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Burch ideals and Burch rings
(Mathematical Sciences Publishers (MSP), 20200918)We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many wellknown concepts, for example integrally closed ideals of finite colength and Cohen–Macaulay ... 
Averaging Gaussian functionals
(Institute of Mathematical Statistics, 20200428)This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a BreuerMajor type ... 
Intermittency for the parabolic Anderson model of Skorohod type driven by a rough noise
(Institute of Mathematical Statistics, 20200714)In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter H ∈ (0, 1/2). By using the FeynmanKac representation for the L^p (Ω) moments of the ... 
Fractional Diffusion in Gaussian Noisy Environment
(MDPI, 20150331)We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: D(α)tu(t,x)=Bu+u⋅W˙H, where D(α)t is the Caputo fractional derivative ... 
On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings
(Mathematical Sciences Publishers (MSP), 20110224)It is wellknown that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we ... 
An adaptive spot placement method on Cartesian grid for pencil beam scanning proton therapy
(IOP Publishing, 20211202)Pencil beam scanning proton radiotherapy (RT) offers flexible proton spot placement near treatment targets for delivering tumoricidal radiation dose to tumor targets while sparing organsatrisk. Currently the spot placement ... 
Unique Factorization Domains in Commutative Algebra
(Department of Mathematics, University of Kansas, 20210520)In this project, we learn about unique factorization domains in commutative algebra. Most importantly, we explore the relation between unique factorization domains and regular local rings, and prove the main theorem: If R ... 
Initialboundary value problems for a reactiondiffusion equation
(American Institute of Physics, 20190827)A novel approach that utilizes Fokas’s unified transform is employed for studying a reactiondiffusion equation with power nonlinearity formulated either on the halfline or on a finite interval with data in Sobolev spaces. ... 
A General Stochastic Volatility Model on VIX Options
(University of Kansas, 20191231)Abstract In this dissertation, we study a general stochastic volatility model for the VIX options (Chicago Board Options Exchange) volatility index, which is a stochastic differential equation with 8 unknown parameters. ... 
On the Generation of Stable Kerr Frequency Combs in the LugiatoLefever Model of Periodic Optical Waveguides
(Society for Industrial and Applied Mathematics, 20190307) 
The Kortewegde Vries equation on an interval
(American Institute of Physics, 20190508)The initialboundary value problem (IBVP) for the Kortewegde Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the wellposedness of the KdV on the halfline, which is ... 
Finitary isomorphisms of Poisson point processes
(Institute of Mathematical Statistics, 20191022)As part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48 (1987) 1–141) proved that any two Poisson point processes are isomorphic as measurepreserving ... 
A weighted cellular matrixtree theorem, with applications to complete colorful and cubical complexes
(Elsevier, 20180326)We present a version of the weighted cellular matrixtree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply ... 
Counting arithmetical structures on paths and cycles
(Elsevier, 20180727)Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag(d)A)r = 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical ... 
Increasing spanning forests in graphs and simplicial complexes
(Elsevier, 20181101)Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F forms an increasing sequence. Hallam and Sagan showed ... 
A positivity phenomenon in Elser's Gaussiancluster percolation model
(Elsevier, 20201218)Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we ... 
Enumerating Parking Completions Using Join and Split
(Electronic Journal of Combinatorics, 20200612)Given a strictly increasing sequence t with entries from [n] := {1, . . . , n}, a parking completion is a sequence c with t + c = n and {t ∈ t  t 6 i} + {c ∈ c  c 6 i} > i for all i in [n]. We can think of t as ... 
Optimal Energy Decay for the Damped KleinGordon Equation
(University of Kansas, 20190831)In this dissertation we study the long time dynamics of damped KleinGordon and damped fractional KleinGordon equations using $C_0$ Semigroup theory and its application. The $C_0$semigroups are used to solve a large ... 
Sharp time asymptotics for the quasigeostrophic equation, the Boussinesq system and near plane waves of reactiondiffusion models
(University of Kansas, 2019531)Through this dissertation we present the sharp time decay rates for three equations, namely quasigeostrophic equation (SQG), Boussinesq system (BSQ) and plane wave of general reactiondiffusion models. In addition, in ... 
Interval parking functions
(Elsevier, 20201116)Interval parking functions (IPFs) are a generalization of ordinary parking functions in which each car is willing to park only in a fixed interval of spaces. Each interval parking function can be expressed as a pair (a, ...