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https://mathoverflow.net/questions/318901 | 11 | Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is *$F$-archimedean* if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$.
>
> Is it true that if $E$ is $F$-archimedean then every interval in $E$ contains an element in $F$? That is, is it true that for every $e<e'$ in $... | https://mathoverflow.net/users/43054 | Are archimedean subextensions of ordered fields dense? | Let $F$ be any real-closed field of uncountable cofinality. That is, every countable subset of $F$ is bounded. One can make such a field in a process of $\omega\_1$-many field extensions; alternatively, the ultrapower of $\mathbb{R}$ by nonprincipal ultrafilter on $\omega$ also has uncountable cofinality.
Let $E=F^\o... | 11 | https://mathoverflow.net/users/1946 | 318908 | 137,820 |
https://mathoverflow.net/questions/317290 | 1 | Consider the following versions of Möbius inversion:
1. Let $(A,+)$ be an abelian group, and let $f$ and $g$ be functions $\mathbb N\rightarrow A$. Then $$\left((\forall n )\;g(n)=\sum\_{d|n}f(d)\right)\;\Longleftrightarrow\;\left((\forall n)\;f(n)=\sum\_{d|n}\mu(n/d)g(d)\right).$$
2. Let $R$ be a commutative ring wi... | https://mathoverflow.net/users/46987 | Two versions of the Möbius inversion formula | One can work in the ring of operators that transform functions $\mathbb N \to A$ to functions $\mathbb N \to A$, for $A$ an abelian group - either a specific abelian group or operators defined on all ableian groups at once.
This is a noncommutative ring, but if you restrict to convolution operators, i.e. operators of... | 1 | https://mathoverflow.net/users/18060 | 318913 | 137,823 |
https://mathoverflow.net/questions/318911 | 16 | I am curious about stacky generalizations of the following GAGA theorem:
>
> **If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is algebraic.**
>
>
>
There is an established theory of analytic stacks (as well as higher analytic stacks). I am cur... | https://mathoverflow.net/users/7108 | GAGA for stacks | For your "special" question, the answer is negative, already when $A$ is an elliptic curve. In fact, a principal $A$-bundle over a smooth projective curve $B$ which is not topologically trivial is *never* algebraic — see the book by Barth, Hulek, Peters, Van de Ven, ch. V, Proposition 5.3. There are many examples of th... | 21 | https://mathoverflow.net/users/40297 | 318914 | 137,824 |
https://mathoverflow.net/questions/317196 | 2 | Denote $M\_I(\lambda)$ be the generalized Verma module with highest weight $\lambda$ and $L(\mu)$ is the simple highest weight module with highest weight $\mu$.
Suppose $\text{Ext}\_{\mathcal{O}^\mathfrak{p}}^i\left(M\_I(\lambda),L(\mu)\right)\neq 0$ for some $i$, does this implies $\lambda\le \mu$ or $\lambda=w\cdot\m... | https://mathoverflow.net/users/110229 | About Extension group and weights in $\mathcal{O}^\mathfrak{p}$ | See proof of Theorem 6.11 of *Representations of semisimple Lie algebras in the BGG category* $\mathcal{O}$ by James E. Humphreys. This theorem proves what you want in the case $\mathfrak{p}$ is a Borel subalgebra ($\mathfrak{p} = \mathfrak{b}$). You just need existence of projective covers in parabolic setting, statem... | 2 | https://mathoverflow.net/users/6818 | 318919 | 137,827 |
https://mathoverflow.net/questions/318920 | 11 | Consider the elliptic surface $E$ with affine equation
$$y^2 = x(x-1)(x-t^2)$$
over the base $\mathbf{P}^1$ with parameter $t$ (with complex scalar field). Then $E$ has four points of bad reduction, namely $0$, $1$, $-1$, and $\infty$. One can check that the reduction type is multiplicative at each bad place. Is th... | https://mathoverflow.net/users/271 | Elliptic curve over projective line with four points of multiplicative reduction | The classification of semistable families of elliptic curves with 4 singular points is due to [Beauville in](https://math.unice.fr/~beauvill/pubs/ellss.pdf) . The trick for the classification is to study, not the monodromy of the family of elliptic curves, but the monodromy of the $j$ invariant map as a cover of the pr... | 12 | https://mathoverflow.net/users/18060 | 318925 | 137,831 |
https://mathoverflow.net/questions/318784 | 3 | I'm working through the paper *Singular semipositive metrics in non-Archimedean geometry* by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. **25** (2016), 77-139, doi:[10.1090/jag/656](https://doi.org/10.1090/jag/656), arXiv:[1201.0187](https://arxiv.org/abs/1201.0187)).
In Section 1, subse... | https://mathoverflow.net/users/117432 | Definition of a vertical ideal sheaf and a vertical fractional ideal sheaf | In general, a fractional ideal sheaf on an integral scheme $X$ is a coherent $\mathcal{O}\_X$-submodule $\mathfrak{a}$ of the function field of $X$, thought of as a constant sheaf. In particular, for every open $U \subseteq X$, $\mathfrak{a}(U)$ is an $\mathcal{O}\_X(U)$-submodule of the function field, not necessarily... | 1 | https://mathoverflow.net/users/47692 | 318947 | 137,844 |
https://mathoverflow.net/questions/318844 | 3 | The usual framework for the Kantorovich duality in optimal transport theory uses Polish spaces as ground spaces for the distributions that should be transported. Are there results available that generalize the setting to pseudometric spaces? Or is this impossible? I will now specify the setting I am mainly interested i... | https://mathoverflow.net/users/133573 | Kantorovich duality with pseudometrics | Yes the Kantorovich Duality holds for continuous cost functions by following the proof in Villani's book without any change. The proof for general cost functions needs compactness of the set of couplings between the marginals which follows from Prokhorov's Theorem but might be true in more general settings.
Being Pol... | 2 | https://mathoverflow.net/users/123897 | 318965 | 137,850 |
https://mathoverflow.net/questions/318843 | 10 | Linked with this question and discussion
([Bilinear product of two summable families](https://mathoverflow.net/questions/289760/bilinear-product-of-two-summable-families)), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall that a
family $(a\_i)\_{i\in I}$ in a t... | https://mathoverflow.net/users/25256 | Unconditionally convergent series in some functional spaces | A good resource for these things is Section IV.10 of Schaefer's *Topological Vector Spaces*, so you should look there for the proofs of the following statements. For $E$ a locally convex space, let $\ell^1[E]$ denote the set of absolutely summable ($\mathbb{N}$-indexed) series and $\ell^1(E)$ the set of unconditionally... | 12 | https://mathoverflow.net/users/61785 | 318967 | 137,851 |
https://mathoverflow.net/questions/318972 | 7 | Today I read the following brief but insightful account of Ramanujan's approach to mathematics: <https://www.imsc.res.in/~rao/ramanujan/images/KSRchap3.pdf> and while reading this I wondered whether we have a lower-bound on the percentage of Ramanujan's conjectures which are correct.
I'm planning to get a copy of Ra... | https://mathoverflow.net/users/56328 | Percentage of Ramanujan's conjectures that were proven correct | [This interview with Prof. Bruce Berndt](https://web.archive.org/web/20130925201456/http://www.frontline.in/static/html/fl1617/16170810.htm) indicates the percentage of correct results from his notebooks to be greater than 99.7%. (See also this [longer writeup.](https://faculty.math.illinois.edu/~berndt/articles/aachen... | 15 | https://mathoverflow.net/users/11260 | 318976 | 137,856 |
https://mathoverflow.net/questions/318971 | 3 | Let $G$ be a Lie group acting on a complex manifold $M$. Let $p$ be an isolated fixed point. Let us look at the representation of $G$ on $T\_pM$. Suppose $T\_pM = \bigoplus V\_i^{\oplus n\_i}$ where $V\_i$ is an irreducible complex representation and $V\_i\ne V\_j$. Let us denote $V\_i^{\oplus n\_i}$ by $W\_i$ and call... | https://mathoverflow.net/users/88385 | Invariant submanifolds tangent to isotypic subrepresentations | By Bochner's theorem, in the complex analytic setting, if $G$ is a reductive complex Lie group acting holomorphically, or is a compact Lie group, then there such submanifolds, because the action is linearisable.
| 3 | https://mathoverflow.net/users/13268 | 318981 | 137,858 |
https://mathoverflow.net/questions/318987 | 3 | Let $X$ be a holomorphic symplectic variety of dimension $2n$ and $\pi: X \to \mathbb{P}^n$ be a Lagrangian fibration. It is known that $\pi$ is smooth outside of the discrimiant divisor $\Delta$. The divisor $\Delta$ is not necessarily irreducible and hence it can be singular. Does somebody know some resctrictions on ... | https://mathoverflow.net/users/98256 | The singularties of the dicriminant loci of the Lagrangian fibration | I am afraid not. As it is usual for discriminants, the singularities are rather complicated. Take for instance a K3 surface $S\subset \mathbb{P}^g$, with Picard group generated by $\mathcal{O}\_S(1)$. There is a holomorphic symplectic manifold $X$ with a Lagrangian fibration $\pi :X\rightarrow (\mathbb{P}^g)^\vee$, suc... | 5 | https://mathoverflow.net/users/40297 | 318988 | 137,859 |
https://mathoverflow.net/questions/318954 | 1 | Let $(M^n,g)$ be a Riemannian manifold, and $T$ a symmetric $(1,1)$-tensor field, i.e., $\langle T(X),Y\rangle = \langle X,T(Y)\rangle $. For convenience, denote $$\Delta\_Tu=\sum\_i\langle \nabla\_{e\_i}\nabla u, Te\_i\rangle $$
and
$$\mathrm{Ric}\_T(X,Y)=\sum\_i\langle R(X,e\_i)(Te\_i), Y\rangle , $$
where $u$ is a s... | https://mathoverflow.net/users/132279 | What is wrong with the derivation? | Let me re-do the computation using index notation.
