This dissertation presents investigations on dynamics of discrete-time quantum walk and some of its applications. Quantum walks has been exploited as an useful tool for quantum algorithms in quantum computing. Beyond quantum computational purposes, it has been used to explain and control the dynamics in various physical systems. In order to use the quantum walk to its fullest potential, it is important to know and optimize the properties purely due to quantum dynamics and in presence of noise. Various studies of its dynamics in the absence and presence of noise have been reported. We propose new approaches to optimize the dynamics, discuss symmetries and effect of noise on the quantum walk. Making use of its properties, we propose the use of quantum walk as an efficient new tool for various applications in physical systems and quantum information processing. In the first and second part of this dissertation, we discuss evolution process of the quantum walks, propose and demonstrate the optimization of discrete-time quantum walk using quantum coin operation from SU(2) group and discuss some of its properties. We investigate symmetry operations and environmental effects on dynamics of the walk on a line and an $n-$cycle highlighting the interplay between noise and topology. Using the properties and behavior of quantum walk discussed in part two, in part three we propose the application of quantum walk to realize quantum phase transition in optical lattice, that is to efficiently control and redistribute ultracold atoms in optical lattice. We also discuss the implementation scheme. Another application we consider is creation of spatial entanglement using quantum walk on a quantum many body system.

In this thesis, mechanical and elastic behaviour of anisotropic materials are inves- tigated in order to understand the optimum mechanical behaviour of them in selected directions. For an anisotropic material with known elastic constants, it is possible to choose the best set of e&curren / ective elastic constants and e&curren / ective eigen- values which determine the optimum mechanical and elastic properties of it and also represent the material in a speci.ed greater material symmetry. For this reason, bounds on the e&curren / ective elastic constants which are the best set of elastic constants and e&curren / ective eigenvalues of materials have been constructed symbollicaly for all anisotropic elastic symmetries by using Hill [4,13] approach. Anisotropic Hooke.s law and its Kelvin inspired formulation are described and generalized Hill inequalities are explained in detail. For di&curren / erent types of sym- metries, materials were selected randomly and data of elastic constants for them were collected. These data have been used to calculate bounds on the e&curren / ective elastic constants and e&curren / ective eigenvalues. Finally, by examining numerical results of bounds given in tables, it is seen that the materials selected from the same symmetry type which have larger interval between the bounds, are more anisotropic, whereas some materials which have smaller interval between the bounds, are closer to isotropy.

This thesis is devoted to use Lie group analysis to obtain all invariant solutions by constructing optimal system of one-dimensional subalgebras of the Lie algebra L5 for a nonlinear wave equation. I will show how the given symmetries ( Eq.2) are admitted by using partial differential equation (Eq.1), In addition to obtain the commutator table by using the same given symmetries. Subsequently, I calculate the transformations of the generators with the Lie algebra L5, which provide the 5-parameter group of linear transformations for the operators. Finally, I construct the invariant solutions for each member of the optimal system.

