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https://mathoverflow.net/questions/444373 | 4 | Apologies in advance if this question is obvious/not research level.
>
> Let $\preceq$ be the consistency strength relationship on theories. Working over $ZF$ or $ZFC$, is there some large cardinal notion $\psi(\alpha)$ with an ordinal parameter such that there exists some ordinal $\kappa$ such that $ZF(C)+\psi(\al... | https://mathoverflow.net/users/92164 | Cofinal inconsistency | The answer is yes.
One can make easy artificial examples. For example, let $\psi(\gamma)$ assert that there are (only) finitely many inaccessible cardinals, but at least $\gamma$ many.
From the assumption of infinitely many inaccessible cardinals, we can prove the consistency of ZFC+$\psi(n)$ for any particular fin... | 5 | https://mathoverflow.net/users/1946 | 444376 | 179,174 |
https://mathoverflow.net/questions/444371 | 9 | Let $\sigma\_0(n)$ be the divisor counting function:
$$\sigma\_0(n) = \sum\_{d \vert n} 1.$$
I ran some numerical experiments that showed when $p$ is prime, the function $\sigma\_0(n)$ is equidistributed mod $p$. That is, for any residue class $a \mod p$,
$$\lim\_{X \to \infty} \dfrac{ \vert \{ n<X: \sigma\_0(n) \equiv... | https://mathoverflow.net/users/394740 | Is the divisor counting function equidistributed mod $p$? | $\newcommand{\Y}{\mathfrak{X}\_p(X)}$I haven't checked all the details on the application of Selberg–Delange below, so it's possible something I am saying is nonsense, but even after accounting for the correction noted by Noam Elkies in the [comments](https://mathoverflow.net/questions/444371/is-the-divisor-counting-fu... | 17 | https://mathoverflow.net/users/37327 | 444381 | 179,177 |
https://mathoverflow.net/questions/444379 | 0 | I was looking at the fresnel integral $S(x)=\int^x\_0\sin(s^2)ds$. From reading I learned that this integral approaches $\frac{1}{2} \sqrt{\frac{\pi}{2}}$ as $x \rightarrow \infty$. Through messing around on desmos, I found an excellent estimate: $S\_2(x)=\frac{\sin(x^2)+2x^2\cos(x^2)}{-4x^3}+\frac{1}{2}\sqrt{\frac{\pi... | https://mathoverflow.net/users/502440 | How to prove approximation for fresnel integral converges | If you want to build an asymptotics, write
$$S(x)=\int^x\_0\sin(s^2)\,ds=\frac{1}{2}\sqrt{\frac{\pi }{2}}-\int\_x^\infty\sin(s^2)\,ds.$$ Let $s=\sqrt x$
$$\int\_x^\infty\sin(s^2)ds=\frac 12\int\_{x^2}^\infty\frac{\sin (t)}{\sqrt{t}}\,dt.$$ Integrate by parts a few times and using the bounds, you will obtain
\begin{gath... | 3 | https://mathoverflow.net/users/42185 | 444384 | 179,178 |
https://mathoverflow.net/questions/444099 | 4 | Let $A$ be a unital $C^{\ast}$-algebra and $\{ f\_i: A \rightarrow A\_i \}\_i$ a finite collection of morphisms of unital $C^{\ast}$-algebras, such that the associated map $A \rightarrow \prod\_i A\_i$ is injective. Let $g: A \rightarrow B$ be a morphism of unital $C^{\ast}$-algebras. Then, there is an induced collecti... | https://mathoverflow.net/users/130868 | Property of pushouts in the category of unital $C^{\ast}$-algebras | Unfortunately, the assertion is, in general, not true even if we just have a single (injective) unital $\*$-homomorphism $f\_1\colon A\to A\_1$ and another (non-injective) unital $\*$-homomorphism $g\colon A\to B$. The reason is that amalgamated free products with non-injective maps can be trivial: $B\*\_A A\_1=0$ in c... | 4 | https://mathoverflow.net/users/75215 | 444385 | 179,179 |
https://mathoverflow.net/questions/443919 | 1 | $\DeclareMathOperator\Hom{Hom}$For which unital $C^{\ast}$-algebras $A$ does it hold that for all compact Hausdorff $S$ we have the bijection:
\begin{align\*}
\Hom(A, C(S)) \cong \Hom(S, \Hom (A, \mathbb{C}))?
\end{align\*}
This holds for all commutative unital $C^{\ast}$, but it seems to hold for all $C^{\ast}$ algebr... | https://mathoverflow.net/users/130868 | Adjunction via Gelfand duality | $\DeclareMathOperator\Hom{Hom}$
Yes, this is true, and the proof is elementary: let us write $\Omega(A):=\Hom(A,\mathbb{C})$ for the space of characters of $A$, viewed as a subspace of the unit ball of the dual $A^\*$, and endowed with the weak\*-topology (i.e., the topology of pointwise convergence). This is a compa... | 1 | https://mathoverflow.net/users/75215 | 444388 | 179,180 |
https://mathoverflow.net/questions/444362 | 1 | If I have a smooth surface $M$ (2D embedded in 3D), under what conditions I can assure that there exists a finite collection of charts $\{U\_i, \phi\_i \}\_{i}$, with $\phi\_i : U\_i \to M$, such that its directional derivatives have always the same magnitud and that they are orthogonal set of coordinates?, i.e.
$$
|... | https://mathoverflow.net/users/130126 | Requirement of parametrization of surfaces | You can always do this, but it's not as simple as using some kind of ODE (such a flow of vector fields) to construct such charts.
First, assume that your surface is connected and simply-connected. Then it's either compact, in which case it's a $2$-sphere, or it's diffeomorphic to $\mathbb{R}^2$. In the compact, case,... | 3 | https://mathoverflow.net/users/13972 | 444395 | 179,182 |
https://mathoverflow.net/questions/444394 | 0 | Let $F$ be a field, $G$ be a finite group and $\alpha \in Z^2(G, F^\*)$ . The twisted group algebra $F^{\alpha}G$ is a $F$-algebra with $F$ vector basis, $\{\bar g : g \in G \},$ and multiplication defined by $\bar x \bar y = \alpha(x,y) \overline{xy}$ for $x,y \in G$ and extended distributively.
Let $F^{\alpha} G$ a... | https://mathoverflow.net/users/502460 | Examples of isomorphic non-equivalent twisted group algebras | Let $G={\mathbb Z}/3 \times {\mathbb Z}/3$ and $\alpha$ be the $2$-cocycle corresponding to the extraspecial group of exponent $3$ and order $27$. Then the twisted group algebras defined using $\alpha$ and $-\alpha$ are isomorphic, by swapping the two copies of ${\mathbb Z}/3$.
| 2 | https://mathoverflow.net/users/460592 | 444396 | 179,183 |
https://mathoverflow.net/questions/444169 | 3 | In [this math.stackexchange question](https://math.stackexchange.com/q/4668840/111012) Adam Rubinson asked (I paraphrase):
>
> Given a natural number $r$, what is the least number $n$ such that every strictly increasing sequence of $n$ real numbers has a subsequence $x\_1,x\_2,\dots,x\_r$ of length $r$ whose sequen... | https://mathoverflow.net/users/43266 | A combinatorial problem about sequences of numbers | As pointed out in a comment by [Vladimir Dotsenko](https://mathoverflow.net/users/1306/vladimir-dotsenko), the problem I asked about was posed and solved in the classical paper by Erdős and Szekeres, A combinatorial problem in geometry, *Compositio Math.* 2 (1935), 463–470 ([pdf](http://www.numdam.org/item/CM_1935__2__... | 1 | https://mathoverflow.net/users/43266 | 444399 | 179,185 |
https://mathoverflow.net/questions/444407 | 9 | Have there been any attempts to extend the "F\_un" analogy to the representation theory of finite groups?
If I might be allowed some speculation:
If combinatorics can be regarded as analagous to linear algebra over the "field with one element", then are characters of finite groups over the "field with one element" ... | https://mathoverflow.net/users/502468 | Representations of finite groups over the "field with one element" | The table of marks, as defined by Burnside, has rows indexed by the transitive permutations $X$ and columns indexed by the conjugacy classes of subgroups $H\leqslant G$. The entry corresponding to $X$ and $H$ is $|X^H|$, the number of fixed points.
The Burnside ring $A(G)$ is the Grothendieck ring of permutation repr... | 13 | https://mathoverflow.net/users/460592 | 444415 | 179,189 |
https://mathoverflow.net/questions/444416 | 4 | $\DeclareMathOperator\cl{cl}$Let $X$ be a topological space and let $Y$ be a dense subspace of $X$. Suppose
that $R\left( X\right) $ denotes all regular closed subsets of $X$.
Question 1: $R\left( Y\right) \longrightarrow R\left( X\right) $, $A\rightarrow \cl\_{X}A$ is bijective.
Question 2: If $A$ is regular close... | https://mathoverflow.net/users/86099 | A question about regular closed sets | The answer to both of these questions is **Yes.** And this result can generalize to point-free topology and I consider this result to be more natural in the context of point-free topology.
The following observations were produced earlier by Mehmet Onat, so let me paraphrase those arguments.
Observation: If $U$ is a... | 4 | https://mathoverflow.net/users/22277 | 444419 | 179,190 |
https://mathoverflow.net/questions/444392 | 0 | Assume $A\in \mathbb{R}^{n\times n}$ with each entry being i.i.d. bounded r.v. in $[a,b]$, is $\Vert A\Vert\_2$ is sub-Gaussian?
Intuitively, since $\{A\_{ij}\}\_{i,j=1,...,n}$ is bounded, then
$$\Vert A \Vert\_2 = \sup\_{\Vert v \Vert = 1} \vert v^TA^TAv\vert = \sup\_{\Vert v \Vert = 1}\vert\sum\_{i,j}v\_iv\_j(\sum\... | https://mathoverflow.net/users/500967 | Spectral norm of matrices of bounded random variables | Your chain of equality is slightly off, the idea is there however. I think that the following one is correct :
\begin{align\*}
\| A \|\_2^2 &= \sup\_{\| v \|\_2=1} v^TA^TAv\\
&= \sup\_{\|v \|\_2 = 1} \left| \sum\_{i,j} v\_i v\_j \left( \sum\_{k} A\_{ki} A\_{kj} \right) \right|\\
&\leq \sup\_{\|v \|\_2 = 1} \sum\_{i,j} ... | 0 | https://mathoverflow.net/users/492816 | 444420 | 179,191 |
https://mathoverflow.net/questions/444134 | 5 | Suppose that a finite group $G$ admits a Frobenius group of
automorphisms $F H$ with kernel $F$ and complement $H$ such that $F$
acts without nontrivial fixed points (that is, such that $C\_G(F)=1$).
It is proved by Belyaev and Hartley in [Centralizers of finite nilpotent subgroups in locally finite groups](https://doi... | https://mathoverflow.net/users/44312 | Finite abelian group admits a Frobenius group of automorphism | Let $V$ be an $\mathbb{F}\_p[X]$-module such that $p\mid |G|$.
Then $V$ is not necessarily completely reducible.
A classical example is: $X\cong C\_p$ and $V\cong C\_p \times C\_p$ and $XV$ is extraspecial $p$-group of order $p^3$.
Richard Lyons's [idea](https://mathoverflow.net/questions/444134/finite-abelian-group-... | 1 | https://mathoverflow.net/users/44312 | 444433 | 179,195 |
https://mathoverflow.net/questions/444421 | 3 | There are two proofs of
$$\sum\_{n=1}^\infty \frac{1}{n^s}=\prod\_{p \text{ prime}}\frac{1}{1-p^{-s}},\quad \Re (s)\gt 1$$
which I'm aware of. I'll call the first one the **Sieve proof** and the second one will be the **Factorization proof**. Both of them use infinitude of primes (at least I think so).