Your goal is to compute $\partial\_r (\Delta\_T r)$ which we can rewrite as
$$ \nabla^c( T^{ab} \nabla^2\_{ab} r) \nabla\_c r = \nabla^c T^{ab} \nabla^2\_{ab} r \nabla\_c r + T^{ab} (\nabla^c \nabla\_a \nabla\_b r) \nabla\_c r$$
The first term on the right is $\... | 2 | https://mathoverflow.net/users/3948 | 318991 | 137,860 |
https://mathoverflow.net/questions/318982 | 1 |
>
> Evaluate $$I=\int\_{0}^{\infty} \frac{t\arg
> \Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$
> where $\Gamma(s)=\int\_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$
>
>
>
Note that $I$ converges since $\Gamma(s)\sim s\log s$. I tried Wolfram Alpha, but it hasn't given me an answer after almost... | https://mathoverflow.net/users/133634 | An integral involving the argument of the Gamma function and the Riemann Hypothesis | We prove that
$$I=-\frac{\pi}{4}(\gamma+\log 4).$$
$$I=\int\_0^\infty\frac{t\arg\Gamma(\frac14+\frac{it}{2})}{(\frac14+t^2)^2}\,dt.$$
$I$ is the imaginary part of the complex integral
$$\int\_0^\infty \frac{t\log\Gamma(\frac14+\frac{it}{2})}{(\frac14+t^2)^2}\,dt$$
using the usual branch of the logarithm of $\Gamma(s)... | 16 | https://mathoverflow.net/users/7402 | 318993 | 137,861 |
https://mathoverflow.net/questions/318917 | 15 | The **Rohrlich-Lang Conjecture for polynomial relations in Gamma values** predicts that all polynomial relations between Gamma values over $\mathbb Q$ come from the functional equations satisfied by the Gamma function. An other statement, somewhat narrower: if a quotient of products of gamma values over $\mathbb Q$ is a... | https://mathoverflow.net/users/29783 | A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture | It turns out that triplication is not needed here:
the recursion $\Gamma(z+1) = z \Gamma(z)$, the reflection formula
$$
\Gamma(z) \Gamma(1-z) = \frac\pi{\sin \pi z},
$$
and the duplication formula
$$
\Gamma(z) \Gamma\bigl(z+\frac12\bigr) = 2^{1-2z} \sqrt\pi \Gamma(2z)
$$
suffice to reduce the gamma product to a trigon... | 11 | https://mathoverflow.net/users/14830 | 319001 | 137,864 |
https://mathoverflow.net/questions/317561 | 28 | Let $K,L\subset\mathbb R^d$ be two ***disjoint*** compact convex sets with non-empty interiors. Can $x=0$ be a point of local minimum for the function $F(x)=\text{vol}\_d(\text{conv(K,L+x))}$?
I was asked this question by Dan Florentin a few weeks ago and I still cannot answer it in high dimensions (for $d=1,2$ the a... | https://mathoverflow.net/users/1131 | Can we always shift two disjoint convex bodies a little bit to decrease the volume of their convex hull? | This fails already for $d=3$.
Consider a tetrahedron, e.g. the convex hull of the points $v\_1,v\_2,v\_3,v\_4$. Let $K$ be the closed subset consisting of $\sum\_{i=1}^4 a\_i v\_i$ with $\sum\_{i=1}^4 a\_i=1$ and $a\_3 +a\_4 \leq 1/2-\epsilon$. Let $L$ be defined similarly, but $a\_1+a\_2 \leq 1/2-\epsilon$. Clearly ... | 21 | https://mathoverflow.net/users/18060 | 319008 | 137,869 |
https://mathoverflow.net/questions/318990 | 12 | In relation to [this question](https://mathoverflow.net/questions/318900/3-fold-of-general-type-homeomorphic-to-rational-3-fold), I would like to ask for examples of (complex) threefolds of general type with no (nontrivial) holomorphic form.
| https://mathoverflow.net/users/40297 | Threefolds of general type with no holomorphic forms? | One simply connected example arises as the total space of a family of Godeaux surfaces over $\mathbb{P}^1\_k$ with sufficiently positive discriminant.
Let $k$ be a field of characteristic different from $5$. Let $\mathbb{P}^1\_k$ be $\text{Proj}\ k[R,S]$. Let $\mathbb{P}^3\_k$ be $\text{Proj}\ k[T\_0,T\_1,T\_2,T\_3]$... | 8 | https://mathoverflow.net/users/13265 | 319009 | 137,870 |
https://mathoverflow.net/questions/318968 | 3 | In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the [spectral theorem](https://core.ac.uk/download/pdf/82151498.pdf) (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, allowing for a convenient notion of rank of a normal matrix over quaternion... | https://mathoverflow.net/users/118731 | Dimension of hermitian rank at most $k$ matrices over quaternions | Let $E$ be the space of hermitian matrices of size $m$ with quaternionic coefficients. This a $\mathbb{R}$-vector space of real dimension $4m(m-1)/2 + m = m(2m-1)$. For any $A \in E$, define the (right) kernel of $A$ : this is the subset of $\mathbb{H}^m$ defined by the condition $A \times x = 0\_{\mathbb{H}^m}$. Since... | 1 | https://mathoverflow.net/users/37214 | 319011 | 137,871 |
https://mathoverflow.net/questions/319022 | 1 | Let $(M,g)$ be a Riemannian manifold with boundary $\partial M$. Let $(g\_t)\_{0\leq t\leq T}$ be a family of Riemannian metrics with $g\_0=g$. Suppose that $\partial M$ has at least two connected components. Suppose that there exists $C>0$ such that $|Rm(g\_t)|\_{g\_t}\leq C$ and $|\mathcal{A}(g\_t)|\_{g^T\_t}\leq C$ ... | https://mathoverflow.net/users/125531 | Control distance of the boundary | No, consider a part of Lobachevsky plane bounded by two asymptotically parallel lines.
| 2 | https://mathoverflow.net/users/1441 | 319026 | 137,876 |
https://mathoverflow.net/questions/319038 | 13 | I am reading "[Some notes on Differentiable stacks](https://www.uni-due.de/%7Ehm0002/stacks.pdf)" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows.
>
> Let $G$ be a Lie group action on a manifold $X$ (left action). We define the quotient stack $[X/G]$ as $[X/G](Y):=\{P\xrightarrow{p... | https://mathoverflow.net/users/118688 | Motivation for definition of Quotient stack | Let's start approaching the question from the simplest possible case $Y=\*$. What should be the points of $[X/G]$?
Recall that the idea here is to generalize the construction of the [*action groupoid*](https://ncatlab.org/nlab/show/action+groupoid) for discrete groups acting on sets to the manifold case. This allows ... | 16 | https://mathoverflow.net/users/43054 | 319039 | 137,878 |
https://mathoverflow.net/questions/317736 | 4 | We consider the following 4 dimensional open manifold $$M=Gl(2,\mathbb{R})\setminus \{\lambda I\_2 \mid \lambda \in \mathbb{R}\}$$ where $I\_2$ is the identity $2\times 2$ matrix.
We consider the $2$ dimensional foliation $\mathcal{F}$ of $M$ tangent to the vector fields $X(A)=A, Y(A)=A^2$ for $A\in M$.
Is there a ... | https://mathoverflow.net/users/36688 | On certain 2 dimensional foliation of $Gl(2,\mathbb{R})$ deleted by scalar matrices | Since $A^2 = \mathrm{tr}(A)\,A - \det(A)\,I\_2$, and $\det(A)\not=0$ for $A\in M$, the leaves of your foliation are the same as the leaves of the foliation determined by the vector fields $X(A) = A$ and $Z(A) = I\_2$, namely, the leaf through $A$ is the connected component containing $A$ of the (open) set $\Pi^\*\_A$ c... | 8 | https://mathoverflow.net/users/13972 | 319041 | 137,880 |
https://mathoverflow.net/questions/319042 | 4 | To a finite $p$-group, we can associate two vectors $(v\_0,v\_1,\dotsc)$:
1. The class vector - $v\_i$ is the number of conjugacy classes of order $p^i$.
2. The character vector - $v\_i$ is the number of complex irreducible representations of dimension $i$ up to isomorphism.
Question: Do these two invariants of the... | https://mathoverflow.net/users/9672 | Do the class vector and character vector of a $p$-group determine each other? | The answer is no. This observation is cataloged at the Group Properties Wiki on pages [1](https://groupprops.subwiki.org/wiki/Conjugacy_class_size_statistics_need_not_determine_degrees_of_irreducible_representations) and [2](https://groupprops.subwiki.org/wiki/Degrees_of_irreducible_representations_need_not_determine_c... | 4 | https://mathoverflow.net/users/118731 | 319045 | 137,882 |
https://mathoverflow.net/questions/319044 | 1 | Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.
Denote by $\Phi$
the root system of $(\mathfrak{g},\mathfrak{h})$ and denote by $\mathfrak{g}\_\alpha$ the root subspace of $\mathfrak{g}$ corresponding
to a root $\alpha$.
We fix a choice of ... | https://mathoverflow.net/users/110229 | About locally finite condition in category $\mathcal{O}^\mathfrak{p}$ | Locally finite in this context means that every element (equivalently every finitely generated submodule) is finite-dimensional vector space. One reason to consider this condition is to see that it holds for (parabolic) Verma modules which are one of the basic modules that you can study (i.e. modules algebraically indu... | 1 | https://mathoverflow.net/users/6818 | 319048 | 137,884 |
https://mathoverflow.net/questions/317349 | 4 | Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x\_0\in S$ and define
$$
B\_\varepsilon=\{x\in X:\|x-x\_0\|\le\varepsilon\}\cap S.
$$
Consider the following problem:
>
> Question 1: Can we find $y\in X$ such that $\|y\... | https://mathoverflow.net/users/53175 | Weak closure of subsets of the unitary sphere of a Banach space | If you look at the definition of a locally uniformly convex Banach space in, for example, my book Ostrovskii, Metric Embeddings, page 291, you will see that any point $x\_0$ at which the condition of this definition is satisfied, has the property described in your Question 1. The converse does not seem to be true, as i... | 3 | https://mathoverflow.net/users/37822 | 319063 | 137,888 |
https://mathoverflow.net/questions/319058 | 18 | Denote $\square\_m=\{\pmb{x}=(x\_1,\dots,x\_m)\in\mathbb{R}^m: 0\leq x\_i\leq1,\,\,\forall i\}$ be an $m$-dimensional cube.
It is all too familiar that $\int\_{\square\_1}\frac{dx}{1+x^2}=\frac{\pi}4$.
>
> **QUESTION.** If $\Vert\cdot\Vert$stands for the Euclidean norm, then is this true?
> $$\int\_{\square\_{2n... | https://mathoverflow.net/users/66131 | Integrating over a hypercube, not a hypersphere | Yes, this is true, it follows from formulas in [Higher-Dimensional Box Integrals](https://projecteuclid.org/download/pdf_1/euclid.em/1317758103), by Jonathan M. Borwein, O-Yeat Chan, and R. E. Crandall.
\begin{align}
&\text{define}\;\;C\_{m}(s)=\int\_{[0,1]^m}(1+|\vec{r}|^2)^{s/2}\,d\vec{r},\;\;\text{we need}\;\;C\_... | 19 | https://mathoverflow.net/users/11260 | 319066 | 137,891 |
https://mathoverflow.net/questions/318827 | 4 | I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds by Bieberbach's Theorem. I am almost sure I missed something trivial, but what is special about studying flat solvmanifol... | https://mathoverflow.net/users/47336 | Flat solvmanifolds? | I guess there could be many answers to this question. First, it is not quite correct to say that there is a complete classification of flat manifolds, i.e. Bieberbach's Theorem does not produce a list of manifolds in each dimension. I don't know for which $n$ the number of $n$-dimensional crystallographic groups is kno... | 6 | https://mathoverflow.net/users/943 | 319071 | 137,893 |
https://mathoverflow.net/questions/319067 | 11 | I'm curious about a general answer for oblique planes, but specifically, I'm interested in the case where one circle's axis is perpendicular to the other's, and its center lies on the other's axis. To be precise, let $C\_1$ be the unit circle in the $XY$ plane, and $C\_2$ be a circle of radius $r$, center $(0, 0, h)$, ... | https://mathoverflow.net/users/133693 | What is the minimal surface connecting two circles that don't lie in parallel planes? | This is a quite subtle question and probably doesn't have an answer without further assumptions (and even then is possibly hard to say much).