Η μελέτη συνήθων διαφορικών εξισώσεων συχνά χρησιμοποιεί μεθόδους γνωστές από την κλασική Μηχανική. Η πιο γνωστή από αυτές ϕέρει το όνομα του εμπνευστή της, του Ιρλανδού Sir William Rowan Hamilton (1805 - 1865), κι αποτελεί μία μαθηματικά πλήρη ϑεωρία για τα λεγόμενα συστήματα Hamilton. Πρόσφατα, όμως, δομές τύπου Hamilton άρχισαν να μελετώνται και σε συστήματα μερικών διαφορικών εξισώσεων, συγκεκριμένα εξισώσεων εξέλιξης. Σκοπός της παρούσας εργασίας είναι η ανάπτυξη της ϑεωρίας Hamilton για τα συστήματα αυτά και ιδιαίτερα για τις περιπτώσεις εκείνες που εμφανίζουν ολοκληρωσιμότητα. Η γραμμή που ϑα ακολουθήσουμε έχει ως κύριο οδηγό τις συμμετρίες των διαφορικών εξισώσεων, ένα πολύ χρήσιμο εργαλείο για την επίλυση οποιασδήποτε διαφορικής εξίσωσης, που πρώτος ανέδειξε ο Νορβηγός Marius Sophus Lie (1842 - 1899). Στο πρώτο κεφάλαιο λοιπόν γίνεται μία εισαγωγή στην ϑεωρία των (γεωμετρικών) συμμετριών, ενώ επίσης παρουσιάζονται τρόποι επίλυσης και γενικότερα αντιμετώπισης ξεχωριστά συνήθων και μερικών διαφορικών εξισώσεων με την χρήση των ομάδων συμμετρίας τους. Το δεύτερο κεφάλαιο ϕιλοδοξεί να αναδείξει την αντιστοιχία μεταξύ των συμμετριών ενός συστήματος διαφορικών εξισώσεων και των νόμων διατήρησης στους οποίους υπακούει το ϕυσικό σύστημα που περιγράφουν. Αυτό είναι και το περιεχόμενο του ϑεωρήματος που διατύπωσε η Γερμανίδα Amalie Emmy Noether (1882 - 1935), το οποίο ισχύει και στην ειδική περίπτωση των συστημάτων Hamilton. Το πρώτο, λοιπόν, ϐήμα προς αυτήν την κατεύθυνση είναι η επέκταση της έννοιας της συμμετρίας στις λεγόμενες γενικευμένες συμμετρίες, με ιδιαίτερη έμφαση στις εξισώσεις εξέλιξης. Το δεύτερο είναι ουσιαστικά μια μικρή εισαγωγή στην ϑεωρία μεταβολών, απαραίτητη όμως και για τα επόμενα κεφάλαια. Την γνωστή ϑεωρία Hamilton για πεπερασμένα συστήματα, συστήματα δηλαδή συνήθων διαϕορικών εξισώσεων πραγματεύεται το τρίτο κεφάλαιο. Σκοπός του κεφαλαίου αυτού δεν είναι η πλήρης περιγραφή της ϑεωρίας, αλλά η διατύπωση των εννοιών εκείνων που μπορούν να γενικευτούν και στην περίπτωση των απειροδιάστατων συστημάτων. Για τον λόγο αυτό έχει προτιμηθεί η κάπως πιο αφηρημένη και σίγουρα όχι τόσο συνηθισμένη περιγραφή στο πλαίσιο της γεωμετρίας Poisson. Αντιμετωπίζοντας τις συμπλεκτικές δομές, οι οποίες επικρατούν στην ϐιβλιογραφία, ως μια υποπερίπτωση των γενικότερων δομών Poisson, έχουμε ουσιαστικά αποφύγει τελείως την χρήση διαφορικών μορφών, στρέφοντας περισσότερο την προσοχή στις ομάδες συμμετρίας Hamilton, μία έννοια-κλειδί για την ολοκληρωσιμότητα των συστημάτων αυτών. Στο τέταρτο κεφάλαιο παρουσιάζουμε το κεντρικό ϑέμα αυτής της εργασίας, δηλαδή τη ϑεωρία Hamilton για απειροδιάστατα συστήματα εξισώσεων εξέλιξης, και ειδικότερα την ολοκληρωσιμότητα τους. Τα ϐασικά μας εργαλεία είναι αυτά που παρουσιάστηκαν νωρίτερα, δηλαδή οι (γενικευμένες) συμμετρίες και οι νόμοι διατήρησης από την μια, και τα διανυσματικά πεδία Hamilton από την άλλη που μας επιτρέπουν την μεταξύ τους αντιστοιχία. Με ϐάση αυτά τα εργαλεία ϐλέπουμε πως η μελέτη πολλών μερικών διαφορικών εξισώσεων ϑυμίζει εκείνων των κλασικών συστημάτων Hamilton της Μηχανικής. Στην παραπάνω αντιστοιχία ϐασίζεται και η έννοια των δι-Χαμιλτονικών συστημάτων, την οποία μελετάμε στο πέμπτο κεφάλαιο. Μέσα από το παράδειγμα της εξίσωσης Korteweg-de Vries αναδεικνύονται τα πλεονεκτήματα της εύρεσης δύο διαφορετικών, ανεξάρτητων εκφράσεων Hamilton, που οδηγούν στην κατασκευή άπειρων συμμετριών ή ακόμα και νόμων διατήρησης. Η διπλή αυτή δομή Hamilton των απειροδιάστατων συστημάτων συνδέεται, όπως ϑα δούμε, με την ολοκληρωσιμότητα είτε με την έννοια του Liouville, είτε με διάφορα άλλα κριτήρια. Γνωστά παραδείγματα παραθέτονται, πέρα από την KdV, όπως η εξίσωση Schroedinger, η modified KdV, κι άλλες μη γραμμικές κυματικές εξισώσεις. Στο έκτο και τελευταίο κεφάλαιο παρουσιάζουμε την περίπτωση, όπου ένα σύστημα επιδέχεται πολλαπλή δομή Hamilton. Τέτοιου είδους συστήματα μας επιτρέπουν να δούμε προϋπάρχουσες έννοιες από την ϑεωρία Hamilton, αλλά κι όχι μόνο, κάτω από μία άλλη σκοπιά. Γι΄ αυτό κι έχουν απασχολήσει την σύγχρονη ϐιβλιογραφία, πάνω στην οποία κάνουμε μία σύντομη επισκόπηση, τόσο στο κομμάτι εκείνο που ασχολείται με τις πρόσφατες εξελίξεις της ϑεωρίας Hamilton, όσο και με την μελέτη γενικότερα της ολοκληρωσιμότητας των μερικών διαφορικών εξισώσεων. / The study of ordinary differential equations has often borrowed well known methods from Classical Mechanics. The most popular one is due to Sir William Rowan Hamilton (1805-1865), which has become a complete mathematical theory for the so-called Hamiltonian systems. Recently, Hamiltonian structures have been developed for systems of partial differential equations, particularly evolution equations. The purpose of this master thesis is to present the Hamiltonian theory for this type of systems, and especially for integrable equations. Our description is based on Symmetries, a useful tool for solving any differential equation, first discovered by Marius Sophus Lie (1842-1899). Thus, an introduction to his theory of point or geometrical symmetries is given in the first chapter, along with some applications, such as integration of ordinary differential equations and group-invariant solutions of partial differential equations. In the second chapter we discuss the connection between the symmetries of a system of differential equations and the conservation laws of the physical problem that they describe. That is the content of Noether’s theorem, which also holds in the particular case of Hamiltonian systems. The first step towards this direction is the generalization of the basic symmetry concept, and the second one is a small introduction to variational problems, also necessary for the next chapters. The well known Hamilton’s theory for finite systems is presented in the third chapter. We do not wish to describe the whole theory in full detail but only focus on these points that will be needed to handle the infinite-dimensional case. Therefore, we introduce the general notion of a Poisson structure, instead of the more familiar symplectic one. Avoiding the use of differential forms almost entirely, we concentrate on the Hamiltonian symmetries and their key role in the reduction theory of these systems. In Chapter 4 lies the heart of the subject, the Hamiltonian approach to a system of evolution equations. We start off by drawing an analogy between first order ordinary differential equations and evolution equations, and then we establish the fundamental concepts of the Hamiltonian franework, i.e. the Poisson bracket and Hamiltonian vector fields. Through another version of Noether’s theorem, we are able to explore, once again, the correspondence between (generalized) symmetries and conservation laws. Thus, we see that the study of several partial differential equations is in some way very close to the one of classical mechanical Hamiltonian systems. Evolution equations possessing, not just one, but two Hamiltonian structures, called bi-Hamiltonian systems, are discussed in the next chapter. The advantages of finding two different, independent Hamiltonian expressions are pointed out through the example of the Korteweg-de Vries equation. We show that such systems have an infinite number of symmetries and, subject to a mild compatibility condition, they also have an infinite number of conservation laws. Therefore they are completely integrable in Liouville’s sense. Several examples are presented, besides the KdV equation, such as the nonlinear Schroedinger, the modified KdV and other nonlinear wave equations. The final chapter is devoted to some of the recent publications, regarding multi-Hamiltonian evolution equations. This type of systems puts the classical Hamiltonian theory of ordinary differential equations in a new perspective and at the same time allows us to draw some connections with other integrability criteria used in the field of partial differential equations.

