is yes for other manifolds like $\mathbb{CP}... | 8 | https://mathoverflow.net/users/118731 | 347656 | 147,220 |
https://mathoverflow.net/questions/347653 | 8 | Let $A,B$ be two generic (in particular invertible) $2\times 2$ upper-triangular complex matrices. They generate a countable group $G$, the commutator subgroup of $G$ is abelian. Are there other relations in $G$? How is this group called?
| https://mathoverflow.net/users/4312 | Which group do two generic $2\times 2$ triangular matrices generate? | I think it is a old result of Magnus that the free metabelian group on $d$ generators can be embedded as group $T\_2$ of $2\times 2$ matrices over $\mathbf{C}$.
It is easy to deduce that
>
> every generic $d$-tuple in $T\_2$ freely generates such a free metabelian group $\Gamma\_d$.
>
>
>
Here generic mean... | 16 | https://mathoverflow.net/users/14094 | 347660 | 147,222 |
https://mathoverflow.net/questions/347076 | 7 | Assume that $G\leq\operatorname{Sym}(X)$ is a permutation group generated by all its point stabilisers, i.e. $G=\langle G\_x \mid x\in X\rangle$. There is no cardinality restriction on $X$. Furthermore, assume that $G$ has *finitely many* orbits, and that $G$ is subdegree-finite, i.e. all point stabilisers have only *f... | https://mathoverflow.net/users/57533 | Permutation groups generated by finitely many point stabilisers | Let $A$ be an infinite abelian group of odd finite exponent. For example we could take $A$ to be the direct product of infinitely many copies of a cyclic group $C\_n$, with $n>1$ odd.
Let $\langle t \rangle$ be a cyclic group of order $2$, define $\phi:\langle t \rangle \to {\rm Aut}(A)$ by $\phi(t): a \mapsto a^{-1}... | 5 | https://mathoverflow.net/users/35840 | 347667 | 147,223 |
https://mathoverflow.net/questions/345177 | 1 | $E\subset\mathbb{R}^n$ is an ellipsoid if $E = E(g):= \{x\in \mathbb{R}^n \mid x^t g x \le 1\}$ for some inner product $g$ on $\mathbb{R}^n$. Given an ellipsoid $E\subset\mathbb{R}^n$, how unique is $g$ such that $E=E(g)$? Is there a formula for $g$ such that $E=E(g)$ (see the note below for what kind of formula I envi... | https://mathoverflow.net/users/43645 | Ellipsoids and their defining inner product | The *Binet-Legendre metric* of an ellipsoid $E\subset \mathbb{R}^n$ is defined as $g\_F$, the metric dual to
$$
g\_F^\*(\xi,\eta)=\frac{n+2}{\operatorname{Vol}(E)}\int\_E \xi(x)\eta(x) dx.
$$
where the volume and integral are computed using a translation invariant Lebesgue measure. Note that rescaling the choice of me... | 1 | https://mathoverflow.net/users/13268 | 347674 | 147,224 |
https://mathoverflow.net/questions/347670 | 1 | Let $H=(V,E)$ be a hypergraph, where $V\neq \emptyset$ is a set, and $E\subseteq {\cal P}(V)$. We say that $H$ is *connected* if whenever $S\subseteq V$ with $\emptyset \neq S \neq V$, there is $e\in E$ with $$e\cap S \neq \emptyset \neq e\cap (V\setminus S).$$
It is easily verified that any graph (finite or infinite) ... | https://mathoverflow.net/users/8628 | Minimally connected hypergraphs | No. Let $V=E=\omega$ (so $E$ consists of all the initial segments of $\omega$). A subgraph $(V,E\_0)$ is connected iff $E\_0$ is infinite, so there is no minimal connected subgraph.
| 2 | https://mathoverflow.net/users/30186 | 347676 | 147,225 |
https://mathoverflow.net/questions/347675 | 2 | I have a smooth projective $k$-scheme $X$ with a local system $F$ (locally constant sheaf) of finite dimensional $k$-vector spaces (on étale topology). My question is whether there exists a finite étale morphism $f \colon Y \rightarrow X$ such that pullback of $F$ is a constant sheaf on $Y$.
