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https://mathoverflow.net/questions/354348 | 7 | I have asked related questions on MSE and received no answer, so: I am looking for proofs of the integrals presented in Erdélyi's *Table of Integral Transforms* vol. i-ii, specifically proofs of the various gamma integrals presented on page 297-299 of volume 2, for instance:
$$\int\_{\mathbb{R}} \frac{dx}{\Gamma(\alp... | https://mathoverflow.net/users/128941 | Reference request: proofs of integrals presented in Erdélyi's *Table of Integral Transforms* | Titchmarsh’s *Fourier integrals* ([1937](//zbmath.org/?q=an:0017.40404), [7.6.4](//archive.org/details/IntroductionToTheTheoryOfFourierIntegrals/page/n196)) has proof and attribution to Ramanujan.
| 10 | https://mathoverflow.net/users/19276 | 354359 | 149,663 |
https://mathoverflow.net/questions/354322 | 3 | Consider the projective symplectic group $\mathrm{PSp}(n+1)$ over $\mathbb{C}$. This parametrizes $(n+1)\times (n+1)$ symplectic matrices modulo scalar multiplication.
Is $\mathrm{PSp}(n+1)$ irreducible?
Consider $4\times 4$ symplectic matrices. A matrix $A$ has a symplectic representative (modulo scalar) if and on... | https://mathoverflow.net/users/14514 | Topology of the projective symplectic group | The symplectic group $G=\mathrm{Sp}(2n)=\mathrm{Sp}(V)$ is connected (say, in characteristic zero, as algebraic group), and hence so is its quotient $\mathrm{PSp}(2n)$. Let $K$ be the ground algebraically closed field, and $(V,\langle\cdot,\cdot\rangle)$ the given symplectic space.
Indeed, we have to check that ever... | 3 | https://mathoverflow.net/users/14094 | 354364 | 149,666 |
https://mathoverflow.net/questions/354372 | 3 | **Recap: bordism group**
An *oriented singular $n$-manifold* in $X$is a map $f:M^n\to X$ where $M$ is a finite disjoint union of $n$-dimensional smooth manifolds.
The empty set is an admissible oriented singular $n$-manifold.
Two oriented singular manifolds $f\_i:M\_i^n\to X$ $i=1,2$ are *bordant* in $X$ if exists ... | https://mathoverflow.net/users/99042 | Bordism groups of $X$, Thom isomorphism and characteristic numbers | For question 2b, the answer is that elements of $H^n(X; A)$ determine bordism invariants
$\Omega\_n^{\mathrm{SO}}(X)\to A$, and if $H^\*(X)$ contains $p$-torsion for $p$ odd, these can't be interpreted as
Stiefel-Whitney or Pontrjagin numbers. A simple example is $\Omega\_1^{\mathrm{SO}}(B\mathbb Z/3)$, the bordism
gro... | 8 | https://mathoverflow.net/users/97265 | 354391 | 149,678 |
https://mathoverflow.net/questions/354392 | 0 | Is there an established name for the matrices that establish the conditions for a linear combination of $n$ functions $\lbrace f\_1(x),\dots,f\_n(x)\rbrace$ being the $n$-times smoothly differentiable continuation of a function $g(x)$ for $x=x\_0$ after adding an appropriate constant:
$$\begin{pmatrix}\frac{d}{dx}f\... | https://mathoverflow.net/users/31310 | Name for matrix associated to smooth continuation | According to [this Wikipedia page](https://en.wikipedia.org/wiki/Wronskian#Definition), the matrix in the question is the derivative of what is sometimes called a *fundamental matrix*. A Google search of that term together with "Wronskian" gives quite a few relevant hits, so the name seems to be in common use.
| 3 | https://mathoverflow.net/users/115044 | 354405 | 149,685 |
https://mathoverflow.net/questions/354410 | 13 | There exist homogeneous spaces such as the [pseudo-arc](https://en.wikipedia.org/wiki/Pseudo-arc), which are compact, connected, and totally path-disconnected. Is there a nontrivial, Hausdorff topological group with the same properties, i.e. that is compact, connected, and totally path-disconnected? What about a metriz... | https://mathoverflow.net/users/5801 | Is there a compact, connected, totally path-disconnected topological group? | (I'm assuming the groups to be Hausdorff to avoid the discussion degenerate into idle banter.)
The answer is yes: $\{1\}$ is such a group.
The answer to the intended question (which is probably whether there's a nontrivial such group) is no.
Andrew M. Gleason. Arcs in locally compact groups. Proc. Nat. Acad. Sci.... | 18 | https://mathoverflow.net/users/14094 | 354439 | 149,701 |
https://mathoverflow.net/questions/354429 | 4 | Let $\mathbb{D}$ be the unit disk on the plane and let $U,V\subset \mathbb{D}$ be open and such that $U\cup V=\mathbb{D}$.
>
> Is there a holomorphic map $\varphi:\mathbb{D}\times \mathbb{D}\to \mathbb{D}$ such that $\varphi(U\times V)=\mathbb{D}$?
>
>
>
Of course one can ask the same question for "ambient dom... | https://mathoverflow.net/users/53155 | Holomorphic union of sets | Edit2: everything works, updating the answer.
Yes. Consider two cases.
Case 1. There is a point on the boundary contained both in the closure of $U$ and $V$. Do a Mobius transform transforming our disk into upper half-plane $\mathbb{H}$ and putting this point to $0$.
Now take the map $(x, y) \mapsto x+y$.
It is... | 4 | https://mathoverflow.net/users/33286 | 354444 | 149,704 |
https://mathoverflow.net/questions/354435 | 18 | What is the simplest diophantine equation for which we (collectively) don't know whether it has *any* solutions? I'm aware of many simple ones where we don't know (whether we know) *all* the solutions, but all of these that I know have some solution.
Yes, I know that "simplest" is subjective. I'd be satisfied if it c... | https://mathoverflow.net/users/51744 | Simplest diophantine equation with open solvability | Determining which integers $n$ are a sum of three cubes is a very famous open problem:
$$a^3 + b^3 + c^3 = n, \quad a,b,c \in \mathbb{Z}.$$
Conjecturally, $n$ is a sum of three cubes iff $n \not \equiv 4,5 \bmod 9$.
Note that this is really a family of Diophantine equations, rather than a single Diophantine equatio... | 21 | https://mathoverflow.net/users/5101 | 354447 | 149,705 |
https://mathoverflow.net/questions/354418 | 1 | Let's suppose that with number $\mu\_1 \in \mathbb{R}$ we associate a Hilbert space $\mathcal{H}\_{\mu\_1}$ with countable basis $|1\rangle \_{\mu\_1}$, $|2\rangle \_{\mu\_1}$, $|3\rangle \_{\mu\_1}$, $\ldots$ Analogically
$\mathcal{H}\_{\mu\_2}$: $\mu\_2 \in \mathbb{R}$ with basis $|1\rangle \_{\mu\_2}$, $|2\rangle ... | https://mathoverflow.net/users/152731 | Result of continuum tensor product of Hilbert spaces | There are several ways you can define a Hilbert space tensor product $\bigotimes\_{t \in X} H\_t$, if each $H\_t$ is a Hilbert space. The "full" tensor product is generated by all functions $h: X \to \bigcup H\_t$ with $h(t) \in H\_t$ for each $t$ and such that $\prod \|h(t)\|$ converges. We write $h$ as $\bigotimes h(... | 6 | https://mathoverflow.net/users/23141 | 354457 | 149,708 |
https://mathoverflow.net/questions/354422 | 2 | I have been trying to understand a bit of the basics of deformations of Galois representations. One point which leaves me curious now is that proving modularity lifting with arbitrary ramification on the Tate module, compared to modulo $p$, seems quite alot to ask. As that is true, it is not so much after all. But is i... | https://mathoverflow.net/users/6575 | Modularity of elliptic curves with only minimal lifting | All the modularity-lifting theorems in the world aren't going to help you if you don't have something modular to lift!
The reason $p=3$ is so important is that $GL\_2(\mathbf{F}\_3)$ is solvable, which allows you to use Langlands--Tunnell to show that the mod 3 representation is modular (at least if the image is lar... | 3 | https://mathoverflow.net/users/2481 | 354466 | 149,713 |
https://mathoverflow.net/questions/354464 | 5 | Let $X,Y$ be two closed subschemes of $\mathbb{A}^n\_{\mathbb{C}}$ for some fixed $n$. Assume that I have an isomorphism of the underlying reduced spaces:
$$f : X\_\text{red} \stackrel{\sim}\longrightarrow Y\_\text{red}$$ which induces an isomorphism of $\mathcal{O}\_{X\_\text{red}}$-module:
$$ f^{\*} \left(\Omega\_{... | https://mathoverflow.net/users/37214 | schemes having same reduced underlying space and same cotangent sheaf are isomorphic? | Consider the simplest example:
$$
X = \mathrm{Spec}(\mathbb{C}[\epsilon]/\epsilon^2), \qquad
Y = \mathrm{Spec}(\mathbb{C}[\epsilon]/\epsilon^3).
$$
Definitely, $X\_{\mathrm{red}} \cong Y\_{\mathrm{red}}$. Also, a simple computation shows that
$$
\Omega\_{X/\mathbb{C}} \cong \mathbb{C},
\qquad
\Omega\_{Y/\mathbb{C}} ... | 14 | https://mathoverflow.net/users/4428 | 354470 | 149,715 |
https://mathoverflow.net/questions/353641 | 7 | Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see [here](https://link.springer.com/article/10.1007/s00208-008-0317-1). Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see [here](https://projecteuclid.org/euclid.gt/1510859209).
I've heard that it is open whether hi... | https://mathoverflow.net/users/111917 | Thickness and hierarchical hyperbolicity | This is not an answer, just some sketchy thoughts that are too long for the comment box. I and some other HHS enthusiasts are very interested in this question being answered; we've tried a fair bit and have set it aside, so I don't think they'll mind me trying to recall what some of the strategies and issues are.
It... | 4 | https://mathoverflow.net/users/76590 | 354477 | 149,716 |
https://mathoverflow.net/questions/354448 | 0 | If we consider the set of all unramified Satake parameters $S$ of all automorphic representations of $\operatorname{GL}\_{n}(\mathbb{A}\_{\mathbb{Q}})$ as $n$ varies, do we know absolute (that is, independent of $n$) lower and upper bounds for the norm of elements of $S$?
