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https://mathoverflow.net/questions/414361 | 0 | Let $b\geq 2$ be an integer. A real $r \in [0,1]$ is said to be *normal with respect to $b$* if every finite string made from the elements $\{0,\ldots,b-1\}$ appears in the $b$-ary expansion of $r$.
Are there integers $b, b'\geq 2$ as well as a real number $r\in[0,1]$ such that $r$ is normal with respect to $b$, but ... | https://mathoverflow.net/users/8628 | Does normalcy in one base imply normalcy in any other base? | A [1960 paper by Wolfgang Schmidt](https://msp.org/pjm/1960/10-2/p22.xhtml) states the following:
>
> We write $r \sim s$, if there exist integers $n$, $m$ with $r^n = s^m$. Otherwise, we put $r \not\sim s$.
>
>
> In this paper we solve the following problem. Under what conditions on $r$, $s$ is every number $\xi... | 7 | https://mathoverflow.net/users/70594 | 414363 | 169,000 |
https://mathoverflow.net/questions/371951 | 4 | Let $\mathcal{C}$ be a braided monoidal category. We have a canonical functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ from $\mathcal{C}$ to the Drinfeld center $\mathcal{Z}(\mathcal{C})$ sending an object $V$ in $\mathcal{C}$ to $(V,c\_{V,\,\\_})$. Here, $c$ is the braiding in $\mathcal{C}$.
When does this functor... | https://mathoverflow.net/users/58211 | Does the functor $\mathcal{C} \to \mathcal{Z}(\mathcal{C})$ have adjoints? | **Short answer**: Yes, it can possibly have an adjoint.
**Longer answer**:
Assume that $\mathcal{C}$ is rigid, and that the coend $L = \int^{X \in \mathcal{C}} X^\* \otimes X$ exists.
It is a coalgebra.
Your assumptions on $\mathcal{C}$ were that it is braided, and in that case, it is well-known that $L$ is even a bi... | 7 | https://mathoverflow.net/users/41706 | 414369 | 169,001 |
https://mathoverflow.net/questions/414305 | 2 | Let $L/K$ be an abelian extension of number fields with Galois group $G$ and let $\chi : G \to \{\pm 1\}$ denote a real linear character of $G$. Denote $L(\chi,s)$ the Artin L-function associated to $\chi$, $n=\mathrm{ord}\_{s=0} L(\chi,s)$ and
$$
L^\ast(\chi,0)=\lim\_{s\to 0} L(\chi,s)s^{-n}
$$
its special value at $s... | https://mathoverflow.net/users/138396 | Sign of the special value at s=0 of Hecke L-functions | If $\chi$ is real-valued, then the question makes sense. Using the functional equation, it reduces to computing the sign of the non-zero real number $L(\chi, 1)$ if $\chi$ is non-trivial, or the residue of $L(\chi, s) = \zeta\_K(s)$ at $s = 1$ if $\chi$ is trivial. In either case, $L(\chi, s)$ tends to $+1$ for $s$ lar... | 4 | https://mathoverflow.net/users/2481 | 414371 | 169,003 |
https://mathoverflow.net/questions/414375 | 2 | During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them.
The function $f:[-1,1] \times \mathbb R \to \mathbb R$ is defined by
$$
f(x,\xi) = \left[ w(x) \xi - 2 x w(x) \sin\left(\frac{\pi}{x}\right) + \pi w(x) \cos\... | https://mathoverflow.net/users/471464 | The regularity theorem, a non-regular minimizer problem | $\newcommand\ol\overline$The steps were:
1. Show that $f$ is infinitely differentiable, $\xi \mapsto f(x,\xi)$ is convex and $f\_{\xi \xi} (x,\xi) > 0$ holds for all $x$ except for $x=0$.
2. Show that the function
\begin{equation}
\overline{u}(x) = \begin{cases} x^2 \sin \frac \pi x,& x \not = 0\\ 0, & x=0\end{cases... | 1 | https://mathoverflow.net/users/36721 | 414377 | 169,005 |
https://mathoverflow.net/questions/414378 | 16 | I usually work in the field of differential geometry, but I have encountered the following problem in my research: Are there infinitely many positive integers $k,l,m\in\mathbb N^{>0}$ such that $$(3+3k+l)^2=m\,(k\,l-k^3-1)\,?$$ Obviously, taking $l=k^2$
and $m=-(3+3k+l)^2$
gives infinitely many integer solutions, but $... | https://mathoverflow.net/users/4572 | Are there infinitely many positive integer solutions to $(3+3k+l)^2=m\,(k\,l-k^3-1)$? | It does have infinitely many positive solutions. Here is just one such series.
Consider the following recurrence sequence:
$$u\_0=1,\ u\_1=2,\ u\_{n+1} = 23 u\_n - u\_{n-1} - 4\qquad (n\geq 1).$$
Let $t,k$ be any two consecutive terms of this sequence, then setting $l:=k^2+t$ produces the following equality:
$$(3... | 23 | https://mathoverflow.net/users/7076 | 414380 | 169,007 |
https://mathoverflow.net/questions/414344 | 1 | Let $(X, \mathcal{T}\_X)$ and $(Y, \mathcal{T}\_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}\_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ be compact. Is $f \restriction C = f|(C \to f[C])$ a quotient map?
Background
----------
In what at first seems l... | https://mathoverflow.net/users/32487 | Is the restriction of a projection to a compact subset a quotient map? | A counterexample with finite topological spaces:
recall that on a set $X$, an "Alexandroff discrete" topology ($\mathrm T\_0$ where all intersections of open sets are open)
is the same thing as a partial order; the minimum open set containing $x$ is the principal order filter $Fx$ in the order.
The dual order gives t... | 3 | https://mathoverflow.net/users/474159 | 414388 | 169,010 |
https://mathoverflow.net/questions/414394 | 5 | Given a (0,1)-matrix $A$, I'll denote by $\mu(A)$ the number of maximal monochromatic polyominoes in $A$ (i.e., the number of connected polyominoes contained in $A$ each of which is either all 0 or all 1 such that each polyomino is as large as possible - so they thus tile $A$). I do not know of any literature on $\mu(A... | https://mathoverflow.net/users/99414 | Average number of tiles of a (0,1)-matrix? | The limit is positive but well below $1$; I show that the limit is in $(0.09448, 0.2646)$, and outline how to approximate it much more closely than this.
The average of $\mu$, call it ${\rm E}(\mu)$,
is twice the average number of maximal all-0 polyominos;
so the limit of ${\rm E}(\mu) / n^2$ is twice the expected nu... | 8 | https://mathoverflow.net/users/14830 | 414405 | 169,013 |
https://mathoverflow.net/questions/414402 | 23 | On MSE this got 5 upvotes but no answers not even a comment so I figured it was time to cross-post it on MO:
Is the Moebius strip a linear group orbit? In other words:
**Does there exists a Lie group $ G $ a representation $ \pi: G \to \operatorname{Aut}(V) $ and a vector $ v \in V $ such that the orbit
$$
\mathcal... | https://mathoverflow.net/users/387190 | Is it possible to realize the Moebius strip as a linear group orbit? | Yes. Here is one way: Consider standard $\mathbb{R}^3$ endowed with the Lorentzian quadratic form $Q = x^2+y^2-z^2$, and let $G\simeq\mathrm{O}(2,1)\subset\mathrm{GL}(3,\mathbb{R})$ be the symmetry group of $Q$. Then $G$ preserves the hyperboloid $H$ of $1$-sheet given by the level set $Q=1$, which is diffeomorphic to ... | 28 | https://mathoverflow.net/users/13972 | 414425 | 169,022 |
https://mathoverflow.net/questions/414382 | 5 | Let $S$ be a minimal compact complex surface of general type with ample canonical class $K\_S$. In [**1**, Theorem 3] the following result is stated:
>
> **Theorem.** Every symmetric power $S^n \Omega\_S$ of the cotangent bundle $\Omega\_S$ is stable with respect to $K\_S$, unless $S$ is
> uniformized by the bi-dis... | https://mathoverflow.net/users/7460 | Semi-stability of $S^n\Omega_S$ with respect to $K_S$ | Ciao Francesco!
The answer to your question is yes, and the work of Bogomolov you are looking for about semistability of the tangent space (for minimal surfaces of general type indeed, no need of ampleness of the canonical bundle) is
"Holomorphic tensors and vector bundles on projective varieties", Math. USSR Izves... | 3 | https://mathoverflow.net/users/9871 | 414427 | 169,023 |
https://mathoverflow.net/questions/414367 | 0 | Let $X$ be a compact metric space, $A,B\subset X$ be subsets and $f\colon X\times X\to \mathbb{R}$ a continuous function that is strictly positive on $A\times B$. Do there exist increasing sequences of subsets $A\_1\subseteq A\_2\subseteq \dots$ and $B\_1\subseteq B\_2\subseteq \dots$ such that:
* $A=\bigcup\_{n\in\m... | https://mathoverflow.net/users/54309 | Exhaustions of product subsets by smaller product subsets | Take $A,B$ complementary dense subsets of $X$ (say, rationals and irrationals in $[0,1]$).
$d(a,b)$ is strictly positive on $A\times B$.
Suppose $A$ increasing join of the $A\_n$, dually $B\_n$, with positive distance between $A\_n,B\_n$ i.e.
between their closures $A'\_n,B'\_n$.
Then Baire assures that for one $n$... | 1 | https://mathoverflow.net/users/474159 | 414439 | 169,027 |
https://mathoverflow.net/questions/414460 | 5 | Is the following a theorem of $\sf ZF(C)$?
**Countable reflection:** If $\phi$ is a sentence in which $W$ is not free, then:
$\phi \to \exists W: |W|= \omega \land \operatorname {Transitive}(W) \land \phi^W$
Where $\phi^W$ is the sentence obtained by merely bounding all quantifiers in $\phi$ by $ \in W$.
That i... | https://mathoverflow.net/users/95347 | Does ZF(C) prove countable reflection? | As Monroe Eskew points out in his comment to the question, the positive answer is well-known for ZFC, thanks to the ZF reflection theorem and the Löwenheim-Skolem Theorem (in the form: every model in a countable language has a countable elementary submodel). See, e.g., part (iii) of Theorem 12.14 in Jech's canonical te... | 18 | https://mathoverflow.net/users/9269 | 414474 | 169,045 |
https://mathoverflow.net/questions/414404 | 1 | If we add to the language of set theory a total one place function symbol $\mathcal P$ standing for powerset operator, and then add to ZF-Power the following axioms:
**Power:** if $\phi$ is a formula in which only the symbol $y$ occurs free, then: $$ X \subseteq A \land X=\{ y \mid \phi\} \to X\in \mathcal P(A) $$
... | https://mathoverflow.net/users/95347 | Does this restriction on powersets in ZF have a proof theoretic ordinal? | Your theory interprets $\mathsf{ZF}^-$. In fact, $\mathsf{ZF}-$ (namely, $\mathsf{ZF}$ without Powerset) interprets $\mathsf{ZF}^-+(V=L)$.
It answers your questions negatively since the proof-theoretic strength of $\mathsf{ZF}^-+(V=L)$ is that of Full Second-order Arithmetic.
The reason is that we can construct $L$ f... | 2 | https://mathoverflow.net/users/48041 | 414476 | 169,046 |
https://mathoverflow.net/questions/414487 | 0 | Title says it all. Sorry my previous question was wrong; I see now; very stupid of me. So this is what I meant to ask. I am looking for a field extension of $\mathbb Q$, let's call it $K$, s.t. $K$ is a proper subset of $\bar{\mathbb{Q}}$, and $X^n-a$ has a root in $K$ whenever $a$ is in $K$.
