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https://mathoverflow.net/questions/248647 | 7 | I have a (maybe dumb) question about the relation between the Künneth theorem and the Theorem-universal coefficient theorem (UCT for short) in $KK$-theory (for the setting see "The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor"- Rosenberg and Schochet).
Let $N$ be the boo... | https://mathoverflow.net/users/nan | Relation between the UCT and Künneth ($C^*$-algebras) | It is absolutely true that these two theorems are conceptually related. But instead of one being a consequence of the other, both rather follow from the same machinery being applied to slightly different setups: left deriving the covariant hom functor vs right deriving the contravariant hom functor. This is clarified i... | 0 | https://mathoverflow.net/users/1291 | 248699 | 113,049 |
https://mathoverflow.net/questions/248486 | 15 | This is an update to my question [Rigorous numerical integration](https://mathoverflow.net/questions/123677/rigorous-numerical-integration/123683#123683) from three years ago.
Is there now a package for rigorous numerical integration that uses interval arithmetic and has access to a well-developed library of special ... | https://mathoverflow.net/users/398 | Numerical integration using interval arithmetic, nowadays | I develop [Arb](http://fredrikj.net/arb/), an arbitrary-precision interval library with special functions support. There is some code included for integration of complex analytic functions using Taylor series ([documentation](http://fredrikj.net/arb/acb_calc.html)), which in principle should scale nicely to high precis... | 21 | https://mathoverflow.net/users/4854 | 248700 | 113,050 |
https://mathoverflow.net/questions/116356 | 8 | Here is a problem in contact topology that was suggested by Petya's answer to [this mathoverflow question](https://mathoverflow.net/questions/116200/a-problem-on-convex-geometry) of mine.
Let $S^\* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. I will think of contact elements as pairs o... | https://mathoverflow.net/users/21123 | From convex geometry to contact topology | This is true for mere topological reasons: By a result due to G. Hirsch, every self-map of a sphere of odd degree sends a pair of antipodal points to a pair of antipodal points. See e.g. Theorem 6.2 in §9 of [A. Granas & J. Dugundji; *Fixed Point Theory*, Springer (2003)]
| 1 | https://mathoverflow.net/users/48067 | 248707 | 113,053 |
https://mathoverflow.net/questions/248685 | 3 | I am not a mathematician, but would like really like to get some confirmation on the things I am doing here.
Let $-\Delta: H^2(\mathbb{R}) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ then every physicist knows that the evolution operator $U(t)=e^{i\Delta t} \in L(L^2,L^2)$ preserves the $L^2$ norm, i.e.
$$||... | https://mathoverflow.net/users/97785 | Free quantum evolution operator on Sobolev space | Question 1: yes, I think so. The generator of any one-parameter strongly continuous unitary group is self-adjoint. This is [Stone's theorem](https://en.wikipedia.org/wiki/Stone%27s_theorem_on_one-parameter_unitary_groups).
Question 2: frankly, it's hard for me to follow your notation here since you use the same symbo... | 2 | https://mathoverflow.net/users/23141 | 248713 | 113,056 |
https://mathoverflow.net/questions/248708 | 4 | If we are given countably many Banach spaces $X\_1, X\_2, \ldots, X\_k, \ldots$, then we can norm their Cartesian product by
$$
\left\| (x\_k)\_{k \in \mathbb N} \right\|\_p := \left( \sum\_k \| x\_k \|\_k^p \right)^{1/p},
$$
or the analogous construction for the case $p = \infty$. Trivially, the elements where thi... | https://mathoverflow.net/users/58976 | Reference on the countable product of Banach spaces | I am posting a CW answer. Feel free to add other references.
Here are some references containing at least some basic facts about the construction from the question (or at least the fact that it yields a Banach space).
**Books**
* Aliprantis, Charalambos D.; Border, Kim C. (2006). *Infinite dimensional analysis: A... | 6 | https://mathoverflow.net/users/8250 | 248715 | 113,057 |
https://mathoverflow.net/questions/231736 | 7 | So i am trying to understand what happens in Implicit (backward) and Explicit (forward) Euler in Stochastic Differential Equations
I'll start with explicit. Say I have the following SDE known as Geometric Brownian Motion
$dX(t)=aX(t)dt + bX(t)dW(t)$, for $a,b$ constants
The explicit scheme goes as follows
$X\_{n+... | https://mathoverflow.net/users/87940 | Euler Schemes in Stochastic Differential Equations | The current state of the art is to *tame* the forward Euler-Maruyama scheme in an explicit way, and completely avoid tricky convergence questions that may come up when using drift-implicit Euler-Maruyama. The *tamed Euler-Maruyama scheme* is easy to implement and provably works if the drift coefficient is globally one-... | 1 | https://mathoverflow.net/users/64449 | 248718 | 113,058 |
https://mathoverflow.net/questions/248691 | 2 | We call a finite simple undirected graph $G=(V,E)$ *critical* if $\chi(G\setminus\{v\}) < \chi(G)$ for all $v\in V$.
If $G$ is critical then for any graph homomorphism $f:G\to G$ we have that $f$ is surjective, and therefore bijective. (The reason is that whenever $f: G\to H$ is a graph homomorphism, then $\chi(G)\le... | https://mathoverflow.net/users/8628 | Critical graphs and endomorphisms | There are lots of examples. They even have a name for when a graph has only surjective endomorphisms, such a graph is called a core. In fact every graph is homomorphically equivalent (has a homomorphism both to and from) a unique core. A good reference for cores is Godsil and Royle's Algebraic Graph Theory. Also, Hahn ... | 2 | https://mathoverflow.net/users/18606 | 248719 | 113,059 |
https://mathoverflow.net/questions/248654 | 5 | Here's a chaser to [this question](https://mathoverflow.net/questions/109127/what-properties-should-a-transform-have-to-deserve-the-descriptor-fourier).
Recall the proof that the number of partition of an integer $n$ into at most $k$ addends is the same as the number of partition of an integer $n$ into integers no la... | https://mathoverflow.net/users/38448 | Does flipping Young diagrams has anything to do with Fourier? | A sort of answer:
There are two ways to realise the Springer correspondence in the language of perverse sheaves: one by restriction (see [BM]), and one by Fourier transform (see [HK]).
[BM] Borho, Walter; MacPherson, Robert
Partial resolutions of nilpotent varieties.
Astérisque 101-102, 23-74 (1983).
[HK] Hotta, ... | 7 | https://mathoverflow.net/users/919 | 248722 | 113,061 |
https://mathoverflow.net/questions/248698 | 2 | In my research, I stumbled upon a particular kind of poset and I was wondering, whether there is something in the literature (I could not find anything so far).
They are distributive lattice $L$ such that their associated subposet of join-irreducibles is a forest (so $L$ is the Birkhoff lattice of a forest). Is ther... | https://mathoverflow.net/users/37708 | Birkhoff Lattice of a forest | This set of lattices is the closure of the set {poset with one element} under two allowed operations: adding a top element or taking a Cartesian product. This implies that the Möbius numbers are in $\{-1,0,1\}$. Any similar property whose behavior under the two operations can be easily followed can also be described on... | 2 | https://mathoverflow.net/users/10881 | 248726 | 113,065 |
https://mathoverflow.net/questions/248711 | 2 | A frame is a complete lattice $\langle L,\mathord{\leqslant}\rangle$ which satisfies the following distributivity law:
$$a\wedge\bigvee\_{i\in I}b\_i=\bigvee\_{i\in I}a\wedge b\_i\,.$$
A frame homomorphism is a mapping $h\colon L\_1\rightarrow L\_2$ which preserves finite operations and such that $h(\bigvee\_{i\in I}... | https://mathoverflow.net/users/22019 | An example of a frame homomorphism which does not preserve Heyting implication | Let $U^{\*}$ denote the pseudocomplement of an element $U$. Then since Heyting algebra homomorphisms preserve pseudocomplements, we shall produce a counterexample by constructing a frame homomorphism which does not preserve pseudocomplements (these frame homomorphisms are quite common).
Let $U\subseteq\mathbb{R}$ be ... | 4 | https://mathoverflow.net/users/22277 | 248731 | 113,067 |
https://mathoverflow.net/questions/248687 | 14 | Does there exist a strictly increasing sequence $\{a\_n\}\_{n\in N}$ of natural numbers such that the following two requirements hold:
1, For all $n\in N$, there is NO subset $M$ of $\{0,\cdots ,n-1\}$ such that $a\_n=\Sigma\{a\_m\ |\ m\in M\}$.
2, $\{a\_n\}\_{n\in N}$ is $o(2^n)$, i.e. $\lim\_{n\to\infty}\frac{a\_... | https://mathoverflow.net/users/97893 | Is there a strictly increasing sequence such that it is o(2^n) and any term cannot equal the sum of any unrepeated predecessors? | Such sequences are called [sum free sequences](https://en.wikipedia.org/wiki/Sum-free_sequence). In the paper ["On a question about sum-free sequences"](http://www.sciencedirect.com/science/article/pii/S0012365X98003227), Deshouillers, Erdős and Melfi construct a sequence where $a\_n$ is $o(n^{3+\epsilon})$. Luczak and... | 18 | https://mathoverflow.net/users/2384 | 248736 | 113,070 |
https://mathoverflow.net/questions/248701 | 4 | If $X'$ is the topological dual of a Banach space, then is that true that a convex set is closed (for the norm on $X'$ given by $\lVert f \rVert\_{X'} := \sup \frac{\langle f , x \rangle}{\lVert x \rVert\_{X}}$) if and only if it is weak\*-closed?
If it is false, is there a class of infinite-dimensional spaces on wh... | https://mathoverflow.net/users/39261 | weak*-closed convex = closed convex? | No it is false in general.
Yes, there is a class of spaces where it is true: these are exactly the reflexive spaces.
Suppose $X$ is not reflexive. Then considering $X$ embedded into $X''$, we have $X \subsetneq X''$. If $\alpha \in X'' \setminus X$, then its kernel is norm closed, weakly closed, convex, but not wea... | 11 | https://mathoverflow.net/users/454 | 248738 | 113,071 |
https://mathoverflow.net/questions/248690 | 3 | Let $(X, T):= (\{0,1\}^{\mathbb Z}, \text{the shift transformation})$ be the full two shift system. Then the set $M(X, T)$ consisting of all invariant (probability) measures on that system is a compact metrizable space under the weak\* topology so that we can talk about residual subsets and meager subsets of $M(X, T)$.... | https://mathoverflow.net/users/41757 | topological size of the set of weakly mixing measures on the full two-shift | Weak mixing is generic. The result is due to K. R. Parthasarathy, *"Indian Journal of Statistics"*, November 1962, Series A vol.24.