Motivation: denoting by $\mathcal{M}\_{a,b}$ ... | https://mathoverflow.net/users/13625 | Do we know absolute bounds for the norm of Satake parameters? | We don't know any upper bound for $|\alpha\_{p,j}|$ that is independent of $p$. On the other hand, we do know that each $|\alpha\_{p,j}|$ is bounded by $p^{1/2}$, hence if $\pi$ is an automorphic representation whose Rankin-Selberg powers $\pi\otimes\dots\otimes\pi$ are all automorphic, then the Satake parameters of $\... | 4 | https://mathoverflow.net/users/11919 | 354484 | 149,718 |
https://mathoverflow.net/questions/354482 | 4 | The quintic del Pezzo $3$-fold $V(5)$ of degree $5$ is defined as the intersection of $Grass(2,5)$ and a codimension $3$ linear subspace. I would like to show the following:
**For every point $p$ in $V(5)$ there is a line contained in $V(5)$ not passing through $p$.**
I know that $\operatorname{Gr}(2,5)$ is define... | https://mathoverflow.net/users/nan | lines on quintic del Pezzo 3-fold of degree 5 | Let $X$ be the quintic del Pezzo 3-fold. The Hilbert scheme of lines on $X$ is 2-dimensional (this follows easily from deformation theory). If there is a point $p \in X$ such that every line passes through $p$, then $X$ is a cone with vertex $p$,
hence either $X$ is singular at $p$, or $X \cong \mathbb{P}^3$. But, of c... | 4 | https://mathoverflow.net/users/4428 | 354485 | 149,719 |
https://mathoverflow.net/questions/347795 | 3 | Per the title, what are some of the oldest books on logic out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
Update: Doesn't have to be mathematical logic per se. I mean logic at large.
| https://mathoverflow.net/users/126532 | Reference request: Oldest books on logic with unsolved exercises? | The update to the question now asks for "logic at large" -- rather than specifically mathematical logic. Then one can go back to before the 20th century, as in:
* [Studies and exercises in formal logic,](https://archive.org/details/studiesandexerc02keyngoog/page/n29/mode/2up%0A) J.N.Keynes, 1884.
* [Questions and exe... | 3 | https://mathoverflow.net/users/11260 | 354492 | 149,722 |
https://mathoverflow.net/questions/346693 | 3 | I would like to know if it in the literature an approximation for
$$\sum\_{\rho}\frac{1}{|\rho|^2}\tag{1}$$
where the sum is over all of the non-trivial zeros of the Ramanujan's zeta function (also known as Ramanujan $\tau$-Dirichlet series). I know a similar series as **Lemma 3.3** from [1], and that there are similar... | https://mathoverflow.net/users/142929 | Approximation of $\sum_{\rho}\frac{1}{|\rho|^2}$, over the non-trivial zeros of the Ramanujan's zeta function | Writing the nontrivial zeros $\rho$ as $\beta+i\gamma$ with $\beta,\gamma\in\mathbb{R}$, we first observe that
$\displaystyle\sum\_{\rho}\frac{1}{|\rho|^2}\leq \sum\_{\gamma}\frac{1}{\gamma^2}$.
We find using the first few zeros of $L(s,\Delta)$ (Ramanujan's zeta function) as computed on [LMFDB](https://www.lmfdb.o... | 5 | https://mathoverflow.net/users/111215 | 354505 | 149,731 |
https://mathoverflow.net/questions/353743 | 3 | A result due to B. Conrad (<http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf>, Theorem A.1) states that the Atkin-Lehner operator $w\_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M\_k(\Gamma\_0(N))$. In other words, if $f\in M\_k(\Gamma\_0(N))$ has coefficients in $\mathbb{Z}[1/Q]$ then $w\_{Q,k}(f)$ has coefficien... | https://mathoverflow.net/users/151669 | Integrality of Atkin-Lehner operator for $\Gamma_1(N)$ | **Theorem**. Let $\ell$ be prime, and $Q, R \ge 1$ such that $(\ell, Q, R)$ are pairwise coprime. Let $N = QR$ and for simplicity assume $N \ge 4$. Then $W\_Q$ preserves $M\_k(\Gamma\_1(N), \mathbf{Z}[1/N, \zeta\_Q])$.
*Proof*. Let $M\_k^{\mathrm{wk}}(\Gamma\_1(N), A)$ denote the space of weakly modular forms (possib... | 4 | https://mathoverflow.net/users/2481 | 354512 | 149,733 |
https://mathoverflow.net/questions/354409 | 5 | Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the manifold $S^{n-2}\times S^1\times S^1$ with coordinates $(x\_1,...,x\_{n-1},\varphi,\theta)$ such that $\sum\_{i=1}^{n-1}x\... | https://mathoverflow.net/users/153319 | Reeb stability counterexample: foliation in $S^{n-2}\times S^1\times S^1$ with non-diffeomorphic leaves | The precise description of the leaves which are not spheres is as follows.
The leaves passing through points with $x\_1\neq 0, \theta=\pi/2, \theta=constant+1/x\_1$ are homeomorphic to $\mathbb{R}^{n-2}$ (you can work out an explicit parametrization).
The leaves passing through points with $x\_1=0, \theta=\pi/2$ are ho... | 2 | https://mathoverflow.net/users/47274 | 354521 | 149,737 |
https://mathoverflow.net/questions/354475 | 3 | Let $X = (X\_1, X\_2, \ldots, X\_n)$ be a sequence of (not necessarily independent) Bernoulli random variables where for each $i$, the success probability $\Pr[X\_i = 1]$ itself is a random variable depending on the sequence $(X\_1, \ldots, X\_{i-1})$. For any assignment $X'=(X'\_1, \ldots, X'\_n)$ define $$\mu^\star(X... | https://mathoverflow.net/users/153090 | Chernoff-type bound for sum of Bernoulli random variables, with outcome-dependent success probabilities | $\newcommand\ep{\delta}$$\newcommand\de{\epsilon}$For $j=0,\dots,n$, let $S\_j:=\sum\_1^j d\_i$, where $d\_i:=X\_i-E\_{i-1}X\_i$ and $E\_{i-1}$ is the conditional expectation given $X\_1,\dots,X\_{i-1}$, with $E\_0:=E$ and $S\_0:=0$. Clearly, $(S\_j)$ is a martingale.
By [Theorem 8.7](https://projecteuclid.org/eucli... | 3 | https://mathoverflow.net/users/36721 | 354544 | 149,743 |
https://mathoverflow.net/questions/354532 | 2 | Suppose that we have two polyhedra $P\_1$ and $P\_2$ in $\mathbb{R}^3$. I would like to define such a metric $\rho(P\_1, P\_2)$ that depends on several factors, but currently I don't know how to do it better.
The distance is intended to be smaller if:
1. "big facets" of $P\_1$ have one-to-one correspondence with "b... | https://mathoverflow.net/users/56107 | Distance between two polyhedra that takes incidence structure into account | Two suggestions of literature that might lead you somewhere useful (but neither of these references in themselves seem to address your several criteria).
The first below uses curvature distributions. The second uses Minkowski's
mixed volumes. Both introductions survey "shape similarity" measures.
>
> Shum, Heung-Ye... | 3 | https://mathoverflow.net/users/6094 | 354545 | 149,744 |
https://mathoverflow.net/questions/354516 | 0 | Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively:
* A step function: $u\_1(x)=\begin{cases}
u\_{L}, x<c\_1, \\[2ex]
u\_{R}, x>c\_1,
\end{cases}$
* A "generalization to two dimensions": $u\_2(x,t)=\begin{cases}
u\_{L}, x<c\_2\cdot t, \\[2ex]
u\_{R}, x>c\_2\cdot t.
\end... | https://mathoverflow.net/users/117762 | Law of a step function and its generalization to two dimensions on an appropriate spaces | Your functions $u\_1$ and $u\_2$ are not completely defined. For instance, $u\_1(c\_1)$ is undefined.
If you do not need to distinguish functions differing only on a set of Lebesgue measure $0$, then you may consider $u\_1$ a point in the Banach space $B\_1:=L^\infty(D)$, and $u\_2$ a point in the Banach space $B\_2... | 1 | https://mathoverflow.net/users/36721 | 354547 | 149,746 |
https://mathoverflow.net/questions/354541 | 1 | Let $f,f\_n \in L^1(\mathbb{R},\mathbb{R}\_+)$ with $\int\_{\mathbb{R}} f = \int\_{\mathbb{R}} f\_n = 1$, $(\sqrt{f\_n})'$ bounded in $L^2$, $\nabla \sqrt{f}\in L^2$ and such that $$\int\_{ p+[0,1/n]} f\_n = \int\_{ p+[0,1/n]} f$$ for any $p \in \mathbb{Z}/n$ and any $n \in \mathbb{N}$. How to prove that $f\_n$ converg... | https://mathoverflow.net/users/153405 | Convergence of local means implies converge ae? | $\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb{Z}}$
We should, and will, assume that
\begin{equation}
\int\_p^{p+1/n}f\_n=\int\_p^{p+1/n}f \tag{0}
\end{equation}
for all $n\in\N$ and all $p\in\Z/n$. (If (0) is assumed only for $p\in\N/n$, then the conclusion will obviously be false in ... | 2 | https://mathoverflow.net/users/36721 | 354565 | 149,753 |
https://mathoverflow.net/questions/354559 | 1 | Let $X$ be a $N\times P$ matrix with random independent and identically distributed entries $x\_{ij}$. I also assume that $\langle x\rangle = 0$ and $\langle x^2\rangle = 1$.
Define the $N\times N$ matrix $C = (1/N)XX^T$.
I am interested in the limit $N\rightarrow\infty$, with $P=\alpha N$ for some finite positive ... | https://mathoverflow.net/users/16615 | Universality of the top eigenvalue of correlation matrices | In general you need more than second moment - you need fourth moment finite, otherwise the top eigenvalue can run off to infinity in your scaling. For this and more see the book of Bai and Silverstein (and the original papers).
| 1 | https://mathoverflow.net/users/35520 | 354566 | 149,754 |
https://mathoverflow.net/questions/354515 | 2 | I received the following interesting point in ([1](https://mathoverflow.net/questions/286332/a-criterion-for-second-countability)). I could not find any reference or clear proof. Any suggestion?
**Theorem.** *A topological space $X$ is hereditary Lindelof if and only if for any subspace $Y\subset X$, the $\sigma$-al... | https://mathoverflow.net/users/84390 | On the hereditary Lindelof topological spaces | If $X$ is hereditarily Lindelof then any open subset of $Y$ is Lindelof and therefore it is the union of countably many basic (for a given predetermined base for $Y$) open sets. Hence The $\sigma$-algebra generated by the base contains (so it is equal to) the Borel $\sigma$-algebra.
The other direction is not true. L... | 3 | https://mathoverflow.net/users/17836 | 354569 | 149,755 |
https://mathoverflow.net/questions/354542 | 47 | I am about to (hopefully!) begin my PhD (in Europe) and I have a question: how did you learn so much mathematics?
Allow me to explain. I am training to be a number theorist and I have only some read Davenport's Multiplicative Number Theory and parts of Vaughan's book on the circle method. I have briefly seen some var... | https://mathoverflow.net/users/146669 | How and when do I learn so much mathematics? | The other answers have some good general advice. Let me try to say something that is specific to the topics of analytic number theory, and number theory generally.
First, there is no such thing as training to be a number theorist. There are many different kinds of number theorists, and very few of him are comfortable... | 39 | https://mathoverflow.net/users/18060 | 354574 | 149,757 |
https://mathoverflow.net/questions/354575 | 2 | I am trying to understand the proof of Theorem 15.2 in the aforementioned book. In this proof, the authors *seem* to infer that, if $\psi(x) - x < x^{\Theta - \epsilon}$ for every $\epsilon > 0$ and sufficiently large $x$, then the function $f(s)=\int\_{1} ^{\infty} (x^{\Theta - \epsilon} - \psi(x) + x)x^{-s-1} \mathrm... | https://mathoverflow.net/users/153423 | On the proof of Theorem 15.2 in Montgomery-Vaughan's Multiplicative number theory | It seems you are not using Lemma 15.1 correctly. For convenience I write $$\frac{1}{s - \Theta + \varepsilon} + \frac{\zeta'(s)}{s\zeta(s)} + \frac{1}{s-1} = \int\_1^{+\infty} (x^{\Theta - \varepsilon} - \psi(x) + x) x^{-s-1} \, \mathrm{d}x. \quad (\*)$$
Now the left-hand side of $(\*)$ is analytic for **real** $s > ... | 8 | https://mathoverflow.net/users/133679 | 354578 | 149,759 |
https://mathoverflow.net/questions/354572 | 3 | Suppose that the Axiom of Determinacy (AD) holds in $L(ℝ)$, and for some statement φ, $α$ is minimal such that $L\_α(ℝ)⊨φ$. Do definable (in $L\_α(ℝ)$) elements of $L\_α(ℝ)$ form an elementary substructure $L\_α(ℝ)$?
