Fractional normal inverse Gaussian diffusion

A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics.     

Disjoint Hamilton cycles in the random geometric graph

We consider the standard random geometric graph process in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of edge‐length. For fixed k?1, weprove that the first edge in the process that creates a k‐connected graph coincides a.a.s. with the first edge that causes the graph to contain k/2 pairwise edge‐disjoint Hamilton cycles (for even k), or (k?1)/2 Hamilton cycles plus one perfect matching, all of them pairwise edge‐disjoint (for odd k). This proves and extends a conjecture of Krivelevich and M ler. In the special case when k = 2, our result says that the first edge that makes the random geometric graph Hamiltonian is a.a.s. exactly the same one that gives 2‐connectivity, which answers a question of Penrose. (This result appeared in three independent preprints, one of which was a precursor to this article.) We prove our results with lengths measured using the ?_p norm for any p>1, and we also extend our result to higher dimensions. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:299‐322, 2011     

A note on simultaneous preconditioning and symmetrization of non‐symmetric linear systems

Motivated by the theory of self‐duality that provides a variational formulation and resolution for non‐self‐adjoint partial differential equations (Ann. Inst. Henri Poincaré (C) Anal Non Linéaire 2007; 24 :171–205; Selfdual Partial Differential Systems and Their Variational Principles. Springer: New York, 2008), we propose new templates for solving large non‐symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well‐known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill‐conditioned, and highly non‐symmetric systems. The numerical and theoretical results are provided to show the efficiency of our approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Piercing random boxes

Consider a set of n random axis parallel boxes in the unit hypercube in ${\bf R}^{d}$ , where d is fixed and n tends to infinity. We show that the minimum number of points one needs to pierce all these boxes is, with high probability, at least $\Omega_d(\sqrt{n}(\log n)^{d/2-1})$ and at most $O_d(\sqrt{n}(\log n)^{d/2-1}\log \log n)$ . © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 365–380, 2011     

A phase transition for the heights of a fragmentation tree

We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behavior of the shattering times, defined as the first instants when all the blocks of the partition process have cardinality less than a fixed integer. Our results may be applied to the study of certain random split trees. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 247‐274, 2011     

一些稀疏图的强边染色（英文）

图G的强边染色是指对图G进行正常边染色使得任意长度为3的路的三条边染不同的颜色.图G的强边色数,记为χ’_s(G),是使得图G是强k边着色的最小正整数kk.2015年,Zang [arXiv:1510.00785]证明了:最大度△(G)=5的图G,χ’_s(G)≤37.本文证明了:最大度△(G)=5且最大平均度小于8/3(或者14/5)的图G,χ’_s(G)≤13 (或者14).另外,本文证明了:最大度△(G)≥3的不含K_2,3-图子式的图G,χ’_s(G)≤4△(G)-6,这个界是紧的.     

基于DCSBM模型的受访者驱动抽样调查估计量改进

大数据背景下,将受访者驱动抽样(RDS)用于网络抽样调查,解决了传统抽样调查难以获得可用抽样框、难以接触被调查者以及难以获得回答等问题,也使得网络调查可以实现概率抽样,得到一定误差范围内的总体参数估计.然而,在实际抽样过程中,同质性问题(即样本单元在推荐同伴时倾向于推荐那些与自己有相同属性的同伴)会导致RDS估计量的方...     

N叉树上的离散时间量子随机行走

通过离散时间量子随机行走的框架,我们研究了在N叉树上的离散时间量子随机行走,该框架不需要硬币空间,仅仅只需要选择一个除了酉性再无其它限制的演化算子,并且包含了使用再生结构的轨道枚举和z变换.作为结果,我们在封闭形式中计算了在根处的振幅的生成函数.     

Additive Hermitian idempotent preservers between operator algebras

Let L be an additive map between (real or complex) matrix algebras sending $n\times n$ Hermitian idempotent matrices to $m\times m$ Hermitian idempotent matrices. We show that there are nonnegative integers $p,q$ with $n(p+q)=r\le m$ and an $m\times m$ unitary matrix U such that$L(A)=U[(I_p?A)\oplus (I_q?A^t)\oplus 0_{m?r}]U^{?},\text{for any}n\times n\text{Hermitian}A\text{with rational trace}.$ We also extend this result to the (complex) von Neumann algebra setting, and provide a supplement to the Dye-Bunce-Wright Theorem asserting that every additive map of Hermitian idempotents extends to a Jordan ?-homomorphism.