Note that in the measurable setting, this is due to Halmos (see his 1956 book *"lectures on ergodic theory"*).
| 2 | https://mathoverflow.net/users/6129 | 248742 | 113,072 |
https://mathoverflow.net/questions/232333 | 2 | A set of relative prime, positive integers $A = [a\_1, \dots, a\_d]$ describe the restricted partition function
$$
p\_A(n) = \# \{(m\_1,\dots,m\_d)\in\mathbb{Z}^d: \textrm{ all }m\_j \geq 0, \sum\_{j=1}^d m\_j a\_j = n \}
$$
I'm interested in a binary version of $p\_A(n)$, namely
$$
q\_A(n) =
\begin{cases}
1 \textr... | https://mathoverflow.net/users/51478 | Relation to Ehrhart polynomial with Uniqueness | Your function $q\_A(n)$ is an indicator whether or not $n$ is representable as a nonnegative combination of $A$ and thus connected to the infamous *Frobenius problem* (also known as the *coin-exchange problem* and the *chicken nuggets progblem*): given $A$ (relatively prime), which is the largest integer that is not re... | 3 | https://mathoverflow.net/users/3193 | 248743 | 113,073 |
https://mathoverflow.net/questions/248740 | 1 | In my research, all the proof comes down to an estimate of the following term
$$\int\_t^{t+h} E|\partial\_x P\_{t+h-\tau}f(X\_\tau)-P\_{t+h-\tau}\partial\_xf(X\_\tau)|^2\,d\tau,\tag{1}$$
where $t>0$ is fixed, $P\_t$ is the transition semigroup associated with a time-homogeneous Ito diffusion, i.e., SDE with time-inde... | https://mathoverflow.net/users/91196 | relationship between transition semigroup and first order spatial derivative | It seems one can obtain the required estimate by a straightforward application of [Dynkin's formula](https://en.wikipedia.org/wiki/Dynkin%27s_formula). In particular, Dynkin's formula allows you to expand $\partial\_x P\_h f(x)$ about $h=0$, $$
\partial\_x P\_h f(x) = \partial\_x f(x) + \int\_0^h \partial\_x \mathbb{E}... | 1 | https://mathoverflow.net/users/64449 | 248745 | 113,074 |
https://mathoverflow.net/questions/248694 | 1 | Let $Q$ be a quiver. The mutation class of $Q$ consists of all quivers which can be obtained from $Q$ by a sequence of mutations. Are there some softwares which compute all non-isomorphic quivers in a the mutation class of a given quiver? Thank you very much.
| https://mathoverflow.net/users/11877 | Softwares which compute all non-isomorphic quivers in a mutation class | This can be done in [sage](http://www.sagemath.org/). See the method "mutation\_class" described [here](http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/cluster_algebra_quiver/quiver.html#sage-combinat-cluster-algebra-quiver-quiver). Below is an example of computing the mutation class of the a directed ... | 2 | https://mathoverflow.net/users/51668 | 248746 | 113,075 |
https://mathoverflow.net/questions/248733 | 2 | Assume the following matrix
$$
C\_p^{(a,b)}:=\left(
\begin{array}{cccccc}
a &a &0 &\cdots &\cdots &0 \\
0 &0 &a &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &\ddots &\ddots &0 \\
0 &\cdots &\cdots &0 &0 &a \\
b &b &\cdots &\cdots &b &b \\
\end{array}
\right)\_{p \t... | https://mathoverflow.net/users/64181 | The structure of the $n$-th power of a special matrix | The characteristic polynomial of $C\_p^{(a,b)}$ is $\lambda^p - (a+b) \lambda^{p-1}$. Therefore, for $m \ge p$ we have $$(C\_p^{(a,b)})^m = (a+b)^{m-p} (C\_p^{(a,b)})^{p-1}$$
It appears that $B = (C\_p^{(a,b)})^{p-1}$ has entries
$$ \eqalign{b\_{1j} &= a^{p-1}\cr
b\_{ij} &= a^{p-i} b (a+b)^{i-2}\ \text{for}\ i \ge 2\c... | 5 | https://mathoverflow.net/users/13650 | 248756 | 113,078 |
https://mathoverflow.net/questions/248748 | -1 | Let $ A \subseteq \mathbf{R}^{n} $ be a closed set and let $ B $ be a closed unit normal bundle over $ A $ ( that means for every $ a \in A $ we have closed subset $ B\_{a} \subseteq \mathbf{S}^{n-1} $ and $ B = \{ (a,u) : u \in B\_{a} \} $ is a closed set).
Consider the set $ F $ of functions $ f : A \rightarrow \ma... | https://mathoverflow.net/users/88920 | existence of continuous functions with values in the fiber of a closed bundle | Unless I'm missing something, the answer is no.
Take $n=2$, $A=\mathbb{R}^2$, and let $a, b$ be distinct points on the unit circle. Now let $B$ be defined as follows:
* $B\_{(x, y)}=\{a\}$ if $x<0$
* $B\_{(x, y)}=\{b\}$ if $x>0$
* $B\_{(x, y)}=\{a, b\}$ if $x=0$.
It's clear that there is no continuous selector f... | 1 | https://mathoverflow.net/users/8133 | 248761 | 113,079 |
https://mathoverflow.net/questions/248763 | 6 | Is there a known set $S=\{x \in G: x^2=1, x\ne1\}$ of elements of a simple compact Lie group $G$ ? By simple compact Lie group I consider $SO\_n$, $SU\_n$, $Sp\_n$, $G\_2$, $F\_4$, $E\_6$, $E\_7$, $E\_8$. (Feel free to extend this definition in case you find it useful). What are the conjugacy classes in this set and th... | https://mathoverflow.net/users/nan | Conjugacy classes of involutions in compact simple Lie group | For the case of a *connected semisimple* compact Lie group $G$, see [this preprint](http://arxiv.org/abs/1506.06252), Section 3, where, following ideas of Kac and Vinberg, we describe set of conjugacy classes of $n$-th roots of a given central element $z$ of $G$. You should take $n=2$ and $z=1$ (or $z=-1$ in your sense... | 7 | https://mathoverflow.net/users/4149 | 248769 | 113,081 |
https://mathoverflow.net/questions/248573 | 1 | Let $\mathcal{C}$ be a [pointed, salient](https://en.wikipedia.org/wiki/Convex_cone#Blunt.2C_pointed.2C_flat.2C_salient.2C_and_proper_cones) cone in $\mathbb{R}^d$. We may also assume that $\mathcal{C}$ is full-dimensional. Consider the set of binary classifiers $$\mathcal{H} = \{\boldsymbol{x}\mapsto\boldsymbol{1}\hsp... | https://mathoverflow.net/users/37202 | VC dimension of cone-restricted linear classifiers | No, I claim that the set of $d$ unit vectors, $e\_1=(1,0,\ldots,0)$, $e\_2=(0,1,0,\ldots,0)$, $\ldots,$ $e\_d=(0,\ldots,0,1)$, along the axes can be shattered, except that for every $\mathcal H$ either the empty set, or all $d$ of the won't be in any $h\in \mathcal H$, so the VC-dimension is $d-1$. (For similar reasons... | 1 | https://mathoverflow.net/users/955 | 248773 | 113,082 |
https://mathoverflow.net/questions/248791 | 8 | Let $K$ be an finite abelian extension of $\mathbf{Q}$ conductor $p$, where $p$ is an odd prime. That is, $K \subset \mathbf{Q}(\mu\_ p)$, the $p$-th cyclotomic field. Let $h\_K$ be the class number of $K$.
If $[K:\mathbf{Q}]=2$, we know that $K=\mathbf{Q}(\sqrt{p})$ or $\mathbf{Q}(\sqrt{-p})$. Gauss's genus theory ... | https://mathoverflow.net/users/95241 | class number of prime degree field with prime conductor | Maybe I am making a mistake here, but let me try:
Let $H$ be the Hilbert class field of $K$. Then $H\cap \mathbb{Q}(\zeta\_p)=K$ as otherwise one prime in there should be totally ramified and unramified at the same time. This shows the triviality of the maximal quotient of the class group $C$ on which the Galois grou... | 10 | https://mathoverflow.net/users/5015 | 248799 | 113,090 |
https://mathoverflow.net/questions/248717 | 10 | Ramsey theory studies whether a *monochromatic* subgraph (more generally, structure) appears when we color the edges of a complete graph with some colors.
I wonder if the following type of question has been studied before, where monochromatic is replaced by bichromatic (which now I use in the sense of having *at most* ... | https://mathoverflow.net/users/955 | How many colors do we need to avoid bichromatic triangles? | Problems of this variety have been studied, beginning with a paper of Erdős and Gyárfás, 'A variant of the classical Ramsey problem'. In that paper, they define a function $f(n,p,q)$ to be the smallest number of colours $k$ needed to produce a $k$-colouring of the edges of $K\_n$ such that every $K\_p$ contains at leas... | 9 | https://mathoverflow.net/users/66275 | 248802 | 113,093 |
https://mathoverflow.net/questions/248778 | 0 | Let $f:X\rightarrow Y$ be smooth family of complex curves over $Y\backslash S$, S-finite set, Y-smooth complex curve. Let $Z \rightarrow Y $ be a ramified covering, ramified over points of $S .$ Let $W$ be a normalization of $X\times\_{Y} Z,$ and we have the induced map $g:W\rightarrow X.$ If $\mathcal{F},\mathcal{G}$ ... | https://mathoverflow.net/users/54760 | Isomorphic direct images of sheaves on product space | In general no. For instance take $Y = P^1$, $X = P^1 \times Y$, and take $S$ to be 4 different points. Then $Z$ is an elliptic curve and $W = P^1 \times Z$. Take $F$ to be (the pullback to $W$ of) a nontrivial line bundle on $Z$ of degree $0$. Then $g\_\*F$ is (the pullback to $X$ of) the pushforward of that line bundl... | 3 | https://mathoverflow.net/users/4428 | 248803 | 113,094 |
https://mathoverflow.net/questions/248808 | 1 | I've noticed that in a 2D manifold, the second Stiefel-Whitney class can always be obtained as the cup product of the first one with itself.
In other words $w\_2=w\_1\smile w\_1$.
Is there a 'natural' way to prove this? Does it appear as a consequence of some deeper relationship between the Stiefel-Whitney classes ... | https://mathoverflow.net/users/97936 | Algebraic relationships between the Stiefel-Whitney classes of a manifold? | The 'natural' way to prove this is to use Wu's formula for the Stiefel-Whitney classes and basic properties of the Steenrod squares.
