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In this paper, we introduce a novel scalarization function for set-valued maps and establish several significant properties that highlight its utility. We extend the concept of regular-global-inf functions, initially studied in the context of single-valued maps, to the realm of set-valued maps. As the first main result, we derive a new Weierstrass-type theorem for set-valued maps satisfying the coercivity condition. Furthermore, utilizing tools from asymptotic analysis, we present an existence theorem for strict weakly l-efficient solutions of set optimization problems under certain non-coercive conditions. To underscore the relevance and applicability of our findings, we provide several corollaries and illustrative examples. [ABSTRACT FROM AUTHOR]

We study asymptotic profiles for singular solutions to a class of critical strongly coupled fourth order systems on the punctured ball. Assuming a superharmonicity condition, we prove that sufficiently close to the isolated singularity, singular solutions behave like the so-called Emden–Fowler solution to the blow-up limit problem. On the technical level, we use an involved spectral analysis to study the Jacobi fields' growth properties in the kernel of the linearization of our system around a blow-up limit solution, which may be of independent interest. Our main theorem positively answers a question posed by Frank and König (2019) [12] concerning the local behavior of singular solutions close to the isolated singularity for scalar solutions in the punctured ball. It also extends to the case of strongly coupled systems, the celebrated asymptotic classification due to Korevaar et al. (1999) [21]. [ABSTRACT FROM AUTHOR]

We present two new algorithms for solving norm equations over global function fields with at least one infinite place of degree one. The first of these is a substantial improvement of a method due to Gaál and Pohst, while the second approach uses index calculus techniques and is significantly faster asymptotically and in practice. Both algorithms incorporate compact representations of field elements which results in a significant gain in performance compared to the Gaál–Pohst approach. We provide Magma implementations, analyze the complexity of all three algorithms under varying asymptotics on the field parameters, and provide empirical data on their performance. [ABSTRACT FROM AUTHOR]

We are concerned with a coefficient inverse problem of a multidimensional heat equation. The objective is to reconstruct the sought coefficient from a sequence of observations of the solution taken at a single point. To do so we first obtain an explicit formula for the sought coefficient, and then see how we can approximate it using few observations only. We also show that asymptotics of the solution help reduce the data processing to overcome the curse of dimensionality. This new and direct reconstruction method is fast and gives an alternative to iterative and Newton's type methods. Numerical examples are provided at the end. [ABSTRACT FROM AUTHOR]

In this paper, we study the topological asymptotic expansion of a topology optimisation problem that is constrained by the Poisson equation with the design/shape variable entering through the right hand side. Using an averaged adjoint approach, we give explicit formulas for topological derivatives of arbitrary order for both an L 2 and H 1 tracking-type cost function in both dimension two and three and thereby derive the complete asymptotic expansion. As the asymptotic behaviour of the fundamental solution of the Laplacian differs in dimension two and three, also the derivation of the topological expansion significantly differs in dimension two and three. The complete expansion for the H 1 cost functional directly follows from the analysis of the variation of the state equation. However, the proof of the asymptotics of the L 2 tracking-type cost functional is significantly more involved and, surprisingly, the asymptotic behaviour of the bi-harmonic equation plays a crucial role in our proof. [ABSTRACT FROM AUTHOR]

The aim of the article is to create a method for studying the asymptotics of solutions to second-order differential equations with irregular singularities. The method allows us to prove the convergence of formal series included in the asymptotics of solutions for a wide class of second-order differential equations in the neighborhoods of their irregular singular points, including the point at infinity, which is generally irregular. The article provides a number of applications of the method for studying the asymptotics of solutions to both ordinary differential equations and partial differential equations. [ABSTRACT FROM AUTHOR]

