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This article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi‐closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black–Scholes price for a barrier option with respect to the underlying price and computing the one‐dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte‐Carlo simulation and find the SEV effect on the Black–Scholes barrier option prices. [ABSTRACT FROM AUTHOR]

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1 / n and 1 / n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations. [ABSTRACT FROM AUTHOR]

The paper provides a precise error estimate for an asymptotic expansion of a certain stochastic control problem related to relative entropy minimization. In particular, it is shown that the expansion error depends on the regularity of functionals on path space. An efficient numerical scheme based on a weak approximation with Monte Carlo simulation is employed to implement the asymptotic expansion in multidimensional settings. Throughout numerical experiments, it is confirmed that the approximation error of the proposed scheme is consistent with the theoretical rate of convergence.

In this paper we study planar polynomial Kolmogorov's differential systems \[ X_\mu\quad\begin{cases}{\dot{x}=f(x,y;\mu),}\\{\dot{y}=g(x,y;\mu),} \end{cases} \] with the parameter $\mu$ varying in an open subset $\Lambda\subset\mathbb{R}^N$. Compactifying $X_\mu$ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle $\Gamma$, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all $\mu\in\Lambda.$ We are interested in the cyclicity of $\Gamma$ inside the family $\{X_\mu\}_{\mu\in\Lambda},$ i.e., the number of limit cycles that bifurcate from $\Gamma$ as we perturb $\mu.$ In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with $N=3$ and $N=5$, and in both cases we are able to determine the cyclicity of the polycycle for all $\mu\in\Lambda,$ including those parameters for which the return map along $\Gamma$ is the identity.

This paper considers a simplified three dimensional Ericksen-Leslie System for nematic liquid crystal flows in the unbounded domain $ \Omega: = \mathbb R^+\times \mathbb R^2 $ or the smooth bounded domain $ \Omega $. The hydrodynamic system consists of the Navier-Stokes type equations for the fluid velocity coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We first establish the global existence of Sobolev regular solution with finite energies in Sobolev space $ H^{s}(\Omega)\times H^{s}(\Omega) $, where the index $ s $ of the Sobolev space can be any large fixed integer, but $ s\neq+\infty $. Then we give an asymptotic expansions of a family of Sobolev regularity solutions for such system in $ \Omega $.

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. [ABSTRACT FROM AUTHOR]

In this paper, testing the equality of two mean vectors is considered when each dataset has a monotone pattern of missing observations. A simplified Hotelling's T2-type test statistic in a two-sample problem under monotone missing data is given. Transformed test statistics of this simplified T2 statistic based on Bartlett adjustment to improve the chi-squared approximation are derived. Furthermore, the accuracy of the approximations is investigated using a Monte Carlo simulation. Finally, the numerical power of the proposed test is presented. [ABSTRACT FROM AUTHOR]

We examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviours in semi-infinite domains. The first problem is that of the 'Sakiadis' boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air–liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution—given as distance from the wall as a function of meniscus height—has long been known (Batchelor, G. K. (1967) An Introduction to Fluid Dynamics, chapter 1: The physical properties of fluids. Cambridge). Here, we provide an explicit solution as meniscus height versus distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that—in both problems—the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviours to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviours can be useful when proposing variable transformations to overcome power series divergence. [ABSTRACT FROM AUTHOR]

This study proposes an artificial neural network (ANN) based option pricing framework under the SABR (stochastic alpha beta rho) and free boundary SABR volatility models. Unlike previous research, we do not directly apply the ANN technique to train and predict option implied volatilities. Instead, we use the technique to train and predict the differences between the implied volatilities calculated by asymptotic approximation and those computed by Monte Carlo simulation. Since the analytical solution for an option written on an underlying asset price in the SABR model is intractable, approximation techniques are used to derive an approximate closed-form solution in practice. However, these approximation formulas worsen as maturity increases and the underlying asset's volatility is high. The accuracy decreases rapidly in wings, which represents deep in-the-money and out-of-the-money cases. By combining the approximation formulas and ANN framework, our new option pricing method offers the following improvements: (1) the training becomes more robust, and the predictions produce more stable and accurate results: (2) it significantly speeds up the training procedure: and (3) the accuracy and efficiency of approximation are improved in the wings without sacrificing performance. [ABSTRACT FROM AUTHOR]

