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Dissertations / Theses on the topic 'Lattices theory'
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1
Race, David M. (David Michael). "Consistency in Lattices." Thesis, North Texas State University, 1986. https://digital.library.unt.edu/ark:/67531/metadc331688/.
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Let L be a lattice. For x ∈ L, we say x is a consistent join-irreducible if x V y is a join-irreducible of the lattice [y,1] for all y in L. We say L is consistent if every join-irreducible of L is consistent. In this dissertation, we study the notion of consistent elements in semimodular lattices.
2
Radu, Ion. "Stone's representation theorem." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3087.
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The thesis analyzes some aspects of the theory of distributive lattices, particularly two representation theorems: Birkhoff's representation theorem for finite distributive lattices and Stone's representation theorem for infinite distributive lattices.
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Gragg, Karen E. (Karen Elizabeth). "Dually Semimodular Consistent Lattices." Thesis, North Texas State University, 1988. https://digital.library.unt.edu/ark:/67531/metadc330641/.
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A lattice L is said to be dually semimodular if for all elements a and b in L, a ∨ b covers b implies that a covers a ∧ b. L is consistent if for every join-irreducible j and every element x in L, the element x ∨ j is a join-irreducible in the upper interval [x,l]. In this paper, finite dually semimodular consistent lattices are investigated. Examples of these lattices are the lattices of subnormal subgroups of a finite group. In 1954, R. P. Dilworth proved that in a finite modular lattice, the number of elements covering exactly k elements is equal to the number of elements covered by exactly k elements. Here, it is established that if a finite dually semimodular consistent lattice has the same number of join-irreducibles as meet-irreducibles, then it is modular. Hence, a converse of Dilworth's theorem, in the case when k equals 1, is obtained for finite dually semimodular consistent lattices. Several combinatorial results are shown for finite consistent lattices similar to those already established for finite geometric lattices. The reach of an element x in a lattice L is the difference between the rank of x*, the join of x and all the elements covering x, and the rank of x; the maximum reach of all elements in L is the reach of L. Sharp lower bounds for the total number of elements and the number of elements of a given reach in a semimodular consistent lattice given the rank, the reach, and the number of join-irreducibles are found. Extremal lattices attaining these bounds are described. Similar results are then obtained for finite dually semimodular consistent lattices.
4
Cheng, Y. "Theory of integrable lattices." Thesis, University of Manchester, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.568779.
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This thesis deals with the theory of integrable lattices in "solitons" throughout. Chapter 1 is a general introduction, which includes an historical survey and a short surrunary of the "solitons" theory and the present work. In Chapter 2, we discuss the equivalence between two kinds of lattice AKNS spectral problems - one includes two potentials, while the other includes four. The two nonlinear lattice systems associated with those two spectral problems, respectively is also proved to be equivalent to each other. In Chapter 3, we derive a class of nonlinear differential-difference equations (NDDEs) and put them into the Hamiltonian systems. Their complete integrability are proved in terms of so called "r-matrix". In the end of this Chapter, we study the symmetry properties and the related topics for lattice systems. In particular, we give detail for the Toda lattice systems. Chapter 4 is concerned with the Backlund transformations (BTs) and nonlinear superposition formulae (NSFs) for a class of NDDEs. A new method is presented to derive the generalized BTs and to prove that these BTs are precisely and really the auto-BTs. The three kinds of NSFs are derived by analysis of so called "elementary BTs". In Chapter 5, we investigate some relations between our lattices and the well-studied continuous systems. The continuum limits of our lattice systems and the discretizations of the continuous systems are discussed. The other study is about how we can consider a BT of continuous systems as a NDDE and then how a BT of such a NDDE can be reduced to the three kinds of NSFs of the continuous systems. The last Chapter is a study of integrable lattices under periodic boundary conditions. It provides a mathematical foundation for the study of integrable models in statistical mechanics. We are particularly interested in the lattice sine-Gordon and sinh-Gordon models. We not only prove the integrability of these models but also derive all kinds of classical phase shifts and some other physically interesting relations.
5
Heeney, Xiang Xia Huang. "Small lattices." Thesis, University of Hawaii at Manoa, 2000. http://hdl.handle.net/10125/25936.
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This dissertation introduces triple gluing lattices and proves that a triple gluing lattice is small if the key subcomponents are small. Then attention is turned to triple gluing irreducible small lattices. The triple gluing irreducible [Special characters omitted.] lattices are introduced. The conditions which insure [Special characters omitted.] small are discovered. This dissertation also give some triple gluing irreducible small lattices by gluing [Special characters omitted.] 's. Finally, K-structured lattices are introduced. We prove that a K-structured lattice L is triple gluing irreducible if and only if [Special characters omitted.] . We prove that no 4-element antichain lies in u 1 /v1 of a K-structured small lattice. We also prove that some special lattices with 3-element antichains can not lie in u1 /v1 of a K-structured small lattice.<br>viii, 87 leaves, bound : ill. ; 29 cm.<br>Thesis (Ph. D.)--University of Hawaii at Manoa, 2000.
6
Craig, Andrew Philip Knott. "Lattice-valued uniform convergence spaces the case of enriched lattices." Thesis, Rhodes University, 2008. http://hdl.handle.net/10962/d1005225.
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Using a pseudo-bisymmetric enriched cl-premonoid as the underlying lattice, we examine different categories of lattice-valued spaces. Lattice-valued topological spaces, uniform spaces and limit spaces are described, and we produce a new definition of stratified lattice-valued uniform convergence spaces in this generalised lattice context. We show that the category of stratified L-uniform convergence spaces is topological, and that the forgetful functor preserves initial constructions for the underlying stratified L-limit space. For the case of L a complete Heyting algebra, it is shown that the category of stratified L-uniform convergence spaces is cartesian closed.
7
Jipsen, Peter. "Varieties of lattices." Master's thesis, University of Cape Town, 1988. http://hdl.handle.net/11427/15851.
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Bibliography: pages 140-145.<br>An interesting problem in universal algebra is the connection between the internal structure of an algebra and the identities which it satisfies. The study of varieties of algebras provides some insight into this problem. Here we are concerned mainly with lattice varieties, about which a wealth of information has been obtained in the last twenty years. We begin with some preliminary results from universal algebra and lattice theory. The next chapter presents some properties of the lattice of all lattice sub-varieties. Here we also discuss the important notion of a splitting pair of varieties and give several characterisations of the associated splitting lattice. The more detailed study of lattice varieties splits naturally into the study of modular lattice varieties and non-modular lattice varieties, dealt with in the second and third chapter respectively. Among the results discussed there are Freese's theorem that the variety of all modular lattices is not generated by its finite members, and several results concerning the question which varieties cover a given variety. The fourth chapter contains a proof of Baker's finite basis theorem and some results about the join of finitely based lattice varieties. Included in the last chapter is a characterisation of the amalgamation classes of certain congruence distributive varieties and the result that there are only three lattice varieties which have the amalgamation property.