In more detail, if $M$ is a closed $n$-manifold (connected, but not necessarily orientable) then by mod 2 Poincaré duality there are unique classes $v\_i\in H^i(M;\mathbb{Z}\_2$) such ... | 10 | https://mathoverflow.net/users/15046 | 248818 | 113,099 |
https://mathoverflow.net/questions/248783 | 6 | Due to Mazur, Akbulut and Kirby and many others, there are many examples of integer homology 3-spheres which bound contractible 4-manifolds given by attaching a single 2-handle to $S^1 \times D^3$ which algebraically intersects the one-handle once but maybe geometrically many times. Many of these can be distinguished f... | https://mathoverflow.net/users/50754 | Distinguishing homology $S^1 \times S^2$'s which bound homotopy $S^1$'s | A standard construction would be to take a 3-manifold $Y$ given by 0-surgery on a knot $K$. If $K$ is the boundary of a slice disk $D \subset B^4$, then the complement, say $W$ of a neighborhood of $D$ is a homology $S^1 \times B^3$ with boundary $Y$. But you are asking for more, since $W$ should be a homotopy $S^1 \ti... | 7 | https://mathoverflow.net/users/3460 | 248823 | 113,103 |
https://mathoverflow.net/questions/248798 | 7 | Suppose I have $n$ points $x\_1,\dots,x\_n$ that are all independent uniform samples in the unit square, and I'd like to find a short path (in terms of Euclidean length) that touches all of them (a traveling salesman path, in other words). Obviously, one (inefficient) way to do this would be to write down every permuta... | https://mathoverflow.net/users/70190 | Finding a short path using $(0.99n)!$ permutations | That is highly unlikely. Take any such system $S$ of permutations. Take any configuration of points obtained by sampling and note that we can also get it with the points re-enumerated in any way. Now let us look at how many reasonably short path enumerations are there at all. For each short path, there is a sequence of... | 9 | https://mathoverflow.net/users/1131 | 248828 | 113,107 |
https://mathoverflow.net/questions/247845 | 21 | The EHP sequence consists of maps $S^n\stackrel{E}{\to} \Omega S^{n+1} \stackrel{H}{\to} \Omega S^{2n+1}$. This is a homotopy fibration sequence if $n$ is odd. For even $n$ it is a homopy fibration sequence after localizing at $2$. Applying $\Omega^n$, one obtains the sequence $\Omega^n S^n{\to} \Omega^{n+1} S^{n+1} {\... | https://mathoverflow.net/users/6668 | Is the J homomorphism compatible with the EHP sequence? | **Added 9/7/16:**
I just got access to the paper:
James, I. M. On the iterated suspension. Quart. J. Math., Oxford Ser. (2) 5, (1954). 1–10
which is an explicit reference to Greg's questions on the level of homotopy groups. See section 2.
–––––––––––
${}^\dagger$**Added 9/2/16:** I realize now that what I wrote... | 12 | https://mathoverflow.net/users/8032 | 248839 | 113,111 |
https://mathoverflow.net/questions/248835 | 4 | Is there a number $c>0$ such that:
For any $n$ there is a polynomial $p(x) = a\_nx^n +\cdots + a\_0$ where the coefficients are $-1, 0$ or $1$ such that the number of repetition of the root $x=1$ in $p$ is at least equal to $c\sqrt{\frac{n}{\log n}}$.
| https://mathoverflow.net/users/85969 | On the number of repeated roots | Yes, there is a cute combinatorial argument by Bloch and Polya
>
> "On the roots of certain algebraic equations", Proc. London Math. Soc. 33 (1932), 102-114.
>
>
>
Consider the points $c=(c\_1,\dots,c\_k)\in \mathbb Z^k$ defined by $c\_i=\sum\_{r=0}^{n} \binom{r}{i-1}b\_r$ for $i=1,2,\dots,k$, where the $b\_r$... | 13 | https://mathoverflow.net/users/2384 | 248841 | 113,113 |
https://mathoverflow.net/questions/248784 | 22 | Keating and Snaith have a famous conjecture on the asymptotics of the
integral $\int\_0^T |\zeta(\frac 12+it)|^{2k}\, dt$, where $\zeta$
denotes the Riemann zeta function. See page 510 of the book review by
Brian Conrey of H. Iwaniec, *Lectures on the Riemann zeta function* in
*Bull. Amer. Math. Soc.* **53** (2016), 50... | https://mathoverflow.net/users/2807 | On an asymptotic formula of Keating and Snaith involving the Riemann zeta function | There is a connection! *(Though see the edit below.)* Keating and Snaith make their conjecture by modeling the distribution of $\zeta(s)$ by the distribution of the characteristic polynomial of a random unitary matrix, distributed according to Haar measure. This is connected to symmetric function theory by Schur-Weyl d... | 17 | https://mathoverflow.net/users/5621 | 248843 | 113,114 |
https://mathoverflow.net/questions/248857 | 1 | If $m=p^k$ is a prime power then I know:
$$\exists x\in \mathbb{Z}:x^n\equiv a\bmod p^k\iff a^{\frac{p-1}{\gcd(n,p-1)}}\equiv 1\bmod p^{j}$$
$$\text{ where: }j=\min\left(v\_p(n)+1+[p\mid n][p=2],k\right)$$
Thus if I have the prime factorization of $m$ then by the Chinese remainder theorem I can just verify the abov... | https://mathoverflow.net/users/38626 | Given any $(a,m,n)\in \mathbb{Z}\times\mathbb{N}^2$ with $\gcd(a,m)=1$ is there a quick way to determine if $x^n\equiv a\bmod m$ is solvable? | The Goldwasser-Micali probabilistic cryptosystem is based on exactly this principle. Let $N=pq$ and let $a$ be an integer with $\left(\frac{a}{p}\right)=\left(\frac{a}{q}\right)=-1$, i.e., $a$ is a non-residue mod $p$ and mod $q$. The numbers $N$ and $a$ are public knowledge. In order to encrypt a single bit $\beta$, c... | 7 | https://mathoverflow.net/users/11926 | 248862 | 113,119 |
https://mathoverflow.net/questions/248471 | 2 | Let
* $U,H$ be separable $\mathbb R$-Hilbert spaces
* $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
* $U\_0:=Q^{1/2}U$
* $(\Omega,\mathcal A,(\mathcal F\_t)\_{t\ge 0},\operatorname P)$ be a filtered probability space
* $(W\_t)\_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,(\mathca... | https://mathoverflow.net/users/91890 | Existence of a solution to an infinite dimensional Stratonovich SDE | In order for (2) to hold, it seems that Ito's formula needs to hold for the stochastic process $X(s)$, $0 \le s \le t$. This already requires that: (i) this process is well-defined; and (b) the Frechet partial derivatives $\partial\_t u$, $\partial\_x u$, and $\partial\_{xx} u$ are continuous and locally bounded. To ch... | 2 | https://mathoverflow.net/users/64449 | 248864 | 113,120 |
https://mathoverflow.net/questions/158881 | 23 | I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|\_1 \leq \|A\|\_p\|B\|\_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But it seems that this does not hold in general, in fact
$$A = \begin{bmatrix}1 & 2\\ 0 & 0 \end{bmatrix}, \; B = \begin... | https://mathoverflow.net/users/41123 | Hölder's inequality for matrices | There are (at least two) "generalizations" of Hölder inequality to the non-commutative case. One is the so called tracial matrix Hölder inequality:
$$
|\langle A, B \rangle\_{HS} |= |\mathrm{Tr} (A^\dagger B) | \le \| A\|\_p \,\, \| B\|\_q
$$
where $\| A\|\_p$ is the Schatten $p$-norm and $1/p+1/q=1$. You can find... | 24 | https://mathoverflow.net/users/74539 | 248865 | 113,121 |
https://mathoverflow.net/questions/248767 | 4 | Let $\Omega\_1,\Omega\_2,\dots$ be a sequence of finite nonempty sets endowed with discrete topology. The product space
$$\Omega:=\Omega\_1\times \Omega\_2\times\cdots=\prod\_{n\geq 1}\Omega\_n$$ can be metrizised by the ultrametric $d((\omega\_n),(\nu\_n)):=\inf\{n:\omega\_n\neq \nu\_n\}^{-1}$ with $d((\omega\_n),(\o... | https://mathoverflow.net/users/94251 | Lebesgue Density Theorem: From convergence in probability to a.s. convergence | The answer is "No, of course". Let us denote $(\omega,\omega',n)=(\omega\_1,\dots,\omega\_n,\omega'\_{n+1},\dots)$. Take any large $m$ and consider the event $A\_m$ that $\sum\_{n=0}^{2m}\omega\_n=m$. Then $P(A\_m)\approx m^{-1/2}$ but for every $n\_0$, we have $P((\omega,\omega',n)\in A\_m\text{ for some }n\ge n\_0)\g... | 4 | https://mathoverflow.net/users/1131 | 248867 | 113,122 |
https://mathoverflow.net/questions/248866 | 14 | If $\Sigma\_1$ and $\Sigma\_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma\_1 \hookrightarrow \Sigma\_2$ are two orientation-preserving embeddings that are homotopic, then they are also isotopic. This is a consequence of a result of Maxime Fortier Bourque on conformal embeddings, but it ... | https://mathoverflow.net/users/5010 | Who proved that two homotopic embeddings of one surface in another are isotopic? | First, for simple closed curves, this was known long before Freedman-Hass-Scott. For closed surfaces, it was first proved by Baer in
Baer, R., [Kurventypen auf Flächen](https://eudml.org/doc/149605). J. reine angew. Math., 156 (1927), 231–246.
and
Baer, R., [Isotopie von Kurven auf orientierbaren](https://eudml.o... | 15 | https://mathoverflow.net/users/317 | 248869 | 113,123 |
https://mathoverflow.net/questions/248853 | 4 | Suppose $M$ is a quasi-projective variety, $G$ is a finite group acting on $M$. Let $X$ be the quotient $M/G$ (we assume $X$ to be singular) and $\pi: M\to X$ be the natural projection.
We have $(\pi)^{G}\_{\*}\circ \pi^{\*}=id$, and in the case when $G$ is acting freely, the two functors are left and right adjoints ... | https://mathoverflow.net/users/48616 | Functors between equivariant derived category and derived category of the quotient | The functors are still adjoint and the same relation holds. For details, see for example <http://arxiv.org/pdf/1406.4409.pdf> (especially the proof of Theorem 3.0.14) and references therein.
| 4 | https://mathoverflow.net/users/4428 | 248872 | 113,125 |
https://mathoverflow.net/questions/248819 | 3 | Let $E= \mathbb{F}\_p(\!(u)\!)$, $E^s$ a separable closure of $E$ and write $G\_E= \mathrm{Gal}(E^s/E)$ for the absolute Galois group of $E$. Take a lift of the $u$-adic valuation on $E$ to $E^s$ and write $\nu \colon E^s \rightarrow \mathbb{Q}$ for the resulting valuation. Let $\hat{E^s}$ be a completion of $E^s$ with... | https://mathoverflow.net/users/33573 | Fix field of a certain galois group action | This question is completely answered in
J. Ax – “Zeros of polynomials over local fields—The Galois action”, J. Algebra 15 (1970), p. 417–428.
You get the completion of the perfection of E.
| 4 | https://mathoverflow.net/users/5743 | 248881 | 113,129 |
https://mathoverflow.net/questions/248863 | 23 | Are there any general conjectures/properties (in the Langlands Program) for automorphic representations of $GL\_n$ which are still open for $n=1$?
| https://mathoverflow.net/users/47195 | Any open Langlands Conjectures for GL_1? | The question is subject to interpretation, because the Langlands program was never a clearly delimited set of conjectures to begin with, and moreover many things were added during the following decades, and the answer depends if you want to consider these innovations as part of the Langlands program or not.