A positive radial singular solution for Δ u + f (u) = 0 with a general supercritical growth is constructed. An exact asymptotic expansion as well as its uniqueness in the space of radial functions are also established. These results can be applied to the bifurcation problem Δ u + λ f (u) = 0 on a ball. Our method can treat a wide class of nonlinearities in a unified way, e.g., u p log u , exp (u p) and exp (⋯ exp (u) ⋯) as well as u p and e u . Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]

In this paper, we study a learning problem in which a forecaster only observes partial information. By properly rescaling the problem, we heuristically derive a limiting PDE on Wasserstein space which characterizes the asymptotic behavior of the regret of the forecaster. Using a verification type argument, we show that the problem of obtaining regret bounds and efficient algorithms can be tackled by finding appropriate smooth sub/supersolutions of this parabolic PDE. [ABSTRACT FROM AUTHOR]

This paper characterizes those functions given by convergent q-Lidstone series expansion. We give the necessary and sufficient conditions so that the entire function f (z) has such an expansion, in which case convergence is uniform on compact sets. [ABSTRACT FROM AUTHOR]

We present a method to compute efficiently and easily solutions of systems of linear neutral delay differential equations with highly oscillatory forcing terms. This method is based on asymptotic expansions in inverse powers of a perturbed oscillatory parameter. Each term of the asymptotic expansion is derived by recursion. The cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided and show that with few terms of the asymptotic expansion, the solutions are approximated with high accuracy. [ABSTRACT FROM AUTHOR]

This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method's implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations. [ABSTRACT FROM AUTHOR]

. In this paper, we propose and validate a multiscale model for the description of particle diffusion in presence of trapping boundaries. We start from a drift-diffusion equation in which the drift term describes the effect of bubble traps and is modeled by a short-range potential with an attractive term and a repulsive core. The interaction of the particles attracted by the bubble surface is simulated by the Lennard--Jones potential that simplifies the capture due to the hydrophobic properties of the ions. In our model, the effect of the potential is replaced by a suitable boundary condition derived by mass conservation and asymptotic analysis. The potential is assumed to have a range of small size \varepsilon . An asymptotic expansion in the \varepsilon is considered, and the boundary conditions are obtained by retaining the lowest-order terms in the expansion. Another aspect we investigate is the saturation effect coming from high concentrations in the proximity of the bubble surface. The validity of the model is carefully checked with several tests in one and two dimensions and different geometries. [ABSTRACT FROM AUTHOR]

In this work, we propose a method for solving Kolmogorov hypoelliptic equations based on Fourier transform and Feynman–Kac formula. We first explain how the Feynman–Kac formula can be used to compute the fundamental solution to parabolic equations with linear or quadratic potential. Then applying these results after a Fourier transform we deduce the computation of the solution to a first class of Kolmogorov hypoelliptic equations. Then we solve partial differential equations obtained via Feynman–Kac formula from the Ornstein–Uhlenbeck generator. Also, a new small time approximation of the solution to a certain class of Kolmogorov hypoelliptic equations is provided. We finally present the results of numerical experiments to check the practical efficiency of this approximation. [ABSTRACT FROM AUTHOR]

Generally, it is well known that the constant elasticity of variance (CEV) model fails to capture the empirical results verifying that the implied volatility of equity options displays smile and skew curves at the same time. In this study, to overcome the limitation of the CEV model, we introduce a new model, which is a generalization of the CEV model, and show that it can capture the smile and skew effects of implied volatility. Using an asymptotic analysis for two small parameters that determine the volatility shape, we obtain approximated solutions for option prices in the extended model. In addition, we demonstrate the stability of the solution for the expansion of the option price. Furthermore, we show the convergence rate of the solutions in Monte‐Carlo simulation and compare our model with the CEV, Heston, and other extended stochastic volatility models to verify its flexibility and efficiency compared with these other models when fitting option data from the S&P 500 index. [ABSTRACT FROM AUTHOR]

