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We present an inference scheme of long timescale, non-exponential kinetics from molecular dynamics simulations accelerated by stochastic resetting. Standard simulations provide valuable insight into chemical processes but are limited to timescales shorter than ∼ 1 μ s. Slower processes require the use of enhanced sampling methods to expedite them and inference schemes to obtain the unbiased kinetics. However, most kinetics inference schemes assume an underlying exponential first-passage time distribution and are inappropriate for other distributions, e.g., with a power-law decay. We propose an inference scheme that is designed for such cases, based on simulations enhanced by stochastic resetting. We show that resetting promotes enhanced sampling of the first-passage time distribution at short timescales but often also provides sufficient information to estimate the long-time asymptotics, which allows the kinetics inference. We apply our method to a model system and a peptide in an explicit solvent, successfully estimating the unbiased mean first-passage time while accelerating the sampling by more than an order of magnitude. [ABSTRACT FROM AUTHOR]

A word w = w 1 ⋯ w n over the set of positive integers is a Motzkin word whenever w 1 = 1 , 1 ≤ w k ≤ w k − 1 + 1 , and w k − 1 ≠ w k for k = 2 , ... , n. It can be associated to a n -column Motzkin polyomino whose i -th column contains w i cells, and all columns are bottom-justified. We reveal bijective connections between Motzkin paths, restricted Catalan words, primitive Łukasiewicz paths, and Motzkin polyominoes. Using the aforementioned bijections together with classical one-to-one correspondence with Dyck paths avoiding U D U s, we provide generating functions with respect to the length, area, semiperimeter, value of the last symbol, and number of interior points of Motzkin polyominoes. We give asymptotics and closed-form expressions for the total area, total semiperimeter, sum of the last symbol values, and total number of interior points over all Motzkin polyominoes of a given length. We also present and prove an engaging trinomial relation concerning the number of cells lying at different levels and first terms of the expanded (1 + x + x 2) n. [ABSTRACT FROM AUTHOR]

In this paper, our attention is given to the semidiscrete Gardner equation, which is a discrete analogue of the continuous Gardner equation describing the long wave propagation in a two-layer fluid. Firstly, a generalized (n , N - n) -fold Darboux transformation (DT) for this semi-discrete Gardner equation is constructed based on its recognized Lax pair. Secondly, the resulting DT is used to obtain exact solutions including ordinary soliton, rational soliton (RS) and their hybrid solutions within the non-zero seed background, and then analyze their asymptotic states as well as physical characteristics. Numerical simulations are also carried out to exhibit the dynamic characteristics of certain exact solutions. Thirdly, the soliton surface corresponding to this semi-discrete equation is investigated. Finally, employing continuum limit theory, we map the semi-discrete equation to the continuous equation, and obtain corresponding continuum limit for its Lax pair and DT. The findings given in this paper are conducive to a more profound understanding of the physical properties depicted by this equation. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ | u | α u , where 0 < α < 2 , and λ is a complex constant satisfying Im λ > α | Re λ | 2 α + 1 . For arbitrary large initial data, we present the uniform time decay estimates when 4 / 3 ≤ α < 2 , and the large time asymptotics of the solution when 7 + 145 12 < α < 2 . The proof is based on the vector fields method and a semiclassical analysis method. [ABSTRACT FROM AUTHOR]

The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a fractional Laplacian and a variable coefficient wave number μ , as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix sequences, with the main novelty regarding a complete picture of the case where μ = μ (x , y) is nonconstant. We report numerical evidence supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed. [ABSTRACT FROM AUTHOR]

In this article, we investigate the precise asymptotics for complete convergence and complete moment convergence for the spatial averages of the solution to the stochastic heat equation over a Euclidean ball as the radius of the ball diverges to infinity. Some general results on precise asymptotics are obtained, which can describe the relations among the boundary function, weighted function, convergence rate, and limit value. [ABSTRACT FROM AUTHOR]

