Infinite Time Research Articles

Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims

The Parisian ruin time, which is the first time the insurer's surplus process has an excursion below level zero that exceeds a prescribed time length, has been extensively analyzed in recent years mainly in the Lévy model and its special cases. However, the cumulative Parisian ruin time, which is the first time the total time spent by the surplus process below level zero exceeds a certain time length, has been rarely considered in the literature. In this paper, we study the cumulative Parisian ruin problem in a renewal risk model with general interclaim times and exponential claims. Explicit formulas for the infinite-time cumulative Parisian ruin probability is first derived under a deterministic Parisian clock and then under an Erlang clock, where the latter case can also serve as an approximation of the former. The finite-time cumulative Parisian ruin probability is subsequently analyzed as well when the time horizon is another Erlang random variable. Our formulas are applied in various numerical examples where the interclaim times follow gamma, Weibull, or Pareto distribution. Consequently, we demonstrate that the choice of the interclaim distribution does have a significant impact on the cumulative Parisian ruin probabilities when one deviates from the exponential assumption.

A modelling approach for the assessment of energy recovery and impact on the water line of sludge pre-treatments

A simple, easy-to-use, first-order model was elaborated to predict the methane production and the release of ammoniacal nitrogen (N–NH3) to the digestate in full-scale anaerobic digestion (AD) processes. The study used long-term, semi-continuous AD tests, carried out with samples of primary sludge (PS), raw waste activated sludge (WAS), WAS after a thermo-alkali pre-treatment (90 °C, 90 min, 4 g NaOH/100 g TS) and mixed sludge (PS/treated WAS), to calibrate and validate the model. The results of both the experimental activities and the phase of model tuning demonstrated that the proposed model was capable to provide reliable information to completely characterize the AD process, thus overcoming the limitations due to discontinuity of experimental tests. Furthermore, it was demonstrated that low-temperature thermo-alkali pre-treatments could increase the values of the model parameters, namely methane production after an infinite time (B0, +70%) and hydrolysis constant (k, +450%), and made them comparable to those obtained by the application of commercial, high-energy demanding treatments (e.g. Cambi). Finally, the issue concerning the release of N–NH3 to digestate was deemed to be very worthy to being investigated because, after pre-treatments, the cost for nitrogen removal in the water line, through the traditional processes of nitrification – denitrification, could increase even by 140%.

WHAM-FTOXβ – An aquatic toxicity model based on intrinsic metal toxic potency and intrinsic species sensitivity

We developed a model that quantifies aquatic cationic toxicity by a combination of the intrinsic toxicities of metals and protons and the intrinsic sensitivities of the test species. It is based on the WHAM-FTOX model, which combines the calculated binding of cations by the organism with toxicity coefficients (αH, αM) to estimate the variable FTOX, a measure of toxic effect; the key parameter αM,max (applying at infinite time) depends upon both the metal and the test species. In our new model, WHAM-FTOXβ, values of αM,max are given by the product αM* × β, where αM* has a single value for each metal, and β a single value for each species. To parameterise WHAM-FTOXβ, we assembled a set of 2182 estimates of αM,max obtained by applying the basic model to laboratory toxicity data for 76 different test species, covering 15 different metals, and including results for metal mixtures. Then we fitted the log10αM,max values with αM* and β values (a total of 91 parameters). The resulting model accounted for 72% of the variance in log10αM,max. The values of αM* increased markedly as the chemical character of the metal changed from hard (average αM* = 4.4) to intermediate (average αM* = 25) to soft (average αM* = 560). The values of log10β were normally distributed, with a 5–95 percentile range of -0.73 to +0.56, corresponding to β values of 0.18 to 3.62. The WHAM-FTOXβ model entails the assumption that test species exhibit common relative sensitivity, i.e. the ratio αM,max / αM* is constant across all metals. This was tested with data from studies in which the toxic responses of a single organism towards two or more metals had been measured (179 examples for the most-tested metals Ni, Cu, Zn, Ag, Cd, Pb), and statistically-significant (p < 0.003) results were obtained.

