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https://mathoverflow.net/questions/283933 | 0 | In most of the literature, I come across marked Poisson Poisson process with just one marks. Can someone give me details on Marked Poisson process with multiple marks?
| https://mathoverflow.net/users/114144 | Marked Poisson point process with multiple marks? | I have been interested in the same question recently and, just as you, I did not find much in literature. What kind of details do you need?
Here I report a few basic properties one can easily find, as a starting point.
First denote with $\lambda$ the exponential rate of the process and with $\varphi$ the probabilit... | 1 | https://mathoverflow.net/users/69207 | 287757 | 126,989 |
https://mathoverflow.net/questions/287751 | 2 | In the paper *Faster Algorithms for Max-Product Message Passing* by McAuley and Caetano (see e.g. [here](https://cseweb.ucsd.edu/~jmcauley/pdfs/jmlr11.pdf) or [here](https://arxiv.org/abs/0910.3301)), several statements are made which seem mathematically questionable to me.
For completeness sake some of the relevant ... | https://mathoverflow.net/users/93694 | Is there any "fundamental" distinction between min-plus, max-plus, min-product, and max-product algebras? | Consider the following four semirings, listed in the order underlying set, addition, additive identity, multiplication, multiplicative identity:
1. $A = (\mathbb{R} \cup \{ \infty \}, \text{min}, \infty, +, 0)$
2. $B = (\mathbb{R} \cup \{ - \infty \}, \text{max}, -\infty, +, 0)$
3. $C = (\mathbb{R}\_{> 0} \cup \{ \in... | 15 | https://mathoverflow.net/users/290 | 287760 | 126,990 |
https://mathoverflow.net/questions/285454 | 5 | This question has its seed in [this one](https://mathoverflow.net/questions/283716/what-is-the-matter-with-hecke-operators) by Gory, which found an enlightening answer but one of the comments kept me wondering. I am beginning to discover Hecke operators, and there appears to be an ubiquitous relation between Hecke eige... | https://mathoverflow.net/users/43737 | Relation between Hecke operators and coefficient of L-functions | A normalized version of your guess is right. First note that the $T\_{p^n}$'s satisfy the relation
$$ T\_{p^{n+1}} = T\_p T\_{p^n} - p T\_{p^{n-1}} $$
(e.g., Bump Prop 4.6.4). This gives you a recursion relation among Hecke eigenvalues of $\phi$. E.g., $T\_{p^2} = T\_p T\_p - p T\_1$ says if the eigenvalue for $T\_p$ i... | 6 | https://mathoverflow.net/users/6518 | 287761 | 126,991 |
https://mathoverflow.net/questions/287669 | 4 | Let $G$ be an arbitrary graph on $n$ vertices and $\mathcal L$ be its Laplacian.
I need to show that
\begin{equation}\tag{$\*$}
\langle \mathcal Lx,\mathcal L(|x|^{p-2}x)\rangle\_{\mathbb R^n}\ge 0\qquad \hbox{for all }x\in \mathbb R^n
\end{equation}
and small $p\in (2,\infty)$ (my guess is that $p\approx 3$ will do),... | https://mathoverflow.net/users/26039 | An elementary inequality for graph Laplacians | $2<p<p(G)$ with $p(G)>2$ is, indeed, possible. Moreover, it is Analysis 1. Calculus 1 is an extremely difficult subject, never try to learn it yourself or to explain it to your students. It is way above a normal person's head! (It is second in difficulty only to Business Calculus, which is just beyond the comprehension... | 2 | https://mathoverflow.net/users/1131 | 287765 | 126,992 |
https://mathoverflow.net/questions/287770 | 5 | There is a statement in the literature (see the paragraph between equations (18) and (19) in <http://aip.scitation.org/doi/10.1063/1.523863>), which I would like to generalise, but I don't have a nice proof of the original claim. The statement is the following:
>
> Given any symmetric tensor field $T\_{ab}$ on a 3-... | https://mathoverflow.net/users/118177 | Existence of second order potential for PDE | First of all, you have a sign wrong in your formula for the curvature. The curvature tensor you gave has positive constant sectional curvature +1 while you claim that you want negative sectional curvature (i.e., hyperbolic space), which would flip the sign of $R$. Second, when you speak of spherical harmonics, I believ... | 7 | https://mathoverflow.net/users/13972 | 287772 | 126,995 |
https://mathoverflow.net/questions/287733 | 1 | Is there a nontrivial abelian category $C$ such that both $C$ and $C^{op}$ satisfy AB3-AB5? ("nontrivial" means that there are nonzero objects in $C$)
| https://mathoverflow.net/users/49822 | Abelian category which is nice and its dual is also | The answer is no.
This is mentioned (with a hint for a proof) on page 129 of A. Grothendieck, *Sur quelques points d'algèbre homologique,* Tohoku Math. J. (2) 9, (1957), 119-221.
For a complete proof (albeit in somewhat different terminology) see Proposition III.1.10 in B. Mitchell, *Theory of categories,* Academic... | 4 | https://mathoverflow.net/users/11025 | 287778 | 126,998 |
https://mathoverflow.net/questions/287361 | 14 | I was wondering if anyone could give me some intuition as to why, for a smooth projective variety $X$ over $\mathbf{C}$ of complex dimension $d$, the Tate twist on $H^n(X(\mathbf{C}),\mathbf{Z})$ to be incorporated to have a pairing with $H\_{d-n}(X(\mathbf{C}),\mathbf{Z})$ into $\mathbf{Z}(-d)$, is related to (tensor ... | https://mathoverflow.net/users/nan | Tate twists and cohomology of $\mathbf{P}^1$ | The Tate twist is what we need to express Poincaré duality without making any choice. Such a choice appears in the choice of an orientation of the affine line minus the origin, and have shadows in the description of the Thom isomorphism, hence, in the description/construction of Gysin maps, trace maps, and so forth. A ... | 15 | https://mathoverflow.net/users/1017 | 287783 | 126,999 |
https://mathoverflow.net/questions/287779 | 2 | Let $V$ be an infinite dimensional vector space, $A$ its algebra of linear endomorphisms (also known as a full matrix ring). The algebra $A$ is primitive because $V$ is a faithful simple (left) $A$-module.
**Question 1:** does $A$ admit a non-isomorphic to $V$ faithful simple module?
I understand that $A$ has many ... | https://mathoverflow.net/users/5301 | Nontrivial primitivity of full matrix ring | Question 1:
The subset of transformations with finite dimensional images forms the (left and right) socle of this ring, a nontrivial two-sided ideal. The socle is the intersection of all essential right ideals, so the essential maximal right ideals cannot provide simple modules which are faithful.
You're left with... | 1 | https://mathoverflow.net/users/19965 | 287784 | 127,000 |
https://mathoverflow.net/questions/287550 | 7 | Let $X$ be a Banach space. By the weak operator topology on $B(X)$, we mean the locally convex topology implemented by the following semi-norms:
$$B(X)\to[0,\infty) : T\to|\langle Tx,x^\*\rangle|$$
where $x\in X$ and $x^\*\in X^\*$.
Q. True or false: Assume that with respect to the weak operator topology, the unit b... | https://mathoverflow.net/users/84390 | An equivalent condition for separability of $X^*$ | I *think* this works. Suppose $B(X)$ with WOT is second countable, so there is a countable base to the topology $(U\_n)$. This means that if $(T\_i)$ is a net in $B(X)$, then $T\_i\rightarrow 0$ WOT if and only if, for each $n$ with $0\in U\_n$, there is $i\_n$ so that $i\geq i\_n \implies T\_i\in U\_n$.
Each WOT ope... | 3 | https://mathoverflow.net/users/406 | 287786 | 127,001 |
https://mathoverflow.net/questions/287762 | 3 | Recently, I read a [paper](https://dml.cz/bitstream/handle/10338.dmlcz/106296/CommentatMathUnivCarol_025-1984-2_5.pdf) in which the author construct a space $X$ which is dense in a $\sigma$-product $S$ of closed unit intervals. The space $X$ is CCC (denotes countable chain condition); it is also, clearly Tychonoff (see... | https://mathoverflow.net/users/39873 | How could I see quickly that this space is not normal? | **Theorem.** A dense subspace $X$ of the Tychonoff cube $[0,1]^\kappa$ is not normal if $X$ contains an uncountable closed discrete subspace $D$ such that each $x\in D$ has finite support $\mathrm{supp}(x)=\{i\in\kappa:x(i)\ne0\}$.
*Proof.* By the $\Delta$-lemma, the uncountable set $D$ contains an uncountable subset... | 5 | https://mathoverflow.net/users/61536 | 287796 | 127,006 |
https://mathoverflow.net/questions/285935 | 6 | The Rock-paper-scissors flow is the following reaction-diffusion system
$$r\_t = \Delta r + rs-rp,$$
$$p\_t = \Delta p + pr-ps,$$
$$s\_t = \Delta s + sp-sr.$$
We can assume $r,p,s\geq 0$, $r+p+s$ is constant, and $1=\int\_M (r+p+s) dV$.
Are there travelling wave solutions in one spacial variable of the form
... | https://mathoverflow.net/users/58787 | Pursuit solutions to the Rock-paper-scissors flow and delay differential equations | Damn it! Such a nice equation with such an interesting behavior that I just started to understand after a few days of thinking, and the stupid positivity condition spoils all the fun. Why cannot we just trade this world for one in which burning a negative amount of fuel in a negative amount of oxidizer would produce a ... | 4 | https://mathoverflow.net/users/1131 | 287801 | 127,009 |
https://mathoverflow.net/questions/287785 | 33 | Let us consider the following conjecture:
**Conjecture**: There are no integer solutions of the equation $$x^{y-z}z^{x-y}=y^{x-z}$$ with $x,y,z$ distinct positive integers greater than or equal to $2$.
I came across this result when studying some diophantine equations. Several attempts were made to find a solution,... | https://mathoverflow.net/users/74668 | About the validity of a new conjecture about a diophantine equation | The conjecture is true, in fact the equation has no solution in distinct positive real numbers. To see this, let us write the equation in the more symmetric form
$$ x^y y^z z^x = x^z y^x z^y. \tag{$\ast$}$$
We get the same equation after interchanging $x$ and $y$, or $y$ and $z$, i.e., after permuting the variables arb... | 87 | https://mathoverflow.net/users/11919 | 287803 | 127,010 |
https://mathoverflow.net/questions/287788 | 3 | While computing conditional expectations of certain functionals of a Poisson white noise field (details are long and probably irrelevant), I've stumbled upon the need to use the following identity involving Stirling numbers of the second kind:
$$
\ell{k\brace \ell} = \sum\_{j=\ell}^k {k\choose j-1} (-1)^{k-j} {j\brace ... | https://mathoverflow.net/users/14756 | Proof of identity involving Stirling numbers of the second kind | You can give a short proof by interpreting the identity as an instance of inclusion-exclusion. The left hand side counts the number of ways of partitioning $S=\{1,2,\dots,k\}$ into $\ell$ parts and then picking one of the parts as the *designated* one.
