parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/449864 | 2 | Let $\pi$ be an automorphic representation of $\textrm{GL}\_n$. Associated to $\pi$, we can define the standard $L$-function $L(s, \pi)$.
My question is: what is the difference between $L(s, \pi)$ and the $L$-function attached to a **cuspidal** automorphic representation of $\textrm{GL}\_n$ ? For example, are there a... | https://mathoverflow.net/users/167708 | Question on automorphic $L$-functions | Let us restrict to automorphic representations of $\mathrm{GL}\_n$ over $\mathbb{Q}$ with arbitrary $n$ and unitary central character.
If $\pi$ is an irreducible cuspidal representation, then $L(s,\pi)$ is entire unless $\pi\cong|\det|^{it}$ in which case $L(s,\pi)=\zeta(s+it)$ has a simple pole at $s=1-it$. Let us c... | 2 | https://mathoverflow.net/users/11919 | 449938 | 181,034 |
https://mathoverflow.net/questions/449858 | 17 | What are the torsion units of the ring $R\_n:=\mathbb{Z}[x]/(1+x+x^2+\cdots+x^{n-1})$? Since $x^n = 1$ in $R\_n$ it is clear that all elements of the form $\pm x^i$ are torsion units. Is this all of them?
Of course, if $n$ is prime then $R\_n$ is the ring of integers of the $n^{th}$ cyclotomic field and the result is... | https://mathoverflow.net/users/507811 | Torsion units of the ring $\mathbb{Z}[x]/(1+x+x^2+\cdots+x^{n-1})$ | I think, they are all of them. Let me be more concrete and accurate than in the initial answer. But this also makes the answer sometimes boring. Shortcuts are welcome.
Let $f$ be a torsion unit, that is, $f$ is represented by a polynomial with integer coefficients (again denoted by $f$) of degree at most $n-2$ such t... | 13 | https://mathoverflow.net/users/4312 | 449940 | 181,035 |
https://mathoverflow.net/questions/449618 | 3 | **Definition 1**: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal.
**Definition 2**: A matrix $A$ is half-skew-centrosymmetric if there exist two square matrices $B$ and $C$ of order $n$ such that
\begin{equation}
A = \begin{bmatrix}
B & R... | https://mathoverflow.net/users/369335 | On the half-skew-centrosymmetric Hadamard matrices | Let $H\_n$ be an $n×n$ Hadamard matrix and $R\_n$ the $n×n$ reverse identity matrix.
The matrix $X= \begin{pmatrix}
H\_n & R\_nH\_n \\
H\_n & -R\_nH\_n
\end{pmatrix}$ has entries of length $1$ and $$XX^\* = 2nI\_{2n} + ((nI\_n - R\_nH\_nH\_n^\*R\_n) \otimes R\_2)$$ which is simply $2nI\_{2n}$ so it is a Hadamard matr... | 2 | https://mathoverflow.net/users/490128 | 449943 | 181,036 |
https://mathoverflow.net/questions/449898 | 2 | Let $e\_d$ be the $d$-th standard-basis vector in the Hilbert space $H=l\_2(\mathbb{N})$.
Let $h(n) = J\_2(n)$ be the second Jordan totient function, defined by:
$$J\_2(n) = n^2 \prod\_{p|n}(1-1/p^2)$$
Define:
$$\phi(n) = \frac{1}{n} \sum\_{d|n}\sqrt{h(d)} e\_d.$$
Then we have:
$$ \left < \phi(a),\phi(b) \right... | https://mathoverflow.net/users/165920 | A geometric proof that there are infinitely many primes? | You have claimed twice that
$$
d(n)= \prod\_{k = 1}^n \prod\_{p\mid k}(1-p^{-2})
$$
If this is true, your desired claim immediately holds, namely $d(n+1) <d(n)$.
This is because the quotient
$$
\frac{d(n+1)}{d(n)} = \frac{\prod\_{k = 1}^{n+1}\prod\_{p\mid k}(1-p^{-2})}{\prod\_{k = 1}^n\prod\_{p\mid k}(1-p^{-2})} ... | 4 | https://mathoverflow.net/users/101207 | 449944 | 181,037 |
https://mathoverflow.net/questions/449954 | 3 | Assume we have a Borel probability measure $\mu$ in $\mathbb{R}^n$ and a sequence of $\mu$ distributed I.I.D. random variables $x\_n$. Is there a limit formula for $supp(\mu)$, something like closure of limit points of $x\_n$ or similar, which allows kind of asymptotic recovering of $supp(\mu)$ from observed data.
| https://mathoverflow.net/users/34984 | Recovering measure support from the sequence of I.I.D random variables | By f.i. <https://encyclopediaofmath.org/wiki/Support_of_a_measure> $supp(\mu)$ is the smallest closed set $C \subset \mathbb{R}^n$ such that $\mu(C^c) = 0$, $C^c$ the complement of $C$. Let $y := (y\_n)\_{n \in \mathbb{N}}$ be any sample of $(x\_n)$ (independent realizations of $\mu$) and $A := \{y\_n \colon n \in \mat... | 4 | https://mathoverflow.net/users/100904 | 449958 | 181,041 |
https://mathoverflow.net/questions/449948 | 0 | I am aware that in a finite dimensional vector space, any two norms are equivalent.
However, I cannot really figure out how "universal" the equivalence constants are.
To be specific, let us think of the space $L^2\Bigl([0,1],\mathbb{R} \Bigr)$ of periodic real-valued functions on $[0,1]$. Denote its inner product b... | https://mathoverflow.net/users/56524 | Norm equivalence in finite dimensions - is the equivalence "universal" if the dimension is fixed? | I am turning my comment into an answer.
If I've understood your question correctly, the constants $(c,C)$ cannot be "universal" in your sense for rather trivial set-theoretical reasons.
Suppose that the norm inequality is valid for every 3-dimensional vector subspace for the same constant choice of constants $(c,C)... | 4 | https://mathoverflow.net/users/1849 | 449960 | 181,042 |
https://mathoverflow.net/questions/449922 | 2 | If we coin a theory in $\mathcal L\_{\omega\_1, \omega}$ that begins with constructing pure true well founded finite sets, then the set of all true well founded hereditarily finite sets, then builds up stages of $L$ on top of it using the infinitary machinery (depicted below), then restrict iteration to be secured only... | https://mathoverflow.net/users/95347 | Is this theory finitary first order complete? | Let $\mathbb{K}$ be the class of well-founded models of $\mathsf{ZFC+V=L}$ + "There is no inaccessible cardinal." This is a subclass of the model class of your theory, but under mild hypotheses its associated theory $$Th(\mathbb{K}):=\bigcap\_{\mathfrak{M}\in\mathbb{K}} Th(\mathfrak{M})$$ is not complete. For example, ... | 4 | https://mathoverflow.net/users/8133 | 449963 | 181,043 |
https://mathoverflow.net/questions/449961 | 1 | Are there set theories that extend some complete infinitary language $\mathcal L\_{\kappa, \lambda}$, prove all axioms of $\sf ZFC$, and are finitary $\textbf{FOL}$ complete? That is, every sentence in $\mathcal L(=,\in)\_{\omega,\omega}$ is decidable.
| https://mathoverflow.net/users/95347 | Are there strong set theories written in infinitary language, that are finitary FOL complete? | To avoid triviality (e.g. "The true $\mathcal{L}\_{\kappa,\lambda}$-theory of $V$") let's look specifically for theories which $(1)$ consist of adding to $\mathsf{ZFC}$ a single infinitary sentence and $(2)$ have that single sentence being "reasonably simple." As long as we stick to $\mathcal{L}\_{\omega\_1,\omega\_1}$... | 3 | https://mathoverflow.net/users/8133 | 449965 | 181,044 |
https://mathoverflow.net/questions/449934 | 2 | Let $p:E\to B$ be a locally trivial fibration of connected, non-compact smooth manifolds. Let $U\subset E$ be a connected open subset and $p|\_U:U\to p(U)$ has connected diffeomorphic fibers.
Can we conclude that $p|\_U$ is again a locally trivial fibration?
This question is related to my other question on fibering... | https://mathoverflow.net/users/126243 | Restriction of a fibration to an open subset with connected diffeomorphic fibers | Here is a counterexample inspired by algebraic geometry:
**Example.** Let $E \to B$ be the first projection $\mathbf C^2 \to \mathbf C$ (or $\mathbf R^4 \to \mathbf R^2$, if you like), and let $U \subseteq \mathbf C^2$ be the complement of the divisor $\{(x,y) \in \mathbf C^2\ |\ xy = 1\}$ and the origin $\{(0,0)\}$.... | 4 | https://mathoverflow.net/users/82179 | 449966 | 181,045 |
https://mathoverflow.net/questions/449949 | 4 | I’ve been self-studying axiomatic systems for classical logic for a while. The standard Hilbert/Mendelssohn/Lukasiewicz axiomatizations were a bit tough for me to get used to without using the Deduction Theorem, but now I’m confident with those systems.
I recently learned about Meredith’s axiom for classical logic:
... | https://mathoverflow.net/users/498245 | How to use Meredith’s axiom for classical logic? | See <https://us.metamath.org/mpeuni/meredith.html> and the links there for the proofs you want.
| 4 | https://mathoverflow.net/users/3684 | 449971 | 181,047 |
https://mathoverflow.net/questions/449663 | 5 | A (very?) naïve question, but I didn't get an answer on math.se: so here goes ….
In his original [ETCS](https://doi.org/10.1073/pnas.52.6.1506) paper, Lawvere states a categorial version of choice for sets in this form: "If the domain of $f$ has elements, then there exists $g$ such that $fgf = f$". So let's say, gene... | https://mathoverflow.net/users/14111 | Versions of Choice in categories | If we take Choice-1 in Lawvere's original form with "has (global) elements" in place of "is not initial", then Choice-2 follows from Choice-1 in any [Boolean category](https://ncatlab.org/nlab/show/Boolean+category). Factor $f:X\to Y$ as an epi $h : X \twoheadrightarrow Z$ followed by a mono $m:Z\hookrightarrow Y$. The... | 4 | https://mathoverflow.net/users/49 | 449974 | 181,049 |
https://mathoverflow.net/questions/449978 | 10 | The standard algorithm to compute the factorial function $N!$ via repeated multiplications has complexity $\mathcal{O}(N)$, in the model in which each operation costs 1, no matter how many digits the operands have. Intuitively, it would seem that this cost could not be improved.
However, how can we be sure? What if t... | https://mathoverflow.net/users/161776 | Can $N!$ be computed in less than $\mathcal{O}(N)$ operations? | The thing that you are asking is the required number of integer operations for computing $N!$ starting only from $1$ and $N$, which is usually referred to as *the straight-line complexity* of $N!$ denoted by $\tau(N!)$. (or [BSS model](https://en.wikipedia.org/wiki/Blum%E2%80%93Shub%E2%80%93Smale_machine), though I am ... | 11 | https://mathoverflow.net/users/171820 | 449983 | 181,050 |
https://mathoverflow.net/questions/449964 | 6 | Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align\*}
{\color{red}{\frac{1}{\varphi(n)}\sum\_{k\geq1}\frac{(-1)^k}{\binom{3n+k+2}{2n+2}}}}
&={\color{blue}{\sum\_{j=1}^n\frac{(-1)^j(2... | https://mathoverflow.net/users/66131 | A need for analytic continuation of a finite sum function | As for the sum $$\sum\_{k\geq1}(-1)^k\binom{k}{\frac23}^{-1}$$
one can evaluate it by means of the Beta function integral, like in this recent [computation](https://mathoverflow.net/questions/449776/closed-form-of-an-infinite-series/449810#449810).
$$\sum\_{k\geq1}(-1)^k\binom{k}{\frac23}^{-1}=\frac23\sum\_{k\geq1}(-1)... | 5 | https://mathoverflow.net/users/6101 | 449986 | 181,052 |
https://mathoverflow.net/questions/449846 | 2 | Consider the following random walk $(y\_t)\_{t \in \mathbb Z\_+}$:
$$y\_t = y\_{t-1} + u\_t,\quad (u\_t)\_{t \in \mathbb Z\_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z\_+)$$
where $y\_0, u\_1, u\_2,...$ are independent.
