1
vote
3answers
1k views

Seemingly complex logic/set-theoretic puzzle

I got this puzzle some time ago and it has been bugging me since, I cant solve it - but it is supposedly solvable, I am interested in a solution or any tips on how to proceed. In front of you is an ...
2
votes
2answers
528 views

Function with all but mixed second partial derivatives twice differentiable?

Let $f(x,y)$ be a a real valued function on an open subset of $\mathbf{R}^2$ with continuous partial derivatives $\frac{\partial^2 f}{\partial x^2}$ and $\frac{\partial^2}{\partial y^2}$. Is $f$ twice ...
2
votes
1answer
355 views

Decidability survives new constants

Let $L$ be a finite first order language and let $M$ be an $L$-structure with universe $\mathbb{N}$ that interprets all $L$-symbols as recursive sets (so $M$ is a recursive $L$-structure). Let ...
1
vote
0answers
95 views

Six operations passage from $X_0$ to $X$ reference request

Let $X_0$ be a variety over $\mathbb F_q$ and denote by $X$ its basechange to the algebraic closure. Consider the constructible derived categories $D^b_c(X_0,\mathbb E)$ and $D^b_c(X,\mathbb E)$, ...
4
votes
4answers
835 views

Quotient Surface of A Hyperelliptic Involution

Let $X$ be a hyperelliptic Riemann surface, and let $J$ be the hyperelliptic involution. Then consider the quotient surface $X/ < J > ,$ my question is whether $X/ < J > $ is a Riemann ...
8
votes
2answers
589 views

Subtlety in the definition of the Kobayashi metric

When defining the Kobayashi metric on a connected complex analytic space $X$, one makes the following auxiliary definition: A holomorphic chain from $x\in X$ to $y\in X$ is a finite sequence of ...
0
votes
2answers
196 views

Properties of the Euler Discretization of a diffusion

Let $X$ be a continuous 1-d diffusion: $$ dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x. $$ W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties. Let ...
18
votes
3answers
1k views

Minimal surface in a ball

Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space and $r$ is the distance from $\Sigma$ to the center of the ball. Is it true that $$\mathop{\rm area} \Sigma\ge ...
2
votes
0answers
157 views

The balls and bins model: bounding the marginal contributions in the m>>n regime

Consider the standard balls and bins process, where $m$ balls are thrown into $n$ bins, and consider the case where $m >> n$. Denote the load on bin $i$ by the RV $L_i$. Given a set $S ...
4
votes
2answers
625 views

How is this observation related to Koszul duality?

Let $X$ be a smooth variety, $\mathcal D$ the sheaf of algebraic differentail operators, $\Omega$ the algebraic deRham complex and $\mathcal M$ a quasi coherent $\mathcal O_X$-module. Now there is a ...
8
votes
2answers
795 views

Solving the equation $xax=b$ in a C*-algebra.

Let $a, b\in A_+$ be positive elements of some C*-algebra $A$. Assume furthermore that $a$ is invertible. Is it true that $$ \exists! x\in A_+\quad:\quad xax=b\quad ? $$ Already in the case ...
12
votes
3answers
1k views

A nontrivial surface on which any two curves intersect

One interesting property of the projective plane is that any two plane curves intersect. (More generally, if $V$ and $W$ are subvarieties of any projective space, and codim $V$ + codim $W \geq 0$, ...
1
vote
1answer
224 views

Question on localization technique

In the book "Local cohomology : An algebraic introduction with geometric application", page 289 there is a proof of the following theorem : Assume that $R=\bigoplus_{n}R_{n}$ is positive graded ...
2
votes
0answers
239 views

uniqueness in $\infty$ categories.

I am studying the basics of infinity categories and am rather confused about the issues of uniqueness (my main example is the $\infty$ derived category of chain complexes of vector spaces over a field ...
5
votes
2answers
146 views

Are there better upper bounds on the rank of the commutant of a fusion module than the global dimension?

Suppose I have a fusion category $\mathcal{C}$ and an indecomposable module category $\mathcal{M}$ over it. The commutant $\mathcal{C}_\mathcal{M}^*$ is the category of module endofunctors, and gives ...
11
votes
3answers
488 views

sums of fractional parts of linear functions of n

As $\alpha$ and $\gamma$ range uniformly over $[0,1]$, what is the typical (e.g. median or root-mean-square) order of magnitude of $C_m (\alpha,\gamma)$ := $\sum_{1 \leq k \leq m} \left( {\rm ...
0
votes
1answer
140 views

How to handle a scalar product in an integral?