Let $A\_i$ denote the number of partitions of $S$ into $\ell$ par... | 6 | https://mathoverflow.net/users/2384 | 287808 | 127,014 |
https://mathoverflow.net/questions/287812 | 5 | Let $F$ be a $p$-adic field. Let $G$ be a connected reductive group and $\rho$ an irreducible admissible representation of $G(F)$. Let $P$ be a parabolic subgroup of $G$ and suppose further that $\rho$ is essentially square integrable. It seems like the Jacquet module $\mathrm{Jac}\_P(\rho)$ is essentially square integ... | https://mathoverflow.net/users/97316 | Jacquet Module of an Essentially Square Integrable Representation | Yes, your claim is true.
It follows from proposition 43 of the [Bernstein notes](http://www.math.harvard.edu/~gaitsgde/Jerusalem_2010/GradStudentSeminar/p-adic.pdf) on representation of $p$-adic groups.
That proposition gives a characterization of essentially square integrable representations (called square integrabl... | 3 | https://mathoverflow.net/users/101091 | 287822 | 127,019 |
https://mathoverflow.net/questions/287819 | 4 | I am looking for an example of a prime, p, for which there exists two $p$-ordinary rational elliptic curves $E$, $F$ for which, at every prime $l$ not dividing $N=p \operatorname{Cond}(E) \operatorname{Cond}(F)$:
$$\#E(\mathbb{F}\_l) \equiv \#F( \mathbb{F}\_l) \mod p $$
Such a congruence could be detected by a Hida... | https://mathoverflow.net/users/47195 | Example of two p-Ordinary Elliptic Curves congruent to each other | There are many such curves. For $p = 2$, one can just take two curves $E$ and $F$ with $E : y^{2} = f(x)$ and $F : y^{2} = g(x)$ where $f(x)$ and $g(x)$ define the same number field. Also, if $E$ and $F$ are isogenous, then the above statement is true for any $p$ (but the corresponding modular forms are the same then).... | 6 | https://mathoverflow.net/users/48142 | 287827 | 127,022 |
https://mathoverflow.net/questions/287834 | 10 | Is there a compact Riemann surface other than the sphere with an atlas consisting of open subsets biholomorphic to $\mathbb{C}$? Is there a compact Riemann surface other than the sphere which possesses an open subset biholomorphic to $\mathbb{C}$?
| https://mathoverflow.net/users/36688 | Riemann surfaces with an atlas all of whose open sets are biholomorphic to $\mathbb{C}$? | Using the uniformization theorem, we can prove that the only compact connected Riemann surface admitting an open set biholomorphic to $\mathbb{C}$ is the Riemann sphere $\mathbb{P}^1(\mathbb{C})$. This answers negatively both questions of the OP.
Indeed, let $X$ be a compact connected Riemann surface with an open set... | 19 | https://mathoverflow.net/users/6506 | 287838 | 127,024 |
https://mathoverflow.net/questions/287833 | 10 | Let $(E\_b)\_{b\in B}$ be a family of vector bundles on a smooth projective variety $X$, parameterized by a smooth curve $B$. Let $\mathrm{o}\in B$. Assume that $E\_b$ is decomposable (= direct sum of two lower rank bundles) for $b\neq \mathrm{o}$. Can we conclude that $E\_{\mathrm{o}}$ is decomposable? If not, what wo... | https://mathoverflow.net/users/40297 | Limit of decomposable bundles | What ulrich writes is correct. I will delete this answer if ulrich posts his answer. For me, the fastest way to think about this is via smoothness and irreducibility of the stack of locally free sheaves of specified rank and degree on a curve $X$.
For a fixed smooth, projective curve $X$ over a field $k$, there is a... | 10 | https://mathoverflow.net/users/13265 | 287842 | 127,025 |
https://mathoverflow.net/questions/287843 | 0 | This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.
Suppose that $\lambda\_1 \geq \lambda\_2 \geq \dots \lambda\_k$ and $\rho$ be the irreducible representation of $S\_n$ associated with Young tableau $(\lambda\_1,\dots,\lambda\_k)$ where $n=\sum ... | https://mathoverflow.net/users/18785 | Dimension of irreducible representation associated to a Young tableau | This is a classical application of Frobenius formula and Vandermonde determinant. We have $$\dim V\_{\lambda}=\frac{n!}{l\_1! \cdots l\_k!} \prod\_{i<j}(l\_i-l\_j),$$
where $l\_i=\lambda\_i+k-i$. See Section 4.1 of
W. Fulton, J. Harris: *Representation Theory* (*a first course*), GTM **129** (1991).
| 8 | https://mathoverflow.net/users/7460 | 287844 | 127,026 |
https://mathoverflow.net/questions/287847 | 1 | I'm working on a special kind of graphs. To prove some uniqueness, I need to prove that the polynomial
\begin{equation}
x^{8}-7x^{6}+14x^{4}-8x^{2}+1
\end{equation}
does not have any root of the form
\begin{equation}
2\cos\frac{(2k+1)\pi}{2n} \quad k\in \lbrace 0,1,\cdots , n-1 \rbrace , n \in \mathbb{E}.
\end{equation... | https://mathoverflow.net/users/111007 | How to prove the following polynomial does not have root of a special form? | The degree of the minimal polynomial of your cos is bounded below by phi(n)/2 since it generates a real field inside the cyclotomic field. If phi(n)/2 is bigger than 8 you are done. For small n you can check one by one with computer (or write the minimal polynomial and see it doesn’t divide that polynomial or compute d... | 4 | https://mathoverflow.net/users/2042 | 287848 | 127,027 |
https://mathoverflow.net/questions/287866 | 3 | This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to [this observation](https://mathoverflow.net/questions/287794/noncentral-matrix-beta-distributions-of-type-i-and-ii), but I open a new question because yet I'm not sure it is exactly the same).
Let $... | https://mathoverflow.net/users/21339 | integral involving hypergeometric function of matrix argument | The zonal polynomial $C\_\kappa$ satisfies $C\_\kappa(SR) = C\_\kappa(S^\frac12 R S^\frac12)$, for $S$ symmetric positive definite and $R$ symmetric, according to (1.5.3) in Gupta & Nagar's book. This implies that the same relation holds for the hypergeometric functions, in view of their definition in terms of the zona... | 1 | https://mathoverflow.net/users/21339 | 287877 | 127,035 |
https://mathoverflow.net/questions/287876 | 2 | Let $A=(a\_{ij})$ be an infinite doubly stochastic matrix. Is the following statement true ?
$$
\lim\_{n\to\infty}\frac{1}{n}\sum\_{i=1}^n\sum\_{j=1}^na\_{ij} >0
$$
Any reference or comment on this is much appreciated.
| https://mathoverflow.net/users/118240 | A question on the partial sum of infinite doubly stochastic matrix | The answer is no. Indeed, let
\begin{equation\*}
s\_n:=\sum\_{i=1}^n\sum\_{j=1}^na\_{ij},\quad b\_n:=\frac{s\_n}{n}=\frac{1}{n}\sum\_{i=1}^n\sum\_{j=1}^na\_{ij}.
\end{equation\*}
Then $0\le s\_n\le\sum\_{i=1}^n\sum\_{j=1}^\infty a\_{ij}=n$, and hence we always have
\begin{equation\*}
0\le\liminf\_{n\to\infty}b\_n\... | 6 | https://mathoverflow.net/users/36721 | 287880 | 127,037 |
https://mathoverflow.net/questions/287873 | 6 | I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then in a paper, which I read, it is used that $K\_{\upsilon}|K$ is separable, where $K\_{\upsilon}|K$ is the completion. Thi... | https://mathoverflow.net/users/118130 | Why is $K_{\upsilon}|K$ separable for a global field $K$? | By definition an extension of fields $K'/K$ is separable when $K' \otimes\_K F$ is reduced for all field extensions $F/K$, and by limit considerations it is the same to say that all finitely generated subextensions are separable. But for finitely generated extensions, separability can be tested via scalar extension aga... | 5 | https://mathoverflow.net/users/81332 | 287884 | 127,039 |
https://mathoverflow.net/questions/246985 | 3 | Recall that an bounded operator $T$ on a Hilbert space $\mathcal H$ is said to be $n$-hypercontraction for $n\in\mathbb N$ if
$$ I- {n \choose 1} T^\*T + {n \choose 2} {T^\*}^2T^2-\cdots+ (-1)^{n}{n \choose n}{T^\*}^nT^n \geq 0.$$
Let $T$ be an bounded operator on a Hilbert space $\mathcal H$ such that its spectrum $\... | https://mathoverflow.net/users/82713 | Hypercontractions and automorphisms of the unit disc | Over a year has passed since the original question was posted. Hopefully the OP is still interested in the answer.
By the recursive formula for binomial coefficients one has
\begin{align\*}
\left(\sum\_{k=0}^n (-1)^k{n\choose k}T^{\*k}T^k\right) + T^\*\left(\sum\_{k=0}^n (-1)^k{n\choose k}T^{\*k}T^k\right)T = \sum\_{... | 1 | https://mathoverflow.net/users/76593 | 287893 | 127,043 |
https://mathoverflow.net/questions/287889 | 5 | If $f$ is entire and we assume a decay rate on $\mathbb{R}^+$, does this dictate a growth rate on $\mathbb{C}$?
For example, is there a function that decays as $e^{-x^2}$ on $\mathbb{R}^+$ and is of exponential type?
What I am actually interested in is whether there exists an entire function such that $|f(z)|\le Me... | https://mathoverflow.net/users/18812 | Does the rate of decay of an entire function dictate the global growth rate? | This is a typical walk-to-the-library problem. I used Boas, but probably other standard books would have worked too.
Boas proves the following results: (1) if $f$ is of order $1$, then $\limsup m(r)M(r)^{1+\epsilon}=\infty$ (Theorem 3.3.1); (2) if $f$ is of order $0<\rho<1$, then we obtain $m(r\_n)\gtrsim M(r\_n)^{-a... | 7 | https://mathoverflow.net/users/48839 | 287894 | 127,044 |
https://mathoverflow.net/questions/286366 | 14 | In [this paper](http://www.ams.org/journals/proc/1953-004-03/S0002-9939-1953-0060806-9/S0002-9939-1953-0060806-9.pdf) R.H. Bing has constructed his famous example of a countable connected Hausdorff space.
The Bing space $\mathbb B$ is the rational half-plane $\{(x,y)\in\mathbb Q\times \mathbb Q:y\ge 0\}$ endowed wit... | https://mathoverflow.net/users/61536 | Is Bing's countable connected space topologically homogeneous? | The Bing space is topologically homogeneous.