First of all, when the circles are too far apart there are no connected minimal surfaces spanning them, just the two flat disk solutions.
When the circles are coaxial and lyi... | 13 | https://mathoverflow.net/users/127803 | 319075 | 137,894 |
https://mathoverflow.net/questions/319002 | 5 | It is a standard exercise (see Jech's "Set Theory" Exercise 13.8) to prove that ordinal addition and multiplication are $\Delta\_1$ expressible functions. The proof for addition comes from noting that $\alpha+\beta$ is order isomorphic to the disjoint union $(\{1\}\times \alpha)\cup (\{2\}\times \beta)$ under the lexic... | https://mathoverflow.net/users/3199 | Ordinal Exponentiation Levy Hierarchy | First, I claim that one can express in $\Delta\_1$ form the following property of $\alpha,\gamma, g$, which I'll abbreviate as $P(\alpha,\gamma,g)$: $\alpha$ and $\gamma$ are ordinals, $g$ is a function with domain $\gamma$, and, for each $\beta<\gamma$, $g(\beta)$ is the set of finitely supported functions $f:\beta\to... | 7 | https://mathoverflow.net/users/6794 | 319077 | 137,895 |
https://mathoverflow.net/questions/319081 | 1 | Let $I=\bigoplus\_n M\_n(\Bbb C)$,can we construct a non-unital $C^\*$ algebra $A$ such that $I$ is essential in $A$ and $A/I\cong K(H)$ for some separable infinite Hilbert space.
---
[note added by YC: this question is a minor variant on previous questions by the OP, see [construct a non-unital nuclear $C^\*$ al... | https://mathoverflow.net/users/63864 | construct a non-unital nuclear $C^*$ algebra | For each $k$, each $x \in M\_k$, and each $n \geq k$, let $x^{(n)} \in M\_n$ be the matrix whose upper left $k \times k$ corner is $x$ and which is zero elsewhere. For $n < k$ let $x^{(n)} = 0$.
Let $A$ be the norm closure of the \*-algebra $A\_0$ generated by $\bigoplus M\_n$ and, for all $k \in \mathbb{N}$ and $x \... | 3 | https://mathoverflow.net/users/23141 | 319083 | 137,896 |
https://mathoverflow.net/questions/318959 | 10 | Let $R$ be a Riemann surface that admits a non-constant bounded holomorphic function. Then is it true that any two points of $R$ can be separated by a bounded holomorphic function? This is easy to see when $R$ is a planar domain.
| https://mathoverflow.net/users/36038 | Bounded holomorphic functions on a Riemann surface separating points | The answer is no.
See
Stanton, Charles M., Bounded analytic functions on a class of open Riemann surfaces. Pacific J. Math. 59 (1975), no. 2, 557–565.
Stanton shows that a branched cover $R$ of the disk is separated by $H^\infty(R)$ if and only if the branch points lie over a set that is the zero set of a Blaschk... | 10 | https://mathoverflow.net/users/1335 | 319088 | 137,897 |
https://mathoverflow.net/questions/319086 | 2 | I have a problem for which I either need a proof or a counterexample.
We are given two discrete random variables $x\_1$ and $x\_2$ in $[0, n]$ where $F\_1(x)$ is the probability of $x\_1\leq x$, and similarly $F\_2(x)$ is the probability of $x\_2 \leq x$. My goal is to show that there always exists a number $c > 0$ t... | https://mathoverflow.net/users/130779 | Given two probability density functions find a number that satisfies a given equation | The OP has just been edited from an apparently absolutely continuous version to a discrete one. So, I gather that the integrals are now to be understood with respect to the counting measure, that is, as the sums.
However, there is no essential difference here between the absolutely continuous and discrete versions, ... | 1 | https://mathoverflow.net/users/36721 | 319091 | 137,900 |
https://mathoverflow.net/questions/319073 | -1 | Let $\zeta$ denote the Riemann zeta function. In this answer: <https://mathoverflow.net/a/314066/133634>, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int\_1^\infty (\sum\_{1 \le n \le x} 1) x^{-s-1}dx$$ for $\Re(s)>1$, and then claims (without proof) that this representation yields
$$\frac{\ze... | https://mathoverflow.net/users/133634 | On a certain representation of the Riemann zeta function | Let $0<\Re(s)<1$ unless otherwise indicated. Then
$$\int\_0^\infty (\sum\_{1 \le n \le x} 1-x) x^{-s-1}\,dx=-\int\_0^1 x^{-s}+\int\_1^\infty (\sum\_{1 \le n \le x} 1-x) x^{-s-1}\,dx.$$
On the right hand side, the first integral equals $1/(s-1)$. The second integral equals $\zeta(s)/s-1/(s-1)$ as can be seen by proving ... | 2 | https://mathoverflow.net/users/11919 | 319095 | 137,901 |
https://mathoverflow.net/questions/319093 | 10 | Let $(M,\omega)$ be a symplectic manifold, and let $\mathscr{J}$ be the set of compatible almost complex structures on $M$.Finally let $A \in H^2(M,\mathbb{Z})$. Then we can consider the moduli space of parametrised J-holomorphic curves defined as follows.
$\mathscr{M}(A,J) = \{ (u,J)~|~ J\in \mathscr{J} ~ \text{ and... | https://mathoverflow.net/users/92483 | Moduli space of curves | Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). The latter is more accurate if you want to correctly model the topology of the moduli space (with a fixed number of mar... | 7 | https://mathoverflow.net/users/12310 | 319096 | 137,902 |
https://mathoverflow.net/questions/319094 | 2 | So I'm looking at the proof of the ZBC lemma in Odifreddi's Classical Recursion Theory volume 2 page 808 and I don't see why $ 0' \oplus C$ produced computes $B'$ as claimed. The positive requirements try and code
$$
P^C\_e: \; x \in C^{[e]} \iff (\exists z > x) B^{[e]}(x) \not= B^{[e]}(z)
$$
Now if these requiremen... | https://mathoverflow.net/users/23648 | Correct Proof Of ZBC Theorem From Odifreddi? Also Extension Question | A proof of Harrington’s ZBC Lemma can be found in Theorem 2.5 of
*Hinman, Peter G.; Slaman, Theodore A.*, [**Jump embeddings in the Turing degrees**](http://dx.doi.org/10.2307/2274700), J. Symb. Log. 56, No. 2, 563-591 (1991). [ZBL0745.03036](https://zbmath.org/?q=an:0745.03036).
It [consists](https://math.berkele... | 2 | https://mathoverflow.net/users/4600 | 319097 | 137,903 |
https://mathoverflow.net/questions/319102 | 0 | Let $\mathrm{M}\in\lbrace0,1\rbrace^n$ be the adjacency matrix of a graph $\mathrm{G}\left(V,E\subseteq\lbrace\lbrace u,v\rbrace| u,v\in V\rbrace\right)$ of order $n$.
Let $\mathrm{G}$ additionally be metric in the sense, that $$\lbrace v\_i, v\_j, v\_k\rbrace\subseteq V\ \ \wedge\ \ w\_{uv}\in\lbrace 0,1\rbrace\ \ ... | https://mathoverflow.net/users/31310 | Estimating Maximal-Clique of Metric Graphs via the Rank of their Adjacency Matrix | I also assume that the graph contains at least one edge, else the maximal clique contains 1 vertex while the rank is 0. In any graph (not necessarily metric) which contains at least one edge the rank of the adjacency matrix is not less than the size of maximal clique (look to corresponding minor).
Your metric conditi... | 1 | https://mathoverflow.net/users/4312 | 319113 | 137,905 |
https://mathoverflow.net/questions/319084 | 13 | *I asked [this question on MSE](https://math.stackexchange.com/q/3043380/496634) but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level.*
A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In [this question](https://math... | https://mathoverflow.net/users/130100 | Is there a substitution that relates every Fermat curve to an elliptic curve? | The following reference seems to answer the question of which Fermat curves admit a non-constant map to an elliptic curve:
>
> Neal Koblitz, David Rohrlich, *Simple factors in the Jacobian of a Fermat curve*, Canadian Journal of Mathematics **30** No. 6 (1978) pp. 1183–1205, doi:[10.4153/CJM-1978-099-6](https://doi... | 19 | https://mathoverflow.net/users/6506 | 319114 | 137,906 |
https://mathoverflow.net/questions/317701 | 2 | Suppose I have a system of polynomials which are homogeneous but of distinct degrees that I want to solve simultaneously:
$$F\_1(z\_1,\ldots,z\_n)=\cdots=F\_m(z\_1,\ldots,z\_n)=0.$$
Let $X(\mathbb F)$ denote the solutions to this system over the field $\mathbb F$. For the application I have in mind, the system is o... | https://mathoverflow.net/users/132495 | Relate solutions to a polynomial system in complex numbers to solutions in a finite field | If $X(\mathbb{C})$ is empty, or all solutions satisfy some further equation $G(x\_1, \ldots, x\_n)=0$, then the same holds true for $X(\mathbb{F}\_p)$, provided that $p$ is sufficiently large. To see this note that you can check these properties algorithmically by computing resultants, Gröbner bases, or Macaulay determ... | 2 | https://mathoverflow.net/users/37555 | 319118 | 137,909 |
https://mathoverflow.net/questions/319005 | 4 | Let $G$ be a sufficiently nice topological or Lie group (e.g. compact), and let $H$ be a closed subgroup. This data determines a principal $H$ bundle $G \rightarrow G/H$ defined by the projection $g \mapsto gH$.
In this case, we have Haar measures on $G$ and $G/H$ which allow us to perform invariant integration. I am... | https://mathoverflow.net/users/57456 | Invariant integration on principal bundles | $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\GL}{GL}$ $\DeclareMathOperator{\Aut}{Aut}$
You should start with a simple case
$$M=\bR^n,\;\; G=\GL\_m(\bR),\;\;P=G\times M. $$
In this case
$$
\Aut(P)=C^\infty\big(\; \bR^n,\;\GL\_k(\bR)\;\big).
$$
The action of gauge transformation $\gamma:\bR^n\to\GL\_m(\bR... | 1 | https://mathoverflow.net/users/20302 | 319119 | 137,910 |
https://mathoverflow.net/questions/318770 | 3 | Let $(X\_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f\_{s,t} : X\_t\to X\_s$ the maps, and $\Lambda\_n := f\_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $X\_0$ with value $\Lambda$, an abelian group.
Clearly $\Lambda\_n$ is the constant sheaf on $X\_n$ with value $\... | https://mathoverflow.net/users/nan | Sheaves on solenoids | I think that the sheaf $\mathcal{F}\_\Lambda$ is the constant sheaf with stalk $\Lambda$.
