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We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient η, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when η → 0+ (small dissipation) and when η → +∞ (large dissipation). Surprisingly, at the limit η → +∞, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to η. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper, we corroborate the main theorems establishing the justification of Koiter's model, established by Ciarlet and his associates, for all the types of linearly elastic shells via a set of numerical experiments. [ABSTRACT FROM AUTHOR]

The effects of spatial heterogeneity on the dynamics of reaction-diffusion models have been studied extensively. In particular, global dynamics of general spatially heterogeneous (but temporally static) Lotka–Volterra competition-diffusion systems were completely clarified by He and Ni in 2016. However, the evolutionary impacts of temporal periodicity combined with spatial heterogeneity in population ecology remain a challenging issue. In this work, we consider a population model of two competing species in a both spatially varying and temporally periodic environment, where the two species only differ in their random dispersal rates but are otherwise ecologically identical. In a pioneering 2001 work on this model by Hutson et al., by constructing various choices of resource functions and dispersal rates of the two species, the authors demonstrated that all the following three types of dynamics are possible: (i) stable coexistence of the two species; (ii) the slower diffuser invades the faster one but not vice versa; (iii) the faster diffuser invades the slower one but not vice versa. This is in drastic contrast with the spatially heterogeneous but temporally static case, where Dockery et al. showed in 1998 that the slower diffuser always wipes out the faster one. In this paper, we completely and explicitly characterize the asymptotic stability of both semitrivial periodic solutions in terms of the two dispersal rates and the resource function, when either dispersal rate is sufficiently small or large. In particular, the direction of selection on the dispersal rate during the evolution can be elucidated in these instances. Some novel analytical methods are developed to investigate asymptotic behaviors of the underlying time-periodic parabolic eigenvalue problem and its adjoint problem. We hope that these methods are of independent interest in the area of time-periodic parabolic equations. [ABSTRACT FROM AUTHOR]

We present an effective model of "space-coiled metacrystals"" composed of a periodic array of sound rigid blocks into which long slots have been coiled up. The periodic cell of the block contains a coiled slot whose straight parts are at wavelength scale, which enables the appearance of Bragg resonances. These resonances, which prevent high transmission, compete with the Fabry--P\'erot resonances of the entire slot, which foster perfect transmission. This results in complex scattering properties driven by the characteristics of the turning regions that act as atoms in a onedimensional coiled crystal. Using appropriate scaling and combining two-scale homogenization with matched asymptotic techniques, the modeling of such metacrystals is proposed. The resulting model is validated through a comparison with full-wave numerics in both harmonic and transient regimes. [ABSTRACT FROM AUTHOR]

Rod and cone photoreceptors in the retina of vertebrates are the primary sensory neurons underlying vision. They convert light into an electrical current using a signal transduction pathway that depends on Ca 2 + feedback. It is known that manipulating the Ca 2 + kinetics affects the response shape and the photoreceptor sensitivity, but a precise quantification of these effects remains unclear. We have approached this task in mouse retina by combining numerical simulations with mathematical analysis. We consider a parsimonious phototransduction model that incorporates negative Ca 2 + feedback onto the synthesis of cyclic GMP, and fast buffering reactions to alter the Ca 2 + kinetics. We derive analytic results for the photoreceptor functioning in sufficiently dim light conditions depending on the photoreceptor type. We exploit these results to obtain conceptual and quantitative insight into how response waveform and amplitude depend on the underlying biophysical processes and the Ca 2 + feedback. With a low amount of buffering, the Ca 2 + concentration changes in proportion to the current, and responses to flashes of light are monophasic. With more buffering, the change in the Ca 2 + concentration becomes delayed with respect to the current, which gives rise to a damped oscillation and a biphasic waveform. This shows that biphasic responses are not necessarily a manifestation of slow buffering reactions. We obtain analytic approximations for the peak flash amplitude as a function of the light intensity, which shows how the photoreceptor sensitivity depends on the biophysical parameters. Finally, we study how changing the extracellular Ca 2 + concentration affects the response. [ABSTRACT FROM AUTHOR]

This paper reports the analytical results on the incompressible micropolar fluid flowing through a thin (or long) cylindrical pipe filled with porous medium. We start from the Brinkman-type system governing the filtration of the micropolar flow and perform the asymptotic analysis in the critical case characterized by the strong coupling between the velocity and microrotation. The error estimates are also derived providing the rigorous justification of the proposed effective model. [ABSTRACT FROM AUTHOR]

