Abstract Versions of L′Hôpital′s Rule for Holomorphic Functions in the Framework of Complex B-Modules

AbstractAbstract versions of L′Hôpital′s rule are proved for the "ratio" f(z)(g(z))−1, where f : S → X, g : S → A are vector-valued holomorphic functions defined in a region of the complex plane containing S, A being a complex unilal Banach algebra, and X a complex Banach module over A. Both cases, (i) (g(z))−1[formula] 0, and (ii) f(z) [formula] 0, g(z) [formula] 0, as z[formula] α, α being either finite or infinite, are considered when f′(z)(g′(z))−1 has a finite limit. Applications are given to the asymptotics of linear second-order differential equations in Banach algebras

The PDD method for solving linear, nonlinear, and fractional PDEs problems

We review the Probabilistic Domain Decomposition (PDD) method for the numerical solution of linear and nonlinear Partial Differential Equation (PDE) problems. This Domain Decomposition (DD) method is based on a suitable probabilistic representation of the solution given in the form of an expectation which, in turns, involves the solution of a Stochastic Differential Equation (SDE). While the structure of the SDE depends only upon the corresponding PDE, the expectation also depends upon the boundary data of the problem. The method consists of three stages: (i) only few values of the sought solution are solved by Monte Carlo or Quasi-Monte Carlo at some interfaces; (ii) a continuous approximation of the solution over these interfaces is obtained via interpolation; and (iii) prescribing the previous (partial) solutions as additional Dirichlet boundary conditions, a fully decoupled set of sub-problems is finally solved in parallel. For linear parabolic problems, this is based on the celebrated Feynman-Kac formula, while for semilinear parabolic equations requires a suitable generalization based on branching diffusion processes. In case of semilinear transport equations and the Vlasov-Poisson system, a generalization of the probabilistic representation was also obtained in terms of the Method of Characteristics (characteristic curves). Finally, we present the latest progress towards the extension of the PDD method for nonlocal fractional operators. The algorithm notably improves the scalability of classical algorithms and is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Numerical examples conducted in 1D and 2D, including some for the KPP equation and Plasma Physics, are given.info:eu-repo/semantics/acceptedVersio

Gaussian processes in complex media: new vistas on anomalous diffusion

Normal or Brownian diffusion is historically identified by the linear growth in time of the variance and by a Gaussian shape of the displacement distribution. Processes departing from the at least one of the above conditions defines anomalous diffusion, thus a nonlinear growth in time of the variance and/or a non-Gaussian displacement distribution. Motivated by the idea that anomalous diffusion emerges from standard diffusion when it occurs in a complex medium, we discuss a number of anomalous diffusion models for strongly heterogeneous systems. These models are based on Gaussian processes and characterized by a population of scales, population that takes into account the medium heterogeneity. In particular, we discuss diffusion processes whose probability density function solves space- and time-fractional diffusion equations through a proper population of time-scales or a proper population of length-scales. The considered modeling approaches are: the continuous time random walk, the generalized gray Brownian motion, and the time-subordinated process. The results show that the same fractional diffusion follows from different populations when different Gaussian processes are considered. The different populations have the common feature of a large spreading in the scale values, related to power-law decay in the distribution of population itself. This suggests the key role of medium properties, embodied in the population of scales, in the determination of the proper stochastic process underlying the given heterogeneous medium.This research was supported by the Basque Government through the BERC 2014–2017 and BERC 2018–2021 programs, and by the Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditations SEV- 2013-0323 and SEV-2017-0718 and through project MTM2016- 76016-R MI

Formation of Magnetic Fields on Grand Scale Distances

The emergence of significant magnetic fields in cosmic plasmas over large scale distances is an important issue to deal with as known and potentially applicable theories, such as those based on the Weibel instability, suffers from the difficulty of involving unrealistically small distances (e.g. $c/omega_pe$). The presently proposed theory, to avoid this difficulty, starts from considering the electron density and temperature fluctuations [1] which can be excited in circumbinary disks sustained by pairs of black holes. These low frequency fluctuations can drive a ``magneto-thermal alternator’’ of the kind introduced in Ref. [2] which can produce a slowly varyingly and sheared magnetic field structure. The shearing component of this field can then be amplified by a magneto-thermal reconnection process [2] up to more significant amplitudes. This however requires an event that would produce a strong local electron pressure gradient. An important feature of magneto-thermal reconnection is that the width of the layer where reconnection takes place can grow with the involved macroscopic distances [2] unlike the case of the collisionless tearing mode whose analysis was given in Ref. [3]. *Sponsored by the Kavli Foundation and CNR. [1] B. Coppi, Fundamental Pl. Phys., 100007 (2023). [2] B. Coppi, and B. Basu, Phys. Lett. A, 397, 127265 (2021). [3] B. Coppi, L. Sugiyama, J. Mark and G. Bertin, Ann. Phys. 119, 2 (1979)

Time-periodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions

The mean field Kuramoto model describing the synchronization of a population of phase oscillators with a bimodal frequency distribution is analyzed (by the method of multiple scales) near regions in its phase diagram corresponding to synchronization to phases with a time periodic order parameter. The richest behavior is found near the tricritical point were the incoherent, stationarily synchronized, ``traveling wave'' and ``standing wave'' phases coexist. The behavior near the tricritical point can be extrapolated to the rest of the phase diagram. Direct Brownian simulation of the model confirms our findings.Comment: Revtex,16 pag.,10 fig., submitted to Physica