I am still not familiar ... | https://mathoverflow.net/users/148020 | pullback of a local system | Yes, there is such a correspondence. This is mentioned in Milne's étale cohomology and in other places. You actually want the kernel to be an open subgroup and not just finite index, as only these correspond to finite étale covers
But the correspondence covers only continuous representations, where $k$ is normally gi... | 4 | https://mathoverflow.net/users/18060 | 347681 | 147,227 |
https://mathoverflow.net/questions/347683 | 0 | Let $p\_{odd}(n)$ be the number of partitions of $n$ into odd parts ([see here](https://oeis.org/A000700)). For instance, one has the generating function
$$\prod\_{k\geq1}\frac1{1-q^{2k-1}}.$$
>
> **QUESTION.** What is the size of this set
> $$A\_N:=\{n\in\{1,2,\ldots,N\}: \text{$p\_{odd}(n)$ is odd}\}$$
> for la... | https://mathoverflow.net/users/66131 | Size of parities in counting partitions into odd parts | Note that the generating function is
$$
\prod\_{k=1}^{\infty} \frac{1}{1-q^{2k-1}} = \prod\_{k=1}^{\infty} \frac{1-q^{2k}}{1-q^k} = \prod\_{k=1}^{\infty} (1+q^k) \equiv \prod\_{k=1}^{\infty} (1-q^k) \mod 2.
$$
Now use Euler's pentagonal number theorem, which says that the RHS is
$$
1+ \sum\_{k=1}^{\infty} (-1)^k ... | 9 | https://mathoverflow.net/users/38624 | 347685 | 147,228 |
https://mathoverflow.net/questions/347686 | 0 | Let us say we have two probability measures, $X$ and $Y$ on sample spaces $\Sigma\_X$ and $\Sigma\_Y$ (which are finite sets, with the largest sigma algebra on each space) and suppose we get measure $Y$ via a function $f: g(X) \rightarrow Y$. What is the [Relative Entropy](https://en.wikipedia.org/wiki/Kullback%E2%80%9... | https://mathoverflow.net/users/10007 | What is the Relative Entropy between distributions $X$ and $Y$, when $Y$ is a function of $X$? | As R W noted, the relative entropy depends only on the individual (marginal) distributions of $X$ and of $Y$; it does not depend on the joint distribution of $X$ and $Y$. So, the condition that $Y$ is a function of $X$ is quite irrelevant as far as the relative entropy is concerned, not matter what the function is.
... | 2 | https://mathoverflow.net/users/36721 | 347691 | 147,229 |
https://mathoverflow.net/questions/347696 | 3 | Let $\mathcal{C}$ be a small category; let $\mathcal{S}$ be any family of maps in $\text{Psh}\left(\mathcal{C}\right)$.
Call $X\in \text{Psh}\left(\mathcal{C}\right) $ an $\mathcal{S}$-sheaf when $\text{Hom}\_{\text{Psh}\left(\mathcal{C}\right)}(-,X)=h\_X$ is such that $h\_X(s)$ is an isomorphism for every $s\in \ma... | https://mathoverflow.net/users/139854 | Universal property of the category of $\mathcal{S}$-sheaves and the definition of Topos | 1. (&3?) Can follow from the theory of orthogonality classes. A reference is **Thm 1.38** and **1.39** in **Locally presentable and accessible categories** by *Adamek and Rosicky.*
2. I do not have a reference, but this is really a standard technique. In my mind, this follows from the theory of Kan extensions. A place ... | 5 | https://mathoverflow.net/users/104432 | 347700 | 147,231 |
https://mathoverflow.net/questions/347684 | 1 | The question can be described in the following way:
Suppose I have a finite language $\mathcal{L}$ over alphabet $\Sigma$.
I have a string that is composed of a concatenated series of $n$ instances of $w\_i$ chosen randomly(uniformly, independently) from $\mathcal{L}$. I want to find the number of non-overlapping ... | https://mathoverflow.net/users/142777 | Distribution of non-overlapping words in randomly generated text | I'm not sure about the general case, but your particular one is relatively easy.
Let $r$ be a word composed of $n$ instances of $u:=a$ and $v:=ab$ chosen randomly.