A generic way to verify asymptotic optimality of semi-open-loop policies for a wide class of MDPs with large lead times. In many real-life inventory models, order lead times can result in uncertain effects of inventory decisions. However, as the lead time grows large, one would naturally postulate that the effect of the delayed order depends weakly on the current inventory level and, thus, intuit that decoupling the delayed order with the current inventory level may provide good heuristics. Motivated by these examples, in "Asymptotic Optimality of Semi-open-Loop Policies in Markov Decision Processes with Large Lead Times," Bai et al. consider a generic Markov decision process (MDP) with one delayed control and one immediate control. For MDPs defined on general spaces with uniformly bounded cost functions and a fast mixing property, they construct a periodic semi-open-loop policy for each lead time value and show that these policies are asymptotically optimal as the lead time goes to infinity. For MDPs defined on Euclidean spaces with linear dynamics and convex structures, they impose another set of conditions under which constant delayed-control policies are asymptotically optimal. We consider a generic Markov decision process (MDP) with two controls: one control taking effect immediately and the other control whose effect is delayed by a positive lead time. As the lead time grows, one naturally expects that the effect of the delayed action only weakly depends on the current state, and decoupling the delayed action from the current state could provide good controls. The purpose of this paper is to substantiate this decoupling intuition by establishing asymptotic optimality of semi-open-loop policies, which specify open-loop controls for the delayed action and closed-loop controls for the immediate action. For MDPs defined on general spaces with uniformly bounded cost functions and a fast mixing property, we construct a periodic semi-open-loop policy for each lead time value and show that these policies are asymptotically optimal as the lead time goes to infinity. For MDPs defined on Euclidean spaces with linear dynamics and convex structures (convex cost functions and constraint sets), we impose another set of conditions under which semi-open-loop policies (actually, constant delayed-control policies) are asymptotically optimal. Moreover, we verify that these conditions hold for a broad class of inventory models, in which there are multiple controls with nonidentical lead times. Funding: Research of the first three authors was partly supported by the National Science Foundation [Grant CMMI-1635160]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2021.0088. [ABSTRACT FROM AUTHOR]

The linear stability of a compressible flow in a pipe is examined using a modal analysis. A steady fully developed flow of a calorifically perfect gas, driven by a constant body acceleration, in a pipe of circular cross section is perturbed by small-amplitude normal modes and the temporal stability of the system is studied. In contrast to the incompressible pipe flow that is linearly stable for all modal perturbations, the compressible flow is unstable at finite Mach numbers due to modes that do not have a counterpart in the incompressible limit. We obtain these higher modes for a pipe flow through numerical solution of the stability equations. The higher modes are distinguished into an "odd" and an "even" family based on the variation of their wave-speeds with wave-number. The classical theorems of stability are extended to cylindrical coordinates and are used to obtain the critical Mach numbers below which the higher modes are always stable. The critical Reynolds number is calculated as a function of Mach number for the even family of modes, which are the least stable at finite Mach numbers. The numerical solution of the stability equations in the high Reynolds number limit demonstrates that viscosity is essential for destabilizing the even family of modes. An asymptotic analysis is carried out at high Reynolds numbers to obtain the scalings, and solutions for the eigenvalues in the high Reynolds number limit for the lower and upper branches of the stability curve. [ABSTRACT FROM AUTHOR]

The objective of this work is to study the asymptotic justification of the two- dimensional equations for membrane shells with boundary conditions of von Kármán's type. More precisely, we consider a three-dimensional model for a nonlinearly elastic membrane shell of Saint Venant–Kirchhoff material, where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type. Using technics from formal asymptotic analysis with the thickness of the shell as a small parameter, we show that the scaled three-dimensional solution still leads to the so-called two-dimensional equations of von Kármán membrane shell. [ABSTRACT FROM AUTHOR]

The paper addresses the mathematical justification of a model describing steady vibrations for a planar thermoelastic body with an incorporated thin inclusion. The body is composed of three parts: two adherents and an adhesive layer between them, and we begin with a general mathematical formulation of a problem. By means of the modern methods of asymptotic analysis, we rigorously investigate the behavior of solutions as the thickness of the adhesive tends to zero. As a result, we construct the model that corresponds to the limit case. It turned out that the adhesive is reduced to the inclusion, which is thin (of zero thickness) and relatively hard (compared to the rigidity of the surrounding body). Furthermore, we supplement the obtained results with numerical experiments demonstrating the consistency of the theoretical conclusions. [ABSTRACT FROM AUTHOR]

