8
votes
2answers
470 views

Bit String Bijection

I am searching for a bijection between two types of bit strings (strings of 0's and 1's) both of even length (2n). The restriction on the first type of bit string is that they must have the same ...
1
vote
0answers
6 views

Conditions for a set being closed under taking complement of a ball twice

Given a subset $S$ of a finite metric space $F$ with a distance function $d(,)$ and a number $\delta$ let $N_\delta(S) = \{x \in F| d(x,S)\ge \delta\}$. Is there a characterization of conditions ...
0
votes
0answers
10 views

A question on the Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$ where the ...
4
votes
1answer
189 views

Non-negativity of invariant polynomials

Certifying the non-negativity of a symmetric polynomial is much easier than certifying the non-negativity of an arbitrary polynomial function: for instance, in (1) is proved that the complexity of ...
0
votes
0answers
8 views

Preimage of singular points of smooth map between vector space and $SU(n)$

(Moved from Math SE as no answer was forthcomming: http://math.stackexchange.com/q/1294521/161684) Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ (which is taken to be surjective) ...
0
votes
0answers
10 views

Implementation of almost integer to cryptography

Can there be any implementation of almost integers to create intractable problems relevant to public key cryptography?
-1
votes
0answers
15 views

Matrix decomposition ( Kronecker product decomposition)

How to solve the following matrix equation: $Q = A \bigotimes B$ (Given $Q$)? Can we prove that if $Q \in \mathbb{R}^{n^2\times n^2}$ and $Q$ is symmetric, there always exist $A,B$ such that $Q = A ...
0
votes
1answer
14 views

solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G(\int k(x,y)f(y)dy)$ $(*)$ where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...
6
votes
1answer
90 views

Is every closed curve in 3D a geodesic on a genus-0 surface?

Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$. Q. Does there always exist a smooth, embedded, genus-zero surface $S \subset \mathbb{R}^3$ such that $\gamma$ is a ...
3
votes
2answers
130 views

Conformal map and Jordan curve

Here is my question : Suppose you have a simple (analytic) closed curve $\gamma$ in an open simply connected domain $\Omega \neq \mathbb{C}$. Does there exist a conformal bijection $f : \Omega ...
3
votes
0answers
31 views

Singularities of Pfaffian hypersurfaces

Let $X\subset\mathbb{P}^4$ be an hypersurface of degree six given by the Pfaffian of a $6\times 6$ matrix $M$ whose entries are quadratic forms in the homogeneous coordinates of $\mathbb{P}^4$. I am ...
1
vote
2answers
45 views

Distinct 2D RCFTs with the same underlying MTC

A 2d rational conformal field theory (RCFT) gives rise to a modular tensor category (MTC) equipped with a Frobenius algebra object (see, for example, http://arxiv.org/abs/hep-th/0204148). Is there an ...
0
votes
0answers
9 views

Link Between Birkhoff Ergodic Theorem and Strong LLN for Harris Recurrent Markov chain

Is it possible to derive strong law of large numbers for a Harris recurrent stationary Markov chain form Birkhoff Ergodic Theorem? As I know that there is a link between SLLN for iid sample and ...
0
votes
0answers
7 views

Any reference in absorbing boundary conditions for non-abelian gauge fields?

Is there any paper on absorbing boundary conditions for non-abelian gauge fields? Currently I only saw some on elastic wave equations and some on EM fields.
0
votes
0answers
11 views

A nullspace identity for operator exponentials

Let $X$ be a complex Banach space. Does validity of $$ \mbox{ker}\left(e^{2\pi \imath \, T} - 1\right) = \sum\nolimits_{k\in \mathbb{Z}} \mbox{ker} (T-k) \quad \forall \, T \in B(X,X) $$ imply that ...
3
votes
0answers
17 views

divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1

For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...
0
votes
0answers
13 views

Estimation of connection ignoring the inverse parallel transport in manifolds open in Euclidean space

Let $(M,g)$ be a Riemannian manifold, with parallel transport $P_{t_1,t_2}$ from time $t_1$ to time $t_2$. We know that, along a curve $c$: $$ \nabla_{c} V(t)= lim_{h\to 0} ...
1
vote
1answer
56 views

Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, $f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′)$. Suppose $f$ and $g$ are supermodular, non-negative and increasing in both arguments. ...
0
votes
0answers
73 views

Global and local maxima in a weighted sum of logarithms of linear functionals?