Or is this question unkn... | https://mathoverflow.net/users/475698 | Does there exist a proper intermediate field between ℚ and ℚ̅ closed under taking nth roots? | Yes because the algebraic closure of $\mathbb{Q}$ contains all roots of all polynomials. But what we learn in elementary Galois theory is that there are polynomials that cannot be solved using any finite number of applications of the operations of taking $n$th roots for every $n$ in $\mathbb{N}$ and field operations. S... | 4 | https://mathoverflow.net/users/471081 | 414490 | 169,050 |
https://mathoverflow.net/questions/414465 | 11 | Is every $ \mathbb{R}P^{2n} $ bundle over the circle trivial?
Are there exactly two $ \mathbb{R}P^{2n+1} $ bundles over the circle?
This is a cross-post of (part of) my MSE question
<https://math.stackexchange.com/questions/4349052/diffeomorphisms-of-spheres-and-real-projective-spaces>
which has been up for a c... | https://mathoverflow.net/users/387190 | $ \mathbb{R}P^n $ bundles over the circle | Your answer is correct if appropriately understood, but it's a little subtle. Here I should note that I'm interpreting your question as a purely homotopy theoretic one (in particular ignoring smooth structure), and that by "bundle" you mean "Serre bundle". If you care about smooth bundles, see Tom Goodwillie's answer.
... | 8 | https://mathoverflow.net/users/7108 | 414496 | 169,052 |
https://mathoverflow.net/questions/413526 | 4 | Let $\left(\mathcal{M}^2,g\_\mathcal{M};X\right)$ and $\left(\mathcal{N}^2,g\_{\mathcal{N}};Y\right)$ be two smooth two-dimensional, simply connected Riemannian manifolds (with or without boundary), equipped with non-nonvanishing Killing fields $X$ and $Y$, respectively.
Does there exist a mapping $f:\mathcal{M}^2\ri... | https://mathoverflow.net/users/171439 | Mappings between 2-manifolds with symmetries with fixed singular values | To understand the local conditions, it's convenient to establish canonically associated local coordinate expressions for the quantities involved. Thus, let $(M^2,g,X)$ and $(N^2,h,Y)$ be as described and suppose that we want to test whether, for a given $p\in M$ and $q\in N$, there exists an open $p$-neighborhood $U\su... | 4 | https://mathoverflow.net/users/13972 | 414499 | 169,053 |
https://mathoverflow.net/questions/414477 | 3 | Suppose that I want to send a message (consisting of bits) over a channel where from $n$ transferred bits as many as $n/2-\varepsilon n$ might be flipped, i.e., the distance of the code is $n-2\varepsilon n$. How does the number of encodable messages $m$ change as a function of $n$ and $\varepsilon$? I do not want to a... | https://mathoverflow.net/users/955 | Coding over very noise channel | For a binary code, as $n$ grows you cannot do better than the repetition code $$C=\{11\cdots1,00\cdots0\},$$ with two codewords as soon as the minimum distance required $d>n/2.$ See Theorem 4 of Venkat Guruswami's notes, for example, available [here][1]. The result for your case is
that if a binary code $C$ must satisf... | 3 | https://mathoverflow.net/users/17773 | 414508 | 169,058 |
https://mathoverflow.net/questions/414322 | 6 | Recent references on the matter at hand include, a lecture slide [The Konvalinka-Amdeberhan conjecture
and plethystic inverses](https://people.brandeis.edu/%7Egessel/homepage/slides/K-A%20conjecture.pdf) and a preprint on [Counting tanglegrams with species](https://arxiv.org/pdf/1509.03867.pdf) by I. Gessel; the initia... | https://mathoverflow.net/users/66131 | Tanglegrams and functional equations of M. Somos | In my lecture slides that Tewodros cites, I studied symmetric functions that I called $g\_m$, where $m$ is a positive integer, that have constant term 1 and satisfy
$$ -L\_m[g\_m] = p\_1,$$
where $L\_m$ is the Lyndon symmetric function given by
$$L\_m = \frac{1}{m} \sum\_{d\mid m} \mu(d) p\_d^{m/d}.$$
Here $\mu$ is ... | 6 | https://mathoverflow.net/users/10744 | 414516 | 169,060 |
https://mathoverflow.net/questions/414514 | 5 | In proposition 2.7. of the condensed notes of professors Scholze and Clausen it is said that the category of extremally disconnected sets is a site, but in the definition of a site in the Stacks Project (<https://stacks.math.columbia.edu/tag/00VH>) it is necessary for a site to have fibre products sometimes (Axiom (3) ... | https://mathoverflow.net/users/130868 | The site of extremally disconnected sets | Usage varies. Let's at least stipulate that "site" is synonymous with "category equipped with a Grothendieck topology".
Some, but not all, authors, require a site to have pullbacks, because this assumption simplifies the definition a bit. But e.g. the [nlab](https://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/Gro... | 8 | https://mathoverflow.net/users/2362 | 414520 | 169,062 |
https://mathoverflow.net/questions/413220 | 7 | I am working on $\mathbb{Z}/18\mathbb{Z}$ elliptic curves over cubic fields. The curves are created using the formulas on p. 584 of
>
> D. Jeon, C. H. Kim, Y. Lee, *Families of elliptic curves over cubic number fields with prescribed torsion subgroups*, Mathematics of Computation, V. 80, 273, January 2011, p. 579-5... | https://mathoverflow.net/users/95511 | ℤ/18ℤ elliptic curves over cubic fields | We already have an accepted answer, but since i had already started an answer and was at the half of the route beyond getting the essence of the structure, i completed it now, since it may be useful in similar contexts.
---
On the mathematical side the situation is as follows, recalled for the convenience of the ... | 5 | https://mathoverflow.net/users/122945 | 414529 | 169,066 |
https://mathoverflow.net/questions/414479 | 9 | *This question is a follow-up to another question of mine, with different language - see the link below.*
Say that an infinite regular cardinal $\kappa$ is **Fraissean** iff the logic $\mathcal{L}\_{\kappa,\omega}$ has the following property (called "SED" in the below-linked question):
>
> For every finite signat... | https://mathoverflow.net/users/8133 | Logics detecting their own equivalence notions, take two: $\mathcal{L}_{\omega_2,\omega}$ | (Working in ZFC.)
$\omega\_2$ is not Fraissean. In fact, it is not Fraissean with respect to $\Sigma$, where $\Sigma$ is the signature with a single binary relation $<$.
To see this we use a variant of the argument you linked in the question. Suppose otherwise, and let $\Sigma'$ and $\eta$ witness this. Let $\gamma$ ... | 12 | https://mathoverflow.net/users/160347 | 414533 | 169,067 |
https://mathoverflow.net/questions/414495 | 4 | Note: I asked this question a few months ago [here](https://math.stackexchange.com/questions/4303093/relation-between-two-permutation-metrics), but received no answer.
Consider the following two metrics on permutations of $\{1,2,\dots,n\}$:
$d\_\text{swap}(\sigma,\tau)$ is the minimum number of swaps of adjacent el... | https://mathoverflow.net/users/475708 | Relation between two permutation metrics | Both statistics are preserved under right multiplication by a permutation, so you can reduce to the case where $\sigma$ is the identity.
In that case, $d\_{\rm swap}$ is the inversion number or equivalently the Coxeter length of the permutation, and $d\_{\rm sum}$ is the total displacement, studied by Diaconis and Gr... | 5 | https://mathoverflow.net/users/3077 | 414537 | 169,068 |
https://mathoverflow.net/questions/414511 | 3 | Is there a family of continuous functions $(f\_n)\_{n \in \mathbb{N}}$ on $[0,1]$ whose span is dense in $L^1[0,1]$ for the $L^1$-norm, but not dense in $L^2[0,1]$ for the $L^2$-norm?
---
Some preliminary considerations: I suspect the answer is yes, since there exist a family of $L^2$ functions dense in $L^1$ but... | https://mathoverflow.net/users/105382 | Functions dense in $L^1[0,1]$ but not in $L^2[0,1]$ | $\newcommand{\ep}{\varepsilon}$User Z. M [provided](https://mathoverflow.net/questions/414511/functions-dense-in-l10-1-but-not-in-l20-1#comment1062924_414511) an elegant and brief (and yet complete) detailization of Fedor Petrov's [comment](https://mathoverflow.net/questions/414511/functions-dense-in-l10-1-but-not-in-l... | 3 | https://mathoverflow.net/users/36721 | 414538 | 169,069 |
https://mathoverflow.net/questions/414453 | 11 | Here's a mix of heuristic and precise questions as I try to grapple with topos theory.
I try to think of topoi as two notions of "$1$" being glued at the hip. One is the "building block" $1$, generating the naturals, the ordinals, the cardinals... with all its usual arithmetic properties. This generates the set theor... | https://mathoverflow.net/users/475672 | Is every topos a sheaf topos with values in a well-pointed one? | A solution to your first question is given in "Sketches of an Elephant" by Johnstone, Example A4.4.2(d). If a topos $\mathcal{E}$ is the topos of sheaves on some internal site in a topos $\mathcal{S}$, then there is a natural geometric morphism $p : \mathcal{E} \to \mathcal{S}$. It turns out that there are toposes that... | 7 | https://mathoverflow.net/users/37368 | 414558 | 169,071 |
https://mathoverflow.net/questions/414547 | 3 | $\newcommand{\loc}{\mathrm{loc}}$Let $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n),\mu)$ denote the Euclidean space $\mathbb{R}^n$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ equipped with the Lebesgue measure $\mu$ and let $H$ be a separable Hilbert space. Let $L^1\_{\loc}(\mathbb{R}^n,H)$ denote the space ... | https://mathoverflow.net/users/36886 | Schauder basis of $L^1_{\mathrm{loc}}(\mathbb{R}^n,H)$ | Instead of the balls, you can equivalently look at convergence on cubes $Q\_k:=k+[0,1)^n$ for $k\in\mathbb{Z}^n$. More precisely: Convergence in your metric is equivalent to convergence to w.r.t. the set of seminorms
$$f\mapsto\int\_{B\_0(r)} \|f(x)\|\_H \,dx$$
for all $r\in\mathbb{N}$ which is equivalent to convergenc... | 1 | https://mathoverflow.net/users/3041 | 414560 | 169,072 |
https://mathoverflow.net/questions/414561 | 6 | Let $G$ be a free group of finite rank. Consider any commutative ring $R$ containing $\mathbb{Z}$. Consider the group ring $RG$.
Q) What can we say about the Hochschild cohomology groups of $RG$ with coefficients in $R$?
I can find the $HH^1(RG)$ in terms of outer derivations but again I don't know the complete des... | https://mathoverflow.net/users/9485 | Hochschild cohomology of group ring of a free group | Hochschild cohomology of group rings is reducible to ordinary group cohomology. (Not really if you're interested in multiplicative structure, but that's another story). Everyting works over any commutative base ring, you can assume it's $\Bbb Z$ if you'd like.
Namely, there's an adjunction between $kG-bimod$ and $kG-... | 8 | https://mathoverflow.net/users/81055 | 414565 | 169,073 |
https://mathoverflow.net/questions/414542 | 2 | I would like to try my luck here for the following question after failing to elicit an answer to [it on math.stackexchange.com](https://math.stackexchange.com/q/4363044/64809).
