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Dormancy stabilizes non-transitive competitive dynamics
Authors:
José Chacón,
Adrián González-Casanova,
Imanol Nuñez,
Rafael Peña-Miller,
José Luis Pérez,
Johnny Yang
Abstract:
Competitive interactions can maintain diversity, yet coexistence is often fragile in well-mixed populations, where stochastic fluctuations can lead to extinction. This is the case in non-transitive systems, such as rock-paper-scissors dynamics, where no single type dominates globally. While spatial structure can stabilize these systems by providing refuges in space, it remains unclear whether anal…
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Competitive interactions can maintain diversity, yet coexistence is often fragile in well-mixed populations, where stochastic fluctuations can lead to extinction. This is the case in non-transitive systems, such as rock-paper-scissors dynamics, where no single type dominates globally. While spatial structure can stabilize these systems by providing refuges in space, it remains unclear whether analogous mechanisms can operate in time in well-mixed environments. Here, we develop a population-genetic framework showing that dormancy can act as a temporal refuge, preserving lineages and preventing collapse to fixation under interaction-driven fluctuations. We introduce a discrete-time Wright-Fisher model that combines generalized seed-banks with frequency-dependent interactions, allowing individuals to inherit their type from potential parents sampled across multiple past generations. This construction provides a tractable framework in which dormancy stores and later reintroduces lost types. In the case of either weak or moderate selection, we prove a multidimensional diffusion limit for the resulting type-frequency process and use it to analyze complex selective interactions. In non-transitive systems, dormancy stabilizes trajectories that would otherwise collapse through stochastic extinction, extends fixation times, and sustains coexistence. These effects cannot be explained solely by an increase in effective population size. Our results show that dormancy introduces temporal memory that qualitatively alters competitive dynamics, stabilizing otherwise fragile systems and enabling long-term coexistence.
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Submitted 17 August, 2026;
originally announced August 2026.
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Fluctuation theory for spectrally negative Lévy processes killed by additive functionals
Authors:
Kei Noba,
José-Luis Pérez
Abstract:
In this paper, we study fluctuation identities for spectrally negative Lévy processes killed by a general class of additive functionals. We consider positive co-natural additive functionals (PcNAFs), which include as special cases both absolutely continuous functionals and finite mixtures of local times. Our main result shows that the associated fluctuation identities, such as two-sided exit probl…
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In this paper, we study fluctuation identities for spectrally negative Lévy processes killed by a general class of additive functionals. We consider positive co-natural additive functionals (PcNAFs), which include as special cases both absolutely continuous functionals and finite mixtures of local times. Our main result shows that the associated fluctuation identities, such as two-sided exit problems and resolvent measures, retain the same structure as in the classical case and can be expressed in terms of generalized scale functions. These scale functions are characterized as the unique solutions to Volterra-type integral equations driven by Radon measures, thereby extending the results of Li and Palmowski and Li and Zhou. Our approach is based on representing the additive functional as a mixture of local times with respect to its Revuz measure, combined with classical fluctuation identities and an approximation scheme for general Radon measures using Poisson random measures.
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Submitted 21 April, 2026;
originally announced April 2026.
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Quantifier elimination for lovely pairs of strongly geometric fields
Authors:
Pablo Cubides Kovacsics,
Felipe Estrada,
Juan Pérez,
David Rincón
Abstract:
Let $T$ be a complete strongly geometric theory of fields with quantifier elimination. We show that the theory of lovely pairs of $T$ has quantifier elimination in Delon's definitional expansion by predicates for linear independence and function symbols for the corresponding coordinate functions. Apart from recovering Delon's original results for pairs of algebraically closed fields and dense pair…
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Let $T$ be a complete strongly geometric theory of fields with quantifier elimination. We show that the theory of lovely pairs of $T$ has quantifier elimination in Delon's definitional expansion by predicates for linear independence and function symbols for the corresponding coordinate functions. Apart from recovering Delon's original results for pairs of algebraically closed fields and dense pairs of algebraically closed valued fields, we obtain as particular cases, quantifier elimination for theories of dense pairs of real closed and $p$-adically closed fields.
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Submitted 6 March, 2026;
originally announced March 2026.
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Continuous-time multi-armed bandits under random intervention times
Authors:
Kei Noba,
José Luis Pérez,
Kazutoshi Yamazaki,
Qingyuan Zhang
Abstract:
This paper examines multi-armed bandits in which actions are taken at random discrete times. The model consists of $J$ independent arms. When an arm is operated, it must remain active for a random duration, modeled by the inter-arrival time of a (possibly arm-dependent) renewal process. For arms evolving as a Lévy process, we provide an explicit characterization of the Gittins index, which is know…
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This paper examines multi-armed bandits in which actions are taken at random discrete times. The model consists of $J$ independent arms. When an arm is operated, it must remain active for a random duration, modeled by the inter-arrival time of a (possibly arm-dependent) renewal process. For arms evolving as a Lévy process, we provide an explicit characterization of the Gittins index, which is known to yield an optimal strategy. Furthermore, when the inter-arrival times are exponential and the arms evolve as either a spectrally negative Lévy process, a reflected spectrally negative Lévy process, or a diffusion process, the Gittins index is explicitly characterized in terms of the scale function or diffusion characteristics, respectively. Numerical experiments are performed to support the theoretical results.
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Submitted 3 March, 2026;
originally announced March 2026.
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Optimal Liquidation in a Defaultable Market
Authors:
Daniel Hernández-Hernńdez,
Harold A. Moreno-Franco,
José-Luis Pérez
Abstract:
In this paper we address the problem of optimal liquidation of a large portfolio composed by securities exposed to default risk. The default time is described in terms of a Brownian motion representing the evolution of the value of the firm, whose assets are available in the market for investors. Considering that selling a large number of assets has a significant impact in the price, and hence in…
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In this paper we address the problem of optimal liquidation of a large portfolio composed by securities exposed to default risk. The default time is described in terms of a Brownian motion representing the evolution of the value of the firm, whose assets are available in the market for investors. Considering that selling a large number of assets has a significant impact in the price, and hence in the portfolio's value, the control problem involved to describe the optimal strategy to liquidate a large position is analyzed. Under suitable assumptions in the model, an explicit solution is given to the value function and a precise description of the optimal strategy is obtained.
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Submitted 2 February, 2026;
originally announced February 2026.