A basis of open sets in a solenoidal space is given by the sets $f\_k^{-1}(U\_k)$ where $U\_k \subseteq X\_k$ is open and $f\_k \colon X \to X\_k$ is the canonical projection. Now if you take $U\_k \subseteq X\_k$ so small that ... | 1 | https://mathoverflow.net/users/54441 | 319128 | 137,912 |
https://mathoverflow.net/questions/319107 | 10 | Let $M$ be an oriented closed smooth manifold, and let $C=M\times[0,1]$, the cylinder over $M$. Let $g$ be an arbitrary Riemannian metric on $C$ (in particular, $g$ may look nothing like a product metric). Let $f:C\to\mathbb{R}$ be the unique $g$-harmonic function satisfying $$f|\_{M\times\{i\}}=i,\quad i=0,1.$$
**Ques... | https://mathoverflow.net/users/132436 | Can a harmonic function on a topological cylinder have critical points? | If $M$ has dimension at least $2$ the harmonic function $f$ can have critical points. This can be quickly deduced from Calabi's characterisation of harmonic $1$-forms.
**Example.** Suppose $M=S^2$. Then it is not hard to construct a Morse function $f$ on $M\times [0,1]$ satisfying the following properties
1) $f(M,... | 7 | https://mathoverflow.net/users/943 | 319133 | 137,914 |
https://mathoverflow.net/questions/318986 | 1 | Let $\kappa < \lambda < \mu$ be infinite cardinals. Is there a collection ${\cal U}\subseteq {\cal P}(\mu)$ of subsets of $\mu$ with the following properties?
1. for all $U\in {\cal U}$ we have $|U| = \lambda$;
2. every $S\subseteq \mu$ with $|S| = \kappa$ is contained in exactly one member of ${\cal U}$; and
3. for ... | https://mathoverflow.net/users/8628 | On a combinatorial set covering property | Extending the problem, admit the possibility of equality of cardinals, so consider that $\kappa \leq \mu$. If indeed $\kappa=\mu$, then the collection of subsets with one member, namely $\mu$, suffices. But that is the only case.
For we borrow the argument from the comments. Let $U$ be a proper subset of $\mu$ that i... | 1 | https://mathoverflow.net/users/3402 | 319138 | 137,916 |
https://mathoverflow.net/questions/319131 | 3 | Given a positive integer $n$, the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance) $d^H\_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H\_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$
Given an integer $n>0$ and a set $S\subseteq \{0,1\}^n$ with $|S| = n$, is it possible to find a map $f:S... | https://mathoverflow.net/users/8628 | Stretching map of $n$ points from $\{0,1\}^n$ to $\{0,1\}^{n+1}$ with respect to their Hamming distance | Here is an infinite family of counterexamples. Let $n = 2^r$. The $2^r$ distinct rows of a $2^r \times 2^r$ Hadamard matrix with entries from $\{0,1\}$ are all at distance $2^{r-1}$. However the maximum number of binary words all at distance $2^{r-1} + 1$ in $\{0,1\}^{2^r+1}$ is just $2$. *Proof.* We can suppose that $... | 6 | https://mathoverflow.net/users/7709 | 319140 | 137,917 |
https://mathoverflow.net/questions/319072 | 0 | I have been working with the chromatic polynomials of *circulant graphs of prime order* $p$ with two distinct parameters, i.e.
$P\_{p,i,j}(x):= P(C\_{p}(i,j),x)$ with $1 \leq i \neq j \leq \ n/2.$
In this regard, I have a couple of questions. Let $p$ be a prime number as above, and $1 \leq i' \neq j' \leq \ n/2$ ... | https://mathoverflow.net/users/133696 | Chromatic Polynomials of Circulant Graph With Two Parameters | I assume that $C\_p(i,j)$ means the graph with vertices $0,\dots,p-1$ and edges between each pair of vertices with difference $i$ or $j$ mod $p$. If that is the case, then identity 1 does not appear to be true in general; Sage gives different chromatic polynomials for $C\_{11}(1,2)$ and $C\_{11}(1,3)$. (By the way, you... | 5 | https://mathoverflow.net/users/8604 | 319142 | 137,918 |
https://mathoverflow.net/questions/311094 | 0 | If you don't mind, please consider the eigenvalue problem
$$ (1-x^2)u''+ \lambda u=0 \ \ \ \forall x\in (-1,1), $$
$$ u(\pm 1) = 0. $$
Observe that for suitable values of $\lambda$, the ODE resembles a Gegenbauer differential equation (on the boundary of the typical parameter range) that yields an orthonormal basis of ... | https://mathoverflow.net/users/33849 | Function approximation via an orthonormal basis (with singular weight) | I hate to answer my own question but since nobody else has ... I now realise that this is essentially a ``singular" Jacobi/Gegenbauer equation ... i.e. it is not technically a Jacobi/Gegenbnauer equation due to the singular weight but it qualitatively satisfies lots of relations satisfied by Jacobi polynomials, but wit... | 0 | https://mathoverflow.net/users/33849 | 319151 | 137,922 |
https://mathoverflow.net/questions/319116 | 5 | Pick a sequence of real numbers $x\_i$ as follows. Put $x\_0=1$. If $x\_i$ is chosen, then pick $x\_{i+1}\in[0, x\_i]$ according to the uniform distribution. Obviously we have $x\_i\rightarrow 0$ with probability 1. Put $I\_i=[x\_i, x\_{i-1}]$.
Next we pick $n$ random numbers $y\_1, \ldots, y\_n$ in $[0, 1]$ independ... | https://mathoverflow.net/users/37555 | Estimating the size of the remainder in a random partition | Here is a heuristic that I am sure can be made rigorous. The $x\_i$'s can be written recursively as $x\_{i+1}=U\_{i+1}x\_i$, where the $U\_i$ are independent Unif[0,1] random variables. In particular, $x\_n=U\_n\cdots U\_1$, so that $\log x\_n=\log U\_n+\ldots+\log U\_1$. By the strong law of large numbers, $(1/n)\log ... | 3 | https://mathoverflow.net/users/11054 | 319157 | 137,924 |
https://mathoverflow.net/questions/319110 | 5 | Let $G$ be a virtually abelian group. Are there any general results on the existence or non-existence of Cartan subalgebras in the generated group $C^\*$-algebra or group von Neumann algebra?
---
| https://mathoverflow.net/users/64444 | Cartan subalgebras in the group algebras of virtually abelian groups | For von Neumann algebras in the discrete case Cartan subalgebras always exist. Indeed, if we only assume that the group is amenable then its von Neumann algebra is finite and injective, so it is of the form $\bigoplus\_{n \in \mathbb{N}} A\_n \otimes R\_n$, where $R\_1=\mathcal{R}$ is the unique hyperfinite $\mathrm{II... | 1 | https://mathoverflow.net/users/24953 | 319164 | 137,928 |
https://mathoverflow.net/questions/319156 | 1 | How to recognize, by "analytic" methods, if a $C^0$ vector field $v:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is the gradient of a function $h:\mathbb{R}^n \rightarrow \mathbb{R}$, given that the verification of the path independence of the line integrals may be unpractical and that one cannot verify the symmetry $\partia... | https://mathoverflow.net/users/66889 | How to recognize if a continuous vector field in the Euclidean space is a gradient | Let $(\phi\_k)$ be any sequence of mollifiers such that $v\*\phi\_k\to v$ uniformly for each $C^0$ vector field $v$. Then we have
>
> **Theorem.** A $C^0$ field $v$ is a gradient field iff for all $k,i,j$ we have $\partial\_{i}v\_k^j =\partial\_{j} v\_k^i$, where $v\_k:=v\*\phi\_k$.
>
>
>
*Proof.* Suppose th... | 1 | https://mathoverflow.net/users/36721 | 319166 | 137,930 |
https://mathoverflow.net/questions/319108 | 2 | I have a question about properties of transient diffusion process.
In the case of $d$-dimensional Brownian motion $B=(B\_t,P\_x)$ ($d \ge 3$), we can prove that
\begin{align}
(1)&\quad 0<P\_{x}(\sigma\_{K}<\infty)\quad \text{ for any }x \in \mathbb{R}^d,\\
(2)&\quad \lim\_{|x| \to \infty,\ x \in \mathbb{R}^d}P\_{x}(\... | https://mathoverflow.net/users/68463 | Hitting probability of a transient diffusion process | I believe that the infimum is always zero. This follows by the following argument, valid for any standard Markov process with infinite lifetime.
Let $T$ be the hitting time of $K$, and write $f(x) = P\_x(T\_K < \infty)$. Suppose, contrary to our claim, that $f(x) \ge c > 0$ for all $x$, and define $g(x) = (f(x) - c) ... | 2 | https://mathoverflow.net/users/108637 | 319177 | 137,932 |
https://mathoverflow.net/questions/317080 | 9 | Quillen famously constructed a fibration sequence $BGL(\mathbb{F}\_q)^+ \to BU \to BU$ to compute the algebraic K-groups of finite fields, where the second map is $\psi^\ell-1$ for $\ell$ a generator of $\mathbb{Z}\_q^\times$. Does this lift to the level of the K-theory spectra?
I don't know much about delooping, but... | https://mathoverflow.net/users/120548 | Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$? | The infinite loop space/spectrum level statements were written down in
May, Quinn, Ray, Tornehave:
"$E\_\infty$ Ring Spaces and $E\_\infty$ Ring Spectra" (1977) [http://www.math.uchicago.edu/~may/BOOKS/e\_infty.pdf](https://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf) ,
see Theorem VIII.3.2 and its proof on pages ... | 5 | https://mathoverflow.net/users/9684 | 319183 | 137,934 |
https://mathoverflow.net/questions/319146 | 1 | I have a question regarding the proof of theorem 4.6 in <https://arxiv.org/abs/1007.3899> (Hall's conjecture).
Let $S^2$ be the class of Borel subsets in $\mathbb{R}^2$ with finite and positive Lebesgue measure $|\, .\, |$. For a given set $E\in S^2$ we denote with $B\_E$ the ball centered in the origin with $|E|=|B\... | https://mathoverflow.net/users/129823 | convergence and a mean curvature condition imply convexity | Actually one does not use the convergence of $Q(E\_j)$, but one implicitly use (iii) of theorem 3.2: your sets are the interiors of Jordan curves that are $C^1$ close to the unit circle. Then positive curvature implies convexity, and since the curvature goes to $1$ uniformly, it is eventually positive.
To see that po... | 0 | https://mathoverflow.net/users/4961 | 319199 | 137,937 |
https://mathoverflow.net/questions/319028 | 5 | Let $\mathbb{H}$ be the Poincare half plane with the hyperbolic metric. Let $Gl(2,\mathbb{R})$ be equipped with a left invariant metric?
>
> Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to $\mathbb{H}$? If yes, what is a precise formula for such a Riemannian submersion?
