Computer simulation studies of vector spin glasses

Die vorliegende Doktorarbeit befasst sich mit klassischen Vektor-Spingläsern eine Art von ungeordneten Magneten - auf verschiedenen Gittertypen. Da siernbedeutsam für eine experimentelle Realisierung sind, ist ein theoretisches Verständnis von Spinglas-Modellen mit wenigen Spinkomponenten und niedriger Gitterdimension von großer Bedeutung. Da sich dies jedoch als sehr schwierigrnerweist, sind neue, aussichtsreiche Ansätze nötig. Diese Arbeit betrachtet daher den Limesrnunendlich vieler Spindimensionen. Darin entstehen mehrere Vereinfachungen im Vergleichrnzu Modellen niedriger Spindimension, so dass für dieses bedeutsame Problem Eigenschaften sowohl bei Temperatur Null als auch bei endlichen Temperaturenrnüberwiegend mit numerischen Methoden ermittelt werden. Sowohl hyperkubische Gitter als auch ein vielseitiges 1d-Modell werden betrachtet. Letzteres erlaubt es, unterschiedliche Universalitätsklassen durch bloßes Abstimmen eines einzigen Parameters zu untersuchen. "Finite-size scaling''-Formen, kritische Exponenten, Quotienten kritischer Exponenten und andere kritische Größen werden nahegelegt und mit numerischen Ergebnissen verglichen. Eine detaillierte Beschreibung der Herleitungen aller numerisch ausgewerteter Gleichungen wird ebenso angegeben. Bei Temperatur Null wird eine gründliche Untersuchung der Grundzustände und Defektenergien gemacht. Eine Reihe interessanter Größen wird analysiert und insbesondere die untere kritische Dimension bestimmt. Bei endlicher Temperatur sind der Ordnungsparameter und die Spinglas-Suszeptibilität über die numerisch berechnete Korrelationsmatrix zugänglich. Das Spinglas-Modell im Limes unendlich vieler Spinkomponenten kann man als Ausgangspunkt zur Untersuchung der natürlicheren Modelle mit niedriger Spindimension betrachten. Wünschenswert wäre natürlich ein Modell, das die Vorteile des ersten mit den Eigenschaften des zweiten verbände. Daher wird in Modell mit Anisotropie vorgeschlagen und getestet, mit welchem versucht wird, dieses Ziel zu erreichen. Es wird auf reizvolle Wege hingewiesen, das Modell zu nutzen und eine tiefergehende Beschäftigung anzuregen. Zuletzt werden sogenannte "real-space" Renormierungsgruppenrechnungen sowohl analytisch als auch numerisch für endlich-dimensionale Vektor-Spingläser mit endlicher Anzahl von Spinkomponenten durchgeführt. Dies wird mit einer zuvor bestimmten neuen Migdal-Kadanoff Rekursionsrelation geschehen. Neben anderen Größen wird die untere kritische Dimension bestimmt.

Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

Crystalline Confinement

We show that exotic phases arise in generalized lattice gauge theories known as quantum link models in which classical gauge fields are replaced by quantum operators. While these quantum models with discrete variables have a finite-dimensional Hilbert space per link, the continuous gauge symmetry is still exact. An efficient cluster algorithm is used to study these exotic phases. The (2+1)-d system is confining at zero temperature with a spontaneously broken translation symmetry. A crystalline phase exhibits confinement via multi stranded strings between chargeanti-charge pairs. A phase transition between two distinct confined phases is weakly first order and has an emergent spontaneously broken approximate SO(2) global symmetry. The low-energy physics is described by a (2 + 1)-d RP(1) effective field theory, perturbed by a dangerously irrelevant SO(2) breaking operator, which prevents the interpretation of the emergent pseudo-Goldstone boson as a dual photon. This model is an ideal candidate to be implemented in quantum simulators to study phenomena that are not accessible using Monte Carlo simulations such as the real-time evolution of the confining string and the real-time dynamics of the pseudo-Goldstone boson.

Entire functions uniformly bounded on balls of a Banach space

Let X be an in�finite-dimensional complex Banach space. Very recently, several results on the existence of entire functions on X bounded on a given ball B1 � X and unbounded on another given ball B2 � X have been obtained. In this paper we consider the problem of �finding entire functions which are uniformly bounded on a collection of balls and unbounded on the balls of some other collection. RESUMEN. Sea X un espacio de Banach complejo de dimensión infinita. En este trabajo, los autores estudian el problema de encontrar una función entera en X que esté uniformemente acotada en una colección de de bolas en X y que no esté acotada en las bolas de otra colección.

Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l ∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel–Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system’s data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504–1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system’s coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

Logic-Based Outer Approximation for the Design of Discrete-Continuous Dynamic Systems with Implicit Discontinuities

We address the optimization of discrete-continuous dynamic optimization problems using a disjunctive multistage modeling framework, with implicit discontinuities, which increases the problem complexity since the number of continuous phases and discrete events is not known a-priori. After setting a fixed alternative sequence of modes, we convert the infinite-dimensional continuous mixed-logic dynamic (MLDO) problem into a finite dimensional discretized GDP problem by orthogonal collocation on finite elements. We use the Logic-based Outer Approximation algorithm to fully exploit the structure of the GDP representation of the problem. This modelling framework is illustrated with an optimization problem with implicit discontinuities (diver problem).

Entangled subspaces and quantum symmetries

Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum j with a spin 1/2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.

Coherent state construction of representations of osp(2|2) and primary fields of osp(2|2) conformal field theory

Representations of the superalgebra osp(2/2)(k)((1)) and current superalgebra. osp(2/2)k in the standard basis are investigated. All finite-dimensional typical and atypical representations of osp(2/2) are constructed by the vector coherent state method. Primary fields of the non-unitary conformal field theory associated with osp(2/2)(k)((1)) in the standard basis are obtained for arbitrary level k. (C) 2004 Elsevier B.V. All rights reserved.