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Genus two embedded minimal surfaces in $\mathbb{S}^3$ with dihedral symmetry
Authors:
José M. Espinar,
Joaquín Pérez
Abstract:
We prove that the Lawson surface $ξ_{2,1}$ is the unique closed embedded minimal surface of genus $2$ in $\mathbb{S}^3$ whose isometry group contains the dihedral group $D_4$ generated by two reflections across orthogonal totally geodesic two-spheres and a half-turn about a great circle. This weakens the full-symmetry hypotheses in previous characterizations of Lawson surfaces and leads to a subst…
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We prove that the Lawson surface $ξ_{2,1}$ is the unique closed embedded minimal surface of genus $2$ in $\mathbb{S}^3$ whose isometry group contains the dihedral group $D_4$ generated by two reflections across orthogonal totally geodesic two-spheres and a half-turn about a great circle. This weakens the full-symmetry hypotheses in previous characterizations of Lawson surfaces and leads to a substantially different geometric problem.
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Submitted 20 August, 2026; v1 submitted 20 November, 2025;
originally announced November 2025.
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Partially exchangeable Markov chains and characterisation of multitype Lambda-coalescents
Authors:
Adrián González Casanova,
Noemi Kurt,
Imanol Nuñez Morales,
José Luis Pérez
Abstract:
In this paper, we study consistent and partially exchangeable sequences of Markov chains on a finite state space. We provide a characterisation of the admissible transition rates via a decomposition into individual and coordinated motion of particles. As a consequence, we find a characterisation of multitype Lambda-coalescents with multiple switches. Moreover, we provide convergence and duality re…
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In this paper, we study consistent and partially exchangeable sequences of Markov chains on a finite state space. We provide a characterisation of the admissible transition rates via a decomposition into individual and coordinated motion of particles. As a consequence, we find a characterisation of multitype Lambda-coalescents with multiple switches. Moreover, we provide convergence and duality results for the corresponding process of limiting relative frequencies that we call the de Finetti measure process, and discuss a number of examples from the recent literature.
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Submitted 7 November, 2025;
originally announced November 2025.
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An Inventory System with Two Supply Modes and Lévy Demand
Authors:
José Luis Pérez,
Kazutoshi Yamazaki,
Qingyuan Zhang
Abstract:
This study considers a continuous-review inventory model for a single item with two replenishment modes. Replenishments may occur continuously at any time with a higher unit cost, or at discrete times governed by Poisson arrivals with a lower cost. From a practical standpoint, the model represents an inventory system with random deal offerings. Demand is modeled by a spectrally positive Lévy proce…
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This study considers a continuous-review inventory model for a single item with two replenishment modes. Replenishments may occur continuously at any time with a higher unit cost, or at discrete times governed by Poisson arrivals with a lower cost. From a practical standpoint, the model represents an inventory system with random deal offerings. Demand is modeled by a spectrally positive Lévy process (i.e., a Lévy process with only positive jumps), which greatly generalizes existing studies. Replenishment quantities are continuous and backorders are allowed, while lead times, perishability, and lost sales are excluded. Using fluctuation theory for spectrally one-sided Lévy processes, the optimality of a hybrid barrier policy incorporating both kinds of replenishments is established, and a semi-explicit expression for the associated value function is computed. Numerical analysis is provided to support the optimality result.
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Submitted 29 October, 2025;
originally announced October 2025.
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Minimal Surfaces of Finite Genus: Classification, Dynamics and Laminations
Authors:
Joaquín Pérez
Abstract:
This article explains a program to study complete and properly embedded minimal surfaces in $\mathbb{R}^3$ developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with the same title and supplies details and references to the original papers. After recalling the role of the classical Riemann minimal examples in minimal surface…
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This article explains a program to study complete and properly embedded minimal surfaces in $\mathbb{R}^3$ developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with the same title and supplies details and references to the original papers. After recalling the role of the classical Riemann minimal examples in minimal surface theory, we explain our four-step classification of properly embedded minimal surfaces of genus zero and infinite topology in $\mathbb{R}^3$: the periodic case, the quasi-periodicity of the two-limit-ended case, the non-existence of one-limit-ended examples, and the final classification. We then review the lamination techniques (limit-leaf stability, local removable singularity, and singular structure theorems), the dynamics theorem, bounds on topology and index for complete embedded minimal surfaces of finite total curvature, and the resolution of the embedded Calabi-Yau problem for finite genus and countably many ends. Throughout we emphasize the interaction between topology, flux, curvature estimates, and the structure of related moduli spaces. We end this article with a list of some open problems.
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Submitted 14 October, 2025;
originally announced October 2025.
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Jordan type of full Perazzo algebras
Authors:
Pedro Macias Marques,
Rosa M. Miró-Roig,
Josep Pérez
Abstract:
In this paper, we compute all possible Jordan types of linear forms $\ell$ in any full Perazzo algebra $A$. In some cases we are also able to compute the corresponding Jordan degree type, which is a finer invariant.
In this paper, we compute all possible Jordan types of linear forms $\ell$ in any full Perazzo algebra $A$. In some cases we are also able to compute the corresponding Jordan degree type, which is a finer invariant.
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Submitted 31 October, 2025; v1 submitted 30 June, 2025;
originally announced June 2025.
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On optimal periodic dividend and capital injection strategies for general Lévy models
Authors:
Dante Mata,
Kei Noba,
José-Luis Pérez
Abstract:
We consider a version of de Finetti's dividend problem, with the bail-out contraint to keep the surplus non-negative, and where dividend payments can only be made at the arrival times of an independent Poisson process. For a general Lévy process with positive and negative jumps, we show the optimality of a periodic-classical reflection strategy that pays the excess above a given level at each Pois…
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We consider a version of de Finetti's dividend problem, with the bail-out contraint to keep the surplus non-negative, and where dividend payments can only be made at the arrival times of an independent Poisson process. For a general Lévy process with positive and negative jumps, we show the optimality of a periodic-classical reflection strategy that pays the excess above a given level at each Poisson arrival time, and also reflects below at 0 in the classical sense.
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Submitted 10 May, 2025;
originally announced May 2025.
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Betti numbers of full Perazzo algebras
Authors:
Rosa Maria Miró-Roig,
Josep Pérez
Abstract:
In this paper we prove that any full Perazzo algebra $A_F$, whose Macaulay dual generator is a Perazzo form $F\in K[X_0,\dots,X_n,U_1,\dots,U_m]_d$ with $n+1 = \binom{d+m-2}{m-1}$, is the doubling of a 0-dimensional scheme in $\PP^{n+m}$ and we compute the graded Betti numbers of a minimal free resolution of $A_F$.
In this paper we prove that any full Perazzo algebra $A_F$, whose Macaulay dual generator is a Perazzo form $F\in K[X_0,\dots,X_n,U_1,\dots,U_m]_d$ with $n+1 = \binom{d+m-2}{m-1}$, is the doubling of a 0-dimensional scheme in $\PP^{n+m}$ and we compute the graded Betti numbers of a minimal free resolution of $A_F$.
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Submitted 21 January, 2025;
originally announced January 2025.