8
Bystrik, Anna. "On Delocalization Effects in Multidimensional Lattices." Thesis, University of North Texas, 1998. https://digital.library.unt.edu/ark:/67531/metadc278868/.
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A cubic lattice with random parameters is reduced to a linear chain by the means of the projection technique. The continued fraction expansion (c.f.e.) approach is herein applied to the density of states. Coefficients of the c.f.e. are obtained numerically by the recursion procedure. Properties of the non-stationary second moments (correlations and dispersions) of their distribution are studied in a connection with the other evidences of transport in a one-dimensional Mori chain. The second moments and the spectral density are computed for the various degrees of disorder in the prototype lattice. The possible directions of the further development are outlined. The physical problem that is addressed in the dissertation is the possibility of the existence of a non-Anderson disorder of a specific type. More precisely, this type of a disorder in the one-dimensional case would result in a positive localization threshold. A specific type of such non-Anderson disorder was obtained by adopting a transformation procedure which assigns to the matrix expressing the physics of the multidimensional crystal a tridiagonal Hamiltonian. This Hamiltonian is then assigned to an equivalent one-dimensional tight-binding model. One of the benefits of this approach is that we are guaranteed to obtain a linear crystal with a positive localization threshold. The reason for this is the existence of a threshold in a prototype sample. The resulting linear model is found to be characterized by a correlated and a nonstationary disorder. The existence of such special disorder is associated with the absence of Anderson localization in specially constructed one-dimensional lattices, when the noise intensity is below the non-zero critical value. This work is an important step towards isolating the general properties of a non-Anderson noise. This gives a basis for understanding of the insulator to metal transition in a linear crystal with a subcritical noise.
9
Madison, Kirk William. "Quantum transport in optical lattices /." Digital version accessible at:, 1998. http://wwwlib.umi.com/cr/utexas/main.
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10
Ocansey, Evans Doe. "Enumeration problems on lattices." Thesis, Stellenbosch : Stellenbosch University, 2013. http://hdl.handle.net/10019.1/80393.
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Thesis (MSc)--Stellenbosch University, 2013.<br>ENGLISH ABSTRACT: The main objective of our study is enumerating spanning trees (G) and perfect matchings
PM(G) on graphs G and lattices L. We demonstrate two methods of enumerating
spanning trees of any connected graph, namely the matrix-tree theorem and as a special
value of the Tutte polynomial T(G; x; y).
We present a general method for counting spanning trees on lattices in d 2 dimensions.
In particular we apply this method on the following regular lattices with d = 2:
rectangular, triangular, honeycomb, kagomé, diced, 9 3 lattice and its dual lattice to
derive a explicit formulas for the number of spanning trees of these lattices of finite sizes.
Regarding the problem of enumerating of perfect matchings, we prove Cayley’s theorem
which relates the Pfaffian of a skew symmetric matrix to its determinant. Using
this and defining the Pfaffian orientation on a planar graph, we derive explicit formula for
the number of perfect matchings on the following planar lattices; rectangular, honeycomb
and triangular.
For each of these lattices, we also determine the bulk limit or thermodynamic limit,
which is a natural measure of the rate of growth of the number of spanning trees (L)
and the number of perfect matchings PM(L).
An algorithm is implemented in the computer algebra system SAGE to count the
number of spanning trees as well as the number of perfect matchings of the lattices
studied.<br>AFRIKAANSE OPSOMMING: Die hoofdoel van ons studie is die aftelling van spanbome (G) en volkome afparings
PM(G) in grafieke G en roosters L. Ons beskou twee metodes om spanbome in ’n samehangende
grafiek af te tel, naamlik deur middel van die matriks-boom-stelling, en as ’n
spesiale waarde van die Tutte polinoom T(G; x; y).
Ons behandel ’n algemene metode om spanbome in roosters in d 2 dimensies af te
tel. In die besonder pas ons hierdie metode toe op die volgende reguliere roosters met
d = 2: reghoekig, driehoekig, heuningkoek, kagomé, blokkies, 9 3 rooster en sy duale
rooster. Ons bepaal eksplisiete formules vir die aantal spanbome in hierdie roosters van
eindige grootte.
Wat die aftelling van volkome afparings aanbetref, gee ons ’n bewys van Cayley se
stelling wat die Pfaffiaan van ’n skeefsimmetriese matriks met sy determinant verbind.
Met behulp van hierdie stelling en Pfaffiaanse oriënterings van planare grafieke bepaal
ons eksplisiete formules vir die aantal volkome afparings in die volgende planare roosters:
reghoekig, driehoekig, heuningkoek.
Vir elk van hierdie roosters word ook die “grootmaat limiet” (of termodinamiese limiet)
bepaal, wat ’n natuurlike maat vir die groeitempo van die aantaal spanbome (L) en die
aantal volkome afparings PM(L) voorstel.
’n Algoritme is in die rekenaaralgebra-stelsel SAGE geimplementeer om die aantal
spanboome asook die aantal volkome afparings in die toepaslike roosters af te tel.
11
Short, A. J. "Localisation, diffraction and hyperfinite lattices in quantum theory." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403987.
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12
Goosen, Gerrit. "Relational representations for bounded lattices with operators." Thesis, Stellenbosch : University of Stellenbosch, 2010. http://hdl.handle.net/10019.1/4343.
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Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.<br>ENGLISH ABSTRACT: Within lattice theory, an interesting question asked is whether a given abstract
lattice may be represented concretely as subsets of a closure system on a topological
space. This is true for boolean algebras, bounded distributive lattices
and arbitrary bounded lattices. In particular, there are a multitude of ways
to represent bounded lattices. We present some of these ideas, as well as an
analysis of the differences between them. We further investigate the attempts
that were made to extend the above representations to lattices endowed with
operators, in particular the work done on bounded distributive lattices with operators.
We then make a new contribution by extending this work to arbitrary
bounded lattices with operators. We also show that the so-called sufficiency
operator has a relational representation in the bounded lattice case.<br>AFRIKAANSE OPSOMMING: Binne die raamwerk van tralie teorie word die vraag soms gevra of ’n gegewe
tralie konkreet veteenwoordig kan word as subversamelings van ’n afsluitingssisteem
op ’n topologiese ruimte. Die voorgenoemde is waar vir, onder andere,
boolse algebras, begrensde distributiewe tralies en algemene begrensde
tralies. Daar is veral vir begrensde tralies menigte maniere om hul te verteenwoordig.
