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https://mathoverflow.net/questions/189411 | 0 | I am looking at a research paper that mentions integral operators (which in this case is brought up in reference to shading equations that are integral operators) and it says that we can create a matrix representation of integral operators using Galerkin projections. I am having difficulty finding specific literature o... | https://mathoverflow.net/users/62833 | Galerkin Projection on Integral Operators | The idea of the Galerkin method is similar for partial differential equations and for integral equations, hence, any book on numerics of PDEs (that does finite elements) should be a good reference, and also every book on numerical methods for integral equations, e.g.
* "Integral Equations: Theory and Numerical Treat... | 1 | https://mathoverflow.net/users/9652 | 189425 | 93,711 |
https://mathoverflow.net/questions/189424 | 2 | **Conjecture**
Let $U$ be an (infinite) set. Let $f$ be an $N$-ary (where $N$ is an
arbitrary index set) relation on $U$ (that is a set of functions $N
\rightarrow U$).
Let $\mathcal{L}\_0$, $\mathcal{L}\_1$ be $N$-indexed families of filters (on
$U$).
I denote $\operatorname{up} \mathcal{L} = \prod\_{i \in N}
... | https://mathoverflow.net/users/4086 | Interweaving two indexed families of filters | Something's wrong in the last formula of the question. The elements of $f$ are functions $N\to U$, but the elements of $\prod\_{i\in N}(\mathcal L\_{c(i)}(i))$ are functions on $N$ whose values are subsets of $U$, elements of the filters $\mathcal L\_{c(i)}(i)$, not elements of $U$. I'm going to assume that you meant t... | 3 | https://mathoverflow.net/users/6794 | 189429 | 93,714 |
https://mathoverflow.net/questions/189432 | 6 | In Alan Robinson's paper, *Classical Obstructions and S-algebras*, he provides conditions for a ring spectrum to have an $A\_n$ and $\mathbb{E}\_\infty$-structure.
>
> **Have the obstructions for an object of an $\infty$-operad $\mathcal{C}^\otimes$ from having an $\mathbb{E}\_n$-algebra structure been studied? Mo... | https://mathoverflow.net/users/nan | Obstructions for $E_n$-algebras | Let me expand a little on what Qiaochu and Craig mentioned.
If you want an obstruction theory for building an uber-gadget, you'll need (i) an algebraic approximation to such gadgets, and (ii) a way to resolve every uber-gadget by special uber-gadgets where the algebraic approximation remembers everything about the th... | 14 | https://mathoverflow.net/users/6936 | 189445 | 93,720 |
https://mathoverflow.net/questions/189407 | 0 | Can anybody solve this:
For a constant positive integer $n\geq6$
find $k$ and positive integers $a\_{1},a\_{2},...,a\_{k}$
that maximize the expression
$$\sum\_{i=1}^{k}\left[-4a\_{i}^{3}+\left(3n-3\right)a\_{i}^{2}+\left(3n+1\right)a\_{i}\right],$$
with $a\_{1}+a\_{2}+\dots+a\_{k}=n$.
Some of my experimental... | https://mathoverflow.net/users/37757 | Finding maximum of a function with unfixed number of variables | Allow the variables to be zero, and take $k=n$.
If $a\_i,a\_j$ are changed to $a\_i-1,a\_j+1$, the function changes by $6(a\_j-a\_i+1)(n-2a\_i-2a\_j-1)$. From this, everything follows.
If there are four non-zero values, the smallest two of them sum to at most $n/2$, so it is advantageous to move them apart until on... | 1 | https://mathoverflow.net/users/9025 | 190447 | 93,721 |
https://mathoverflow.net/questions/189446 | 3 | The Dedekind zeta function for a number field $K$ is defined as
$\zeta\_K(s)=\sum\_{I\subset O\_K} (N\_{K/\mathbb{Q}}(I))^{-s}$.
By attaching a Hecke character $\psi$, we can define $L(s,\psi)=\sum\_{I\subset O\_K} \psi(I)(N\_{K/\mathbb{Q}}(I))^{-s}$.
Is there any way to calculate the special value of such $L(s,\ps... | https://mathoverflow.net/users/32631 | Special values of Hecke L-function | I think the following Magma code gives you what you want:
```
> K:=QuadraticField(3);
> G:=HeckeCharacterGroup(1*Integers(K),[1,2]);
> psi:=G.1;
> L:=LSeries(psi);
> Evaluate(L,1);
```
Note that in fact $L(\psi)=L(\chi\_{-3})\cdot L(\chi\_{-4})$.
```
> K:=NumberField(Polynomial([-1,-3,2,1]));
> [Sign(Real(e)) :... | 6 | https://mathoverflow.net/users/52997 | 190448 | 93,722 |
https://mathoverflow.net/questions/190455 | 2 | Consider the projection map
$$\pi: X = V(t\_0 f + t\_1 gh) \to \mathbf P^1,$$
where $[t\_0: t\_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x\_0, \dots, x\_n)$ is a homogeneous polynomial of some degree $d$ such that $V(f) \subset \mathbf P^n$ is smooth and $g$ and $h$ satisfy $\deg g+\deg h = d$ and the... | https://mathoverflow.net/users/18116 | Number of singular fibers in families of hypersurfaces | Definitely there is an upper bound for the degree, namely the degree $(n+1)(d-1)^{n-1}$ of the discriminant hypersurface $\Delta$ in the projective space $\mathbb{P} S\_d$ associated to the finite dimensional $k$-vector space $S\_d = k[x\_0,\dots,x\_n]\_d$ of degree $d$ homogeneous polynomials in $x\_0,\dots,x\_n$. If ... | 4 | https://mathoverflow.net/users/13265 | 190459 | 93,728 |
https://mathoverflow.net/questions/190467 | 2 | Fix a prime number $p$. Is there a sequence $\{a\_k\}\_{k \in \mathbb{N}}$ of real numbers with $$\lim\_{k \to \infty} a\_k = 0$$
such that for any finite $p$-group $G$, and any subgroup $H \leq G$ with $[G : H] \geq k$ we have that $$\frac{|\bigcup\_{g \in G} gHg^{-1}|}{|G|} \leq a\_k ?$$
| https://mathoverflow.net/users/38889 | Union of conjugates in p-groups | I think the answer is no for all $p$.
For $p=2$, let $G$ be a dihedral group, and $H$ a noncentral subgroup of order $2$. Then the the fraction in question is about $1/4$.
For odd $p$ and $k>0$, let $G\_k$ be the metacyclic group $$\langle a,b \mid a^{p^{k+1}}=1, b^{p^k}=1, a^b=a^{1+p}\rangle,$$
and $H = \langle b ... | 10 | https://mathoverflow.net/users/35840 | 190473 | 93,735 |
https://mathoverflow.net/questions/190464 | 12 | Let $P,Q$ be partially ordered sets (posets). We consider the set $\text{Hom}(P,Q)$ of order-preserving functions $f:P\to Q$. (We call a function $f:P\to Q$ order preserving if $x\leq y$ in $P$ implies $f(x)\leq f(y)$ in $Q$.)
There is a natural ordering relation on $\text{Hom}(P,Q)$ given by $f\leq g$ if and only i... | https://mathoverflow.net/users/nan | Is the homomorphism poset directed if the codomain is directed? | The answer is negative.
Let $P=\omega\_1$, and $Q$ be the disjoint union of
* $A=A\_0\mathbin{\dot\cup}A\_1$, where each $A\_i$ is a copy of $\omega\_1$,
* $B$, which consists of finite subsets of $A$ that intersect both $A\_0$ and $A\_1$, ordered by inclusion.
If $a\in A$ and $b\in B$, we put $a\le b$ iff there ... | 10 | https://mathoverflow.net/users/12705 | 190475 | 93,736 |
https://mathoverflow.net/questions/190454 | 0 | As I said previously, I have some problems in the theory of valuations and places.
Let L/K be a finite (say) Galois extension, F a place of L, and v a valuation of L.
I denote by l and k the residue field of F and of the restriction of F to K resp.
It is known that the map from the decomposition group D of F to the ... | https://mathoverflow.net/users/62826 | Homologue of the Inertia group and of the Frobenius theorem for the group of values of a valuation | D consists of the automorphisms which preserve the valuation v of L, by definition of decomposition group. In the case of a discrete valuation, D has a filtration by "ramification groups" given by valuation of (sx/x - 1) for s in D (where x is a uniformizer for v). See Serre, "Local Fields".
| 1 | https://mathoverflow.net/users/59248 | 190484 | 93,739 |
https://mathoverflow.net/questions/190472 | 3 | The question is about well-posedness of heat equation
$$
\frac{\partial\Theta}{\partial t}=\alpha^2\Delta\Theta+p(t)\delta(x-u(t))\delta(y-v(t)),~~ (x,y,t)\in\Omega\times[0,T],
$$
subjected to boundary conditions
$$
\Theta=0,~~ (x,y,t)\in\partial\Omega\times[0,T],
$$
and initial condition
$$
\Theta(x,y,0)=\Theta\_0(x,y... | https://mathoverflow.net/users/63858 | Well-posedness of heat equation with distributional right hand side | Let $E\_t:=e^{t\alpha^2\Delta}$ be the solution operator to the heat equation
$$ \frac{\partial \Theta}{\partial t} = \alpha^2 \Delta \Theta$$
subject to the initial and boundary contitions above. Then the solution to your equation is given by
$$ \Theta\_t = \Theta\_0 +\int\_0^t E\_{t-s} T \,\mathrm{d} s.$$
Here th... | 0 | https://mathoverflow.net/users/16702 | 190485 | 93,740 |
https://mathoverflow.net/questions/190487 | 9 | We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$.
It may even be possible to write an explicit formula for the general Mobius action on any given circle:
$$ |z-z\_0|=r \mapsto \left|\frac{az+b}{cz+d} -z\_0 \right|=r$$
... | https://mathoverflow.net/users/1358 | Geometry of the space of circles in the Euclidean plane | Things will simplify if you just consider the circles on the Riemann sphere $S^2 = \mathbb{C}\cup\{\infty\}$, for your space is simply the space of circles on the sphere (with the lines in $\mathbb{C}$ just becoming the circles through $\infty$). Thus, your extended space of circles becomes $M^3 = \mathrm{PSL}(2,\mathb... | 21 | https://mathoverflow.net/users/13972 | 190491 | 93,741 |
https://mathoverflow.net/questions/190495 | 9 | It is well known that the moduli space of flat connections over a closed manifold $M$ can be identified with the representation space $Hom(\pi\_1(M), G) / G$.