1. Sieve proof... | https://mathoverflow.net/users/502484 | Do all rigorous proofs of Euler's product for $\zeta (s)$ use infinitude of primes? | Neither proof uses the infinitude of primes. Here is why ($p$ will always denote a prime number).
1. Sieve proof.
Proceed as you indicated, but don't assume that $q$ is prime. We still have that
$$\left|\left(\prod\_{p\le q}\left(1-\frac{1}{p^s}\right)\right)\zeta (s)-1\right|=\left|\sum\_{\substack{n>1\\\forall p:... | 14 | https://mathoverflow.net/users/11919 | 444435 | 179,196 |
https://mathoverflow.net/questions/444447 | 2 | I'm trying to find formulas for the finite difference approximation "Five-points-stencil" of the first derivative for non-constant grid spacing. It's needed for the outermost left and right points. I tried to search for it but can't find it.
| https://mathoverflow.net/users/500658 | Finite difference approximation | Given pairwise distinct real numbers $x\_0,\dots,x\_4$, one can approximate $f'(x\_0)$ by a linear combination $a\_0f(x\_0)+\cdots+a\_4f(x\_4)$ so that
$$g\_j'(x\_0)=a\_0g\_j(x\_0)+\cdots+a\_4g\_j(x\_4)$$
for $g\_j(x):=x^j$ and $j\in\{0,\dots,4\}$.
Solving the resulting system of equations for $a\_0,\dots,a\_4$, we g... | 2 | https://mathoverflow.net/users/36721 | 444457 | 179,200 |
https://mathoverflow.net/questions/444448 | 2 | I am new to this area and I am a bit confused by the literature. Is there a strong maximum principle for CR functions over domains in a CR manifold, please? If so, could someone please state it (together with its hypotheses etc.) and provide a reference to a proof (or maybe just prove it if it is not too long)? In the ... | https://mathoverflow.net/users/81645 | Is there a maximum principle for CR functions over domains inside CR manifolds? | It's not really about real-analytic or smooth. It is really about the CR structure of the manifold. In your case, the manifold is given by $\operatorname{Im} z\_{n+1} = 0$, so any CR function is just a function holomorphic in the first $n$ variables, as you noted. Hence, any CR function on your manifold that achieves a... | 3 | https://mathoverflow.net/users/2783 | 444464 | 179,203 |
https://mathoverflow.net/questions/444432 | 4 | I am looking for a proof of the following statement without using full power of [Chevalley's theorem](https://stacks.math.columbia.edu/tag/00FE) on constructible sets. We say a domain $A$ is $0$*-open* if $\{(0)\}$ is open in $\operatorname{Spec}(A)$. Equivalently, there is an element $x\in A$ such that $A\_x$ is a fie... | https://mathoverflow.net/users/132430 | Finite type injective ring map between domains preserves the open point $(0)$ | Here's a more down to earth argument that uses the weak Nullstellensatz¹ instead of Chevalley's theorem:
**Lemma.** *Let $\phi \colon A \hookrightarrow B$ be an injective ring homomorphism of finite type between integral domains, and assume there exists a nonzero element $y \in B$ such that $B\_y$ is a field. Then th... | 2 | https://mathoverflow.net/users/82179 | 444471 | 179,208 |
https://mathoverflow.net/questions/444321 | 4 | It is known that
\begin{equation\*}
\tan x=\sum\_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B\_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation\*}
and
\begin{equation\*}
\ln\tan x=\ln x+\sum\_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k-1}-1\bigr)}{k(2k)!}|B\_{2k}|x^{2k}, \quad 0<x<\frac{\pi}{2},
\end{equation... | https://mathoverflow.net/users/147732 | What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$? | Let
\begin{equation\*}
f(x)=\begin{cases}
\ln\dfrac{\tan x-x}{x^3}, & 0<|x|<\dfrac{\pi}{2};\\
-\ln3, & x=0.
\end{cases}
\end{equation\*}
Then the even function $f(x)$ has the Maclaurin power series expansion
\begin{equation}\label{ln-tan-x-cubic-ser-expans}
\begin{aligned}
f(x)&=-\ln3-\sum\_{k=1}^{\infty}\frac{3^{2k}D\... | -1 | https://mathoverflow.net/users/147732 | 444485 | 179,210 |
https://mathoverflow.net/questions/444484 | 3 | I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal.
>
> Is there any $\textbf{closed}$ prime ideal of $C[0, 1]$ which is not maximal?
>
>
>
Any references or ideas?
| https://mathoverflow.net/users/129638 | Closed prime ideal in $C[0, 1]$ | No. Note that an ideal in a commutative ring with identity is prime if and only if the quotient ring is an integral domain.
Now consider $C[0,1]$. It is known that the closed ideals in this Banach algebra are all of the form $J\_F = \{ f\in C[0,1] \colon {f\vert}\_F =0 \}$ for some closed subset $F\subseteq [0,1]$, a... | 6 | https://mathoverflow.net/users/763 | 444488 | 179,212 |
https://mathoverflow.net/questions/444472 | 1 | Classical case:
Let $\{a,b\}$ be linearly independent set over $\mathbb Q$ and $\{e^{at},e^{bt}\}$ be linearly independent set over $\mathbb Q[[t]]$. Suppose $P(x,y)$ is a polynomial over $\mathbb Q$. Then it seems $P(e^{a},e^{b})=0$ implies $P=0$, because
if $P(x,y)=Ax^2+Bxy+Cy^2 \in \mathbb Q[x,y]$. Then,
\begin{al... | https://mathoverflow.net/users/493164 | Does $P(\exp_p(a),\exp_p(b))=0$ imply $P=0$, where $\exp_p(\cdot)$ is $p$-adic exponential? | As written, both the classical and $p$-adic statements are false. Let $a=\log(9)$ and $b=\log(17)$ (classical or $2$-adic log). Then they are linearly independent over $\mathbb{Q} $ since otherwise we would find integers $n, m$ with $9^n=17^m$. But $P(\exp(a), \exp(b))=0$ for the nontrivial polynomial $P(x, y) = 17x-9y... | 3 | https://mathoverflow.net/users/39747 | 444495 | 179,216 |
https://mathoverflow.net/questions/444497 | 2 | Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu\_K$ the normalized Haar measure on $K$:
$$
\mu\_K(K)=1.
$$
Let us denote by $\widetilde{\mu\_K}$ the measure on $G$ defined as the functional on ${\mathcal C}(G)$ by the formula
$$
\widetilde{\mu\_K}(u)=\int\_K u(t) \ \mu\_K(dt), \qquad u\in{\m... | https://mathoverflow.net/users/18943 | Haar measures of compact subgroups | The answer to both questions seems yes to me.
First, only assume that $K$ is a normal subgroup of $G$. Then, $(\text{Ad } g)\_{g \in L}$ defines a continuous action of $L$ by automorphisms of $K$ and we can define the semidirect product group $K \rtimes L$, which is just the set $K \times L$ with product
$$(x,g) \cdo... | 1 | https://mathoverflow.net/users/159170 | 444504 | 179,220 |
https://mathoverflow.net/questions/444500 | 5 | We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f\_{n} \in C\_c^{\infty}(\mathbb R)$ such that
$$\lVert f-f\_n\rVert\_{C^{\alpha}([0,1])} \le \frac{1}{n}$$
and $\lvert f\_n(x)\rvert \le \l... | https://mathoverflow.net/users/496243 | Approximation of Hölder continuous functions "from below" | It is not possible: your condition $|f\_n|<|f|$ implies
that the zero set of $f\_n$ is contained in the zero set of $f$. So $f(x)=|x|^{\alpha}$ cannot be approximated by a smooth function $f\_n$, since $f\_n(x)=(c+o(1))x$, and $\sup|f(x)-f\_n(x)|\geq (1+o(x))|x|^\alpha$ for some small $|x|$.
| 7 | https://mathoverflow.net/users/25510 | 444506 | 179,222 |
https://mathoverflow.net/questions/444481 | 1 | Is the integral
$$
t^2\left(\iint\_{\mathbb{R}^2}|\xi|e^{\frac{-a\xi^4t}{\xi^2+b}}\,d\xi\_1\,d\xi\_2\right)$$ bounded when $t\rightarrow\infty$? Here
* $\xi=(\xi\_1,\xi\_2)\in\mathbb{R}^2$,
* $|\xi|=\sqrt{\xi\_1^2+\xi\_2^2}$,
* $a$ and $b$ are fixed positive numbers.
| https://mathoverflow.net/users/502529 | Whether the integral $t^2(\iint_{\mathbb{R}^2}|\xi|e^{\frac{-a\xi^4t}{\xi^2+b}} \,d\xi_1 \,d\xi_2)$ is bounded? | Assuming that by $\xi^2$ and $\xi^4$ you meant, respectively, $|\xi|^2$ and $|\xi|^4$, switching to polar coordinates, and making the substitution $r=ut^{-1/4}$, we have
$$
I(t):=t^2\left(\iint\_{\mathbb{R}^2}|\xi|e^{\frac{-a\xi^4t}{\xi^2+b}}\,d\xi\_1\,d\xi\_2\right)
=2\pi t^2\int\_0^\infty dr\,r^2\exp\Big(-\frac{tar^4... | 1 | https://mathoverflow.net/users/36721 | 444511 | 179,224 |
https://mathoverflow.net/questions/444515 | 4 | A locally compact quantum group (in the sense of Vaes-Kustermans) consists of the data $(M, \Delta, \varphi, \psi)$ with $M$ a von Neumann algebra, $\Delta: M \to M \overline{\otimes} M$ a normal unital $\*$-homomorphism satisfying coassociativity and $\varphi, \psi: M\_+ \to [0, \infty]$ normal, semifinite, faithful w... | https://mathoverflow.net/users/216007 | Every locally compact group gives rise to a locally compact quantum group | In the general, not necessarily $\sigma$-compact setting, one has to interpret $L^\infty(G,\lambda)$ as the von Neumann algebra of locally measurable functions that are bounded outside a locally null set. This is very well explained in Section 2.3 of Folland's "A course in abstract harmonic analysis".
Also note that ... | 5 | https://mathoverflow.net/users/159170 | 444520 | 179,227 |
https://mathoverflow.net/questions/444525 | 4 | According to [this comment](https://mathoverflow.net/questions/444490/ask-for-a-proof-of-an-inequality-involving-the-bernoulli-numbers#comment1147661_444490) and [this comment](https://mathoverflow.net/questions/444490/ask-for-a-proof-of-an-inequality-involving-the-bernoulli-numbers#comment1147671_444490), a positive a... | https://mathoverflow.net/users/36721 | On the monotonicity of the ratio of two logarithmic expressions | The values $5/6$ and $1$ for $r(0+):=\lim\_{x\downarrow0}r(x)$ and $r(\frac\pi2-):=\lim\_{x\uparrow\pi/2}r(x)$ are easy to get by the l'Hospital rule for limits.
It remains to show that $r=f/g$ is increasing on $(0,\pi/2)$.
Note that $f(0+):=\lim\_{x\downarrow0}f(x)=0$ and $g(0+)=0$. Next, for real $t>0$ we have $0... | 5 | https://mathoverflow.net/users/36721 | 444526 | 179,229 |
https://mathoverflow.net/questions/444524 | 2 | Side note: so far neither Bard nor ChatGPT has managed to do this correctly, even when I show the errors.
I have 4N players ( N = 4 or N = 5 suffices) and want to set up three rounds of play. In each round, there will be N games played (four players per game). I want to set up the groupings so that no two players app... | https://mathoverflow.net/users/50013 | Optimal algorithm for a "round robin" doubles tournament? | For $N=5$ and more generally $N$ relatively prime to $6$, it is easy to make $N$ rounds.
Name the players $(i,j)$ for $0 \leq i \leq 3$ and $0 \leq j \leq N-1$. In the $a$-th round, take the elements of the $b$-th quadruple to be $\{ (0,b \bmod N), (1,a+b \bmod N), (2, 2a+b \bmod N), (3, 3a+b \bmod N) \}$.
Suppose,... | 3 | https://mathoverflow.net/users/297 | 444528 | 179,230 |
https://mathoverflow.net/questions/444531 | 11 | Mertens' first theorem states that
$$
\sum\_{p \leq n} \frac{\log p}{p} = \log n + O(1).
$$
I read in [this paper](https://tenenb.perso.math.cnrs.fr/PPP/PropStatFriables.pdf) that the following variant is "classical":
$$
\sum\_{p \leq n} \frac{\log p}{p - 1} = \log n - \gamma + o(1).
$$
Could anyone provide a reference... | https://mathoverflow.net/users/502584 | Mertens-like theorem | This lies beyond Mertens, in the sense that this variant actually implies the Prime Number Theorem, as will be explained below, while Mertens' theorem is weaker than the PNT.
I sketch below a complex analytic proof of the variant, due to Landau.
---
Let $\Lambda$ be the von Mangoldt function, defined as $\Lambd... | 16 | https://mathoverflow.net/users/31469 | 444534 | 179,231 |
https://mathoverflow.net/questions/444533 | 1 | I have a question about the following statement from the article
* Alves, M., Rivera, J.M., Sepúlveda, M., Villagrán, O.V. and Garay, M.Z., *The asymptotic behavior of the linear transmission problem in viscoelasticity*, Math. Nachr. **287** (2014) pp 483-497, doi:[10.1002/mana.201200319](https://doi.org/10.1002/mana... | https://mathoverflow.net/users/481556 | $\operatorname{Coth}(\alpha_n a) \to i$ when $n \to \infty $ | $\newcommand\la\lambda\newcommand\al\alpha\newcommand\ka\kappa\newcommand\th\theta$First of all, you copied the expression for $\al$ incorrectly. On p. 489 of the paper linked by you (the fourth display from the bottom of the page), we find
$$\al=r(\la)e^{i\th/2},$$
where $\mathbb R\ni\la=\la\_n\to\infty$,
$$r(\la):=\f... | 2 | https://mathoverflow.net/users/36721 | 444538 | 179,232 |
https://mathoverflow.net/questions/444480 | 1 | I'm interested in the computations of the Goresky-Hingston product (defined <https://arxiv.org/abs/0707.3486>)
on the cohomology of the relative free loop space on the circle (or better yet, their extension to absolute cohomology). They give a full computation for all $S^n, n\geq 3$, and give some description of all ma... | https://mathoverflow.net/users/166758 | Goresky-Hingston product on cohomology of the relative free loop space on $S^1$ | Here is a sort of provocative, quick, partial answer. Take it as a pointer. Let $R$ be a commutative ring and consider $R[x]$ the polynomial ring on one variable $x$. For any $f \in R[x]$ we may consider the following discrete version of the derivative:
$$D\_q(f)= \frac{f(qx)-f(x)}{qx-x}$$
where $q$ here is thought of ... | 2 | https://mathoverflow.net/users/5450 | 444544 | 179,235 |
https://mathoverflow.net/questions/444393 | 17 | Thinking on the theory NBG (of von Neumann–Bernays–Gödel) I arrived at the conclusion that it is contradictory using an argument resembling Russell's Paradox. I am sure that I made a mistake in my arguments (because NBG cannot be contradictory, as all mathematicians believe), but I can not find the exact place where th... | https://mathoverflow.net/users/61536 | A contradiction in the Set Theory of von Neumann–Bernays–Gödel? | The comments to the question, especially those by Emil Jeřábek and Joel Hamkins make it clear that the proposed inconsistency proof breaks down because the recursive construction carried out in the proposed proof of inconsistency cannot be implemented in NBG.
As pointed out first by Mostowski in his 1950 paper [Some ... | 21 | https://mathoverflow.net/users/9269 | 444549 | 179,238 |
https://mathoverflow.net/questions/444551 | 8 | This question is motivated by my preceding [MO-question](https://mathoverflow.net/q/444393/61536) on (in)consistency of [NBG theory of classes](https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory).
Let $\varphi(x,Y,C)$ be a formula of NBG with free parameters $x,Y,C$ and all quanti... | https://mathoverflow.net/users/61536 | The existence of definable subsets of finite sets in NBG | The answer is in the negative. Let $\mathcal{M}$ be an $\omega$-nonstandard model of ZF, and $\mathfrak{X}$ be the collection of parametrically definable subsets of $\mathcal{M}$. Let $I$ be the cut defined in my answer to [the other MO question](https://mathoverflow.net/questions/444393/a-contradiction-in-the-set-theo... | 14 | https://mathoverflow.net/users/9269 | 444553 | 179,239 |
https://mathoverflow.net/questions/444507 | 9 | Let $\alpha\in H^2(B\operatorname{PSU}(N) ; \mathbb{Z}\_N)$ be the obstruction class for lifting a $\operatorname{PSU}(N)$-bundle to an $\mathrm{SU}(N)$-bundle. Note that $\operatorname{PSU}(N)\cong \operatorname{SU}(N)/\mathbb{Z}\_N$ is the quotient group of $\operatorname{SU}(N)$ by its center, and $B\operatorname{PS... | https://mathoverflow.net/users/17644 | Non-triviality of a Postnikov class in $H^3\left(B \operatorname{PSU}(N) ; \mathbb{Z}_q\right)$ | Consider the map of short exact sequences
$$\begin{array}{ccccccccc} 0 & \rightarrow & \mathbb{Z} & \xrightarrow{\cdot N} & \mathbb{Z} & \xrightarrow{mod N} & \mathbb{Z}\_N & \rightarrow & 0\\
\downarrow & & \downarrow & & \downarrow & & \downarrow{=} & & \downarrow\\
0 & \rightarrow & \mathbb{Z}\_q & \xrightarrow{1 ... | 5 | https://mathoverflow.net/users/104342 | 444559 | 179,241 |
https://mathoverflow.net/questions/444561 | 4 | A group is called minimal non-abelian if it is non-abelian and all proper subgroups are abelian.
Does this notion also exist with Lie groups or algebras? As an example, consider the Lie algebra defined by the generators $\{e\_1,e\_2\}, [e\_1,e\_2]=e\_2$. Do you have a paper at hand?
| https://mathoverflow.net/users/11504 | Minimal non-abelian groups -> Lie groups/algebras | These Lie algebras were called *semiabelian* by some authors. Some old papers are [On the structure of simple-semiabelian
Lie-algebras](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-111/issue-2/On-the-structure-of-simple-semiabelian-Lie-algebras/pjm/1102710571.pdf) by Farnsteiner and [On simp... | 6 | https://mathoverflow.net/users/18739 | 444562 | 179,242 |
https://mathoverflow.net/questions/442177 | 9 | Let $G$ be a directed graph.
Call a vertex $v$ in $G$ *central* if there exists $\Theta(n^2)$ distinct pairs of vertices $(u,w)$ such that $v$ lies on some path from $u$ to $w$. We do not care whether these paths are shortest or not, nor whether other paths avoiding $v$ exist.
Question: Do strongly connected digrap... | https://mathoverflow.net/users/16485 | Does a strong digraph always admit a vertex that lies on some path between $\Theta(n^2)$ pairs of vertices? | In [this manuscript](https://arxiv.org/abs/2304.03567), Bessy, Thomassé, and Viennot proved the following stronger property:
Let $D$ be a strongly connected directed graph, then there is a vertex $v$ in $D$ such that there exists an in-tree towards $v$ and an out-tree from $v$ that are vertex-disjoint (apart from sha... | 5 | https://mathoverflow.net/users/16485 | 444571 | 179,246 |
https://mathoverflow.net/questions/444575 | 5 | Let $R$ be a local ring with field of fractions $K$, maximal ideal $\mathfrak{m}$ and residue field $\kappa = R/\mathfrak{m}$. Let $R^\mathrm{sh}$ be a strict henselization of $R$, and let $L$ be the field of fractions of $R^\mathrm{sh}$.
1. Is $L$ a maximal unramified extension of $K$ with respect to $\mathfrak{m}$?... | https://mathoverflow.net/users/37368 | Alternative description of strict henselization | (Assuming $R$ is not complete)
For 2, the answer is no, as suggested in a now-deleted comment of LSpice.
For $k$ of characteristic not $2$, with $R$ the localization of $k[t]$ at $0$, the element $\frac{1}{ \sqrt{1+t}+1 }$ is in the strict henselization but not integral. This is the root of the polynomial $$(1-x)^2... | 9 | https://mathoverflow.net/users/18060 | 444576 | 179,248 |
https://mathoverflow.net/questions/415167 | 3 | I am reading Stopple's *A Primer of Analytic Number Theory*. On page 234, the Mellin Transform $\mathcal{M} f$ of a function $f$ is defined as
$$\mathcal{M} f (s) = \int\_1^{\infty} f(x) x^{-s - 1} dx$$
His goal is to use the injectivity of the Mellin Transform (if $\mathcal{M} f = \mathcal{M} g$ then $f = g$) to pro... | https://mathoverflow.net/users/78173 | Injectivity of the Mellin Transform and discontinuities | The answer is implicit in what you wrote. One can define an equivalence relation $f\sim g$ if $f$ and $g$ are equal except on a set of measure 0, then define Mellin transform on equivalence classes instead of functions. For context, the book was intended for undergraduates who have seen no more than a typical calculus ... | 3 | https://mathoverflow.net/users/6756 | 444591 | 179,251 |
https://mathoverflow.net/questions/444592 | 8 | Let $f\_n(x)=1+x+x^{\sqrt{2}}+x^{\sqrt{3}}+x^{\sqrt{4}}+\cdots+x^{\sqrt{n}}$ be a sequence of functions on the interval $[0, 1]$. Is there a good closed form approximation for such a function ( whatever "good" means), analogous to the well-known expression for the sum of geometric progression.
Such an approximation i... | https://mathoverflow.net/users/34984 | Approximation of pseudogeometric progression | For natural $n$ and $x\in(0,1)$, one has
$$L\_n(x)\le f\_n(x)\le U\_n(x), \tag{10}\label{10}$$
where
$$U\_n(x):=1+I\_n(x), \tag{20}\label{20}$$
$$I\_n(x):=\int\_0^n x^{\sqrt t}\,dt
=2\frac{1+x^{\sqrt{n}} \left(\sqrt{n} \ln x-1\right)}{\ln^2 x},$$
$$L\_n(x):=I\_{n+1}(x). \tag{30}\label{30}$$
Next, fixing any real $c>0... | 10 | https://mathoverflow.net/users/36721 | 444596 | 179,254 |
https://mathoverflow.net/questions/444598 | 8 | We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1].$
I am wondering if there exists an explicit construction of a sequence $f\_{n} \in C\_c^{\infty}(\mathbb R)$ such that
$$\lVert f-f\_n\rVert\_{C^{\beta}([0,1])} \le \frac{1}{n}$$
for fixed $\beta<\alpha$ and $\lvert f\_n(x)\rv... | https://mathoverflow.net/users/496243 | Smooth approximation of Hölder functions "from below" | $\newcommand\R{\mathbb R}\newcommand{\al}{\alpha}\newcommand{\de}{\delta}\newcommand{\J}{\mathcal J}
\newcommand{\be}{\beta}\newcommand{\ep}{\varepsilon}$Yes, such a construction exists.