The proof of this fact can be found [here](https://arxiv.org/abs/1712.01964). It is a bit long (to be reproduced here) and uses the standard back-and-forth argument.
| 6 | https://mathoverflow.net/users/61536 | 287897 | 127,046 |
https://mathoverflow.net/questions/287911 | 2 | Let $G$ be a finite group, and let $M(G)=H\_2(G,\mathbb{Z})$ be its Schur multiplier. Are there any known bounds on the number of generators of $M(G)$ in terms of $G$? For example, if $G$ is abelian of rank $r$, then $M(G)$ can be generated by $r(r-1)/2$ elements. Is there anything that can be said for more general fin... | https://mathoverflow.net/users/41644 | Number of generators for the Schur multiplier of a finite group | If a finite group has a presentation with $g$ generators and $r$ relations, then the Schur multiplier is generated by $r-g$ elements. There's been lots of work studying groups and presentations where this is minimal ("efficient" presentations and groups).
| 7 | https://mathoverflow.net/users/22989 | 287913 | 127,051 |
https://mathoverflow.net/questions/287587 | 4 | In the article <http://www.sciencedirect.com/science/article/pii/S0021869301991306?via%3Dihub> (see also the MO thread [Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\infty$?](https://mathoverflow.net/questions/282441/why-does-there-exist-a-non-split-sequence-with-the-condition-that-m... | https://mathoverflow.net/users/61949 | Property of non-Gorenstein algebras | I think I found a proof that in a general Artin algebra one has that the projective dimension of $D(A)$ is equal to the Gorenstein projective dimension of $D(A)$. The proof is simple in case I made no mistake. But it relies on something which I expected to be wrong so I have not thought about this at the beginning.
A... | 2 | https://mathoverflow.net/users/61949 | 287914 | 127,052 |
https://mathoverflow.net/questions/287910 | 1 | Let $G$ be a finitely generated group, then its action groupoid $BG$ is a simplicial set. In fact $BG$ is the nerve of a groupoid where the set of objects is given by a point $\*$ and the set of maps is given by $G$. Now assume that $G=F\_{2}$, i.e. the free group on two generators $a$ and $b$. I would like to construc... | https://mathoverflow.net/users/41970 | A groupoid which is homotopy equivalent to $BG$ | Let $X = BG \times [0; 1]$, then $\ast\_a = (\ast; 0)$, $\ast\_b = (\ast; 1)$, $\gamma$ is the image of $[0;1]$, $f\_a$ is any choice of path in the class of $a$ and $f\_b$ is any choice of path in class $b$ conjugated by $\gamma$, i.e. $\gamma \bullet b \bullet \gamma^{-1}$.
Equivalently, this is the action groupoid... | 4 | https://mathoverflow.net/users/10605 | 287915 | 127,053 |
https://mathoverflow.net/questions/287888 | 10 | For an uncountable cardinal $\kappa$, we are interested in the least size of a cofinal subset of the partial order $([\kappa]^\omega, \subseteq)$. It is obvious that this cofinality is at least $\kappa$ and a rather simple induction shows that $cof([\aleph\_n]^\omega, \subseteq)=\aleph\_n$ for each natural number $n \g... | https://mathoverflow.net/users/17836 | What is the cofinality of $([\kappa]^\omega, \subseteq)$? | I think he means the consistency of "there exists a cardinal $\kappa$ with uncountable cofinality so that $cf([\kappa]^\omega, \subseteq) > \kappa$" requires large cardinal axioms.
To motivate this, note that for example if $0^\sharp$ does not exist, then for any cardinal $\kappa > 2^{\aleph\_0}$ of uncountable cofin... | 10 | https://mathoverflow.net/users/11115 | 287918 | 127,054 |
https://mathoverflow.net/questions/287908 | 8 | Let $A=(a\_{ij})$ be an infinite doubly stochastic matrix. Does there necessarily exist a subsequence $\{n\_k\}\_{k=1}^\infty$ such that
$$ \lim\_{k\to\infty}\frac{1}{n\_k}\sum\_{i=1}^{n\_k}\sum\_{j=1}^{n\_k}a\_{ij} >0?$$
In a previous post [A question on the partial sum of infinite doubly stochastic matrix](https:/... | https://mathoverflow.net/users/118240 | On the limit of partial sum of infinite doubly stochastic matrix | No. Enumerate all positive integers which are not powers of $2$: $3=n\_1<n\_2<n\_3<\dots$ and partition positive integers into two-element sets $\{n\_k,2^{k-1}\}$. Let $a\_{i,j}=1$ if the set $\{i,j\}$ is such a two-element set, and let $a\_{i,j}=0$ otherwise. We get a symmetric bistochastic matrix, and $1$'s are only ... | 11 | https://mathoverflow.net/users/4312 | 287922 | 127,056 |
https://mathoverflow.net/questions/287902 | 5 | Let $\mathbf{G}$ be a connected reductive group over $\mathbb{F}\_q$, let $G=\mathbf{G}(\mathbb{F}\_q)$, and let $k$ be a large enough finite extension of $\mathbb{F}\_q$.
If $\mathbf{G}=\mathrm{SL}\_2$, then in page-109 of Bonnafe's [book](https://www.springer.com/us/book/9780857291561) it is said that "It turns ou... | https://mathoverflow.net/users/56217 | The defining characteristic representations of Lie type groups | This result is false for a general connected reductive algebraic group $\mathbf{G}$ but is true if $\mathbf{G}$ is simple and simply connected. This was proved by Steinberg in Theorem 1.3 of the following paper
* R. Steinberg, *[Representations of algebraic groups](https://projecteuclid.org/euclid.nmj/1118801156)*, N... | 7 | https://mathoverflow.net/users/22846 | 287938 | 127,062 |
https://mathoverflow.net/questions/287925 | 5 | I have a countable sequence of finite suspension spectra $X\_i$, whose $BP$-homology is a $BP\_\*(BP)$-comodule. Let's assume $BP\_\*(X\_i) = \Sigma^{d\_i} BP\_\* / (v\_0^{k\_0}, \dots v\_i^{k\_i}),$ for some $d\_n$ big enough that makes them being suspension spectra.
Then, if we let $X = \bigvee\_i X\_i$,
$$BP\_\*... | https://mathoverflow.net/users/93775 | Homology of a limit of spectra + Cofiber | So far as I can tell, it depends.
The map $\oplus BP\_\*(X\_i) \to \prod BP\_\*(X\_i)$ is injective and the cokernel $M$ is $BP\_\*$ of the cofiber. In particular, for any $N$ we have an isomorphism
$$
M \cong \prod\_{i \geq 0} BP\_\*(X\_i) \Big/ \bigoplus\_{i \geq 0} BP\_\*(X\_i) \cong \prod\_{i \geq N} BP\_\*(X\_i)... | 5 | https://mathoverflow.net/users/360 | 287940 | 127,063 |
https://mathoverflow.net/questions/287398 | 7 | While reading Fulton's Intersection theory, I came across the following comment.
Let $X$ be a projective scheme over an algebraically closed field. Assume we have been given a map $g : \mathbb{P}^1 \rightarrow S^nX$. Then this map factors though $S^nC$, where $C$ is a smooth curve with a proper map $C \rightarrow X$ ... | https://mathoverflow.net/users/58056 | Family of zero dimensional subschemes | Let me try the following argument.
First, as you explain one can reduce the general case to the case when $X$ is a curve $C$, possibly singular and reducible. This is because the preimage of $g(\mathbb{P}^1)$ under the quotient morphism $\phi: X^n \to S^n(X)$ has dimension dimension one (or zero if $g$ is constant),... | 1 | https://mathoverflow.net/users/111491 | 287942 | 127,064 |
https://mathoverflow.net/questions/287945 | 9 | In my research I came upon a recursively defined sequence, and I'm pretty sure it converges to $\sqrt{2}$ though I can't prove it easily. I don't think it is a difficult question but I'm not sure.
Consider the following sequence of functions over $\mathbb{R}$, where it makes sense:
$f\_0(x)=0$,
$\displaystyle f\_... | https://mathoverflow.net/users/88188 | Maximum zero converges to $\sqrt{2}$ | Modify the function so the equation becomes $f\_n(x)=0$. That is
$$f\_0(x)=\frac{1}{2 x}-x \quad\quad\quad f\_n(x)=\frac{1}{2 \left(\frac{1}{2 x}-f\_{n-1}(x)\right)}-x+\frac{1}{2 x}$$
Multiply both side of the equation by $-U\_n(\sqrt{1/2}x)x$
$$f\_0(x)=-U\_0(\sqrt{1/2}x)x\left(\frac{1}{2 x}-x\right) \quad\quad\q... | 3 | https://mathoverflow.net/users/68622 | 287953 | 127,068 |
https://mathoverflow.net/questions/287941 | 2 | Let $\mathcal{I}^3\subset\mathbb{R}^4$
be the standard hyperboloid model for hyperbolic $3$-space and consider the usual $\mathrm{SO}(3,1)$
action of $\mathrm{PSL}(2,\mathbb{C})$
on $\mathcal{I}^3$.
Given a matrix
$\gamma=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}\in\mathrm{PSL}(2,\mathbb{C})$,
there is a unique geode... | https://mathoverflow.net/users/14835 | Find the fixed geodesic of an orientation-preserving isometry of the $3D$ hyperboloid model | The fixed points in the upper half space model map to vectors on the lightcone, which span the plane that intersects with the $\mathcal{I}^3$ at $g$. The details to compute this are [here](https://math.stackexchange.com/questions/499042/how-to-convert-between-the-hyperboloid-model-and-the-poincare-patch-for-mathbb).
| 2 | https://mathoverflow.net/users/11142 | 287960 | 127,072 |
https://mathoverflow.net/questions/287947 | 6 | For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of the cosines of rational multiples of $\pi$?
Products of linear combinations of cosines of rational multiples of $\pi$... | https://mathoverflow.net/users/3319 | Is every square root of an integer a linear combination of cosines of $\pi$-rational angles? | Someone should actually record the formula. If $p$ is a prime $\equiv 1 \bmod 4$, then
$$\sqrt{p} = \sum\_{k=1}^{p-1} \left( \frac{k}{p} \right) \cos \frac{2 k \pi}{p}$$
where $\left( \tfrac{k}{p} \right)$ is the quadratic residue symbol. Note that $\left( \tfrac{k}{p} \right) = \left( \tfrac{p-k}{p} \right)$, so ever... | 23 | https://mathoverflow.net/users/297 | 287963 | 127,074 |
https://mathoverflow.net/questions/287957 | 8 | Let $S$ be a finitely generated free abelian semigroup (or monoid), and let $T \subset S$ be a sub-semigroup (sub-monoid). Does the [Nielsen-Schreier theorem](https://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier_theorem) hold in this case, that is, will $S$ still be free? Moreover, will $S$ be finitely generated as i... | https://mathoverflow.net/users/90430 | Nielsen-Schreier theorem for monoids | For the question of finite generation, here's a a counterexample that appears in D.B. McAlister and L. O'Carroll,
*Finitely generated commutative semigroups.* Glasgow Math. J. 11 1970 134–151. MR0269765 (42 #4660:
If $S$ is the free commutative semigroup on $a$ and $b$, and $K=\{a^nb^m\mid n,m>0\}$, then $K$ is not f... | 14 | https://mathoverflow.net/users/3959 | 287971 | 127,077 |
https://mathoverflow.net/questions/287794 | 1 | In Gupta & Nagar's book *Matrix variate distributions*, the noncentral Beta type I(B) distribution with parameters $a$, $b$ and noncentrality parameter $\Theta$ is defined by $U={(S\_1+S\_2)}^{-\frac12}S\_1{(S\_1+S\_2)}^{-\frac12}$ where $S\_1 \sim W(2a, I, \Theta)$ is independent of $S\_2\sim W(2b, I)$ (Wishart distri... | https://mathoverflow.net/users/21339 | Noncentral matrix beta distributions of type I and II | The density of $U$ is (see Gupta & Nagar page 189)
$$
f(U) = C\_f {\det(U)}^{a - \frac12(p+1)} {\det(I\_p-U)}^{b - \frac12(p+1)} \\
\int\_{S>0} {\det(S)}^{a+b-\frac12(p+1)}
\exp\left(-\mathrm{tr}\left(\frac{S}{2}\right)\right)
{}\_0\!F\_1\left(a, \frac{1}{4}\Theta\_1 S^\frac12 US^\frac12\right) \mathrm{d}S \\
= K\_f ... | 0 | https://mathoverflow.net/users/21339 | 287973 | 127,078 |
https://mathoverflow.net/questions/287977 | 7 | Given a model $V$ of set theory and an inner model $W \subseteq V$, a real $x \in V \cap \omega^\omega$ is *infinitely often equal over* $W$ if for each real $y \in \omega^\omega$ there are infinitely $n$ with $x(n) = y(n)$. Cohen reals have this property and it was an open question for a while as to whether every forc... | https://mathoverflow.net/users/114946 | Does Laver Forcing add an infinitely often equal real? | The answer is no.