>
>
>
| https://mathoverflow.net/users/36688 | Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to the Poincare half plane? | Define the hyperbolic metric $g$ on $\mathbb{H}^2$ by
$$
g = \frac{1}{2y^2} (d x^2 + d y^2).
$$
The Iwasawa decomposition yields an isomorpism $SL(2)/SO(2)=\mathbb{H}^2$ which is given by
$$
\pmatrix{a\_{11} & a\_{12}\\ a\_{21} & a\_{22}} \mapsto \frac{1}{(a\_{21}^2 + a\_{22}^2)}(a\_{11} a\_{21} + a\_{12} a\_{22}, 1).... | 5 | https://mathoverflow.net/users/17047 | 319223 | 137,945 |
https://mathoverflow.net/questions/172729 | 16 | It is well known that one can specify a [complex structure](http://en.wikipedia.org/wiki/Complex_manifold) on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an [integrable almost complex structure](http://en.wikipedia.org/wiki/Almost_complex_manifold... | https://mathoverflow.net/users/2622 | Analog of Newlander–Nirenberg theorem for real analytic manifolds | I think this question can be addressed in a few ways. Two great answers have already been given:
1. Real analytic is a regularity condition, while holomorphic is more algebraic, so you'd need a somewhat different condition than NN.
2. Every $C^1$ manifold can be given a real analytic structure, so you don't even need... | 4 | https://mathoverflow.net/users/27404 | 319227 | 137,949 |
https://mathoverflow.net/questions/319220 | 2 | Let $E$ an **infinite dimensional** complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.
**Definition:** Let $T \in \mathcal{L}(E)$. The Moore-Penrose inverse of $T$, denoted by $T^{+}$, is defined as the unique linear extension of $(\bar{T})^{-1}$ in
$$D(T^{+}) = \mathca... | https://mathoverflow.net/users/113054 | A formula related to the Moore-Penrose pseudo-inverse of Hilbert space operators | What may be tripping you up here is that $T^+$, as you have defined it, could be unbounded, if the range of $T$ is not closed. So the answer to
>
> Is the formula (1) true only for matrix or even for Hilbert space operators?
>
>
>
is ... sort of, depending on what you mean by taking a limit. I'm not sure there... | 3 | https://mathoverflow.net/users/23141 | 319236 | 137,951 |
https://mathoverflow.net/questions/316656 | 1 | Let $ \xi \_t(\omega), t\in[0,\infty)$, be a random process and let $ \xi \_t(\omega)\in \{\mathfrak F\_t\}$ be some filtration. Even if $ \xi \_t(\omega) $ is $ \mathfrak F\_t $ measurable then $\int\_0^t\xi \_s(\omega)L(ds)$ is not necessarily $ \mathfrak F\_t $ measurable. I am looking for a process that will demons... | https://mathoverflow.net/users/131512 | An example of a measurable random process with non-measurable integral | The following example is a variant of an example of Sierpinski that Martin Väth told me; it depends on the Continuum Hypothesis. Let $<^\*$ be a well-ordering of $[0,1]$. By CH, $\{s: s<^\*t\}$ is always at most countable. Let $M\subset [0,1]$ and define $k: [0,1]\times [0,1]\to \mathbb{R}$ by $k(s,\omega)=1$ if $\omeg... | 2 | https://mathoverflow.net/users/127871 | 319237 | 137,952 |
https://mathoverflow.net/questions/319217 | 16 | Given a function $f: \mathbb{R}\to\mathbb{R}$, we define its left modulus of continuity, $L(f): \mathbb{R} \times (0, \infty)\to [0,\infty]$ by
$$L(f)(x, e) := \sup \{d \ge 0 \,:\, f((x, x+d)) \subseteq [f(x) - e, f(x) + e]\} $$
Similarly define the right modulus of continuity,
$R(f): \mathbb{R} \times (0, \infty)... | https://mathoverflow.net/users/132446 | Are continuous functions almost completely determined by their modulus of continuity? | I think they are. Assume the contrary, then $f$ is not constant and therefore there exists a point $a$ such that $f$ is not constant neither on $[a,+\infty)$ nor on $(-\infty,a]$. Call such points admissible. The admissible points form an interval (possibly infinite). Without loss of generality $0$ is admissible point ... | 6 | https://mathoverflow.net/users/4312 | 319240 | 137,953 |
https://mathoverflow.net/questions/319202 | 6 | Suppose an algebraic variety $V$ is given as the solutions to $q$ polynomial equations of degree $\le k$ with real coefficients
$$p\_1(x\_1,\dots,x\_m)=0,\dots,p\_q(x\_1,\dots,x\_m)=0$$
for $x\in\mathbb R^m$. It is a theorem due to Milnor that the sum of Betti numbers of $V$ is bounded by $k(2k-1)^{m-1}$. In particular... | https://mathoverflow.net/users/133752 | Number of connected components of degree 2 affine algebraic varieties | No. There are better bounds for particular cases (for instance for $3$ quadrics as given in the reference below), and in general it remains an open problem to find sharp bounds on the maximal number of components of a real affine/projective variety of given degree. The paper [here](http://www.fen.bilkent.edu.tr/~degt/p... | 4 | https://mathoverflow.net/users/118731 | 319241 | 137,954 |
https://mathoverflow.net/questions/319207 | 2 | We are looking for graphs with certain properties that have a specific genus. We constructed a simple family, but now realised that we actually only have an upper bound for the genus. Is there an easy way to show that the genus is actually the desired value?
Take the cartesian product of two sufficiently large cycles... | https://mathoverflow.net/users/15684 | Genus for specific family of graphs | Since the genus is additive over connected components, it is sufficient to find $g$ vertex-disjoint subdivisions of $K\_{3,3}$, one in each copy of $C\_{10}\times C\_{10}$. This will show that the genus of the whole graph is at least $g$. The same approach may be used after blowing-up the vertices into cycles.
| 4 | https://mathoverflow.net/users/24076 | 319248 | 137,958 |
https://mathoverflow.net/questions/319255 | 8 | Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: k\in \mathbb N\})=\dfrac 1{n \log n},\forall n>1$ ?
| https://mathoverflow.net/users/127118 | On the existence of a particular type of finite measure on $\mathbb N$ | Other than the fact that the measure's value on $\{1\}$ is completely unspecified by the given constraint, there is only one possibility for such a measure (call it $\lambda$) if it exists:
the [Möbius inversion formula for infinite series](https://math.stackexchange.com/questions/2391847/m%C3%B6bius-inversion-for-infi... | 13 | https://mathoverflow.net/users/5091 | 319260 | 137,961 |
https://mathoverflow.net/questions/318918 | 1 | Recall the (Jacobson-)Witt Lie algebras in positive characteristic: $W(n,1)$ is the Lie algebra of derivations of $\Bbbk[X\_1,\dots,X\_n]/(X\_1^p,\dots,X\_n^p)$. (For simplicity; more generally, I'm interested in the $W(n,m\_1,\dots,m\_n)$ which are derivations of a ring of divided power series truncated at $X\_1^{p^{m... | https://mathoverflow.net/users/10481 | Centralizers in Jacobson-Witt Lie algebras | For $W(n,1)={\rm Der}(\mathcal{O}\_n)$, defined over an algebraically filed $k$ of characteristic $p>2$, the smallest dimension of centralizers equals $n$ (here $\mathcal{O}\_n$ is the $k$-algebra $k[X\_1,\ldots, X\_n]/(X\_1^p,\ldots, X\_n^p)$ of truncated polynomials in $n$ variables). Let $\mathcal{L}$ be a finite di... | 3 | https://mathoverflow.net/users/24386 | 319270 | 137,964 |
https://mathoverflow.net/questions/319287 | 3 | Let $(X,\tau)$ be connected such that $\emptyset$ and $X$ are the only open connected subsets. Does this imply that $\tau = \{\emptyset, X\}$?
| https://mathoverflow.net/users/8628 | Connected topological space $X$ such that $\emptyset, X$ are the only open connected subsets | Let $X\_0$ be any topological space in which no nonempty open set is connected, e.g., $X\_0 = \mathbb{Q}$ with the usual topology. Then let $X = X\_0 \cup \{\infty\}$ with open sets $\{$all open subsets of $X\_0\} \cup \{X\}$. The whole space is connected because the only open set containing $\infty$ is the whole space... | 10 | https://mathoverflow.net/users/23141 | 319295 | 137,972 |
https://mathoverflow.net/questions/319293 | 7 | Let $X$ be a spectrum. Is there a canonical construction/functor that would associate to this spectrum, an inverse spectrum $X'$, in the sense that $$\pi\_\*(X)\cong \pi\_{-\*}(X')?$$
To be more precise, such a spectrum $X'$ can always be constructed by attaching cells to produce the right homotopy groups, but is the... | https://mathoverflow.net/users/128857 | Inverting homotopy groups of spectra | Not exactly what you asked for, but there is the Brown-Comenetz dual of $X$, $I\_{\mathbb{Q}/\mathbb{Z}}X$, which has the property that its homotopy groups are Pontryagin dual to the homotopy groups of $X$ (in the negative degree):
$\pi\_{-\ast}(I\_{\mathbb{Q}/\mathbb{Z}}X) \simeq Hom(\pi\_\ast(X),\mathbb{Q}/\mathbb{... | 15 | https://mathoverflow.net/users/7762 | 319297 | 137,974 |
https://mathoverflow.net/questions/319306 | 6 | A quick question.
>
> Let $j : {\sf cat} \to {\sf Cat}$ be the inclusion of small categories in locally small categories; is $j$ a dense functor?
>
>
>
If it is not (as I expect, since it's hard for a generic large category $X$ to be determined by the canonical colimit over $(j\downarrow X))$), is it true that... | https://mathoverflow.net/users/7952 | Is the embedding ${\sf cat}\subset {\sf Cat}$ dense? | Yes it is. In fact, the full subcategory of $Cat$ on the walking commutative triangle is dense. So $cat$, being a full supercategory of a dense category, is dense.
| 10 | https://mathoverflow.net/users/2362 | 319307 | 137,979 |
https://mathoverflow.net/questions/319303 | 10 | Suppose $f:\mathbf{N} \to [0,1]$ satisfies $$f(x)f(y) + f(x+y)\leq 1\qquad(1)$$ for all $x,y$. Let $$d\_n = \frac{1}{n} \sum\_{x=0}^{n-1} f(x).$$ It is easy to prove that $$\limsup d\_n \leq 1/\varphi,$$ where $1/\varphi$ is the positive solution to $t^2+t=1$. (Start with $(1)$, take an average and $\limsup$ in $x$, th... | https://mathoverflow.net/users/20598 | Maximize $f(0)+\cdots+f(n-1)$ subject to $f(x)f(y) + f(x+y) \leq 1$ | This system seems simple enough to throw to a CAS or some optimizer for small $n$. The inequalities are sequential: once you have understood behaviour for $n$, you can add a few more constraints for $n+1$ and try to understand the expanded system. To illustrate, I use $x\_n$ in place of $f(n)$.
The first round is to ... | 3 | https://mathoverflow.net/users/3402 | 319312 | 137,981 |
https://mathoverflow.net/questions/319264 | 36 | I am interested in the following question:
>
> Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$?
>
>
>
I suspect the answer is no, but I don't know how to prove it.
Currently, all I have is that the map cannot be locally 1-to-1. For if $f$ is locally 1... | https://mathoverflow.net/users/92293 | Does there exist a continuous 2-to-1 function from the sphere to itself? | There is no such function as I will prove below using the work of [Civin (1943)](https://doi.org/10.1215/S0012-7094-43-01006-3) and [Kerékjártó (1919)](https://doi.org/10.1007/BF01463232).