Constrained Optimization Problem Is is possible to describe, and locate efficiently, the maxima of the function $f$, as described below in the parameters $\mathbf{x} = (x_1,...,x_m)$. The constraints ...
1
vote
0answers
30 views

Euclidean minimum spanning trees intersecting each unit square

The recent question "Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell" can be restated in terms of "minimum spanning trees intersecting each (closed) lattice square of an ...
4
votes
1answer
252 views

A property of the Frechet filter and every ultrafilter

(Joint question with Piotr Szewczak.) Definitions and notation. By filter we mean a filter on $\omega$ containing the cofinite sets at least. For a filter $\mathcal{F}$, let ...
-4
votes
0answers
32 views

Periodicity of any fermat number modulo a prime [on hold]

It's simple to prove the recursive formula for Fermat numbers $F_n$ : $F_{n+1} = ( F_n - 1 )^2 +1 $. From this , if one define the sequence $a_n = F_n \pmod p$ , where $p$ is a odd prime , there's a ...
1
vote
1answer
20 views

Efficient algorithm for finding normals of a high dimensional convex hull with few facets

I am looking for the most efficient algorithm to use to, given a set of points in $d$ dimensional space, find the normals of the convex hull of these points, given that I know that the number of ...
-2
votes
0answers
46 views

For natural numbers 1 to n, is the square of their sum equal to the sum of their cubes? [on hold]

I've had this rolling around my head for over a decade now. It first occurred to me in high school. I never knew where to ask, but I thought this might be a good place. Given a sequence of natural ...
27
votes
3answers
574 views

How did Cole factor $2^{67}-1$ in 1903

I just heard a This American Life episode which recounted the famous anecdote about Frank Nelson Cole factoring $N:=2^{67}-1$ as $193707721 \times 761838257287$. There doesn't seem to be a historical ...
5
votes
0answers
169 views
+100

Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...
3
votes
2answers
258 views

Non-bijective conformal maps between annuli

I need to answer the following question, hopefully in the negative. Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk ...
-2
votes
0answers
98 views

Are there “adelic” L-functions?

Following Tom163's answer to this question, I would like to know whether L-functions defined through adelic representations (as defined in https://projecteuclid.org/euclid.em/1317758108) have been ...
3
votes
2answers
165 views

For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k-1}}+1}{2}$?

Question: For $k>3$ does there exist an odd prime $q_k$ such that $p_k=2^kq_k+1$ is prime and $p_k$ divides $a_k=\dfrac{3^{2^{k-1}}+1}{2}$?\ If $k=3$ the answer is Yes because for $q_3=5$ we ...
-1
votes
1answer
48 views

Exponential map and convergence

I posted this question on Math Stack Exchange, but nobody answered so I decided to ask this question here. Suppose that $M$ is smooth compact manifold and let $y \to x$. Let also $f \in C^{\infty}(M)$ ...
10
votes
2answers
483 views

Does van der Waerden's Theorem hold for $\omega_1$?

One way to phrase van der Waerden's Theorem is: For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...
1
vote
1answer
33 views

On a tower of strongly normal extensions

Where I could see the following statement? Let $K\subset L\subset M$ be a tower of the strongly normal extensions of differential fields. If $M$ is weakly normal over $K$, then $M$ is strongly ...
7
votes
1answer
104 views

The multiplication on $THH$ of finite fields

Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...
2
votes
1answer
70 views

Real slices of Minkowski space, using a complex quadratic form

Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$ where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$ is a quadratic form of signature $(3,1)$. Lying within this is a hyperboloid model ...
0
votes
1answer
146 views

Is g( ) rational if it looks that way on a large rational subset?

