Multiplicative Systems on Ultra-Metric Spaces

AMS Subj. Classification: MSC2010: 42C10, 43A50, 43A75

Spaces with Noetherian cohomology

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? This note provides, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens-Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements, with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

A moment problem for random discrete measures

Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η =

GALAXY EVOLUTION IN A COMPLEX ENVIRONMENT: A MULTI-WAVELENGTH STUDY OF HCG 7

The environment where galaxies are found heavily influences their evolution. Close groupings, like the ones in the cores of galaxy clusters or compact groups, evolve in ways far more dramatic than their isolated counterparts. We have conducted a multi-wavelength study of Hickson Compact Group 7 (HCG 7), consisting of four giant galaxies: three spirals and one lenticular. We use Hubble Space Telescope (HST) imaging to identify and characterize the young and old star cluster populations. We find young massive clusters (YMCs) mostly in the three spirals, while the lenticular features a large, unimodal population of globular clusters (GCs) but no detectable clusters with ages less than a few Gyr. The spatial and approximate age distributions of the similar to 300 YMCs and similar to 150 GCs thus hint at a regular star formation history in the group over a Hubble time. While at first glance the HST data show the galaxies as undisturbed, our deep ground-based, wide-field imaging that extends the HST coverage reveals faint signatures of stellar material in the intragroup medium (IGM). We do not, however, detect the IGM in H I or Chandra X-ray observations, signatures that would be expected to arise from major mergers. Despite this fact, we find that the H I gas content of the individual galaxies and the group as a whole are a third of the expected abundance. The appearance of quiescence is challenged by spectroscopy that reveals an intense ionization continuum in one galaxy nucleus, and post-burst characteristics in another. Our spectroscopic survey of dwarf galaxy members yields a single dwarf elliptical galaxy in an apparent stellar tidal feature. Based on all this information, we suggest an evolutionary scenario for HCG 7, whereby the galaxies convert most of their available gas into stars without the influence of major mergers and ultimately result in a dry merger. As the conditions governing compact groups are reminiscent of galaxies at intermediate redshift, we propose that HCGs are appropriate for studying galaxy evolution at z similar to 1-2.

Further probing the nature of FSR 1767

With Two-Micron All-Sky Survey (2MASS) photometry and proper motions, Bonatto et al. suggested that FSR 1767 is a globular cluster (GC), while with J and K NTT/SOFI photometry Froebrich, Meusinger & Scholz concluded that it is not a star cluster. In this study, we combine previous and new evidence that are consistent with a GC. For instance, we show that the horizontal branch (HB) and red giant branch (RGB) stars, besides sharing a common proper motion, have radial density profiles that consistently follow the King`s law independently. Reddening maps around FSR 1767 are built using the bulge RGB as reference and also Schlegel`s extinction values to study local absorptions. Both approaches provide similar maps and show that FSR 1767 is not located in a dust window, which otherwise might have produced the stellar overdensity. Besides, neighbouring regions of similar reddening as FSR 1767 do not present the blue HB stars that are a conspicuous feature in the colour-magnitude diagram of FSR 1767. We report the presence of a compact group of stars located in the central parts of FSR 1767. It appears to be a detached post-collapse core, similar to those of other nearby low-luminosity GCs projected towards the bulge. We note that while the NTT/SOFI photometry of the star cluster FSR 1716 matches perfectly that from 2MASS, it shows a considerable offset for FSR 1767. We discuss the possible reasons why both photometries differ. We confirm our previous structural and photometric fundamental parameters for FSR 1767, which are consistent with a GC.

On the algebraic K theory of the massive D8 and M9-branes

In this paper we review some basic relations of algebraic K theory and we formulate them in the language of D-branes. Then we study the relation between the D8-branes wrapped on an orientable, compact manifold W in a massive Type IIA, supergravity background and the M9-branes wrapped on a compact manifold Z in a massive d = 11 supergravity background from the K-theoretic point of view. By interpreting the D8-brane charges as elements of K-0(C(W)) and the (inequivalent classes of) spaces of gauge fields on the M9-branes as the elements of K-0(C(Z) x ((k) over bar*) G) where G is a one-dimensional compact group, a connection between charges and gauge fields is argued to exists. This connection could be realized as a composition map between the corresponding algebraic K theory groups.

