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https://mathoverflow.net/questions/427958 | 2 | Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX\_t = \sigma(X\_t) \, dW\_t \;, \quad X\_0 = x\_0\;.$$
where $\sigma:\mathbb R \to \mathbb R$ is a Lipschitz continuous function, and $x\_0 \in \mathbb R$ is a fixed constant.
For every $\varepsilon > 0$, let $A\_\va... | https://mathoverflow.net/users/173490 | Short time limits for SDE | Let $\tau = \inf\{ t>0 : W\_t = 1 \}$. The conjecture is **true** and the essence of the proof outlined below appears to be the following peculiar property of the hitting time $\tau$: $$
\lim\_{\epsilon \searrow 0} P\_{\epsilon}[\tau > \epsilon-\epsilon^{3/2}] = 1 \;. $$
(This can be computed directly using the fact th... | 2 | https://mathoverflow.net/users/64449 | 428708 | 173,788 |
https://mathoverflow.net/questions/428467 | 3 | Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$.
**Is it true that the equation $$Du=f$$
is solvable in any open ball, when $f,u\in C^\infty$?**
| https://mathoverflow.net/users/16183 | Solvability of general linear PDE with constant coefficients | The answer is yes. More generally, a (non-zero) constant coefficient differential operator $P$ induces a surjective map $P:C^\infty(X)\to C^\infty(X)$ for an open set $X\subset \mathbb{R}^n$ if and only if $X$ is $P$-convex for supports. A reference for this result is Theorem 10.6.6 and Corollary 10.6.8 in Hörmander's ... | 4 | https://mathoverflow.net/users/78745 | 428709 | 173,789 |
https://mathoverflow.net/questions/428657 | 0 | Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X\_t)\_{t \in \mathbb{Z}} \, : \, X \hbox{ is strictly stationary, } \mathbb{E} X\_t = 0 \hbox{ and } \mathbb{E}[X\_t^2]< \infty, \, \forall\, t \in ... | https://mathoverflow.net/users/478920 | Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not) | $\newcommand{\Z}{\mathbb Z}\newcommand{\PP}{\mathcal D}\newcommand{\R}{\mathbb R}$Your function $d$ is not a metric, for two reasons: (i) there may be many processes $(X\_t)\_{t\in\Z}$ with the same distribution $P$ and (ii) your function $d$ does not take into account the values of $X\_t$ for negative $t\in\Z$. So, yo... | 1 | https://mathoverflow.net/users/36721 | 428716 | 173,793 |
https://mathoverflow.net/questions/428714 | 3 | The Rademacher functions are an explicit iid sequence with Bernoulli law. Does it exist an explicit construction of an iid sequence with uniform law?
| https://mathoverflow.net/users/7294 | Uniform iid sequence | $\newcommand\N{\mathbb N}$Let $(r\_k\colon k\in\N)$ be the sequence of the [Rademacher functions](https://en.wikipedia.org/wiki/Rademacher_system). Re-enumerate this sequence into a two-way array $(r\_{i,j}\colon(i,j)\in\N^2)$. So, the $r\_{i,j}$'s are iid random variables (r.v.'s, defined on the standard probability s... | 6 | https://mathoverflow.net/users/36721 | 428718 | 173,795 |
https://mathoverflow.net/questions/428633 | 8 | Are there complete finitely axiomatizable first order theories (with equality) with arbitrarily high computational complexity?
Here, arbitrarily high (computational) complexity means that for every computable predicate $P$ one can choose a theory $T$ (as above) such that $P$ is polynomial time computable from $T$.
... | https://mathoverflow.net/users/113213 | Decidable theories with arbitrary complexity | There are complete decidable finitely axiomatizable theories of arbitrary high computational complexity.
As Dmytro already sketched in his question there are complete decidable theories of arbitrary high computational complexity. Now we want to switch to finitely axiomatizable theories. This could be achieved using a... | 7 | https://mathoverflow.net/users/36385 | 428749 | 173,803 |
https://mathoverflow.net/questions/428642 | 3 | Let $K$ be a finite extension field of $\mathbb{Q}\_p$. Let us consider a semiabelian variety $G$ defined over $K$, i.e there exists an extension of an abelian variety $B$ and a torus $T$ defined over $K$
$$0 \rightarrow T \rightarrow G \rightarrow A \rightarrow 0$$
It is know [that](https://mathoverflow.net/questi... | https://mathoverflow.net/users/146212 | $p$-power torsion of semiabelian variety | $\newcommand{\Spec}{\mathrm{Spec}}\newcommand{\oL}{\overline{L}}\newcommand{\bG}{\mathbb{G}}\newcommand{\bZ}{\mathbb{Z}}\newcommand{\cL}{\mathcal{L}}$Not in general. The sequence of $p$-divisible groups $0\to T[p^{\infty}]\to G[p^{\infty}]\to A[p^{\infty}]\to 0$ is exact as a sequence of sheaves of abelian groups on th... | 2 | https://mathoverflow.net/users/39304 | 428753 | 173,805 |
https://mathoverflow.net/questions/428750 | 8 | In [[KM63]](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/kervmiln.pdf), Kervaire and Milnor introduced the *group of homotopy spheres*. Its elements are h-cobordism classes of smooth homotopy $n$-spheres under the summation induced by connected sum. Further, the trivial element is $S^n$ and this group is denoted by $\... | https://mathoverflow.net/users/475366 | Kervaire-Milnor group of homotopy spheres and smooth Poincaré conjecture | That a homotopy 4-sphere is h-cobordant to $S^4$ is in principle a step towards proving the 4-dimensional Poincaré conjecture. But it's known from Donaldson's work that the h-cobordism theorem is false for simply connected closed $4$-manifolds. Indeed the step that fails is in cancelling handles that homologically canc... | 12 | https://mathoverflow.net/users/3460 | 428758 | 173,806 |
https://mathoverflow.net/questions/428755 | 10 | I was recently compiling some notes for an undergrad-level course on number theory, and I went over the proof of the fact that $(\mathbb{Z}/p\mathbb{Z})^\times$ is cyclic for any prime $p$: it's a finite abelian group and thus the direct sum of cyclic groups, and the fact that $X^r - 1\in (\mathbb{Z}/p\mathbb{Z})[X]$ h... | https://mathoverflow.net/users/61829 | State of the art for primitive roots | Artin's conjecture on primitive roots has a qualitative version and quantitative version. The qualitative version says if $a \in \mathbf Z$ is not $-1$ or a perfect square then $a \bmod p$ is a primitive root mod $p$ for infinitely many $p$. The quantitative version says for such $a$ that the number of $p \leq x$ such ... | 28 | https://mathoverflow.net/users/3272 | 428764 | 173,809 |
https://mathoverflow.net/questions/428785 | 6 | I'm trying to produce a toy version of the RH Weil conjecture. Solving this could help me to get a good start at understanding where the $1/2$'s come in here, ideally without having to prove the Hard Lefschetz theorem.
I'm restricting attention to the scheme $X=\mathbb{F}\_p \mathbb{P}^1$. Write $x = \text{Spec}(\mat... | https://mathoverflow.net/users/30211 | Zeta function of $X = \mathbb{F}_p \mathbb{P}^1$ | Deligne proved the hard Lefschetz theorem as a consequence of the Weil conjectures, not vice versa.
The RH of the Weil conjectures is a statement about the Weil zeta function. For the projective line, this is easy to prove without étale cohomology at all, only by counting points.
But of course you could ask for a p... | 10 | https://mathoverflow.net/users/18060 | 428786 | 173,816 |
https://mathoverflow.net/questions/428781 | 18 | Is there a $C^\infty$-smooth embedding $\gamma : I \to \mathbb{R}^3$ so that there is no real analytic $2$-dimensional submanifold $M \subset \mathbb{R}^3$ with $\gamma(I)\subset M$?
| https://mathoverflow.net/users/1540 | Smooth curve in $\mathbb{R}^3$ not contained in real analytic surface? | Here is an example: Let $\gamma:\mathbb{R}\to\mathbb{R}^3$ be defined by $\gamma(t) = \bigl(t,\exp(-1/t^2),0\bigr)$ for $t<0$, $\gamma(0) = (0,0,0)$ and $\gamma(t) = \bigl(t,0,\exp(-1/t^2)\bigr)$ for $t>0$. Then I claim that there is no nonsingular real-analytic surface $M\subset\mathbb{R}^3$ that contains the image of... | 20 | https://mathoverflow.net/users/13972 | 428800 | 173,821 |
https://mathoverflow.net/questions/428797 | 4 | Let $A$ be a $C^\*$ algebra. A $C^\*$ subalgebra $C\subset A$ is said to be $C^\*$ algebraic complemented of $A$ if there exist a $C^\*$ subalgebra $D\subset A$ with $A=C\oplus D$ and the obvios mapping $A\mapsto C\oplus D$ preserves all algebraic structure.
In this question we consider the particular case $A=\ell^\i... | https://mathoverflow.net/users/36688 | A $C^*$ algebraic analogy of the concept of complemented subspace in the particular case of $\ell^\infty$ | Suppose $l^\infty \cong C \oplus D$ (a C${}^\*$-direct sum). Then both $C$ and $D$ must have units, which will be projections in $l^\infty$, and hence must be of the form $1\_X$ and $1\_{\mathbb{N}\setminus X}$ for some $X \subseteq \mathbb{N}$. Then every element of $C$ must be supported on $X$ and every element of $D... | 7 | https://mathoverflow.net/users/23141 | 428804 | 173,822 |
https://mathoverflow.net/questions/428619 | 2 | Let $R$ be a regular local ring and $M$ a finitely generated reflexive $R$-module. When $R$ has dimension 2, then $M$ is a free $R$-module. This is discussed in [Reflexive modules over a 2-dimensional regular local ring](https://mathoverflow.net/questions/22943/reflexive-modules-over-a-2-dimensional-regular-local-ring)... | https://mathoverflow.net/users/61621 | Structure of reflexive modules over regular local rings | If $R$ is a Gorenstein (or even Cohen-Macaulay and Gorenstein in codimension $1$) local ring, then $M$ is reflexive if and only if it is a second syzygy (see e.g. [this answer](https://math.stackexchange.com/q/3487112)). As regular rings are Gorenstein, this directly extends the result you quote for regular local rings... | 2 | https://mathoverflow.net/users/155965 | 428813 | 173,825 |
https://mathoverflow.net/questions/428807 | 4 | Given an integral positive-definite rank $n$ quadratic form $f$, one can use the algorithm in Conway and Sloane (Chapter 15, [SPLaG](https://link.springer.com/book/10.1007/978-1-4757-6568-7)) to efficiently determine if the genus of $f$ contains more than one spinor genus. My question is: given two (integral positive-d... | https://mathoverflow.net/users/164802 | Computing spinor equivalence for positive definite forms | Yes: one can write down the explicit local transformations at all primes where they are not both unimodular and evaluate the spinor norms and the local automorphism groups. Magma in fact claims to implement such an algorithm, but I don't have personal experience with it.
| 7 | https://mathoverflow.net/users/6084 | 428826 | 173,832 |
https://mathoverflow.net/questions/428824 | 1 | If $G = (V, E)$ is a simple, undirected graph and $T \subseteq V$, let $$N(T) = \{v \in V: \{v, t\}\in E \text{ for some }t\in T\}.$$
Given $v\in V$ we let $N\_0(v) = \{v\}$ and $N\_{k+1}(v) = N\_k(v) \cup N(N\_k(v))$ for all $k\geq 1$. The *iterated degree sequence* of $v$, denoted by $(\text{deg}\_k(v))\_{k\in\omeg... | https://mathoverflow.net/users/8628 | Do graphs with identical degree matrix have the same chromatic number? | There are such examples. Take two graphs with the same degrees but different chromatic number (for example, a cycle of length 6 and two triangles). Add a vertex of full degree to both.
| 5 | https://mathoverflow.net/users/4312 | 428827 | 173,833 |
https://mathoverflow.net/questions/428738 | 1 | For $n\geq 2$, $P\mathbb{R}^n$ is a simple example of finite polyhedron with finitely generated simple fundamental group. I was wondering if someone could give me an example of a finite polyhedron with infinite finitely generated simple fundamental group. Thanks in advance.
