Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations

Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10

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.

The non-linear acousto-mechanical and thermodynamic response of spherical absorbers to laser pulses of various temporal profiles

This study is to theoretically investigate shockwave and microbubble formation due to laser absorption by microparticles and nanoparticles. The initial motivation for this research was to understand the underlying physical mechanisms responsible for laser damage to the retina, as well as the predict threshold levels for damage for laser pulses with of progressively shorter durations. The strongest absorbers in the retina are micron size melanosomes, and their absorption of laser light causes them to accrue very high energy density. I theoretically investigate how this absorbed energy is transferred to the surrounding medium. For a wide range of conditions I calculate shockwave generation and bubble growth as a function of the three parameters; fluence, pulse duration and pulse shape. In order to develop a rigorous physical treatment, the governing equations for the behavior of an absorber and for the surrounding medium are derived. Shockwave theory is investigated and the conclusion is that a shock pressure explanation is likely to be the underlying physical cause of retinal damage at threshold fluences for sub-nanosecond pulses. The same effects are also expected for non-biological micro and nano absorbers. ^

Exterior Differential Systems and Euler-Lagrange Partial Differential Equations

The book also covers the Second Variation, Euler-Lagrange PDE systems, and higher-order conservation laws.

SYSTEMS OF DIFFERENTIAL EQUATIONSTHAT ARE COMPETITIVE OR COOPERATIVE II:CONVERGENCE ALMOST EVERYWHERE*MORRIS W. HIRSCH

A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all of the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The main results in this article are the following. A cooperative system cannot have nonconstant attracting periodic solutions. In a cooperative system whose Jacobian matrices are irreducible the forward orbit converges for almost every point having compact forward orbit closure. In a cooperative system in 2 dimensions, every solution is eventually monotone.  Applications are made to generalizations of positive feedback loops.

Quantitative Derivation of Effective Evolution Equations for the Dynamics of Bose-Einstein Condensates

This thesis proves certain results concerning an important question in non-equilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schroedinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schroedinger equation arises in the mean field limit as number of particles N → ∞ and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics. In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.

Análise estequiométrica da dinâmica não-linear de redes de reações químicas

Dissertação (mestrado)—Universidade de Brasília, Instituto de Química, Programa de Pós-Graduação em Química, 2016.

Semiclassical analysis of systems of Schrödinger equations

The scalar Schrödinger equation models the probability density distribution for a particle to be found in a point x given a certain potential V(x) forming a well with respect to a fixed energy level E_0. Formally two real inversion points a,b exist such that V(a)=V(b)=E_0, V(x)<0 in (a,b) and V(x)>0 for xb. Following the work made by D.Yafaev and performing a WKB approximation we obtain solutions defined on specific intervals. The aim of the first part of the thesis is to find a condition on E, which belongs to a neighbourhood of E_0, such that it is an eigenvalue of the Schrödinger operator, obtaining in this way global and linear dependent solutions in L2. In quantum mechanics this condition is known as Bohr-Sommerfeld quantization. In the second part we define a Schrödinger operator referred to two potential wells and we study the quantization conditions on E in order to have a global solution in L2xL2 with respect to the mutual position of the potentials. In particular their wells can be disjoint,can have an intersection, can be included one into the other and can have a single point intersection. For these cases we refer to the works of A.Martinez, S. Fujiié, T. Watanabe, S. Ashida.

Distributed non-cooperative robust economic predictive control for dynamically coupled linear systems.

In this thesis, a tube-based Distributed Economic Predictive Control (DEPC) scheme is presented for a group of dynamically coupled linear subsystems. These subsystems are components of a large scale system and control inputs are computed based on optimizing a local economic objective. Each subsystem is interacting with its neighbors by sending its future reference trajectory, at each sampling time. It solves a local optimization problem in parallel, based on the received future reference trajectories of the other subsystems. To ensure recursive feasibility and a performance bound, each subsystem is constrained to not deviate too much from its communicated reference trajectory. This difference between the plan trajectory and the communicated one is interpreted as a disturbance on the local level. Then, to ensure the satisfaction of both state and input constraints, they are tightened by considering explicitly the effect of these local disturbances. The proposed approach averages over all possible disturbances, handles tightened state and input constraints, while satisfies the compatibility constraints to guarantee that the actual trajectory lies within a certain bound in the neighborhood of the reference one. Each subsystem is optimizing a local arbitrary economic objective function in parallel while considering a local terminal constraint to guarantee recursive feasibility. In this framework, economic performance guarantees for a tube-based distributed predictive control (DPC) scheme are developed rigorously. It is presented that the closed-loop nominal subsystem has a robust average performance bound locally which is no worse than that of a local robust steady state. Since a robust algorithm is applying on the states of the real (with disturbances) subsystems, this bound can be interpreted as an average performance result for the real closed-loop system. To this end, we present our outcomes on local and global performance, illustrated by a numerical example.