U-statistics are fundamental in modeling statistical measures that involve responses from multiple subjects. They generalize the concept of the empirical mean of a random variable X to include summations over each m-tuple of distinct observations of X. W. Stute introduced conditional U-statistics, extending the Nadaraya–Watson estimates for regression functions. Stute demonstrated their strong pointwise consistency with the conditional expectation r (m) (φ , t) , defined as E [ φ (Y 1 , ... , Y m) | (X 1 , ... , X m) = t ] for t ∈ X m . This paper focuses on estimating functional single index (FSI) conditional U-processes for regular time series data. We propose a novel, automatic, and location-adaptive procedure for estimating these processes based on k-Nearest Neighbor (kNN) principles. Our asymptotic analysis includes data-driven neighbor selection, making the method highly practical. The local nature of the kNN approach improves predictive power compared to traditional kernel estimates. Additionally, we establish new uniform results in bandwidth selection for kernel estimates in FSI conditional U-processes, including almost complete convergence rates and weak convergence under general conditions. These results apply to both bounded and unbounded function classes, satisfying certain moment conditions, and are proven under standard Vapnik–Chervonenkis structural conditions and mild model assumptions. Furthermore, we demonstrate uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship. This result is independently valuable and has potential applications in areas such as set-indexed conditional U-statistics, the Kendall rank correlation coefficient, and discrimination problems. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the mathematical properties of methods based on a fictitious domain approach combined with reduced-order interface coupling conditions, which have been recently introduced to simulate 3D-1D fluid-structure or structure-structure coupled problems. To give insights on the approximation properties of these methods, we investigate them in a simplified setting by considering the Poisson problem in a two-dimensional domain with non-homogeneous Dirichlet boundary conditions on small inclusions. The approximated reduced problem is obtained using a fictitious domain approach combined with a projection on a Fourier finite-dimensional space of the Lagrange multiplier associated to the Dirichlet boundary constraints, obtaining in this way a Poisson problem with defective interface conditions. After analyzing the existence of a solution of the reduced problem, we prove its convergence towards the original full problem, when the size of the holes tends towards zero, with a rate which depends on the number of modes of the finite-dimensional space. In particular, our estimates highlight the fact that to obtain a good convergence on the Lagrange multiplier, one needs to consider more modes than the first Fourier mode (constant mode). This is a key issue when one wants to deal with real coupled problems, such as fluid-structure problems for instance. Next, the numerical discretization of the reduced problem using the finite element method is analyzed in the case where the computational mesh does not fit the small inclusion interface. As is standard for these types of problem, the convergence of the solution is not optimal due to the lack of regularity of the solution. Moreover, convergence exhibits a well-known locking effect when the mesh size and the inclusion size are of the same order of magnitude. This locking effect is more apparent when increasing the number of modes and affects the Lagrange multiplier convergence rate more heavily. To resolve these issues, we propose and analyze a stabilized method and an enriched method for which additional basis functions are added without changing the approximation space of the Lagrange multiplier. Finally, the properties of numerical strategies are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

The author addresses the question, under which conditions an active jammer will be successful in corrupting a wireless communication link, when both the desired link and the jamming link are subject to multipath signal fading. To this end, a simple closed‐form analytical expression is derived for the resulting block‐error probability (BLEP) under Nakagami‐m$m$ fading with in general unequal fading parameters for the desired and the jamming link. As a by‐product, a novel expression for the corresponding probability of a capacity outage is obtained. An asymptotic analysis and numerical evaluation of the BLEP expression show that in the single‐antenna case the jammer will indeed be successful in corrupting the wireless link, unless there is a rather high signal‐to‐noise ratio (SNR) advantage for the desired link. This even holds true, when signal fading for the jamming link is significantly more severe than for the desired link. Finally, the case of multiple antennas is considered and the benefits of array processing and diversity reception at the desired receiver are explored. It is found that receive array processing can significantly improve the resulting BLEP, provided that the desired link is characterized by favorable fading conditions. On the other hand, diversity reception can significantly improve the BLEP, if the desired link has favorable SNR conditions. [ABSTRACT FROM AUTHOR]

Reconfigurable intelligent surfaces (RISs), which consist of numerous passive reflecting elements, have emerged as a prominent technology to enhance energy and spectral efficiency for future wireless networks. RISs have the capability to intelligently reconfigure the incident wave, reflecting it towards the intended target without requiring energy for signal processing. Consequently, they have become a promising solution to support the demand for high-throughput satellite communication (SatCom) and enhanced coverage for areas inaccessible to terrestrial networks. This paper presents an asymptotic analysis of an RIS-assisted SatCom system. In this system, an unmanned aerial vehicle equipped with an RIS operates as a mobile reflector between a satellite and users. In particular, a passive beamformer is designed with the aim of asymptotically attaining optimal performance, considering the limitations imposed by practical SatCom systems. Moreover, the closed-form expressions for the ergodic achievable rate and outage probability are derived considering the proposed passive beamforming technique. Furthermore, we extend the system model to a multicast system and asymptotically analyze the optimality of the proposed scheme, leveraging the derived asymptotic results in the unicast system. The results of the simulations confirm that our analyses can precisely and analytically assess the performance of the RIS-assisted SatCom system, confirming the asymptotic optimality of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model. In this paper, we derive the effective model describing a thin-domain flow with permeable boundary through which the fluid is injected into the domain. We start with incompressible Stokes system and perform the rigorous asymptotic analysis. Choosing the appropriate scaling for the injection leads to a compressible effective model. [ABSTRACT FROM AUTHOR]