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Multitype $Λ$-coalescents and continuous state branching processes
Authors:
Adrián González Casanova,
Noemi Kurt,
Imanol Nuñez Morales,
José Luis Pérez
Abstract:
We provide new connections between multitype $Λ$-coalescents and multitype continuous state branching processes via duality and a homeomorphism on their parameter space. The approach is based on a sequential sampling procedure for the frequency process of independent CSBPs, and provides forward and backward processes for multitype population models under $Λ$-type reproduction. It provides some ins…
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We provide new connections between multitype $Λ$-coalescents and multitype continuous state branching processes via duality and a homeomorphism on their parameter space. The approach is based on a sequential sampling procedure for the frequency process of independent CSBPs, and provides forward and backward processes for multitype population models under $Λ$-type reproduction. It provides some insight on different approaches to generalise $Λ$-coalescents to the multitype setup.
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Submitted 31 December, 2024;
originally announced January 2025.
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On the gluing of formal schemes
Authors:
R. A. Calixto,
T. H. Freitas,
V. H. Jorge Pérez
Abstract:
The main focus of this paper is to show that the gluing of formal schemes is also a formal scheme. The algebraic approach established here also leads us to conclude when the gluing of $k$-formal schemes is a $k$-formal scheme. In addition, we derive that the gluing of formal schemes is always singular, regardless of whether we know the structure of the schemes involved.
The main focus of this paper is to show that the gluing of formal schemes is also a formal scheme. The algebraic approach established here also leads us to conclude when the gluing of $k$-formal schemes is a $k$-formal scheme. In addition, we derive that the gluing of formal schemes is always singular, regardless of whether we know the structure of the schemes involved.
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Submitted 10 December, 2024;
originally announced December 2024.
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Model Theory of Hilbert Spaces with a Discrete Group Action
Authors:
Alexander Berenstein,
Juan Manuel Pérez
Abstract:
In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is existentially closed, has quantifier elimination, is $\aleph_0$-categorical, $\aleph_0$-stable and SFB. On the other hand, when the group involved is countably infinite…
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In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is existentially closed, has quantifier elimination, is $\aleph_0$-categorical, $\aleph_0$-stable and SFB. On the other hand, when the group involved is countably infinite, the theory of the Hilbert space expanded by the representation of this group is $\aleph_0$-categorical up to perturbations. Additionally, when the expansion is model complete, we prove that it is $\aleph_0$-stable up to perturbations.
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Submitted 19 August, 2025; v1 submitted 5 September, 2024;
originally announced September 2024.
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The frequency process in a non-neutral two-type continuous-state branching process with competition and its genealogy
Authors:
Imanol Nuñez,
José Luis Pérez
Abstract:
We consider a population growth model given by a two-type continuous-state branching process with immigration and competition, introduced by Ma. We study the relative frequency of one of the types in the population when the total mass is forced to be constant at a dense set of times. The resulting process is described as the solution to an SDE, which we call the culled frequency process, generaliz…
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We consider a population growth model given by a two-type continuous-state branching process with immigration and competition, introduced by Ma. We study the relative frequency of one of the types in the population when the total mass is forced to be constant at a dense set of times. The resulting process is described as the solution to an SDE, which we call the culled frequency process, generalizing the $Λ$-asymmetric frequency process introduced by Caballero et al. We obtain conditions for the culled frequency process to have a moment dual and show that it is given by a branching-coalescing continuous-time Markov chain that describes the genealogy of the two-type CBI with competition. Finally, we obtain a large population limit of the culled frequency process, resulting in a deterministic ordinary differential equation (ODE). Two particular cases of the limiting ODE are studied to determine if general two-type branching mechanisms and general Malthusians can lead to the coexistence of the two types in the population.
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Submitted 6 June, 2024;
originally announced June 2024.
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On the Bailout Dividend Problem with Periodic Dividend Payments and Fixed Transaction Costs
Authors:
Harold A. Moreno-Franco,
Jose-Luis Pérez
Abstract:
We study the optimal bailout dividend problem with transaction costs for an insurance company, where shareholder payouts align with the arrival times of an independent Poisson process. In this scenario, the underlying risk model follows a spectrally negative Lévy process. Our analysis confirms the optimality of a periodic $(b_{1},b_{2})$-barrier policy with classical reflection at zero. This strat…
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We study the optimal bailout dividend problem with transaction costs for an insurance company, where shareholder payouts align with the arrival times of an independent Poisson process. In this scenario, the underlying risk model follows a spectrally negative Lévy process. Our analysis confirms the optimality of a periodic $(b_{1},b_{2})$-barrier policy with classical reflection at zero. This strategy involves reducing the surplus to $b_1$ when it exceeds $b_{2}$ at the Poisson arrival times and pushes the surplus to 0 whenever it goes below zero.
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Submitted 24 March, 2024;
originally announced March 2024.
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Orbifolds and the modular curve
Authors:
Juan Martín Pérez,
Florent Schaffhauser
Abstract:
We provide an account of the construction of the moduli stack of elliptic curves as an analytic orbifold. While intimately linked to Thurston's point of view on the subject (discrete groups acting properly and effectively on differentiable manifolds), the construction of the modular orbi-curve and its universal family of elliptic curves ends up requiring a bit more technology, in order to allow fo…
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We provide an account of the construction of the moduli stack of elliptic curves as an analytic orbifold. While intimately linked to Thurston's point of view on the subject (discrete groups acting properly and effectively on differentiable manifolds), the construction of the modular orbi-curve and its universal family of elliptic curves ends up requiring a bit more technology, in order to allow for non-effective actions. The paper is entirely expository and makes no claims to originality: its main goal is to be self-contained enough in order to be useful to young researchers who are entering the field and are interested in the interactions between differential and algebraic geometry.
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Submitted 28 December, 2023;
originally announced December 2023.
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Detection of an Arbitrary Number of Communities in a Block Spin Ising Model
Authors:
Miguel Ballesteros,
Ramsés H. Mena,
José Luis Pérez,
Gabor Toth
Abstract:
We study the problem of community detection in a general version of the block spin Ising model featuring M groups, a model inspired by the Curie-Weiss model of ferromagnetism in statistical mechanics. We solve the general problem of identifying any number of groups with any possible coupling constants. Up to now, the problem was only solved for the specific situation with two groups of identical s…
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We study the problem of community detection in a general version of the block spin Ising model featuring M groups, a model inspired by the Curie-Weiss model of ferromagnetism in statistical mechanics. We solve the general problem of identifying any number of groups with any possible coupling constants. Up to now, the problem was only solved for the specific situation with two groups of identical size and identical interactions. Our results can be applied to the most realistic situations, in which there are many groups of different sizes and different interactions. In addition, we give an explicit algorithm that permits the reconstruction of the structure of the model from a sample of observations based on the comparison of empirical correlations of the spin variables, thus unveiling easy applications of the model to real-world voting data and communities in biology.
