Habitar al borde del agua : villas del siglo XX

Las casas del siglo XX construidas al borde del mar –escenario único y origen de su expresión- siguen la topografía del suelo que habitan en su descenso al agua, y organizan espacios que obtienen miradas al horizonte marino. El horizonte nos confronta a lo desconocido. La vista del mar incita al deseo de recorrerlo, al deseo de viajar. Con origen en el otium romano, la presencia del agua define un modo de vida apacible –epicúreo- que los viajeros de la arquitectura -que nos acompañan en la tesis- traducen en sus refugios más íntimos. Experimentan con los cambios en los conceptos y en las técnicas, que se trasladan fácilmente a la arquitectura de la casa al borde del agua desde los inicios del Movimiento Moderno. Sus espacios del habitar frente al mar nos permiten descubrir estrategias comunes en sus respuestas más modernas. El arquitecto ante el proyecto frente al mar llega a la ubicación elegida, mira hacia el horizonte, y desplazándose ladera arriba se coloca en un punto estratégico que elige; después, delante de su tablero, recorre el camino inverso, coloca el lugar y sobre él dibuja los elementos que configurarán los espacios de la casa buscando esa mirada al mar. Las situaciones y ubicaciones posibles son infinitas; se definen algunas consonancias espaciales comunes en los suelos que se ocupan debidas a la presencia del mar que asociamos entre sí. De la comparación entre todas las casas -emblemas del XX-, surgen múltiples variantes de la mirada y de espacios al abierto, y modos de fabricar entornos con criterios comunes para dominar la visión del mar. Interiores que se abren al panorama, espacios cuyas ventanas buscan su mirada en la extensión del horizonte, rescisiones y aperturas. Reconocemos condicionantes en el territorio a los que las villas responden, categorías arquitectónicas que dan respuesta frente al mar en la búsqueda del moderno, la topografía, la mirada y el espacio al abierto. Las casas comparten la idea del dominio del paisaje desde el punto más alto, y en algunos suelos se invierte la tipología por la topografía, confirmando así un criterio común basado en la lectura del suelo como consecuencia de la búsqueda del espacio de la mirada. Los espacios al abierto se significan en todas ellas, son espacios al -aire libre- abiertos, unos envueltos, otros porticados, puertas del horizonte que se abren al exterior, en el techo de la casa, otros cubiertos y abiertos, espacios entre interior y exterior, en plataformas con bancales o patios envolventes, recintos o habitaciones abiertas. Descubrimos un logro del XX en los espacios positivos o negativos que traducen o juegan con el entorno, que ocupan o sustraen de los contornos construidos y que obtienen espacios intermedios en la búsqueda de la relación con el mar. Las herramientas que se utilizan son los dibujos de los autores, de las casas visitadas, el elenco de viajeros y sus viajes, el conocimiento desde el estudio de los proyectos. A través de la comparación por aproximaciones parciales, los dibujos nos definen la mirada al mar, el modo de ocupación y la forma de relación con el paisaje. La arquitectura del habitar frente al mar en el XX, hecha para y por arquitectos, topografía el suelo y construye la mirada, fabricando espacios al abierto en la relación entre la casa y el entorno marítimo. ABSTRACT Houses of the 20th century built by the sea – a unique setting which gives rise to their expression – follow the topography of the land they occupy in its descent towards the sea, and they organize spaces which give views of the maritime horizon. The horizon brings us face to face with the unknown. The sea view provokes a desire to cross it, to travel. The presence of the sea defines a peaceful, epicurean way of life, with origins in the Roman otium, which architectural travellers – who accompany us through the thesis – translate into their most intimate retreats. They experiment with changes in concepts and techniques, which are easily transferred to the architecture of the seaside house since the beginnings of the Modern Movement. Their living spaces allow us to discover common strategies in the most modern responses. The architect with a seaside project arrives at the site, looks towards the horizon, then walks uphill and chooses a strategic point; then with his drawing board he retraces his steps, he sets the position and then draws in the elements that make up the house that seeks a sea view. The number of potential situations and locations is infinite; certain common spatial accordances are defined in land which is occupied due to the presence of the sea. Comparison of all the houses – 20th century emblems – throws up multiple variations of view and open spaces, and ways of creating settings with common criteria so as to command the vision of the sea. Interiors which open up to the panorama, spaces whose windows seek their view in the expanse of the horizon, openings and closures. We recognise determinant factors in the territory to which the villas respond, architectural categories which give a seaside solution to the search for the modern, the topography, the view, and the open space. The houses share the idea of dominating the landscape from the highest point, and in some areas typology and topography are inverted, thus confirming a common criteria based on the reading of the ground as a conse quence of the search for the view space. Open spaces stand out in all the villas – spaces open to the outdoor air - some are wrapped, some arcaded, doors to the horizon which open up to the exterior, on the roof of the house. There are open and covered spaces, spaces between the exterior and interior, on platforms with banks and surrounding patios, enclosures and open rooms. We discover an achievement of the 20th century in the positive and negative spaces which translate and play with the setting, which occupy or are extracted from built contours and which obtain intermediate spaces in the search for the relationship with the sea. The tools used are the author’s drawings of the houses visited, the cast of travelling companions and their travels, the knowledge gained from study of the projects. Through comparison by means of partial approaches, the drawings define the view of the sea, the occupation mode and the way of relating to the landscape. Architecture for living by the sea in the 20th century, carried out both by and for the architects, shapes the land and constructs the view, creating open spaces in the relationship between the house and the sea surroundings.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Antibodies with infinite affinity

