Integral Manifolds and Bounded Solutions of Singularly Perturbed Systems of Impulsive Differential Equations

Sufficient conditions for the existence of bounded solutions of singularly perturbed impulsive differential equations are obtained. For this purpose integral manifolds are used.

Application of Lyapunov's Direct Method to the Existence of Integral Manifolds of Impulsive Differential Equations

The present paper investigates the existence of integral manifolds for impulsive differential equations with variable perturbations. By means of piecewise continuous functions which are generalizations of the classical Lyapunov’s functions, sufficient conditions for the existence of integral manifolds of such equations are found.

Integral Manifolds and Perturbations of the Nonlinear Part of Systems of Autonomous Differential Equations with Impulses at Fixed Moments

* This investigation was supported by the Bulgarian Ministry of Science and Education under Grant MM-7.

On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Integral formulae for a Riemannian manifold with a distribution

We obtain a new series of integral formulae for symmetric functions of curvature of a distribution of arbitrary codimension (an its orthogonal complement) given on a compact Riemannian manifold, which start from known formula by P.Walczak (1990) and generalize ones for foliations by several authors: Asimov (1978), Brito, Langevin and Rosenberg (1981), Brito and Naveira (2000), Andrzejewski and Walczak (2010), etc. Our integral formulae involve the co-nullity tensor, certain component of the curvature tensor and their products. The formulae also deal with a number of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For foliated manifolds of constant curvature the obtained formulae give us the classical type formulae. For a special choice of functions our formulae reduce to ones with Newton transformations of the co-nullity tensor.

SUPPORT OF MAXIMIZING MEASURES FOR TYPICAL C(0) DYNAMICS ON COMPACT MANIFOLDS

Given a compact manifold X, a continuous function g : X -> IR, and a map T : X -> X, we study properties of the T-invariant Borel probability measures that maximize the integral of g. We show that if X is a n-dimensional connected Riemaniann manifold, with n >= 2, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager. We also show that, if X is the circle, then the ""topological size"" of the set of endomorphisms for which there are g maximizing measures with support on a periodic orbit depends on properties of the function g. In particular, if g is C(1), it has interior points.

On maximizing measures of homeomorphisms on compact manifolds

We prove that given a compact n-dimensional connected Riemannian manifold X and a continuous function g : X -> R, there exists a dense subset of the space of homeomorphisms of X such that for all T in this subset, the integral integral(X) g d mu, considered as a function on the space of all T-invariant Borel probability measures mu, attains its maximum on a measure supported on a periodic orbit.

Quantum field theory of (p,0)-forms on kaehler manifolds

In questa tesi abbiamo studiato la quantizzazione di una teoria di gauge di forme differenziali su spazi complessi dotati di una metrica di Kaehler. La particolarità di queste teorie risiede nel fatto che esse presentano invarianze di gauge riducibili, in altre parole non indipendenti tra loro. L'invarianza sotto trasformazioni di gauge rappresenta uno dei pilastri della moderna comprensione del mondo fisico. La caratteristica principale di tali teorie è che non tutte le variabili sono effettivamente presenti nella dinamica e alcune risultano essere ausiliarie. Il motivo per cui si preferisce adottare questo punto di vista è spesso il fatto che tali teorie risultano essere manifestamente covarianti sotto importanti gruppi di simmetria come il gruppo di Lorentz. Uno dei metodi più usati nella quantizzazione delle teorie di campo con simmetrie di gauge, richiede l'introduzione di campi non fisici detti ghosts e di una simmetria globale e fermionica che sostituisce l'iniziale invarianza locale di gauge, la simmetria BRST. Nella presente tesi abbiamo scelto di utilizzare uno dei più moderni formalismi per il trattamento delle teorie di gauge: il formalismo BRST Lagrangiano di Batalin-Vilkovisky. Questo metodo prevede l'introduzione di ghosts per ogni grado di riducibilità delle trasformazioni di gauge e di opportuni “antifields" associati a ogni campo precedentemente introdotto. Questo formalismo ci ha permesso di arrivare direttamente a una completa formulazione in termini di path integral della teoria quantistica delle (p,0)-forme. In particolare esso permette di dedurre correttamente la struttura dei ghost della teoria e la simmetria BRST associata. Per ottenere questa struttura è richiesta necessariamente una procedura di gauge fixing per eliminare completamente l'invarianza sotto trasformazioni di gauge. Tale procedura prevede l'eliminazione degli antifields in favore dei campi originali e dei ghosts e permette di implementare, direttamente nel path integral condizioni di gauge fixing covarianti necessari per definire correttamente i propagatori della teoria. Nell'ultima parte abbiamo presentato un’espansione dell’azione efficace (euclidea) che permette di studiare le divergenze della teoria. In particolare abbiamo calcolato i primi coefficienti di tale espansione (coefficienti di Seeley-DeWitt) tramite la tecnica dell'heat kernel. Questo calcolo ha tenuto conto dell'eventuale accoppiamento a una metrica di background cosi come di un possibile ulteriore accoppiamento alla traccia della connessione associata alla metrica.

