Inom Mirzaev

Inom Mirzaev, David M. Bortz

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowksi coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our Github repository (github.com/MathBioCU).

Inom Mirzaev, David M. Bortz

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 floc size distribution 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, the mathematical model considered in this work accounts for basic biological phenomena in a population of microorganisms including growth, death, sedimentation, predation, surface erosion, renewal, fragmentation and aggregation. The central objective of this work is to prove existence of positive steady states 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 develop a numerical scheme based on spectral collocation method to approximate these positive stationary solutions. We explore the stationary solutions of the model for various biologically relevant parameters and give valuable insights for the efficient removal of suspended particles.

Inom Mirzaev, Anthony Schulte, Michael Conover et al.

Word embedding spaces are powerful tools for capturing latent semantic relationships between terms in corpora, and have become widely popular for building state-of-the-art natural language processing algorithms. However, studies have shown that societal biases present in text corpora may be incorporated into the word embedding spaces learned from them. Thus, there is an ethical concern that human-like biases contained in the corpora and their derived embedding spaces might be propagated, or even amplified with the usage of the biased embedding spaces in downstream applications. In an attempt to quantify these biases so that they may be better understood and studied, several bias metrics have been proposed. We explore the statistical properties of these proposed measures in the context of their cited applications as well as their supposed utilities. We find that there are caveats to the simple interpretation of these metrics as proposed. We find that the bias metric proposed by Bolukbasi et al. 2016 is highly sensitive to embedding hyper-parameter selection, and that in many cases, the variance due to the selection of some hyper-parameters is greater than the variance in the metric due to corpus selection, while in fewer cases the bias rankings of corpora vary with hyper-parameter selection. In light of these observations, it may be the case that bias estimates should not be thought to directly measure the properties of the underlying corpus, but rather the properties of the specific embedding spaces in question, particularly in the context of hyper-parameter selections used to generate them. Hence, bias metrics of spaces generated with differing hyper-parameters should be compared only with explicit consideration of the embedding-learning algorithms particular configurations.