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We consider a C ∞ family of planar vector fields { X μ ˆ } μ ˆ ∈ W ˆ having a hyperbolic saddle and we study the Dulac map D (s ; μ ˆ) and the Dulac time T (s ; μ ˆ) from a transverse section at the stable separatrix to a transverse section at the unstable separatrix, both at arbitrary distance from the saddle. Since the hyperbolicity ratio λ of the saddle plays an important role, we treat it as an independent parameter, so that μ ˆ = (λ , μ) ∈ W ˆ = (0 , + ∞) × W , where W is an open subset of R N. For each μ ˆ 0 ∈ W ˆ and L > 0 , the functions D (s ; μ ˆ) and T (s ; μ ˆ) have an asymptotic expansion at s = 0 and μ ˆ ≈ μ ˆ 0 with the remainder being uniformly L -flat with respect to the parameters. The principal part of both asymptotic expansions is given in a monomial scale containing a deformation of the logarithm, the so-called Ecalle-Roussarie compensator. In this paper we are interested in the coefficients of these monomials, which are functions depending on μ ˆ that can be shown to be C ∞ in their respective domains and "universally" defined, meaning that their existence is stablished before fixing the flatness L and the unfolded parameter μ ˆ 0. Each coefficient has its own domain and it is of the form ((0 , + ∞) ∖ D) × W , where D a discrete set of rational numbers at which a resonance of the hyperbolicity ratio λ occurs. In our main result, Theorem A, we provide explicit expressions for some of these coefficients and to this end a fundamental tool is the employment of a sort of incomplete Mellin transform. With regard to these coefficients we also prove that they have poles of order at most two at D × W and we give the corresponding residue, that plays an important role when compensators appear in the principal part. Furthermore we prove a result, Corollary B, showing that in the analytic setting each coefficient given in Theorem A is meromorphic on (0 , + ∞) × W and has only poles, of order at most two, along D × W. [ABSTRACT FROM AUTHOR]

The asymptotic behavior for a heriditary recursion$ \begin{equation*} x_1>a \, \, \text{and} \, \, x_{n+1} = \frac{1}{n^s}\sum\limits_{k = 1}^nf\left(\frac{x_k}k\right)\text{ for every }n\geq1 \end{equation*} $is studied, where $ f $ is decreasing, continuous on $ (a, \infty) $ ($ a < 0 $), and twice differentiable at $ 0 $. The result has been known for the case $ s = 1 $. This paper analyzes the case $ s > 1 $. We obtain an asymptotic sequence that is quite different from the case $ s = 1 $. Some examples and applications are provided.

Some properties of the constant in asymptotic expansion of iterates of a cubic function were investigated. This paper analyzed the monotonicity, differentiability of the constant with respect to the initial value and the functional equation that is satisfied.

Through combined applications of the transfer-matrix method and asymptotic expansion technique, we formulate a theory to predict the three-dimensional response of micropolar plates. No ad hoc assumptions regarding through-thickness assumptions of the field variables are made, and the governing equations are two-dimensional, with the displacements and microrotations of the mid-plane as the unknowns. Once the deformation of the mid-plane is solved, a three-dimensional micropolar elastic field within the plate is generated, which is exact up to the second order except in the boundary region close to the plate edge. As an illustrative example, the bending of a clamped infinitely long plate caused by a uniformly distributed transverse force is analyzed and discussed in detail. [ABSTRACT FROM AUTHOR]

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results. [ABSTRACT FROM AUTHOR]

In this paper, we apply the steepest descent method to the Schläfli-type integral representation of the three-parameter Mittag-Leffler function (well-known as the Prabhakar function). We find the asymptotic expansions of this function for its large parameters with respect to the real and complex saddle points. For each parameter, we separately establish a relation between the variable and parameter of function to determine the leading asymptotic term. We also introduce differentiations of the three-parameter Mittag-Leffler functions with respect to parameters and modify the associated asymptotic expansions for their large parameters. As an application, we derive the leading asymptotic term of fundamental solution of the time-fractional sub-diffusion equation including the Bessel operator with large order. [ABSTRACT FROM AUTHOR]