I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it. Here's a simplification of the problem: $ ...
0
votes
1answer
204 views

Can one always Decide whether a Systems of Nonlinear Equations with Bilinear terms is Feasible?

I have come to a point in my PhD research were i need to prove that a particular decision procedure is decidable or not. And if i can solve the sub-problem described below, i shall have proved it. The ...
10
votes
1answer
586 views

Traces on Hecke algebras and the Jones polynomial

In his famous paper "Hecke algebra representations of braid groups and link polynomials," (Annals 1987), Jones uses a compatible family of traces $tr_z$ on the Iwahori-Hecke algebras $H(q,n)$ of type ...
0
votes
1answer
888 views

Finding linearly independent columns of a large sparse rectangular matrix

I have a problem that necessitates solving a large non-negative least-squares problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols) and nearly binary. However, A is not ...
11
votes
0answers
214 views

What are the difficulties in proving almost-everywhere stability of Gaussian elimination?

It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of Wilkinson states ...
3
votes
5answers
363 views

Strings and “co-subsequences”

Let $S$ be a string over some alphabet $\Sigma$. It is well known that a substring of $S$ is commonly defined as a sequence of contiguous elements from $S$, while a subsequence of $S$ is a sequence ...
5
votes
1answer
412 views

Elementary end extension of a countable model for ZF

Theorem 2.2.18 in Chang and Kiesler uses omitting types to show that any countable model of ZF has an elementary end extension. Can we control the countable order type of such a model? for example, if ...
1
vote
2answers
638 views

Converting a recursive definition to an explicit one

Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$? I've listed out the first few terms: for $x=0,1,2,3,4,5,6, 7$ we have $a_x ...
3
votes
2answers
480 views

Linear Algebra Over $F_{2}$

Suppose we call a subset S of $F^{n}$ ($F$ is the field with two elements) good if for any $x$ and $y$ (possibly $x=y$) we have $[x,y]=1$ where $[ , ]$ denotes the obvious bilinear form on F. What's ...
1
vote
1answer
416 views

Cotangent space of the sphere

In analyzing the spherical pendulum the cotangent space of the sphere is defined as $ T^*S^2 = \lbrace (q,p) \in \mathbb{R}^3 \times \mathbb{R}^3; |q| = 1, q \cdot p = 0 \rbrace$ my problem with ...
6
votes
3answers
752 views

Sheaf with free stalks

Say we are given a complex manifold $X$ and an $\mathcal{O}_X$-module $\mathcal{F}$. Assume that for any point $P\in X$ the stalk $\mathcal{F}_P$ is a free $(\mathcal{O}_X)_P$-module of finite rank. ...
5
votes
1answer
477 views

Formality of de Rham algebra for two-dimensional closed surfaces

Is it possible to embed de Rham cohomology of a two-dimensional closed surface of genus $g\geq 2$ into the differential graded algebra of differential forms (with de Rham differential and wedge ...
2
votes
1answer
266 views

Solvability for constant-coefficient partial differential operators

Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential ...
5
votes
1answer
536 views

Decay of Relative Growth in Conway's Game of Life

Intro The question is about Game of Life. Let us denote the set of points obtained from initial configuration $A$ after $m$ steps as $A(m)$ (we are only interested in finite initial configuration, ...
10
votes
1answer
824 views

Transcendental numbers: yet another classification

Let $\mathbb{A^+}$ be the set of non-negative algebraic numbers. Consider the set of "polynomials" : $$\mathbb{P} = \lbrace a_0 + a_1x^{r_1} + a_2x^{r_2} + a_3x^{r_3} +\cdots + a_nx^{r_n}| a_0, a_i, ...
7
votes
1answer
272 views

What is the homotopy type of a free simplicial ring?

Is there a good description of the homotopy type of a free simplicial ring (or simplicial $R$-algebra) on a given simplicial set, in terms of the homotopy type of that simplicial set? (This is mostly ...
4
votes
3answers
3k views

Minimize trace of inverse of convex combination of matrices.

Hello! (First question--please forgive me if its unclear.) I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive ...
1
vote
2answers
203 views

How to prove H^2(g,J(g)) is nonzero for a semisimple Lie algebra g, where J(g) is the augmentation ideal of g?