Here by a simple group I mean in the group ... | https://mathoverflow.net/users/114476 | Examples of finite polyhedra with finitely generated simple fundamental group | As suggested in the comments, what you are asking for is essentially the presentation complex of a finitely presented, infinite, simple group. Thus it suffices to exhibit a presentation for such a group. Some are known, but not many.
Probably the easiest examples are Thompson's groups $T$ and $V$. Google gives me a l... | 4 | https://mathoverflow.net/users/1463 | 428843 | 173,838 |
https://mathoverflow.net/questions/428840 | 6 | This question is basically a lift to MO of a part of [an old MSE question](https://math.stackexchange.com/questions/4221319/if-a-structure-has-a-definable-ordered-pairing-function-must-it-have-a-definabl). That question asked, roughly, for the model-theoretic relationship between ordered and unordered pairing functions... | https://mathoverflow.net/users/8133 | Are ordered and unordered pairing functions "definably equivalent?" | The answer is no. Consider absolutely free algebra $\mathfrak{A}=(A;0,\langle \cdot,\cdot\rangle)$ with one binary function $\langle \cdot,\cdot\rangle$ and one generator $0$. Clearly, $\langle \cdot,\cdot\rangle$ is a pairing function on $\mathfrak{A}$. As I will show below there are no definable unordered pairing fun... | 4 | https://mathoverflow.net/users/36385 | 428845 | 173,840 |
https://mathoverflow.net/questions/428838 | 8 | What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but *not* strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \mathbb{N}$, the corresponding $R$-algebra of convergent power series $R\{x\_1, ..., x\_n\}$ is Noetherian) ?
| https://mathoverflow.net/users/143390 | Noetherian but not strongly Noetherian | The only example I know of occurs in [*On Hausdorff completions of commutative rings in rigid geometry*](https://www.sciencedirect.com/science/article/pii/S0021869311000524) by Fujiwara, Gabber, Kato (and according to the intro the example is due to Gabber). The example is the subject of $\S$8.3, in their language they... | 8 | https://mathoverflow.net/users/82848 | 428846 | 173,841 |
https://mathoverflow.net/questions/428170 | 0 | I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match:
1. the distribution of spacings between Satake parameters of an L-function $F$ at all unramified primes on the unit circle
2. the distribution of the spacings between non trivial zeros of $F$ unde... | https://mathoverflow.net/users/13625 | Spacings of Satake parameters under Ramanujan conjecture | If the first item refers to the set of Satake parameters for $X \le p <2 X$ for large $X$, then that distribution will not match the others. This is because the Satake parameters for different primes do not "see" each other: they will not have (quadratic) repulsion as is expected or known in the other two cases.
| 3 | https://mathoverflow.net/users/10220 | 428861 | 173,845 |
https://mathoverflow.net/questions/428860 | 3 | What is the number of $3$-CNF (conjunctive normal form) formulas with $n$ sentential variables and what is the fraction of satisfiable ones? I consider two formulas the same if they are syntactically the same modulo repeated clauses. Thus $$(x\_1\lor x\_1\lor \lnot x\_1)\text{ and } (x\_1\lor x\_1\lor \lnot x\_1)\land ... | https://mathoverflow.net/users/489947 | The number of $3$-CNF formulas in $n$-variables and the fraction of satisfiable ones | Regarding the fraction of satisfiable 3-CNF formulas in $n$ variables, it is widely believed that there is a phase transition that occurs depending on how many clauses there are compared to the number of variables. To be precise, it is conjectured (but not yet proven) that if there are more than $\alpha n$ clauses, the... | 5 | https://mathoverflow.net/users/2233 | 428877 | 173,850 |
https://mathoverflow.net/questions/428855 | 0 | Let $S\_{d-1}$ denote the unit sphere in $\mathbb{R}^d$ and let $(Z\_x)\_{x \in S\_{d-1}}$ be a gaussian process with mean zero and covariance structure given by the square of the scalar product, i.e.
$$
\operatorname{Cov}[Z\_x, Z\_y] = \langle x,y \rangle^2 = (x \cdot y)^2 \ .
$$
I am interested in the distribution of... | https://mathoverflow.net/users/409412 | Maximum of a certain Gaussian field | Your field is the $2$-spin, that is can be represented as $Z\_x=\sum\_{i,j} J\_{ij} x\_i x\_j$, where $J\_{i,j} $ are iid Gaussian (up to symmetry, ie $J\_{i,j}=J\_{j,i}$ and the diagonal has twice the variance of off-diagonal. In short, $J$ is a GOE matrix. In particular, the maximum you seek is the top eigenvalue of ... | 1 | https://mathoverflow.net/users/35520 | 428879 | 173,851 |
https://mathoverflow.net/questions/428852 | 10 | Let $G$ be a countable group. A Følner sequence is a sequence of finite subsets $(F\_n)\_n$ such that
$$\lim\_{n\to\infty} \frac{|KF\_n \mathbin\triangle F\_n|}{|F\_n|} = 0$$
for each fixed finite subset $K \subset G$. Such a sequence exists if and only if the group is [amenable](https://en.wikipedia.org/wiki/Amena... | https://mathoverflow.net/users/489544 | If $(F_n)_n$ is a Følner sequence satisfying Tempelman's condition, is $(F_n^{-1}F_n)_n$ also Følner? | The answer to the first question is negative, even for the integers $G = \mathbb{Z}$. The point is that Følner sequences $F\_n$ are stable under modification by small sets, but the difference set $F\_n^{-1} F\_n$ can be significantly affected by such a modification. For a concrete example, take
$$ F\_n = \{0,\dots,n^... | 11 | https://mathoverflow.net/users/766 | 428884 | 173,852 |
https://mathoverflow.net/questions/428878 | 15 | What are the feasible $2^n$-tuples of entropies $h(S) := H(X\_{i\_1},\dots,X\_{i\_{|S|}})$ where $X\_1,\dots,X\_n$ are discrete random variables with some (unknown) joint probability distribution as $S=\{i\_1,\dots,i\_{|S|}\}$ ranges over the subsets of $\{1,\dots,n\}$? Here $X\_{i\_1},\dots,X\_{i\_{|S|}}$ denotes the ... | https://mathoverflow.net/users/3621 | Information inequalities | Yes. The set of $2^n$ (or $2^n-1$ excluding the empty set) dimensional vectors formed by entropies is called the *entropic region* [1]. Inequalities on the entropic region not implied by the nonnegativity of conditional mutual information are called *non-Shannon-type inequalities*. The first such inequality for $n=4$ w... | 21 | https://mathoverflow.net/users/489962 | 428887 | 173,854 |
https://mathoverflow.net/questions/428875 | 4 | Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$.
If $E$ and $K$ have exactly $2n$ contact points, say $(\pm x\_i)\_{i=1}^{n}$, do $(x\_i)\_{i=1}^n$ form an orthonormal basis for the Euclidean norm indiuced by $E$?
Naively... | https://mathoverflow.net/users/69275 | Contact points for John's ellipsoid | Looks true. A necessary and sufficient condition for these points (let $E$ be a standard ball) is that the identity operator $I$ is a non-negative linear combination of projectors $P\_i$ on lines through $x\_i$: $$I=\sum c\_i P\_i.\quad\quad\quad\quad\quad(\heartsuit)$$
If $x\_i$'s are linearly dependent, multiply $(\h... | 8 | https://mathoverflow.net/users/4312 | 428890 | 173,856 |
https://mathoverflow.net/questions/428891 | 6 | **Background**: Let $G$ be a reductive $\mathbb F\_q$-group and let $X$ be the variety of Borel subgroups of $G$. By the Bruhat decomposition, the $G$-orbits in the space $X\times X$ (with diagonal action) are indexed by elements $w$ of the universal Weyl group $W$ of $G$. Let $\cal O(w)$ be the orbit corresponding to ... | https://mathoverflow.net/users/174855 | (Why) are Deligne-Lusztig varieties nonempty? | $\newcommand\FF{\overline{\mathbb F}}\DeclareMathOperator\Fr{Fr}$The Lang (or Lang–Steinberg) map $G(\FF) \to G(\FF)$ given by $g \mapsto g^{-1}\Fr(g)$ is surjective. In particular, there is some $g \in G(\FF)$ such that $g^{-1}\Fr(g)$ represents $w$. If $B$ is a rational Borel containing the reference torus, then $g B... | 13 | https://mathoverflow.net/users/2383 | 428892 | 173,857 |
https://mathoverflow.net/questions/428882 | 4 | There is a notion of 'oidification' in category theory which characterises many object versions of mathematical objects. For example:
* magmas $\rightarrow$ magmoids
* loops $\rightarrow$ loopoids
* groups $\rightarrow$ groupoids
* rings $\rightarrow$ ringoids
And in reverse, one object magmoids are magmas and so o... | https://mathoverflow.net/users/35706 | What is the definition of a heapoid? | The claim that heapoids exist was added to the nLab page in [revision 3](https://ncatlab.org/nlab/revision/heap/3) by [Toby Bartels](http://tobybartels.name/), so you could ask him what he had in mind.
I can speculate that a heapoid would have
* a set of objects.
* families of morphims $f:x\to y$.
* for any $f:x\to... | 7 | https://mathoverflow.net/users/49 | 428898 | 173,859 |
https://mathoverflow.net/questions/428902 | 0 | Let $u$ be a subharmonic function in a domain $\Omega$ pf $\mathbb{C}$. The functions $u\_{j} := \max(u, -j)$ still subharmonic. Let $\mu := \Delta u$ and $\mu\_{j} := \Delta u\_{j}$ be the associated Riesz measures (which are positive). Let $B$ be a borelian of $\Omega$.