A dynamic game model for no-take marine reserves

Representing about 8% of the world’s oceans, marine protected areas (MPAs) are effective instruments for mitigating the damage that overfishing can do to ecological and economic systems. Yet, less than 2% of the ocean is proclaimed an exclusive no-take zone. Expecting their number and size to increase significantly in the coming years, meticulous implementation of MPAs is central to their success. Using a dynamic game model we explore the impact of designated no-take marine reserves on fish stock and on fishing strategies. Our findings suggest that optimality can be achieved in a context of a combination of three marine zones, i.e., an open access fishery, a marine protected area and a fish bank. Our methodology consists of considering an infinite time horizon discrete fishing game with a migratory species between the three regions. We analyze fishing strategies and characterize the model equilibria for different behaviors, the fully non-cooperative scheme, the social optimum and the cartel setting. Our analytical results are supported by the use of a numerical simulation that compares different scenarios, including the current state of the world in contrast to different sizes of MPAs and fish bank areas.

Additional Food Causes Predators to Explode — Unless the Predators Compete

The literature posits that an introduced predator population is able to drive its target pest population to extinction, if supplemented with high quality additional food of sufficient quantity. We show this approach actually leads to infinite time blow-up of the predator population, so is unpragmatic as a pest management strategy. We propose an alternate model in which the additional food induces predator competition. Analysis of this model indicates that depending on the competition parameter [Formula: see text], one can have global stability of the pest-free state, bistability dynamics, or up to three interior equilibria. As [Formula: see text] and the additional food quantity [Formula: see text] are varied standard codimension one and codimension two bifurcations are observed. We also use structural symmetries to construct several nonstandard bifurcations such as saddle-node-transcritical bifurcation (SNTC) in codimension two and a cusp-transcritical bifurcation (CPTC), also in codimension two. We further use symmetry to construct a novel pitchfork-transcritical bifurcation (PTC) in codimension two, thus explicitly characterizing a new organizing center of the model. Our findings indicate that increasing additional food in predator–pest models can hinder bio-control, contrarily to some of the literature. However, additional food that also induces predator competition, leads to rich dynamics and enhances bio-control.

Controllability results to non-instantaneous impulsive with infinite delay for generalized fractional differential equations

This paper discusses controllability results for active types with infinite-time delay of non-instantaneous impulsive fractional differential equations. The model is constructed based on the generalized Caputo (Caputo-Katugampola) fractional derivative and the control function with non-local Katugampola fractional integral as a boundary condition. Our principal results are established by giving some sufficient hypotheses, utilizing well-known fractional calculus truths and using Krasnoselskii’s fixed point theorem. The infinite time delay has been treated with the abstract phase space techniques and fulfilling the ensuing axioms due to Hale and Kato. It turns out that under some sufficient conditions, the problem has at least one controllable solution. An implementation of our theoretical results is demonstrated by a numerical example.

Optimal transmission scheduling for remote state estimation in CPSs with energy harvesting two-hop relay networks

This paper considers the optimal transmission power scheduling for accurate remote state estimation in cyber–physical systems (CPSs). The plant is modeled as a discrete-time stochastic linear system with sensor measurements transmitted to the remote estimator over an intelligent relay. A dynamic cooperative team is formulated in which the sensor and the relay sharing the same cost function are equipped with corresponding energy harvesters, and need to decide on the transmission power levels to minimize the average error covariance at the remote estimator. Since the nodes make simultaneous decisions to achieve a common goal, the sensor and the relay do not have access to the transmission power that will be selected by another node in advance, and hence they cannot build a value estimation of the cost function directly at each time point, which poses a challenge to determine the optimal transmission policies of both sides. To overcome this challenge, we provide a local-action value function and prove that both nodes select the transmission power with respect to the local-action value function without any loss of performance. In addition, a Markov decision process (MDP) and a multi-agent reinforcement learning algorithm are presented to obtain the optimal transmission power schedule over an infinite time horizon. Finally, simulation examples are provided to corroborate and illustrate the theoretical results.

Blood-brain barrier water exchange measurements using FEXI: Impact of modeling paradigm and relaxation timeeffects.