Suppose g is a fiinte dimensional semisimple lie algebra over a field with characteristic 0. This question is related to Whitehead's second lemma, which says for finite dimensional g-module M, ...
0
votes
1answer
227 views

Variation on Fatou's lemma for Sobolev norms

Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions $$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$ If I am not ...
25
votes
1answer
2k views

Degeneration of the Hodge spectral sequence

Let $f\colon X \to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb Q$-scheme), then Deligne has shown: $R^af_*\Omega^b_{X/S}$ is locally free for ...
2
votes
1answer
630 views

isotopy doesn't make sense (Milnor)

hello, I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots For each $c$ and $p$ in $\mathbb{R}^{n-1}$ such that $\|c-p\| > < r$, there is ...
1
vote
1answer
371 views

Existence of solution for this parabolic PDE

The parabolic PDE $$\langle u', v \rangle + a(u,v) = \langle f, v \rangle$$ has a unique solution $u \in L^2(0,T; H^1)$ with $u' \in L^2(0,T;H^{-1})$ if $a$ is a bounded and coercive bilinear form ...
1
vote
2answers
159 views

Extremal point and probability

Let $(X,\mathcal{F},\mathbf{P})$ be a probability space and $f \colon X \mapsto \mathbf{R}^n$ an integrable function. We assume that $f$ takes its values in a closed convex set $C$ of $\mathbf{R}^n$ ...
7
votes
1answer
864 views

Real analytic function, injective, non surjective and preserving the rationals ?

I'd like to prove the non-existence of a real analytic function, injective, non-surjective that sends rationals to rationals. Is it a classical result ? If not, any hints on how to prove it ? Thanks ...
2
votes
3answers
528 views

sum of $1/ \phi(n)^2$

Hello, I am reading some material on circle method. Right now I am at its application to the binary Goldbach problem. To obtain a certain bound the fact $\sum_{n> X} 1/ \phi(n)^2 = O(1/X)$ is ...
1
vote
1answer
537 views

cohomology of torsion sheaves and nilpotent sheaves

Let $X$ be a scheme and $\mathcal{F}$ be a sheaf on $X$ which is torsion $\mathcal{O}_X-$module (i.e., every local section is annihilated by an element of the ring $\mathcal{O}_X(U)$) or nilpotent ...
16
votes
3answers
1k views

Gosper's Mathematics

Sometimes I bump into more of the astonishing results of Gosper (some examples follow) and I gather that a lot of them come from hypergeometrics and special functions. Have there been any attempts ...
1
vote
0answers
269 views

Wiener-Hopf Integral/Lindley's Equation

Lindley's equation is well known within queueing theory and is as follows $F(y) = - \int_0^\infty F(x)dH(y-x)$ However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which ...
1
vote
0answers
106 views

Elliptic problem on half space; infinite boundary values; Liouville theorem

In a the study of a boundary value problem the following problem is arising: $-\Delta v(x)= e^{v(x)}$ in $ R^N_+$ $v= - \infty$ $\qquad $ on $ \partial R^N_+$ $ \qquad $ $ v \le 0$ in $ R^N_+$. ...
0
votes
0answers
278 views

counting k-cliques not also (k+1) on random graphs

consider the set of graphs with $n$ vertices and exactly half of all $\binom n 2$ possible edges. looking for a formula that counts the number of these graphs that have a $k$-clique but not a ...
1
vote
1answer
167 views

Generalizing the Reshitikhin-Turaev construction possible?

OK, I have to ask a dumb question again: Where do Lie groups enter in the construction of the Reshitikhin-Turaev invariant? The parts of the proof I understand are that 6j symbols take care of ...
15
votes
4answers
1k views

What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?

There is a standard way to construct the sheafification of a presheaf on a Grothendieck topology which involves matching families. Details may be found here: ...
21
votes
2answers
910 views

Different interpretations of moduli stacks

I'm taking my first steps in the language of stacks, and would like something cleared up. The intuitive idea of moduli spaces is that each point corresponds to an object of what we're trying to ...
0
votes
0answers
116 views

Can we construct CHU as an internal category in a monoidal category?

I have recently read Abramsky and Heunen's paper on Operational structures and categorical physics. I have been looking at operational structures as internal categories in a monoidal category like we ...

15 30 50 per page