Indeed, take any real $\ep>0$. For a real $C\ge0$ and $\al\in(0,1]$, let us say that a function $g$ is $(C,\al)$-Hölder on a set $S... | 4 | https://mathoverflow.net/users/36721 | 444611 | 179,257 |
https://mathoverflow.net/questions/444610 | 2 | I have a matrix $A$ $(n\times n)$ with eigenvalues $\lambda\_i$, then I add another matrix to it as: $A+xx^\top$ where $x$ $(n\times 1)$ is a column vector.
and also $A=yy^\top$ with $y$ a $(n-1)$ rank matrix, so A is symmetric.
Is there any way to calculate the new eigenvalues and eigenvectors of $(xx^\top+yy^\top... | https://mathoverflow.net/users/502666 | Eigenvalues of a rank-one update of a symmetric matrix | Given eigenvectors and eigenvalues of the symmetric matrix $A$, you can transform to a basis where $A$ is a diagonal matrix $D$; the vector $x$ in that basis transforms to $\tilde{x}$. Then the problem is that of computing the eigenvalues of a rank-one update $D+\tilde{x}\tilde{x}^\top$ of a diagonal matrix. An efficie... | 5 | https://mathoverflow.net/users/11260 | 444618 | 179,258 |
https://mathoverflow.net/questions/444608 | 16 | I am trying to follow the beautiful notes by Peter Scholze on condensed mathematics (<https://www.math.uni-bonn.de/people/scholze/Condensed.pdf>)
I am noting that the hard time that I am getting is a lack of basis in Grothendieck topologies, sites, and so on, things that are not from the theory of condensed math itse... | https://mathoverflow.net/users/43027 | Reference request for condensed math | Dagur Ásgeirsson has written [a text](https://dagur.sites.ku.dk/files/2022/01/condensed-foundations.pdf) to fill this gap:
>
> We discuss in some detail the prerequisites for each of the first four chapters of Scholze's "Lectures on Condensed Mathematics". Some proofs are
> given in more detail or slightly altered,... | 20 | https://mathoverflow.net/users/11260 | 444624 | 179,261 |
https://mathoverflow.net/questions/444626 | 2 | Let $G(n,l)$ denote the set of connected graphs with $n$ vertices and $l$ edges and let $G\_0(n,l)$ denote the elements of $G(n,l)$ without leaves. It is easily seen that $G\_0(n,n-1)$ is empty, since $G(n,n-1)$ is the set of trees, and $G\_0(n,n)$ consists of all circular grpahs of length $n$.
**Question:** How can ... | https://mathoverflow.net/users/409412 | Characterization of graphs without leaves | The elements of $G\_0(n,n+k)$ for $k\geq 1$ can be characterized as follows. Consider any partition $e\_1+e\_2+\cdots+e\_v = 2k$ of $2k$ into $v\leq n$ parts and any multigraph $S$ with $v$ vertices of degrees $2+e\_1, 2+e\_2, \ldots,2+e\_v$ and $v+k$ edges. Then any graph $H\in G\_0(n,n+k)$ is obtained by distributing... | 2 | https://mathoverflow.net/users/47484 | 444628 | 179,262 |
https://mathoverflow.net/questions/444552 | 2 | I would be very grateful for any references I might be led to, from a categorical point of view for the functors:
* $\textsf{Spec}\_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which leads to $\mathcal{O}(\textsf{Spec}\_{\mathscr{Z}\textrm{arisky}}(-))$,
* $\textsf{Spec}\_{\mathscr{G}\textrm{elfand}... | https://mathoverflow.net/users/502369 | Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections |
>
> I have not yet found the statement of your answer by localization and construction of the sheaf, in your own publications and available texts, or elsewhere.
>
>
>
To the best of my knowledge, there is nothing published on this topic yet.
I once wrote some very rough unfinished notes on how to treat the algeb... | 2 | https://mathoverflow.net/users/402 | 444646 | 179,270 |
https://mathoverflow.net/questions/444630 | 1 | Let $G$ be a locally compact abelian group and $f:G \to \mathbb{C}$ a function. Its Fourier transform (when it exists) is defined to be
$$\widehat{f}(\chi) = \int\_G f(g) \bar{\chi}(g) \mathrm{d} g,$$
Here $\chi \in \widehat{G} := \mathrm{Hom}(G, S^1)$ and $\mathrm{d} g$ is a choice of Haar measure on $G$.
Why does $... | https://mathoverflow.net/users/5101 | Why complex conjugate in definition of the Fourier transform? | This is an $L^2$ product: consider for instance the most classical Fourier expansion of $\mathbb Z$ periodic functions (or distributions) of one real variable.
You have for $f$ locally square integrable
$$
f(x)=\sum\_{k\in \mathbb Z}\langle f, e\_k\rangle\_{L^2} e\_k(x),
$$
that is
$
f(x)=\sum\_{k\in \mathbb Z}\left(\i... | 2 | https://mathoverflow.net/users/21907 | 444652 | 179,272 |
https://mathoverflow.net/questions/431456 | 6 | Edit: I'm specializing this question to the compact case I'll ask about the noncompact case as a new question.
Let $ G $ be a compact connected semisimple Lie group.
Do there always exist two finite order elements of $ G $ which generate a dense subgroup?
Example:
$$
\frac{i}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -... | https://mathoverflow.net/users/387190 | Semisimple compact Lie group topologically generated by two finite order elements | You might want to look at MR0034766 (Kuranishi 1949, [link](https://projecteuclid.org/journals/kodai-mathematical-journal/volume-1/issue-5-6/Two-elements-generations-on-semi-simple-Lie-groups/10.2996/kmj/1138833534.full)). It deals with the connected semisimple case, which might be what you're after.
There's also a 1... | 5 | https://mathoverflow.net/users/460592 | 444664 | 179,277 |
https://mathoverflow.net/questions/444422 | 7 | A classical theorem of Kronecker says that the sequence $(\{\alpha\_1 n\}, \{\alpha\_2 n\},\dots,\{\alpha\_d n\})$ ($n \in \mathbb{N}$) is uniformly distributed in $[0,1)^d$ provided that $1,\alpha\_1,\alpha\_2,\dots,\alpha\_d$ are linearly independent over $\mathbb{Q}$. More generally, for a polynomial $p(x)$ (or a $d... | https://mathoverflow.net/users/14988 | Is the sequence $\{n/\{ \sqrt{2}n\} \}$ uniformly distributed in $[0,1)$? | I'll show that $\{ n / \{ \alpha n \} \}$ is equidistributed under a certain Diophantine condition on $\alpha$ which holds generically (and in particular for $\sqrt 2$), but the proof goes through verbatim for $\{ \beta n / \{ \alpha n \} \}$.
A very reasonable guess is that $n$ and $\{ n \alpha \}$ are "independent"... | 3 | https://mathoverflow.net/users/88679 | 444668 | 179,278 |
https://mathoverflow.net/questions/444665 | 8 | One way to stratify the large cardinal hierarchy between I3 and I1 is by using second-order elementary embeddings. We may view $j\colon V\_{\lambda+1}\to V\_{\lambda+1}$ as a second-order embedding $j\colon (V\_\lambda,V\_{\lambda+1})\to (V\_\lambda,V\_{\lambda+1})$, and it is known that $\mathrm{I2}(\lambda)$ is equiv... | https://mathoverflow.net/users/48041 | The consistency of $\Sigma_1$-elementary embeddings $j\colon V_{\lambda+2}\to V_{\lambda+2}$ over $\mathsf{ZFC}$ | The answer is no, it is not consistent. If $j : V\_{\lambda+2}\to V\_{\lambda+2}$ is $\Sigma\_1$-elementary, then $\mathcal U = \{A\subseteq P(V\_\lambda) : j[V\_\lambda]\in j(A)\}$ is a normal fine ultrafilter concentrating on $\sigma$ such that $\text{ot}(\sigma\cap \lambda) = \lambda.$ Then taking an ultrapower of t... | 9 | https://mathoverflow.net/users/102684 | 444676 | 179,284 |
https://mathoverflow.net/questions/444653 | 21 | There is an easy proof of the PNT, just in a few lines, in the book by Julian Havil, "Gamma", pages 201-202. Specifically, Von Mangoldt's formula, which is very easy to derive:
$$
\psi(x) = x - \ln(2\pi) - \frac{1}{2}\ln(1 - x^{-2}) - \sum\_{\zeta(\rho) = 0} \frac{x^{\rho}}{\rho},
$$
implies the PNT, since from this eq... | https://mathoverflow.net/users/502718 | What is the difference between elementary and non-elementary proofs of the Prime Number Theorem? | To complement Will Sawin's answer, in the specific context of the prime number theorem, there are historically well-established notions of "elementary" and "non-elementary" proofs, stemming from Hardy's 1921 lecture:
>
> No elementary proof of the prime number theorem is known, and one may ask whether it is reasona... | 29 | https://mathoverflow.net/users/56624 | 444677 | 179,285 |
https://mathoverflow.net/questions/444694 | 7 | For *weakly cohesive toposes*, there exists a notion of *contractability*, and toposes with a subobject classifier $\Omega$ that is contractible are of special interest (see [here](https://ncatlab.org/nlab/show/sufficiently+cohesive+topos#cohesively_connected_truth)).
It occured to me that one could view probability ... | https://mathoverflow.net/users/3824 | Topos with $\Omega = [0,1]$? | Here's an example which, I'm afraid, is not very interesting (and may not match your notion of “suitable sense”): let $X$ be the topological space $\mathopen]0,1\mathclose[$ (the open interval) with the (not at all separated) topology given by the $U\_x := \mathopen]0,x\mathclose[$ for $x \in [0,1]$ (with $U\_0 = \varn... | 6 | https://mathoverflow.net/users/17064 | 444696 | 179,291 |
https://mathoverflow.net/questions/444625 | 1 | Let $(F,|.|)$ be a complete algebraically closed field. Let $x$ be the point of type 5 corresponding to the unit open disc of the adic affine line over $F$. Can we obtain a concrete description of the complete residue field of $x$ (as for point of type 2 or 3)? Does the local ring of $x$ coincide with the Robba ring?
... | https://mathoverflow.net/users/117853 | Complete residue field of a point of type 5 | A point of type 5 corresponds to a valuation of rank 2, so I am not sure if the meaning of completion is completely clear here. I think that the usual thing is to complete with respect to the associated rank 1 valuation. Then, you would get back the complete residue field at the associated type 2 point (the Gauss point... | 5 | https://mathoverflow.net/users/4069 | 444698 | 179,292 |
https://mathoverflow.net/questions/444615 | 3 | For a Lie group $G$, we can define a principal $G$ bundle as a submersion of manifolds $\pi:P \to X$ equipped with a free right $G$-action on $P$ that is transitive on the fibres over $X$.
What goes wrong with an analogous definition for 2-groups? For now, we can think of 2-groups as weak monoidal groupoid with monoi... | https://mathoverflow.net/users/18080 | Categorifying the definition of a principal $G$ bundle | The definition you are looking for is precisely Def. 6.1.5 in:
*Nikolaus, Thomas; Waldorf, Konrad*, [**Four equivalent versions of nonabelian gerbes**](https://doi.org/10.2140/pjm.2013.264.355), Pac. J. Math. 264, No. 2, 355-420 (2013). [ZBL1286.55006](https://zbmath.org/?q=an:1286.55006).