We know that the process is not stationary and non-ergodic. On the other hand, if $|... | https://mathoverflow.net/users/479236 | A question about convergence of stochastic processes converging to a random walk | Assume, naturally, that for each $n$ we have $y\_0^n\to y\_0$ (as $n\to\infty$) in distribution and $y\_0^n$ is independent of $(u^n\_t)$.
Then for each $T=0,1,\dots$ we have $Y^n\_T\to Y\_T$ in distribution, where $Y^n\_T:=(y^n\_0,\dots,y^n\_T)$ and $Y^n\_T:=(y\_0,\dots,y\_T)$. This follows because (say) for all $t=... | 1 | https://mathoverflow.net/users/36721 | 449995 | 181,055 |
https://mathoverflow.net/questions/449999 | 0 | In the Milnor and Moore paper, "[On the structure of Hopf algebras](https://doi.org/10.2307/1970615)" proposition 1.7 said the following:
>
> 1. $A$ a connected $K$-algebra.
> 2. $N$ a left $A$ module that is connected as a $K$-graded module i.e. there is an isomorphism $\eta\_N:K\to N\_0$ which results in an
> aug... | https://mathoverflow.net/users/77914 | Milnor and Moore paper "On the structure of Hopf algebra" proposition 1.7 , filtration and grading | Although $E^0(A\otimes C)$ can be identified with $A\otimes C$ as objects, this is not compatible with morphisms. If $g\colon A\otimes C\to A\otimes C$ is a map of graded groups, then the map $g\colon A\_i\otimes C\_j\to(A\otimes C)\_{i+j}$ can be expressed as a sum of morphisms $g\_k\colon A\_i\otimes C\_j\to A\_{i+k}... | 2 | https://mathoverflow.net/users/10366 | 450008 | 181,060 |
https://mathoverflow.net/questions/450010 | 0 | Let $A$ and $B$ be two compact convex sets (which may be assumed to be polytopes) in $\mathbb R^n$ such that $A\cap B\ne\emptyset$. Is it then always true that either $A\cap\text{ext}B\ne\emptyset$ or $B\cap\text{ext}A\ne\emptyset$, where $\text{ext}$ denotes the set of all extreme points of a set?
| https://mathoverflow.net/users/36721 | On the extreme points of two convex sets | No: consider the line segments $\{0\}\times[-1,1]$ and $[-1,1]\times\{0\}$ in $\mathbb{R}^2$.
| 3 | https://mathoverflow.net/users/2363 | 450011 | 181,062 |
https://mathoverflow.net/questions/449894 | 3 | Let $A\_{ijl}(t,x) : [0,\infty) \times \mathbb{R}^n \to [\mathbb{R}^n]^3$ be a smooth tensor field. That is, $i,j,l \in \{1,2,3, \cdots, n\}$
Further assume that $A\_{ijl}(t,x)=A\_{jil}(t,x)$ for all $(t,x) \in [0,\infty) \times \mathbb{R}^n$.
Also, let $0=\lambda\_1< \lambda\_2 < \cdots < \lambda\_n$ be some fixed... | https://mathoverflow.net/users/56524 | An ODE for tensor - possibility of the equation together with the initial condition at $t=0$ deciding the solution for all $t>0$ | These equations are not sufficient to determine $\sum\_l A\_{ijl}$ for $n>1$.
Here is a counterexample of a solution of the problem in the OP which contradicts the conjectured solution:
set $n=2$, $\lambda\_1=0$, $\lambda\_2=1$; all elements of $A\_{ijl}$ are identically zero, except
$$A\_{111}=1-t-t^2/2,\;\;A\_{... | 1 | https://mathoverflow.net/users/11260 | 450020 | 181,065 |
https://mathoverflow.net/questions/449993 | 5 | Question:
Do all connected Lie groups have dense torsion-free subgroups?
Context :
Let $ R\_\alpha \in SO\_2(\mathbb{R}) $ be a rotation by $ \alpha/2\pi $. If $ \alpha $ is irrational, then $ R\_\alpha $ generates a dense torsion free subgroup of $ SO\_2(\mathbb{R}) $.
Let $ R\_{\alpha,z} \in SO\_3(\mathbb{R})... | https://mathoverflow.net/users/387190 | Does every connected Lie group have a dense torsion-free subgroup? | Let $G$ be a connected Lie group. We know that $G$ has a real analytic structure for which the law is analytic, and we fix it. Also fix a left Haar measure and a smooth positive function of integral 1, thus defining a fully-supported probability $\mu$ on $G$.
Let $F\_n$ be the free group on $n$ given generators, and ... | 12 | https://mathoverflow.net/users/14094 | 450022 | 181,067 |
https://mathoverflow.net/questions/449991 | 3 | Let $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer.
It is conjectured that 50％ of twist of elliptic curve $E\_D$ has rank $0$ and $50％$ has rank $1$.
But is some particular case known or conjectured?
I'm particularly interested in the case $E:y^2=x^3+17x$.
Are there infinitely many tw... | https://mathoverflow.net/users/144623 | Infinitely many elliptic curve with twist rank more than $1$ in specific case | Let me turn my comment into an answer. There are indeed infinitely many such twists with a non-torsion point. By Nagell–Lutz, it suffices to produce infinitely many different squarefree integers $D \equiv 5 \pmod 8$ for which there exist $x,y \in \mathbf Q$ such that $y^2 = x^3+17D^2x$ and $x$ and $y$ are not both inte... | 2 | https://mathoverflow.net/users/82179 | 450029 | 181,069 |
https://mathoverflow.net/questions/450035 | 2 | Let $K$ be a number field, $\overline{K}$ an algebraic closure, and $X$ be a positive dimensional finite type $K$-scheme.
Could there exist a proper subfield $L\subset\overline{K}$ such that the natural inclusion
$$X(L)\hookrightarrow X(\overline{K})$$
is surjective?
If so, what sorts of conditions can we put on $X... | https://mathoverflow.net/users/88840 | Varieties whose residue fields do not generate the algebraic closure of the ground field | No this never happens: Without loss of generality, $X$ is affine. The case $X=\mathbb{A}^n\_K$ is obvious, the general case then follows from Noether normalization and going up.
| 5 | https://mathoverflow.net/users/50351 | 450039 | 181,071 |
https://mathoverflow.net/questions/450044 | 0 | Let $M$ and $N$ be Riemannian manifolds such that $\pi:M\to N$ is a surjective Riemannian submersion, i.e. for each $x\in M$,
$$\langle \pi\_{\*x}(v),\pi\_{\*x}(w) \rangle\_{\pi(x)} = \langle p(v), p(w) \rangle\_x$$
where $p: T\_xM = \text{Ker}(\pi\_{\*x}) \oplus \text{Ker}(\pi\_{\*x})^\perp \to \text{Ker}(\pi\_{\*x})^... | https://mathoverflow.net/users/506774 | Local isometric embedding right inverse to a Riemannian submersion | You can do so if and only if the *horizontal distribution* $x \mapsto \ker(\pi\_\*)^\perp$ is integrable. This is equivalent to the vanishing of O'Neill's $A$-tensor for the Riemannian submersion:
$$
\langle \nabla\_X Y, U \rangle = 0,
$$
for all vector fields $X,Y,U$ with $X,Y$ horizontal (i.e. tangent to $\ker( \pi... | 6 | https://mathoverflow.net/users/14708 | 450046 | 181,073 |
https://mathoverflow.net/questions/431453 | 1 | Let $ G $ be a compact topological group which is quasisimple in the sense that
$$
[G,G]=G
$$
and
$$
G/Z(G)
$$
is simple as an abstract group. Must $ G $ be a Lie group?
This is a follow-up question to
<https://math.stackexchange.com/questions/4537401/compact-simple-group-which-is-not-a-lie-group>
By Peter-Weyl th... | https://mathoverflow.net/users/387190 | Is every compact quasisimple group a Lie group? | The group $G/Z(G)$ being simple, is a Lie group by Peter-Weyl. Hence it is either finite or connected. If it is finite, $G$ has center of finite index, hence has a finite derived subgroup. Since $G$ is perfect, this means that $G$ is finite.
Now suppose that $G/Z(G)$ is connected. Let $H$ be any Lie quotient of $G$ t... | 3 | https://mathoverflow.net/users/14094 | 450050 | 181,074 |
https://mathoverflow.net/questions/449975 | 5 | I do not understand the proof of Variant 4.2.3.16 of *Higher Topos Theory* by Jacob Lurie, and I need help.
---
Variant 4.2.3.16 asserts the following:
>
> ($\diamond$) Let $K$ be a finite simplicial set. There is a cofinal map $N(A)\to K$, where $A$ is a finite poset.
>
>
>
The proof proceeds as follows... | https://mathoverflow.net/users/144250 | Cofinal maps from posets (HTT, 4.2.3.16) | [Rephrasing my comment as an answer]
While I cannot speak for the actual proof, two points are worth noting. First, a similar construction appears in Kerodon (<https://kerodon.net/tag/02QA>) but there (1) $\widetilde{K} \to K$ is required to be a trivial fibration but (2) it doesn't restrict correctly to when $K$ is ... | 2 | https://mathoverflow.net/users/76636 | 450057 | 181,077 |
https://mathoverflow.net/questions/450061 | 2 | Assume that C is a stable infinity category; $SH\_{fin}$ is the homotopy category of finite spectra. Is there a canonical bi-functor (action? module structure?) $SH\_{fin}\times hC \to hC$?
Is there any "canonical" formulation of this statement; what about the properties of this bi-functor? It possibly follows from P... | https://mathoverflow.net/users/2191 | Does the homotopy category of finite spectra act on stable homotopy categories? | Yes: Since $\mathcal{C}$ is stable, $\operatorname{Fun}(\mathcal{C},\mathcal{C})$ is stable, too. In particular, it has finite colimits, so $\operatorname{Ind}\operatorname{Fun}(\mathcal{C},\mathcal{C})$ has all colimits. So we get a unique colimit-preserving functor $\mathcal{S} \to \operatorname{Ind}\operatorname{Fun... | 7 | https://mathoverflow.net/users/39747 | 450068 | 181,079 |
https://mathoverflow.net/questions/449932 | 2 | Let $A$ be a finite-dimensional $\*$-algebra over $\mathbb R$. We say that an element $x \in A$ is *positive definite* if $x$ admits an inverse and if $x = y y^\*$ for some $y \in A$. Does every such $x$ admit a $z \in A$ such that $x = z^2$? If someone has references for me to read, I would appreciate that.
I think ... | https://mathoverflow.net/users/75761 | Do positive-definite elements in finite-dimensional $*$-algebras over $\mathbb R$ always admit square roots? | I'll assume that a $\*$-algebra just means an $\mathbb{R}$-algebra with a linear operation satisfying $(xy)^\*=y^\*x^\*$ and $x^{\*\*}=x$ (as at <https://ncatlab.org/nlab/show/star-algebra>). If so, you can just take $A=\mathbb{R}\times\mathbb{R}$ with $(x\_0,x\_1)^\*=(x\_1,x\_0)$. Then take $x=(-1,-1)$ and $y=(1,-1)$ ... | 6 | https://mathoverflow.net/users/10366 | 450072 | 181,082 |
https://mathoverflow.net/questions/450075 | 0 | What are the rational solutions to the equation
$$
y^3 = x^4 + x,
$$
in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$?
