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Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions
(American Meteorological Society, 20160322)For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H>12, it is known that the existing (naive) Euler scheme has the rate of convergence n1−2H. Since the limit H→12 ... 
Fluctuations of TASEP and LPP with general initial data
(Institute of Mathematical Statistics, 20150813)We prove Airy process variational formulas for the onepoint probability distribution of (discrete time parallel update) TASEP with general initial data, as well as last passage percolation from a general lattice path to ... 
A monotone Sinai theorem
(Institute of Mathematical Statistics, 20160202)Sinai proved that a nonatomic ergodic measurepreserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy ... 
Quantitative stable limit theorems on the Wiener space
(Institute of Mathematical Statistics, 2016)We use Malliavin operators in order to prove quantitative stable limit theorems on the Wiener space, where the target distribution is given by a possibly multidimensional mixture of Gaussian distributions. Our findings ... 
On the intermittency front of stochastic heat equation driven by colored noises
(Institute of Mathematical Statistics, 20160301)We study the propagation of high peaks (intermittency fronts) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in RdRd. The noise is assumed to have a general homogeneous ... 
Stability of Explicit OneStep Methods for P1Finite Element Approximation of Linear Diffusion Equations on Anisotropic Meshes
(Society for Industrial and Applied Mathematics, 20160526)We study the stability of explicit onestep integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and ... 
The Partitionability Conjecture
(American Mathematical Society, 201702)In 1979 Richard Stanley made the following conjecture: Every CohenMacaulay simplicial complex is partitionable. Motivated by questions in the theory of face numbers of simplicial complexes, the Partitionability Conjecture ... 
Certain plane configurations
(University of Kansas, 1930) 
Multiplicities in Commutative Algebra
(University of Kansas, 20160831)This dissertation explores the notion of multiplicity and its generalizations within the theory of commutative algebra. Chapter 2 is dedicated to calculating the limits which give rise to BuchsbaumRim multiplicities. We ... 
Composing Scalable Nonlinear Algebraic Solvers
(Society for Industrial and Applied Mathematics (SIAM), 20151105)Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for ... 
A FETIDP TYPE DOMAIN DECOMPOSITION ALGORITHM FOR THREEDIMENSIONAL INCOMPRESSIBLE STOKES EQUATIONS
(Society for Industrial and Applied Mathematics, 20150303)The FETIDP (dualprimal finite element tearing and interconnecting) algorithms, proposed by the authors in [SIAM J. Numer. Anal., 51 (2013), pp. 1235–1253] and [Internat. J. Numer. Methods Engrg., 94 (2013), pp. 128–149] ... 
Nonexistence of solitonlike solutions for defocusing generalized KdV equations
(Texas State University, Department of Mathematics, 20150224)We consider the global dynamics of the defocusing generalized KdV equation $$ \partial_t u + \partial_x^3 u = \partial_x(u^{p1}u). $$ We use Tao's theorem [5] that the energy moves faster than the mass to prove a ... 
On L2 modulus of continuity of Brownian local times and Riesz potentials
(Institute of Mathematical Statistics, 20150505)This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on three closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, ... 
Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency
(Institute of Mathematical Statistics, 20160604)This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. ... 
Density convergence in the BreuerMajor theorem for Gaussian stationary sequences
(Bernoulli Society for Mathematical Statistics and Probability, 2015)Consider a Gaussian stationary sequence with unit variance X={Xk;k∈N∪{0}}. Assume that the central limit theorem holds for a weighted sum of the form Vn=n−1/2∑n−1k=0f(Xk), where f designates a finite sum of Hermite ... 
Effects of (Small) Permanent Charge and Channel Geometry on Ionic Flows via Classical PoissonNernstPlanck Models
(Society for Industrial and Applied Mathematics, 20150115)In this work, we examine effects of permanent charges on ionic flows through ion channels via a quasionedimensional classical PoissonNernstPlanck (PNP) model. The geometry of the threedimensional channel is presented ... 
Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations
(Society for Industrial and Applied Mathematics, 20150917)We study the stability and instability of periodic traveling waves for Kortewegde Vriestype equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer ... 
On summation of finite series
(University of Kansas, 1929) 
Application of stochastic differential equations to option pricing
(University of Kansas, 20160531)The financial world is a world of random things and unpredictable events. Along with the innovative development of diversity and complexity in modern financial market, there are more and more financial derivative emerged ... 
Some Studies on Parameter Estimations
(University of Kansas, 20160531)Parameter estimation has wide applications in such fields as finance, biological science, weather prediction, oil deposit detection, etc. Researchers are particularly interested in reconstructing some unknown parameters ...