This follows easily from the Laver property. Every new real which is bounded by a ground model real (say: by the identity function) is contained in a small slalom from the ground model, hence will be eventually different from lots of ground model reals.
| 9 | https://mathoverflow.net/users/14915 | 287980 | 127,080 |
https://mathoverflow.net/questions/287982 | 2 | I have read [(In French)](http://les.mathematiques.free.fr/pdf/chap10_11_12_13.pdf)that the differential of a function depends on the topology and not the norm, the latter is rather easy to grasp, the first is hard for me to construct.
Norms being equivalent in finite dimensional spaces, I am looking for an example (... | https://mathoverflow.net/users/3898 | Differentials in different topologies | Let $V:=\mathcal S(\mathbb R)$, the space of smooth functions with rapid decay.
Let $\gamma:\mathbb R\to V$ be the map which sends $t$ to the function $x\mapsto e^{-(x-t)^2}$.
Then $\gamma$ is differentiable.
Pick a Hamel basis of $V$ which includes all the elements $\gamma(t)$, and which also includes all the ... | 6 | https://mathoverflow.net/users/5690 | 287984 | 127,082 |
https://mathoverflow.net/questions/286327 | 11 | ((In conclusion)
It was hard to choose which answer to accept. I decided for the one which addressed most of the various aspects of the question.
)
(Later addon)
I now decided to put a bounty on this, and want to put particular accent on one aspect of the question that puzzles me most.
As discussed in greater... | https://mathoverflow.net/users/41291 | Can Homotopy Type Theory or algebraic geometry deal with homotopy fibers in terms of families? | Yes, there is a certain sense in which your statements are true. As Mike Shulman and Qiaochu Yuan said, the strict fiber of a map cannot be defined in HoTT and doesn't make sense, but you can work from the other direction: instead of trying to associate a family of types to a map start with an arbitrary type family $P:... | 6 | https://mathoverflow.net/users/10605 | 287995 | 127,086 |
https://mathoverflow.net/questions/288000 | 10 | Consider for $X = 1,2, \ldots$ the set $\mathcal{S}\_X$ of trigonometric polynomials $f(t) := \sum\_{|k| \leq X} c\_k e^{2\pi i kt}$ on the circle $\mathbb{T} := \mathbb{R}/\mathbb{Z}$ of degree $\leq X$ (Fourier transform supported on $\{-X,\ldots, X\}$), such that $f(0) = 1$ and $c\_0 = 0$ (the last assumption is pro... | https://mathoverflow.net/users/26522 | An extremal problem related either to an uncertainty principle on the circle, or else to the prime number theorem | A compactness argument shows that for sufficiently large $X$ one has the bound
$$ \sup\_{x \in {\mathbb T} \backslash [-1/X,1/X]} |f(x)| \gg \sup\_{x \in [-1/X,1/X]} |f(x)|$$
whenever $f$ is a trigonometric polynomial of degree at most $X$; this would imply that $B\_X \gg X$. (Perhaps there is a normalising factor of $... | 11 | https://mathoverflow.net/users/766 | 288003 | 127,089 |
https://mathoverflow.net/questions/288004 | 1 | Let $(H,||\cdot||\_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||\_K$, which satisfies
$$||x||\_H\leq ||x||\_K$$
for all $x\in K$. Let $(S,\mu)$ be a measure space and $f:S\rightarrow K$ a strongly measurable function in the sense... | https://mathoverflow.net/users/78729 | Bochner integrability within a subspace | The Bochner integral is a red herring here. Let $S = \mathbb{N}$ and let $\mu$ be counting measure. Then a "strongly measurable function" is just a sequence, and it is "Bochner integrable" iff it is absolutely summable (in $H,K$ respectively).
If the $H$ and $K$ norms are not equivalent (which is necessarily the cas... | 3 | https://mathoverflow.net/users/4832 | 288010 | 127,090 |
https://mathoverflow.net/questions/288017 | -2 | A number $\alpha$ is said to satisfy the Diophantine condition with
exponent $\beta$ iff for some constant $C>0$ the estimate
$$
\left| \alpha - \frac{p}{q} \right| > \frac{C}{q^{2+\beta}}
$$
holds for every rational fraction $p/q \in \mathbb{Q}$.
A question:
1. Is it true that if $\alpha$ satisfies the Diophantine... | https://mathoverflow.net/users/14009 | Inverse to a Diophantine number | Of course, with the same $\beta$. Note that in the definition we may restrict only to the fractions $p/q$ satisfying inequality $|\alpha-p/q|<|\alpha|/2$. Next, $$\left|\frac1{\alpha}-\frac qp\right|=\left|\alpha-\frac pq\right|\cdot|\alpha^{-1}|\cdot \left|\frac qp\right|.$$
If $|\alpha^{-1}-q/p|<|\alpha^{-1}|/2$, the... | 4 | https://mathoverflow.net/users/4312 | 288018 | 127,093 |
https://mathoverflow.net/questions/288008 | 3 | May I ask what is the basic motivation behind studying bounded cohomology?
Is there a simple reason why bounded cohomology is more interesting / useful than usual cohomology?
Also, is bounded cohomology only interesting for infinite spaces (with infinite simplices/singular simplices)? From the definition, it seems ... | https://mathoverflow.net/users/83274 | Bounded Cohomology Motivation | The canonical homomorphism from bounded cohomology to cohomology has often nontrivial kernel. For example it is infinite-dimensional (in second cohomology) for non-elementary hyperbolic groups. So invariants that are trivial in cohomology can be nontrivial in bounded cohomology.
An example: group actions on the circl... | 5 | https://mathoverflow.net/users/39082 | 288022 | 127,095 |
https://mathoverflow.net/questions/288006 | 0 | A b-metric is defined similar to a metric in which the triangle inequality is replaced by the inequality
$$d(x,z)\leq s\Big[d(x,y)+d(x,z)\Big]\quad\forall\ x,y,z$$
where $s\geq1$.
There is an example of a b-metric which is not continuous.
My question: If the b-metric space is complete, can we conclude that the b-me... | https://mathoverflow.net/users/111987 | continuity of b-metric | Nope. Let $X$ be the set $\{0\} \cup \{\frac{1}{n}: n \in \mathbb{N}\} \subset \mathbb{R}$ together with one additional point $e$. The distance between two points neither of which is $e$ is just their usual distance in $\mathbb{R}$. Also set $d(e,e) = 0$, $d(e,0) = 2$, and $d(e,\frac{1}{n}) = 1$ for all $n$. This is a ... | 2 | https://mathoverflow.net/users/23141 | 288039 | 127,101 |
https://mathoverflow.net/questions/287951 | 2 | Given any Euclidean triangle of vertexes $v\_1,v\_2,v\_3$ and any point $P$ in the interior of the triangle, consider the following system of 6 real equations in $c\_{12},c\_{13},c\_{23},d\_1,d\_2,d\_3$:
$$c\_{12}(v\_1-v\_2)+c\_{13}(v\_1-v\_3)=d\_1(v\_1-P)$$
$$c\_{12}(v\_2-v\_1)+c\_{23}(v\_2-v\_3)=d\_2(v\_2-P)$$
$$c\... | https://mathoverflow.net/users/nan | Question on strictly positive solutions of a system involving an Euclidean triangle | Ok, here goes.
WLOG, $P=0$. Then there exist positive $a\_k$ such that $\sum\_k a\_kv\_k=0$ (this is just a restatement of the condition that the origin is inside). We want to find $c\_{km}=c\_{mk}>0$ and $d\_k>0$ such that $\sum\_{m:m\ne k}c\_{km}(v\_k-v\_m)=d\_kv\_k$. Projecting to the hyperplane (line in $\mathbb ... | 2 | https://mathoverflow.net/users/1131 | 288048 | 127,103 |
https://mathoverflow.net/questions/286830 | 9 | In Hamiltonian dynamics and symplectic geometry a *twisted cotangent bundle* is the cotangent space $T^\*N$ of a closed (compact without boundary) $n$-manifold $N$ equipped with a twisted symplectic structure: $T^\*N$ carries the canonical symplectic structure $\omega=d\lambda$, where $\lambda$ is the Liouville 1-form.... | https://mathoverflow.net/users/47228 | What are the compact Lagrangian submanifolds of a twisted cotangent bundle? | Let's begin by pointing out the following: you will not find monotone examples for the simple reason that a nontrivial such deformations creates a class of nonzero symplectic area, while the Chern class is always vanishing. The best you could hope for is Calabi-Yau, and such examples indeed exist. However, I know of no... | 4 | https://mathoverflow.net/users/48067 | 288055 | 127,105 |
https://mathoverflow.net/questions/287343 | 4 | Let $K$ be a simplicial complex: it consists from the set (called the set of vertices) and a family of subsets of set of vertices satisfying the property of being closed under taking subsets (those set are called simplices: condition translates that the subset of a simplex is again simplex). Homology of simplicial comp... | https://mathoverflow.net/users/24078 | Homology of simplicial complex versus homology of simplicial _set_ | There is a problem with your definition of the boundary map $\partial :
C\_q(K)\to C\_{q-1}(K)$. The formula $\partial \_i[v\_0,\cdots,v\_q]=[v\_0,\cdots,v\_{i-1},v\_{i+1},\cdots,v\_q]$ depends on an ordering of the vertices of the simplex $[v\_0,\cdots,v\_q]$, but such an ordering is not part of the definition of a s... | 8 | https://mathoverflow.net/users/23571 | 288063 | 127,108 |
https://mathoverflow.net/questions/288041 | 3 | For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:
1) $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL\_2$
2) $E$ has complex multiplication by a field $k$ and the Mumford-Tate group $E$ is a torus (of dimension two) in $GL\_2$ induced by $k$.
In ... | https://mathoverflow.net/users/nan | Mumford-Tate groups of abelian surfaces | The list of of all possible Hodge (special Mumford-Tate) groups of complex abelian varieties up to dimension 4 (and for simple abelian varieties up to dimension 5) is contained in <https://arxiv.org/pdf/math/9901113.pdf> <https://link.springer.com/article/10.1007%2Fs002080050333> .
| 3 | https://mathoverflow.net/users/9658 | 288064 | 127,109 |
https://mathoverflow.net/questions/287979 | 9 | Let $V=\bigoplus\_{d\in\mathbb N}V(d)$ be a Möbius-covariant vertex algebra with $V(0)=\mathbb C$.
