Asymptotic Representation Research Articles

The unpolarized two-loop massive pure singlet Wilson coefficients for deep-inelastic scattering

We calculate the massive two–loop pure singlet Wilson coefficients for heavy quark production in the unpolarized case analytically in the whole kinematic region and derive the threshold and asymptotic expansions. We also recalculate the corresponding massless two–loop Wilson coefficients. The complete expressions contain iterated integrals with elliptic letters. The contributing alphabets enlarge the Kummer-Poincaré letters by a series of square-root valued letters. A new class of iterated integrals, the Kummer-elliptic integrals, are introduced. For the structure functions F2 and FL we also derive improved asymptotic representations adding power corrections. Numerical results are presented.

On the Asymptotics of Solutions of Second-Order Differential Equations with Rapidly Varying Nonlinearities

We establish conditions for the existence of one class of monotone solutions of two-term nonautonomous differential equations of the second order with rapidly varying nonlinearities, as well as the asymptotic representations of these solutions and their first-order derivatives as t ↑ ω (ω ≤ + ∞).

On a relationship between randomly and non-randomly thresholded empirical average excesses for heavy tails

Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov’s central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.

Asymptotic Representations of the Solutions of Differential Equations with Regularly and Rapidly Varying Nonlinearities

We establish the asymptotic properties of some types of solutions of the second-order differential equations whose right-hand sides contain a sum of terms with regularly and rapidly varying nonlinearities.

The probability of the correct majority made decision

Aim: the probability of correctness of the collegial decision, which is made by a majority vote of some collective (board), consisting of an odd number of members is investigated, if the probability of correctness of the individual decision of each member of Board is known.
 Мaterials and methods: Bernoulli scheme, asymptotic representation, estimation via geometric series, power series expansion, the formula of Wallis, a power scale of averages, average of Kolmogorov.
 Result: it is established, that if for each member of the board the probability of correctness of the individual decision is more than ½, then with an unlimited increase in the number of members of the Board the probability of correctness of the collegial decision tends to 1. The asymptotic representation and a number of bilateral estimates characterizing the speed of this aspiration are obtained. For heterogeneous Board (that is a Board, whose members make the right individual decision with different probability) introduced the concept of collegial average as an average characteristics, which can replaced the individual probability of each member of the board with the preservation of the probability of a collegial decision. The existence and uniqueness of the collegial average are proved.
 We derive a collegial inequality showing that the collegial average of some a set of numbers is not less than the geometric average of the same numbers with the equality takes place in the case and only if all the numbers are equal to each other. The collegial inequality serves as an analogue and complement to known set of inequalities establishing a connection between different averages (for example, Cauchy inequality for arithmetic average and geometric average).
 Conclusion: thus, the results of the study fully meet the aim of determining the probability of correctness of collegial decision taken by a majority of votes under the assumptions. As a result we obtain an asymptotic representation and bilateral estimates characterizing the rate of striving for the correct solution. For a heterogeneous board, the existence uniqueness of the concept of collegial average as an average characteristics is introduced and strictly proved, which can be replaced by an individual probability of each with preserving the probability of correctness of the collegial decision. It is established that the collegial average is not less than the geometric average. Possible applications of the results obtained can be the quantitative evaluation of election procedures and the solution of problems associated with improving the reliability of recognition of weak signals of control sensors of various transport systems, including high-speed transport systems on magnetic suspension.

Catastrophe theory and caustics of radially symmetric beams

The work is devoted to the study of the caustics of radial beams. Analytical expressions for caustic surfaces of wave fronts created by radially symmetric diffractive optical elements are found. The result is presented in a curvilinear coordinate system consistent with the caustic surface. An asymptotic representation of the Kirchhoff integral near the optical axis is obtained, ensuring the correct calculations in the non-paraxial case.

Asymptotic Representations for the Solutions of Second-Order Differential Equations with Rapidly and Regularly Varying Nonlinearities

We determine asymptotic representations for the solutions of one class of second-order differential equations with rapidly and regularly varying nonlinearities.

Asymptotics of Jack characters

Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack characters from the viewpoint of the asymptotic representation theory. In particular, we give explicit formulas for their asymptotically top-degree part, in terms of bicolored oriented maps with an arbitrary face structure. We also study their multiplicative structure and their structure constants and we prove that they fulfill approximate factorization property, a convenient tool for proving Gaussianity of fluctuations of random Young diagrams.

Estimation of the extreme value index in a censorship framework: Asymptotic and finite sample behavior

We revisit the estimation of the extreme value index for randomly censored data from a heavy tailed distribution. We introduce a new class of estimators which encompasses earlier proposals given in Worms and Worms (2014) and Beirlant et al. (2018), which were shown to have good bias properties compared with the pseudo maximum likelihood estimator proposed in Beirlant et al. (2007) and Einmahl et al. (2008). However the asymptotic normality of the type of estimators first proposed in Worms and Worms (2014) was still lacking. We derive an asymptotic representation and the asymptotic normality of the larger class of estimators and consider their finite sample behavior. Special attention is paid to the case of heavy censoring, i.e. where the amount of censoring in the tail is at least 50%. We obtain the asymptotic normality with a classical k rate where k denotes the number of top data used in the estimation, depending on the degree of censoring.

