# All Questions

**8**

votes

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**5**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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, ...