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https://mathoverflow.net/questions/209376 | 17 | In §5.3 of Kontsevich's *[Formality Conjecture](http://www.ihes.fr/~maxim/TEXTS/Formality%20conjecture.pdf)* he writes:
>
> This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is described by the following non-linear partial diffe... | https://mathoverflow.net/users/61197 | Kontsevich's flow on the space of Poisson structures | By looking at scaling, both $f$ and $g$ have to be cubic in $u$. Moreover, monomials in $f$ contain 3 derivatives in $y$ and 2 derivatives in $x$; similarly for $g$.
There are 12 possible monomials in each $f$ and $g$. E.g. the ones in $g$ are $u\_{xxxyy}u^2, u\_{xxxy}u\_yu, u\_{xxyy}u\_xu, u\_{xxx}u\_y^2,u\_{xxx}u\_... | 3 | https://mathoverflow.net/users/18512 | 209882 | 99,994 |
https://mathoverflow.net/questions/209859 | 3 | Let $P(n)$ be the set of subsets $P$ of $\mathbb{N}$ with the properties
1. All elements of $P$ are relative prime to each other.
2. The product of all $k \in P$ is greater or equal to $n$.
Now let $f(n) = \min\_{P \in P(n)} \sum\_{k \in P} k$.
What can be said about the size of $f(n)$ (in relation to $n$)?
A s... | https://mathoverflow.net/users/3816 | Products of relative prime numbers with least sum | Notice that the product prod P ( bounded below by n) represents the order of a permutation with cycle structure given by P and
sitting in S\_m, where m=f(n). So considering the largest order of elements occurring in finite symmetric groups should give you a good idea of the growth rate of f(n). Ben Barber has given wha... | 2 | https://mathoverflow.net/users/35626 | 209891 | 99,997 |
https://mathoverflow.net/questions/209893 | 0 | In the setting of algebraic groups:
I understand that a central extension of a group $G$ by an abelian group $A$ is a exact sequence of groups :$0\rightarrow A\rightarrow \tilde{G}\rightarrow G\rightarrow 1$ such that $A\subset Z(\tilde{G})$ and I understand that a $G-$torsor in a scheme $X$ is an $G-$action on $X$ s... | https://mathoverflow.net/users/47300 | Torsors and Central Extensions | For $A$ an abelian group, $A$-torsors on $X$ are classified by $H^1(X,A)$. (I'm just going to take this to be the definition of $H^1(X,A)$.)
There is a natural map:
$$ H^1(X,A) \times H^1(Y,A) \to H^1(X \times Y, A)$$
For certain types of $X,Y$ and certain $A$, we can show that this map is an isomorphism. Wehen t... | 5 | https://mathoverflow.net/users/18060 | 209899 | 99,998 |
https://mathoverflow.net/questions/209906 | 3 | If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G\_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}\_n$ over $\mathrm{Spec}(\mathbf{Z})$? Or is life not so easy? I feel like if I actually knew the construction of the irreps of $G\_{\mathbf{C}... | https://mathoverflow.net/users/69613 | Representations of complex semi-simple algebraic group "defined over $\mathbf{Z}$"? | This is an application of the geometric method of constructing highest-weight representations via line bundles on the "variety of Borel subgroups", together with the theory of split semisimple Chevalley groups (existence of Borel $\mathbf{Z}$-subgroups containing a split maximal $\mathbf{Z}$-torus, existence of $G/B$ a... | 11 | https://mathoverflow.net/users/70739 | 209910 | 100,000 |
https://mathoverflow.net/questions/209926 | 5 | Let $G$ be a compact (connected) Lie group with a maximal torus $T$. For each (analytically) integral weight $\lambda$ the Weyl character formula
$$\Theta\_{\lambda}(H)=\frac{\sum\_{w\in W(G)}\epsilon(w)e^{w(\lambda+\rho)(H)}}{\sum\_{w\in W(G)}\epsilon(w)e^{w(\rho)(H)}}$$
defines a function that descends to the set of ... | https://mathoverflow.net/users/47762 | Understanding the Weyl Character Formula | One way to interpret the Weyl character formula is as the Euler characteristic of the [BGG resolution](https://books.google.com/books?id=8GCP4Ng6risC&pg=PA108&lpg=PA108&dq=BGG+resolution&source=bl&ots=oJhgf9VOTp&sig=_LeaI5STRo8hb4ZRT6k9luqCTLM&hl=en&sa=X&ei=9dyIVYfkLIu_sQSWp67ADQ&ved=0CE8Q6AEwBg#v=onepage&q=BGG%20resol... | 8 | https://mathoverflow.net/users/66 | 209929 | 100,007 |
https://mathoverflow.net/questions/209925 | 15 | I am conducting research in Combinatorial Game Theory (CGT). Although I have done a considerable amount of reading, I do not completely understand why the bit-xor function also known as the nim-sum appears in Nim. To be more clear, I will explain my current level of understanding. I completely grasp the mechanics of th... | https://mathoverflow.net/users/75293 | Why does the bitxor function appear in Nim? | There's something of a good explanation. You accept the Sprague-Grundy theorem, which implies in particular that Nim positions must form a group - any two Nim positions are equivalent to one Nim position, and each Nim position plus itself gets you back to the identity.
OK, but which group? I guess you also know it's ... | 13 | https://mathoverflow.net/users/18060 | 209934 | 100,009 |
https://mathoverflow.net/questions/209936 | 14 | This is a [re-post from MSE](https://math.stackexchange.com/questions/1332444/transcendence-of-gamma1-3-gamma1-4) as I did not get even a single comment there.
[Wikipedia](https://en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function) mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by... | https://mathoverflow.net/users/15540 | Transcendence of $\Gamma(1/3), \Gamma(1/4)$ | [Algebraic Independence of Values of Exponential and Elliptic Functions](http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0339.0350.ocr.pdf), G. V. Chudnovsky (1978)
(a.i. = algebraically independent)
| 16 | https://mathoverflow.net/users/11260 | 209937 | 100,011 |
https://mathoverflow.net/questions/209945 | 3 | Let $\Bbb{S}\_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices, and let $A,B\in\Bbb{S}\_{++}^n$.
Is it possible to express the logarithm of $A^{-1}B$ as a difference of the form $P-Q$, where $P,Q$ are $n\times n$ real matrices, not necessarily SPD, but P must depend solely on $... | https://mathoverflow.net/users/42645 | Prove or disprove a matrix logarithm equation | Solution is inspired by the comment of Federico Poloni. Suppose there exist such functions, i.e. $$log(A^{-1}B)=P(A)-Q(B).$$ for all SPD-matrices $A,B$. Putting $A=B$ yields $P(A)=Q(A)$ for all SPD-matrix $A$. Now if the equation above holds we would have $$log(A^{-1}B)+log(B^{-1}C)+log(C^{-1}A)=0.$$
However simply cho... | 4 | https://mathoverflow.net/users/35593 | 209947 | 100,016 |
https://mathoverflow.net/questions/209247 | 1 | Is there a compact Reiemannian manifold $M$ for which the following complex $C^{\*}$ algebra does not have a nontrivial idempotent:
$A=Hom(E,E)$ where $E$ is the complexification of $TM$.
Of course any such $M$ can not have an almost complex structure.
| https://mathoverflow.net/users/36688 | Almost complex structure and nontrivial idempotents | No: if $M^n$ is $n$-dimensional then $E = TM \otimes\_{\mathbb{R}} \mathbb{C}$ is $n$-complex dimensional so classified by a map $M \to BU(n)$. By obstruction theory this lifts to $BU(n-1)$ (the obstructions are in the groups $H^{i+1}(M^n; \pi\_i(S^{2n-1}))$, which all vanish).
Hence there is an isomorphism of vecto... | 4 | https://mathoverflow.net/users/318 | 209952 | 100,017 |
https://mathoverflow.net/questions/209950 | 7 | Is there any way to get on top of the procedure for constructing Bruhat-Tits buildings for non-split groups over a non-archimedean local field $k$, by Galois descent, other than reading both the Bruhat-Tits articles "Groupes réductifs sur un corps local"? In particular is there any exposition which is written in Englis... | https://mathoverflow.net/users/15482 | Understanding how to construct Bruhat-Tits buildings for non-split groups by Galois descent | Jiu-Kang Yu has an article "Bruhat-Tits Theory and Buildings" which has appeared in "Ottawa Lectures on Admissible Representations of Reductive $p$-adic Groups". It doesn't directly address your question, instead it contains a guide to the literature on Bruhat-Tits buildings.
There is a [26 page pdf](http://citeseerx... | 7 | https://mathoverflow.net/users/425 | 209954 | 100,018 |
https://mathoverflow.net/questions/209915 | 7 | Let $B^n$ denote the unit ball in $\mathbb{R}^n$ (wrt the standard euclidean metric) and $\bar{B}^n$ denote the unit closed ball. Suppose that $\Sigma$ is a a smooth embedded hypersurface with boundary in $\bar{B}^4$ which is topologically $\bar{B}^3$ and so that $\Sigma$ meets $\partial \bar{B}^4$ transversely and $\p... | https://mathoverflow.net/users/66607 | Can a 3-ball divide a standard 4-ball into two exotic 4-balls? | I think this is equivalent to the smooth Schoenflies conjecture; the executive summary is that this is true because smooth balls (in any dimension) are isotopic to standard ones.
Here are some details, starting with some preliminary remarks. Your $\bar{B}^3$ is diffeomorphic to a closed 3-ball, so I'll take that as ... | 5 | https://mathoverflow.net/users/3460 | 209960 | 100,021 |
https://mathoverflow.net/questions/209966 | 2 | I am a bit confused regarding the possible constructions/realizations of symplectic groups. Basically I am looking for the following:
A linear algebraic group $\mathbb{G}$ defined over $\mathbb{Q}$ such that $\mathbb{G}(\mathbb{C})$ is (isomorphic to) the group $\mathrm{Sp\_{2m}}(\mathbb{C})$ while $\mathbb{G}(\mathb... | https://mathoverflow.net/users/37942 | Compact form of symplectic groups defined over the rationals | Yes, and you can make it split away from whatever even finite set $S$ of places of $\mathbf{Q}$ you wish that contains the archimedean place, and such a form of ${\rm{Sp}}\_{2n}$ is uniquely determined as well. This is seen via Galois cohomology and class field theory (a not exactly constructive method, but Springer's ... | 7 | https://mathoverflow.net/users/70739 | 209976 | 100,029 |
https://mathoverflow.net/questions/209978 | 0 | I am working with an article, where it says:
>
> "that the discrete time stationary sequence $\{Y\_j\}\_{j\in Z}$ is
> mixing and hence ergodic."
