Eduardo Chillida, arquitecto

Es sabido que tras abandonar la carrera de arquitectura Chillida marcha a Paris a comenzar su carrera como escultor. De vuelta al País Vasco el hierro es el material en el que encuentra un camino propio. Toda la obra de su primera década está muy alejada en el aspecto formal de la arquitectura. Sin embargo, las líneas de fuerza que los hierros configuran muestran un interés espacial que queda manifiesto en una obra de 1953 denominada Consejo al espacio I. A partir de aquí su obra gira en torno al vacío. Las formas cambiarán con los materiales pero no el propósito. En sus dibujos, las manos expresan, más allá de su condición figurativa, la búsqueda del espacio cóncavo que los dedos encierran. El espacio que encuentra en la palma de la mano es equivalente al que construye con dedos gigantes de madera u hormigón. Chillida observa sus obras con una mirada cuya idea de escala se distancia del concepto de dimensión. Adquieren así una posibilidad de crecer que facilita imaginar sus espacios como arquitectura. Tras el hierro, el trabajo en madera y alabastro aproxima -en el aspecto formal- la obra de Chillida a la arquitectura. Los títulos de numerosas obras hacen referencia a ella o a conceptos con ella relacionados. Elogio de la arquitectura, Homenaje a la arquitectura, Arquitectura heterodoxa, Modulación del espacio, construcción heterodoxa, Alrededor del vacío, Mesa del arquitecto o Casa de luz, son algunos de ellos. La introducción del vacío en el alabastro da comienzo a un proceso tendente a que el espacio interior tenga una importancia inversamente proporcional a su presencia en la forma exterior. Un proceso de progresivo hermetismo donde pequeños espacios interiores son expresados mediante grandes masas envolventes. El espacio interior es el principal motivo por el que vemos la obra de Eduardo Chillida como arquitectura. La condición de interior, apreciable igualmente en sus grandes obras en el espacio público, hace que estas no constituyan únicamente hitos visuales sino espacios de protección con los que cuerpo interactúa estableciendo una nueva relación con el paisaje, el horizonte o el cosmos. La búsqueda de un interior vacío tiene como consecuencia la evolución hacia la desaparición de la forma exterior. Tal evolución comienza con el diálogo entre el bolo natural de alabastro y el vacío tallado de Homenaje a Goethe, y, como muestra de la inter-escalabilidad de la obra de Chillida, concluye con la introducción de un vacío oculto en la montaña sagrada de Tindaya. El gran vacío de Tindaya nos hace mirar la obra de pequeño formato a través de su filtro de aumento. Nos permite entender que el límite entre arquitectura y escultura es difuso en la obra del escultor vasco. Que la arquitectura puede estar en el origen de su escultura. Que su escultura puede ser el germen de muchas arquitecturas. ABSTRACT It is well known that after leaving his architectural studies Chillida went to Paris in order to begin his career as a sculptor. Back again to the Basque Country, iron is the material in which he finds his own way. In terms of form, his work from the very first ten years is far away from architecture. However, the strength lines set by the iron show a spatial will that is clearly evident in a 1953 piece called Advice to space I. From there on, his work focuses on void. Different materials will set different forms but the purpose will remain the same. In his drawings, hands are expressing, beyond its figurative condition, the search of the concave space that fingers are enclosing. The space founded in the palm of the hand is equivalent to the one built with giant wood or concrete fingers. Chillida faces his work with a look where the idea of scale takes distance to the concept of dimension. His works gets then a possibility to grow that allow us to imagine his spaces as architecture. Following iron, wood and alabaster pieces, in the formal aspect, approaches Chillida´s work to architecture. The titles of many sculptures are referred to it or to the concept related to it. In praise of architecture, Homage to architecture, Heterodox architecture, Modulation of space, Heterodox construction, Around the void, Architect’s table, or House of light, are some of them. The introduction of void in alabaster begins a process leading to the interior space has a presence inversely proportional to its importance in the external form. A process of progressive secrecy where small interior spaces are expressed through large enveloping masses. The interior space is the main reason why we see the work of Eduardo Chillida as architecture. The condition of inner space, equally noticeable in his great works in public space, makes this not only constitute visual landmarks, but protection spaces that body interacts with establishing a new relationship with the landscape, the horizon or the cosmos. The search of an inner void leads to an evolution towards the disappearance of the external form. The evolution begins in the dialogue between the natural bolus of alabaster and the carved void of Homage to Goethe, and as a sign of inter-scalability of the work of Chillida, it concludes with the introduction of a hidden void in the sacred mountain of Tindaya. The great void of Tindaya makes us look at a small format work trough the filter of his increase filter. It allows us to understand that the boundary between architecture and sculpture is diffuse in the work of the Basque sculptor. That architecture can be at the origin of his sculpture. That his sculpture may be the seed of many architectures.