This definition is based ... | 7 | https://mathoverflow.net/users/3473 | 444700 | 179,293 |
https://mathoverflow.net/questions/444662 | 10 | Igusa defined a genus 2 Siegel modular form $\chi\_{10}$, which vanishes on the Humbert surface $G\_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, inside the Siegel modular threefold, see for instance page 218 of van der Geer's Hilbert Modular Surfaces).
This should be relat... | https://mathoverflow.net/users/85392 | Igusa's $\chi_{10}$ and Borcherds products | The product expansion and the weakly holomorphic (Jacobi) form it lifts from appear as Example 2.4 in Gritsenko, Nikulin - *Automorphic Forms and Lorentzian Kac--Moody Algebras II*, Internat. J. Math. 9(2) (1998), 201--275.
[(arXiv)](https://arxiv.org/abs/alg-geom/9611028).
The square root of $\chi\_{10}$ is denote... | 4 | https://mathoverflow.net/users/502756 | 444704 | 179,295 |
https://mathoverflow.net/questions/444701 | 11 | I came up with this question when trying to give a more detailed answer to a question by Tim Campion in [a comment](https://mathoverflow.net/questions/424356/examples-of-statements-that-are-valid-in-every-spatial-topos?rq=1#comment1121603_435325) to Ingo Blechschmidt's answer to [Examples of statements that are valid i... | https://mathoverflow.net/users/41291 | Do all toposes satisfy the internal Zorn's lemma? | Assuming the law of excluded middle internally your formulation of Zorn's lemma is equivalent to the axiom of choice by the usual argument. Now there are Boolean Grothendieck topos in which the axiom of choice fails. Hence they don't satisfies Zorn lemma either.
I think the simplest example of this is to take $G$ to ... | 9 | https://mathoverflow.net/users/22131 | 444719 | 179,301 |
https://mathoverflow.net/questions/444723 | 10 | [This is a cross-post](https://math.stackexchange.com/questions/4602412/derivative-without-extrema-is-monotone) from Math.SE.
The question was asked there 3 months ago but didn't receive much attention aside from one comment asking for clarification. I feel like it might be non-trivial and so I decided to repost the qu... | https://mathoverflow.net/users/494118 | Derivative without extrema is monotone | The following is copied from A. Bruckner, *Differentiation of Real Functions* (2d ed, 1994), at the end of chapter 6 (“The Zahorski Classes”): I didn't check the details carefully, but it seems to answer your question. (Remarks in brackets are mine. All typos are also likely to be mine.)
>
> A continuous function d... | 13 | https://mathoverflow.net/users/17064 | 444724 | 179,304 |
https://mathoverflow.net/questions/444727 | 5 | Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)\_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}\_a,\mathbb{G}\_m)=\widehat{\mathbb{G}}\_a$. (Here we see all the groups as abelian sheaves on $(\mathsf{Sch}/k)\_\text{fppf}... | https://mathoverflow.net/users/131975 | Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$? | This is false, see Remark 2.2.16 of [Rosengarten - Tate Duality In Positive Dimension Over Function Fields](https://arxiv.org/pdf/1805.00522.pdf#page34) in which a nontrivial local extension of $\mathbb{G}\_a$ by $\mathbb{G}\_m$ is constructed. However, the same paper shows that the first Ext-sheaf vanishes in positive... | 6 | https://mathoverflow.net/users/101861 | 444742 | 179,308 |
https://mathoverflow.net/questions/444300 | 4 | Let $A$ be a supersingular elliptic curve over $\mathbb{Z}/p\mathbb{Z}$ and $\mathcal{O}$ an order in an imaginary quadratic field contained in the quaternion algebra $\operatorname{End}(A)$, then by Deuring's lifting lemma, $A$ lifts to an elliptic curve $E$ over $\bar{\mathbb{Q}}$ with complex multiplication by an or... | https://mathoverflow.net/users/470091 | Quantitative lifting for mod-p elliptic curves to characteristic zero CM elliptic curves | Let $E$ be a supersingular elliptic curve in characteristic $p$. Then
$A = \mathrm{End}(E)$ is a maximal order in the quaternion algebra
$D/\mathbf{Q}$ ramified exactly at $p$ and $\infty$. If $\alpha \in A \smallsetminus \mathbf{Z}$, then Deuring's theorem guarantees that the pair $(E,\alpha)$ can be lifted to a chara... | 2 | https://mathoverflow.net/users/491858 | 444748 | 179,310 |
https://mathoverflow.net/questions/444709 | 5 | I have a concrete question about continuous functions on $X = [0,1]^\omega$ (with the product topology).
The map $f:X\to [0, 1]$ given by $(x\_i)\mapsto \prod x\_i$ is well-defined and Borel but not continuous.
Suppose instead that a *continuous* function $g: X\to [0, 1]$ satsifies $g \ge f$ on $X$. What can be sai... | https://mathoverflow.net/users/502762 | Continuous functions on $[0,1]^\omega$ and a product lower bound | The second question also has the negative answer:
Take any sequence $z=(z\_n)\_{n\in\omega}\in[0,1]^\omega$ with $f(z)=0$. On the Hilbert cube $[0,1]^\omega$, consider the metric $d(x,y)=\max\_{n\in\omega}\frac{|x\_n-y\_n|}{2^n}$.
Let $c=(c\_n)\_{n\in\omega}$ be the constant sequence with $c\_n=\frac12$ for all $n$... | 6 | https://mathoverflow.net/users/61536 | 444758 | 179,312 |
https://mathoverflow.net/questions/444756 | 4 | Specifically, I'm wondering, if X and Y are Hausdorff, and Y is compactly generated, does it follow that C(X,Y), with the compact-open topology, is compactly generated?
Edit: answered as written, but curious about other conditions that do imply the compact-open topology is compactly generated. E.g. if we strengthen t... | https://mathoverflow.net/users/83073 | Under what conditions is the compact-open topology compactly generated? | Not necessarily: consider the compactly generated space $Y=\mathbb R^\infty=\varinjlim \mathbb R^n$, which is the direct limit of Euclidean spaces. Then for the countable discrete space $X=\omega$ the function space $C(X,Y)$ is homeomorphic to $(\mathbb R^\infty)^\omega$ and hence is not sequential and so is not compac... | 9 | https://mathoverflow.net/users/61536 | 444759 | 179,313 |
https://mathoverflow.net/questions/444757 | 0 | Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that
$${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$
$${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha\_1 a +\alpha\_2 b ))$$
for any $0\ne \alpha\_1,\alpha\_2\in \mathbb{R}$, where ${\rm Tr}$ denotes the standard trace on $B(H)$ and $l(\c... | https://mathoverflow.net/users/91769 | Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$? | This already fails for $2 \times 2$ matrices. Take the rank one matrices $a = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $b = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$. For all $\alpha \neq 0$ and $\beta \neq 0$, the matrix $\alpha a + \beta b$ is invertible, so that $\text{Tr}(l(\alpha a + \beta b)) = 2$. A... | 3 | https://mathoverflow.net/users/159170 | 444766 | 179,314 |
https://mathoverflow.net/questions/444768 | 1 | Let $G = (V, E)$ be a finite, simple, undirected graph with $V \cap E = \emptyset$. The *total graph* $T(G)$ is defined on the vertex set $V \cup E$ and its edge set is given by $$E(T(G)) = E \cup \big\{\{e, f\}: e, f \in E\text{ and } |e\cap f| = 1\big\}\cup \big\{\{v, e\}: v\in V, e\in E, v\in e\big\}.$$
**Question... | https://mathoverflow.net/users/8628 | Hamiltonian path in total graph | The edges incident to $v \in V$ form a clique in $T(G)$, so the Hamiltonian path $v\_1, v\_2, v\_3, \ldots, v\_n$ in $G$ can be lifted into a Hamiltonian path in $T(G)$ as follows:
1. Insert the edges to get $v\_1, \{v\_1, v\_2\}, v\_2, \{v\_2, v\_3\}, v\_3, \ldots, v\_n$.
2. In arbitrary order, insert every edge whi... | 3 | https://mathoverflow.net/users/46140 | 444769 | 179,315 |
https://mathoverflow.net/questions/444750 | 1 | $X$ is in the form of exponential family i.e.
$$\mathbb{P\_\theta}x = h(x)e^{\langle \theta,T(x)\rangle-\phi(\theta)}$$
where $\theta\in \mathbb{R}^d$. If $\nabla\phi(\theta)$ is L-Lipschitz i.e.
$$\vert\nabla\phi(\theta\_1) - \nabla\phi(\theta\_2)\vert \leq L\vert\theta\_1-\theta\_2\vert,$$
how can we prove $Z = \lang... | https://mathoverflow.net/users/500967 | Is the main part of certain exponential family sub-Gaussian? | $\newcommand\th\theta\newcommand\la\lambda\newcommand\R{\mathbb R}$Note that for $t\in\R^d$ we have
$$M\_\th(t):=E\_\th e^{t\cdot T(X)}
=\int\_{\R^d}dx\,h(x)e^{(t+\th)\cdot T(x)-\phi(\th)}
=e^{\phi(t+\th)-\phi(\th)},$$
where $\cdot$ denotes the dot product. So,
$$E\_\th T(X)=\nabla M\_\th(0)=\nabla\phi(\th)$$
and, f... | 1 | https://mathoverflow.net/users/36721 | 444776 | 179,317 |
https://mathoverflow.net/questions/444781 | 6 | #### Background/Motivation
A *theory* T over a signature(language) Σ is a set of formulae over Σ. These formulae are called the *non-logical axioms* of T.
To talk about what is provable in T we can agree on Hilbert calculus and first-order logic.
A theory $ T' $ over $ Σ' $ is said to be an *extension* of a theory ... | https://mathoverflow.net/users/502824 | Is the union of two conservative extensions of a theory conservative? | Yes, this is true (and somewhat nontrivial). That is, if $T$ is a theory in a language $\Sigma$, and $T\_1$ and $T\_2$ are conservative extensions of $T$ in languages $\Sigma\_1$ and $\Sigma\_2$ (respectively) such that $\Sigma\_1\cap\Sigma\_2=\Sigma$, then $T\_1\cup T\_2$ is a conservative extension of $T$. This is a ... | 11 | https://mathoverflow.net/users/12705 | 444784 | 179,321 |
https://mathoverflow.net/questions/444778 | 11 | Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus\_{n=0}^\infty R\_n$ with $R\_0=k$ (and $R=k[R\_1]$). The Hilbert function $h\_R:\mathbb{N}\rightarrow \mathbb{N}$ is given by $h\_R(n)=\dim\_k R\_n$. When $n\gg 0$, it is known that $h\_R(n)$ agrees with a polnyomial $p... | https://mathoverflow.net/users/159030 | Hilbert polynomials of graded algebras evaluated at negative numbers | This answer is essentially the same as that of Phil Tosteson, written
before I saw that post. I also mention a non-Cohen-Macaulay example at
the end.
If $R$ is Cohen-Macaulay (but not necessarily generated in degree
one), then $R$ has associated with it a *canonical module*
$\Omega(R)$ which can be graded so its Hilb... | 11 | https://mathoverflow.net/users/2807 | 444790 | 179,325 |
https://mathoverflow.net/questions/444658 | 10 | Consider the homotopy category $\mathrm{hoSp}$ of spectra. It has a full subcategory $\mathrm{hoSp}\_{\geq 0}$ of connective spectra, equivalently of infinite loop spaces, equivalently $E\_\infty$-group spaces.