**Question :** is it true that
$$
\int\_{B} 1... | https://mathoverflow.net/users/339613 | Convergence of Riesz measure of SH function | This is not true. Take $u(z)=(2\pi)^{-1}\log|z|$, so that the Riesz measure $\mu=\delta\_0$. Then the Riesz measure $\mu\_j$ of
$u\_j\max\{ u,-j\}$ is the uniform measure on the circle
$\{ z:|z|=e^{-j}\}$. Now let $B$ be the open right half-plane,
so $\mu\_j(B)=1/2$ while $\mu(B)=0$. If you the choose the closed right ... | 2 | https://mathoverflow.net/users/25510 | 428907 | 173,860 |
https://mathoverflow.net/questions/428912 | 2 | **Real-life motivation.** My eldest son attended a football (soccer) class with $17$ other students, so $18$ students in total. There were $6$ students with year of birth (yob) $n$, then $6$ more with yob $n+1$ and the remaining $6$ had yob $n-1$. In the evening, my son told me that at some point, all $18$ students sat... | https://mathoverflow.net/users/8628 | Assigning numbers $\{1,\ldots,k+1\}$ in a balanced way in a $k$-regular graph | The satisfying graphs are precisely the [covering graphs](https://en.wikipedia.org/wiki/Covering_graph) of $K\_{k+1}$. General theory of covering spaces implies that, since $K\_{k+1}$ is connected, each point in the graph has the same number of preimages under a covering map, so every satisfying graph indeed has the nu... | 4 | https://mathoverflow.net/users/30186 | 428916 | 173,862 |
https://mathoverflow.net/questions/428924 | 11 | Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left topological semigroup.
1. What are the invertible (left/right/both) elements of $\beta(G)$ as a semigroup?
2. Is right i... | https://mathoverflow.net/users/113200 | Stone–Čech compactification as a semigroup | Corollary 4.33 of Hindman and Strauss's book on Algebra in the Stone Cech Compactification says that if $S$ is an infinite cancellative (discrete) semigroup, then the nonprincipal ultrafilters in $\beta S$ form a two-sided ideal. In particular every left invertible element is invertible and the units of $\beta S$ and $... | 8 | https://mathoverflow.net/users/15934 | 428930 | 173,864 |
https://mathoverflow.net/questions/428927 | 2 | Let $W$ be the Weyl group of a semisimple algebraic group $G$. $I$ be the simple roots. $J\subset I$ generate a parabolic subgroup of $W$ denote by $W\_J$. $w^J$
is the shortest representative of $w$ in $W/W\_J$. Suppose we have $u^J\le v^J$, here $\le$ is the Bruhat order. Let $M=\{ x\in W\_J| u^{J}x\le v^{J}\}$. My q... | https://mathoverflow.net/users/147080 | Parabolic subgroup of Weyl group | No. In $A\_2$ with $I=\{s\_1,s\_2\}$, take $J=\{s\_1\}$ and $u^J=id$ and $v^J=s\_1s\_2$. Then $M=W\_J$ because $s\_1\le s\_1s\_2$.
| 6 | https://mathoverflow.net/users/5519 | 428932 | 173,865 |
https://mathoverflow.net/questions/427328 | 4 | Let $\zeta\_n = e^{i2\pi/n}$. What is the group of all units in the integral cyclotomic ring $\mathbb{Z}[\zeta\_n]$?
Here I like to know all the group elements for small $n$'s. For $n=1$ and $n=2$, the group is given by $\{1,-1\}$. For $n=3$, the group appears to be $\{\pm 1,\pm \zeta\_3, \pm \zeta\_3^2\}$. For $n=4$... | https://mathoverflow.net/users/17787 | The group of all units of integral cyclotomic ring | As you suspect, identifying all the units in cyclotomic rings of order greater than $4$ is a non-trivial problem (I am ignoring order $6$ because, of course $\mathbb Z[\zeta\_6]$ is the same as $\mathbb Z[\zeta\_3]$, and similarly for all other orders twice an odd number.)
We can get just a taste of what is involved ... | 3 | https://mathoverflow.net/users/86625 | 428934 | 173,866 |
https://mathoverflow.net/questions/428936 | 2 | For $\Omega$ a bounded open set of $\mathbf{R}^d$, denote $\mathrm{d}\_\Omega:x\mapsto \mathrm{d}(x,\partial\Omega)$ the distance-to-boundary function.
If $\Omega$ is convex, a short argument [recalled here](https://mathoverflow.net/questions/291308/convexity-of-distance-to-boundary-function) by Anton Petrunin proves... | https://mathoverflow.net/users/27767 | Concavity/convexity of distance-to-boundary function | The signed distance function $f=\pm\textrm{dist}\_{\partial \Omega}$ is concave everywhere;
here $f=\textrm{dist}\_{\partial \Omega}$ inside and $f=-\textrm{dist}\_{\partial \Omega}$ outside.
The proof is straightforward, for inside you know it, for outside it is very similar:
Assume $B(x,r\_x)\cap \Omega\ni p$ and... | 2 | https://mathoverflow.net/users/1441 | 428946 | 173,868 |
https://mathoverflow.net/questions/428926 | 5 | To motivate things, let me start with a special case of the question I'm interested in. Let $\mathsf{In}(x)\equiv$ "$x$ is an inaccessible cardinal."
>
> **Question 1**: Is it consistent with the theory $$\mbox{$\mathsf{ZFC}$ + "There is a transitive model of $\mathsf{ZFC}$ + 'There are $2$ inaccessibles'"}$$ that ... | https://mathoverflow.net/users/8133 | Fragility of large cardinals with respect to transitive end extensions | Question 1: Yes, in fact if $M\models$ZFC+"There is a transitive model of $T$" where $T$ is the theory ZFC + "There are two distinct inaccessibles", then there is such an $(A,\alpha)$ - just let $A$ be the least model of ZFC + "There is an inaccessible". Then $A=L\_\gamma$ for some ordinal $\gamma$, and there is a uniq... | 5 | https://mathoverflow.net/users/160347 | 428948 | 173,869 |
https://mathoverflow.net/questions/428931 | 6 | Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics?
Tried my luck with Google's search engine, didn't show much info.
I guess you can try to use these features of GT and combinatorics in QFT (there's a section in Kaku's book on QFT on forests and skeletons from GT).
And what ... | https://mathoverflow.net/users/13904 | Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrodynamics? | 1. The inviscid Hopf-Burgers equation is at the heart of combinatorics of convex polytopes surrounding compositional inversion. See, for example, the MO-Q "[Why is there a connection between enumerative geometry and nonlinear waves?](https://mathoverflow.net/questions/145555/why-is-there-a-connection-between-enumerativ... | 5 | https://mathoverflow.net/users/12178 | 428957 | 173,870 |
https://mathoverflow.net/questions/428862 | 1 | The permanent of an $n$-by- $n$ matrix $A=\left(a\_{i j}\right)$ is defined as
$$
\operatorname{perm}(A)=\sum\_{\sigma \in S\_{n}} \prod\_{i=1}^{n} a\_{i, \sigma(i)}
$$
The sum here extends over all elements $\sigma$ of the symmetric group $S\_{n}$ i.e. over all permutations of the numbers $1,2, \ldots, n$.
$$
\operato... | https://mathoverflow.net/users/489949 | Deciding if given number is a permanent of matrix | First, there is a subtlety: the permanent of *nonnegative* integer matrices is computable in [**#P**](https://complexityzoo.net/Complexity_Zoo:Symbols#sharpp), and it is **#P**-complete even for $\{0,1\}$-matrices. However, the permanent of general integer matrices is only in [**GapP**](https://complexityzoo.net/Comple... | 2 | https://mathoverflow.net/users/12705 | 428959 | 173,872 |
https://mathoverflow.net/questions/428928 | 4 | If we have $Z\subset X$ a closed irreducible subscheme of an integral scheme $X$ (which you can take to have various further niceness properties if you want), one can take its generic point $\eta\_Z$ and localize at it, to get a local ring with quotient field the quotient field of $Z$.
However, intuitively, I'd like ... | https://mathoverflow.net/users/120548 | When is it possible to localize a scheme along a closed subscheme? | Let $W \subset X$ denote the set of points specializing to $Z$ with the induced topology. Denote $\mathcal{O}\_W$ the pullback of $\mathcal{O}\_X$. Then $W = (W, \mathcal{O}\_W)$ is a locally ringed space and we can ask:
Question: is $W$ a scheme?
Usually not: for example if $X = \mathbf{P}^2\_k$ and $Z$ is a line,... | 5 | https://mathoverflow.net/users/152991 | 428961 | 173,873 |
https://mathoverflow.net/questions/428720 | 4 | I am wondering if there is a reference for the following:
Let $G$ be a finite group, and suppose that $f\colon M\rightarrow N$ is a continuous and $G$-equivariant map. Here $M$ and $N$ are finite dimensional $G$-manifolds, where $M$ possibly has boundary (I only care when $M$ is compact if that helps). Suppose also t... | https://mathoverflow.net/users/489804 | Equivariant Whitney approximation | This is Corollary 1.12 in
*Wasserman, A. G.*, [**Equivariant differential topology**](http://dx.doi.org/10.1016/0040-9383(69)90005-6), Topology 8, 127-150 (1969). [ZBL0215.24702](https://zbmath.org/?q=an:0215.24702).
The proof is essentially the same as the one given by Peter Michor in [his answer](https://mathover... | 3 | https://mathoverflow.net/users/8103 | 428971 | 173,875 |
https://mathoverflow.net/questions/428974 | 1 | Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell\_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int\_{w=0}^\infty h(w) \cdot 1\_{w \mathbb B^n} \ dw$ with $h(w) \geq 0$, where $w \mathbb B^n = \{\mathbf x \in \mathbb R^n:|... | https://mathoverflow.net/users/490036 | Integration by parts for indicator of a sphere to indicator of a ball | $\newcommand\R{\mathbb R}$Let $f$ be a radial pdf on $\R^n$, so that
$$f(x)=g(|x|)\tag{1}\label{1}$$
for some function $g\colon[0,\infty)\to\R$ and all $x\in\R^n$, where $|x|:=|x|\_2$. Then your desired respresentation
$$f(x)=\int\_0^\infty h(w) \cdot 1\_{w \mathbb B^n} \ dw\tag{2}\label{2}$$ with $h\ge0$ can be rewrit... | 2 | https://mathoverflow.net/users/36721 | 428979 | 173,877 |
https://mathoverflow.net/questions/428960 | 5 | Let $f: \mathbb R^n \to \mathbb R$ be a bounded measurable function, and $g: \mathbb R \to \mathbb R$ an absolutely continuous function.
**Question:** Is it true that if $x \in \mathbb R^n$ is a Lebesgue point of $f$, then $x$ is a Lebesgue point of $g \circ f$?
*Notes:*
1. Here $g \circ f$ is the composite funct... | https://mathoverflow.net/users/173490 | Does postcomposition with an absolutely continuous function preserve Lebesgue points? | We only need $g$ to be continuous. For $\varepsilon>0$ we can find $u\in C^1(\mathbb{R})$ such that $$\sup\_{s\in\mathbb{R}}|g(s)-u(s)|<\varepsilon .$$
For all $h>0$
$$\frac{1}{2h}\int\_{x-h}^{x+h}|g(f(t))-g(f(x))|d{t}\le 2\varepsilon+\frac{K}{2h}\int\_{x-h}^{x+h}|f(t)-f(x)|d{t}$$
where $K=\sup\_{s\in H}|u'(s)|<\inft... | 3 | https://mathoverflow.net/users/481665 | 428986 | 173,879 |
https://mathoverflow.net/questions/428983 | 1 | I am studying exceptional isomorphisms recently, which arise due to the coincidence in Dynkin diagram.