Fourier duality as a quantization principle

The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary background for the implementation of Fourier duality on general locally compact groups. Kac algebras - and the duality they incorporate - are consequently examined as candidates for a general quantization framework extending the usual formalism. Using as a test case the simplest nontrivial phase space, the half-plane, it is shown how the structures present in the complete-plane case must be modified. Traces, for example, must be replaced by their noncommutative generalizations - weights - and the correspondence embodied in the Weyl-Wigner formalism is no longer complete. Provided the underlying algebraic structure is suitably adapted to each case, Fourier duality is shown to be indeed a very powerful guide to the quantization of general physical systems.

How far is C-0(Gamma, X) with Gamma discrete from C-0(K, X) spaces?

For a locally compact Hausdorff space K and a Banach space X we denote by C-0(K, X) the space of X-valued continuous functions on K which vanish at infinity, provided with the supremum norm. Let n be a positive integer, Gamma an infinite set with the discrete topology, and X a Banach space having non-trivial cotype. We first prove that if the nth derived set of K is not empty, then the Banach-Mazur distance between C-0(Gamma, X) and C-0(K, X) is greater than or equal to 2n + 1. We also show that the Banach-Mazur distance between C-0(N, X) and C([1, omega(n)k], X) is exactly 2n + 1, for any positive integers n and k. These results extend and provide a vector-valued version of some 1970 Cambern theorems, concerning the cases where n = 1 and X is the scalar field.

Reproducing kernel Hilbert spaces associated with kernels on topological spaces

We analyze reproducing kernel Hilbert spaces of positive definite kernels on a topological space X being either first countable or locally compact. The results include versions of Mercer's theorem and theorems on the embedding of these spaces into spaces of continuous and square integrable functions.