In this article, densities (and their derivatives) of subordinators and inverse subordinators are considered. Under minor restrictions, generally milder than the existing in the literature, employing a useful modification of the saddle point method, we obtain the large asymptotic behaviour of these densities (and their derivatives) for a specific region of space and time and quantify how the ratio between time and space affects the explicit speed of convergence. The asymptotics is governed by an exponential term depending on the Laplace exponent of the subordinator and the region represents the behaviour of the subordinator when it is atypically small (the inverse one is larger than usual). As a result, a route to the derivation of novel general or particular fine estimates for densities with explicit constants in the speed of convergence in the region of the lower envelope/the law of iterated logarithm is available. Furthermore, under mild conditions, we provide a power series representation for densities (and their derivatives) of subordinators and inverse subordinators. This representation is explicit and based on the derivatives of the convolution of the tails of the corresponding Lévy measure, whose smoothness is also investigated. In this context, the methods adopted are based on Laplace inversion and strongly rely on the theory of Bernstein functions extended to the cut complex plane. As a result, smoothness properties of densities (and their derivatives) and their behaviour near zero immediately follow. [ABSTRACT FROM AUTHOR]

Reinforcement Learning holds great promise for data-driven decision-making in various social contexts, including healthcare, education, and business. However, classical methods that focus on the mean of the total return may yield misleading results when dealing with heterogeneous populations typically found in large-scale datasets. To address this issue, we introduce the K-Value Heterogeneous Markov Decision Process, a framework designed to handle sequential decision problems with latent population heterogeneity. Within this framework, we propose auto-clustered policy evaluation for estimating the value of a given policy and auto-clustered policy iteration for estimating the optimal policy within a parametric policy class. Our auto-clustered algorithms can automatically identify homogeneous subpopulations while simultaneously estimating the action value function and the optimal policy for each subgroup. We establish convergence rates and construct confidence intervals for the estimators. Simulation results support our theoretical findings, and an empirical study conducted on a real medical dataset confirms the presence of value heterogeneity and validates the advantages of our novel approach. for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

This article develops a unified and computationally efficient method for change-point estimation along the time dimension in a nonstationary spatio-temporal process. By modeling a nonstationary spatio-temporal process as a piecewise stationary spatio-temporal process, we consider simultaneous estimation of the number and locations of change-points, and model parameters in each segment. A composite likelihood-based criterion is developed for change-point and parameter estimation. Under the framework of increasing domain asymptotics, theoretical results including consistency and distribution of the estimators are derived under mild conditions. In contrast to classical results in fixed dimensional time series that the localization error of change-point estimator is O p (1) , exact recovery of true change-points is possible in the spatio-temporal setting. More surprisingly, the consistency of change-point estimation can be achieved without any penalty term in the criterion function. In addition, we further establish consistency of the change-point estimator under the infill asymptotics framework where the time domain is increasing while the spatial sampling domain is fixed. A computationally efficient pruned dynamic programming algorithm is developed for the challenging criterion optimization problem. Extensive simulation studies and an application to the U.S. precipitation data are provided to demonstrate the effectiveness and practicality of the proposed method. for this article are available online. [ABSTRACT FROM AUTHOR]

Huber regression (HR) is a popular flexible alternative to the least squares regression when the error follows a heavy-tailed distribution. We propose a new method called the enveloped Huber regression (EHR) by considering the envelope assumption that there exists some subspace of the predictors that has no association with the response, which is referred to as the immaterial part. More efficient estimation is achieved via the removal of the immaterial part. Different from the envelope least squares (ENV) model whose estimation is based on maximum normal likelihood, the estimation of the EHR model is through Generalized Method of Moments. The asymptotic normality of the EHR estimator is established, and it is shown that EHR is more efficient than HR. Moreover, EHR is more efficient than ENV when the error distribution is heavy-tailed, while maintaining a small efficiency loss when the error distribution is normal. Moreover, our theory also covers the heteroscedastic case in which the error may depend on the covariates. The envelope dimension in EHR is a tuning parameter to be determined by the data in practice. We further propose a novel generalized information criterion (GIC) for dimension selection and establish its consistency. Extensive simulation studies confirm the messages from our theory. EHR is further illustrated on a real dataset. for this article are available online. [ABSTRACT FROM AUTHOR]