We notice that each non-overlapping occurrence of $W$ in $r$ starts with an instance of $v$, unless this instance comes immediately after an odd number ... | 2 | https://mathoverflow.net/users/7076 | 347702 | 147,232 |
https://mathoverflow.net/questions/347690 | 0 | This week I wondered about diophantine problems that involve the volume of certain cubes and frustums, see the Wikipedia [*Frustum.*](https://en.wikipedia.org/wiki/Frustum) I wondered if each one of these problems have infinitely many solutions, I would like to know if it is possible to determine if some of these have ... | https://mathoverflow.net/users/142929 | Diophantine equations that involve cubes and the volume of square frustums | The Problem 1 you specify defines a cubic surface in $\mathbb{P}^{3}$, and there is a lot known about the rational points on such a surface. (For example, Chapter 2 of the book "Rational and nearly rational varieties" by Kollár, Smith and Corti is entirely about cubic surfaces.) A smooth cubic surface can fail to have ... | 6 | https://mathoverflow.net/users/48142 | 347718 | 147,235 |
https://mathoverflow.net/questions/2323 | 10 | I have in mind the following question for some time. Is there an example of a compact symplectic manifold with a Hamiltonian $S^1$-action with isolated fixed points, that does not admit a compatible $S^1$-invariant Kahler strucutre? One would say, of course there should be such an example. But I have not seen any...
... | https://mathoverflow.net/users/943 | Hamiltonian $S^1$ actions with isolated fixed points | Nick Lindsay and have just proved that such a manifold indeed exists. And surprise, surprise, this is Tolman's manifold. See Theorem 1.3 and Corollary 1.4 of our paper: <https://arxiv.org/abs/1912.02785>
| 3 | https://mathoverflow.net/users/943 | 347731 | 147,239 |
https://mathoverflow.net/questions/347723 | 2 | Let $A \in \mathbb{R}^{n \times n}$ be a bidiagonal matrix with non zero elements on its diagonal and super diagonal. Let $B$ be defined as
$$B=\begin{bmatrix}0&{A} \\{A}^T &0 \end{bmatrix}$$.
Find permutation matrix $P$ such that $C=PBP^T$ is a tridiagonal matrix.
I noticed that permutation matrices with only row... | https://mathoverflow.net/users/148659 | Using permutation matrix to convert a matrix into tridiagonal matrix | The permutation matrix $P$ corresponding to the permutation
$$ \sigma \colon \begin{cases} i \mapsto 2i & \text{ if } i \in [1,n] \\ i \mapsto 2i - 2n - 1 & \text{ if } i \in [n+1,2n] \end{cases} $$
does the job.
**Edit:**
As requested, I am providing some details. Notice that the original matrix $B$ has zero diagona... | 2 | https://mathoverflow.net/users/12858 | 347735 | 147,241 |
https://mathoverflow.net/questions/347689 | 11 | Let $R$ be the ring of integers in a (complete) algebraic closure of $\mathbb Q\_p$ with maximal ideal $\mathfrak p$. Suppose I have an Abelian surface $\mathcal A/R$ such that over every $R/\mathfrak p^n$, there exist elliptic curves $E\_n, E\_n'$ over $R$ with $\mathcal A$ isogenous to $E\_n\times E\_n'$ over $R/\mat... | https://mathoverflow.net/users/58001 | Lifting a splitting of an Abelian variety to characteristic 0 | $\newcommand{\cA}{\mathcal{A}}\newcommand{\cB}{\mathcal{B}}\newcommand{\bZ}{\mathbb{Z}}$No, that does not imply that $\cA$ splits over $R$. In fact, if $\cA\_1=\cA\times\_R R/p$ is isogenous to a product of elliptic curves then all $\cA\_n=\cA\times\_R R/p^n$ are isogenous to products of elliptic curves:
>
> **Lemm... | 6 | https://mathoverflow.net/users/39304 | 347759 | 147,249 |
https://mathoverflow.net/questions/347733 | 1 | I am looking for the most efficient algorithm that can solve this problem:
Given a directed graph with real-valued edge weights, find a set of directed cycles (no two cycles can share a vertex) that have the maximum sum of weights.
| https://mathoverflow.net/users/147231 | Finding the max-value set of cycles in a weighted digraph | This is known as the *maximum weight cycle packing* problem. See, for example, [this paper](https://pdfs.semanticscholar.org/3017/ac96269be829de49761d0d67fc8570b8d640.pdf). The *kidney exchange* and *barter exhange* problems are also relevant.
| 1 | https://mathoverflow.net/users/7076 | 347762 | 147,251 |
https://mathoverflow.net/questions/347771 | 4 | Let $X$ be a separable Banach space embedded canonically in $X^{\*\*}$. Is there a retraction from the unit ball $B\_{X^{\*\*}}$ of $X^{\*\*}$ onto the unit ball $B\_X$ of $X$?