I believe... | 43 | https://mathoverflow.net/users/9317 | 248889 | 113,135 |
https://mathoverflow.net/questions/248884 | 9 | The original definition of operads involves maps
$$ \gamma\colon C(n) \times \prod\_{i=1}^n C(k\_i) \to C(\sum\_ik\_i) $$
There is an alternative definition in terms of maps
$$ \circ\_i \colon C(n) \times C(m) \to C(n+m-1). $$
It is not hard to outline an argument that the two definitions are equivalent (subject to so... | https://mathoverflow.net/users/10366 | Detailed proof that $\circ_i$ operads are the same as operads | There is a detailed discussion of the "partial" composition operations of an operad in Volume 1, Chapter 2 of Fresse's monograph:
* Benoit Fresse, *Homotopy of Operads & Grothendieck-Teichmüller Groups*. ([book project web page](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/)) ([author pdf for volume 1](http:... | 7 | https://mathoverflow.net/users/1015 | 248896 | 113,138 |
https://mathoverflow.net/questions/248845 | 3 | Given $g\in L^2(\mathbb{R}^3)$, consider the following function ( defined for $r>0$ ):
$$c(r):=\int\_{\mathbb{R}^3}\frac{g(x)}{|x|^2+r}dx$$
I'm interested in the behavior of $c(r)$ for large $r$. A simple application of Cauchy-Schwarz inequality gives
$$|c(r)|\lesssim r^{-\frac14}$$
However, I think that something bett... | https://mathoverflow.net/users/54552 | Asymptotic behavior of an integral transform | Once you asked, here goes. Let $h(r)=c(r^2)$. Then you need to look at the operator
$$
\Psi: f\mapsto \int\_0^\infty f(\rho)\frac{\rho}{\rho^2+r^2}\,d\rho.
$$
Note two things:
1) The kernel is positive and dominated by $1/(r+\rho)$ (just consider the cases $\rho\le r$ and $\rho\ge r$ separately. Hence, it is not wor... | 5 | https://mathoverflow.net/users/1131 | 248899 | 113,139 |
https://mathoverflow.net/questions/248897 | 7 | Let $P\_{c}(z)=z^2+c$. It seems from the software that the map between the parameter $c$ and the Julia set $J(P\_c)$ is an injective map. Is there some reference about it? Any comments and reference will be appreciated.
| https://mathoverflow.net/users/11966 | A question about Julia set for quadratic family | I expect that you will find the answer (and a lot more) in the following paper, since it's easy to figure out which quadratic polynomials (if any) commute with one another:
Commuting polynomials and polynomials with same Julia set,
Pau Atela, Jun Hu,
*Int. J. Bifurcation Chaos* **06**, 2427 (1996). DOI: <http://dx.d... | 8 | https://mathoverflow.net/users/11926 | 248904 | 113,142 |
https://mathoverflow.net/questions/248890 | 6 | Let $\Delta\_N := \{0 \leq \tau\_1 \leq \dots \leq \tau\_N \leq 1\}$ be the $N$-simplex. Let $a\_j: [0, 1] \rightarrow \mathcal{A}$, $j=1, \dots, N$ be continuous functions with values in some (non-commutative) finite-dimensional Banach algebra (e.g. some matrix algebra). Consider the $\mathcal{A}$-valued integral
$$
... | https://mathoverflow.net/users/16702 | Combinatorics of integral over simplices | For a $\sigma\in S\_N$, let $\Delta\_N^{\sigma}=\{(\tau\_1, \ldots, \tau\_N)\mid 0\le \tau\_{\sigma\_1}\le\cdots\le\tau\_{\sigma\_N}\le 1\}$. Clearly, $\Delta\_N^{\sigma}$ is a space homeomorphic to $\Delta\_N$, and there is a decomposition $$I^N=\bigcup\_{\sigma\in S\_N} \Delta\_N^{\sigma}.$$
The points of $I^N$ that ... | 2 | https://mathoverflow.net/users/6668 | 248911 | 113,145 |
https://mathoverflow.net/questions/248900 | 2 | Let $\mathfrak{n}$ be a $2k$ dimensional $2$-step nilpotent Lie algebra and suppose that its center is $k$ dimensional. Does $\mathfrak{n}$ admit symplectic structure?
Let $\{f\_1,\dots,f\_k\}$ be a basis of the center of $\mathfrak{n}$ and complete it to a basis of $\mathfrak{n}$ $\{e\_1,\dots,e\_k,f\_1,\dots,f\_k\}... | https://mathoverflow.net/users/51380 | Symplectic $2$-step nilpotent Lie algebras | Consider the Lie algebra with basis $(e\_1,\dots,e\_7,z\_1,\dots,z\_7)$, with nonzero brackets (up to skew-symmetry):
$$z\_1=[e\_1,e\_2]=[e\_3,e\_4]=[e\_1,e\_6]=[e\_5,e\_7];$$
$$z\_2=[e\_2,e\_5],z\_{3}=[e\_2,e\_6],z\_{4}=[e\_2,e\_7],z\_5=[e\_3,e\_5],z\_{6}=[e\_3,e\_6],z\_{7}=[e\_3,e\_7].$$
Note that it's 2-step nilpo... | 1 | https://mathoverflow.net/users/14094 | 248939 | 113,159 |
https://mathoverflow.net/questions/248917 | 0 | This question is related to the existence of Einstein metrics on tangent bundles where the metric is induced by the [isotropic almost complex structures](http://link.springer.com/article/10.1007/BF02568316) on the tangent bundle. I'm trying this on the hyperbolic space and I got the following equations as the necessary... | https://mathoverflow.net/users/86401 | Solutions of this system of PDE's | Your problem can be simplified considerably if you use the fact that the first equation and the second equation imply that $\alpha$ has the form $\alpha = f(x\_2,r)$ where $r = {y\_1}^2+{y\_2}^2$. Then the third equation implies that, in fact, $\alpha$ must have the form
$$\alpha(x\_1,x\_2,y\_1,y\_2) = g\left(\frac{r}... | 8 | https://mathoverflow.net/users/13972 | 248942 | 113,162 |
https://mathoverflow.net/questions/248876 | 2 | It seems clear to me that there should be some analogue to the notion of a region of convergence for a Volterra series.
However, it seems as though there are now many different subtle ways for things not to converge, and possibly even more so if we're talking about distributions rather than functions.
How does this... | https://mathoverflow.net/users/24611 | Do Volterra series have a region of convergence, as do Taylor series? | The convergence issue of a Volterra series, basically a Taylor series for *functionals*, is summarized as follows by [Scholarpedia:](http://www.scholarpedia.org/article/Volterra_and_Wiener_series)
>
> Due to its power series character, the convergence of an infinite
> Volterra series cannot be guaranteed for arbit... | 4 | https://mathoverflow.net/users/11260 | 248959 | 113,166 |
https://mathoverflow.net/questions/248684 | 6 | For example,
>
> Under the Hodge conjecture the Motivic galois group coincides with Mumford-Tate group.
>
>
> The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as conjecture D (for singular cohomology) over fields of characteristic 0.
>
>
>
| https://mathoverflow.net/users/83957 | Interesting implications on the theory of motives if the Hodge conjecture holds | * If we assume the Hodge or the Tate conjecture, then the
functor H∗MR is fully faithful on the category of Grothendieck motives (with homological or, under these assumptions equivalently, numerical equivalence).*Hence it gives a linear algebra description of the conjectural abelian category of pure motives.*
* Under t... | 6 | https://mathoverflow.net/users/83957 | 248976 | 113,171 |
https://mathoverflow.net/questions/248961 | 3 | I have a question arising from von Neumann's C\*-algebra formulation of quantum mechanics. In it, a state is a $\mathbb{C}$-linear functional on a C\*-algebra $A$
$\rho:A\rightarrow\mathbb{C}$
satisfying the following conditions:
* $\rho$ is *positive*: for every $a\in A$, we have $ \rho(a^\*a) \geq 0 \in \mathbb... | https://mathoverflow.net/users/24340 | $C^*$ algebras and states | If you want a criterion which is not tautological, that is, beyond the very definition of equivalence of \*-representations, there are (at least) two situations where there is a criterion for equivalence of GNS representations, namely:
1. $\rho$ and $\rho'$ are *pure* states (equivalently, the GNS representations $\... | 3 | https://mathoverflow.net/users/11211 | 248978 | 113,173 |
https://mathoverflow.net/questions/248985 | 2 | I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is:
*does there exist an Itô formula for a Lévy process with values in a Hilbert space*?
| https://mathoverflow.net/users/36886 | Itô Formula for Hilbert space-valued Lévy processes | Yes, there is an Itô formula for Hilbert space valued semimartingales. On this matter, see Appendix D of
>
> S. Peszat and J. Zabczyk, **Stochastic Partial Differential Equations
> with Levy Noise**, Encyclopedia of Mathematics, Cambridge University
> Press, 2007.
>
>
>
| 2 | https://mathoverflow.net/users/64449 | 248992 | 113,179 |
https://mathoverflow.net/questions/248974 | 2 | The so-called Symm's integral equation on an interval $[a,b]$ is defined by
$$\int\_a^bu(y)\log|x-y|dy=f(x),\,\,x\in[a,b],$$ and $f$ is a given function.
In the introduction of a [paper](https://www.researchgate.net/publication/231939080_Collocation_with_Chebyshev_polynomials_for_Symm%27s_integral_equation_on_an_int... | https://mathoverflow.net/users/48438 | Regularity of solutions to certain integral equation | Let's first make the interval equal to $(-1,1)$ by the change of variable
$$
x = \frac{a+b}{2} + \frac{b-a}{2}\, t ;
$$
then the equation becomes
$$
\int\_{-1}^1 u(s)[c+\log |s-t|]\, ds = g(t) , \quad\quad\quad\quad (1)
$$
with $c=\log (b-a)/2$ and $g=2f/(b-a)$ (and I use the sloppy but convenient notation where $u(s)$... | 3 | https://mathoverflow.net/users/48839 | 248995 | 113,180 |
https://mathoverflow.net/questions/248994 | 16 | It seems natural to consider $\lim\_{q \rightarrow 1^-} \sum\_{n \in S} q^n - \sum\_{n \in T} q^n$, when it exists, as a way of comparing the sizes of two sets $S,T \subseteq {\bf N}$ that have the same density; for instance, $\{0,2,4,...\}$ and $\{1,3,5,...\}$ both have density 1/2, but the first set might be said to ... | https://mathoverflow.net/users/3621 | Comparing sizes of sets of natural numbers | Different notions of summability have indeed been considered before (see in particular Abel summation). For example, in the study of primes a famous observation of Chebyshev is that there are more primes of the form $3\pmod 4$ than $1\pmod 4$. Usually this is interpreted in the sense of the number of primes up to $x$ t... | 20 | https://mathoverflow.net/users/38624 | 248998 | 113,181 |
https://mathoverflow.net/questions/248935 | 2 | I have come across various instances of interpolation estimates of the following type:
$ \| f \|\_{L^r(\mathbb{R^n})} \lesssim \| |\nabla|^{-s\_1} f \|\_{L^{r\_1}(\mathbb{R^n})}^\theta \| |\nabla|^{s\_2} f \|\_{L^{r\_2}(\mathbb{R^n})}^{1-\theta}$,
where
$\tfrac{1}{r} = \tfrac{\theta}{r\_1}+\tfrac{1-\theta}{r\_2}$... | https://mathoverflow.net/users/36601 | An interpolation estimate involving fractional derivatives | I would add this as a comment, but I dont have enough reputation.