Nested simulation concerns estimating functionals of a conditional expectation via simulation. In this paper, we propose a new method based on kernel ridge regression to exploit the smoothness of the conditional expectation as a function of the multidimensional conditioning variable. Asymptotic analysis shows that the proposed method can effectively alleviate the curse of dimensionality on the convergence rate as the simulation budget increases, provided that the conditional expectation is sufficiently smooth. The smoothness bridges the gap between the cubic root convergence rate (that is, the optimal rate for the standard nested simulation) and the square root convergence rate (that is, the canonical rate for the standard Monte Carlo simulation). We demonstrate the performance of the proposed method via numerical examples from portfolio risk management and input uncertainty quantification. This paper was accepted by Baris Ata, stochastic models and simulation. Funding: The authors acknowledge financial support from the National Natural Science Foundation of China [Grant NSFC 12101149] and the Hong Kong Research Grants Council [Grants GRF 17201520 and GRF 17206821]. Supplemental Material: The e-companion and data files are available at https://doi.org/10.1287/mnsc.2022.00204. [ABSTRACT FROM AUTHOR]

We consider a two-player finite horizon linear–quadratic Stackelberg differential game. For this game, we study the case where the control cost of a leader in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. We look for open-loop optimal players' controls of this game. Using the game's solvability conditions, the obtaining such controls is reduced to the solution to a proper boundary-value problem. Due to the smallness of the leader's control cost, this boundary-value problem is singularly perturbed. Asymptotic behavior of the solution to this problem is analyzed. Based on this analysis, the asymptotic behavior of the open-loop optimal players' controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players' controls are designed. An illustrative example is presented. [ABSTRACT FROM AUTHOR]

In cluster analysis one often finds several partitions of a data set using different clustering methods and algorithms set with a variety of hyperparameters and tunings. The number of clusters K is one of the most relevant of such hyperparameters. Cluster selection is the task of choosing the desired partitions. The Bootstrap Quadratic Scoring is a recently introduced method where the cluster selection is performed by optimizing a score attached to a partition that is based on the quadratic discriminant function. Previously, we proposed the estimation of this cluster score via bootstrap resampling and investigated the proposed estimator based on numerical experiments and real data applications. However, that earlier work did not provide theoretical guarantees. In this paper, we fill that gap. We study the asymptotic behavior of the scoring method and show that the proposed estimator converges to well-defined population counterparts. [ABSTRACT FROM AUTHOR]

We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a highly efficient computational model compared to the current cycle-by-cycle simulation strategy to compute the viscoelastic responses of asphalt concretes and pavement structures under large numbers of load cycles. An explicit constitutive relation for viscoelastic solids in multiple time scales was developed based on the temporal homogenization. The original initial-boundary value problem was divided into a global part in the slow time scale and a local part in the fast time scale. Two simulation studies were presented to validate the computational accuracy and efficiency of the proposed model: (1) a cylindrical asphalt concrete subject to a uniaxial cyclic compression load, and (2) a pavement structure subject to a locally cyclic loading. The laboratory test results and field measurements were compared with the modeled responses to validate the models before comparing with the reference solutions. Results indicate that the temporal homogenization-based viscoelastic model saves considerable computational cost and maintains a satisfactory accuracy. The absolute values of relative error of the modeled responses between the time homogenization and reference solutions are lower than 1% and 4% for the cylindrical asphalt concrete and pavement structure under locally cyclic loadings, respectively. Based on the proposed computational approach, only 4 min are needed to model the response of a cylindrical asphalt concrete subject to 104 repeated load cycles under a uniaxial compression load. The computational time is reduced from 7 h of the reference solution to 38 min of the temporal homogenization solution to model 103 load cycles of a viscoelastic pavement structure. Practical Applications: This study introduces a highly efficient computational model to compute the viscoelastic responses of asphalt concretes and pavement structures under large numbers of load cycles. Multiple time scales were applied to an explicit constitutive relation of asphalt concretes to obtain the formula of global and local initial-boundary value problems. Its computational accuracy and efficiency were verified by comparing the temporal homogenization-based solutions with the testing results and reference solutions for a cylindrical sample and a pavement structure. It is the basis for the long-term pavement performance prediction after including the fatigue damage analysis in the present temporal homogenization framework. By successfully implementing a mechanistic framework for the long-term pavement performance prediction, the pavement design can more rely on the material inherent properties instead of using redundant empirical transfer functions. It is promising that the local calibrations of the current empirical performance transfer functions can be minimized as the proposed pavement long-term performance predictions depend on the material inherent properties and constitutive models. [ABSTRACT FROM AUTHOR]

