Artículos (Matemática Aplicada II)

URI permanente para esta colecciónhttps://hdl.handle.net/11441/10899

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The slope problem in discrete iteration
(AMS, 2024-11-21) Contreras Márquez, Manuel Domingo; Cruz Zamorano, Francisco José; Rodríguez Piazza, Luis; Análisis Matemático; FQM104: Análisis Matemático
The slope problem in holomorphic dynamics in the unit disk goes back to Wolff in 1929. However, there have been several contributions to this problem in the last decade. In this article the problem is revisited, comparing the discrete and continuous cases. Some advances are derived in the discrete parabolic case of zero hyperbolic step, showing that the set of slopes has to be a closed interval which is independent of the initial point. The continuous setting is used to show that any such interval is a possible example. In addition, the set of slopes of a family of parabolic function is discussed, leading to examples of functions with some regularity whose set of slopes is non-trivial.
Boundedness properties for Sobolev inner products
(Elsevier, 2003-04-02) Castro Smirnova, Mirta María; Durán Guardeño, Antonio José; Matemática Aplicada II; FQM262: Teoría de la Aproximación
Abstract Sobolev orthogonal polynomials with respect to measures supported on subsets of the complex plane are considered. The connection between the following properties is studied: the multiplication operator Mp(z)=zp(z) defined on the space P of algebraic polynomials with complex coefficients is bounded with respect to the norm defined by the Sobolev inner product, the supports of the measures are compact and the zeros of the orthogonal polynomials lie in a compact subset of the complex plane. In particular, we prove that the boundedness of the multiplication operator M always implies the compactness of the supports.
Study of a homoclinic canard explosion from a degenerate center
(Elsevier, 2022-10) Qin, Bo-Wei; Chung, Kwok-Wai; Algaba, Antonio; Rodríguez Luis, Alejandro José; Matemática Aplicada II; Ministerio de Economía y Competitividad (MINECO). España; Ministerio de Ciencia, Innovación y Universidades (MICIU). España; Junta de Andalucía; National Natural Science Foundation of China
Canard explosion is an appealing event occurring in singularly perturbed systems. In this phenomenon, upon variation of a parameter within an exponentially small range, the amplitude of a small limit cycle increases abruptly. In this letter we analyze the canard explosion in a limit cycle related to a degenerate center (with zero Jacobian matrix). We provide a second-order approximation of the critical value of the parameter for which the canard explosion occurs. Numerical results are compared with the analytical predictions and excellent agreements are found. As in this problem the canard explosion ends in a homoclinic connection, a very good approximation for the homoclinic curve in the parameter plane is also obtained.
Memory effects in a vibrated thin granular layer
(EDP Sciences, 2025-12-01) Vega Reyes, Francisco; Rodríguez Rivas, Álvaro; García de Soria Lucena, María Isabel; Maynar Blanco, Pablo; Física Atómica, Molecular y Nuclear; Matemática Aplicada II; Física Aplicada I
We present in this work the first experimental evidence of the termal memory effect in agran-ular fluid. In particular, we observe here the Kovacs memory effect (an anomalous evolution of at least one macroscopic variable) in the granular temperatura of the fluidized granular monolayer. The experimental set-up consists here in a vertically shaken granular monolayer. The evolution of the granular temperatura curves clearly displays the characteristic Kovacs humps. Furthermore, it appears that, at experimental level, the shaken monolayer displays the so-called anomalous Kovacs effec; i. e., and upwards hump for cool down protocol (or a downwards hump for a heating up process). The experimental results are also supported by molecular dynamics simulation data which use a realisti ccomputational model for both the dynamics and tribology properties of the oscillatory top and bottom walls that are present in our laboratory.
Topological Loewner theory on Riemann surfaces
(Elsevier, 2021-01) Contreras Márquez, Manuel Domingo; Díaz Madrigal, Santiago; Matemática Aplicada II; European Union; FQM133: Grupo de Investigación en Análisis Funcional
We prove that topological evolution families on a Riemann surface S are rather trivial unless S is conformally equivalent to the unit disc or the punctuated unit disc. We also prove that, except for the torus where there is no non-trivial continuous Loewner chain, there is a topological evolution family associated to any topological Loewner chain and, conversely, any topological evolution family comes from a topological Loewner chain on the same Riemann surface.