Cocoa is an important crop that is predominantly grown in the western part of Africa. However, there have been fluctuations and declining trends in production and several factors have been identified to be responsible for this. A significant factor is the effect of climate variation, which could result in a low farm-level yield. Therefore, to understand the contribution of climate variability on the farm-level yield, we construct and analyse a time-delayed model to capture the effect of rainfall on cocoa production. This work uses a system of differential equations to model the crop transition from the flowering stage to pod formation, pod ripening and then to harvesting. We introduce a periodic forcing function into the model of flowering to account for the impact of seasonal rainfall variations. This leads to a novel nonlinear parametrically forced ODE for the flowering with periodically varying coefficients, which is coupled to a time-delayed model for the ripened pod formation and then harvesting. We perform an analysis of all parts of the system proving that it has a periodic solution when (parametrically) forced periodically, and we then conduct an asymptotic analysis on this periodic solution to show how its rich behaviour depends on the parameters of the climatic forcing in the model. [ABSTRACT FROM AUTHOR]

The dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis. [ABSTRACT FROM AUTHOR]

Probabilistic predictions for regression problems are more popular than point predictions and interval predictions, since they contain more information for test labels. Conformal predictive system is a recently proposed non-parametric method to do reliable probabilistic predictions, which is computationally inefficient due to its learning process. To build faster conformal predictive system and make full use of training data, this paper proposes the predictive system based on locally weighted jackknife prediction approach. The theoretical property of our proposed method is proved with some regularity assumptions in the asymptotic setting, which extends our earlier theoretical researches from interval predictions to probabilistic predictions. In the experimental section, our method is implemented based on our theoretical analysis and its comparison with other predictive systems is conducted using 20 public data sets. The continuous ranked probability scores of the predictive distributions and the performance of the derived prediction intervals are compared. The better performance of our proposed method is confirmed with Wilcoxon tests. The experimental results demonstrate that the predictive system we proposed is not only empirically valid, but also provides more information than the other comparison predictive systems. [ABSTRACT FROM AUTHOR]

In this paper, we show that the spectral problem associated to stretching modes in a thin folded plate can be derived from the three-dimensional eigenvalue problem of linear elasticity through a rigourous convergence analysis as the thickness of the plate goes to zero. We show, using a nonstandard asymptotic analysis technique, that each stretching frequency of an elastic thin folded plate is the limit of a family of high frequencies of the three-dimensional linearized elasticity system in the folded plate, as the thickness approaches zero. [ABSTRACT FROM AUTHOR]

This article provides an in-depth exploration of Big-Oh and small-oh notations, shedding light on their practical implications in the analysis of algorithm complexity. Big-Oh notation offers a valuable tool for estimating an upper bound on the growth rate of an algorithm's running time, whereas small-oh notation delineates a lower limit on this growth rate. The piece delves into a comprehensive examination of various complexity classes that emerge through the application of Big-Oh notation, underscoring the significance of small-oh notation as it complements and enriches complexity analysis. In the realm of programming and computer science, the employment of these notations holds paramount importance. They empower developers and researchers to make informed decisions regarding algorithm selection and optimization. It is crucial to recognize that while complexity analysis is a vital facet of effective programming, ongoing research endeavors may yield more refined methodologies and approaches within this domain. By understanding and harnessing the power of Big-Oh and small-oh notations, professionals can effectively evaluate algorithm efficiency and scalability. This knowledge equips them with the ability to design and implement algorithms that meet specific performance criteria, which is pivotal in the ever-evolving landscape of technology and computation. As pushing the boundaries of what is possible in the field of algorithm design is being continued, these notations remain invaluable tools for navigating the complex terrain of algorithmic analysis and optimization. By embracing Big-Oh and small-oh notations, professionals can finely assess algorithmic efficiency, ensuring they meet performance criteria in the evolving technological landscape. These notations remain indispensable for algorithmic analysis.