I saw two forms of expressing the exceptional isomorphisms, one is isomorphisms between the spin group and the corresponding group, established by spin or half-spin representations; the other is isomorphism from sym... | https://mathoverflow.net/users/350297 | Exterior square of $\operatorname{Sp}(4,\mathbb{C})$ is isomorphic to $\operatorname{SO}(5,\mathbb{C})$ | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Ker{Ker}$Let's start with the double cover $\SL(4) \to \SO(6)$. Let $V$ be the standard $4$-dimensional representation of $\text{SL}(4)$, and let $\bigwedge^2 V$ be its tensor square. We have a non-degenerate symmetri... | 7 | https://mathoverflow.net/users/297 | 428990 | 173,883 |
https://mathoverflow.net/questions/428944 | 3 | First let me recall Stone duality in terms of propositional logic.
Let $L$ and $K$ be propositional signatures (i.e., sets of propositional variables). Let $T$ be a propositional theory over $L$ and $S$ a propositional theory over $K$. An *interpretation* of $T$ in $S$ is a map $I\colon L\to \{\text{$K$-sentences}\}$... | https://mathoverflow.net/users/476804 | An extension of Stone duality | Let $(L,W)$ be a propositional class, so $W\subseteq 2^L$. The topology you assign to $W$ is exactly the subspace topology inherited from $2^L$, where $2$ gets the discrete topology and $2^L$ gets the product topology. So the essential image of your functor is just the subcategory of $\mathsf{Top}$ consisting of all sp... | 2 | https://mathoverflow.net/users/2126 | 428993 | 173,885 |
https://mathoverflow.net/questions/428991 | 18 | I've been trying to read [Kapustin–Witten - Electric–Magnetic Duality And The Geometric Langlands Program](https://arxiv.org/abs/hep-th/0604151) recently, as someone whose mathematical interests are in the Langlands program. I have some physics background, but not including string theory. I'm looking to understand "bra... | https://mathoverflow.net/users/85392 | What are "branes", and why do they form a category? | Let me start by putting your questions into a bit more context. Kapustin and Witten's story occurs within string theory, a theory of 1-dimensional extended objects. Strings may be "closed," forming loops, or "open," with two boundary points. As a string evolves in time, it traces out a 2-dimensional "worldsheet," which... | 16 | https://mathoverflow.net/users/490054 | 429006 | 173,887 |
https://mathoverflow.net/questions/428938 | 16 | In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction maps surject," one says something about Čech cohomologies vanishing. The problem is then to set up the theory of flasque she... | https://mathoverflow.net/users/490010 | Zorn's lemma for Grothendieck sites | The statement you formulate is not true in this generality.
The idea of the following counterexample is to exploit the fact that the assumption "all restriction maps along morphisms of the site are surjective", applied to a thin site (no parallel morphisms), does not ensure that the sheaf is flabby (flasque) when reg... | 14 | https://mathoverflow.net/users/166281 | 429010 | 173,888 |
https://mathoverflow.net/questions/428661 | 7 | In this article:
>
> Canfell, M. J. "Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition." Journal of algebra 176.2 (1995): 480-503.
>
>
>
the author defines a ring $R$ to have the unique generator property on right ideals if for all pairs $a,b\in R$, $aR=bR$ implies $a=bu$ for some ... | https://mathoverflow.net/users/19965 | Symmetry of unique generator property | Lam informed me that, as far as he knew, this problem was still open. However, the example below shows that the condition is **not** left-right symmetric.
Let
$$
R=\mathbb{F}\_2\langle a,b,c\, :\, a^2=ab=ac=ba=b^2=bc=0,\ ca=b,\ cb=a\rangle.
$$
The ideal generated by $a$ and $b$ is nilpotent of index $2$; call that ... | 7 | https://mathoverflow.net/users/3199 | 429013 | 173,889 |
https://mathoverflow.net/questions/429009 | 7 | So rank $1$ local Langlands is special in as that it is given by the Artin map
$$\text{GL}\_1(K)\to G\_K^{ab},$$
whereas in the higher rank (to the best of my knowledge) there doesn't exist a map
$$\text{GL}\_n(K)\to G\_K$$
which realizes the rank $n$ local Langlands. In fact, I think I've read that there exists a good... | https://mathoverflow.net/users/152554 | Non-existence of "higher" Artin map | There is no way of reformulating local Langlands for $n > 1$ in terms of such a map.
Local Langlands is a bijection between irreducible smooth representations of $\operatorname{GL}\_n(K)$, and $n$-dimensional Frobenius-semisimple Weil–Deligne representations. However, if $n > 1$ then an irreducible smooth $\operatorn... | 10 | https://mathoverflow.net/users/2481 | 429018 | 173,892 |
https://mathoverflow.net/questions/428935 | 8 | $\DeclareMathOperator\ncl{ncl}$When I attended a geometric group theory summer school, a question asked by the speaker reminded me of an old [question](https://mathoverflow.net/questions/171776/two-relator-products-of-cyclic-groups) but shaped in a different manner:
>
> Given two nontrivial groups $A,B$ and $w \in ... | https://mathoverflow.net/users/114032 | Nontriviality of one-relator products | Here is a partial answer: The Kervaire-Laudenbach Conjecture states that, for any group $A$, $(A\ast\mathbb{Z})/\operatorname{ncl}(w)$ is non-trivial. This was proven by Klyachko for torsion free groups $A$; see Fenn and Rourke, *Klyachko's methods and the solution of equations over torsion-free groups*, Enseign. Math.... | 5 | https://mathoverflow.net/users/6503 | 429023 | 173,894 |
https://mathoverflow.net/questions/428962 | 3 | Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P\_a$ the following property: given $x\_1,\dots,x\_a\in X$ general points there exists an irreducible curve $C\subset X$ such that $x\_1,\dots,x\_a\in C$, $C$ spans a projective space of dimension $2a-1$ and $\deg(C) > 2a-1... | https://mathoverflow.net/users/14514 | Varieties connected by curves in projective spaces of small dimension | I am writing an answer to the question in the comments. Let $a$ be a positive integer. Let $X\subset \mathbb{P}^N$ be a linearly nondegenerate, smooth, projective variety. Let $\alpha \in \text{Hom}(\text{Pic}(X)/\text{Pic}^0(X),\mathbb{Z})$ be the class of a rational curve in $X$. Denote by $a$ the $\mathcal{O}\_{\mat... | 1 | https://mathoverflow.net/users/13265 | 429028 | 173,895 |
https://mathoverflow.net/questions/428870 | 5 | Let G be a graph with $n \ge 3$ nodes and with average degree $3\le k\le n$. What are the largest and smallest girth G can have?
| https://mathoverflow.net/users/36886 | Largest girth of a graph of average degree k | You can find your answer in the paper:
**Generalized Girth Problems in Graphs and Hypergraphs**
By Uriel Feige and Tal Wagner.
Let $A\_2(n,k)$ denotes the optimal upper bound for girth of graphs with $n$ nodes and average degree $k$. So, it is proved that we have
$$A\_2(n,k)=\Theta(\log\_{k-1}(n))$$ and it is bes... | 3 | https://mathoverflow.net/users/19885 | 429032 | 173,897 |
https://mathoverflow.net/questions/425477 | 6 | Let $(M, g)$ be a compact Riemannian manifold and $f: M \rightarrow \mathbb{R}$ be a Morse-Bott function, i.e. the set a critical points of $f$, $Crit(f)$, has connected components which are smooth manifolds and such that $T\_x Crit(f) = \ker \nabla^2\_x f$,
(where $\nabla^2\_x f: T\_x M \rightarrow T\_xM$ is the lin... | https://mathoverflow.net/users/172459 | The negative gradient flow of a Morse-Bott function on a compact manifold converges to a critical point? | I will assume that "converges in the critical set of $f$" is asking that if $f$ is MB then $\phi\_t(y)$ (the flowlines) converge as $t\to\infty$ to $y\_\infty$ a critical point (of course, depending on $y$).
---
For any function $f$ on $(M,g)$ a closed Riemannian manifold, note that
$$
\tag{\*} \frac{d}{dt} f(\ph... | 6 | https://mathoverflow.net/users/1540 | 429039 | 173,899 |
https://mathoverflow.net/questions/429011 | 5 | *Although self-contained, this question is a follow-up to [this earlier one](https://mathoverflow.net/questions/428926/fragility-of-large-cardinals-with-respect-to-transitive-end-extensions/428948#428948). Also, thanks to Fedor Pakhomov for fixing a trivial early version of this question!*
Work in $\mathsf{ZFC}$ + "T... | https://mathoverflow.net/users/8133 | Upwards-fragility of inaccessibles (again) | There are no sentences $\sigma$ with this property.
For any $\sigma$ having transitive models we could
consider the $<\_L$-least well-founded model $E\_\sigma$ of
$\mathsf{ZFC}+\sigma$. Let $M\_\sigma$ be the transitive collapse of $E\_\sigma$. To answer the question it is sufficient to prove the following claim.
*... | 7 | https://mathoverflow.net/users/36385 | 429054 | 173,902 |
https://mathoverflow.net/questions/429003 | 20 | The category of smooth manifolds (SmoothMfld) can be thought of the Cauchy completion of the category $U$ of open subsets of Euclidean spaces (with smooth maps) [1]. This fact is shocking to me as it provides an **intrinsic** definition of smooth manifolds.
If this fact can be generalized to manifolds equipped with o... | https://mathoverflow.net/users/124549 | Manifolds as Cauchy completed objects | (Expanding on Phil Tosteson’s [comment](https://mathoverflow.net/questions/429003/manifolds-as-cauchy-completed-objects#comment1103552_429003).) **No: the Cauchy-completion characterisation doesn’t hold for the PL, topological, or complex-analytic cases.**
The key technical point is that split idempotents are always ... | 19 | https://mathoverflow.net/users/2273 | 429055 | 173,903 |
https://mathoverflow.net/questions/346571 | 5 | The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is smooth, by page 73 of Hyperbolic Manifolds and Kleinian Groups. The authors does not provide a proof for this fact. This does not look very obvious as the $r$-neighborhood of a convex set in a Riemannian manifold may not be smooth. One... | https://mathoverflow.net/users/136031 | Smoothness of boundary of $r$-neighborhood of convex core | Just coming across this 3 years too late, but thought I'd mention for posterity that I don't think the boundary of the r-nbhd is smooth. It's not so hard to prove it's $C^{1,1}$. Intuitively, it's $C^1$ because every point on the boundary of the r-nbhd is on the boundary of a r-ball centered in the convex core. That en... | 3 | https://mathoverflow.net/users/74169 | 429056 | 173,904 |
https://mathoverflow.net/questions/429052 | 13 | Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}\_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet\_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with coefficients in $A$. Recall that a class $\alpha \in \mathrm{H}^\bullet\_{\mathrm{gp}}(G; A)$ is *essential* if it is nonzero, ... | https://mathoverflow.net/users/78 | Which finite groups have low-degree essential cohomology? | Call a finite $p$ group $G$ *$p$--central* if every element of order $p$ is central.