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Submitted 29 November, 2023;
originally announced November 2023.
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Refraction strategies in stochastic control: optimality for a general Lévy process model
Authors:
Kei Noba,
José Luis Pérez,
Kazutoshi Yamazaki
Abstract:
We revisit an absolutely-continuous version of the stochastic control problem driven by a Lévy process. A strategy must be absolutely continuous with respect to the Lebesgue measure and the running cost function is assumed to be convex. We show the optimality of a refraction strategy, which adjusts the drift of the state process at a constant rate whenever it surpasses a certain threshold. The opt…
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We revisit an absolutely-continuous version of the stochastic control problem driven by a Lévy process. A strategy must be absolutely continuous with respect to the Lebesgue measure and the running cost function is assumed to be convex. We show the optimality of a refraction strategy, which adjusts the drift of the state process at a constant rate whenever it surpasses a certain threshold. The optimality holds for a general Lévy process, generalizing the spectrally negative case presented in Hernández-Hernández et al.(2016).
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Submitted 16 August, 2023;
originally announced August 2023.
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An optimal multibarrier strategy for a singular stochastic control problem with a state-dependent reward
Authors:
Mauricio Junca,
Harold Moreno-Franco,
Jose Luis Perez
Abstract:
We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish sufficient conditions for determining the optimality of the one-barrier strategy when the uncontrolled process $X$ follows a spectrally negative Lévy process with…
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We consider a singular control problem that aims to maximize the expected cumulative rewards, where the instantaneous returns depend on the state of a controlled process. The contributions of this paper are twofold. Firstly, to establish sufficient conditions for determining the optimality of the one-barrier strategy when the uncontrolled process $X$ follows a spectrally negative Lévy process with a Lévy measure defined by a completely monotone density. Secondly, to verify the optimality of the $(2n+1)$-barrier strategy when $X$ is a Brownian motion with a drift. Additionally, we provide an algorithm to compute the barrier values in the latter case.
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Submitted 16 July, 2024; v1 submitted 3 August, 2023;
originally announced August 2023.
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On Coefficient Module of Arbitrary Modules
Authors:
M. D. Ferrari,
V. H. Jorge Perez,
P. H. Lima
Abstract:
Let $(R, \mathfrak{m})$ be a $d$-dimensional Noetherian local ring that is formally equidimensional, and let $M$ be an arbitrary $R$-submodule of the free module $F = R^p$ with an analytic spread $s:=s(M)$. In this work, inspired by Herzog-Puthenpurakal-Verma in \cite{herzog}, we show the existence of an unique largest $R$-module $M_{k}$ with $\ell_R(M_{k}/M)<\infty$ and…
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Let $(R, \mathfrak{m})$ be a $d$-dimensional Noetherian local ring that is formally equidimensional, and let $M$ be an arbitrary $R$-submodule of the free module $F = R^p$ with an analytic spread $s:=s(M)$. In this work, inspired by Herzog-Puthenpurakal-Verma in \cite{herzog}, we show the existence of an unique largest $R$-module $M_{k}$ with $\ell_R(M_{k}/M)<\infty$ and $M\subseteq M_{s}\subseteq\cdots\subseteq M_{1}\subseteq M_{0}\subseteq q(M),$ such that $°(P_{M_{k}/M}(n))<s-k,$ where $q(M)$ is the relative integral closure of $M,$ defined by $q(M):=\overline{M}\cap M^{sat},$ where $M^{sat}=\cup_{n\geq 1}(M:_F\mathfrak{m}^n)$ is the saturation of $M$. We also provide a structure theorem for these modules. Furthermore, we establish the existence of coefficient modules between $I(M)M$ and $M$, where $I(M)$ denotes the $0$-th Fitting ideal of $F/M$, and discuss their structural properties. Finally, we present some applications and discuss some properties.
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Submitted 12 July, 2023;
originally announced July 2023.
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Optimal dividends and capital injection: A general Lévy model with extensions to regime-switching models
Authors:
Dante Mata López,
Kei Noba,
José-Luis Pérez,
Kazutoshi Yamazaki
Abstract:
This paper studies a general Lévy process model of the bail-out optimal dividend problem with an exponential time horizon, and further extends it to the regime-switching model. We first show the optimality of a double barrier strategy in the single-regime setting with a concave terminal payoff function. This is then applied to show the optimality of a Markov-modulated double barrier strategy in th…
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This paper studies a general Lévy process model of the bail-out optimal dividend problem with an exponential time horizon, and further extends it to the regime-switching model. We first show the optimality of a double barrier strategy in the single-regime setting with a concave terminal payoff function. This is then applied to show the optimality of a Markov-modulated double barrier strategy in the regime-switching model via contraction mapping arguments. We solve these for a general Lévy model with both positive and negative jumps, greatly generalizing the existing results on spectrally one-sided models.
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Submitted 24 October, 2024; v1 submitted 21 June, 2023;
originally announced June 2023.
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The ancestral selection graph for a $Λ$-asymmetric Moran model
Authors:
Adrián González Casanova,
Noemi Kurt,
José Luis Pérez
Abstract:
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is en…
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Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure $Λ$ for each of the two types. Our approach consists of constructing a $Λ$-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the "partial order of adaptation", we can couple these measures. This allows us to construct the central object of this paper, the $Λ-$asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time $Λ$-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
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Submitted 5 January, 2024; v1 submitted 31 May, 2023;
originally announced June 2023.
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Debunking Cantor: New Set-Theoretical and Logical Considerations
Authors:
Juan A Perez
Abstract:
For more than a century, Cantor's theory of transfinite numbers has played a pivotal role in set theory, with ramifications that extend to many areas of mathematics. This article extends earlier findings with a fresh look at the critical facts of Cantor's theory: i) Cantor's widely renowned Diagonalization Argument (CDA) is fully refuted by a set of counter-examples that expose the fallacy of this…
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For more than a century, Cantor's theory of transfinite numbers has played a pivotal role in set theory, with ramifications that extend to many areas of mathematics. This article extends earlier findings with a fresh look at the critical facts of Cantor's theory: i) Cantor's widely renowned Diagonalization Argument (CDA) is fully refuted by a set of counter-examples that expose the fallacy of this proof. ii) The logical inconsistencies of CDA are revisited, exposing the short-comings of CDA's implementation of the method of proof by contradiction. iii) The denumerability of the power set of the set of the natural numbers, P(N), is substantiated by a proof that takes full account of all the infinite subsets of N. Such a result confirms the denumerability of the set of the real numbers, R, and with it the countable nature of the continuum. iv) Given that the denumerable character of (probably) all infinite sets makes their comparison in terms of one-to-one correspondences a rather pointless exercise, a new concept of relative cardinality is introduced which facilitates a quantitative evaluation of their different magnitudes.