The inclusion $\mathrm{hoSp}\_{\geq 0} \hookrightarrow \mathrm{hoSp}$ has a right adjoint, sending a spectr... | https://mathoverflow.net/users/78 | Which spectra have a universal connective quotient? | This answer is about the $\infty$-categorical variant. This is a fancy way to say: on *spaces* of maps, the natural map
$$
Map(T',A) \to Map(T,A)
$$
is an equivalence for any connective $A$. Note that for any bounded-below spectrum $B$, we can use $Map(-,B) \simeq \Omega^n Map(-,\Sigma^n B)$ to conclude that the natura... | 9 | https://mathoverflow.net/users/360 | 444797 | 179,328 |
https://mathoverflow.net/questions/444754 | 2 | Let $m$ be a positive integer satisfying $\dfrac{m(m+1)}{4}\in \mathbb{Z}$. Show that there exists a positive integer $t$ and $t$ positive integers $m\_1,m\_2,\cdots,m\_t$ such that $$\begin{cases} \sum\limits\_{k=1}^{t}m\_k=m\\ \sum\limits\_{k=1}^{t}\dfrac{m\_k(m\_k+1)}{2}=\dfrac{m(m+1)}{4} \end{cases}.$$
P.S. This qu... | https://mathoverflow.net/users/502801 | The existence of solutions of a system of indeterminate equations | In the comments OP proposed a greedy algorithm to represent a given positive integer $A$ as the sum of triangular numbers whose indices sum to $m$, and applied it to $A = \frac{m(m+1)}4$. I will prove that it indeed always produces a solution, and thus the given system is soluble for any positive integer $m$ with $\fra... | 2 | https://mathoverflow.net/users/7076 | 444800 | 179,329 |
https://mathoverflow.net/questions/444796 | 1 | (all morphism here means birational)
(the ground field is "small", but I don't think it should matter)
Here is the picture. I have a morphism $f:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$. I want to lift this up to a morphism $\tilde{f}:C\rightarrow E$ where $C$ is a curve and $E$ is an elliptic curve with projection... | https://mathoverflow.net/users/502838 | Are morphism from $\mathbb{P}^{1}$ to itself often liftable to a morphism from a curve to an elliptic curve with bounded degree? | $\newcommand{\CC}{\mathbb C}$If there is a commutative diagram as requested, then there is also one with $C$ and $\pi\_C$ replaced by $C'$ and $\pi\_{C'}$, where $\pi\_{C'}$ has degree $2$ by letting $C'$ be a connected component of the fiber product of $f$ and $\pi\_E$.
Or in terms of function fields, with $x$ a tra... | 3 | https://mathoverflow.net/users/18739 | 444801 | 179,330 |
https://mathoverflow.net/questions/444804 | 2 | Let $G$ be a simple graph on $n$ vertices.
Prove that, for every $k\in\{2,\dots,n\}$, there exist $k$ distinct vertices in $G$ whose degrees differ by at most $k-2$.
| https://mathoverflow.net/users/501463 | There exist $k$ vertices with degree differences at most $k - 2$ | This result was originally proved by [Erdős-Chen-Rousseau-Schelp (1993)](https://www.sciencedirect.com/science/article/pii/S0195669883710231) using the [Erdős-Gallai theorem](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Gallai_theorem). A more direct proof (which seems to be identical to the one in 1001's response)... | 6 | https://mathoverflow.net/users/11919 | 444810 | 179,334 |
https://mathoverflow.net/questions/444811 | 0 | Recently I have made some interesting observations on the limit $$\lim\_{k\rightarrow \infty}{\sum\_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}. $$ When this limit exist denoteits convergence point by $\zeta$. If we define $f\_k$ to be the partial sums of this series:
$$f\_k... | https://mathoverflow.net/users/353746 | proving inequality in Riemann zeta function | The inequality you want to prove is false. For example,
$$f\_5(1/2,1)=0.6096\dots,$$
while
$$\lim\_{k\to\infty}f\_k(1/2,1)=0.6398\dots$$
| 2 | https://mathoverflow.net/users/11919 | 444812 | 179,335 |
https://mathoverflow.net/questions/442718 | 5 | Let $\mathcal{U}\subset \mathbb{R}\times \mathbb{C}$ a neighborhood of $(0,0)$, and $f:\mathcal{U}\to \mathbb{C}$ differentiable in the first variable and holomorphic in the second variable, with $f(0,0)=0$. I want to locally express the zeros of $f$ as one or more curves $z=z(x)$ with $z(0)=0$. The hypothesis I want t... | https://mathoverflow.net/users/125758 | Implicit function theorem with singularities of any order | [Rouché's theorem](https://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem) provides a simple solution.
Let $f\_1(x,z)=z^k+x^h$ and $f\_2(x,z)=h.o.t.$. For each $x$, the functions $f\_1$ and $f\_2$ are holomorphic in $z$. We know $f\_2(x,z)=o(z^k)+o(x^h)$, so there exist $z\_0,x\_0>0$ such that for $|z|\leq z\_0$ and $... | 1 | https://mathoverflow.net/users/125758 | 444814 | 179,336 |
https://mathoverflow.net/questions/444822 | 8 | Are there any uses of generating functions within logic, in particular to count how many models exists for a given theory $T$, say in FOL?
The concrete problem I'm hoping to apply this to involves counting the number of states witnessed by a given program. In this application some ad-hoc combinatorial arguments can b... | https://mathoverflow.net/users/165086 | Use of generating functions in logic | Not sure if the lambda calculus or 2-SAT counts for you as "logics" but here is a couple of recent papers presenting bijections between these formal structures of mathematical logic and "more classical" combinatorial objects. The bijections transfer certain properties of the former objects to the properties and pattern... | 5 | https://mathoverflow.net/users/31830 | 444830 | 179,340 |
https://mathoverflow.net/questions/444829 | 5 | Given a diagram of sets $D\colon\mathcal{C}\to\mathsf{Set}$, we have a bijection ([Proof](https://math.stackexchange.com/a/3838971))
$$\operatorname{colim}(D) \cong \pi\_0 (\textstyle\int\_\mathcal{C}D).$$
1. Is there any known application or significance of the homotopy groups of $\int\_\mathcal{C}D$ (meaning the ho... | https://mathoverflow.net/users/130058 | Homotopy groups of categories of elements as higher colimits | To answer these questions, the best is to note that $|\int\_C D|$, the geometric realization of this total category, is equivalently the colimit of $D$, *viewed as a functor with values in the $\infty$-category of spaces*, namely, along the inclusion $Set\to Spaces$.
So
1. Insofar, as you believe that this colimit ... | 13 | https://mathoverflow.net/users/102343 | 444836 | 179,341 |
https://mathoverflow.net/questions/444837 | 0 | If $G = (V,E)$ is a finite, connected, simple, undirected graph, is there a [Hamiltonian path](https://en.wikipedia.org/wiki/Hamiltonian_path) in the [line graph](https://en.wikipedia.org/wiki/Line_graph) $L(G)$ of $G$?
| https://mathoverflow.net/users/8628 | Hamiltonian path in the line graph of a connected graph | No. Consider the graph $G = (V, E)$ with
$$V = \{0,1,2,3,4,5,6\}\quad \text{and} \quad E = \{\{0,1\}, \{1, 2\}, \{0,3\}, \{3, 4\}, \{0,5\}, \{5, 6\}\}.$$
Note that its line-graph has three vertices of degree 1. Therefore $L(G)$ has no Hamiltonian path.
| 5 | https://mathoverflow.net/users/502833 | 444838 | 179,342 |
https://mathoverflow.net/questions/376467 | 90 | Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\gamma$ must be a straight line?
| https://mathoverflow.net/users/167834 | Does this property characterize straight lines in the plane? | It seems $\gamma:\mathbb{R}\to\mathbb{R}^2$ (I assume $\gamma$ is injective and continuous) is indeed a line. My argument is very similar to the [one](https://mathoverflow.net/a/377074/172802) by Ilkka Törmä (I thought I could write a shorter one, but it grew longer than expected). As in their answer, I will take the r... | 2 | https://mathoverflow.net/users/172802 | 444859 | 179,351 |
https://mathoverflow.net/questions/444865 | 1 | The general $4$-deg and some $8$-deg (such as the [Schein octic](https://mathoverflow.net/q/145145/12905)) when a linear transformation is done so their $x^{n-1}$ term vanishes can have a neat solution as,
$$x = \sqrt{z\_1}+\sqrt{z\_2}+\sqrt{z\_3}$$
$$x = \sqrt{z\_1}+\sqrt{z\_2}+\dots+\sqrt{z\_7}$$
where the $z\_... | https://mathoverflow.net/users/12905 | On reducing degree-$12$ equations with Mathieu group $M_{11}$ to its degree-$11$ resolvent? | The answer is yes: $M\_{11}$ in its action of degree $12$ has a subgroup of index $11$ (some people call it $M\_{10}$) such that there is a $6$-element subset $A$ of the $12$ points which $M\_{11}$ acts on such that $A^g=A$ or $A\cap A^g=\emptyset$ for all $g\in M\_{10}$. From that the claim follows immediately.
| 5 | https://mathoverflow.net/users/18739 | 444866 | 179,352 |
https://mathoverflow.net/questions/444401 | 17 | Does a irrational number $x > 1$ exist such that $\{x^n \} \le \frac{1}{2}$ for
all positive integers $n$ ?
$x=1+ \sqrt 2$ holds for $n$ odd, but not in even
| https://mathoverflow.net/users/174530 | Fractional part power | The OP asks for an instance of what Dubickas [1] has called a ${\cal Z}$-number: A real number $x>1$ for which there exists a real $\xi\neq 0$ such that $\{\xi x^n\}<1/2$ for every integer $n$.
An example [2] of an irrational ${\cal Z}$-number is $x=\tfrac{1}{2}(7+\sqrt{41})$, when $\{\xi x^n\}$ with $\xi=\tfrac{1}{4... | 11 | https://mathoverflow.net/users/11260 | 444868 | 179,353 |
https://mathoverflow.net/questions/444798 | 1 | My research has brought me to the following linear parabolic second order PDE:
$$ \frac{\partial^2}{\partial x^2}\Psi(t,x)=c(t,x)\frac{\partial}{\partial t}\Psi(t,x) $$
for $c(t,x)=-\frac{t}{x}$ and $x\in (0,\infty)$ and time $t\in (0, 1).$
Using the ansatz $\Psi(t,x)=X(x)Y(t)$ this can be simplified to ODE's and... | https://mathoverflow.net/users/411249 | Physical relevancy of two curious PDE's | Sorry to bring bad news. Usually, physicists are interested in the Cauchy problem: given initial data, determine the evolution of the state.
Alas, it is well known that PDE's whose order in the time variable (here 2) exceeds the order in the space variable (here 0) are not globally well-posed. Well, you can invoque C... | 2 | https://mathoverflow.net/users/8799 | 444872 | 179,355 |
https://mathoverflow.net/questions/444699 | 8 | I know Jensen developed a forcing notions in $L$ that adds a unique, minimal and $\Delta\_3^1$ $L$-generic real. In his paper [Definable sets of minimal degree](https://zbmath.org/?q=an:0245.02055) he says that Solovay had already shown the consistency relative to $\mathsf{ZF}$ of $``V$ is the constructible closure of ... | https://mathoverflow.net/users/141146 | Forcing a unique $\Delta_3^1$ generic real | Since the first question has already been answered in the EDIT at the end of the question, I will focus on the second question.