Context: This is the simplest-looking example of a Picard curve, that is, curve $y^3=P(x)$, where $P(x)$ is a polynomial of degree 4. The rank of its Jacobian is ... | https://mathoverflow.net/users/89064 | $y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves | Let $x = \frac{a}{d}$, $y = \frac{b}{d}$, $\gcd(a, d, b) = 1$.
$$b^3d = a^4 + ad^3$$
Suppose $p \mid a$ and $p \mid d$ for prime $p$. Then $b$ is not divisible by $p$.
$$\nu\_p(d) = \nu\_p(a^4 + ad^3) = \nu\_p(a) + \nu\_p(a^3 + d^3) = 4\nu\_p(a)$$
The last equality is true because $\nu\_p(a) < \nu\_p(d)$. Hence, we can... | 5 | https://mathoverflow.net/users/507773 | 450081 | 181,085 |
https://mathoverflow.net/questions/449920 | 2 | If someone has gone through the Griffiths' paper ``On the periods of certain rational integrals: I,'' could you help me to understand Lemma 8.10?
I don't get why $\eta\in Z^{q,k+1}(l-1)$; although $\eta$ is surely a closed form of type $(q,0)+\cdots+(q-k-1,k+1)$ with a pole of order $l-1$, I think we have to say more t... | https://mathoverflow.net/users/507853 | A specific question on the Griffiths' paper: the reduction of the pole order | In fact, it just follows from that $\eta$ has a pole of order $l-1$ and is $\textit{closed}$: We have the identity $$f^{l-2}df\wedge\eta=\frac{1}{l-1}d(f^{l-1})\wedge\eta=\frac{1}{l-1}d(f^{l-1}\wedge\eta),$$
which is $C^{\infty}$ since $f^{l-1}\wedge\eta$ is $C^{\infty}$.
| 2 | https://mathoverflow.net/users/74322 | 450085 | 181,088 |
https://mathoverflow.net/questions/450048 | 2 | First of all: I apologise in advance for if my question will be arid, wrong written or even nonsensical.
I was at a talking with a professor last week, and the question of "Entanglement and Algebraic Geometry" came out.
What emerged really fascinated me, so I'm here to ask more clarifications, explanations or even re... | https://mathoverflow.net/users/88816 | Entanglement, quadrics and $\mathbb{P}^2(\mathbb{C}^3)$ | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$Let me just make a couple of comments on the question to clean up the confusion about the correct dimension of the projective space — it should be $\mathbb{P}^3(\mathbb{C})$. The state space of a qubit is the Hilbert space $\mathbb{C}^2$, and the $n$-qubit is ob... | 6 | https://mathoverflow.net/users/50846 | 450090 | 181,091 |
https://mathoverflow.net/questions/450095 | 4 | Let $X\_k$ be i.i.d. Cauchy random variables with parameters $0,1$. For each $N$ define the process $Y\_N$ by $$Y\_N(t)=\frac{1}N\sum\_{k=1}^{\lfloor tN\rfloor}X\_k+\text{piecewise linear interpolation}.$$
Note that for each grid point, the sum of Cauchy random variables is another Cauchy random variable. I am intere... | https://mathoverflow.net/users/479223 | Hölder continuity of process from Donsker like theorem with Cauchy random variables | Indeed this falls a bit outside of the standard theory. As mentioned in this MSE answer to [Does a random walk with infinite mean ever converge to anything?](https://math.stackexchange.com/questions/1577043/does-a-random-walk-with-infinite-mean-ever-converge-to-anything), in particular for Cauchy too in the second answ... | 5 | https://mathoverflow.net/users/99863 | 450097 | 181,092 |
https://mathoverflow.net/questions/450025 | 17 | Let $X$ and $Y$ be sets. It is undecidable in ZFC whether $2^{|X|} = 2^{|Y|}$ implies $|X| = |Y|$ (in Cohen's original model for ZFC + $\neg$CH, one has $2^{\aleph\_0} = 2^{\aleph\_1}$). What if we restrict our attention to the *finite* parts of $X$ and $Y$?
**Question.** Do $X$ and $Y$ have the same cardinality if t... | https://mathoverflow.net/users/16537 | Do $X$ and $Y$ have the same cardinality if their families of finite subsets do? | It can be proved in $\mathsf{ZF}$ that ``for all cardianls $\mathfrak{a},\mathfrak{b}$, if $\mathrm{fin}(\mathfrak{a})=\mathrm{fin}(\mathfrak{b})$, then $\mathfrak{a}=\mathfrak{b}$'' implies $\mathsf{AC}$. Here $\mathrm{fin}(\mathfrak{a})$ denotes the cardinality of the set of all finite subsets of a set which is of ca... | 16 | https://mathoverflow.net/users/101817 | 450104 | 181,095 |
https://mathoverflow.net/questions/449946 | 1 | Let $G$ be a $p$-adic reductive group and $Z\subseteq G$ the center. Let $\pi$ be an irreducible admissible representation of $G$. By Schur's lemma, it is easy to show that there is a group homomorphism $\omega\_\pi:Z\to \mathbb{C}^\times$, namely the central character. But how does one to prove that this group homomor... | https://mathoverflow.net/users/32746 | Continuity of central character | Let $K\subset G$ be a compact open subgroup such that $V^K\ne0$. Now $Z$ acts on $V^K$ by the central character $\omega\_\pi$, and the action is trivial on $Z\cap K$. Thus $\omega\_\pi$ is trivial on the compact open subgroup $Z\cap K$, hence is continuous.
| 1 | https://mathoverflow.net/users/123673 | 450105 | 181,096 |
https://mathoverflow.net/questions/449600 | 1 | Let $G$ be a $k$-partite $k$-uniform hypergraph with at least $dn^k$ many edges. I want a lower bound on the number of $K\_{2, 2,\, \ldots\,,2}$ in $G$, preferably something like $\gamma n^{2k}$ for some constant $\gamma$ independent of $n$.
My ideas: I can get down to a minimum degree subhypergraph from the edge-den... | https://mathoverflow.net/users/485879 | Counting $K_{2, 2, \,\ldots\,,2}$ in a $k$-partite $k$-uniform hypergraph | You can use the doubling method (a.k.a. Cauchy-Schwarz)
For simplicity, suppose we are working in the graph case. So, let $G$ be a graph with bipartition $X\cup Y$ with $|X|=|Y|=n$ and suppose that $e(G)=\Omega(n^2)$. Let $f(x,y)$ be the indicator function of the edges of $G$ and let
$$A=\sum\_{x\in X, y\in Y}f(x,... | 3 | https://mathoverflow.net/users/507998 | 450115 | 181,097 |
https://mathoverflow.net/questions/319554 | 8 | Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is
>
> Suppose $C \subset \mathbb{F}\_2^n $ is a code such $d(C)\ge d$. Let
> $\beta(x) = 1+ \sum\_{k=1}^{n} y\_k K\_k (x)$ be a polyno... | https://mathoverflow.net/users/94546 | How did they come up with the MRRW bound? | I don't know what they were thinking in the past, but there are more modern points of view these days.
1. We can view the Delsarte LP as the symmetrization of a very large LP (or as the symmetrization of the [Lovász theta function](https://en.wikipedia.org/wiki/Lov%C3%A1sz_number) SDP on the hypercube graph with edge... | 2 | https://mathoverflow.net/users/48204 | 450137 | 181,105 |
https://mathoverflow.net/questions/450133 | 7 | Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the natural morphisms
$$\mathrm{H}^i(\pi\_1(X,\overline{x}),\mathscr{F}\_x) \to \mathrm{H}^i\_\text{ét}(X,\mathscr{F})$$
... | https://mathoverflow.net/users/131975 | Is anything known about de Rham $K(\pi,1)$'s? | I had a derelict project (joint with Javier Fresan) trying to study this notion. We didn't prove much and we got stuck with the more interesting questions.
The isomorphism you wrote always holds for $i=1$. An easy way to see this is to compare both sides to extensions of $\mathcal{O}$ by $\mathcal{E}$.
If $X$ is pr... | 9 | https://mathoverflow.net/users/3847 | 450138 | 181,106 |
https://mathoverflow.net/questions/450111 | 2 | Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that
$f(z) = \int\_{\partial D} f(\zeta) \omega(\zeta, z)$,
where $\omega(\zeta, z)$ is the Bochner-Martinelli kernel. I am wondering if a $... | https://mathoverflow.net/users/167284 | Inverse of Bochner–Martinelli formula | The result you need is a now classical result of Aronov and Kytmanov [see for example [1] chapter 4 §15.1, p. 161, theorem 15.1]: if $D$ is a bounded domain in $\Bbb C^n$ with piecewise smooth boundary and $f\in C^1(\bar D)$ then
$$
f\in \mathscr{O}(D) \iff f(z) = \int\_{\partial D} f(\zeta) \omega(\zeta, z)
$$
Therefo... | 3 | https://mathoverflow.net/users/113756 | 450154 | 181,111 |
https://mathoverflow.net/questions/450162 | 5 | I note that Mathematica could yield the identity
$$\int\_0^1\frac{\log(1+x^2(x-1)/2)}{x^2(x-1)}dx=\frac{\pi(\pi-4)-12\log^22+24\log2}{16}.\tag{1}\label{1}$$
But I don't know how Mathematica got this.
**Question.** How to prove \eqref{1} manually?
Your comments are welcome!
| https://mathoverflow.net/users/124654 | For a manual evaluation of a definite integral | With Mathematica, one can actually find an antiderivative $F=G+H$ of the function $f$, where
$$f(x):=\frac{\ln(1+x^2(x-1) /2)}{x^2(x-1)},$$
$$G(x):=-\text{Li}\_2\left(\frac{1-x}{2}\right)-\frac{1}{2}
\text{Li}\_2\left(-(x-1)^2\right)+\text{Li}\_2(-x)+\text{Li}\_2\left(\left(\frac{1}{2}-\frac{i}{2}\right) x\right)+\tex... | 9 | https://mathoverflow.net/users/36721 | 450164 | 181,115 |
https://mathoverflow.net/questions/449709 | 20 | I recently discovered a proof of the following. Let $p$ be a prime that's $1 \bmod {3}$. Suppose that $p$ is not represented by the principal quadratic form $(1,9,81)$ of discriminant $-243$ (The first three $p$ that are represented by this form are $61$, $67$, and $73$). Then neither $3p$ nor $3p^2$ is a sum of two ra... | https://mathoverflow.net/users/6214 | Writing $3p$ when $p \equiv 1 \pmod{3}$ as a sum of two rational cubes. Is this result new? And what about its converse? | For question 1.
The condition $p$ is represented by $(1,9,81)$ or equivalently by $(1,1,61)$ is equivalent to the condition that $p\equiv 1\mod 3$ and $3 \mod p$ is not a cube (this is exercise 9.10 in Cox's book 'Primes of the forms...').
And Satge's paper 'Groupes de Selmer et corps cubiques' already proved under t... | 6 | https://mathoverflow.net/users/144225 | 450175 | 181,117 |
https://mathoverflow.net/questions/449839 | 7 | $\DeclareMathOperator\dom{dom}$Sorry to bother the community again with these type of questions about power series, I am ready to delete the question if it is not suitable.