In this paper, we show that the spectral problem associated to stretching modes in a thin folded plate can be derived from the three-dimensional eigenvalue problem of linear elasticity through a rigourous convergence analysis as the thickness of the plate goes to zero. We show, using a nonstandard asymptotic analysis technique, that each stretching frequency of an elastic thin folded plate is the limit of a family of high frequencies of the three-dimensional linearized elasticity system in the folded plate, as the thickness approaches zero.

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.

In this paper, we show that the high frequency modes of a thin clamped plate and the associated eigenfunctions converge, as the thickness of the plate goes to zero, to the eigenvalues and the eigenfunctions of a two-dimensional eigenvalue problem associated to the stretching displacements of the plate.

The Cahn–Hilliard equation with degenerate mobility is used in several areas including the modeling of living tissues, following the theory of mixtures. We are interested in quantifying the pressure jump at the interface between phases in the case of incompressible flows. To do so, we depart from the spherically symmetric dynamical compressible model and include an external force. We prove existence of stationary states as limits of the parabolic problems. Then we prove the incompressible limit and characterize compactly supported stationary solutions. This allows us to compute the pressure jump in the small dispersion regime and in particular the force dependent curvature effect.

Abstract Probabilistic predictions for regression problems are more popular than point predictions and interval predictions, since they contain more information for test labels. Conformal predictive system is a recently proposed non-parametric method to do reliable probabilistic predictions, which is computationally inefficient due to its learning process. To build faster conformal predictive system and make full use of training data, this paper proposes the predictive system based on locally weighted jackknife prediction approach. The theoretical property of our proposed method is proved with some regularity assumptions in the asymptotic setting, which extends our earlier theoretical researches from interval predictions to probabilistic predictions. In the experimental section, our method is implemented based on our theoretical analysis and its comparison with other predictive systems is conducted using 20 public data sets. The continuous ranked probability scores of the predictive distributions and the performance of the derived prediction intervals are compared. The better performance of our proposed method is confirmed with Wilcoxon tests. The experimental results demonstrate that the predictive system we proposed is not only empirically valid, but also provides more information than the other comparison predictive systems.

In this work, we study the influence of orientation on the stability conditions of the reaction front where the monomer is solid and the polymer is liquid. The mathematical model includes the heat equation, the concentration equation and the Navier–Stokes equation under the Boussinesq approximation. We use the method proposed by Zeldovich and Frank-Kamenetskii to perform asymptotic analysis. We then perform a stability analysis. The linearized problem is solved numerically using a multiquadric radial basis function method (MQ-RBF) to find the stability boundary. This will allow us to deduce the influence of each control parameter of the problem on this stability, in particular the angle of inclination of the experimental tube. [ABSTRACT FROM AUTHOR]

We study the time-domain acoustic wave propagation in the presence of a microbubble. This microbubble is characterized by a mass density and bulk modulus which are both very small compared to the ones of the background vacuum. The goal is to estimate the amount of pressure that is created very near (at a distance proportional to the radius of the bubble) to the bubble. We show that, at that small distance, the dominating field is reminiscent of the wave created by a point-like obstacle modeled formally by a Dirac-like heterogeneity with support at the location of the bubble and the contrast between the bubble and background material as the scattering coefficient. As a conclusion, we can tune the bubble's material properties so that the pressure near it reaches a desired amount. Such a design might be useful for the purpose of acoustic cavitation, where one needs enough, but not too much, pressure to eliminate unwanted anomalies. The mathematical analysis is done using time-domain integral equations and asymptotic analysis techniques. A well known feature here is that the contrasting scales between the bubble and the background generate resonances (mainly the Minnaert one) in the time-harmonic regime. Such critical scales, and the generated resonances, are also reflected in the time-domain estimation of the acoustic wave. In particular, reaching the desired amount of pressure x'prime near the location of the bubble is possible only with such resonating bubbles. [ABSTRACT FROM AUTHOR]

In this paper, a finite-horizon two-person Nash equilibrium game for a linear time-dependent differential system with delays (point-wise and distributed) in the state variable and the players' control variables is considered. The behaviour of each player is evaluated by its own quadratic functional to be minimized by this player. The feature of this game is that a weight matrix of the control in the functional of one player is singular (but, in general, non-zero). Due to this feature, the game itself is singular. Using the regularization method and the asymptotic analysis of the regularized Nash equilibrium game, an open-loop Nash equilibrium solution to the considered singular game is derived. Illustrative example is presented. Along with this example, two examples on non-uniqueness of open-loop Nash equilibrium solutions to the singular game are presented. [ABSTRACT FROM AUTHOR]