If I understand you correctly, then your $|| |\nabla|^{-s\_1}||\_{L^{r\_1}(\mathbb{R}^n)}$ is just the norm of the Bessel potential space $H^{-s\_1}\_{r\_1}(\mathbb{R}^n)$. For the complex interpolation method we have (using your notation)
$(H^{s\_2}... | 1 | https://mathoverflow.net/users/97669 | 249007 | 113,184 |
https://mathoverflow.net/questions/248988 | 2 | Let $R=k[x\_1,\ldots,x\_6]$ be a polynomial ring and $I=(x\_1x\_5-x\_2x\_4,x\_2x\_6-x\_3x\_5)$ be an ideal.
>
> How to show that, $(I^2:x\_1x\_5-x\_2x\_4)=I$ ?
>
>
>
| https://mathoverflow.net/users/68302 | Properties of colon ideal | Put $P=x\_1x\_5-x\_2x\_4$, $Q=x\_2x\_6-x\_3x\_5$. Assume a polynomial $A$ satisfies $AP\in I^2$. Then we have a relation $$(\*)\quad AP=UP^2+VQ^2+WPQ.$$ In particular, $P$ divides $VQ^2$. Since $P$ is prime and does not divide $Q$, it must divide $V$. Simplifying $(\*)$ by $P$ gives the result.
| 6 | https://mathoverflow.net/users/7666 | 249011 | 113,186 |
https://mathoverflow.net/questions/249002 | 0 | Is there a simple, finite, undirected graph $G=(V,E)$ with the following properties?
1. $\omega(G) < \chi(G)$ (where $\omega(G)$ is the size of the largest clique in $G$), and
2. up to isomorphism, the only proper minor $M$ of $G$ such that there is a graph homomorphism $f:G\to M$ is the complete graph $K\_n$ where $... | https://mathoverflow.net/users/8628 | Graph homomorphism to minor | I think a 5-cycle meets your needs.
Clique number 2, chromatic number 3. Check.
(It does have a homomorphism to itself, but I assume you meant proper minor.)
It has no homomorphism to a 4-cycle or a 4-path (the two possible minors with 4 edges) and so that leaves a hom to a triangle. Check.
| 5 | https://mathoverflow.net/users/1492 | 249013 | 113,187 |
https://mathoverflow.net/questions/249012 | 4 | Are there any models of Martin-Löf's intensional type theory in which univalence or function extensionality fails?
In the [HoTT book](https://homotopytypetheory.org/book/), axioms like $\mathsf{LEM}\_{\infty}$ (in Section 3.4) are proved to be inconsistent with univalence. I suppose that there might be some models in... | https://mathoverflow.net/users/89916 | HoTT without Funext, Univalence | “HoTT” isn’t generally currently considered as referring to a single specific formal system — it’s a similar situation to, say, “constructive mathematics”, for which there are various different more or less well-studied formal systems.
The core of most systems currently used for HoTT is Martin-Löf’s intensional type ... | 13 | https://mathoverflow.net/users/2273 | 249014 | 113,188 |
https://mathoverflow.net/questions/249001 | 1 | I asked a similar question yesterday. But I found out that I should make some major changes to it. So I decided to ask a new version of my previous question.
Prove or find a counterexample to the following statement.
Let $G$ be a finite group with at least two minimal subgroups and $I\_1, I\_2, ... ,I\_n$ be a fami... | https://mathoverflow.net/users/97247 | Proper subgroups which have the same minimal subgroups(revision) | There are so many conditions that it is hard to find examples satisfying the hypotheses, which makes it harder to find counterexamples, but I am sure they exist.
I believe there is a counterexample that is an extension of $P=C\_{17^2} \times C\_{17^2}$ by $C\_9$ with fixed point free action of $C\_9$ on $P$. For the ... | 2 | https://mathoverflow.net/users/35840 | 249016 | 113,189 |
https://mathoverflow.net/questions/249015 | 15 | Let $G$ be a compact group and $T$ a maximal torus on $G$. Suppose $f$ is an analytic function defined on $T$. Is there an analytic function $F$ on $G$ whose restriction agrees with $f$ on $T$?
| https://mathoverflow.net/users/12233 | Can an analytic function defined on a maximal torus be extended analytically to all the Lie group? | This seems to be consequence of the paper
Cartan, Henri: Variétés analytiques réelles et variétés analytiques complexes. Bull. Soc. Math. France 85 1957 77–99
Cartan shows more generally (see sections 6 and 7), that a real analytic function on a real analytic submanifold of $\mathbb R^n$ can be extended to a real a... | 21 | https://mathoverflow.net/users/89948 | 249022 | 113,192 |
https://mathoverflow.net/questions/249032 | 6 | Suppose that $M$ is a transitive class, denote by $\mathrm{HOD}(M)$ the class of all those sets which are hereditarily definable from ordinals and parameters in $M$.
Some trivial examples include $\mathrm{HOD}(V)=V$, and $\rm HOD(Ord)=HOD$. Note that this model might not satisfy the axiom of choice, for example if $M... | https://mathoverflow.net/users/7206 | Definability of defining classes | Yes, this can happen.
Let's start in $L$ and first perform Easton forcing to add a Cohen
subset to every regular cardinal, giving rise to the forcing
extension $L[G]$. Next, over $L[G]$ we force to code every set into
the GCH pattern, for example, by iterating the forcing that forces
GCH or its negation with a lotter... | 6 | https://mathoverflow.net/users/1946 | 249035 | 113,194 |
https://mathoverflow.net/questions/249029 | 5 | Excuse me if this question is trivial or trivially false, or not at this sites level. Lets work over an algebraically closed field of characteristic $p>0$, say $k$. The action of the Frobenius on $H^r\_{crys}(X/W(k))$ gives us a sequence of positive rational numbers $a\_i$ by taking the order of the eigenvalues.
Now ... | https://mathoverflow.net/users/41103 | Are the Eigenvalues of the Frobenius on Crystalline cohomology bounded by degree? | I assume that you mean $H^r\_{crys}(X/W(k))$ because $H^r\_{crys}(X/k)$ is just the de Rham cohomology over $k$.
Next, over arbitrary base field this definition of $a\_i$'s(they are called slopes of Frobenius) does not make sense because Frobenius is only $p$-linear and its eignevalues a priori depend on the choice o... | 11 | https://mathoverflow.net/users/39304 | 249040 | 113,196 |
https://mathoverflow.net/questions/249045 | 4 | Consider the matrix $D$, adjacency matrix of an undirected graph $G$ on $n$ vertices, with the notation that $d\_{i,i}=0,\forall i$.
The matrices $A\_i$ are constructed from Identity matrices, $I\_{n\*n}$ with $i^{th}$ column replaced with $i^{th}$ column of $D$.
I want to consider matrix $A=\prod\_{s\in S} A\_{s}$... | https://mathoverflow.net/users/98053 | No arbitrary product of matrices has eigenvalue 1? | In the case $n=4$ you could have $D = \pmatrix{0 & 0 & 0 & 1\cr 0 & 0 & 1 & 0\cr
0 & 1 & 0 & 0\cr 1 & 0 & 0 & 0}$, in which case $A\_1 A\_2 A\_3 A\_4 A\_1 A\_2 A\_3 A\_4 = \pmatrix{0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 1 & 1 & 0\cr 1 & 0 & 0 & 1\cr}$ has eigenvalues $0$ and $1$, both with multiplicity $2$.
EDIT: In f... | 11 | https://mathoverflow.net/users/13650 | 249048 | 113,198 |
https://mathoverflow.net/questions/248945 | 22 | In the Boston conference on Fermat's Last Theorem (Summer 1995), [Barry Mazur said](https://vimeo.com/album/1940496/video/42658745) (around 15m into the video) about class field theory that
>
> If you are a number-theorist and you want to cheer yourself up, you
> can say that we really, we completely understand t... | https://mathoverflow.net/users/2821 | What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$? | I am not sure it makes sense to ask "what is the p-adic local Langlands conjecture for $\mathrm{GL}\_1$". Nobody has succeeded in even formulating a reasonable candidate for a p-adic LLC for $\mathrm{GL}\_n$, so you can't just plug $n = 1$ in and see what it says. Morally, the conjecture should be something like this: ... | 18 | https://mathoverflow.net/users/2481 | 249051 | 113,199 |
https://mathoverflow.net/questions/249053 | 2 | For $A\subseteq \mathbb{N}$ we define the upper density by $$\mu(A)=\limsup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
A nice property of this map $\mu:{\cal P}(\mathbb{N})\to [0,1]$ is that it is translation-invariant (that is for $A\subseteq \mathbb{N}$ we have $\mu(A) = \mu(n+A)$ for all $n\in \mathbb{N}$, wh... | https://mathoverflow.net/users/8628 | Density measure on $\mathbb{N}^2$ | Write $p\_1$ and $p\_2$ for the two projections from $\mathbb N\times \mathbb N\to \mathbb N$.
Define $\mu\_2(U) = \mu(p\_1[U]) \cdot \mu(p\_2[U])$. This satisfies your demands.
But this is far from unique. You do not demand that $\mu\_2$ is monotone, so you might also define $\mu\_2(U)=0$ whenever $U$ is not of ... | 4 | https://mathoverflow.net/users/14915 | 249059 | 113,202 |
https://mathoverflow.net/questions/249068 | 4 | Let $M$ be a metric space and let $U \subset M$ be open. Then the Hausdorff dimension of $U$ is defined in the usual way. If there is a single dimension number $d$ that is the Hausdorff dimension of every open set $U$ in $M$ I say the metric space $M$ has uniform Hausdorff dimension $d$.
What I want to know is whethe... | https://mathoverflow.net/users/58103 | Is there a concept of uniform Hausdorff dimension? | This is not an answer, just a comment:
I have never heard a name for the property you cite.
There is a significantly stronger property that is commonly used and does have a name: A metric space is called **Ahlfors $n$-regular** if there is a constant $K$ such that for each closed ball $B(x,r)$ in the space ($0<r\le... | 5 | https://mathoverflow.net/users/98074 | 249074 | 113,206 |
https://mathoverflow.net/questions/249069 | 18 | As we know, Voevodsky constructed several candidates for the triangulated category of motives using different constructions and topologies (h, qfh, etale, and Nisnevich).
>
> I would like to know what motivated these specific topologies, and the
> relationship between these different categories (I know the ones
> ... | https://mathoverflow.net/users/98070 | Voevodsky's Triangulated Categories of Motives and their Relationships | I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint.
Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with integral coefficients. Furthermore, motives with transfers(!; see below) with rational coefficients "do not depend on ... | 16 | https://mathoverflow.net/users/2191 | 249076 | 113,207 |
https://mathoverflow.net/questions/249072 | 7 | Where $a<b$, say that the four “types” of non-empty bounded intervals are:
$(a,b)$, $[a,b]$, $(a,b]$, and $[a,b)$.