In this paper, we consider the product ... of n independent identically distributed gamma random variables ... n. We derive an asymptotic formula for the survival probability P(n > x) as x --> θwith the first two remaining terms. [ABSTRACT FROM AUTHOR]

We used the method of matched asymptotic expansions to examine the behavior of the Giesekus fluid near to the static contact line singularity in extrudate swell. This shear-thinning viscoelastic fluid had a solution structure in which the solvent stresses dominated the polymer stresses near to the singularity. As such, the stress singularity was Newtonian dominated, but required viscoelastic stress boundary layers to fully resolve the solution at both the die wall and free surface. The sizes and mechanism of the boundary layers at the two surfaces were different. We gave a similarity solution for the boundary layer at the die wall and derived the exact solution for the boundary layer at the free-surface. The local behavior for the shape of the free-surface was also derived, which we showed was primarily determined by the solvent stress. However, the angle of separation of the free surface was determined by the the global flow geometry. It was this which determined the stress singularity and then in turn the free-surface shape. [ABSTRACT FROM AUTHOR]

The asymptotic behavior for a heriditary recursion x 1 > a and x n + 1 = 1 n s ∑ k = 1 n f (x k k) for every n ≥ 1 is studied, where f is decreasing, continuous on (a , ∞) ( a < 0), and twice differentiable at 0. The result has been known for the case s = 1. This paper analyzes the case s > 1. We obtain an asymptotic sequence that is quite different from the case s = 1. Some examples and applications are provided. [ABSTRACT FROM AUTHOR]

Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not critical according to our classification. Furthermore, we gave two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in [13]. [ABSTRACT FROM AUTHOR]

Introduction: The SIR (Susceptible-Infected-Recovered) model is one of the simplest and most widely used frameworks for understanding epidemic outbreaks. Methods: A second-order dynamical system for the R variable is formulated using an infinite exponential series expansion, and a recursion relation is established between the series coefficients. A numerical approximation scheme for the R variable is also developed. Results: The proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model's time constants, and an excitation timescale, dictated by the population size. The DE solution is applied to estimate model parameters for a well-known epidemiological dataset—the boarding school flu outbreak. Discussion: From a theoretical standpoint, the primary contribution of this work is the derivation of an infinite exponential, Dirichlet, series for the model variables. Truncating the series yields a finite approximation, known as a Prony series, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale. This apparent complexity can be approximated well by the DE solution, which appears to be of main practical interest. [ABSTRACT FROM AUTHOR]

In "Real-Time Spatial–Intertemporal Pricing and Relocation in a Ride-Hailing Network: Near-Optimal Policies and The Value of Dynamic Pricing," Chen, Lei, and Jasin consider a dynamic pricing problem faced by a ride-hailing service provider who manages a fixed number of servers and serves price-sensitive customers within a network. Servers serve arriving customers by relocating from the requested origins to destinations within a certain travel time. The authors first propose a static pricing policy based on the optimal solution to a deterministic relaxation of the original stochastic problem. They show that the proposed static policy matches the best possible asymptotic performance of any static policy. The authors further propose a dynamic pricing policy that adaptively changes the prices in a way that reduces the impact of past demand randomness on the balance of future distributions of servers and customers across the network. They show that the dynamic pricing policy achieves significantly better asymptotical performance. The proposed policies and their performance guarantees are further extended to a case where the firm jointly decides the relocation of vacant servers Motivated by the growth of ride-hailing services in urban areas, we study a (tactical) real-time spatial–intertemporal dynamic pricing problem where a firm uses a pool of homogeneous servers (e.g., a fleet of taxis) to serve price-sensitive customers (i.e., a rider requesting a trip from an origin to a destination) within a finite horizon (e.g., a day). We consider a revenue maximization problem in a model that captures the stochastic and nonstationary nature of demands, and the nonnegligible travel time from one location to another location. We first show that the relative revenue loss of any static pricing policy is at least in the order of n − 1 / 2 in a large system regime where the demand arrival rate and the number of servers scale linearly with n, which highlights the limitation of static pricing control. We also propose a static pricing control with a matching performance (up to a multiplicative logarithmic term). Next, we develop a novel state-dependent dynamic pricing control with a reduced relative revenue loss of order n − 2 / 3 . The key idea is to dynamically adjust the prices in a way that reduces the impact of past "errors" on the balance of future distributions of servers and customers across the network. Our extensive numerical studies using both a synthetic data set and a real data set from the New York City Taxi and Limousine Commission, confirm our theoretical findings and highlight the benefit of dynamic pricing over static pricing, especially when dealing with nonstationary demands. Interestingly, we also observe that the revenue improvement under our proposed policy primarily comes from an increase in the number of customers served instead of from an increase in the average prices compared with the static pricing policy. This suggests that dynamic pricing can be potentially used to simultaneously increase both revenue and the number of customers served (i.e., service level). Finally, as an extension, we discuss how to generalize the proposed policy to a setting where the firm can also actively relocate some of the available servers to different locations in the network in addition to implementing dynamic pricing. Funding: Y. (M.) Lei was partially supported by the Natural Sciences and Engineering Research Council of Canada [Fund 1378108, Grant RGPIN-2021-02973]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2425. [ABSTRACT FROM AUTHOR]