---
For $r\ge -1$, the exponential of the negative Renyi entropy is defined as
$$M(p):=\Big(\sum\_i p\_i^{1+r}\Big)^{\frac1r},$$
for a pr... | https://mathoverflow.net/users/32660 | Convexity of the exponential of the negative Renyi entropy | Take any $r\in(-1,0)$, any vector $(p\_i)\_{i=1}^n$ with $p\_i>0$ for all $i$, and any vector $(h\_i)\_{i=1}^n\in\mathbb R^n$. For all real $t$ close enough to $0$, let
$$g(t):=M(p+th).$$
Then
$$g''(0)=(1+r)\Big(\sum\_{i=1}^n p\_i^{r-1}h\_i^2\,\sum\_{i=1}^n p\_i^{1+r}
+\frac{1-r^2}{r^2}\,\Big(\sum\_{i=1}^n p\_i^r h\_i\... | 1 | https://mathoverflow.net/users/36721 | 414566 | 169,074 |
https://mathoverflow.net/questions/414551 | 8 | In the first section of "A procedure for killing homotopy groups of differentiable manifolds", Milnor gives the surgery construction as follows. Let $W$ be an $n=p+q+1$ dimensional manifold. Given a smooth, orientation preserving embedding:
$$f:S^p\times D^{q+1}\to W$$
we may obtain a new manifold as the disjoint s... | https://mathoverflow.net/users/475761 | Regarding the surgery construction in "A procedure for killing homotopy groups of differentiable manifolds" by Milnor | There is no error in the paper as far as I can tell. One *thinks* of the surgery as taking out the interior of the image of $S^p\times D^{q+1}$, and glueing in a copy of $D^{p+1}\times S^q$ along the common boundary $S^p\times S^q$. However, that does only define a topological space and not an orientable differentiable... | 8 | https://mathoverflow.net/users/85592 | 414570 | 169,075 |
https://mathoverflow.net/questions/414572 | 3 | The following is taken from [a post by Terence Tao on the Chowla conjecture and the Sarnak conjecture](https://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture/):
Given a bounded sequence ${f: {\bf N} \rightarrow {\bf C}}$, define the topological entropy of the sequence to be the leas... | https://mathoverflow.net/users/64462 | An example of deterministic sequence from Terence Tao's blog | Let $S\_1$ be a subset of $[0,1]$ with consecutive elements separated by a distance of at most $\epsilon/2\pi $ and let $S\_2$ be a subset of $[0,1]$ with consecutive elements separated by at most $\epsilon / (2 \pi m)$.
We can take $|S\_1| < 2\pi\epsilon^{-1} +1$ and $|S\_2| < 2\pi m \epsilon^{-1} +1$.
Let $\gamma... | 7 | https://mathoverflow.net/users/18060 | 414578 | 169,077 |
https://mathoverflow.net/questions/414563 | 12 | Let $\chi$ be a primitive Dirichlet character of conductor $q$. I want to compute numerically $$G(k)=\sum\_{n\bmod q}\chi(n)e^{2\pi i n(n-k)/(2q)}$$
for all $k$ with $0\le k<2q$ with $k\equiv q\pmod2$ (thanks to this last condition the sum $G(k)$ is well defined). I have two questions:
1. For now I compute these valu... | https://mathoverflow.net/users/81776 | Computation of modified Gauss sums | For the second question, if $p$ is an odd prime dividing $q$, $p$ divides $k$ with at least the multiplicity with which it divides $q$, and $\chi\_p(-1)=-1$, where $\chi\_p$ is the $p$-adic part of $\chi$, then $G(k)=0$.
This generalizes what you wrote in the $\chi$ quadratic case. In that case, the multiplicity whic... | 5 | https://mathoverflow.net/users/18060 | 414580 | 169,078 |
https://mathoverflow.net/questions/414584 | 4 | Let $X = Q\_1\cap Q\_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y\_0,y\_1,y\_2,y\_3,y\_4$.
Set $Q\_1 = \{F\_1 = 0\}$ and $Q\_2 = \{F\_2 = 0\}$ and assume that the monomial $y\_0^2$ does not appear in $F\_1$ so that $Q\_1$ is rational and tha... | https://mathoverflow.net/users/14514 | Del Pezzo surfaces of degree four and complete intersections of two quadrics | No.
Let $Q\_1,Q\_2$ be arbitrary quadrics. Let $a$ be the coefficient of $y\_0^2$ in $F\_1$, $b$ the coefficient of $y\_0^2$ in $F\_2$, $c$ the coefficient of $y\_0 y\_1$ in $F\_1$, $d$ the coefficient of $y\_0 y\_1$ in $F\_2$.
Then the coefficient of $y\_0^2$ in $b F\_1 - a F\_2$ and the coefficient of $y\_0 y\_1$... | 7 | https://mathoverflow.net/users/18060 | 414587 | 169,079 |
https://mathoverflow.net/questions/414590 | 5 | Fix a prime $p$. A p-adic field is a finite extension of $\mathbb{Q}\_p$.
Question 1: Let $K$ be a $p$-adic field and fix $n$. Is there $m$ such that if $\alpha \in \mathbb{Q}\_p$ is an $m$th power in $K$ then $\alpha$ is an $n$th power in $\mathbb{Q}\_p$? $\quad$ (Probably $m$ is a multiple of $n$.)
I think that y... | https://mathoverflow.net/users/152899 | Powers in finite extensions of the p-adics | Yes.
I'll solve the more general question 2.
Let $v\_p(n)$ be the highest power of $p$ dividing $n$. Then if $\alpha \in L$ is congruent to $1$ modulo $p^{v\_p(n)+1}$, then $\alpha$ is an $n$th power in $L$.
Every $\alpha \in L$ can be written as $\pi^j u$ where $j\in \mathbb Z$ and $u$ is a unit in $L$. It suffi... | 4 | https://mathoverflow.net/users/18060 | 414592 | 169,081 |
https://mathoverflow.net/questions/414579 | 3 | Let $R$ be a unital ring. Let $\mathbf{A}\_\bullet$ and $\mathbf{C}\_\bullet$ be positive chain complexes of $R$-modules. If $\mathbf{A}\_\bullet$ consists of flat $R$-modules then there is *homology Künneth spectral sequence*
$$E^2\_{p,q}:=\bigoplus\_{s+t=q}\mathrm{Tor}\_p^R(H\_s(\mathbf{A}\_\bullet),H\_t(\mathbf{C}\_... | https://mathoverflow.net/users/121307 | Künneth spectral sequence for cohomology of chain complexes of $R$-modules | This is a special case of the hyper(co)homology spectral sequence from Chapter XVII, Section 2 of Cartan-Eilenberg (1953), for the functor $T(C,A) = \mathrm{Hom}\_R(A,C)$. The $E\_2$-term is given in equation (4) on page 368, essentially as
$$
E\_2^{p,q} = \prod\_{s+t=q} \mathrm{Ext}^p\_R(H\_s(\mathbf{A}\_\bullet), H^t... | 4 | https://mathoverflow.net/users/9684 | 414593 | 169,082 |
https://mathoverflow.net/questions/414526 | 1 | In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?
Particularly, is there an element $w$ of the field such that the standard part (the zeroth element of the corresponding series) of $w^n$ is $B\_n$ (Bernoulli numbers)?
| https://mathoverflow.net/users/10059 | In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence? | Yes, it is true, and there even are such elements in the field of formal Laurent series. Specifically, let $a\_n,n\geq 1$ be any sequence of real numbers. We then take a Levi-Civita series
$$z=\varepsilon^{-1}+\sum\_{i=0}^\infty b\_i\varepsilon.$$
We want to show the $b\_i$ can be chosen so that the constant term of $z... | 1 | https://mathoverflow.net/users/30186 | 414599 | 169,085 |
https://mathoverflow.net/questions/414352 | 2 | Let $(M^{2n},\omega)$ be a symplectic manifold of dimension $2n$. Let $L^k\_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ be the map given by $L^k\_{\omega}(\alpha)=\alpha\wedge\omega^k$. Then is it true that $L^k\_{\omega}$ is injective for all $q\leq n-k$ and surjective for all $q\geq n-k$?
It seems that this is true, ... | https://mathoverflow.net/users/153479 | $\omega$ is a symplectic form then $L^k_{\omega}:\Omega^q(M)\to \Omega^{q+2k}(M)$ is injective for all $q\leq n-k$ and surjective for all $q\geq n-k$ | The injectivity case is well-known and follows quite easily from the statement (usually attributed to Lefschetz) that $L^k\_\omega:\Omega^{n-k}(M)\to \Omega^{n+k}(M)$ is a isomorphism for $0\le k\le n$.
This is a purely linear algebra statement and only relies on $\omega$ being nondegenerate, i.e., that $\omega^n$ be... | 5 | https://mathoverflow.net/users/13972 | 414605 | 169,088 |
https://mathoverflow.net/questions/414610 | 5 | Let $\operatorname{Col}(\omega,<\kappa)$ denote the Lévy collapse of an inaccessible cardinal $\kappa$. A variant of the Factor Lemma is as follows:
>
> **Lemma.** Suppose that $\kappa$ is an inaccessible cardinal and that $\mathbb{P}$ is a poset of size $<\kappa$. Let $G$ be $\operatorname{Col}(\omega,<\kappa$)-ge... | https://mathoverflow.net/users/146831 | Locating generic filters in the Lévy collapse | Note that the lemma doesn't show that $h$ is in $V[G]$, it *assumes* this. But yes, if $h\in V[G]$ is a subset of a set $X\in V$ such that $|X| < \kappa$, then for some $\beta < \kappa$, $h\in V[G\restriction \beta]$. The argument to follow is by no means original, but I don't remember where I saw it. Fix a name $\dot ... | 10 | https://mathoverflow.net/users/102684 | 414612 | 169,091 |
https://mathoverflow.net/questions/414596 | 6 | I posted this on MSE a couple months ago and it got three upvotes but no answers or even comments so I decided to cross-post it here:
For every pair $ a,b $ of real numbers define the operator $ U\_{a,b} $ on $ L^2(\mathbb{R}) $ sending $ \psi \in L^2(\mathbb{R}) $ to $ U\_{a,b}\psi $ defined by the equation
$$
[U\_{... | https://mathoverflow.net/users/387190 | Characterize this subspace of the bounded operators on $ L^2(\mathbb{R}) $ | Any linear combination $L$ of $U\_{a,b}$'s can be written $(L\psi)(x) = \sum\_{k=1}^n \alpha\_ke^{ib\_kx}\psi^{\to a\_k}(x)$, where $\psi^{\to a\_k}(x) = \psi(x + a\_k)$. Fix $L$.
Let $N \in \mathbb{N}$ be such that $Nb\_k$ is close to an integer multiple of $2\pi$, for all $k$. Then $$(L\psi^{\to N})(x) = \sum \alph... | 7 | https://mathoverflow.net/users/23141 | 414613 | 169,092 |
https://mathoverflow.net/questions/323445 | 5 | Suppose that $X$ is a finite dimensional Hilbert space. Let $A\_{1},\dots,A\_{r}:X\rightarrow X$ be linear operators. Then define the multi-spectral radius of $(A\_{1},\dots,A\_{r})$ to be
$$\limsup\_{n\rightarrow\infty}(\sum\_{a\_{1},\dots,a\_{n}\in\{1,\dots,r\}}\|A\_{a\_{1}}\dots A\_{a\_{n}}\|)^{1/n}.$$
If $r=1$, t... | https://mathoverflow.net/users/22277 | Spectral radius for multiple linear operators | Yes. There is a characterization of the multi-spectral radius in terms of the ordinary spectral radius.
