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Abelian Actions on Pseudo-real Riemann Surfaces
(Springer, 2023-04-08) Bujalance García, Emilio; Cirre Torres, Francisco Javier; J. Rodríguez
A compact Riemann surface is called pseudo-real if it admits orientation-reversing automorphisms but none of them has order two. In this paper, we find necessary and sufficient conditions for the existence of an action on a pseudo-real surface of genus g 2 of an abelian group containing orientation-reversing automorphisms. Several consequences are obtained, such as the solution of the minimum genus problem for such abelian actions.
Absorbing Markov decision processes
(EDP Sciences, 2024-02-09) Dufour, François; Prieto Rumeau, Tomás
In this paper, we study discrete-time absorbing Markov Decision Processes (MDP) with measurable state space and Borel action space with a given initial distribution. For such models, solutions to the characteristic equation that are not occupation measures may exist. Several necessary and sufficient conditions are provided to guarantee that any solution to the characteristic equation is an occupation measure. Under the so-called continuity-compactness conditions, we first show that a measure is precisely an occupation measure if and only if it satisfies the characteristic equation and an additional absolute continuity condition. Secondly, it is shown that the set of occupation measures is compact in the weak-strong topology if and only if the model is uniformly absorbing. Several examples are provided to illustrate our results.
Analysis of smart thermostat thermal models for residential building
(Elsevier, 2022-06-02) Arias, J.; Khan, A.A.; Rodriguez Uría, J.; Sama Meige, Miguel Ángel
This work studies the thermal behavior of residential buildings by using the data provided by smart thermostats and weather forecast data. We consider an equivalent circuit model depending on four parameters related to the heater power, the solar energy, heat capacity, and the thermal resistance of the building. We employ a random ordinary differential equation to overcome the natural model uncertainty. We develop a differential equation constrained least-squares-based parameter identification approach for uncertainty quantification. We provide results related to the analytic properties of the parameter-to-solution map involving a derivative characterization. Furthermore, using a discretization scheme adapted to the given data, we derive discrete formulas for the deterministic identification model and compute the statistical moments of the random model. Following a machine learning approach, we propose an algorithm that consists of three phases. The first is a training phase where we identify the parameter uncertainties on a training dataset. In the second phase, we establish a normal distribution of the parameters using these uncertainties. In the final phase, we simulate the temperature on a test dataset by solving the random model. We test this algorithm on real data from two residential buildings. The detailed numerical experiments show the feasibility and the efficacy of the developed framework.
Asymptotically Linear Problems and Antimaximum Principle for the Square Root of the Laplacian
(2016-03-10) Arcoya, David; Colorado, Eduardo; Leonori, Tommaso; https://orcid.org/0000-0002-7284-2413; https://orcid.org/0000-0002-1067-5752; https://orcid.org/0000-0002-0848-4463
This work deals with bifurcation of positive solutions for some asymptotically linear problems, involving the square root of the Laplacian (-Delta)(1/2). A simplified model problem is the following: {(-Delta)(1/2)u = lambda m(x)u + g(u) in Omega, u = 0 on partial derivative Omega, with Omega subset of R-N a smooth bounded domain, N >= 2, lambda > 0, m is an element of L-infinity(Omega), m(+) not equivalent to 0 and g is a continuous function which is super-linear at 0 and sub-linear at infinity. As a consequence of our bifurcation theory approach we prove some existence and multiplicity results. Finally, we also show an anti-maximum principle in the corresponding functional setting.
Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations
(American Institute of Mathematical Sciences (AIMS), 2015-12) Leonori, Tommaso; Peral, Ireneo; Primo, Ana; Soria, Fernando; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0003-2297-9910; https://orcid.org/0000-0003-1804-3175; https://orcid.org/0000-0001-5753-807X
In this work we consider the problems { script Lu = f in Ω, u = 0 in ℝN\Ω, and { ut + script Lu = f in QT ≡ Ω x (0,T), u(x,t) = 0 in (ℝN\Ω) x (0,T), u(x, 0) = 0 in Ω, where script L is a nonlocal differential operator and Ω is a bounded domain in ℝN, with Lipschitz boundary. The main goal of this work is to study existence, uniqueness and summability of the solution u with respect to the summability of the datum f. In the process we establish an Lp-theory, for p ≥ 1, associated to these problems and we prove some useful inequalities for the applications.