Here we report an approach to the design and production of antibody/ligand pairs, to achieve functional affinity far greater than avidin/biotin. Using fundamental chemical principles, we have developed antibody/ligand pairs that retain the binding specificity of the antibody, but do not dissociate. Choosing a structurally characterized antibody/ligand pair as an example, we engineered complementary reactive groups in the antibody binding pocket and the ligand, so that they would be in close proximity in the antibody/ligand complex. Cross-reactions with other molecules in the medium are averted because of the low reactivity of these groups; however, in the antibody/ligand complex the effective local concentrations of the complementary reactive groups are very large, allowing a covalent reaction to link the two together. By eliminating the dissociation of the ligand from the antibody, we have made the affinity functionally infinite. This chemical manipulation of affinity is applicable to other biological binding pairs.

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.

Exact and analytic-numerical solutions of lagging models of heat transfer in a semi-infinite medium

Different non-Fourier models of heat conduction have been considered in recent years, in a growing area of applications, to model microscale and ultrafast, transient, nonequilibrium responses in heat and mass transfer. In this work, using Fourier transforms, we obtain exact solutions for different lagging models of heat conduction in a semi-infinite domain, which allow the construction of analytic-numerical solutions with prescribed accuracy. Examples of numerical computations, comparing the properties of the models considered, are presented.

On stable uniqueness in linear semi-infinite optimization

This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-infinite optimization (LSIO) problem, in the sense of maintaining the strong uniqueness property under sufficiently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (affinely) stable problems, i.e., those LSIO problems whose feasible set (its affine hull, respectively) remains constant under sufficiently small perturbations.

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Stationarity and regularity of infinite collections of sets. Applications to infinitely constrained optimization

This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements—normals and/or subdifferentials.

Calmness modulus of linear semi-infinite programs

Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored, too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.

Substantial enlargement of angular existence range for Dyakonov-like surface waves at semi-infinite metal-dielectric superlattice

We investigated surface waves guided by the boundary of a semi-infinite layered metal-dielectric nanostructure cut normally to the layers and a semi-infinite dielectric material. Using the Floquet-Bloch formalism, we found that Dyakonov-like surface waves with hybrid polarization can propagate in dramatically enhanced angular range compared to conventional birefringent materials. Our numerical simulations for an Ag-GaAs stack in contact with glass show a low to moderate influence of losses.

Constraint qualifications in linear vector semi-infinite optimization

Linear vector semi-infinite optimization deals with the simultaneous minimization of finitely many linear scalar functions subject to infinitely many linear constraints. This paper provides characterizations of the weakly efficient, efficient, properly efficient and strongly efficient points in terms of cones involving the data and Karush–Kuhn–Tucker conditions. The latter characterizations rely on different local and global constraint qualifications. The global constraint qualifications are illustrated on a collection of selected applications.

Robust Solutions of MultiObjective Linear Semi-Infinite Programs under Constraint Data Uncertainty

The multiobjective optimization model studied in this paper deals with simultaneous minimization of finitely many linear functions subject to an arbitrary number of uncertain linear constraints. We first provide a radius of robust feasibility guaranteeing the feasibility of the robust counterpart under affine data parametrization. We then establish dual characterizations of robust solutions of our model that are immunized against data uncertainty by way of characterizing corresponding solutions of robust counterpart of the model. Consequently, we present robust duality theorems relating the value of the robust model with the corresponding value of its dual problem.

Primal Attainment in Convex Infinite Optimization Duality

This article provides results guarateeing that the optimal value of a given convex infinite optimization problem and its corresponding surrogate Lagrangian dual coincide and the primal optimal value is attainable. The conditions ensuring converse strong Lagrangian (in short, minsup) duality involve the weakly-inf-(locally) compactness of suitable functions and the linearity or relative closedness of some sets depending on the data. Applications are given to different areas of convex optimization, including an extension of the Clark-Duffin Theorem for ordinary convex programs.

Walking Through Cantor's Paradise and Escher's Garden: Epistemological Reflections on the Mathematical Infinite (II)

Infinity is not an easy concept. A number of difficulties that people cope with when dealing with problems related to infinity include its abstract nature, understanding infinity as an ongoing, never ending process, understanding infinity as a set of an infinite number of elements and appreciating well-known paradoxes. Infinity can be understood in several ways with often incompatible meanings, and can involve value judgments or assumptions that are neither explicit nor desired. To usher in its definition, we distinguish several aspects, teleological, artistic (Escher); some definitive, some potential, and others actual. This article also deals with some still unresolved aspects of the concept of infinity.

Comparative study of RPSALG algorithm for convex semi-infinite programming

The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.