Suppose that $A$ is a unital Banach algebra. If $a\_{1},\dots,a\_{r}\in A$, then define the multi-spectral radius of $a\_{1},\dots,a\_{r}$ to be
$\limsup\_{n\rightarrow\infty}(\sum\_{i\_{1},\dots,i\_{n}\in\{1,\dots... | 0 | https://mathoverflow.net/users/22277 | 414615 | 169,093 |
https://mathoverflow.net/questions/414614 | 5 | Assume $V=L$.
Let $\alpha$ be the least ordinal such that there is a $\Diamond\_{\omega\_1}$-sequence in $L\_\alpha$.
It's obvious that $\omega\_1 < \alpha < \omega\_2$.
Do we have some better estimates of $\alpha$?
Also, what about $\Diamond^+$, squares and morasses?
| https://mathoverflow.net/users/170286 | Height of diamond | It depends on what diamond sequence you have.
$V=L$ proves there is a $\diamondsuit$-sequence of $L$-rank $\omega\_1+1$: recall the famous way of proving the existence of $\diamondsuit$-sequence. The definition goes as follows (for example, Theorem 13.21 of [Jech - Set theory (3rd edition)](https://doi.org/10.1007/3-... | 8 | https://mathoverflow.net/users/48041 | 414625 | 169,095 |
https://mathoverflow.net/questions/414603 | 2 | Suppose that $E/\mathbf{Q}$ is an elliptic curve and $K$ is an imaginary quadratic field. Let $\mathbf{Q}\_{\infty}$ denote the cyclotomic $\mathbf{Z}\_p$ extension of $\mathbf{Q}$, and let $K\_{\infty}$ denote the cyclotomic $\mathbf{Z}\_p$ extension of $K$,
If we know that the $\mu$-invariant for the Selmer over $K... | https://mathoverflow.net/users/394740 | Does $\mu=0$ for an imaginary quadratic field $K$ imply $\mu=0$ for $\mathbf{Q}$? | Under your setting
\begin{equation}
\operatorname{Sel}(E/K\_{\infty})\simeq\operatorname{Sel}(E/\mathbb Q\_{\infty})\oplus\operatorname{Sel}(E\otimes\chi/\mathbb Q\_{\infty})
\end{equation}
where $\chi$ is the quadratic character attached to $K/\mathbb Q$ and all these $\mathbb Z\_{p}[[X]]$-modules are torsion. Hence
\... | 3 | https://mathoverflow.net/users/2284 | 414630 | 169,096 |
https://mathoverflow.net/questions/414627 | 1 | $\DeclareMathOperator\Hom{Hom}$Let $G$ be a (pro-)cyclic group (topologically) generated by $\phi$, and let $V,W$ be two finite-dimensional (continuous) $k$-linear semisimple representations of $G$, where $k$ is some algebraically closed (complete) field.
Let $$P\_V(t) = \prod\_{i=1}^n (t - a\_i), P\_W(t) = \prod\_{j... | https://mathoverflow.net/users/196880 | Dimension of $\mathrm{Hom}_G(V, W)$ in terms of characteristic polynomial | If $k$ is of characteristic zero or characteristic $p$ and $G$ is a pro-$p'$-group, then yes. In that case, everything in sight is semisimple by Maschke's theorem, the eigenvalues $a\_i$ and $b\_j$ give you the decomposition into irreducibles, and every pair of equal eigenvalues contributes one degree of freedom by Sch... | 7 | https://mathoverflow.net/users/3041 | 414635 | 169,097 |
https://mathoverflow.net/questions/414488 | 3 | Let $0< \alpha \ll 1$. I'm trying to minimize $\int\_0^\pi |f'|^2 dx$ over the functions $f \in W\_0^{1,2}([0,\pi])$ (or at least find "good" lower bound in terms of $\alpha$) such that satisfy the following constraints:
$$
\begin{cases}
\int\_0^\pi f^2 dx = 1, \\
\int\_{\pi/3}^{2\pi/3} f^2 dx \leq \alpha.
\end{cases}
... | https://mathoverflow.net/users/173610 | Non convex optimization problem in $W_0^{1,2}$ | You can treat this as a problem with two Lagrange multipliers. Then by standard methods, a minimizer $f$ has to exist (by convexity in $f'$) and has to be a weak solution to
$$-f'' + \lambda f + \mu f \chi\_{[\pi/3,2\pi/3]} = 0$$
with $\lambda, \mu \in \mathbb{R}$ and $\mu \leq 0$ because the constraint is one-sided.
... | 1 | https://mathoverflow.net/users/51695 | 414638 | 169,098 |
https://mathoverflow.net/questions/414624 | 2 | Let's say we have two metric spaces, $(X,\rho)$ and $(Y,\tau)$. The continuity of $f:X\to Y$ is obvious and natural to define. What about semi-continuity? Without a natural ordering on $Y$, perhaps "upper" and "lower" won't make sense anymore. But are there other, weaker notions of continuity such that $f$ is continuou... | https://mathoverflow.net/users/12518 | Metric analogue of upper/lower semicontinuity | A "weak" generalization of continuity for a function $f:X\rightarrow Y$ with values in a metric space:
for each fixed $y\in Y$, consider the real valued function $f\_y:x\mapsto d(f(x),y)$;
for $f$ to be continuous each $f\_y$ must be continuous
(i.e. both upper and lover semicontinuous).
"Weak" here means that it is ... | 6 | https://mathoverflow.net/users/474159 | 414641 | 169,099 |
https://mathoverflow.net/questions/414629 | 5 | Let $K$ be a knot in $S^3$. Let $N(K)$ be a tubular neighborhood of $K$, a solid torus. On $\partial N(K)$, we may specify a *preferred* longitude $\lambda$, i.e., a simple closed curve whose linking number with $K$ is $0$. Also, we can choose a canonical meridian $\mu$ whose linking number with $K$ is $1$.
A $p/q$-*... | https://mathoverflow.net/users/475366 | Integral surgeries on $3$-manifolds | Two places in which this is discussed are Gompf and Stipsicz's book *4-manifolds and Kirby calculus* (Sections 5.2 and 5.3) and Ozbagci and Stipsicz's *Surgery on contact 3-manifolds and Stein surfaces* (Chapter 2). The reason this is not discussed in many knot theory references is that this is more of a 3.5-dimensiona... | 5 | https://mathoverflow.net/users/13119 | 414642 | 169,100 |
https://mathoverflow.net/questions/414617 | 0 | **Summary**
I would like to pass from a sequence of probability measures whose "limit" satisfies a desired property to a new probability measure that satisfies this property.
**Details**
We work on a probability space $(\Omega,\mathcal{F},P)$. Let $C\subseteq L^{\infty}$ be convex, contain $-L^{\infty}\_+$, and b... | https://mathoverflow.net/users/nan | Construction of a probability measure from a sequence of probability measures | $\newcommand\R{\mathbb R}$This is impossible to do in such generality.
E.g., suppose that $\Omega=\R$, $\mathcal{F}$ is the Borel $\sigma$-algebra over $\R$, $P$ is the standard normal distribution,
$$C=\{g\in L^\infty\colon g\le f\},$$
$f$ is the standard normal pdf, and $Q^n$ is the normal distribution with mean $n... | 2 | https://mathoverflow.net/users/36721 | 414649 | 169,103 |
https://mathoverflow.net/questions/414568 | 13 | Grothendieck topoi have cohomology: the abelian category of abelian group objects in a topos has enough injectives, hence one can consider the right derived functors of the global sections functor from abelian group objects to abelian groups. (For details see Chapter 8 in Johnstone's book *Topos theory*.)
Also, Groth... | https://mathoverflow.net/users/475784 | Do pretopoi have cohomology and homotopy groups? | There's a long story that can be told here but I will try to be brief.
In one sense, the answer is yes – you can certainly define cohomology and homotopy groups and so on for pretoposes and have them coincide with the classical definitions for Grothendieck toposes – but in another sense the answer is no – because you a... | 5 | https://mathoverflow.net/users/11640 | 414654 | 169,104 |
https://mathoverflow.net/questions/414552 | 5 | [Edit: added details on the Reed-Muller codes]
Are there explicit (non random) constructions of probability measures on $D\_N = \{0,1\}^N$ with support of size $O(N)$ and with all nontrivial Fourier coefficients less than $\frac 1 2$ is absolute value ?
I would expect that such questions are very well understood, b... | https://mathoverflow.net/users/10265 | Probability measure on the boolean cube with small support and small Fourier transform | Just to close the lid on my comment: the desired construction is known as an $\epsilon$-biased space, where here $\epsilon = 1/2$. The best known construction seems to be by Ta-Shma from 2017 (<https://eccc.weizmann.ac.il/report/2017/041/>), which yields a subset with support size $O(N/\epsilon^{2+o(1)})$. This is near... | 7 | https://mathoverflow.net/users/170770 | 414657 | 169,105 |
https://mathoverflow.net/questions/409371 | 2 | I have been reading the paper *"Coherent orientations for periodic orbits problems in symplectic geometry"* by Floer and Hofer, trying to understand how we can orient the moduli spaces that appear in Floer Homology in a way that is coherent.
So to do this we first fix on the problem on considering the Fredholm operat... | https://mathoverflow.net/users/155363 | Associativity of orientations of determinant bundles in Floer homology | The issue here are the isomorphisms in your suggested proof. There are choices and conventions involved. To give names to the isomorphisms clarifies what needs to be checked.
$$
\iota\_{K,L}:Det(K\sharp\_{\rho}L)\to Det(K)\otimes Det(L).
$$
You need to check that you've set things up so that:
$$
\iota\_{K\sharp\_{\rh... | 3 | https://mathoverflow.net/users/12605 | 414658 | 169,106 |
https://mathoverflow.net/questions/414661 | -1 | Let $(X, \lVert \cdot \rVert)$ be a separable normed space. Can we always guarantee that there is a nonempty compact set $K \subseteq B\_X$, where $B\_X$ is a closed unit ball in $X$ such that:
$$\forall \Lambda \in S\_{X^\*} \quad \sup\_{k \in K} \Lambda(k) > 0,$$
where $S\_{X^\*}$ denotes the unit sphere in the d... | https://mathoverflow.net/users/170491 | "Large" compact sets in separable normed space | Let $S=\{x\_n:n\in\mathbb N\}$ be a dense subset and choose
$\varepsilon\_n>0$ such that $y\_n=\varepsilon\_n x\_n\to 0$ (e.g., $\varepsilon\_n=1/n\|x\_n\|$). Then $K=\{ty\_n:n\in\mathbb N, |t|=1\}\cup\{0\}$ is compact and for every every $\Lambda\in X^\*\setminus \{0\}$ we have $\sup\{\Lambda(k):k\in K\}\ge |\Lambda(y... | 2 | https://mathoverflow.net/users/21051 | 414664 | 169,107 |
https://mathoverflow.net/questions/414663 | 1 | Assume we are over $\mathbb C$. Let $C$ be a complete algebraic curve, and $E$ an algebraic vector bundle. Its Hilbert polynomial is
$$p(t)=rt+r(1-g)+d$$
where $r=\mathrm{rank}(E)$ and $d=\deg(E)$ and $g$ is the genus of $C$. So a condition on the Hilbert polynomial is a condition on the rank, degree and genus.