In a 1997 Comm. Math. Helv. paper, A. Adem and D. Karagueuzian proved that every $p$--central group has essential mod $p$ cohomology. I got at this theorem in a different way in [N.J.Kuhn, Adv. Math. 216 (2007), 387--442], with a res... | 7 | https://mathoverflow.net/users/102519 | 429062 | 173,906 |
https://mathoverflow.net/questions/428970 | 0 | For any integer $k\geq 2$, a $k$-regular linear set system is a set ${\cal E}\subseteq {\cal P}(\omega)$ such that $|e| = k$ for all $e\in {\cal E}$, and moreover, for all $a\neq b\in\omega$ there is exactly one $e\in {\cal E}$ such that $\{a,b\}\subseteq e$. ([There is indeed](https://mathoverflow.net/a/428639/8628) a... | https://mathoverflow.net/users/8628 | Are $k$-regular linear set systems vertex-transitive? | **Introduction**
I'll provide plenty of non-transitive examples (*just be patient, please*).
Let me introduce a useful and more general notion of *selector system* instead of *linear set system*; then, I will specialize it to the *linear* ones. even in the special case, we can equivalently consider an arbitrary inf... | 1 | https://mathoverflow.net/users/110389 | 429067 | 173,908 |
https://mathoverflow.net/questions/429068 | 6 | Is there a compact metric space $X$ of covering dimension $2$ without a Lipschitz surjection on $[0,1]^2$?
For a space $X$ with Hausdorff dimension greater than $2$, we have a negative answer (see Theorem 1.1 in article <https://arxiv.org/pdf/1203.0686.pdf>).
| https://mathoverflow.net/users/490101 | Lipschitz mappings, covering dimension | Let $\ n\in\mathbb N.\ $ It's convenient to consider the injective metric
in $\ \mathbb R^n,\ $ it's Lipschitz equivalent to the Euclidean metrics.
By dimension, let's mean the *topological dimension* dim (say, *covering* -- for metric compact spaces, *topological* has only one standard meaning).
---
**Theorem*... | 7 | https://mathoverflow.net/users/110389 | 429069 | 173,909 |
https://mathoverflow.net/questions/428532 | 2 | **Some motivation:**
Let $W$ be a standard Brownian motion, and $X$ an integrable process with respect to $W$, i.e. progressively measurable with respect to the natural filtration of $W$ and square integrable on compacts almost surely.
It is known that if $X$ is continuous, then for any sequence of partitions $\mat... | https://mathoverflow.net/users/173490 | Divergence of Riemann sums in the Itô integral | Here is an example of an integrand $X$ and a deterministic sequence of partitions such that almost surely, the limit fails to exist.
Indeed, consider $X = W$. The associated Riemann sums are
$$\lim\_{n \to \infty} \sum\_{[u, v] \in \mathcal P\_n} W\_u (W\_{v} - W\_{u}).$$
We may rewrite this as
$$\frac{1}{2} W^... | 3 | https://mathoverflow.net/users/479223 | 429074 | 173,911 |
https://mathoverflow.net/questions/429073 | 2 | I’ve recently realised there is a subtlety in Girsanov’s theorem that I don’t really understand.
Consider a standard one dimensional Brownian motion $W$, and consider the SDE
$$dZ\_t = \mu(t, Z\_t) \, dt + \sigma(t, Z\_t) \, dW\_t \, , \,Z\_0 = x\_0 \, \, \, \text{(Equation 1)}$$
for some $x\_0 \in \mathbb R$, wh... | https://mathoverflow.net/users/173490 | A question related to Girsanov’s theorem | I managed to find an answer to this problem - indeed the answer is yes. The key ingredient is a theorem in Protter’s *Stochastic Integration and Differential Equations* (Chapter 2, Theorem 14) which states the following:
>
> For every fixed process $X$, integrable with respect to a semimartingale $S$, the stochasti... | 3 | https://mathoverflow.net/users/173490 | 429077 | 173,912 |
https://mathoverflow.net/questions/429076 | 4 | Let $(M, g)$ be a compact Riemannian manifold. Suppose $M$ admits a smooth free circle action (Denote the circle group by $G$. The action $G$ on $M$ is not necessarily isometric) and the orbit space $B$ is another closed manifold, i.e., we have a smooth principal circle bundle $G\hookrightarrow M\rightarrow B$.
My qu... | https://mathoverflow.net/users/169860 | Smooth circle actions on Riemannian manifolds and harmonicity of quotient map | I don't think so. I suspect that this action does not exists for generic metric $g$ on manifolds which admit circle actions.
Here is the thing that I have in mind.
Consider the two torus. This clearly has a free $G=S^1$ action. Now equip the torus with a metric $g$, and consider the scalar curvature $f$ of the metr... | 4 | https://mathoverflow.net/users/12156 | 429080 | 173,913 |
https://mathoverflow.net/questions/429043 | 3 | Prove: There exist real numbers a,b, such that 1<a<b and for any integer $n\ge 2$, there exist positive integers $x\_1\ge x\_2\ge ...\ge x\_{\lfloor{\frac{n}{2}}\rfloor}$such that $n!=\Pi\_{k=1}^{\lfloor\frac{n}{2}\rfloor}x\_k$ and $x\_k\in (a^\frac{n}{k},b^{\frac{n}{k}})$.
I've tried to list all the prime factors of... | https://mathoverflow.net/users/472630 | Factorising n! fitting $x^\frac{n}{k}$ | Your initial guess about prime factors can produce a solution. More precisely, it is known that
$$
n!=\prod\_{p\leq n} p^{v\_p(n!)},
$$
where the product is taken over prime numbers and
$$
v\_p(n!)=\left[\frac{n}{p}\right]+\left[\frac{n}{p^2}\right]+\ldots
$$
Let now for any $1\leq k\leq n/2$
$$
x\_k=\prod\_{\alpha, p:... | 7 | https://mathoverflow.net/users/101078 | 429083 | 173,915 |
https://mathoverflow.net/questions/428863 | 0 | I am trying to characterize a set of distributions that satisfy two conditions. It is easy to characterize distributions fitting each of those conditions separately, but I am unable to make progress on characterizing the intersection. Specifically:
Let's have $X\_0,X\_1$ independent random variables over the interval... | https://mathoverflow.net/users/105908 | A set of equations for two (cumulative distribution) functions and their inverses | So I figured it out.
first substituting $x = F\_1(s)$ into condition 2:
$$
s = \sqrt{2 F\_1(s) - 2 F\_0^{-1}(F\_1(s)) + (F\_0^{-1}(F\_1(s)))^2} \\
s^2 = 2 F\_1(s) - 2 F\_0^{-1}(F\_1(s)) + (F\_0^{-1}(F\_1(s)))^2
$$
This is a quadratic equation for $F\_0^{-1}(F\_1(s))$ we solve it (only one branch of the solutions ... | 0 | https://mathoverflow.net/users/105908 | 429089 | 173,916 |
https://mathoverflow.net/questions/429087 | 1 | How to calculate this sum $$\sum^\infty\_{n=0}\frac{1}{n^4+n^2+1}.$$
Thank you in advance
| https://mathoverflow.net/users/172078 | Closed formula for this sum $\sum^\infty_{n=0}\frac{1}{n^4+n^2+1}.$ | $\newcommand{\Ga}{\Gamma}$Martin Sleziak gave references to several answers to this question. All of those answers seem to involve complex analysis.
Here is how this sum can be evaluated (almost) without complex analysis: Using partial fraction decomposition, as e.g. in [this answer](https://math.stackexchange.com/a/... | 3 | https://mathoverflow.net/users/36721 | 429090 | 173,917 |
https://mathoverflow.net/questions/429098 | 3 | Let $\Omega \subset \mathbb R^2$ be a *bounded* $C^2$ domain. Is $\Omega$ then finitely connected? As I learned [recently](https://mathoverflow.net/questions/421658/what-is-a-finitely-connected-domain#comment1083842_421658bounded) a domain in $\mathbb R^2$ is finitely connected iff “[its] complement has finitely many c... | https://mathoverflow.net/users/44919 | Is every planar bounded $C^2$ domain finitely connected? | Yes. It is finitely connected. The boundary of your domain is a compact one-dimensional manifold and therefore it consists of a finite number of curves diffeomorphic to circles. You can find a proof of classification of compact one dimensional manifolds in:
**J.W. Milnor,** *Topology from the differentiable viewpoint... | 6 | https://mathoverflow.net/users/121665 | 429103 | 173,924 |
https://mathoverflow.net/questions/429116 | 1 | A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e. for each $\lambda\in \Omega$ the map $f\mapsto f(\lambda)$ is a continuous linear functional on $\mathscr H$. The Ries... | https://mathoverflow.net/users/113054 | Is $N(A^n)=N^n(A)\;$ for a self-adjoint operator $A$? | No. If $A=P$ is a projection, then $P^2=P$, but in general you won't have $N(P)=0$ or $1$. In fact, you could just take $\Omega=\{ 1,2\}$, so $\mathcal H\cong\mathbb C^2$, and $P$ as the projection onto $(1,1)$.
| 3 | https://mathoverflow.net/users/48839 | 429117 | 173,925 |
https://mathoverflow.net/questions/429139 | 1 | Let $\mathbb{H}$ be the upper half plane model of hyperbolic geometry. Let $\Gamma$ be the Fuchsian group such that $\mathbb{H}/\Gamma$ is the compact orientable surface of genus $2$.
Suppose $\Gamma = \langle g\_1,g\_2,g\_3,g\_4 \mid g\_1g\_2g\_1^{-1}g\_2^{-1}g\_3g\_4g\_3^{-1}g\_4^{-1}=1 \rangle$ where $g\_1,g\_2,g\... | https://mathoverflow.net/users/490039 | Area of fundamental domain of Fuchsian group and index of a Fuchsian group in the triangle group | EDIT: My answer (below) is for the original question. The current question has been modified to include my answer.
---
The area of a fundamental domain for $\Gamma$ is $4\pi = -2\pi \chi(\mathbb{H} / \Gamma)$. There are various proofs; most people think of this as a consequence of the Gauss-Bonnet theorem.
--... | 4 | https://mathoverflow.net/users/1650 | 429140 | 173,930 |
https://mathoverflow.net/questions/429152 | 3 | Let $X \subset \mathbb{C}^n$ be a smooth complex affine variety of dimension $m$. Let
$$I\_X \subset \mathbb{C}[z\_1, \ldots, z\_n]$$
be its ideal. Note that it is not always possible to find a set of generators
$$I\_X = \langle g\_1, \ldots, g\_r \rangle$$
such that $r = n - m$ is the codimension. In general, we only ... | https://mathoverflow.net/users/479013 | Locally, the minimal number of ideal generators is the codimension | I am just writing an answer to correct my mistake!
This is true for smooth subvarieties, roughly by the Jacobian criterion. For each closed point, the rank of the Jacobian matrix at that point equals the codimension $m$. So if you choose $m$ defining relations whose partial derivative vectors give $m$ linearly indepe... | 6 | https://mathoverflow.net/users/13265 | 429165 | 173,938 |
https://mathoverflow.net/questions/429161 | 7 | I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.
