Center for Mathematical Modeling

We investigate stable solutions of elliptic equations of the type where n ≥ 2, s ∈ (0, 1), λ ≥0 and f is any smooth positive superlinear function. The operator (− Δ) s stands for the fractional Laplacian, a pseudo-differential operator of order 2s. According to the value of λ, we study the existence and regularity of weak solutions u.

Let $J \in C(\mathbb{R})$, $J\ge 0$, $\int_{\mbox{\tinyR}} J = 1$ and consider the nonlocal diffusion operator $\mathcal{M}[u] = J \star u - u$. We study the equation $\mathcal{M} u + f(x,u) = 0$, $u \ge 0$, in $\mathbb{R}$, where f is a KPP-type nonlinearity, periodic in x. We show that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, J is symmetric, then the nontrivial solution is unique.

We consider a classical equation known as the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Superscript 4"> <mml:semantics> <mml:msup> <mml:mi> ϕ </mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\phi ^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> model in one space dimension. The kink, defined by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis x right-parenthesis equals hyperbolic tangent left-parenthesis x slash StartRoot 2 EndRoot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>tanh</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H(x)=\tanh (x/{\sqrt {2}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.

We describe an emerging initiative - the 'Functional Annotation of All Salmonid Genomes' (FAASG), which will leverage the extensive trait diversity that has evolved since a whole genome duplication event in the salmonid ancestor, to develop an integrative understanding of the functional genomic basis of phenotypic variation. The outcomes of FAASG will have diverse applications, ranging from improved understanding of genome evolution, to improving the efficiency and sustainability of aquaculture production, supporting the future of fundamental and applied research in an iconic fish lineage of major societal importance.

Nous introduisons la méthode d’Analyse en Composantes Principales Géodésiques (GPCA) dans l’espace des mesures de probabilités à support sur la droite réelle, admettant un moment d’ordre deux, et muni de la métrique de Wasserstein. Nous discutons des avantages de cette approche par rapport à une ACP fonctionnelle standard de densités de probabilités dans l’espace de Hilbert des fonctions de carrés intégrable. Nous établissons la consistence de cette méthode en montrant que la GPCA empirique converge vers sa version population lorsque la taille de l’échantillon tend vers l’infini. Une propriété clé dans l’étude de la GPCA est l’isométrie entre l’espace de Wasserstein et un sous-espace convexe fermé de l’ensemble des fonctions de carrés intégrable, par rapport à une mesure de référence appropriée. De ce fait, nous considérons le problème général de l’ACP dans un sous-ensemble convexe fermé d’un espace de Hilbert séparable, qui sert de base à l’analyse de la GPCA. Nous proposons différents exemples illustratifs à partir de modèles statistiques simples pour montrer les bénéfices de cette approche pour l’analyse de données. La méthode est également appliquée à un exemple réel sur les pyramides des âges.

We prove the nondegeneracy of the extremals of the fractional Sobolev inequality as solutions of a critical semilinear nonlocal equation involving the fractional Laplacian.

Americanae nace como un proyecto conjunto que surge dentro de la Red Europea de Información y Documentación sobre América Latina (REDIAL), y que ha afrontado la Biblioteca de la Agencia Española de Cooperación Internacional para el Desarrollo (AECID). Esta nueva biblioteca virtual hace más accesibles los libros digitales de tema americanista a los investigadores y usuarios interesados de cualquier parte del mundo.

Chemical weathering of continental rocks plays a central role in regulating the carbon cycle and the Earth's climate However, the role of climate variability on chemical weathering is still strongly debated. Here we focus on the Himalayan range and use the lithium isotopic composition of clays in fluvial terraces to show a tight coupling between climate change and chemical weathering over the past 40 ka. Between 25 and 10 ka ago, weathering rates decrease despite temperature increase and monsoon intensification. This suggests that at this timescale, temperature plays a secondary role compared to runoff and physical erosion, which inhibit chemical weathering by accelerating sediment transport and act as fundamental controls in determining the feedback between chemical weathering and atmospheric carbon dioxide.

The COVID-19 pandemic has recently caused so much anxiety and speculation around the world. This phenomenon was mainly driven by the drastic increase in the number of infected people with the COVID-19 virus worldwide. Here we propose a simple model to predict the endemic in Indonesia. The model is based on the Richard’s Curve that represents a modified logistic equation. Based on the similar trends of initial data between Indonesia and South Korea, we use parameter values that are obtained through parameter estimation of the model to the data in South Korea. Further, we use a strict assumption that the implemented strategy in Indonesia is as effective as in South Korea. The results show that endemic will end in April 2020 with the total number of cases more than 8000.

Timely detection and accurate description of extreme events, such as natural disasters and other crisis situations, are crucial for emergency management and mitigation. Extreme-event detection is challenging, since one has to rely upon reports from human observers appointed to specific geographical areas, or on an expensive and sophisticated infrastructure. In the case of earthquakes, geographically dense sensor networks are expensive to deploy and maintain. Therefore, only some regions-or even countries-are able to acquire useful information about the effects of earthquakes in their own territory. An inexpensive and viable alternative to this problem is to detect extreme real-world events through people's reactions in online social networks. In particular, Twitter has gained popularity within the scientific community for providing access to real-time “citizen sensor” activity. Nevertheless, the massive amount of messages in the Twitter stream, along with the noise it contains, underpin a number of difficulties when it comes to Twitter-based event detection. We contribute to address these challenges by proposing an online method for detecting unusual bursts in discrete-time signals extracted from Twitter. This method only requires a one-off semisupervised initialization and can be scaled to track multiple signals in a robust manner. We also show empirically how our proposed approach, which was envisioned for generic event detection, can be adapted for worldwide earthquake detection, where we compare the proposed model to the state of the art for earthquake tracking using social media. Experimental results validate our approach as a competitive alternative in terms of precision and recall to leading solutions, with the advantage of implementation simplicity and worldwide scalability.