It is... | https://mathoverflow.net/users/105537 | For a vector bundle over a curve, is there a condition on the Hilbert polynomial for no non-zero section? | If the bundle is semi-stable, then $H^0(E) \neq 0$ implies $\deg(E) \geq 0$. Indeed, a global section of $E$ induces a non-zero map of shaves $\mathcal{O}\_C \to E$, and semi-stability implies $$0= \mu (\mathcal{O}\_C) \leq \mu(E)=\frac{1}{r}\deg(E).$$ For strictly unstable bundles one cannot say anything. For instance... | 2 | https://mathoverflow.net/users/7460 | 414665 | 169,108 |
https://mathoverflow.net/questions/414521 | 3 | Suppose that $f\in \mathbb{Z}[x\_1,\dots,x\_n]$ and $f$ is a homogenous polynomial of degree $d$. Can we always construct $f$ such that the hypersurface $S\_f=\{x \in \mathbb{Z}^n:f(x)=0\}$ exhibits the failure of the Hasse principle? In particular, I am trying to understand why the non-singularity condition in results... | https://mathoverflow.net/users/392272 | Examples of non-singular hypersurfaces exhibiting Hasse principle failures | Daniel Loughran's suggestion to study the paper [Random Diophantine Equations by Poonen and Voloch](https://math.mit.edu/%7Epoonen/papers/random.pdf) was a good one.
Te conjecture 3.2 of the paper Daniel Loughran linked, together with Appendix A (both due to Colliot-Thélène), implies that the Hasse principle holds fo... | 4 | https://mathoverflow.net/users/18060 | 414681 | 169,110 |
https://mathoverflow.net/questions/414639 | 2 | This question arose by reading the paper **[1]**, in particular, the remark at p. 737:
>
> As an example, consider a non-isotrivial smooth projective morphism $f \colon X \to Y$ from a smooth projective surface to a smooth
> projective curve. Suppose that there exists a $y \in Y$ such that the
> Kodaira-Spencer map... | https://mathoverflow.net/users/7460 | Smooth, non-isotrivial fibration with vanishing Kodaira-Spencer map at a point | Any ramified base change of a smooth, non-isotrivial fibration should give an example.
You can give a simple proof using one of the various characterizations of the Kodaira spencer map:
<https://en.wikipedia.org/wiki/Kodaira%E2%80%93Spencer_map#In_scheme_theory>
Namely, if $f:X\to Y$ is a smooth fibration over a cu... | 5 | https://mathoverflow.net/users/112142 | 414682 | 169,111 |
https://mathoverflow.net/questions/414522 | 19 | Does anybody know any biographical information about [М. М. Артюхов](http://www.mathnet.ru/php/person.phtml?&personid=34845&option_lang=eng) (e.g., first name, affiliation)?
It seems he discovered a criterion for primality equivalent to the Solovay–Strassen one in 1966, in this paper: [Некоторые критерии простоты чис... | https://mathoverflow.net/users/658 | M. M. Artyukhov / М. М. Артюхов | In the books [1], pp. 41-42 and [2], p. 67, it is stated that his name (and patronymic) is Mikhail Mikhailovich, confirming [Anatoly Kochubei's comment](https://mathoverflow.net/questions/414522/m-m-artyukhov-%d0%9c-%d0%9c-%d0%90%d1%80%d1%82%d1%8e%d1%85%d0%be%d0%b2/414688#comment1063272_414525) to [Kostya\_I's answer](... | 13 | https://mathoverflow.net/users/113756 | 414688 | 169,113 |
https://mathoverflow.net/questions/414594 | 9 | I'm reading [Amplitudes and the Riemann Zeta Function](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.241602), which recently appeared in Physical Review Letters. It's received some [publicity](https://phys.org/news/2022-01-quantum-zeta-epiphany-physicist-approach.html), including [my own campus](https:/... | https://mathoverflow.net/users/6756 | Scattering amplitudes and the Riemann zeta function | Thanks to @reuns for the answer in the comments. I've asked him to post as an answer, and I will accept it if he does. Meanwhile, his comments encouraged me to look again, and here is another approach, quite easy (Lemma 4.11 in Equivalents of the Riemann Hypothesis vol. II)
Let $\Xi(s)=\xi(1/2+is)$, so $\Xi(-s)=\Xi(s... | 9 | https://mathoverflow.net/users/6756 | 414689 | 169,114 |
https://mathoverflow.net/questions/414619 | 0 | Consider the following """easier""" conjectures:
**C1**. every sum of two semiprimes $n = pq + rs$, $p,q,r,s$ primes, can be expressed as $n = (a + b)/2$; with $a,b$ primes.
**C2**. every number $p(q + r)$, $p,q,r$ primes (sum of two squarefree semiprimes that share a factor), can be expressed as $p(q + r) = (a + b... | https://mathoverflow.net/users/35419 | Name of conjectures similar to Goldbach conjecture | I don't believe either has a name in the literature.
Also, I don't know how much easier these conjectures would be. They don't seem to have any useful structure for the problem to take advantage of. C1 is almost surely equivalent to the Goldbach conjecture itself: probably every number greater than 33 (resp., 82) is ... | 3 | https://mathoverflow.net/users/6043 | 414703 | 169,119 |
https://mathoverflow.net/questions/414714 | 0 | Let $A\in \mathbb{R^{n\times n}}$ be a symmetric negative difinite matrix and
$D\in \mathbb{R}^{n\times n}$ be a diagonal matrix $D = \mathrm{diag}\{d\_i\}, (d\_i < 0)$.
From Weyl's inequality, the maximum eigenvalue of the sum of these matrices $S = A + D$ can be evaluated as follows.
$\sigma\_{1} \leq \alpha\_{1} +... | https://mathoverflow.net/users/475899 | Change in the largest eigenvalue due to perturbation of diagonal components of a symmetric matrix | You don't really have any stricter conditions, and in fact it is simple to reduce the general case to yours: every matrix is diagonal in *some* basis, and every matrix is negative definite if you subtract a suitable multiple of the identity to it.
So no, there can be no better bounds than the general case.
| 2 | https://mathoverflow.net/users/1898 | 414716 | 169,123 |
https://mathoverflow.net/questions/414710 | 1 | Let $V$ be a TRO i.e. closed subspace of $B(H,K)$ such that $xy^\*z \in V$ for all $x,y,z \in V$. Let $C(V)$ and $D(V)$ denotes the $C^{\ast}$-algebra generated by $VV^{\ast}$ and $V^\*V$ respectively. We define $A(V)$, the linking $C^\*$-algebra of $V$ as follows:
$$A(V) = \begin{bmatrix}
C(V) & V\\
V^\* & D(V)
\e... | https://mathoverflow.net/users/129638 | Let $V$ be a TRO such that $A(V)= \mathbb{C}$, what can we say about $V$? | So, I think when we write
$$ A(V) = \left[ \begin{matrix} C(V) & V \\ V^\* & D(V) \end{matrix} \right] $$
we implicitly mean taking the linear span. Thus $A(V) = \mathbb C$ means that $A(V)$ is spanned by a single matrix, which is necessarily a projection, as $A(V)$ is a $C^\*$-algebra. So $V$ must certainly be one-dim... | 1 | https://mathoverflow.net/users/406 | 414721 | 169,124 |
https://mathoverflow.net/questions/414268 | 0 | Are there any central simple algebras admitting a standard basis?
By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for Quaternion algebras $(a, b \mid F)$, satisfying relations $i^2=a, j^2=b, k=ij=-ji$ for $a, b \in F^\times$.
I saw [Cayley–Di... | https://mathoverflow.net/users/475480 | Are there any central simple algebras admitting a standard basis? | As suggested by [@Kimball](https://mathoverflow.net/questions/414268/are-there-any-central-simple-algebras-admitting-a-standard-basis/414750#comment1063039_414268) I develop my [comment](https://mathoverflow.net/questions/414268/are-there-any-central-simple-algebras-admitting-a-standard-basis/414750#comment1062278_4142... | 3 | https://mathoverflow.net/users/40297 | 414725 | 169,126 |
https://mathoverflow.net/questions/414729 | 2 | It is well-known that $[0,1]$ is not a nontrivial disjoint union of closed intervals -- e.g.:
<https://math.stackexchange.com/questions/1195179/the-interval-0-1-is-not-the-disjoint-countable-union-of-closed-intervals>
Using fat Cantor sets, I can construct disjoint unions of closed intervals in $[0,1]$ with arbitrari... | https://mathoverflow.net/users/12518 | Disjoint union of closed sets | Yes, sure: If you have a finite union of closed intervals, the complement in [0,1] is a finite union of open (in [0,1]) intervals. Hence, you find in this complement a finite union of closed intervals such that, if you add these intervals to your previous collection, the measure of the complement is only half of the me... | 6 | https://mathoverflow.net/users/165275 | 414732 | 169,129 |
https://mathoverflow.net/questions/414604 | 4 | Let $\mathcal{A} \subset \mathbb{N}$ be an infinite sequence with positive density, in the sense that
$$
\tag{1}
\lim\_{x\to\infty} \frac{|\mathcal{A} \cap x|}{x} = c > 0,
$$
and define the remainder term $r\_p(x)$ by
$$
r\_p(x) = \sum\_{\substack{n\leq x\\ n\in \mathcal{A}\\ p\mid n}} 1 - \frac{|\mathcal{A}\cap x|}{p}... | https://mathoverflow.net/users/307675 | Remainder terms of congruence sums in sets of positive density | Let $\mathcal A$ be the set of all $n\in \mathbb N$ with a prime factor $p>\sqrt{n}$. First, the number of elements $a\in \mathcal A$ with $a\leq x$ is $\gg x$. Indeed, this large prime factor is unique for any $a$, for a given $p\leq x$ there are $\left[\frac{\min(x,p^2)}{p}\right]$ integers $a\leq x$ with $p>\sqrt{a}... | 4 | https://mathoverflow.net/users/101078 | 414734 | 169,130 |
https://mathoverflow.net/questions/414684 | 4 | Is the following true? For every $n \geq 1, k\geq 2$, there is a set $S \subseteq [n]^k$ of size $|S| = n^2$ such that every two $k$-tuples in $S$ have at most one common entry.
Does anyone know if this is true? Is there a reference?
| https://mathoverflow.net/users/141963 | k-partite design | Assuming that "at most common entry" means that any two tuples $(x\_1,\ldots,x\_k)$ and $(y\_1,\ldots,y\_k)$ *match in at most one position*, that is, there is at most one $i$ such that $x\_i=y\_i$.
The claim is false for $n=2$ and any $k \ge 4$. We would need $n^2=4$ tuples $x,y,z,w$. Then each of $y$ and $z$ must d... | 1 | https://mathoverflow.net/users/171662 | 414741 | 169,132 |
https://mathoverflow.net/questions/414730 | 10 | Can this number $\ln \omega$ be written in $\{L|R\}$ form? What's its birthday?
| https://mathoverflow.net/users/10059 | In surreal numbers, what is $\ln \omega$? | In general this is taken to mean the value of Gonshor's logarithm at $\omega$. This was defined in the tenth chapter of his 1986 book *An introduction to the theory of surreal numbers* where you can find a justification for my answer.
In an informal way, the function $\ln$ is the "simplest" function that is eventuall... | 15 | https://mathoverflow.net/users/45005 | 414742 | 169,133 |
https://mathoverflow.net/questions/349352 | 4 | Let $X$ be a Banach space. Suppose $f:X^\*\to\mathbb R\cup\{\infty\}$ is convex, has weak\*-compact effective domain, and is weak\*-continuous on its effective domain. In particular, $f$ is weak\*-lower semicontinuous on $X^\*$.