When we insist on uniformly continuous retraction, the answer is no, I think, but what if we simply ask for continuous retractions?
| https://mathoverflow.net/users/148734 | Are unit balls in Banach spaces retracts of bidual balls? | Indeed, if $B\subset C$ are closed convex subsets of a Banach space, like in this case, the identity map $B\to B$ extends to a continuous retraction $C\to B$, by the [Dugundji extension theorem](https://projecteuclid.org/euclid.pjm/1103052106).
Since a Banach space has Lipschitz partition of unity, the resulting ret... | 3 | https://mathoverflow.net/users/6101 | 347773 | 147,258 |
https://mathoverflow.net/questions/347774 | 2 | Let $\Pi$ be the set of odd prime numbers and let $\mathcal P(\Pi)$ be the Boolean algebra of subsets of $\Pi$.
For a number $x$ denote by $\Pi(x)$ the set of odd prime divisors of $x$.
>
> **Problem.** Does each singleton $\{p\}\subset \Pi$ belong to the Boolean algebra generated by the family $\{\Pi(2^n-1):n\g... | https://mathoverflow.net/users/61536 | The Boolean algebra generated by sets of prime divisors of the numbers $2^n-1$ | $2^{11}-1=2047=23\cdot 89$ and neither prime divides a smaller such number. So $23 \mid 2^n-1$ exactly when $n=11m.$ Similarly $89 \mid 2^n-1$ exactly when $n=11m.$ So $\{23,89\}$ is in the Boolean Algebra but neither of its singleton subsets is.
Here is a list for $n \leq 33$ of the set of primes which divide $2^n-... | 3 | https://mathoverflow.net/users/8008 | 347780 | 147,261 |
https://mathoverflow.net/questions/347764 | 2 | Can we find for any real number $x$ the sequence of rationals $q\_n(x)$ with properties:
* $\lim\limits\_{n\to\infty} q\_n(x)=x$
* $q\_n(x+y)=q\_n(x)+q\_n(y)$
* $q\_n(xy)=q\_n(x)q\_n(y)$
?
| https://mathoverflow.net/users/118366 | Rational representation of reals | Assume $q(x)\neq 0.$ Then $q(2x)=q(x+x)=q(x)+q(x)=2q(x)$ but also $q(2x)=q(2)q(x)$ so $q\_n(2)=2$ for all $n.$ But then $2=q(2)=q(\sqrt{2}^2)=q(\sqrt{2})^2$ leading to $q(\sqrt{2})=\sqrt{2}$ which is not rational.
For any rational $r$ one has $q\_n(r)=r$ for all $n.$
| 4 | https://mathoverflow.net/users/8008 | 347782 | 147,263 |
https://mathoverflow.net/questions/346862 | 2 |
>
>
> >
> > *Definition*
> >
> >
> > Let $W$ be the function , defined as $W(a,b)=r$
> >
> >
> > given $a,b\in \mathbb{Z\_+}$ and $a>1$
> >
> >
> > Take $m$ to be the integer s.t. $a^{m+1} \ge b > a^{m}$, i.e. $m = \lceil \log{b}/\log{a} \rceil - 1$.
> >
> >
> > Convert number $a^{m+1} - b$ in base $a$ and... | https://mathoverflow.net/users/149083 | Sum of the digits in base $p+1$ | Define $X\_a$ be the set as, $\{2,3,...,a-1,a\}$
let $D(b,m)$ be the sum of the base-$b$ digits of $m$.