Let me review what we need from the work of [Civin (1943)](https://doi.org/10.1215/S0012-7094-43-01006-3). Let $f$ be a continuou... | 31 | https://mathoverflow.net/users/11919 | 319326 | 137,983 |
https://mathoverflow.net/questions/319261 | 5 | Our goal is to sample from the Laplace distribution conditioned on a linear subspace. Here are the details of this problem.
Let
$$p(x) \propto \exp(-\|x\|\_1/\sigma)$$
be the pdf of the Laplace distribution, where $x = (x\_1, \dots, x\_d)\in \mathbb{R}^d$. Consider the conditional distribution $p(x\,|\,\sum^d\_{i = 1... | https://mathoverflow.net/users/133774 | What is the pdf of Laplace distribution conditioned on a plane? How can I sample from it? | Let $n:=d$. By rescalling, without loss of generality $\sigma=1$.
Let $X\_1,\dots,X\_n$ be iid random variables (r.v.'s) with the standard Laplace distribution, so that the joint pdf of $X:=(X\_1,\dots,X\_n)$ is
\begin{equation}
f\_X(x)=\frac1{2^n}\,\exp\Big\{-\sum\_1^n|x\_i|\Big\}
\end{equation}
for $x=(x\_1,\dots,... | 2 | https://mathoverflow.net/users/36721 | 319330 | 137,985 |
https://mathoverflow.net/questions/319339 | 0 | Suppose $f(x, e): \mathbb{R} \times (0, \infty)\to [0,\infty]$ is right continuous in $x$, and monotone increasing in $e$. Is $f$ jointly measurable?
| https://mathoverflow.net/users/132446 | Are these conditions enough to ensure joint measurability? | This is a direct consequence of the more general Lemma 6.4.6 in V.I. Bogachev, Measure Theory, Vol. 2 (2007).
| 2 | https://mathoverflow.net/users/100904 | 319350 | 137,990 |
https://mathoverflow.net/questions/319292 | 1 | This is when studying about Chern classes from Kobayashi and Nomizu.
Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$.
Let $p:P\rightarrow M$ be associated principal $G$ bundle. Let $\mathfrak{g}=\mathfrak{gl}(r,\mathbb{C})$ denote the Lie algebra of $G$.
... | https://mathoverflow.net/users/118688 | determinant of curvature (notation issue) | Curvature $\Omega$ is a $\mathfrak{g}$ valued $2$-form on $P$ i.e., for each $p\in P$, we have $\Omega(p):T\_pP\times T\_pP\rightarrow \mathfrak{g}$.
As $\mathfrak{g}$ is $\mathfrak{gl}(r,\mathbb{C})$, given $(v\_1,v\_2)\in T\_pP\times T\_pP$, we get a $r\times r$ matrix $\Omega(p)(v\_1,v\_2)=[a\_{ij}]$.
Once we ... | 1 | https://mathoverflow.net/users/118688 | 319353 | 137,993 |
https://mathoverflow.net/questions/319338 | 6 | Let $p\_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p\_n \alpha\}$ is equidistributed. Is it known whether the sequence $\{p\_n^2 \alpha \}$ is equidistributed?
| https://mathoverflow.net/users/133802 | Equidistribution of $\{p_n^2α\}$ | Yes - this follows from a general theorem of Bergelson, Kolesnik, Madritsch, Son, and Tichy (Theorem 2.1 in <https://people.math.osu.edu/bergelson.1/BKMS_PrimePowers.pdf>):
Let $\xi(x)=\sum\_{j=1}^m\alpha\_j x^{\theta\_j}$ be a polynomial with real coefficients $\alpha\_i\in\mathbb{R}$ such that $0<\theta\_1<\cdots <... | 12 | https://mathoverflow.net/users/385 | 319354 | 137,994 |
https://mathoverflow.net/questions/319288 | 3 | Let $\mathcal{M}$ be the field of meromorphic functions of one (complex) variable and $w = w(z)$ an analytic function satisfying a polynomial equation
$P(w; z) := w^n + a\_{n-1}(z) w^{n-1} + \cdots + a\_1(z) w + a\_0(z) = 0$,
where $a\_0(z), \ldots, a\_{n-1}(z)$ are in $\mathcal{M}$ (actually, is suffices to consid... | https://mathoverflow.net/users/133759 | On finite extensions of the field of meromorphic functions | First, as you noticed, it is enough to consider the case that the equation has the form
$$w^n+a\_{n-1}(z)w^{n-1}+\ldots+a\_0(z)=0,$$
where the coefficients are entire. Then $w$ is holomorphic on its Riemann surface, let us call this Riemann surface $S$. From your condition follows that $S$ is a compact Riemann surface... | 3 | https://mathoverflow.net/users/25510 | 319362 | 137,998 |
https://mathoverflow.net/questions/319347 | 0 | Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at $\bar x$. Does this imply that $\bar x$ is a Lebesgue point for $u$?
| https://mathoverflow.net/users/nan | Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$ | **The answer is no**. There are many counterexamples, but the following one is particularly interesting:
Let
$$
f(x)=\begin{cases}
\sin (1/x), & 0<x<1,\\
0 & \text{otherwise.}
\end{cases}
$$
Then $f\in L^1(\mathbb{R})$ has compact support equal $[0,1]$ and $0$ is not a Lebesgue point of $f$. However,
$$
\frac{1}{t}\... | 0 | https://mathoverflow.net/users/121665 | 319363 | 137,999 |
https://mathoverflow.net/questions/310759 | 4 | This question regards colourings on edges and vertices on countable directed multigraphs.
We start with an example. Let $G=\mathbb Z^2$. We define two functions $a\_h$ and $a\_v$ from $\mathbb Z^2$ to $\mathbb C$ with the property that
for all $\epsilon>0$,
$$
\{(n,m)\in\mathbb Z^2 : |a\_h(n,m)a\_v(n+1,m)-a\_v(n,m)a... | https://mathoverflow.net/users/47948 | Asymptotic colouring of edges and vertices, and untwisting cocycles | There indeed exists $a$ on $\mathbf{Z}^2$, valued in $\mathbf{U}$, the unit circle in the complex plane, such that no $d$ (valued in $\mathbf{C}^\*$) works.
---
When all functions ($a\_h$, $a\_v$, $d$) are assumed to have values in $\mathbf{U}$, the question can be rephrased so as only to rely on the structure o... | 1 | https://mathoverflow.net/users/14094 | 319368 | 138,001 |
https://mathoverflow.net/questions/319046 | 3 | Let $C$ be an algebraic curve of genus $g \geq 2$ defined over a number field $K$, having good reduction outside of a finite set $S$ of primes in $K$. A *singly branched* cover $C'$ of $C$ is a curve such that there exists a map $\pi : C' \rightarrow C$ and a point $x \in C(K)$ such that $\pi$ is only branched at $x$. ... | https://mathoverflow.net/users/10898 | Comparing the height of a curve and a singly branched cover | If $f:C'\to C$ is a singly branched cover, the (Faltings) heights of $C$ and $C'$ are certainly related.
First, by Lemma 6.1 in [1], the inequality $$h(C) \leq h(C') + \log(2\pi) g(C') \log(\deg f)$$ holds. Since $g(C') \leq 3 g(C) \deg f$, this implies that
$$ h(C) \leq h(C') + 100 g(C) \log(\deg f) \deg f.$$
Pr... | 2 | https://mathoverflow.net/users/4333 | 319371 | 138,003 |
https://mathoverflow.net/questions/319372 | 6 | I've seen the following theorem attributed to Lurie:
>
> Theorem. There is an equivalence of $(\infty,1)$-categories between $E\_n$ algebras and locally constant factorization algebra on $\mathbf{R}^n$.
>
>
>
And the reference is usually given as Lurie's Higher Algebra. However it's a 1500+ pages book and sear... | https://mathoverflow.net/users/82513 | Precise reference for the equivalence of $E_n$ algebras and locally constant factorization algebra? | The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra
>
> **Theorem 5.4.5.9.** Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Composition with the map
> $$\mathrm{Disk}(M)^⊗ → \mathbb{E}\_M^⊗$$
> of Remark 5.4.5.8 induces a fully faithful embedding
> $$θ:\mathrm{Alg}\_{\mathbb{E}... | 8 | https://mathoverflow.net/users/43054 | 319373 | 138,004 |
https://mathoverflow.net/questions/319379 | 17 | I'm hoping the answer to this is well-known.
Let $X$ be an ordered set (i.e. poset). An **inflationary operator** $f$ on $X$ is a function $f: X \to X$, not necessarily order-preserving, such that $f(x) \geq x$ for all $x \in X$. I'm interested in the following result:
>
> **Theorem** Let $X$ be an ordered set in... | https://mathoverflow.net/users/586 | How is this fixed point theorem related to the axiom of choice? | I'll deduce Zorn's Lemma from your fixed-point theorem. Suppose $P$ is a poset violating Zorn's Lemma; so all chains in $P$ have upper bounds, but there's no maximal element. Consider the poset $Q=P\times\omega$ with the lexicographic ordering; that is, in $P$ replace every element by a chain ordered like the set $\ome... | 19 | https://mathoverflow.net/users/6794 | 319382 | 138,007 |
https://mathoverflow.net/questions/319141 | 6 | Let dgCat be the category of small dg-categories. The well-known [Dwyer-Kan model structure](https://ncatlab.org/nlab/show/model+structure+on+dg-categories) makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-morphisms dg-functors, and $2$-morphisms (degree $0$,... | https://mathoverflow.net/users/24965 | Does the Dwyer-Kan model structure make dgCat a model $2$-category? | No. If dgCat were a model 2-category, then the 2-functor from dgCat to Cat that sends a dg-category $A$ to its underlying category (which has the same objects as $A$, and whose morphisms are the $0$-cycles in the hom chain complexes of $A$) would send DK-equivalences to equivalences of categories, since it is represent... | 8 | https://mathoverflow.net/users/57405 | 319384 | 138,008 |
https://mathoverflow.net/questions/319383 | 7 | If $A$ and $B$ are $n\times n$ matrices, then it easily follow from the definition of the determinant by sum over permutations, and from the Young inequality that
$$
|\det (A+B)|\leq C(n)(\Vert A\Vert^n+\Vert B\Vert^n),
$$
where $\Vert \cdot\Vert$ stands for the Hilbert-Schmidt norm of a matrix. I am looking for strong... | https://mathoverflow.net/users/121665 | An upper estimate for $|\det(A+B)|$ | Okay, here is the proof that $C(n) = \frac{2^{n-1}}{\sqrt{n}^n}$.
Firstly, it is enough to find best constant $c(n)$ in $|\det(A)| \le c(n)||A||^n$. Indeed, we have
$$|\det(A+B)| \le c(n)||A+B||^n \le c(n)(||A|| + ||B||)^n \le 2^{n-1}c(n) (||A||^n + ||B||^n),$$
and for $A = B$ we have here equality.