The best approximation of a given function in L2-norm by Lipschitz functions with gradient constraint
(De Gruyter, 2024-04-24) Buccheri, Stefano; Leonori, Tommaso; Rossi, Julio D.; https://orcid.org/0000-0002-0667-233X; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0002-5905-4412
The starting point of this paper is the study of the asymptotic behavior, as p → ∞, of the following minimization problem: min{1 p ∫ Ω |∇v|p + 1 2 ∫ (v − f)2, v ∈ W1,p(Ω)}. Ω We show that the limit problem provides the best approximation, in the L2-norm, of the datum f among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover, such an approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the (N − 1)-Hausdorff measure of the jump set of the function.
Comparison principles for p-Laplace equations with lower order terms
(Springer Nature, 2016-08-06) Leonori, Tommaso; Porretta, Alessio; Riey, Giuseppe
We prove comparison principles for quasilinear elliptic equations whose simplest model is (Formula presented.), where Δpu=div(|Du|p-2Du) is the p-Laplace operator with p> 2 , λ≥ 0 , H(x, ξ) : Ω × RN→ R is a Carathéodory function and Ω ⊂ RN is a bounded domain, N≥ 2. We collect several comparison results for weak sub- and super-solutions under different setting of assumptions and with possibly different methods. A strong comparison result is also proved for more regular solutions.
Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth
(American Institute of Mathematical Sciences (AIMS), 2019-05) Leonori, Tommaso; Magliocca, Martina
In this paper we deal with uniqueness of unbounded solutions to the following problem (formula pergented) where QT = (0, T) × Ω is the parabolic cylinder, Ω is an open subset of RN, N ≥ 2, 1 < p < N, and the right hand side H(t, x, ξ): (0, T) × Ω × RN → R exhibits a superlinear growth with respect to the gradient term. © 2019 American Institute of Mathematical Sciences.
Constrained Markov decision processes with non-constant discount factor
(Springer Nature, 2024-05-30) Jasso Fuentes, Héctor; Prieto Rumeau, Tomás
This paper studies constrained Markov decision processes under the total expected discounted cost optimality criterion, with a state-action dependent discount factor that may take any value between zero and one. Both the state and the action space are assumed to be Borel spaces. By using the linear programming approach, consisting in stating the control problem as a linear problem on a set of occupation measures, we show the existence of an optimal stationary Markov policy. Our results are based on the study of both weak-strong topologies in the space of occupation measures and Young measures in the space of Markov policies
Continuous and discrete periodic asymptotic behavior of solutions to a competitive chemotaxis PDEs system
(Elsevier, 2021-04) Negreanu, Mihaela; Vargas Ureña, Antonio Manuel
In this paper we study the continuous and full discrete versions of a parabolic-parabolic-elliptic system with periodic terms that serves as a model for some chemotaxis phenomena. This model appears naturally in the interaction of two biological species and a chemical. The presence of the periodic terms has a strong impact on the behavior of the solutions. Some conditions on the system’s data are given that guarantee the global existence of solutions that converge to periodical solutions of an associated ODE’s system. Further, we analyze the discretized version of the model using a Generalized Finite Difference Method (GFDM) and we confirm that the properties of the continuous model are also preserved for the resulting discrete model. To this end, we prove the conditional convergence of the numerical model and study some practical examples.
