ON ISOMORPHIC CLASSIFICATIONS OF SPACES OF COMPACT OPERATORS

We prove an extension of the classical isomorphic classification of Banach spaces of continuous functions on ordinals. As a consequence, we give complete isomorphic classifications of some Banach spaces K(X,Y(n)), eta >= omega, of compact operators from X to Y(eta), the space of all continuous Y-valued functions defined in the interval of ordinals [1, eta] and equipped with the supremum norm. In particular, under the Continuum Hypothesis, we extend a recent result of C. Samuel by classifying, up to isomorphism, the spaces K(X(xi), c(0)(Gamma)(eta)), where omega <= xi < omega(1,) eta >= omega, Gamma is a countable set, X contains no complemented copy of l(1), X* has the Mazur property and the density character of X** is less than or equal to N(1).

ON SPACES OF COMPACT OPERATORS ON C(K, X) SPACES

This paper concerns the spaces of compact operators kappa(E,F), where E and F are Banach spaces C([1, xi], X) of all continuous X-valued functions defined on the interval of ordinals [1, xi] and equipped with the supremun norm. We provide sufficient conditions on X, Y, alpha, beta, xi and eta, with omega <= alpha <= beta < omega 1 for the following equivalence: (a) kappa(C([1, xi], X), C([1, alpha], Y)) is isomorphic to kappa(C([1,eta], X), C([1, beta], Y)), (b) beta < alpha(omega). In this way, we unify and extend results due to Bessaga and Pelczynski (1960) and C. Samuel (2009). Our result covers the case of the classical spaces X = l(p) and Y = l(q) with 1 < p, q < infinity.

COUNTABLE GROUPS OF ISOMETRIES ON BANACH SPACES

A group G is representable in a Banach space X if G is isomorphic to the group of isometrics on X in some equivalent norm. We prove that a countable group G is representable in a separable real Banach space X in several general cases, including when G similar or equal to {-1,1} x H, H finite and dim X >= vertical bar H vertical bar or when G contains a normal subgroup with two elements and X is of the form c(0)(Y) or l(p)(Y), 1 <= p < +infinity. This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space X and a countable bounded group G of isomorphisms on X containing -Id, there exists an equivalent norm on X for which G is equal to the group of isometrics on X. We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometrics. It follows that every real Banach space of dimension at least 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.

POLAR ACTIONS ON CERTAIN PRINCIPAL BUNDLES OVER SYMMETRIC SPACES OF COMPACT TYPE

We study polar actions with horizontal sections on the total space of certain principal bundles G/K -> G/H with base a symmetric space of compact type. We classify such actions up to orbit equivalence in many cases. In particular, we exhibit examples of hyperpolar actions with cohomogeneity greater than one on locally irreducible homogeneous spaces with nonnegative curvature which are not homeomorphic to symmetric spaces.

Intercellular transport of epidermis-expressed MADS domain transcription factors and their effect on plant morphology and floral transition

P>During the lifetime of an angiosperm plant various important processes such as floral transition, specification of floral organ identity and floral determinacy, are controlled by members of the MADS domain transcription factor family. To investigate the possible non-cell-autonomous function of MADS domain proteins, we expressed GFP-tagged clones of AGAMOUS (AG), APETALA3 (AP3), PISTILLATA (PI) and SEPALLATA3 (SEP3) under the control of the MERISTEMLAYER1 promoter in Arabidopsis thaliana plants. Morphological analyses revealed that epidermal overexpression was sufficient for homeotic changes in floral organs, but that it did not result in early flowering or terminal flower phenotypes that are associated with constitutive overexpression of these proteins. Localisations of the tagged proteins in these plants were analysed with confocal laser scanning microscopy in leaf tissue, inflorescence meristems and floral meristems. We demonstrated that only AG is able to move via secondary plasmodesmata from the epidermal cell layer to the subepidermal cell layer in the floral meristem and to a lesser extent in the inflorescence meristem. To study the homeotic effects in more detail, the capacity of trafficking AG to complement the ag mutant phenotype was compared with the capacity of the non-inwards-moving AP3 protein to complement the ap3 mutant phenotype. While epidermal expression of AG gave full complementation, AP3 appeared not to be able to drive all homeotic functions from the epidermis, perhaps reflecting the difference in mobility of these proteins.