In the paper, we apply a new approach to the comparison of the distributions of sums of random variables to the case of Poisson random sums. This approach was proposed in our previous work (Bening, Korolev, 2022) and is based on the concept of statistical deficiency. Here, we introduce a continuous analog of deficiency. In the case under consideration, by continuous deficiency, we will mean the difference between the parameter of the Poisson distribution of the number of summands in a Poisson random sum and that of the compound Poisson distribution providing the desired accuracy of the normal approximation. This approach is used for the solution of the problem of determination of the distribution of a separate term in the Poisson sum that provides the least possible value of the parameter of the Poisson distribution of the number of summands guaranteeing the prescribed value of the (1 − α) -quantile of the normalized Poisson sum for a given α ∈ (0 , 1) . This problem is solved under the condition that possible distributions of random summands possess coinciding first three moments. The approach under consideration is applied to the collective risk model in order to determine the distribution of insurance payments providing the least possible time that provides the prescribed Value-at-Risk. This approach is also used for the problem of comparison of the accuracy of approximation of the asymptotic (1 − α) -quantile of the sum of independent, identically distributed random variables with that of the accompanying infinitely divisible distribution. [ABSTRACT FROM AUTHOR]

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Independently and under the milder hypothesis that f is even and monotone over [0,π], matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: (a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; (b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is vastly better for the computation of the extreme eigenvalues; a mild deterioration in the quality of the numerical experiments is observed when dense Toeplitz matrices are considered, having generating function of low smoothness and not satisfying the simple-loop assumptions. [ABSTRACT FROM AUTHOR]

The discrete Bell distribution and its associated regression model were introduced recently in the statistical literature. The Bell distribution has proved to be a useful alternative to the traditional Poisson distribution, mainly to deal with overdispersion. Likelihood‐based inference on the Bell regression parameters relies on asymptotic assumptions like the sample size going to infinity. In this paper, we focus on the small‐sample case and consider higher order asymptotic refinements in this class of regression models. In particular, we derive the second‐order biases of the maximum likelihood estimators, which are used to define bias‐corrected estimators. The preventive method to bias reducing is also considered. We provide a simple matrix formula for the skewness of the distributions of the maximum likelihood estimators, and also derive a simple matrix formula for the second‐order variance–covariance matrix of the maximum likelihood estimators in this class of regression models. We use Monte Carlo simulations to verify the performance of the proposed methods. Our simulation results suggest that the analytical expressions we derive deliver more reliable results in small‐sized samples. An empirical application to a real data set is considered for illustrative purposes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The flow under an Eyring-Powell description has attracted interest to model different scenarios related with non-Newtonian fluids. The goal of the present study is to provide analysis of solutions to a one-dimensional Eyring-Powell fluid in Magnetohydrodynamics (MHD) with general initial conditions. Firstly, regularity and bounds of solutions are shown as a baseline to support the construction of existence and uniqueness results. The existence analysis is based on the definition of a Hamiltonian that constitutes the underlying theory to obtain stationary profiles of solutions that are validated with a numerical approach. Afterwards, non-stationary profiles of solutions are explored based on an asymptotic approximation to a Hamilton-Jacobi type of equation. To this end, an exponential scaling is considered together with perturbation methods. Finally, a region of validity for such exponential scaling is provided. [ABSTRACT FROM AUTHOR]

Operators of exponential type have kernel functions which satisfy a certain homogeneous differential equation related to a certain function p. They were introduced in 1978 by Ismail and May. We derive the complete asymptotic expansion for the operators of exponential type related to p(x) = x3 with an explicit representation of the coefficients. The results apply to all locally integrable real functions f on [0, +∞) satisfying the growth condition f(t) = O (ect) as t → +∞, for some c > 0. The expansion is shown to be valid also for simultaneous approximation with respect to the first derivative. [ABSTRACT FROM AUTHOR]