**Definition:** I say a function $h$ is $(m,n)$-representable by power series iff $0\in \dom(h)$, $h(0)=0$, and $h$ is a restriction (to a smalle... | https://mathoverflow.net/users/32135 | Composition of power series is power series? | Maybe I misunderstood the question, is this a counterexample?
Let $c>1$, put $f(x) = \sin x$ and $g(x) = \tfrac{x}{c^2+x^2}$. Then $f$ is representable by a power series on $\mathbb{R}$, $g$ is representable by a power series on $(-c,c)$ and $f(\mathbb{R}) \subset (-c,c)$.
However, the meromorphic function $g \circ... | 9 | https://mathoverflow.net/users/297 | 450185 | 181,120 |
https://mathoverflow.net/questions/450093 | 2 | One of the problems that has come up during my research concerns $K\_4$-simple groups (simple groups with $4$ prime divisors). The only (potentially) infinite family of groups satisfying this condition is $PSL(2,q)$, and I was interested in exactly what values of $q$ satisfied this condition. After writing a quick Pyth... | https://mathoverflow.net/users/507796 | Sparsity of q in groups PSL(2,q) that are K_4-simple | Let me expand my earlier comment to a partial answer of "why sparsity?". It is impossible for $|{\rm PSL}(2,p^{m})|$ to have four or fewer different prime factors when
$p$ is an odd prime which is neither Fermat nor Mersenne and $m > 1$ is an integer. In fact, if $m >1$ is not a power of $2$, we will see that
whenever ... | 5 | https://mathoverflow.net/users/14450 | 450187 | 181,121 |
https://mathoverflow.net/questions/450182 | 6 | I put forward a hypothesis in number theory, it is as follows.$ \sigma\_1(n)=\sigma\_1(m)=p$, where $\sigma\_1$ is the divisor sum function, $n,m\in \mathbb N$, and $p$ is prime. I recently noticed and assumed that the pattern $ \sigma\_1(16)=\sigma\_1(25)=31$ is the only one. And what do you think about this?
| https://mathoverflow.net/users/508058 | Are there infinite numbers of the form $\sigma_1(n)=\sigma_1(m)=p$, or is there only one? | As mentioned in the comments, it suffices to assume that $m,n$ are prime powers. Then we are looking at an equation of the form
$$\displaystyle u^a + \cdots + u + 1 = v^b + \cdots + v + 1.$$
The expressions on either side are cyclotomic polynomials. If either one is reducible (over $\mathbb{Z}$), then it is exceedi... | 5 | https://mathoverflow.net/users/10898 | 450188 | 181,122 |
https://mathoverflow.net/questions/450189 | 1 | This is soft question. I decided to ask my question on MathOverflow rather than on academia StackExchange because I believe that the community here is more equipped to answer the question.
The [Graduate Journal of Mathematics](https://gradmath.org/) mentions in its mission-statement that it takes inspiration from a s... | https://mathoverflow.net/users/493164 | What causes some mathematical journals to discontinue? | At <https://futurelms.wordpress.com/2016/01/14/save-the-lms-journal-of-computation-and-mathematics/> you will find discussion of the decision to close the LMS Journal of Computation and Mathematics. My knowledge is based solely on a brief scan of the documents linked from that page. It seems that some people felt that ... | 5 | https://mathoverflow.net/users/10366 | 450194 | 181,125 |
https://mathoverflow.net/questions/450200 | 0 | Let $F$ be the set of all integers $n>1$ such that in the [Fibonacci sequence](https://en.wikipedia.org/wiki/Fibonacci_sequence) modulo $n$, the value $0$ occurs infinitely often. What is the value of $\lim\sup\_{n\to\infty}\frac{|F\cap\{0,\ldots,n\}|}{n+1}$?
| https://mathoverflow.net/users/8628 | Density of "Fibonacci friends" | The value $0$ always occurs infinitely often regardless of the value $n>1$. To see this, consider the mapping $L\_n:\mathbb{Z}\_n^2\rightarrow\mathbb{Z}\_n^2$ defined by $L\_n(x,y)=(y,x+y)$. Then the mapping $L\_n$ is invertible. Then if we set $F\_{0,n}=[0]\_n,F\_{1,n}=[1]\_n$ and $F\_{i+2,n}=F\_{i+1,n}+F\_{i,n}$, the... | 10 | https://mathoverflow.net/users/22277 | 450201 | 181,128 |
https://mathoverflow.net/questions/450030 | 5 | Denote by $D(r)$ the disc at the origin of radius $r>1$. Denote by $P\_m$ the set of polynomials of degree $m$. Since $P\_m$ is finite dimensional, there is a constant $C(m,r)$ such that
$$
\|p\|\_{L^{\infty}(D(r))}
\leq
C(m,r)
\|p\|\_{L^1(-1,1)}
$$
for all $p \in P\_m$.
I'd like some estimates for $C(m,r)$, especial... | https://mathoverflow.net/users/73890 | The constant of the reverse Hölder inequality for polynomials | I presume that $D(r)$ refers to a disk in the complex plane, in which case the maximum value of a polynomial (like any analytic function) is achieved on the boundary of $D(r)$. If $\|p\|\_{L^\infty(D(r))} \le C \|p\|\_{L^1(-1,1)}$, it means that the ball $\|p\|\_{L^1(-1,1)} \le 1/C$ fits entirely into the ball $\|p\|\_... | 4 | https://mathoverflow.net/users/2622 | 450211 | 181,131 |
https://mathoverflow.net/questions/450228 | 1 | Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known about this quantity... If someone could provide me a reference it would be appreciated. Thank you
ps I assume $\deg f >... | https://mathoverflow.net/users/84272 | Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$ | The assumption that $f(X)$ and $g(X)$ are relatively prime means that there is
a positive integer $R\_{f,g}=\operatorname{Resultant}(f,g)$ and polynomials $a(X)$ and $b(X)$ in $\mathbb Z[X]$ so that
$$ a(X)f(X) + b(X)g(X) = R\_{f,g}. $$
Hence if $x\in\mathbb Z$ satisfies $f(x)\mid g(x)$, then $f(x)\mid R$. Since $\bigl... | 6 | https://mathoverflow.net/users/11926 | 450229 | 181,136 |
https://mathoverflow.net/questions/450219 | 2 | Let $f\geq 0$ be a Lipschitz function and let $(L\_t)\_{t\geq 0}$ be an $\alpha$-stable Lévy process ($0<\alpha<2$, possibly multivariate). Consider the process given by $$dX\_t=-\nabla f(X\_t)dt+\sigma dL\_t$$ where $\sigma>0$ is a constant. This SDE is reminiscient of Langevin Dynamics, where we usually let the proce... | https://mathoverflow.net/users/498406 | Stationary Distribution of Langevin Dynamics driven by Lévy Process | There are several generalizations of the Brownian case results to general L'evy processes. For instance, under a classical log-concave assumption on $U$ we have a convergence to equilibrium in Wassertein distance to a unique stationary distribution. More precisely, consider the following SDE
$$
dX^x\_t=-\nabla U(X^x\... | 3 | https://mathoverflow.net/users/48356 | 450232 | 181,139 |
https://mathoverflow.net/questions/450222 | 6 | [A previous question on the categorical nature of ultraproducts](https://mathoverflow.net/questions/11261/is-the-ultraproduct-concept-fundamentally-category-theoretic) had great answers, mostly categorically characterizing ultraproducts in the category of $L$-structures and *homomorphisms* for a fixed signature $L$. Th... | https://mathoverflow.net/users/310424 | Ultraproducts in the category of structures and elementary embeddings | Since you asked about ultraproducts, and not ultrapowers, let me argue that the answer must be negative. The reason is that the category of $L$-structures under elementary embeddings is partitioned into disconnected components by the theories of those structures, since elementary embeddings must preserve the theory of ... | 8 | https://mathoverflow.net/users/1946 | 450237 | 181,141 |
https://mathoverflow.net/questions/450225 | 1 | Last October, I learned from [Benjamin Steinberg's answer](https://mathoverflow.net/questions/432657/structure-theorem-for-a-class-of-idempotent-monoids-where-xy-x-or-xy-y/432691#432691) to another question of mine that a semigroup $S$ is called *breakable* if $xy \in \{x, y\}$ for all $x, y \in S$. Let's now say that ... | https://mathoverflow.net/users/16537 | Cycles in almost breakable semigroups | Here is a self-contained version of the argument that there are no cycles, avoiding using the structure of bands. Suppose that $S$ is almost breakable.
Claim 1. If $SxS=SyS$, then both $xy,yx\in \{x,y\}$ and $xyx=x$, $yxy=y$.
Pf. Without loss of generality, assume that $yx=y$ (the other cases follow from renaming o... | 1 | https://mathoverflow.net/users/15934 | 450244 | 181,142 |
https://mathoverflow.net/questions/450243 | 2 | Let G be an undirected, simple graph containing distinct vertices x and y. Let P,Q,R be three distinct paths in G from x to y. We can assume the graph G is only those paths (any vertex in G is in one of P,Q, and R and same with any edge in G).
Assume there is no vertex (other than x and y) such that P,Q, and R contai... | https://mathoverflow.net/users/114995 | If you have three paths from vertex x to vertex y, when are you guaranteed a cycle which contains both x and y? | Yes, by Menger theorem. There exists either a vertex $z\notin \{x, y\}$ after removing which there is no path from $x$ to $y$ remained, or two disjoint paths from $x$ to $y$.
| 1 | https://mathoverflow.net/users/4312 | 450250 | 181,144 |
https://mathoverflow.net/questions/450241 | 13 | This came up in the comments to [an answer of Joel's](https://mathoverflow.net/questions/450222/ultraproducts-in-the-category-of-structures-and-elementary-embeddings). Suppose $\mathcal{M}\_i$ ($i\in I$) are elementarily equivalent structures in the same fixed signature and $\mathcal{U}$ is an ultrafilter on $I$. Must ... | https://mathoverflow.net/users/8133 | Can ultraproducts avoid all "factor structures"? | Here is another example, which is inspired by James's answer, but I find this one a little simpler.
Let $T$ be the theory of an equivalence relation $\sim$ with infinitely many classes, all infinite, plus countably many constants $c\_0,c\_1,c\_2,\ldots$, taken from different equivalence classes. This is a complete th... | 19 | https://mathoverflow.net/users/1946 | 450257 | 181,146 |
https://mathoverflow.net/questions/450163 | 3 | Kolmogorov tightness criterion says that if $X\_N$ is a sequence of continuous process with $X\_N(0)=0$ and $E[[X\_N(t)-X\_N(s)|^p]\leq C\_p |t-s|^{1+\beta}$ then for all $\gamma\in (0,\beta/p)$ we have that the laws of $X\_N$ are tight on the Holder space $C^\gamma$.
There are some easy examples where the assumption... | https://mathoverflow.net/users/479223 | Version of Kolmogorov tightness criterion without moments | Unfortunately, this does not hold, in the sense that you cannot conclude tightness even for any particular $\gamma’ < \gamma$. This very simple modification of my example [here](https://mathoverflow.net/questions/449705/garsia-rodemich-rumsey-without-markov) is a counterexample (I’ve just replaced the $\gamma$ in the e... | 3 | https://mathoverflow.net/users/173490 | 450260 | 181,148 |
https://mathoverflow.net/questions/450254 | 1 | Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\begin{align}
\begin{bmatrix}
x\_1 & x\_2 & \cdots & x\_{n-1} & x\_n \\
-x\_n & x\_1 & \cdots & x\_{n-2} & x\_{n-1... | https://mathoverflow.net/users/369335 | One question about nega-cyclic Hadamard matrices | Such matrices do not exist as from the parity consideration already first two rows cannot be orthogonal.
| 2 | https://mathoverflow.net/users/7076 | 450274 | 181,152 |
https://mathoverflow.net/questions/450273 | 7 | It is known that the only elementary abelian $2$-groups (finite and nonfinite) in $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are in fact finite and cyclic – that is to say, they are of order $2$.