Two-layer fluid models are used for investigating certain nonlinear phenomena in fluid mechanics, medical science and thermodynamics. This paper investigates the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation in a two-layer liquid. Gram-type solutions are derived via the Kadomtsev-Petviashvili hierarchy reduction. Based on the Gram-type solutions, three-breather solutions and three kinds of hybrid the solutions that behave as the interactions among the V-shaped soliton, kink soliton and breather are derived. Three-breather solutions describe that a breather splits into two breathers or two breathers fuse into a breather. For the three kinds of the hybrid solutions, asymptotic analyses indicate that: (1) the V-shaped soliton is constructed via the interaction between two kink solitons; (2) the V-shaped soliton reduces to a kink soliton at certain conditions; (3) certain interaction results in a decrease (or increase) in the constant background of the V-shaped soliton and kink soliton. Based on the asymptotic analysis for a set of the hybrid solutions, we find the hybrid solutions which describe that a breather splits into two breathers while the breathers are connected to the V-shaped soliton. Furthermore, we show the hybrid solutions which describe that three breathers fuse into the V-shaped soliton. [ABSTRACT FROM AUTHOR]

In this paper, the Sasa-Satsuma equation in fluid dynamics and nonlinear optics is investigated. Starting from the first-order Darboux transformation, we construct an N-fold generalized Darboux transformation (GDT), where N is a positive integer. Through the obtained N-fold GDT, we derive three kinds of the semirational solutions, which describe the second-order degenerate solitons, third-order degenerate solitons and interaction between the second-order degenerate solitons and one soliton, respectively. We graphically illustrate the above three kinds of semirational solutions and investigate them through the asymptotic analysis, from which we find that the characteristic lines of the semirational solutions are composed of the straight lines and curves. Expressions of the characteristic lines, positions, amplitudes, slopes, positions and phase shifts of the asymptotic solitons are presented through the asymptotic analysis. The above discussions might be extended to the higher-order solitons, and to the relevant analysis on the degenerate breathers. [ABSTRACT FROM AUTHOR]

This article provides an in-depth exploration of Big-Oh and small-oh notations, shedding light on their practical implications in the analysis of algorithm complexity. Big-Oh notation offers a valuable tool for estimating an upper bound on the growth rate of an algorithm's running time, whereas small-oh notation delineates a lower limit on this growth rate. The piece delves into a comprehensive examination of various complexity classes that emerge through the application of Big-Oh notation, underscoring the significance of small-oh notation as it complements and enriches complexity analysis. In the realm of programming and computer science, the employment of these notations holds paramount importance. They empower developers and researchers to make informed decisions regarding algorithm selection and optimization. It is crucial to recognize that while complexity analysis is a vital facet of effective programming, ongoing research endeavors may yield more refined methodologies and approaches within this domain. By understanding and harnessing the power of Big-Oh and small-oh notations, professionals can effectively evaluate algorithm efficiency and scalability. This knowledge equips them with the ability to design and implement algorithms that meet specific performance criteria, which is pivotal in the ever-evolving landscape of technology and computation. As pushing the boundaries of what is possible in the field of algorithm design is being continued, these notations remain invaluable tools for navigating the complex terrain of algorithmic analysis and optimization. By embracing Big-Oh and small-oh notations, professionals can finely assess algorithmic efficiency, ensuring they meet performance criteria in the evolving technological landscape. These notations remain indispensable for algorithmic analysis. [ABSTRACT FROM AUTHOR]