Recall that a vector $v\in V$ is called *quasi-primary* if $L\_1v=0$.
For $v\in V(d)$, we write $Y(v,z)=\sum\_{n\in\mathbb Z} z^{-n-d}v\_{(n)}$.
With that convention, $v\_{(n)}$ is an operator $V(k)\to V(k-n)$.
... | https://mathoverflow.net/users/5690 | Annihilation operators in a vertex algebra | The answer seems to be yes for quasi-primary $v$ if $V$ has a suitable invariant bilinear form. Then one can identify $v\_{(n)} w$ with its pairing with the vacuum, and obtains it as the appropriate coefficient of
$$(\mathbf{1}, Y(v, x)w) = (-x^{-2})^d (Y(v, x^{-1})\mathbf{1}, w) = (-x^{-2})^d (e^{x^{-1} L(-1)} v, w) =... | 4 | https://mathoverflow.net/users/118337 | 288065 | 127,110 |
https://mathoverflow.net/questions/286967 | 5 | **Update**:
The chat room for this question: <http://chat.stackexchange.com/rooms/69953/discussion-between-fedja-and-abiessu>
The problem statement remains unchanged (below).
---
Consider the following problem:
>
> Given the following sets with $u \in\Bbb Z^+$: $$\begin{align}A\_u&=\{x^2:x\in [2^{u-1},2^u... | https://mathoverflow.net/users/117368 | A counting problem related to twin primes | **Edit** (5/3/18): A significant improvement to the comparison has been found. See the chat <http://chat.stackexchange.com/rooms/69953/discussion-between-fedja-and-abiessu> for more details.
The sieve function $x^2=s^3+2s^2+st+t$ in $A\_u$ can be replaced with another function $x^2=m^4+mn+n$. This new function produc... | 0 | https://mathoverflow.net/users/117368 | 288066 | 127,111 |
https://mathoverflow.net/questions/288059 | 7 | I am a physicist and I am wondering whether the following integral over Haar measure （edit: say $U$ is unitary, orthogonal or symplectic matrix)
\begin{align}
\int dU \: \exp\left( \mathrm{tr}(UX) + \mathrm{tr}(X^\dagger U^\dagger) \right)
\end{align}
have an explicit expression in terms of the matrix X. For example, i... | https://mathoverflow.net/users/116462 | Haar Measure Integral | Depending on what you mean by "explicit", in the unitary case this can be read off
from a generalization of the Harish--Chandra-Itzykson-Zuber formula. To see that, note that your integral can be rewritten as
$$J=\int\_{U\_N}\int\_{U\_N} \exp(\Re (\mbox{tr} V YU)) dU dV,$$
where $Y$ is a diagonal real matrix whose entr... | 6 | https://mathoverflow.net/users/35520 | 288071 | 127,113 |
https://mathoverflow.net/questions/288051 | 16 | In enumerative combinatorics, a *bijective proof* that $|A\_n| = |B\_n|$ (where $A\_n$ and $B\_n$ are finite sets of combinatorial objects of size $n$) is a proof that constructs an explicit bijection between $A\_n$ and $B\_n$.
Bijective proofs are often prized because of their beauty and because of the insight that th... | https://mathoverflow.net/users/3106 | Automated search for bijective proofs | As mentioned in the comments, the [FindStat](http://findstat.org) project is aiming at what you want. Concerning the size: it contains currently about 1000 'combinatorial statistics', that is maps $s:\mathcal C\_n\to \mathbb Z$ on some (graded) set of 'combinatorial' objects $\mathcal C\_n$ and about 150 'combinatorial... | 10 | https://mathoverflow.net/users/113201 | 288075 | 127,114 |
https://mathoverflow.net/questions/287810 | 4 | i hope this is the right place for this.
In <https://math.berkeley.edu/~ogus/preprints/anonhodge.pdf> on page 15 Ogus and Vologodsky state the following:
Let $\pi\_T: \textbf{T} \rightarrow X$ be a vector group and let $T$ be its sheaf of sections. Thus $T$ is a locally free sheaf of $\mathcal{O}\_X$-modules with du... | https://mathoverflow.net/users/89594 | Vector groups in Ogus-Vologodskys Nonabelian Hodge theory in characteristic p | Arthur Ogus was so kind as to explain this to me in an Email:
The use of the phrase "vector group" was motivated by the similarities to Lie theory (as stated in my question).
As to the construction: For every $\xi \in T$ we get a map $ Sym^k( \Omega) \rightarrow Sym^{k-1} ( \Omega)$ by interior multiplication by $\... | 2 | https://mathoverflow.net/users/89594 | 288077 | 127,115 |
https://mathoverflow.net/questions/288052 | 18 | So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved by matrices whose columns are probability vectors. And this got me thinking: But what are the matrices preserving the $\... | https://mathoverflow.net/users/113001 | What are the matrices preserving the $\ell^1$-norm? | As pointed out by YCor in the comments, the following theorem is true:
**Theorem 1** Let $p \in [1,\infty] \setminus \{2\}$. If a matrix $A \in \mathbb{R}^{n \times n}$ is an isometry on $\mathbb{R}^n$ with respect to the $p$-norm, then $A$ is a signed permutation matrix, i.e. a permutation matrix where some of the o... | 26 | https://mathoverflow.net/users/102946 | 288084 | 127,117 |
https://mathoverflow.net/questions/288057 | 1 | I have asked [this question on math.stackexchange.com](https://math.stackexchange.com/q/2525587) but received no response; hoping someone on here can help.
Suppose a function $f$, representing what I call a "dynamic transposition cipher" taking one string of text $str$ as input, is defined so that it outputs another ... | https://mathoverflow.net/users/118328 | Number of iterations required for a transposition cipher to yield the original input | First off, the listed values for $g(n)$ are incremented by 1. For example, for $n=6$, it takes 5 (not 6) iterations of $f$ to take the input string to itself: $f^{(5)}(\texttt{ABCDEF})=\texttt{ABCDEF}$.
It is convenient to index the positions in $n$-symbol string from $0$ to $n-1$. Then the function $f$ defines a per... | 2 | https://mathoverflow.net/users/7076 | 288096 | 127,119 |
https://mathoverflow.net/questions/287983 | 4 | Let $X$
be a complete finite-volume orientable hyperbolic $3$-manifold,
and let $\Gamma$
be a Kleinian representation of $\pi\_1(X)$.
Let $K\Gamma:=\mathbb{Q}\big(\{\mathrm{tr}\mid\gamma\in\Gamma\}\big)$,
i.e. its *trace field*.
Let $\Gamma^{(2)}:=\langle\gamma^2\mid\gamma\in\Gamma\rangle$,
the group generated by squar... | https://mathoverflow.net/users/14835 | Trace field of a hyperbolic $3$-manifold with a totally geodesic subsurface | The answer to this follows easily from section 9.5 of Machlachlan and Reid -- which somehow I managed not to see every other time I checked the book! The thing stated in my question is not true, but the following is true.
Suppose $X$
is arithmetic and contains an immersed closed totally-geodesic surface $S$.
Then $\G... | 0 | https://mathoverflow.net/users/14835 | 288113 | 127,125 |
https://mathoverflow.net/questions/288115 | 4 | For me it looks like computing the Vietoris-Rips complex from a data cloud is very similar to the clique problem in graph theory, which it NP-hard.
How do the two differ and what is the computational complexity for finding the complex?
| https://mathoverflow.net/users/101335 | Complexity of computing the Vietoris-Rips complex | The [Vietoris-Rips complex](https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex) at scale $\epsilon$ of a set of points $P$ in some metric space $X$ ("data cloud") is the [clique complex](https://en.wikipedia.org/wiki/Clique_complex) for the graph whose vertex set is $P$ and where pairs of distinct points $p\ne... | 4 | https://mathoverflow.net/users/353 | 288120 | 127,128 |
https://mathoverflow.net/questions/288097 | 6 | Do locally cartesian closed $\infty$-categories form a presentable $\infty$-category? It seems like they should, and that the inclusion $\text{LCC}\rightarrow\text{Cat}$ preserves limits and maybe even colimits.
Here is a possible strategy for proof. There is a functor $\text{Cat}\rightarrow\text{Fun}(\Delta^1,\text{... | https://mathoverflow.net/users/47658 | (Co)limits of locally cartesian closed categories | In [this preprint](https://arxiv.org/abs/1610.08459) Isaev describes a combinatorial model category which presents the $(\infty, 1)$-category of finitely complete $(\infty, 1)$-categories. The underyling category is the category of simplicial sets equipped with collections of distinguished cones over diagrams indexed b... | 5 | https://mathoverflow.net/users/12547 | 288140 | 127,134 |
https://mathoverflow.net/questions/288111 | 1 | Let $M \in \{0, 1\}^{n\times n}$.
Given a *constant* integer $c \ge 2$, let the number of $1$s in each row be equal to $\frac{n}{c}$ (assuming $c$ is a divisor of $n$). Let $\mathcal{M}\_c$ be the set of such matrices.
Given a *constant* $\beta \in (0,1)$ and a matrix $M \in \mathcal{M}\_c$, we say that "a row vector... | https://mathoverflow.net/users/115803 | Combinatorial 0-1 vector problem | Here is an example of $\approx n^{1/3}$ vectors of length $n/c$ with overlaps strictly less than $n/c^2$. Probably, one can do $\approx n$ too, but I do not see any neat construction.
Take a prime $p$ of the form $p=cq-1$. Now, for the index set, use $\mathbb Z\_p^2\times\{1,2,\dots,p+1\}$, so $n=p^2(p+1)$. For the $... | 3 | https://mathoverflow.net/users/1131 | 288169 | 127,142 |
https://mathoverflow.net/questions/287912 | 1 | Suppose I have a binary vector `a = (0,1,1,1,0,0)` of length $N$. I want to detect in a linear way whether `a` has any gaps in consecutive `1`'s. `a` as defined above has no gaps. But `b = (0,1,0,1,0,0)` has gaps.
Quadratic solution:
$$\sum\_i^N a\_i - \sum\_i^{N-1}a\_i a\_{i+1} \le 1$$
Linear solution:
$$?$$
Is ... | https://mathoverflow.net/users/90497 | Detection of gaps in binary vector through linear methods | OK, sorry for the delay. I just don't have too much free time nowadays.
Let $a\_0,\dots,a\_{n-1}$ be your string. Note that for any $z=e^{it}$, we have
$$
P\_a(z)=a\_0+a\_1z+a\_2z^2+\dots+a\_{n-1}z^{n}=\frac 1{z-1}\sum\_s(z^{n\_s}-z^{m\_s})
$$
where $[m\_s,n\_s-1]$ ($s=1,2,\dots,p$) are the groups of successive $1$'... | 3 | https://mathoverflow.net/users/1131 | 288183 | 127,147 |
https://mathoverflow.net/questions/288155 | 12 | There is obviously a set of situations where one lack an algorithm to compute the exact solution of an equation via symbolic manipulation only, for example `x = sin(x)`.