Deterministic KPZ-type equations with nonlocal “gradient terms”
(Springer Nature, 2023-12-03) Abdellaou, Boumediene; Fernández, Antonio J.; Leonori, Tommaso; Younes, Abdelbadie; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0001-6236-2769
The main goal of this paper is to prove existence and non-existence results for deterministic Kardar–Parisi–Zhang type equations involving non-local “gradient terms”. More precisely, let Ω ⊂ RN, N≥ 2 , be a bounded domain with boundary ∂Ω of class C2. For s∈ (0 , 1) , we consider problems of the form (Formula presented.) where q> 1 and λ> 0 are real parameters, f belongs to a suitable Lebesgue space, μ∈ L∞(Ω) and D represents a nonlocal “gradient term”. Depending on the size of λ> 0 , we derive existence and non-existence results. In particular, we solve several open problems posed in [Abdellaoui in Nonlinearity 31(4): 1260-1298 (2018), Section 6] and [Abdellaoui in Proc Roy Soc Edinburgh Sect A 150(5): 2682-2718 (2020), Section 7]. © 2022, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.
Discrete-time control with non-constant discount factor
(Springer Nature, 2020-06-20) Jasso Fuentes, Héctor; Menaldi, José Luis; Prieto Rumeau, Tomás
This paper deals with discrete-time Markov decision processes (MDPs) with Borel state and action spaces, and total expected discounted cost optimality criterion. We assume that the discount factor is not constant: it may depend on the state and action; moreover, it can even take the extreme values zero or one. We propose sufficient conditions on the data of the model ensuring the existence of optimal control policies and allowing the characterization of the optimal value function as a solution to the dynamic programming equation. As a particular case of these MDPs with varying discount factor, we study MDPs with stopping, as well as the corresponding optimal stopping times and contact set. We show applications to switching MDPs models and, in particular, we study a pollution accumulation problem.
Discrete-time hybrid control in Borel spaces
(Springer Nature, 2018-05-18) Jasso Fuentes, Héctor; Menaldi, José-Luis; Prieto Rumeau, Tomás
A discrete-time hybrid control model with Borel state and action spaces is introduced. In this type of models, the dynamic of the system is composed by two sub-dynamics affecting the evolution of the state; one is of a standard-type that runs almost every time and another is of a special-type that is active under special circumstances. The controller is able to use two different type of actions, each of them is applied to each of the two sub-dynamics, and the activations of these sub-dynamics are possible according to an activation rule that can be handled by the controller. The aim for the controller is to find a control policy, containing a mix of actions (of either standard- or special-type), with the purpose of minimizing an infinite-horizon discounted cost criterion whose discount factor is dependent on the state-action history and may be equal one at some stages. Two different sets of conditions are proposed to guarantee (i) the finiteness of the cost criterion, (ii) the characterization of the optimal value function and (iii) the existence of optimal control policies; to do so, we employ the dynamic programming approach. A useful characterization that signalizes the accurate times between changes of sub-dynamics in terms of the so-named contact set is also provided. Finally, we introduce two examples that illustrate our results and also show that control models such as discrete-time impulse control models and discrete-time switching control models become special cases of our present hybrid model.
Doubling constants and spectral theory on graphs
(Elsevier, 2023-02-08) Durand Cartagena, Estibalitz; Soria, Javier; Tradacete, Pedro
We study the least doubling constant among all possible doubling measures defined on a (finite or infinite) graph G. We show that this constant can be estimated from below by 1+r(AG), where r(AG) is the spectral radius of the adjacency matrix of G, and study when both quantities coincide. We also illustrate how amenability of the automorphism group of a graph can be related to finding doubling minimizers. Finally, we give a complete characterization of graphs with doubling constant smaller than 3, in the spirit of Smith graphs.