Synchronization in populations of globally coupled oscillators with inertial effects

A model for synchronization of globally coupled phase oscillators including ``inertial'' effects is analyzed. In such a model, both oscillator frequencies and phases evolve in time. Stationary solutions include incoherent (unsynchronized) and synchronized states of the oscillator population. Assuming a Lorentzian distribution of oscillator natural frequencies, $g(\Omega)$, both larger inertia or larger frequency spread stabilize the incoherent solution, thereby making harder to synchronize the population. In the limiting case $g(\Omega)=\delta(\Omega)$, the critical coupling becomes independent of inertia. A richer phenomenology is found for bimodal distributions. For instance, inertial effects may destabilize incoherence, giving rise to bifurcating synchronized standing wave states. Inertia tends to harden the bifurcation from incoherence to synchronized states: at zero inertia, this bifurcation is supercritical (soft), but it tends to become subcritical (hard) as inertia increases. Nonlinear stability is investigated in the limit of high natural frequencies.Comment: Revtex, 36 pages, submit to Phys. Rev.

Molecular cytogenetics (FISH, GISH) of Coccinia grandis: A ca. 3 myr-old species of Cucurbitaceae with the largest Y/autosome divergence in flowering plants

The independent evolution of heteromorphic sex chromosomes in 19 species from 4 families of flowering plants permits studying X/Y divergence after the initial recombination suppression. Here, we document autosome/Y divergence in the tropical Cucurbitaceae Coccinia grandis, which is ca. 3 myr old. Karyotyping and C-value measurements show that the C. grandis Y chromosome has twice the size of any of the other chromosomes, with a male/female C-value difference of 0.094 pg or 10% of the total genome. FISH staining revealed 5S and 45S rDNA sites on autosomes but not on the Y chromosome, making it unlikely that rDNA contributed to the elongation of the Y chromosome; recent end-to-end fusion also seems unlikely given the lack of interstitial telomeric signals. GISH with different concentrations of female blocking DNA detected a possible pseudo-autosomal region on the Y chromosome, and C-banding suggests that the entire Y chromosome in C. grandis is heterochromatic. During meiosis, there is an end-to-end connection between the X and the Y chromosome, but the X does not otherwise differ from the remaining chromosomes. These findings and a review of plants with heteromorphic sex chromosomes reveal no relationship between species age and degree of sex chromosome dimorphism. Its relatively small genome size (0.943 pg/2C in males), large Y chromosome, and phylogenetic proximity to the fully sequenced Cucumis sativus make C. grandis a promising model to study sex chromosome evolution. Copyright © 2012 S. Karger AG, Base

A specific insertion of a solo-LTR characterizes the Y-chromosome of Bryonia dioica (Cucurbitaceae)

Background: Relatively few species of flowering plants are dioecious and even fewer are known to have sex chromosomes. Current theory posits that homomorphic sex chromosomes, such as found in Bryonia dioica (Cucurbitaceae), offer insight into the early stages in the evolution of sex chromosomes from autosomes. Little is known about these early steps, but an accumulation of transposable element sequences has been observed on the Ychromosomes of some species with heteromorphic sex chromosomes. Recombination, by which transposable elements are removed, is suppressed on at least part of the emerging Y-chromosome, and this may explain the correlation between the emergence of sex chromosomes and transposable element enrichment. Findings: We sequenced 2321 bp of the Y-chromosome in Bryonia dioica that flank a male-linked marker, BdY1, reported previously. Within this region, which should be suppressed for recombination, we observed a solo-LTR nested in a Copia-like transposable element. We also found other, presumably paralogous, solo-LTRs in a consensus sequence of the underlying Copia-like transposable element. Conclusions: Given that solo-LTRs arise via recombination events, it is noteworthy that we find one in a genomic region where recombination should be suppressed. Although the solo-LTR could have arisen before recombination was suppressed, creating the male-linked marker BdY1, our previous study on B. dioica suggested that BdY1 may not lie in the recombination-suppressed region of the Y-chromosome in all populations. Presence of a solo-LTR near BdY1 therefore fits with the observed correlation between retrotransposon accumulation and the suppression of recombination early in the evolution of sex chromosomes. These findings further suggest that the homomorphic sex chromosomes of B. dioica, the first organism for which genetic XY sex-determination was inferred, are evolutionarily young and offer reference information for comparative studies of other plant sex chromosomes

Familiarization: A theory of repetition suppression predicts interference between overlapping cortical representations

Repetition suppression refers to a reduction in the cortical response to a novel stimulus that results from repeated presentation of the stimulus. We demonstrate repetition suppression in a well established computational model of cortical plasticity, according to which the relative strengths of lateral inhibitory interactions are modified by Hebbian learning. We present the model as an extension to the traditional account of repetition suppression offered by sharpening theory, which emphasises the contribution of afferent plasticity, by instead attributing the effect primarily to plasticity of intra-cortical circuitry. In support, repetition suppression is shown to emerge in simulations with plasticity enabled only in intra-cortical connections. We show in simulation how an extended ‘inhibitory sharpening theory’ can explain the disruption of repetition suppression reported in studies that include an intermediate phase of exposure to additional novel stimuli composed of features similar to those of the original stimulus. The model suggests a re-interpretation of repetition suppression as a manifestation of the process by which an initially distributed representation of a novel object becomes a more localist representation. Thus, inhibitory sharpening may constitute a more general process by which representation emerges from cortical re-organisation