In this thesis we explore various aspects of Calabi-Yau (CY) manifolds and com- pactifications of the heterotic string over them. At first we focus on classifying symmetries and computing Hodge numbers of smooth CY quotients. Being non- simply connected, these quotients are an integral part of CY compactifications of the heterotic string, aimed at producing realistic string vacua. Discrete symmetries of such spaces that are generically present in the moduli space, are phenomenologically important since they may appear as symmetries of the associated low energy theory. We classify such symmetries for the class of smooth Complete Intersection CY (CICY) quotients, resulting in a large number of regular and R-symmetry examples. Our results strongly suggest that generic, non-freely acting symmetries for CY quotients arise relatively frequently. A large number of string derived Standard Models (SM) were recently obtained over this class of CY manifolds indicating that our results could be phenomenologically important. We also specialise to certain loci in the moduli space of a quintic quotient to produce highly symmetric CY quotients. Our computations thus far are the first steps towards constructing a sizeable class of highly symmetric smooth CY quotients. Knowledge of the topological properties of the internal space is vital in determining the suitability of the space for realistic string compactifications. Employing the tools of polynomial deformation and counting of invariant Kähler classes, we compute the Hodge numbers of a large number of smooth CICY quotients. These were later verified by independent cohomology computations. We go on to develop the machinery to understand the geometry of CY manifolds embedded as hypersurfaces in a product of del Pezzo surfaces. This led to an interesting account of the quotient space geometry, enabling the computation of Hodge numbers of such CY quotients. Until recently only a handful of CY compactifications were known that yielded low energy theories with desirable MSSM features. The recent construction of rank 5 line bundle sums over smooth CY quotients has led to several SU(5) GUTs with the exact MSSM spectrum. We derive semi-analytic results on the finiteness of the number of such line bundle models, and study the relationship between the volume of the CY and the number of line bundle models over them. We also imply a possible correlation between the observed number of generations and the value of the gauge coupling constants of the corresponding GUTs. String compactifications with underlying SO(10) GUTs are theoretically attractive especially since the discovery that neutrinos have non-zero mass. With this in mind, we construct tens of thousands of rank 4 stable line bundle sums over smooth CY quotients leading to SO(10) GUTs.