In this paper, a modification of a renewal-reward process with dependent components is mathematically constructed and the stationary characteristics of this process are studied. Stochastic processes with dependent components have rarely been studied in the literature owing to their complex mathematical structure. We partially fill the gap by studying the effect of the dependence assumption on the stationary properties of the process . To this end, first, we obtain explicit formulas for the ergodic distribution and the stationary moments of the process. Then we analyze the asymptotic behavior of the stationary moments of the process by using the basic results of the renewal theory and the Laplace transform method. Based on the analysis, we obtain two-term asymptotic expansions of the stationary moments. Moreover, we present two-term asymptotic expansions for the expectation, variance, and standard deviation of the process . Finally, the asymptotic results obtained are examined in special cases. [ABSTRACT FROM AUTHOR]

We considered the Mellin transform assigned to the convolution wave kernel associated to the Laplace-Beltrami operator on higher rank Riemannian symmetric spaces of the non-compact type. The occurrence of the analyticity strip of this transform can be deduced directly from the pointwise kernel estimates. Using the zeta function techniques, we established its meromorphic extension to the entire complex plane C with simple poles on the real line. [ABSTRACT FROM AUTHOR]

We considered the Mellin transform assigned to the convolution wave kernel associated to the Laplace-Beltrami operator on higher rank Riemannian symmetric spaces of the non-compact type. The occurrence of the analyticity strip of this transform can be deduced directly from the pointwise kernel estimates. Using the zeta function techniques, we established its meromorphic extension to the entire complex plane $ {{\Bbb C}} $ with simple poles on the real line.

The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville's transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville's transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically. [ABSTRACT FROM AUTHOR]

The asymptotic expansion of the eigenvalue of Sturm-Liouville problem is presented. The problem has a symmetric double well potential that is continuous, symmetrical to both the midpoint and quarter point of the related interval and non-increasing on the quarter interval. [ABSTRACT FROM AUTHOR]

The higher-order nonlinear Schrödinger equation with combined nonlinearities is derived by an asymptotic reduction from the (2+4) Korteweg–de Vries model for weakly nonlinear wave packets for the context of interfacial waves in a three-layer symmetric media. Focusing properties and modulation instability effects are discussed for the considered physical context. Instability growth rate, maximum of the increment and the boundaries of the instability interval are derived in terms of three-layer density stratification, their structure on the parameter planes of relative layer depth, carrier wavenumber and envelope amplitude, are considered in detail. [ABSTRACT FROM AUTHOR]

We consider the model of Mindlin–Timoshenko for a multi-structure composed of an elastic plate surrounded by a thin layer of uniform thickness. From the viewpoint of numerical simulation, the treatment of the behavior of this structure is difficult because of the presence of the thin coating. In order to overcome this difficulty, we use the asymptotic expansion method to identify an approximate model that does not involve the thin layer geometrically but which accounts for its effect through new approximate boundary conditions. These conditions are set on the junction interface between the two sub-structures and depend on the thickness and the physical characteristics of the thin layer. Moreover, we give optimal error estimates between the exact and the approximate solutions of the considered transmission problem, which validate this approximation. [ABSTRACT FROM AUTHOR]

We study a new class of time-periodic solutions of singularly perturbed systems of reaction–diffusion equations in the case of a fast and a slow equation, which are usually called Tikhonov-type systems. A boundary layer asymptotics of solutions is constructed, the existence of solutions with this asymptotics is proved, and conditions for the Lyapunov asymptotic stability of these solutions treated as solutions of the corresponding initial–boudary value problems are obtained. [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. [ABSTRACT FROM AUTHOR]

Given the important role an asymptotic expansion of Whittaker functions or Shalika functions plays in the study of L-functions via the Rankin–Selberg method, we first establish an asymptotic expansion of (k, c)-functions for GL n (F) over a p-adic field F via Casselman–Shalika’s argument. As an application, we then show the holomorphy of Ginzburg–Kaplan’s local “tensor product” L-functions for GL c (F) × GL k (F) . In the appendix, following Wallach–Soudry’s argument, we also provide an analogous asymptotic expansion of (k, c)-functions for GL n (F) with F an Archimedean local field. [ABSTRACT FROM AUTHOR]

