Suppose that m drivers each choose a preferred parking space in a linear car park with n spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $m=n$.

We investigate various probabilistic properties of a uniform parking function. Through a combinatorial construction termed a parking function multi-shuffle, we give a formula for the law of multiple coordinates in the generic situation $m \lesssim n$. We further deduce all possible covariances: between two coordinates, between a coordinate and an unattempted spot, and between two unattempted spots. This asymptotic scenario in the generic situation $m \lesssim n$ is in sharp contrast with that of the special situation $m=n$.

A generalization of parking functions called interval parking functions is also studied, in which each driver is willing to park only in a fixed interval of spots. We construct a family of bijections between interval parking functions with n cars and n spots and edge-labeled spanning trees with $n+1$ vertices and a specified root.

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$ , which appear in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study the second shifted difference of partitions, $f(\,j,n) := p(n) -2p(n-j) +p(n-2j)$ , and give another easy-to-use estimate of $f(\,j,n)$ . As applications of these, we prove a shifted convexity property of $p(n)$ , as well as giving new estimates of the k-rank partition function $N_k(m,n)$ and non-k-ary partitions along with their differences.

The self-interaction force of dislocation curves in metals depends on the local arrangement of the atoms and on the non-local interaction between dislocation curve segments. While these non-local segment–segment interactions can be accurately described by linear elasticity when the segments are further apart than the atomic scale of size $\varepsilon$ , this model breaks down and blows up when the segments are $O(\varepsilon)$ apart. To separate the non-local interactions from the local contribution, various models depending on $\varepsilon$ have been constructed to account for the non-local term. However, there are no quantitative comparisons available between these models. This paper makes such comparisons possible by expanding the self-interaction force in these models in $\varepsilon$ beyond the O(1)-term. Our derivation of these expansions relies on asymptotic analysis. The practical use of these expansions is demonstrated by developing numerical schemes for them, and by – for the first time – bounding the corresponding discretisation error.

When a liquid fills the semi-infinite space between two concentric cylinders which rotate at different steady speeds, how about the shape of the free surface on top of the fluid? The different fluids will lead to a different shape. For the Newtonian fluid, the meniscus descends due to the centrifugal forces. However, for the certain non-Newtonian fluid, the meniscus climbs the internal cylinder. We want to explain the above phenomenon by a rigorous mathematical analysis theory. In the present paper, as the first step, we focus on the Newtonian fluid. This is a steady free boundary problem. We aim to establish the well-posedness of this problem. Furthermore, we prove the convergence of the formal perturbation series obtained by Joseph and Fosdick in Arch. Ration. Mech. Anal. 49 (1973), 321–380.

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.

We extend the existing small-time asymptotics for implied volatilities under the Heston stochastic volatility model to the multifactor volatility Heston model, which is also known as the Wishart multidimensional stochastic volatility model (WMSV). More explicitly, we show that the approaches taken in Forde and Jacquier (2009) and Forde, Jacqiuer and Lee (2012) are applicable to the WMSV model under mild conditions, and obtain explicit small-time expansions of implied volatilities.

We extend previous large deviations results for the randomised Heston model to the case of moderate deviations. The proofs involve the Gärtner–Ellis theorem and sharp large deviations tools.

Transform inversions, in which density and survival functions are computed from their associated moment generating function $\mathcal{M}$, have largely been based on methods which use values of $\mathcal{M}$ in its convergence region. Prominent among such methods are saddlepoint approximations and Fourier-series inversion methods, including the fast Fourier transform. In this paper we propose inversion methods which make use of values for $\mathcal{M}$ which lie outside of its convergence region and in its analytic continuation. We focus on the simplest and perhaps richest setting for applications in which $\mathcal{M}$ is either a meromorphic function in its analytic continuation, so that all of its singularities are poles, or else the singularities are isolated essential. Asymptotic expansions of finite- and infinite-orders are developed for density and survival functions using the poles of $\mathcal{M}$ in its analytic continuation. For finite-order expansions, the expansion error is a contour integral in the analytic continuation, which we approximate using the saddlepoint method based on following the path of steepest descent. Such saddlepoint error approximations accurately determine expansion errors and, thus, provide the means for determining the order of the expansion needed to achieve some preset accuracy. They also provide an additive correction term which increases accuracy of the expansion. Further accuracy is achieved by computing the expansion errors numerically using a contour path which ultimately tracks the steepest descent direction. Important applications include Wilks’ likelihood ratio test in MANOVA, compound distributions, and the Sparre Andersen and Cramér–Lundberg ruin models.

The Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

We elucidate the long-term behavior of failure rates for a broad class of frailty models in survival analysis. The class properly includes the proportional hazard frailty model, the additive frailty model, and the accelerated failure time frailty model. A complete asymptotic expansion is derived and compared with the corresponding result for the limiting behavior obtained by Finkelstein and Esaulova (2006a). Several examples are provided to facilitate the comparison and to illustrate both the applicability and the limitations of our approach.

We study the distribution of the discrete spectrumof the Schrödinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$.

In this paper we develop a prelimit analysis of performance measures for importance sampling schemes related to small noise diffusion processes. In importance sampling the performance of any change of measure is characterized by its second moment. For a given change of measure, we characterize the second moment of the corresponding estimator as the solution to a partial differential equation, which we analyze via a full asymptotic expansion with respect to the size of the noise and obtain a precise statement on its accuracy. The main correction term to the decay rate of the second moment solves a transport equation that can be solved explicitly. The asymptotic expansion that we obtain identifies the source of possible poor performance of nevertheless asymptotically optimal importance sampling schemes and allows for a more accurate comparison among competing importance sampling schemes.

We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta (1, b)-coalescent by exploiting the method of sequential approximations.

A continuum hydrodynamic model has been used to characterize flowing active nematics. The behavior of such a system subjected to a weak steady shear is analyzed. We explore the director structures and flow behaviors of the system in flow-aligning and flow tumbling regimes. Combining asymptotic analysis and numerical simulations, we extend previous studies to give a complete characterization of the steady states for both contractile and extensile particles in flow-aligning and flow-tumbling regimes. Another key prediction of this work is the role of the system size on the steady states of an active nematic system: if the system size is small, the velocity and the director angle files for both flow-tumbling contractile and extensile systems are similar to those of passive nematics; if the system is big, the velocity and the director angle files for flow-aligning contractile systems and tumbling extensile systems are akin to sheared passive cholesterics while they are oscillatory for flow-aligning extensile and tumbling contractile systems.

Aiming at the isoparametric bilinear finite volume element scheme, we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids. Furthermore, we prove that the approximate derivatives are convergent of order two. Finally, numerical examples verify the theoretical results.

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

In this article a variational reduction method, how to handle the case of heterogenousdomains for the Transport equation, is presented. This method allows to get rid of therestrictions on the size of time steps due to the thin parts of the domain. In the thinpart of the domain, only a differential problem, with respect to the space variable, is tobe approximated numerically. Numerical results are presented with a simple example. Thevariational reduction method can be extended to thin domains multi-branching in 3dimensions, which is a work in progress.

We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product.As an application we obtain expansions for P(a < Y < b) for Y ~ Np(0,V) andfor repeated integrals of the density of Y.When V−1y > 0 in R3 the expansion for P(Y < y) reduces toone given by [H. Ruben J. Res. Nat. Bureau Stand. B 68 (1964) 3–11]. in terms of the moments of Np(0,V−1).This is shown to be a special case of an expansion in terms of the multivariate Hermite polynomials.These are given explicitly.

We study the H –1-norm of the function 1 on tubular neighbourhoods of curves in ${\mathbb R}^{2}$ . We take the limit of small thickness ε, and we prove two different asymptotic results. The first is an asymptotic development for a fixed curve in the limit ε → 0, containing contributions from the length of the curve (at order ε 3), the ends (ε 4), and the curvature (ε 5). The second result is a Γ-convergence result, in which the central curve may vary along the sequence ε → 0. We prove that a rescaled version of the H –1-norm, which focuses on the ε 5 curvature term, Γ-converges to the L 2-norm of curvature. In addition, sequences along which the rescaled norm is bounded are compact in the W 1,2-topology. Our main tools are the maximum principle for elliptic equations and the use of appropriate trial functions in the variational characterisation of the H –1-norm. For the Γ-convergence result we use the theory of systems of curves without transverse crossings to handle potential intersections in the limit.