(Here, $\overline{\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$ inside $\mathbb{C}$.)
Question: Does one n... | https://mathoverflow.net/users/12884 | Involutions in the absolute Galois group (and the Axiom of Choice) | No, you shouldn't need any choice for this, and it should still be true if you replace $\overline{\mathbb{Q}}$ with any other algebraic closure of $\mathbb{Q}$. Let $K$ be a field (which in our application will be $\overline{\mathbb{Q}}$) and let $G$ be a finite group of automorphisms of $K$. Then $K/K^G$ is always a d... | 11 | https://mathoverflow.net/users/297 | 450279 | 181,154 |
https://mathoverflow.net/questions/450280 | 4 | Let $0 < \varepsilon < 1$. A natural number $x$ is called an $\varepsilon$-square if $x = ab$, $a, b \in \mathbb{N}$ and $(1 - \varepsilon)b \le a \le b$. Denote by $f(N)$ the number of $\varepsilon$-squares on the interval $[1, N]$. Is it true that $\lim\_{N\to\infty} \frac{f(N)}{N} = 0$?
There is at least $cN$, for... | https://mathoverflow.net/users/507773 | Density of a set of natural numbers which are the product of close numbers | If $x\leq N$ is an $\varepsilon$-square, then $x=ab$ and $a\leq b\leq \sqrt{x/(1-\varepsilon)}=:K$. So, $x$ should appear in $K\times K$ multiplication table. It is known, however, that most numbers below $K^2$ are not in this multiplication table, see [Erdős–Tenenbaum–Ford constant](https://en.wikipedia.org/wiki/Erd%C... | 6 | https://mathoverflow.net/users/101078 | 450283 | 181,156 |
https://mathoverflow.net/questions/450233 | 0 | I came across this partial sum which I cannot find reasonable bounds on; I feel this must be known in the literature, but I do not know where to look. Here is the problem:
Let $s\in (0,1)$ and consider the binomial sum
$$
\boldsymbol{(\*)}\qquad \sum\_{k=1}^N\, \binom{N}{k}\, \frac1{k^s}
$$
are there any known good u... | https://mathoverflow.net/users/491352 | Two-Sided Bounds on Binomial Sum | $\newcommand{\Si}{\Sigma}$We have to lower- and upper-bound
\begin{equation\*}
\Si\_N:=\sum\_{k=1}^N\,\binom Nk\, \frac1{k^s}.
\end{equation\*}
Note that
\begin{equation\*}
\Si\_N=2^N EX^{-s}\,1(X\ge1), \tag{10}\label{10}
\end{equation\*}
where $X$ is a random variable with the binomial distribution with parameter... | 2 | https://mathoverflow.net/users/36721 | 450284 | 181,157 |
https://mathoverflow.net/questions/450272 | 2 | I have this integral that comes from my research with some Fourier Transforms of spectrum functions:
$$ G(\tau) = \int\_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c\_1 - c\_2 e^{-c\_3 x} ) } dx $$
where $c\_1, c\_2, c\_3, \Lambda, n > 0$.
The only way for me now is to use a series expansion of the term $e^{i \ta... | https://mathoverflow.net/users/489481 | Is this integral solvable analytically? | there is a closed form solution for
$$I\_n = \int\_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c\_1 - c\_2 e^{-c\_3 x} ) } dx$$
for integer $n$, for example, for $n=0$:
$$I\_0=\frac{1}{\lambda (c\_3+\lambda)}e^{i c\_1 \tau}$$
$$\qquad\times \left[(c\_3+\lambda) \, \_1F\_2\left(\frac{\lambda}{2 c\_3};\tfrac{1}{2},\tfr... | 3 | https://mathoverflow.net/users/11260 | 450286 | 181,159 |
https://mathoverflow.net/questions/450218 | 1 | Consider the spatially homogenous Boltzmann equation $$\partial\_t f\_t = Q^+(f\_t,f\_t) - f\_t.$$ A semi-explicit representation formula for solutions of this Boltzmann equation can be written as (see for instance [Villani's monograph](https://cedricvillani.org/sites/dev/files/old_images/2012/07/B01.Handbook.pdf))
$$
... | https://mathoverflow.net/users/163454 | Wild's sum for Boltzmann's equation | Completely ignoring all convergence issues, this really is just following your nose.
Plugging in the representation you give, you want to check
$$ \sum\_{n = 1}^\infty (1 - e^{-t})^{n-1} Q\_n^+(f\_0) \overset{?}{=} f\_0 + \int\_0^t e^{s} Q^+(f\_s,f\_s) ~ds $$
Using the representation you give again to replace $f\... | 2 | https://mathoverflow.net/users/3948 | 450289 | 181,161 |
https://mathoverflow.net/questions/450086 | 8 | Suppose $M\_1^\#$ exists and is $\omega\_1$-iterable.
Is it consistent that we can go to a generic extension $V[G]$ where $M\_1^\#$ is no longer $\omega\_1$-iterable?
Or "worse" $M\_1^\#$ is no longer 2-iterable in Neeman's sense?
I suspect the answer is yes and that it will be relatively obvious. So more general... | https://mathoverflow.net/users/9324 | Destroying the iterability of $M_1^\#$ | Andreas Lietz has already pointed out that $\omega\_1$-iterability can fail in a generic extension. One can also consistently get $(\omega+1)$-iterability to fail:
Suppose there is a transitive model of ZFC + "$M\_1^\#$ exists and is $(0,\omega\_1)$-iterable", and let $M$ be such with minimal ordinal height. It is ea... | 5 | https://mathoverflow.net/users/160347 | 450299 | 181,165 |
https://mathoverflow.net/questions/450294 | 5 | Let $s\_{\lambda}(x\_1,\dots,x\_k)$ be the [Schur polynomial](https://en.wikipedia.org/wiki/Schur_polynomial) associated to the partition $\lambda=(\lambda\_1\geq\lambda\_2\geq\cdots\geq\lambda\_k>0)$.
Among the many things involved with these polynomial, I was exploring the number of (distinct) monomials appears in ... | https://mathoverflow.net/users/66131 | Enumerating monomials in Schur polynomials | A monomial $x\_1^{a\_1}\cdots x\_{n}^{a\_n}$ will appear in the expansion of the Schur polynomial $s\_{\lambda}(x)$ if and only if $(a\_1,a\_2,...,a\_n)\le (\lambda\_1,\lambda\_2\dots, \lambda\_n)$ in the dominance (majorization) order. This is equivalent to saying that $(a\_1,a\_2,...,a\_n)$ is a lattice point in the ... | 5 | https://mathoverflow.net/users/2384 | 450312 | 181,169 |
https://mathoverflow.net/questions/450308 | 0 | [Nancy Cartwright](https://en.wikipedia.org/wiki/Nancy_Cartwright_(philosopher)) introduced an interesting distinction with regard to modeling of physical phenomena. According to Cartwright, a mathematical theory is not applied directly to such phenomena. Rather, one first builds a basic mathematical model of the pheno... | https://mathoverflow.net/users/28128 | Nancy Cartwright's dichotomy | Cartwright's case study, model building for the theory of superconductivity, has been explored further in the Ph.D.thesis [The Role of Concrete Models in the Revolution in Superconductivity](https://uwspace.uwaterloo.ca/bitstream/handle/10012/9818/Chattoraj__Ananya.pdf;sequence=1) (A. Chattoraj, 2015).
More generally... | 3 | https://mathoverflow.net/users/11260 | 450313 | 181,170 |
https://mathoverflow.net/questions/450300 | 2 | This is probably a simple question, maybe more suited for MSE. In the coarea formula, you have
$$\int\_{{\mathbb{R}}^n} g (x) |\nabla f(x)|\, dx= \int\_\mathbb{R} \left(\int\_{\{f=t\}} g d \mathcal{H}^{n-1} \right)dt ,$$
with $f$ Lipshitz and $g$ Borel (positive maybe?). Anyway the question is: how do you define th... | https://mathoverflow.net/users/109382 | Definition of integral over level sets in coarea formula | Check out section 3.4 in
*Evans, Lawrence Craig; Gariepy, Ronald F.*, Measure theory and fine properties of functions, Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4822-4238-6/hbk). 309 p. (2015). [ZBL1310.28001](https://zbmath.org/?q=an:1310.28001).
I won't reproduce all the details, but a key s... | 6 | https://mathoverflow.net/users/3948 | 450314 | 181,171 |
https://mathoverflow.net/questions/449311 | 1 | Let $M$ be a closed complex Kähler manifold, $dim\_{\mathbb C} M = n\geq 2$, with a Kähler form $\omega$. Assume $U\subset M$ is a Stein domain with a smooth boundary and $f: U\to [0;1]$ is a smooth exhausting function for $U$ such that $f^{-1} (1) = \partial U$ and $L:=f^{-1} (0)$ is a smooth totally real closed subma... | https://mathoverflow.net/users/102829 | Restrictions of strictly $\omega$-plurisubharmonic functions to a Stein domain in a closed Kahler manifold | An answer courtesy of Vincent Guedj (for the terminology see the book "Degenerate Complex Monge-Ampère Equations" by V.Guedj and A.Zeriahi):
No, $C$ cannot be made arbitrarily large - there is an upper bound on it depending only on $L$.
The proof goes as follows.
Since $L$ is totally real, it is not locally pluri... | 0 | https://mathoverflow.net/users/102829 | 450319 | 181,172 |
https://mathoverflow.net/questions/450253 | 4 | Assume that we have heavy-tailed distribution $F(x)$ such that
\begin{align}
F(x)=\mathbb{P}[X\geq x]=x^{-0.5}.
\end{align}
Then, we produce $N$ independent samples $X\_1,X\_2,\ldots,X\_N$ from this distribution. Assuming that $n(N)\leq N$ is a function of $N$, we pick the $n(N)$ largest amounts among $X\_1,X\_2,\ldots... | https://mathoverflow.net/users/68835 | Does a subset with small cardinality represent the whole set? | The probability that all samples are less than $N^{19/10}$ is $(1-N^{-19/20})^{N}$ that tends to 0. The expected number of samples greater than $N^{1/2}$ is $N^{3/4}$, thus, the probability that we have more than $N^{4/5}$ such samples is by Chebyshev inequality at most $N^{-1/20}$,also tends to 0. Therefore, with prob... | 4 | https://mathoverflow.net/users/4312 | 450332 | 181,176 |
https://mathoverflow.net/questions/450317 | 4 | I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a profinite group becomes important.
Suppose we have two profinite groups $G\_1$ and $G\_2$, with proper, dense subgroups $\Ga... | https://mathoverflow.net/users/508126 | Profinite groups with isomorphic proper, dense subgroups are isomorphic | Let $G$ be a compact group and $H$ a dense subgroup. I claim that $H$, as topological group, determines $G$. For simplicity, let me assume that $G$ is metrizable.