In this note, we spotlight a generalization of Weierstrass integral operators introduced by Eugeniusz Wachnicki. The construction involves modified Bessel functions. The operators are correlated with diffusion equation. Our main result consists in obtaining the asymptotic expansion of derivatives of any order of Wachnicki's operators. All coefficients are explicitly calculated, and distinct expressions are provided for analytical functions. [ABSTRACT FROM AUTHOR]

This article deals with an effcient approximation method named successive complementary expansion method (SCEM) for solving singularly perturbed differential-difference equations with mixed shifts. It is compared with the method of matched asymptotic expansion (MMAE) and the parameter uniform upwind finite difference scheme for solving such a model. The comparison shows, unlike the MMAE, the SCEM method requires no matching procedure. It requires less computation when compared to the upwind finite difference scheme on the Shishkin mesh. The error analysis is carried out to prove the robustness of the method. Some numerical experiments are provided, which show the effectiveness of the proposed method.

We consider a Hamiltonian of a system of two fermions on a two-dimensional lattice Z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}^2$$\end{document} with a potential of a certain type. The Schrödinger operator H(k),k∈T2,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(\textbf{k}),\textbf{k}\in \mathbb {T}^2,$$\end{document} corresponding to the considered system has two invariant subspaces L2oe(T2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2^{oe}(\mathbb { T }^2)$$\end{document} and L2eo(T2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2^{eo}(\mathbb {T}^2)$$\end{document} in the space of odd functions L2o(T2),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2^{o}(\mathbb {T}^2),$$\end{document} where the restrictions of the operator H(k)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H(\textbf{k})$$\end{document} in L2oe(T2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${L_2^{oe}(\mathbb {T}^2)}$$\end{document} and L2eo(T2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${L_2^{eo}(\mathbb {T}^2)}$$\end{document} are denoted by Hoe(k)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{oe}(\textbf{k})$$\end{document} and Heo(k)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{eo}(\textbf{k})$$\end{document}, respectively. It is proved that for kβ=(π-2β,π),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{k}_{\beta }=(\pi -2\beta ,\pi ),$$\end{document} the operator Hoe(kβ),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{oe}(\textbf{k}_{\beta }),$$\end{document} has a finite number of eigenvalues for any β∈(0,π),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\,\,\beta \in (0,\pi ),$$\end{document} and for any β∈[0,π]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta \in [0,\pi ]$$\end{document} operator Heo(kβ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{eo}(\textbf{k}_{\beta })$$\end{document} has an infinite number of eigenvalues. It is shown that if β→0,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta \rightarrow 0,$$\end{document} then the number of eigenvalues N(β)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N(\beta )$$\end{document} of the operator Hoe(kβ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{oe}(\textbf{k}_{\beta })$$\end{document} tends to infinity and the following asymptotic formula limβ→0N(β)|logsinβ|=2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\underset{\beta \rightarrow 0}{\lim }\dfrac{N(\beta )}{|\log \sin \beta |}=2$$\end{document} holds. [ABSTRACT FROM AUTHOR]

We derive an asymptotic expansion for the quadratic variation of a stochastic process satisfying a stochastic differential equation driven by a fractional Brownian motion, based on the theory of asymptotic expansion of Skorohod integrals converging to a mixed normal limit. In order to apply the general theory, it is necessary to estimate functionals that are a randomly weighted sum of products of multiple integrals of the fractional Brownian motion, in expanding the quadratic variation and identifying the limit random symbols. To overcome the difficulty, we utilized the theory of exponents of functionals, which was introduced by the authors in Yamagishi and Yoshida (2023). [ABSTRACT FROM AUTHOR]

While evaluating the near field of dipole antennas, it is noted that different software suites yield values of the electric and the magnetic fields, at the surface of antennas, that can be substantially different, especially at the tips and at the feed gap. The close near field of dipoles has not been yet analysed in detail. An asymptotic expansion method for the near fields at the surface of these antennas has been developed and compared to the computed fields. The results show that the software of the computed field should not use a uniform mesh. The mesh should be much tighter at the metal extremities than in the dipole body. The proposed technique can be used to check complex field computations, producing diverging values, with simple analytical equations.