>
>
>
where $Y\_t$ is defined as
$Y\_t = \int\_{-\infty}^{t} h\_k(t-s)dL\_s$
and $L$ is a Lévy process.
I have been unable to find a definition of mixing and er... | https://mathoverflow.net/users/75307 | Ergodic and mixing processes | Yes, the results you quote are general statements on mixing and ergodicity, which can be translated to stochastic process as follows.
In the source you mention (and many other sources), mixing is defined as
$\lim\_{n \rightarrow \infty} \mu(T^{-n}(A) \cap B) = \mu(A) \mu(B), $
where $T$ is a transformation of in... | 4 | https://mathoverflow.net/users/22157 | 209985 | 100,035 |
https://mathoverflow.net/questions/209988 | 2 | A Lie algebra $\mathfrak{g}$ generates its universal enveloping algebra $\mathrm{U}\mathfrak{g}$, which has the structure of a Hopf algebra. Modules of $\mathrm{U}\mathfrak{g}$ are exactly the of modules of $\mathfrak{g}$ itself.
In the case $\mathfrak{g} = \mathfrak{su}\_2$, the simple modules are labeled uniquely (... | https://mathoverflow.net/users/66403 | Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure? | The problem is that $U\_J\frak{su}\_2$ as you define it, that is $\bigoplus\limits\_{j=0}^{2J}V\_{2j+1}$, is NOT the space of all differential operators of order at most $J$. For the latter, Formula (2) in the paper suggests that it is the sum $\bigoplus\limits\_{j=0}^J S^j(\mathfrak{su}\_2)$, so its dimension is equal... | 3 | https://mathoverflow.net/users/1306 | 210003 | 100,041 |
https://mathoverflow.net/questions/210001 | 0 | I would appreciate any reference that contains either a translation or proof of the main theorem in [this](http://www.numdam.org/item?id=BSMF_1955__83__279_0) paper. Thank you in advanced.
| https://mathoverflow.net/users/16533 | Marcel Berger's "Sur les groupes d'holonomie homogènes de variétés à connexion affine et des variétés riemanniennes." | Perhaps Robert Bryant's "[Recent advances in the theory of holonomy](http://www.numdam.org/item?id=SB_1998-1999__41__351_0)" would be a good place to start.
| 2 | https://mathoverflow.net/users/394 | 210006 | 100,044 |
https://mathoverflow.net/questions/209940 | 6 | Can anyone point me to an example of a problem that (more or less) originated in computational geometry whose solution requires the use of Lagrange multipliers (or Kuhn-Tucker conditions, or dual variables in a linear program, etc.)? I have not been able to find such an example in any of the literature that I own.
| https://mathoverflow.net/users/70190 | Do computational geometers use Lagrange multipliers? | A convex optimization method for constructing a set of points in the plane with prescribed (combinatorial) Delaunay triangulation is given in
Euclidean structures on simplicial surfaces and hyperbolic volume
I Rivin - Annals of Mathematics, 1994
| 3 | https://mathoverflow.net/users/11142 | 210008 | 100,045 |
https://mathoverflow.net/questions/209905 | 6 | Let $P$ denote the perimeter function. It's not hard to prove that for any rectangle $R$ in $\mathbb{R}^2$, $R$ can be partitioned into a countable collection of squares $\{Q\_k\}\_{k=1}^{\infty}$ such that $R = \cup\_kQ\_k$, $\mathring{Q\_k} \cap \mathring{Q\_l}=\emptyset$, and $$\Sigma\_{k=1}^{\infty}P(Q\_k)\le 6P(R)... | https://mathoverflow.net/users/51546 | partition of a convex set into squares | Well, let me prove that the answer is negative even for triangle. Rotating coordinate system we may suppose that vertical lines are not parallel to sides of triangle and to sides of all squares (as there are only countably many of them), also, if the triangle has two perpendicular sides, vertical lines can not meet bot... | 1 | https://mathoverflow.net/users/4312 | 210026 | 100,049 |
https://mathoverflow.net/questions/210032 | 3 | Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space?
In other words, is it true that conditional expectation with respect to any sub-$\sigma$-algebra is the same as conditional expectation with respect to some random ... | https://mathoverflow.net/users/75342 | Sub-$\sigma$-algebras and conditional expectation | Yes - this is one of the key results of the Rokhlin theory. Namely, any complete sub-$\sigma$-algebra of a Lebesgue space can be realized as the preimage $\sigma$-algebra of a quotient map.
| 2 | https://mathoverflow.net/users/8588 | 210034 | 100,055 |
https://mathoverflow.net/questions/209902 | 2 | Let $Y$ be a marked simplicial set, whose underlying simplicial set is also denoted by $Y$. Let $X$ be a scaled simplicial set such that the [decalage](http://ncatlab.org/nlab/show/decalage) of its underlying simplicial set is $Y$. $X$ is then a scaled simplicial set, whose thin $2$-simplices are precisely the marked $... | https://mathoverflow.net/users/nan | Building $(\infty,2)$-categories from $\infty$-categories | The following is not an answer, just an observation suggesting that maybe the question should be rephrased. I claim that if $X$ is a fibrant scaled simplicial set whose decalage is a fibrant marked simplcial set then both $X$ and $Y$ must in fact be $\infty$-groupoid (i.e., all the triangles in $X$ are thin, all the ed... | 2 | https://mathoverflow.net/users/51164 | 210036 | 100,056 |
https://mathoverflow.net/questions/210038 | 0 | Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f\_n)$ of simple measurable functions, such that (i) $0 \le f\_n(\omega) \uparrow f(\omega)$ if $f(\omega) \ge 0$, and (ii) $0 \ge f\_n(\omega) \ge f(\omega)$ if... | https://mathoverflow.net/users/37580 | Approximation of general measurable maps by simple functions | There are results of this sort for functions with values in a Banach space---see, for example, p. 42 of "Vector measures" by Diestel and Uhl.
| 1 | https://mathoverflow.net/users/75303 | 210052 | 100,058 |
https://mathoverflow.net/questions/173316 | 7 | There follows a possible construction, and I would like to know if it or a similar construction has been done before (as I suspect), so that I can reference it, or if it obviously does not work! Any useful comments welcomed.
What is wanted is the noncommutative analogue of taking an open subset of a compact Hausdorff... | https://mathoverflow.net/users/29625 | allowing `discontinuous functions' into a C* algebra | G.R. Allan defined "GB\*-algebras" as locally convex $\ast$-algebras $A$ such that
1. $1+xx^\ast$ is invertible for all $x\in A$
2. the set $\mathcal{B}$ of absolutely convex, closed, bounded submonoids of $(A,1,\cdot)$ has a largest element $B$.
3. Some form of completeness is satisfied.
In the commutative case th... | 1 | https://mathoverflow.net/users/3041 | 210063 | 100,060 |
https://mathoverflow.net/questions/210066 | 1 | Let $f:X \to Y$ be a proper, flat morphism of projective scheme and $Y$ is an irreducible, non-singular surface. Assume further that there exists a Zariski open subset $U$ of $Y$ whose complement is of codimension $2$ such that the total space $f^{-1}(U)$ is non-singular and such that $f:f^{-1}(U) \to U$ has geometrica... | https://mathoverflow.net/users/54369 | Sections of proper, flat morphism | I am certain that I have answered this before. Let $Y$ be $\mathbb{P}^2$. Let $\Omega\_{\mathbb{P}^2/k}$ be the sheaf of relative differentials. This is locally free of rank $2$. Via the Euler sequence and the Whitney sum formula, the total Chern class of $\Omega\_{\mathbb{P}^2/k}$ equals
$$
c\_t(\Omega\_{\mathbb{P}^2/... | 2 | https://mathoverflow.net/users/13265 | 210070 | 100,061 |
https://mathoverflow.net/questions/210065 | 3 | For any $n$ dimensional closed manifold $M^n$, can we find an open covering $\{U\_i\}\_{i\in[2^n]}$ such that $M=\cup U\_i$ and each $U\_i\cong \mathbb R^n$? How about complex manifolds (replacing $\mathbb R^n$ by $\mathbb C^n$)?
| https://mathoverflow.net/users/48006 | Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces? | In the smooth, PL or topological category, you can cover a $n$-dimensional connected manifold by $n+1$ charts. Furthermore, for $k \leq n-3$ a $k$-connected $n$-dimensional manifold can be covered by at most $\lceil (n+1)/(k+1) \rceil$ charts. The reference for both of these statements is Luft's "Covering of manifolds ... | 8 | https://mathoverflow.net/users/798 | 210071 | 100,062 |
https://mathoverflow.net/questions/210073 | 3 | Let $K$ be a number field and let $A$ be an abelian variety of dimension $g$ over $K$. Let $L$ be a CM field and suppose that $[L:{\bf Q}]=2g$. Suppose that there exists an embedding $\iota:L\hookrightarrow{\rm End}\_K(A)\otimes{\bf Q}$, where ${\rm End}\_K(A)$ is the ring of $K$-endomorphisms of $A$. I shall say that ... | https://mathoverflow.net/users/17308 | CM abelian varieties over the rationals | The action of $L$ on global 1-forms would give an embedding of $L$ into the algebra $M$ of $g$-by-$g$ matrices over $\mathbb{Q}$ (since we're in characteristic 0). But any maximal commutative $\mathbb{Q}$-subalgebra of $M$ is $g$-dimensional.
| 7 | https://mathoverflow.net/users/949 | 210076 | 100,064 |
https://mathoverflow.net/questions/210041 | 16 | Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?
| https://mathoverflow.net/users/51203 | A probability distribution in n dimensional space which its projection on any line is a uniform distribution? | No if $n\geq 4$. By translating we may assume that the mean is $0$, and by scaling we may assume that the variance-covariance matrix is the scalar matrix $(1/3)I$. Then all projections have mean $0$ and variance $1/3$, hence are uniform in the interval $[-1,1]$.
So if we take a random projection, the value will also ... | 9 | https://mathoverflow.net/users/18060 | 210078 | 100,065 |
https://mathoverflow.net/questions/210021 | 1 | Is there an example of a family $(X\_i)\_{i\in I}$ of [metacompact](https://en.wikipedia.org/wiki/Metacompact_space) spaces, such that their [box product](https://en.wikipedia.org/wiki/Box_topology) $\prod\_{i\in I}^{\textrm{Box}}X\_i$ is not metacompact?
| https://mathoverflow.net/users/8628 | Example of a collection of metacompact spaces with non-metacompact box-product | The Sorgenfrey plane is not metacompact by [Example 2 in these notes](https://dantopology.wordpress.com/tag/metacompact-space/). However, the Sorgenfrey plane is the product of two paracompact (and hence metacompact) spaces. Therefore metacompact spaces are not even closed under taking finite products.