Furthermore, Atiyah and Bott (1983) showed that over a closed surface the moduli space of (central) Yang-Mills connections can be seen as the space of represe... | https://mathoverflow.net/users/17047 | Moduli spaces of connections as representation spaces | Narasimhan and Seshadri proved that the moduli space of semi-stable vector bundles of rank $r$ on a compact Riemann surface is canonically isomorphic to the moduli space of unitary representations of degree $r$. This has been extended to projective manifolds by Donaldson, then to compact Kähler manifolds by Uhlenbeck-Y... | 10 | https://mathoverflow.net/users/40297 | 190498 | 93,744 |
https://mathoverflow.net/questions/190469 | 1 | Let $M$ be a compact manifold.
In their paper "On the Floer Homology of Cotangent Bundles", A. ABBONDANDOLO and M. SCHWARZ define the Floer homology of $T^\*M$ by looking at 1-periodic Hamiltonian orbits on the space of smooth loops on $T^\*M$ and show that the resulting Floer homology is isomorphic to the singular hom... | https://mathoverflow.net/users/61515 | floer homology and viterbo's theorem | I think that the Floer complex decomposes as a direct sum of over the *conjugacy classes* of $\pi\_1(M)$, no? Since the manifold on which we are doing infinite-dimensional Morse theory is of unbased, not based, loops.
In Theorem 3.1 of [this](http://arxiv.org/abs/math/0408280) version of the paper you refer to, the l... | 2 | https://mathoverflow.net/users/20391 | 190506 | 93,747 |
https://mathoverflow.net/questions/190504 | 7 | Set theory ZFC- is ZFC without power set, but with replacement. It does not imply the collection axiom scheme, as discussed in <http://jdh.hamkins.org/what-is-the-theory-zfc-without-power-set/>
Does consistency of ZFC- imply consistency of that plus collection? I do not mind taking all of ZFC- as metatheory.
| https://mathoverflow.net/users/38783 | Consistency of the collection axiom scheme compared to replacement | The answer is yes, because you can go to $L$.
More specifically, if $M$ is a model of ZFC-, defined with
replacement, then consider $L^M$, the class of sets that $M$
thinks arise in the constructible hierarchy. It is not difficult
to see that $M$ thinks $L\_\alpha$ exists for every ordinal
$\alpha$ that it has, and u... | 7 | https://mathoverflow.net/users/1946 | 190509 | 93,749 |
https://mathoverflow.net/questions/190508 | 4 | Fix $n$ and consider the polytope given by the inequalities
$$x\_i\leq x\_j, \text{ and } 0 \leq x\_i \leq a\_i \text{ for all } 1\leq i<j \leq n,$$
where $a\_i \leq a\_i\leq \dots \leq a\_n$ are fixed positive integers.
Has this family of polytopes been studied anywhere? Do they have a name?
Is there a general form... | https://mathoverflow.net/users/1056 | Marked chain polytope, has this been studied? | Set $x\_i=y\_1+\cdots+y\_i$ and $a\_i=x\_1+\cdots+x\_i$. We then we get the so-called "Pitman-Stanley polytope" (see Pitman and Stanley's *[A polytope related to empirical distributions, plane trees, parking functions, and the associahedron](https://arxiv.org/abs/math/9908029))*.
| 9 | https://mathoverflow.net/users/2807 | 190511 | 93,750 |
https://mathoverflow.net/questions/190477 | 6 | Zbierski "Models for Higher Order Arithmetics" (BULL. DE L'ACAD. POLONAISE DES SCIENCES Serie des sciences math., astr. et phys. - Vol. XIX, No. 7, 1971) defines ZF$\_n$ as ZFC with the power set axiom limited to $n$ successive power sets starting from the natural numbers $\mathbb{N}$. Note this includes the axiom of c... | https://mathoverflow.net/users/38783 | Higher order arithmetic and fragments of ZFC | The usual way to prove something like this is to take a model of $(n+1)$-st order arithmetic $\mathcal{A}$ and interpret a model of set theory $\mathcal{V}$ in $\mathcal{A}$ such a the usual interpretation of $(n+1)$-st order arithmetic in $\mathcal{V}$ leads to a model isomorphic to $\mathcal{A}$. There are a few reci... | 3 | https://mathoverflow.net/users/2000 | 190516 | 93,753 |
https://mathoverflow.net/questions/190478 | 6 | Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and $H$ be a self-adjoint Schrödinger operator $H = -\frac{d^2}{dx^2} + V$ and $V \in C^{\infty}$. Now assume that the Schrödi... | https://mathoverflow.net/users/nan | Domains of raising and lowering operators in QM | In the applications of the method that I know of, one usually doesn't make explicit use of the *operators* $A, A^\*$; rather, this is a method to go from the original potential $V=W^2-W'$ to a new potential $W^2+W'$ that has an extra eigenvalue.
However, if the original operator $H$ has Dirichlet boundary conditions ... | 2 | https://mathoverflow.net/users/48839 | 190517 | 93,754 |
https://mathoverflow.net/questions/190489 | 4 | Let $q = q(z)\,dz^2$ be a quadratic differential on the unit disk $D(1)$, normalized so that $\int\_{D(1)}|q| = 1$. I have a fairly convoluted argument that for any smaller disk $D(r)$ with $r < 1$, there is an upper bound on the area:
$$
\int\_{D\_r} |q| \le r^2.
$$
(The optimal quadratic differential is just $dz^2$.)... | https://mathoverflow.net/users/5010 | Bound areas of disks with respect to a quadratic differential | The family of averages of $|q|$ on $D(r)$ is indeed non-decreasing. To see this, let $A(r)$ be the average of $|q|$ on $D(r)$. Let $r<\rho$ Then $A(r)=v(r)$, where
$$v(z)=\frac{1}{\pi}\int\_{0}^{2\pi} \int\_{0}^{1}|q(zse^{it})|s \, ds \, dt.$$
Now, such a $v$ is subharmonic on $D(\rho)$ (see Ransford's book "Potential ... | 2 | https://mathoverflow.net/users/1162 | 190522 | 93,755 |
https://mathoverflow.net/questions/190486 | 7 | By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}\_2(V) := \{\{a,b\}: a\neq b \in V\}$. A *graph homomorphism* between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, w\}\in E(G)$ implies $\{f(v), f(w)\} \in E(H)$.
We say that a graph homomorphism $u:G\to H$ is *universal* ... | https://mathoverflow.net/users/8628 | Universal graph homomorphisms | Ok, try again.
Let $G\_k=W\_{2k+5}\cup W\_{2k+1}$, where $W\_k$ is the wheel with $k$ spokes. Then for $k\ge 3$, every homomorphism $G\_k\to G\_{k-1}$ fixes the hub of the smaller wheel as there is no homomorphism $W\_{2k+1}\to W\_{2k+3}$, and every homomorphism $W\_{2k+1}\to W\_{2k-1}$ fixes the hub.
Now let $\phi... | 7 | https://mathoverflow.net/users/12487 | 190524 | 93,756 |
https://mathoverflow.net/questions/190518 | 2 | Let $\mathbb{F}\_q$ be a finite field with $q=p^f$ elements. I need to know when $\mathrm{PSL}\_2(\mathbb{F}\_q)$ contains the group $D\_{(q+1)/2}$, where by $D\_n$ I mean the dihedral group of order $2n$. Also if such a classification of $q$ exists, can we write explicit generators of $D\_{(q+1)/2}$?
Thanks
| https://mathoverflow.net/users/8419 | Dihedral subgroups of $\mathrm{PSL}_2(\mathbb{F}_q)$ | The answer is indeed "always" when $q$ is odd, though the analysis when $q \equiv 1$ (mod $4$) and $q \equiv -1$ (mod $4$) differs. If $q \equiv 1$ (mod $4$), then $G = {\rm PSL}(2,q)$ has a cyclic Hall subgroup $H$ of odd order $\frac{q+1}{2}$ which is the centralizer of all its non-identity subgroups. All odd order s... | 6 | https://mathoverflow.net/users/14450 | 190527 | 93,757 |
https://mathoverflow.net/questions/189377 | 3 | In the question [Fixed points and universal maps for posets](https://mathoverflow.net/questions/180252/fixed-points-and-universal-maps-for-posets), we find the following definition: if $P, Q$ are partially ordered sets, an order-preserving map $u:P\to Q$ is said to be *universal* if for every order-preserving map $f:P\... | https://mathoverflow.net/users/8628 | "Universal maps" as a universal property | The full categorical definition of universal morphism was given in Włodzimierz Holsztyński, Universal Mappings and Fixed Points Theorems, Bull Acad Polon Sci, v.XV, No 7, 1967, pp.433-438:
**DEFINITION.** A morphism $\ u: Y\rightarrow X\ $ in category $\ K\ $ is *universal* $\ \Leftarrow:\Rightarrow\ $ for any morp... | 4 | https://mathoverflow.net/users/8385 | 190532 | 93,758 |
https://mathoverflow.net/questions/188855 | 5 | Let $\mathcal{M}$ be a rank-$d$ matroid on $[n]$. Say a matroid $\mathcal{N}$ is a *relaxation* of $\mathcal{M}$ if $\mathrm{rank}(\mathcal{N})=d$, $\mathrm{groundset}(\mathcal{N})=[n]$, and every independent set of $\mathcal{M}$ is an independent set of $\mathcal{N}$ (observe that this notion of relaxation is in the l... | https://mathoverflow.net/users/15054 | Matroids relaxations of a given matroid | You are describing what are usually called "weak maps" between matroids.
