A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms u(tt) - Delta u - gamma(t)Delta u(t) + beta(epsilon)(t)u(t) = f(u), in a bounded domain Omega subset of R(n), under some restrictions on beta(epsilon)(t), gamma(t) and growth restrictions on the nonlinear term f. The function beta(epsilon)(t) depends on a parameter epsilon, beta(epsilon)(t) -> 0. We will prove, under suitable assumptions, local and global well-posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors {A(epsilon)(t) : t is an element of R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at epsilon = 0. (C) 2010 Elsevier Ltd. All rights reserved.

Lower semicontinuity of attractors for non-autonomous dynamical systems

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

Non-autonomous semilinear evolution equations with almost sectorial operators

Inspired by the theory of semigroups of growth a, we construct an evolution process of growth alpha. The abstract theory is applied to study semilinear singular non-autonomous parabolic problems. We prove that. under natural assumptions. a reasonable concept of solution can be given to Such semilinear singularly non-autonomous problems. Applications are considered to non-autonomous parabolic problems in space of Holder continuous functions and to a parabolic problem in a domain Omega subset of R(n) with a one dimensional handle.

A Trotter-Kato product formula for a class of non-autonomous evolution equations

In this article dedicated to Professor V. Lakshmikantham on the occasion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system UA+B(t, s)(0 <= s <= t <= T) generated by the sum -(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing G(B) for the algebra of all linear bounded operators on B, we can express UA+B(t, s)(0 <= s <= t <= T) as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by -A(t) and -B(t), thereby getting a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND D-t epsilon[0,D-T](A(t)+B(t)) everywhere dense in B. We then mention several possible applications of our product formula to various classes of non-autonomous parabolic initial-boundary value problems, as well as to evolution problems of Schrodinger type related to the theory of time-dependent singular perturbations of self-adjoint operators in quantum mechanics. We defer all the proofs and all the details of the applications to a separate publication. (C) 2008 Elsevier Ltd. All rights reserved.

On dichotomic maps for non autonomous discrete equations

A new procedure is given for the study of stability and asymptotic stability of the null solution of the non autonomous discrete equations by the method of dichotomic maps, which it includes Liapunov's Method asa special case. Examples are given to illustrate the application of the method.

Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

STRUCTURE AND BIFURCATION OF PULLBACK ATTRACTORS IN A NON-AUTONOMOUS CHAFEE-INFANTE EQUATION

The Chafee-Infante equation is one of the canonical infinite-dimensional dynamical systems for which a complete description of the global attractor is available. In this paper we study the structure of the pullback attractor for a non-autonomous version of this equation, u(t) = u(xx) + lambda(xx) - lambda u beta(t)u(3), and investigate the bifurcations that this attractor undergoes as A is varied. We are able to describe these in some detail, despite the fact that our model is truly non-autonomous; i.e., we do not restrict to 'small perturbations' of the autonomous case.

Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.

Solutions of second-order and fourth-order ODEs on the half-line

We start by studying the existence of positive solutions for the differential equation u '' = a(x)u - g(u), with u ''(0) = u(+infinity) = 0, where a is a positive function, and g is a power or a bounded function. In other words, we are concerned with even positive homoclinics of the differential equation. The main motivation is to check that some well-known results concerning the existence of homoclinics for the autonomous case (where a is constant) are also true for the non-autonomous equation. This also motivates us to study the analogous fourth-order boundary value problem {u((4)) - cu '' + a(x)u = vertical bar u vertical bar(p-1)u u'(0) = u'''(0) = 0, u(+infinity) = u'(+infinity) = 0 for which we also find nontrivial (and, in some instances, positive) solutions.

IOPT Petri Net State Space Generation Algorithm with Maximal-Step Execution Semantics

This paper presents an algorithm to efficiently generate the state-space of systems specified using the IOPT Petri-net modeling formalism. IOPT nets are a non-autonomous Petri-net class, based on Place-Transition nets with an extended set of features designed to allow the rapid prototyping and synthesis of system controllers through an existing hardware-software co-design framework. To obtain coherent and deterministic operation, IOPT nets use a maximal-step execution semantics where, in a single execution step, all enabled transitions will fire simultaneously. This fact increases the resulting state-space complexity and can cause an arc "explosion" effect. Real-world applications, with several million states, will reach a higher order of magnitude number of arcs, leading to the need for high performance state-space generator algorithms. The proposed algorithm applies a compilation approach to read a PNML file containing one IOPT model and automatically generate an optimized C program to calculate the corresponding state-space.