Since their original introduction, strongly complementary observables have been a fundamental ingredient of the ZX calculus, one of the most successful fragments of Categorical Quantum Mechanics (CQM). In this thesis, we show that strong complementarity plays a vastly greater role in quantum theory. Firstly, we use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Secondly, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.

Differential equations, in particular the nonlinear ones, are commonly used in formulating most of the fundamental laws of nature as well as many technological problems, among others. This makes the need for methods in finding closed form solutions to such equations all-important. In this thesis we study Lie symmetry methods for some nonlinear ordinary differential equations (ODE). The study focuses on identifying and using the underlying symmetries of the given first order nonlinear ordinary differential equation. An extension of the method to higher order ODE is also discussed. Several illustrative examples are presented. / Differentialekvationer, framförallt icke-linjära, används ofta vid formulering av fundamentala naturlagar liksom många tekniska problem. Därmed finns det ett stort behov av metoder där det går att hitta lösningar i sluten form till sådana ekvationer. I det här arbetet studerar vi Lie symmetrimetoder för några icke-linjära ordinära differentialekvationer (ODE). Studien fokuserar på att identifiera och använda de underliggande symmetrierna av den givna första ordningens icke-linjära ordinära differentialekvationen. En utvidgning av metoden till högre ordningens ODE diskuteras också. Ett flertal illustrativa exempel presenteras.

Orientador: Prof. Dr. Igor Leite Freire / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática, 2016. / Neste trabalho estudamos diversos aspectos de algumas classes de equações ou sistemas de equações. Simetrias de Lie, de Noether, leis de conservação derivadas do Teorema de Noether e soluções invariantes são obtidas para uma classe de equações diferenciais ordinárias. Também consideramos equações e sistemas do tipo Camassa-Holm, alguns dos quais foram obtidos como soluções de um problema inverso. Para todos são encontradas as simetrias de Lie e, para alguns, obtemos leis de conservação utilizando o Teorema de Ibragimov. Além disso, para casos particulares das equações deduzidas via problema inverso, investigamos a existência de soluções peakon e multipeakon. Finalmente, consideramos uma família de equações evolutivas, a qual admite soluções peakon e membros integráveis. / In this work we study several aspects of some families of differential equations and systems. Lie point symmetries, Noether symmetries, conservation laws obtained from Noether Theorem and invariant solutions are derived for a class of ordinary differential equations. We also consider Camassa-Holm type equations and systems, some of which deduced from an inverse problem. For all of them we obtain Lie point symmetry classifications and, for some, conservation laws using Ibragimov¿s Theorem. Furthermore, for particular cases of the equations obtained as an inverse problem, we investigate the existence of peakon and multipeakon solutions. Finally, we consider a family of evolution equations, which admits peakon solutions and integrable members.