Let $F$ be any infinite field, $U\subset F^n$ be an open, dense (in Zariski topology) subset, $x_1,x_2,…,x_n$ be an algebraic independent system of variables over $F$ , $f,f_1,f_2,…,f_n \in ...
1
vote
1answer
29 views

Morphisms $P \to M$ in the derived category of a dg-category, if $P$ is h-projective

Let $\mathbf A$ be a dg-category. Denote by $\mathsf{C}_{\mathrm{dg}}(\mathbf A)$ the dg-category of right $\mathbf A$-modules, and by $\mathsf{C}(\mathbf A) = Z^0(\mathsf{C}_{\mathrm{dg}}(\mathbf ...
3
votes
2answers
102 views

How to find an ODE with prescribed terminal values?

Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$. Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
6
votes
1answer
52 views

Determinant of some covariance matrix (Gaussian kernel process)

Let $x_1,\dots,x_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x_1=0$. Consider the symmetric matrix $M(x)=(m_{ij}(x))_{1\leq i,j\leq p}$ where $m_{ij}(x) = \exp(-\frac{1}{2}\Vert x_i - ...
0
votes
1answer
56 views

Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all: Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...
6
votes
3answers
187 views

Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...
50
votes
5answers
4k views

Are there any serious investigations of whether “mathematicians do their best work when they're young”?

There is no shortage of anecdotes and conjectures on both sides of this widespread belief, but good supporting data either way is harder to find. Can anyone provide any references for serious ...
0
votes
1answer
96 views

Schatten $p$-classes for small $p$

Suppose $\mathcal H$ is a separable Hilbert space and $T$ is a compact self-adjoint operator on $\mathcal H$. Let $\{e_n\}$ be an orthonormal basis for $\mathcal H$. Fix $1<p<2$. Does ...
2
votes
2answers
127 views

$\mathcal S'(\mathbb R^d)$ is separable

I Think the statement is true, but I struggle to find a reference for the fact that the space of tempered distributions equipped with the weak-* topology is separable. Thank you for your help!
2
votes
2answers
129 views

Examples of toric threefolds

I am looking for examples of smooth projective toric threefolds $\mathbb P_\Delta$ such that the rational polytope $\Delta$ has only pentagonal faces and hexagonal faces. I quickly searched for ...
9
votes
2answers
418 views

Flows of vector fields and diffeomorphisms isotopic to the identity

Let $M$ be a compact manifold and $\varphi : M \longrightarrow M$ be a diffeomorphism which is isotopic to the identity. Does there exist a vector field $ X $ on $M$ such that $\varphi$ is the flow at ...
5
votes
1answer
167 views

real representation of a product group

Let $G_1$ and $G_2$ be compact Lie groups. We know that each finite-dimensional complex irreducible representation of $G_1\times G_2$ is the tensor product of an irreducible representation of $G_1$ ...
0
votes
1answer
296 views

Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...
0
votes
0answers
16 views

On the induced norms of stochastic operator and its adjoint operator

The background: when studying the paper published in Automatica named '$H_{\infty}$ control and filtering of discrete-time stochastic systems with multiplicative noise' (volume 37, pp. 409-417), I ...
3
votes
3answers
268 views

what is exactly the difference between the Selberg class and the set of Artin L-functions?

The question is in the title: from what I read in the answer to another question, Artin L-functions are conjecturally cuspidal automorphic L-functions for some algebraic group that can be transfered ...
0
votes
0answers
13 views

Truncated Robbins-Monro

I'm reading Han-Fu Chen's book "Stochastic Approximation and Its Applications", and in Chapter 1, he's got a statement of a theorem and proof on a truncated Robbins-Monro algorithm. In this version, ...

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