By transfinite... | 3 | https://mathoverflow.net/users/22277 | 210084 | 100,067 |
https://mathoverflow.net/questions/209229 | 6 | The following fact, which I've heard being called "soft version of Moschovakis's lemma" (see top answer [here](https://mathoverflow.net/questions/129036/counterintuitive-consequences-of-the-axiom-of-determinacy)) is the following:
>
> Under AD, if there is a surjection $\Bbb R\rightarrow\alpha$, then there is a sur... | https://mathoverflow.net/users/30186 | Proof of a soft version of Moschovakis's lemma | The argument you are looking for is given in Kanamori's book, see Theorem 28.15.
For the more nuanced version of the lemma, see section 7D in Moschovakis's descriptive set theory book (particularly 7.D.5-8), or section 3.1 in the Koellner-Woodin chapter of the Handbook.
| 5 | https://mathoverflow.net/users/6085 | 210087 | 100,068 |
https://mathoverflow.net/questions/210080 | 8 | At the beginning I thought that the following statement could be an easy exercise after Stallings' theorem, but I found myself incapable of proving it:
*Any countable f.g. simple group has one end.*
It is obvious that a f.g. simple group cannot have two ends, as $\mathbb Z$ has many quotients. If it has infinitely ... | https://mathoverflow.net/users/47274 | On the number of ends of a countable simple group | Let $G$ be a group with infinitely many ends. According to Stallings' theorem, $G$ splits non trivially over a finite subgroup. Now, consider the action of $G$ on the associated Bass-Serre tree $T$. Because the edge stabilizers are finite, it is clear that the action $G \curvearrowright T$ is acylindrical. Furthermore,... | 10 | https://mathoverflow.net/users/43559 | 210091 | 100,070 |
https://mathoverflow.net/questions/209889 | 3 | I need to solve an optimization problem that involves an expected value like
$$F(n,x) = \sum\_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$
Here $f(k,x)$ is actually a probability coming from a normal distribution. And what I need, in particular, is to solve a first order condition that involves the derivative o... | https://mathoverflow.net/users/51312 | Expected value (probability) maximization with binomial distribution | The maximum converges to $1$ as $n\to\infty$. Indeed, one can write
$$F(n,x)=P(X\_n<w)\quad\text{and}\quad G(n,x)=P(X\_n>w-d),
$$
where $X\_n:=\dfrac{(a-b)Y\_n+Z}n$, $Y\_n$ and $Z$ are independent random variables (r.v.'s), $Y\_n$ has the binomial distribution with parameters $(n,p)$, $Z$ is standard normal, $a:=A/\s... | 3 | https://mathoverflow.net/users/36721 | 210092 | 100,071 |
https://mathoverflow.net/questions/209904 | 2 | [nLab](http://ncatlab.org/nlab/show/van+Kampen+colimit) uses the following definition of van Kampen colimits --- a colimit in a category $\mathbb{C}$ is called van Kampen iff it is preserved by the internal indexing functor $\mathbb{C}/(-) \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$ defined as:
$$X \mapsto \mathbb... | https://mathoverflow.net/users/13480 | Van Kampen colimits | Here's a summary of the comments above (which does not answer the question on the origin of the term, that I have no idea).
A colimit in a category $C$ with pullbacks is van Kampen if the indexing functor $C/(-): C^{op} \to Cat$ transforms it into a *weak* 2-limit, or bilimit, or homotopy limit. These are sometimes c... | 4 | https://mathoverflow.net/users/20233 | 210095 | 100,073 |
https://mathoverflow.net/questions/210044 | 5 | The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see (1.6.4) and (3.4.6)).
Let $\mathcal G$ be the fibred category of **group** algebraic spaces over a scheme $S$. Is $\mat... | https://mathoverflow.net/users/75349 | The stack of group algebraic spaces | Here's a sketch proof. First, recall that inside the category of presheaves, sheaves are closed under taking limits. Second, inside the set of maps between the underlying algebraic spaces, the group homomorphisms are constructed by taking an equaliser:
$$
Hom\_{Grp}(G,H) \to Hom\_{Sp}(G,H) \rightrightarrows Hom\_{Sp}(G... | 3 | https://mathoverflow.net/users/4177 | 210096 | 100,074 |
https://mathoverflow.net/questions/210101 | 0 | We say that a space $X$ has *covering property (C)* if the following holds:
(C) For any open cover ${\cal U}$ of $X$ there is a closed discrete set $D\subseteq X$ and a map $\varphi: D\to {\cal U}$ such that
* $d\in\varphi(d)$ for all $d\in D$;
* $\bigcup \varphi(D) = X$.
If a space $X$ has property $(C)$, how ab... | https://mathoverflow.net/users/nan | If $X$ has the "discrete" covering property, how about $X^2$? | This (rather exotic) covering property is often called "Property D". [Here](http://auburn.edu/~gruengf/papers/dsurv7.pdf) you find a nice survey of $D$-spaces.
On p. 11, assuming $\textrm{(CH)}$, a space $Y$ is mentioned that has property $D$, but $Y^2$ doesn't.
So in ${\sf ZFC}$ you definitely won't be able to pro... | 1 | https://mathoverflow.net/users/8628 | 210102 | 100,075 |
https://mathoverflow.net/questions/210103 | 3 | Let $C$ be a smooth curve of genus$\geq3$ over $\mathbb{C}$, so there are $2^{g-1}(2^g-1)$ odd theta characteristics and $2^{g-1}(2^g+1)$ even theta characteristics. Do we know how many of them has $h^0=0$, $h^0=1$, $h^0=2$...? If $h^0\geq2$, are they base-point free?
In particular, I would like to know the following... | https://mathoverflow.net/users/41734 | Theta characteristics of genus$\geq3$ curve | A lot is known about this problem.
Firt of all, the answer to your first question is **no**. In fact, we can prove the following easy result:
>
> **Proposition.** A curve $C$ of genus $3$ is hyperelliptic if and only if it contains a theta characteristic $L$ such that $h^0(L) \geq 2$.
>
>
> *Proof.* Assume tha... | 7 | https://mathoverflow.net/users/7460 | 210105 | 100,077 |
https://mathoverflow.net/questions/210100 | 9 | Recall that a variety $X$ over a field $k$ is called *rigid* if $H^1(X, T\_X) = 0$. I am interested in understanding this property under specialisation.
>
> Let $R$ be a discrete valuation ring and let $\pi: X \to \mathrm{Spec }R$ be a smooth proper morphism with geometrically irreducible generic fibre. Assume that... | https://mathoverflow.net/users/5101 | Specialisation of rigid varieties | I am just posting my comment above as an answer. Of course the only rigid curve is $\mathbb{P}^1$, and this specializes to $\mathbb{P}^1$. Building on Mori's work, Siu proved that every smooth, projective specialization of $\mathbb{P}^n$ is $\mathbb{P}^n$. The analogous result for quadric hypersurfaces in $\mathbb{P}^n... | 14 | https://mathoverflow.net/users/13265 | 210109 | 100,078 |
https://mathoverflow.net/questions/210051 | 0 | Assume $u\_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u\_n$ and a new function $\bar u$ such that
1. $\int\_\Omega|\nabla \bar u\_n|^2dx\leq \int\_\Omega|\nabla u\_n|^2dx$ and $\int\_\Omega|\nabla \bar u|^2dx\leq \i... | https://mathoverflow.net/users/62560 | How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient? | I think that the construction that you are proposing by harmonic extension does not work. Indeed consider on the unit disk $\mathbb{B}^2 \subset \mathbb{R}^2$ the function $u\_n : \mathbb{B}^2 \to \mathbb{R}$ defined for $x = (x\_1, x\_2) \in \mathbb{B}^2$
$$
u\_n (x\_1, x\_2) = \frac{\operatorname{Re} \bigl((x\_1 + i... | 1 | https://mathoverflow.net/users/42047 | 210110 | 100,079 |
https://mathoverflow.net/questions/210113 | 7 | I can't find a way to prove that the following equation has only one solution :
$$
X = \frac{2^Q - 1}{2^{P+Q} - 3^P}
$$
with $X,P,Q$ integers $> 0$.
One trivial solution is $X = 1, P = 1, Q = 1$.
Does anyone has an idea ?
Best regards
| https://mathoverflow.net/users/75381 | How to prove that this equation has only one solution? | I'm more used to the formulation in the following form:
$$ X(2^{P+Q} - 3^P)=2^Q-1 \\
2^Q(2^Px -1) = 3^Px -1 $$
and then
$$ 2^Q = {3^P \cdot X - 1 \over 2^P \cdot X - 1} \tag 1
$$
and Ray Steiner has proved in 1976 in the context of the Collatz-problem (using Rhin's result given in the other answer), that there is only... | 7 | https://mathoverflow.net/users/7710 | 210117 | 100,081 |
https://mathoverflow.net/questions/209541 | 13 | **Update:** Further work with Adam (who answers below) and Piotr led to a rather satisfactory result about the problem that motivated the problem below, see our recent paper
[The Haar Measure Problem](https://arxiv.org/abs/1709.02312). In particular, we answer there a problem mentioned in the discussion below.
The f... | https://mathoverflow.net/users/2415 | Avoiding countable subgroups of a group homeomorphic to the Cantor space | Yes, there exists.
Since $G$ is profinite, we may write it as $G=\lim\_{n\in\mathbb{N}} G\_n$, where $G\_n$ are finite and the projections $\pi\_n:G\to G\_n$ are onto.
Since $H$ is countable, we may write it as $H=\bigcup\_{n\in\mathbb{N}} H\_n$, where $H\_n\subseteq H\_{n+1}$ are finite, $H\_0$ nonempty.
Let $\bar H... | 8 | https://mathoverflow.net/users/16678 | 210125 | 100,084 |
https://mathoverflow.net/questions/210046 | 6 | Let $I\subseteq \mathbb{R}$ be an interval.
Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE:
$$
M\_{xx}M\_{yy}-M\_{xy}^{2}+\frac{M\_{y}M\_{yy}}{y}=0.
$$
My question is to describe/characterize solutions of this PDE.