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Submitted 14 March, 2023;
originally announced May 2023.
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Alpha-stable branching and beta-Frequency processes, beyond the IID assumption
Authors:
Adrián González Casanova,
Imanol Nuñez,
J. -L. Pérez
Abstract:
Birkner et al. obtained necessary and sufficient conditions for the frequency between two independent and identically distributed continuous-state branching processes time-changed by a functional of the total mass process to be a Markov process. Foucart et al. extended this result to continuous-state branching processes with immigration. We generalize these results by dropping the independent and…
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Birkner et al. obtained necessary and sufficient conditions for the frequency between two independent and identically distributed continuous-state branching processes time-changed by a functional of the total mass process to be a Markov process. Foucart et al. extended this result to continuous-state branching processes with immigration. We generalize these results by dropping the independent and identically distributed assumption. Our result clarifies under which conditions a multi-type $Λ$-coalescent can be constructed from a multi-type branching process by a time change using the total mass. Finally, we address a problem formulated by Griffiths, by clarifying the relation between 2-type $α$-stable continuous-state branching processes and 2-type $β$-Fleming--Viot processes with mutation and selection.
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Submitted 9 March, 2023;
originally announced March 2023.
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Lévy bandits under Poissonian decision times
Authors:
José-Luis Pérez,
Kazutoshi Yamazaki
Abstract:
We consider a version of the continuous-time multi-armed bandit problem where decision opportunities arrive at Poisson arrival times, and study its Gittins index policy. When driven by spectrally one-sided Lévy processes, the Gittins index can be written explicitly in terms of the scale function, and is shown to converge to that in the classical Lévy bandit of Kaspi and Mandelbaum (1995).
We consider a version of the continuous-time multi-armed bandit problem where decision opportunities arrive at Poisson arrival times, and study its Gittins index policy. When driven by spectrally one-sided Lévy processes, the Gittins index can be written explicitly in terms of the scale function, and is shown to converge to that in the classical Lévy bandit of Kaspi and Mandelbaum (1995).
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Submitted 18 January, 2023;
originally announced January 2023.
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Geometry of CMC surfaces of finite index
Authors:
William H. Meeks III,
Joaquin Perez
Abstract:
Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M\looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ and with index at most $I$. We will obtain geometric estimates for such…
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Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M\looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ and with index at most $I$. We will obtain geometric estimates for such an $M\looparrowright X$ as a consequence of the Hierarchy Structure Theorem in [9]. The Hierarchy Structure Theorem (see Theorem 2.2 below) will be applied to understand global properties of $M\looparrowright X$, especially results related to the area and diameter of $M$. By item E of Theorem 2.2, the area of such a non-compact $M\looparrowright X$ is infinite. We will improve this area result by proving the following when $M$ is connected; here $g(M)$ denotes the genus of the orientable cover of $M$:
1. There exists $C_1=C_1(I,r_0,K_0,H_0)>0$ such that Area$(M)\geq C_1(g(M)+1)$.
2. There exists $C>0,G(I)\in \mathbb{N}$ independent of $r_0,K_0,H_0$ and also $C$ independent of $I$ such that if $g(M)\geq G(I)$, then Area$(M)\geq \frac{C}{(\max\{1,\frac{1}{r_0},\sqrt{K_0}, H_0\})^2}(g(M)+1)$.
3. If the scalar curvature $ρ$ of $X$ satisfies $3H^2+\frac{1}{2}ρ\geq c$ in $X$ for some $c>0$, then there exist $A,D>0$ depending on $c,I,r_0,K_0,H_0$ such that Area$(M)\leq A$ and Diameter$(M)\leq D$. Hence, $M$ is compact and, by item 1, $g(M)\leq A/C -1$.
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Submitted 27 March, 2023; v1 submitted 29 December, 2022;
originally announced December 2022.
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Hierarchy structures in finite index CMC surfaces
Authors:
William H. Meeks III,
Joaquin Perez
Abstract:
Given $\varepsilon_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq \varepsilon_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M \looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ with index at most $I$. For such…
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Given $\varepsilon_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq \varepsilon_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M \looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ with index at most $I$. For such $M \looparrowright X$, we prove Structure Theorem 1.2 which describes how the interesting ambient geometry of the immersion is organized locally around at most $I$ points of $M$ where the norm of the second fundamental form takes on large local maximum values.
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Submitted 27 March, 2023; v1 submitted 27 December, 2022;
originally announced December 2022.
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Loewner Theory for Bernstein functions II: applications to inhomogeneous continuous-state branching processes
Authors:
Pavel Gumenyuk,
Takahiro Hasebe,
José-Luis Pérez
Abstract:
This paper continues the research project launched in [Constr. Approx. (2025) https://doi.org/10.1007/s00365-023-09675-9] and aimed at studying time-inhomogeneous one-dimensional branching processes (mainly on a continuous but also on a discrete state space) with the help of recent achievements in Loewner Theory dealing with evolution families of holomorphic self-maps in simply connected domains o…
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This paper continues the research project launched in [Constr. Approx. (2025) https://doi.org/10.1007/s00365-023-09675-9] and aimed at studying time-inhomogeneous one-dimensional branching processes (mainly on a continuous but also on a discrete state space) with the help of recent achievements in Loewner Theory dealing with evolution families of holomorphic self-maps in simply connected domains of the complex plane. Under a suitable stochastic continuity condition, we show that the families of the Laplace exponents of branching processes on$~[0,\infty]$ can be characterized as topological (i.e. depending continuously on the time parameters) reverse evolution families whose elements are Bernstein functions. For the case of a stronger regularity w.r.t. time, we establish a Loewner-Kufarev type ODE for the Laplace exponents and characterize branching processes with finite mean in terms of the vector field driving this ODE. Similar results are obtained for families of probability generating functions of branching processes on the discrete state space $\{0,1,2,\ldots\}\cup\{\infty\}$. In addition, we find a necessary and sufficient condition for "spatial" embeddability of such branching processes into branching processes on$~[0,\infty]$. Finally, we give some probabilistic interpretations of the Denjoy-Wolff point at$~0$ and at$~\infty$.
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Submitted 12 December, 2025; v1 submitted 22 November, 2022;
originally announced November 2022.