Genus for knots and links in renormalizable templates with several branch nodes

We apply kneading theory to describe the knots and links generated by the iteration of renormalizable nonautonomous dynamical systems with reducible kneading invariants, in terms of the links corresponding to each factor. As a consequence we obtain explicit formulas for the genus for this kind of knots and links.

Da 'portugalidade' à lusofonia

Tese de Doutoramento em Ciências da Comunicação (Especialidade em Teoria da Cultura)

Role of Notch signaling in the skin and cornea

Résumé : La voie de signalisation Notch est essentielle pour la différentiation de l'épiderme lors du développement embryonnaire de la peau. Il a été démontré que l'inactivation de Notch1 dans la peau de souris conduit à une hyperplasie de l'épiderme ainsi qu'à la formation subséquente de carcinomes basocellulaires ainsi que de plaques cornéennes. L'inactivation de Notch1 dans la cornée combinée à des lésions mécaniques démontre que les cellules progénitrices de la cornée se différentient en un épithélium hyperplasique et kératinisé comme la peau. Ce changement de destinée cellulaire conduit à une cécité cornéenne et implique des processus non-autonomes aux cellules épithéliales, caractérisés par la sécrétion de FGF-2 par l'épithélium Notch1-/- suivi d'une vascularisation et d'un remaniement du stroma sous-jacent. La déficience en vitamine A est connu comme cause de lésions cornéennes humaines (xérophtalmie sévère). En accord, nous avons trouvé que la signalisation Notch1 était liée au métabolisme de la vitamine A par la régulation de l'expression de CRBP1, nécessaire pour générer un pool de rétinol intracellulaire. La perte de Notch1 dans l'épiderme, l'autre récepteur de la famille présent dans la peau marine, ne conduit pas à un phénotype manifeste. Cependant, l'inactivation dans l'épiderme de Notch1 et Notch2 ensemble, ou de RBP-J, induit une dermatite atopique (DA) sévère chez les souris. De même, les patients souffrants de DA mais pas ceux souffrant de psoriasis ou de lichen plan, ont une réduction marquée de l'expression des récepteurs Notch dans la peau. La perte de Notch dans les keratinocytes conduit à une activation de la voie NF-κB, ce qui ensuite induit la production de TSLP, une cytokine profondément impliquée dans la pathogenèse de la DA. Nous démontrons génétiquement que TSLP est responsable de la DA ainsi que du développent d'un syndrome myéloprolifératif non-autonome aux cellules induit par le G-CSF. Cependant, ces souris avec une inactivation dans l'épiderme de Notch1 et Notch2 et aussi incapables de répondre au TSLP développent des tumeurs invasive sévères caractérisées par une haute activité de signalisation ß-catenin. TSLPR est identifié comme un potentiel suppresseur de tumeur non-autonome aux cellules tumorales; la transplantation de cellules hématopoïétiques TSLPR-/- dans des souris déficientes pour Notch est suffisant pour causer des tumeurs. Summary : The Notch pathway is essential for proper epidermal differentiation during embryonic skin development. It has previously been demonstrated that Notch1 inactivation in marine skin results in epidermal hyperplasia and subsequent formation of basal cell carcinoma-like (BCC-like) tumors as well as corneal plaques. Inducible ablation of Notch1 in the cornea combined with mechanical wounding show that Notch1 deficient corneal progenitor cells differentiate into a hyperplasic, keratinized, skin-like epithelium. This cell fate switch leads to corneal blindness and involves cell non-autonomous processes, characterized by secretion of FGF-2 through Notch1-/- epithelium followed by vascularisation and remodelling of the underlying stroma. Vitamin A deficiency is known to induce a similar corneal defect in humans (severe xerophthalmia). Accordingly, we found that Notch1 signaling is linked to vitamin A metabolism by regulating the expression of CRBP1, required to generate a pool of intracellular retinol. Epidermal loss of Notch2, the other Notch receptor present in marine skin, doesn't lead to any overt phenotypes. However, postnatal epidermis-specific inactivation of both Notch1 and Notch2, or of RBP-J, induces the development of a severe form of atopic dermatitis (AD) in mice. Likewise, patients suffering from AD, but not psoriasis or lichen planas, have a marked reduction of Notch receptor expression in the skin. Loss of Notch in keratinocytes leads to an activation of NF-κB signaling which in turn induces the production of Thymic stromal lymphopoietin (TSLP), a cytokine deeply implicated in the pathogenesis of AD. We genetically demonstrate that TSLP is responsible for AD as well as the development of a cell non-autonomous G-CSF induced myeloproliferative disorder (MPD) in mice. However, these mice with conditional epidermal inactivation of Notch1 and Notch2 as well as incapable to respond to TSLP develop severe invasive tumors characterized by high ß-catenin signaling activity. TSLPR is identified as a potential cell non-autonomous tumor suppressor; transplantation of TSLPR-/- hematopoietic cells into epidermal Notch deficient mice is sufficient to cause tumors.