This computational study investigates a class of singularly perturbed second-order boundary-value problems having dual (twin) boundary layers and simple turning points. It is well-known that the classical discretization methods fail to resolve sharp gradients arising in solving singularly perturbed differential equations as the perturbation (diffusion) parameter decreases, i.e., ε → 0+. To this end, this paper proposes a semi-analytic hybrid method consisting of a numerical procedure based on finite differences and an asymptotic method called the Successive Complementary Expansion Method (SCEM) to approximate the solution of such problems. Two numerical experiments are provided to demonstrate the method’s implementation and to evaluate its computational performance. Several comparisons with the numerical results existing in the literature are also made. The numerical observations reveal that the hybrid method leads to good solution profiles and achieves this in only a few iterations.

The concern of this study is to continue the investigation of convergence properties of Urysohn type generalized sampling operators, which are defined by the author in [Dolomites Res. Notes Approx. 2021, 14 (2), 58-67]. In details, the paper centers around to investigation of the asymptotic properties together with some Voronovskaya-type theorems for the linear and nonlinear counterpart of Urysohn type generalized sampling operators.

In this work, our main focus is to establish asymptotic expansions for the triple gamma function in terms of the triple Bernoulli polynomials. As application, an asymptotic expansion for hyperfactorial function is also obtained. Furthermore, using these asymptotic expansions, Padé approximants related to the triple gamma function are derived as a consequence. The results obtained are new, and their importance is demonstrated by deducing several interesting remarks and corollaries. [ABSTRACT FROM AUTHOR]

A flow of Eyring–Powell type constitutes a remarkable area of analysis to model non‐Newtonian processes in fluids. The associated diffusion term comes from the general kinetic theory of liquids and permits to account for a wider diffusivity, which is applicable for qualitatively low to higher shear stresses. The goal of the present article is to introduce a generalization of an Eyring–Powell fluid by the introduction of a porous reaction term (of Darcy–Forchheimer type) and a perturbation with a higher order operator. In particular, we consider that our model is an extension of a classical Eyring–Powell fluid in the same manner as introduced for other equations (see the extended Fisher–Kolmogorov model). The obtained equation is novel and requires analysis about existence, regularity and uniqueness of solutions. Stationary solutions are explored under the definition of a Hamiltonian. In addition, profiles of solutions are obtained with an exponential scaling that ends in a Hamilton–Jacobi equation. Eventually, some numerical assessments are introduced to validate the hypothesis done, and to discuss about the accuracy of the analytical approach followed. [ABSTRACT FROM AUTHOR]

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α and the others of weight 1. We denote by n the order of the graph and suppose that n tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on Re (α). After that, through the rest of the paper we suppose that 0 < α < 1. It is easy to see that the eigenvalues belong to [ 0 , 4 ] and are asymptotically distributed as the function g (x) = 4 sin 2 ⁡ (x / 2) on [ 0 , π ]. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of [ 0 , 4 ]. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every n ≥ 3. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue. [ABSTRACT FROM AUTHOR]

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term O (n 1 − p / 2) , for some p ≥ 4 , is shown in both cases. From the results, it is obtained that asymptotic expansions for the Cram e ´ r–von Mises statistics of the uniform distribution U (0 , 1) hold with the remainder term O n 1 − p / 2 for any p ≥ 4 . The scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of the kernel function u (x , y) . The key condition for the convergence is the nuclearity of a linear operator T u defined by the kernel function. The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables. [ABSTRACT FROM AUTHOR]