Suppose I know $f$ is subdifferentiable at $x^\*\in \text{dom}(f)$, i.e. the subdifferent... | https://mathoverflow.net/users/145424 | Subgradient in a predual under weak* continuity | Finally, I was able to cook up a counterexample. We choose $X = c\_0$ (zero sequences equipped with supremum norm). Thus, the dual spaces are (isometric to) $X^\* = \ell^1$ and $X^{\*\*} = \ell^\infty$.
We define
$$
C := \{ x \in \ell^1 \mid \forall n \in \mathbb N : |x\_n| \le 1/n^2 \}$$
and $f \colon \ell^1 \to \ma... | 3 | https://mathoverflow.net/users/32507 | 414752 | 169,136 |
https://mathoverflow.net/questions/414708 | 0 | Let $G(V,E)$ be a symmetric graph with $n$ vertices and $m$ edges that has a $2\text{-factor}$ with edge set $F$, i.e. $F$ are the edges of an undirected vertex-disjoint cycle cover of $G$.
**Question:**
given only $F$ represented as an unordered sequence $\big( (u\_1,v\_1),\,\dots,\,(u\_n,v\_n)\big)$, what is the... | https://mathoverflow.net/users/31310 | Reconstructing a 2-factor from its edge set | Construct an undirected graph on the edge set $F$. This graph has each vertex with even degree, i.e. it's an Eulerian graph. Construct an Eulerian cycle in each connected component, e.g. following the [Hierholzer's algorithm](https://en.wikipedia.org/wiki/Eulerian_path#Hierholzer%27s_algorithm), which takes linear time... | 2 | https://mathoverflow.net/users/7076 | 414754 | 169,138 |
https://mathoverflow.net/questions/395980 | 2 | Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ideals of $H$ that is strictly increasing with respect to inclusion, where a principal left ideal of $H$ is a set of the f... | https://mathoverflow.net/users/16537 | Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR | A "close analogue" of (what I'm referring to as) *Cohn's theorem on atomic factorizations in cancellative monoids* (that is, Theorem 1 in the OP) is given by the unnumbered corollary on the bottom of p. 589 in P.M. Cohn's
* *Torsion modules over free ideal rings*, Proc. London Math. Soc. III. Ser. **17** (1967), 577-... | 1 | https://mathoverflow.net/users/16537 | 414758 | 169,139 |
https://mathoverflow.net/questions/414718 | 2 | Consider a stochastic process $X$ defined by
$$X\_t:=1+\int\_0^t b(s,X\_s) \, ds+ W\_t,\quad \forall t\ge 0,$$
where $(W\_t)\_{t\ge 0}$ is a standard Brownian motion. Suppose that $b:\mathbb R\_+ \times \mathbb R \to \mathbb R$ is Lipschitz and of linear growth so that $X$ is uniquely defined. Under what kind of co... | https://mathoverflow.net/users/nan | Search for conditions of the positive probability that a stochastic process never hits zero | $\newcommand\ep\varepsilon$The case of interest when $b(t,x)=b(t)$ depends only on $t$ is comparatively simple.
Indeed, let
$$g(t):=\sqrt{(2t+1/2)\ln\ln(3+t)}$$
for real $t\ge0$.
By the law of the iterated logarithm,
$$\sup\_{s\in[t,\infty)}\frac{W\_s}{g(s)}\to1$$
as $t\to\infty$ almost surely and hence in probabilit... | 1 | https://mathoverflow.net/users/36721 | 414771 | 169,142 |
https://mathoverflow.net/questions/414705 | 1 | $\DeclareMathOperator\SU{SU}\DeclareMathOperator\SL{SL}$I'm working through some of the constructions in [*Introduction to Arithmetic Groups* by Dave Witte Morris](https://arxiv.org/abs/math/0106063), and I'm confused by the construction of example 6.3.1 on page 121. For reference, here's the setup:
Take $a,b \in \ma... | https://mathoverflow.net/users/151664 | Clarification on arithmetic groups example | YCor gave a brief explanation in the comments. Based on your comment about seeking intuition, I think it would be helpful to explain concretely the backstory to the calculation YCor did:
Given any kind of arithmetically-defined group $\Gamma$, you want to find a real algebraic group / Lie group $G$ in which $\Gamma$ ... | 4 | https://mathoverflow.net/users/18060 | 414775 | 169,144 |
https://mathoverflow.net/questions/414773 | 2 | I've found through evidence and have conjectured on a math publication that:
$$\Big\lfloor\int\_1^\infty (k^{1/(k^{1+1/\sqrt{x}})} - 1)dk\Big\rfloor = \Big\lfloor\sum\_{k=1}^{\infty}k^{1/(k^{1+1/\sqrt{x}})} -1\Big\rfloor = x $$
where $ x \in \mathbb{N}, x>1$.
It is very hard to compute these values. Repeated Shan... | https://mathoverflow.net/users/nan | A conjecture relating an integral and a sum, the floor function and squares | First of all,
$$k^{1/k^t}=e^{(\log k)/k^t} = \sum\_{n = 0}^{\infty} \frac{\left( \log k \right)^n}{n! k^{n t}}$$
and therefore
$$\intop\_{1}^{\infty} \left( k^{1/k^t} - 1 \right) d k = \sum\_{n = 1}^{\infty} \frac{1}{n!} \intop\_{1}^{\infty} \left( \log k \right)^n k^{- n t} d k =$$
$$\sum\_{n = 1}^{\infty} \frac{1}{n!... | 3 | https://mathoverflow.net/users/88679 | 414777 | 169,145 |
https://mathoverflow.net/questions/390655 | 4 | Suppose that $p\ge 5$ is a prime, $n$ a positive integer divisible by $p-1$,
and $L<\mathbb F\_p^n$ a subspace of dimension $d=n/(p-1)$. Do there exist
vectors $l\_1,\dotsc,l\_n\in L$ such that the matrix with $l\_1,\dotsc,l\_n$ as
its columns has a nonzero permanent? Clearly, the answer is negative if $L$
is contained... | https://mathoverflow.net/users/9924 | Subspaces of vanishing permanent | Let $v\_1,\dots, v\_d$ be a basis for $L$. Then the permanent of the matrix obtained from $p-1$ repetitions each of $v\_1,\dots, v\_d$ is a polynomial function in the entries of $v\_1,\dots, v\_d$. Keeping all the entries but the last one in each $v\_i$ constant, we get a linear function. Since it is linear, among all ... | 1 | https://mathoverflow.net/users/18060 | 414778 | 169,146 |
https://mathoverflow.net/questions/414733 | 5 | Are there situations in which the [polygamma](https://en.wikipedia.org/wiki/Polygamma_function) pops up naturally in a mathematical physics context? In particular: are there examples of potentials having some interest for which the dependence on the distance is expressed in terms of $\psi^{(n)}$?
**Update**: While Ca... | https://mathoverflow.net/users/167834 | Polygamma function in mathematical physics | Abou-Salem, L. I., *A study on baryons spectroscopy using digamma-function as interacting potential*, <https://arxiv.org/abs/1311.6743> studies using the digamma function as an interaction potential (for quarks in baryons).
| 6 | https://mathoverflow.net/users/75890 | 414779 | 169,147 |
https://mathoverflow.net/questions/414623 | 4 | $\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel\_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quad... | https://mathoverflow.net/users/475824 | An analogy of product formula for homogeneous space? | (**Edit**: I revise most of my question as my first answer overlooked that $d\_1$ and $d\_2$ are odd.)
$\DeclareMathOperator{\res}{res}$
Let $E$ be an elliptic curve over $\mathbb{Q}$ with $E[2]\subset E(\mathbb{Q})$. Let $S$ be the normal $2$-Selmer group and let $S'$ be the relaxed Selmer group, that is the subset ... | 1 | https://mathoverflow.net/users/5015 | 414781 | 169,149 |
https://mathoverflow.net/questions/414782 | 0 | Can all elements of the Levi-Civita field be represented as power series of a single element
$$p=\varepsilon^{-1}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$$
where the numerators of the terms are given in <https://oeis.org/A118050> and the denominators are in <ht... | https://mathoverflow.net/users/10059 | Levi-Civita field in unusual basis | Let $p$ be any Laurent series in $\varepsilon$ of the form $\varepsilon^{-1}+\sum\_{n=0}^\infty a\_n\varepsilon^n$, like the one in the question. Then infinite Laurent series in $p$ itself never converge (not in the ring of Laurent series, or Levi-Civita field, or Hahn series, or any related such field), because positi... | 3 | https://mathoverflow.net/users/30186 | 414785 | 169,150 |
https://mathoverflow.net/questions/412741 | 1 | In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is diffeomorphic to a cartesian product of its maximal compact $ K $ with a contractible piece $ D/K $. So (again assuming $ G $ is con... | https://mathoverflow.net/users/387190 | Is the manifold of complex points of a quotient of compact groups just the tangent bundle? | This fact about the tangent space being the complexification is known to be true if $ G\_\mathbb{R}/H\_\mathbb{R} $ is a compact symmetric space. In other words, if $ H\_\mathbb{R} $ is the fixed points of an involution of $ G\_\mathbb{R} $. This includes, for example, all the spheres written as
$$
S^n \cong SO\_{n+1}(... | 0 | https://mathoverflow.net/users/387190 | 414792 | 169,153 |
https://mathoverflow.net/questions/414666 | 2 | I am interested in vector bundles over a nonsingular complete algebraic curve $C$ over $\mathbb C$. For a vector bundle $E$, its Harder-Narasimhan filtration is a filtration of subbundles
$$0=E\_0\subset E\_1\subset\cdots\subset E\_n=E$$
such that each $E\_i/E\_{i-1}$ is semitable and $\frac{\deg(E\_1/E\_0)}{\mathrm{ra... | https://mathoverflow.net/users/105537 | Do we know anything about Harder-Narasimhan filtrations of tensor products of vector bundles? | It is a result of Narasimhan and Seshadri that if $V$ is semistable and $W$ is semistable then $V \otimes W$ is semistable.
If $E$ has a filtration with associated graded $E\_i/ E\_{i-1}$, and $F$ has a filtration with associated graded $F\_j/ F\_{j-1}$, then $E \otimes F$ has a filtration with associated graded $(E\... | 4 | https://mathoverflow.net/users/18060 | 414793 | 169,154 |
https://mathoverflow.net/questions/414701 | 2 | I would like to prove that there are two sets $A,B\subset \mathbb{N}$ such that
* $A |\_T B$
* $\emptyset' \equiv\_T A\oplus B$
* for every $e$, if $\{e\}^{A\oplus B}=\emptyset'$ then the map sending $(B,n)$ to the prefix of $B$ used in the computation $\{e\}^{A\oplus B}(n)$ is not computable.
I believe the existen... | https://mathoverflow.net/users/130978 | Computing the halting problem with no computable bound on the use function | The answer is yes.
Since Chaitin's $\Omega$ is $wtt$-reducible to $\emptyset'$, we may replace $\emptyset'$ with $\Omega$.
Now let $A<\_T \emptyset'$ be a $K$-trivial but promptly simple set. Then there is an incomplete c.e. set $B$ so that $\emptyset'\equiv\_T A \oplus B$.
Suppose that there is such an $e$. Sinc... | 1 | https://mathoverflow.net/users/14340 | 414795 | 169,155 |
https://mathoverflow.net/questions/414756 | 5 | Let $X$ be a Polish space and let $G\in\mathbf{\Sigma}^1\_1(X^2)$ be a graph on $X$, that is an irreflexive and symmetric relation on $X$.