Define $f(a,k)=\frac{D(a,a^{k+1}-S(a,k))}{a-1}$
**Theorem**:
Given $a\in \mathbb{Z}\_{\ge 4}$ and $m\in \mathbb{Z}\_{\ge 1}$, If $a-1\mid S(a-1,2m)$ and $a-1>2m+1$ then $(f(a,2m))\_a\in X\_a$
**incomplete P... | 0 | https://mathoverflow.net/users/149083 | 347786 | 147,265 |
https://mathoverflow.net/questions/347804 | 4 | In the book "Index Transforms" by S.B. Yakubovich (World Scientific, 1996), I encountered the notion of a "smooth function of the order two" with compact support on $\mathbb R\_+$, denoted $C^{(2)}(\mathbb R\_+)$, cf. [here](https://books.google.com/books?id=vODsCgAAQBAJ&pg=PA49&lpg=PA49&dq=yakubovich%20smooth%20functi... | https://mathoverflow.net/users/105173 | Notion of a "smooth function of the order two" (Yakubovich, "Index Transforms") | The symbols $C^k$, with $k=0,\dots,\infty$ are today universally acknowledged and can be used with no need of other specification; yet sometimes (and more often $20$ years ago) for the sake of clarity, or style, one would still insert them in expressions like *regular/regularity* or *smooth/smoothness of order/class $C... | 2 | https://mathoverflow.net/users/6101 | 347807 | 147,268 |
https://mathoverflow.net/questions/347748 | 2 | Let $X$ be a completely regular Hausdorff space, $C\_b(X)$ the Banach space of bounded continuous function and $M(X) \subseteq M(\beta X) = C(\beta X)' = C\_b(X)'$ the spaces of Radon measures on $X$. For the strong topologies it then holds
$$ \beta(C\_b(X), M(X)) \subseteq \beta(C\_b(X), M(\beta X)) = \lVert \cdot \... | https://mathoverflow.net/users/58682 | Identification of some strong topology | The set of all Dirac measures is bounded and its polar in the space of bounded continuous functions is the unit ball of the supremum-norm -- do I miss something?
| 3 | https://mathoverflow.net/users/21051 | 347815 | 147,269 |
https://mathoverflow.net/questions/347787 | 5 | In 2009, Moser published a breakthrough [paper](https://dl.acm.org/citation.cfm?id=1536462) constructifying the Lovász Local Lemma (LLL). His talk at STOC was described in a [blog post](https://blog.computationalcomplexity.org/2009/06/kolmogorov-complexity-proof-of-lov.html) by Fortnow that proves a slightly weakened r... | https://mathoverflow.net/users/149542 | Formalizing Entropy Compression (as used to constructify the Lovász Local Lemma) | In my formulation of the argument, the length of the string $r$ (which I call $R$) *is* fixed: one does indeed only read a prefix of this string, but the remaining unread bits of the string are saved as part of the output (and referred to in my writeup as $R'$). The precise length of $r$ is not terribly relevant (it ca... | 4 | https://mathoverflow.net/users/766 | 347817 | 147,270 |
https://mathoverflow.net/questions/347816 | -3 | The principal inverse function of the gamma function is denoted by $\Gamma^{-1}$. See the paper: [Uchiyama - The principal inverse of the gamma function](https://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2).
$\Gamma^{-1}$ is an increasing and concave function defined on $(0.8856,+\infty)$.
I am as... | https://mathoverflow.net/users/74668 | The existence of an interval $I\subset (0.8856,+\infty)$ such that the derivative $(\Gamma^{-1})’(x)$ is greater than $1$ | Am I missing something? $(\Gamma^{-1})'(y) = \frac{1}{\Gamma '(x)} = \frac{1}{\Gamma(x) \cdot \psi(x)}$ for $y = \Gamma(x)$. The existence of $I$ now immediately follows from the continuity of $(\Gamma^{-1})'$.
| 3 | https://mathoverflow.net/users/100904 | 347819 | 147,271 |
https://mathoverflow.net/questions/347814 | 7 | Let $L$ be a finite dimensional Lie algebra over $\mathbb{R}$, and $K$ a subalgebra of $L$. Then, by Lie's correspondence theorems, there exists a unique (up to isomorphisms) simply connected Lie group $G$ having $L$ as a Lie algebra. There also exists a unique connected Lie subgroup $H$ of $G$ having $K$ as its Lie al... | https://mathoverflow.net/users/32135 | Algebraic condition that distinguishes embedded from immersed lie subgroups | See the [closed-subgroup theorem](https://www.wikiwand.com/en/Closed-subgroup_theorem), in particular the [conditions for being closed](https://www.wikiwand.com/en/Closed-subgroup_theorem#Conditions_for_being_closed).
| 3 | https://mathoverflow.net/users/11142 | 347834 | 147,277 |
https://mathoverflow.net/questions/347866 | 4 | Suppose I play the following game against the Opponent. My moves are rational numbers $p\_i$ and the Opponent's moves are real numbers $\epsilon\_i>0$.