Note that a sequence $(h\_n)$ in $H$ converges in $G$ if and if $h\_n^{-1}h\_m\to 1$ when $n,m\to\infty$, and two such sequences $(h\_n)$, $(h'\_n)$ have t... | 4 | https://mathoverflow.net/users/14094 | 450335 | 181,178 |
https://mathoverflow.net/questions/450357 | 1 | Let $P$ and $Q$ be two distributions over a sample space $\Omega$ which I would like to show are close under some choice of distance function. So far I have managed to show that there exists a subset $S\subseteq \Omega$ such that:
* $P(S)$ is large, say, at least $(1-\varepsilon)$
* the conditional distributions of $... | https://mathoverflow.net/users/508178 | What is this distributional closeness? | $\newcommand\Om\Omega$No. E.g., for natural $n$, suppose that $\Om=[n]:=\{1,\dots,n\}$, $S=[n-1]$, $P(x)=\frac1n$ for $x\in\Om$, $Q(x)=\frac1{n^2}$ for $x\in S$, and $Q(n)=1-\frac{n-1}{n^2}$.
Then your conditions hold for $\varepsilon=\frac1n$, $P$ is uniform over $\Om$, but (for large $n$) almost all $Q$-mass is at ... | 1 | https://mathoverflow.net/users/36721 | 450358 | 181,182 |
https://mathoverflow.net/questions/450094 | 3 | I am currently going through Shimura's paper on half-integer weight modular forms. I would like to understand given a -expansion of half-integral weight modular forms of arbitrary level and character, how to compute the effect of the Hecke operator and the Atkin-Lehner operator/Fricke involution in SAGE/Magma. I am a b... | https://mathoverflow.net/users/86441 | Computations of half-integer forms in SAGE/Magma | This can be done using PARI/GP, which can deal with spaces of modular forms of half-integral weight. Given a modular form $f$ of weight $k$ (possibly half-integral), the command *mfslashexpansion* can compute the $q$-expansion of $f |\_k g$ for any $g \in \mathrm{GL}\_2^+(\mathbf{Q})$. This relies on a floating-point m... | 7 | https://mathoverflow.net/users/6506 | 450364 | 181,183 |
https://mathoverflow.net/questions/450349 | 1 | *This question (and a second part) have been [asked at MSE](https://math.stackexchange.com/questions/4718733/an-infinitely-conditioned-state-in-a-c-algebra) and gone through two bounties without an answer. I have been beating my head at it for a while without success*.
Let $\mathcal{A}$ be a unital $\mathrm{C}^\*$-al... | https://mathoverflow.net/users/35482 | Conditioning a $\mathrm{C}^*$-algebra state with infinite precision | No, this limit doesn't exist in general. Here's a commutative counterexample.
Let $A = L^\infty[0,1]$, $f(x) = x$, $\phi =$ integration against Lebesgue measure. Let $g$ be the indicator function of the set $\bigcup [10^{-(2n + 1)}, 10^{-2n}] = [.1, 1] \cup [.001, .01] \cup [.00001, .001] \cup \cdots$.
Taking $\lam... | 3 | https://mathoverflow.net/users/23141 | 450365 | 181,184 |
https://mathoverflow.net/questions/450321 | 1 | Let $X=\text{Sp}(A)$ be an affinoid $K$ space, where $K$ is a $p$-adic field. If $f\_0, f\_1,..., f\_s \in A$ generate the unit ideal then we can define the rational subdomain $U= X(f\_0, f\_1..., f\_s) = \{ x \in X: \vert f\_i(x) \vert \leq \vert f\_0(x) \vert \text{ for } i=1...s \}$ of $X$ with coordinate ring $\mat... | https://mathoverflow.net/users/498675 | Injectivity of sheaf restriction maps for wide open neighbourhoods of rational subdomains | If your space has several connected components, then a rational domain may well isolate one of them and you could get a non-injective map.
For an explicit example, you can choose $A = \mathbb{Q}\_p \langle pT \rangle/(T(pT-1))$. Its spectrum has exactly two points: $0$ and $1/p$. To find a example of a non-injective ... | 0 | https://mathoverflow.net/users/4069 | 450373 | 181,187 |
https://mathoverflow.net/questions/450331 | 3 | If I have a braided tensor category that's unitary and modular, then how does the unitarity and modularity constrain the fusion multiplicities?
I know that if $a,b,c \in ob({C})$ satisfy the fusion rule $a \otimes b = \oplus\_c N^c\_{ab} c $ then $N^c\_{ab} \in \mathbb{Z}\_+$ for all possible non-trivial fusion chann... | https://mathoverflow.net/users/146495 | Does unitarity and modularity constrain fusion multiplicities to be 0,1? | This is false for $D(G)$, when $G$ is sufficiently complicated. For a finite group $G$, the representation category of $D(G)$ has irreducible objects parametrized by pairs $(g, V)$ where $g$ is a conjugacy class representative in $G$ and $V$ is an irreducible representation of the centralizer of $g$. The monoidal struc... | 7 | https://mathoverflow.net/users/121 | 450379 | 181,190 |
https://mathoverflow.net/questions/450239 | 2 | Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ admits a Dualizing complex.
Let $Y$ be a Gorenstein normal scheme over Spec$(k)$ of finite Krull-dimension.
If there e... | https://mathoverflow.net/users/386496 | Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay? | I started writing this last night, but didn't finish. In addition to [Jason Starr's answer](https://mathoverflow.net/a/450348/33088), you can use my new vanishing theorems to remove the assumption that $R$ is essentially of finite type over a field. Note that the special case when $R$ is essentially of finite type over... | 3 | https://mathoverflow.net/users/33088 | 450383 | 181,191 |
https://mathoverflow.net/questions/450337 | 0 | Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n / S\_n$ is a complex manifold. My question is: is the universal covering of $C^{(n)}$ provided by $\mathcal{H}^{(n)}$? Or ... | https://mathoverflow.net/users/505150 | Universal covering of symmetric product | In fact, the universal cover of $C^{(n)}$ will not be $\mathcal H^n$ once $n \gg 0$. Indeed, if $C$ is a compact Riemann surface of genus $g \geq 2$ (so the universal cover is $\mathcal H$) with a base point $x$, then the Riemann–Roch theorem implies that the map
\begin{align\*}
C^{(n)} &\to \operatorname{Jac}\_C = \op... | 6 | https://mathoverflow.net/users/82179 | 450393 | 181,194 |
https://mathoverflow.net/questions/450395 | 1 | Let $p:I\to Cat\_{\infty}$ be a diagram of infinity categories, where $I$ is a small Kan complex. Let $C:=\lim p$ be the limit of $p$. For any two objects $x,y\in C$ and $i\in I$, let $x\_i,y\_i\in C\_i=p(i)$ be the corresponding objects in $C\_i$. Then we have a new diagram $q\_{x,y}: I\to \mathcal{S}$ given by $q\_{x... | https://mathoverflow.net/users/153842 | Limits of infinity categories and mapping spaces | Yes. To see this, let us make the preliminary observation that it suffices to prove that this holds for products and pullbacks since we can decompose a general limit into these two special cases.
Let us begin with products. If $I$ is discrete then we want to show that $\prod\_i \hom\_{\mathcal{C\_i}}(x\_i,y\_i) \cong... | 3 | https://mathoverflow.net/users/76636 | 450398 | 181,195 |
https://mathoverflow.net/questions/450368 | 8 | A special case of a theorem of Brian Scott (from [*On the existence of totally inhomogeneous spaces*](https://www.ams.org/journals/proc/1975-051-02/S0002-9939-1975-0375262-5/S0002-9939-1975-0375262-5.pdf)) is that there is a size-continuum set $S\subset\mathbb{R}$ such that if $x,y\in S$ are distinct then $S\setminus\{... | https://mathoverflow.net/users/8133 | Can totally inhomogeneous sets of reals coexist with determinacy? | In *Rigid Borel sets and better quasiorder theory* (Logic and combinatorics, Proc. AMS-IMS-SIAM Conf., Arcata/Calif. 1985, Contemp. Math. 65, 199-222 (1987), [zbMath review here](https://zbmath.org/0646.03045)) Fons van Engelen, Arnold Miller, and John Steel showed that the only rigid Borel sets are the singletons; the... | 8 | https://mathoverflow.net/users/5903 | 450402 | 181,198 |
https://mathoverflow.net/questions/450412 | 0 | Let $p \in \mathbb{Z}$ be prime and $K / \mathbb{Q}$ be a finite Galois extension. The Galois group $G$ of $K$ acts on the primes of $\mathcal{O}\_K$ over $p$. Do we know any statistical information about the distribution of isomorphism classes of these actions as $p$ ranges over all unramified primes? By this I mean t... | https://mathoverflow.net/users/91041 | Statistics of action of Galois group of number field on primes over unramified rational primes | This is an elaboration of Chris Wuthrich's comment. Let $p$ be unramified (i.e. $p$ does not divide the discriminant of the Galois extension $K / \mathbb{Q}$), and let $\mathfrak{P}$ be a prime in $\mathcal{O}\_K$ over $p$ with decomposition group $D\_\mathfrak{P}\leq G$. The primes in $\mathcal{O}\_K$ over $p$ corresp... | 4 | https://mathoverflow.net/users/11919 | 450414 | 181,200 |
https://mathoverflow.net/questions/450417 | 9 | Let $F\_n$ be the free group on letters $\{x\_1,\ldots,x\_n\}$ and let $X\_n$ be the (reduced) outer space of rank $n$. Points of $X\_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected metric graph of total edge-length $1$ with no valence $1$ or $2$ vertices and no separating edges and $\mu\colon F\... | https://mathoverflow.net/users/508240 | Morse theory on outer space via the lengths of finitely many conjugacy classes | You don't misunderstand, it's a subtle point that I'm sure I'll get wrong here too. You might find the proof of a slightly more general statement in Krstić and Vogtmann's "Equivariant Outer Space and automorphisms of free-by-finite groups" illuminating: basically the idea is that for any rose (I believe in fact for any... | 13 | https://mathoverflow.net/users/135175 | 450421 | 181,202 |
https://mathoverflow.net/questions/450297 | 4 | My question comes from a computation in the paper [Central limit theorem for Maxwellian molecules and truncation of Wild expansion](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=827b55efbcc4a2eaf1c12526590ab0d067f0f650). Specially, consider the following Boltzmann equation
$$\frac{\partial f}{\partial ... | https://mathoverflow.net/users/163454 | Iterated Duhamel's formula for solutions of Boltzmann equation | I took a closer look at the manuscript. If one lets $f\_{[n]}$ denote the quantity implicitly defined by (1.15), then it appears to me that this is indeed slightly different from $f\_{(n)}$ in that some terms in the expansion of $f\_{(n)}$ are missing in $f\_{[n]}$, leading to the inequalities
$$ f\_{[n]} \leq f\_{(n)}... | 5 | https://mathoverflow.net/users/766 | 450422 | 181,203 |
https://mathoverflow.net/questions/450441 | 2 | Let $M$ be a von Neumann algebra, $P(M)$ be its projection lattice, and $\mathcal{F}$ a proper filter on $P(M)$. Does there exist a state $\varphi$ (not necessarily normal) s.t. $\varphi(p) = 1$ for all $p \in \mathcal{F}$? Does this hold even more generally, say for the projection lattices of $AW^\*$-algebras?
| https://mathoverflow.net/users/504602 | Defining states on von Neumann algebras from filters on the projection lattices | Filters are directed downward. Given a filter $F$, for every $p\in F$ let $\phi\_p$ be a state that takes the value $1$ on $p$, then find a cluster point of the net $(\phi\_p)\_{p\in F}$. This will be a state that takes the value $1$ on everything in $F$. I think this works fine for $AW{}^\*$-algebras.
| 4 | https://mathoverflow.net/users/23141 | 450444 | 181,206 |
https://mathoverflow.net/questions/450428 | 4 | Trying to find and answer to this question, I have encountered two more-studied problems.