Given a cardinal $\kappa$ we say that $G$ has chromatic number $\kappa$, in symbols $\chi(G)=\kappa$, if there is a function $\varphi\colon X\to Y$ for some Polish space $Y$ such ... | https://mathoverflow.net/users/49381 | On the chromatic number of an analytic graph | There is a counterexample due to Todorcevic. See Proposition 9.2 in Kechris, Solecki, Todorcevic, Borel Chromatic Numbers, Advances in Mathematics, Vol. 141, Issue 1, 1999, Pages 1-44
| 6 | https://mathoverflow.net/users/475361 | 414797 | 169,156 |
https://mathoverflow.net/questions/414789 | 14 | Given a metric space $(X, d)$, we can consider the set of all quasi-isometries $f: X \to X$, and quotient out by the equivalence relation identifying $f$ and $g$ if $\sup\_{x \in X}d(f(x), g(x))$ is finite. Doing so, we obtain a set of equivalence classes $\mathcal{QI}(X)$ that is a group under composition.
In the sa... | https://mathoverflow.net/users/474271 | Quasi-isometry groups of metric spaces | A first observation is that $\mathcal{QI}$ is quite complicated for most natural spaces. For instance, any two linear maps $x \mapsto \lambda x, x \mapsto \lambda' x$ are equal in $\mathcal{QI}(\mathbb{R})$ if and only if $\lambda = \lambda'$. A corollary of this is that $\mathcal{QI}(\mathbb{N})$ is uncountable.
In ... | 11 | https://mathoverflow.net/users/136267 | 414798 | 169,157 |
https://mathoverflow.net/questions/414737 | 1 | Let $G=(V\_G,E\_G)$ and $H=(V\_H,E\_H)$ be two undirected graphs. When considering the *two-dimensional Weisfeiler-Leman* ($2$-$\mathsf{WL}$) procedure (the one corresponding to the three-variable fragment of first-order logic with counting quantifiers [[1](https://link.springer.com/article/10.1007/BF01305232)]), one t... | https://mathoverflow.net/users/9839 | Distinguishing graphs by 2-dimensional Weisfeiler-Leman vertex colourings | These notions of distinguishability are, in fact, equivalent. If 2-WL can distinguish $G$ from $H$, then their multisets of 2-WL labels are actually disjoint. And thus, in particular, there cannot be two vertices with the same 2-WL label. See Corollary 1 in "*On the Combinatorial Power of the
Weisfeiler-Lehman Algorith... | 2 | https://mathoverflow.net/users/475964 | 414802 | 169,158 |
https://mathoverflow.net/questions/414810 | 1 | Consider a family of smooth, atomless CDFs, $F\_x(\cdot)$, for each $x \in \mathbb R$. Suppose that $F\_x(\cdot)$ are FOSD ranked in $x$. That is, for any $x, x'$ such that $x \ge x'$, $F\_x(\cdot) \le F\_{x'}(\cdot)$.
Let $K > 0$ be a fixed scalar. I want to know whether there exists a function $g(\cdot)$ and a numb... | https://mathoverflow.net/users/78761 | Expectation of a function according to a family of distributions | $\newcommand\R{\mathbb R}$This is true for some first order stochastic dominance (FOSD) families but not all of them.
Indeed, if $F\_x$ does not depend on $x$ and $c:=E\_x1\_{[y\_\*,\infty)}>0$, then just let $g:=\dfrac Kc\,1\_{[y\_\*,\infty)}$.
On the other hand, suppose that $F\_x$ is the cdf of $Z+x$, where $Z$ ... | 1 | https://mathoverflow.net/users/36721 | 414825 | 169,162 |
https://mathoverflow.net/questions/414818 | 1 | Over the past few decades, a vast research area in number theory is surrounded by the $p$-adic number field $\mathbb{Q}\_p$ and its extensions.
My question is on different perspective.
What are the lists of some specialized journals that publishes works in $p$-adic number theory ?
I have in mind the following jou... | https://mathoverflow.net/users/122445 | Some good journals in $p$-adic number theory | Your question is somewhat broad, since local and p-adic fields permeate number theory and other parts of mathematics. If you want to get an idea of which journals publish articles about aspects of p-adic fields, you can look on MathSciNet. For example, there's an entire category
* 11S Algebraic number theory: local a... | 10 | https://mathoverflow.net/users/11926 | 414830 | 169,164 |
https://mathoverflow.net/questions/414713 | 11 | Let $G$ be a finite group, and $R(G)$ its representation ring over $\mathbb{C}$. We have the Adams operations $\psi^k:R(G)\rightarrow R(G)$, given on the level of characters by: $$\chi\_{\psi^k{V}}(g)=\chi\_V(g^k).$$
Since the power sums can be expressed as polynomials in the homogenous symmetric functions $h\_n$, thes... | https://mathoverflow.net/users/128502 | A question about the adjoint of the Adams operations on representation rings | If I understand the question correctly, then $\nu^{2}(1)$ isn't in the image of the natural map $A(G) \to R(G)$ when $G = Q\_{8},$ the quaternion group of order $8$.
The number of square roots of the identity in $G$ is $2$, the number of square roots of the central involution $z$ in $G$ is $6$, and the number of squa... | 6 | https://mathoverflow.net/users/14450 | 414833 | 169,165 |
https://mathoverflow.net/questions/414848 | 11 | In Besse's "Einstein manifolds", p. 177, he states that, until that moment, no general classification of homogeneous Einstein manifolds was know, even in the compact case. More specifically, he poses a problem: classify the compact simply connected homogeneous manifolds $M=G/K$ which admit a $G$-invariant Einstein metr... | https://mathoverflow.net/users/101432 | Classification of homogeneous Einstein manifolds | Yes, the question is still open.
I suggest to read [this quite recent paper](https://arxiv.org/abs/2107.06609) by Kerr and Böhm. It reviews some of the most important advances in the problem and includes several open problems.
| 13 | https://mathoverflow.net/users/20052 | 414852 | 169,174 |
https://mathoverflow.net/questions/414767 | 3 | Let's define a coprime graph as a simple graph (undirected graph without any self-loops or multiple-edges) in which for all edges $(, )$, the property $\gcd(\mathrm{degree}\_u, \mathrm{degree}\_v) = 1$ is true (i.e., degrees of vertices $$ and $$ are coprime).
I conjecture that, such a graph with maximum number of ed... | https://mathoverflow.net/users/475929 | Maximum number of edges in a "coprime graph" | OK, I can now confirm that even your modified conjecture is false, although the first counterexample has $13$ vertices.
First we need to know which complete $k$-partite graphs on $13$ vertices are coprime and how many edges they have. Both of these things can be determined directly from the sizes of the partite-sets.... | 5 | https://mathoverflow.net/users/1492 | 414864 | 169,178 |
https://mathoverflow.net/questions/414869 | 1 | I am a mathematician studying the dynamics of the $N$-Body density matrix $\rho\_{N}(x;y)$ for $n$ particles, defined by
$$\rho\_{N, t}^{(n)} (x\_1,..,n\_n; y\_1,...,y\_n) =
\begin{cases}
\int \rho\_{N,t}(x\_1,...,x\_n,x\_{n+1},...,x\_N; y\_1,...,y\_n,x\_{n+1},...,x\_N) dx\_{n+1}...d\_{x\_N},\,\,\,\,\, 1 \leq n \le N... | https://mathoverflow.net/users/471464 | Integration of Wigner transform | I think your coefficients have typo's, but in any case, you first want to make sure that your reduced density matrix is properly normalized to unit trace. For one particle, that would mean that
$$\int\rho^{(1)}\_N(x,x)\,dx=1.$$
The Wigner transform can be then be defined as
$$w\_N^{(1)}(x,v)= \frac{1}{2\pi} \int e^{- i... | 1 | https://mathoverflow.net/users/11260 | 414871 | 169,179 |
https://mathoverflow.net/questions/414868 | 4 | Is the following problem known? Suppose one is given some of the entries of an $n \times n$ matrix $A$ over $\mathbb{R}$, so that the given entries are symmetric. Can one assign values to the remaining entries so that the resulting matrix is positive definite?
Has this problem been studied from a theoretical or algor... | https://mathoverflow.net/users/141963 | Completing partial matrix to a positive definite matrix | There is an extensive literature. Here are some entry points:
* [Positive definite completions of partial Hermitian matrices](https://www.sciencedirect.com/science/article/pii/0024379584902076)
* [The positive definite completion problem revisited](https://doi.org/10.1016/j.laa.2008.04.020)
* [Matrix Completion Probl... | 4 | https://mathoverflow.net/users/11260 | 414872 | 169,180 |
https://mathoverflow.net/questions/414870 | 0 | Let $(X,d)$ be a separable and connected metric space. My question is rather short and to the point: do there exist $\{x\_n\}\_{n=0}^{\infty}\subseteq X$ such that
$$
\left\{d(x\_n,\cdot)-d(x\_0,\cdot)\right\}\_{n=1}^{\infty},
$$
is a Schauder basis of $C\_b(X)$? If so, what is this "basis" called in the literature?
| https://mathoverflow.net/users/469470 | When does $C_b(X)$ admit a Schauder Basis? | Note that in order for $C\_b(X)$ to have a Schauder basis, $X$ has to be compact. Indeed, $C\_b(X)$ is naturally isomorphic to $C(\beta X)$ and the latter is non-separable (because $\beta X$ is non-metrisable) [as long as $X$ is a non-compact metric space](https://math.stackexchange.com/a/217422/17929).
Thus, you are... | 2 | https://mathoverflow.net/users/15129 | 414873 | 169,181 |
https://mathoverflow.net/questions/414736 | 3 | Let $M$ be a complex manifold. Consider a connection $\nabla$ on the holomorphic tangent bundle $T^{1,0}M$. The *torsion* of $\nabla$ is defined as the torsion of the induced connection $D$ on the real tangent bundle, $$T\_\nabla(\alpha,\beta) = T\_D(\alpha,\beta) := D\_\alpha \beta - D\_\beta \alpha - [\alpha,\beta]$$... | https://mathoverflow.net/users/49151 | Torsion free (1,0)-connections on the holomorphic tangent bundle? | I will write in terms of the holomorphic frame bundle, i.e. the bundle of choices of complex linear bases of tangent spaces, a holomorphic principal $\operatorname{GL}\_n$-bundle.
Any sum $\gamma=\sum h\_a \gamma\_a$ with $\sum h\_a=1$ of $(1,0)$-connection forms on the holomorphic frame bundle is a $(1,0)$-connection.... | 3 | https://mathoverflow.net/users/13268 | 414874 | 169,182 |
https://mathoverflow.net/questions/414746 | 8 | Exponential sums are a powerful tool in additive combinatorics and number theory. In my understanding, when it comes to estimate the cardinality of a certain set, exponential sums are (essentially) used in this way: (1) the indicator function of the set is replaced by an appropriate exponential sum; (2) the sum of the ... | https://mathoverflow.net/users/475914 | Why are exponential sums so bad at solving this very easy problem? | The basic example of exponential sums in Number Theory is to count solutions to an equation such as $f(x\_1,\cdots,x\_n)\equiv 0$ $\mod p$ where $p$ is prime this is because $p$-adic solutions are a necessary condition for solutions in integers and also for getting methods like the [Hardy\_Littlewood circle method](htt... | 7 | https://mathoverflow.net/users/7113 | 414881 | 169,184 |
https://mathoverflow.net/questions/414858 | 3 | Consider some deterministic, monotonic, eroding binary cellular automata on some lattice $\mathbb{Z}^d$, and consider the set of initial states $I(L)$ in which all of the vertices are $0$ except for finitely many, and these finitely many $1$'s are within some hypercube of side length $L$ centered at the origin. I am in... | https://mathoverflow.net/users/153549 | Binary cellular automata: How slowly can an eroder remove $1$'s? | The answer is no.