Ons bied sommige van hierdie idees voor, asook ’n analiese van
die verskille daarin teenwoordig. Verder ondersoek ons ook sommige van die
maniere waarop tralies tesame met operatore verteenwoordig kan word. Ons
sal spesiale aandag gee aan distributiewe tralies met operatore, soos gedoen in,
met die idee om die voorgenoemde uit te brei na algemene begrensde tralies met
operatore. Ons toon dan verder aan dat die sogenoemde voldoende operator
ook ’n relasionele verteenwoordiging het in die begrensde tralie geval.
13
Catterall, Simon Marcus. "Numerical studies of field theories on random lattices." Thesis, University of Oxford, 1988. http://ora.ox.ac.uk/objects/uuid:48c22ec6-0096-4a2c-be45-cbb20ba7a5b6.
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In this thesis we shall be concerned with the study of models which arise as a consequence of adopting discrete regularisations for various Euclidean space quantum field theories. Specifically, we employ a random triangulation of the continuum space, and define the fields only over nodes or links of the mesh. Lattice field theories, together with the Renormalisation Group, are introduced in the first chapter. Continuum physics is shown to depend on the positions and stabilities of zeroes of the β-function, which in turn requires a knowledge of the critical behaviour of the associated statistical model. In Chapter 2. we examine a theory of Dirac fermions in 2 + 1 dimensions on a random lattice. We investigate the behaviour of the 2-pt function and fermion condensate in the absence of any background gauge field. The results indicate certain doubling problems, generic to regular lattice formulations of fermion field theories, are evaded, at least at tree graph level. We then go on to examine the fermion vacuum currents in the presence of background fields with non-zero winding number. We are able to demonstrate the existence of a Chern-Simon's topological term in the gauge field effective action which yields parity violating vacuum currents. The magnitude of these are in agreement with certain continuum calculations. The final chapter concerns the properties of random surfaces. The particular class of models chosen originate as discretisations of Polyakov's string. The partition function is approximated by a sum over all possible random triangulations and an integral over vertex positions. The sum over random lattices is intended to mimick the functional integral over intrinsic metrics encountered in the continuum, and the model may also be pictured as 2D quantum gravity coupled to a scalar field. We consider the phase structure of the models when two forms of extrinsic curvature are added to the standard action. Monte-Carlo simulation indicates that with one type of curvature term a strong 2<sup>nd</sup> order phase transition exists at finite coupling, leading to a new continuum limit for the model possessing long-range correlation properties. With the other type a much weaker higher order transition is observed. In this case the surface will be crumpled at long distance. We discuss the implications of these results for continuum surfaces.
14
Harkins, Andrew. "Combining lattices of soluble lie groups." Thesis, University of Newcastle Upon Tyne, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341777.
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Janzen, David. "The smallest irreducible lattices in the product of trees /." Thesis, McGill University, 2007. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=101884.
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We produce a nonpositively curved square complex, X, containing exactly four squares. Its universal cover, X̃ ≅ T4 x T 4, is isomorphic to the product of two 4-valent trees. The group, pi1X, is a lattice in Aut (X̃) but π1X is not virtually a nontrivial product of free groups. There is no such example with fewer than four squares. The main ingredient in our analysis is that X̃ contains an "anti-torus" which is a certain aperiodically tiled plane.
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Matsoha, Moroli David Vusi. "Lattices of properties of countable graphs and the Hedetniemi Conjecture." Diss., University of Pretoria, 2013. http://hdl.handle.net/2263/33313.
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Lattices of hereditary properties of nite graphs have been extensively studied. We investigate the lattice L of induced-hereditary
properties of countable graphs. Of interest to us will be some of
the members of L. Much of our focus will be on hom-properties. We analyze their behaviour and consider their link to solving
the long standing Hedetniemi Conjecture. We then discuss universal
graphs and construct a universal graph for hom-properties.
We then use these universal graphs to prove a theorem by Szekeres and
Wilf. Lastly we off er a new proof of a theorem by Du ffus, Sands and
Woodrow.<br>Dissertation (MSc)--University of Pretoria, 2013.<br>Mathematics and Applied Mathematics<br>Unrestricted
17
Codogni, Giulio. "Satake compactifications, lattices and Schottky problem." Thesis, University of Cambridge, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.648555.
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Kotecha, Vinay. "Solitons on lattices and curved space-time." Thesis, Durham University, 2001. http://etheses.dur.ac.uk/3845/.
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This thesis is concerned with solitons (solutions of certain nonlinear partial differential equations) in certain cases when the underlying space is either a lattice or curved. Chapter 2 of the thesis is concerned with the outcome of collisions between a kink (a 1-dimensional soliton) and an antikink for certain topological discrete (TD) systems. The systems considered are the TD sine-Gordon and the TD ø(^4) For the TD sine-Gordon system it is found that the kink can support an internal shape mode which plays an important role during the collisions. In particular, this mode can be excited during collisions and this leads to spectacular resonance effects. The outcome of any particular collision has sensitive dependence on the initial conditions and could be either a trapped kink-antikink state, a "reflection" or a "transmission”. Such resonance effects are already known to exist for the conventional discrete ø(^4) system, and the TD ø(^4) system is no different, though the results for the two are not entirely similar. Chapter 3 considers the question of the existence of explicit travelling kink solutions for lattice systems. In particular, an expression for such a solution for the integrable lattice sine-Gordon system is derived. In Chapter 4, by reducing the Yang-Mills equations on the (2 + 2)-dimensional ultrahyperbolic space-time, an integrable Yang-Mills-Higgs system on (2 + 1) dimensional de Sitter space-time is derived. It represents the curved space-time version of the Bogomolny equations for monopoles on R(^3) . Using twister methods, various explicit solutions with gauge groups U(l) and SU(2) are constructed. A multi-solution SU(2) solution is also presented.
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Sellapillay, Kevissen. "Quantum dynamics on lattices." Electronic Thesis or Diss., Aix-Marseille, 2022. http://www.theses.fr/2022AIXM0429.
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Le domaine de l’informatique quantique a connu ces dernières années un fort développement. Un nouveau phénomène a récemment été découvert sur atomes de Rydberg, le scarring, qui s’est révélé être un nouveau mécanisme de brisure d’ergodicité: certains états initiaux n’explorent pas l’entièreté de l’espace de Hilbert mais restent localisés. Ce type de mécanisme a un potentiel intérêt pour encoder de l’information dans des systèmes quantiques.Dans cette thèse, on étudie des dynamiques discrètes en espace temps, qui sont des modèles quantiques simples à plusieurs corps. Des exemples de ces dynamiques discrètes sont les marches quantiques et les automates cellulaires quantiques. Elles sont écrites en termes de portes quantiques, ce qui les rend adaptées aux plateformes expérimentales.Dans la première partie de la thèse, on s’est intéressé à ces dynamiques discrètes dans le but de simuler un système physique. Plus précisément, nous proposons un automate cellulaire quantique et montrons sa convergence vers une théorie quantique des champs, le modèle de Schwinger en 1+1 dimensions.Dans la seconde partie de la thèse, nous investiguons ces dynamiques quantiques hors équilibre et étudions leurs propriétés de thermalisation sous le point de vue de l’information quantique.Nous proposons une généralisation de la marche quantique telle qu’elle interagit avec des spins situés sur les liens d’un réseau, et nous étudions les propriétés de propagation de l’aimantation et de l’intrication.Nous étudions ensuite un automate cellulaire quantique et découvrons que parmi les états chaotiques générés par la dynamique une hiérarchie peut émerger<br>Quantum computing has seen tremendous growth in the last few years. Progress in experimental techniques has allowed theoretical ideas to be tested and new quantum many-body phenomena to be discovered. One example of such a discovery is the scarring phenomenon in Rydberg atoms dynamics which turned out to be a new mechanism for ergodicity breaking : certain initial states do not explore the whole available Hilbert space but remain localized. Such mechanisms might be interesting for encoding information in a quantum system. In this thesis, we study discrete space time dynamics which are simple quantum many-body models. Examples of such discrete dynamics are quantum walks or quantum cellular automata. They are written in the gate language, which makes them adaptated to experimental platforms.In the first part of the thesis, we are interested about such discrete dynamics for the purpose of simulating a physical system. More specifically, we propose a quantum cellular automaton and show its convergence towards a quantum field theory, the Schwinger model in 1+1D. In the second part of the thesis, we investigate these quantum dynamics out of equilibrium and study their thermalization properties through the lens of quantum information.We propose a generalization of a quantum walk such that it interacts with spins on a lattice, and we study the propagation of magnetization and entanglement.We then study a quantum cellular automaton and show evidence that among the chaotic states that are generated by the dynamics some order emerges
20
Zschalig, Christian. "Characterizations of Planar Lattices by Left-relations." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2009. http://nbn-resolving.de/urn:nbn:de:bsz:14-ds-1240834941828-67021.
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Recently, Formal Concept Analysis has proven to be an efficient method for the analysis and representation of information. However, the possibility to visualize concept hierarchies is being affected by the difficulty of drawing attractive diagrams automatically. Reducing the number of edge crossings seems to increase the readability of those drawings. This dissertation concerns with a mandatory prerequisite of this constraint, namely the characterization and visual representation of planar lattices. The manifold existing approaches and algorithms are thereby considered under a different point of view. It is well known that exactly the planar lattices (or planar posets) possess an additional order ``from left to right''. Our aim in this work is to define left-relations and left-orders more precisely and to describe several aspects of planar lattices with their help. The three approaches employed structure the work in as many parts: Left-relations on lattices allow a more efficient consideration of conjugate orders since they are uniquely determined by the sorting of the meet-irreducibles. Additionally, the restriction on the meet-irreducibles enables us to achieve an intuitive description of standard contexts of planar lattices similar to the consecutive-one property. With the help of left-relations on diagrams, planar lattices can indeed be drawn without edge crossings in the plane. Thereby, lattice-theoretically found left-orders can be detected in the graphical representation again. Furthermore, we modify the left-right-numbering algorithm in order to obtain attribute-additive and plane drawings of planar lattices. Finally, we will consider left-relations on contexts. They turn out to be fairly similar structures to the Ferrers-graphs. Planar lattices can be characterized by a property of these graphs, namely the bipartiteness. We will constructively prove this result. Subsequently, we can design an efficient algorithm that finds all non-similar plane diagrams of a lattice<br>Die Formale Begriffsanalyse hat sich in den letzten Jahren als effizientes Werkzeug zur Datenanalyse und -repräsentation bewährt. Die Möglichkeit der visuellen Darstellung von Begriffshierarchien wird allerdings durch die Schwierigkeit, ansprechende Diagramme automatisch generieren zu können, beeinträchtigt. Offenbar sind Diagramme mit möglichst wenig Kantenkreuzungen für den menschlichen Anwender leichter lesbar. Diese Arbeit beschäftigt sich mit mit einer diesem Kriterium zugrunde liegenden Vorleistung, nämlich der Charakterisierung und Darstellung planarer Verbände. Die schon existierenden vielfältigen Ansätze und Methoden werden dabei unter einem neuen Gesichtspunkt betrachtet. Bekannterweise besitzen genau die planaren Verbände (bzw. planare geordnete Mengen) eine zusätzliche Ordnung &quot;von links nach rechts&quot;. Unser Ziel in dieser Arbeit ist es, solche Links-Relationen bzw. Links-Ordnungen genauer zu definieren und verschiedene Aspekte planarer Verbände mit ihrer Hilfe zu beschreiben. Die insgesamt drei auftretenden Sichtweisen gliedern die Arbeit in ebensoviele Teile: Links-Relationen auf Verbänden erlauben eine effizientere Behandlung konjugierter Ordnungen, da sie durch die Anordnung der Schnitt-Irreduziblen schon eindeutig festgelegt sind. Außerdem erlaubt die Beschränkung auf die Schnitt-Irreduziblen eine anschauliche Beschreibung von Standardkontexten planarer Verbände ähnlich der consecutive-one property. Mit Hilfe der Links-Relationen auf Diagrammen können planare Verbände tatsächlich eben gezeichnet werden. Dabei lassen sich verbandstheoretisch ermittelte Links-Ordnungen in der graphischen Darstellung wieder finden. Weiterhin geben wir in eine Modifikation des left-right-numbering an, mit der planare Verbände merkmaladditiv und eben gezeichnet werden können. Schließlich werden wir Links-Relationen auf Kontexten betrachten. Diese stellen sich als sehr ähnlich zu Ferrers-Graphen heraus. Planare Verbände lassen sich durch eine Eigenschaft dieser Graphen, nämlich die Bipartitheit, charakterisieren. Wir werden dieses Ergebnis konstruktiv beweisen und darauf aufbauend einen effizienten Algorithmus angeben, mit dem alle nicht-ähnlichen ebenen Diagramme eines Verbandes bestimmt werden können
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Kwuida, Leonard. "Dicomplemented Lattices." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2004. http://nbn-resolving.de/urn:nbn:de:swb:14-1101148726640-29266.
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Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassen<br>The aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a &quot;negation&quot; on formal concepts, &quot;concept algebras&quot; are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on &quot;Contextual Logic&quot;, a Formal Concept Analysis approach, based on concepts as units of thought
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Boykin, Charles Martin. "The Study of Translation Equivalence on Integer Lattices." Thesis, University of North Texas, 2003. https://digital.library.unt.edu/ark:/67531/metadc4345/.
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This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify certain variables as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of specific variables . Some properties that are equivalent to hyperfiniteness in the certain space are also developed. Lastly, we will give the new result that there is a continuous injective embedding from certain defined variables.
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Parviainen, Robert. "Connectivity Properties of Archimedean and Laves Lattices." Doctoral thesis, Uppsala : Matematiska institutionen, Univ. [distributör], 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4251.
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Goodwin, Michelle. "Lattices and Their Application: A Senior Thesis." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/cmc_theses/1317.
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Lattices are an easy and clean class of periodic arrangements that are not only discrete but associated with algebraic structures. We will specifically discuss applying lattices theory to computing the area of polygons in the plane and some optimization problems. This thesis will details information about Pick's Theorem and the higher-dimensional cases of Ehrhart Theory. Closely related to Pick's Theorem and Ehrhart Theory is the Frobenius Problem and Integer Knapsack Problem. Both of these problems have higher-dimension applications, where the difficulties are similar to those of Pick's Theorem and Ehrhart Theory. We will directly relate these problems to optimization problems and operations research.
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Semegni, Jean Yves. "On the computation of freely generated modular lattices." Thesis, Stellenbosch : Stellenbosch University, 2008. http://hdl.handle.net/10019.1/1207.
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McKinney, Sarah. "Dynamics of Bose-Einstein condensates in optical lattices /." Thesis, Connect to this title online; UW restricted, 2004. http://hdl.handle.net/1773/9805.
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Johnston, I. J. "Modularity and distributivity in directed multilattices." Thesis, Queen's University Belfast, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.373530.
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Zschalig, Christian. "Characterizations of Planar Lattices by Left-relations." Doctoral thesis, Technische Universität Dresden, 2008. https://tud.qucosa.de/id/qucosa%3A23687.
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Recently, Formal Concept Analysis has proven to be an efficient method for the analysis and representation of information. However, the possibility to visualize concept hierarchies is being affected by the difficulty of drawing attractive diagrams automatically. Reducing the number of edge crossings seems to increase the readability of those drawings. This dissertation concerns with a mandatory prerequisite of this constraint, namely the characterization and visual representation of planar lattices. The manifold existing approaches and algorithms are thereby considered under a different point of view. It is well known that exactly the planar lattices (or planar posets) possess an additional order ``from left to right''. Our aim in this work is to define left-relations and left-orders more precisely and to describe several aspects of planar lattices with their help. The three approaches employed structure the work in as many parts: Left-relations on lattices allow a more efficient consideration of conjugate orders since they are uniquely determined by the sorting of the meet-irreducibles. Additionally, the restriction on the meet-irreducibles enables us to achieve an intuitive description of standard contexts of planar lattices similar to the consecutive-one property. With the help of left-relations on diagrams, planar lattices can indeed be drawn without edge crossings in the plane. Thereby, lattice-theoretically found left-orders can be detected in the graphical representation again. Furthermore, we modify the left-right-numbering algorithm in order to obtain attribute-additive and plane drawings of planar lattices. Finally, we will consider left-relations on contexts. They turn out to be fairly similar structures to the Ferrers-graphs. Planar lattices can be characterized by a property of these graphs, namely the bipartiteness. We will constructively prove this result. Subsequently, we can design an efficient algorithm that finds all non-similar plane diagrams of a lattice.<br>Die Formale Begriffsanalyse hat sich in den letzten Jahren als effizientes Werkzeug zur Datenanalyse und -repräsentation bewährt. Die Möglichkeit der visuellen Darstellung von Begriffshierarchien wird allerdings durch die Schwierigkeit, ansprechende Diagramme automatisch generieren zu können, beeinträchtigt. Offenbar sind Diagramme mit möglichst wenig Kantenkreuzungen für den menschlichen Anwender leichter lesbar. Diese Arbeit beschäftigt sich mit mit einer diesem Kriterium zugrunde liegenden Vorleistung, nämlich der Charakterisierung und Darstellung planarer Verbände. Die schon existierenden vielfältigen Ansätze und Methoden werden dabei unter einem neuen Gesichtspunkt betrachtet. Bekannterweise besitzen genau die planaren Verbände (bzw. planare geordnete Mengen) eine zusätzliche Ordnung &quot;von links nach rechts&quot;. Unser Ziel in dieser Arbeit ist es, solche Links-Relationen bzw. Links-Ordnungen genauer zu definieren und verschiedene Aspekte planarer Verbände mit ihrer Hilfe zu beschreiben. Die insgesamt drei auftretenden Sichtweisen gliedern die Arbeit in ebensoviele Teile: Links-Relationen auf Verbänden erlauben eine effizientere Behandlung konjugierter Ordnungen, da sie durch die Anordnung der Schnitt-Irreduziblen schon eindeutig festgelegt sind. Außerdem erlaubt die Beschränkung auf die Schnitt-Irreduziblen eine anschauliche Beschreibung von Standardkontexten planarer Verbände ähnlich der consecutive-one property. Mit Hilfe der Links-Relationen auf Diagrammen können planare Verbände tatsächlich eben gezeichnet werden. Dabei lassen sich verbandstheoretisch ermittelte Links-Ordnungen in der graphischen Darstellung wieder finden. Weiterhin geben wir in eine Modifikation des left-right-numbering an, mit der planare Verbände merkmaladditiv und eben gezeichnet werden können. Schließlich werden wir Links-Relationen auf Kontexten betrachten. Diese stellen sich als sehr ähnlich zu Ferrers-Graphen heraus. Planare Verbände lassen sich durch eine Eigenschaft dieser Graphen, nämlich die Bipartitheit, charakterisieren. Wir werden dieses Ergebnis konstruktiv beweisen und darauf aufbauend einen effizienten Algorithmus angeben, mit dem alle nicht-ähnlichen ebenen Diagramme eines Verbandes bestimmt werden können.
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LANCELLOTTI, BENEDETTA. "Linear source lattices and their relevance in the representation theory of finite groups." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2018. http://hdl.handle.net/10281/199015.
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Molti dei problemi ancora aperti nella teoria delle rappresentazioni dei gruppi finiti riguardano la struttura locale-globale dei gruppi.
Sia G un gruppo finito, p un primo che ne divide l'ordine e (K,O,F) un sistema p-modulare di spezzamento. Lo studio locale-globale delle rappresentazioni di G cerca gli invarianti di G che possono essere individuati nei suoi sottogruppi locali, i.e, nel normalizzatore N di un p-sottogruppo D di G, e viceversa. Uno strumento chiave in questo contesto è la corrispondenza di Green, che stabilisce una biezione tra gli OG-reticoli indecomponibili che hanno D (o un suo coniugato in G) come vertice e gli ON-reticoli indecomponibili con vertice D.
Lo scopo principale della tesi è lo studio dei reticoli con sorgente lineare e il loro rapporto con le rappresentazioni irriducibili di G e N su K e su F. Gli oggetti principali utilizzati per questo fine sono l'anello di Grothendieck L(G) degli OG-reticoli con sorgente lineare e il suo sottoanello dei reticoli con sorgente banale.
Il primo capitolo raccoglie le definizioni e i risultati principali della teoria delle rappresentazioni utilizzati nella tesi. Una particolare attenzione è data alle proprietà dei reticoli con sorgente lineare e alla loro individuazione.
Nel Capitolo 2 sono costruite le sezioni canoniche del prodotto tensore con K (risp. con F) definito dall'anello degli OG-reticoli con sorgente lineare (risp. con sorgente banale) all'anello delle KG-rappresentazioni (risp. FG-rappresentazioni). Questo risultato è stato ottenuto seguendo due strategie. La prima prevede la costruzione di mappe duali considerando le "species" degli anelli coinvolti. Il punto di forza di questo approccio è il legame con le tavole delle rappresentazioni definite da Benson, d'altra parte però le mappe considerate prendono valori sulla complessificazione degli anelli. La seconda strategia, che risolve questo problema, consiste nell'utilizzare le formule canoniche di induzione introdotte da Boltje. Infine viene dimostrato che queste due strategie portano allo stesso risultato.
Il terzo capitolo è diviso in due parti. Nella prima viene formalmente introdotto l'anello dei reticoli essenziali con sorgente lineare, come conseguenza della definizione di opportune forme bilineari. Nella seconda parte viene analizzato il rapporto tra i reticoli con sorgente banale e vertice massimo e le KG-rappresentazioni irriducibili in due casi particolari: gruppi con sottogruppi normali di indice p e gruppi con sottogruppi di Sylow di ordine p.
Nell’ultimo capitolo viene indagato il legame tra la congettura di Alperin-McKay e il gruppo di Grothendieck Lmx(B) dei reticoli con sorgente lineare e vertice massimo in un blocco B di OG. Considerando una delle forme bilineari definite nel capitolo 3 e una opportuna sezione della proiezione canonica di L(B) in Lmx(B), è possibile formulare due nuove congetture (1 e 2), che implicano la congettura di Alperin-McKay e una sua riformulazione di Isaacs e Navarro. Inoltre la congettura 1 e la congettura di Alperin-McKay implicano la riformulazione proposta dai matematici sopracitati.
Il risultato principale di questo capitolo è la verifica della congettura 1 in alcuni casi non banali. Per esempio per blocchi "slendid equivant" al loro corrispondente di Brauer. Per un un risultato di R.Rouquier questo vale per tutti i blocchi con gruppo di difetto ciclico. In particolare, questo mostra un inedito legame tra la “splendid form” della congettura di Broué e la riformulazione di Isaacs e Navarro della congettura di Alperin-McKay.<br>Many investigated and interesting problems in the representation theory of finite groups concern the global and local structure of the groups. Let G be a
finite group, p a prime which divides the order of G and (K,O,F) a splitting p-modular system. The local-global study of the representations of G looks for the invariants of G that can be seen in its local subgroups, i.e., the normalizer N of a p-subgroup D of G, and vice versa.
A very strong tool in this context is the Green correspondence, which establishes a bijection between the indecomposable OG-lattices with D as a vertex and the indecomposable ON lattices with vertex D. The main scope of this thesis is the study of linear source lattices and their connection with the irreducible representations of G and N both over K and F. The main objects involved for this goal is the Grothendieck ring of linear source OG-lattices L(G) with its subring of trivial source lattices.
The first chapter is dedicated to the main results of the representation theory used through all the thesis. Special emphasis is laid on linear source lattices and their detection.
In Chapter 2 the canonical sections of the surjective maps given by the tensor product with K for linear source lattices and with F for trivial source lattices are constructed. This result has been obtained following two strategies. The first involves the construction of dual maps defined considering the species of the rings. The strength of this approach is its link to the representation tables defined by Benson; but the maps constructed take values on the complexification of the rings. The canonical induction formulas introduced by R. Boltje turn out to be the solution to bypass this problem. The final result of this part is the proof that these two approaches lead to the same maps.
Chapter 3 is divided in two parts. Studying a ring of modules a natural question is if it is possible to define a meaningful bilinear form. In this context the ring of essential linear source lattices arises.
In the first part of Chapter 3 it is formally introduced and its species are studied. In the last part the link between trivial source lattices with maximal vertex and irreducible characters is analyzed in two particular cases: groups with normal subgroups of index p and groups with Sylow subgroups of order p.
In the last chapter a connection between the Alperin-McKay conjecture and the Grothendieck group Lmx(B) of linear source lattices with maximal vertex in a
block B of OG is established. Considering a bilinear form defined in Chapter 3 and a section of the canonical projection of L(B) in Lmx(B), it is possible to state two new conjectures (1 and 2). If both of them are affirmative, then they yield the Alperin-McKay conjecture and one of its refinements due to M. Isaacs and G. Navarro. Moreover, Conjecture 1 and Alperin-McKay conjecture imply its refinement stated by the previously mentioned mathematicians. The main result of this chapter is the proof of Conjecture 1 in some non trivial cases. E.g., for a block splendid equivalent to its Brauer correspondent (for some defect group) Conjecture 1 is positively verified. By a result of R. Rouquier this applies to the case of blocks with cyclic defect groups. This result establishes a new connection between the refinement due to Isaacs and Navarro and the "splendid form" of Broué conjecture.
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Hausmann, Markus [Verfasser]. "Symmetric products, subgroup lattices and filtrations of global K-theory / Markus Hausmann." Bonn : Universitäts- und Landesbibliothek Bonn, 2016. http://d-nb.info/1113688408/34.
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Huff, Cheryl Rae. "Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices." Thesis, University of North Texas, 1999. https://digital.library.unt.edu/ark:/67531/metadc278330/.
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The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(Σ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and Lewis.
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Underwood, Devin Lane. "Microwave cavity lattices for quantum simulation with photons." Thesis, Princeton University, 2015. http://pqdtopen.proquest.com/#viewpdf?dispub=3686679.
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<p> Historically our understanding of the microscopic world has been impeded by limitations in systems that behave classically. Even today, understanding simple problems in quantum mechanics remains a difficult task both computationally and experimentally. As a means of overcoming these classical limitations, the idea of using a controllable quantum system to simulate a less controllable quantum system has been proposed. This concept is known as quantum simulation and is the origin of the ideas behind quantum computing. </p><p> In this thesis, experiments have been conducted that address the feasibility of using devices with a circuit quantum electrodynamics (cQED) architecture as a quantum simulator. In a cQED device, a superconducting qubit is capacitively coupled to a superconducting resonator resulting in coherent quantum behavior of the qubit when it interacts with photons inside the resonator. It has been shown theoretically that by forming a lattice of cQED elements, different quantum phases of photons will exist for dierent system parameters. In order to realize such a quantum simulator, the necessary experimental foundation must rst be developed. Here experimental eorts were focused on addressing two primary issues: 1) designing and fabricating low disorder lattices that are readily available to incorporate superconducting qubits, and 2) developing new measurement tools and techniques that can be used to characterize large lattices, and probe the predicted quantum phases within the lattice. </p><p> Three experiments addressing these issues were performed. In the rst experiment a Kagome lattice of transmission line resonators was designed and fabricated, and a comprehensive study on the effects of random disorder in the lattice demonstrated that disorder was dependent on the resonator geometry. Subsequently a cryogenic 3-axis scanning stage was developed and the operation of the scanning stage was demonstrated in the final two experiments. The rst scanning experiment was conducted on a 49 site Kagome lattice, where a sapphire defect was used to locally perturb each lattice site. This perturbative scanning probe microscopy provided a means to measure the distribution of photon modes throughout the entire lattice. The second scanning experiment was performed on a single transmission line resonator where a transmon qubit was fabricated on a separate substrate, mounted to the tip of the scanning stage and coupled to the resonator. Here the coupling strength of the qubit to the resonator was mapped out demonstrating strong coupling over a wide scanning range, thus indicating the potential for a scanning qubit to be used as a local quantum probe.</p>
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Goswick, Lee Michael. "On the dynamical, geometric, and arithmetic properties of Euclidean lattices." Birmingham, Ala. : University of Alabama at Birmingham, 2007. https://www.mhsl.uab.edu/dt/2007p/goswick.pdf.
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Thesis (Ph. D.)--University of Alabama at Birmingham, 2007.<br>Additional advisors: Nikolai Chernov, S. S. Ravindran, Alan Sprague, Min Sun. Description based on contents viewed Feb. 6, 2008; title from title screen. Includes bibliographical references.
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Kallio, Karn. "Static potential between adjoint quarks on coarse lattices in SU(2) gauge theory." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0022/MQ51373.pdf.
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Osborne, Charles Allen. "Some Aspects of the Theory of the Adelic Zeta Function Associated to the Space of Binary Cubic Forms." Diss., Temple University Libraries, 2010. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/74040.
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Mathematics<br>Ph.D.<br>This paper examines the theory of an adelization of Shintani's zeta function, especially as it relates to density theorems for discriminants of cubic extensions of number fields.<br>Temple University--Theses
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Meyer, Nicolas David. "Determination of Quadratic Lattices by Local Structure and Sublattices of Codimension One." OpenSIUC, 2015. https://opensiuc.lib.siu.edu/dissertations/1026.
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For definite quadratic lattices over the rings of integers of algebraic number fields, it is shown that lattices are determined up to isometry by their local structure and sublattices of codimension 1. In particular, a theorem of Yoshiyuki Kitaoka for $\mathbb{Z}$-lattices is generalized to definite lattices over algebraic number fields.
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Gatica, Perez Daniel. "Extensive operators in lattices of partitions for digital video analysis /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/5874.
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Santos, Francisco Ednilson Alves dos [Verfasser]. "Ginzburg-Landau theory for bosonic gases in optical lattices / Francisco Ednilson Alves dos Santos." Berlin : Freie Universität Berlin, 2011. http://d-nb.info/1026266092/34.
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Prabaharan, Kanagarajah. "Topics in ergodic theory : existence of invariant elements and ergodic decompositions of Banach lattices /." The Ohio State University, 1991. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487688973685025.
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Taylor, Imogen T. F. "Control and synchronisation of coupled map lattices : interdisciplinary modelling of synchronised dynamic behaviour (insects in particular)." Thesis, University of Derby, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275687.
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Sale, Andrew W. "The length of conjugators in solvable groups and lattices of semisimple Lie groups." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:ea21dab2-2da1-406a-bd4f-5457ab02a011.
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The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their L<sub>p</sub> compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.
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Maeda, Kazuki. "Theory of Discrete and Ultradiscrete Integrable Finite Lattices Associated with Orthogonal Polynomials and Its Applications." 京都大学 (Kyoto University), 2014. http://hdl.handle.net/2433/188859.
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Klein, Alexander. "Special purpose quantum information processing with atoms in optical lattices." Thesis, University of Oxford, 2007. http://ora.ox.ac.uk/objects/uuid:bc67ec3e-3cc7-4d13-ae11-b436b2ca897b.
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Atoms in optical lattices are promising candidates to implement quantum information processing. Their behaviour is well understood on a microscopic level, they exhibit excellent coherence properties, and they can be easily manipulated using external fields. In very deep optical lattices, each atom is restricted to a single lattice site and can be used as a qubit. If the lattice is shallow enough such that the atoms can move, their properties can be used to simulate certain condensed matter phenomena such as superconductivity. In this thesis, we show how technical problems of optical lattices such as restricted decoherence times, or fundamental shortcomings such as the lack of phonons or strong spin interactions, can be overcome by using current or near-future experimental techniques. We introduce a scheme that makes it possible to simulate model Hamiltonians known from high-temperature superconductivity. For this purpose, previous simulation schemes to realise the spin interaction terms are extended. We especially overcome the condition of a filling factor of exactly one, which otherwise would restrict the phase of the simulated system to a Mott-insulator. This scheme makes a large range of parameters accessible, which is difficult to cover with a condensed matter setup. We also investigate the properties of optical lattices submerged into a Bose-Einstein condensate (BEC). A weak-coupling expansion in the BEC-impurity interaction strength is used to derive a model that describes the lattice atoms in terms of polarons, i.e.~atoms dressed by Bogoliubov phonons. This is analogous to the description of electrons in solids, and we observe similar effects such as a crossover from coherent to incoherent transport for increasing temperatures. Moreover, the condensate mediates an attractive off-site interaction, which leads to macroscopic clusters at experimentally realistic parameters. Since the atoms in the lattice can also be used as a quantum register with the BEC mediating a two-qubit gate, we derive a quantum master equation to examine the coherence properties of the atomic qubits. We show that the system exhibits sub- and superdecoherence and that a fast implementation of the two-qubit gate competes with dephasing. Finally, we show how to realise the encoding of qubits in a decoherence-free subspace (DFS) using optical lattices. We develop methods for implementing robust gate operations on qubits encoded in a DFS exploiting collisional interactions between the atoms. We also give a detailed analysis of the performance and stability of the gate operations and show that a robust implementation of quantum repeaters can be achieved using our setup. We compare the robust repeater scheme to one that makes use of conventional qubits only, and show the conditions under which one outperforms the other.
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Wild, Marcel Wolfgang. "Dreieckverbande : lineare und quadratische darstellungstheorie." Thesis, University of Zurich, 1987. http://hdl.handle.net/10019.1/70322.
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Prof. Marcel Wild completed his PhD with Zurick University and this is a copy of the original works<br>The original works can be found at http://www.hbz.uzh.ch/<br>ABSTRACT: A linear representation of a modular lattice L is a homomorphism from L into the lattice Sub(V) of all subspaces of a vector space V. The representation theory of lattices was initiated by the Darmstadt school (Wille, Herrmann, Poguntke, et al), to large extent triggered by the linear representations of posets (Gabriel, Gelfand-Ponomarev, Nazarova, Roiter, Brenner, et al). Even though posets are more general than lattices, none of the two theories encompasses the other. In my thesis a natural type of finite lattice is identified, i.e. triangle lattices, and their linear representation theory is advanced. All of this was triggered by a more intricate setting of quadratic spaces (as opposed to mere vector spaces) and the question of how Witt’s Theorem on the congruence of finite-dimensional quadratic spaces lifts to spaces of uncountable dimensions. That issue is dealt with in the second half of the thesis.
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Sercombe, Damian. "A family of uniform lattices acting on a Davis complex with a non-discrete set of covolumes." Thesis, The University of Sydney, 2015. http://hdl.handle.net/2123/16026.
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Let (W,S) be a Coxeter system with Davis complex Σ. The polyhedral automorphism group G of Σ is a locally compact group under the compact-open topology. If G is a discrete group (as characterised by Haglund-Paulin), then the set Vu(G) of uniform lattices in G is discrete. Whether the converse is true remains an open problem. Under certain assumptions on (W,S), we show that Vu(G) is non-discrete and contains rationals (in lowest form) with denominators divisible by arbitrarily large powers of any prime less than a fixed integer. We explicitly construct our lattices as fundamental groups of complexes of groups with universal cover Σ. We conclude with a new proof of an already known analogous result for regular right-angled buildings.
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Usatine, Jeremy. "Arithmetical Graphs, Riemann-Roch Structure for Lattices, and the Frobenius Number Problem." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/hmc_theses/57.
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If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the construction for Lorenzini's Riemann-Roch structure and how it relates to the Riemann-Roch theorem for finite graphs shown by Matthew Baker and Serguei Norine.
We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of R. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of R.
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Kelvey, Robert J. Kelvey. "Properties of groups acting on Twin-Trees and Chabauty space." Bowling Green State University / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1479423366082688.
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Hasselfield, Matthew. "Localization of a particle due to dissipation in 1 and 2 dimensional lattices." Thesis, Vancouver : University of British Columbia, 2006. http://hdl.handle.net/2429/74.
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We study two aspects of the problem of a particle moving on a lattice while subject to dissipation, often called the "Schmid model." First, a correspondence between the Schmid model and boundary sine-Gordon field theory is explored, and a new method is applied to the calculation of the partition function for the theory. Second, a traditional condensed matter formulation of the problem in one spatial dimension is extended to the case of an arbitrary two-dimensional Bravais lattice.
A well-known mathematical analogy between one-dimensional dissipative quantum mechanics and string theory provides an equivalence between the Schmid model at the critical point and boundary sine-Gordon theory, which describes a free bosonic field subject to periodic interaction on the boundaries. Using the tools of conformal field theory, the partition function is calculated as a function of the temperature and the renormalized coupling constants of the boundary interaction. The method pursues an established technique of introducing an auxiliary free boson, fermionizing the system, and constructing the boundary state in fermion variables. However, a different way of obtaining the fermionic boundary conditions from the bosonic theory leads to an alternative renormalization for the coupling constants that occurs at a more natural level than in the established approach.
Recent renormalization group analyses of the extension of the Schmid model to a two-dimensional periodic potential have yielded interesting new structure in the phase diagram for the mobility. We extend a classic one-dimensional, finite temperature calculation to the case of an arbitrary two-dimensional Bravais lattice. The duality between weak-potential and tightbinding lattice limits is reproduced in the two-dimensional case, and a perturbation expansion in the potential strength used to verify the change in the critical dependence of the mobility on the strength of the dissipation. With a triangular lattice the possibility of third order contributions arises, and we obtain some preliminary expressions for their contributions to the mobility.
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Tsuchiya, Shunji. "Theory of Bose-Einstein condensates in optical lattices = Hikari kōshichū no Bōsu-Ainshutain gyōshukutai ni kansuru rironteki kenkyū /." Electronic version of text Electronic version of summary Electronic version of examination, 2005. http://www.wul.waseda.ac.jp/gakui/honbun/4003/.
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Thesis (Sci. D.)--Waseda University, 2005.<br>Accompanied by summary (6 p. ; 30 cm.) in Japanese. Includes bibliographical references (p. 83-88). "List of publications [by Shunji Tsuchiya]": p. 89-90.
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Schauch, Tim Konstantin [Verfasser], and Urs [Akademischer Betreuer] Hartl. "Weak admissibility of Hodge-Pink lattices in terms of Geometric Invariant Theory / Tim Konstantin Schauch ; Betreuer: Urs Hartl." Münster : Universitäts- und Landesbibliothek Münster, 2014. http://d-nb.info/1138285722/34.
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