Asymptotic behavior of orbits of holomorphic semigroups
(Elsevier, 2020-01) Bracci, Filippo; Contreras Márquez, Manuel Domingo; Díaz Madrigal, Santiago; Gaussier, Hervé; Zimmer, Andrew; Matemática Aplicada II; Ministerio de Economía y Competitividad (MINECO). España; FQM133: Grupo de Investigación en Análisis Funcional
Let (ϕt) be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let Ω be the starlike at infinity domain image of the Koenigs function of (ϕt). In this paper we characterize the type of convergence of the orbits of (ϕt) to the Denjoy-Wolff point in terms of the shape of Ω. In particular we prove that the convergence is non-tangential if and only if the domain Ω is “quasi-symmetric with respect to vertical axis”. We also prove that such conditions are equivalent to the curve [0,∞)∋t↦ϕt(z) being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of Ω.
Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step
(Springer, 2025) Contreras Márquez, Manuel Domingo; Díaz Madrigal, Santiago; Gumenyuk, Pavel; Matemática Aplicada II; Ministerio de Ciencia e Innovación (MICIN). España
Let ϕ : D → D be a parabolic self-map of the unit disc D having zero hyperbolic step. We study holomorphic self-maps of D commuting with ϕ. In particular, we answer a question from Gentili and Vlacci (1994) by proving that ψ ∈ Hol(D, D) commutes with ϕ if and only if the two self-maps have the same Denjoy –Wolff point and ψ is a pseudo-iterate of ϕ in the sense of Cowen. Moreover, we show that the centralizer of ϕ, i.e. the semigroup Z∀(ϕ) := {ψ ∈ Hol(D, D) : ψ ◦ ϕ = ϕ ◦ ψ} is commutative. We also prove that if ϕ is univalent, then all elements of Z∀(ϕ) are univalent as well, and if ϕ is not univalent, then the identity map is an isolated point of Z∀(ϕ). The main tool is the machinery of simultaneous linearization, which we develop using holomorphic models for iteration of non-elliptic self-maps originating in works of Cowen and Pommerenke.
Well-balanced physics-based finite volume schemes for Saint-Venant–Exner-type models of sediment transport
(Elsevier, 2025) Bürguer, Raimund; Fernández Nieto, Enrique Domingo; Garres-Díaz, José; Moya Abuhadba, Jorge Johnny; Matemática Aplicada I; Matemática Aplicada II; Junta de Andalucía; European Union (UE)
The Saint-Venant–Exner (SVE) model is widely used for the description of sediment transport including bedload, erosion, and deposition processes. A modified version of the SVE model, which includes sediment concentration, incorporates exchange of sediment between the fluid and an erodible bed and a non-hydrostatic pressure for the fluid along with non-equilibrium entrainment and deposition velocities, is introduced. Gravitational effects on erosion are described by an effective shear stress formulation. This modified SVE model is derived from a general approach with density variations. It preserves the mass of both the sediment and the fluid, and satisfies a dissipative energy balance. On the other hand, well-balanced finite volume schemes adapted for SVE models are derived since standard well-balanced schemes for the Saint-Venant system with fixed bottom are in general no more well-balanced when applied to the SVE model. The latter property is due to the uncontrolled numerical diffusion associated with the bed evolution equation. Two novel techniques to achieve the well-balanced property for the modified SVE model are proposed. The first is a new polynomial-viscosity-matrix-based (PVM) scheme, denoted “PVM-2I”, that modifies the numerical approximation of the bed evolution equation according to its related characteristic speed. The second is a physically motivated correction of the numerical diffusion term for the Rusanov and Harten–Lax–van Leer (HLL) schemes. The proposed schemes are positivity-preserving for the water height. Numerical solutions are compared with exact solutions with gravitational effects, with a novel exact solution in non-equilibrium conditions, and with experimental data. It is illustrated how the use of standard non-well-balanced schemes leads to a large artificial (unphysical) erosion and completely degraded solutions. This undesirable behaviour is avoided by the proposed well-balanced schemes. Moreover, it is demonstrated that for dam-break flows the inclusion of non-hydrostatic pressure improves the prediction of the water surface and sediment evolution, while for overtopping flow erosion tests, accounting for erosion–deposition exchanges between the bedload and suspended sediment layers leads to better agreement with experimental data.
The Pascal matrix, commuting tridiagonal operators and Fourier algebras
(Elsevier, 2026) Riley Casper, William; Zurrián, Ignacio Nahuel; Matemática Aplicada II; Ministerio de Ciencia e Innovación (MICIN). España
We consider the (symmetric) Pascal matrix, in its finite and infinite versions, and prove the existence of symmetric tridiagonal matrices commuting with it by giving explicit expressions for these commuting matrices. This is achieved by studying the associated Fourier algebra, which as a byproduct, allows us to show that all the linear relations of a certain general form for the entries of the Pascal matrix arise from only three basic relations. We also show that pairs of eigenvectors of the tridiagonal matrix define a natural eigenbasis for the binomial transform. Lastly, we show that the commuting tridiagonal matrices provide a numerically stable means of diagonalizing the Pascal matrix.