In the dynamics of linear structures, the impulse response function is of fundamental interest. In some cases one examines the short term response wherein the disturbance is still local and the boundaries have not yet come into play, and for such short-time analysis the geometrical extent of the structure may be taken as unbounded. Here we examine the response of slender beams to angular impulses. The Euler–Bernoulli model, which does not include rotary inertia of cross sections, predicts an unphysical and unbounded initial rotation at the point of application. A finite length Euler–Bernoulli beam, when modeled using finite elements, predicts a mesh-dependent response that shows fast large-amplitude oscillations setting in very quickly. The simplest introduction of rotary inertia yields the Rayleigh beam model, which has more reasonable behavior including a finite wave speed at all frequencies. If a Rayleigh beam is given an impulsive moment at a location away from its boundaries, then the predicted behavior has an instantaneous finite jump in local slope or rotation, followed by smooth evolution of the slope for a finite time interval until reflections arrive from the boundary, causing subsequent slope discontinuities in time. We present a detailed study of the angular impulse response of a simply supported Rayleigh beam, starting with dimensional analysis, followed by modal expansion including all natural frequencies, culminating with an asymptotic formula for the short-time response. The asymptotic formula is obtained by breaking the series solution into two parts to be treated independently term by term, and leads to a polynomial in time. The polynomial matches the response from refined finite element (FE) simulations. [ABSTRACT FROM AUTHOR]

In the present article, we extend the results of the neural network interpolation operators activated by smooth ramp functions proposed by Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016). We give different results from Yu (Acta Math. Sin.(Chin. Ed.) 59:623-638, 2016) we discuss the high-order approximation result using the smoothness of φ and a related Voronovskaya-type asymptotic expansion for the error of approximation. In addition, we showcase the related fractional estimates result and the fractional Voronovskaya type asymptotic expansion. We investigate the approximation degree for the iterated and complex extensions of the aforementioned operators. Finally, we provide numerical examples and graphs to effectively illustrate and validate our results.

We study the dynamics of the flow X solution to the ODE: X ′ (t , x) = b (X (t , x)) with X (0 , x) = x ∈ R d , where b is a regular Z d -periodic vector field in R d. We provide conditions on b to get the fine asymptotic expansion: | X (t , x) − x − t ζ (x) | ≤ M < ∞. To this end, we try to express X (t , x) − x − t ζ (x) as Φ (X (t , x)) − Φ (x) , which yields the desired expansion when Φ is bounded. Then, assuming that the 2D Kolmogorov theorem and some extension for d > 2 hold, we establish several regimes depending on the commensurability of the rotation vectors ζ (x) for which the expansion of X is valid. Moreover, we prove that for any 2D flow with a non vanishing smooth b inducing a unique incommensurable rotation vector ξ , the fine expansion holds in R 2 if, and only if, ξ 1 / ξ 2 is a Diophantine number. The case where ξ is commensurable is also investigated. Finally, several examples illustrate the different results, including the case of a vanishing b which blows up the asymptotic expansion in some direction. In particular, the case of some Euler flows is investigated. [ABSTRACT FROM AUTHOR]

We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. We perturbed it by a first order differential operator arbitrarily depending on a small multi-dimensional parameter and we study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multipliers. The second part of our results says that the convergence in the mentioned spaces of multipliers is equivalent to the convergence of certain local mean values over small pieces of the considered domains. These results are supported by series of examples. We also provide a series of ways of generating new non-periodically oscillating perturbations for which our results are applicable. [ABSTRACT FROM AUTHOR]

The higher-order nonlinear Schrödinger equation with combined nonlinearities is derived by an asymptotic reduction from the (2+4) Korteweg–de Vries model for weakly nonlinear wave packets for the context of interfacial waves in a three-layer symmetric media. Focusing properties and modulation instability effects are discussed for the considered physical context. Instability growth rate, maximum of the increment and the boundaries of the instability interval are derived in terms of three-layer density stratification, their structure on the parameter planes of relative layer depth, carrier wavenumber and envelope amplitude, are considered in detail.