Submitted by Ana Paula Inforsato (ana.inforsato@gmail.com) on 2018-05-21T19:31:12Z No. of bitstreams: 1 Ana Paula final.pdf: 2233592 bytes, checksum: 79e4da8a53734795b00c77eb03daa0af (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-05-22T13:00:36Z (GMT) No. of bitstreams: 1 inforsato_ap_me_rcla.pdf: 2232239 bytes, checksum: 6e5ef796e47228af1164f8da713ea8de (MD5) / Made available in DSpace on 2018-05-22T13:00:36Z (GMT). No. of bitstreams: 1 inforsato_ap_me_rcla.pdf: 2232239 bytes, checksum: 6e5ef796e47228af1164f8da713ea8de (MD5) Previous issue date: 2018-05-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho tratamos da classificação dos grupos de friso. Para realizar este objetivo abordamos elementos básicos da estrutura algébrica de grupo bem como apresentamos transformações geométricas, entre estas destacamos: translações, reflexões, rotações e reflexão com deslizamento. Além disso, localizamos este assunto como tópico da estrutura curricular do Ensino Fundamental e executamos uma atividade em sala de aula em que os alunos criaram frisos ornamentais. / In this work we deal with the classification of frieze groups. In order to accomplish this objective we approach basic elements of the algebraic group structure as well as present geometric transformations, among which we highlight: translations, reflections, rotations and glide reflection. In addition, we locate this subject as a topic of the curricular structure of Elementary School and perform a classroom activity in which the students created ornamental friezes.

Nesta tese, mostramos que o modelo estocástico binário para expressão gênica, por um gene auto-regulado, possui solução completa. A solução dependente do tempo é escrita via expansão em termos das funções de Heun confluentes. Apresentamos um exemplo de dinâmica estocástica desse gene. Para tal, desenvolvemos uma relação de recorrência entre derivadas arbitrárias das funções de Heun confluentes. Mostramos também que o regime estacionário deste modelo possui simetria de Lie SO(2, 1) tipo Lorentz. Esta simetria é análoga à simetria do momento angular, porém com um sinal errado. O invariante desta álgebra define a meia-vida relativa do regime dinâmico do gene. O equivalente do momento angular azimutal é uma medida indireta do nível de atividade do gene. Os operadores levantamento e abaixamento conectam diferentes processos estocásticos de expressão proteínica. As flutuações destes processos estocásticos são classificadas em termos das relações entre os etiquetadores de um elemento da representação da álgebra. No arcabouço da teoria dos grupos, o modelo estocástico para um gene externamente regulado aparece como um caso particular do modelo para um gene auto-regulado. Mostramos, por fim, uma comparação entre estas duas estratégias de regulação. Demonstramos que um gene auto-regulado pode expressar proteínas em regimes sub Poisson, Poisson ou super Poisson. Por seu turno, o gene externamente regulado somente expressa proteínas em regimes Poisson ou super Poisson. Portanto, num processo estocástico, a auto-regulação mostra-se como uma forma de controle mais precisa. Também mostramos que a dinâmica de genes auto-regulados possui meia-vida mais curta que a de genes externamente regulados. Ou seja, a auto-regulação permite respostas mais rápidas à perturbações externas. / In this thesis we show that the stochastic binary model to protein synthesis by na auto-regulated gene is completely solvable. The time-dependent solution is written in terms of the confluent Heun functions. We present an example of probability dynamics to this gene. To get that, we developed a recurrence relation between arbitrary derivatives of the confluent Heun functions. We also show the existence of a Lorentz-like Lie symmetry SO(2, 1). This is an analogous to the angular momentum symmetry but presenting one wrong sign in its preserved form. This invariant defines the relative half-life of the dynamical regime of the gene. The equivalent of the azimuth angular momentum measures indirectly the activity level of the gene. The ladder operators connect distinct stochastic processes of protein synthesis. The fluctuations of these processes are classified in terms of the relation between labeling numbers of a representation of the algebra. In the group theory formalism, the stochastic model to an externally regulated gene is a particular case of the model to an auto-regulated gene. We compare these two gene regulation strategies, and show that the auto-regulated gene can synthesize proteins into the super Poisson, Poisson, and sub Poisson fluctuating regimes. The externally regulated gene only presents the super Poisson and Poisson regimes. Therefore, the auto-regulation is responsible for a more precise control of gene expression. We also show that the dynamics of the auto-regulated genes has a shorter half-life. Thus, the auto-regulation permits faster responses of the system to external perturbation.