Problem definition: We study customer service chat (CSC) systems, in which agents can serve multiple customers simultaneously, with generally distributed service and patience times. The multitasking capability of agents introduces idiosyncratic challenges when making routing and staffing decisions. Methodology/results: To determine the dynamic matching of arriving customers with available agents, we first formulate a routing linear program (LP) based on system primitives. Inspired by the optimal solution of the routing LP, we design a parsimonious dynamic routing policy that is independent of arrival rate and service capacity information. We also use the optimal solution to develop closed-form approximations for crucial performance metrics and show that a similar LP can be utilized to make staffing decisions. Through extensive simulation experiments, we showcase the efficacy of our approximations and staffing decisions. Furthermore, under our proposed policy, the CSC system exhibits a unique stationary fluid model in which the steady-state performance measures align with our approximations. Managerial implications: The extant literature primarily focuses on Markovian systems with exponential distributions. In this paper, customers' service and patience times are allowed to be generally distributed to agree with practical settings. Our findings indicate that the distributions have a significant impact on routing policies, staffing decisions, and system performance. Funding: Z. Long was supported by the National Natural Science Foundation of China [Grants 72101112, 72132005, and 72271119] and Jiangsu Province, China [Grant BK20210171]. J. Zhang was supported by the Hong Kong Research Grants Council, General Research Fund [Grants 16208120 and 16214121]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0114. [ABSTRACT FROM AUTHOR]

The subject of mathematical study and modelling in this paper is an inbound call center that receives calls initiated by customers. A closed exponential queueing network with customer retrials and impatient customers is used as a stochastic model of call processing. A brief review of published results on the application of queueing models in the mathematical modeling of customer service processes in call centers is discussed. The network model is described. The possible customer states, customer routing, parameters, and customer service features are given. The allocation of customers by network nodes at a fixed time fully describes the situation in the call center at that time. The state of the network model under study is represented by a continuous-time Markov chain on finite state space. The model is studied in the asymptotic case under the critical assumption of a large number of customers in the queueing network. The mathematical approach used makes it possible to use the passage to the limit from a Markov chain to a continuous-state Markov process. It is proved that the probability density function of the model state process satisfies the Fokker – Planck – Kolmogorov equation. Using the drift coefficients of the Fokker – Planck – Kolmogorov equation, a system of ordinary differential equations for calculating the expected number of customers in each network node over time can be written. The solution of this system allows for predicting the dynamics of the expected number of customers at the model nodes and regulating the parameters of the call center operation. The asymptotic technique used is applicable both in transient and steady states. The areas of implementation of research results are the design of call centers and the analysis of their workload.