Operations Management in a Repair Facility Capital goods are machines or products that are used in the production of goods or service deliveries. Maintenance service providers keep spare parts on stock so that, whenever a critical component of a capital good breaks down, the broken part is replaced with a spare part to prevent long and costly downtime. In "Joint Inventory and Scheduling Control in a Repair Facility," Özkan and van Houtum study inventory and repair scheduling decisions of a maintenance service provider for repairable capital goods. In case of a stock-out, the service provider should decide whether to back-order the demand or execute an emergency repair, which is an urgent but expensive repair operation for a broken part. The authors derive a simple and intuitive decision rule stating if the emergency repairs are necessary to achieve a close-to-optimal system performance. Moreover, they propose a simple, intuitive, and easy-to-implement heuristic control policy that performs well in numerical experiments. We study inventory and repair scheduling decisions of a maintenance service provider for repairable capital goods. Because of high downtime costs, the service provider keeps spare parts on stock to replace broken parts quickly. The service provider should determine the inventory level of spare parts for each component and the repair scheduling policy. Furthermore, in case of a stock-out, the service provider should decide whether to back-order the demand or execute an emergency repair, which is an urgent but expensive repair operation for a broken part followed by a fast form of installation. The objective is to minimize the long-run average inventory holding, back-order, and emergency repair costs. We formulate the repairable network as a closed queueing system and consider an asymptotic regime in which the repair facility is in the conventional heavy-traffic regime. Then, we formulate and solve a Brownian control problem (BCP). From the optimal BCP solution, we derive a simple and intuitive decision rule stating if the emergency repairs are necessary to achieve a close-to-optimal system performance. Moreover, we propose a simple, intuitive, and easy-to-implement heuristic control policy and demonstrate its close-to-optimal performance via numerical experiments. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.2459. [ABSTRACT FROM AUTHOR]

We consider the diffusion asymptotics of a coupled model arising in radiative transfer in a unit disk in R 2 with one-speed velocity. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled a nonhomogeneous heat equation satisfied by the material temperature. For the well-prepared data, we prove the existence and the nonequilibrium diffusion limit of solutions to the initial boundary problem for the coupled model. [ABSTRACT FROM AUTHOR]

In the framework of 2D-elasticity, an equilibrium problem for an inhomogeneous body with a curvilinear inclusion located strictly inside the body is considered. The elastic properties of the inclusion are assumed to depend on a small positive parameter δ characterizing its width and are assumed to be proportional to δ − 1 . Moreover, it is supposed that the inclusion has a curvilinear rough boundary. Relying on the variational formulation of the equilibrium problem, we perform the asymptotic analysis, as δ tends to zero. As a result, a variational model of an elastic body containing a thin curvilinear rod is constructed. Numerical calculations give a relative error between the initial and limit problems depending on δ. [ABSTRACT FROM AUTHOR]

The SIR (susceptible–infected–recovered) is one of the simplest models for epidemic outbreaks. The present paper demonstrates the parametric solution of the model in terms of quadratures and derives a double exponential analytical asymptotic solution for the I-variable, which is valid on the entire real line. Moreover, the double exponential solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the parametric solution. A second, refined, asymptotic solution involving exponential gamma kernels was also demonstrated. The approach was applied to the coronavirus disease 2019 (COVID-19) pandemic in six European countries—Belgium, Italy, Sweden, France, Spain and Bulgaria in the period 2020-2021. [ABSTRACT FROM AUTHOR]

In this paper, we revisit large variable asymptotic expansions of tronquée solutions of the Painlevé I equation, obtained via the Riemann–Hilbert approach and the method of steepest descent. The explicit construction of an extra local parametrix around the recessive stationary point of the phase function, in terms of complementary error functions, makes it possible to give detailed information about exponential-type contributions beyond the standard Poincaré expansions for tronquée and tritronquée solutions. [ABSTRACT FROM AUTHOR]

Curved channels with constant cross sections are constructed which support a trapped surface wave. Since corresponding eigenvalues are embedded in the continuous spectrum of the water wave problem and therefore possess the natural instability, the construction procedure requires "fine-tuning" of several parameters in the (small) curvature of the channel as well as geometrical restrictions on the cross section. In particular, the mirror symmetry of the cross section with respect to the vertical axis disrupts the procedure, but examples of suitable non-symmetric cross sections are provided. [ABSTRACT FROM AUTHOR]

The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number,  R a c , and the wave number,  a c , are obtained in closed form. An asymptotic analysis is performed, to obtain  R a c  for the case of adiabatic boundaries, for which  a c ≃ 0 . A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known. [ABSTRACT FROM AUTHOR]

In this paper, we show that the high frequency modes of a thin clamped plate and the associated eigenfunctions converge, as the thickness of the plate goes to zero, to the eigenvalues and the eigenfunctions of a two-dimensional eigenvalue problem associated to the stretching displacements of the plate. [ABSTRACT FROM AUTHOR]