So a bijection $\varphi: E(\mathcal{N}) \to E(\mathcal{M})$ is called a *weak map* if for every independent set $I$ of $\mathcal{M}$ the inverse image $\varphi^{-1}(I)$ is independent in $\mathcal{N}$. In your case, you are asking when the *iden... | 4 | https://mathoverflow.net/users/1492 | 190537 | 93,762 |
https://mathoverflow.net/questions/190530 | 13 | Is there an elliptic curve over $\mathbb{C}[t, t^{-1}]$ that has a nonconstant $j$-invariant? What is an equation for such a curve, if it exists?
| https://mathoverflow.net/users/63877 | Are there nonisotrivial elliptic curves over $\mathbb{G}_m$? | Let me give an analytic argument, to complement Noam's algebraic one. Suppose $E\to \mathbb{C}^\*$ is an elliptic curve. Then pulling back $E$ along the universal covering map (also known as the exponential map) $\mathbb{C}\to \mathbb{C}^\*$, one obtains an elliptic curve $\tilde E\to \mathbb{C}$. Choosing a basis for ... | 23 | https://mathoverflow.net/users/6950 | 190538 | 93,763 |
https://mathoverflow.net/questions/190528 | 12 | Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a "non trivial" solution of the Cauchy functional equation, i.e. $f$ is not of the form $f(x)=cx$ for any $c\in\mathbb{R}$ and satisfies the relation
$$f(x+y)=f(x)+f(y)\quad\forall\,x,y\in\mathbb{R}$$
Define
$$A=\{x\in\mathbb{R}:f(x)\geq 0\},\quad B=\{x\in\mathbb{R}:f(x)< 0\}$... | https://mathoverflow.net/users/54552 | Partitions of $\mathbb{R}^+$ into subset closed by sum and product | Yes, it is possible to construct such a partition. See [this](http://math.stanford.edu/~dankane/algebraicallycloseddecomposition.pdf) note by Daniel M. Kane. Any proof would have to involve the axiom of choice, otherwise you can't partition the positive reals into subsets closed under addition, let alone both addition ... | 8 | https://mathoverflow.net/users/2384 | 190539 | 93,764 |
https://mathoverflow.net/questions/185247 | 6 | There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C}-$vector space.
Is there any similar pictorial models for affine Grassmannian of other types? Any references?
| https://mathoverflow.net/users/7780 | Lattice model for Affine Grassmannians of non type A | For any $ G$, you can always embed the affine Grassmannian $Gr\_G$ into the space of $\mathbb C [[t]]$-lattices in $ \mathfrak{g} \otimes \mathbb C ((t))$. I'm pretty sure that this appears in Lusztig's paper.
Alternatively, for any $ G $, there is an embedding $ Gr\_G \rightarrow \prod\_{\lambda} \{ \text{lattices i... | 6 | https://mathoverflow.net/users/438 | 190547 | 93,767 |
https://mathoverflow.net/questions/190552 | 1 | Let $G$ be a finite, simple, loopless graph with $|V(G)|=n$. We define its *edge density* as $$ed(G) := \frac{|E(G)|}{n \choose 2}.$$
Moreover we set $$d\_n := \text{max}\big\{ed(G): G \text{ is a triangle-free graph with } V(G) = \{1,\ldots, n\} \big\}.$$
Does $\lim\_{n\to\infty}d\_n$ exist and if yes, what is its... | https://mathoverflow.net/users/8628 | Edge density of triangle-free graphs | [Turan's theorem](http://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem) says that this limit exists and is equal to $\frac{1}{2}$. The graphs that achieve this are the complete bipartite graphs. Moreover, the theorem gives the answer for the more general question of $K\_{r+1}$-free graphs, where maximum edge density com... | 6 | https://mathoverflow.net/users/2384 | 190554 | 93,769 |
https://mathoverflow.net/questions/190549 | 20 | I have verified the following double sum is always an integer for $s$ up to $1000$ via Maple.
But I can not prove it. Proofs, hints, or references are all welcome.
Thanks!
$$\sum\_{m=s}^{2s}\sum\_{k=0}^{s} {2s\choose s}{s\choose k}{m\choose k}{k\choose m-s} \frac{1}{(s+1)(2k-1)(2m-2k-1)}$$
What I have known is that:
... | https://mathoverflow.net/users/6104 | How to prove that the following double sum is always an integer？ | Here is an attempt at an answer. We assume that the recurrence from my comment above holds [with a small correction] (a proof was obtained by Kevin using Zeilberger's algorithm, see the comment below):
$$ (7s+8)(s+4)(s+3)^2 a\_{s+3} - 4(56s^2+127s+57)(s+3)(s+2) a\_{s+2} $$
$$ - 16(7s^4-6s^3-121s^2-210s-90) a\_{s+1} + 1... | 17 | https://mathoverflow.net/users/21146 | 190559 | 93,772 |
https://mathoverflow.net/questions/189447 | 2 | According to Chapter 4 of Beilinson, Bernstein, and Deligne's "Faisceaux Pervers" (Asterisque 100, 1980) the inverse image $Rf^\*$ with respect to a finite morphism $f$ is right t-exact with respect to the perverse t-structure. Is there an example that shows that it is not necessarily t-exact? Is there a class of finit... | https://mathoverflow.net/users/12395 | Example to show that the inverse image under a finite morphism is not t-exact with respect to the perverse t-structure | By Noether normalisation any affine variety admits a finite map $f$ to affine space. Hence it is enough to find an example of an affine variety for which the constant sheaf is not perverse. (In this case $f^\* \mathbb{Q}\_{\mathbb{A}^n}[n]$ will provide a counter-example.)
Now it is easy to find examples where the co... | 4 | https://mathoverflow.net/users/919 | 190561 | 93,773 |
https://mathoverflow.net/questions/190546 | 6 | Let $\lambda$ be a partition of $ n$. Let $ SYT(\lambda) $ denote the set of standard Young tableaux of shape $ \lambda $.
For $ i = 1, \dots, n $, let me define permutations $ S\_i $ of the set $ SYT(\lambda) $. Let $ S\_n(T) $ be the Schutzenberger involution of $ T $. For $ i < n $, let $ S\_i(T) $ be the $i$th "... | https://mathoverflow.net/users/438 | Transitivity for Schutzenberger involutions on standard Young tableaux | Yes. It has been a while since I did this calculation but I think this is the gist of it:
Let $ s\_{1q} $ be the permutation induced by the partial Schützenberger involution $ S\_q(\cdot) $. Then define the permutations
$$
s\_{pq} := s\_{1q} s\_{1(q-p+1)} s\_{1q}.
$$
The $Q$-symbol of the RSK correspondence gives a b... | 4 | https://mathoverflow.net/users/48296 | 190563 | 93,774 |
https://mathoverflow.net/questions/189420 | 0 | Given a $z\in \mathbb{C}^N$, the DFT of $z$ is given for every $k\in [0,N-1]\_\mathbb{N}$ by
$$DFT\_z(k)=\frac{1}{N} \sum\_{j=0}^{N-1} z\_j\, \omega^{-k j}$$ where I have denoted by $\omega$ the $N$-th root of unit. That discrete transform is weel known and studied.
I was wondering if it is also well known and studie... | https://mathoverflow.net/users/41568 | A kind of Discrete Fourier Transform | There is no $O(N \log N)$ algorithm, because your computer will take $N^2$ computations to even read the entries of the matrix $z$. However, you definitely get $O(N^2 \log N)$ since this is the usual DFT done $N$ times and suitably restricted.
| 1 | https://mathoverflow.net/users/121 | 190564 | 93,775 |
https://mathoverflow.net/questions/22257 | 11 | A [determinantal process](http://terrytao.wordpress.com/2009/08/23/determinantal-processes/) on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x\_1, \dots, x\_n$ lying on the random set is $\det (K(x\_i, x\_j))\_{(i,j)}$. Examples of determinantal processes include the eigenvalu... | https://mathoverflow.net/users/1358 | Sampling from Sine Kernel and Airy Kernel | Just in case someone is still interested. The algorithm of Hough, Krishnapur, Peres and Virag was implemented here
<http://arxiv.org/abs/1404.0071>
in the case of eigenvalues of random matrices. The methodology uses Chebyshev approximations to do inverse sampling of the marginal distributions. The code is available... | 4 | https://mathoverflow.net/users/63889 | 190574 | 93,780 |
https://mathoverflow.net/questions/189410 | 1 | Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G\_1\sqcup G\_2$, where $G\_1 \to U$ is finite and $G\_2\to U$ lies over Spec $K$?
| https://mathoverflow.net/users/62832 | Decomposing quasi-finite separated group schemes | [At OP's suggestion, I am moving my comment to an answer. In the original question $U$ was the spectrum of a Henselian DVR $R$ with $K = \mathrm{Frac}(R)$.]
Define $G\_1$ to be the union of those connected components of $G$ whose images cover $U$. Then use [EGA $\text{IV}\_4$, 18.5.11 c)] to verify that $G\_1$ is a c... | 1 | https://mathoverflow.net/users/5498 | 190577 | 93,781 |
https://mathoverflow.net/questions/190578 | 5 | This is more or less a followup of [this](https://mathoverflow.net/questions/21667/are-there-any-homotopical-spaces) question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no faithful functor
$\bf{HoTop}$ $\to$ $\bf{Set}$... | https://mathoverflow.net/users/4721 | Can one make a category concrete by "enlarging the universe"? | By the Yoneda embedding, any category is a full subcategory of its category of presheaves. For this to be valid, the collection of objects of the category has to be a set. So if you enlarge the universe to make this the case then you can make any category concrete.
That being said, I "don't believe in" (the usefulnes... | 6 | https://mathoverflow.net/users/2733 | 190579 | 93,782 |
https://mathoverflow.net/questions/190571 | 7 | Suppose that $X/\mathbb{Q}\_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}\_k^M$ of the monodromy filtration on the $i$th cohomology groups $H^i\_\mathrm{et}(X\_{\overline{\mathbb{Q}}\_p},\mathbb{Q}\_\ell)$ are pure of w... | https://mathoverflow.net/users/13647 | "Weight-monodromy" for open varieties | Tony Scholl gave a talk at a conference in Warwick in 2013 on exactly this topic (his talk was called "Remarks on monodromy and weights"). He explained how to formulate a precise version of weight-monodromy for arbitrary varieties over p-adic fields (not necessarily proper or smooth).
The idea is that for *any* field... | 9 | https://mathoverflow.net/users/2481 | 190583 | 93,783 |
https://mathoverflow.net/questions/190586 | 13 | During a lecture I gave on Catalan numbers, I pointed out that that it
is possible to give a continuum number of combinatorial
interpretations of these numbers. See the solution to (f$^5$) on
page 54 of <http://math.mit.edu/~rstan/ec/catadd.pdf>. After the lecture
someone from the audience (I don't know who) asked me i... | https://mathoverflow.net/users/2807 | Lots of combinatorial interpretations of Catalan numbers | For a genuine answer, one needs a genuine definition of what counts as a combinatorial interpretation. One needs to exclude silly things like the following "interpretation $X$" for any set $X$ of real numbers: The number of pairs $(p,X)$ where the first component $p$ is a proper sequence of $2n$ parentheses and the sec... | 18 | https://mathoverflow.net/users/6794 | 190592 | 93,785 |
https://mathoverflow.net/questions/188968 | 10 | I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers and texts on the Burnside problem.