Now let $A$ be ... | 4 | https://mathoverflow.net/users/104330 | 319387 | 138,011 |
https://mathoverflow.net/questions/319397 | 2 | The [Erdős Discrepancy Problem](https://en.wikipedia.org/wiki/Sign_sequence#Erd%C5%91s_discrepancy_problem) is whether in any two-coloring of the naturals for any $C$ there is a sequence $d, 2d, \ldots nd$ such that the difference of red and blue numbers in it is more than $C$.
This was recently shown to be true by Tao... | https://mathoverflow.net/users/955 | Homogeneous van der Waerden | This is false already for $k=2,n=4$. Color an integer $m$ according to the parity of the exponent of $2$ in the prime factorization. Among $i+1,i+2,i+3,i+4$ at least one number is odd, and at least one is divisible by $2$ and not by $4$. Those two numbers have the parity of the exponent different, hence so do the corre... | 10 | https://mathoverflow.net/users/30186 | 319400 | 138,016 |
https://mathoverflow.net/questions/319394 | 7 |
>
> $A$ is a C$^\*\! $-algebra and $(x\_n)\_{n\in \mathbb{N}} \subseteq A $.
> If $\ $ $yx\_n\to 0 $ for all $y\in A$, Is it true that $x\_n$ is weakly
> convergent to $0$ ?
>
>
>
For unitals this is trivial. For characters like
$w\in \Omega (A)$ we have $w(x\_n)\to 0$ but if for all functionals, I don't know... | https://mathoverflow.net/users/130589 | If $\ $ $yx_n\to 0 $ for all $y$ in a C$^*$-algebra, Is it true that $x_n$ is weakly convergent to $0$? | Yes, it's true. By the GNS construction, every bounded linear functional on $A$ is of the form $A\ni a\mapsto \langle \pi(a)\xi,\eta \rangle$ for some non-degenerate \*-representation $\pi$ on $H$ and $\xi,\eta\in H$. By the Cohen--Hewitt factorization theorem, $H=\pi(A)H$ (no need to take the closure). Consequently, $... | 7 | https://mathoverflow.net/users/7591 | 319403 | 138,017 |
https://mathoverflow.net/questions/319392 | 32 | I have encountered very strange commutative nonassociative algebras without unit, over a characteristic zero field, and I cannot figure out where do they belong. Has anybody seen these animals in any context?
For each natural $n$, the $n$-dimensional algebra $A\_n$ with the basis $x\_1$, ..., $x\_n$ has the multiplic... | https://mathoverflow.net/users/41291 | Have you ever seen this bizarre commutative algebra? | This algebra is defined on the permutation module of the symmetric group. It was studied by K. Harada and R. Griess in the 1970s and a proof that its automorphism group is the symmetric group can also be found in Haraada's [paper](https://repository.dl.itc.u-tokyo.ac.jp/?action=pages_view_main&active_action=repository_... | 47 | https://mathoverflow.net/users/9471 | 319405 | 138,019 |
https://mathoverflow.net/questions/310181 | 5 | Recall that a module $M$ over a ring $R$ is reflexive in case the natural evaluation map $f\_M:M \rightarrow M^{\*\*}$ (where $M^{\*}=Hom\_R(M,R)$) is an isomorphism, where $f\_M(m)=g$ with $g(h)=h(m)$, when $h \in Hom\_R(M,R)$.
For example every projective module is reflexive. Assume that all modules in the following ... | https://mathoverflow.net/users/61949 | Existence of non-trivial reflexive modules | The algebra with quiver
$\require{AMScd}$
\begin{CD}
\bullet @>>>\bullet@>>>\bullet\\
@AAA&@AAA\\
\bullet@>>>\bullet
\end{CD}
and radical square zero has global dimension three and (according to my calculations) no non-projective reflexive modules.
| 2 | https://mathoverflow.net/users/22989 | 319407 | 138,020 |
https://mathoverflow.net/questions/319410 | 1 | Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ be three solenoidal vector fields i.e. $\nabla\cdot\mathbf{A}=\nabla\cdot\mathbf{B}=\nabla\cdot\mathbf{C}=0$.
Can we prove that
$\mathbf{A}\times\nabla\times(\mathbf{B}\times\mathbf{C})+\mathbf{C}\times\nabla\times(\mathbf{A}\times\mathbf{B})+\mathbf{B}\times\nabla\times(... | https://mathoverflow.net/users/126346 | A vector Jacobi identity | I don't think the identity is true. A counterexample:
$$\mathbf{A}=(-x,y,0),\;\;
\mathbf{B}=(0,1,0),\;\;
\mathbf{C}=(0,0,1).$$
All three divergences vanish while
$$\mathbf{A}\times\nabla\times(\mathbf{B}\times\mathbf{C})+\mathbf{C}\times\nabla\times(\mathbf{A}\times\mathbf{B})+\mathbf{B}\times\nabla\times(\mathbf{C}\ti... | 3 | https://mathoverflow.net/users/11260 | 319413 | 138,021 |
https://mathoverflow.net/questions/319370 | 6 | Let $M$ be a Riemannian manifold and let $n$ a positive integer. Denote by $F\_n(M) \subset M^n$ the space of all $n$-tuples of pairwise distinct points from $M$. The isometries of $M$ act co-ordinate wise on $M^n$, and this action restricts to an action on $F\_n(M)$. What is known about the quotient of $F\_n(M)$ by th... | https://mathoverflow.net/users/133812 | Configurations of $n$ points modulo isometries of the ambient space | Let us consider the case when $M$ is a hyperbolic plane, $M=\mathbb H^2$ and restrict to orientation preserving isometries of $\mathbb H^2$. Let's identify $\mathbb H^2$ with the open radius $1$ disk $D\subset\mathbb C$, centred at $0$. Then we have
$$D^n/PSL(2,\mathbb R)\cong (D)^{n-1}/S^1,\,\,\,\bf \*$$
where $S^1... | 3 | https://mathoverflow.net/users/943 | 319416 | 138,022 |
https://mathoverflow.net/questions/310138 | 3 | Recall that an $A$-module $M$ is reflexive in case the natural evaluation map $M \rightarrow M^{\*\*}$ is an isomorphism, where $M^{\*}=Hom\_A(M,A)$.
Question:
>
> Given a finite dimensional algebra $A$ of finitistic dimension at least two, is the finitistic dimension of $A$ equal to the supremum of finite projec... | https://mathoverflow.net/users/61949 | Finitistic dimension via reflexive modules | My answer to your other question <https://mathoverflow.net/a/319407/22989> gives an example of a finite dimensional algebra with global dimension three and no non-projective reflexive modules, which also gives a negative answer to this question.
| 1 | https://mathoverflow.net/users/22989 | 319420 | 138,023 |
https://mathoverflow.net/questions/319412 | -3 | From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|\_\infty<\lceil\sqrt p\rceil$ holds where $t(a,b)\equiv(x,y)\bmod p$.
1. Is there no distinct coprime $a,b$ in $(0,\lceil\sqrt p\rceil)$ such that there is $t\i... | https://mathoverflow.net/users/10035 | Negative Dirichlet Pigeonhole Principle | No. Observe that $a/b=x/y$, from which it follows that $ay=xb$. Since $a,b,x,y$ are all in $(0,\sqrt{p})$, it follows that the equality $ay=xb$ holds not merely mod $p$, but in $\mathbb{Z}$. Since $a,b$ are coprime, it follows that $a\mid x$ and $b\mid y$. Hence, $\lvert x\rvert\geq \lvert a\rvert$ and $\lvert y\rvert\... | 1 | https://mathoverflow.net/users/806 | 319423 | 138,024 |
https://mathoverflow.net/questions/316014 | 3 | In a [Maker-Breaker game](https://en.wikipedia.org/wiki/Maker-breaker_game), there is a finite set of elements $X$, and a family $F$ of subsets of $X$ called the "winning sets". Two players, Maker and Breaker, take turns picking untaken elements from $X$. Maker wins by holding a full winning-set while Breaker wins by h... | https://mathoverflow.net/users/34461 | Satisfier-Falsifier games | This question was bumped to the homepage by Community, it sounded interesting to me so I did a little searching. The only thing I found was the paper
>
> L. Zhao and M. Müller. Game-SAT: A preliminary report. In Seventh International Conference on Theory and Applications of Satisfiability Testing (SAT 2004), pages ... | 1 | https://mathoverflow.net/users/51668 | 319433 | 138,028 |
https://mathoverflow.net/questions/319444 | 1 | Suppose $Y=\overline{\cup E\_n}$ is a closed subspace of a Banach space, where each $E\_n$ is a $n$-dimensional subspace, $K$-complemented in $X$, and for any $n$, $E\_n\subseteq E\_{n+1}$. Can one conclude that $Y$ is complemented in $X$?
| https://mathoverflow.net/users/129564 | Complemented subspace constructed from finite pieces | No. Take $X=\ell\_\infty({\bf N})$ and take $E\_n = \operatorname{span}(e\_1,\dots, e\_n)$. Then $Y=c\_0({\bf N})$ which is well-known – by a non-trivial argument – to be uncomplemented in $X$ (in the sense of Banach spaces). Look up "Phillips's Lemma".
| 2 | https://mathoverflow.net/users/763 | 319449 | 138,031 |
https://mathoverflow.net/questions/319451 | 7 | This is a follow up to: [Complemented subspace constructed from finite pieces](https://mathoverflow.net/questions/319444/complemented-subspace-constructed-from-finite-pieces)
Suppose $Y=\overline{\cup E\_n}$ is a closed subspace of a *separable* Banach space X, where each $E\_n$ is a $n$-dimensional subspace, $K$-com... | https://mathoverflow.net/users/129564 | Complemented subspaces constructed from finite pieces- part II | The answer to your question in the last paragraph is "Yes", it is a contents of the well-known Sobczyk Theorem, see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I.
The answer to the first question is still negative. You can find an example of this type in W.B. Johnson, J. Lindenstrauss, Examples of L1 spaces... | 11 | https://mathoverflow.net/users/37822 | 319452 | 138,032 |
https://mathoverflow.net/questions/319453 | 3 | What one can say about a category in which, any two objects are isomorphic. I know that this is a strange question.
Thanks
| https://mathoverflow.net/users/85359 | Category with one isomorphism class | The question in the title and the body are slightly different -- a category with all objects isomorphic either has one isomorphism class of objects, or *zero* isomorphism classes of objects (i.e. it could be empty). Probably the right class of categories to consider is the one from the title -- categories with one isom... | 5 | https://mathoverflow.net/users/2362 | 319454 | 138,033 |
https://mathoverflow.net/questions/181506 | 2 | Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd\_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free pro-$p$ group of rank $d(G)$. Must $L$ be free pro-$p$ ?
Here, $d(G)$ is the smallest cardinality of a generating se... | https://mathoverflow.net/users/38889 | An epimorphism into a profinite group | No.
Let $G = D\_p \times F$,
where $D\_p = \langle a,b \mid\ a^p = 1,\ b^2 = 1,\ a^b \ (:=b^{-1}ab)=a^{-1}\rangle$
is the dihedral group of order $2p$
and $F$ is the free pro-$p$ group on two generators, say, $x,y$.