Let $\langle X,< \rangle$ and $\langle Y,< \rangle$ be dense linear orderings without endpoints such that
1. $\langle \mathbb{R},< \rangle$ is order-embeddable into $\langle X,< \rangle$ and into $\l... | https://mathoverflow.net/users/83742 | Incomparable dense linear orderings extending $\langle \mathbb{R},< \rangle$ | As Wojowu says, the long ray and the reverse long ray are each locally like the reals, but one has an increasing $\omega\_1$-sequence and no decreasing $\omega\_1$-sequence, and the other a decreasing and no increasing $\omega\_1$-sequence, and this prevents an embedding in either direction. So they are incomparable.
... | 5 | https://mathoverflow.net/users/1946 | 249082 | 113,210 |
https://mathoverflow.net/questions/214792 | 4 | I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems unavailable. So is there any way to define a solution (a solution with singularity) properly?
(A local classical solut... | https://mathoverflow.net/users/64461 | Is there any theory of Hamilton-Jacobi system? | Unfortunately, there is (to the best of my knowledge) no general theory for systems of Hamilton-Jacobi equations. That is to say, it is easy to come up with various notions of viscosity solution for systems, but no general uniqueness results are known. The main obstacle is how to extend the maximum principle to systems... | 2 | https://mathoverflow.net/users/18406 | 249086 | 113,211 |
https://mathoverflow.net/questions/249071 | 12 | Is there a finitely generated soluble group with uncountably many maximal subgroups? Any classification of such groups?
| https://mathoverflow.net/users/98061 | Finitely generated soluble groups with uncountably many maximal subgroups | Yes, there is a finitely generated soluble group with uncountably many maximal subgroups.
Fix an odd prime $p$. Denote by $F\_p$ the field on $p$ elements, $C\_2$ the cyclic group on $2$ elements. Let $A$ be the set of functions from $\mathbf{Z}$ to $\{-1,1\}$. Let $M$ be the free $F\_p$-module on the generators $(e\... | 17 | https://mathoverflow.net/users/14094 | 249095 | 113,213 |
https://mathoverflow.net/questions/248999 | 4 | Suppose $X$ is a projective smooth, geometrically connected, hyperelliptic curves over a field k, we may ask $X(k)\neq\varnothing$. I want to know how to compute the $H^{0}(X,\Omega\_{X}^{1})$.
And there is theorem of Noether, which claims that:
Let $X$ be a non-hyperelliptic curve, then the map:
$H^{0}(X,K\_{X})^{\o... | https://mathoverflow.net/users/88343 | kahler differential on hyperelliptic curves | I just wanted to mention another way to recover the basis of holomorphic differentials and the failure of surjectivity given in the comments : an hyperelliptic curve of genus $g\geq 2$ can be expressed as a 2-to-1 cover of $\mathbb{P}^1$ ramified along $2g+2$ points, $f\colon X\to \mathbb{P}^1$. In particular, using do... | 4 | https://mathoverflow.net/users/45285 | 249097 | 113,214 |
https://mathoverflow.net/questions/249060 | 5 | The roots of trinomial equations $x^p+x-q=0$ ($p\in\mathbb{N}$) can be expressed in terms of the *hypergeometric functions*. I am wondering if at least one real root, for instance given by the following iterative solution $x\_n=\frac{q - (1 - p) x\_{n-1}^p}{ 1+p x\_{n-1}^{(p - 1)}}$, $x\_0=q^{1/p}$ can be expressed in ... | https://mathoverflow.net/users/41145 | Series solution of the trinomial equation | An analogous formula does hold, although the corresponding functions are not hypergeometric if $p$ is irrational.
For given $p\in\mathbb{R}$, $p>1$, consider the power series
$$h(z)=\sum\_{k=0}^{\infty} \frac{(-1)^k}{pk-k+1}\binom{pk}{k}\, z^k$$
with radius of convergence $R=(p-1)^{p-1}/p^p.$
Then, for $0\le y\le ... | 14 | https://mathoverflow.net/users/6101 | 249098 | 113,215 |
https://mathoverflow.net/questions/249078 | 2 | I'm studying Fourier analysis and have a question about approximate identities.
Let $k\_{\epsilon}$ be an approximate identity on $L^{1}(\mathbf{T})$. We know that $k\_{\epsilon}\*f\to f$ in $L^{1}$ as $\epsilon\to 0$.
Question: Can we construct a $k\_{\epsilon}$ such that for every $f\in L^{1}$, $k\_{\epsilon}\*f$... | https://mathoverflow.net/users/98076 | Approximate identities and pointwise convergence | No. Given any $k\_{\epsilon}$, we can easily build an $f$ so that $I(\epsilon):=k\_{\epsilon}\*f(0)$ oscillates roughly between $-1$ and $1$ as $\epsilon\to 0$. My $f$ will only take the two values $\pm 1$.
Just take $f=1$ everywhere except on a tiny interval centered at $0$, and take this interval so small that $I(1... | 2 | https://mathoverflow.net/users/48839 | 249106 | 113,218 |
https://mathoverflow.net/questions/249103 | 0 | I'm trying to understand Harbater's "Mock Covers and Galois Extensions".
There, a "mock cover" of a domain $S$ is a finite map $p : $ Spec $T\rightarrow $ Spec $S$, where $T$ is a torsion-free $S$-algebra, and such that the restriction of $p$ to any irreducible component of Spec $T$ is an isomorphism onto Spec $S$.
... | https://mathoverflow.net/users/88840 | How does normalization behave on closed subschemes? | In general, you can say very little about the fibers of a normalization. It is possible that the total space is very nice yet the fibers have horrible singularities. (OK, not extremely horrible, but definitely not normal is not only possible, but the expected behavior for a special fiber).
In your case the situation ... | 3 | https://mathoverflow.net/users/10076 | 249107 | 113,219 |
https://mathoverflow.net/questions/249109 | 4 | In my research in linear algebra, I have come across a useful result stating that the Schur complement of a principal non-singular submatrix of an M-matrix is also an M-matrix, but I have never found a source with a proof for this fact. Could someone please point me to a source of the proof? I thank all helpers.
| https://mathoverflow.net/users/69446 | Requesting reference for result from linear algebra on Schur complements of M-matrices | Here is a "proof sketch"; please go through the references to iron out the details.
1. An M-matrix is a special case of an H-matrix (i.e., a matrix $A$ such that there exists a nonsingular diagonal matrix $D$ for which $DA$ is strictly column diagonally dominant).
2. Strictly column diagonally dominant matrices are c... | 3 | https://mathoverflow.net/users/8430 | 249118 | 113,225 |
https://mathoverflow.net/questions/249117 | 5 | While doing some simulations dealing with the triadic odometer $T$, I observed that the results were the same when I replace $T$ with $T^2$. Therefore I'm wondering whether $T$ is isomorphic to $T^2$. Is it true?
| https://mathoverflow.net/users/21339 | Is the triadic odometer isomorphic to its square? | Yes. More generally, if $T$ is a $k$-odometer, and $r$ is relatively prime to $k$, then $T^r$ is conjugate (that is, measure-theoretically isomorphic) to $T$. Relative primeness is necessary in order that $T^r$ be ergodic.
One way to prove this (among several) is to use the characterization of odometers by their spe... | 7 | https://mathoverflow.net/users/42278 | 249122 | 113,228 |
https://mathoverflow.net/questions/248975 | 5 | Let $\mathbf A=\left[\begin{matrix}\mathbf A\_{11}&\mathbf A\_{12}\\ \mathbf A\_{21}&\mathbf A\_{22}\end{matrix}\right]$ be a positive semi-definite matrix, $\mathbf A\_{ij}\in\mathbb C^{n\times n}$ and $rank(\mathbf A)=n$.
**Prove or disprove** that
$$
\lambda\_{max}(\mathbf A) + \lambda\_{max}(\mathbf A\_{11}+\math... | https://mathoverflow.net/users/97587 | Upper bound of the largest eigenvalue of a PSD block matrix in terms of blocks | Write $A=\sum w\_i\otimes w\_i$ where $w\_i$ are orthogonal and $\|w\_1\|=\max\_i\|w\_i\|$. Also write $w\_i=(u\_i,v\_i)$. Then the inequality in question is just
$$
\|u\_1\|^2+\|v\_1\|^2+\max\_{e:\|e\|=1}\sum\_i[|(u\_i,e)|^2+|(v\_i,e)|^2]
\\
\le \left\|\begin{pmatrix}\sum\_i\|u\_i\|^2&\sum\_i(u\_i,v\_i)\\\sum\_i(v\_i... | 3 | https://mathoverflow.net/users/1131 | 249135 | 113,233 |
https://mathoverflow.net/questions/249149 | 2 | For any $k$, the class of graphs of treewidth at most $k$ can be characterized by a finite set of forbidden minors.
For treewidth $1$ and $2$, the set is of size $1$. Then for treewidth $3$, the set is of size $4$, and beyond that the size of the set grows very rapidly.
For a given $k$, is there a known upper boun... | https://mathoverflow.net/users/31016 | Size of forbidden minors for treewidth | Yes, an upperbound was proved in [Upper Bounds on the Size of Obstructions and Intertwines](http://ac.els-cdn.com/S0095895697917883/1-s2.0-S0095895697917883-main.pdf?_tid=78779910-7428-11e6-a86b-00000aab0f01&acdnat=1473162965_e2b1c9896d06ad788ff7004e3435616b) by Lagergren. The paper is behind a paywall, but the relevan... | 4 | https://mathoverflow.net/users/2233 | 249154 | 113,237 |
https://mathoverflow.net/questions/249142 | 1 | I'm trying to understand the proof of the Barr-Diaconescu theorem about Boolean covers for Grothendieck sites. Precisely, the versions you can find in Jardine's book "Local Homotopy Theory" or in Mac Lane - Moerdijk "Sheaves in Geometry and Logic", which are essentially the same. That is,
**Theorem.** For every Groth... | https://mathoverflow.net/users/1246 | Missing an elementary step about the frame of strings in Barr-Diaconescu theorem | I think that the missing step is to build the locale $X$ from your poset of strings. But you should not look for a frame structure in the poset; rather, you are supposed to apply Theorem 5.1 in Maclane-Moerdijk to get that locale. If the site of the topos is a poset, the proof of that theorem (which uses in turn Giraud... | 1 | https://mathoverflow.net/users/12976 | 249155 | 113,238 |
https://mathoverflow.net/questions/248255 | 2 | I'm looking for a proof of convergence of stochastic gradient descent applied to a non-convex smooth function. I'm generally interested in just asymptotic convergence, preferably to a critical point, but not necessary to a (local) minimizer.