Atlas visual de cincuenta y siete años de arquitectura

En 1929 aparece el primer número de AA (L’architecture d’aujourd’hui), en 1932 existe un número-Monografía sobre los hermanos Perret, escrita por Pierre Vago, en 1946 se consolida como revista mensual y bajo la fundación de Andre Bloc. El primer número de AA que tengo en mi biblioteca es el número 34 (febrero-Marzo) de 1951. Mediante la lectura observada de una colección como AA, la determinación de unos capítulos representativos y la elección de imágenes de su tiempo se explican estos 57 años de arquitectura, cuyos resultados de un proceso temporal disfrutamos desde hace aproximadamente veinte años. A principio de los años cincuenta un grupo de jóvenes arquitectos, denso e intercomunicado en los congresos CIAM se propone situar la realidad de la arquitectura en los principios y realidades de su tiempo y de los que se intuyen futuro. Resultados de la Segunda Gran Guerra no son solo tragedias humanas sino gran investigación y desarrollo concretada industrialmente, enormes movimientos de personas en Europa y una sociedad enormemente optimista en USA, todo esto producirá las grandes transformaciones sociales de los 60’ y sus concreciones tecnológicas, políticas y desarrollo. Muchos arquitectos, publicaciones, concursos o decisiones políticas o privadas han producido el catalogo de arquitecturas de estos años, desde el CIAM IX hasta el POMPIDOU, desde la casa GEHRY hasta el KUNSTHAL, desde BRASILIA hasta SIDNEY, desde COPLEJIDAD Y CONTRADICCIÓN hasta DELIRIOS DE NY, desde OSAKA hasta MUNICH. En todas han existido un esfuerzo enorme por concretar la realidad de nuestras aspiraciones desde las puramente ideológicas de introspección social, hasta las concreciones de imagen directa. Varias líneas he abierto en mi proceso de investigación, las he llamado “anillos” porque todas estas líneas tienen similitud con los “anillos de crecimiento” de los árboles en cuanto a como se presentan en la estructura de formación y ha la cantidad de información no solo interna sino externa que aportan sobre la estructura árbol, su medio y la historia. Igual que podemos saber las temperaturas o las pluviometría que cubrieron Europa en la edad media solo estudiando los anillos de crecimiento de nuestro árboles (su grosor), de igual forma repasando LOS CONCURSOS y sus resultados que existieron en los últimos cincuenta años, podemos entender las aspiraciones y concreciones de las sociedades y sus arquitectos en este tiempo. Cuatro capítulos, los mas determinantes son los elegidos para dar cuerpo a una TESIS de tamaño capaz: Las ideas, el futuro, las referencias y el presente son los capítulos que de forma visual intentan explicar el fruto arquitectónico, sus aspiraciones y sus concreciones. Las ideas sin ninguna duda, pertenecen a los padres de nuestro tiempo, son las del Team X, la reflexión sobre lo perecedero, las realidades programáticas, densidades o lo publico-privado son solo planetas en el universo de sus ideas. El futuro lo trazaron aquellos que empezaron a investigar, concretar o reflexionar sobre la incidencia tanto de los procesos industriales con sus nuevos materiales como de las nuevas concreciones urbanas que los movimientos migratorios producirían en las ciudades. Las referencias son las bibliotecas de carácter informativo-visual que han generado nuestro inventario icónico. El presente son las imágenes de referencia de nuestro tiempo-mediático, no solo las produce un arquitecto (en este caso R.Koolhas), pero si que es verdad que en las imágenes arquitectónicas de OMA se concreta todo el catalogo de arquitecturas del presente. ENGLISH SUMMARY The fisrt AA (l´Architecture d´Aujourd´hui) issue was published in 1929, three years later, in 1932, a monographic issue on Perret brothers was written by Pierre Vago and in 1946 the magazine was strongly established as a monthly publication under the direction of André Bloc. The oldest copy I own on my bookshelves is nº 34 printed in February-March 1951. While carefully reading a collection such as AA we are able to extract representative chapters and images that can explain a linear process lasting 57 years of fruitful architectural production of which consequences we have been enjoying the past twenty years. In the early fifties a compact group of young architects linked by the CIAM congress decided to encompass architectural reality to the needs and principles of their time. Not only big human tragedies arose from the Second World War but also some of the fastest industrial inventions due to a powerful will to development, that altogether with european migrations and a high standard of optimism in the United States headed to the peak transformations of the sixties and their technological and political development. A bunch of architects, magazines and architectural competitions sided by political and private decisions produced the architectural catalogue of those years, from CIAM IX to the Pompidou art centre, from Gerhy´s house to the Kunsthal museum, from Brasilia to Sidney, from “Complexity and Contradiction“ to “Delirious NY” and from Osaka all the way to Munich. All of them carried a vast effort towards the concretion of will, from social introspection to a more effective development of images. Several paths run across my investigation, namely “the rings”, as they tend to behave as a growing structure like a tree trunk, providing internal and external information not only of the vegetal element but also of the environment and events crossing its time. In the same direction as we are able to predict the weather in the Middle Age by means of studying our forests, we can use the architectural competitions ant their results for the past fifty years to understand the will and ambitions of these developing societies and their architects. To give shape to a sizeable thesis the selected information has been packed in four chapters: Ideas, Future, References and Present, each of them structured as an Image bank visualizing the architectural product, its will and specific ambition. The first group, Ideas, is devoted entirely to the step-fathers of present architecture, with ideas that belong to TEAM X and embrace the reflection about the transitory, the programmatic reality, density or the public-private debate as wandering planets of their ideal universe. The second group is dedicated to a Future that was traced by those engaged on industrial processes and new material investigation together with some others exploring new urban concretions brought to existence by the pressure of the after war migrations. The third group, References, have been shaped as a stock-list containing all our iconic cross-references. The last group, Present, brings together the icons of this media-time we live in and not only those produced by one single architect (Rem Koolhaas) even if his production embodies all architectural references at the moment.

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.

Regular and Other Kinds of Extensions of Topological Spaces

∗ This work was partially supported by the National Foundation for Scientific Researches at the Bulgarian Ministry of Education and Science under contract no. MM-427/94.

Criterion of Normality of the Completely Regular Topology of Separate Continuity

2000 Mathematics Subject Classification: 54C10, 54D15, 54G12.

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(S(N))→Spec(S(M)) and Spec(j)2:Spec(S*(N))→Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iM:Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(S*(N))→Spec(S*(M)). If we denote Z:=ClSpec(S*(M))(M N), it holds that the restriction map Spec(j)2|:Spec(S*(N)) Spec(j)2-1(Z)→Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals "close" to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2-1(p) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2-1(p). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.