Krein’s Theorem in the Context of Topological Abelian Groups

A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of G is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups G which endowed with the weak topology associated to their character groups G∧, do not have the qcp. Thus, Krein’s Theorem, a well known result in the framework of locally convex spaces, cannot be fully extended to locally quasi-convex groups. Some features of the qcp are also studied

A survey on reflexivity of abelian topological groups,

a r t i c l e i n f o a b s t r a c t In this survey we deal with results on reflexivity of certain classes of groups, with special emphasis on the smaller class which better reflects the properties of LCA groups, namely that of strongly reflexive groups. A topological abelian group is said to be strongly reflexive if all its closed subgroups and its Hausdorff quotients as well as the closed subgroups and the Hausdorff quotients of its dual group are reflexive

Acceso a pruebas complementarias para el diagnóstico de enfermedades infecciosas en las consultas de pediatría de atención primaria

Introducción En los procesos infecciosos la realización de pruebas complementarias puede aumentar la precisión diagnóstica, la adecuación de los tratamientos, así como dar a conocer la epidemiología y patrón de resistencias bacterianas de la comunidad. El Grupo de Patología Infecciosa de la Asociación Española de Pediatría de Atención Primaria (GPI-AEPap) diseñó este estudio para conocer la accesibilidad a pruebas complementarias (PC) y sus resultados que tienen los pediatras que trabajan en atención primaria en el ámbito de la salud pública. Material y métodos Estudio observacional transversal descriptivo de ámbito nacional, a través de una encuesta de cumplimentación voluntaria, distribuida on line a todos los socios de AEPap y a los suscriptores de la lista de distribución PEDIAP entre los meses de abril y mayo de 2017. Resultados Se obtienen 517 respuestas. Se analizan datos del entorno profesional, así como los referidos a la solicitud de pruebas complementarias básicas (hemograma, bioquímica, sistemático de orina), utilización de Test Rápido Detección de Antígeno para Streptococo grupo A (TRDA), sobre cultivos bacterianos, serologías, pruebas diagnósticas de tosferina y tuberculosis (Mantoux) y pruebas de imagen. Conclusiones Hay variabilidad entre CCAA y áreas asistenciales. Se detectan claras áreas de mejora en la accesibilidad a diferentes PC, tiempo de recogida y envío de muestras, demora en la recepción, en los resultados y en tiempos de espera para pruebas de imagen no urgentes. Esto interfiere en la capacidad de intervención y resolución del pediatra de atención primaria. Introduction: The performing of complementary tests in infectious processes can increase the diagnostic precision, the adequacy of treatments, as well as determining the epidemiology and pattern of bacterial resistance of the community. The Infectious Pathology Group of the Spanish Association of Primary Care Paediatrics (GPI-AEPap) has designed this study in order to determine the availability of complementary tests (CT) for paediatricians working in Primary Care of the public health system as well as their results. Material and methods: Observational cross-sectional descriptive national study was carried out using a voluntary self-report questionnaire distributed online to all AEPap members and to the subscribers of the PEDIAP distribution list between the months of April and May 2017. Results: A total of 517 responses were obtained. An analysis was made of the data from the professional environment, as well as those related to the request for basic supplementary tests (blood count, biochemistry, and routine urine analysis), the use of Rapid Antigen Detection Test for group A Streptococcus, bacterial cultures, serology, diagnostic tests for pertussis and tuberculosis (Mantoux), as well as imaging tests. Conclusions: There is variability between Autonomous Communities and healthcare areas. Areas for improvement were found in the accessibility to different CT, collection time and sending of samples, delay in receiving results, as well as waiting times for non-urgent imaging tests. These affect the intervention and resolution capacity of the primary care paediatrician

Mechanical Bonds and Topological Effects in Radical Dimer Stabilization

While mechanical bonding stabilizes tetrathiafulvalene (TTF) radical dimers, the question arises: what role does topology play in catenanes containing TTF units? Here, we report how topology, together with mechanical bonding, in isomeric [3]- and doubly interlocked [2]catenanes controls the formation of TTF radical dimers within their structural frameworks, including a ring-in-ring complex (formed between an organoplatinum square and a {2+2} macrocyclic polyether containing two 1,5-dioxynaphthalene (DNP) and two TTF units) that is topologically isomeric with the doubly interlocked [2]catenane. The separate TTF units in the two {1+1} macrocycles (each containing also one DNP unit) of the isomeric [3]catenane exhibit slightly different redox properties compared with those in the {2+2} macrocycle present in the [2]catenane, while comparison with its topological isomer reveals substantially different redox behavior. Although the stabilities of the mixed-valence (TTF2)^(•+) dimers are similar in the two catenanes, the radical cationic (TTF^(•+))_2 dimer in the [2]catenane occurs only fleetingly compared with its prominent existence in the [3]catenane, while both dimers are absent altogether in the ring-in-ring complex. The electrochemical behavior of these three radically configurable isomers demonstrates that a fundamental relationship exists between topology and redox properties

An approach to duality on abelian precompact groups

We prove that every dense subgroup of a topological abelian group has the same ‘convergence dual’ as the whole group. By the ‘convergence dual’ we mean the character group endowed with the continuous convergence structure. We draw as a corollary that the continuous convergence structure on the character group of a precompact group is discrete and therefore a non-compact precompact group is never reflexive in the sense of convergence. We do not know if the same statement holds also for reflexivity in the sense of Pontryagin; at least in the category of metrizable abelian groups it does

Two geometric constants for operators acting on a separable Banach space

Let A be an operator from a separable Banach space X into another Banach space Y. For every M-basis (an) of X the authors define two numbers: hA = hA,(an) = infn||A|[an,...]||and HA = HA,(an) = supnm(A|[an,...]), where [ ] stands for closed linear span and m stands for minimum modulus, i.e. m(A)=inf||x||=1||Ax||. First they prove that reflexivity of X can be characterized by the stability of HA,(an) under changes of the M-basis. In the case X is a separable reflexive Banach space these constants are related with s-numbers. The authors show that HA is the infimum of the Gelʹfand numbers of A and hA is a lower bound of the Bernstein numbers of A defined by J. Zemánek [Studia Math. 80 (1984), no. 3, 219–234]. They prove that a separable Banach space X is reflexive if and only if the infimum of the Gelʹfand numbers of every operator A from X into a Banach space Y can be computed in terms of one sequence of closed, nested, finite codimensional subspaces with null intersection. Several relationships between these numbers and the spectral theory are discussed. Finally, in the framework of a separable Hilbert space X and a selfadjoint operator A on X, it is shown that HA and hA are respectively the maximum and the minimum of the limit points of the spectrum of A. If the operator is not selfadjoint, HA and hA are exactly the maximum and minimum of the limit points of the spectrum of (A*A)1/2

A class of angelic sequential non-Frechet-Urysohn topological groups

Feechet-Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative: for instance, the square of a compact F-U space is not in general Frechet-Urysohn [P. Simon, A compact Frechet space whose square is not Frechet, Comment. Math. Univ. Carolin. 21 (1980) 749-753. [27]]. Van Douwen proved that the product of a metrizable space by a Frechet-Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following: (1) If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact. (2) Leaning on (1) we point out a big class of hemicompact sequential non-Frechet-Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex. (3) Similar results are also obtained in the framework of locally convex spaces. Another class of sequential non-Frechet-Urysohn complete topological Abelian groups very different from ours is given in [E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445-460. [32]]