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Geometry of branched minimal surfaces of finite index
Authors:
William H. Meeks III,
Joaquin Perez
Abstract:
Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of Fischer-Colbrie and Ros explain that such surfaces are precisely the complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature and finite total branching o…
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Given $I,B\in\mathbb{N}\cup \{0\}$, we investigate the existence and geometry of complete finitely branched minimal surfaces $M$ in $\mathbb{R}^3$ with Morse index at most $I$ and total branching order at most $B$. Previous works of Fischer-Colbrie and Ros explain that such surfaces are precisely the complete minimal surfaces in $\mathbb{R}^3$ of finite total curvature and finite total branching order. Among other things, we derive scale-invariant weak chord-arc type results for such an $M$ with estimates that are given in terms of $I$ and $B$. In order to obtain some of our main results for these special surfaces, we obtain general intrinsic monotonicity of area formulas for $m$-dimensional submanifolds $Σ$ of an $n$-dimensional Riemannian manifold $X$, where these area estimates depend on the geometry of $X$ and upper bounds on the lengths of the mean curvature vectors of $Σ$. We also describe a family of complete, finitely branched minimal surfaces in $\mathbb{R}^3$ that are stable and non-orientable; these examples generalize the classical Henneberg minimal surface.
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Submitted 8 November, 2022; v1 submitted 7 November, 2022;
originally announced November 2022.
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On the Polishness of the inverse semigroup $Γ(X)$ on a compact metric space $X$
Authors:
Jerson Pérez,
Carlos Uzcátegui
Abstract:
Let $Γ(X)$ be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space $X$. There is a topology, denoted $τ_{hco}$, that makes $Γ(X)$ a topological inverse semigroup. We address the question of whether $τ_{hco}$ is Polish. For a 0-dimensional compact metric space $X$, we prove that $(Γ(X), τ_{hco})$ is Polish by showing that it is topologically isomorphic to a…
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Let $Γ(X)$ be the inverse semigroup of partial homeomorphisms between open subsets of a compact metric space $X$. There is a topology, denoted $τ_{hco}$, that makes $Γ(X)$ a topological inverse semigroup. We address the question of whether $τ_{hco}$ is Polish. For a 0-dimensional compact metric space $X$, we prove that $(Γ(X), τ_{hco})$ is Polish by showing that it is topologically isomorphic to a closed subsemigroup of the Polish symmetric inverse semigroup $I(\N)$. We present examples, similar to the classical Munn semigroups, of Polish inverse semigroups consisting of partial isomorphism on lattices of open sets.
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Submitted 22 July, 2023; v1 submitted 5 November, 2022;
originally announced November 2022.
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A geometrically intrinsic Lagrangian-Eulerian scheme for 2D Shallow Water Equations with variable topography and discontinuous data
Authors:
Eduardo Abreu,
Elena Bachini,
John Perez,
Mario Putti
Abstract:
We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value problem that describes the evolution of the balance la…
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We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value problem that describes the evolution of the balance laws governing the geometrically intrinsic shallow water equations. The evolved solution set is then projected back to the original surface grid to complete the proposed Lagrangian-Eulerian formulation. The resulting scheme maintains monotonicity and captures shocks without providing excessive numerical dissipation also in the presence of non-autonomous fluxes such as those arising from the geometrically intrinsic shallow water equation on variable topographies. We provide a representative set of numerical examples to illustrate the accuracy and robustness of the proposed Lagrangian-Eulerian formulation for two-dimensional surfaces with general curvatures and discontinuous initial conditions.
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Submitted 8 September, 2022;
originally announced September 2022.
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New characterizations of ruled real hypersurfaces in complex projective space
Authors:
Juan de Dios Pérez,
David Pérez-López
Abstract:
We consider real hypersurfaces $M$ in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull constant $k$ and any symmetric tensor field of type (1,1) $L$ on $M$ we can define two tensor fields of type (1,2) on $M$, $L_F^{(k)}$ and $L_T^{(k)}$, related to both connections. We study the behaviour of the structure operator $φ$ with res…
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We consider real hypersurfaces $M$ in complex projective space equipped with both the Levi-Civita and generalized Tanaka-Webster connections. For any nonnull constant $k$ and any symmetric tensor field of type (1,1) $L$ on $M$ we can define two tensor fields of type (1,2) on $M$, $L_F^{(k)}$ and $L_T^{(k)}$, related to both connections. We study the behaviour of the structure operator $φ$ with respect to such tensor fields in the particular case of $L=A$, the shape operator of $M$, and obtain some new characterizations of ruled real hypersurfaces in complex projective space.
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Submitted 25 August, 2022;
originally announced August 2022.
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Perturbation theory of transfer function matrices
Authors:
Vanni Noferini,
Lauri Nyman,
Javier Pérez,
María C. Quintana
Abstract:
Zeros of rational transfer function matrices $R(λ)$ are the eigenvalues of associated polynomial system matrices $P(λ)$, under minimality conditions. In this paper we define a structured condition number for a simple eigenvalue $λ_0$ of a (locally) minimal polynomial system matrix $P(λ)$, which in turn is a simple zero $λ_0$ of its transfer function matrix $R(λ)$. Since any rational matrix can be…
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Zeros of rational transfer function matrices $R(λ)$ are the eigenvalues of associated polynomial system matrices $P(λ)$, under minimality conditions. In this paper we define a structured condition number for a simple eigenvalue $λ_0$ of a (locally) minimal polynomial system matrix $P(λ)$, which in turn is a simple zero $λ_0$ of its transfer function matrix $R(λ)$. Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yield a structured perturbation theory for simple zeros of rational matrices $R(λ)$. To capture all the zeros of $R(λ)$, regardless of whether they are poles or not, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of $λ_0$ being not a pole of $R(λ)$ since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur's unstructured condition number for eigenvalues of matrix polynomials, and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.
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Submitted 15 July, 2022; v1 submitted 14 July, 2022;
originally announced July 2022.
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On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes
Authors:
Dante Mata,
Harold A. Moreno-Franco,
Kei Noba,
José-Luis Pérez
Abstract:
This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative…
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This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative \lev process with a final payoff at an exponential terminal time and characterise the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.
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Submitted 3 July, 2022;
originally announced July 2022.
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Generalized Henneberg stable minimal surfaces
Authors:
David Moya,
Joaquín Pérez
Abstract:
We generalize the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on a positive integer (called the complexity), which essentially measures the number of branch points. The classical Henneberg surface $H_1$ is characterized as the uniq…
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We generalize the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on a positive integer (called the complexity), which essentially measures the number of branch points. The classical Henneberg surface $H_1$ is characterized as the unique example in the subfamily of the simplest complexity $m=1$, while for $m\geq 2$ multiparameter families are given. The isometry group of the most symmetric example $H_m$ with a given complexity $m\in \mathbb{N}$ is either isomorphic to the dihedral isometry group $D_{2m+2}$ (if $m$ is odd) or to $D_{m+1}\times \mathbb{Z}_2$ (if $m$ is even). Furthermore, for $m$ even $H_m$ is the unique solution to the Björling problem for a hypocycloid of $m+1$ cusps (if $m$ is even), while for $m$ odd the conjugate minimal surface $H_m^*$ to $H_m$ is the unique solution to the Björling problem for a hypocycloid of $2m+2$ cusps.