One has to resort to numerical analysis and iteration methods (for example) to estimate a value resolving the above.
My question is: is there an ... | https://mathoverflow.net/users/118398 | Proof there is no algorithm to compute the intersection of a line and sinusoidal wave? | Your question is addressed in my paper [What is a closed-form number?](http://alum.mit.edu/www/tchow/closedform.pdf)
The first step is to decide which "symbols" or functions you accept as furnishing a "symbolic solution." In my paper I focus on perhaps the most restrictive set, namely exp and log and the arithmetic o... | 15 | https://mathoverflow.net/users/3106 | 288192 | 127,149 |
https://mathoverflow.net/questions/288154 | 2 | I am reading Chapter 3 of the book "Singularities of the minimal model program", and I have some problems on calculating certain quantities.
Let $X$ be a projective variety, $L$ be an ample line bundle on $X$, and $K\_X \sim\_{\mathbb{Q}} rL$ for some $r\in \mathbb{Q}$. Then one can define the affine cone ($\S$ 3.8 l... | https://mathoverflow.net/users/29730 | Some calculations on cones | Let's use the following notation $C=C\_a(X,L)$ and $B=BC\_a(X,L)$.
(1) This is the usual statement about the normal bundle of the exceptional divisor restricted to the exceptional. You can look at the computation at the end of II.8 in [Hartshorne] to see that this restriction is always $\mathscr O(-1)$. In this situa... | 1 | https://mathoverflow.net/users/10076 | 288195 | 127,151 |
https://mathoverflow.net/questions/287422 | 7 | Let $S\subset\mathbb{R}^n$, with $n\geq 3$, such that for any hyperplane $L$ one has $L\cap S$ closed, semialgebraic, and convex. Is it true that $S$ itself is semialgebraic?
A colleague explained to me that closedness is necessary, as one can take an open ball in $\mathbb{R}^n$ and paste a sufficiently nasty curve i... | https://mathoverflow.net/users/11100 | Do convex closed semialgebraic hyperplane cross-sections imply semi-algebraicity? | Here I record an answer based on very kind comments above (and offline).
The construction starts from a closed ball $B\subset\mathbb{R}^3$ and an infinite sequence $\mathcal{L}:=\{L^+\_k\}$ of open half-spaces, satisfying the condition
$$B\cap L^+\_i\cap L^+\_j=\emptyset\quad\text{ if and only if $i\neq j$.}\qquad\qqu... | 3 | https://mathoverflow.net/users/11100 | 288197 | 127,152 |
https://mathoverflow.net/questions/288198 | 10 | **Question**. Do these identities involving even-index Catalan numbers have a known combinatorial interpretation? They look as though they should. I haven’t seen one in the literature.
$$\sum\_{a+b=n}C\_{2a}C\_{2b}=4^nC\_n$$
$$\sum\_{a+b=n}C\_{2a}{4b\choose 2b}=4^n{2n\choose n}$$
---
**Added**. Thanks to Rich... | https://mathoverflow.net/users/8217 | Curious Catalan convolutions | For the first identity, see additional problem A33 in my book *Catalan Numbers*. References are given to bijective proofs by Andrews and Nagy.
| 12 | https://mathoverflow.net/users/2807 | 288200 | 127,154 |
https://mathoverflow.net/questions/288150 | -4 | In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them):
\begin{align}
& f(2) = 1+t, \\
& f(3) = 1+4t+3t^2, \\
& f(4) = 1+6t+12t^2+7t^3, \\
& f(5) = 1+8t+20t^2+28t^3+15t^4.
\end{align}
Is it possible to find the general formula of $f(n)$ using these 4 polynomials? Some patte... | https://mathoverflow.net/users/11877 | What are the patterns of the sequence of polynomials? | One way is
$f(n)=(1+t)\left( \sum\_{k=0}^{n-2}(2(2^k-1)(n-k-1)+1)t^k \right)$
| 3 | https://mathoverflow.net/users/112259 | 288201 | 127,155 |
https://mathoverflow.net/questions/86924 | 14 | Consider lattice paths consisting of $2n$ steps, each of which is either $(1,1)$ or $(1,-1)$. The number of such lattice paths that return to the horizontal axis only at times that are a multiple of $4$ is given by $2^n \binom{n}{n/2}$. Can someone provide a combinatorial proof of this fact?
---
Background: A few... | https://mathoverflow.net/users/9716 | Combinatorial proof for the number of lattice paths that return to the axis only at times that are a multiple of 4 | For the record, this question is answered in a pair of papers:
Gábor V. Nagy, [A combinatorial proof of Shapiroʼs Catalan convolution](https://www.sciencedirect.com/science/article/pii/S0196885812000772), In Advances in Applied Mathematics, Volume 49, Issues 3–5, 2012, Pages 391-396
Péter Hajnal and Gábor V. Nagy. ... | 7 | https://mathoverflow.net/users/8217 | 288203 | 127,156 |
https://mathoverflow.net/questions/288202 | 5 | Is it possible given a set of Betti numbers to construct a (possibly set of) simplicial complex with the given Betti-described topology? I understand there can be an infinity of simplicial complexes with the same Betti numbers but does a "minimal" simplicial complex construction algorithm exist?
Such an algorithm wi... | https://mathoverflow.net/users/61077 | Simplicial complex construction from given Betti numbers? | One way to make things "minimal" (given the lack of any further information) is to construct a simplicial complex whose cup products are all trivial, so the (co)homology generators don't interact with each other in exciting ways. That is to say, just build a simplicial wedge of spheres. All you need to do is have the a... | 5 | https://mathoverflow.net/users/18263 | 288204 | 127,157 |
https://mathoverflow.net/questions/288191 | 0 | Suppose $f$ a function holomorphic on the unit bidisk $\mathbb{D}\times \mathbb{D}$, such that $f$ is $\mathcal{C}^{\infty}$ on $]-1,1[\times\partial\mathbb{D}$, and has holomorphic extension on $\mathcal{U}\times \mathbb{D}$, where $\mathcal{U}$ is some neighborhood of $\partial\mathbb{D}$. Does $f$ has an extension i... | https://mathoverflow.net/users/110900 | a counter-example of a holomorphic extension | Take your favorite $C^\infty$ function $F(x)$ on the line that is not analytic and write $F(x)=\sum\_{k\ge 1}P\_k(x)$ where $P\_k$ are polynomials of degree $k$ such that for all $m\ge 0, d>0$, we have $|P\_k^{(m)}(x)|\le C\_{m,d}k^{-d}$ for all $x\in[-2,2],k\ge 1$. Then $|P\_k(z)|\le C\_{0,0}(1+|z|)^{k}$ for all $z\in... | 1 | https://mathoverflow.net/users/1131 | 288212 | 127,161 |
https://mathoverflow.net/questions/288127 | 23 | Assume that $(M,g)$ is a Riemannian manifold.
Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of Laplace operator acting on the space of smooth $k$-forms for $k\neq 0$?
| https://mathoverflow.net/users/36688 | Eigenvalues of Laplace operator | It sounds to me like you have heard, and are trying to remember the statement of, the following fact about the *Hodge Laplacian* $d^\*d+dd^\*$ on forms.
Slogan: eigenforms come in pairs.
Sketch proof: If $(d^\*d+dd^\*)\eta=-\lambda\eta$, then $(d^\*d+dd^\*)(d\eta)=dd^\*d\eta=d(d^\*d+dd^\*)\eta=-\lambda d\eta$.
St... | 7 | https://mathoverflow.net/users/2819 | 288216 | 127,162 |
https://mathoverflow.net/questions/288189 | 3 | In this question, I will be working in ZF.
Let $h(\kappa)$ for a cardinal $\kappa$ (not necessarily an ordinal) be the smallest ordinal $\alpha$ such that there is no surjection from a set of size $\kappa$ onto $\alpha$.
$h(\mathfrak{c})$ is defined as $\Theta$.
Let an uncountable ordinal $\alpha$ be a **high lim... | https://mathoverflow.net/users/115951 | Strong limit cardinals in AD | First, let me point out that the notion of "high limit cardinal" is misleading, and perhaps not quite what you want it to be.
The reason is that strong limit cardinals are defined, in the absence of choice, as $\aleph$ numbers $\alpha$, such that for all $\beta<\alpha$, there is no surjection from $V\_\beta$ onto $\a... | 8 | https://mathoverflow.net/users/7206 | 288219 | 127,163 |
https://mathoverflow.net/questions/288217 | 5 | Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely for any open subset $U\subset M$, $F(U)$ is the space of generalized functions on $U$.
>
> Is it true that $H^i\_Z... | https://mathoverflow.net/users/16183 | Cohomology of sheaf of Schwartz distributions with support in a submanifold | The long exact sequence cotains $H^0(M,F)\to H^0(M\smallsetminus Z,F|\_{M\smallsetminus Z})\to H\_Z^1(M,F)\to H^1(X,F)$.
The last term is zero, as $F$ is a fine sheaf. But the first arrow is not surjective in general, so the first cohomology group is not trivial. The same argument shows that for $i>1$ the cohomologies ... | 3 | https://mathoverflow.net/users/nan | 288226 | 127,166 |
https://mathoverflow.net/questions/287529 | 1 | Let a cardinal $\kappa$ be $n$-shadow iff $\kappa$ does not have cofinality $\omega$ and for any $n$-th order sentence $\varphi$ in the language $\mathcal{L}\_\in$, $\varphi\Leftrightarrow V\_\kappa\models\varphi$. If this is unclear, for a standard $M\models\text{ZFC}$ and some $x\in M$ which is a cardinal in $M$, $x$... | https://mathoverflow.net/users/111429 | A Weak form of Extendibility and Inner Model Theory | You consider the strength of having a $1$-shadow cardinal in an
inner model. But I claim that no proper inner model $M$ has any
$1$-shadow cardinals at all.
Assume $M\subsetneq V$ is a proper inner model, by which I mean
that $M$ is a transitive class model of ZF, containing all
ordinals, and $M\neq V$. Suppose that ... | 2 | https://mathoverflow.net/users/1946 | 288235 | 127,169 |
https://mathoverflow.net/questions/288234 | 6 | The inclusion $I\colon \mathbf{Grpd}\hookrightarrow\mathbf{Cat}$ of groupoids into categories has both a left and a right adjoint $L,R\colon \mathbf{Cat}\to \mathbf{Grpd}$, with $R(C)$ being largest groupoid contained in $C$ and $L(C) = C[C^{-1}]$ being $C$ with all morphisms brutally inverted. Going into $\infty$-cate... | https://mathoverflow.net/users/80467 | Left adjoint of $I\colon \mathrm{Kan}\hookrightarrow\mathrm{WeakKan}$? | The inclusion of $\infty$-groupoids into $(\infty,1)$-categories certainly has a left adjoint. The functor $\mathbf{Kan} \hookrightarrow \mathbf{WKan}$ does not have a left adjoint, but the higher left adjoint can be modeled by any [fibrant replacement functor](https://ncatlab.org/nlab/show/Kan+fibrant+replacement).
... | 8 | https://mathoverflow.net/users/62782 | 288239 | 127,170 |
https://mathoverflow.net/questions/288246 | 4 | Let $M\colon \partial\_- M \to \partial\_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M\_i, i \in \{0,\dots 4\}$.