*Extension:* Assume ZF+AD (or if needed $\text{AD}^+$), and let $W\_α$ consist of all sets of reals o... | https://mathoverflow.net/users/113213 | Pointwise definable models of determinacy | The least ordinal $\kappa$ such that $L\_\kappa(\mathbb R)$ satisfies KP is a counterexample. This essentially reduces to Theorem 1.3 of Tony Martin's "The Largest Countable This, That, and the Other." Of course, we are assuming some determinacy; $\text{AD}^{L(\mathbb R)}$ is more than enough, and Martin used the optim... | 5 | https://mathoverflow.net/users/102684 | 354586 | 149,762 |
https://mathoverflow.net/questions/354592 | 2 | I am not an expert of this area but I need help to answer this question: For a compact metric space $X$, let $C(X,X)$ be the space of all continuous functions $X\to X$, equipped with the uniform metric
$$d(f,g)=\sup\_x d(f(x),g(x)).$$
Can we say the space $C(X,X)$ is compact?
| https://mathoverflow.net/users/44949 | Compactness of $C(X,X)$ | Indeed this space is (almost, see the comment by YCor) never compact except for trivial cases (i.e. the compact set $X$ having only finitely many points).
For more information on the topology (which is colloquially known as the compact-open topology) of the space you are asking about, see e.g. the book Engelking: Gener... | 2 | https://mathoverflow.net/users/46510 | 354595 | 149,766 |
https://mathoverflow.net/questions/354327 | 76 | In this post, we look for the existing atlas-like websites providing well-presented classifications or database about some specific areas of mathematics. Here are some examples:
* GroupNames: <https://people.maths.bris.ac.uk/~matyd/GroupNames>
>
> Finite groups of order ≤500, group names, extensions, presentatio... | https://mathoverflow.net/users/34538 | Atlas-like websites on specific areas of mathematics | This [catalogue of mathematical datasets](https://mathdb.mathhub.info/) could be of some interest to you - at least some of the entries are atlas-like websites. It includes several of the websites mentioned above, and I'm slowly adding more to it.
| 25 | https://mathoverflow.net/users/35913 | 354606 | 149,769 |
https://mathoverflow.net/questions/354526 | 1 | (I previously asked a similar question on [cstheory.SE](https://cstheory.stackexchange.com/questions/46422/apredictable-sets); I have simplified the notion, which presumably changes it but does not change the key properties I'm interested in.)
This is about a strange recursion-theoretic notion I encountered, I am una... | https://mathoverflow.net/users/123634 | Impredictable subsets of $\mathbb{N}$ | No, the halting set is not impredictable. By the recursion theorem, there is an infinite computable sequence of values such that I control their entry into $K$ (the halting problem). Number these values as $(x\_{e, p})\_{e < p \in \omega}$. Let $\phi$ be such that $\phi(p) > x\_{e, p}$ for all $e < p$. The point is tha... | 3 | https://mathoverflow.net/users/32178 | 354613 | 149,771 |
https://mathoverflow.net/questions/354608 | 16 | Is there a constant $c>0$ such that for $X,Y$ two iid variables supported by $[0,1]$,
$$
\liminf\_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c
$$
I can prove the result if they have a density, of if they have atoms, but not in the general case.
| https://mathoverflow.net/users/16934 | How often two iid variables are close? | If $\epsilon \geqslant \tfrac{1}{n}$, then
$$ \mathbb{P}(|X-Y|<\epsilon) \geqslant \sum\_{i=1}^n \mathbb{P}(X, Y \in [\tfrac{i-1}{n}, \tfrac{i}{n}]) = \sum\_{i=1}^n (\mathbb{P}(X \in [\tfrac{i-1}{n}, \tfrac{i}{n}]))^2 . $$
It follows that
$$ \mathbb{P}(|X-Y|<\epsilon) \geqslant \frac{1}{n} \biggl(\sum\_{i=1}^n \mathbb{... | 27 | https://mathoverflow.net/users/108637 | 354614 | 149,772 |
https://mathoverflow.net/questions/208066 | 6 | Fix a compact convex subset $C \subset \mathbb{R}^n$ with nonempty interior. For any subspace $S \subset \mathbb{R}^n$, let $P\_S$ denote the orthogonal linear projection onto $S$. I'd like to claim that for almost every (in either a measure theory or topological sense) nontrivial subspace $S$ of a given dimension, the... | https://mathoverflow.net/users/54756 | Linear projections of convex sets with unique preimages of boundary points | I asked this question several years ago, and I recently found the answer in a paper by Ewald, Larman, and Rogers called "The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space". This was published in 1970 in Mathematika, and the main result in the paper is... | 2 | https://mathoverflow.net/users/54756 | 354632 | 149,778 |
https://mathoverflow.net/questions/354626 | 5 | The question I have is quite specific. So in the hope that this post might help others in the future, my problem boils down to solving the following integral:
$$I\_\tau=\int \prod\_{i, j=1}^{N} d J\_{i j} \exp \left\{-\frac{N}{2(1-\tau^2)}\left( \sum\_{i, j, k} J\_{k i} A\_{i j} J\_{k j}-\tau\sum\_{ij}J\_{ij}J\_{ji}\... | https://mathoverflow.net/users/142153 | A general formula for Gaussian integrals over matrix elements | Only the symmetric part of $A$ contributes to the integrand, so we may assume $A$ is symmetric and diagonalize it as $A=O\Lambda O^T$ with $O$ orthogonal and $\Lambda={\rm diag}\,(\lambda\_1,\lambda\_2,\ldots\lambda\_N)$. The orthogonal matrix may be removed by a change of variables, so without loss of generality we ca... | 5 | https://mathoverflow.net/users/11260 | 354639 | 149,782 |
https://mathoverflow.net/questions/354194 | 27 | Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis [Topos annelés et schémas relatifs](https://doi.org/10.1007/978-3-662-59155-0) under Grothendieck's guidance and appear in many of later works of the Grothendieck school, such as Berthelot's [Cohomologie Cristallin... | https://mathoverflow.net/users/130058 | Motivation for relative schemes: why should one work with schemes over a ringed topos? | The comments to your question discuss the variation of relative schemes over a topos, vs relative schemes over a site. But it seems your question
stood at the more basic level of the relevance of relative schemes over something else than a scheme.
In Grothendieck's philosophy, a relative scheme $f\colon X\to S$ func... | 15 | https://mathoverflow.net/users/10696 | 354654 | 149,784 |
https://mathoverflow.net/questions/354656 | 0 | Let $X$ be $k$ variety or more genrally a $k$-scheme. Denote the algebraic closure of $k$ by $\overline{k}$. it's a fact that $X(\overline{k}):=Hom(\operatorname{Spec} \ \overline{k}, X)$ as set is dense in the underlying topological space of $X$.
Obviously the Galois group $Gal(\overline{k}/k)$ acts on $X(\overline{... | https://mathoverflow.net/users/108274 | Galois action on $\overline{k}$-valued points extends to action on $k$-scheme $X$ | No, the Galois action on $X(\overline k)$ corresponds to a trivial action on the scheme $X$. This you can already see if $X=\mathbf A^1\_k=\mathop{\rm Spec}(k[T])$ is the affine line.
Then $X(\overline k)=\overline k$, with its obvious Galois action. However, the scheme $X$ has two kind of points: the generic point, an... | 5 | https://mathoverflow.net/users/10696 | 354657 | 149,785 |
https://mathoverflow.net/questions/269602 | 7 | Let $k$ be a field (of characteristic zero).
For $k[x\_1,\dotsc,x\_n]$ it is known that the affine and triangular automorphisms generate $G\_n$, the group of automorphisms of $k[x\_1,\dotsc,x\_n]$,
see, for example, [van den Essen's book "Polynomial automorphisms and the Jacobian conjecture"](http://www.springer.com/g... | https://mathoverflow.net/users/72288 | The group of $k$-automorphisms of $k[x_1,\ldots,x_n,x_1^{-1}]$ | I repeat my comment as an answer as suggested.
Your argument for $n=2$ generalises and shows that $x\_1$ is always mapped to some $\lambda x\_1^m$ for $\lambda\in k^\times$ and $m\in\{\pm1\}$. Therefore if you consider the subring $S:=k[x\_1^{\pm1}]$, a general $k$-linear automorphism of $R:=k[x\_1^{\pm 1},x\_2,\ldot... | 7 | https://mathoverflow.net/users/3041 | 354663 | 149,789 |
https://mathoverflow.net/questions/354662 | 5 | In the spirit of [this related question](https://math.stackexchange.com/questions/2499051/is-there-a-contradiction-hiding-in-this-alternative-set-theory-with-3-axioms), consider a set theory with the following axioms:
Axiom of extension:
$$ \forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \rightarrow ... | https://mathoverflow.net/users/74578 | How strong is this set theory? | Even if $φ$ is restricted to not use $C$, the theory is inconsistent! Here is the simple 2-line proof.
Let $R$ be such that $∀x\ ( x∈R ⇔ C(x) ∧ x∉x )$ by Comprehension.
Then $C(R)$ by Construction. Thus $R∈R ⇔ R∉R$. Contradiction.
| 14 | https://mathoverflow.net/users/50073 | 354674 | 149,796 |
https://mathoverflow.net/questions/354591 | 4 | Let $K=K(v)$ be a valued field of characteritic $0$ with non trivial valuation $v:K\rightarrow\mathbb{R}\cup\{\infty\}$. I'm looking for a proof of following characterization of Henselian property:
$K$ is Henselian iff $K=K\_v\cap \overline{K}$ where $K\_v$ is the completion wrt distance $\vert \ \vert\_v$ and $\ove... | https://mathoverflow.net/users/108274 | Henselian valued fields for characteristic $0$: a characterization | In general, i.e. for any valued field $(K,v)$, the implication $K = K\_v \cap \overline{K}$ (or more precisely that $(K,v)$ have no immediate algebraic extension, i.e. that $(K,v)$ be *algebraically maximal*, otherwise you seem to be already assuming that $v$ extends uniquely to $\overline{K}$) implies that $K$ is hens... | 4 | https://mathoverflow.net/users/45005 | 354679 | 149,798 |
https://mathoverflow.net/questions/354629 | 4 | I just picked up the paper "The classification of quotient singularities which are complete intersections" by Haruhisa Nakajima and Kei-Ichi Watanabe, which is in the book
* Greco, Silvio, and Rosario Strano, eds. Complete intersections: lectures given at the 1st 1983 session of the Centro Internationale Matematico E... | https://mathoverflow.net/users/12419 | Why do Nakajima and Watanabe claim the induced action of a finite linear group on the invariant subring of the reflection subgroup is linearizable? | I think one should consider the ideal $I = S^H\_+$ of invariants with zero constant coefficient. To give algebra-generators for $S^H$ is the same as to give generators of $I/I^2$ as a vector space. So here dim$\_K(I/I^2) = n$. $G/H$ acts linearly on $I$, $I^2$ and on $I/I^2$. Since we are working on the nonmodular case... | 3 | https://mathoverflow.net/users/82616 | 354681 | 149,799 |
https://mathoverflow.net/questions/350547 | 4 | Let $M$ be a closed additive submonoid of $\mathbb{R}^n$ with $n\geq1$. Suppose also that there exists $r>0$ such that every ball of radius $r$ intersects $M$. I wonder if we can obtain more information on $M$. For example, if $n=1$, it is easy to see that $M$ has to be a subgroup of $\mathbb{R}$, thanks to the classic... | https://mathoverflow.net/users/142808 | Closed cobounded additive submonoid of $\mathbb{R}^n$ | Yes, it has to be a subgroup. Fix $v\in M$. We need to prove that $-v\in M$. It is sufficient to find an element of $M$ arbitrarily close to $-v$.
