No description available.

One attractive feature of lattice-based cryptosystems is the existence of security reductions relating the difficulty of breaking the cryptosystem to the difficulty of solving variants of the shortest vector problem (Regev, STOC 2005; Peikert, ePrint 2008). As there are no known polynomial-time algorithms which solve these lattice problems, this implies the asymptotic security of the cryptosystem. However, current lattice-based cryptosystems using the learning with errors (LWE) problem select parameters for which the reduction to the underlying lattice problem gives no meaningful assurance of concrete security. We analyze the runtime of the algorithm constructed in the reductions and select parameters for a cryptosystem under which the reductions give 128-bit security. While the resulting LWE-based cryptosystem is somewhat cumbersome, requiring a dimension of n = 1460, this is less than 2 times the dimension in the recently proposed Frodo cryptosystem (Bos et al., ACM CCS 2016), and could be implemented without catastrophic damage to communication times. We also investigate the runtime necessary for a reduction to give meaningful security assurances for current cryptosystems. / Thesis / Master of Science (MSc) / The advent of quantum computing poses a serious threat to modern cryptography, as most cryptosystems in use today are vulnerable to attacks by quantum algorithms. Recently proposed cryptosystems based on lattices are conjectured to be resistant to attacks by quantum computers. These cryptosystems also have a conditional security guarantee: if the cryptosystem can be broken by an attack, then a reduction exists which uses that attack to solve variants of the shortest vector problem (Regev, STOC 2005; Peikert, ePrint 2008). As these problems have no known efficient solutions, breaking the cryptosystem should be hard. However this guarantee only holds if the cryptosystem is constructed using parameters which satisfy conditions given in the reduction. Current proposals do not do this, and so cannot claim even a conditional security guarantee. We analyze two reductions and select parameters for a cryptosystem which satisfy these conditions. We also investigate the runtime necessary for a reduction to give meaningful security assurances for current cryptosystems.

The mean quadratic elongation, < A >, is used to measure the distortions of close packed arrays of anions in crystals. It is also used to trace how such distortions vary with temperature (T) and pressure (P). A calculation of < ..\ > for arrays of anions in materials that adopt the rutile and olivine structure-types shows that the arrays exhibit increased distortions with increasing P whereas those in materials that adopt the dolomite structure-type exhibit a decrease in distortion. On the other hand, the close packed arrays in the dolomite and calcite structure-types become more distorted with increasing T. The array of oxide anions in rutile show an increased distortion while those in stishovite show a decreased distortion with increasing T. An analysis of apparent bulk moduli and linear thermal expansion coefficients calculated for the constituent polyhedra shows that the distortion of a material with a close packed structure is governed by its unoccupied as well as its occupied polyhedra. It also indicates that the rearrangement of the anions in such materials induced by an increase in P is governed by the apparent bulk moduli of its constituent polyhedra rather than by a tendency of the anions to adopt a more perfect close packed array. / Master of Science

Ultracold atoms in optical lattices offer an important tool for studying dynamics in many-body interacting systems in a pristine environment. This thesis focuses on three theoretical works motivated by recent optical lattice experiments. In the first, we theoretically study the center of mass dynamics of states derived from the disordered Bose-Hubbard model in a trapping potential. We find that the edge states in the trap allow center of mass motion even with insulating states in the center. We identify short and long-time mechanisms for edge state transport in insulating phases. We also argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our zero temperature results offer important insights into mechanisms of transport of atoms in trapped optical lattices while putting bounds on center of mass dynamics expected at non-zero temperature. In the second work, we study the domain wall expansion dynamics of strongly interacting bosons in 2D optical lattices with disorder in a recent experiment {[}J.-y. Choi et al., Science 352, 1547 (2016)]. We show that Gutzwiller mean-field theory (GMFT) captures the main experimental observations, which are a result of the competition between disorder and interactions. Our findings highlight the difficulty in distinguishing glassy dynamics, which can be captured by GMFT, and many-body localization, which cannot be captured by GMFT, and indicate the need for further experimental studies of this system. The last work features our study of phase diagrams of the 2D Bose-Hubbard model in an optical lattice with synthetic spin-orbit coupling. We investigate the transitions between superfluids with different phase patterns, which may be detected by measuring the spin-dependent momentum distribution. / Ph. D. / Ultracold atoms in optical lattices, a periodic potential generated by laser beams, offer an important tool for quantum simulations in a pristine environment. Motivated by recent optical lattice experiments with the implementation of disorder and synthetic spin-orbit coupling, we utilize Gutzwiller mean-field theory (GMFT) to study the dynamics of disordered state in an optical lattice under the sudden shift of the harmonic trap, the domain wall expansion of strongly interacting bosons in 2D lattices with disorder, and spin-orbit-driven transitions in the Bose-Hubbard model. We argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our findings show that evidence for many-body localization claimed in experiments [J.-y. Choi et al., Science 352, 1547 (2016)] must lie in the differences between GMFT and experiments. We also find that strong spin-orbit coupling alone can generate superfluids with finite momentum and staggered phase patterns.