Here $\Omega$ is any open subset of some $\mathbb{R}^n$ and $C^\infty(\Omega)$ is the space of complex-valued $C^\infty$ functions on $\Omega$. ${C^\in... | https://mathoverflow.net/users/56524 | How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$? | The standard topology on $C^\infty(\Omega)$ is that of uniform convergence on compact subsets of $\Omega$ of all derivatives which is given by the seminorms $\|f\|\_{K,n}=\sup\{|\partial^\alpha f(x)|: x\in K, |\alpha|\le n\}$ with $K\subseteq \Omega$ compact and $n\in \mathbb N$. This topology makes $C^\infty(\Omega)$ ... | 8 | https://mathoverflow.net/users/21051 | 429173 | 173,939 |
https://mathoverflow.net/questions/429176 | 4 | Let $V$ be a vector space and $\|\cdot \|\_1$ and $\|\cdot\|\_2$ two norms on $V$.
Let $\|\cdot\|\_+$ be given by
$$ \|v\|\_+ := \inf\_{v = v\_1 + v\_2} \|v\_1\|\_1 + \|v\_2\|\_2 $$
It is well-known that $\|\cdot\|\_+$ is a norm on $V$. ${}{}{}{}$
Is it true that the (closed) unit ball of $\|\cdot\|\_+$ is the clos... | https://mathoverflow.net/users/3948 | Unit ball of the sum space | Let $B\_+,B\_1,B\_2$ denote the closed unit balls w.r. to $\|\cdot\|\_+,\|\cdot\|\_1,\|\cdot\|\_2$, respectively. Let $C$ be the convex hull of $B\_1\cup B\_2$. Let $\bar C$ be the closure of $C$ (w.r. to the "norm" $\|\cdot\|\_+$ -- which is actually only a semi-norm in general, as pointed out by [Jochen Wengenroth](h... | 6 | https://mathoverflow.net/users/36721 | 429180 | 173,940 |
https://mathoverflow.net/questions/428726 | 2 | Suppose a group $G$ acts on an infinite set $X$ and $X$ has no non-empty $G$-paradoxical subsets. Is it possible for $X$ to have non-trivial $G$-paradoxical subsets modulo finite sets? I.e., can there be infinite subsets $E\_1,...,E\_n$ such that $E\_i \cap E\_j$ is finite when $i\ne j$, and $\tau\_1,\dots,\tau\_n \in ... | https://mathoverflow.net/users/26809 | Paradoxical decomposition modulo finite sets | No it's not possible.
Recall the notation. For $G$ acting on a set $X$, subsets $Y,Y'$ of $X$ are called $G$-congruent if $\exists g\in G$ such that $Y'=gY$, and piecewise $G$-congruent if there are partitions $(Y\_i)\_{1\le i\le n}$ of $Y$, $(Y'\_i)\_{1\le i\le n}$ of $Y$ such that $Y\_i$ and $Y'\_i$ are $G$-congrue... | 2 | https://mathoverflow.net/users/14094 | 429188 | 173,942 |
https://mathoverflow.net/questions/429168 | 7 | $\newcommand\CAR{\mathit{CAR}}\newcommand\Cl{\mathbb C\mathit l}$This question will be rather long and it will be my attempt to finally clarify many issues concerning CCR, CAR and Clifford algebras together with the Fock spaces and the classification of the corresponding von Neumann algebras appearing in this way. I mu... | https://mathoverflow.net/users/24078 | CCR vs. CAR vs. Clifford algebras, infinite tensor products and type of the corresponding von Neumann algebra | Wow, that's a lot of questions. For a more in-depth discussion you might look at my book *Mathematical Quantization*, particularly Section 2.5 and Chapters 4 and 7. But anyway, let me start with the most interesting questions, 2b and 3a.
Question 2b. Are Fock spaces infinite tensor products? Yes. First, remember that... | 6 | https://mathoverflow.net/users/23141 | 429222 | 173,953 |
https://mathoverflow.net/questions/429214 | 2 | **Background**
Say we have an optimization problem $$\min\_x f(x) = g(x) + h(x)$$
where $g$ is differentiable and convex, and $h$ are convex but not necessarily differentiable. If $g$ is the mean squared error function, we can use proximal algorithms to solve optimizations of this type, using the proximal operator,... | https://mathoverflow.net/users/479197 | Can we use the solution to two optimisation problems to solve a third, bigger, one? | I think what you want is the alternating direction method of multipliers (ADMM), which is a type of proximal method.
The formulas you've written down are really close to ADMM.
Rather than try to minimize the difficult objective $f(x) = g(x) + h(x)$ directly, we instead introduce an auxiliary variable $y$ and minimize
... | 2 | https://mathoverflow.net/users/49417 | 429226 | 173,956 |
https://mathoverflow.net/questions/427802 | 5 | Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F\_1/\mathbf{Q}$ and $F\_2/\mathbf{Q}$ be finite, abelian, disjoint extensions, and let $F=F\_1.F\_2$ denote their compositum.
>
> **Question:** Under what conditions ca... | https://mathoverflow.net/users/10547 | Additivity of Elliptic Curve Rank over Compositum of Fields | If you're willing to expand your question a bit, there are interesting relations to be found. More precisely, let $E/K$ be an elliptic curve defined over a number field, and let $L/K$ be a Galois extension such that $G(L/K)$ has a non-trivial idempotent relation (also sometimes called a Brauer relation). For example, t... | 3 | https://mathoverflow.net/users/11926 | 429239 | 173,958 |
https://mathoverflow.net/questions/429242 | 0 | Let $\Omega$ be a smooth bounded domain, $H^1(\Omega) :=\{u: u, Du\in L^2(\Omega)\},$
and $H^1\_0(\Omega)$ is the closure of $C^{\infty}\_{c}(\Omega)$ in $H^1(\Omega)$. Define:
* $\sup\_{\partial\Omega } u:=\inf\{a :(u-a)^+\in H^1\_0(\Omega)\}$
* $ess \sup\_{\Omega} u:= \inf\{a :(u-a)^+=0, a.e.~ in~ \Omega\}$
The w... | https://mathoverflow.net/users/166368 | Sobolev space and weak maximum principle | I may have overlooked something, but I think $\operatorname{esssup}\_{\partial \Omega} \operatorname{Tr} u = \sup\_{\partial \Omega} u$ for all $u \in H^1(\Omega)$.
This follows at once from the well-known characterization $H^1\_0(\Omega) = \{u \in H^1(\Omega) \mid \operatorname{Tr} u = 0\}$ and the fact that $(\operat... | 1 | https://mathoverflow.net/users/490262 | 429245 | 173,959 |
https://mathoverflow.net/questions/429240 | 4 | Let $H$ be a Hilbert space, which we interpret as a space of quantum states.
* If $U(t):H\to H$ is a unitary **norm-continuous** one-parameter group with $U(0)=I$, (essentially) *Cauchy's functional equation* says it has the form $$U(t)=e^{itA}$$ for some **bounded** self-adjoint Hamiltonian $A$.
* If $U(t):H\to H$ i... | https://mathoverflow.net/users/142740 | "Open systems" version of Stone's Theorem for one-parameter groups of quantum operations | This has been generalized by Brian Davies to the general case, the article is
Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equation. Rep. Math.
Phys. 11(2), 169–188 (1977)
A more modern introduction are chapters 5-7 in this book
<https://www.cambridge.org/core/books/quantum-stochastics/0E5... | 4 | https://mathoverflow.net/users/490271 | 429246 | 173,960 |
https://mathoverflow.net/questions/429241 | 2 | Let $f: X \rightarrow S$ be a finitely presented morphism of schemes and let $$E = \{s \in S \mid \text{ $X\_s$ is a point with residue field $\kappa(s)$ } \}$$
Is $E$ a constructible set?
The basic motivation for my asking this is in to check, for dominant $f$, whether there exists an affine monomorphism $S' \righ... | https://mathoverflow.net/users/97635 | Constructibility of the locus of points where the fiber is an isomorphism modulo nilpotents | Let $k$ be a field of characteristic $p>0$, $X=S=\operatorname{Spec}k[t]$ the affine line, and $f:X\to X$ the $p$-th power map. Then $E$ is dense but does not contain the generic point, so it is not constructible.
**[Added August 29]** On the other hand, if $S$ is a noetherian $\mathbb{Q}$-scheme, then $E$ is constru... | 6 | https://mathoverflow.net/users/7666 | 429249 | 173,961 |
https://mathoverflow.net/questions/428895 | 6 | Suppose that $H = H(X)$ is some quantity growing with $X$. Are there any bounds on $$F(X, H) = \min\_{X < n\le X + H} \omega(n)?$$
It isn't hard to obtain a lower bound $\max\_{x\sim X} F(X, H)\gg \frac{\log X}{H\log\log X}$. Also, for $H\gg X^\delta$, I believe it follows from a standard lower bound sieve the bound $F... | https://mathoverflow.net/users/40983 | Upper bound on minimum number of prime factors in short intervals | There is some useful information in the paper [P. Erdôs and I. Kátai. On the growth of some additive functions of small intervals. Acta Math. Hungar. 33 (1979), 345-359.](https://P.%20Erd%C3%B4s%20and%20I.%20K%C3%A1tai.%20On%20the%20growth%20of%20some%20additive%20functions%20of%20small%20intervals.%20Acta%20Math.%20Hu... | 1 | https://mathoverflow.net/users/5712 | 429255 | 173,962 |
https://mathoverflow.net/questions/428557 | 4 | $\newcommand\R{\mathbb R}$Let $0<d\_1<\cdots<d\_k<\infty$ and let $m\_1,\dots,m\_k$ be any integers $\ge1$. Let $n:=m\_1+\dots+m\_k-1$.
Let $d$ denote the Euclidean distance in $\R^n$.
>
> Do then there always exist pairwise disjoint subsets $S\_1,\dots,S\_k$ of $\R^n$ of respective cardinalities $m\_1,\dots,m\_k$ ... | https://mathoverflow.net/users/36721 | About Euclidean distances | Finite [ultrametric spaces](https://en.wikipedia.org/wiki/Ultrametric_space) [allow isometric embeddings into $L\_2$](https://core.ac.uk/download/pdf/82490074.pdf).
| 3 | https://mathoverflow.net/users/32454 | 429256 | 173,963 |
https://mathoverflow.net/questions/429257 | 3 | Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
| https://mathoverflow.net/users/490039 | Reference for triangle groups | It seems that you are interested in “Fenchel’s conjecture” stating that (essentially) all two-orbifolds are finitely (orbifold) covered by surfaces. The case of triangle orbifolds is the hardest, and was solved by Ralph Fox (with, it seems, a few errors). See the 1983 paper by Chau titled [*A note concerning Fox’s pape... | 4 | https://mathoverflow.net/users/1650 | 429267 | 173,965 |
https://mathoverflow.net/questions/429273 | 6 | Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be **periodic** for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is continuous. In private, I've been saying that a point $x \in X$ is **topologically periodic** for $f$ if $x \in \ov... | https://mathoverflow.net/users/586 | What is the name for a point that is periodic to within $\varepsilon$? | Such a point is called **recurrent**.
---
It was stated by Alp Uzman in a comment that the usual definition of recurrent is that $x$ is in its own $\omega$-limit set. This is the definition I had in mind, and under a suitable separation axiom it's obvious that this is equivalent to what was asked, which is why I ... | 13 | https://mathoverflow.net/users/123634 | 429276 | 173,971 |
https://mathoverflow.net/questions/429270 | 4 | Let $f: X\rightarrow S$ be a flat quasi-projective morphism, where $X$ is a smooth variety, and $S$ is a discrete valuation ring. Then we know that $f$ is proper morphism if and only if it satisfies the well-known valuative criterion of properness. The evaluative criterion of properness says the following.