It is known ("On attaching 3-handles to a 1-connected manifold" by Bruce Trace) that if $M$ is simply con... | https://mathoverflow.net/users/13767 | Are there Kirby diagrams with 3-handles? | 3-handles do matter. The reason why they're not usually there is because people often care about closed 4-manifolds; in this case, there is an essentially unique way of attaching all 3-handles to the boundary of (in your notation) $M\_2$, which is just $\#^n(S^1\times S^2)$ for some $n$. The result is due to Laudenbach... | 5 | https://mathoverflow.net/users/13119 | 288248 | 127,173 |
https://mathoverflow.net/questions/288231 | 4 | Is there an integer $n\ge1$ such that every prime $p\equiv1\pmod{9}$ is representable in the form $x^2+ny^2$?
| https://mathoverflow.net/users/108143 | Is there an $n\ge1$ such that every prime $p\equiv1\pmod{9}$ is representable in the form $x^2+ny^2$? | Yes, $n=3$ works, and no other integer works. The "no other" part follows easily by inspecting $p=19$ and $p=37$. The "works" part was conjectured by Fermat in 1654 and proved by Euler in 1772. In modern terminology, the proof is quite simple, and it goes as follows (I give an outline):
A prime $p\nmid 2n$ is represe... | 8 | https://mathoverflow.net/users/11919 | 288254 | 127,176 |
https://mathoverflow.net/questions/287729 | 6 | A subgroup $H$ of an algebraic group $G$ is said to be Zariski-dense if its Zariski closure is all of $G$ (or alternatively, if every polynomial which vanishes on all elements of $H$ vanishes identically).
**My question:** Is any irreducible subgroup of $SL(2,\mathbb{C})$ Zariski-dense? (It's easy to see that the co... | https://mathoverflow.net/users/59235 | Are irreducible subgroups Zariski-dense? | To summarize the discussion. Let $G$ be a subgroup of $\mathrm{SL}\_2(\mathbf{C})$, $H$ its Zariski closure.
>
> **Proposition.** Equivalent statements:
>
>
> (i) $G$ acts irreducibly on $\mathbf{C}^2$;
>
>
> (ii) $H$ acts irreducibly on $\mathbf{C}^2$;
>
>
> (iii) $G$ fixes no point on $\mathbb{P}^1\_\mathbf... | 5 | https://mathoverflow.net/users/14094 | 288267 | 127,180 |
https://mathoverflow.net/questions/288249 | 9 | Let $G$ be a topological group, and $M$ a connected compact smooth manifold. I'm studying
$$ \pi\_0 (map (BG,M)). $$
For $G$ a finite group, we know that this is just a point by the Sullivan conjecture on maps from classifying spaces which was proven by Miller. (This does not require smoothness of $M$.)
On the oth... | https://mathoverflow.net/users/26470 | Sullivan conjecture for compact Lie groups | You were right to single out Lie groups as potentially interesting. In [Topology 5 (1966), 241-243], Brayton Gray showed that the homotopy group of maps $[BS^1, S^3]$ was uncountable. Indeed, he showed that the subgroup of phantom maps -- maps null on every finite subcomplex - was uncountable. Then Alex Zabrodsky, in [... | 20 | https://mathoverflow.net/users/102519 | 288268 | 127,181 |
https://mathoverflow.net/questions/288259 | 19 | Enumerate the rationals as $b\_1,b\_2,\dots$ and define the (set) function:
$$f(x) = (x-b\_1)^2 + (x-b\_1)^2(x-b\_2)^2 + \dots.$$
At any particular $x$, only finitely many terms are non zero so this is perfectly well defined as a (set) function but surely, it is not equal to any polynomial! (or is it?) How do I show th... | https://mathoverflow.net/users/58001 | How to show that the following function isn't a polynomial over Q? | For each positive integer $n$ and any rational $x$, we have
$$f(x)\geq (x-b\_1)^2(x-b\_2)^2\dots(x-b\_n)^2.$$
For large $x$, we then have $f(x)\gg x^{2n}$, which implies that if $f$ is a polynomial, it must have degree $≥2n$.
| 28 | https://mathoverflow.net/users/7666 | 288273 | 127,182 |
https://mathoverflow.net/questions/288272 | 4 | I am reading Fulton's but I cannot find a useful result for my problem that seems to be something well known.
Let $\mathcal{J}$ be the Jacobian of a hyperelliptic curve of genus $2$. Consider $D\_1,D\_2\in \text{Div}(\mathcal{J})$ (two curves inside $\mathcal{J}$) and let $D\_2'$ be a translation by a $2$-Torsion poi... | https://mathoverflow.net/users/91023 | Intersection number of divisors on abelian surfaces and its invariance under translation by 2-Torsion points | The answer is *yes* if $D\_2$ is an ample curve. More generally, the following result holds:
>
> Two *ample* line bundles on an abelian variety
> are algebraically equivalent if and only if they differ by a translation.
>
>
>
This is essentially Corollary 2.5.4 in Birkhenake-Lange book *Complex Abelian Varie... | 3 | https://mathoverflow.net/users/7460 | 288274 | 127,183 |
https://mathoverflow.net/questions/288256 | 0 | Informally, right-angled hyperbolic hexagon is a hyperbolic triangle with vertices outside infinity. I think there should be a Gram matrix for it, and **what does it looks like?**
(The Gram matrix here defines the existence of a simplex)
Also, for hyperbolic truncated tetrahedron or hyperbolic truncated n-simplex wha... | https://mathoverflow.net/users/117580 | What’s the form of Gram matrix for right-angled hexagon | Why do you mean by "what is the form"? And which Gram matrix? (there are two). But if you mean the usual Gram matrix, then the $ij$ element is $-\cosh d(s\_i, s\_j),$ where $s\_i$ is the $2i$-th side of the hexagon.
| 1 | https://mathoverflow.net/users/11142 | 288276 | 127,185 |
https://mathoverflow.net/questions/288237 | 2 | In an extensive category, the following conditions are equivalent for an object.
1. In a coproduct decomposition, exactly one of the summands is initial;
2. The covariant functor it represents preserves coproducts.
An object satisfying these equivalent conditions is called *connected*.
Now consider the functor $\... | https://mathoverflow.net/users/69037 | Connectedness in extensive categories in terms of set of connected components | I wouldn't say that there is a bijection between $\mathrm{Hom}(X,2)$ and binary coproduct decompositions of $X$ since the former is a set and the latter is a category. If we ignore this fact, then we can show that the natural function $\coprod\limits\_{A,B \in \mathrm{Ob(C)}} A \amalg B \simeq X \to \mathrm{Hom}(X,2)$ ... | 3 | https://mathoverflow.net/users/62782 | 288284 | 127,188 |
https://mathoverflow.net/questions/288290 | 5 | This is a followup to the question here: [How to show that the following function isn't a polynomial over Q?](https://mathoverflow.net/questions/288259/how-to-show-that-the-following-function-isnt-a-polynomial-over-q).
As before, let $b\_1,b\_2,\dots$ be an enumeration of $\mathbb Q$. The question might be sensitive... | https://mathoverflow.net/users/58001 | (Variation of an old question) Are these functions polynomials? | $f\_3(x)$ is not a polynomial function, which can be seen as follows. Call $b\_n$ a champion if it exceeds all the earlier terms $b\_1,\dots,b\_{n-1}$. Clearly, there are infinitely many champions, and they tend to infinity. Let $m$ be fixed, and consider the champions $b\_n$ with $n>m$. We get, as $b\_n$ tends to infi... | 4 | https://mathoverflow.net/users/11919 | 288292 | 127,192 |
https://mathoverflow.net/questions/288303 | 6 | Suppose that $G$ is a finite group, $M$ is a right $G$-set and $N$ is a left $G$-set. Then we have a simplicial set $B(M,G,N)$ whose $n$-simplicies are $M \times G^n \times N$.
Now suppose that $H \subseteq G$ is a subgroup. There is a natural inclusion $B(\*,H,\*) \hookrightarrow B(\*,G,G/H)$. Both of these spaces ... | https://mathoverflow.net/users/4002 | Computing an explicit homotopy inverse for $B(*,H,*) \hookrightarrow B(*,G,G/H)$ | Yes, there is an explicit algorithm for doing this. Pick a set of representatives $a\_i \in G$ for the left cosets of $G/H$. Then the inverse map is as follows.
Given any element $(g\_n,\dots,g\_1, g\_0H)$ in $G^n \times G/H$, let $a\_i$ be the chosen representative of the left coset $g\_i g\_{i-1} \dots g\_0 H$. The... | 14 | https://mathoverflow.net/users/360 | 288304 | 127,197 |
https://mathoverflow.net/questions/288214 | 9 |
>
> Let $X$ be a smooth variety over a perfect field $k$ with $X(k) \neq \emptyset$. Then is the natural map
> \begin{equation}
> \mathrm{Pic}(X) \to (\mathrm{Pic}(X\_{\bar{k}}))^{\mathrm{Gal}(\bar{k}/k)} \qquad (1)
> \end{equation}
> surjective?
>
>
>
Remarks:
1. If $X$ is a proper, then the map (1) is in ... | https://mathoverflow.net/users/5101 | Galois invariant Picard group elements | **Updated. The example by @Lucifer is completely correct.** Thanks to @Count Dracula who explained the example proposed by @Lucifer.
That example is fine. I am keeping the counterexamples below, since they arise in a different way: as Severi-Brauer schemes over multiplicative group schemes over a field (and schemes ma... | 6 | https://mathoverflow.net/users/13265 | 288315 | 127,198 |
https://mathoverflow.net/questions/288309 | 0 | Let $G$ be a finite abelian group (cosidered as a discrete topological group), $A$ a unital separable $C^\*$-algebra. Let $T\colon G\to \operatorname{Aut}(A)$, $T\_g(a)=a$ for all $g\in G$ the trivial action.
Let $Lt\colon G\to \operatorname{Aut}(C(G))$ the left translation, i.e. $Lt$ is given by $Lt\_g(f)(h)=f(g^{-... | https://mathoverflow.net/users/75338 | Dualizing the trivial action on a $C^*$-algebra | Using the isomorphisms you wrote down, the dual action is the left translation action, composed with the automorphism $g\mapsto g^{-1}$.
