Recurrence properties of hypercyclic operators

[EN] We generalize the notions of hypercyclic operators, U-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely A-hypercyclicity. We then state an A-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the A-hypercyclicity for weighted shifts. We also investigate which density properties can the sets N(x, U) = {n is an element of N; T-n x is an element of U} have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.This work is supported in part by MEC and FEDER, Project MTM2013-47093-P, and by GVA, Projects PROMETEOII/2013/013 and ACOMP/2015/005. The second author was a postdoctoral researcher of the Belgian FNRS.Bès, JP.; Menet, Q.; Peris Manguillot, A.; Puig-De Dios, Y. (2016). Recurrence properties of hypercyclic operators. Mathematische Annalen. 366(1):545-572. https://doi.org/10.1007/s00208-015-1336-3S5455723661Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211, 766–793 (2007)Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358, 5083–5117 (2006)Bayart, F., Grivaux, S.: Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94, 181–210 (2007)Bayart, F., Matheron, É.: Dynamics of linear operators, Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge (2009)Bayart, F., Matheron, É.: (Non-)weakly mixing operators and hypercyclicity sets. Ann. Inst. Fourier 59, 1–35 (2009)Bayart, F., Ruzsa, I.: Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory Dynam. Syst. 35, 691–709 (2015)Bergelson, V.: Ergodic Ramsey Theory- an update, Ergodic Theory of $$\mathbb{Z}^d$$ Z d -actions. Lond. Math. Soc. Lecture Note Ser. 28, 1–61 (1996)Bernal-González, L., Grosse-Erdmann, K.-G.: The Hypercyclicity Criterion for sequences of operators. Studia Math. 157, 17–32 (2003)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergodic Theory Dynam. Syst. 27, 383–404 (2007)Bonilla, A., Grosse-Erdmann, K.-G.: Erratum: Ergodic Theory Dynam. Systems 29, 1993–1994 (2009)Chan, K., Seceleanu, I.: Hypercyclicity of shifts as a zero-one law of orbital limit points. J. Oper. Theory 67, 257–277 (2012)Costakis, G., Sambarino, M.: Topologically mixing hypercyclic operators. Proc. Amer. Math. Soc. 132, 385–389 (2004)Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton (1981)Giuliano, R., Grekos, G., Mišík, L.: Open problems on densities II, Diophantine Analysis and Related Fields 2010. AIP Conf. Proc. 1264, 114–128 (2010)Grosse-Erdmann, K.-G.: Hypercyclic and chaotic weighted shifts. Studia Math. 139, 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.: Frequently dense orbits. C. R. Math. Acad. Sci. Paris 341, 123–128 (2005)Grosse-Erdmann, K.G., Peris, A.: Weakly mixing operators on topological vector spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 104, 413–426 (2010)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear chaos, Universitext. Springer, London (2011)Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Amer. Math. Soc. arXiv:1410.7173Puig, Y.: Linear dynamics and recurrence properties defined via essential idempotents of $$\beta {\mathbb{N}}$$ β N (2014) arXiv:1411.7729 (preprint)Salas, H.N.: Hypercyclic weighted shifts. Trans. Amer. Math. Soc. 347, 993–1004 (1995)Salat, T., Toma, V.: A classical Olivier’s theorem and statistical convergence. Ann. Math. Blaise Pascal 10, 305–313 (2003)Shkarin, S.: On the spectrum of frequently hypercyclic operators. Proc. Am. Math. Soc. 137, 123–134 (2009

Ndel1 Promotes Axon Regeneration via Intermediate Filaments

Failure of axons to regenerate following acute or chronic neuronal injury is attributed to both the inhibitory glial environment and deficient intrinsic ability to re-grow. However, the underlying mechanisms of the latter remain unclear. In this study, we have investigated the role of the mammalian homologue of aspergillus nidulans NudE, Ndel1, emergently viewed as an integrator of the cytoskeleton, in axon regeneration. Ndel1 was synthesized de novo and upregulated in crushed and transected sciatic nerve axons, and, upon injury, was strongly associated with neuronal form of the intermediate filament (IF) Vimentin while dissociating from the mature neuronal IF (Neurofilament) light chain NF-L. Consistent with a role for Ndel1 in the conditioning lesion-induced neurite outgrowth of Dorsal Root Ganglion (DRG) neurons, the long lasting in vivo formation of the neuronal Ndel1/Vimentin complex was associated with robust axon regeneration. Furthermore, local silencing of Ndel1 in transected axons by siRNA severely reduced the extent of regeneration in vivo. Thus, Ndel1 promotes axonal regeneration; activating this endogenous repair mechanism may enhance neuroregeneration during acute and chronic axonal degeneration