Here are few nontrivial examples which solve the PDE:
\begin{align}
&1)\quad ... | https://mathoverflow.net/users/50901 | Monge–Ampère with drift | Perhaps the following observations will be of use to you: First, for any (local) solution $M(x,y)$ of your equation on a simply-connected open domain $D\subset \mathbb{R}\times(0,\infty)$ in the $xy$-plane, consider its $1$-graph
$$
(x,y,p,q)=\bigl(x,y,M\_x(x,y),M\_y(x,y)\bigr)
$$
in $xypq$-space. This is a simply-con... | 9 | https://mathoverflow.net/users/13972 | 210127 | 100,085 |
https://mathoverflow.net/questions/210126 | 2 | Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of:
$$\Omega = \begin{bmatrix} \beta\_1^2(1+\theta\_1^2) & \beta\_1 \beta\_2 & ... & \beta\_1 \beta\_{k-1} & \beta\_1 \beta\_k \\ \beta\_2 \beta\_1 & \beta\_2^2(1+ \theta\_2^2) & ... & \beta\_2 \beta\_{k-1} & \beta\_2 \beta\_k \\ ... & ... | https://mathoverflow.net/users/75388 | Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation | Your matrix is diagonal plus rank 1. Use the [Sherman-Morrison formula](https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula) and the [matrix determinant lemma](https://en.wikipedia.org/wiki/Matrix_determinant_lemma).
| 7 | https://mathoverflow.net/users/1898 | 210131 | 100,087 |
https://mathoverflow.net/questions/210111 | 0 | I need help with this excercise
>
> Let $k[X\_1,\ldots,X\_d]$ be the polynomial ring in $X\_1,\ldots,X\_d$ over a field $k$, and let $F\_1,\ldots,F\_m$ be forms of degree $n$. Assume that $(X\_1,\ldots,X\_d)=\sqrt{(F\_1,\ldots,F\_m)}$. Prove that $\overline{(F\_1,\ldots,F\_m)}=(X\_1,\ldots,X\_d)^n$.
>
>
>
Clea... | https://mathoverflow.net/users/75380 | integral closure of m-primary ideals | The asked for inclusion $\overline{(F\_1,\ldots,F\_m)} \subseteq (X\_1,\ldots,X\_d)^n$ follows easily by considering the lowest non-zero homogeneous component. Formally:
If $f = f\_0 + \cdots + f\_m$ is the decomposition of $f \in k[X\_1,...,X\_d], f\neq 0$ into homogeneous components ($\deg f\_k = k)$ then set $\el... | 2 | https://mathoverflow.net/users/18571 | 210134 | 100,089 |
https://mathoverflow.net/questions/210085 | 0 | Let $f:X\rightarrow Y$ be a morphism of normal projective varieties. Let $S\subseteq X$ be a surface admitting a morphism $g:S\rightarrow C$ to a curve $C$ such that any fiber of $g$ is a curve.
Assume that the general fiber of $g$ is contracted to a point by $f$. Is it true that any fiber of $g$ is contracted by $f$... | https://mathoverflow.net/users/nan | Morphisms contracting a family of curves | Actually, this is true in more generality:
* You don't need $f, X$, or $Y$ to be normal or projective (this is of course, trivial, they are all red herrings). You only need $g$ to be proper (I suppose you meant that $S$ is a projective surface, so that's covered).
* If all the fibers of $g$ are connected and **any** ... | 4 | https://mathoverflow.net/users/10076 | 210135 | 100,090 |
https://mathoverflow.net/questions/208180 | 6 | I am looking for a Siegel modular form of genus $2$ (living on the Siegel modular 3-fold $A\_2=\mathrm{Sp}(4,\mathbb{Z})\backslash \mathfrak H\_2$) which becomes "roughly" the product of two eta functions on the locus where the parametrized abelian surface is the product of two elliptic curves. Here "roughly" means up ... | https://mathoverflow.net/users/21014 | A Siegel modular form related to the product of two eta functions | I hesitate to answer my own question, but the answer below may be useful to some people.
The Siegel cusp form $\chi\_{10}$ of weight $10$ of genus $2$ has the following asymptotic behaviour as $z$ goes to $0$
$$
\chi\_{10}(x,y,z) = \eta(x)^{24}\eta(y)^{24}(\pi z)^2+O(z^4),
$$
where $x,y,z$ are the coordinates of the... | 1 | https://mathoverflow.net/users/21014 | 210141 | 100,091 |
https://mathoverflow.net/questions/210139 | 2 | Let $G$ be a finite group.
By a *$G$-lattice* we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a *permutation $G$-lattice* if $L$ has a ${{\mathbf{Z}}}$-basis permuted by $G$.
By a *permutation covering* of a $G$-lattice $L$ we mean a surjective morphism of $G$-lattices $P\t... | https://mathoverflow.net/users/4149 | Permutation covering of a $G$-lattice | The answer for $p>2$ is NO:
Suppose $M \to L$ is a permutation covering with kernel $K\leq M$, and the rank of $M$ smaller than $p^2$. The permutation lattice $M$ decomposes into a direct sum
$$ M = M\_1 \oplus \dotsb \oplus M\_r, $$
where each summand corresponds to an orbit of the underlying $G$-set. These orbi... | 3 | https://mathoverflow.net/users/10266 | 210162 | 100,096 |
https://mathoverflow.net/questions/210090 | 4 | In Conformal Field Theory (in $D$ dimensions) one considers (in particular) correlation functions of the form
$$
\langle O(x)O(y)\rangle,
$$
where $O$ is a scalar primary field. Scale covariance demands
$$
\langle O(\lambda x)O(\lambda y)\rangle=\lambda^{-2\Delta}\langle O(x)O(y)\rangle,
$$
where $\Delta\geq\frac{D-2}{... | https://mathoverflow.net/users/32985 | Conformally covariant distributions | Let me extend my comment. The Osterwald-Schrader axioms were invented to make the physical idea of Wick rotating quantum fields rigorous. Namely, the OS axioms are axioms on the correlation function of some classical fields, which ensure that these correlation functions can be analytical continued, such that they fulfi... | 4 | https://mathoverflow.net/users/10718 | 210163 | 100,097 |
https://mathoverflow.net/questions/210167 | 40 | While trying to answer [this MSE question](https://math.stackexchange.com/q/1338044/19661), I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of positive integers:
$$\begin{align}
\arctan\varphi&=2\arctan1-\frac12\,\... | https://mathoverflow.net/users/9550 | Arctangents of odd powers of the golden ratio | **Q1, Q2:** Such a representation exists for all odd powers of $\varphi$ as we can show by induction. Using the arctangent identity, let us first write:
$\arctan \varphi^{2n+1} = \arctan\frac{\varphi^{2n+1}+v}{1-\varphi^{2n+1}v} - \arctan v$
Next, we set the argument $\frac{\varphi^{2n+1}+v}{1-\varphi^{2n+1}v}$ equ... | 46 | https://mathoverflow.net/users/12301 | 210169 | 100,100 |
https://mathoverflow.net/questions/210185 | 6 | For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2\_{ac}(\mathbb{R}^d)$), let
$$
\mathcal{H}(\rho)=\int\_{\mathbb{R}^d}\rho\log \rho \,dx
$$
be the Boltzmann entropy.
>
> **Question:** Is $\mathcal{... | https://mathoverflow.net/users/33741 | continuity of the Boltzmann entropy in the Wasserstein metric | Let $A\_n=\bigcup\_{i=0}^{n-1}[2i/(2n),(2i+1)/2n)$, $\rho\_n=2\chi\_{A\_n}$, $A=[0,1]$ and $\rho=\chi\_{[0,1]}$.
Then it seems pretty clear that $\mathcal W\_2(\rho\_n,\rho)\to 0$. On the other hand $H(\rho\_n)\not\to H(\rho)$, right?
| 8 | https://mathoverflow.net/users/11054 | 210188 | 100,107 |
https://mathoverflow.net/questions/210173 | 6 | Consider
$$\bar{\rho}:G\_{\mathbb Q}\longrightarrow\operatorname{GL}\_2(\mathbb F\_7)$$
the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then $\bar{\rho}$ is unramified exactly outside $\{2,3,7\}$ and the traces of $\operatorname{Fr}(\ell)$ for $\ell=5,11,13,... | https://mathoverflow.net/users/2284 | Computing an eigencuspform in $S_2(\Gamma_0(1776))$ | I did the computation in Sage, and there is no such form $f$. There are 21 Galois orbits of newforms of level $\Gamma\_0(1776)$ and trivial character, of which the largest has size 3, and none of the reductions modulo any of the primes above 7 in the coefficient fields is congruent to the form associated to $\bar\rho$.... | 15 | https://mathoverflow.net/users/2481 | 210190 | 100,109 |
https://mathoverflow.net/questions/210187 | 7 | Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot.
Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$?
Are there any softwares which compute $\tau(K)$ if we insert Gauss code or PD code of $K$?
For Rasmussen's $s$-invariant, Knottheory pac... | https://mathoverflow.net/users/63984 | What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot? | In principle, a computer can compute $\tau$ for any knot, using [grid diagrams](http://msp.org/gt/2007/11-4/p09.xhtml) (or [variants](http://www.worldscientific.com/doi/abs/10.1142/S0218216510007796) [thereof](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI... | 9 | https://mathoverflow.net/users/13119 | 210191 | 100,110 |
https://mathoverflow.net/questions/209869 | 3 | I am very interested in the maximum number of triangles could a connected graph with $n$ vertices and $m$ edges have. For example, if $m\leq n−1$, this number is $0$, if $m=n$, this number is $1$, if $m=n+1$, this number is $2$, and if $m=n+2$, this number is $4$.
| https://mathoverflow.net/users/75264 | How many triangles can a connected graph with $n$ vertices and $m$ edges have? | It is a bound and since it is very long, I wrote it an answer, may be it can be helpful.
Let $G$ be a connected graph with $n$ vertices and $m$ edges. Suppose the eigenvalues of this graph are $\lambda\_1\geq \lambda\_2\geq\ldots\geq\lambda\_n$. We know that $\sum{\lambda\_i^3}=6\Delta\_G$, where $\Delta\_G$ counts t... | 8 | https://mathoverflow.net/users/19885 | 210198 | 100,113 |
https://mathoverflow.net/questions/210194 | 2 | Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with the nearest neighbours only. However, if such a system is arranged to a circle (or endowed with suitable periodical cond... | https://mathoverflow.net/users/56553 | Information on special matrices similar to Jacobi matrices | Matrices of this form are called "periodic Jacobi matrices", see for example,
-- [The spectrum of Jacobi matrices.](http://link.springer.com/article/10.1007%2FBF01418827)
-- [The construction of Jacobi and periodic Jacobi matrices with prescribed spectra](http://www.jstor.org/stable/2006386).
-- [Continued fracti... | 3 | https://mathoverflow.net/users/11260 | 210200 | 100,115 |
https://mathoverflow.net/questions/210178 | 2 | Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold
$\frac{\int\_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for all t>0.
I tried to find a counterexample but I was not successful. Is the above true or one can find a counterexample?