Choose $u\_1,\ldots,u\_n\in \mathbb{R}^n$ so that $v,u\_1,\ldots,u\_n$ are the vertices of a regular simplex with center at the origin. Choose large $N$ and consider the p... | 2 | https://mathoverflow.net/users/4312 | 354684 | 149,800 |
https://mathoverflow.net/questions/350208 | 1 | let a convex polytope $\mathcal{P}$ in $E^n$ be defined as in the tag-description with the additional requirement that their volume be strictly positive.
let further the Voronoi Cells $VC(f)$ of $\mathcal{P}$ be defined as the set of points of $\mathcal{P}$ that attain one of their minimal distances to the boundary o... | https://mathoverflow.net/users/31310 | Convexity of the Voronoi cells of higher-dimensional polyhedra | Yes. Denote $H\_1,\ldots,H\_m$ hyperplanes of facets $f\_1,\ldots,f\_m$ of $\mathcal{P}$, denote by $\partial \mathcal{P}$ the union of $f\_1,\ldots,f\_m$ (the boundary of $\mathcal{P}$). Let $d(x,A)$ denote the distance from point $x$ to set $A$.
**Proposition**. For $x\in \mathcal{P}$ we have $x\in VC(f\_j)$ if and... | 1 | https://mathoverflow.net/users/4312 | 354689 | 149,802 |
https://mathoverflow.net/questions/354694 | 12 | This question is motivated by the earlier MO question: [Show that $(\sum\_{k=1}^{n}x\_{k}\cos{k})^2+(\sum\_{k=1}^{n}x\_{k}\sin{k})^2\le (2+\frac{n}{4})\sum\_{k=1}^{n}x^2\_{k}$](https://mathoverflow.net/questions/354383/show-that-sum-k-1nx-k-cosk2-sum-k-1nx-k-sink2-le-2) .
It is a cleaned up asymptotic version of that q... | https://mathoverflow.net/users/38624 | Bounding a Fourier coefficient of a non-negative periodic function in terms of its $L^2$-norm | Your conjecture is indeed correct. The proof is based on the following simple yet powerful trick I learned many years ago: $|z| = \sup\_{|v| = 1} \Re (zv)$. Therefore it is enough to only bound from above $\Re \left(\int\_0^1 vf(x)e^{-2\pi ix}dx\right)$ for all $v\in \mathbb{T}$. And now it is clear by Cauchy-Schwarz t... | 15 | https://mathoverflow.net/users/104330 | 354698 | 149,803 |
https://mathoverflow.net/questions/354700 | -1 | I´m working with "Algebraic Number Fields" from Gerald J. Janusz (1. edition from 1973) and I have a question about his notation.
In chapter IV proposition 4.5 he states if K is an algebraic number field and S is the set of primes of K which have relative degree one over Q then S is an infinte set.
Up to this point... | https://mathoverflow.net/users/153482 | Relative degree of a prime over a number field (notation from Algebraic Number Fields from Gerald J. Janusz) | No, Janusz clearly refers to the relative degree over the ring of integers of $\mathbb Q$, that is $\mathbb Z$. It is very common in algebraic number theory to speak of something relative to a number field, when in reality it means relative to its ring of integers. For instance, we commonly talk of ideals of $K$, which... | 0 | https://mathoverflow.net/users/133679 | 354703 | 149,805 |
https://mathoverflow.net/questions/354706 | 1 | Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$. Is the exceptional divisor $E$ necessarily irreducible?
| https://mathoverflow.net/users/nan | Terminal $\mathbb{Q}$-factorial divisorial contractions | Since $f:Y\rightarrow X$ is terminal we may write
$$K\_Y = f^{\*}K\_X + aE$$
with $a>0$. Let $C\subset Y$ be a curve in the ray contracted by $f$. Then
$$K\_Y\cdot C = C\cdot f^{\*}K\_X+aC\cdot E = K\_X\cdot f\_{\*}C+aC\cdot E = aC\cdot E$$
Now, $K\_Y\cdot C<0$ yields $C\cdot E < 0$.
Assume that $E = E\_1\cup E\_2$ h... | 2 | https://mathoverflow.net/users/14514 | 354709 | 149,808 |
https://mathoverflow.net/questions/354696 | 2 | Let $X$ be a non-negative random variable with cdf $F$ and define
$$G(s) = E[\max(0,u(X)-sX)],$$ where $u$ is some real function.
Let $s\_0$ be the unique fixed point of $G$.
Now let $X\_1,\dots,X\_t$ be a sequence of samples drawn independantly from $F$.
Let
$$G\_t(s) = \frac{1}{t}\sum\_{i=1}^t \max(0,u(X\_i)-sX\_... | https://mathoverflow.net/users/153481 | Convergence of estimator given by a fixed point | $\newcommand\N{\{1,2,\dots\}}
\newcommand\NN{\{1,2,\dots,\infty\}}
\newcommand\si{\sigma}
\newcommand{\ep}{\varepsilon}$
Suppose that $E u(X)\_+<\infty$, where $x\_+:=\max(0,x)$. Then the convergence takes place, with the rate $O(1/\sqrt t)$ (as $t\to\infty$) if we also assume that $EX<\infty$, $E u(X)\_+^a<\infty$ for... | 2 | https://mathoverflow.net/users/36721 | 354712 | 149,810 |
https://mathoverflow.net/questions/354704 | 2 | I'm trying to construct Brownian motion using the Kolmogorov extension theorem.
I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a random function in $D([0, \infty), R)$ - (the set of all functions from $R\_+$ to $R$, not just cadlag functions). I am also... | https://mathoverflow.net/users/nan | Continuity of Brownian motion constructed from Kolmogorov extension theorem? | I think it helps to look more closely into the construction. I'm going to use $\Omega = \mathbb{R}^{[0,\infty)}$ instead of $D$ to denote the space of all real-valued functions on $[0,\infty)$, since $D$ is more often used for the Skorokhod space of cadlag functions.
The Kolmogorov extension theorem gives you a proba... | 4 | https://mathoverflow.net/users/4832 | 354715 | 149,812 |
https://mathoverflow.net/questions/354720 | 2 | Let us assume we've a rectangular data matrix $X=[x\_1 \dots x\_n] \in \mathbb{R}^{p \times n}$, where the $x\_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the entries of the matrix $X$ are iid, or also that $x\_i$ are of the form $x\_i = C^{1/2}z\_i$, where $Z:=[z\_1 \dots z\_n]$ h... | https://mathoverflow.net/users/35936 | Marcenko-Pastur and Tracy-Widom laws for sample covariance and Gram matrices when the "features" are correlated: references | If the correlations can be described by a multivariate Gaussian there exist results for the spectral density in the limit of large matrices, see for example [Spectral Moments of Correlated Wishart Matrices](https://www.researchgate.net/publication/7952513_Spectral_Moments_of_Correlated_Wishart_Matrices) (2005).
For t... | 2 | https://mathoverflow.net/users/11260 | 354727 | 149,814 |
https://mathoverflow.net/questions/353430 | 2 | Consider the following oscillatory integral
$$
I(n):=\int\_{-\pi}^\pi\int\_{-\pi}^\pi e^{i n(x+y)}\frac
{(1 - \cos(2x)) (1 - \cos(2y))}
{2k - (\cos x + \cos y)}\ \mathrm{d}x\,\mathrm{d}y.
$$
where $n\in\mathbb{N}$, $n>2$, $k\in\mathbb{R}$, $k>1$, and $i$ is the imaginary unit.
>
> **My question.** Is it possible ... | https://mathoverflow.net/users/62673 | Asymptotic decay rate of an oscillatory integral | The asymptotics for large $n$ is
$$ J(n,\kappa) := \Big(\frac{2}{\pi}\Big)^2 \int\_{-\pi}^\pi \int\_{-\pi}^\pi
\exp{(i\,n(x+y))}\frac{\sin^2x\,\sin^2y} {2\kappa - (\cos{x}+\cos{y}) }\, dx \,dy \sim$$
$$ \sim \frac{8}{\sqrt{\pi \kappa n}}(\kappa^2-1)^{7/4} (\kappa - \sqrt{\kappa^2-1})^{2n}\quad, \quad (\kappa>1)$$
The ... | 3 | https://mathoverflow.net/users/121836 | 354732 | 149,817 |
https://mathoverflow.net/questions/354695 | 4 | Let $p:X^{\natural} \to S$, $q:Y^{\natural} \to S$ be two cartesian fibrations of simplicial sets (where the marking is given by cartesian edges) and assume that we are given an equivalence of cartesian fibrations $X^{\natural} \to Y^{\natural}$.
Given a marking on $S$ we define the marked simplicial set $X^{\dagger}... | https://mathoverflow.net/users/141150 | On equivalences of cartesian fibrations | Yes. Since $X^{\natural} \to S$ and $Y^{\natural} \to S$ are both cartesian fibrations they are fibrant and cofibrant objects in the cartesian model structure over $S$, which is a simplicial model structure. If two such objects are weakly equivalent then there must exist maps $f:X^{\natural} \to Y^{\natural}$ and $g: Y... | 2 | https://mathoverflow.net/users/51164 | 354740 | 149,819 |
https://mathoverflow.net/questions/354618 | 6 | I have a simple question about the generating function for reverse plane partitions:
$$\sum\_{\pi \in RPP(\lambda)} z^{|\pi|}= \prod\_{s \in \lambda} \frac{1}{1-z^{h\_{\lambda}(s)}}$$
There's a natural refinement of the right hand side:
$$
\prod\_{s \in \lambda} \frac{1}{1-t z\_1^{a\_{\lambda}(s)}z\_2^{l\_{\lambd... | https://mathoverflow.net/users/153442 | Refined reverse plane partition generating function | Yes, there is a way to introduce certain statistics that lead to this refinement.
First, I'll assume partitions are given as collections of boxes with coordinates $(i,j)\in \mathbb N^2$. The content of the box $(i,j)$ is the quantity $i-j$. A border strip of a partition $\lambda$ is a subset of boxes of $\lambda$ whi... | 6 | https://mathoverflow.net/users/2384 | 354756 | 149,827 |
https://mathoverflow.net/questions/354744 | 1 | Call $X$ *very hyperlow* if $\mathcal{O}^X \le\_T \mathcal{O}$, where $\mathcal{O}$ is your favorite $\Pi^1\_1$-complete set. Note: Turing reducibility, not hyp-reducibility. Observe that this is a (Turing) degree invariant notion.