Given a DV... | https://mathoverflow.net/users/490296 | Relative valuative criteria of properness for flat morphisms | Let $S = \operatorname{Spec} R$ where $R$ is a DVR with generic point $\eta$ and closed point $0$. A homomorphism $R \to T$ is flat if and only if it is injective and more generally, $f : X \to S$ is flat if and only if $X$ is equal to the scheme theoretic closure of the generic fiber $X\_{\eta}$.
In particular, $T$ ... | 5 | https://mathoverflow.net/users/12402 | 429278 | 173,972 |
https://mathoverflow.net/questions/429285 | 3 | Let $y\in \mathbb{R}$ and $\mathbf{x},\mathbf{z}\in\mathbb{R}^p$ be *random variable* and *random vectors*. Assume $y=f(\mathbf{x}+\mathbf{z},\mathbf{z})$ for some function $f$.
Is the following statement correct?
If $\mathbf{z}\perp \!\!\! \perp y$ and $\mathbf{z}\perp \!\!\! \perp\mathbf{x}$, where $\perp \!\!\! \per... | https://mathoverflow.net/users/99157 | Independent input feature z can be removed: if y=f(x+z,z), then y=g(x)? | A simple counter-example: take $p=1$, $x=+1$ or $x=-1$ with probability 1/2, $z=+1$ or $z=-1$ with probability 1/2, independently of $x$.
Define $y=xz$ [or, if you wish, $f(x+z,z)=(x+z)z-z^2$]; one has $y=+1$ or $-1$ with probability 1/2, independently of either $x$ or $z$; hence, there is no function $g(x)$ such tha... | 4 | https://mathoverflow.net/users/11260 | 429290 | 173,975 |
https://mathoverflow.net/questions/429287 | -9 | **Notation:**
$$ \{x\}\ :=\ x-\lfloor x\rfloor $$
---
APF-functions $\ \tau(n)\ $ for $\ 2<n\in\mathbb N,\ $ and $\ \xi(n)\ $ for $\ 3<n\in\mathbb N,\ $ are defined as follows:
$$ \tau(n)\ :=\ \sum\_{k=2}^{n-1}\,\left\{\frac nk\right\}\qquad\qquad\text{and}\qquad
\qquad\xi(n)\ :=\ \sum\_{k=2}^{\lfloor\sqrt n\r... | https://mathoverflow.net/users/110389 | Summatory functions for fractional parts | Your $\tau(n)$ and $\xi(n)$ are essentially the same as the [divisor summatory function](https://en.wikipedia.org/wiki/Divisor_summatory_function), often denoted by $\sigma(n)$. Indeed, we have
$$\sigma(n)=\sum\_{m=1}^n\sum\_{k\mid m}1=\sum\_{k=1}^n\left[\frac{n}{k}\right]=2\sum\_{k=1}^{\left[\sqrt{n}\right]}\left[\fra... | 7 | https://mathoverflow.net/users/11919 | 429292 | 173,976 |
https://mathoverflow.net/questions/315913 | 2 | Let $M\_1$ and $M\_2$ be two real positive-semidefinite matrices. Is there any algorithm to compute a permutation matrix $P$ to minimize $\| M\_1-PM\_2P^T \|\_F^2$ or equivalently to maximize $trace(M\_1PM\_2P^T)$?
To be simple, for $i=1,2$, further assume $M\_i=Q\_iQ\_i^T$, where $Q\_i$ has orthonormal columns.
| https://mathoverflow.net/users/99157 | matching two positive-semidefinite matrices | It is a graph-matching problem, which is NP-hard. But some solutions available if relaxing the perturbation matrix assumption. See example at <https://doi.org/10.1016/j.patcog.2016.07.015>
| 4 | https://mathoverflow.net/users/84817 | 429300 | 173,981 |
https://mathoverflow.net/questions/429308 | 1 | I asked the same on [math.Stackexchange](https://math.stackexchange.com/q/4520055/721298).
I have $n$ (say $n = 300$) vectors $v\_1,\dots,v\_n$. Each of them has $K$ coordinates (say $K = 30$). For vector $v\_j$ I denote it's coordinates as $v\_{j1},\dots,v\_{jK}$. I want select $M$ (say $M = 5$) vectors ${v\_{\hat s... | https://mathoverflow.net/users/32454 | Given a set of vectors how to pick $M$ such that sum of maximums of coordinates is maximized? | At very least the problem can be approached via (M)ILP by introducing indicator variables $q\_{jk}$ telling whether the corresponding coordinate $v\_{jk}$ is present in the sum of max, and $r\_j$ telling whether the vector $v\_j$ is selected. This results in the following formulation:
$$\begin{cases}
\sum\_{j=1}^n \s... | 1 | https://mathoverflow.net/users/7076 | 429315 | 173,987 |
https://mathoverflow.net/questions/429322 | 3 | Let $\mathbf{y},\mathbf{x},\mathbf{z}$ are real-valued *random vectors* with possibly different dimensions. Assume $\mathbf{y}=f(\mathbf{x},\mathbf{z})$ for some function $f$.
If $\mathbf{z} \perp\!\!\!\perp \{\mathbf{y},\mathbf{x}\}$ (i.e., $\mathbf{z}$ is *statistically* independent of $\mathbf{y}$ and $\mathbf{x}$ *... | https://mathoverflow.net/users/99157 | $\mathbf{y}=f(\mathbf{x},\mathbf{z})=g(\mathbf{x})$ if $\mathbf{z}\perp \!\!\! \perp \{\mathbf{y},\mathbf{x}\}$ jointly? | $\newcommand\R{\mathbb R}$The answer is yes. According to standard convention, write $X,Y,Z$ in place of $\mathbf x,\mathbf y,\mathbf z$, respectively, reserving the corresponding lower-case letters $x,y,z$ for the corresponding values of $X,Y,Z$.
So, $Z$ is independent of $(X,Y)$. We also have $Y=f(X,Z)$ for some Bo... | 4 | https://mathoverflow.net/users/36721 | 429327 | 173,990 |
https://mathoverflow.net/questions/429330 | 5 | A very important theorem in mathematical physics is Poincaré’s recurrence theorem.
As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for every $A$ measurable the set of points of $A$ for which it exists some $N>0$ such that $\phi^Na \notin A$ for $n \geq N$ h... | https://mathoverflow.net/users/139854 | Poincaré recurrence and its implications for statistical physics and the arrow of time | Since the question is about physical implications of Poincaré recurrence one should take both quantum effects and gravitational effects into consideration. Quantum mechanics does not spoil the recurrences, any finite quantum mechanical system evolves quasi-periodically ([Wikipedia](https://en.wikipedia.org/wiki/Poincar... | 11 | https://mathoverflow.net/users/11260 | 429334 | 173,992 |
https://mathoverflow.net/questions/429319 | 5 | Let $A$ be a separable $C^\*$-algebra and let $\omega$ be a state on $A$.
Then there is an "orthogonal" probability measure $\mu$ on the pure state space $P(A)$ of $A$ such that $\omega(x) = \int\_{P(A)} \psi(x) \, d\mu(\psi)$ [Takesaki 1, IV.6.28]).
If I understand correctly the orthogonality of $\mu$ means that the... | https://mathoverflow.net/users/485160 | Separable C* algebras and type I states | I don't think $\pi\_\omega(A)''$ has that form. For example, take $A = M\_2$ and let $\omega$ be the normalized trace. Then $\omega = \frac{1}{2}(\psi\_1 + \psi\_2)$ where $\psi\_i(x) = \langle xe\_i, e\_i\rangle$, for $x \in M\_2$ and $\{e\_1,e\_2\}$ the standard basis of $\mathbb{C}^2$. That is, $\omega$ is the integ... | 7 | https://mathoverflow.net/users/23141 | 429337 | 173,993 |
https://mathoverflow.net/questions/429341 | 7 | I'm reading Michael Shulman's articles on cohomology in HoTT [here](https://homotopytypetheory.org/2013/07/24/cohomology/) and [here](https://ncatlab.org/nlab/show/spectral+sequences+in+homotopy+type+theory), as well as Floris van Doorn's thesis [here](https://florisvandoorn.com/papers/dissertation.pdf).
Given $E: Z ... | https://mathoverflow.net/users/56938 | Long exact sequences for parametrized cohomology | $\require{AMScd}$Yes. It is better to consider the more general situation of a pushout square of types, since pushout squares (unlike cofiber sequences) are stable under base change. Then the difference between parametrized and unparametrized disappears, since one can work in the $\infty$-topos over the pushout.
More... | 7 | https://mathoverflow.net/users/20233 | 429350 | 173,997 |
https://mathoverflow.net/questions/429351 | 3 | Let $\widetilde{M}\_k^{\leq \ell}$ be the space of weight $k$ depth $\leq \ell$ quasimodular forms, and $\widetilde{M}\_{k,\mathbb R}^{\leq \ell}$ be a subspace of $\widetilde{M}\_k^{\leq \ell}$ whose elements have real Fourier coefficients.
Let $D=\frac{1}{2\pi i}\frac{d}{d\tau}=q\frac{q}{dq} : \widetilde{M}\_k^{\leq ... | https://mathoverflow.net/users/123157 | Decomposition of real quasimodular forms of depth 1 | There is an involution of $\widetilde M\_k^{\le l}$, $$i : \; f(\tau) \mapsto \overline{f(-\overline{\tau})}, \quad \sum\_{n=0}^{\infty} a\_n q^n \mapsto \sum\_{n=0}^{\infty} \overline{a\_n} q^n$$ which commutes with $D$, whose fixed points are exactly the quasimodular forms with real Fourier coefficients. So if $f \in... | 3 | https://mathoverflow.net/users/490352 | 429355 | 173,998 |
https://mathoverflow.net/questions/429346 | 3 | Let $f:[0,1]\to [0,1]$ be continuous. Let—
$$B\_n(f)(x)=\sum\_{k=0}^n f(k/n) {n \choose k} x^k (1-x)^{n-k},$$
be the *Bernstein polynomial* of $f$ of degree $n$.
This question relates to the difference between two Bernstein polynomials of the same function $f(x)$, namely the question of finding a simple and expli... | https://mathoverflow.net/users/171320 | Explicit bounds on the difference between Bernstein polynomials | Some comments.