The reference for the paper is I. N. Sanov, "Solution of the Burnside problem for ex... | https://mathoverflow.net/users/39630 | Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4 | As it was pointed out by Alex Dugas, one can find Sanov theorem in Hall's book [here](http://books.google.com.pe/books?id=oyxnWF9ssI8C&lpg=PA429&pg=PA324#v=onepage&q&f=false), see Theorem 18.3.1. According to the author, the proof does not determine precisely the order of $B(n,4)$. However, it is quite easy to show tha... | 8 | https://mathoverflow.net/users/17845 | 190596 | 93,786 |
https://mathoverflow.net/questions/190589 | 0 | I am looking for reference books and research articles which cover analysis of uniformly convex and locally uniformly convex and strictly convex Banach spaces.
| https://mathoverflow.net/users/52860 | Book and Papers for properties of uniformly convex and locally uniformly convex and strictly convex Banach spaces. | I would recommend you a nice book by B. Beauzamy titled "Introduction to Banach spaces and their geometry". See also this article
<http://www.ams.org/journals/tran/1955-078-01/S0002-9947-1955-0066558-7/S0002-9947-1955-0066558-7.pdf>
You can extract useful information about requested topics.
Good luck.
| 0 | https://mathoverflow.net/users/63869 | 190602 | 93,789 |
https://mathoverflow.net/questions/190595 | 4 | $\def\PP{\mathbb{P}}$Let $z\_1$, $z\_2$, ..., $z\_n$ be points in $\PP^{k-1}$. I am interested in equations for when the $z\_i$ lie on a rational normal curve (or degeneration thereof.)
Specifically, let $s: \PP^{k-1} \to \PP^{\binom{k+1}{2} - 1}$ be the $2$-uple Veronese. If the $z\_i$ lie on a rational normal curv... | https://mathoverflow.net/users/297 | Equations for points to lie on a rational normal curve | Let me try the $\mathbb P^3$ case, and only look for a non-degenerate rational normal curve. Apply a linear map to move the first four of your points to be the standard four points ($[1,0,0,0]$,...) (if the four are coplanar, they're not on an rnc). Then apply the standard Cremona involution inverting each coordinate. ... | 5 | https://mathoverflow.net/users/nan | 190607 | 93,791 |
https://mathoverflow.net/questions/190594 | 0 | Suppose we have differentiable functions $F$, $f\_1, \dots, f\_n$, and $g\_1, \dots, g\_n$ satisfy the following relation
$$ F(x+y) = \sum\_{i=1}^n f\_i(x) g\_i(y).$$
What are the possible forms of $F$?
| https://mathoverflow.net/users/17276 | Functions with special separability | Differentiate with respect to $x$ and then with respect to $y$, you get
$$F'(x+y)=\sum\_i f\_i^\prime(x)g\_i(y)=\sum\_i f\_i(x)g\_i^\prime(y).$$
Substitute to the last equality some values $y\_j$ then
$$\sum\_i f\_i^\prime(x)g\_i(y\_j)=\sum\_i f\_i(x)g\_i^\prime(y\_j),\quad j=1,\ldots,n.$$
Assuming that $\det g\_i(y\_j... | 0 | https://mathoverflow.net/users/25510 | 190608 | 93,792 |
https://mathoverflow.net/questions/190616 | 5 | I would like to know information regarding the function $h(n)$ where $h(n)$ is the number of hamiltonian cycles the graph $K\_n$ has after removing the edges that make up a hamiltonian cycle of $K\_n$. Thank you very much in advance.
Regards.
| https://mathoverflow.net/users/24478 | Determining the number of hamiltonian paths of $K_n-C_n$ | This is sequence [A002816](http://oeis.org/A002816) at OEIS. You can find a recurrence and asymptotic expansion there.
There is a summation for it in [this old Stanford research report](http://i.stanford.edu/pub/cstr/reports/cs/tr/80/829/CS-TR-80-829.pdf) (end of page 6).
| 8 | https://mathoverflow.net/users/9025 | 190618 | 93,794 |
https://mathoverflow.net/questions/190617 | 15 | (This question is pure curiosity. Feel free to close it if you feel it is not appropriate for mathoverflow.)
In 2013 Zhang showed that there are infinitely many pairs of primes which are less that 70,000,000 apart. Since then, this result has been significantly improved. (See [here](http://michaelnielsen.org/polymath... | https://mathoverflow.net/users/12978 | Is there a known primitive recursive upper bound on the nth "Zhang prime" | One does not have to dig too deep into the arguments to answer this. Maynard's formulation (see his preprint [here](https://arxiv.org/abs/1311.4600)) shows that if $x$ is sufficiently large, there is a Zhang prime (or even a prime $p$ so that $p+k$ is prime with $k \leq 600$) between $x$ and $2x$. It follows that there... | 14 | https://mathoverflow.net/users/48142 | 190619 | 93,795 |
https://mathoverflow.net/questions/190614 | 16 | What is the strongest known natural large cardinal axiom compatible with $V = L$ (strongest in the sense that it implies all known "small" large cardinal axioms, where a large cardinal axiom is said to be "small" if it doesn't imply $V \neq L$)?
One candidate might be "there is an $\alpha$-Erdos cardinal for every co... | https://mathoverflow.net/users/17218 | Strongest large cardinal axiom compatible with $V = L$? | Maybe a comment:
In the paper ``[A large cardinal in the constructible universe](http://matwbn.icm.edu.pl/ksiazki/fm/fm69/fm6916.pdf)'' Silver shows that if $\kappa\to (\alpha)^{<\aleph\_0}$ for all countable $\alpha,$ then the same is true for $\kappa$ in $L$.
On the other hand, by a result of Rowbottom, $\kappa \... | 13 | https://mathoverflow.net/users/11115 | 190625 | 93,797 |
https://mathoverflow.net/questions/190641 | 0 | Let $F\_0,\ldots,F\_m\in\mathbb{Z}[x\_0,\ldots,x\_n]$ be polynomials with integer coefficients and let $p$ be a prime integer. Consider the two ideals: $$I\_0:=(F\_0,\ldots,F\_m)\subset \mathbb{Q}[x\_0,\ldots,x\_n]$$
and $$I\_p:=(F\_0\,\mathrm{mod}\,p,\ldots,F\_m\,\mathrm{mod}\,p)\subset \mathbb{Z}/(p)[x\_0,\ldots,x\_n... | https://mathoverflow.net/users/15606 | Codimension in zero and positive characteristic | That depends on what you mean by "codimension", but I think most interpretations give an answer of "no". For instance, let $m=n$ be $2$, let $F\_0$ be $p(px\_0-1)$, let $F\_1$ be $(px\_0 - 1)x\_1$, and let $F\_2$ be $(px\_0-1)x\_2$. Then the codimension of $\langle F\_0,F\_1,F\_2 \rangle$ in $\mathbb{Q}[x\_0,x\_1,x\_2]... | 4 | https://mathoverflow.net/users/13265 | 190642 | 93,804 |
https://mathoverflow.net/questions/190637 | 4 | I need some sort of classification (up to isometry) of spherical polygons (i.e. polygons in $\mathbb{S}^2$ whose edges are given by geodesics) subject to the interior angles and the perimeter of the polygon itself. Most likely this question is already solved and can be found in some literature, but I couldn't find some... | https://mathoverflow.net/users/63920 | Classification of spherical polygons | You should specify what you mean by a polygon: a broken line or a surface.
If we are talking about a broken line, then some classification is given here:
MR1703691 Kapovich, Michael; Millson, John J. On the moduli space of a spherical polygonal linkage. Canad. Math. Bull. 42 (1999), no. 3, 307–320.
If we are talkin... | 5 | https://mathoverflow.net/users/25510 | 190645 | 93,805 |
https://mathoverflow.net/questions/119355 | 16 | I am just curious: is there a published proof of the compactness of the [Hilbert cube](http://en.wikipedia.org/wiki/Hilbert_cube) that does not use the Axiom of Choice, or is it well known?
| https://mathoverflow.net/users/20995 | Compactness of the Hilbert cube without the Axiom of Choice | I have found this paper by Peter Loeb:
* Peter A. Loeb, [*A new proof of the Tychonoff Theorem*](http://www.jstor.org/stable/2314411), The American Mathematical Monthly **72** (1965), no. 7, 711--717.
Here is the theorem from this paper that implies that the usual Hilbert cube is compact without using the AC.
> ... | 1 | https://mathoverflow.net/users/20995 | 190651 | 93,808 |
https://mathoverflow.net/questions/190648 | 2 | Suppose we have a Clifford algebra $Cl(V,q)$, $V\simeq \mathbb{R}^n$ and $q$ non-degenerate bilinear form. Then every non-zero element of $V\subset Cl(V,q)$ invertible, but they are not the only ones (of course the product of invertible elements is again an invertible one). In fact, according to *Lawson, H. Blaine; Mic... | https://mathoverflow.net/users/62367 | Equation for non-invertible elements in Clifford algebras | For any finite-dimensional associative unital $\mathbb{R}$-algebra $A$, the set $S$ of noninvertible elements is the zero-set of a polynomial. Namely, let $f:A \to Hom (A,A)$ be the map $a \mapsto ( x \mapsto ax)$ to the linear endomorphisms of $a$. $f$ is an injective algebra homomorphism, since $f(a)=0$ means $ax =0$... | 11 | https://mathoverflow.net/users/9928 | 190652 | 93,809 |
https://mathoverflow.net/questions/190601 | -1 | Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$).
I denote $\mathcal{L}\in \upuparrows f \Leftrightarrow \forall L\in\prod\_{i\in N}\mathcal{L}\_i: f \cap \prod\_{i\in N} L\_i\ne\emptyset$ for every $N$-indexed fam... | https://mathoverflow.net/users/4086 | Lowering from filters to ultrafilters for an infinitary relation | Counter-example:
**Example**
There exists such an (infinite) set $N$ and $N$-ary relation $f$ that
$\mathcal{P} \in \upuparrows f$ but there are no indexed family $a \in
\prod\_{i \in N} \operatorname{atoms} \mathcal{P}\_i$ of atomic filter such that
$\forall A \in \operatorname{up} a : f \cap \prod A \ne \emptys... | -1 | https://mathoverflow.net/users/4086 | 190653 | 93,810 |
https://mathoverflow.net/questions/190585 | -2 | Let $U$ be a set. I denote $\mathfrak{A}$ the lattice of filters on $U$ ordered *reverse* to set theoretic inclusion of filters. I denote $\bigvee$ and $\bigwedge$ correspondingly the supremum and infimum on $\mathfrak{A}$.
Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation.
I denote $\langl... | https://mathoverflow.net/users/4086 | Expressing a value related to an infinitary relation through ultrafilters | There is a counter-example againt my conjecture.
I will denote $a\not\asymp b$ iff there is a non-least element which is below both $a$ and $b$.
Take $\mathcal{P} \in F$ from [the previous counter-example](https://mathoverflow.net/a/190653/4086). We have
$$ \forall a \in \prod\_{i \in \operatorname{dom} F} \operato... | -1 | https://mathoverflow.net/users/4086 | 190664 | 93,813 |
https://mathoverflow.net/questions/190667 | 3 | Let $f\colon X \rightarrow Y$ be a proper morphism with $Y$ Noetherian (and even affine, if you wish), and let $\mathscr{A} = \bigoplus\_{n \ge 0} \mathscr{A}\_n$ be a quasi-coherent graded $\mathscr{O}\_X$-algebra of finite type. Is the quasi-coherent graded $\mathscr{O}\_Y$-algebra $f\_\* \mathscr{A}$ of finite type?... | https://mathoverflow.net/users/53197 | Push-forward of a quasi-coherent graded algebra under a proper map | No, that is not true, and there are several similar (and standard) examples. For one, let $Y$ be $\text{Spec}(k)$, let $X$ be an elliptic curve over $k$ (there are many other varieties one could use), let $f$ be the unique $k$-morphism, let $\mathcal{L}$ be an invertible sheaf on $X$ that is algebraically equivalent to... | 9 | https://mathoverflow.net/users/13265 | 190673 | 93,818 |
https://mathoverflow.net/questions/176544 | 20 | I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this equations are always true" p.361 4.1.8, or "I do not know..." p.189 2.10.26. Are there counterexamples or proofs of these asum... | https://mathoverflow.net/users/56191 | Open problems in Federer's Geometric Measure Theory | For a full proof of the coarea inequality in the context of metric spaces using the results of Davies you may also be interested in the PhD thesis of L. Reichel. See Theorem 7.1 in
<http://e-collection.library.ethz.ch/eserv/eth:289/eth-289-02.pdf>
Regarding currents of low dimension or codimension, a proof of the ... | 9 | https://mathoverflow.net/users/63927 | 190677 | 93,820 |
https://mathoverflow.net/questions/186670 | 6 | I am trying to understand a step for proving that there exists large Hausdorff Lindelöf Spaces with points $G\_\delta$ using forcing. I am following Isaac Gorelic's "The Baire Category And Forcing Large Lindelöf Spaces With Points $G\_\delta$. (<http://www.jstor.org/discover/10.2307/2160344?uid=3737664&uid=2&uid=4&sid=... | https://mathoverflow.net/users/nan | Gorelic's Forcing for large Lindelöf spaces with points $G_\delta$ | I find that actually this
forcing is conceptually clear and not too technical.
Don't get bogged down with a few typos.
Fact 3 is about amalgamation of similar conditions.
Only $g\_\alpha^p$-s are important for amalgamation (since the rest is preassigned in the ground model).
There are only $\omega\_1$-many order typ... | 5 | https://mathoverflow.net/users/63936 | 190692 | 93,825 |
https://mathoverflow.net/questions/190696 | 4 | [I'd be very happy for a better question title, if anyone has any suggestions.]
I have a category $C$, two functors $F,G : C \to \mbox{Cat}$, a natural transformation $\alpha : F \to G$, and a section $\sigma : G \to F$ of $\alpha$. I am interested in the colimits of the composites $C \xrightarrow{F,G} \mbox{Cat} \xr... | https://mathoverflow.net/users/303 | To what extent are homotopy colimits over a weakly contractible category "determined by local data"? | Fix path-connected spaces $A \subset B$ with a retraction $r: B \to A$. Let ${\cal C}$ be the category $\* \leftarrow \* \rightarrow \*$, whose nerve is weakly contractible, and consider the map of diagrams
$$
(CB \leftarrow B \rightarrow CB) \to (CA \leftarrow A \to CA).
$$
Here $C(-)$ denotes the cone and the maps ar... | 4 | https://mathoverflow.net/users/360 | 190708 | 93,828 |
https://mathoverflow.net/questions/190711 | 12 | Are there existential theorems of ZFC, or PA say, with no witnesses?
Ie does there exist a formula $\phi$ such that ZFC $\vdash\exists x \phi(x)$, but for all numerals $\underline{n}$, ZFC $\nvdash \phi(\underline{n})$?
Can you give an example of such a formula?
| https://mathoverflow.net/users/63959 | Existential statement without witness | The answer is yes, provided these theories are consistent. For example, PA proves that there is a number $n$, such that if there is no proof of a contradiction in PA of size at most $n$, then there is no proof of a contradiction in PA at all. This is trivial if you think about it, since either there is a proof of a con... | 23 | https://mathoverflow.net/users/1946 | 190712 | 93,829 |
https://mathoverflow.net/questions/190580 | 4 | Let $\mathcal{C}$ be a small simplicial category and let $F\: : \:\mathcal{C}^{op}\to sSet$ be a simplicial functor, we denote with
$$
\int\_{\mathcal{C}}F
$$
the category where objects are triples $\left( x,\left[n \right],C \right)$ such that $x\in F\_{n}\left(C \right) $ and maps
$$
\left(\phi, f \right)\: : \: \... | https://mathoverflow.net/users/41970 | Adjoint of simplicial left Kan extension | It is generally true that if $\mathcal{E}$ is cotensored, then the underlying adjunction $L \dashv R$ lifts to enriched adjunction iff $R$ preserves cotensors. Let us assume that $\mathbb{V}$ is our base of enrichment. By classical Yoneda lemma, the objects:
$$\operatorname{Map}\_{\mathcal{E}}\left(L\left(F \right) ,E ... | 2 | https://mathoverflow.net/users/13480 | 190714 | 93,830 |
https://mathoverflow.net/questions/190718 | 6 | Suppose that $G$ is an algebraic group over a field $k$. Let $G^o$be the connected component of the identity. Since $G^0$ contains a $k$-rational point (the identity) therefore it is geometrically connected. I am wondering whether the remaining connected components of $G$ are also geometrically connected.
| https://mathoverflow.net/users/63961 | Geometrically connected components of an algebraic group | No, take for example $k=\mathbb{Q}$ and $G=\mu\_p$ for some prime $p>2$. Then $G$ has only two connected components, whereas $G\_\overline{\mathbb{Q}}$ has $p$.
| 12 | https://mathoverflow.net/users/2308 | 190720 | 93,835 |
https://mathoverflow.net/questions/190732 | 8 | Morita equivalence of algebras certainly don't preserve commutativity: even if $A$ is commutative there are plenty of noncommutative algebras which are Morita equivalent with $A$---for example all algebras of the form $M\_n(A)$ are good. What is the simplest example of an algebra which is not Morita equivalent to any c... | https://mathoverflow.net/users/24078 | Purely noncommutative algebra-Morita equivalence | An algebra is Morita equivalent to a commutative algebra iff it's Morita equivalent to its center, since the center is Morita invariant. So any representative of a nontrivial class in the Brauer group of the underlying field $k$ is a counterexample: for example, when $k = \mathbb{R}$ we can take the quaternions $\mathb... | 19 | https://mathoverflow.net/users/290 | 190736 | 93,840 |
https://mathoverflow.net/questions/190721 | 10 | What is known about free non-abelian subgroups in finitely generated associative algebras of polynomial growth (e.g., over finite fields, to avoid finite-dimensional free subgroups)? For example, are there examples of ideals I in the group algebra A of the free group F such that A/I has polynomial growth but the natura... | https://mathoverflow.net/users/63960 | Free subgroups in algebras of polynomial growth | Let $p$ be a prime number. Consider the algebra $A:=M\_2((\mathbb Z/p\mathbb Z)[t])$ and the group $G:=(\mathbb Z/p \mathbb Z) \ast (\mathbb Z/p \mathbb Z) = \langle a,b \mid a^p,b^p \rangle$.
Look at the homomorphism $\varphi \colon G \to A^{\times}$ which is defined by
$$\varphi(a):=\left( \begin{matrix} 1& t \\ ... | 10 | https://mathoverflow.net/users/8176 | 190740 | 93,842 |
https://mathoverflow.net/questions/190738 | 7 | In base 10, the number 3816547290 contains every digit **exactly once**. When I take the first N digits, that substring is divisible by N. For example, 381 is divisible by 3, 38165 is divisible by 5, etc. In base 10, 3816547290 is the only such number which meets the above criteria.
My question is, how can I find the... | https://mathoverflow.net/users/39774 | Numbers with all N-digit prefixes divisible by N | This seems to be a difficult question. <https://oeis.org/A111456> is "Pandigitals in some base with an extra property: each number formed by the first $i$ digits is divisible by $i$ (digits in the pandigital base)." A comment there says, "Finite? There are no more terms up to base 40. A probabilistic argument says high... | 8 | https://mathoverflow.net/users/3684 | 190741 | 93,843 |
https://mathoverflow.net/questions/190745 | 8 | Does every $\mathbb{P}^{19}\subset \mathbb{P}(\mathbb{C}^5\otimes\mathbb{C}^5)$
intersect the Segre variety of rank one matrices in at least a $\mathbb{P}^1$?
A naive dimension count suggests this is possible. The intersection is a
$3$-fold and I would be happy for any qualitative information about it.
| https://mathoverflow.net/users/21399 | Fano variety of lines on the Segre and the Grassmannian | Although dhy's comment does solve this problem, I thought I would point out another approach. Let $U$, $V$ and $W$ be $5$-dimensional $k$-vector spaces. Let $$B: U\times V \to W,$$ be a bilinear pairing. Landsberg's question, essentially, asks whether there exists a $2$-dimensional subspace, $S\subset U$, and a $1$-dim... | 9 | https://mathoverflow.net/users/13265 | 190752 | 93,846 |
https://mathoverflow.net/questions/190755 | 3 | I am working on formalizing software design using category theory.
However the most natural way for me to express what I want is with a Category where multiple morphisms can join into a single morphism.
