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Inom Mirzaev

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About

Inom4.jpg
I am fifth-year PhD student in applied mathematics at CU. I received my B.S. from Middle East Technical University in Ankara, Turkey. Currently, I am working with Prof. David Bortz on microbial flocculation dynamics. I am also a huge soccer fan. I support Barça from La Liga and Manchester City from Barclays League.


Google Scholar ResearchGate Github repository


Contact Information

Email: lastname <at>colorado<dot>edu
Office: STAD 155

Research

Effect of flow and restructuring on microbial aggregate growth

This is a joint work with Eric P. Kightley. Our main goal in this project is to develop an agent-based model of microbial growth which also incorporates properties of the flow field. We plan to extend our current microscale model for more general flow fields. Our future research is guided by the following questions:

  • How do the properties of the flow field affect the growth of microbial clusters?
  • Does incorporating flow field effects promote the aggregate growth?
  • What types of flow fields provide best conditions for the aggregate growth?

Answers to any of these questions would have broad-reaching impacts in many diverse areas such as water treatment, biofuel production, beer fermentation, etc. From the mathematical point of view, there would also be a substantial contribution on the microscale modeling of the microbial flocculation.

Deform flow3.gif

Identifying Conditional Probability Measures

Keywords: identification of probability measures, inverse problem, measure-dependent dynamical system, size-structured populations, flocculation, fragmentation, bacterial aggregates


In this work, we present and investigate an inverse problem for estimating the conditional probability measures from size-distribution measurements. We use the Prohorov metric (convergence in which is equivalent to weak convergence of measures) in a functional-analytic setting and show well-posedness of the inverse problem. We develop an approximation approach for computational implementation and show well-posedness of this approximate inverse problem. We also show the convergence of solutions to the approximate inverse problem to solutions of the original inverse problem. The primary contribution of this work is to extend this theory to conditional probability measures. We also illustrate that the flocculation dynamics of bacterial aggregates in suspension is one realization of systems satisfying the hypotheses in our framework.

Beta distribution with α=β=2 was used to generate test data
Results of the optimization scheme based on these data and an initial density function uniform in x .




Microbial Flocculation Dynamics

Keywords: Flocculation model, nonlinear evolution equations, principle of linearized stability, spectral analysis, structured populations dynamics, semigroup theory


Flocculation is the process whereby particles (i.e., flocs) in suspension reversibly combine and separate. The process is widespread in soft matter and aerosol physics as well as environmental science and engineering. We consider a general size-structured flocculation model, which describes the evolution of flocs in an aqueous environment. Our work provides a unified treatment for many size-structured models in the environmental, industrial, medical, and marine engineering literature. In particular, our model accounts for basic biological phenomena in a population of microorganisms including growth, death, sedimentation, predation, renewal, fragmentation and aggregation. Our central goal in this project is to rigorously investigate the long-term behavior of this generalized flocculation model. Using results from fixed point theory we derive conditions for the existence of continuous, non-trivial stationary solutions. We further apply the principle of linearized stability and semigroup compactness arguments to provide sufficient conditions for local exponential stability of stationary solutions as well as sufficient conditions for instability. The end results of this analytical development are relatively simple inequality-criteria which thus allows for the rapid evaluation of the existence and stability of a non-trivial stationary solution. To our knowledge, this work is the first to derive precise existence and stability criteria for such a generalized model. Lastly, we also provide an illustrating application of this criteria to several flocculation models. Here is an animation showing time evolution of microbial flocculation model:

Flocculation3.gif

Laplacian Dynamics

Keywords: Laplacian dynamics, biochemical networks, synthesis and degradation, Matrix-Tree Theorem, insulin secretion


Analyzing qualitative behaviors of biochemical reactions using its associated network structure has proven useful in diverse branches of biology. As an extension of our previous work with Prof. Gunawardena, we introduce a graph-based framework to calculate steady state solutions of biochemical reaction networks with synthesis and degradation. Our approach is based on a labeled directed graph G and the associated system of linear non-homogeneous differential equations with first order degradation and zeroth order synthesis.

Laplacian.gif

Teaching

I have TA-ed the following courses so far:

  • Differential Equations with Linear Algebra Fall 2012, Spring 2013 and Summer 2013
  • Calculus 3 for Engineers Fall 2013
  • Calculus 2 for Engineers Spring 2014

Publications

"A problem well put is half solved" -- John Dewey

Journal Articles

  • I. Mirzaev, E.C. Byrne & D.M. Bortz. An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations, Inverse Problems 32, no. 9 (15 July 2016) [2]

Preprints

  • I. Mirzaev & D. M. Bortz. (2015). Stability of steady states for a class of flocculation equations with growth and removal. Submitted [5]
  • I. Mirzaev & D. M. Bortz (2015). Criteria for linearized stability for a size-structured population model. arXiv:1507.07127 [6]