To evaluate potential modeling paradigms and the impact of relaxation time effects on human blood-brain barrier (BBB) water exchange measurements using FEXI (BBB-FEXI), and to quantify the accuracy, precision, and repeatability of BBB-FEXI exchange rate estimates at 3 . Three modeling paradigms were evaluated: (i) the apparent exchange rate (AXR) model; (ii) a two-compartment model ( ) explicitly representing intra- and extravascular signal components, and (iii) a two-compartmentmodeladditionally accounting for finite compartmental and relaxation times ( ). Each model had three free parameters. Simulations quantified biases introduced by the assumption of infinite relaxation times in the AXR and models, as well as the accuracy and precision of all three models. The scan-rescan repeatability of all paradigms was quantified for the first time in vivo in 10 healthy volunteers (age range 23-52 years; five female). The assumption of infinite relaxation times yielded exchange rate errors in simulations up to 42%/14% in the AXR/ models, respectively. Accuracy was highest in the compartmental models; precision was best in the AXR model. Scan-rescan repeatability in vivo was good for all models, with negligible bias and repeatability coefficients in grey matter of , , and . Compartmental modelling of BBB-FEXI signals can provide accurate and repeatable measurements of BBB water exchange; however, relaxation time and partial volume effects may cause model-dependent biases.

Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation

In this work, we consider the following focusing inhomogeneous nonlinear Schrödinger equation i∂tu+Δu+|x|−b|u|pu=0,(t,x)∈R×RNwith 0<b<min{2,N} and 4−2bN<p<4−2bN−2. Assume that u0∈H1(RN) and beyond the ground state threshold, then we prove the following two statements, (1) when 4−2bN<p<min{4N,4−2bN−2}, or p=4N when b∈(0,4N), then the corresponding solution blows up in finite time; (2) when 4N<p<4−2bN−2, we prove the finite or infinite time blow-up. Moreover, we can further obtain a precise lower bound of infinite time blow-up rate, that is supt∈[0,T]‖∇u(t)‖L2≳Tκ,for someκ>0.To our knowledge, the statement (1) establishes the first finite time blow-up result for this equation in the intercritical case when the initial data u0 does not have finite variance and is non-radial. The statement (2) gives the first result for the infinite time blow-up rate for this equation.

Controllability for Fractional Evolution Equations with Infinite Time-Delay and Non-Local Conditions in Compact and Noncompact Cases

The goal of this dissertation is to explore a system of fractional evolution equations with infinitesimal generator operators and an infinite time delay with non-local conditions. It turns out that there are two ways to regulate the solution. To demonstrate the presence of the controllability of mild solutions, it is usual practice to apply Krasnoselskii’s theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness. A fractional Caputo approach of order between 1 and 2 was used to construct our model. The families of linear operators cosine and sine, which are strongly continuous and uniformly bounded, are used to achieve the mild solution. To make our results seem to be applicable, a numerical example is provided.

Sliding Mode Control of Energy Storage Systems for Reshaping the Accelerating Power of Synchronous Generators

The rotor angle stability has been a key concern in the stable operation of power systems, especially in the renewable energy age. However, to improve the power system rotor angle stability, none of the existing methods considered achieving it through reshaping the accelerating power of synchronous generators (SGs) and decoupling them after grid disturbances. In this paper, the sliding mode control (SMC) theory is utilized to tackle the coupling among SGs. After careful modeling and controller design, a SMC-based energy storage system (ESS) controller is proposed to temporarily decouple the interacted SGs through reshaping the accelerating power of SGs to the desired value. The proposed controller can: 1) drive the SG’s accelerating power to follow the reference in infinite time; 2) decouple the interacted SG dynamics and stabilize the disturbed SG exponentially. The proposed controller achieves the above two goals without using the communication and hard-to-estimate SG rotor angles. Case studies are carried out on the 2-Area 4-Generator System and the Australian 14-Generator Equivalent System to demonstrate the effectiveness of the proposed controller. The results also show that the proposed controller stabilizes the disturbed power systems with better performance over existing controllers.

On Singular Vortex Patches, I: Well-posedness Issues

The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globallymm-fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long asm≥3.m\geq 3.In this case, all of the angles involved solve aclosedODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show thatanyother type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angleπ2\frac {\pi }{2}for all time. Even in the case of vortex patches with corners of angleπ2\frac {\pi }{2}or with corners which are only locallymm-fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches onR2\mathbb {R}^2with interesting dynamical behavior such as cusping and spiral formation in infinite time.

Stagnation Detection with Randomized Local Search.