... | https://mathoverflow.net/users/42326 | An elementary functional inequality | Well since $g(0) = 0$, we may rewrite the desired inequality as:
$$\left(\int\_0^{t} 1\,ds\right)\left(\int\_0^{t} (g'(s))^2\,ds\right) \ge \left(\int\_0^t g'(s)\,ds\right)^2$$
which is just the Cauchy-Schwarz Inequality.
| 11 | https://mathoverflow.net/users/62680 | 210201 | 100,116 |
https://mathoverflow.net/questions/210210 | 1 | Suppose that
1. $\mu\_k$ is an increasing sequence of numbers such that $0 < \mu\_1 \leq \mu\_2 \leq ..$ with $\mu\_k \to \infty$ as $k \to \infty$
2. $\sum\_{k=1}^\infty |u\_k|^2 < \infty$ and $\sum\_{k=1}^\infty \sqrt{\mu\_k}|u\_k|^2 < \infty$ where $u\_k$ is a given sequence of real numbers
I want to show that ... | https://mathoverflow.net/users/75424 | Is this series involving hyperbolic functions uniformly convergent? | The answer is negative.
Take $\mu\_{k}=k$, $u\_{k}=\frac{1}{k}$ and $t=0$. Conditions 1 and 2 hold. Notice that $\frac{\cosh(2\sqrt{\mu\_{k}}T)}{\sinh^{2}(\sqrt{\mu\_{k}}T)}=1+o(1)$ and the sum diverges.
If you want it to be uniformly convergent you need either $\sum |u\_{k}|^{2}\mu\_{k}<\infty$ or $t\in [\vareps... | 2 | https://mathoverflow.net/users/50901 | 210212 | 100,121 |
https://mathoverflow.net/questions/210025 | 5 | I have a small question about unitization of (unital) $C^\*$-algebras. I first asked on math.stackexchange because it is basic theory, but I still have no suitable answer, the link <https://math.stackexchange.com/questions/1336297/unitization-of-a-unital-c-algebra> .
I will describe my problem: Let $A$ be a $C^\*$-al... | https://mathoverflow.net/users/75338 | unitization-process of unital- and non-unital $C^*$-algebras | This answer is the solution UwF provided in the comments.
Let $A\_1$ act on the normed space $V:=A\oplus\mathbb{C}$ with norm $\|(b,z)\|\_V:=\max\{\|b\|\_A,|z|\}$ via the algebra homomorphism $A\_1\to B(V), (a,\lambda)\mapsto \tilde{L}\_{(a,\lambda)}$ where $\tilde{L}\_{(a,\lambda)}(b,z) := (ab+\lambda b,\lambda z)$.... | 4 | https://mathoverflow.net/users/3041 | 210217 | 100,123 |
https://mathoverflow.net/questions/210219 | 5 | Let $K$ be a number field. Let $A$ and $B$ be abelian varieties over $K$. Assume that $A$ and $B$ are isogenous over $\bar{K}$, the algebraic closure of $K$. We further assume that the endomorphism ring of $A\_{\bar{K}}$ is $\mathbb{Z}$. The question is:
Does there exist a quadratic extension $L$ over $K$ such that $... | https://mathoverflow.net/users/75427 | Definition field of isogeny between abelian varieties | Yes. The group of isogenies form a locally free module of rank $1$ over the endomorphism ring of $A$, hence are generated by a single isogeny of minimal degree $k$. So every isogeny is that isogeny composed with an endomorphism of $k$, so has degree $n^{2g} k$ for $n \in \mathbb Z$. Hence there are two isogenies with a... | 3 | https://mathoverflow.net/users/18060 | 210225 | 100,127 |
https://mathoverflow.net/questions/210234 | 8 | Is there an algorithm to decide whether an algebraic system
\begin{gathered}
{f\_1}({x\_1}, \ldots ,{x\_n}) = 0 \hfill \\
\vdots \hfill \\
{f\_m}({x\_1}, \ldots ,{x\_n}) = 0 \hfill \\
\end{gathered}
where $f\_1,\ldots,f\_m$ are polynomials with given rational coefficients, has a solution in real numbers?
| https://mathoverflow.net/users/51203 | Decidability of an Algebraic System in Real Numbers | Yes, this follows from Tarski's theorem (1951) that the first order theory of real closed fields admits elimination of quantifiers. See also the [Tarski-Seidenberg theorem](https://en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg_theorem).
**P.S.** A consequence of these theorems is that if the given polynomial system... | 8 | https://mathoverflow.net/users/11919 | 210240 | 100,132 |
https://mathoverflow.net/questions/210222 | 3 | Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative
$$
\delta g\_{\mu\nu} = ({\mathcal L}\_\xi G)\_{\mu\nu}=\nabla\_\mu \xi\_\nu+\nabla\_\nu \xi\_\mu
$$
and under a change of metric the Levi-Civita Christoffel symbol changes by
$$
\delta {\Gamma^{\alpha}}\_{\beta \mu}= \frac 12 g^{\a... | https://mathoverflow.net/users/75430 | Diffeomorphism variation of the Christoffel symbol | If you consult Sec 3.1 of Wald's book on GR, you will see how Christoffel symbols can be interpreted as *tensors* parametrizing the difference between the coordinate connection $\partial$ and the Levi-Civita connection $\nabla$, symbolically $\Gamma=\nabla-\partial$ (cf. Wald's (3.1.7)). You are interested in $\Gamma'=... | 6 | https://mathoverflow.net/users/2622 | 210241 | 100,133 |
https://mathoverflow.net/questions/210247 | -3 | I proposed [this](https://math.stackexchange.com/q/1339628/156150) question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ?
Let $H$=$L^2(\mathbb R)$ be a separable, infinite-dimensional Hilbert space, and $B(H) = \{... | https://mathoverflow.net/users/51189 | Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? | If I interpret $H$ to mean $L^2(\mathbb R)$,
and if I interpret $\frac{\mathbb d}{\mathbb d x}$ to mean differentiation by $x$,
then the set of periodic vectors is not dense in $H$.
Excluding functions that grow exponentially, a function is periodic w.r.t. differentiation by $x$ if and only if it is of the form $a \c... | 1 | https://mathoverflow.net/users/5690 | 210249 | 100,135 |
https://mathoverflow.net/questions/210246 | 0 | Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$\_2(\bf{ Z})$' ([link](http://people.mpim-bonn.mpg.de/zagier/files/tex/ConsequencesCohomologySL/fulltext.pdf), p. 6), studies the action of $\Gamma=PSL\_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ \gamma$. Recall the following... | https://mathoverflow.net/users/48554 | On a claim of Zagier on extending a map to cocycle | Define a function $\widehat f$ from $\Gamma$ to the semidirect product $\Gamma \ltimes V$ by $\widehat f(\gamma) = \bigl( \gamma, f(\gamma) \bigr)$. Saying that $f$ is a cocycle is exactly the same as saying that $\widehat f$ is a homomorphism. Therefore, $f$ extends to a cocycle if and only if $\widehat f(S)^2 = 1$ an... | 4 | https://mathoverflow.net/users/68305 | 210256 | 100,138 |
https://mathoverflow.net/questions/210253 | 4 | Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality:
$(∫|u|^{r}dx)^{\frac{1}{r}} \leq GN(N,p,q,r)(∫|∇u|^{p}dx)^{\frac{a}{p}}(∫|u|^{q}dx)^{\frac{1-a}{q}}$.
Can the best constant GN(N,p,q,r) be achieved in some cases? Does anyo... | https://mathoverflow.net/users/40232 | Extremal functions for Gagliardo-Nirenberg inequality | There is a $1$--parameter family of inequalities where the sharp constants and corresponding extremal functions are known. I believe this was first established by Del Pino and Dolbeault. Cordero, Nazaret, and Villani gave a beautiful optimal transportation proof. See their paper for the relevant references. The family ... | 5 | https://mathoverflow.net/users/613 | 210258 | 100,139 |
https://mathoverflow.net/questions/210228 | -1 | Selberg orthonormality conjecture for automorphic L-functions was proven under Ramanujan conjecture, and SOC itself implies unique factorization for those L-functions.
My question is: does the unique factorization imply a weakened form of Ramanujan conjecture?
Thanks in advance.
| https://mathoverflow.net/users/13625 | Does unique factorization for automorphic L-functions imply a weakened form of Ramanujan conjecture? | This seems unlikely to me. Philosophically unique factorization means something like the following: if $R$ is a (say completely reducible) representation of a group $G$ then the trace of $R$ determines the irreducible constituent representations. This will be true if the characters satisfy an orthogonality relation. Se... | 3 | https://mathoverflow.net/users/6518 | 210274 | 100,146 |
https://mathoverflow.net/questions/210276 | 3 | The EGZ theorem states that any multiset of $2n-1$ integers has a subset of size $n$ the sum of whose elements is a multiple of $n$.
Kemnitz-Reiher theorem is a 2-dimensional analogue of EGZ. Here is the formulation:
Let $n$ be a natural number and $S$ a set of $4n − 3$ lattice points in plane. Then there exists a su... | https://mathoverflow.net/users/69929 | higher dimensional analogue of EGZ theorem | In higher dimension things become more complicated. For a finite abelian group $G$ define $\mathfrak{s}(G)$ to be the least integer $N$, such that every sequence $x\_1, \ldots, x\_N$ of elements of $G$ contains a subsequence $x\_{i\_1}, \ldots, x\_{i\_n}$ with sum 0, where $n=\exp(G)=\min \{ n\mid g^n=e \text{ for all ... | 7 | https://mathoverflow.net/users/37555 | 210277 | 100,147 |
https://mathoverflow.net/questions/210282 | 3 | In compressed sensing, we are interested in the restricted isometry property. Suppose the design matrix is $n$ by $p$, consisting of $np$ iid $\mathcal{N}(0, 1/n)$ entries. Assume both $n$ and $p$ are growing to infinity but one may be much larger than the other. Then, my question is, what is the best known upper bound... | https://mathoverflow.net/users/8369 | Vanishing Restricted Isometric Constant | We know that your matrix is $(k,\delta)$-RIP provided $n\geq (C/\delta^2)k\log(p/k)$, where $C$ is an absolute constant. We also know that this bound is tight up to constants (see [this book](http://www.springer.com/us/book/9780817649470) for both results). If you want $\delta$ to vanish asymptotically, the implied bou... | 4 | https://mathoverflow.net/users/29873 | 210301 | 100,154 |
https://mathoverflow.net/questions/210294 | 4 | Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation
$$ \frac{d x (t)}{dt} = f(x(t)) $$
with some initial condition $x(0)=x\_0$ has no solution?
| https://mathoverflow.net/users/51203 | A differential equation with continuous coefficient and no solution in a reflexive Banach space? | You can find a very general example of the desired type in the paper: P. Hajek, M. Johanis, On Peano's theorem in Banach spaces. J. Differential Equations 249 (2010), no. 12, 3342–3351.
| 4 | https://mathoverflow.net/users/37822 | 210302 | 100,155 |
https://mathoverflow.net/questions/210291 | 34 | One way to phrase the
"[concentration-of-measure](https://en.wikipedia.org/wiki/Concentration_of_measure)"
phenomenon is that,
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator."1
>
> ***Q***. How could one explain/justify this intuitively—per... | https://mathoverflow.net/users/6094 | How to explain the concentration-of-measure phenomenon intuitively? | As I see it, the key intuition is passing from the equator orthogonal to a single vector to looking at a whole orthonormal basis.