I have found many relevant results but they all have some additional assump... | https://mathoverflow.net/users/16267 | Stochastic gradient descent convergence for non-convex smooth functions | There is also a more recent literature on convergence of a randomized SGD for non-convex functions: <http://arxiv.org/pdf/1309.5549v1.pdf>
| 3 | https://mathoverflow.net/users/16267 | 249162 | 113,240 |
https://mathoverflow.net/questions/249158 | 2 | **Conjecture:** Let $n\geq4$. Is there a set of $n$ non-planar point $A\_1, A\_2,\cdots , A\_n$ be $n$ on a sphere ([three-dimensional space](https://en.wikipedia.org/wiki/Three-dimensional_space)) satisfying the conditions: if $i-j=l-m$ then $d(A\_i, A\_j) = d(A\_l, A\_m)$ for all $i, j, l, m=1, 2, \cdots, n$? Could y... | https://mathoverflow.net/users/76698 | Exist $A_1, A_2,\cdots , A_n$ be $n$ points on a sphere, satisfy if $i-j=l-m$ then $A_iA_j=A_lA_m$ | The vertices of a [twisted prism](https://en.wikipedia.org/wiki/Prism_(geometry)#Twisted_prism) will achieve this for any even $n$. Choose $0 < r < 1$. Put
$$A\_k = \left( r \cos \frac{2 \pi k}{n}, r \sin \frac{2 \pi k}{n}, (-1)^k \sqrt{1-r^2} \right) .$$
---
YCor points out in comments that, if we want $i-j \bmo... | 5 | https://mathoverflow.net/users/297 | 249166 | 113,242 |
https://mathoverflow.net/questions/248462 | 0 | Let $X\neq \emptyset$ be a set and let ${\cal J} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say that a topology $\tau$ on $X$ is ${\cal J}$-*compatible* if for every $J\in {\cal J}$ there is a continuous surjective map $f:X\to J$, where $J$ carries the subset topology i... | https://mathoverflow.net/users/8628 | Partially ordered set of compatible topologies | If we put $\mathcal J = \{X\}$, then $\mathrm{Cptb}(X,\mathcal J)$ is just the lattice of topologies on $X$. In this case, the answer to your question is known: if $X$ is infinite, then not every topology on $X$ has a "minimal" strict refinement.
For example, let $p \in X$ and let $\tau$ be the one-point compactifica... | 1 | https://mathoverflow.net/users/70618 | 249173 | 113,244 |
https://mathoverflow.net/questions/242468 | 19 | Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?
| https://mathoverflow.net/users/56920 | Origin of the name ''momentum map'' | According to §1.3 and §11.2 of Marsden and Ratiu [1994] (see detailed citation given below), the momentum map concept can be traced back to [Sophus Lie's](https://en.wikipedia.org/wiki/Sophus_Lie) 1890 book, and is an English translation of the French words *application moment.* To quote directly from Marsden and Ratiu... | 26 | https://mathoverflow.net/users/64449 | 249174 | 113,245 |
https://mathoverflow.net/questions/249156 | 4 | I am interested in sufficient conditions for the second homotopy group $\pi\_2(X)$ of a compact connected manifold to be finite. Are there familiar classes of manifolds $X$, for which this is the case?
| https://mathoverflow.net/users/50457 | Finite second homotopy group | You can have for instance a look at the following paper
*Manuel Amann and Anand Dessai*, MR 2600123 [**The $\hat A$-genus of $S^1$-manifolds with finite second homotopy group**](http://dx.doi.org/10.1016/j.crma.2010.01.011), *C. R. Math. Acad. Sci. Paris* **348** (2010), no. 5-6, 283--285.
and at the references gi... | 4 | https://mathoverflow.net/users/7460 | 249175 | 113,246 |
https://mathoverflow.net/questions/245333 | 5 | It has been proved by Toën and Lunts-Schnürer that the dg category $\mathrm{L}\_{qcoh}(X\times Y)$ of quasi-coherent sheaves over the product of two quasi-compact, quasi separated (and flat over a commutative ring $k$) schemes is quasi-equivalent to the dg-category of continuous dg functors $\mathbb{R}\underline{\mathr... | https://mathoverflow.net/users/67157 | Twisted derived Morita theory of schemes | I believe the answer is yes. Let attempt a sketch and give the appropriate references. The general strategy is to prove the main theorems of [Ben-Zvi, Francis, Nadler](https://arxiv.org/abs/0805.0157) in a twisted fashion. The main content of that paper is that the so-called "perfect stacks" are those algebro-geometric... | 3 | https://mathoverflow.net/users/24706 | 249182 | 113,249 |
https://mathoverflow.net/questions/248515 | 5 | Let $X$ be a Banach space (could work for over $\mathbb{R}$ as well?)
Let $A\colon D(A)\subset X\to X$ be a sectorial operator, and $e^{tA}$ be the semigroup generated by $A$.
It is well-known that for $m\in\mathbb{N}$ the real interpolation $(X,D(A^m))\_{\theta,p}$ ($\theta\in(0,1)$, $p\in[1,\infty]$) can be chara... | https://mathoverflow.net/users/97821 | Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ generates? | Yes, there is a characterization like this. See Theorems 1-3 (p.182) in
*Markus Haase*, MR 2183483 [**A functional calculus description of real interpolation spaces for sectorial operators**](http://dx.doi.org/10.4064/sm171-2-4), *Studia Math.* **171** (2005), no. 2, 177--195.
The result you ask for is a special ca... | 4 | https://mathoverflow.net/users/12898 | 249186 | 113,251 |
https://mathoverflow.net/questions/249125 | 7 | Let $G$ be a connected Lie Group. We will denote the outer automorphism group of $G$ by $Out(G)$. Consider $\pi\_0(Diffeo(G))$, the mapping class group of the underlying manifold of $G$, and the map,
$k:Out(G)\rightarrow \pi\_0(Diffeo(G))$
This map is well defined as follows: Given $\phi\_1, \phi\_2\in Aut(G)$ such... | https://mathoverflow.net/users/47707 | Outer Automorphisms and the Mapping Class Group of Lie Groups | Your question is largely intractable with present technology. Little is known about smooth mapping class groups of high-dimensional manifolds. But in dimensions 3 and below, there is considerable knowledge. . . unfortunately there are not many Lie groups in those dimensions.
The mapping class group of $SO\_3$ is tri... | 4 | https://mathoverflow.net/users/1465 | 249198 | 113,255 |
https://mathoverflow.net/questions/249190 | 5 | Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof):
>
> **A. The following isomorphisms hold:**
> $$\lim\_{X' \underset{qis}\to X} Hom\_{K(C)}(X',Y) \widetilde\to
> \lim\_{X' \underset{qis}\to X,Y \underset{qis}\t... | https://mathoverflow.net/users/22810 | Defining hom spaces in the derived category as limits of hom spaces in the homotopy category | This is actually a standard property of the homotopy category of complexes on which construction of the derived category is based, formulated in unusual way.
For a quasi-isomorphism $s:X'\to X$ and $f\in Hom(X',Y)$ denote by $fs^{-1}$ the image of $f$ in the first colimit(in a second we will justify the choice of not... | 6 | https://mathoverflow.net/users/39304 | 249214 | 113,258 |
https://mathoverflow.net/questions/249213 | 0 | Would you tell me some main distinctions between the convex discrete minimization and maximization optimization problems?
In the case of the feasible are bounded then we only need to transform one to another (by multiple -1) and solve only one of them. However, in other cases, we can not do that. Which problem is us... | https://mathoverflow.net/users/87870 | Differences between the convex discrete maximization and minimization problems? | For continuous problems, minimizing a convex function on a convex domain is considered an easy problem, because there is only ever one local minimum, and a local minimum is the global minimum. Finding that minimum (when it exists) can be done by local search methods.
On the other hand, maximizing a convex function o... | 4 | https://mathoverflow.net/users/13650 | 249215 | 113,259 |
https://mathoverflow.net/questions/249172 | 5 | In Gödel's Collected Works (Vol 2), there is a discussion of von Neumann which was brought about by a query, made to Gödel, concerning the existence of a Turing machine which is so complex that its total halting behavior is "asymptotically more complicated to describe than the machine itself".
To this, Gödel replied:... | https://mathoverflow.net/users/64167 | Decoding a Remark of Gödel on Complexity Theory | I just read that as saying U's description is finite in size, but describing U's behavior across all inputs needs an infinite description. There is no finite algorithm to say which inputs halt.
| 3 | https://mathoverflow.net/users/98137 | 249220 | 113,261 |
https://mathoverflow.net/questions/249219 | 29 | Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function.
Then $\mbox{Hess} (f)$ is a two linear map on $M\_{n}(\mathbb{R})\simeq T\_{e}(G)$ where $e$ is the neutral element of $G$, the identity matrix.
What is an explicit formula for this Hessian? (In terms of matrix terminologies)
| https://mathoverflow.net/users/36688 | Is there an explicit formula for the hessian of "Determinant"? | The formula you're looking for can be obtained by differentiating [Jacobi's formula](https://en.wikipedia.org/wiki/Jacobi%27s_formula)
$$ \frac{\mathrm{d}}{\mathrm{d}t} \det A(t) = \det A(t) \cdot \operatorname{tr}\left( A^{-1} \frac{\mathrm{d}A}{\mathrm{d}t} \right) $$
with respect to a second parameter, say $s$:
\beg... | 52 | https://mathoverflow.net/users/61479 | 249223 | 113,263 |
https://mathoverflow.net/questions/249208 | 2 | I have an equation of the form:
$$dX\_t=\mu(X\_t)X\_tdt+\sigma(X\_t)X\_tdZ\_t+dL\_t, \quad X\_0=x\_0\in (0,a]$$
where, $L\_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow the process to get past a barrier $a>0$. Therefore, this process is always between 0 and $a$.
While I was... | https://mathoverflow.net/users/73451 | Existence of solution for reflected SDE | Sufficient conditions for strong existence and uniqueness of solutions to stochastic differential equations with reflection were derived by Dupuis & Ishii (see below for detailed reference) in the following cases:
1. the reflection direction is single-valued and varies smoothly, but the boundary of the domain may be... | 1 | https://mathoverflow.net/users/64449 | 249226 | 113,265 |
https://mathoverflow.net/questions/248957 | 2 | $\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$
$\newcommand{\Lam}{\operatorname{\Lambda}}$
**Motivation (and the "classic" case):**
I am trying to find a **coordinate-free** criterion to determine when a linear map is an **isometric embedding** of an inner product space in a space o... | https://mathoverflow.net/users/46290 | Invariant characterization of isometric embeddings | Here is a proof of $\left( \ast\ast\right) $. As you correctly admitted,
part of the problem is to define the signs that enter into
$\operatorname\*{Cof}A$. Let me actually start from scratch and introduce all
notations that will be needed:
For any $m\in\mathbb{N}$, we shall write $\left[ m\right] $ for the set
$\lef... | 1 | https://mathoverflow.net/users/2530 | 249233 | 113,266 |
https://mathoverflow.net/questions/249210 | 4 | In Coates' *$p$-adic L-functions and Iwasawa's theory*, the first of his congruence hypotheses is that $\delta\_n(\mathfrak{b},\mathfrak{c},\mathfrak{f})\in \mathbb{Z}\_p$, where $\mathfrak{b},\mathfrak{c},\mathfrak{f}$ are ideals of the totally real base field $F$ with $\mathfrak{bc}$ prime to $\mathfrak{f}$ and $\mat... | https://mathoverflow.net/users/58688 | Why are the formulations of Deligne-Ribet/Coates congruences for L-functions equivalent? | I think you are misinterpreting Ribet's claim here.