Are the very hyperlow Turing degrees closed under join?
| https://mathoverflow.net/users/32178 | Are the very hyperlows closed under join? | I don't think so. There is an $\mathscr{O}$-recursive sequence $\{D\_i\}\_{i}$ of dense open sets so that for any real $g$ meeting every member of the sequence, $\mathscr{O}^g\leq\_T g\oplus \mathscr{O}$. Now it is simple to construct two such reals $g\_1$ and $g\_2$ so that $g\_1\oplus g\_2\equiv\_T \mathscr{O}$.
| 2 | https://mathoverflow.net/users/14340 | 354757 | 149,828 |
https://mathoverflow.net/questions/354296 | 1 | Suppose $E$ is an $S^1$-spectra of simplicial Nisnevich sheaves. For any $r\in\mathbb{Z}$, we have a distinguished triangle
$$E\_{\geq r+1}\longrightarrow E\_{\geq r}\longrightarrow F\_r\longrightarrow E\_{\geq r+1}[1]$$
in $SH\_s^{S^1}(k)$, where $\_{\geq r}$ denotes the truncation functor (homological index, which is... | https://mathoverflow.net/users/149491 | Spectral sequence associated with a Postnikov tower (Solved by myself) | The convergence of this spectral sequence is because the Nisnevich topology of schemes has a cohomological dimension. The corresponding filtration is given by
$$F^pH\_n=Im([\Sigma^{\infty}U\_+[n],E\_{\geq -p}]\longrightarrow [\Sigma^{\infty}U\_+[n],E])$$. cf. <https://pdfs.semanticscholar.org/4702/f5e28ad71b82c67e0bb94... | 0 | https://mathoverflow.net/users/149491 | 354765 | 149,831 |
https://mathoverflow.net/questions/354778 | 7 | Suppose, $A$ is a finite alphabet. $L \subset A^\*$ is a language. Let's call $L$ *concatenation-free* iff $\forall u, v \in L$ we have $uv \notin L$.
>
> Does there exist some function $c: \mathbb{N} \to (0; 1)$, such that for any finite language $L \subset A^\*$, there exists a concatenation-free sublanguage $L\... | https://mathoverflow.net/users/110691 | Finite concatenation-free languages | $c(n)=1/3$ works for every $n$. Let $L$ be a finite language, $A$ be the multiset (say, nondecreasing sequence) of lengths of words in $L$. Then there exists a sum-free submultiser (subsequence) $B$ of $A$ of cardinality $\ge |A|/3$. Take $L\_0$ to be the set of all words in $L$ whose lengths are in $B$. $L\_0$ is conc... | 5 | https://mathoverflow.net/users/nan | 354781 | 149,835 |
https://mathoverflow.net/questions/354767 | 0 | Let $\left\{ {{\varphi \_n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H\_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H\_0^1\left( {0,1} \right)} \right)$ and subsequence $\left\{ {{\varphi \_{{n\_k}}}} \right\}$ such that ${\varphi \_{{n\_k}}} \... | https://mathoverflow.net/users/135807 | Existence of subsequences convergence with weak topology | There will certainly be a weak-\* convergent subsequence because $L^\infty(0, \infty; H)$ is the dual of the separable Banach space $L^1(0, \infty; H)$ and so its bounded sets are weak-\* metrizable and relatively compact.
You can't expect an a.e. convergent subsequence though; this is not even true for $L^\infty(0,... | 1 | https://mathoverflow.net/users/4832 | 354784 | 149,837 |
https://mathoverflow.net/questions/354733 | 1 | Lets add a constant symbol $V$ to the signature of the language of set theory. So working in first order logic with equality, add the following axioms about $\in $ and $V$.
**Extensionality:** $\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \to x=y)$
**Set construction (reflection):** if $\phi$ is... | https://mathoverflow.net/users/95347 | Can Ackermann theory minus foundation minus class comprehension permit allowing every proper subclass of $V$ to be a set? | The theory is inconsistent.
Let ZG(x) be the formula ∀u∈x∀v∈u(v∈x)∧∀t(x∈t→∃s∈t(∀∈s(y∉t)))∧(∀t(∃s∈x(s∉t)→∃y∈x(y∉t∧∀u∈(x-t)(u∉y))))∧∀t∈x∃s∀v(v∈s↔(v=tνv=x))∧∀u∈x∃t∈x∀s(s∈t↔(s∈u∧∃r(r∈s)))∧∃t∀s(s∈t↔(s∈x∧∃r(r∈s)))
(That is x is transitive; if x is in t, then t has an ∈-minimal element; if x is not contained in t, then there ... | 2 | https://mathoverflow.net/users/133981 | 354785 | 149,838 |
https://mathoverflow.net/questions/354745 | 5 | Consider algebraic representations of a reductive group $G$ over a field in characteristic $p$. I even want to allow *potentially disconnected* reductive groups, i.e. $G$ could be a finite group. (However I'm also interested if the behavior in the connected case is different.)
If $V$ and $W$ are two such representat... | https://mathoverflow.net/users/125639 | Are modular representations isomorphic if they're isomorphic after raising to the pth power? | Here's one way of constructing counterexamples for finite groups.
Suppose $M$ is a periodic $kG$-module with period $p$: i.e., the $p$th syzygy $\Omega^pM$ is isomorphic to $M$, but $\Omega M\not\cong M$. Then
$$(\Omega M)^{\otimes p}\cong \Omega^pM\otimes M^{\otimes (p-1)}\cong M^{\otimes p},$$
up to projective dire... | 13 | https://mathoverflow.net/users/22989 | 354793 | 149,841 |
https://mathoverflow.net/questions/354788 | 0 | Let $G = (V,E)$ be a finite, connected, simple, undirected graph. By a *roundtrip of $G$* we mean a map $r:\{0,\ldots,n\} \to V$ for some $n\in\mathbb{N}$ with the following properties:
1. $r$ is surjective,
2. $\{r(k), r(k+1)\} \in E$ for all $k \in \{0, \ldots, n-1\}$, and
3. $r(0) = r(n)$.
The integer $n$ is cal... | https://mathoverflow.net/users/8628 | Connected graphs $G$ with $\delta(G) > 1$ and long minimum size roundtrips | The answer is $2$. To see this, let $G$ be the graph which consists of two triangles connected by a path with $k-4$ vertices. Then $G$ has $k$ vertices and the length of a shortest roundtrip is $2k-4$. Since $\lim\_{k \to \infty} \frac{2k-4}{k}=2$, the answer is at least $2$. On the other hand, you have already noted $... | 1 | https://mathoverflow.net/users/2233 | 354794 | 149,842 |
https://mathoverflow.net/questions/238103 | 8 | This question follows up [a previous one](https://mathoverflow.net/q/237688/12419) which was answered by Todd Leason. I want to impose two new requirements on the setup.
Let $k$ be a characteristic zero field. Let $A=k[x\_1,\dots,x\_n]$ be the polynomial algebra with the usual grading. Let $g$ be a graded automorphis... | https://mathoverflow.net/users/12419 | Is the restriction of a graded automorphism linearizable in characteristic zero? | Gregor Kemper answered a [related question](https://mathoverflow.net/questions/354629/why-do-nakajima-and-watanabe-claim-the-induced-action-of-a-finite-linear-group-o) with a technique that can be used to answer this one affirmatively in the case that $g$ has finite order. If $g$ does not have finite order and we drop ... | 2 | https://mathoverflow.net/users/12419 | 354797 | 149,843 |
https://mathoverflow.net/questions/354800 | 2 | Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form
$$
{\rm div}(Av)=f
$$
where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,$v$)?
Thanks!
| https://mathoverflow.net/users/90127 | Estimates on divergence-type operator for the matrix | Of course not, because this equation is far from being elliptic. Actually, it is even under-determined, in the sense that you have only one equation, for $n^2$ unknowns (where the matrix is $n\times n$).
Let me however give you a result in this direction, that I discovered two years ago, which has important consequen... | 4 | https://mathoverflow.net/users/8799 | 354802 | 149,844 |
https://mathoverflow.net/questions/354718 | 3 | Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K\_0(V\_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes and consider their classes $[\mathrm{Spec}(K)]$ and $[\mathrm{Spec}(K')]$ in $K\_0(V\_k)$.
I have two general questi... | https://mathoverflow.net/users/12884 | Field extensions in Grothendieck rings | In characteristic zero $[\mathrm{Spec}(K)] = [\mathrm{Spec}(K')]$ for finite field extensions of $k$ implies that $K$ and $K'$ are isomorphic.
Indeed, by the Larsen-Lunts theorem for smooth projective connected schemes of finite type $[X] = [Y]$ implies that $X$ and $Y$ are stably birational; this applies to $\mathrm... | 7 | https://mathoverflow.net/users/111491 | 354805 | 149,845 |
https://mathoverflow.net/questions/354811 | -1 | Given a prime $\,p\,$ let's consider the following sequence:
$a\_0=p$
$a\_{n+1}=(a\_n-2)\cdot a\_n+2$
>
> **Is it possible to determine whether the sequence $\,a\_n\,$ will reach, sooner or later, another prime number?**
>
>
>
Some examples:
for $\,p=2$, $\;\;a\_1=2\;\;$ (prime)
for $\,p=3$, $\;\;a\_1=... | https://mathoverflow.net/users/150698 | Is it possible to determine whether the sequence $\,a_0=p,\;a_{n+1}=(a_n-2)\cdot a_n+2\,$ will reach another prime number? | Your question can be reformulated as follows.
**Question.** If $p$ is a prime, does there always exist a positive integer $n$ such that $(p-1)^{2^n}+1$ is also a prime?
I believe that this question is out of reach at present (my guess is that the answer is "no", but we will never know). Similar to the well-known qu... | 9 | https://mathoverflow.net/users/11919 | 354812 | 149,848 |
https://mathoverflow.net/questions/354787 | 5 | Does there exist an example of a module $X$ over some ring $R$ together with submodules $T\_i$ such that:
* $X$ is projective,
* $X$ splits as an internal direct sum $X\cong T\_1\oplus T\_2\oplus \ldots \oplus T\_n\oplus S\_n$ (with some $S\_n$) for every $n$,
* $X$ does **not** split off the infinite direct sum $\bi... | https://mathoverflow.net/users/105652 | Projective module which splits off sequence of submodules, but not the sum | Here is an example if we interpret all direct sums as *internal* direct sums.
**Example.** Let $R$ be a discrete valuation ring with uniformiser $\pi$ and fraction field $K$. Let $X = R^{(\mathbf N)}$, and let $T\_i$ be the free rank $1$ submodule with basis $\pi e\_{i+1}-e\_i$. Then the natural map
$$\bigoplus\_{i=1... | 3 | https://mathoverflow.net/users/82179 | 354813 | 149,849 |
https://mathoverflow.net/questions/353898 | 6 | Is there a tuple of parameter-free formulas $\Phi$ and a nonstandard $M\models PA$ such that $\Phi^M\models PA$, the induced $M$-definable initial segment embedding $j\_\Phi^M:M\rightarrow\Phi^M$ is non-surjective, and $M\equiv \Phi^M$?