According to [1] this is proved in [2] and in some paper of Toom from 1979 that does not seem to be listed in their bibliography. I cannot access [2] or any paper of Toom. I am a little unsure of myself here as I never had to write this stuff and don't have access to any of the classical references,... | 3 | https://mathoverflow.net/users/123634 | 414882 | 169,185 |
https://mathoverflow.net/questions/414875 | 9 | Let $X$ be a Banach space and let $\mathrm{Iso}(X)$ be its group of isometries, i.e., the set of surjective linear maps $T: X \to X$ with $\|Tx\| = \|x\|$.
**Q: Is $\mathrm{Iso}(X)$ a topological group under the strong topology?**
While it is easy to show that multiplication is continuous, it is not clear to me how... | https://mathoverflow.net/users/16702 | Group of isometries of Banach spaces a topological group? | That the inverse is continuous for the strong topology would actually be true for any *bounded* subgroup of $GL(X)$, the invertible operators on $X$.
Firstly, as translation is continuous, it suffices to consider continuity at the identity.
Now let $(T\_i)$ be a *bounded* net of invertibles converging strong to $I$... | 10 | https://mathoverflow.net/users/406 | 414892 | 169,188 |
https://mathoverflow.net/questions/414582 | 3 | This has been on MSE for over a month with four upvotes but no answers or even comments so I'm cross-posting:
According to [Examples of two-dimensional Riemannian manifolds that can't be isometrically embedded into $\mathbb{R}^4$](https://mathoverflow.net/questions/231495/examples-of-two-dimensional-riemannian-manifo... | https://mathoverflow.net/users/387190 | Intuition for no isometric embedding of round projective plane into $\mathbb{R}^4 $ | I'm not sure what you would accept as 'intuition' for this result. It's actually a simple consequence of two facts, which actually prove something much stronger:
The first fact is that, for any smooth surface $S\subset\mathbb{R}^n$ with positive Gauss curvature $K$, its mean curvature vector $H$ cannot vanish. This i... | 14 | https://mathoverflow.net/users/13972 | 414894 | 169,189 |
https://mathoverflow.net/questions/342606 | 21 | It's well-known that the norm on a $C^\ast$-algebra is uniquely determined by the underlying $\ast$-algebra by the spectral radius formula. Therefore there should be a way to axiomatize $C^\ast$-algebras directly in terms of the $\ast$-algebra structure, without explicitly talking about a norm.
**Question 1:** How do... | https://mathoverflow.net/users/2362 | Which $\ast$-algebras are $C^\ast$-algebras? | **TL;DR:** A $\*$-algebra embeds in a $C^\*$-algebra iff it is archimedean with no non-trivial infinitesimals. In this case the $\*$-algebra is a $C^\*$-algebra iff a certain norm is complete.
I do explicitly talk about a norm here, but it is explicitly constructed from a natural order structure associated with the $\*... | 4 | https://mathoverflow.net/users/89334 | 414904 | 169,191 |
https://mathoverflow.net/questions/414898 | 5 | Let $M$ be an irreducible 3-manifold with incompressible boundary of genus > 1.
When is $M$ homotopy equivalent to an Eilenberg-MacLane space? Or it is never true?
| https://mathoverflow.net/users/17895 | Irreducible 3-manifold with boundary of genus greater than 1 | M is always aspherical, and hence a Eilenberg-Maclane space.
This is because $\pi\_1(\partial M)$ embeds to $\pi\_1(M)$, by the incompressibility condition. If the genus of the boundary is no less than 1, then $M$ has infinite fundamental group. This forces the universal cover $\tilde M$ to be a non-compact 3-manifol... | 12 | https://mathoverflow.net/users/140203 | 414905 | 169,192 |
https://mathoverflow.net/questions/414877 | 10 | Find all integer solutions to the equation
$$
y(x^2+1)=z^2+1.
$$
There is, for example, an infinite family of solutions $x=u$, $y=(uv\pm1)^2+v^2$, $z=(u^2+1)v \pm u$, $u,v \in {\mathbb Z}$, but there are also solutions outside of this family, e.g. $(x,y,z)=(8,5,18)$ or $(x,y,z)=(12,2,17)$. The question is to describe a... | https://mathoverflow.net/users/89064 | Solve in integers: $y(x^2+1)=z^2+1$ | The equation says that $z^2 + 1 \equiv 0 \mod (x^2+1)$. For each positive integer $x$, you can enumerate the square roots of $-1$ in the integers mod $x^2+1$ as you remark in your last paragraph. This does not require "trial and error": if you can factor $x^2+1$, you can use the Tonnelli-Shanks algorithm
to find the sq... | 5 | https://mathoverflow.net/users/13650 | 414909 | 169,194 |
https://mathoverflow.net/questions/414911 | 10 | I'm interested in cohomology operations (in ordinary cohomology)
$$H^i(-, G)\rightarrow H^{i+1}(-, H)\;,$$
that is, elements of
$$H^{i+1}(K(G, i), H)\;.$$
I know that $K(G, 1)=BG$, so for $i=1$, those cohomology operations are in $H^2(BG, H)$, and therefore given by the Bocksteins of the corresponding central extension... | https://mathoverflow.net/users/115363 | Are all degree-1 cohomology operations Bocksteins? | Yes. For $i\ge1$ you can build $K(G,i)$ from the Moore space $M(G,i)$ by adding cells of dimension $\ge i+2$, so $H\_i(K(G,i); Z) = G$ and $H\_{i+1}(K(G,i); Z) = 0$. Hence $Ext(G, H) \cong H^{i+1}(K(G,i); H)$ by the UCT. The elements of $H^{i+1}(K(G,i); H)$ represent the cohomology operations $H^i(-;G) \to H^{i+1}(-;H)... | 16 | https://mathoverflow.net/users/9684 | 414914 | 169,195 |
https://mathoverflow.net/questions/414902 | 3 | A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.
This statement is from [3-Manifold Groups, page 18 (the link is editted)](https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/3-manifold-groups-final-version... | https://mathoverflow.net/users/140203 | Volume of hyperbolic 3-manifolds with toroidal boundary | I think you are trying to ask the following question.
>
> Suppose that $M$ is a compact connected oriented three-manifold. Suppose that $M^\circ$, the interior of $M$, admits a hyperbolic metric. Then when must this hyperbolic metric have finite volume?
>
>
>
As Ryan points out, if $M$ is closed, then $M^\circ... | 4 | https://mathoverflow.net/users/1650 | 414915 | 169,196 |
https://mathoverflow.net/questions/414244 | 1 | Given two distinct primes $P\_1$ and $P\_2$ picked randomly and uniformly in the interval $[T^2,2T^2]$ consider the set $\chi(P\_1,P\_2)$ of numbers of form $$xP\_1-yP\_2$$ where $x,y$ are in $[0,T^{1+\epsilon}]$ where $T>0$ and $\epsilon\geq0$ is small.
>
> Are there expected to be at least $\frac{T^\mu}{O(\log T)... | https://mathoverflow.net/users/10035 | Expected number of primes of particular size and from a linear form | The set $\chi(P\_1,P\_2)$ has $T^{2+2\epsilon}$ distinct elements in $[-2T^{3+\epsilon},T^{3+\epsilon}]$ (i.e., density of order $T^{\epsilon-1}$),
so we would expect $\chi(P\_1,P\_2)$ to contain about $T^{\mu+\epsilon}$ elements in $[C\_1T^{1+\mu}, C\_2T^{1+\mu}]$ provided that $0<C\_1<C\_2$. Of these, a fraction of a... | 0 | https://mathoverflow.net/users/7691 | 414925 | 169,199 |
https://mathoverflow.net/questions/414929 | 6 | According to the answers in the the following questions: [How to prove the spectrum of the Laplace operator?](https://math.stackexchange.com/questions/790401/how-to-prove-the-spectrum-of-the-laplace-operator?noredirect=1&lq=1) and [What is spectrum for Laplacian in $\mathbb{R}^n$](https://math.stackexchange.com/questio... | https://mathoverflow.net/users/105925 | Eigenvalues and eigenfunctions of the Laplace operator on entire plane | The point spectrum coincides with the spectrum minus 0 if $p>2n/(n-1)$ and it is empty in the remaining cases ($n$ is the dimension). This is proved in G. Talenti: "Spectrum of the Laplace operator acting in $L^p(R^n)$", Indam, Symposia Mathematica vol VII, Academic Press 1971.
| 8 | https://mathoverflow.net/users/150653 | 414932 | 169,202 |
https://mathoverflow.net/questions/414918 | 0 | Let $H$ and $K$ be Hilbert spaces and $D(T)$ a vector subspace of $H$. Let $T: D(T) \to K$ be a densely defined **antilinear** operator. Its adjoint $T^\*: D(T^\*)\to K$ is defined by the relation
$$\langle T^\*\eta, \xi\rangle = \langle T\xi,\eta\rangle$$
for all $\eta \in D(T^\*)$ and all $\xi \in D(T)$, where $D(T^\... | https://mathoverflow.net/users/216007 | Antilinear unbounded operator has closed graph | I'm not sure what you can expect here. Notice that if $T$ is *antilinear* then defining the "graph" as
$$ G(T) = \{ (T\xi, \xi) : \xi\in D(T) \} $$
does not give a subspace: it's not closed under (complex) scalar multiplication. The obvious way to fix this is to consider $T$ as a linear map $H\supseteq D(T)\rightarrow ... | 2 | https://mathoverflow.net/users/406 | 414939 | 169,204 |
https://mathoverflow.net/questions/414924 | 10 | In complex analysis, Hurwitz's theorem roughly states that, under certain conditions, if a sequence of holomorphic functions converges uniformly to a holomorphic function on compact sets, then after a while (above a certain rank ) those functions and the limit function have the same number of zeros in any open disk... | https://mathoverflow.net/users/40906 | What is the most general form of Hurwitz's theorem in complex analysis? | This has nothing to do with convexity. The exact formulation is this:
Let $\Omega$ be an arbitrary region, and $f\_n\to f$ is a sequence of holomorphic functions converging uniformly on compacts in $\Omega$.
If $f\neq 0$ (this is an important condition!), and $D$ is a region,
such that $\overline{D}\subset \Omega$, a... | 11 | https://mathoverflow.net/users/25510 | 414947 | 169,207 |
https://mathoverflow.net/questions/414922 | 3 | Let $X$ and $Y$ be Tychonoff (i.e. completely regular Hausdorff) topological spaces and let $\varphi:X\to Y$ be a continuous surjection that also has a property that $\operatorname{int}\overline{\varphi(U)}\ne\varnothing$, for any open nonempty $U\subset X$ (this property is called skeletal, weakly open, almost open et... | https://mathoverflow.net/users/53155 | Is a certain property of a continuous map preserved under a modification of the topology on the target space? | The answer to this question is negative. A suitable counterexample can be constructed as follows.
Let $Y=\mathbb R$ be the real line with the standard Euclidean topology.
Let $\mathbb Q$ be the subspace of rational numbers in $\mathbb R$. Write $\mathbb R\setminus \mathbb Q$ as the union $\bigcup\_{q\in \mathbb Q}X\_... | 2 | https://mathoverflow.net/users/61536 | 414954 | 169,210 |
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