In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space ℝ d , d = 2 or 3. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence and uniqueness of strong solutions for d = 2 or 3. Moreover, we prove that the solution is global in time for d = 2 and also for d = 3 if the data are small enough.

We consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource, with the objective being to reach a given level of the resource. We allow controls to be bounded measurable functions of time plus possible impulses. For the one-species case, we show that the immediate one-impulse strategy (filling the whole reactor with one single impulse at the initial time) is optimal when the growth function is monotonic. For nonmonotonic growth functions with one maximum, we show that a particular singular arc strategy (precisely defined in section 3) is optimal. These results extend and improve former ones obtained for the class of measurable controls only. For the two-species case with monotonic growth functions, we give conditions under which the immediate one-impulse strategy is optimal. We also give optimality conditions for the singular arc strategy (at a level that depends on the initial condition) to be optimal. The possibility for the immediate one-impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations.

Flavobacteriia are increasingly recognized as key players in the marine carbon cycle, due to their ability to efficiently recycle algal polysaccharides both in the open ocean and in coastal regions. The chemical complexity of algal polysaccharides, their differences between algal groups and variations through time and space, imply that marine flavobacteria must have evolved dedicated degradation mechanisms and regulation of their metabolism during interactions with algae. In the present study, we report the first transcriptome-wide gene expression analysis for an alga-associated flavobacterium during polysaccharide degradation. Zobellia galactanivorans DsijT, originally isolated from a red alga, was grown in minimal medium with either glucose (used as a reference monosaccharide) or one selected algal polysaccharide from brown (alginate, laminarin) or red algae (agar, porphyran, iota- or kappa- carrageenan) as sole carbon source. Expression profiles were determined using whole-genome microarrays. Integration of genomic knowledge with the automatic building of a co-expression network allowed the experimental validation of operon-like transcription units. Differential expression analysis revealed large transcriptomic shifts depending on the carbon source. Unexpectedly, transcriptomes shared common signatures when growing on chemically divergent polysaccharides from the same algal phylum. Together with the induction of numerous transcription factors, this hints at complex regulation events that fine-tune the cell behavior during interactions with algal biomass in the marine environment. The results further highlight genes and loci that may participate in polysaccharide utilization, notably encoding Carbohydrate Active enZymes (CAZymes) and glycan binding proteins together with a number of proteins of unknown function. This constitutes a set of candidate genes potentially representing new substrate specificities. By providing an unprecedented view of global transcriptomic responses during polysaccharide utilization in an alga-associated model flavobacterium, this study expands the current knowledge on the functional role of flavobacteria in the marine carbon cycle and on their interactions with algae.

This paper studies the internal controllability and stabilizabilityof a family of Boussinesq systems recently proposed by J. L. Bona,M. Chen and J.-C. Saut to describe the two-way propagation of smallamplitude gravity waves on the surface of water in a canal. Thespace of the controllable data for the associated linear system isdetermined for all values of the four parameters. As an applicationof this newly established exact controllability, some simplefeedback controls are constructed such that the resultingclosed-loop systems are exponentially stable. When the parametersare all different from zero, the local exact controllability andstabilizability of the nonlinear system are also established.

The prokaryotic oxidation of reduced inorganic sulfur compounds (RISCs) is a topic of utmost importance from a biogeochemical and industrial perspective. Despite sulfur oxidizing bacterial activity is largely known, no quantitative approaches to biological RISCs oxidation have been made, gathering all the complex abiotic and enzymatic stoichiometry involved. Even though in the case of neutrophilic bacteria such as Paracoccus and Beggiatoa species the RISCs oxidation systems are well described, there is a lack of knowledge for acidophilic microorganisms. Here, we present the first experimentally validated stoichiometric model able to assess RISCs oxidation quantitatively in Acidithiobacillus thiooxidans (strain DSM 17318), the archetype of the sulfur oxidizing acidophilic chemolithoautotrophs. This model was built based on literature and genomic analysis, considering a widespread mix of formerly proposed RISCs oxidation models combined and evaluated experimentally. Thiosulfate partial oxidation by the Sox system (SoxABXYZ) was placed as central step of sulfur oxidation model, along with abiotic reactions. This model was coupled with a detailed stoichiometry of biomass production, providing accurate bacterial growth predictions. In silico deletion/inactivation highlights the role of sulfur dioxygenase as the main catalyzer and a moderate function of tetrathionate hydrolase in elemental sulfur catabolism, demonstrating that this model constitutes an advanced instrument for the optimization of At. thiooxidans biomass production with potential use in biohydrometallurgical and environmental applications.

Americanae nace como un proyecto conjunto que surge dentro de la Red Europea de Información y Documentación sobre América Latina (REDIAL), y que ha afrontado la Biblioteca de la Agencia Española de Cooperación Internacional para el Desarrollo (AECID). Esta nueva biblioteca virtual hace más accesibles los libros digitales de tema americanista a los investigadores y usuarios interesados de cualquier parte del mundo.

In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that there exists a large class of finite-dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body into any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers.

We describe the genealogy of individuals with infinite descent in a supercritical continuous-state branching process.

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a measure. We prove that in general the strongest notion is strictly contained in between sequence entropy pairs and topological complexity pairs.

Motivated by stability questions on piecewise-deterministic Markov models of bacterial chemotaxis, we study the long-time behavior of a variant of the classic telegraph process having a nonconstant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both the velocity and position. Sharpness of the obtained convergence rate is discussed.