Envelopes in Banach spaces
(Springer, 2024-07-12) Ferenczi, Valentin; López Abad, Jorge; https://orcid.org/0000-0001-5239-111X
We introduce the notion of isometric envelope of a subspace in a Banach space, establishing its connections with several key elements: (a) we explore its relation to the mean ergodic projection on fixed points within a semigroup of contractions, (b) we draw parallels with Korovkin sets from the 1970s, (c) we investigate its impact on the extension properties of linear isometric embeddings. We use this concept to address the recent conjecture that the Gurarij space and the spaces Lp, p∉2N+4 are the only separable approximately ultrahomogeneous Banach spaces (a certain multidimensional transitivity of the action of the linear isometry group). The similar conjecture for Fraïssé Banach spaces (a strengthening of the approximately homogeneous property) is also considered. We characterize the Hilbert space as the only separable reflexive space in which any closed subspace coincides with its envelope; and we show that the Gurarij space satisfies the same property. We compute some envelopes in the case of Lebesgue spaces, showing that the reflexive Lp-spaces are the only reflexive rearrangement invariant spaces on [0, 1] for which all 1-complemented subspaces are envelopes. We also identify the isometrically unique “full” quotient space of Lp by a Hilbertian subspace, for appropriate values of p, as well as the associated topological group embedding of the unitary group into the isometry group of Lp.
Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality
(De Gruyter, 2019-01-30) Durand Cartagena, Estibalitz; Eriksson Bique, Sylvester; Korte, Riikka; Shanmugalingam, Nageswari
We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.
Existence and uniqueness of ∞-harmonic functions under assumption of ∞-Poincaré inequality
(Springer Nature, 2018-08-22) Durand Cartagena, Estibalitz; Jaramillo, Jesús A.; https://orcid.org/0000-0002-0197-6449; https://orcid.org/0000-0002-2891-5064
Given a complete metric measure space whose measure is doubling and supports an ∞- Poincar´e inequality, and a bounded domain Ω in such a space together with a Lipschitz function f : ∂Ω → R, we show the existence and uniqueness of an ∞-harmonic extension of f to Ω. To do so, we show that there is a metric that is bi-Lipschitz equivalent to the original metric, such that with respect to this new metric the metric space satisfies an ∞- weak Fubini property and that a function which is ∞-harmonic in the original metric must also be ∞-harmonic with respect to the new metric. We also show that if the metric on the metric space satisfies an ∞-weak Fubini property, then the notion of ∞-harmonic functions coincide with the notion of AMLEs proposed by Aronsson. The notion of ∞-harmonicity is in general distinct from the notion of strongly absolutely minimizing Lipschitz extensions found in [13, 25, 26], but coincides when the metric space supports a p-Poincar´e inequality for some finite p ≥ 1.
Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano theorem
(Springer-Verlag, 2013-10-01) Arcoya, David; Leonori, Tommaso; Primo, Ana; https://orcid.org/0000-0002-7284-2413; https://orcid.org/0000-0002-0848-4463; https://orcid.org/0000-0003-1804-3175
In this paper we deal with the existence of positive solutions for the following nonlocal type of problems {-Δu = α/(σωg(u)dx) p f(u) in Ω u>0 in Ω u=0 on ∂ Ω where Ω is a bounded smooth domain in ℝ N (N≥1), f,g are continuous positive functions, σ>0 and pεℝ. We give sufficient conditions on the functions f and g in order to have existence of positive solutions.
Finite difference method for solving fractional differential equations at irregular meshes
(Elsevier, 2021-10-29) Vargas Ureña, Antonio Manuel
This paper presents a novel meshless technique for solving a class of fractional differential equations based on moving least squares and on the existence of a fractional Taylor series for Caputo derivatives. A “Generalized Finite Difference” approach is followed in order to derive a simple discretization of the space fractional derivatives. Consistency, stability and convergence of the method are proved. Several examples illustrating the accuracy of the method are given.
A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids
(Springer Nature, 2023-12-16) Vargas Ureña, Antonio Manuel
In this study, we analyze the convergence of the finite difference method on non-uniform grids and provide examples to demonstrate its effectiveness in approximating fractional differential equations involving the fractional Laplacian. By utilizing non-uniform grids, it becomes possible to achieve higher accuracy and improved resolution in specific regions of interest. Overall, our findings indicate that finite difference approximation on non-uniform grids can serve as a dependable and efficient tool for approximating fractional Laplacians across a diverse array of applications.

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