>
> On turn $n+1$ the past move sequence is $p\_1,\epsilon\_1,\ldots, p\_{n}, \epsilon\_{n}$. I select a point $p\_{n+1}\in \mathbb Q \cap (p\_{n}-\epsilon\_{n},p\... | https://mathoverflow.net/users/58082 | Can I win this variant of the Banach-Mazur Game? | Sure you can win. Let's enumerate rationals as $q\_n, n = 1, 2,\ldots$. Also we can WLOG assume that $\varepsilon\_{i+1} \le \frac{\varepsilon\_i}{20000}$. We will make it so that $d(q\_n, p\_{n+1}) \ge 10\varepsilon\_{n+1}$. Then for $m > n$ we have $d(q\_n, p\_m) \ge 10\varepsilon\_{n+1} - \varepsilon\_{n + 2} - \ldo... | 5 | https://mathoverflow.net/users/104330 | 347870 | 147,284 |
https://mathoverflow.net/questions/347848 | 6 | We know for any principal ideal domain, objects in the bounded derived category are all formal hence we can classify those objects with finitely generated cohomology using structure theorem for finitely generated modules.
Now consider the bounded derived category of $\mathbb C[x]/x^2$-modules, how to classify indeco... | https://mathoverflow.net/users/102104 | Indecomposable objects in bounded derived category of $\mathbb C[x]/x^2$-mod | Up to shifts, every indecomposable object is of one of the forms described in the question.
I don't know an explicit reference, but here's a sketch of a proof.
By induction on the length, it's not hard to prove that every bounded complex of finite rank free modules, such that the image of each differential is conta... | 4 | https://mathoverflow.net/users/22989 | 347871 | 147,285 |
https://mathoverflow.net/questions/347865 | 8 | In the paper *A problem on completeness of exponentials (Annals of Mathematics 178 (2013), 983-1016)*, the author A. Poltoratski studies the following problem:
Let $\mu$ be a finite positive measure on $\mathbb{R}.$ For $a>0$, let
$$
\mathcal{E}\_a = \{ e^{ist} : s \in [0,a] \}
$$
be the set of exponentials with freq... | https://mathoverflow.net/users/149630 | Completeness of exponentials $\mathcal{E} = \{ e^{ist} : s \in \mathbb{R} \}$ in $L^p(\mu)$ | In fact, this is true for every $\mu$ and every $1 \le p < \infty$. It's a pretty standard "textbook" fact, which is probably why you're not finding papers that discuss it.
One possible way to prove it is following Aleksei Kulikov's hint. Suppose they were not dense. Then by Hahn-Banach and the $L^p$-$L^q$ duality, t... | 10 | https://mathoverflow.net/users/4832 | 347872 | 147,286 |
https://mathoverflow.net/questions/347856 | 5 | **Q(1):** Can the category of partial orders be fully embedded in the category of linear orders?
---
Vopěnka's principle, or VP, is a very intriguing axiom with many equivalent forms and consequences spanning universal logic (in the Barwise sense), large cardinals, model theory, and category theory.
*VP for $C$... | https://mathoverflow.net/users/115951 | Can the category of partial orders be fully embedded in the category of linear orders? | Expanding on Jeremy's comments,
Q1) No, the category of partial orders does not admit a faithful functor $F$ into the category of linear orders. The poset $\{l,r\}$ of two incomparable elements has a nonidentity automorphism $n$. If the global elements $l$, $r$ satisfy $F(l) < F(r)$ or $F(r) < F(l)$, then $F(n)$ is n... | 9 | https://mathoverflow.net/users/100508 | 347880 | 147,288 |
https://mathoverflow.net/questions/347788 | 7 | Let $E$ be a multiplicative cohomology theory. Fix a prime p. Call a ring map $\psi^{p}:E\rightarrow E$ an *Adams operation* if it lifts the Frobenius map $E/p\rightarrow E/p$.
