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.

Manifolds, patterns and transitions in a creative life

Using sculpture and drawing as my primary methods of investigation, this research explores ways of shifting the emphasis of my creative visual arts practice from object to process whilst still maintaining a primacy of material outcomes. My motivation was to locate ways of developing a sustained practice shaped as much by new works, as by a creative flow between works. I imagined a practice where a logic of structure within discrete forms and a logic of the broader practice might be developed as mutually informed processes. Using basic structural components of multiple wooden curves and linear modes of deployment – in both sculptures and drawings – I have identified both emergence theory and the image of rhizomic growth (Deleuze and Guattari, 1987) as theoretically integral to this imagining of a creative practice, both in terms of critiquing and developing works. Whilst I adopt a formalist approach for this exegesis, the emergence and rhizome models allow it to work as a critique of movement, of becoming and changing, rather than merely a formalism of static structure. In these models, therefore, I have identified a formal approach that can be applied not only to objects, but to practice over time. The thorough reading and application of these ontological models (emergence and rhizome) to visual arts practice, in terms of processes, objects and changes, is the primary contribution of this thesis. The works that form the major component of the research develop, reflect and embody these notions of movement and change.

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce k-stellated spheres and consider the class W-k(d) of triangulated d-manifolds, all of whose vertex links are k-stellated, and its subclass W-k*; (d), consisting of the (k + 1)-neighbourly members of W-k(d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W-k(d) for d >= 2k. As a consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, and we determine the integral homology type of members of W-k*(d) for d >= 2k + 2. As another application, we prove that, when d not equal 2k + 1, all members of W-k*(d) are tight. We also characterize the tight members of W-k*(2k + 1) in terms of their kth Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for homology manifolds in which the members of W-1(d) provide the equality case. This generalizes a result (the d = 4 case) due to Walkup and Kuhnel. As a consequence, it is shown that every tight member of W-1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuhnel and Lutz asserting that tight homology manifolds should be strongly minimal. (C) 2013 Elsevier Ltd. All rights reserved.

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

Bit-pattern based integral attack

Integral attacks are well-known to be effective against byte-based block ciphers. In this document, we outline how to launch integral attacks against bit-based block ciphers. This new type of integral attack traces the propagation of the plaintext structure at bit-level by incorporating bit-pattern based notations. The new notation gives the attacker more details about the properties of a structure of cipher blocks. The main difference from ordinary integral attacks is that we look at the pattern the bits in a specific position in the cipher block has through the structure. The bit-pattern based integral attack is applied to Noekeon, Serpent and present reduced up to 5, 6 and 7 rounds, respectively. This includes the first attacks on Noekeon and present using integral cryptanalysis. All attacks manage to recover the full subkey of the final round.