The flow of a fluid through a pipe subjected to heating is studied. The fluid is governed by the pressure drop, and the heat exchange between the fluid and the environment is described by Newton's cooling law. The longitudinal heat expansion of the pipe is taken into account. Error estimates for the asymptotic approximation in the extended domain are derived. [ABSTRACT FROM AUTHOR]

This paper reports the analytical results on the non-isothermal stationary fluid flow inside thin vertical annular region formed by two co-axial cylinders. The annulus is packed with the fluid-saturated sparsely packed porous medium which is cooled through the side wall. The flow is governed by the prescribed pressure drop between the top and bottom walls which are maintained at uniform, but different temperatures. The main objective of this work is to propose the approximate model describing the effective flow using rigorous asymptotic analysis with respect to the thickness of the annular region. Starting from the dimensionless Darcy-Brinkman-Boussinesq system endowed with the appropriate boundary conditions, we derive the explicit asymptotic approximation clearly showing the effects of the porous structure and thermal transfer. We also provide the theoretical error analysis in order to indicate the order of accuracy of the proposed model and justify its usage. [ABSTRACT FROM AUTHOR]

This paper deals with the solution and asymptotic analysis for a porous-elastic system with two dynamic control boundary conditions of fractional derivative type. We consider an augmented model. The energy function is presented, and the dissipative property of the system is established. We use the semigroup theory. The existence and uniqueness of the solution are obtained by applying the well-known Lumer-Phillips Theorem. We present two results for the asymptotic behavior: Strong stability of the C0-semigroup associated with the system using Arendt-Batty and Lyubich-Vũ's general criterion and polynomial stability applying Borichev-Tomilov's Theorem. [ABSTRACT FROM AUTHOR]

We consider an elastic body surrounded by thin elastic layers along a part of its boundary. We study the asymptotic behavior of the structure as the maximum thickness of the layers tends to zero.We derive an effective boundary integral energy involving a matrix of Borel measures not charging polar sets and having the same support contained in the boundary. We characterize this matrix for three special cases: periodic layers, layers which are determined by a given nonnegative function h, and layers with abrupt changes along self similar fractals. We then consider an optimal control problem, which consists in determining the shape of the best material distribution around the elastic body, under the maximal work of external loads, and characterize the optimal zones on its boundary where possible elastic layers could take place. [ABSTRACT FROM AUTHOR]

This work deals with the solution and asymptotic analysis for a porous-elastic system with internal damping of the fractional derivative type. We consider an augmented model. The energy function is presented and establishes the dissipativity property of the system. We use the semigroup theory. The existence and uniqueness of the solution are obtained by applying the well-known Lumer-Phillips Theorem. We present two results for the asymptotic behavior: Strong stability of the C 0 -semigroup associated with the system using Arendt-Batty and Lyubich-Vũ's general criterion and polynomial stability applying Borichev and Tomilov's Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we present a new model as an alternative to the classical 3D Richards model for the description of water flow in shallow aquifers. The new model is designed to achieve three goals. First, it provides a good approximation to the Richards model over a wide range of time scales. More specifically we show that both models characterize a flow with the same dominant components when the ratio of the horizontal length to the depth of the aquifer is small. Second, the new model accurately describes the velocity field. In particular, it is not based on the Dupuit hypothesis, which is often used in the context of shallow aquifers. This allows the new model to be well accurate even in the presence of wells and in aquifers with variable bedrock. Third, the new model can be viewed as a coupling of numerous 1D vertical Richards problems with a 2D elliptic one. In practice, this coupling is treated numerically using a Picard fixed point scheme, avoiding the need to solve any three-dimensional problem. This results in a significant reduction in computational cost compared to solving the 3D Richards problem directly. [ABSTRACT FROM AUTHOR]

Asymptotic analysis for an elastic layer under light fluid loading was developed. The ratio of fluid and solid densities was chosen as the main small parameter determining a novel scaling. The leading- and next-order approximations were derived from the full dispersion relation corresponding to long-wave, low-frequency, antisymmetric motions. The asymptotic plate models, including the equations of motion and the impenetrability condition, motivated by the aforementioned shortened dispersion equations, were derived for a plane-strain setup. The key findings included, in particular, the necessity of taking into account transverse plate inertia at the leading order, which is not the case for heavy fluid loading. In addition, the transverse shear deformation, rotation inertia, and a number of other corrections appeared at the next order, contrary to the previous asymptotic developments for fluid-loaded plates not assuming a light fluid loading scenario. [ABSTRACT FROM AUTHOR]

IntroductionThe SIR (Susceptible-Infected-Recovered) model is one of the simplest and most widely used frameworks for understanding epidemic outbreaks.MethodsA second-order dynamical system for the R variable is formulated using an infinite exponential series expansion, and a recursion relation is established between the series coefficients. A numerical approximation scheme for the R variable is also developed.ResultsThe proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model’s time constants, and an excitation timescale, dictated by the population size. The DE solution is applied to estimate model parameters for a well-known epidemiological dataset—the boarding school flu outbreak.DiscussionFrom a theoretical standpoint, the primary contribution of this work is the derivation of an infinite exponential, Dirichlet, series for the model variables. Truncating the series yields a finite approximation, known as a Prony series, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale. This apparent complexity can be approximated well by the DE solution, which appears to be of main practical interest.