The dissertation explores the electronic structures induced by the adsorption of atoms and molecules on surfaces. It focuses on the physical and chemical properties of dispersed metal atom sites on oxide surfaces and the formation of novel molecular structures at hybrid organic-inorganic interfaces. The study uses density functional theory (DFT) calculations to simulate atomic-scale behaviors and aims to contribute to understanding reaction mechanisms and enhancing catalytic activity. Part one investigates the local environments of single platinum atoms on a cerium oxide surface by analyzing their physical and electronic properties. The next part studies the local environments of single platinum atoms dispersed on a cerium oxide surface. Part two compares the water-gas shift reaction for a platinum atom coordinated with a 10-phenanthroline-5,6-dione ligand on titanium oxide with that on a single platinum atom on a titanium oxide surface. It investigates the effect of vacancies in titanium oxide on the electronic structure of platinum and its reactivity. Part three explores the electronic structure of bistable molecules and their potential as molecular switches. Part four analyzes the patterns in scanning tunneling microscope (STM) images of an organic molecule layer on Au(111) and demonstrates electron confinement despite a weak interaction between the molecular layer and the Au surface.

<p>U radu je ispitan sledeći problem: <em>Pod kojim&nbsp;uslovima se može rekonstruisati&nbsp; (sintetisati)&nbsp;intervalno-vrednosni rasplinuti skup iz&nbsp; poznate&nbsp;familije nivo skupova.</em></p><p>U tu svrhu su proučena svojstva mreža&nbsp;intervala za svaki od četiri izabrana mrežna&nbsp;<br />uređenja: poredak po komponentama, neprecizni&nbsp;poredak (skupovna inkluzija), strogi &nbsp;i leksikografski&nbsp;poredak.&nbsp;</p><p>Definisane su i-između i ili-između ravne&nbsp;mreže&nbsp;&nbsp; i ispitana njihova svojstva potrebna za&nbsp;re&scaron;avanje postavljenog problema sinteze za&nbsp;intervalno-vrednosne rasplinute skupove. Za i-između ravne mreže je dokazano da su, u svom&nbsp;konačnom slučaju, slim mreže i dualno, da su ili-između ravne mreže dualno-slim mreže.</p><p>Data je karakterizacija kompletnih konačno&nbsp;prostornih i dualno konačno prostornih mreža.&nbsp;</p><p>Određena je klasa mreža koje se mogu&nbsp;injektivno preslikati u direktan proizvod n&nbsp;<br />kompletnih lanaca tako da su očuvani supremumi i&nbsp;dualno, određena je klasa mreža koje se mogu&nbsp;injektivno preslikati u direktan proizvod n lanaca&nbsp;tako da su očuvani infimumi.&nbsp;</p><p>U re&scaron;avanju problema sinteze posmatrana su&nbsp;dva tipa nivo skupova - gornji i donji nivo skupovi.&nbsp;Potreban i dovoljan uslov za sintezu intervalno-vrednosnog rasplinutog skupa iz poznate familije&nbsp;nivo skupova određen je za mrežu intervala koja je&nbsp;uređena poretkom po komponentama, za oba tipa&nbsp;posmatranih nivo skupova.</p><p>Za mrežu intervala uređenu nepreciznim&nbsp;poretkom, problem je re&scaron;en za donje nivo skupove,&nbsp;dok su za gornje nivo skupove određeni dovoljni&nbsp;uslovi.</p><p>Za mrežu intervala koja je uređena&nbsp;leksikografskim poretkom, takođe su dati dovoljni<br />uslovi i to za oba tipa nivo skupova.