EXISTENCE RESULTS FOR NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

In this paper we discuss the existence of mild, strict and classical solutions for a class of abstract integro-differential equations in Banach spaces. Some applications to ordinary and partial integro-differential equations are considered.

GLOBAL SOLUTIONS FOR ABSTRACT FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

In this paper we study the existence of global solutions for a class of abstract functional differential equation with nonlocal conditions. An application is considered.

Self-organized distribution of periodicity and chaos in an electrochemical oscillator

We report a detailed numerical investigation of a prototype electrochemical oscillator, in terms of high-resolution phase diagrams for an experimentally relevant section of the control (parameter) space. The prototype model consists of a set of three autonomous ordinary differential equations which captures the general features of electrochemical oscillators characterized by a partially hidden negative differential resistance in an N-shaped current-voltage stationary curve. By computing Lyapunov exponents, we provide a detailed discrimination between chaotic and periodic phases of the electrochemical oscillator. Such phases reveal the existence of an intricate structure of domains of periodicity self-organized into a chaotic background. Shrimp-like periodic regions previously observed in other discrete and continuous systems were also observed here, which corroborate the universal nature of the occurrence of such structures. In addition, we have also found a structured period distribution within the order region. Finally we discuss the possible experimental realization of comparable phase diagrams.

Non-linear boundary element formulation with tangent operator to analyse crack propagation in quasi-brittle materials

This work deals with analysis of cracked structures using BEM. Two formulations to analyse the crack growth process in quasi-brittle materials are discussed. They are based on the dual formulation of BEM where two different integral equations are employed along the opposite sides of the crack surface. The first presented formulation uses the concept of constant operator, in which the corrections of the nonlinear process are made only by applying appropriate tractions along the crack surfaces. The second presented BEM formulation to analyse crack growth problems is an implicit technique based on the use of a consistent tangent operator. This formulation is accurate, stable and always requires much less iterations to reach the equilibrium within a given load increment in comparison with the classical approach. Comparison examples of classical problem of crack growth are shown to illustrate the performance of the two formulations. (C) 2009 Elsevier Ltd. All rights reserved.

Linear minimum mean square filter for discrete-time linear systems with Markov jumps and multiplicative noises

In this paper we obtain the linear minimum mean square estimator (LMMSE) for discrete-time linear systems subject to state and measurement multiplicative noises and Markov jumps on the parameters. It is assumed that the Markov chain is not available. By using geometric arguments we obtain a Kalman type filter conveniently implementable in a recurrence form. The stationary case is also studied and a proof for the convergence of the error covariance matrix of the LMMSE to a stationary value under the assumption of mean square stability of the system and ergodicity of the associated Markov chain is obtained. It is shown that there exists a unique positive semi-definite solution for the stationary Riccati-like filter equation and, moreover, this solution is the limit of the error covariance matrix of the LMMSE. The advantage of this scheme is that it is very easy to implement and all calculations can be performed offline. (c) 2011 Elsevier Ltd. All rights reserved.

Molecular cocrystals of carboxylic acids. XXVII - An investigation into the use of triphenylphosphine oxide cocrystals for non-linear optics: the crystal structure of the adduct of triphenylphosphine oxide with N-methylpyrrole-2-carboxylic acid

Four adducts of triphenylphosphine oxide with aromatic carboxylic acids have been synthesized and tested for second-order non-linear optical properties. These were with N-methylpyrrole-2-carboxylic acid (I), indole-2-carboxylic acid (2), 3-dimethylaminobenzoic acid (3), and thiophen-2-carboxylic acid (4). Compound (1) produced clear, colourless crystals (space group P2(1)2(1)2(1) With a 9.892(1), b 14.033(1), c 15.305(1) Angstrom, Z 4) which allowed the structure to be determined by X-ray diffraction.