This work deals with wave propagation into a coaxial cable, which can be modelled by the 3D Maxwell equations or 1D simplified models. The usual model, called the telegrapher's model, is a 1D wave equation of the electrical voltage and current. We derived a more accurate model from the Maxwell equations that takes into account dispersive effects. These two models aim to be a good approximation of the 3D electromagnetic fields in the case where the thickness of the cable is small. We perform some numerical simulations of the 3D Maxwell equations and of the 1D simplified models in order to validate the usual model and the new one. Moreover, we show that, while the usual telegrapher model is of order one with respect to the thickness of the cable, the dispersive 1D model is of order two.

An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.

The classical time-harmonic plane strain problem for a fluid-loaded cylindrical elastic shell is revisited. The results of the low-frequency asymptotic analysis, including explicit formulae for eigenfrequencies, are presented. A refined version of the semi-membrane shell theory is formulated. It is shown that the shell inertia does not affect significantly the lowest eigenfrequencies. It is also demonstrated that the ring stress component has a parabolic linear variation. [ABSTRACT FROM AUTHOR]

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter δ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as δ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model. [ABSTRACT FROM AUTHOR]

Being able to characterise objects at low frequencies, but in situations where the modelling error in the eddy current approximation of the Maxwell system becomes large, is important for improving current metal detection technologies. Importantly, the modelling error becomes large as the frequency increases, but the accuracy of the eddy current model also depends on the object topology and on its materials, with the error being much larger for certain geometries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds not only for frequencies where the eddy current model is valid but also for situations where the eddy current modelling error becomes large and applying the eddy approximation would be invalid. The leading-order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects. [ABSTRACT FROM AUTHOR]

This work deals with wave propagation into a coaxial cable, which can be modelled by the 3D Maxwell equations or 1D simplified models. The usual model, called the telegrapher's model, is a 1D wave equation of the electrical voltage and current. We derived a more accurate model from the Maxwell equations that takes into account dispersive effects. These two models aim to be a good approximation of the 3D electromagnetic fields in the case where the thickness of the cable is small. We perform some numerical simulations of the 3D Maxwell equations and of the 1D simplified models in order to validate the usual model and the new one. Moreover, we show that, while the usual telegrapher model is of order one with respect to the thickness of the cable, the dispersive 1D model is of order two. [ABSTRACT FROM AUTHOR]

The main aim of this article is to study the Poisson type problem for anisotropic p-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo p-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension. [ABSTRACT FROM AUTHOR]

An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint. [ABSTRACT FROM AUTHOR]

Problem definition: Designing effective operational strategies requires a good understanding of customer behavior. The classic economic theory of customer choice has long been the paradigm in the operations literature. However, the rise of online marketplaces such as e-commerce has triggered considerable efforts in academia and industry to develop alternative models that not only provide a good approximation of customer behavior but also are easily scalable for large-scale implementations. In this paper, we consider a multiproduct dynamic pricing problem with limited inventories under the so-called cascade click model, which is one of the most popular click models used in practice and has been intensively studied in the computer science literature. Methodology/results: We present some fundamental results. First, we derive a sufficiently general characterization of the optimal pricing policy and show that it has a different structure than the optimal policy under the standard pricing model. Second, we show that the optimal expected total revenue under the cascade click model can be upper bounded by the objective value of an approximate deterministic pricing problem. Third, we show that two policies that are known to have strong performance guarantees in the standard revenue management setting can be properly adapted (in a nontrivial way) to the setting with cascade click model while retaining their strong performance. Finally, we also briefly discuss the joint ranking and pricing problem and provide an iterative heuristic to calculate an approximate ranking. Managerial implications: Taking into account customers' click-and-search behavior leads to different structures of the optimal pricing policy, and some common insights under the standard pricing models may no longer hold. Moreover, our simulation studies show that pricing under a (misspecified) classic choice model that is oblivious to customers click-and-search behavior can severely impact profitability. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.0504. [ABSTRACT FROM AUTHOR]