Then $L = \langle a \rangle \times F$ is of index $2$ in $G$, pro-$p$, but not free,
and there is a su... | 2 | https://mathoverflow.net/users/133544 | 319455 | 138,034 |
https://mathoverflow.net/questions/319319 | 1 | I want to coin the notion of *heaviness* of a class as a function from classes to ordinals such that $$heaviness(x)=|TC(x)|$$, i.e. the heaviness of a class is the cardinality of its transitive closure.
Now I want to introduce a new unary predicate symbol to the language of set theory, that is the predicate "Heavy"; ... | https://mathoverflow.net/users/95347 | Can Ackermann set theory find a natural interpretation in light\heavy class dichotomy? | This question is more subtle than I originally thought. The answer is that the theory is consistent assuming some large cardinal hypothesis, the existence of $0^\sharp$.
If $0^\sharp$ exists, then there is a club class $C$ of ordinals which are order-indiscernible for $L$. For every limit ordinal $\alpha \in C$, $Hul... | 5 | https://mathoverflow.net/users/11145 | 319462 | 138,037 |
https://mathoverflow.net/questions/319461 | 9 | Is there a countable connected space $(X,\tau)$ such that for all $x\in X$ the space $X\setminus\{x\}$ is not connected any more with the induced subspace topology?
| https://mathoverflow.net/users/8628 | Countable connected space where removing $1$ point destroys connectedness | Let $\mathbb{R}$ be with its usual topology, and let $f:\mathbb{R}\to \mathbb{Z}$ defined by:
$$
f(x)=\left\{\begin{matrix}
2k & x=2k,\text{ where } k\in\mathbb{Z}\\
2k+1 & 2k<x<2k+2,\text{ where } k\in\mathbb{Z}
\end{matrix}\right.
$$
Let $X$ be $\mathbb{Z}$ with the quotient topology induced by $f$.
| 8 | https://mathoverflow.net/users/49822 | 319464 | 138,038 |
https://mathoverflow.net/questions/319239 | 3 | The Series
-----------
Consider the series identity
$$\Phi(s) = \sum\_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum\_{r \in R(n)} \zeta(s,r/n) = \sum\_{n=1}^\infty \frac{\Lambda\_k'(f(n))}{n^s}$$
$$R(n) = \left\{ 1 \leq r \leq n : f(r) \equiv 0 \bmod n \right\}$$
$$\Lambda\_k'(n) = \sum\_{d|n} \mu(d) (\log d)^... | https://mathoverflow.net/users/133655 | Divergence of a series related to Schinzel's hypothesis H | In the 1960's Turán wrote several papers on a function-theoretic sieve. He managed to express the number of prime twins in terms of roots of $L$-series. He began like you did by expressing $\Lambda$ as a sum over divisors. Then he did not consider the behaviour for $s\searrow 1$, but used complex integration to get an ... | 4 | https://mathoverflow.net/users/37555 | 319465 | 138,039 |
https://mathoverflow.net/questions/319470 | 0 | I have two functions, and I want to combine them to define a certain function.
Suppose for every fixed $e$ in $(0, ∞)$, we have a function $g\_e (x): \mathbb{R} \to [0,\infty]$ that is well defined a.e. and right continuous, and for every fixed x in R, we have a function $h\_x (e): (0, ∞) \to [0, ∞]$ that is well def... | https://mathoverflow.net/users/132446 | Given these conditions, can a function be defined that is well defined a.e.? | At least assuming the continuum hypothesis, the answer is *no*. Let $\prec$ be a well-order of $\mathbb R$ of order type $\omega\_1$. Define $g\_e(x)=0$ for $x\succ e$ (and $g\_e(x)$ not defined otherwise) and $h\_x(e)=e$ if $e\succ x$ (and $h\_x(e)$ not defined otherwise). Observe that both $g\_e,h\_x$ are defined eve... | 2 | https://mathoverflow.net/users/30186 | 319472 | 138,040 |
https://mathoverflow.net/questions/319476 | 1 | I have four nonlinear equations I want to find the points of intersection of these equations, and I used the software Mathematica, unfortunately after many hours of waiting it does not give me any result
Do you have an idea how to solve this kind of problem?.
My equations are the following
1) $(52 \alpha ^2+\alpha... | https://mathoverflow.net/users/127642 | Intersection Solutions of four nonlinear equations | This is an expansion of the comment by Noam D. Elkies. Indeed, we can rewrite your equations 3) and 4) as $(\alpha,\beta)=(x^2,x^3)$ and $(\gamma,\delta)=(y^2,y^3)$ for some $x,y$. Substituting into eqs. 1) and 2) these expressions of $\alpha,\beta,\gamma,\delta$ in terms of $x,y$, we reduce your system of four eqs. to... | 2 | https://mathoverflow.net/users/36721 | 319482 | 138,043 |
https://mathoverflow.net/questions/319485 | 5 | Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L\_CM$ exists. Suppose we are given sequences of maps $(p\_{n+1}: X\_{n+1}\to X\_n), (q\_{n+1}: Y\_{n+1}\to Y\_n), (f\_n: X\_n\to Y\_n), n=0, 1,\ldots$ with $q\_{n+1}f\_{n+1}=f\_np\_{n+1}$, so we get a ladder of commu... | https://mathoverflow.net/users/42571 | Limit of weak equivalences in a Bousfield localization | No. For a counterexample to your claim, consider the model category M
of simplicial presheaves on a small site S equipped with the projective
model structure.
Its fibrant objects are presheaves of Kan complexes.
If C is the set of Čech covers of S, then L\_C(M) is the local projective
model structure on simplicial pre... | 3 | https://mathoverflow.net/users/402 | 319493 | 138,046 |
https://mathoverflow.net/questions/319501 | 2 | Let $X$ be a smooth, projective variety and $i:Y \hookrightarrow X$ a smooth divisor. Let $Z \subset X$ be a proper, closed subvariety disjoint from $Y$. Let $\pi:\widetilde{X} \to X$ be the blow-up of $X$ along $Z$. Assume that $\widetilde{X}$ is regular. Denote by $j: Y \hookrightarrow \widetilde{X}$ (as $Z$ does not... | https://mathoverflow.net/users/32151 | Gysin morphism of blow up | This is true. To see this, recall that the Gysin map is defined as follows:
We have an isomorphism $Th(N\_{X/Y})\cong X/(X-Y)$ where $Th$ denotes the Thom space. Then, we have a Thom isomorphism $H^\*(Th(V))\cong H^{\*-dim(V)}(X)$ for a vector bundle $V\to X$ with complex orientation (in particular every complex vecto... | 3 | https://mathoverflow.net/users/115052 | 319502 | 138,048 |
https://mathoverflow.net/questions/319474 | 4 | Let $A=(a\_{ij})\_{1 \le i, j \le n}$ be a matrix such that $\sum\_\limits{i=1}^{n} a\_{ij}=1$ for every $j$, and $\sum\_\limits{j=1}^n a\_{ij} = 1$ for every $i$, and $a\_{ij} \ge 0$. Let
$$\begin{equation}
\begin{pmatrix}
y\_1 \\
\vdots \\
y\_n \\
\end{pmatrix}
=\mathbf{A}
\begin{pmatrix}
x\_1 \\
\vdots \\
... | https://mathoverflow.net/users/133871 | Question about an inequality described by matrices | $$y\_i=\sum\_j a\_{ij} x\_j\geqslant \prod\_j x\_j^{a\_{ij} }$$
by Jensen inequality for logarithm. Now take the product over $i=1,2,\dots,n$.
| 5 | https://mathoverflow.net/users/4312 | 319514 | 138,051 |
https://mathoverflow.net/questions/319510 | 1 | Is every surjective etale morphism from a connected separated scheme to $A^n\_{\mathbb{C}}$ of finite type? Is it finite? We use Stacks project's definitions.
EDIT: From Jason Starr's answer, we learn that such a morphism indeed has to be of finite
type, and since etale morphisms are locally quasi-finite, we infer ... | https://mathoverflow.net/users/132313 | Finiteness of surjective etale morphisms | The question is "really" about quasi-compactness, which is usually assumed as a hypothesis in versions of Zariski's Main Theorem. However, the other strong hypotheses of the OP imply quasi-compactness in this case. The key point is that an open immersion is quasi-compact if the target is Noetherian.
**Lemma.** Let $i... | 4 | https://mathoverflow.net/users/13265 | 319517 | 138,054 |
https://mathoverflow.net/questions/319489 | 2 | Let $\Omega\_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation
$$u\_t=\Delta u + a(x)u, \;\; (t,x)\in \Omega\_T ,$$
$$u|\_{\partial \Omega}=0,$$
$$u(0,\cdot)=u\_0.$$
Assume that the initial condition $u\_0 \in H^2(\Omeg... | https://mathoverflow.net/users/124904 | Bounded solution for parabolic equation | For the heat equation $Lu=u\_t-\Delta u=0$ to guarantee boundedness of $u\_t$ as $t$ tends to zero one has to demand more regularity from the initial function, e.g. $u\_0\in C^{1,1}(\bar\Omega)$ (the first order derivatives are uniformly Lipschitz in $\Omega$).
As for the low order term, differentiating wrt $t$ we h... | 3 | https://mathoverflow.net/users/14551 | 319522 | 138,056 |
https://mathoverflow.net/questions/319513 | 9 | The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.
We have a trivial Kan fibration of simplicial sets $p : S \rightarrow T$ where $T$ is an $\infty$-category. We wish to show that for any two ... | https://mathoverflow.net/users/131022 | Higher Topos Theory Theorem 2.2.5.3 | From naturality we have the following commutative diagram:
$\require{AMScd}
\begin{CD}
Map\_{\mathfrak C[S]}(x,y) @<\sim<< |Hom^R\_S(x,y)|\_{Q\_\bullet} @>\sim>> Hom^R\_S(x,y) \\
@VVV @VVV @VVV\\
Map\_{\mathfrak C[T]}(px,py) @<\sim<< |Hom^R\_T(px,py)|\_{Q\_\bullet} @>\sim>> Hom^R\_T(px,py)
\end{CD}$
From the above ... | 12 | https://mathoverflow.net/users/2362 | 319528 | 138,058 |
https://mathoverflow.net/questions/319525 | 5 | Let $f: R \to R$ be a function such that the closure of its graph contains as a subset the graph of a uniformly continuous function. Does there exist a dense subset $S$ of $R$ such that the restricted function $f|S: S \to R$ is uniformly continuous?
| https://mathoverflow.net/users/132446 | A problem in real analysis of a topological nature | Consider the following modification of the Dirichlet "popcorn" function:
$$f(x) = \begin{cases} 1/q, & \text{$x \in \mathbb{Q}$, $x=p/q$ in lowest terms} \\
-1, & x \notin \mathbb{Q},\, x < 0 \\
-2, & x \notin \mathbb{Q}, \, x > 0.\end{cases}$$
Since every real number can be approximated by rationals with arbitrarily l... | 7 | https://mathoverflow.net/users/4832 | 319531 | 138,059 |
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