The short answer to second question is: Yes, a good source is Sy Friedman's paper *The $\Pi^1\_2$-conjecture*, Journal of the American Mathematical Society, Vol. 3, No. 4 (Oct., 1990), pp. 7... | 8 | https://mathoverflow.net/users/9269 | 444876 | 179,357 |
https://mathoverflow.net/questions/432429 | 1 | Let $X$ be a projective normal variety over $\mathbb C$, I have several questions about semi-stable sheaves:
**Question 1.** Suppose that $E$ is a pure sheaf such that $HN\_\*(E)$ is the Harder-Narasimhan filtration of $E$. Let $H$ be an ample divisor and $D \in |aH|$ be a general element for $a\gg 1$. Then is $HN\_{... | https://mathoverflow.net/users/29730 | Some question about (semi-)stable sheaves | Concerning your **Questions 1** & **2**, since you adress to slope semistable sheaves and Metha-Ramanathan type results, I think every sheaf you mean is in fact torsion free (i.e. pure of dimension $\dim X$), right? Under this assumption, **Question 1** is true if $D$ is replaced by a general complete intersection curv... | 1 | https://mathoverflow.net/users/502863 | 444884 | 179,360 |
https://mathoverflow.net/questions/444846 | 4 | Lemma 3.2.1 in [Baker, González-Jiménez, González, Poonen, "Finiteness theorems for modular curves of genus at least 2", Amer. J. Math. 127 (2005), 1325–1387.](https://math.mit.edu/%7Epoonen/papers/finiteness.pdf)
[enter image description here](https://i.stack.imgur.com/ecXLP.png)
I don't understand why "all sup... | https://mathoverflow.net/users/502709 | Why all supersingular elliptic curves over $\bar{\mathbb{F}_p}$ are isogenous? | This can be proved in several ways (this can be found in the literature as Lemma 42.1.11 in Voight's book on Quaternion algebras, or Proposition 5.2 in <https://arxiv.org/pdf/2005.01537v1.pdf>). Typically the proof relies on Tate's isogeny theorem (1966) which states, among other things, that if two elliptic curves ove... | 2 | https://mathoverflow.net/users/84923 | 444889 | 179,362 |
https://mathoverflow.net/questions/444886 | 4 | Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^\*$ as the set of minimizers of $f$ and assume $X^\*$ is unbounded. Is it possible that $\|g\_x\|$ is unbounded when $d(x,X^\*)$ is bounded where $g\_x\in\partial f(x)$. Here $\partial f(x)$ denotes the set of subgradient vectors of $f$ at $x$ an... | https://mathoverflow.net/users/490600 | An upper bound of gradient norm for convex functions near minimizer | The function $q$ might be unbounded even in 1-neighborhood of $X^\*$; here is an example of such function $f$ on the $(x,y)$-plane.
Let $\phi(t)=|t|-t$.
Choose a sequence $x\_n\to \infty$.
For each $n$ consider function $f\_n(x,y)=\phi[-2\cdot x\_n\cdot(x-x\_n)+ (y-x\_n^2)]$.
Let $f=\max\_n\{f\_n\}$.
| 3 | https://mathoverflow.net/users/1441 | 444895 | 179,364 |
https://mathoverflow.net/questions/444893 | 4 | [Rademacher’s formula](https://en.m.wikipedia.org/wiki/Partition_function_(number_theory)) for the partition function allows fast computation using high precision arithmetic, but requiring a lot of memory. [Here](https://fredrikj.net/blog/2014/03/new-partition-function-record) is an example computation of $p(10^{20})$ ... | https://mathoverflow.net/users/22930 | Fast computation of the partition function modulo a prime | To the best of my knowledge, there is no known method to compute $p(n)$ modulo a small prime using less than $n^{1/2+o(1)}$ time or memory, except for those cases where a Ramanujan-like congruence applies.
| 5 | https://mathoverflow.net/users/4854 | 444897 | 179,365 |
https://mathoverflow.net/questions/444870 | 2 | Suppose an $n$-dimensional process $(X\_t)\_{0 \leq t \leq 1}$ satisfies an SDE of the form:
$$dX\_t = u\_t(X\_t) \,dt + dB\_t, ~~X\_0 = 0$$
where $(B\_t)\_{t\geq 0}$ is a Brownian motion with $B\_1 \sim N(0,K)$, and $K$ is positive definite.
Does anyone have a reference for simple conditions on $u\_t$ that will ... | https://mathoverflow.net/users/99418 | When does a solution to SDE have full support? | I am writing down a very similar answer to [this one](https://mathoverflow.net/a/443534/129074), that I wrote for the similar question [*SDE with non-degenerate diffusion visits every point*](https://mathoverflow.net/q/438425/129074). I am not sure how closely it answers your question, since it is not a reference for t... | 3 | https://mathoverflow.net/users/129074 | 444902 | 179,368 |
https://mathoverflow.net/questions/444885 | 1 | Suppose there are $n$ IID random variables denoted as $X=(X\_1,\dots, X\_n)$, they follow Laplace distribution with parameter $\lambda$, denoted as $Lap(\lambda)$. That is,
$$f(x)=\frac{1}{2\lambda}\exp (-\frac{|x|}{\lambda})$$
$$
F(x)= \begin{cases}\frac{1}{2} \exp \left(\frac{x}{\lambda}\right) & \text { if } x<0 \\ ... | https://mathoverflow.net/users/327644 | Expected (maximum minus minimum) of Laplacian random variables | $\newcommand\la\lambda$By rescaling, it is enough to consider the case $\la=1$ (multiplying the resulting expectation by $\la$ at the end). By symmetry, $Y:=\max X$ and $-\min X$ are identically distributed. So, for $M\_n:=M$ we have
\begin{equation\*}
EM\_n=2EY. \tag{10}\label{10}
\end{equation\*}
Next, $Y=\max(0,Y)-... | 2 | https://mathoverflow.net/users/36721 | 444903 | 179,369 |
https://mathoverflow.net/questions/444899 | 3 | Let $U\subset\mathbb{R}^{d}$ be an open set. The singular support of a distribution $u\in\mathcal{D}^{\prime}(U)$ is defined to be the compliment of the set of points, which have a neighbourhood in which $u$ can be identified with a smooth function. In a paper [Fourier integral operators. I](https://doi.org/10.1007/BF0... | https://mathoverflow.net/users/199422 | Singular support: equivalent definition | The basic idea is the one put forward by Pierre PC's comment above. More precisely, let $u,\varphi,x$ be as in the last paragraph of the OP. There is no loss of generality in assuming that $\varphi(x)=\lambda>0$. Then by continuity of $\varphi$ at $x$ there is an open neighborhood $V\ni x$, $V\subset U$ such that $\var... | 5 | https://mathoverflow.net/users/11211 | 444909 | 179,370 |
https://mathoverflow.net/questions/444882 | 10 | Using the definitions from Peter May's [A Concise Course in Algebraic Topology](https://www.math.uchicago.edu/%7Emay/CONCISE/ConciseRevised.pdf), a topological space $X$ is weak Hausdorff if for every compact Hausdorff space $K$ and continuous function $f:K\to X$, $f(K)$ is closed in $X$. A subset $A$ of $X$ is said to... | https://mathoverflow.net/users/502850 | Space with compactly closed diagonal but which is not weak Hausdorff | The space $X$ constructed in Theorem 1.5 of this [preprint](https://arxiv.org/abs/2211.12579) has the required properties. This space contains a non-closed compact metrizable subspace $K$, so is not weakly Hausdorff.
On the other hand, the space $X$ is $k\_2$-metrizable, which means that $X$ is the image of a metriza... | 11 | https://mathoverflow.net/users/61536 | 444913 | 179,372 |
https://mathoverflow.net/questions/444906 | 5 | In the paper, *A polytope related to empirical distributions, plane trees, parking functions, and the associahedron*, [Pitman and Stanley](https://arxiv.org/pdf/math/9908029.pdf) studied the $n$-dimensional polytope
$$\Pi\_n(\mathbf x)=\{y\in\Bbb{R}^n: y\_i\geq0, y\_1+\cdots+y\_i\leq x\_1+\cdots+x\_i, \text{for all $1\... | https://mathoverflow.net/users/66131 | Catalan sequences vs composition sequences | Given the corrected definition of $\mathcal{B}\_n$ in the Postscriptum, the equality does hold and can be proved by induction on $n$.
For $n=1$, the equality is trivial. For $n>1$, let's expand the determinant in the left-hand size of the equality over the first column to get
$$\det\left(\frac{\chi(j-i+1\geq0)}{(j-i+... | 7 | https://mathoverflow.net/users/7076 | 444917 | 179,374 |
https://mathoverflow.net/questions/444588 | 5 | Let $\lambda$ denote the Liouville function, and let $L(x):=\sum\_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. It is a straightforward exercise to show that for any $\varepsilon>0$, one has $L(x) =\Omega\_{\pm }(x^{c-\varepsilon}... | https://mathoverflow.net/users/501735 | Asymptotics of the Liouville sum at the primes | This should be true. By a Corollary II of a [result of Pintz](http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4214.pdf) (with not too much work, one can get this to work for the Liouville function in place of Mobius), we have that
$$\sum\_{Y/(100\log Y)\le n\le Y} |L(n)|\gg Y^{1 + c - \varepsilon},$$
in your notation. We al... | 4 | https://mathoverflow.net/users/40983 | 444921 | 179,376 |
https://mathoverflow.net/questions/444925 | 14 | Let $f(x\_1, \cdots, x\_n) \in \mathbb{R}[x\_1, \cdots, x\_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive number $\kappa$ such that for all $\mathbf{x} \in \mathbb{R}^n \setminus K$ we have $f(\mathbf{x}) > \kappa$. Clearly, $... | https://mathoverflow.net/users/10898 | Real polynomials that go to infinity in all directions: how fast do they grow? | The assumption "$\lim\_{R\to\infty}m\_f(R)\to\infty$" already includes $\bf P$ as remarked in comments, and it is the same as $\lim\_{\|x\|\to\infty} f(x)=+\infty$, that is: **$f$ is a coercive polynomial**. You are asking whether a coercive polynomial in $n$ variables needs to go to infinity at least quadratically: th... | 27 | https://mathoverflow.net/users/6101 | 444938 | 179,381 |
https://mathoverflow.net/questions/444943 | 7 | Let $Sets$ be the category of finite sets and all maps. I have come accross several example of functors $F : Sets^{op} \to Sets$ which satisfy the condition below:
-- There exists an integer $k$ such that, whenever $X$ is a finite set, with subsets $U\_1, U\_2, U\_3...$ whose union is $X$ *and with* $|U\_i| \ge k$, t... | https://mathoverflow.net/users/37021 | not quite the sheaf condition | Consider the Grothendieck topology in which a cover of $X$ is a way of expressing $X$ as a union of sets $U$ with $|U|\le k$. It seems to me that you are talking about sheaves for this topology.
This looks wrong (because I have "$\le$" where you had "$\ge$"), but I think it's right.