A Third-Order Finite Volume Semi-Implicit Method for the Shallow Water-Exner Model
(Springer, 2025) Fernández Nieto, Enrique Domingo; Garres-Díaz, José; Macca, Emanuele; Russo, Giovanni; Matemática Aplicada I; Matemática Aplicada II; Junta de Andalucía; European Union; Ministerio de Ciencia e Innovación (MICIN). España
In this work, third-order semi-implicit schemes on staggered meshes for the shallow water and Saint-Venant-Exner systems are presented. They are based on a third-order extension of the technique introduced in Casulli & Cheng [15]. The stability conditions for these schemes depend on the velocity and not on the celerity, allowing us to reduce computational efforts, especially in subcritical flow simulations, which is the regime we are mainly interested in. The main novelty consists in the third-order approximation of the pressure gradient term in the momentum equation through appropriate polynomial reconstructions. Concretely, CWENO conservative reconstruction is considered for the water thickness h and a centered fourth-degree polynomial is adopted interpolating the cell averages of the free surface n. For time discretization, a third-order IMEX scheme is applied. In addition, a novel time-dependent semi-analytical solution for Saint-Venant-Exner system is introduced and compared with the numerical ones. Several tests are performed, including accuracy tests showing third-order accuracy, well-balance tests, and simulations of slow bedload processes for large time.
On cycle-free accessible union stable network structures
(Springer, 2025) Algaba Durán, Encarnación; Van Den Brink, René; Dietz, Chris; Matemática Aplicada II; Ministerio de Ciencia e Innovación (MICIN). España
We investigate set theoretic properties that characterize the collection of connected coalitions in cycle-free undirected graphs among the accessible union stable network structures. It turns out that the only additional requirement is that every non-unitary feasible coalition can be written in a unique way as a union of non-unitary supports. As a consequence, a fairness axiom for solutions for cooperative games on cycle-free accessible union stable network structures can be defined that, together with the well-known component efficiency, characterizes the Shapley value on the class of cycle-free accessible union stable network structures. Since this fairness axiom combines ideas behind the traditional fairness axiom and balanced contributions, we refer to it as balanced fairness.
A three population Lotka-Volterra competition model with two populations interacting through an interface
(2026-02-01) Álvarez Caudevilla, Pablo; Brändle Cerqueira, Cristina; Molina Becerra, Mónica; Suárez Fernández, Antonio; Matemática Aplicada II; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Ciencia, Innovación y Universidades (MICIU). España; TIC130: Investigación en Sistemas Dinámicos en Ingeniería; FQM131: Ecuaciones diferenciales, Simulación Num. y Desarrollo Software
In this work we consider three species competing with each other in the same habitat. One of the species lives in the entire habitat, competing with the other two species, while the other two inhabit two disjoint regions of the habitat. These two populations just interact on a region/interface which acts as a geographical barrier. This barrier condition causes a drastic change in species behaviour compared to the classical Lotka-Volterra competitive model, showing very rich and new different situations depending on the several parameters involved in the system.
Local spectral theory for subordinated operators: The Cesàro operator and beyond
(2025-08) Gallardo Gutiérrez, Eva Antonia; González Doña, Francisco Javier; Matemática Aplicada II; Ministerio de Ciencia e Innovación (MICIN). España; Consejo Superior de Investigaciones Científicas (CSIC)
We study local spectral properties for subordinated operators arising from C0-semigroups. Specifically, if T = (Tt)t≥0 is a C0-semigroup acting boundedly on a complex Banach space and Hν is the subordinated operator associated to T , where ν is a sufficiently regular complex Borel measure supported on [0,∞), it is shown that Hν = ∫∞0𝑇𝑡 𝑑𝜈(𝑡) does not enjoy the Single Valued Extension Property (SVEP) and has dense glocal spectral subspaces in terms of the spectrum of the generator of T . Likewise, the adjoint H∗ν has trivial spectral subspaces and enjoys the Dunford property. As an application, for the classical Cesàro operator C acting on the Hardy spaces Hp (1 < p < ∞), it follows that the local spectrum of C at any non-zero Hp-function or the spectrum of the restriction of C to any of its nontrivial closed invariant subspaces coincides with the spectrum of C. Finally, we characterize the local spectral properties of subordinated operators arising from hyperbolic semigroups of composition operators acting on Hp (1 < p < ∞), which will depend only on the geometry of the associated Koenigs domain.