Quantum doubles from a class of noncocommutative weak Hopf algebras

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products is constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf algebra is built. Examples of quantum doubles from a Clifford monoid as well as a noncommutative and noncocommutative weak Hopf algebra are given, generalizing quantum doubles from a group and a noncommutative and noncocommutative Hopf algebra, respectively. Moreover, some characterizations of quantum doubles of finite dimensional biperfect weak Hopf algebras are obtained. (C) 2004 American Institute of Physics.

Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control

When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that are not universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator expansion contains only terms coupling an odd number of systems, as identified by Bremner [Phys. Rev. A 69, 012313 (2004)]. We show that in all other cases entangling many-qudit Hamiltonians are universal for simulation.

Primary Fields and Screening Currents of gl (2/2) non-unitary conformal field theory

The non-semisimple gl(2)k current superalgebra in the standard basis and the corresponding non-unitary conformal field theory are investigated. Infinite families of primary fields corresponding to all finite-dimensional irreducible typical and atypical representations of gl(212) and three (two even and one odd) screening currents of the first kind are constructed explicitly in terms of ten free fields. (C) 2004 Elsevier B.V All rights reserved.

Ornstein-Uhlenbeck operator in convex domains of Banach spaces

Studiamo l'operatore di Ornstein-Uhlenbeck e il semigruppo di Ornstein-Uhlenbeck in un sottoinsieme aperto convesso $\Omega$ di uno spazio di Banach separabile $X$ dotato di una misura Gaussiana centrata non degnere $\gamma$. In particolare dimostriamo la disuguaglianza di Sobolev logaritmica e la disuguaglianza di Poincaré, e grazie a queste disuguaglianze deduciamo le proprietà spettrali dell'operatore di Ornstein-Uhlenbeck. Inoltre studiamo l'equazione ellittica $\lambdau+L^{\Omega}u=f$ in $\Omega$, dove $L^\Omega$ è l'operatore di Ornstein-Uhlenbeck. Dimostriamo che per $\lambda>0$ e $f\in L^2(\Omega,\gamma)$ la soluzione debole $u$ appartiene allo spazio di Sobolev $W^{2,2}(\Omega,\gamma)$. Inoltre dimostriamo che $u$ soddisfa la condizione di Neumann nel senso di tracce al bordo di $\Omega$. Questo viene fatto finita approssimazione dimensionale.

Using ancillary statistics in on-line learning algorithms

Neural networks are usually curved statistical models. They do not have finite dimensional sufficient statistics, so on-line learning on the model itself inevitably loses information. In this paper we propose a new scheme for training curved models, inspired by the ideas of ancillary statistics and adaptive critics. At each point estimate an auxiliary flat model (exponential family) is built to locally accommodate both the usual statistic (tangent to the model) and an ancillary statistic (normal to the model). The auxiliary model plays a role in determining credit assignment analogous to that played by an adaptive critic in solving temporal problems. The method is illustrated with the Cauchy model and the algorithm is proved to be asymptotically efficient.

Finitely coordinated models for low-temperature phases of amorphous systems

We introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite-coordination effects on the low-temperature behaviour of finite-dimensional systems. Our models use a description in terms of small deviations of particle coordinates from a set of reference positions, particularly appropriate for the description of low-temperature phenomena. A Born-von Karman-type expansion with random coefficients is used to model effects of frozen heterogeneities. The key quantity appearing in the theoretical description is a full distribution of effective single-site potentials which needs to be determined self-consistently. If microscopic interactions are harmonic, the effective single-site potentials turn out to be harmonic as well, and the distribution of these single-site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels. For structural glasses characterized by frustration and anharmonicities in the microscopic interactions, the distribution of single-site potentials involves anharmonicities of all orders, and both single-well and double-well potentials are observed, the latter with a broad spectrum of barrier heights. The appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single-site potentials, as previously observed for fully connected systems. Double-well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well-known universal glassy low-temperature anomalies when quantum effects are taken into account. © 2007 IOP Publishing Ltd.

The design of environmental test rigs

Product reliability and its environmental performance have become critical elements within a product's specification and design. To obtain a high level of confidence in the reliability of the design it is customary to test the design under realistic conditions in a laboratory. The objective of the work is to examine the feasibility of designing mechanical test rigs which exhibit prescribed dynamical characteristics. The design is then attached to the rig and excitation is applied to the rig, which then transmits representative vibration levels into the product. The philosophical considerations made at the outset of the project are discussed as they form the basis for the resulting design methodologies. It is attempted to directly identify the parameters of a test rig from the spatial model derived during the system identification process. It is shown to be impossible to identify a feasible test rig design using this technique. A finite dimensional optimal design methodology is developed which identifies the parameters of a discrete spring/mass system which is dynamically similar to a point coordinate on a continuous structure. This design methodology is incorporated within another procedure which derives a structure comprising a continuous element and a discrete system. This methodology is used to obtain point coordinate similarity for two planes of motion, which is validated by experimental tests. A limitation of this approach is that it is impossible to achieve multi-coordinate similarity due to an interaction of the discrete system and the continuous element at points away from the coordinate of interest. During the work the importance of the continuous element is highlighted and a design methodology is developed for continuous structures. The design methodology is based upon distributed parameter optimal design techniques and allows an initial poor design estimate to be moved in a feasible direction towards an acceptable design solution. Cumulative damage theory is used to provide a quantitative method of assessing the quality of dynamic similarity. It is shown that the combination of modal analysis techniques and cumulative damage theory provides a feasible design synthesis methodology for representative test rigs.