Ferromagnetic materials are considered to have the applications in data storage, data processing and telecommunication. A Kraenkel-Manna-Merle system, which describes the nonlinear electromagnetic short waves in a ferromagnetic saturator, is investigated in this paper. With respect to the magnetization related to the saturated ferromagnetic material and external magnetic field, a generalized Darboux transformation (GDT) is constructed and utilized to derive the solitons, multi-pole solitons and their interactions. Analytic expressions of the double-pole solitons are offered and analyzed via the asymptotic analysis. Then, amplitudes, characteristic lines, slopes and phase shifts of the asymptotic solitons are presented. With the multiple spectral parameters involved in the GDT, interactions among the solitons and multi-pole solitons are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we extend the definition of the qx-asymptotic functions, for an extended real-valued function defined on the infinite-dimensional topological normed spaces without lower semicontinuity or quasi-convexity condition. As the main result, by using some asymptotic conditions, we obtain sufficient optimality conditions for the existence of solutions to equilibrium problems, under weaker assumptions of continuity and convexity, when the feasible set is an unbounded subset of infinite-dimensional space. Also, as a corollary, we obtain necessary and sufficient optimality conditions for the existence of solutions to equilibrium problems with an unbounded feasible set. Finally, as an application, we establish a result for the existence of solutions to minimization problems. [ABSTRACT FROM AUTHOR]

We study a mathematical model of gene transcription and protein synthesis with negative feedback. We consider a system of equations taking into account the formation of dimers (i.e., complex formed by two protein monomers), the way in which dimers bind to DNA and time delay in translation process. For the model consisting of three ordinary differential equations with time delay, we derive conditions for stability of the positive steady state and for the existence of the Hopf bifurcation. We check the influence of delay in transcription process on the model dynamics, and we formulate conditions determining the type of the Hopf bifurcation and stability of arising limit cycle. [ABSTRACT FROM AUTHOR]

An asymptotic analysis is conducted of the effect surface vibrations may play in mitigating the resistance present in the system consisting of a fluid contained between a pair of parallel flat plates in relative motion. Previous numerical computations of this situation, which has applications in both biological and technological contexts, have suggested that small vibrations that move at a fast phase speed can lead to substantial reductions in the resistance. The effect seems to be a somewhat intricate function of the wavelength of the vibration modes. Our analysis uncovers the properties of the various phenomena that are at work and describes the interplay between them. The techniques employed here lend themselves to the extension to other situations including problems in which propulsion effects in fluid systems can be induced by vibrations. [ABSTRACT FROM AUTHOR]

The pressure rise caused by end-gas auto-ignition in spark-ignited engines is discussed using numerical simulation and theoretical approaches. The main objective of this study is to explain the mechanism by which the end-gas expansion during auto-ignition suppresses the pressure rise in spark-ignited engines, and theoretically to demonstrate using asymptotic analysis that the pressure rise depends on the Damköhler number. A one-dimensional direct numerical simulation (DNS) of end-gas auto-ignition is performed, and the modelling assumptions for it are discussed based on the DNS results. The Damköhler number, defined as the acoustic time scale and the characteristic time scale of the chemical reaction, is introduced in the modelling. The end-gas auto-ignition model is solved numerically, and it is shown that the pressure rise increases with the Damköhler number. Additionally, it is shown that the tendency of the pressure rise is due to the balance between the propagation rate of the expansion wave generated in the end gas and the reaction rate at auto-ignition, which varies with the Damköhler number. To derive the analytical solution of the relationship between the pressure rise and Damköhler number, the end-gas auto-ignition model is simplified based on the numerical results. The simplified model for end-gas auto-ignition is then solved using Newton's method, and the analytical solution of the pressure rise is derived. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of singularly perturbed convection-diffusion problems on one-dimensional pipe networks. In the vanishing diffusion limit, the number and type of boundary conditions and coupling conditions at network junctions change, which gives rise to singular layers at the outflow boundaries of the pipes. A hybrid discontinuous Galerkin method is proposed, which provides a natural upwind mechanism for the convection-dominated case. Moreover, the method provides a viable approximation for the limiting pure transport problem. A detailed analysis of the singularities of the solution and the discretization error is presented, and an adaptive strategy is proposed, leading to order optimal error estimates that hold uniformly in the singular perturbation limit. The theoretical results are confirmed by numerical tests. [ABSTRACT FROM AUTHOR]