(Here by "$\Phi^M\models PA$" I mean "$\Phi$ defines an interpretation of a struc... | https://mathoverflow.net/users/8133 | Interpreting proper elementarily equivalent end extensions? |
>
> This answer is an attempt at explaining my critical posted comments on Hamkins' proposed answer; it also expands my posted comments to the MO question. I will explain:
> **(a)** the gap in Hamkins' answer, **(b)** how it can be fixed (at the cost of considerably strengthening the hypotheses of the question), and
... | 4 | https://mathoverflow.net/users/9269 | 354818 | 149,852 |
https://mathoverflow.net/questions/354825 | 8 | Let $G$ be a (**Edit:** path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?) Here I'm taking the **free** loop space, and the compact-open topology on it. I know it is true for strong hypothe... | https://mathoverflow.net/users/4177 | For which G is BLG weak homotopy equivalent to LBG? | [UPDATE: There were some mistakes in the first version. Here is a more careful account.]
I'll work everywhere with CGWH spaces, so I have a Cartesian closed category.
Note that $BLG$ is always path-connected, but $\pi\_0(LBG)=\pi\_0(G)/\text{conjugacy}$, so we need to assume that $G$ is path-connected. (The questio... | 16 | https://mathoverflow.net/users/10366 | 354827 | 149,853 |
https://mathoverflow.net/questions/316732 | 2 | Let’s say I have a set $A$. I build the free commutative monoid $M$ generated by $A$.
1. When can a well-order on $A$ be extended to $M$, in a way that is compatible with its monoid structure? I am interested both in the case when $A$ is finite and infinite.
2. Under which conditions would such an extension be unique... | https://mathoverflow.net/users/106304 | Generating totally ordered free commutative monoids | For your first question, the answer is positive with lexicographic ordering (as said by Chris).
Your monoid is $\mathbb{N}^{(A)}$ (i.e. the set of mappings $\alpha: A\to \mathbb{N}$ with finite support, additive version) or $M=\{A^{\alpha}\}\_{\alpha\in \mathbb{N}^{(A)}}$ for a multiplicative version using the [multii... | 4 | https://mathoverflow.net/users/25256 | 354834 | 149,856 |
https://mathoverflow.net/questions/354837 | 0 | I came along the statement that for $x \geq z$, if $U(x)$ is a renewal function, there exists a constant $K$ such that
\begin{align}
U(x) - U(x-z) \leq U(z) \leq K (z+1).
\end{align}
This is not clear to me. For instance, Blackwell's renewal theorem gives $U(x) - U(x-z) = O(1)$, but not the above... It seems to be a r... | https://mathoverflow.net/users/152588 | Renewal functions inequalities | Renewal functions are subadditive. For a reference, [this article](https://www.jstor.org/stable/20535671) (in Example 1) points to [Dal, Section 4], [Fell, Ch XI]:
* [Dal] D. J. Daley, *Upper bounds for the renewal function via Fourier methods*, Annals of Probability 6 (1987), 876-884. MR0494547
* [Fel] W. Feller, *A... | 1 | https://mathoverflow.net/users/108637 | 354841 | 149,857 |
https://mathoverflow.net/questions/354845 | 3 | This may be a rather elementary question, but I haven't been able to figure it out on my own, and the literature appears to be eerily silent on the topic.
Since $\mathbb{Q}\_p$ is a locally compact group with $\mathbb{Z}\_p$ as a compact subset, there exists a unique Haar measure $\mu$ on $\mathbb{Q}\_p$ with $\mu(\m... | https://mathoverflow.net/users/45250 | Connection between Volkenborn integral and Haar measure on $\mathbb{Q}_p$ | This does not just vary with translation, it's non-canonical. The idea is that knowing the size of the compact open subsets $p^n\mathbb{Z}\_p$ should give you a measure (i.e., an integral) by the following, if it exists:
$$ \int\_{\mathbb{Z}\_p} f d\mu = \lim\_{n \to \infty} \sum\_{a=0}^{p^n-1} f(\text{any representa... | 4 | https://mathoverflow.net/users/141571 | 354847 | 149,860 |
https://mathoverflow.net/questions/354714 | 1 | I am reading an article on wavelet connection coefficients (G. Beylkin, "[On the representation of operators in bases of compactly supported wavelets](https://epubs.siam.org/doi/10.1137/0729097)", 1992 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1191143))) and I came across Equation (3.31):
\begin{equation}... | https://mathoverflow.net/users/153487 | Wavelet momentum identity | The identity in the OP does not hold for any $m$, but only for $m< N$ where $N$ is the number of vanishing moments of the wavelet function. To complete the Poisson-summation derivation, one needs the socalled [Strang-Fix condition,](http://cas.ensmp.fr/~chaplais/FTP/Mathematical_Notes/Strang_and_Fix.pdf) which says tha... | 3 | https://mathoverflow.net/users/11260 | 354855 | 149,863 |
https://mathoverflow.net/questions/334536 | 8 | Does the (simply connected compact) Lie group $E\_7$ contain a finite subgroup $G \subset E\_7$ such that the $56$-dimensional irrep of $E\_7$ splits over $G$ as $28 \oplus \overline{28}$, but the $133$-dimensional adjoint representation remains simple when restricted to $G$? By "$28 \oplus \overline{28}$" I mean of co... | https://mathoverflow.net/users/78 | Can the defining rep of $E_7$ split over a finite subgroup while the adjoint remains simple? | I know this is an old question, but I can answer it in the negative. First, I have mostly completed a list of the Lie primitive subgroups of $E\_7(k)$ for all $k$, including $k=\mathbb C$, and there is no such example.
But even before that, a paper of Liebeck and Seitz, entitled 'Subgroups of exceptional algebraic gr... | 6 | https://mathoverflow.net/users/152674 | 354863 | 149,866 |
https://mathoverflow.net/questions/354873 | 1 | For the chromatic number $\chi(G)$ of a simple, undirected graph, there is a ["compactness" theorem by Erdős and De Bruijn](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory)) stating that if an infinite graph $G$ has finite chromatic number, then there is a finite subgraph $G\_0\subseteq... | https://mathoverflow.net/users/8628 | A converse of the Erdős-De Bruijn Theorem? | Apparently yes. By googling, I found the assertion in [this 1951 paper of de Bruijn and Erdős](http://combinatorica.hu/~p_erdos/1951-01.pdf), and the first page contains several further references.
| 4 | https://mathoverflow.net/users/5091 | 354874 | 149,870 |
https://mathoverflow.net/questions/354898 | 2 | Given a nonempty finite subset $S$ of the unit sphere of $d$-dimensional complex Hilbert space, let $\lambda(S) = \frac{1}{\lvert S \rvert^2} \sum\_{x,y \in S} \lvert \langle x,y \rangle \rvert^2$ be the mean squared absolute value of the inner product of two vectors chosen from $S$.
A basic observation from experime... | https://mathoverflow.net/users/60487 | Mean squared absolute value of inner product of unit vectors | The Welch bound gives
$$\lambda(S) = \frac{1}{\lvert S \rvert^2}
\sum\_{x,y \in S} \lvert \langle x,y \rangle \rvert^2 \geq
\frac{(\sum\_{x \in S} \lvert \langle x,x \rangle \rvert)^2}{d \lvert S \rvert^2}=\frac{1}{d}$$
which is what you want.
There are Welch bound equality sets (do a google search) but achievin... | 2 | https://mathoverflow.net/users/17773 | 354900 | 149,876 |
https://mathoverflow.net/questions/354903 | 12 | I am looking for the source and context of this quote, found e.g. at [St Andrews](http://mathshistory.st-andrews.ac.uk/Biographies/Rudio.html):
>
> Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was not yet known how to ***let the formulas speak for th... | https://mathoverflow.net/users/19276 | Source of a quote by Ferdinand Rudio | The quote is from a speech Rudio gave at the Town Hall in Zürich on the 6th December 1883; The German original is published in Felix Stähelin, *Reden und Vorträge* (1956, I have not found it online).
An English translation of the full speech is [here.](http://mathshistory.st-andrews.ac.uk/Extras/Rudio_Euler.html) Th... | 21 | https://mathoverflow.net/users/11260 | 354904 | 149,877 |
https://mathoverflow.net/questions/354916 | 3 | I have read that $H^i(K(\mathbb{R},1)$) has rank $2^\omega$ for any $i\in \mathbb{N}$ (see Thurston's comment here [Nontrivial finite group with trivial group homologies?](https://mathoverflow.net/questions/52552/nontrivial-finite-group-with-trivial-group-homologies#comment130307_52552)) therefore $K(\mathbb{R},1)$ is ... | https://mathoverflow.net/users/99042 | $G$ uncountable implies $K(G,1)$ is not a finite CW complex | For any finite CW-complex $X$ and any basepoint $x \in X$, the fundamental group $\pi\_1(X,x)$ is finitely presented. (This is a consequence of the Seifert-van Kampen theorem.) In particular, the group itself is a quotient of a finitely generated free group, and hence must be a countable set.
However, if $G$ is a Lie... | 10 | https://mathoverflow.net/users/360 | 354917 | 149,882 |
https://mathoverflow.net/questions/354908 | 1 | Let $X$ and $Y$ be two quasi compact, separated schemes over $k$, and consider the fibre product $X \times Y$. If we call $p\_1$ and $p\_2$ the two projections, and we take perfect complexes $F\_1, F\_2 \in \mathfrak{Perf}(X)$, $G\_1, G\_2 \in \mathfrak{Perf}(Y)$, then by flat base change we have an isomorphism
$$
\tex... | https://mathoverflow.net/users/91572 | Morphisms on fibre products | This is not true. For example take $F\_1 = \bigoplus\_{n \in \mathbf{N}} \mathcal{O}\_X$, $F\_2 = \mathcal{O}\_X$, $G\_1 = \bigoplus\_{m \in \mathbf{N}} \mathcal{O}\_Y$ and $G\_2 = \mathcal{O}\_Y$. Moreover, assume $X = \text{Spec}(k)$ and $Y = \text{Spec}(k)$. Then we see that the left hand side is
$$
(\prod\nolimits\... | 2 | https://mathoverflow.net/users/152991 | 354921 | 149,883 |
https://mathoverflow.net/questions/354919 | 1 | I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e.
$$
d(t)\*w(t)=p(t)
$$
where $\*$ denotes **convolution**.The impulse response $w(t)$ may be calculated by going into the frequency domain:
$$
w(t)=F^{-1}\left[\frac{F[p(t)]\overline{F[d(t)]}}{F[d(t)]\overli... | https://mathoverflow.net/users/153615 | The derivative of a filter with respect to a output signal | I don't understand how do you get your expression for the impulse response $w(t)$: however I'd solve the problem considering $w(t)$ as a functional $w(t)=\mathfrak{w}[p](t)$ of $p(t)$ and then calculating the functional derivative, i.e.
$$
\frac{\partial w}{\partial p}=\frac{\delta w}{\delta p}=\frac{\delta\mathfrak{w}... | 0 | https://mathoverflow.net/users/113756 | 354923 | 149,884 |
https://mathoverflow.net/questions/354920 | 8 | In "Partial Horn logic and cartesian categories", E. Palmgren and S. J. Vickers state without proof that "The theory of categories is not algebraic." Is there a reference, or an elementary argument, for this fact? In particular, I'm interested in the setting where "algebraic" refers to the multisorted, potentially infi... | https://mathoverflow.net/users/152679 | Why is the theory of small categories not algebraic? | This follows from two Facts:
**1)** A category monadic over Set/S is always an [exact category](https://ncatlab.org/nlab/show/exact+category). That is it has quotient by equivalence relation that are effective and universal. It is in particular a [regular category](https://ncatlab.org/nlab/show/regular+category). Thi... | 22 | https://mathoverflow.net/users/22131 | 354926 | 149,886 |
https://mathoverflow.net/questions/354772 | 1 | I am looking for a proof of and/or a reference for the result that Markov's principle can be proved in the framework of constant domain logic. By constant domain logic, I mean intuitionistic logic plus the axiom
$\forall x(P(x) \,\vee\, Q) \to \forall xP(x) \,\vee\,Q \quad$ where x is not free in $Q$.