We quantify the coupling asymptotics for the Lyapunov exponent of the Schrödinger cocycle over strongly expanding maps $ x \mapsto bx\ (\mod 1) $ x ↦ bx (mod 1) on $ \mathbb {T} $ T , for a large class of potential functions. We refine the previous lower bound results for these Lyapunov exponents, to get asymptotic results for all $ E \in \mathbb {R} $ E ∈ R and for sufficiently expanding circle maps. [ABSTRACT FROM AUTHOR]

In his seminal 1976 paper, Pósa showed that for all p\geq C\log n/n, the binomial random graph G(n,p) is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well is it typically possible to cover all edges of G(n,p) with Hamilton cycles? How many cycles are necessary? In this paper we show that for p\geq C\log n/n, we can cover G\sim G(n,p) with precisely \lceil \Delta (G)/2\rceil Hamilton cycles. Our result is clearly best possible both in terms of the number of required cycles, and the asymptotics of the edge probability p, since it starts working at the weak threshold needed for Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szabó [Random Structures Algorithms 44 (2014), pp. 183–200] and improves upon previous work of Hefetz, Kühn, Lapinskas and Osthus [Combinatorica 34 (2014), pp. 573–596], and of Ferber, Kronenberg and Long [Israel J. Math. 220 (2017), pp. 57–87], essentially closing a long line of research on Hamiltonian packing and covering problems in random graphs. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to study a non-local scalar field equation in bounded domains. More precisely we consider the semi-linear fractional elliptic problem: \[ (-\Delta)^{s} u = \lambda u^{p-1}-u^{m-1}\hbox{ in } \Omega,\, u>0 \hbox{ in } \Omega, \hbox{ and } u= 0 \hbox{ in }\mathbb{R}^N\setminus \Omega,\] (− Δ) s u = λ u p − 1 − u m − 1 in Ω , u > 0 in Ω , and u = 0 in R N ∖ Ω , where Ω is a regular bounded domain of $ \mathbb {R}^N $ R N , m>p>2 and $ \lambda >0 $ λ > 0. We discuss the existence, non-existence, and multiplicity of solutions, for the largest possible range of the parameters $ \lambda,\,p,\, m $ λ , p , m. Optimal results are obtained in the sub-critical and critical regimes, and partial results are obtained in the super-critical case. Furthermore, a careful asymptotic analysis when m tends to ∞ will be provided and we identify the limiting profiles as solutions to a free boundary type problem with a nonlinear right-hand side. [ABSTRACT FROM AUTHOR]

In this paper, we explore the applications of Tail Variance (TV) as a measure of tail riskiness and the confidence level of using Tail Conditional Expectation (TCE)-based risk capital. While TCE measures the expected loss of a risk that exceeds a certain threshold, TV measures the variability of risk along its tails. We first derive analytical formulas of TV and TCE for a large variety of probability distributions. These formulas are useful instruments for relevant research works on tail risk measures. We then propose a distribution-free approach utilizing TV to estimate the lower bounds of the confidence level of using TCE-based risk capital. In doing so, we introduce sharpened conditional probability inequalities, which halve the bounds of conventional Markov and Cantelli inequalities. Such an approach is easy to implement. We further investigate the characterization of tail risks by TV through an exploration of TV's asymptotics. A distribution-free limit formula is derived for the asymptotics of TV. To further investigate the asymptotic properties, we consider two broad distribution families defined on tails, namely, the polynomial-tailed distributions and the exponential-tailed distributions. The two distribution families are found to exhibit an asymptotic equivalence between TV and the reciprocal square of the hazard rate. We also establish asymptotic relationships between TCE and VaR for the two families. Our asymptotic analysis contributes to the existing research by unifying the asymptotic expressions and the convergence rate of TV for Student-t distributions, exponential distributions, and normal distributions, which complements the discussion on the convergence rate of univariate cases in [28]. To show the usefulness of our results, we present two case studies based on real data from the industry. We first show how to use conditional inequalities to assess the confidence of using TCE-based risk capital for different types of insurance businesses. Then, for financial data, we provide alternative evidence for the relationship between the data frequency and the tail categorization by the asymptotics of TV. [ABSTRACT FROM AUTHOR]

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata's Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes. [ABSTRACT FROM AUTHOR]