Characterization of the hyperbolic step of parabolic functions
(2024-10-24) Contreras Márquez, Manuel Domingo; Cruz Zamorano, Francisco José; Rodríguez Piazza, Luis; Matemática Aplicada II; Análisis Matemático; Ministerio de Ciencia e Innovación (MICIN). España; Junta de Andalucía; FQM133: Grupo de Investigación en Análisis Funcional; FQM104: Análisis Matemático
A classical problem in complex dynamics on hyperbolic domains is to characterize the hyperbolic step of parabolic functions. This topic has been studied by several authors, leading to different results and providing characterizations that depend on the behaviour of the iterates of such functions. In this work we provide new characterizations in terms of intrinsic properties of the functions.
Flow Methods for Cooperative Games with Generalized Coalition Configuration
(Springer, 2025) Algaba Durán, Encarnación; Rémila, Eric; Solal, Philippe; Matemática Aplicada II; Ministerio de Ciencia e Innovación (MICIN). España
A cooperative game with a coalition structure is formed by a TU-game and a partition of the agent set. For this class of games, the Owen value is computed as a two-step procedure where the relevant coalitions are those formed by the union of some elements of the partition and a coalition of another element of the partition. In this paper, we consider a broader class of games where the partition is replaced by a collection of (not necessarily pairwise disjoint) coalitions over the agent set and where, in each element of this collection, cooperation among the agents is restricted. Agents then organize themselves into a profile of feasible coalitions. This class of games can be applied to several situations such as the problem of allocating aircraft landing fees in the presence of airlines and codeshare flights. We begin by defining and axiomatically characterizing the class of flow methods, which are marginal values whose coefficients induce a unit flow on the graph of feasible coalition profiles. We then define Owen-type values constructed from flow methods. We show that these values are flow methods whose flow is decomposable into two flows. Finally, we introduce two axioms from which we characterize the flows that can be decomposed in this way, and hence the flow methods constructed by our Owen-type procedure. The last part of the paper studies some special cases.
A new kind of T-point in the Lorenz system with a different bifurcation set
(Elsevier, 2025-10) Algaba Durán, Antonio; Fernández Sánchez, Fernando; Merlino Morlesín, Manuel; Rodríguez Luis, Alejandro José; Matemática Aplicada II; Ministerio de Ciencia, Innovación y Universidades (MICIU). España; TIC130: Investigación en Sistemas Dinámicos en Ingeniería
In this work we find a new kind of T-point (or Bykov point) in the Lorenz system. At this codimension-two degeneracy, a heteroclinic cycle connects the origin (when it is a real saddle) and non-trivial equilibria (when they are saddle-focus). We observe that it presents a noteworthy geometric difference from the “classical” T-point, known since the 1980s in the Lorenz system. Because the dominant eigenvalue of the two-dimensional manifold at the origin changes, a variation in the direction of the corresponding heteroclinic orbit occurs near this equilibrium. Simultaneously, there is an important change in the bifurcation set, not previously found in the literature. While at the classical T-point the homoclinic and heteroclinic curves of non-trivial equilibria arise as half-lines in the same direction (as predicted by the well-known model of Glendinning and Sparrow), now these global bifurcation curves emerge in opposite directions. To justify this change we build a theoretical model with suitable Poincaré sections in a tubular environment of the heteroclinic cycle. Finally, by introducing a fourth parameter into the Lorenz system (a new quadratic term in its third equation), we show how the classical T-point can also lead to the new bifurcation set. This transition through a nongeneric situation (which occurs when the Jacobian matrix at the origin has a double eigenvalue) implies the existence of a codimension-three degenerate T-point. We find this bifurcation in the Lorenz-like system considered and illustrate how the bifurcation sets evolve by analyzing parallel parameter planes on both sides of the degeneracy.