This result i... | https://mathoverflow.net/users/136180 | Markov's principle from constant domain logic | Emil Jeřábek answered in a comment:
$\forall x(A(x) \vee \neg A(x)) \:\to\: \forall x (\exists y A(y) \vee \neg A(x))
\:\to\: \exists y A(y) \vee \forall x(\neg A(x))$
So, that’s even stronger than MP: decidability is preserved by existential quantification.
| 0 | https://mathoverflow.net/users/136180 | 354930 | 149,887 |
https://mathoverflow.net/questions/354808 | 3 |
>
> What follows, up to the horizontal line, is taken from [Rogers "Arbitrage with fractional Brownian motion"](http://www.long-memory.com/fractional-brownian-motion/Rogers1997.pdf).
>
>
>
Consider an interval $[0,T]$ on which is defined the fractional Brownian motion $B$, and consider its partitions $\pi\_n = \... | https://mathoverflow.net/users/136012 | Prove that fractional Brownian motion is not a semimartingale using the p-variation | Assume $B$ is a semimartingale, then it has finite quadratic variation.
Recall that if $s < b$ then $V\_b \le V\_s$.
* If $H<1/2$ we can choose $p>2$ s.t. $pH<1 \implies V\_p = \infty \implies \infty\le V\_2 \implies V\_2 = \infty$, i.e. the quadratic variation ($p=2$) is infinite too: contradiction.
* If $H>1/2$ w... | 4 | https://mathoverflow.net/users/136012 | 354931 | 149,888 |
https://mathoverflow.net/questions/354885 | 2 | Let $X\_1,...,X\_n$ be iid normal random variables.
I am looking for a strategy to establish the following limit for fraction of expectation values
$$\lim\_{N \rightarrow \infty} \frac{E(\prod\_{1\le i < j\le n} \vert X\_i-X\_j \vert^{1/n})}{E(\prod\_{1\le i < j\le n-1} \vert X\_i-X\_j \vert^{1/n})}=1.$$
Does an... | https://mathoverflow.net/users/119875 | Convergence of fraction of expectation values | The [Mehta integral](https://en.wikipedia.org/wiki/Selberg_integral#Mehta's_integral) is
$$M\_n(\gamma):=E\prod\_{1\le i<j\le n}|X\_i-X\_j|^{2\gamma}
=\prod\_{j=1}^n\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}.$$
So, your fraction under the limit sign is
$$\frac{M\_n(1/(2n))}{M\_{n-1}(1/(2n)}=\frac{\Gamma(3/2)}{\Gamma(... | 4 | https://mathoverflow.net/users/36721 | 354934 | 149,890 |
https://mathoverflow.net/questions/354906 | -1 | **Disclaimer.** This is a follow up to a question I asked and answered on SE <https://math.stackexchange.com/q/3579311/168758>. The question was about upper-bounds. Here I'm interested in lower bounds, and they seem harder to get...
Question
========
So, let $\mathfrak S\_n$ be the symmetric group of permutations o... | https://mathoverflow.net/users/78539 | On bounding a certain discrepancy between probability distributions on the symmetric group | There is no non-trivial lower bound as it is well possible that $P\neq Q$, whereas $p\_{ij}=q\_{ij}$ for all pairs of $i$ and $j$. The reason is that (as you point out), these numbers are nothing but the values of the measure $P$ (resp., $Q$) on the sets $E\_{ij}$, and the collection of the sets $\{E\_{ij}\}$ is just n... | 1 | https://mathoverflow.net/users/8588 | 354935 | 149,891 |
https://mathoverflow.net/questions/354922 | 9 | I'm a topologist and not an algebraic geometer, but the following question arose in my work.
Let $X$ be a quasiprojective algebraic variety over $\mathbb{C}$ and let $G$ be a finite group acting on $X$. Since $X$ is quasiprojective, we have the quotient variety $X/G$.
Question: if $X$ is smooth, must $X/G$ be norma... | https://mathoverflow.net/users/153616 | Quotient of a normal quasi-projective variety by a finite group | Just an algebraic interpretation of @Simpleton's answer in the case of finite group actions. Let $B$ an integrally closed domain with the field of fractions $L$. Let $G$ be a finite subgroup of ${\rm{Aut}}(L)$. Then the extension $L/L^G$ is Galois and $B^G=B\cap L^G$ is a domain (the superscript denotes the invariant s... | 5 | https://mathoverflow.net/users/128556 | 354937 | 149,892 |
https://mathoverflow.net/questions/354944 | 9 | Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert f \rVert\_\infty := \max\_{x \in K} \lVert f(x) \rVert. $$
In the case, $X = \mathbb C$, it is well known that the dual s... | https://mathoverflow.net/users/91108 | Dual space of continuous Banach-space-valued functions | The natural language to work in here is that of tensor norms. Here I follow [Ryan, Introduction to tensor products of Banach spaces](https://books.google.co.uk/books/about/Introduction_to_Tensor_Products_of_Banac.html?id=7xRlVTVSNpQC&redir_esc=y). Section 3.2 shows we can identify $C(K;X)$ with the *injective* Banach s... | 8 | https://mathoverflow.net/users/406 | 354946 | 149,895 |
https://mathoverflow.net/questions/354950 | -2 | Let us consider two sequences of real numbers $a\_n$ and $b\_n$, about which we only know that:
$$\sum\_{1}^{\infty}a\_n = 0$$
and that all $b\_n > 0$, with $b\_{n+1} > b\_n$. Can it be proved that there cannot exist a $b\_n$ sequence with said features, such that also
$$\sum\_{1}^{\infty}a\_n b\_n = 0 \;\;?$$
or is ... | https://mathoverflow.net/users/15020 | On the series of the product of the terms of two sequences whose respective series are one convergent and the other not | Consider the series $a=1-1+1/4-1/4+1/9-1/9...+1/n^2-1/n^2...$ and $b=1+1+2+2+3+3+4+4...+n+n...$ . Their "product" is $1-1+1/2-1/2....1/n-1/n...$ converges to 0.If you really want $b\_{n+1}\gt b\_n$, then instead $...n+n...$ consider the series $...n+(n+1/n^2)...$.
| 1 | https://mathoverflow.net/users/nan | 354951 | 149,896 |
https://mathoverflow.net/questions/354941 | 8 | What is the basic math behind the Virus community spread mathematical modeling,and how the time variable;(in these models),interacts with knowledge (data)?.
I am not asking about how the virus is transmitted or how it replicates.
| https://mathoverflow.net/users/152623 | Virus community spread mathematical modeling | this wikipedia article [Mathematical modelling of infectious disease](https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease) may be a good starting point; **epidemiology mathematical models** is a combination of terms that does the magic with e.g. google.
[Introduction to Mathematical Models of t... | 6 | https://mathoverflow.net/users/31310 | 354953 | 149,897 |
https://mathoverflow.net/questions/354958 | -3 | I need to know how to find the contents of a sphone; however I have not been able to find an equation for it online. I noted that the equation for a cone is 1/3(h)(A base). So I thought that perhaps the formula for a sphone could be the volume of its base \* h \* 1/3 since a sphone is a continuous series of spheres ter... | https://mathoverflow.net/users/153630 | How to find the content of a sphone | The [sphone](http://hi.gher.space/wiki/Sphone) is a 4-dimensional generalization of a cone (height $h$), where the base is a sphere (radius $r$). It is one of a collection of 4-dimensional objects, see [this overview.](https://bendwavy.org/klitzing/explain/round.htm#top) The surface equation is
$$|(x\_1^2+x\_2^2+x\_3^2... | 1 | https://mathoverflow.net/users/11260 | 354961 | 149,902 |
https://mathoverflow.net/questions/354957 | 4 | Let A be a semi-simplicial space and $k^\*$ be a generalised cohomology theory as in [This paper](https://www.maths.ed.ac.uk/~v1ranick/papers/segalclass.pdf) proposition 5.1. Using the natural filtration of the realisation of $A$ and then using the staircase diagramme of long exact sequences of pairs, we get the first ... | https://mathoverflow.net/users/152579 | Understanding the proof of Proposition 5.1 of Segal's paper: Classifying spaces and spectral sequences | Segal is assuming the reader has knowledge of a basic fact about chain complexes formed from simplicial (or cosimplicial) abelian groups: the sub (or quotient) complex of `degenerate' chains is acyclic.
He points out that the $E\_1$ term of the spectral sequence he is describing is naturally mapping to a complement ... | 5 | https://mathoverflow.net/users/102519 | 354965 | 149,903 |
https://mathoverflow.net/questions/354975 | 0 | In T. Skolem 1922 the author publishes a weak version of the Skolem-Löwenheim theorem which we call WLS and which according to [Wikipedia](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) says that every countable theory which is satisfiable in a model is also satisfiable in a countable model. My un... | https://mathoverflow.net/users/37385 | Weak Skolem-Löwenheim and completeness | *For simplicity, below all languages are finite.*
At least at an abstract enough level, *neither* implication holds. When we go a bit more into the details, there is some truth to "completeness yields WLS," but it's not too robust - and the other direction (contra your claim) I don't see at all.
---
It's easies... | 2 | https://mathoverflow.net/users/8133 | 354987 | 149,909 |
https://mathoverflow.net/questions/354995 | 1 | Let $\mathcal{G}$ be the set of all finite connected simple graphs minus the complete graphs. For any $G\in \mathcal{G}$, let $\lambda\_{\geq0}(G)$ denotes the smallest positive adjacency eigenvalue of the graph $G$.
Let $$\tau\_{\mathcal{G}}=\sum\_{G\in \mathcal{G}}{\lambda\_{\geq 0}(G)}.$$
Is it true that $\tau\_... | https://mathoverflow.net/users/19885 | Total behaviour of graph spectrum | The value is infinite. For example, take the friendship graph $F\_k$ (which is $k$ triangles all sharing a common vertex). It's smallest non-negative eigenvalue is 1, so by considering each of these $G$ in the sum we see that the value must be infinite.
| 3 | https://mathoverflow.net/users/106377 | 354998 | 149,913 |
https://mathoverflow.net/questions/354980 | 2 | Fix some positive integers $p,n,k$. Let $w$ be chosen uniformly at random from $[k]^n$ (the set of $n$ length words/sequences where each entry is in $\{1,\ldots,k\}$). Let $A\_i$ be the event that $w\_r=i$ for at least $p$ values of $r$. Can one prove that, for all $s$, $$\Pr\left(\bigwedge\_{i=1}^{s-1} A\_i\ \large|\ ... | https://mathoverflow.net/users/106377 | An "obvious" probability lemma about random words | The property in question is a special case (with the probabilities of all the $k$ outcomes equal to one another) of the known $NA$ (negative association) property of the multinomial distribution; see e.g. [this sentence in the bottom paragraph on page 5](https://arxiv.org/abs/1803.09663v1):
>
> $NA$ property of mu... | 4 | https://mathoverflow.net/users/36721 | 355025 | 149,917 |
https://mathoverflow.net/questions/355018 | 1 | Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism!