It is of course well-known that $K$-theory has Adams operations. (If it weren't for $K$-theory, these operations would have a different name.... | https://mathoverflow.net/users/148857 | Cohomology theory with only one Adams operation? | Adams operations exist in quite wide generality. For any even periodic ring spectra $E$ and $F$, we have associated formal groups $G\_E$ and $G\_F$ over base schemes $S\_E$ and $S\_F$. There is a moduli scheme $\text{Hom}(G\_E,G\_F)$ parametrising pairs $(f,\widetilde{f})$ consisting of a map $f\colon S\_E\to S\_F$ and... | 5 | https://mathoverflow.net/users/10366 | 347887 | 147,291 |
https://mathoverflow.net/questions/347859 | 0 | **Motivation.** Recently I was watching two people play a game that involved arranging sticks in a number of heaps and moving them around in certain allowed ways that I think I was able to infer from observation. (They told me the name of the game in their language, but I can't remember it.)
**Problem.** Let $\mathbb... | https://mathoverflow.net/users/8628 | Majority-driven manipulations of integer vectors | $M\_n$=1 and so $\lim M\_n/n=0\ne 1$.
(We assume your definitions mean that $\bf 1\_\bf n\in {\it S\_n}$.)
We claim that $S\_n=\{\bf 1\_\bf n\}$.
To derive a contradiction, let $(\lambda\_1,\dots,\lambda\_n)\in S\_n\setminus\{\bf 1\_\bf n\}$. WLOG, it is obtained by a majority move from $\bf 1\_\bf n$. Also, WL... | 2 | https://mathoverflow.net/users/51389 | 347890 | 147,294 |
https://mathoverflow.net/questions/347885 | 0 | Given the following function of random variables
$$g = \frac{1}{n} \sum\_{k=1}^{n}{|h\_k|\exp\left( j \theta\_k \right)},$$
where $h\_1, \cdots, h\_n$ are i.i.d. random variables following the complex Gaussian distribution $\mathcal{CN}(0,\beta)$ and $\theta\_1, \cdots, \theta\_n$ are i.i.d. random variables with pro... | https://mathoverflow.net/users/103291 | PDF of $g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)}$? | For the complex Gaussian distribution the real and imaginary parts of $h\_k$ are i.i.d. with a normal distribution; the absolute value $|h\_k|$ has distribution $P(|h\_k|)=|h\_k|\exp(-|h\_k|^2/2)$ and the argument $\phi\_k={\rm arg}\,h\_k$ is uniformly distributed in $(0,2\pi)$, independently of $|h\_k|$. So to generat... | 2 | https://mathoverflow.net/users/11260 | 347891 | 147,295 |
https://mathoverflow.net/questions/347825 | 4 | I hope this question isn't too obfuscated (or easy)!
Given a set $S$, let $S\_\perp$ denote $\{X \in \mathcal P(S) \mid \forall x, y \in X.\, x=y\}$, the elements of which are *subsingletons*. In the following $Y\_\perp^X$ means $(Y\_\perp)^X$.
The question is: *does a partial map between $X$ and $Y$ that doesn't h... | https://mathoverflow.net/users/75761 | Does a map over subsingletons determine a subsingleton over maps? | Under Unique Choice, this implies ¬¬LEM.
Let $X=\Omega$ be the set of truth values, let $Y=2\subseteq\Omega$, and let $\phi$ be equality.
Then $\phi$ is a partial function, and $\neg \neg (x = \bot \lor x = \top)$ holds for any $x$, so the statement would imply that $\neg\neg \exists f:2^\Omega.\forall x:\Omega.x=f... | 5 | https://mathoverflow.net/users/100508 | 347892 | 147,296 |
https://mathoverflow.net/questions/347906 | 11 |
>
> Do either $~S\_4^+(a)~=~\displaystyle\sum\_{n=0}^\infty(n+a){2a\choose n}^4~$ or $~S\_4^-(a)~=~\displaystyle\sum\_{n=-2a}^\infty(n+a){2a\choose-n}^4~$ possess a *meaningful* closed form expression**1** in terms of the general parameter $a\not\in\mathbb Z$ ?