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the 1-dimensional semi-linear fractional equations: | D | α u + u − f (u) = 0 , with α ∈ (0 , 2) , a prescribed coefficient p ⁎ (α) , and a non-linearity f (u) = | u | p − 1 u for p ∈ (1 , p ⁎ (α)) , or f (u) = u p with an integer p ∈ [ 2 ; p ⁎ (α)). Asymptotic developments of order 1 at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion α and of the non-linearity p. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory. [ABSTRACT FROM AUTHOR]

In eddy current (EC) testing applications, ECs σ E ( E : electric field, σ : conductivity) are induced in tested metal parts by a low-frequency (LF) source idealized as a closed current loop in air. In the case of highly conducting (HC) parts, a boundary integral equation (BIE) of the first kind under the magneto-quasi-static approximation - which neglects the displacement current - was shown in a previous work to coincide with the leading order of an asymptotic expansion of the Maxwell BIE in a small parameter reflecting both LF and HC assumptions. The main goal of this work is to generalize the latter approach by establishing a unified asymptotic framework that is applicable to configurations that may involve multiple moderately-conducting (σ = O (1)) and non-conducting objects in addition to (possibly multiply-connected) HC objects. Leading-order approximations of the quantities relevant to EC testing, in particular the impedance variation, are then found to be computable from a reduced set of primary unknowns (three on HC objects and two on other objects, instead of four per object for the Maxwell problem). Moreover, when applied to the Maxwell BIE, the scalings suggested by the asymptotic approach stabilize the condition number at low frequencies and remove the low-frequency breakdown effect. The established asymptotic properties are confirmed on 3D numerical examples for simple geometries as well as two EC testing configurations, namely a classical benchmark and a steam generator tube featured in pressurized water reactors of nuclear power plants. [ABSTRACT FROM AUTHOR]

The Kelvin–Voigt model for a thin stratified two-dimensional visco-elastic strip is analyzed both in the quasistatic and in the dynamic cases. The Neumann boundary conditions on the upper and the lower parts of the boundary and periodicity conditions with respect to the longitudinal variable are stated. A complete asymptotic expansion of the solution is constructed in both cases, by using the dimension reduction combined with a homogenization technique. The error between the exact solution and the asymptotic one is evaluated in each case and the obtained results fully justify the asymptotic construction. The results were partially (quasistatic case) announced in the short note in C.R. Acad. Sci. Paris; the present article contains the complete proofs and generalizations in the dynamic case. [ABSTRACT FROM AUTHOR]

Let Dv (z) be the Parabolic Cylinder function. We study the v-zeros of the function v → Dv (z) with respect to the real variable z. We establish a formula for the derivative of a zero and deduce some monotonicity results. Then we also give an asymptotic expansion for v-zeros for large positive z. [ABSTRACT FROM AUTHOR]

We present a complete asymptotic expansion of the compound mean M⊗N of two symmetric homogeneous means M and N and derive an efficient algorithm for computing coefficients in this expansion. This new approach is applied to obtain a simple formula for computing coefficients in the expansion of the arithmetic-geometric mean. We also give some other applications to the compounds of the classical means. [ABSTRACT FROM AUTHOR]

We offer a new asymptotic expansion with an explicit remainder estimate in the central limit theorem. The results obtained are essentially based on new forms of asymptotic expansions in the central limit theorem. We also present a more accurate estimation of the CLT-expansions remainder which is rigorously proved and backed up numerically. It is shown that our approach can be used for further refinement of allied asymptotic expansions. [ABSTRACT FROM AUTHOR]

Recent mathematical models of reliability computer systems and telecommunication networks are based on distributions with heavy tails. This paper falls into the category of exploring the classical models with heavy tails: Gnedenko-Weibull, Burr, Benktander distributions. The moment's asymptotics for residual time have been derived especially for there heavy-tailed distributions. A simulation study is presented to give a practice characterization of the distributions with heavy tails. [ABSTRACT FROM AUTHOR]