Suppose we pick a random unit vector $(x\_1,\dots,x\_n)$. What we want to know is why $x\_1$ is probably near zero, since this is equivalent to being near the equator relative to the first... | 60 | https://mathoverflow.net/users/4720 | 210304 | 100,156 |
https://mathoverflow.net/questions/210288 | 4 | Suppose that we have:
1) triangulated categories $C,D$, each equipped with a $t$-structure.
2) triangulated functor $F: C \to D$ which is $t$-exact.
3) $F$ reflects isomorphisms, i.e. is conservative.
Point 3) is equivalent to $F$ reflecting zero objects.
It seems that the restriction of $F$ to the hearts is fait... | https://mathoverflow.net/users/2095 | A conservative, non faithful functor between triangulated categories | Here is also a simple example where the restriction of $F$ to the heart is faithful but $F$ itself is not. Let $Vec$ be the abelian category of complex vector spaces and let $Rep(\mathbb{Z})$ be the abelian category of complex representations of the free cyclic group $(\mathbb{Z},+)$. The forgetful functor $U: Rep(\mat... | 5 | https://mathoverflow.net/users/51164 | 210305 | 100,157 |
https://mathoverflow.net/questions/210303 | 4 | Sorry if the question doesn't feed this site, I'm reposting it from [MSE](https://math.stackexchange.com/questions/1340592/transitivity-of-discriminant-for-flat-algebras). Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a known fact), all I f... | https://mathoverflow.net/users/75449 | Transitivity of discriminant for flat algebras | The question has some minor misstatements. You didn't really want to mention $R$ anywhere: the base ring is $A$, over which $A'$ should be assumed to be "finite locally free" (equivalently, "finite flat and finitely presented"); this is the same as finite flat if $A$ is noetherian. Likewise for $A' \rightarrow A''$ (wh... | 5 | https://mathoverflow.net/users/70739 | 210306 | 100,158 |
https://mathoverflow.net/questions/210239 | 35 | In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very likely true that
$$\det \left(A^{\frac{1}{2}}(A+B)A^{\frac{1}{2}}+B^{\frac{1}{2}}(A+B)B^{\frac{1}{2}}\right) \ge \det(A+B... | https://mathoverflow.net/users/54458 | A curious determinantal inequality | Let $C := A^{1/2} (A+B) A^{1/2} + B^{1/2} (A+B) B^{1/2}$; this is a positive semi-definite matrix with the same trace as $(A+B)^2$. We show that the eigenvalues of $C$ are majorised by the eigenvalues of $(A+B)^2$, that is to say that the sum of the top $k$ eigenvalues of $C$ is at most the sum of the top $k$ eigenvalu... | 29 | https://mathoverflow.net/users/766 | 210314 | 100,162 |
https://mathoverflow.net/questions/210279 | 4 |
>
> A set $X\subset \mathbb{R}$ is called *nice* if for every $\epsilon > 0$ there are a
> positive integer $k$ and $k$ bounded intervals $I\_1,I\_2,...,I\_k$ such that
> $X \subset I\_1 \cup I\_2 \cup \cdots \cup I\_k$ and
> $\sum\limits\_{j=1}^k |I\_j|^{\epsilon} < \epsilon$.
>
> Prove that there exist se... | https://mathoverflow.net/users/70464 | Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$ | Following up on Bjørn's suggestion:
Let $S\subset\mathbb{N}$ be a set such that both $S$ and $\mathbb{N}\setminus S$ have lower density zero. For example, let $S=\bigcup\_{n\in\mathbb{N}} [(2n)!,(2n+1)!)$. Set
$$
X' = \{ \sum\_n a\_n 2^{-n}: n\in S, a\_n\in\{0,1\}\},
$$
$$
Y' = \{ \sum\_n a\_n 2^{-n}: n\notin S, a\_n... | 7 | https://mathoverflow.net/users/11009 | 210317 | 100,163 |
https://mathoverflow.net/questions/210261 | 6 | Can anyone direct me to any algorithms or theorems that describe the difficulty of solving a non-smooth convex optimization problem for the special case where the full subdifferential is available? All of the results that I can find are designed for the case where one has an oracle that can generate a single subgradien... | https://mathoverflow.net/users/70190 | Convex optimization with full subdifferential information | The difficulty with non-differentiable convex functions in optimization stems from the fact that an arbitrary subgradient need not be a descent direction, and hence classical first-order methods cannot be guaranteed to work. (But see Lewis' and Overton's works on nonsmooth quasi-Newton methods, e.g., <http://people.ori... | 6 | https://mathoverflow.net/users/30516 | 210325 | 100,164 |
https://mathoverflow.net/questions/210323 | 2 | This post is a sequel of [Diameter of symmetric group](https://mathoverflow.net/q/102737/34538).
Let $\Sigma$ a generating subset of $S\_n$, $\Gamma(S\_n, \Sigma)$ the Cayley graph and $d\_{\Sigma}$ the diameter of $\Gamma(S\_n, \Sigma)$.
Let $s\_n = min\_{\Sigma}(\vert \Sigma \vert \times d\_{\Sigma})$.
*Quest... | https://mathoverflow.net/users/34538 | Asymptotic of min(#generators times diameter), for a Cayley graph of Sn | For a generating subset $S\subset S\_n$ of size $d$, the $n$-ball has size $\le (2d)^n$, which implies that the diameter $D(S)$ satisfies $D\ge\log\_{2d}(n!)\simeq n\log n$.
So $|S|D(S)\ge \frac{d}{\log d}\log(n!)\ge \log(n!)\sim n\log n$, and thus $s\_n=\min\_S |S|D(S)\succeq n\log n$.
This is attained, according... | 5 | https://mathoverflow.net/users/14094 | 210327 | 100,165 |
https://mathoverflow.net/questions/210340 | 22 | How would you go about writing an abstract for a Math paper? I know that an abstract is supposed to "advertise" the paper. However, I do not really know how to get started. Could someone tell me how they go about writing an abstract?
| https://mathoverflow.net/users/75293 | How to write an abstract for a math paper? | 1. Avoid notation if possible. Notation makes it really hard to search electronically.
2. Put the subject in context, e.g., "In a recent paper, T. Lehrer introduced the concept of left-bifurcled rectangles. He conjectured no such rectangles exist when the number of bifurcles $n$ is odd."
3. State your results, in non-t... | 55 | https://mathoverflow.net/users/44797 | 210342 | 100,173 |
https://mathoverflow.net/questions/210293 | 2 | Let $k$ be an ordered field of cofinality $cf(k)$ whose Cauchy $cf(k)$-sequences are convergent.$^{(1)}$
Let $\mathcal{R}(k)$ be its real closure.
As an algebraic extension of $k$, it has the same cofinality.$^{(2)}$
I wonder if $\mathcal{R}(k)$ has the same Cauchy completeness property as $k$, and if so, if this ... | https://mathoverflow.net/users/45005 | Cauchy completeness of the real closure | Note that for a nonarchimedean ordered field, this notion of completeness is equivalent to its completeness as a [valued field](https://en.wikipedia.org/wiki/Valuation_(algebra)) under the natural valuation induced by the order (that is, with valuation ring $\{a\in k:\exists n\in\mathbb N\,(-n\le a\le n)\}$).
Finite ... | 5 | https://mathoverflow.net/users/12705 | 210353 | 100,178 |
https://mathoverflow.net/questions/210358 | 6 | Let $\gamma(G)$ denote the domination number of a graph, and $G\,\square\,H$ denote the cartesian product of two graphs. Then $K\_8\,\square\, K\_8$ is the rook graph, whose vertices are the squares of a chessboard, with edges between squares a rook can move between. We can similarly define the rook graph of any square... | https://mathoverflow.net/users/59232 | Four Dimensional Rook Domination | This might be an answer for n=4. There are 24 points. I think 23 is impossible.
Just a guess, however.
```
0 0 0 0
0 0 0 1
0 1 1 2
0 2 2 3
0 2 3 3
0 3 1 2
1 0 1 3
1 1 2 0
1 1 3 1
1 2 0 2
1 3 2 1
1 3 3 0
2 ... | 5 | https://mathoverflow.net/users/nan | 210368 | 100,185 |
https://mathoverflow.net/questions/208547 | 7 | Is there a self-homeomorphism of a genus 2 (closed, orientable) surface, which has finite order and exactly 5 fixed points?
Of course, the same question can be asked replacing 2 by $g$ and $5$ by any number $k$. An upper bound for possible values of $k$ is (generalized Lefschetz fixed point theorem) $2g+2$.
For $g... | https://mathoverflow.net/users/14233 | Automorphism of genus 2 surface with 5 fixed points | Although Jason and Dylan seemed to have answered this question in the comments, I decided to work out what the generalized version of this kind of statement is.
Let $\sigma$ be an automorphism of a Riemann surface. Any automorphism that is orientation-reversing with a fixed point has a fixed curve, so let's assume $\... | 2 | https://mathoverflow.net/users/18060 | 210371 | 100,186 |
https://mathoverflow.net/questions/210072 | 1 | There may well be an answer to this question in a simpler category than that of finite dimensional quantum groups and in that case this question is more suitable to math.stack and I apologise in advance if this is the case.
Consider a finite dimensional quantum group $A=F(\mathbb{G})$ with a Haar state $h:A\rightarro... | https://mathoverflow.net/users/35482 | 'Test Functions' to Lower Bound the Norm of Elements of Dual Quantum Group | In a more general setting than that of the original question: suppose we have a faithful normal state $h$ on a von Neumann algebra $M$. Suppose furthermore that $h$ is tracial, meaning that $h(xy)=h(yx)$ for all $x,y\in M$. (Warning! there are important examples of compact quantum groups where the Haar state is faithfu... | 1 | https://mathoverflow.net/users/763 | 210391 | 100,192 |
https://mathoverflow.net/questions/210384 | 1 | How many symmetric and non-symmetric $n\times n$ matrices with $0/1$ entries are there such that every row is distinct and every column is distinct? (I am looking for a proof as well).