Thinking diagrammatically, what I am talking about is multiple arrows (from different objects) merging into a sin... | https://mathoverflow.net/users/63981 | A Category-ish Structure with Morphism Domains containing Multiple Objects? | This structure is known as a multicategory. There are many references; e.g., see [Wikipedia](http://en.m.wikipedia.org/wiki/Multicategory) for a basic introduction, and read the nlab's page <http://ncatlab.org:8080/nlab/show/multicategory> as well. One very nice reference is the book [*Higher Operads, Higher Categories... | 12 | https://mathoverflow.net/users/nan | 190756 | 93,847 |
https://mathoverflow.net/questions/189320 | 3 | Maybe the question is too general, but nevertheless:
under what conditions on algebraic variety $X$, algebraic equivalence of divisors coincide with linear equivalence?
What are typical classes of varieties which have this property? As far as know, it is true for $\mathbb{P}^2$, but what makes this variety special... | https://mathoverflow.net/users/43459 | Algebraic equivalence vs linear equivalence | CW answer from the comment by Felipe Voloch:
>
> For smooth projective complex varieties, this is the same as the Picard variety (or equivalently the Albanese) is zero, almost by definition. As the dimension of the Picard variety is the first Betti number, here is your criterion: $b\_1=0$. — Felipe Voloch Dec 9 at ... | 2 | https://mathoverflow.net/users/21815 | 190769 | 93,853 |
https://mathoverflow.net/questions/190768 | 14 | In the study of singular spaces, topological methods like intersection cohomology have played an important role. They have led to the development of technology like perverse sheaves and these find widespread application in, for instance, representation theory. There are, however, another approach to singular spaces tha... | https://mathoverflow.net/users/63986 | Intersection Cohomology and $L^2$ cohomology | An important special case is Zucker's conjecture: the intersection cohomology of the minimal compactification $\bar{X}$ of an Hermitian locally symmetric variety $X$ is the L^2 cohomology of $X$ (with the metric coming from the symmetric space above). It was proved independently by Looijenga and Saper-Stern.
See thi... | 11 | https://mathoverflow.net/users/7878 | 190772 | 93,854 |
https://mathoverflow.net/questions/190713 | 2 | How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)?
What is known about the generation of random convex polygons (or polyhedra)?
In view of the comments, I will address the choice of method of... | https://mathoverflow.net/users/31310 | Measuring the Randomness and Statistics of Convex Polygons | I write my comment as an answer because it is a bit longer.
As Joseph O'Rourke indicated, the question needs to be made more precise because it is not clear (to me) what it asks. One way of generating a random polygon is to choose $n$ points independently according to some probability measure on $\newcommand{\bR}{\ma... | 2 | https://mathoverflow.net/users/20302 | 190773 | 93,855 |
https://mathoverflow.net/questions/190765 | 2 | Does there exist an integral domain $R$ of characteristic $p > 0$ that is perfect (i.e., $x \mapsto x^p$ is bijective on $R$) but not integrally closed in its field of fractions?
| https://mathoverflow.net/users/63877 | A perfect domain that is not integrally closed? | Let $S$ be the local ring of nodal curve, $R$ = inverse limit $Frob: S \to S$. For example:
* $k$ a perfect field,
* $f(x,y)=y^{p+1}-x^{p+1}(1+x)$,
* $R=k[x^{1/p^{\infty}},y^{1/p^{\infty}}]/(f^{1/p^{\infty}})$.
Here's a complete local example:
* $k$ a perfect field,
* $f(x,y)=y^{p}-x^{p}y-x^{p+1}$,
* $R=k[[x^{1/p... | 6 | https://mathoverflow.net/users/59248 | 190776 | 93,857 |
https://mathoverflow.net/questions/190729 | 1 | For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r\_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u\_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$
consider the (energy) functional
$$
E: C^2(T,\mathbb{S}^2) \to \mathbb{R},~~~ v\mapsto \int \limits \_T (\nabla v)^2\, dV~=~ -\int \limits \_T v\cdot \D... | https://mathoverflow.net/users/20804 | Smooth unit vector field on a tetrahedron to interpolate vertex constraints | I dont think there is an explizit expression for the solution. A $C^\infty$ approximation is
$$v(x):=\mathop{\text{argmin}}\_{y\in S^2} \sum\_{i=1}^4 \lambda\_i(x) d^2(y,u\_i)$$
where $\lambda\_i$ are the barycentric coordinates and $d$ is the spherical distance. This function is well-defined if $(u\_i)\_{i=1}^4$ lie i... | 2 | https://mathoverflow.net/users/35593 | 190779 | 93,859 |
https://mathoverflow.net/questions/189334 | 16 | I'm looking for a couple good textbooks covering differential algebra. I'm a prospective Ph.D. student, and this is potentially applicable to my specialization. As such, I'm not afraid of depth; I've got a few years to work through it all.
Specifically, I'm interested in the connections between differential algebra a... | https://mathoverflow.net/users/62805 | Differential Algebra Book | Except for Buium's book these suggestions mostly cover only algebraic versions of linear differential equations and this is only a limited view of the theory developed by Kolchin and others. Kaplansky remains, I think, the best introduction to the basic algebra in rings with differential operators. There is also Kolchi... | 7 | https://mathoverflow.net/users/20674 | 190790 | 93,864 |
https://mathoverflow.net/questions/190791 | 3 | Let $S$ be a surface in $\mathbb{R}^{3}$ such that every regular curve $\gamma\subset S$ has nowhere vanishing curvature, that is $\kappa(z)\neq 0$ for all $z\in \gamma$. Does this imply that $S$ is a part of a sphere?
| https://mathoverflow.net/users/36688 | A surface on which all regular curves have nowhere vanishing curvature | I'm going to assume that you mean the following property
>
> $S \subset \mathbb{R}^3$ has property (\*) if for any regular curve $\gamma\subset S$, the curvature vector $\vec{\kappa}$ of $\gamma$ as a curve in $\mathbb{R}^3$ is nowhere vanishing.
>
>
>
(if you mean the curvature of $\gamma$ as a curve in $S$,... | 5 | https://mathoverflow.net/users/1540 | 190794 | 93,865 |
https://mathoverflow.net/questions/190551 | 2 | Let us consider nine general points $p\_1,...,p\_9\in\mathbb{P}^2$ and the line $L = \left\langle p\_1,p\_2\right\rangle$. Take the standard Cremona $f\_1$ centred in $p\_3,p\_4,p\_5$, then $C\_1 = f\_1(L)$ is a conic through $p\_1,p\_2,p\_3,p\_4,p\_5$ (where I keep denoting by $p\_i$ the images of the contracted lines... | https://mathoverflow.net/users/nan | Standard plane Cremona transformation | We begin with the line $C\_2 = \left\langle p\_1,p\_2\right\rangle$, and consider the standard Cremona $f\_2:\mathbb{P}^2\dashrightarrow\mathbb{P}^2$ centred in $p\_7,p\_8,p\_9$. The curve $C\_3 = f\_2(C\_2)$ is a conic through $p\_1,...,p\_5$. We proceed recursively by taking at the step $i$ the standard Cremona $f\_{... | 2 | https://mathoverflow.net/users/14514 | 190795 | 93,866 |
https://mathoverflow.net/questions/190792 | 4 | Before I ask my question I will provide a brief introduction.
I came across the notion of Rademacher type while reading Assaf Naor's article *An introduction to the Ribe program*, which can be found here <http://arxiv.org/pdf/1205.5993v1.pdf>.
A Banach space $(X,||\cdot||\_X)$ is said to have *Rademacher type* $p... | https://mathoverflow.net/users/47343 | Rademacher type of a Banach space is always less than or equal to 2 | If you you use the second moment instead of the first moment to define type you can use the parallelogram law (or orthogonality of symmetric Bernoulli random variables) to get:
\begin{equation} \frac{1}{2^n}\sum\_{\varepsilon\_1,\ldots,\varepsilon\_n \in \{-1,+1\}} \left|\sum\_{i=1}^n \varepsilon\_i\right|^2 = n.\end... | 5 | https://mathoverflow.net/users/2554 | 190798 | 93,867 |
https://mathoverflow.net/questions/190804 | -1 | The following argument is from a [paper](http://arxiv.org/pdf/1305.3402.pdf) about the [Bendixson-Dulac Theorem](http://en.wikipedia.org/wiki/Bendixson%E2%80%93Dulac_theorem).
>
> Consider a smooth differential equation on the plane
> $$
> x'=g(x,y),\quad y'=h(x,y).
> $$
> Suppose there exists a function $D(x,y)$... | https://mathoverflow.net/users/nan | Integrating factors and integrability of an ODE system | $X=g\partial\_x + h\partial\_y$ is the vector field whose flow lines are wanted.
$\omega=hdx - gdy$ is a 1-form with kernel the span of $X$.
Also $D\omega$ has kernel the span of $X$ for any function $D$ which does not vanish anywhere. If $d(D\omega)=0$ (this is your condition) then $D\omega$ is a closed 1-fo... | 4 | https://mathoverflow.net/users/26935 | 190806 | 93,871 |
https://mathoverflow.net/questions/190809 | 13 | By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle answer queries of form $a\in b?$ for $\in$ being relation on $\Bbb N$ modelling $ZF$). My question then is: is it possibl... | https://mathoverflow.net/users/30186 | Is forcing computable? | Dan's comment below answers the question that was actually asked. Let me explain one way to look at it. If $M$ is any model of ZF and $T$ is any consistent computably axiomatizable theory, then we may look at the version of $T$ inside $M$. There will be some longest initial segment of the axiomatization of $T$ that $M$... | 11 | https://mathoverflow.net/users/1946 | 190810 | 93,873 |
https://mathoverflow.net/questions/176059 | 9 | I asked this question in MSE, but I did not received any answer, so I repeat it here:
<https://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property>
Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mat... | https://mathoverflow.net/users/36688 | A question on fixed point theory | Here is a partial affirmative answer using mod 2 Steenrod operations; the simplest case of this (for $n$ and $k$ even) is just a correction of the slightly incorrect answer originally posted by Włodzimierz Holsztyński. The result is that if $k+1$ and $n+1$ are both odd multiples of $2^d$ for some integer $d\geq 0$, the... | 4 | https://mathoverflow.net/users/75 | 190828 | 93,880 |
https://mathoverflow.net/questions/190771 | 1 | Trying to get a different angle for the question [Fixed points and universal maps for posets](https://mathoverflow.net/questions/180252/fixed-points-and-universal-maps-for-posets), I want to compare universal maps to a different kind of functions.