Prime factors

We use Voiculescu’s free probability theory to prove the existence of prime factors, hence answering a longstanding problem in the theory of von Neumann algebras.

On representations of finite type

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A “coupling constant” (in (0,1]) is defined for primary states of “finite type” of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.

Quantum groups with invariant integrals

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.

Equações de onda generalizadas e quantização funtorial para teorias de campo escalar livre

Nesta dissertação apresentamos um método de quantização matemática e conceitualmente rigoroso para o campo escalar livre de interações. Trazemos de início alguns aspéctos importantes da Teoria de Distribuições e colocamos alguns pontos de geometria Lorentziana. O restante do trabalho é dividido em duas partes: na primeira, estudamos equações de onda em variedades Lorentzianas globalmente hiperbólicas e apresentamos o conceito de soluções fundamentais no contexto de equações locais. Em seguida, progressivamente construímos soluções fundamentais para o operador de onda a partir da distribuição de Riesz. Uma vez estabelecida uma solução para a equação de onda em uma vizinhança de um ponto da variedade, tratamos de construir uma solução global a partir da extensão do problema de Cauchy a toda a variedade, donde as soluções fundamentais dão lugar aos operadores de Green a partir da introdução de uma condição de contorno. Na última parte do trabalho, apresentamos um mínimo da Teoria de Categorias e Funtores para utilizar esse formalismo na contrução de um funtor de segunda quantização entre a categoria de variedades Lorentzianas globalmente hiperbólicas e a categoria de redes de álgebras C* satisfazendo os axiomas de Haag-Kastler. Ao fim, retomamos o caso particular do campo escalar quântico livre.

Sandbox experiments on basin inversion: testing the influence of basin orientation and basin fill

Analogue modelling experiments using brittle materials are performed to study the inversion of extensional structures. Asymmetric grabens of two different orientations are first created during a phase of extension and progressively filled. They are subsequently shortened in the same direction. The aim of our experiments is to determine factors affecting the style of deformation during inversion. We specifically investigate variations in thickness and distribution of strong and weak layers constituting the graben fill and in initial basin orientation. The main advantage of our experimental set-up is that we have a complete control on graben location, width, infill and orientation before inversion. The experiments show that shortening results only in limited reactivation of pre-existing normal faults. In general, forward thrusts and backthrusts cut across normal faults into the footwall of the graben. The forward thrusts either propagate parallel to the enveloping surface of faulted blocks or they cut across basin-limiting normal faults at various angles. The graben fill is mechanically extruded by displacement along forward thrusts that accommodate most of the shortening. Both pre-existing faults and weak graben fill act as zones of weakness during inversion and determine the orientation and location of both backthrusts and forward thrusts. The results of our experiments conform well to natural examples of inverted graben structures.

Rings of operators.

"Prepared with the assistance of a grant from the Research Corporation."

Linear operators and their spectra with applications to quantum mechanics.

From lectures given at the New York university Institute for mathematica and mechanics, by R. Cournat and others.

Analysis of edge detection algorithms on DIAL /

Mode of access: Internet.

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.

Pseudo memory effects, majorization and entropy in quantum random walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.

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.

Solution of the dual reflection equation for A(n-1)((1)) solid-on-solid model

We obtain a diagonal solution of the dual reflection equation for the elliptic A(n-1)((1)) solid-on-solid model. The isomorphism between the solutions of the reflection equation and its dual is studied. (C) 2004 American Institute of Physics.

Fungible dynamics: There are only two types of entangling multiple-qubit interactions

What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? It has been shown that all two-body Hamiltonian evolutions can be simulated using any fixed two-body entangling n-qubit Hamiltonian and fast local unitaries. By entangling we mean that every qubit is coupled to every other qubit, if not directly, then indirectly via intermediate qubits. We extend this study to the case where interactions may involve more than two qubits at a time. We find necessary and sufficient conditions for an arbitrary n-qubit Hamiltonian to be dynamically universal, that is, able to simulate any other Hamiltonian acting on n qubits, possibly in an inefficient manner. We prove that an entangling Hamiltonian is dynamically universal if and only if it contains at least one coupling term involving an even number of interacting qubits. For odd entangling Hamiltonians, i.e., Hamiltonians with couplings that involve only an odd number of qubits, we prove that dynamic universality is possible on an encoded set of n-1 logical qubits. We further prove that an odd entangling Hamiltonian can simulate any other odd Hamiltonian and classify the algebras that such Hamiltonians generate. Thus, our results show that up to local unitary operations, there are only two fundamentally different types of entangling Hamiltonian on n qubits. We also demonstrate that, provided the number of qubits directly coupled by the Hamiltonian is bounded above by a constant, our techniques can be made efficient.