Extremal problems on sum-free sets and coverings in tridimensional spaces

Given a prime power q, define c (q) as the minimum cardinality of a subset H of F 3 q which satisfies the following property: every vector in this space di ff ers in at most 1 coordinate from a multiple of a vector in H. In this work, we introduce two extremal problems in combinatorial number theory aiming to discuss a known connection between the corresponding coverings and sum-free sets. Also, we provide several bounds on these maps which yield new classes of coverings, improving the previous upper bound on c (q)

Hamiltonian mappings and circle packing phase spaces.

We introduce three area preserving maps with phase space structures which resemble circle packings. Each mapping is derived from a kicked Hamiltonian system with one of the three different phase space geometries (planar, hyperbolic or spherical) and exhibits an infinite number of coexisting stable periodic orbits which appear to ‘pack’ the phase space with circular resonances.

On S-asymptotically omega-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions f : [0, infinity] -> X for which exists omega > 0 such that lim(t ->infinity) (f (t + omega) - f (t)) = 0 (in the sequel called S-asymptotically omega-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically omega-periodic functions. We also study the existence of S-asymptotically omega-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces. (C) 2008 Elsevier Inc. All rights reserved.

Existence Results for Second Order Differential Equations with Nonlocal Conditions in Banach Spaces

This work is concerned with implicit second order abstract differential equations with nonlocal conditions. Assuming that the involved operators satisfy sonic compactness properties, we establish the existence of local mild solutions, the existence of global mild solutions and the existence of asymptotically almost periodic solutions.

Controllability of Volterra-Fredholm type systems in Banach spaces

We show the results in Chalishajar [Controllability of mixed Volterra-Fredholm-type integro-differential systems in Banach space, J. Franklin Inst. 344(1) (2007) 12-21] and Chang and Chalishajar [Controllability of mixed Volterra-Fredholm type integro-differential systems in Banach space, J. Franklin Inst., doi:10.1016/j. jfranklin.2008.02.002] are only valid for ordinary differential control systems. As a result the examples provided cannot be recovered as applications of the abstract results. (C) 2008 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Two novel reports of semidry stigmatic surface in Asteraceae

Research documents related to the morphology and function of style branches and stigmatic surface of Asteraceae are still rather few, and the literature reports are thus controversial. We report in the present study that the stigmatic surfaces of two non-related species of Asteraceae (Lessingianthus grandiflorus and Lucilia lycopodioides) have features of semidry stigmas. Sporodermis of both species was also analyzed so that we could understand how the stigmatic surface works during pollen deposition and rehydration. Stylar branches and pollen grains (sporodermis) were studied using scanning and transmission electron microscopy (SEM and TEM) and histochemistry techniques. The inner and marginal bands of stylar branches in these species display intermediary features between the dry and wet types of stigma: the cuticle characterizes the dry stigma and cells with secretory activity characterize the wet stigma; these showed differences from what has been described to the Asteraceae family, where stigmatic surface of species from several tribes is considered dry. Pollen grains are medium-size to large with exine ornamentation (echinate and echinolophate) and abundant secretion which latter characterizes pollenkitt. We can assume that two processes might help pollen grain hydration on stigmatic surface in Lessingianthus grandiflorus and Lucilia lycopodioides: (1) the presence of pollenkitt, as observed in the secretory content inside exine cavities and around pollen grains; and (2) the secretory activity of stigmatic surface cells, whose secretion accumulates among intercellular and subcuticular spaces and leads to cuticle disruption during the floral receptive phase. Our results suggest that ultrastructural and histochemical studies should be considered when describing stigmatic surface and that the ""semidry"" feature within Asteraceae should be investigated still more in detail, so that the taxonomic or adaptation value of this trait in the family can be verified. (C) 2010 Elsevier GmbH. All rights reserved.