Research of PhD Students

PhD Thesis Title

An Iterative Method for Polyadic Decomposition for Tensors
Most tensor applications exploit the low structural and informational complexity of the data via flattening or vectorizing of the tensor data. My main goal is to develop an iterative method of finding the singular values of a matrix and to extend this method to obtain a similar Canonical Polyadic Decomposition of 3-D or higher dimensional tensors.

PhD Research Advisers

Ali Nadim (IMS, CGU), Marina Chugunova (IMS, CGU), Lorne Olfman (CISAT, CGU)

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Research area

Mathematical Modeling of Microscale Biology
Develop mathematical models of macromolecular biophysics that apply to recently-discovered biological phenomena that are based on multivalent binding, as opposed to receptor lock-and-key binding. Examples include SARS-CoV2 virus binding to extracellular sugars and the recent discovery of glycoRNA molecules in human biology.

PhD Research Advisers

Marina Chugunova (IMS, CGU), Ali Nadim (IMS, CGU), and Jim Sterling (KGI)

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Research area

Extremal Eigenvalue Problems on Remannian Manifolds
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry.

PhD Thesis Adviser

Chiu-Yen Kao (CMC)

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Research area

Designing a Gradient Diffusion System to Investigate Chemotaxis in a 3D Collagen Natrix
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been a growing interest in the Steklov problem from the viewpoint of spectral geometry.

PhD Thesis Adviser

Perla Ayala (CSULB)

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Research area

Applications of Interpolation Theory to Problems in Approximation Theory
In an approximation process, one has to have a practical way of measuring the error of approximation of a given function by another much nicer function. The possibility of applying interpolation techniques to approximation theory was initiated by Jaak Peetre in 1963. The main realization starts with recognizing that under some mild conditions (namely the inequalities of Jackson and Bernstein are satisfied), every approximation space is a real interpolation space. This means that the K-method of real interpolation theory becomes available as a tool in approximation theory, which can then be used to obtain, for example, Bernstein and Jackson theorems concerning the best approximation of functions in $L_{p}(\mathbb{R}^{n})$ by entire functions of exponential type, approximation of compact operators by operators of finite rank, approximation of differential operators by difference operators.

PhD Thesis Adviser

Asuman Aksoy (CMC)

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Title of PhD Thesis

On the Average Coherence of Lattices
Coherence is a measure of non-orthogonality in signal processing. Recently, my advisor applied this concept to lattices. The reason for this is that some other measures of non-orthogonality are directly linked to lattice packing density, an important research question in many applied fields. A related measure of non-orthogonality is average coherence, which we introduce to lattices for the first time. Additionally, we investigate various geometric properties of several different important classes of arithmetic lattices with a general view towards maximal and average coherence, as well as some related parameters.

PhD Thesis Adviser

Lenny Fukshansky (CMC)

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PhD Thesis Title

Existence and Uniqueness of the Solution of the Traffic Flow Partial Differential Equation on Multi-Lane Freeways
In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (drivers, and their vehicles) and infrastructure (including highways and traffic control devices), with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems.

PhD Thesis Adviser

Henry Schellhorn (CGU)

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PhD Thesis Title

Diagnostic Techniques in Solid Propellant Combustion
The utilization of novel and established diagnostic techniques for characterizing and quantifying the processes of mixing and combustion of solid rocket propellant. The examination of propellant surface and interfacial chemistries through holography and particle image velocimetry.

PhD Thesis Adviser

Joseph Kalman (CSULB)

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Research Area

Jet Flow with Coil Insert: LES Study of Entrainment, Mixing and Flow Structure.

PhD Research Adviser

Hamid Rahai (CSULB)

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Research Area

Abstract Algebra, focusing on Quasigroups
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that “division” is always possible. Quasigroups differ from groups mainly in that they are not necessarily associative.

PhD Research Adviser

Gizem Karaali (Pomona College)

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Research Area

Dynamic Temporal Link Prediction
My research is focusing on dynamic link prediction problem on graph data though graph neural network, and the interpretability of predictions.

PhD Research Adviser

Allon Percus (IMS, CGU), Yan Li (CISAT, CGU)

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Research Area

Geometry of numbers, polynomials, Diophantine approximation, Theory of height functions
Hilbert’s 10th problem asks for an algorithm to decide whether a given Diophantine equation has an integer solution. By a celebrated result of Matiyasevich, such an algorithm does not exist in general. On the other hand, for linear Diophantine equations solutions are classically given by the Euclidean algorithm. Further, there are also known algorithms for quadratic polynomials. An important approach to the problem of finding such algorithms over rings and fields of arithmetic interest for different classes of polynomials is through the use of search bounds with respect to height, a standard measure of arithmetic complexity. We study the existence of search bounds on height of solutions for large classes of multilinear polynomials. This research direction also has natural connections to the problem of extending a unimodular matrix to a matrix in GL_n.

PhD Research Adviser

Lenny Fukshansky (CMC)

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PhD Thesis Title

Force Regulation in Contractile Cells by Chemical and Mechanical Signaling
Contractile cells, such as heart cells and epithelial cells, generate force that has implications in health and disease. For example, hearts that cannot generate sufficient force to eject blood to the body suffer heart failure, while in other cells, like cancer cells, increasing the force that a cell generates leads to increased metastatic potential. We use computational models to gain mechanistic insight into the molecular processes that regulate force in contractile cells.

PhD Thesis Adviser

Parag Katira (SDSU)