Suggestion from the Past?

Mathematics Subject Classification: 26A33 (main), 35A22, 78A25, 93A30

Linear Fractional PDE, Uniqueness of Global Solutions

Mathematics Subject Classification: 26A33, 47A60, 30C15.

A Fractional Analog of the Duhamel Principle

Mathematics Subject Classification: 35CXX, 26A33, 35S10

Operational Methods in the Environment of a Computer Algebra System

This article presents the principal results of the doctoral thesis “Direct Operational Methods in the Environment of a Computer Algebra System” by Margarita Spiridonova (Institute of mathematics and Informatics, BAS), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 23 March, 2009.

Lp Microlocal Properties for Multi-Quasi-Elliptic Pseudodifferential Operators

2010 Mathematics Subject Classification: Primary 35S05; Secondary 35A17.

On Modified Method of Simplest Equation for Obtaining Exact Solutions of Nonlinear PDEs: Case of Elliptic Simplest Equation

2010 Mathematics Subject Classification: 74J30, 34L30.

Sub- and Super-solutions of a Nonlinear PDE, and Application to a Semilinear SPDE

2010 Mathematics Subject Classification: 35R60, 60H15, 74H35.

Extended Maximum Principles

2000 Mathematics Subject Classification: 35B50, 35L15.

A Simple Example of Localized Parametric Resonance for the Wave Equation

2000 Mathematics Subject Classification: 35L05, 35P25, 47A40.

Mathematical Modeling of Perifusion Cell Culture Experiments

In perifusion cell cultures, the culture medium flows continuously through a chamber containing immobilized cells and the effluent is collected at the end. In our main applications, gonadotropin releasing hormone (GnRH) or oxytocin is introduced into the chamber as the input. They stimulate the cells to secrete luteinizing hormone (LH), which is collected in the effluent. To relate the effluent LH concentration to the cellular processes producing it, we develop and analyze a mathematical model consisting of coupled partial differential equations describing the intracellular signaling and the movement of substances in the cell chamber. We analyze three different data sets and give cellular mechanisms that explain the data. Our model indicates that two negative feedback loops, one fast and one slow, are needed to explain the data and we give their biological bases. We demonstrate that different LH outcomes in oxytocin and GnRH stimulations might originate from different receptor dynamics. We analyze the model to understand the influence of parameters, like the rate of the medium flow or the fraction collection time, on the experimental outcomes. We investigate how the rate of binding and dissociation of the input hormone to and from its receptor influence its movement down the chamber. Finally, we formulate and analyze simpler models that allow us to predict the distortion of a square pulse due to hormone-receptor interactions and to estimate parameters using perifusion data. We show that in the limit of high binding and dissociation the square pulse moves as a diffusing Gaussian and in this limit the biological parameters can be estimated.

C-MEMS Based Micro Enzymatic Biofuel Cells

Miniaturized, self-sufficient bioelectronics powered by unconventional micropower may lead to a new generation of implantable, wireless, minimally invasive medical devices, such as pacemakers, defibrillators, drug-delivering pumps, sensor transmitters, and neurostimulators. Studies have shown that micro-enzymatic biofuel cells (EBFCs) are among the most intuitive candidates for in vivo micropower. In the fisrt part of this thesis, the prototype design of an EBFC chip, having 3D intedigitated microelectrode arrays was proposed to obtain an optimum design of 3D microelectrode arrays for carbon microelectromechanical systems (C-MEMS) based EBFCs. A detailed modeling solving partial differential equations (PDEs) by finite element techniques has been developed on the effect of 1) dimensions of microelectrodes, 2) spatial arrangement of 3D microelectrode arrays, 3) geometry of microelectrode on the EBFC performance based on COMSOL Multiphysics. In the second part of this thesis, in order to investigate the performance of an EBFC, behavior of an EBFC chip performance inside an artery has been studied. COMSOL Multiphysics software has also been applied to analyze mass transport for different orientations of an EBFC chip inside a blood artery. Two orientations: horizontal position (HP) and vertical position (VP) have been analyzed. The third part of this thesis has been focused on experimental work towards high performance EBFC. This work has integrated graphene/enzyme onto three-dimensional (3D) micropillar arrays in order to obtain efficient enzyme immobilization, enhanced enzyme loading and facilitate direct electron transfer. The developed 3D graphene/enzyme network based EBFC generated a maximum power density of 136.3 μWcm-2 at 0.59 V, which is almost 7 times of the maximum power density of the bare 3D carbon micropillar arrays based EBFC. To further improve the EBFC performance, reduced graphene oxide (rGO)/carbon nanotubes (CNTs) has been integrated onto 3D mciropillar arrays to further increase EBFC performance in the fourth part of this thesisThe developed rGO/CNTs based EBFC generated twice the maximum power density of rGO based EBFC. Through a comparison of experimental and theoretical results, the cell performance efficiency is noted to be 67%.

Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation

We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.

Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation

We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.

Funzioni di Sobolev

In questa tesi cercherò di analizzare le funzioni di Sobolev su R}^{n}, seguendo le trattazioni Measure Theory and Fine Properties of Functions di L.C. Evans e R.F.Gariepy e l'elaborato Functional Analysis, Sobolev Spaces and Partial Differential Equations di H. Brezis. Le funzioni di Sobolev si caratterizzano per essere funzioni con le derivate prime deboli appartenenti a qualche spazio L^{p}. I vari spazi di Sobolev hanno buone proprietà di completezza e compattezza e conseguentemente sono spesso i giusti spazi per le applicazioni di analisi funzionale. Ora, come vedremo, per definizione, l'integrazione per parti è valida per le funzioni di Sobolev. È, invece, meno ovvio che altre regole di calcolo siano allo stesso modo valide. Così, ho inteso chiarire questa questione di carattere generale, con particolare attenzione alle proprietà puntuali delle funzioni di Sobolev. Abbiamo suddiviso il lavoro svolto in cinque capitoli. Il capitolo 1 contiene le definizioni di base necessarie per la trattazione svolta; nel secondo capitolo sono stati derivati vari modi di approssimazione delle funzioni di Sobolev con funzioni lisce e sono state fornite alcune regole di calcolo per tali funzioni. Il capitolo 3 darà un' interpretazione dei valori al bordo delle funzioni di Sobolev utilizzando l'operatore Traccia, mentre il capitolo 4 discute l' estensione su tutto R^{n} di tali funzioni. Proveremo infine le principali disuguaglianze di Sobolev nel Capitolo 5.

Nonlinear Dynamics Near the Cell Membrane of Chara Corallina

The phenomenon of patterned distribution of pH near the cell membrane of the algae Chara corallina upon illumination is well-known. In this paper, we develop a mathematical model, based on the detailed kinetic analysis of proton fluxes across the cell membrane, to explain this phenomenon. The model yields two coupled nonlinear partial differential equations which describe the spatial dynamics of proton concentration changes and transmembrane potential generation. The experimental observation of pH pattern formation, its period and amplitude of oscillation, and also its hysteresis in response to changing illumination, are all reproduced by our model. A comparison of experimental results and predictions of our theory is made. Finally, a mechanism for pattern formation in Chara corallina is proposed.