We prove specific biases in the number of occurrences of parts belonging to two different residue classes a and b, modulo a fixed nonnegative integer m, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size n that belong to these sets of partitions and have a symmetric residue class bias (i.e., for 1 ≤ a < m / 2 and b = m - a ), as n tends to infinity. [ABSTRACT FROM AUTHOR]

AbstractDetermining an appropriate sample size is a critical element of study design, and the method used to determine it should be consistent with the planned analysis. When the planned analysis involves Bayes factor hypothesis testing, the sample size is usually desired to ensure a sufficiently high probability of obtaining a Bayes factor indicating compelling evidence for a hypothesis, given that the hypothesis is true. In practice, Bayes factor sample size determination is typically performed using computationally intensive Monte Carlo simulation. Here, we summarize alternative approaches that enable sample size determination without simulation. We show how, under approximate normality assumptions, sample sizes can be determined numerically, and provide the R package bfpwr for this purpose. Additionally, we identify conditions under which sample sizes can even be determined in closed-form, resulting in novel, easy-to-use formulas that also help foster intuition, enable asymptotic analysis, and can also be used for hybrid Bayesian/likelihoodist design. Furthermore, we show how power and sample size can be computed without simulation for more complex analysis priors, such as Jeffreys-Zellner-Siow priors or non-local normal moment priors. Case studies from medicine and psychology illustrate how researchers can use our methods to design informative yet cost-efficient studies. [ABSTRACT FROM AUTHOR]

We study the semiclassical distribution of resonances of a 2 × 2 matrix Schrödinger operator (1.1) near a fixed energy level where the underlying classical trajectories Γ1 of P1 and Γ2 of P2 are, respectively, periodic and non-trapping. The aim is to compute the imaginary part of the resonances appearing near the eigenvalues created by P1 when Γ1 intersects Γ2 with finite contact order m. Recent results [M. Assal, S. Fujiié and K. Higuchi, Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories, Int. Math. Res. Not. 2024 (2024) 6879–6905] and [M. Assal, S. Fujiié and K. Higuchi, Transition of the semiclassical resonance widths across a tangential crossing energy-level, J. Math. Pures Appl. 191 (2024) 103634] assert that the width of resonances is of polynomial order in the semiclassical parameter with exponent 1 + 2/(m + 1), if the interaction U = r0(x) + ihr1(x)Dx is elliptic at the crossing point. Here, we remove this ellipticity assumption and prove that the exponent is 1 + 2(k + 1)/(m + 1), where k is a “vanishing order” of the interaction at the crossing points, suitably defined in terms of the vanishing orders of r0 and r1 depending on whether the crossing point is a turning point or not. [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: Cancer is a collective name for a group of diseases consisting of dozens of different types of malignant tumors, characterized by rapid and uncontrolled proliferation of cells in the body. Cancer can start almost anywhere in the human body such as the breast, prostate, colorectal, brain, bones, lungs, bladder etc. The main differences between the different types of cancer are related to the organ in which the tumor develops and the type of cells that compose the tumor. Method: This paper focused on the breast cancer. Breast cancer is a malignant tumor that originates in the breast tissue. It is the most common malignant tumor in women. There are several types of breast cancer, but in all types early diagnosis and treatment is crucial. In this study, the treatment of breast cancer involving a combination of two drugs was investigated: the oral estrogen receptor inhibitor AZD9496 and the CDK4/6 protein inhibitor Palbociclib. The mathematical model that described the interaction between the cancer cells, the treatment, and the immune system cells includes a system of nonlinear ordinary differential equations of the firs order. In general, dynamic variables of a given system change each at a different rate. And it is not possible to know from the mathematical model which variable is fast and which is slow. Therefore, in order to reveal the hierarchy of the system of equations ,a numerical algorithm called the singularly perturbed vector field (SPVF) was applied. This algorithm transform the mathematical model to a new coordinate system in which the rate of change of each dynamic variable of the system can be known. Results and Discussion: After writing the mathematical model in new coordinates, the equilibrium point was obtained analytically. The stability of the equilibrium points is investigated, which is essential from a practical perspective. Investigating the stability of the equilibrium points allows determination of when the tumor does not continue to develop and thereby allows adjustment of treatment continuation. [ABSTRACT FROM AUTHOR]