If only every row (or column) is distinct is needed, the answer is easy.
As suggested in comments, <https://oeis.org/A088310> provi... | https://mathoverflow.net/users/10035 | Counting matrices of special types | For generic (not necessarily symmetric) $m\times n$ matrices over a set of $k$ elements, the number of those with pairwise distinct columns and rows is
$$\sum\_{i=0}^m\sum\_{j=0}^n s(m,i)\cdot s(n,j)\cdot k^{i\cdot j},$$
where $s(,)$ are Stirling numbers of first kind with sign.
**UPDATE**. For symmetric $n\times n$ ... | 3 | https://mathoverflow.net/users/7076 | 210398 | 100,194 |
https://mathoverflow.net/questions/136933 | 9 | A [difference set](http://en.wikipedia.org/wiki/Difference_set) of a group $G$ is a subset $D\subseteq G$ with the property that there exists an integer $\lambda>0$ such that for every non-identity member $g$ of $G$, there exist exactly $\lambda$ ordered pairs $(a,b)\in D\times D$ such that $g=ab^{-1}$. Note that $D=G$... | https://mathoverflow.net/users/29873 | Does $(\mathbb{Z}/n\mathbb{Z})^2$ ever admit a difference set when $n$ is odd? | Such difference sets exist. There exist (nontrivial) difference sets with
$|G| = q^{d+1}[1+(q^{d+1}-1)/(q-1)]$,
$|D| = q^d(q^{d+1}-1)/(q-1)$,
$\lambda = q^d(q^d-1)/(q-1)$,
whenever $q$ is a prime power (R. L. McFarland, A family of difference sets in non-cyclic groups, JCT A, 15 (1973), pp. 1-10). More precise... | 9 | https://mathoverflow.net/users/75499 | 210400 | 100,196 |
https://mathoverflow.net/questions/209874 | 5 | It is well-known that long ago, Wigner classified the unitary irreducible representations of the Poincare group in dimension 4.
I am looking for a convenient reference describing all unitary irreducible representations of the Poincare group in dimensions 2 and 3. (I know how it can be done in principle. But I am loo... | https://mathoverflow.net/users/56920 | Unitary irreps of the Poincare group in dimension <4 | I finally found a complete analysis for dimension 3 in
D.R. Grigore, The projective unitary irreducible representations of the Poincaré group in 1+2 dimensions, J. Math. Phys. 34 (1993), 4172-4189.
(<http://arxiv.org/abs/hep-th/9304142>)
Now I also found a complete analysis for dimension 2 in
S.K. Bose, Projecti... | 6 | https://mathoverflow.net/users/56920 | 210411 | 100,199 |
https://mathoverflow.net/questions/210401 | 7 | In other words, given a sequence $(s\_n)$, how can we tell if there exist irrationals $u>1$ and $v>1$ such that
$$s\_n = \lfloor un\rfloor + \lfloor vn\rfloor$$
for every positive integer $n$?
A few thoughts: Graham and Lin (*Math. Mag.* 51 (1978) 174-176) give a test for $(s\_n)$ to be a single Beatty sequence ... | https://mathoverflow.net/users/61426 | When is a sequence the sum of two Beatty sequences? | Let's use the notation $\{ x\}$ for the fractional part of a number $x$.
Assume $u, v$, and $u/v$ are all irrational.
Then, $\{un\}$ and $\{vn\}$ behave as independent uniform random variables. (This is proved by Fourier analysis, vindicating James Cranch's suggestion) $s\_{n+1}-s\_n$ depends on $\{un\}$ and $\{vn\... | 5 | https://mathoverflow.net/users/18060 | 210419 | 100,201 |
https://mathoverflow.net/questions/108202 | 5 | A bit of light googling turns up several sources asserting that the model structure on simplicial commutative algebras over a ring is left proper (for example, 2.9 in Charles Rezk's paper [Every homotopy theory of simplicial algebras admits a proper model](http://arxiv.org/abs/math/0003065)). Does a proof of this fact ... | https://mathoverflow.net/users/1100 | Left Properness of Simplicial Commutative Algebras | For simplicial commutative rings this is proved in Lemma 3.1.2
of Schwede's “Spectra in model categories and applications
to the algebraic cotangent complex”,
and the proof there immediately extends to algebras.
| 3 | https://mathoverflow.net/users/402 | 210420 | 100,202 |
https://mathoverflow.net/questions/210387 | 2 | Let us consider two arbitrary Hermitian square matrices $\mathbf{A,B}$ with the same dimension. Given $\mathbf{v}$ the eigenvector associated to the maximum eigenvalue of $\mathbf{A}$:
>
> Are there any conditions appart from $\mathbf{A} = \mathbf{B}$ or $\mathbf{B} = \mathbf{v}\mathbf{v}^H$ so that it can be ensur... | https://mathoverflow.net/users/11825 | Shared maximum eigenvector | This is true iff the largest eigenvalue of $A+B$ is the sum of that of $A$ and that of $B$.
| 3 | https://mathoverflow.net/users/391 | 210433 | 100,205 |
https://mathoverflow.net/questions/210313 | 4 | Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha\_i, \check{\alpha\_i}, i \in I$, respectively. Let $L$ be an integrable highest weight module.
Write $C$ for the dominant Weyl chamber, i.e. the locus $ \{ \lambda \in \mathfrak{h}^\*... | https://mathoverflow.net/users/30971 | Is every weight of an integrable highest weight module in the Tits cone? | Yes. The Tits cone, as the name implies, is a cone: in particular, it's convex. Any weight for a highest weight module is an affine linear combination of finitely many extremal weights (I'll leave that as an exercise; one hint is to prove it by induction on the number of simple roots you need to get to your weight from... | 3 | https://mathoverflow.net/users/66 | 210434 | 100,206 |
https://mathoverflow.net/questions/210432 | 32 | [Reverse mathematics](https://en.wikipedia.org/wiki/Reverse_mathematics) (RM) is that area that tries to pin down exactly which axioms are necessary to prove theorems, given some weak base theory. Harvey Friedman has pointed out several times (on the [FOM mailing list](http://www.cs.nyu.edu/mailman/listinfo/fom)) that ... | https://mathoverflow.net/users/4177 | What is the reverse mathematical strength of the fundamental theorem of algebra? | Tanaka and Yamazaki (in the volume *Reverse Mathematics 2001*, see [review](http://www-compsci.swan.ac.uk/~csulrich/ftp/reviewRevMath2001.pdf)) show that a substantial portion of field theory can be done in the weak base theory RCA$\_0$, by proving in RCA$\_0$ the fundamental theorem of algebra as well as quantifier el... | 25 | https://mathoverflow.net/users/4600 | 210436 | 100,208 |
https://mathoverflow.net/questions/210050 | 0 | This post was inspired by an exchange with the indian woman mathematician [Ajit Iqbal Singh](https://en.wikipedia.org/wiki/Ajit_Iqbal_Singh).
The chinese remainder theorem can be stated as follows:
Let $n\_1, \dots, n\_r \ge 2$ be positive integers such that $n\_i \wedge n\_j = 1$ iff $i \neq j$, then the system o... | https://mathoverflow.net/users/34538 | Chinese remainder theorem for cyclic subfactor planar algebras | **No**, we show above an inclusion of finite groups counter-example.
First of all, the statement for an inclusion of finite groups $(H \subset G)$ can be reformulate as follows:
Let $K\_1, \dots , K\_r$ be strict intermediate subgroups (i.e. $H \subsetneq K\_i \subsetneq G$, $\forall i$), such that $\langle K\_i ... | 0 | https://mathoverflow.net/users/34538 | 210443 | 100,211 |
https://mathoverflow.net/questions/210161 | 1 | I am interested in the solvability of
$$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial\_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ \Omega$ (a bounded smooth domain in $ R^N$). I have tried the variational approach but cannot obtain the correct boundary co... | https://mathoverflow.net/users/66623 | Elliptic pde with bilaplacian; boundary conditions. | You will not get a direct variational structure (because of the boundary conditions) but there is a mixed approach that will work on your case: Set $-\Delta u=v$ and obtain the following system:
$$
\begin{equation}
\left\{
\begin{array}{rl}
-\Delta v+u=f & \text{in }\Omega , \\
v=0 & \text{on }\partial \Omega%
\end{arr... | 2 | https://mathoverflow.net/users/40644 | 210451 | 100,213 |
https://mathoverflow.net/questions/210456 | 1 | Let $G$ be a connected, linear, semi-simple algebraic group over an algebraically closed field of characteristic zero and $P$ be the maximal parabolic subgroup. We know that the quotient $Z=G/P$ is a projective scheme. Is it true that the Picard group of $Z$ is isomorphic to $\mathbb{Z}$?
| https://mathoverflow.net/users/58203 | Picard group of a quotient of a group by its maximal parabolic subgroup | Yes. We can assume that $G$ is simply connected, and in this case, any line bundle on $G/P$ is equivariant for $G$ (if $G$ isn't simply connected, you get silly things like $\mathcal{O}(1)$ on $\mathbb{P}^n$ isn't equivariant for $PSL\_{n+1}$). For any maximal parabolic subgroup, $P/[P,P]\cong \mathbb{G}\_m$, and every... | 6 | https://mathoverflow.net/users/66 | 210463 | 100,217 |
https://mathoverflow.net/questions/210290 | 4 | The only rough path that I've ever seen discussed are the ones associated with Brownian motion. I could use a "rough path" for any nice function, defeating the point. In particular are there interesting rough paths that are $\alpha$-Holder continuous for $\frac13 \lt \alpha \lt \frac12$?
| https://mathoverflow.net/users/nan | What's an example of a rough path that's not Ito/Stratonovich-Brownian rough path? | [An explicit rough path construction for continuous paths with arbitrary Hölder exponent](https://hal.inria.fr/hal-00370570/document), by Jeremie Unterberger (2009) [[published](http://link.springer.com/article/10.1007%2Fs00220-010-1064-1) with a different title]:
>
> We construct an explicit geometric rough path o... | 3 | https://mathoverflow.net/users/11260 | 210470 | 100,219 |
https://mathoverflow.net/questions/210464 | 5 | Is there a formula to compute the determinant of block tridiagonal matrices when the determinants of the involved matrices are known?