Let $\mathcal{L}$ be an invertible sheaf on $Y$ and $\mathcal{G}:=f^\*\mathcal{L}$ it's pullback. Assume that $\mathcal{G}$ is very ample... | https://mathoverflow.net/users/108274 | Pullback map on global sections surjective | If the map $f^\* \colon H^0(Y,\mathcal{L}) \to H^0(X,f^\*\mathcal{L})$ is surjective then there is a commutative diagram
$$
\require{AMScd}
\begin{CD}
X @>>> \mathbb{P}(H^0(X,f^\*\mathcal{L})^\vee)
\\
@VfVV @VVV
\\
Y @>>> \mathbb{P}(H^0(Y,\mathcal{L})^\vee),
\end{CD}
$$
where the right vertical arrow is an embedding an... | 2 | https://mathoverflow.net/users/4428 | 355027 | 149,918 |
https://mathoverflow.net/questions/355029 | 1 | $\newcommand{\Ann}{\operatorname{Ann}}\newcommand{\Max}{\operatorname{Max}}$I am looking for an example of a commutatvive ring $R$ with $1$ having two ideals $I$ and $J$ such that $I\cap J\not=0$, $\sqrt{\Ann(I)}, \sqrt{\Ann(J)}\in \Max(R)$, and $I+J$ is indecomposable ideal of $R$.
Where, $I+J$ is indecomposable if ... | https://mathoverflow.net/users/153680 | When $I+J$ is a special indecomposable ideal of $R$ | $\newcommand{\Ann}{\operatorname{Ann}}$Let $A=K[x]/(x^n)$ and $I=\langle x^a\rangle$ and $J=\langle x^b\rangle$ for $1 \leq a \leq b \leq n-1$. Their intersection is nonzero.
$\Ann(I)=\langle x^{n-a}\rangle, \Ann(J)=\langle x^{n-b}\rangle$ and their radicals are $\langle x^1\rangle$, which is maximal.
$I+J=\langle x^{\... | 1 | https://mathoverflow.net/users/61949 | 355031 | 149,920 |
https://mathoverflow.net/questions/355026 | 0 | I'm attempting Exercise 5.33 of Le Gall's Brownian motion, Martingales and Stochastic Calculus.
Let $B\_t$ be a 3-dimensional Brownian motion starting from $x$.
Part 6 asks me to show that
$$|B\_t| = |x| + \beta\_t +\int\_0^t\dfrac{ds}{|B\_s|} \quad (\*)$$
where $$\beta\_t = \sum\_{i=1}^3 \int\_0^t \dfrac{B^i\_s... | https://mathoverflow.net/users/nan | Transience of 3-dimensional Brownian motion | You're on the right track near the end. As a non-negative supermartingale, $|B\_t|^{-1}$ converges almost surely; call the limit $X$. On the event $\{X \ne 0\}$, we have $|B\_t|$ converging to the finite limit $1/X$. But intuitively it is absurd for a Brownian motion to do that (it is trying to "wiggle", not "settle do... | 0 | https://mathoverflow.net/users/4832 | 355034 | 149,921 |
https://mathoverflow.net/questions/354892 | 5 | Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection
\begin{equation\*}
m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\},
\end{equation\*}
where $\lvert \, \cdot \, \rvert$ denotes the 2-norm.
The minimal selection theorem in question can be stated... | https://mathoverflow.net/users/153602 | Generalization of minimal selection theorem | I think the Berge Maximum theorem could be applicable?
<https://en.wikipedia.org/wiki/Maximum_theorem>
See page 116;
[C. Berge, Topological Spaces: including a treatment of multi-valued
functions, vector spaces, and convexity. CC, 1997]
| 1 | https://mathoverflow.net/users/130184 | 355041 | 149,924 |
https://mathoverflow.net/questions/355045 | 4 | The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A\_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius, so being a filtered colimit in Rings, the perfection functor commutes with **finite** products. But, in general, does perfection commutes with products? If not, why in a scheme $... | https://mathoverflow.net/users/92322 | Does perfection of rings commute with products? | No, even restricting to domains: it doesn't even commute to infinite powers.
Indeed, let $A$ be a non-perfect domain. So $A\_{\mathrm{pf}}$ is an overring of $A$, such that for every $x\in A\_{\mathrm{pf}}$ there exists some minimal $n=N(x)\ge 0$ such that $x^{p^n}\in A$. Then $x\mapsto N(x)$ is surjective onto non-n... | 6 | https://mathoverflow.net/users/14094 | 355048 | 149,925 |
https://mathoverflow.net/questions/355061 | 12 | Let $G$ be a discrete group and let $(X,x\_0)$ be a based Eilenberg-MacLane space for $G$, so there is a fixed isomorphism $\pi\_1(X,x\_0) = G$ and the universal cover $\widetilde{X}$ is contractible. The homology of $X$ is thus the same as the homology of $G$.
Choose some $\gamma \in G$, and let $c\_{\gamma} \in \te... | https://mathoverflow.net/users/153616 | Realizing inner automorphisms on Eilenberg-MacLane spaces | Let's assume that the space $X$ is reasonable in the sense that the inclusion $x\_0 \hookrightarrow X$ is a cofibration, i.e. has the homotopy extension property. This will hold, for instance, if $X$ is a CW complex and $x\_0$ is a vertex.
Represent $\gamma$ by a path $\rho\colon [0,1] \rightarrow X$ with $\rho(0)=\r... | 11 | https://mathoverflow.net/users/317 | 355067 | 149,932 |
https://mathoverflow.net/questions/355068 | 1 | Suppose we are given a rooted tree $T$, and a set of vertices $M$ that separates the root of $T$ from its leaves. In other words, every path from the root of $T$ to a leaf contains a vertex in $M$. Is there a standard term for such a set $M$?
| https://mathoverflow.net/users/24226 | What do you call a set of vertices that separates the root from the leaves? | Call the root node $r$. If you join each leaf node to a dummy sink node $t$, then $M$ would be an $(r,t)$ [*vertex separator*](https://en.wikipedia.org/wiki/Vertex_separator), also known as *vertex cut* or *separating set*.
Even without the dummy node, $M$ is an $(r,\ell)$ vertex separator for each leaf node $\ell$.
... | 2 | https://mathoverflow.net/users/141766 | 355070 | 149,933 |
https://mathoverflow.net/questions/354872 | 2 | The following notion of upper Banach density was defined (Definition 2.1(c)) by Hindman and Strauss in their paper '[Density in arbitrary semigroups](https://www.researchgate.net/publication/266702090_Density_in_Arbitrary_Semigroups)':
**Definition:** Let $S$ be a semigroup, let $\mathcal{F}=\{F\_i\}\_{i\in \mathcal{... | https://mathoverflow.net/users/83956 | Additivity of the upper Banach density | For every family $\mathcal F\subset\mathcal P\_f(\Bbb N) $ both a set $\{1, 3,4, 7,8,9, 13,14,15,16,\dots\} $ and its complement have upper Banach density $d^\*\_{\mathcal F}$ equal to $1$.
| 2 | https://mathoverflow.net/users/43954 | 355087 | 149,938 |
https://mathoverflow.net/questions/355086 | 8 | Suppose $U$ is a normal ultrafilter on $\kappa$ of Mitchell order zero, and let $j\_U : V \to M$ be the associated embedding. Does there exist a nonstationary $X \subseteq \kappa^+$ such that $X \in M$ and $M \models X$ is stationary?
Note that if $W$ is a normal measure derived from an embedding $i : V \to N$ where ... | https://mathoverflow.net/users/11145 | Stationary correctness of ultrapowers by low order measures | It is consistent that $\kappa$ has a unique normal measure (so, in particular, Mitchell minimal) and that measure's ultrapower $M$ satisfies $\mathcal{P}(\kappa^+)\subseteq M$, so, by your note, the ultrapower is stationary-correct.
The model is just the Friedman--Magidor model for a $\kappa$ with a unique normal mea... | 9 | https://mathoverflow.net/users/1058 | 355106 | 149,942 |
https://mathoverflow.net/questions/355101 | 1 | Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with [Schauder basis](https://en.wikipedia.org/wiki/Schauder_basis) $\{b\_i\}\_{i =1}^{\infty}$. Is the set
$$
D^+\triangleq \left\{\sum\_{j=1}^n \beta\_j d\_j\otimes b\_j: \, d\_j \in D, \, k\_... | https://mathoverflow.net/users/36886 | Density and the projective tensor product | This seems to be easy, mayby I am missing something: Given dense sets $D$ of $X$ and $E$ of $B$, a seminorm $p$ on $X$, $\varepsilon >0$, and $z=\sum\limits\_{j=1}^n x\_j \otimes y\_j \in X\otimes B$ it is enough to $\varepsilon/n$-approximate with respect to $p\otimes\_\pi \|\cdot\|$ each term of the sum by an element... | 2 | https://mathoverflow.net/users/21051 | 355110 | 149,943 |
https://mathoverflow.net/questions/355120 | 3 | Let $X\_1,X\_2$ be Banach subspaces of a locally convex space $X$. Then the subset
$$
X\_1+X\_2 = \left\{
x\in X:\, x= \beta\_1 x\_1 + \beta\_2 x\_2 \, \beta\_i \in \mathbb{R},\, x\_i \in X\_i
\right\},
$$
a is a normed space with respect to the norm
$$
\|x\| := \inf\left\{
\sum\_{i=1}^2|\beta\_i|\|x\_i\|\_i: x = x\_1 ... | https://mathoverflow.net/users/36886 | Reference request: completion of Banach norm on sum | You find it in the manuscript of Alessandra Lunardi on Interpolation Theory. A free version is available here: <http://people.dmi.unipr.it/alessandra.lunardi/LectureNotes/SNS1999.pdf>
In these notes the argument is (somewhat compressed) on page 3.
If you want a ZBL reference, then see page ix in
*Lunardi, Alessan... | 6 | https://mathoverflow.net/users/12898 | 355123 | 149,948 |
https://mathoverflow.net/questions/355119 | 1 | Suppose that a team of $ n $ players (numbered from 1 to $ n $) sit around a circular table and there are their numbers on their T-shirts.
Let $ a\_1,a\_2,..., a\_n $ be the sequence of numbers around the table and so $ a\_n $ sits beside $ a\_1$.
What is the maximum value of $|a\_1-a\_n|+\sum\_{k=1}^{n-1 }|a\_{k+1}... | https://mathoverflow.net/users/97333 | Maximum distance of the numbers of players around a table | For $n$ even let $\{b\_{2i-1}, b\_{2i}\} = \{a\_{2i-1}, a\_{2i}\}$ with $b\_{2i-1} < b\_{2i}$. Then $\{b\_1, \ldots, b\_n\} = \{1, \ldots, n\}$ and
$$
\begin{align\*}
\sum\_{i=1}^{n/2} |a\_{2i-1} - a\_{2i}|
&= \sum\_{i=0}^{n/2} (b\_{2i} - b\_{2i-1}) \\
&= \sum\_{i=1}^{n/2} b\_{2i} - \sum\_{i=0}^{n/2} b\_{2i-1} \\
&\le... | 2 | https://mathoverflow.net/users/25485 | 355130 | 149,950 |
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