EDIT: This was nonsense. I shoul... | 8 | https://mathoverflow.net/users/6666 | 444948 | 179,383 |
https://mathoverflow.net/questions/444963 | 1 | Consider the matrix
$$A\_2:= \begin{pmatrix} a & b\_1 \\ b\_2 & a\end{pmatrix}.$$
Let $\sigma\_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then
$$\sigma\_2 A\_2 \sigma\_2 = \begin{pmatrix} a & -b\_2 \\ -b\_1 & a\end{pmatrix}.$$
I wonder if I have the matrix
$$A\_3:= \begin{pmatrix} a & b\_1 & 0 \\ b\_... | https://mathoverflow.net/users/496243 | Matrix transformation that always works? | The answer is no. Indeed, let $T:=\sigma\_3=(t\_{ij}\colon i,j\in\{1,2,3\})$, $A:=A\_3$, and
$$B:=\begin{pmatrix} a & -b\_2 & 0 \\ -b\_1 & a & -c\_2 \\ 0 & -c\_1 & a \end{pmatrix}.$$
Then the equality in question would imply that
$$TA=BT \tag{1}
\label{1}$$
for all $a,b\_1,b\_2,c\_1,c\_2$. Solving system \eqref{1} of l... | 2 | https://mathoverflow.net/users/36721 | 444967 | 179,387 |
https://mathoverflow.net/questions/444970 | 0 | Let $\{ g\_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as
\begin{equation} S f := \sum\_{n=1}^\infty (f,g\_n)g\_n \end{equation}
is a Hilbert space isomorphism i.e., continuous and bijective. Then one can define the canonica dual frame of $g\_n$ as the frame $S^{-1}g\_... | https://mathoverflow.net/users/153260 | A property of the canonical dual frame in a Hilbert space | I assume your inner product is conjugate linear in the second factor, based on your assertion that the operator $S$ is a Hilbert space iso. Then
$$ \langle f\_n, g\_n\rangle = \langle f\_n, S f\_n \rangle = \langle f\_n, \sum\_k \langle f\_n, g\_k\rangle g\_k\rangle = \sum\_k |\langle f\_n, g\_k\rangle|^2 $$
is real an... | 1 | https://mathoverflow.net/users/3948 | 444976 | 179,390 |
https://mathoverflow.net/questions/444950 | 3 | (The following definitions are meant to be standard and are reproduced for completeness of the question.) A **frame** is a partially ordered set in which every finite subset has a greatest lower bound (“meet”, denoted $x\_1\wedge\cdots\wedge x\_n$, or $\top$ for the empty meet) and every subset has a least upper bound ... | https://mathoverflow.net/users/17064 | Computing the Heyting operation on the frame of nuclei | First, every nucleus is the join of those of the form $j^x\land j\_y$.
Namely,
$$
j=\bigvee\_xj^x\land j\_{jx}
$$
Hm, this is too confusing. Let me change notation and write: $u\_a$ instead of $j^a$; $v\_a$ instead of $j\_a$. Thus,
$$
j=\bigvee\_xu\_x\land v\_{jx}.
$$
It then follows
$$
j\_1\Rightarrow j\_2=\bigwedge... | 1 | https://mathoverflow.net/users/41291 | 444977 | 179,391 |
https://mathoverflow.net/questions/444964 | 2 | Do there exist integers $(x,y,z)\neq (0,0,0)$ such that
$$
(a) \quad 2x^5+3y^5=6z^5
$$
$$
(b) \quad x^5+3y^5=7z^5
$$
Here is a short motivation. Equation $ax^d + by^d=cz^d$ is trivial for $d=1$, solved by Lagrange for $d=2$, investigated in depth by Selmer <https://doi.org/10.1007/BF02395746> for $d=3$. For $d=4$, at... | https://mathoverflow.net/users/89064 | Existence of rational points on generalized Fermat quintics | Both curves have no rational points.
Curves of this shape map to genus 2 curves of the form $y^2 = a x^5 + b$
(one can make $a = 1$ if one likes), by quotienting out by the group
of automorphisms generated by $(x:y:z) \mapsto (\zeta x:\zeta^{-1} y:z)$
where $\zeta$ is a primitive fifth root of unity (one can pick any... | 11 | https://mathoverflow.net/users/21146 | 444982 | 179,393 |
https://mathoverflow.net/questions/444951 | 3 | I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann Hypothesis and some statistical randomness across the ordinates $\gamma$ of the nontrivial zeros of the Riemann zeta functi... | https://mathoverflow.net/users/153908 | What heuristic arguments support Montgomery's conjecture for primes in short intervals? | The motivation for many conjectures about the distribution of primes $p\in[x,x+h]$ in intervals of length $o(\sqrt{x}\log x)$ comes from studying the mean square of $|\psi(x+h)-\psi(x)-h|$. For example, for $\theta\in[0,1]$, Selberg considered the integral
$$J(x,\theta):= \frac{1}{x}\int\_{x}^{2x}|\psi(t+\theta t)-\p... | 4 | https://mathoverflow.net/users/111215 | 444993 | 179,395 |
https://mathoverflow.net/questions/444996 | 18 | I recently learnt that the proof of the classical theorem "$\mathsf{AD}$ $\implies$ $\aleph\_1$ is measurable" uses computability theory tools (or at least one of its proofs does so). I'm interested in more examples of this. More precisely:
1. What are some other well-known results in set theory that use non-trivial ... | https://mathoverflow.net/users/146831 | Theorems in set theory that use computability theory tools, and vice versa | Here are several examples.
* There is a natural affinity between forcing and many constructions in the Turing degrees. Specifically, many constructions of degrees by meeting requirements in succession can be seen as meeting dense sets in a suitable forcing notion, with the result that the computability construction a... | 21 | https://mathoverflow.net/users/1946 | 444997 | 179,397 |
https://mathoverflow.net/questions/444687 | 5 | I am sorry that the following question is elementary. I have not received an answer from my post at [Math Stack Exchange](https://math.stackexchange.com/questions/4676078/is-the-interior-of-the-tensor-product-of-two-convex-cones-equal-to-the-tensor-pr).
In the following question, all cones are convex and contain the ... | https://mathoverflow.net/users/136356 | Is the interior of the tensor product of two convex cones equal to the tensor product of their respective interiors? | Yes, $(C \otimes D)^\circ = C^\circ \otimes D^\circ$ is correct.
Let us start by proving the inclusion $C^\circ \otimes D^\circ \subseteq (C \otimes D)^\circ$. To this end, it is enough to show that $C^\circ \otimes D^\circ$ is open. For a given point $z = x\_1 {y\_1}^T + \dotsb + x\_q {y\_q}^T \in C^\circ \otimes D^... | 3 | https://mathoverflow.net/users/27013 | 445012 | 179,401 |
https://mathoverflow.net/questions/444989 | 2 | I'm working on a problem about Artin groups, and to simplify this problem I want to take a quotient that allow us to go to an easier Artin group, but I'm not sure if the quotient is well defined. This is the statement of my problem:
Let $A\_\Gamma$ be an Artin group, where $\Gamma$ is a complete graph, $f:A\_\Gamma\t... | https://mathoverflow.net/users/482329 | Quotient of an Artin group is an Artin group | (Edit: Missed that the graph should be complete, now it is.)
If I'm understanding all this correctly, the answer is "no". Take three vertices, $a$, $b$, and $c$. Take an edge labeled 4 from $a$ to $b$, an edge labeled 4 from $a$ to $c$, and an edge labeled 2 from $b$ to $c$. Take $f$ to be $f(a)=f(b)=1$, $f(c)=0$. So... | 3 | https://mathoverflow.net/users/164670 | 445018 | 179,403 |
https://mathoverflow.net/questions/444949 | 1 | **Question**
Given a random field $X(t)$ where the parameter space $T\subset\mathbb{R}\_N$. Is there result regarding the concentration of the random field? For example
$\mathbb{P}\{\|X(t)-\mathbb{E}\{X(t)\}\|\geq t\}\leq \text{small}$ where $\|\cdot\|$ is certain norm, maybe $L\_p$ norm.
**Take the following ran... | https://mathoverflow.net/users/147940 | concentration of random field to its expectation function | $\newcommand\R{\mathbb R}$For real $x\_1,x\_2,x\_3$, let
\begin{equation}
g(x\_1,x\_2,x\_3):=\sin(x\_1-x\_2),\quad h(x\_1,x\_2,x\_3):=\sin(x\_2-x\_3).
\end{equation}
The request is to bound
\begin{equation}
P(Y\ge t)
\end{equation}
from above, where
\begin{equation}
Y:=\|w\_1 g+w\_2 h\|\_{L^p}
\end{equation}
and... | 1 | https://mathoverflow.net/users/36721 | 445031 | 179,407 |
https://mathoverflow.net/questions/445028 | 2 | Is it posible to find the article "Pósa Lajos, A prímszámok egy tulajdonságáról (Hungarian), Mat. Lapok 11 (1960), 124-129" on the internet?
| https://mathoverflow.net/users/169583 | Pósa's inequality for a product of consecutive primes | Yes, here is the archive of [Matematikai Lapok](http://real-j.mtak.hu/view/journal/Matematikai_Lapok.html).
| 2 | https://mathoverflow.net/users/11919 | 445041 | 179,409 |
https://mathoverflow.net/questions/445044 | 2 | Given an integer $n\geq2$, consider the following integral
$$I\_n:=\int\_0^1nx^{n-1}\sqrt{\left\vert \frac{\log(1-x)}{\log n}\right\vert} \, dx.$$
**QUESTION.** Is this true? It appears to be so.
$$\lim\_{n\rightarrow\infty}I\_n=1.$$
| https://mathoverflow.net/users/66131 | Asymptotics of an integral requested | Change variables to $y=n(1-x)$, which gives
$$
I\_n=\int\limits\_{0}^{n}\left(
1-\frac{y}{n}\right)^{n-1}
\sqrt{1-\frac{\log y}{\log n}}\ {\rm d}y\ ,
$$
then dominated convergence should allow passing to the limit inside the integral
so that
$$
\lim\_{n\rightarrow \infty} I\_n=\int\limits\_{0}^{\infty} e^{-y}\ {\rm d}y... | 9 | https://mathoverflow.net/users/7410 | 445048 | 179,412 |
https://mathoverflow.net/questions/445043 | 6 | Let $u\in L^1(\mathbb{S}^{d-1})$. I want to show that
\begin{align\*}
\lim\_{|\xi|\to \infty}
\int\_{\mathbb{S}^{d-1}}(1-\cos(\xi\cdot w))u(w)d \sigma\_{d-1}(w)
= \int\_{\mathbb{S}^{d-1}}u(w)d \sigma\_{d-1}(w).
\end{align\*}
Basically, the question can be reduced into showing that
\begin{align\*}
\lim\_{|\xi|\to \inft... | https://mathoverflow.net/users/112207 | Computing a limit on the unit sphere: Riemann Lebesgue? | The key fact here is the (surprising, initially, but well known) (power) decay of $\widehat{\sigma}(\xi)$. If $u\in C^{\infty}(S)$, we can extend to a function $u\_0\in C^{\infty}\_0(\mathbb R^d)$, and then
$$
\widehat{u\,d\sigma}=\widehat{u\_0\,d\sigma}=\widehat{u\_0}\*\widehat{\sigma}
$$
still decays. See [here](http... | 9 | https://mathoverflow.net/users/48839 | 445055 | 179,414 |
https://mathoverflow.net/questions/445058 | 3 | Consider an ideal $I = \langle f\_1,\dotsc,f\_n\rangle$ in the ring $k[x\_1,\dotsc,x\_m]$.
Define the $i$-th elimination ideal to be $I\_i = I \cap k[x\_{i+1},\dotsc,x\_m]$.
For any two polynomials $f$ and $g$ in $I$, the resultant $\operatorname{Res}(f,g,x\_1)$ belongs to the first elimination ideal $I\_1$.
1. Is th... | https://mathoverflow.net/users/48526 | Resultants and elimination theory | A bit long for a comment, so posted as an answer, although this is really a comment.
For 1) I would rather try the following. Assuming the $f$'s have the same degree in $x\_1$,
introduce formal variables $s\_1,\dotsc,s\_n$ and $t\_1,\dotsc,t\_n$, and take the binary resultant $\operatorname{Res}(s\_1f\_1+\dotsb+s\_nf... | 3 | https://mathoverflow.net/users/7410 | 445061 | 179,416 |
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