Our goal is to prove the Euler–Maclaurin summation formula using only the Taylor formula. Furthermore, the Euler–Maclaurin summation formula will be considered for the case of functions of the class C 2 k ([ a , b ]) . A stronger version of the estimation of the rest for functions of class C n ([ a , b ]) is given. We will also present the application of the obtained Euler–Maclaurin summation formulas to determine the asymptotics of Riemann sums for uniform partitions of intervals of integration. This paper also introduces the concepts of the asymptotic smoothing of the graph of a given function, as well as the asymptotic uniform distribution of the positive and negative values of the given function (generalizing the concept of the symmetric distribution of these values with respect to the x-axis, as in the case of the Peano–Jordan measure). [ABSTRACT FROM AUTHOR]

We give an analytic description for the infinitesimal generator constructed in Applebaum and Estrade (Ann Probab 28(1):166-184, 2000) for Lévy flights on a broad class of closed Riemannian manifolds including all negatively-curved manifolds, the flat torus and the sphere. Various properties of the associated semigroup and the asymptotics of the expected stopping time for Lévy flight based random searches for small targets, also known as the "narrow capture problem", are then obtained using our newfound understanding of the infinitesimal generator. Our study also relates to the Lévy flight foraging hypothesis in the field of biology as we compute the expected time for finding a small target by using the Lévy flight random search. Compared to the random search time for Brownian motion on surfaces done in Nursultanov et al. (arXiv:2209.12425), our result suggests that Lévy flight may not always be the optimal strategy, consistent with the conclusion obtained in Palyulin et al. (Proc Natl Acad Sci 111(8):2931-2936, 2014) for the one dimensional case. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating an unknown parameter vector θ ∈ R n , given noisy observations Y = θ θ T / n + Z of the rank-one matrix θ θ T , where Z has independent Gaussian entries. When information is available about the distribution of the entries of θ , spectral methods are known to be strictly sub-optimal. Past work characterized the asymptotics of the accuracy achieved by the optimal estimator. However, no polynomial-time estimator is known that achieves this accuracy. It has been conjectured that this statistical-computation gap is fundamental, and moreover that the optimal accuracy achievable by polynomial-time estimators coincides with the accuracy achieved by certain approximate message passing (AMP) algorithms. We provide evidence towards this conjecture by proving that no estimator in the (broader) class of constant-degree polynomials can surpass AMP. [ABSTRACT FROM AUTHOR]

In this work, a non-integer-order epidemic system modelling the nonlinear dynamics of Ebola virus disease is formulated in the sense of Caputofractional derivative. The existence and uniqueness of solution of the model is established. More importantly, the theoretical analysis carried out is aimed at establishing the local and global asymptotic stability properties of the disease-free steady state of the epidemic model using the fractional Routh-Hurwitz criterion and Lyapunov functional technique, respectively. It is proved that the steady state is locally and globally asymptotically stable at the value of the key epidemiological threshold quantity lower than unity. The result is numerically validated for different values of fractional order to show the asymptotic behavior of the disease dynamics. This result is significant for fighting and preventing Ebola epidemic in the population, since the Caputo derivative operator allows for effective description of the disease dynamics with memory, where the future evolution of the disease is governed by its prior history. [ABSTRACT FROM AUTHOR]

In this work we analytically investigate the alignment mechanism of self-propelled ellipse-shaped cells in two spatial dimensions interacting via overlap avoidance. By considering a two-cell system and imposing certain symmetries, we obtain an analytically tractable dynamical system, which we mathematically analyse in detail. We find that for elongated cells there is a half-stable steady state corresponding to perfect alignment between the cells. Whether cells move towards this state (i.e., become perfectly aligned) or not is determined by where in state space the initial condition lies. We find that a separatrix splits the state space into two regions, which characterise these two different outcomes. We find that some self-propulsion is necessary to achieve perfect alignment, however too much self-propulsion hinders alignment. Analysing the effect of small amounts of self-propulsion offers an insight into the timescales at play when a trajectory is moving towards the point of perfect alignment. We find that the two cells initially move apart to avoid overlap over a fast timescale, and then the presence of self-propulsion causes them to move towards a configuration of perfect alignment over a much slower timescale. Overall, our analysis highlights how the interaction between self-propulsion and overlap avoidance is sufficient to generate alignment. [ABSTRACT FROM AUTHOR]