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Submitted 27 July, 2022; v1 submitted 3 July, 2022;
originally announced July 2022.
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On the structure Lie operator of a real hypersurface in the complex quadric
Authors:
Juan de Dios Pérez,
David Pérez-López,
Young Jin Suh
Abstract:
The almost contact metric structure that we have on a real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ allows us to define, for any nonnull real number $k$, the $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. Associated to this connection we have Cho and torsion operators, $F_X^{(k)}$ and $T_X^{(k)}$, respectively, for any vector field $X$ tangent t…
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The almost contact metric structure that we have on a real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ allows us to define, for any nonnull real number $k$, the $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. Associated to this connection we have Cho and torsion operators, $F_X^{(k)}$ and $T_X^{(k)}$, respectively, for any vector field $X$ tangent to $M$. From them and for any symmetric operator $B$ on $M$ we can consider two tensor fields of type (1,2) on $M$ that we will denote by $B_F^{(k)}$ and $B_T^{(k)}$, respectively. We will classify real hypersurfaces $M$ in $Q^m$ for which any of those tensors identically vanishes, in the particular case of $B$ being the structure Lie operator $L_ξ$ on $M$.
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Submitted 25 June, 2022;
originally announced June 2022.
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Loewner Theory for Bernstein functions I: evolution families and differential equations
Authors:
Pavel Gumenyuk,
Takahiro Hasebe,
José-Luis Pérez
Abstract:
One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play t…
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One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-analytic proof of a qualitative description equivalent to Silverstein's representation formula for the infinitesimal generators of one-parameter semigroups of Bernstein functions. We also establish several sufficient conditions for families of holomorphic self-maps, satisfying the algebraic part in the definition of an evolution family, to be absolutely continuous and hence to be described as solutions to the generalized Loewner - Kufarev differential equation. Most of these results are then applied in the sequel paper [https://doi.org/10.48550/arXiv.2211.12442] to study continuous-state branching processes.
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Submitted 28 March, 2023; v1 submitted 9 June, 2022;
originally announced June 2022.
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Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber product rings
Authors:
T. H. Freitas,
V. H. Jorge PÉrez,
R. Wiegand,
S. Wiegand
Abstract:
In this paper we explore consequences of the vanishing of ${\rm Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a local ring such that $R/(\underline x)$ is a non-trivial fiber product ring, for some regular sequence $\underline x$ of $R$. Equivalently, the maximal ideal of $R/(\underline x)$ decomposes as a direct sum of two nonzero ideals. Gorenstein qua…
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In this paper we explore consequences of the vanishing of ${\rm Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a local ring such that $R/(\underline x)$ is a non-trivial fiber product ring, for some regular sequence $\underline x$ of $R$. Equivalently, the maximal ideal of $R/(\underline x)$ decomposes as a direct sum of two nonzero ideals. Gorenstein quasi-fiber product rings are AB-rings and are Ext-bounded. We show in Theorem 3.31 that quasi-fiber product rings satisfy a sharpened form of the Auslander-Reiten Conjecture. We also make some observations related to the Huneke-Wiegand conjecture for quasi-fiber product rings.
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Submitted 2 May, 2022;
originally announced May 2022.
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Pettis property for Polish inverse semigroups
Authors:
Karen Arana,
Jerson Perez,
Carlos Uzcategui
Abstract:
We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections between subsets of N. We also study whether our ex…
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We study a property about Polish inverse semigroups similar to the classical theorem of Pettis about Polish groups. In contrast to what happens with Polish groups, not every Polish inverse semigroup have the Pettis property. We present several examples of Polish inverse subsemigroup of the symmetric inverse semigroup I(N) of all partial bijections between subsets of N. We also study whether our examples satisfy automatic continuity.
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Submitted 26 March, 2022;
originally announced March 2022.
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Two-type branching processes with immigration, and the structured coalescents
Authors:
María Emilia Caballero,
Adrián González Casanova,
José Luis Pérez
Abstract:
We consider a population constituted by two types of individuals; each of them can produce offspring in two different islands (as a particular case the islands can be interpreted as active or dormant individuals). We model the evolution of the population of each type using a two-type Feller diffusion with immigration, and we study the frequency of one of the types, in each island, when the total p…
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We consider a population constituted by two types of individuals; each of them can produce offspring in two different islands (as a particular case the islands can be interpreted as active or dormant individuals). We model the evolution of the population of each type using a two-type Feller diffusion with immigration, and we study the frequency of one of the types, in each island, when the total population size in each island is forced to be constant at a dense set of times. This leads to the solution of a SDE which we call the asymmetric two-island frequency process. We derive properties of this process and obtain a large population limit when the total size of each island tends to infinity. Additionally, we compute the fluctuations of the process around its deterministic limit. We establish conditions under which the asymmetric two-island frequency process has a moment dual. The dual is a continuous-time two-dimensional Markov chain that can be interpreted in terms of mutation, branching, pairwise branching, coalescence, and a novel mixed selection-migration term. Also, we conduct a stability analysis of the limiting deterministic dynamical system and present some numerical results to study fixation and a new form of balancing selection. When restricting to the seedbank model, we observe that some combinations of the parameters lead to balancing selection. Besides finding yet another way in which genetic reservoirs increase the genetic variability, we find that if a population that sustains a seedbank competes with one that does not, the seed producers will have a selective advantage if they reproduce faster, but will not have a selective disadvantage if they reproduce slower: their worst case scenario is balancing selection.
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Submitted 1 May, 2024; v1 submitted 15 September, 2021;
originally announced September 2021.
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Lie derivatives and structure Jacobi operator on real hypersurfaces in complex projective spaces II
Authors:
Juan de Dios Pérez,
David Pérez-López
Abstract:
Let $M$ be a real hypersurface in complex projective space. The almost contact metric structure on $M$ allows us to consider, for any nonnull real number $k$, the corresponding $k$-th generalized Tanaka-Webster connection on $M$ and, associated to it, a differential operator of first order of Lie type. Considering such a differential operator and Lie derivative we define, from the structure Jacobi…
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Let $M$ be a real hypersurface in complex projective space. The almost contact metric structure on $M$ allows us to consider, for any nonnull real number $k$, the corresponding $k$-th generalized Tanaka-Webster connection on $M$ and, associated to it, a differential operator of first order of Lie type. Considering such a differential operator and Lie derivative we define, from the structure Jacobi operator $R_ξ$ on $M$ a tensor field of type (1,2), $R_{ξ_T}^{(k)}$. We obtain some classifications of real hypersurfaces for which $R_{ξ_T}^{(k)}$ is either symmetric or skew symmetric.
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Submitted 8 September, 2021;
originally announced September 2021.