In particular, I am interested in the case
$$A = \begin{pmatrix} J\_n & I\_n & 0 & \cdots & \cdots & 0 \\ I\_n & J\_n & I\_n & 0 & \cdots & 0 \\ 0 & I\_n & J\_n & I\_n & \ddots & \vd... | https://mathoverflow.net/users/50081 | Determinant of block tridiagonal matrices | The Kronecker product idea brought up in Algebraic Pavel's comment on the original maths stack exchange question seems like a good way to approach the particular case of interest to you. Specifically, assuming $A$ is $m n \times m n$, i.e., there are $m$ block rows and columns, then
$$A = J\_m \otimes I\_n + I\_m \otim... | 5 | https://mathoverflow.net/users/70005 | 210492 | 100,227 |
https://mathoverflow.net/questions/210462 | 1 | Let $f:X \to Y$ be a smooth morphism between projective varieties. Suppose $Y$ is a homogeneous space. Under what additional condition on $f$, can we conclude that every fibers of $f$ are isomorphic?
| https://mathoverflow.net/users/58203 | Smooth morphism to homogeneous spaces and fibers | For your particular case, there is a sufficient hypothesis that the morphism is étale locally a product. Let $k$ be a field. Let $G$ be a smooth $k$-group scheme. Let $$\Psi\_X:G\times\_{\text{Spec}(k)} X \to X\times\_{\text{Spec}(k)} X, \ \ \Psi\_X(g,x) = (g\cdot x,x), $$
$$\Psi\_Y:G\times\_{\text{Spec}(k)} Y \to Y\ti... | 1 | https://mathoverflow.net/users/13265 | 210494 | 100,228 |
https://mathoverflow.net/questions/210487 | 12 | Let $K\subset \mathbb C$ be a closed subset of the complex plane, not necessarily bounded.
Let $U$ be the interior of $K$.
Let $f:K\to \mathbb C$ be a continuous bounded function, whose restriction to $U$ is holomorphic.
Assume furthermore that for every closed curve $\gamma\subset K$, the integral $\int\_\gamma... | https://mathoverflow.net/users/38805 | Is Every Holomorphic Near an Entire? | As mentioned in the comments, this is true if $K$ is compact and the complement of $K$ in the Riemann sphere is connected : it is the content of Mergelyan's Theorem on uniform polynomial approximation of holomorphic functions.
**EDIT** If $K$ is only assumed to be closed, this is also true with the additional assumpt... | 13 | https://mathoverflow.net/users/1162 | 210499 | 100,231 |
https://mathoverflow.net/questions/210346 | 5 | Let $X$ be a rigid analytic space over a non-Archimedean field $k$. If $U\_1,\ldots,U\_n\subseteq X$ are affinoid opens, then it's usually not clear whether or not the admissible open $U=U\_1\cup\cdots\cup U\_n$ is affinoid. But we have (due to the sheaf axioms, and the fact that the $U\_i$ constitute an admissible cov... | https://mathoverflow.net/users/4351 | Topology of ring of global sections of finite union of affinoid opens in a rigid analytic space | So that this question doesn't remain unanswered, I will provide an elaboration of grghxy's answer in the comments. The quasi-separatedness of $X$ ensures that $U$ is admissible open with $U=\bigcup\_{i=1}^n U\_i$ an admissible covering (and in fact that any finite covering of $U$ by affinoid opens of $X$ is admissible)... | 2 | https://mathoverflow.net/users/4351 | 210500 | 100,232 |
https://mathoverflow.net/questions/210489 | 21 | Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care most about the situation where $\mathcal C$ is *finite $\mathbb C$-linear* in the sense of [arXiv:1406.4204](http://arxiv.... | https://mathoverflow.net/users/78 | Does projective imply flat? | I believe the following is a counterexample. Let $\mathcal{A}$ and $\mathcal{B}$ be closed symmetric monoidal abelian categories such that the unit object $1\in\mathcal{B}$ is projective and let $F:\mathcal{A}\to\mathcal{B}$ be a non-exact strong symmetric monoidal functor which has a right adjoint $G:\mathcal{B}\to\ma... | 20 | https://mathoverflow.net/users/75 | 210504 | 100,234 |
https://mathoverflow.net/questions/210502 | 1 | Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k\*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a\_1 < a\_2 < \cdots < a\_n.$$ Let $D\_n$ be the set of all the differences $a\_{i+1} - a\_i$ for $k = 1,2, \ldots, n-1.$ Can someone cite a reference or give a proof that $D\_n$ conta... | https://mathoverflow.net/users/61426 | Differences of consecutive ordered fractional parts | The case $h=0$ is known as the "Three-Distance Theorem"; just google for numerous references or look [here](https://sumidiot.wordpress.com/2009/12/23/the-steinhaus-conjecture) for discussion and nice pictures, or [here](http://www.theoremoftheday.org/NumberTheory/ThreeDistance/TotDThreeDistance.pdf) for an interesting ... | 3 | https://mathoverflow.net/users/9924 | 210505 | 100,235 |
https://mathoverflow.net/questions/210514 | 6 | When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their exact definition or motivation. Could someone explain its motivation and definition to me? A source would also be helpfu... | https://mathoverflow.net/users/75293 | What is the motivation and purpose of the Floretion group? | A rather comprehensive collection of information on floretions, specifically in the context of Oeis, is [Sequences related to floretions.](http://www.mrob.com/pub/seq/floretion.html)
>
> In essence, most of the "floretion" sequences come from an iterated
> function that begins with some initial value and produces o... | 4 | https://mathoverflow.net/users/11260 | 210517 | 100,239 |
https://mathoverflow.net/questions/210475 | 5 | Let $G$ be a the fundamental group of a hyperbolic knot complement.
Then $G$ is hyperbolic relative to a subgroup $P\cong \mathbb Z \oplus \mathbb Z$.
The knot complement has a $2$-dimensional spine with contractible universal cover illustrating that $G$ has geometric dimension $2$.
Question: Does there exist a $2$... | https://mathoverflow.net/users/75544 | Hyperbolic knot complement groups and relative dimension | Proposition 9.6 of M. Kapovich's 'Homological dimension and critical exponent of Kleinian groups' (it's Proposition 9.5 of the [arXiv](http://arxiv.org/abs/math/0701797) version) asserts that the cohomological dimension of the pair $(G,\mathcal{F})$ is equal to $\mathrm{dim}~\Lambda(G)+1$, where $\Lambda(G)$ is as usua... | 7 | https://mathoverflow.net/users/1463 | 210519 | 100,240 |
https://mathoverflow.net/questions/210424 | 7 | Let $F$ denote the Fourier transform over some group. What is known about the following quantity?
$$\gamma:=\inf\_{x\neq 0}\frac{\|Fx\|\_1}{\|F|x|\|\_1}$$
Here, $|x|$ denotes the pointwise absolute value of $x$. We know $\gamma\leq1$ since $x$ can be pointwise nonnegative. In the case where $F$ is the DFT, we also ... | https://mathoverflow.net/users/29873 | Can phase significantly concentrate a function's spectrum? | Observe that if $n$ and $m$ are relatively prime, $\gamma\_{nm}\leq \gamma\_n \gamma\_m$ because you can combine functions on $\mathbb Z/n$ and $\mathbb Z/m$ using the Chinese remainder theorem.
So if we show $\gamma\_n \leq 1- \epsilon$ for some $\epsilon$ and all sufficiently large $n$, then we can make $\gamma\_n$... | 5 | https://mathoverflow.net/users/18060 | 210524 | 100,241 |
https://mathoverflow.net/questions/210508 | 4 | Let $A$ be an abelian variety defined over $\overline{\mathbb{Q}}$ and with complex multiplication by a CM field $K$. Looking at the action of $K$ on $H^0(A, \Omega^1\_A)$ one gets a CM type of $K$, that is, a subset $\Phi$ of $\mathrm{Hom}(K, \overline{\mathbb{Q}})$ such that $\Phi \amalg \bar{\Phi}=\mathrm{Hom}(K, \o... | https://mathoverflow.net/users/75556 | isogeny clases of CM abelian varieties | You can find what you are looking for in Milne's notes on complex multiplication:
<http://jmilne.org/math/CourseNotes/CM.pdf>
The overall idea is that for a CM-field $K$ you can consider a pair $(A,i)$, where $A$ is an abelian variety together with an isomorphism $i : K \to \mathrm{End}^0(A) := \mathrm{End}(A) \oti... | 6 | https://mathoverflow.net/users/7263 | 210525 | 100,242 |
https://mathoverflow.net/questions/210523 | 1 | Let $X\_0$ be a smooth projective variety over the complex numbers and let $X$ be an infinitesimal deformation of $X\_0$ over the ring of dual numbers.
If the canonical bundle of $X\_0$ is ample (resp. anti-ample), does it follow that the canonical bundle of $X$ over $\mathbb C[\epsilon]$
is ample (resp. anti-ample)... | https://mathoverflow.net/users/75562 | The canonical bundle of an infinitesimal deformation | This is true in a much more general setting:
>
> **Fact** Let $X$ be a proper variety over a noetherian ring $A$ and let $\mathscr L$ be a line bundle on $X$. Then $\mathscr L$ is ample if and only if $\mathscr L\_{\mathrm{red}}\simeq \mathscr L\otimes \mathscr O\_{X\_{\mathrm{red}}}\ $ is ample on $X\_{\mathrm{red... | 4 | https://mathoverflow.net/users/10076 | 210530 | 100,245 |
https://mathoverflow.net/questions/210518 | -1 | How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can satisfy. Given properties of a function, can these properties be used to prove that a function is a polynomial. I would like to... | https://mathoverflow.net/users/75293 | What conditions imply that a function over $\mathbb{Z}$ is a polynomial? | This is the same answer as given several times already. It is prefaced with a few facts on Taylor series to make it seem familiar. First recall that a real function $f(x)$ well enough behaved at $x=0$ has a Taylor series $$f(x)=\sum\_0^{\infty}a\_k\frac{x^k}{k!} $$ valid in some interval. And
>
> f(x) is a polynom... | 2 | https://mathoverflow.net/users/8008 | 210541 | 100,248 |
https://mathoverflow.net/questions/210537 | 1 | I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my [earlier post](https://math.stackexchange.com/questions/959502/partial-proof-of-second-hardy-littlewood-conjecture-modified) I conjectured that the inequality ($\pi(x)$ denotes the prime c... | https://mathoverflow.net/users/nan | A conjecture on the prime counting function | Look for a large gap in the distribution of primes. For this conjecture, the gap between $n!+2$ and $n!+n$ will suffice. Set $y = n!+2$ (which is composite) and set $m$ (which will be $\frac{x+y}{2}$, so $x$ will be $2m-y$ eventually) to be the largest composite so that there are no primes between $m$ and $y$. Then the... | 6 | https://mathoverflow.net/users/3206 | 210542 | 100,249 |
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