On the non-commutative fractional Wishart process

We investigate the process of eigenvalues of a fractional Wishart process defined by $N=B^{?}B$, where B is the matrix fractional Brownian motion recently studied in [18]. Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in (1/2,1)$, we derive a stochastic differential equation in the Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally, a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the non-commutative fractional Wishart process, which constitutes the family of fractional dilations of the free Poisson distribution.     

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1)

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator Q, provided that $H\in (1/2,1)$ and $\mathrm{tr}(Q)$ is sufficiently small.     

Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation ${(?\mathrm{\Delta })}^{\alpha }$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta }$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q({\mathrm{BMO}}_x^{?\gamma })$ with $q>2$ and $\gamma \in (0,2\alpha ?1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.     

Overlarge sets of resolvable idempotent quasigroups

An idempotent quasigroup $(X,°)$ of order $v$ is called resolvable (denoted by RIQ$\left(v\right)$) if the set of $v(v?1)$ non-idempotent 3-vectors $\{(a,b,a°b):a,b\in X,a\ne b\}$ can be partitioned into $v?1$ disjoint transversals. An overlarge set of idempotent quasigroups of order $v$, briefly by OLIQ$\left(v\right)$, is a collection of $v+1$ IQ$\left(v\right)$s, with all the non-idempotent 3-vectors partitioning all those on a $(v+1)$-set. An OLRIQ$\left(v\right)$ is an OLIQ$\left(v\right)$ with each member IQ$\left(v\right)$ being resolvable. In this paper, it is established that there exists an OLRIQ$\left(v\right)$ for any positive integer $v\ge 3$, except for $v=6$, and except possibly for $v\in \{10,11,14,18,19,23,26,30,51\}$. An OLIQ${}^{?}\left(v\right)$ is another type of restricted OLIQ$\left(v\right)$ in which each member IQ$\left(v\right)$ has an idempotent orthogonal mate. It is shown that an OLIQ${}^{?}\left(v\right)$ exists for any positive integer $v\ge 4$, except for $v=6$, and except possibly for $v\in \{14,15,19,23,26,27,30\}$.     

The local eigenvalues of a bipartite distance-regular graph

We consider a bipartite distance-regular graph $\Gamma$ with vertex set $X$, diameter $D\ge 4$, and valency $k\ge 3$. For $0\le i\le D$, let ${\Gamma }_i\left(x\right)$ denote the set of vertices in $X$ that are distance $i$ from vertex $x$. We assume there exist scalars $r,s,t\in \mathbb{R}$, not all zero, such that $r|{\Gamma }_1\left(x\right)\cap {\Gamma }_1\left(y\right)\cap {\Gamma }_2\left(z\right)|+s|{\Gamma }_2\left(x\right)\cap {\Gamma }_2\left(y\right)\cap {\Gamma }_1\left(z\right)|+t=0$ for all $x,y,z\in X$ with path-length distances $\partial (x,y)=2,\partial (x,z)=3,\partial (y,z)=3$. Fix $x\in X$, and let ${\Gamma }_2^2$ denote the graph with vertex set $\tilde{X}=\{y\in X\mid \partial (x,y)=2\}$ and edge set $\tilde{R}=\{yz\mid y,z\in \tilde{X},\partial (y,z)=2\}$. We show that the adjacency matrix of the local graph ${\Gamma }_2^2$ has at most four distinct eigenvalues. We are motivated by the fact that our assumption above holds if $\Gamma$ is $Q$-polynomial.     

A continuum of periodic solutions to the planar four-body problem with two pairs of equal masses

In this paper, we apply the variational method with Structural Prescribed Boundary Conditions (SPBC) to prove the existence of periodic and quasi-periodic solutions for the planar four-body problem with two pairs of equal masses $m_1=m_3$ and $m_2=m_4$. A path $q(t)$ on $[0,T]$ satisfies the SPBC if the boundaries $q(0)\in \mathbf{A}$ and $q(T)\in \mathbf{B}$, where A and B are two structural configuration spaces in ${({\mathbf{R}}^2)}^4$ and they depend on a rotation angle $\theta \in (0,2\pi )$ and the mass ratio $\mu =\frac{m_2}{m_1}\in {\mathbf{R}}^{+}$.We show that there is a region $\mathrm{\Omega }?(0,2\pi )\times R^{+}$ such that there exists at least one local minimizer of the Lagrangian action functional on the path space satisfying the SPBC $\{q(t)\in H^1([0,T],{({\mathbf{R}}^2)}^4)|q(0)\in \mathbf{A},q(T)\in \mathbf{B}\}$ for any $(\theta ,\mu )\in \mathrm{\Omega }$. The corresponding minimizing path of the minimizer can be extended to a non-homographic periodic solution if θ is commensurable with π or a quasi-periodic solution if θ is not commensurable with π. In the variational method with the SPBC, we only impose constraints on the boundary and we do not impose any symmetry constraint on solutions. Instead, we prove that our solutions that are extended from the initial minimizing paths possess certain symmetries.The periodic solutions can be further classified as simple choreographic solutions, double choreographic solutions and non-choreographic solutions. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution when $(\theta ,\mu )=(\frac{4\pi }{5},1)$. Remarkably the unequal-mass variants of the stable star pentagon are just as stable as the equal mass choreographies.     

Refined weight of edges in normal plane maps

The weight $w\left(e\right)$ of an edge $e$ in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge $e=uv$ is of type $(i,j)$ if $d\left(u\right)\le i$ and $d\left(v\right)\le j$. In 1940, Lebesgue proved that every NPM has an edge of one of the types $(3,11)$, $(4,7)$, or $(5,6)$, where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-connected planar graph has an edge $e$ with $w\left(e\right)\le 13$, which bound is sharp. Borodin (1989), answering Erd?s’ question, proved that every NPM has either a $(3,10)$-edge, or $(4,7)$-edge, or $(5,6)$-edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencová and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge $e$ with $w\left(e\right)\le 12$. Recently, we confirmed this conjecture by proving that every NPM has either a simplicial 3-vertex adjacent to a vertex of degree at most 10, or an edge of types $(3,9)$, $(4,7)$, or $(5,6)$.By a $k^{\left(?\right)}$-vertex we mean a $k$-vertex incident with precisely $?$ triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: $(3^{\left(3\right)},10)$, $(3^{\left(2\right)},9)$, $(3^{\left(1\right)},7)$, $(4^{\left(4\right)},7)$, $(4^{\left(3\right)},6)$, $(5^{\left(5\right)},6)$, or $(5,5)$, where all bounds are best possible. In particular, this implies that the bounds in $(3,10)$, $(4,7)$, and $(5,6)$ can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively.