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Varieties of residuated latticesGalatos, Nikolaos. January 1900 (has links)
Thesis (Ph. D. in Mathematics)--Vanderbilt University, 2003. / Title from PDF title screen. Includes bibliographical references and index.
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Modeling and computing based on latticesZhao, Haifeng, 1980- 07 February 2011 (has links)
This dissertation presents three studies addressing various modeling and computational aspects of lattice structures. The first study is concerned with characterization of the threshold behavior for very slow (subcritical) crack growth. First, it is shown that this behavior requires the presence of a healing mechanism. Then thermodynamic analysis of brittle fracture specimens near the threshold developed by Rice (1978) is extended to specimens undergoing microstructural changes. This extension gives rise to a generalization of the threshold concept that mirrors the way the resistance R-curve generalizes the fracture toughness. In the absence of experimental data, the resistance curve near the threshold is constructed using a lattice model that includes healing and rupture mechanisms. The second study is concerned with transmission of various boundary conditions through irregular lattices. The boundary conditions are parameterized using trigonometric Fourier series, and it is shown that, under certain conditions, transmission through irregular lattices can be well approximated by that through classical continuum. It is determined that such transmission must involve the wavelength of at least 12 lattice spacings; for smaller wavelength classical continuum approximations become increasingly inaccurate. Also it is shown that this restriction is much more severe than that associated with identifying the minimum size for representative volume elements. The third study is concerned with extending the use of boundary algebraic equations to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as boundary algebraic equations provide seamless bridging between discrete and continuum models. It is shown that, in contrast to regular lattices, boundary algebraic equations for irregular lattices require a statistical rather than deterministic treatment. Furthermore, boundary algebraic equations for irregular lattices contain certain terms that require the same amount of computational effort as the original problem. / text
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L2-invariants of nonuniform lattices in semisimple Lie groupsKammeyer, Holger 17 April 2013 (has links)
Wir berechnen L²-Invarianten bestimmter nichtuniformer Gitter in halbeinfachen Lie-Gruppen mithilfe der Borel-Serre-Kompaktifizierung arithmetisch definierter lokalsymmetrischer Räume. Als Hauptergebnisse erhalten wir neue Abschätzungen für Novikov-Shubin-Zahlen und das Verschwinden der L²-Torsion für Gitter in Gruppen mit geradem Fundamentalrang. Wir diskutieren Anwendungen auf Gromovs Null-im-Spektrum-Vermutung sowie auf eine Proportionalitätsvermutung für die L²-Torsion maßäquivalenter Gruppen. Im Schlussteil der Arbeit beschreiben wir einen Anpassungsvorgang für Chevalley-Basen komplexer halbeinfacher Lie-Algebren. Zu einer gegebenen Realform liefert dieser eine Basis mit halb- und ganzzahligen Strukturkonstanten, die wir durch das Wurzelsystem mit Involution ausdrücken.
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Exploring Matter-wave Dynamics with a Bose-Einstein CondensateChang, Rockson 08 January 2014 (has links)
Bose-Einstein condensates of dilute gases provide a rich and versatile platform to study both single-particle and many-body quantum phenomena. This thesis describes several experiments using a Bose-Einstein condensate of Rb-87 as a model system to study novel matter-wave effects that traditionally arise in vastly different systems, yet are difficult to access. We study the scattering of a particle from a repulsive potential barrier in the non-asymptotic regime, for which the collision dynamics are on-going. Using a Bose-Einstein condensate interacting with a sharp repulsive potential, two distinct transient scattering effects are observed: one due to the momentary deceleration of particles atop the barrier, and one due to the abrupt discontinuity in phase written on the wavepacket in position-space, akin to quantum reflection. Both effects lead to a redistribution of momenta, resulting in a rich interference pattern that may be used to reconstruct the single-particle wavefunction. In a second experiment, we study the response of a particle in a periodic potential to an applied force. By abruptly applying an external force to a Bose-Einstein condensate in a one-dimensional optical lattice, we show that the initial response of a particle in a periodic potential is in fact characterized by the bare mass, and only over timescales long compared to that of interband dynamics is the usual effective mass an appropriate description. This breakdown of the effective mass description on fast timescales is difficult to observe in traditional solid state systems due to their large bandgaps and fast timescale of interband dynamics. Both these experiments make use of the condensate's long coherence length, and the ability to shape and modulate the external potential on timescales fast compared to the particle dynamics, allowing for observation of novel matter-wave effects.
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Exploring Matter-wave Dynamics with a Bose-Einstein CondensateChang, Rockson 08 January 2014 (has links)
Bose-Einstein condensates of dilute gases provide a rich and versatile platform to study both single-particle and many-body quantum phenomena. This thesis describes several experiments using a Bose-Einstein condensate of Rb-87 as a model system to study novel matter-wave effects that traditionally arise in vastly different systems, yet are difficult to access. We study the scattering of a particle from a repulsive potential barrier in the non-asymptotic regime, for which the collision dynamics are on-going. Using a Bose-Einstein condensate interacting with a sharp repulsive potential, two distinct transient scattering effects are observed: one due to the momentary deceleration of particles atop the barrier, and one due to the abrupt discontinuity in phase written on the wavepacket in position-space, akin to quantum reflection. Both effects lead to a redistribution of momenta, resulting in a rich interference pattern that may be used to reconstruct the single-particle wavefunction. In a second experiment, we study the response of a particle in a periodic potential to an applied force. By abruptly applying an external force to a Bose-Einstein condensate in a one-dimensional optical lattice, we show that the initial response of a particle in a periodic potential is in fact characterized by the bare mass, and only over timescales long compared to that of interband dynamics is the usual effective mass an appropriate description. This breakdown of the effective mass description on fast timescales is difficult to observe in traditional solid state systems due to their large bandgaps and fast timescale of interband dynamics. Both these experiments make use of the condensate's long coherence length, and the ability to shape and modulate the external potential on timescales fast compared to the particle dynamics, allowing for observation of novel matter-wave effects.
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Residually small varieties and commutator theory.Swart, Istine Rodseth. January 2000 (has links)
Chapter 0
In this introductory chapter, certain notational and terminological conventions
are established and a summary given of background results that are
needed in subsequent chapters.
Chapter 1
In this chapter, the notion of a "weak conguence formula" [Tay72], [BB75] is
introduced and used to characterize both subdirectly irreducible algebras and
essential extensions. Special attention is paid to the role they play in varieties
with definable principal congruences.
The chapter focuses on residually small varieties; several of its results take
their motivation from the so-called "Quackenbush Problem" and the "RS Conjecture".
One of the main results presented gives nine equivalent characterizations
of a residually small variety; it is largely due to W. Taylor. It is followed
by several illustrative examples of residually small varieties.
The connections between residual smallness and several other (mostly categorical)
properties are also considered, e.g., absolute retracts, injectivity, congruence
extensibility, transferability of injections and the existence of injective
hulls. A result of Taylor that establishes a bound on the size of an injective
hull is included.
Chapter 2
Beginning with a proof of A. Day's Mal'cev-style characterization of congruence
modular varieties [Day69] (incorporating H.-P. Gumm's "Shifting Lemma"),
this chapter is a self-contained development of commutator theory in
such varieties. We adopt the purely algebraic approach of R. Freese and R.
McKenzie [FM87] but show that, in modular varieties, their notion of the commutator
[α,β] of two congruences α and β of an algebra coincides with that
introduced earlier by J. Hagemann and C. Herrmann [HH79] as well as with
the geometric approach proposed by Gumm [Gum80a],[Gum83].
Basic properties of the commutator are established, such as that it behaves
very well with respect to homomorphisms and sufficiently well in products
and subalgebras. Various characterizations of the condition "(x, y) Є [α,β]”
are proved. These results will be applied in the following chapters. We show
how the theory manifests itself in groups (where it gives the familiar group
theoretic commutator), rings, modules and congruence distributive varieties.
Chapter 3
We define Abelian congruences, and Abelian and affine algebras. Abelian
algebras are algebras A in which [A2, A2] = idA (where A2 and idA are the
greatest and least congruences of A). We show that an affine algebra is polynomially
equivalent to a module over a ring (and is Abelian). We give a proof that
an Abelian algebra in a modular variety is affine; this is Herrmann's Funda-
mental Theorem of Abelian Algebras [Her79]. Herrmann and Gumm [Gum78],
[Gum80a] established that any modular variety has a so-called ternary "difference
term" (a key ingredient of the Fundamental Theorem's proof). We derive
some properties of such a term, the most significant being that its existence
characterizes modular varieties.
Chapter 4
An important result in this chapter (which is due to several authors) is the
description of subdirectly irreducible algebras in a congruence modular variety.
In the case of congruence distributive varieties, this theorem specializes to
Jόnsson's Theorem.
We consider some properties of a commutator identity (Cl) which is a necessary
condition for a modular variety to be residually small. In the main
result of the chapter we see that for a finite algebra A in a modular variety,
the variety V(A) is residually small if and only if the subalgebras of A satisfy
(Cl). This theorem of Freese and McKenzie also proves that a finitely generated
congruence modular residually small variety has a finite residual bound,
and it describes such a bound. Thus, within modular varieties, it proves the
RS Conjecture.
Conclusion
The conclusion is a brief survey of further important results about residually
small varieties, and includes mention of the recently disproved (general) RS
Conjecture. / Thesis (M.Sc.)-University of Natal, Durban, 2000.
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A Study of Periodic and Aperiodic Ferromagnetic Antidot LatticesBhat, Vinayak S 01 January 2014 (has links)
This thesis reports our study of the effect of domain wall pinning by ferromagnetic (FM) metamaterials [1] in the form of periodic antidot lattices (ADL) on spin wave spectra in the reversible regime. This study was then extended to artificial quasicrystals in the form of Penrose P2 tilings (P2T). Our DC magnetization study of these metamaterials showed reproducible and temperature dependent knee anomalies in the hysteretic regime that are due to the isolated switching of the FM segments. Our dumbbell model analysis [2] of simulated magnetization maps indicates that FM switching in P2T is nonstochastic. We have also acquired the first direct, two-dimensional images of the magnetization of Permalloy films patterned into P2T using scanning electron microscopy with polarization analysis (SEMPA). Our SEMPA images demonstrate P2T behave as geometrically frustrated networks of narrow ferromagnetic film segments having near-uniform, bipolar (Ising-like) magnetization, similar to artificial spin ices (ASI). We find the unique aperiodic translational symmetry and diverse vertex coordination of multiply-connected P2T induce a more complex spin-ice behavior driven by exchange interactions in vertex domain walls, which differs markedly from the behavior of disconnected ASI governed only by dipolar interactions.
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Resonant tunnelling spectroscopy of vertical GaAs/AlGaAs structuresHolder, Jonathan Paul January 1999 (has links)
No description available.
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Finite Element Analyses of Failure Mechanisms and Structure-Property Relationships in Microtruss MaterialsBele, Eral 10 December 2012 (has links)
Microtruss materials are assemblies of struts or columns arranged periodically in space. The majority of past research efforts have focused on the key issue of microtruss architectural optimization. By contrast, this study focuses on the internal material structure at the level of the individual struts. Microstructural, geometrical, and material design techniques are used to improve their mechanical properties.
The finite element method is used to verify and create predictive analytical models, explain the dependence of strut properties on geometry, material properties and failure mechanisms, and extend the strut design analysis into suggestions for the improvement of fabrication methods. Three strut design methods are considered. First, microstructural design is performed by considering the influence of strut geometry on the strain energy imparted during stretch bending. By using the perforation geometry to modify the location and magnitude of this strain energy, microtruss materials with lower density and higher strength can be fabricated. Second, structural sleeves of aluminum oxide and electrodeposited nanocrystalline nickel are used to reinforce architecturally optimized aluminum alloy microtruss assemblies, creating hybrid materials with high weight-specific strength. The mechanical properties are controlled by the interaction between material and mechanical failure; this interaction is studied through finite element analyses and a proposed analytical relationship to provide suggestions for further improvements. Finally, hollow cylindrical struts are fabricated from electrodeposited nanocrystalline nickel. The high strength to weight ratio achieved in these struts is due to the microstructural and cross-sectional efficiency of the material.
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Finite Element Analyses of Failure Mechanisms and Structure-Property Relationships in Microtruss MaterialsBele, Eral 10 December 2012 (has links)
Microtruss materials are assemblies of struts or columns arranged periodically in space. The majority of past research efforts have focused on the key issue of microtruss architectural optimization. By contrast, this study focuses on the internal material structure at the level of the individual struts. Microstructural, geometrical, and material design techniques are used to improve their mechanical properties.
The finite element method is used to verify and create predictive analytical models, explain the dependence of strut properties on geometry, material properties and failure mechanisms, and extend the strut design analysis into suggestions for the improvement of fabrication methods. Three strut design methods are considered. First, microstructural design is performed by considering the influence of strut geometry on the strain energy imparted during stretch bending. By using the perforation geometry to modify the location and magnitude of this strain energy, microtruss materials with lower density and higher strength can be fabricated. Second, structural sleeves of aluminum oxide and electrodeposited nanocrystalline nickel are used to reinforce architecturally optimized aluminum alloy microtruss assemblies, creating hybrid materials with high weight-specific strength. The mechanical properties are controlled by the interaction between material and mechanical failure; this interaction is studied through finite element analyses and a proposed analytical relationship to provide suggestions for further improvements. Finally, hollow cylindrical struts are fabricated from electrodeposited nanocrystalline nickel. The high strength to weight ratio achieved in these struts is due to the microstructural and cross-sectional efficiency of the material.
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