The first is to find when a Banach space admits an **equivalent** uniformly convex norm. [The answer](https://mathoverflow.net/a/30458/58082) is that for example separable spaces always do, but nonseparable spaces might not. Thi... | https://mathoverflow.net/users/58082 | Does every Banach space admit a continuous (not necessarily equivalent) strictly convex norm? | If $\|\cdot\|\_1$ is a continuous strictly convex norm on $(X,\|\cdot\|\_0)$, then $\|x\|\_2=\|x\|\_0 + \|x\|\_1$ defines an strictly convex norm on $X$ equivalent to $\|\cdot\|\_0$. Therefore, a space like $\ell\_\infty/c\_0$ that does not admit an equivalent strictly convex norm, it does not admit a continuous strict... | 13 | https://mathoverflow.net/users/39421 | 450445 | 181,207 |
https://mathoverflow.net/questions/434602 | 9 | I'm looking for a reference that covers things like the lemma below - it doesn't have to be the exact statement I'm going to give, anything in the general ballpark would probably be useful. Or if you know a very short proof of the lemma - that would be interesting too.
So, I have a map $\pi: E \to U$ in a category $C... | https://mathoverflow.net/users/22131 | Reference request: a lemma on universes and polynomial monads | This is mentioned in Remark 13 in
>
> Steve Awodey: *Natural models of homotopy type theory*, January 2017, [arXiv:1406.3219](https://arxiv.org/abs/1406.3219)
>
>
>
and then followed up in detail in:
>
> Steve Awodey, Clive Newstead: *Polynomial pseudomonads and dependent type theory*, February 2018, [arXi... | 9 | https://mathoverflow.net/users/123877 | 450446 | 181,208 |
https://mathoverflow.net/questions/450410 | 4 | Let $C^{j\_3 m\_3}\_{j\_1 m\_1 j\_2 m\_2}$ be the standard Clebsch–Gordan coefficients of $\operatorname{SU}(2)$. They obey the orthogonality relation
$$ \sum\_{j\_3} \sum\_{m\_3} \left(C^{j\_3 m\_3}\_{j\_1 m\_1 j\_2 (m\_3 - m\_1)} \right)^2 = 1.$$
My question is about what can be said if I remove the sum over $j\_3$. ... | https://mathoverflow.net/users/104213 | Single sum of squares of Clebsch–Gordan coefficients | Consider the sum of [Clebsch–Gordan coefficients](https://en.wikipedia.org/wiki/Clebsch%E2%80%93Gordan_coefficients),$^\ast$
$$J= \sum\_{m\_3=-j\_3}^{j\_3} \left(C^{j\_3,m\_3}\_{j\_1, m\_1; j\_2, (m\_3 - m\_1)} \right)^2$$
with $2j\_1,2j\_2,2j\_3\in\mathbb{N}$ and $2m\_1\in\mathbb{Z}$. For an nonvanishing sum we also n... | 7 | https://mathoverflow.net/users/11260 | 450447 | 181,209 |
https://mathoverflow.net/questions/450360 | 3 | Based on a method that apparently seems to be widely used in computational chemistry (cf <https://en.wikipedia.org/wiki/Anisotropic_Network_Model>)
Trying to build a very simple model with 3 atoms linked together in an equilateral triangle, Suppose the 2d coordinates of the 3 atoms are as follows:
$(x\_0, y\_0) = (... | https://mathoverflow.net/users/22279 | Elastic network model Hessian rigid body motion 0 eigenvalues | Your construction of the stiffness matrix (Hessian or Kirchhoff matrix) is not correct. The $2\times 2$ diagonal blocks should not be zero,$^\ast$ these are minus the sums of the other blocks in the same row, $H\_{ii}=-\sum\_{j\neq i}H\_{ij}$, where $H\_{ij}=\begin{pmatrix}
(x\_j-x\_i)^2&(x\_j-x\_i)(y\_j-y\_i)\\
(y\_j-... | 3 | https://mathoverflow.net/users/11260 | 450453 | 181,212 |
https://mathoverflow.net/questions/450220 | 9 | Let $x>0$ and consider the integral
$$I(x):=\int\_0^\infty \frac{e^{i r}}{r^{\frac{1}{2}}} \int\_0^\infty \frac{e^{-s}}{s^{\frac{1}{2}}} \frac{r}{sx+\sqrt{sxr}+r} \, ds \, dr.$$
I am trying to determine the asymptotic behavior of $I(x)$ as $x\rightarrow+\infty$.
Note that $\lim\_{x\rightarrow+\infty}I(x)=0$.
Here i... | https://mathoverflow.net/users/116555 | Asymptotic behavior of a certain oscillatory integral | We can evaluate $I(x)$ explicitly, and then asymptotically.
Indeed, using the substitution $s=ru/x$, we get
\begin{equation\*}
I(x)=\frac1{\sqrt x}\lim\_{R\to\infty}J\_R(x), \tag{1}\label{1}
\end{equation\*}
where
\begin{equation\*}
\begin{aligned}
J\_R(x)&:=\int\_0^R dr\,e^{ir}\int\_0^\infty \frac{du}{\sqrt u\,(u+... | 8 | https://mathoverflow.net/users/36721 | 450458 | 181,213 |
https://mathoverflow.net/questions/450186 | 4 | Let $ O(n) $ be the manifold of orthornormal matrix, i.e.
$$
O(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=I\}.
$$
Then $ O(n) $ is a submanifold of $ \mathbb{R}^{n\times n} $. On $ O(n) $, there is a Riemannian metric on $ O(n) $ induced by Euclidean metric of $ \mathbb{R}^{n\times n} $. I want to consider the geodesics on i... | https://mathoverflow.net/users/241460 | Geodesics on orthogonal matrix | A direct computation shows that for each $T\in O(n)$, the map $L\_T : \mathbb{R}^{n\times n} \to \mathbb{R}^{n\times n}$ given by
$$
A \mapsto TA,
$$
is an isometry. This isometry preserves the submanifold $O(n)$, thus it's also an isometry of $O(n)$ when the latter is endowed with the induced metric. The same appl... | 2 | https://mathoverflow.net/users/14708 | 450463 | 181,214 |
https://mathoverflow.net/questions/450462 | 1 | This question is about the content of [this paper](https://arxiv.org/abs/math/0301343) by J. Bourgain, N. Katz, T. Tao.
---
In the final step (page 18) of the proof of Szemerédi-Trotter type theorem, we have already known
$$|A''+A''|\lesssim N^{\frac{1}{2}+C\epsilon}$$
Similarly, by the Balog-Szemerédi-Gowers... | https://mathoverflow.net/users/508148 | Szemerédi–Trotter type theorem in finite field | I think you're correct that (18) should read:
$$|A'' + A''| \lesssim N^{1/2+C\epsilon}.$$
The display before (16) tells you that for each $x\_1 \in A' $, (and hence $x\_1 \in A'' \subseteq A'$) one has:
$$|\{(t,x\_0) \in B \times A : (1-t)x\_0 + t x\_1 \in A; t \neq 0, 1 \}| \gtrsim N^{1-C \epsilon}$$
Since $|A... | 1 | https://mathoverflow.net/users/630 | 450465 | 181,215 |
https://mathoverflow.net/questions/450466 | 3 | I'm trying to derive a very basic result stated in several books on random matrix theory (e.g. Terry Tao's book and Potters & Bouchaud's book).
Given a symmetric matrix $A \in \mathbb{R}^{N \times N}$, with eigenvalues $\lambda\_1, \dots, \lambda\_n$, define a complex-valued function $g : \mathbb{C} \mapsto \mathbb{C... | https://mathoverflow.net/users/59128 | Taylor expansion of Stieltjes Transform | You certainly can't expand around $z=0$ if "$z$ has to be sufficiently large". Expand around $1/z =0$:
$$
\frac{1}{N} \sum\_{n=1}^{N}\frac{1}{z-\lambda\_{n} } =
\frac{1}{N} \frac{1}{z} \sum\_{n=1}^{N}\frac{1}{1-\lambda\_{n}/z }
=\frac{1}{N} \frac{1}{z} \sum\_{n=1}^{N} \sum\_{k=0}^{\infty } \frac{\lambda\_{n}^{k} }{z^k ... | 8 | https://mathoverflow.net/users/134299 | 450468 | 181,216 |
https://mathoverflow.net/questions/450429 | 5 | I am interested in the interplay between the Playfair and Proclus Axioms in linear spaces.
By a *[linear space](https://mathworld.wolfram.com/LinearSpace.html)* I understand a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ such that the following axioms are satisfied:
(L1)... | https://mathoverflow.net/users/61536 | Does Playfair imply Proclus? | I think the following construction gives a counterexample. It stems from the observation that the Playfair axiom is quite weak in the case where all lines only have three points (it produces some pairs of parallel lines, but doesn't force any new intersections between lines).
The simplest geometry in which lines have... | 6 | https://mathoverflow.net/users/766 | 450472 | 181,217 |
https://mathoverflow.net/questions/450457 | -2 | If $u : \mathbb{T}^3 \to \mathbb{R}$ is a smooth function on the $3$-dimensional torus $\mathbb{T}^3$, I wonder it is possible to reverse the Sobolev interpolation inequality in the sense that
\begin{equation}
\lVert u \rVert\_2 \lVert \Delta u \rVert\_2 \leq C\lVert \nabla u \rVert\_2^2
\end{equation}
for some const... | https://mathoverflow.net/users/56524 | Inverse of Sobolev interpolation inequality : $\lVert u \rVert_2 \lVert \Delta u \rVert_2 \leq C\lVert \nabla u \rVert_2^2$? | This inequality is incorrect for an essential reason. Assume it is true for smooth functions, then using an approximation by convolution we would conclude that it is true for Sobolev spaces and hence it would imply that functions in $W^{1,2}$ belong to $W^{2,2}$ so $W^{1,2}=W^{2,2}$. Then by an inductive argument we wo... | 4 | https://mathoverflow.net/users/121665 | 450473 | 181,218 |
https://mathoverflow.net/questions/450471 | 4 | We say a field $F$ has the property $\*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $\*$. On the other hand if $ F $ is a finite field then by Chevalley-Warning theorem the equation $ x^2 + y^2 =-1 $ always has a solution.
I want to know... | https://mathoverflow.net/users/215016 | Fields in which $ -1 $ can't be written as sum of two square elements | In the notation of Lam's *Quadratic forms over fields*, the *Stufe* (a German word) or *level* (its English translation) $s(F)$ of a field is the minimal $n$ such that $-1$ is the sum of $n$ squares.
A theorem of Pfister says that $s(F)$ is always a power of $2$ (or $\infty$).
So you are taking of fields $F$ such t... | 9 | https://mathoverflow.net/users/105957 | 450476 | 181,219 |
https://mathoverflow.net/questions/450470 | 0 | Given a prime $P$, an integer $A$ $(0\leq A<P$), and a set of legal positions (encoded as a binary mask $\text{mask}$), is there an efficient algorithm to find a number $B$ that has the same modulus as $A$ and (when $B$ is represented in its binary form) has ones only in legal positions. In other words, $B$ satisfies t... | https://mathoverflow.net/users/508296 | Algorithm to find a number B with same modulus as A with prime P and specific binary positions set to zero | If you interpret the bit mask as encoding a finite set $S = \{2^{b\_i}\}$ of powers of $2$, you are precisely asking whether there exists a subset of $S$ which sums to $A$ modulo $p$. This is known as the modular subset-sum problem, for algorithms, see for example
<https://arxiv.org/pdf/2008.10577.pdf>
Since there ar... | 2 | https://mathoverflow.net/users/39747 | 450480 | 181,221 |
https://mathoverflow.net/questions/450482 | 0 | For a script I have to evaluate the associated Legendre polynomial of second kind $Q^0\_{n}(z)$.