To see this, first notice that the $C^\*$-algebra $A$ plays no role in any of the definitions, so it can safely be disregarded in the computation. We're thus considering the action ... | 1 | https://mathoverflow.net/users/85913 | 288319 | 127,199 |
https://mathoverflow.net/questions/288236 | 5 | Given a knot $K \subset S^3$ one can form its double branched cover $\Sigma\_2(K)$ and consider the pull-back knot $\widetilde{K} \subset \Sigma\_2(K)$ of $K$ to $\Sigma\_2(K)$ (the locus fixed by the involution of $\Sigma\_2(K)$). Here a few questions I would like to ask
* How do I compute the Alexander polynomial ... | https://mathoverflow.net/users/58934 | Pull-back of knots in branched covers and the Alexander polynomial | Corollary 4.2 from the paper "Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation" by Herald, Kirk and Livingston gives the followwing recipe to compute Alexander polynomial of the lift of $K$ to the $n$-fold cover $\Sigma\_n(K)$:
$$\Delta\_{\widetilde{K}\_n}(t) = \prod\_{i=0}^{n-1} \D... | 3 | https://mathoverflow.net/users/21892 | 288323 | 127,200 |
https://mathoverflow.net/questions/288320 | 1 | Define $\binom{a\_1,...,a\_n}{b\_1,...,b\_m}=\Pi\_i(a\_i!)/\Pi\_j(b\_j!)$. (WLOG null-pad to have $m=n$ and sort descendingly.) It's natural to ask when the value is an integer. Since I am working with "majorization" currently...but alas, $\binom{3,1}{2,2}=3/2$ and $\binom{6,1,1}{5,3,0}=1$ are counterexamples for both ... | https://mathoverflow.net/users/11504 | "Super" multinomial coefficients | Mordell, L. J. (1959),
Integer Quotients of Products of Factorials.
Journal of the London Mathematical Society, s1-34: 134–138.
doi:10.1112/jlms/s1-34.2.134
Errata:
On a Problem of Hardy and Littlewood Volume s1-34, Issue 4, 485,
Errata:
Integer Quotients of Products of Factorials Volume s1-34, Issu... | 1 | https://mathoverflow.net/users/454 | 288325 | 127,202 |
https://mathoverflow.net/questions/288299 | 3 | Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following relations hold:
$(f\cdot g)\cdot h=f\cdot (g\cdot h)$
$(f\cdot g)\triangleright X=f\triangleright(g\triangleright X)$
$... | https://mathoverflow.net/users/104743 | A differential graded Lie algebra with the Hochschild differential | Not in general, no. However $H^\*(V,W)$ is a representation of the Lie algebra $H^\*(V,V)$, by letting, for $f \in \hom(V^{\otimes m}, W)$ and $g \in \hom(V^{\otimes n}, V)$,
$$f \circ g := \sum\_{i=1}^m \pm f \circ\_i g,$$
just like in the definition of the Lie bracket on $H^\*(V,V)$. This module structure already app... | 1 | https://mathoverflow.net/users/36146 | 288328 | 127,204 |
https://mathoverflow.net/questions/288336 | 8 | Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere)
$$
F(y)=\int\_a^b f(t,y(t),y'(t))\,dt
$$
such that
$$
\inf\_{y\in Lip([a,b])}F(y)<\inf\_{y\in C^1([a,b])}F(y)
$$
that is, it shows the Lavrentiev phenomenon between $C^1$ and Lipschitz.
| https://mathoverflow.net/users/24152 | Lavrentiev phenomenon between $C^1$ and Lipschitz | Then trivially yes. Just take $F(y)=1+\sum\_{q\in \mathbb Q}\frac{a\_q}{|q-y|^2}$ with $a\_q>0$ such that the series converges a.e. Then change all $+\infty$ values of $F$ to $1$. Now take $[a,b]=[-1,1]$ and define
$$
f(t,y,\xi) =
\begin{cases}
0 &\text{if }y=|t|\\
F(y) &\text{otherwise}
\end {cases}.
$$
A Lipschitz f... | 8 | https://mathoverflow.net/users/1131 | 288359 | 127,211 |
https://mathoverflow.net/questions/288346 | 3 | I am reposting the second question from [here](https://mathoverflow.net/questions/288290/variation-of-an-old-question-are-these-functions-polynomials) (after clarifying it) on the recommendation of user "GH from MO".
Let $b\_1,b\_2,\dots$ be an enumeration of $\mathbb Q$.
**Question 2:** Suppose I define $$G(x,y)... | https://mathoverflow.net/users/58001 | Is the following function a polynomial? | The answer is no. There are uniquely determined linear polynomials $a\_m(x):=x-c\_m$ over $\mathbb{Q}$ such that $g(x)$ is identically zero on $\mathbb{Q}$. Indeed, for such polynomials the condition means that
$$ \sum\_{0\leq m\leq n-1}(b\_n-c\_m)\cdot(b\_n-b\_1)\dots(b\_n-b\_m) = 0,\qquad n\geq 1. $$
Here, the second... | 4 | https://mathoverflow.net/users/11919 | 288361 | 127,212 |
https://mathoverflow.net/questions/41913 | 18 | I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print:
* Richard K. Guy. *[Unsolved problems in combinatorial games](http://library.msri.org/books/Book29/files/unsolved.pdf)*. ... | https://mathoverflow.net/users/6085 | David Gale's subset take-away game | The conjecture that this game is always a second player win has recently been disproved by Brouwer and Christensen (already for $n=7$), <https://arxiv.org/abs/1702.03018>.
| 7 | https://mathoverflow.net/users/57973 | 288370 | 127,217 |
https://mathoverflow.net/questions/287816 | 14 |
>
> Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
>
>
> $$B^\top - T B^\top = B + B T$$
>
>
> 1. Prove that $\mbox{tr}(TB) \leq 0$.
> 2. What are necessary and sufficient conditions on $B$ such th... | https://mathoverflow.net/users/116256 | For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$ | First let us check that $T$ exists and is unique. Let $\mathrm{Sym}\_n$ be the space of symmetric matrices (with real coefficients), $\mathrm{M}\_n$ the space of all matrices and $\mathrm{Alt}\_n$ the space of antisymmetric matrices.
Claim: the map $\mathrm{Alt}\_n \rightarrow \mathrm{M}\_n / \mathrm{Sym}\_n, T \maps... | 8 | https://mathoverflow.net/users/118504 | 288373 | 127,220 |
https://mathoverflow.net/questions/288332 | 4 | In order to understander the nonlinear elliptic equation with natural boundary condition,
$$\sigma\_2(D^2u)=0 \text{ in } \Omega$$
I wish to understand the following integral,
$$E(u,\Sigma)=\int\_\Sigma \det(D^2u|\_\Sigma) \, de\_1\wedge de\_2$$
Where $u\in C^2(\Omega)$ is a solution of some elliptic equation, so we ha... | https://mathoverflow.net/users/91939 | How to understand the integral? | If I understand your question correctly, the answer is affirmative for Ricci-flat (i.e. flat) surfaces (not necessarily embedded in a Euclidean space) according to equation (14) of
*Reilly, Robert C.*, [**Applications of the Hessian operator in a Riemannian manifold**](http://dx.doi.org/10.1512/iumj.1977.26.26036), I... | 5 | https://mathoverflow.net/users/766 | 288376 | 127,221 |
https://mathoverflow.net/questions/288301 | 1 | Let $\Psi(x,B)$ denote the number of $B$-smooth numbers less than $x$. [Wikipedia](https://en.wikipedia.org/wiki/Smooth_number#Distribution) gives the following "good estimate" for small, fixed $B$:
$$\Psi(x,B) \sim \frac{1}{\pi(B)!} \prod\_{p\le B}\frac{\log x}{\log p}$$
However, no citation or further discussion ... | https://mathoverflow.net/users/51384 | Estimate for de Bruijn function with small fixed smoothness bound | This is discussed in chapter III.5 of Tenenbaum *Intro to analytic and probabilistic number theory* (Cambridge, 1995). The estimate up to a factor $1+O(B^2/\log x\log B)$ uniform in $2 \le B \le \sqrt{\log x\log\log x}$ is attributed to Ennola 1969.
| 2 | https://mathoverflow.net/users/61870 | 288378 | 127,222 |
https://mathoverflow.net/questions/288322 | 7 | Let $S\subset\mathbb{P}^g$ be a polarised smooth projective K3 surface of genus $g$ (and degree $d=2g-2$) over $\mathbb{C}$. Denote by $\phi: S\to G(3,g+1)$ the Gauss map, taking a point $s\in S$ to its tangent 2-plane $\mathbb{T}\_{S,s}\cong\mathbb{P}^2$ in $\mathbb{P}^g$. This is known to always be finite and biratio... | https://mathoverflow.net/users/386 | Gauss map of general K3 surface | I think that a sufficient condition for the Gauss map to be a closed embedding is that each tangent plane $\mathbb{T}\_{S,s}$ intersects the surface $S$ only at the point $s$ and with the expected multiplicity $3$.
In turn, a sufficient condition to guarantee this is that the polarization is $3$-very ample: a paper o... | 4 | https://mathoverflow.net/users/45285 | 288382 | 127,224 |
https://mathoverflow.net/questions/288306 | 7 | Let $\zeta$ be a primitive $2^k$th root of unity, and consider the ring $ \mathbb{Z}[\zeta, \frac{1}{2}] \subset \mathbb{C}$. Are there any elements of absolute value 1 other than the powers of $\zeta$?
Note that if $\frac{1}{2}$ were replaced by $\frac{1}{5}$, then the number $\frac{4}{5} + \frac{3}{5} i$ would be a... | https://mathoverflow.net/users/39521 | Elements of absolute value 1 in cyclotomic extension of dyadic rationals | The answer is yes. Let $K=\mathbb{Q}(\zeta)$ and suppose that there is an element $u\in\mathbb{Z}[\frac{1}{2},\zeta]$ such that for some embedding $\sigma\_0\colon K\hookrightarrow \mathbb{C}$ the absolute value of $\sigma\_0(u)$ is $1$. Then I claim that $u\in\mathbb{Z}[\frac{1}{2},\zeta]^\times$ and that for all embe... | 3 | https://mathoverflow.net/users/18238 | 288392 | 127,232 |
https://mathoverflow.net/questions/288136 | 16 | Consider the following discrete optimization problem: given a collection of $m$-dimensional vectors $\{ v\_1, \dots, v\_n \}$ with entries in $\{-1, +1\}$, find an $m$-dimensional vector $x$ with entries in $\{0,1\}$ that maximizes the number of vectors $v\_i$ having positive dot product with $x$.
For example, for th... | https://mathoverflow.net/users/118386 | NP-hardness of finding 0-1 vector to maximize rows of {-1, +1} matrix | Here is a simple embedding of 3-SAT into the current setup (the question is just if we can get *all* vectors good).
Call the first column special with $1$'s.
Split the other variables into pairs $(a,b)\in\{(0,0),(1,1),(1,0),(0,1)\}$.
Our first task will be to eliminate any $(1,0)$ or $(0,1)$ options. For that,
us... | 8 | https://mathoverflow.net/users/1131 | 288402 | 127,235 |
https://mathoverflow.net/questions/288360 | 1 | Let $\varphi\_{1},\varphi\_{2}:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be two
smooth general position (Morse) functions having the same set of critical
points $\left\{ p\_{1},...,p\_{n}\right\} \subset\mathbb{S}^{1}$ ($n$ is even)
and both $\varphi\_{1}$ and $\varphi\_{2}$ have a local maximum at $p\_{1}$.
Suppose that $\... | https://mathoverflow.net/users/118503 | Topological similarity of solutions to Dirichlet problem | Certainly not. Imagine six critical points: $3$ minima and $3$ maxima so that all minima are about $-1$ and all maxima are about $+1$ (but may be all different if you want).
Now assume that the first minimum is "thick", so the corresponding value of about $-1$ spreads almost until the adjacent maxima. If out of the oth... | 4 | https://mathoverflow.net/users/1131 | 288404 | 127,236 |
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