First recall that for posets $P,Q$ an order-preserving map $u:P\to Q$ ... | https://mathoverflow.net/users/8628 | Universal and left-factoring order-preserving maps | I claim that not every universal map is left-factoring. Let me first give a proposition that allows us to construct many universal maps, and then I will give a counterexample from the proposition. A poset $X$ shall be called a DCPO (directed complete partial order) if every directed subset of $X$ has a least upper boun... | 1 | https://mathoverflow.net/users/22277 | 190829 | 93,881 |
https://mathoverflow.net/questions/190821 | 3 | Let $f(x\_1,\dots,x\_{16})=(x\_1+x\_2+x\_3+x\_4)(x\_5+x\_6+x\_7+x\_8)(x\_9+x\_{10}+x\_{11}+x\_{12})(x\_{13}+x\_{14}+x\_{15}+x\_{16})\in\Bbb R[x]$.
Let $\mathcal{Z}$ be the zero set of $f$ in $\mathcal{C\_{16}}=\{0,1\}^{16}$.
Total degree of $f$ is $4$.
Can one show there is no polynomial $g$ of degree $\mathsf{de... | https://mathoverflow.net/users/10035 | Polynomials of low degree that clone polynomials of higher degree | Let $\{e\_k\}\_{1\le k\le 16}$ denote the standard basis of $\mathbb{R}^{16}$, and $x:=(x\_1,\dots,x\_{16})$. Let's consider the difference operator in the $k$-th variable, $\delta\_k:\mathbb{R}[x]\to\mathbb{R}[x ]$, that is $\delta\_kp(x):=p(x+e\_k)-p(x)$. So $$\delta\_{13}\delta\_9 \delta\_5 \delta\_1p(x)=\sum\_{\eps... | 1 | https://mathoverflow.net/users/6101 | 190842 | 93,884 |
https://mathoverflow.net/questions/190832 | 7 | $G$ is a cyclic group iff
$$ \forall H < G, \ \exists k, \ H = \{a^k : a \in G\}. $$
Is it right?
| https://mathoverflow.net/users/22954 | A proposition on cyclic group | A bit of searching revealed the following reference for this statement in question
>
> [F. Szasz, On cyclic groups](http://matwbn.icm.edu.pl/ksiazki/fm/fm43/fm43123.pdf), Fund. Math., 43(1956), 238-240
>
>
>
In the following more recent paper the authors proved a refinement. Let $k$ denote the number of subgro... | 8 | https://mathoverflow.net/users/2384 | 190853 | 93,887 |
https://mathoverflow.net/questions/190812 | 2 | Suppose $G$ is a finite group which acts on a $C^\*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$.
I thank you for the attention and the help.
| https://mathoverflow.net/users/47294 | Coaction of a group | The way I would write this is as follows and I would appreciate users calling to attention any errors.
Take a linear action $\Phi:A\times G\rightarrow A$ and extend it to a bilinear map
$$\Phi\_1:A\times \mathbb{C}G\rightarrow A.$$
Now using the universal property we have a linear map
$$\tilde{\Phi}:A\otimes \mathbb... | 4 | https://mathoverflow.net/users/35482 | 190857 | 93,888 |
https://mathoverflow.net/questions/190859 | 2 | I would like to ask a question about automorphisms of free products of groups.
More specifically, let $G = G\_1 \ast ... \ G\_n \ast F\_r$ where $F\_r$ is free group on r generators. We can define the group $Out(G)$ of outer automorphisms of $G$, as the quotient of automorphisms with the inner automorphisms.
Then we ... | https://mathoverflow.net/users/42001 | A subgroup of outer automorphisms group of a free product | **EDITED:** Thanks to @Mathieu for finding a bug.
The group you describe is actually isomorphic to $\text{Aut}(F\_r)$ (which, of course, has a well known relation to $\text{Out}(F\_r)$).
To see why, you have defined a subgroup of $\text{Aut}(G)$ which I will denote $A\_0$, consisting of all automorhisms which take... | 3 | https://mathoverflow.net/users/20787 | 190868 | 93,893 |
https://mathoverflow.net/questions/190863 | 9 | I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping.
Anyone has any tips?
| https://mathoverflow.net/users/64029 | Conformal map of polygon with circle segments | The mapping function is a solution of the Schwarz differential equation
$$\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^2=R(z),$$
where $R$ is a rational function with poles at the preimages of the vertices. The poles of $f$ are of second order, and the coefficients at the second order terms are determined by ... | 10 | https://mathoverflow.net/users/25510 | 190873 | 93,895 |
https://mathoverflow.net/questions/190837 | 29 | I would like to ask about, does there exists an entire function which is bounded on every line parallel to $x$ - axis , but unbounded on the $x$ - axis.
| https://mathoverflow.net/users/54245 | Entire function bounded at every line | Yes, there are such functions. Take a very narrow region $D$ containing the positive ray,
with nice boundary and such that $D$ intersects every any horizontal line other than the real line by a bounded interval.
Let $g$ be a conformal map of $D'$ onto the right half-plane, where
$D'\subset D$ is another similar region.... | 25 | https://mathoverflow.net/users/25510 | 190874 | 93,896 |
https://mathoverflow.net/questions/181242 | 1 | In [this MSE question/thread](https://math.stackexchange.com/questions/931782/a-conjecture-on-products-composition-of-pell-forms), I have been discussing the equation
$$
(x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{$\star$}
$$
where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to be false. Now, as a... | https://mathoverflow.net/users/19844 | Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution? | As posted in [my comment above,](https://mathoverflow.net/questions/181242/does-the-diophantine-equation-x2ay2u2bv2-p2cq2-admit-a-complete#comment454532_181242) the case $a=b=c=1$ is relatively trivial to solve, using existing (nearly "classical") solutions to the 2.2.4 Diophantine sums-of-squares equation $$X\_1^2 + X... | 1 | https://mathoverflow.net/users/19844 | 190884 | 93,899 |
https://mathoverflow.net/questions/190778 | 8 | Is there a nontrivial profinite group $G$ with a binary transitive relation $<$ such that
1. $x<y$ implies $x\neq y$, and for any different $x,y \in G$ either $x < y$ or $y < x$ and such that for any $x,y,z \in G$ we have that $x < y$ implies that $zx < zy$ (i.e., $<$ defines a left-invariant strict total order)
2. ... | https://mathoverflow.net/users/38889 | Is there a left-orderable profinite group? | There is no such ordering, which is compatible with the profinite topology in the following way: If $x<y$, then there are small neighbourhoods $U, V$ of $x, y$, such that $u<v$ for all $u\in U, v\in V$.
To see this note that If $x>1$, then $x^n>1$ for all $n>0$ and $x^n<1$ for all $n<0$. But in the pro-finite topolog... | 7 | https://mathoverflow.net/users/37555 | 190886 | 93,900 |
https://mathoverflow.net/questions/190802 | 15 | The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real valued on the critical line $s=1/2+it$, there is a zero of the derivative $\xi^\prime$ between each successive pair of zero... | https://mathoverflow.net/users/6756 | Zeros of the derivative of Riemann's $\xi$-function | In exercise 1 on page 443 of their book "Multiplicative Number Theory," Montgomery & Vaughan outline a proof of the statement:
"Assuming the Riemann Hypothesis, $\xi'(s)=0 \implies \mathrm{Re}(s)=1/2$."
Assuming RH, let $s=\sigma+it$ and let $\rho=\frac{1}{2}+i\gamma$ denote a zero of $\xi(s)$. The main idea of th... | 11 | https://mathoverflow.net/users/3659 | 190888 | 93,902 |
https://mathoverflow.net/questions/185354 | 7 | The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf\_{n\to\infty}p\_{n+k}-p\_{n}\sim k\log k}$?
Thanks in advance.
Edit December 16th 2014: I'd be interested in references about this topic as well.
| https://mathoverflow.net/users/13625 | What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$? | The situation is similar to that in your previous question [what would be the consequences on the distribution of primes of $\Lambda=\infty$?](https://mathoverflow.net/questions/184479/what-would-be-the-consequences-on-the-distribution-of-primes-of-lambda-infty) : by itself, not very much, because one only needs an arb... | 23 | https://mathoverflow.net/users/766 | 190892 | 93,903 |
https://mathoverflow.net/questions/190893 | 3 | The solvable ***Emma Lehmer quintic*** is given by,
$$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
with discriminant $D = (7 + 10 n + 5 n^2 + n^3)^2(25 + 25 n + 15 n^2 + 5 n^3 + n^4)^4$.
For prime $p=25 + 25 n + 15 n^2 + 5 n^3 + n^4$, w... | https://mathoverflow.net/users/12905 | Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$? | For question 1, the answer is yes, as shown by Emma Lehmer herself. (See [the paper here](http://www.ams.org/journals/mcom/1988-50-182/S0025-5718-1988-0929551-0/home.html), in particular, equation (5.8) on page 539.) In particular, Lehmer states that one can take
$$
a = \frac{\left(\frac{n}{5}\right) - n^{2}}{5}, \qu... | 5 | https://mathoverflow.net/users/48142 | 190896 | 93,905 |
https://mathoverflow.net/questions/190668 | 14 | A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture).
(The question was first asked at [math.SE](https://math.stackexchange.com/questions/958395/maximizing-curious-symmetric-function-from-simple-combinatorics "math.... | https://mathoverflow.net/users/63938 | Conjecture on maximum of symmetric combinatoric function | Indeed, I have just made things overly complicated in my approach. Thanks to JiK, I now see it fairly clearly.
Let $(\{S\},P\_S)$ be our probability space and let $X\_k$ be the event $S\ni k$. Let $u\_k=EX\_k$. Note that at each $S$ there are exactly $m$ values $X\_k=1$ and the rest are $0$. What we need to control i... | 17 | https://mathoverflow.net/users/1131 | 190900 | 93,908 |
https://mathoverflow.net/questions/190899 | 8 | The higher cohomology of coherent sheaves vanish on Stein spaces (both complex and p-adic). In the case when the space ($X$) is a curve and we're working in the complex world, this shows that all holomorphic line bundles are trivial (The exponential exact sequence and the fact that $H^2(X,\mathbb{Z})= 0$). Is the same ... | https://mathoverflow.net/users/64044 | p-adic Stein spaces | The answer is no. In fact, a theorem of Lazard shows that the open unit disk has a non-trivial Picard group when the ground field is a non maximally complete ultrametric valued field.
See Proposition 6 in [Les zéros d'une fonction analytique d'une variable sur un corps valué complet](http://www.numdam.org/item?id=PMI... | 11 | https://mathoverflow.net/users/10696 | 190903 | 93,910 |
https://mathoverflow.net/questions/190933 | 1 | Let $\mathcal{C}$ be any category and let $A, B$ be objects.
A *retraction* is a morphism $r: A\to B$ such that there is $s:B\to A$ such that $r\circ s:B\to B$ is the identity.