Until now, I was using an implementation that is based on the definition in equation 8.702 in the Book "[Table of Integral, Series and Products](http://fisica.ciens.ucv.ve/%7Esvincenz/TISPISGIMR.pdf)" by Gradshteyn.
Whe... | https://mathoverflow.net/users/508182 | How are the Legendre Polynomials of second kind for negative degrees defined? | It helps to rewrite the expression from Gradshteyn,
$$Q\_\nu^0(z)=\frac{ \Gamma \left(\frac{1}{2}\right) \Gamma (\nu+1)\, \_2F\_1\left(\frac{\nu}{2}+1,\frac{\nu}{2}+\frac{1}{2};\nu+\frac{3}{2};\frac{1}{z^2}\right)}{2^{\nu+1}z^{\nu+1} \Gamma \left(\nu+\frac{3}{2}\right)},$$
in terms of the [regularized hypergeometric fu... | 2 | https://mathoverflow.net/users/11260 | 450486 | 181,222 |
https://mathoverflow.net/questions/440658 | 3 | It is well known that some dispersive non--linear equations admit traveling wave solutions
$$
u(t,x)=u\_0(x-ct)\in L^2\_x\,, \qquad (t,x)\in \mathbb{R}\times \mathbb{R}\,\text{ or }\, \mathbb{R}\times\mathbb{T}\,,
$$
where $u\_0$ is the profile and $c$ is a real constant.
Sometimes these traveling waves can be obtaine... | https://mathoverflow.net/users/498602 | Does there exist always an $L^2$ threshold below (or above) which a traveling waves of a nonlinear dispersive PDE cannot exist? | I recently came across a [paper on arXiv](https://arxiv.org/abs/2307.01592) that addresses the question. I am sharing it here if it may be of interest to someone else.
The author seems to consider a nonlocal nonlinear schrödinger equation, referred to as *the Calogero-Sutherland DNLS equation*. And she finds periodic... | 1 | https://mathoverflow.net/users/498602 | 450490 | 181,223 |
https://mathoverflow.net/questions/449278 | 7 | I am trying to work through a supposedly simple counterexample given in papers by [Love](https://academic.oup.com/jlms/article-abstract/s1-26/1/1/966397?redirectedFrom=fulltext) and [Gehring](https://www.jstor.org/stable/1990790) regarding a $p$-power generalization of bounded variation and absolute continuity.
Let $... | https://mathoverflow.net/users/118997 | A counterexample showing $BV_p \neq AC_p$ | So first of all, what is claimed in Love's paper is slightly different, It says that for a sufficient large choice of the parameter $c>0$ the function
$$ g(x): = \sum\_{n=0}^\infty c^{-n/p}\cos(c^n \pi x) $$ is of vounded $p$-variation but not $AC\_p$. One way to see it is that to notice that the Weierstrass type funct... | 1 | https://mathoverflow.net/users/153260 | 450494 | 181,226 |
https://mathoverflow.net/questions/450492 | 7 | Let $A, B, C$ be finite Abelian groups fitting in a short exact sequence
$$
1 \rightarrow A\overset{\iota}{\rightarrow} B\overset{\pi}{\rightarrow} C\rightarrow 1
$$
This determines a class $[\epsilon]\in H^2(C,A)$ measuring the failure of the sequence to split:
$$
s(c\_1)+s(c\_2)=s(c\_1+c\_2)+\iota(\epsilon (c\_1,c\_... | https://mathoverflow.net/users/495347 | Pontryagin dual of a group-cohomology class | Ok, I think I worked this out based on my last comment, writing down the usual double complex for $\operatorname{Ext}(A^\vee,C^\vee)$ using a projective resolution of $A^\vee$ and an injective resolution of $C^\vee$ at the same time, and doing the diagram chase. Here's how the resulting map works:
Given $\varepsilon:... | 7 | https://mathoverflow.net/users/39747 | 450505 | 181,230 |
https://mathoverflow.net/questions/450498 | 2 | (Reposted from [MSE](https://math.stackexchange.com/questions/4720776/weighted-sobolev-spaces-and-decay) after no responses)
Introduce the following weighted Sobolev space norm on $\mathbb{R}^n$ (common in the study of hyperbolic PDE):
$$
\|u\|\_{H\_{k,\delta}}^2 = \sum\_{0 \leq i \leq k} \int\_{\mathbb{R}^n} \langle x... | https://mathoverflow.net/users/147016 | Weighted Sobolev Spaces and Decay | Question 1 (that higher derivatives are not used) is **yes**.
Question 2 (getting decay without weights) is **no**.
Without weights, let $u$ be a compactly supported smooth function. Let $f\_k(x) = u(x - k v) + u(x + kv)$ where $v$ is a unit vector. The family $f\_k$ is uniformly bounded in any classical $H^s$ spac... | 2 | https://mathoverflow.net/users/3948 | 450506 | 181,231 |
https://mathoverflow.net/questions/325055 | 2 | If $X$ is a self-distributive algebra, then define $x^{[n]}$ for all $n\geq 1$ by letting $x^{[1]}=x$ and $x^{[n+1]}=x\*x^{[n]}$. The motivation for this question comes from the following fact about self-distributivity on one generator.
>
> **Theorem:** Suppose that $X$ is a self-distributive algebra generated by o... | https://mathoverflow.net/users/22277 | Attraction in Laver tables | I have a method of generating counterexamples even if $(X,\*,1)$ is critically simple. In this post, all algebras will be assumed to be finite reduced permutative
self-distributive algebras. To do this, we provide an example of an algebra algebra $X$ generated by $x,y$ where there does not exist a algebra $Y$ generated... | 2 | https://mathoverflow.net/users/22277 | 450507 | 181,232 |
https://mathoverflow.net/questions/450353 | 0 | Given a rational polyhedral fan $\Sigma$ in $\mathbb{R}^d$ (say with full-dimensional support), its barycentric subdivision $\mathrm{bar}(\Sigma)$ is obtained by performing star-subdivision at the barycenters
$$
\rho\_\sigma = \sum\_{\tau \in \sigma(1)} \rho\_\tau
$$
of its cones $\sigma$ (where $\rho\_\tau$ are the pr... | https://mathoverflow.net/users/69630 | Iterated barycentric subdivision cofinal in system of subdivisions? | It turns out that the answer is "No" for dimension $d \geq 3$.
Indeed, for the counter-example in dimension $3$ take $\Sigma$ given by $\sigma\_3$ and its faces, and consider the fan $\Sigma'$ obtained by subdividing $\Sigma$ along the hyperplane $H=-x\_1 + 2 x\_2 + x\_3 = 0$. The fact that $H$ has both positive and ... | 0 | https://mathoverflow.net/users/69630 | 450511 | 181,233 |
https://mathoverflow.net/questions/450521 | 9 | So, I ask whether from the ZFC axioms one can prove X that *every uncountable set has strictly more than continuum many subsets*, or whether X is independent of the ZFC axioms. Note that (within ZFC) the continuum hypothesis implies X and hence "not X" is not provable in ZFC if ZFC is consistent.
The above question a... | https://mathoverflow.net/users/12643 | Within ZFC, is $2^{\aleph_0}<2^{\aleph_1}$ provable/independent? | The assertion that $2^{\aleph\_0}=2^{\aleph\_1}$ is known as [Luzin's hypothesis](https://encyclopediaofmath.org/wiki/Luzin_hypothesis), and was presented by Luzin as an alternative to Cantor's continuum hypothesis.
This is now known to be independent of ZFC by the method of forcing (assuming ZFC is consistent).
Na... | 22 | https://mathoverflow.net/users/1946 | 450522 | 181,235 |
https://mathoverflow.net/questions/450518 | 2 | Let $T \in \mathbb{R}$ be large and $\rho$ be a non-trivial zero of the Riemann zeta function. Assume that $|\rho|=|\rho\_T| \approx T$ and let $\varepsilon\_T \approx \frac{\log \log T}{\log T}$. Is it true that
$$|\zeta(1-\rho\_T + \varepsilon\_T)|=o(1)$$
uniformly for $T \geq T\_0$?
| https://mathoverflow.net/users/507786 | On the upper bound for $|\zeta(s)|$ near the zeta zeros | If $\rho\_T$ is a zero of $\zeta(s)$, so is $1-\rho\_T$ by the functional equation, and $|1-\rho\_T|\approx T$ (depending on what you mean by $\approx$), so the $1-\ldots$ is superfluous.
A Taylor expansion of $\zeta(s)$ at $\rho\_T$ gives
$$
\zeta(\rho\_T+\epsilon)=\zeta^\prime(\rho\_T)\cdot \epsilon+O(\epsilon)^2.
... | 9 | https://mathoverflow.net/users/6756 | 450526 | 181,237 |
https://mathoverflow.net/questions/450531 | 4 | Assume spaces are regular.
A space is [$\sigma$-compact](https://topology.pi-base.org/properties/P000017) if and only if the second player in the [Menger game](https://en.wikipedia.org/wiki/Selection_principle#The_Menger_game) has a winning Markov strategy (relying on only the most recent move of the opponent and the... | https://mathoverflow.net/users/73785 | Does there exist a non-hemicompact regular space for which the 2nd player in the $K$-Rothberger game has a winning Markov strategy? | I believe no such example exists. In fact, finite selections do the trick, so I'll detail the case for both finite and single selections (though it seems the argument for single selections below requires $T\_1$).
>
> **Claim.** For a regular space $X$, if the second player has a Markov winning strategy in the $k$-M... | 4 | https://mathoverflow.net/users/57800 | 450536 | 181,243 |
https://mathoverflow.net/questions/450539 | 4 | Let $G$ be a finite nonabelian simple group.
We call $G$ a $K\_3$-group if $|G|=p^aq^br^c$ where $p,q,r$ are distinct primes and $a,b,c$ are positive integers.
**My question is: Is there a CFSG-free proof of the classification of simple $K\_3$-groups?**
Any explanation, references, suggestion and examples are appre... | https://mathoverflow.net/users/44312 | CFSG-free proof for classifying simple $K_3$-group | By the Feit-Thompson theorem, one of the primes has to be two. Then by John Thompson's N-group classification (1970-1973), the three primes have to come from a very short list of triples. The various cases were then worked out by Geoff Mason, David Wales, and Jeff Leon in the seventies. This should give you enough info... | 6 | https://mathoverflow.net/users/460592 | 450542 | 181,245 |
https://mathoverflow.net/questions/450544 | 1 | Let $$f(z)=(1+z)^{3/4}-\left(\frac{3}{8}+\frac{\sqrt{3}}{4}\right)^{1/4}-\frac{\left(3 z+\sqrt{6} \sqrt{-1+z^2}\right)^{3/4}}{\left(2 \left(2+\sqrt{3}\right)\right)^{3/4}}.$$ Is there a simple proof that this function is positive for $z\ge 1$?
| https://mathoverflow.net/users/504719 | An inequality for a real function | Make the substitution $u=(1+z)^{3/4}$, so that $u\ge2^{3/4}$, $z=u^{4/3}-1$, and the inequality in question becomes
$$F(u):=f(u^{4/3}-1)>0. \tag{1}\label{1}$$
For $z=u^{4/3}-1>1$
$$F''(u)=\frac{5 u^2 \sqrt{z-1}+\sqrt{6} \left(2 z^2+3 z-1\right)}{\left(2
\left(2+\sqrt{3}\right)\right)^{3/4} (z-1)^{3/2} (z+1) \left(\s... | 1 | https://mathoverflow.net/users/36721 | 450580 | 181,255 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.