&nbsp;</p><p>Za mrežu intervala uređenu strogim poretkom&nbsp;problem nije re&scaron;avan, jer izlazi izvan okvira ovog&nbsp;rada.</p><p><br />Dobijeni rezultati su primenjeni za re&scaron;avanje&nbsp;sličnog problema sinteze za intervalno-vrednosne&nbsp;intuicionističke rasplinute skupove&nbsp; za mrežu&nbsp;intervala uređenu poretkom po komponentama.&nbsp;</p><p>Rezultati ovog istraživanja su od teorijskog&nbsp;značaja u teoriji mreža i teoriji rasplinutih skupova,&nbsp;ali postoji mogućnost za primenu u matematičkoj&nbsp;morfologiji i obradi slika.</p> / <p>In this thesis&nbsp; the following problem was investigated: Under which conditions an interval-valued fuzzy set can be reconstructed from the given family of cut sets.</p><p>We consider interval-valued fuzzy sets as&nbsp; a special type of lattice-valued fuzzy sets and&nbsp; we studied properties of lattices of intervals using four different lattice&nbsp; order: componentwise ordering, imprecision ordering (inclusion of sets), strong and&nbsp;lexicographical ordering.</p><p>We proposed new definitions&nbsp; of meet-between planar and join - between planar lattices, we investigated their properties and used them for solving problem of synthesis&nbsp; in&nbsp; interval-valued fuzzy sets.</p><p>It has been proven that finite meet- between planar lattices and slim lattices are equivalent, and dually:&nbsp;&nbsp;&nbsp; finite join-&nbsp; between planar lattices and dually slim lattices are equivalent.</p><p>Complete finitely&nbsp; spatial lattices and complete dually finitely spatial lattices are fully&nbsp;characterized&nbsp; in this setting. Next, we characterized&nbsp; lattices which can be order<br />embedded into a Cartesian product of&nbsp; n&nbsp; complete chains such that all suprema are preserved under the embedding.</p><p>And dually, we characterized lattices which can be order embedded into a Cartesian product of n complete chains such that all infima are preserved under the embedding.</p><p>We considered two types of cut sets &ndash; upper cuts and lower cuts.</p><p>Solution of the&nbsp; problem of synthesis of interval-valued fuzzy sets are given for lattices of intervals under componentwise ordering for both types of cut sets. Solution of problem of synthesis of&nbsp; interval-valued fuzzy sets&nbsp; are&nbsp; given for lower cuts for lattices of intervals under imprecision ordering.&nbsp; Sufficient conditions are given for lattices of intervals under imprecision ordering and family of upper cuts.</p><p>Sufficient conditions are also given for lattices of intervals under lexicographical ordering.</p><p>The problem of synthesis of interval-valued fuzzy sets for lattices of&nbsp; intervals under strong ordering is beyond the scope of this thesis.</p><p>A similar problem of synthesis of&nbsp; interval-valued intuitionistic fuzzy sets is solved for lattices of intervals under componentwise ordering.</p><p>These results are&nbsp; mostly of theoretical importance in lattice theory and fuzzy sets theory, but also they could&nbsp; be applied in mathematical morphology and in&nbsp; image processing.</p>

No description available.