A morphism $l: A\to B$ is said to be *left-factoring* if for any $Z\in \mathbf{Ob}(\mathcal{C})$ and any morphism $f: Z\to B$ there is $h: Z... | https://mathoverflow.net/users/nan | Retractions and left-factoring morphisms | Yes - these terms are equivalent.
First, it is easy to see that any left-factoring morphism is a retraction: Suppose $l: A \to B$ and pick $Z:= B$ in your definition above and $f:= \mathsf{id}\_B$. Since $l$ is left-factoring there is $h:B\to A$ sich that $\mathsf{id}\_B = l\circ h$. So $l$ is a retraction.
Convers... | 3 | https://mathoverflow.net/users/8628 | 190934 | 93,920 |
https://mathoverflow.net/questions/190913 | 5 | Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set
of such a polynomial gives a real curve in $\mathbb{R} \mathbb{P}^2$. Let me
define the space $X$ to be the space of real curves $[f]$ and a marked point
$p$, ... | https://mathoverflow.net/users/4463 | Is the space of real conics with a singular point an orientable manifold? | Yes, it is a smooth manifold. No, it is not orientable.
For the first, just think geometrically, i.e., without bases: Fix a $3$-dimensional vector space $V$ and consider the homogeneous quadratic polynomials on it, which is a $6$-dimensional vector space isomorphic to $\mathsf{S}^2(V^\ast)$. Let $\mathbb{P}V$ be the ... | 10 | https://mathoverflow.net/users/13972 | 190942 | 93,921 |
https://mathoverflow.net/questions/190945 | 4 | Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found it hard to get the actual historical source of an example where this line of thought fails.
More precisely, given a smo... | https://mathoverflow.net/users/18974 | Cap product à la Poincaré | You need to add the word "oriented", in order to assign signs consistently to the intersections, and "compact" so that there will be finitely many intersections. To see the issue for unoriented manifolds, think about $\mathbb{RP}^2$. We have $H\_0(\mathbb{RP}^2) \cong \mathbb{Z}$ and $H\_1(\mathbb{RP}^2) \cong \mathbb{... | 5 | https://mathoverflow.net/users/297 | 190951 | 93,924 |
https://mathoverflow.net/questions/190895 | 4 | Consider the SDE:
\begin{equation}
dv\_t = k(\theta - v\_t) dt + \xi \sqrt{v\_t} dW^{v}\_{t}
\end{equation}
It describes a process $v\_t$ which is a strictly positive if the drift is stronger enough, i.e. if drift parameters ($k$, the speed of mean-reverting, and $\theta$, mean-reverting level) and the Vol-of-Vol $\xi$... | https://mathoverflow.net/users/61342 | Do we need Feller condition if the process jumps? | The Feller condition is hardly satisfied in the market, mainly because having $k\theta >\frac 12 \xi^2$ implies a higher mean reversion which reduces the variance of the stochastic volatility, this in turn reduces the convexity of the smile, and in order to increase this convexity you have to increase your vol of vol $... | 4 | https://mathoverflow.net/users/30889 | 190956 | 93,926 |
https://mathoverflow.net/questions/190952 | 4 | I´m studying Huybrechts book "Fourier-Mukai transforms in algebraic geometry" and I came up with the following: as an example of semi orthogonal decomposition of a triangulated category $\mathcal{D}$ it is given $\mathcal{D}\_1=\mathcal{D}´^{\perp}$, $\mathcal{D}\_2=\mathcal{D}´$ where $\mathcal{D}´$ is an admissible (... | https://mathoverflow.net/users/56703 | Is the orthogonal complement of an admissible subcategory admissible itself? | If $D$ is arbitrary then $D\_1$ is not necessarily admissible. However, if $D$ is saturated, then $D\_1$ is admissible. See the paper of Bondal and Kapranov for a proof.
| 3 | https://mathoverflow.net/users/4428 | 190957 | 93,927 |
https://mathoverflow.net/questions/188920 | 3 | Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which are indexed by the Weyl group elements?
I was thinking about constructing a curve that connects two $T-$fixed points... | https://mathoverflow.net/users/7780 | Closure relations between Bruhat cells on the flag variety | The place I've read about this is in the paper "Schubert cells, and the cohomology of the spaces $G/P$" by Berstein, Gelfand, and Gelfand. The flag variety can be identified with the projectivization of a representation of $G$, $V\_{\lambda}$. Then you compute the limit of the 1-psg $exp(t\cdot e\_{\alpha})$ as it acts... | 2 | https://mathoverflow.net/users/64068 | 190961 | 93,931 |
https://mathoverflow.net/questions/190978 | 3 | In a question and an answer at MO, Joel David Hamkins showed that (if ZFC is consistent) there are models of ZFC in which $V\neq HOD$ and every $\Sigma\_2$-definable set has a definable member.
Let $\mathfrak{M}$ be such a model.
My question is: Can such a model $\mathfrak{M}$ satisfy further the following:
(\*) Ev... | https://mathoverflow.net/users/38200 | A question on models of set theory and Lebesgue measure | This is impossible; there is no model of ZFC like that. The reason is that the set of non-measurable sets of reals (or non-Baire sets, respectively) is definable, and moreover $\Sigma\_2$ definable; so under the first part of your conditions, it would have a definable member, which would violate the second part of your... | 5 | https://mathoverflow.net/users/1946 | 190980 | 93,937 |
https://mathoverflow.net/questions/190902 | 17 | Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "ordinal-categorical" if, whenever $M$ and $N$ are standard models of $T$ sharing the same ordinals, one has $M = N$. For example, if $T$ proves $V = L$, then $T$ is ordinal-categorical. I think the same is true if $T$ proves $V = L(0^\sharp)$.... | https://mathoverflow.net/users/17218 | Classifying set theories whose standard models sharing the same ordinals are equal |
>
> In this edit, the statement of Friedman's theorem is reformulated (the previous formulation was incorrectly stated). Thanks to Dmytro Taranovsky and Farmer Schultzenberg for pointing out the blooper. *See also Remark 2 for recent progress (January 2023) on this topic by Schultzenberg and Taranovsky.*
>
>
>
A... | 24 | https://mathoverflow.net/users/9269 | 190982 | 93,938 |
https://mathoverflow.net/questions/190989 | 5 | How do we interpret Barry Mazur's quote of
>
> Modular forms are functions on the complex plane that are inordinately symmetric. They satisfy so many internal symmetries that their mere existence seem like accidents. But they do exist.
>
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>
I figure the symmetries in a (elliptic) modular form are just what y... | https://mathoverflow.net/users/64080 | Modular forms and "too many symmetries" | My interpretation of Mazur's quote was in terms of the history of the discovery of modular forms. Of course trigonometric functions came first, and then a whole variety of other special functions in the eighteenth and nineteenth century, culminating in the study of elliptic functions. Since elliptic functions exhibit s... | 16 | https://mathoverflow.net/users/2384 | 190992 | 93,940 |
https://mathoverflow.net/questions/190983 | 6 | This is a crosspost of [this question](https://math.stackexchange.com/questions/1071068/a-more-advanced-version-of-aluffi) from MSE.
Paulo Aluffi's Book, *Algebra, Chapter 0* aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce groups and rings using only very b... | https://mathoverflow.net/users/53127 | A More Advanced Version of Aluffi's Chapter 0 | The closest I know for what you're looking for is Bergman's *An Invitation to General Algebra and Universal Constructions*, which can be downloaded [here](https://math.berkeley.edu/~gbergman/245/3.0.pdf). It is a simultaneous introduction to both universal algebra and universal constructions in category theory. (Univer... | 6 | https://mathoverflow.net/users/3711 | 190993 | 93,941 |
https://mathoverflow.net/questions/190990 | 2 | I'm looking for the following limit
$$\lim\_{x\rightarrow 0} \frac{\sqrt{\frac{\text{BesselK}^{(2,0)}(0,x)}{\text{BesselK}(0,x)}}}{\log (x)}$$
I believe the limit is finite, and is near -0.578. However, it is hard to get more digits numerically, and I'm interested in knowing if there is a closed form behind this co... | https://mathoverflow.net/users/8737 | Limit involving modified Bessel Function of the second kind | Using series representations, it turns out to be $-\frac{1}{\sqrt{3}}$
| 2 | https://mathoverflow.net/users/8737 | 190995 | 93,942 |
https://mathoverflow.net/questions/190935 | 2 | I am studying *analytic spreads* from *Bruns-Herzog's* book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too.
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> Is there is a intuitive interpretation of an analytic spread that can help for a better understanding of it?
>... | https://mathoverflow.net/users/47763 | intuitive interpretation of analytic spread | Question 1: Interpretation
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Ok, so here is how I like to think of it from a geometric perspective.
**Definition:** (Under moderate hypotheses) *analytic spread* of an ideal $J$ is the minimum number of generators of an ideal with the same integral closure as $J$.
**Definition:** Two idea... | 7 | https://mathoverflow.net/users/3521 | 190998 | 93,944 |
https://mathoverflow.net/questions/190965 | 30 | In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi\_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are conveniently tabulated, and reproduced on the Wikipedia page [Homotopy groups of spheres](http://en.wikipedia.org/wiki/Homotop... | https://mathoverflow.net/users/8103 | Unstable homotopy groups of spheres beyond Toda's range | I don't know the answer to your question, but I asked Fred Cohen. He had this to say:
Most of the computations are in Mahowald's work
with the EHP sequence. This gives infinite
families at p = 2 with Rob Thompson's extensions to p > 2.
Specific extensions of 2-primary components in fixed stems
are in
* N. Oda, O... | 25 | https://mathoverflow.net/users/1465 | 191005 | 93,948 |
https://mathoverflow.net/questions/191006 | 5 | I am looking for a construction that can be stated as the following coding problem: a binary code with good distance ($d = \Omega(n)$ where codeword length is $n$) that "resists local decoding" in the sense that, for some $k$, reading $k$ bits is *never* sufficient to decode (even were the channel noiseless).
In othe... | https://mathoverflow.net/users/29697 | Balanced binary code that "resists" local decoding? | For each $k$ take $n=2^k$ and the hyperplanes of the $k$-dimensional affine space over $\mathbb{F}\_2$. There are $n$ points and $2^{k+1}-2$ hyperplanes; the codewords are the indicator functions of the latter. Then $k-1$ bits never suffice, but $k$ bits always do give you unique codeword.
| 3 | https://mathoverflow.net/users/11100 | 191014 | 93,950 |
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