>
>
>
[Ramanujan](http://math.stackexchange.com/que... | https://mathoverflow.net/users/39602 | Closed Form for $\sum\limits_{n=-2a}^\infty(n+a){2a\choose-n}^4,~a\not\in\mathbb Z$ | Some experimentation suggests that the second series is given by
$$
S\_4^-(a) = \sum\_{n=-2a}^\infty (n+a) \binom{2a}{-n}^4 = \frac{1}{4\cos(2\pi a) \,\Gamma(2a+1)^2\,\Gamma(-4a)},
$$
which agrees with Ramanujan's at $a = -1/8$, but I have no proof...
| 12 | https://mathoverflow.net/users/47484 | 347922 | 147,304 |
https://mathoverflow.net/questions/347937 | 0 | I have to obtain an asymptotic solution for small real positive $x$ for the ratio of Spherical Hankel functions ($n=0,1,2....)$
${h^{(2)}\_n(x)}/{h^{(1)}\_n(x)}$
I found that series should be
$-1 + i C\_n x^{2n+1}+O(x^{2n+2})$
but how can I get the accurate coefficient at least for the first non-zero order?
| https://mathoverflow.net/users/142364 | The ratio of Hankel functions | $$\frac{h^{(2)}\_n(x)}{h^{(1)}\_n(x)}=-1+\frac{i \pi x^{2n+1}}{2^{2n}\Gamma \left(n+\frac{1}{2}\right) \Gamma \left(n+\frac{3}{2}\right)}+{\cal O}(x^{2n+2}).$$
For $n\geq 1$ the next term is actually of order $x^{2n+3}$.
| 2 | https://mathoverflow.net/users/11260 | 347940 | 147,307 |
https://mathoverflow.net/questions/347939 | -1 | Do there exist a real vector space $X$ *complete* with respect to norms $|\cdot|$ and $\|\cdot\|$ and a sequence $(x\_n)\_{n\in \mathbb N} \subset X$ such that there exist $x,y\in X$: $x\ne y$, $|x\_n - x|\to 0$ and $\|x\_n -y\|\to 0$ as $n\to \infty$?
Without requiring $X$ to be complete with respect to $|\cdot|$ an... | https://mathoverflow.net/users/44463 | Sequence converging to different limits with respect to two different _complete_ norms | $|\cdot|$ is complete. Indeed, suppose $(x\_n)$ is a Cauchy sequence in $(X,|\cdot|)$. Then $(Ax\_n)$ is a Cauchy sequence in $(X, \|\cdot\|)$ which is complete and hence has a limit $y$. Since $A$ was constructed to be a linear isomorphism, $y=Ax$ for some $x\in X$. Therefore $|x\_n-x|=\|A(x\_n-x)\|=\|Ax\_n-y\|$ tends... | 1 | https://mathoverflow.net/users/30186 | 347941 | 147,308 |
https://mathoverflow.net/questions/347944 | 8 | Let $G$ be a finite group and $D(G)$ its quantum double. Its finite dimensional complex representations are classified in this [Dijkgraaf et al. Quasi-Quantum Groups Related To Orbifold Models](https://pdfs.semanticscholar.org/7d76/3f033462bd49fe945ff9125e050139beb68f.pdf). However, in the paper, the authors claimed th... | https://mathoverflow.net/users/124549 | Classification of $\operatorname{Rep} D(G)$ | There are some classic results on the **classification of the irreducible $D(G)$-modules**:
If the field is the complex numbers $\mathbb{C}$, it has been shown that a representation of the finite group $G$, induced from an irreducible representation of the centralizer subgroup of an element $g$ of $G$, generates an ... | 6 | https://mathoverflow.net/users/85967 | 347947 | 147,311 |
https://mathoverflow.net/questions/347898 | 13 | Someone told me that it is possible to solve the Yamabe problem using Ricci flow. The proof I know of is the one originally proposed by Yamabe and then completed by Trudinger, Aubin and Schoen (in particular, the final step by Schoen made use of the positive mass theorem which had earlier been proved by Schoen and Yau)... | https://mathoverflow.net/users/119114 | Is there a solution of the Yamabe problem using Ricci flow? | The references provided by Carlo Beenakker solve the problem only in special cases. The article "Global existence and convergence of Yamabe flow" by R. Ye assumes -- for example -- conformal flatness. The problem was solved completely within the last years. Important progress in dimension 3,4 and 5 was obtained by
* ... | 12 | https://mathoverflow.net/users/110127 | 347954 | 147,313 |
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