The estimate $\|B\_{2n} f-B\_n f\|\_\infty \leq c \omega (f, n^{-1/2})$, ($\omega$ is the modulus of continuity of $f$) follows from $\|f-B\_n f\|\_\infty \leq c \omega (f, n^{-1/2})$. Both estimates can be improved to $\frac C n$ if $f \in C^1([0,1]$. However, since
$$ n(f-B\_nf) \to -\frac{x(1-x)}{2}f'... | 0 | https://mathoverflow.net/users/150653 | 429359 | 173,999 |
https://mathoverflow.net/questions/429356 | 2 | Let $I \subset \mathbb{R}$ be an interval, and $f\_n, f: I \rightarrow \mathbb{R}$ a convex function; then its Legendre transform is the function $f^{\ast}: I^{\ast} \rightarrow \mathbb{R}$ defined by
$$
f^{\ast}\left(x^{\ast}\right)=\sup \_{x \in I}\left(x^{\ast} x-f(x)\right), \quad x^{\ast} \in I^{}
$$
where sup den... | https://mathoverflow.net/users/127839 | Continuity of Legendre transform | Conditions in terms of the function $f$ that are simultaneously necessary and sufficient for such convergence are given in [this paper](https://www.heldermann.de/JCA/JCA26/JCA261/jca26005.htm) and in its [arXiv version](https://arxiv.org/abs/1307.3806). The form of those conditions depends on whether either or both end... | 1 | https://mathoverflow.net/users/36721 | 429369 | 174,002 |
https://mathoverflow.net/questions/429332 | 6 | Given a $d$-ball $\mathcal{S}^{d}$, let $P\_n$ a set of $n$ points selected uniformly at random on the boundary $\mathcal{S}^{d-1}$ of $\mathcal{S}^{d}$. Let $\mathcal{C}\_n$ the convex hull of $P\_n$. We denote by $V(\mathcal{S}^{d})$ and $V(\mathcal{C}\_n)$ the volume of $\mathcal{S}^d$ and $\mathcal{C}\_n$ respectiv... | https://mathoverflow.net/users/115803 | $d$-ball approximation for $d\gg 1$ with a convex hull of random points on its boundary | It's true deterministically in $P\_n$. In fact, here's a proof that there is some $c$ so that if $n \leq e^{cd}$ then we have that $V(C\_n)/V(S^d) \to 0$.
We'll work with the ball of radius $1$. For a given configuration $P\_n$, let $p$ denote the probability that a point chosen uniformly from the sphere lies in the ... | 4 | https://mathoverflow.net/users/69870 | 429382 | 174,005 |
https://mathoverflow.net/questions/429376 | 4 | In a comment to this [question](https://mathoverflow.net/questions/429009/non-existence-of-higher-artin-map), David Loeffler asked if one can show that the (local) Artin map
$$K^\times \to G\_K^{ab}$$
does not have a lift to $G\_K$. Probably this wouldn't be canonical, but I can't show that such a lift doesn't exist.
... | https://mathoverflow.net/users/152554 | Existence of lift of (local) Artin map | No, there does not, except possibly a few small corner cases. The problem is finding somewhere for the torsion in $K^\times$ to go.
If $K$ is a finite extension of $\mathbf{Q}\_p$, then there exists a number field $\mathscr{K}$ and a prime $v$ of $\mathscr{K}$ above $p$ such that $\mathscr{K}\_v = K$; and hence we ob... | 5 | https://mathoverflow.net/users/2481 | 429386 | 174,007 |
https://mathoverflow.net/questions/429126 | 6 | Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. Let $<.,>$ denote a $G$-invariant inner product on $\mathfrak{g}$.
Let $(M,\omega)$ be a symplectic compact manifold endowed with a hamiltonian action of $G$, and let $\mu : M \longrightarrow \mathfrak{g}^\*,$ be a moment map associated to this action. W... | https://mathoverflow.net/users/172459 | The norm-squared of a moment map behaves like a Morse-Bott function | As requested I am submitting my comment as an answer.
The desired statement is proved in [Gradient flow of the norm squared of a moment map](https://www.e-periodica.ch/cntmng?pid=ens-001%3A2005%3A51%3A%3A14) by Eugene Lerman who attributes the proof to Duistermaat.
| 4 | https://mathoverflow.net/users/33141 | 429390 | 174,009 |
https://mathoverflow.net/questions/428728 | 1 | Let $T'$ be the Tsirelson space dual in the Figiel-Johson construction, and $B\_{T'}$ the unit closed ball. Then, $B\_{T'}$ satisfies the next properties:
(i) $B\_{T'}\subset B\_{l^\infty}.$ Each vector basis $e\_n$ belongs to $B\_{T'}.$
(ii) $\forall \mathbf{x} \in B\_{T'},\ \mathbf{y} \in \mathbb{R}^{\mathbb{N}},... | https://mathoverflow.net/users/489810 | About the constructions of Tsirelson space | IIRC Figiel and I reasoned this way:
The dual norm to Tsrilson's original norm has the property that for any vector $x$ in $c\_{00}$,
$$
\|x\| \ge \|x\|\_{c\_0} \vee (1/2) \sup \sum\_{k=1}^n \|E\_k x\|,
$$
where the sup is over all admissible $(E\_k)$.
But, by construction, the FJ norm is the smallest unconditional nor... | 4 | https://mathoverflow.net/users/2554 | 429401 | 174,012 |
https://mathoverflow.net/questions/429399 | 1 | "Rule 150" is a fascinating one-dimensional and simple [cellular automaton](https://en.wikipedia.org/wiki/Cellular_automaton) giving raise to some quite chaotic behaviour. This is the starting point of this question.
Let $\{0,1\}^\mathbb{Z}$ denote the set of all functions $x:\mathbb{Z}\to \{0,1\}$. Let $+$ denote ad... | https://mathoverflow.net/users/8628 | Possible finite periodicities of "Rule 150" in the infinite setting | As already mentioned, this cellular automaton is Rule 150.
Rule 150 is an example of a bipermutive cellular automaton, meaning that it is both left permutive (the value of $f(x)(i)$ can be permuted by permuting the value of $x(i-1)$) and right permutive (the value of $f(x)(i)$ can be permuted by permuting the value o... | 9 | https://mathoverflow.net/users/178593 | 429414 | 174,017 |
https://mathoverflow.net/questions/429312 | 7 | Let $A$ be a C\*-algebra that has no one-dimensional irreducible representations, that is, there is no (closed) two-sided ideal $I\subseteq A$ such that $A/I\cong\mathbb{C}$.
Let $J$ denote the (not necessarily closed) two-sided ideal generated by additive commutators in $A$:
$$
J:=\{ \sum\_{k=1}^n a\_k[b\_k,c\_k]d\_... | https://mathoverflow.net/users/24916 | Commutator ideal in nonunital C*-algebra | The answer is NO. Rordam and Robert [MR3072284](https://mathscinet.ams.org/mathscinet-getitem?mr=3072284) have found
a sequence $(A\_n)\_n$ of simple unital infinite dimensional C\*-algebras
such that $\prod A\_n$ has a nonzero character.
(Thanks are due to Yasuhiko Sato for informing me of this.)
Thus the following is... | 9 | https://mathoverflow.net/users/7591 | 429423 | 174,018 |
https://mathoverflow.net/questions/429405 | 2 | I have a question about a proof from Jacob Lurie's [Higher Topos Theory](https://people.math.harvard.edu/%7Elurie/papers/highertopoi.pdf) (p 45):
**Proposition 1.2.12.5.** Let $\mathcal{C}$ be a simplicial set. Every strongly final object (see below what means) of
$\mathcal{C}$ is a final object of $\mathcal{C}$. The... | https://mathoverflow.net/users/108274 | Strongly final vertex $Y$ in a simplicial set gives a retraction of $\mathcal{C}^{ \triangleright }$ onto $\mathcal{C}$ | I think this can be done pretty concretely. According to Definition 1.2.8.1, there is an obvious inclusion $i:C \to C\ast \Delta^0 = \mathcal{C}^{ \triangleright }$. Now you need the retraction.
The key is Proposition 1.2.9.2, which implies that $Hom\_{sSet}(C,C/Y) = Hom\_p(C\ast \Delta^0,C)$ where $p: \Delta^0 \to C... | 1 | https://mathoverflow.net/users/11540 | 429425 | 174,019 |
https://mathoverflow.net/questions/429363 | 2 | Let $\mathscr{H}^m$ be the $m$ dimensional Hausdorff measure in $\mathbb{R}^n$, $m\leq n$. Is it true that for $\mathscr{H}^m$-almost every point $p$ on a Lipschitz manifold $M$ of dimension $m$ embedded in $\mathbb{R}^n$ there is a neighborhood $B\_\epsilon(p)\subset \mathbb{R}^n$ such that $M\cap B\_\epsilon(p)=graph... | https://mathoverflow.net/users/351083 | $(1+\epsilon)$-bilipschitz parametrization of Lipschitz manifold | The answer to my question is no, and a counter-example is provided by the staircase function of the fat-Cantor set, which is a Lipschitz function $f:[0,1]\to\mathbb{R}$ with the property that $\{f'=0\}$ is dense and $\{f'\geq1\}$ has positive measure. Therefore, for $\epsilon>0$ small enough, a function $u$ as in the b... | 1 | https://mathoverflow.net/users/351083 | 429438 | 174,022 |
https://mathoverflow.net/questions/429302 | 2 | Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^\*$ in $K$. Then for each place $v$ of $F$, we have the local Kummer map
$$ \kappa\_v: E(F\_v) \, \otimes \mathbf{Q}\_p/\mathbf{Z}\... | https://mathoverflow.net/users/394740 | Image of Kummer map for CM Elliptic curves | There are three cases:
1. If $v$ is a finite place of $F$ not lying over $\pi$ or $\overline{\pi}$, then the image of the Kummer map $\kappa\_v$ is zero.
2. If $v$ is a finite place of $F$ lying over $\overline{\pi}$, it turns out that the image of the Kummer map $\kappa\_v$ is *still* zero.
3. The tricky case is whe... | 0 | https://mathoverflow.net/users/394740 | 429445 | 174,023 |
https://mathoverflow.net/questions/429440 | 2 | Is there a standard or "simple" contact structure on the $3$-dimensional torus $T^3$, like there are for example for the Eucliden space and the $3$-sphere?
My first thought was to consider a structure descending to $T^3$ from the standard structure defined by the form
$$ \omega\_{std} = dz + x\wedge dy$$
on $\mathbb{... | https://mathoverflow.net/users/142154 | Standard contact forms on the torus | Consider the $1$-form on $\mathbb{R}^3$ given by
$$ \alpha = \cos(2\pi z) dx + \sin(2\pi z) dy, $$
where $(x,y,z)$ are the standard coordinates on $\mathbb{R}^3$. This is a contact form on $\mathbb{R}^3$ with Reeb vector field given by $$R\_{\alpha} = \cos(2\pi z) \partial\_x + \sin(2\pi z) \partial\_y.$$
This vector... | 3 | https://mathoverflow.net/users/13022 | 429453 | 174,025 |
https://mathoverflow.net/questions/429198 | 1 | Let $X$, $Y$, $Z$ be discrete random variables, with $Y$ and $Z$ independent. Does the following equality hold **if $Z$ is independent also of $X$**?
$$
\max\_{f\_{Y,Z}} \big\{ \ I(X; f\_{Y,Z}(Y,Z)) \ \big\} = \max\_{f\_X, f\_Y} \big \{ \ I(X; f\_Y(Y), f\_Z(Z)) \ \big \}
$$
where the maximization is taken over all **no... | https://mathoverflow.net/users/101100 | Maximization of information over set of non-injective functions (Equality) | No. Let $Y,Z$ be iid Bernoulli(1/2), and let $X=Y+Z$ mod 2, which induces pairwise independence. The only non-injective deterministic functions $f\_Y,f\_Z$ are constants, rendering the RHS zero. For the LHS, we can take $f\_{Y,Z}(Y,Z)$ equal to the the binary AND of $Y$ and $Z$, which is not injective, and not independ... | 1 | https://mathoverflow.net/users/99418 | 429463 | 174,030 |
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