Recently a mechanism called stagnation detection was proposed that automatically adjusts the mutation rate of evolutionary algorithms when they encounter local optima. The so-called SD-(1+1) EA introduced by Rajabi and Witt (Algorithmica 2022) adds stagnation detection to the classical (1+1) EA with standard bit mutation. This algorithm flips each bit independently with some mutation rate, and stagnation detection raises the rate when the algorithm is likely to have encountered a local optimum. In this paper, we investigate stagnation detection in the context of the k-bit flip operator of randomized local search that flips k bits chosen uniformly at random and let stagnation detection adjust the parameter k. We obtain improved runtime results compared to the SD-(1+1) EA amounting to a speedup of at least (1-o(1))2πm, where m is the so-called gap size, i. e., the distance to the next improvement. Moreover, we propose additional schemes that prevent infinite optimization times even if the algorithm misses a working choice of k due to unlucky events. Finally, we present an example where standard bit mutation still outperforms the k-bit flip operator with stagnation detection.

L2-blowup estimates of the wave equation and its application to local energy decay

We consider the Cauchy problems in [Formula: see text] for the wave equation with a weighted [Formula: see text]-initial data. We derive sharp infinite time blowup estimates of the [Formula: see text]-norm of solutions in the case of [Formula: see text] and [Formula: see text]. Then, we apply it to the local energy decay estimates for [Formula: see text], which is not studied so completely when the [Formula: see text]th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the [Formula: see text]-norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].

Infinite Time Ruin Probability in the Individual Risk Model with Dependent Structure for Light and Heavy Tailed Distributions

Infinite Time Ruin Probability in the Individual Risk Model with Dependent Structure for Light and Heavy Tailed Distributions

Problems of Optimal Resource Harvesting for Infinite Time Horizon

Problems of Optimal Resource Harvesting for Infinite Time Horizon

Robust servo linear quadratic regulator controller based on state compensation and velocity feedforward of the spherical robot: Theory and experimental verification

There are few studies on the lateral motion of spherical robots. In this article, a new algorithm is proposed to solve the problem of low control accuracy of the lateral motion. A single lateral motion model is established, and the optimal solution of the linear quadratic regulator in infinite time domain is obtained. Aiming at the problems of longitudinal velocity and lateral angle coupling and low control precision, state compensation and velocity feedforward are carried out, and an improved robust servo linear quadratic regulator control algorithm is proposed. Experiments show that the proposed lateral control algorithm has strong adaptability and robustness to changing speeds and lateral angles, and the control effect is stable and reliable.

Robust exponential synchronization results for uncertain infinite time varying distributed delayed neural networks with flexible delayed impulsive control

This paper discusses the synchronization results like robust exponential synchronization (RES) and exponential synchronization (ES) of uncertain infinite time-varying flexible delayed impulsive neural networks with and without distributed delay. First, we introduced a new lemma based on fundamental solution matrix for solving nonlinear neural networks by utilizing average impulsive interval (AII) and average impulsive delay (AID). Further, we examined some novel RES and ES results for proposed neural networks via the method of variation of constants. Finally, numerical examples are given to validate the proposed results, and the effectiveness of flexible delayed impulsive control is demonstrated using graphical representations.

Dynamic Controller That Operates Over Homomorphically Encrypted Data for Infinite Time Horizon

In this article, we present a dynamic feedback controller that computes the next state and the control signal over encrypted data using homomorphic properties of cryptosystems, whose performance is equivalent to the linear dynamic controllers over real-valued data. Assuming that the input as well as the output of the plant is encrypted and transmitted back to the controller, it is shown that the state matrix of any linear time-invariant controller can be always converted to a matrix of integer components. This allows the dynamic feedback controller to operate for infinite time horizon without decryption or reset of its internal state. For implementation in practice, we illustrate the use of a cryptosystem that is based on the Learning With Errors problem, which allows both multiplication and addition over encrypted data. It is also shown that the effect of injected random numbers during encryption for security can be maintained within a small bound by way of the closed-loop stability.

Optimal Accumulation of Factors with Linear-Homogeneous Production Functions on Infinite Time Horizon

Optimal Accumulation of Factors with Linear-Homogeneous Production Functions on Infinite Time Horizon