We consider acoustic wave propagation through a periodic array of the inclusions of arbitrary shape. The inclusion size is much smaller than the array period while the wavelength is fixed. We derive and rigorously justify the dispersion relation for general frequencies and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in different directions with different frequencies so that the dispersion relation cannot be defined uniquely. Examples are provided for the spherical inclusions. [ABSTRACT FROM AUTHOR]

The article contains a survey of publications studying problems characterized by the presence of fast variables with various rates of change (time scales). We consider the passage to the limit from the solution of a perturbed problem to the solution of a degenerate one, asymptotic solutions of initial and boundary value problems, stability and controllability, asymptotic solutions of optimal control problems, and problems with "hidden" multi-tempo variables. In addition, problems with control constraints, game problems, and stochastic systems are given. The last section presents practical problems with multi-tempo fast motions. [ABSTRACT FROM AUTHOR]

In the present paper, we study some nonlinear autonomous systems of three nonlinear ordinary differential equations (ODE) with small parameter such that two variables are fast and the remaining variable is slow. In the limit as , from this "complete dynamical system" we obtain the "degenerate system," which is included in a one-parameter family of two-dimensional subsystems of fast motions with parameter in some interval. It is assumed that there exists a monotone function that, in the three-dimensional phase space of a complete dynamical system, defines a parametrization of some arc of a slow curve consisting of the family of fixed points of the degenerate subsystems. Let have two points of the Andronov–Hopf bifurcation in which some stable limit cycles arise and disappear in the two-dimensional subsystems. These bifurcation points divide into the three arcs; two arcs are stable, and the third arc between them is unstable. For the complete dynamical system, we prove the existence of a trajectory that is located as close as possible to both the stable and unstable branches of the slow curve as tends to zero for values of within a given interval. [ABSTRACT FROM AUTHOR]

A new model is proposed for supercritical-fluid extraction of oil from vegetable raw materials. The internal cellular structure of raw materials, the localization of oil in the cells, and its diffusion through transport channels are taken into account in detail. The raw material is characterized by the following two parameters: the coefficient of diffusion through transport channels and the coefficient of mass transfer through the cell membrane. An analytical solution to the problem is obtained for raw materials with high initial reserves of oil. The main extraction regimes predicted within the model are investigated. It is shown that the published schematics of the compressible core and intact and destroyed cells are limiting ones for the new model. Thus, the proposed approach to the description of extraction generalizes the classical schematics and describes raw materials with intermediate properties. [ABSTRACT FROM AUTHOR]

We present some asymptotic expansions and inequalities for Γ(x+1)1/xΓ(x+2)1/(x+1) and Γ(x + 1)1/x. [ABSTRACT FROM AUTHOR]

We examine a family of linear partial differential equations both singularly perturbed in a complex parameter ϵ and singular in complex time t at the origin. These equations entail forcing terms which combine polynomial and logarithmic type functions in time and that are bounded holomorphic on horizontal strips in one complex space variable. A set of sectorial holomorphic solutions are built up by means of complete and truncated Laplace transforms w.r.t t and ϵ and Fourier inverse integral in space. Asymptotic expansions of these solutions relatively to t and ϵ are investigated and two distinguished Gevrey type expansions in monomial and logarithmic scales are exhibited. [ABSTRACT FROM AUTHOR]

We consider a long‐term portfolio choice problem with two illiquid and correlated assets, which is associated with an eigenvalue problem in the form of a variational inequality. The eigenvalue and the free boundaries implied by the variational inequality correspond to the portfolio's optimal long‐term growth rate and the optimal trading strategy, respectively. After proving the existence and uniqueness of viscosity solutions for the eigenvalue problem, we perform an asymptotic expansion in terms of small correlations and obtain semi‐analytical approximations of the free boundaries and the optimal growth rate. Our leading order expansion implies that the free boundaries are orthogonal to each other at four corners and have C1 regularity. We propose an efficient numerical algorithm based on the expansion, which proves to be accurate even for large correlations and transaction costs. Moreover, following the approximate trading strategy, the resulting growth rate is very close to the optimal one. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]