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Non-zero-sum optimal stopping game with continuous versus periodic exercise opportunities
Authors:
José Luis Pérez,
Neofytos Rodosthenous,
Kazutoshi Yamazaki
Abstract:
We introduce a new non-zero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modelling the value of an asset, one player observes and can act on the process continuously, while the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one g…
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We introduce a new non-zero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modelling the value of an asset, one player observes and can act on the process continuously, while the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, while the other one gets nothing. We study how each player balances the maximisation of gains against the maximisation of the likelihood of stopping before the opponent. In such a setup, driven by a Lévy process with positive jumps, we not only prove the existence, but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behaviour of the players' strategies as well as the quantification of the value of available exercise opportunities.
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Submitted 15 May, 2024; v1 submitted 17 July, 2021;
originally announced July 2021.
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Convergence of a Lagrangian-Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems
Authors:
Eduardo Abreu,
Arthur Espírito Santo,
Wanderson Lambert,
John Pérez
Abstract:
In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis. We discuss theoretical developments on the conception of no-flow curves for hyperbolic problems within scientific computing. The resulting algorithms have been p…
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In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis. We discuss theoretical developments on the conception of no-flow curves for hyperbolic problems within scientific computing. The resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions. We present experiments to a study based on the use of the Wasserstein distance to show the effectiveness of the no-flow curves approach in the cases of shock interaction with an entropy wave related to the inviscid Burgers' model problem and to a 2x2 nonlocal traffic flow symmetric system of type Keyfitz--Kranzer.
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Submitted 2 February, 2022; v1 submitted 15 June, 2021;
originally announced June 2021.
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Introducción a la teoría de complejidad topológica
Authors:
Cesar A. Ipanaque Zapata,
Rodolfo J. Gálvez Pérez
Abstract:
In this work we will review the notion of topological complexity, introduced by Michael Farber in 2003. We will use this theory of topological complexity to solve the motion planning problem of a mobile robot that navigates in the Euclidean plane avoiding colliding with an obstacle. Specifically, we calculate topological complexity and design explicit algorithms.
In this work we will review the notion of topological complexity, introduced by Michael Farber in 2003. We will use this theory of topological complexity to solve the motion planning problem of a mobile robot that navigates in the Euclidean plane avoiding colliding with an obstacle. Specifically, we calculate topological complexity and design explicit algorithms.
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Submitted 17 May, 2021;
originally announced May 2021.
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On the coefficients in an asymptotic expansion of $(1+1/x)^x$
Authors:
T. M. Dunster,
Jessica M. Perez
Abstract:
The function $g(x)= (1+1/x)^{x}$ has the well-known limit $e$ as $x\rightarrow{\infty}$. The coefficients $c_{j}$ in an asymptotic expansion for $g(x)$ are considered. A simple recursion formula is derived, and then using Cauchy's integral formula the coefficients are approximated for large $j$. From this it is shown that $|c_{j}|\rightarrow{1}$ as $j\rightarrow{\infty}$.
The function $g(x)= (1+1/x)^{x}$ has the well-known limit $e$ as $x\rightarrow{\infty}$. The coefficients $c_{j}$ in an asymptotic expansion for $g(x)$ are considered. A simple recursion formula is derived, and then using Cauchy's integral formula the coefficients are approximated for large $j$. From this it is shown that $|c_{j}|\rightarrow{1}$ as $j\rightarrow{\infty}$.
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Submitted 10 August, 2021; v1 submitted 8 May, 2021;
originally announced May 2021.
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Ruled Real Hypersurfaces in the Indefinite Complex Projective Space
Authors:
Marilena Moruz,
Miguel Ortega,
Juan de Dios Pérez
Abstract:
The main two families of real hypersurfaces in complex space forms are Hopf and ruled. However, very little is known about real hypersurfaces in the indefinite complex projective space $\cpn$. In a previous work, Kimura and the second author introduced Hopf real hypersurfaces in $\cpn$. In this paper, ruled real hypersurfaces in the indefinite complex projective space are introduced, as those whos…
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The main two families of real hypersurfaces in complex space forms are Hopf and ruled. However, very little is known about real hypersurfaces in the indefinite complex projective space $\cpn$. In a previous work, Kimura and the second author introduced Hopf real hypersurfaces in $\cpn$. In this paper, ruled real hypersurfaces in the indefinite complex projective space are introduced, as those whose maximal holomorphic distribution is integrable, and such that the leaves are totally geodesic holomorphic hyperplanes. A detailed description of the shape operator is computed, obtaining two main different families. A method of construction is exhibited, by gluing in a suitable way totally geodesic holomorphic hyperplanes along a non-null curve. Next, the classification of all minimal ruled real hypersurfaces is obtained, in terms of three main families of curves, namely geodesics, totally real circles and a third case which is not a Frenet curve, but can be explicitly computed. Four examples of minimal ruled real hypersurfaces are described.
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Submitted 18 May, 2022; v1 submitted 21 February, 2021;
originally announced February 2021.
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Topologies on the symmetric inverse semigroup
Authors:
J. Perez,
C. Uzcategui
Abstract:
The symmetric inverse semigroup $I(X)$ on a set $X$ is the collection of all partial bijections between subsets of $X$ with composition as the algebraic operation. We study a minimal Hausdorff inverse semigroup topologies on $I(X)$. When $X$ is countable, we show some Polish semigroup topologies on $I(X)$.
The symmetric inverse semigroup $I(X)$ on a set $X$ is the collection of all partial bijections between subsets of $X$ with composition as the algebraic operation. We study a minimal Hausdorff inverse semigroup topologies on $I(X)$. When $X$ is countable, we show some Polish semigroup topologies on $I(X)$.
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Submitted 5 December, 2020;
originally announced December 2020.
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Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis
Authors:
Fernando De Terán,
Carla Hernando,
Javier Pérez
Abstract:
In the framework of Polynomial Eigenvalue Problems, most of the matrix polynomials arising in applications are structured polynomials (namely (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve Polynomial Eigenvalue Problems is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix pol…
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In the framework of Polynomial Eigenvalue Problems, most of the matrix polynomials arising in applications are structured polynomials (namely (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve Polynomial Eigenvalue Problems is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as {\em companion linearizations}. It is well known, however, that is not possible to construct companion linearizations that preserve any of the previous structures for matrix polynomials of even degree. This motivates the search for more general companion forms, in particular {\em companion $\ell$-ifications}. In this paper, we present, for the first time, a family of (generalized) companion $\ell$-ifications that preserve any of these structures, for matrix polynomials of degree $k=(2d+1)\ell$. We also show how to construct sparse $\ell$-ifications within this family. Finally, we prove that there are no structured companion quadratifications for quartic matrix polynomials.
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Submitted 12 October, 2020;
originally announced October 2020.