On étudie le problème de la transmission de l'information à travers le canal AWGN en utilisant des réseaux. On commence par considérer des constellations infinies. Une nouvelle famille de réseaux obtenus par Construction A à partir de codes linéaires non binaires est proposée. Ces réseaux sont appelés LDA ("Low-Density Construction A") et sont caractérisés par des matrices de parité p-aires creuses, qui les mettent en relation directe avec les codes LPDC. Deux résultats sur leur possibilité d'atteindre la capacité de Poltyrev sont provés ; cela est d'abord démontré pour des poids des lignes logarithmiques des matrices de parité associées, puis pour des poids constants. Le deuxième résultat est basé sur certaines propriétés d'expansion des graphes de Tanner correspondants à ces matrices. Un autre sujet de ce travail concerne les constellations finies de réseaux. une nouvelle preuve est donnée du fait que des réseaux aléatoires obtenus par Construction A generale atteignent la capacité avec décodage de type "lattice decoding". Cela prolonge et améliore le travail de Erez et Zamir (2004), Ordentlich et Edrez (2012) Ling et Belfiore (2013). Cette preuve est basée sur les constellations de Coronoï et la multiplication par le coefficient de Wiener ("MMSE scaling") du siganl en sortie du canal. Finalement, ce résultat est adapté au cas des réseaux LDA, qui eux aussi atteignent la capacité avec le même procédé de transmission. Encore une fois, il est nécessaire d'exploiter les propriétés d'expansion des graphes de Tanner. A la fin de la dissertation, on présente un algorithme de décodage itératif et de type "message-passing" approprié au décodage des LDA en grandes dimensions. / The probleme of transmission of information over the AWGN channel using lattices is addressed. Firstly, infinite constellations are considered. A nex family of integer lattices built by means of construction A with non-binary linear condes is introduced. These lattices are called LPA (Low-Density Construction A) and are characterised by sparse p-ary parity-chedk matrices, that put them in direct relation with LPDC codes. Two results about the Poltyrev-capacity-archieving qualities of this family are proved, respectively for logarithmic row degree and constant row degree of the associated parity-check matrices. The second result is based on some expansion poperties of the Tanner graphs related to these matrices. Another topic of this work concerns finite lattice constellations. A new proff that heneral random Construction A lattices achieve capacity under lattice deconding is provided, continuing and pimproving the work of Erez and Zamir (2004), Ordentlich an Erez (2012), and Ling and Belfiore (2013). This proof is based on Voronoi lattice constellations and MMSE scaling of the channel output. Finally, this approach is adapted to the LDA case abd ut us scgiwn tgat LDA lattices achive capacity with the ame transmission scheme, too. Once again, it is necessary to exploit the expansion properties of the Tanner graphs. At he end of the dissertation, an iterative message-passing algorithm suitable for decoding LDA lattices in high dimensions is presented.