In this article, we consider two opposite and intergitated 2D-combs, each one with ε-periodically distributed fingers, each finger having cross-section of order ε and fixed height. Via an asymptotic analysis based on the two-scale convergence method, we study, as ε vanishes, the electrical potential in the vacuum between these two combs when one comb is grounded while the voltage in the other one is assumed of order ε, and we prove a corrector result. [ABSTRACT FROM AUTHOR]

The search for one-step alternatives to the generalized method of moment (GMM) has identified broad classes of potential estimators such as generalized empirical likelihoods (GEL), empirical cressie-read (ECR), exponentially tilted empirical likelihood (ETEL), and minimum discrepancy (MD) estimators. While empirical likelihood (EL) dominates other ECR estimators in terms of higher-order asymptotics, it lacks robustness to model misspecification. ETEL was shown to combine higher-order efficiency and robustness to misspecification but demands strong moment generating function existence conditions. We show, both theoretically and via simulations, how to achieve the same goal under weaker moment existence conditions within the class of MD estimators. [ABSTRACT FROM AUTHOR]

We prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end, we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with kth$k\text{th}$‐order remainders that satisfy uniform Ck$C^k$‐estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for k=0$k = 0$ known from previous work of the second‐named author. For k>0$k > 0$, the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder. [ABSTRACT FROM AUTHOR]

This research explores refined boundary conditions for a traction-free surface in a non-local micropolar half-space, combining non-local and micropolar elasticity effects to study Rayleigh wave propagation in an isotropic, homogeneous medium. This study revisits the solution for Rayleigh waves obtained within the framework of Eringen's non-local differential model. It highlights that the equivalence between the non-local differential and integral formulations breaks down for a micropolar half-space and can only be restored under specific additional boundary conditions. For mathematical tractability, equivalence is assumed for a defined subset of stresses. Asymptotic analysis is further employed to capture the effects of the boundary layer within the non-local micropolar half-space. This technique finally derives the refined boundary conditions and as an application to Rayleigh wave propagation problem yields two distinct non-local corrected dispersion relations, with one mode unique to the micropolar characteristics. Phase velocity curves are plotted to describe the dispersion in Rayleigh waves propagating in various modes. Sensitivity analysis reveals the dependence of phase velocity on the non-local and micropolar parameters with Poisson's ratio, thus providing valuable insights for material design and wave propagation control. [ABSTRACT FROM AUTHOR]

Forced low-frequency vibrations of a functionally graded thin-walled cylindrical shell submerged in a compressible fluid are investigated. The problem undergoes asymptotic analysis aimed at elucidating smallest resonances. The associated explicit estimates are derived, including the formulae for the imaginary quantities corresponding to the radiation of vibration energy. The robust approximate relations for stress and displacement components are also obtained. Numerical results are presented for two particular setups demonstrating the effect of transverse inhomogeneity on the dynamic behavior of the FGM cylinder. [ABSTRACT FROM AUTHOR]

This paper concerns the asymptotics of certain parabolic–elliptic chemotaxis-consumption systems with logistic growth and constant concentration of chemoattractant on the boundary. First we prove that in two dimensional bounded domains there exists a unique global classical solution which is uniformly bounded in time, and then, we show that if the concentration of chemoattractant on the boundary is sufficiently low, then the solution converges to the positive steady state as time goes to infinity. [ABSTRACT FROM AUTHOR]

Motivated by the questions posed by W. V. Li and A. Wei [Proc. Amer. Math. Soc. 137 (2009), pp. 195–204] and the conjecture of E. Lundberg and A. Thomack [ On the average number of zeros of random harmonic polynomials with iid coefficients: precise asymptotics, Preprint, https://arxiv.org/ abs/2308.10333, 2023], we study the expected number of zeros of random harmonic polynomials H_{n,m}(z)= p_{n}(z)+\overline {q_{m}(z)} with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is O(n) when \deg p = \alpha \deg q, where 0\leq \alpha <1. This result extends the previous estimates when m is a fixed constant or m=n to more general case. [ABSTRACT FROM AUTHOR]