In Ribet's article he defines $\Delta\_c(1 -k, \epsilon)$ for an **arbitrary function** $\epsilon: G\_f \to V$ where $V$ is a $\mathbf{Q}\_p$-vector space. It needn't be a group homomorphism into the unit group of a field extension of $\mathbf{Q}\_p$, even though th... | 4 | https://mathoverflow.net/users/2481 | 249234 | 113,267 |
https://mathoverflow.net/questions/248055 | 7 | Here an almost simple group is a finite group whose socle (product of all minimal normal subgroups) is a nonabelian simple group. As an extension of its socle, an almost simple group could be split or non-split. For example, there are four groups with socle $L=PSL(2,9)$ other than $L$, i.e. $S\_6$, $PGL(2,9)$, $M\_{10}... | https://mathoverflow.net/users/26700 | When is an almost simple group a split extension of its socle? | See the following paper:
A. Lucchini, F. Menegazzo, M. Morigi.
On the existence of a complement for a finite simple group in its automorphism group. Special issue in honor of Reinhold Baer (1902–1979).
Illinois J. Math. 47 (2003), no. 1-2, 395–418. MR2031330
| 4 | https://mathoverflow.net/users/23827 | 249235 | 113,268 |
https://mathoverflow.net/questions/249184 | 2 | I'm trying to understand the lecture notes comparison theorems in Riemannian geometry of Eschenburg and have a problem about the proof of Rauch I. Let $E$ be an euclidean vector space and $S(E)$ the space of its self adjoint endomorphisms with a partial ordering by putting $ A \leq B $ if $ \langle Ax,x\rangle \leq \la... | https://mathoverflow.net/users/97941 | comparison of eigenvalue of solution of Riccati equation | A classical proof can be found here:
*H. L. Royden*, MR 948079 [**Comparison theorems for the matrix Riccati equation**](http://dx.doi.org/10.1002/cpa.3160410512), *Comm. Pure Appl. Math.* **41** (1988), no. 5, 739--746.
Actually, more general version of this statement holds for more general version of the matrix R... | 2 | https://mathoverflow.net/users/13915 | 249241 | 113,270 |
https://mathoverflow.net/questions/249229 | 1 | Consider the double layer potential $K: L^2(S^2)\to L^2(S^2)$
$$(Kf)(x)=\int\_{S^2}f(y)\frac{\partial}{\partial v\_y}E(x,y)dS\_y,$$
where $E(x,y)=||x-y||^{-1}$ and $\frac{\partial}{\partial v\_y}$ means the outer normal derivative on $S^2$.
Show that the eigenvalues of $K$ are $\lambda\_k=\frac{-1}{2k+1}$, $k=0,1,...... | https://mathoverflow.net/users/98145 | Eigenvalues of the double layer potential | This is a classic problem solved by Poincaré in [La méthode de Neumann et le problème de Dirichlet](http://link.springer.com/article/10.1007/BF02418028) (1897). You can find the solution explained in section 8.2 of [Poincaré's variational problem in potential theory](http://www.math.ucsb.edu/~mputinar/poincare.pdf) (20... | 4 | https://mathoverflow.net/users/11260 | 249247 | 113,273 |
https://mathoverflow.net/questions/249123 | 0 | The [product formula](http://groupprops.subwiki.org/wiki/Product_formula) on finite groups states that for $H\_1, H\_2$ subgroups of $G$, then
$$ |H\_1H\_2| \cdot |H\_1 \cap H\_2|=|H\_1| \cdot |H\_2| $$
This statement could be generalized to any finite index irreducible subfactor planar algebra $\mathcal{P}$ by:
Fo... | https://mathoverflow.net/users/34538 | Generalization of the product formula on subfactors | There are various ways to generalize the equation. I give two different ones here:
(1) In terms of bimodules: Take a finite index irreducible subfactor $N \subset M$. Suppose $P\_1$ and $P\_2$ are two intermediate subfactors and $P=P\_1 \cap P\_2$. Then we have the following equation for $N-N$ bimodules:
$$\dim(P\_1\... | 3 | https://mathoverflow.net/users/57468 | 249255 | 113,277 |
https://mathoverflow.net/questions/249128 | 7 | I would like to know the explicit expression of 2-cocycle from a 2nd cohomology group $H^2[Q\_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$ with $U(1)\equiv \mathbb{R}/\mathbb{Z}$ coefficient, or namely $H^2[Q\_8 \times \mathbb{Z}\_2, U(1)]$, here we denote the cyclice group $\mathbb{Z}/2\mathbb{Z}=\mathbb{Z}\_2$, and the $Q... | https://mathoverflow.net/users/27004 | Explicit 2-cocycle from a 2nd cohomology group $H^2[Q_8 \times \mathbb{Z}/2\mathbb{Z}, U(1)]$ | For any two finite groups $G$ and $H$ the Kuenneth formula will give you a homomorphism $$H^1(G,U(1))\otimes\_{\mathbb{Z}} H^1(H,U(1))\to H^2(G\times H,U(1)).$$
We can describe this map explicitly. Let $\phi:G\to U(1)$ and $\psi:H\to U(1)$ be group homomorphisms. For some $n$ and some $n$-th root of unity $\zeta$ we ca... | 2 | https://mathoverflow.net/users/41644 | 249262 | 113,279 |
https://mathoverflow.net/questions/249265 | 1 | In [this paper](http://link.springer.com/article/10.1007/BF02941339), page 149, the super Jacobi identity is given by
\begin{align}
J(x, y,z) := (-1)^{|x||z|}[[x, y],z] +(-1)^{|z||y|}[[z,x], y]+(-1)^{|y||x|}[[y,z],x] = 0.
\end{align}
But in [this article](https://en.wikipedia.org/wiki/Lie_superalgebra), the super Jacob... | https://mathoverflow.net/users/11877 | Two definitions of the super Jacobi identity | By super skew-symmetry $[x, y] = - (-1)^{|x||y|}[y, x]$. Also note that $\left|[x, y]\right| = |x| + |y| \bmod 2$, in particular, $(-1)^{|[x, y]|} = (-1)^{|x|+|y|}$. So we see that
\begin{align\*}
[x, [y, z]] &= -(-1)^{|x||[y, z]|}[[y, z], x]\\
&= -(-1)^{|x||y| + |x||z|}[[y, z], x]\\
&= -(-1)^{|x||z|}\left(-(-1)^{|x|... | 9 | https://mathoverflow.net/users/21564 | 249276 | 113,284 |
https://mathoverflow.net/questions/249239 | 7 | (First posted in [StackExchange](https://math.stackexchange.com/q/1912968/365891))
I am interested in the Löwenheim numbers associated with quantifiers that are prevalent in ordinary language use, such as "more" and "most".
Definition: The *Löwenheim number* of a logic L, $\ell$(L), is the least cardinal $\mu$ suc... | https://mathoverflow.net/users/98152 | Lowenheim numbers for ordinary language quantifiers | $\def\L#1{\mathcal L(#1)}\let\fii\varphi\let\eq\leftrightarrow\DeclareMathOperator\I{I}\DeclareMathOperator\most{Most}\DeclareMathOperator\more{More}$I don’t know what other quantifiers from ordinary language are there in the wild, but the three mentioned in the question have the same Löwenheim number:
>
> **Theore... | 5 | https://mathoverflow.net/users/12705 | 249284 | 113,287 |
https://mathoverflow.net/questions/249266 | 5 | The Carmichael Numbers can be factorized in polynomial time.
Are weak fermat pseudoprimes (w.r.t. a given base) easy to factorize as well?
What are some large/broad classes of numbers that are easy to factorize (i.e. in polynomial time). This part is pretty broad, so i would be thankful if someone can provide some... | https://mathoverflow.net/users/74367 | Classes of Numbers that are easy to factorize using Classical Computers? | Six classes of "lucky numbers", that can be factorized easily, are discussed on page 107 and following of [Integer Factoring](https://www.fdi.ucm.es/profesor/m_alonso/Documentos/factorizacion/arjlensfac.pdf) by Arjen Lenstra (2000). These include: trial division by a small prime (very effective, since 88% of all positi... | 8 | https://mathoverflow.net/users/11260 | 249292 | 113,290 |
https://mathoverflow.net/questions/249286 | 1 | I have an equation of the form:
$$dX\_t=\mu(X\_t)dt+\sigma(X\_t)dZ\_t+dL\_t, \quad X\_0=x\_0\in (-\infty,a]$$
where, $L\_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow the process to get past a barrier $a>0$.
While I was able to find many results concerning the structure I ... | https://mathoverflow.net/users/73451 | Reflected SDE with non-Lipschitz coefficients | The given SDE is a special case of a one-dimensional SDE with a **drift measure**
$$
dX = \int\_{\mathbb{R}} d \Lambda\_X(t,x) \mu(dx) + \sigma(X(t)) dZ(t) \tag{$\star$}
$$
where from left: $\Lambda\_Y(t,x)$ is the (symmetric) local time of $X(t)$ at the level $x$, $\mu$ is a measure that we will specify shortly, $\sig... | 1 | https://mathoverflow.net/users/64449 | 249294 | 113,291 |
https://mathoverflow.net/questions/249275 | 0 | Can someone suggest a reference to study Hodge locus, period mappings and period domains on moduli space of principally polarised abelian varieties?
More precisely, consider the moduli space of principally polarized abelian varieties of dimension $g$, denoted $A\_g$. Using the Gauss-Manin connection, one defines a na... | https://mathoverflow.net/users/43198 | Reference for Hodge loci on moduli space of principally polarised abelian varieties | It is NOT an isomorphism, by dimension count: the LHS has dimension $g(g+1)/2$, whereas the RHS has dimension $g^2$.
It is injective though. A (polarized) abelian variety is the same as a polarized weight 1 Hodge structure. Such a Hodge structure on $H^1(X\_b,\mathbb C)$ is determined by the subspace $H^{1,0}(X\_b)$... | 5 | https://mathoverflow.net/users/30554 | 249296 | 113,292 |
https://mathoverflow.net/questions/247832 | 8 | In the opening chapters of Hartley Rogers, Jr.'s book *Theory of Recursive Functions and Effective Computability*, the proofs of the unsolvability of the halting problem and related unsolvability results invoke Church's Thesis. Can Church's Thesis be avoided?
Edit added on September 7 while putting a bounty on the qu... | https://mathoverflow.net/users/37385 | The Halting Problem and Church's Thesis | Let me point out that there are really a family of Church-Turing
theses assertions.
On the one hand, for what is sometimes described as the *weak*
Church-Turing thesis, one imagines an idealized human agent, not
constrained by resources of time or memory (or supplies of paper
and pencil), but carrying out the kind of... | 13 | https://mathoverflow.net/users/1946 | 249304 | 113,294 |
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