Deformation Lemma, Ljusternik-Schnirellmann Theory and Mountain Pass Theorem on C1-Finsler Manifolds

∗Partially supported by Grant MM409/94 Of the Ministy of Science and Education, Bulgaria. ∗∗Partially supported by Grant MM442/94 of the Ministy of Science and Education, Bulgaria.

Integral Representations of the Logarithmic Derivative of the Selberg Zeta Function

AMS Subj. Classification: MSC2010: 11F72, 11M36, 58J37

Integral Piezoactuator System with Optimum Placement of Functionally Graded Material - A Topology Optimization Paradigm

Piezoactuators consist of compliant mechanisms actuated by two or more piezoceramic devices. During the assembling process, such flexible structures are usually bonded to the piezoceramics. The thin bonding layer(s) between the compliant mechanism and the piezoceramic may induce undesirable behavior, including unusual interfacial nonlinearities. This constitutes a drawback of piezoelectric actuators and, in some applications, such as those associated to vibration control and structural health monitoring (e. g., aircraft industry), their use may become either unfeasible or at least limited. A possible solution to this standing problem can be achieved through the functionally graded material concept and consists of developing `integral piezoactuators`, that is those with no bonding layer(s) and whose performance can be improved by tailoring their structural topology and material gradation. Thus, a topology optimization formulation is developed, which allows simultaneous distribution of void and functionally graded piezoelectric materials (including both piezo and non-piezoelectric materials) in the design domain in order to achieve certain specified actuation movements. Two concurrent design problems are considered, that is the optimum design of the piezoceramic property gradation, and the design of the functionally graded structural topology. Two-dimensional piezoactuator designs are investigated because the applications of interest consist of planar devices. Moreover, material gradation is considered in only one direction in order to account for manufacturability issues. To broaden the range of such devices in the field of smart structures, the design of integral Moonie-type functionally graded piezoactuators is provided according to specified performance requirements.

Calculation of line integrals in boundary integral methods

We propose quadrature rules for the approximation of line integrals possessing logarithmic singularities and show their convergence. In some instances a superconvergence rate is demonstrated.

A finite integral transform technique for solving the diffusion-reaction equation with Michaelis-Menten kinetics

An approximate analytical technique employing a finite integral transform is developed to solve the reaction diffusion problem with Michaelis-Menten kinetics in a solid of general shape. A simple infinite series solution for the substrate concentration is obtained as a function of the Thiele modulus, modified Sherwood number, and Michaelis constant. An iteration scheme is developed to bring the approximate solution closer to the exact solution. Comparison with the known exact solutions for slab geometry (quadrature) and numerically exact solutions for spherical geometry (orthogonal collocation) shows excellent agreement for all values of the Thiele modulus and Michaelis constant.

Is the irritable bladder associated with anterior compartment relaxation? A critical look at the 'integral theory of pelvic floor dysfunction'

The 'integral theory of pelvic floor dysfunction', first proposed by Petros and Ulmsten in 1990, claims that anterior vaginal wall relaxation is associated with symptoms of urgency, frequency, nocturia and urge incontinence. A retrospective study was designed to test this hypothesis. Imaging data and urodynamic reports from 272 women suffering from symptoms of lower urinary tract dysfunction were evaluated. Opening of the retrovesical angle, bladder neck descent, urethral rotation and descent of a cystocele during Valsalva were used to quantify anterior vaginal wall laxity None of the tested parameters were associated with symptoms and signs of detrusor overactivity. On the contrary, patients with higher grades of urethral and bladder descent were less likely to suffer from nocturia and urge incontinence and were less likely to leave sensory urgency and detrusor instability diagnosed on urodynamic testing. The findings of this study therefore do not support this hypothesis of the 'integral theory'.