First introduced by Société Générale Corporate and Investment Banking in 2007, timer options are financial instruments whose payoffs rely on a random date of the exercise related to the realized variance of the underlying asset. This is contrary to vanilla options exercised at a fixed expiration date. However, option holders are vulnerable to credit risks arising from the uncertainty that counterparties may not implement their contractual obligation, particularly in the over-the-counter market. Hence, in this article, motivated by the credit risk model proposed by Johnson and Stulz (JFinac 42:281–300, 1987), we deal with the pricing of the timer option considering the counterparty default risk by utilizing the technique of asymptotic analysis. Moreover, we investigate the pricing accuracy of our analytic formulas, comparing them with the solutions from the Monte Carlo method, and examine the impact of stochastic volatility on the credit risk or the variance budget on the option value based on our pricing formula for vulnerable timer options. [ABSTRACT FROM AUTHOR]

We introduce a new approach for comparing the predictive accuracy of two nested models that bypasses the difficulties caused by the degeneracy of the asymptotic variance of forecast error loss differentials used in the construction of commonly used predictive comparison statistics. Our approach continues to rely on the out of sample mean squared error loss differentials between the two competing models, leads to nuisance parameter-free Gaussian asymptotics, and is shown to remain valid under flexible assumptions that can accommodate heteroskedasticity and the presence of mixed predictors (e.g., stationary and local to unit root). A local power analysis also establishes their ability to detect departures from the null in both stationary and persistent settings. Simulations calibrated to common economic and financial applications indicate that our methods have strong power with good size control across commonly encountered sample sizes. [ABSTRACT FROM AUTHOR]

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by x t ∈ ( 1 3 , 1) , in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV. [ABSTRACT FROM AUTHOR]

We propose a model uncertainty approach to heavy traffic asymptotics that allows for a high level of uncertainty. That is, the uncertainty classes of underlying distributions accommodate disturbances that are of order 1 at the usual diffusion scale as opposed to asymptotically vanishing disturbances studied previously in relation to heavy traffic. A main advantage of the approach is that the invariance principle underlying diffusion limits makes it possible to define uncertainty classes in terms of the first two moments only. The model we consider is a single-server queue with multiple job types. The problem is formulated as a zero sum stochastic game played between the system controller, who determines scheduling and attempts to minimize an expected linear holding cost, and an adversary, who dynamically controls the service time distributions of arriving jobs and attempts to maximize the cost. The heavy traffic asymptotics of the game are fully solved. It is shown that an asymptotically optimal policy for the system controller is to prioritize according to an index rule, and for the adversary, it is to select distributions based on the system's current workload. The workload-to-distribution feedback mapping is determined by a Hamilton–Jacobi–Bellman equation, which also characterizes the game's limit value. Unlike in the vast majority of results in the heavy traffic theory and as a direct consequence of the diffusive size disturbances, the limiting dynamics under asymptotically optimal play are captured by a stochastic differential equation where both the drift and the diffusion coefficients may be discontinuous. Funding: R. Atar is supported by the Israeli Science Foundation [Grant 1035/20]. [ABSTRACT FROM AUTHOR]

We prove a sample-path large deviation principle (LDP) with sublinear speed for unbounded functionals of certain Markov chains induced by the Lindley recursion. The LDP holds in the Skorokhod space D[0,1] equipped with the M1′ topology. Our technique hinges on a suitable decomposition of the Markov chain in terms of regeneration cycles. Each regeneration cycle denotes the area accumulated during the busy period of the reflected random walk. We prove a large deviation principle for the area under the busy period of the Markov random walk, and we show that it exhibits a heavy-tailed behavior. Funding: The research of B. Zwart and M. Bazhba is supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 639.033.413]. The research of J. Blanchet is supported by the National Science Foundation (NSF) [Grants 1915967, 1820942, and 1838576] as well as the Defense Advanced Research Projects Agency [Grant N660011824028]. The research of C.-H. Rhee is supported by the NSF [Grant CMMI-2146530]. [ABSTRACT FROM AUTHOR]