Interface logistic problems: Large diffusion and singular perturbation results
(Elsevier, 2025-06) Álvarez Caudevilla, Pedro; Brändle, Cristina; Molina Becerra, Mónica; Suárez Fernández, Antonio; Matemática Aplicada II; Ecuaciones Diferenciales y Análisis Numérico; Ministerio de Economía y Competitividad (MINECO). España; Ministerio de Ciencia, Innovación y Universidades (MICIU). España; TIC130: Investigación en Sistemas Dinámicos en Ingeniería; FQM131: Ecuaciones diferenciales, Simulación Num. y Desarrollo Software
In this work we consider an interface logistic problem where two populations live in two different regions, separated by a membrane or interface where it happens an interchange of flux. Thus, the two populations only interact or are coupled through such a membrane where we impose the so-called Kedem–Katchalsky boundary conditions. For this particular scenario we analyse the existence and uniqueness of positive solutions depending on the parameters involved in the system, obtaining interesting results where one can see for the first time the effect of the membrane under such boundary conditions. To do so, we first ascertain the asymptotic behaviour of several linear and nonlinear problems for which we include a diffusion coefficient and analyse the behaviour of the solutions when such a diffusion parameter goes to zero or infinity. Despite their own interest, since these asymptotic results have never been studied before, they will be crucial in analysing the existence and uniqueness for the main interface logistic problems under analysis. Finally, we apply such an asymptotic analysis to characterize the existence of solutions in terms of the growth rate of the populations, when both populations possess the same growth rate and, also, when they depend on different parameters.
Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc
(Wiley, 2025-02-17) Contreras Márquez, Manuel Domingo; Díaz Madrigal, Santiago; Gumenyuk, Pavel; Matemática Aplicada II; Ministerio de Ciencia e Innovación (MICIN). España; FQM133: Grupo de Investigación en Análisis Funcional
Let 𝜑 be a univalent non-elliptic self-map of the unit disc 𝔻 and let (𝜓𝑡) be a continuous one-parameter semigroup of holomorphic functions in 𝔻 such that 𝜓1 ≠ 𝗂𝖽𝔻 commutes with 𝜑. This assumption does not imply that all elements of the semigroup (𝜓𝑡) commute with 𝜑. In this paper, we provide a number of sufficient conditions that guarantee that 𝜓𝑡 ◦𝜑 = 𝜑 ◦ 𝜓𝑡 for all 𝑡 > 0: This holds, for example, if 𝜑 and 𝜓1 have a common boundary (regular or irregular) fixed point different from their common Denjoy–Wolff point 𝜏, orwhen 𝜓1 has a boundary regular fixed point 𝜎 ≠ 𝜏 at which 𝜑 is isogonal, or when (𝜑 − 𝗂𝖽𝔻)∕(𝜓1 − 𝗂𝖽𝔻) has an unrestricted limit at 𝜏. In addition, we analyze how 𝜑 behaves in the petals of the semigroup (𝜓𝑡).
Optimal Bounds for POD Approximations of Infinite Horizon Control Problems Based on Time Derivatives
(Springer, 2025) De Frutos, Javier; García-Archilla, Bosco; Novo, Julia; Matemática Aplicada II; Agencia Estatal de Investigación. España; European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER); Junta de Castilla y León
In this paper we consider the numerical approximation of infinite horizon problems via the dynamic programming approach. The value function of the problem solves a Hamilton–Jacobi–Bellman equation that is approximated by a fully discrete method. It is known that the numerical problem is difficult to handle by the so called curse of dimensionality. To mitigate this issue we apply a reduction of the order by means of a new proper orthogonal decomposition (POD) method based on time derivatives. We carry out the error analysis of the method using recently proved optimal bounds for the fully discrete approximations. Moreover, the use of snapshots based on time derivatives allows us to bound some terms of the error that could not be bounded in a standard POD approach. Some numerical experiments show the good performance of the method in practice.
High-order well-balanced schemes for shallow models for dry avalanches
(2025) Castro Díaz, M. J.; Escalante, C.; Garres-Díaz, José; Morales de Luna, T.; Matemática Aplicada II; Ministerio de Ciencia, Innovación y Universidades (MICIU). España; Junta de Andalucía; FQM120: Modelado Matemático y Simulación de Sistemas Medioambientales
In this work we consider a depth-averaged model for granular flows with a Coulomb-type friction force described by the rheology. In this model, the so-called lake-at-rest steady states are of special interest, where velocity is zero and the slope is under a critical threshold defined by the angle of repose of the granular material. It leads to a family with an infinite number of lake-at-rest steady states. We describe a well-balanced reconstruction procedure that allows to define well-balanced finite volume methods for such problem. The technique is generalized to high-order space/time schemes. In particular, the second and third-order schemes are considered in the numerical tests section. An accuracy test is included showing that second and third-order are achieved. A well-balanced test is also considered. The proposed scheme is well-balanced for steady states with non-constant free surface, and it is exactly well-balanced for those steady states given by a simple characterization.

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