CRTC Potentiates Light-independent timeless Transcription to Sustain Circadian Rhythms in Drosophila

Light is one of the strongest environmental time cues for entraining endogenous circadian rhythms. Emerging evidence indicates that CREB-regulated transcription co-activator 1 (CRTC1) is a key player in this pathway, stimulating light-induced Period1 (Per1) transcription in mammalian clocks. Here, we demonstrate a light-independent role of Drosophila CRTC in sustaining circadian behaviors. Genomic deletion of the crtc locus causes long but poor locomotor rhythms in constant darkness. Overexpression or RNA interference-mediated depletion of CRTC in circadian pacemaker neurons similarly impairs the free-running behavioral rhythms, implying that Drosophila clocks are sensitive to the dosage of CRTC. The crtc null mutation delays the overall phase of circadian gene expression yet it remarkably dampens light-independent oscillations of TIMELESS (TIM) proteins in the clock neurons. In fact, CRTC overexpression enhances CLOCK/CYCLE (CLK/CYC)-activated transcription from tim but not per promoter in clock-less S2 cells whereas CRTC depletion suppresses it. Consistently, TIM overexpression partially but significantly rescues the behavioral rhythms in crtc mutants. Taken together, our data suggest that CRTC is a novel co-activator for the CLK/CYC-activated tim transcription to coordinate molecular rhythms with circadian behaviors over a 24-hour time-scale. We thus propose that CRTC-dependent clock mechanisms have co-evolved with selective clock genes among different species.ope

Distributionally chaotic families of operators on Fréchet spaces

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis (CPAA) following peer review. The definitive publisher-authenticated version Conejero, J. A., Kostić, M., Miana, P. J., & Murillo-Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces.Communications on Pure and Applied Analysis, 2016, vol. 15, no 5, p. 1915-1939, is available online at: http://dx.doi.org/10.3934/cpaa.2016022The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C-0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C-0-semigroups, C-regularized semigroups, and alpha-timesintegrated semigroups on Frechet spaces. We also add a study of rescaled distributionally chaotic C-0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragon, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. The fourth author is supported by a grant of the FPU Program of Ministry of education of Spain.Conejero, JA.; Kostic, M.; Miana Sanz, PJ.; Murillo Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure and Applied Analysis. 15(5):1915-1939. https://doi.org/10.3934/cpaa.2016022S1915193915

Syndromics: A Bioinformatics Approach for Neurotrauma Research

Substantial scientific progress has been made in the past 50 years in delineating many of the biological mechanisms involved in the primary and secondary injuries following trauma to the spinal cord and brain. These advances have highlighted numerous potential therapeutic approaches that may help restore function after injury. Despite these advances, bench-to-bedside translation has remained elusive. Translational testing of novel therapies requires standardized measures of function for comparison across different laboratories, paradigms, and species. Although numerous functional assessments have been developed in animal models, it remains unclear how to best integrate this information to describe the complete translational “syndrome” produced by neurotrauma. The present paper describes a multivariate statistical framework for integrating diverse neurotrauma data and reviews the few papers to date that have taken an information-intensive approach for basic neurotrauma research. We argue that these papers can be described as the seminal works of a new field that we call “syndromics”, which aim to apply informatics tools to disease models to characterize the full set of mechanistic inter-relationships from multi-scale data. In the future, centralized databases of raw neurotrauma data will enable better syndromic approaches and aid future translational research, leading to more efficient testing regimens and more clinically relevant findings

Role of the lesion scar in the response to damage and repair of the central nervous system

Traumatic damage to the central nervous system (CNS) destroys the blood-brain barrier (BBB) and provokes the invasion of hematogenous cells into the neural tissue. Invading leukocytes, macrophages and lymphocytes secrete various cytokines that induce an inflammatory reaction in the injured CNS and result in local neural degeneration, formation of a cystic cavity and activation of glial cells around the lesion site. As a consequence of these processes, two types of scarring tissue are formed in the lesion site. One is a glial scar that consists in reactive astrocytes, reactive microglia and glial precursor cells. The other is a fibrotic scar formed by fibroblasts, which have invaded the lesion site from adjacent meningeal and perivascular cells. At the interface, the reactive astrocytes and the fibroblasts interact to form an organized tissue, the glia limitans. The astrocytic reaction has a protective role by reconstituting the BBB, preventing neuronal degeneration and limiting the spread of damage. While much attention has been paid to the inhibitory effects of the astrocytic component of the scars on axon regeneration, this review will cover a number of recent studies in which manipulations of the fibroblastic component of the scar by reagents, such as blockers of collagen synthesis have been found to be beneficial for axon regeneration. To what extent these changes in the fibroblasts act via subsequent downstream actions on the astrocytes remains for future investigation