The paper develops a new deep learning based scheme for solving high-dimensional nonlinear forward-backward stochastic differential equations (FBSDE) and associated partial differential equations. Firstly, the original BSDE is split into the linear dominant BSDE part and the nonlinear residual BSDE part. Then the linear BSDE part is approximated with high accuracy using a weak approximation technique. To approximate the nonlinear BSDE part, Deep BSDE solver is applied with asymptotic expansions which work as control variates. A sharp error estimate provides how the new scheme improves the original Deep BSDE method. Numerical experiments for high-dimensional nonlinear models show the validity and the effectiveness of the new scheme in financial application. [ABSTRACT FROM AUTHOR]

Initial-boundary value problems including space-time fractional PDEs have been used to model a wide range of problems in physics and engineering fields. In this paper, a non-self adjoint initial boundary value problem containing a third order fractional differential equation is considered. First, a spectral problem for this problem is presented. Then the eigenvalues and eigenfunctions of the main spectral problem are calculated. In order to calculate the roots of their characteristic equation, the asymptotic expansion of the roots is used. Finally, by suitable choice of these asymptotic expansions, related eigenfunctions and Mittag-Lefler functions, the analytic and numerical solutions to the main initial-boundary value problem are given.

Abstract 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.

The Clarkson-McLeod solutions of the fourth Painlevé equation behave like κ D α − 1 2 2 (2 x) as x → + ∞ , where κ is some real constant and D α − 1 2 (x) is the parabolic cylinder function. Using the Deift-Zhou nonlinear steepest descent method, we derive the asymptotic behaviors for this class of solutions as x → − ∞. This completes a proof of Clarkson and McLeod's conjecture on the asymptotics of this family of solutions. The total integrals of the Clarkson-McLeod solutions and the asymptotic approximations of the σ -form of this family of solutions are also derived. Furthermore, we find a determinantal representation of the σ -form of the Clarkson-McLeod solutions via an integrable operator with the parabolic cylinder kernel. [ABSTRACT FROM AUTHOR]

Both theoretical and applied mathematics depend heavily on inequalities, which are rich in symmetries. In numerous studies, estimations of various functions based on the characteristics of their symmetry have been provided through inequalities. In this paper, we study the monotonicity of certain functions that involve Gamma functions. We were able to obtain some of the bounds of Γ (v) that are more accurate than some recently published inequalities. [ABSTRACT FROM AUTHOR]

The celebrated Wallis sequence Wn, which is defined by ..., is known to have the limit π/2 as n → ∞. Without using the Bernoulli numbers Bn, the authors present several asymptotic expansions and a recurrence relation for determining the coefficients of each asymptotic expansion related to the Wallis sequence Wn and the newly-introduced constants D and E, which are analogous to the Glaisher-Kinkelin constant A and the Choi-Srivastava constants B and C. [ABSTRACT FROM AUTHOR]

On the set M of mean functions, the symmetric mean of M with respect to mean M 0 can be defined in several ways. The first one is related to the group structure on M , and the second one is defined trough Gauss' functional equation. In this paper, we provide an answer to the open question formulated by B. Farhi about the matching of these two different mappings called symmetries on the set of mean functions. Using techniques of asymptotic expansions developed by T. Burić, N. Elezović, and L. Mihoković (Vukšić), we discuss some properties of such symmetries trough connection with asymptotic expansions of means involved. As a result of coefficient comparison, a new class of means was discovered, which interpolates between harmonic, geometric, and arithmetic mean. [ABSTRACT FROM AUTHOR]

In this paper we present estimation formulas for the expectations of power means of large data and associate them with means of probability distribution and means of random sample. The proposed method follows from the asymptotic expansion of power means which is applicable for sufficiently large data and it is especially useful when value of such expectation is hard to obtain. We will show the accuracy of these approximations for random samples which have uniform and normal distribution and analyse their behaviour for large sample volume. [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 $.

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]

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 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]

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]