A fish-specific transposable element shapes the repertoire of p53 target genes in zebrafish.

Transposable elements, as major components of most eukaryotic organisms' genomes, define their structural organization and plasticity. They supply host genomes with functional elements, for example, binding sites of the pleiotropic master transcription factor p53 were identified in LINE1, Alu and LTR repeats in the human genome. Similarly, in this report we reveal the role of zebrafish (Danio rerio) EnSpmN6_DR non-autonomous DNA transposon in shaping the repertoire of the p53 target genes. The multiple copies of EnSpmN6_DR and their embedded p53 responsive elements drive in several instances p53-dependent transcriptional modulation of the adjacent gene, whose human orthologs were frequently previously annotated as p53 targets. These transposons define predominantly a set of target genes whose human orthologs contribute to neuronal morphogenesis, axonogenesis, synaptic transmission and the regulation of programmed cell death. Consistent with these biological functions the orthologs of the EnSpmN6_DR-colonized loci are enriched for genes expressed in the amygdala, the hippocampus and the brain cortex. Our data pinpoint a remarkable example of convergent evolution: the exaptation of lineage-specific transposons to shape p53-regulated neuronal morphogenesis-related pathways in both a hominid and a teleost fish.

Atopic dermatitis-like disease and associated lethal myeloproliferative disorder arise from loss of Notch signaling in the murine skin.

BACKGROUND: The Notch pathway is essential for proper epidermal differentiation during embryonic skin development. Moreover, skin specific loss of Notch signaling in the embryo results in skin barrier defects accompanied by a B-lymphoproliferative disease. However, much less is known about the consequences of loss of Notch signaling after birth. METHODOLOGY AND PRINCIPAL FINDINGS: To study the function of Notch signaling in the skin of adult mice, we made use of a series of conditional gene targeted mice that allow inactivation of several components of the Notch signaling pathway specifically in the skin. We demonstrate that skin-specific inactivation of Notch1 and Notch2 simultaneously, or RBP-J, induces the development of a severe form of atopic dermatitis (AD), characterized by acanthosis, spongiosis and hyperkeratosis, as well as a massive dermal infiltration of eosinophils and mast cells. Likewise, patients suffering from AD, but not psoriasis or lichen planus, have a marked reduction of Notch receptor expression in the skin. Loss of Notch in keratinocytes induces the production of thymic stromal lymphopoietin (TSLP), a cytokine deeply implicated in the pathogenesis of AD. The AD-like associated inflammation is accompanied by a myeloproliferative disorder (MPD) characterized by an increase in immature myeloid populations in the bone marrow and spleen. Transplantation studies revealed that the MPD is cell non-autonomous and caused by dramatic microenvironmental alterations. Genetic studies demontrated that G-CSF mediates the MPD as well as changes in the bone marrow microenvironment leading to osteopenia. SIGNIFICANCE: Our data demonstrate a critical role for Notch in repressing TSLP production in keratinocytes, thereby maintaining integrity of the skin and the hematopoietic system.