L'ordre faible sur un groupe de Coxeter W est un ordre partiel sur les éléments de W, intervenant dans de nombreux domaines de la combinatoire algébrique. Dans cette thèse, on propose un nouveau modèle général pour l'étude de cet ordre ainsi que d'autres ensembles ordonnés affiliés, et on explore diverses conséquences aussi bien algébriques que combinatoires de cette construction. On commence, dans le chapitre 3, par étudier une version restreinte de ce modèle. Plus précisément, on explique comment on peut associer un ensemble ordonné (aussi appelé « poset » à tout graphe orienté, simple, acyclique et muni d'une valutation sur ses sommets (aussi appelé « graphe valué »). On montre ensuite que ces posets sont en général des semi-treillis inférieurs, des treillis quand le graphe est fini, et on donne une formule explicite pour les valeurs de leurs fonctions de Möbius. On prouve ensuite que l'ordre faible sur les groupes de Coxeter de type A, B et A, le « flag weak order », ainsi que le treillis des idéaux supérieurs et inférieurs de tout poset fini peuvent être décrit avec notre modèle. Cette description amène naturellement à associer une série quasi-symétrique à chaque élément de An et An et on montre que cette série est en fait la série de Stanley associée. On présente dans le chapitre 4 les résultats centraux de la thèse, en effet on y introduit la généralisation de la construction faite au chapitre précédent au cas de tout graphe valué, c'est-à-dire sans condition s'acyclicité et de simplicité. On s'affranchit également de certaines contraintes imposées par la définition du chapitre 3, ce qui nous permet d'associer à tout graphe valué un treillis complet, et non plus un semi-treillis. En particulier, les semi-treillis du chapitre 3 se retrouvent naturellement plongés dans un treillis complet. Ceci nous amène à nous intéresser à des conjectures de Dyer portant sur l'étude d'une extension de l'ordre faible sur tout groupe de Coxeter (entre autres, il est conjecturé que ces extensions sont des treillis complets). On construit alors, à l'aide de notre formalisme, des extensions de l'ordre faible ayant beaucoup des propriétés conjecturalement attachées aux extensions de Dyer, et contenant ces dernières comme sous-poset. On conjecture que l'une de ces extensions coïncide avec celle de Dyer, et on fournit des outils pour le tester. Finalement, on étudie diverses conséquences de notre théorie : la construction d'extensions des semi-treillis cambriens (fin du chapitre 4), la construction d'un nouveau modèle combinatoire pour le treillis de Tamari et m-Tamari (chapitre 5), et enfin on propose une application à la combinatoire des tableaux (chapitre 6) / Weak order on a Coxeter group W is a partial order on W appearing in many areas of algebraic combinatorics. In this thesis, we propose a new general model for the study of the weak order and other related partially ordered sets (also called “posets”) and we explore various algebraic and combinatorial consequences of this construction. We begin with studying a restricted version of this model in Chapter 3. More precisely, we explain how one can associate a poset to any simple acyclic digraph together with a valuation on its vertices (also called “valued digraph”). We then prove that these posets are complete meet semi-lattices in general, complete lattices when the underlying digraph is finite, and we give an explicit formula to compute the value of their Möbius functions. Then, we show that the weak order on Coxeter groups of type A, B and A, the flag weak order, and the up-set (resp. down-set) lattices of any finite poset can be described within this theory. This description naturally leads to associate a quasi-symmetric function to any element of An And An, and we demonstrate that this function is in fact the corresponding Stanley symmetric function. In Chapter 4 we introduce the main results of this thesis. Indeed, we introduce in this chapter the generalization of the construction made in Chapter 3 to the case of any valued digraph, that is without the simplicity and acyclicity condition. Furthermore, this new definition allows us to get rid of some constraints of the definition of Chapter 3, allowing us to associate a complete lattice to each valued digraph. In particular, the meet semi-lattices of Chapter 3 are naturally extended into complete lattices. This leads us to the study of some conjectures of Dyer about the properties of an extension of the weak order having a lot of the properties conjecturally attached to Dyer’s extensions, and we prove that each one of our extensions contains Dyer’s extension as a sub-poset. We make the conjecture that one of this extension coincide with the one of Dyer, and we provide tools in order to test this conjecture. Finally, we study various consequences of out theory : we provide extensions of Cambrian semi-lattices into complete lattices (end of Chapter 4), we construct a new combinatorial model for Tamari and m-Tamari lattices (Chapter 5), and we finish with an application to tableaux combinatorics (Chapter 6)

Orientador: Antonio Aparecido de Andrade / Banca: Edson Donizete de Carvalho / Banca: Clotilzio Moreira dos Santos / Resumo: Neste trabalho, descrevemos cíclicas de reticulados algébricos Zn-rotacionados de dimensão ímpar. Essas construções são obtidas através da imersão Rn, via homomorfismo canônico, de determinados Z-módulos livres de posto finito contidos em subcorpos de extensões ciclotômicas do tipo Q(ζp), Q(ζp2), Q(ζpq)e Q(ζpq2), com p e q primos ímpares. Caracterizamos os reticulados e apresentamos propriedades e aplicações na Teoria da Informação. / Abstract: In this work we describe cyclic constructions of odd dimension. These constructions are obtained by immersion in Rn via the canonical homomorphism, of certain Z-free modules of finite rank contained in subfield cyclotomic extensions of type Q(ζp), Q(ζp2), Q(ζpq)e Q(ζpq2), com p e q odd prime. Featuring the obtained lattices and presenting properties and applications in Information Theory. / Mestre