Inconsistency transmission and variance reduction in two-stage quantile regression

In this paper, we propose a new variance reduction method for quantile regressions with endogeneity problems, for alpha-mixing or m-dependent covariates and error terms. First, we derive the asymptotic distribution of two-stage quantile estimators based on the fitted-value approach under very general conditions. Second, we exhibit an inconsistency transmission property derived from the asymptotic representation of our estimator. Third, using a reformulation of the dependent variable, we improve the efficiency of the two-stage quantile estimators by exploiting a tradeoff between an inconsistency confined to the intercept estimator and a reduction of the variance of the slope estimator. Monte Carlo simulation results show the fine performance of our approach. In particular, by combining quantile regressions with first-stage trimmed least-squares estimators, we obtain more accurate slope estimates than 2SLS, 2SLAD and other estimators for a broad set of distributions. Finally, we apply our method to food demand equations in Egypt.

Finite-part integration of the generalized Stieltjes transform and its dominant asymptotic behavior for small values of the parameter. II. Non-integer orders

This paper constitutes the second part on the subject of finite part integration of the generalized Stieltjes transform Sλ[f]=∫0∞f(x)(ω+x)−λdx about ω = 0, where now λ is a non-integer positive real number. Divergent integrals with singularities at the origin are induced by writing (ω + x)−λ as a binomial expansion about ω = 0 and interchanging the order of operations of integration and summation. The prescription of finite part integration is then implemented by interpreting these divergent integrals as finite part integrals which are rigorously represented as complex contour integrals. The same contour is then used to express Sλ[f] itself as a complex contour integral. This led to the recovery of the terms missed by naïve term-wise integration which themselves are finite parts of divergent integrals whose singularity is at the finite upper limit of integration. When the function f(x) has a zero at the origin of order m = 0, 1, … such that m − λ < 0, the correction terms missed out by the naïve term by term integration give the dominant contribution to Sλ[f] as ω → 0. Otherwise, the correction term is sub-dominant to the leading convergent terms in the naïve term by term integration. We apply these results by obtaining exact and asymptotic representations of the Kummer and Gauss hypergeometric functions by evaluating their known Stieltjes integral representations. We then apply the method of finite part integration to obtain the asymptotic behavior of a generalization of the Stieltjes integral which is relevant in the calculation of the effective index of refraction of a shallow potential well.

Estimates with Asymptotically Uniformly Minimal $d$-Risk

The definition of a decision function with asymptotically ($n\to\infty$) uniformly minimal $d$-risk is presented in the framework of the general theory of statistical inference. Using this definition, we prove that the maximum likelihood estimate has asymptotically uniformly minimal $d$-risk. This extends one result by I. N. Volodin and A. A. Novikov [Theory Probab. Appl., 38 (1994), pp. 118--128] for shrinking priors to the general class of continuous distributions. The proof uses the asymptotic representation of the posterior risk function, as obtained in [A. A. Zaikin, J. Math. Sci. (N.Y.), 229 (2018), pp. 678--697].

Telescopic Shearing of Non-Newtonian Fluids with a Saturation Stress

A closed form solution for quasi-static telescopic shearing of non-Newtonian fluids between two concentric cylinders is derived and then analysed. The paper focuses on qualitative features of the solution. In particular, it is shown that the qualitative behaviour of the solution in the vicinity of the inner friction surface is controlled by the dependence of the quadratic invariant of the stress tensor on the quadratic invariant of the strain rate tensor as the latter approaches infinity. In particular, no solution at sticking may exist if the quadratic invariant of the stress tensor approaches a finite value as the quadratic invariant of the strain rate tensor approaches infinity. In this case, it is necessary to construct the solution at sliding. This solution is singular, and the exact asymptotic representation of the quadratic invariant of the strain rate tensor in the vicinity of the friction surface depends on the exact dependence of the quadratic invariant of the stress tensor on the quadratic invariant of the strain rate tensor as the latter approaches infinity.

Asymptotic representation theory and the spectrum of a random geometric graph on a compact Lie group

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma _{\mathrm{geom} }(N,L)$ whose vertices are $N$ random points $g_1,\ldots ,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma _{\mathrm{geom} }(N,L)$, when $N$ goes to infinity. 1. If $L$ is fixed and $N \to + \infty $ (Gaussian regime), then the largest eigenvalues of $\Gamma _{\mathrm{geom} }(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions. 2. If $L = O(N^{-\frac{1} {\dim G}})$ and $N \to +\infty $ (Poissonian regime), then the geometric graph $\Gamma _{\mathrm{geom} }(N,L)$ converges in the local Benjamini–Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma _{\mathrm{geom} }(N,L)$. In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl’s character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.

Предельные множества Азарина функций и асимптотическое представление интегралов

Предельные множества Азарина функций и асимптотическое представление интегралов

Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature.

We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators (-Δ±β 2) in d-dimensional, R-radius hyperbolic and hyperspherical geometry, which represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. In particular, we compute closed-form expressions for fundamental solutions of (-Δ±β 2) on , (-Δ±β 2) on , and present two candidate fundamental solutions for (-Δ - β 2) on . Flat-space limits, with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. To derive Gegenbauer polynomial expansions of our fundamental solutions for Helmholtz operators on hyperspheres and hyperboloids, we derive a collection of infinite series addition theorems for Ferrers and associated Legendre functions which are generalizations and extensions of the addition theorem for Gegenbauer polynomials. Using these addition theorems, in geodesic polar coordinates for dimensions greater than or equal to three, we compute Gegenbauer polynomial expansions for these fundamental solutions, and azimuthal Fourier expansions in two-dimensions.

Parameterisations of slow invariant manifolds: application to a spray ignition and combustion model

A wide range of dynamic models, including those of heating, evaporation and ignition processes in fuel sprays, is characterised by large differences in the rates of change of variables. Invariant manifold theory is an effective technique for investigation of these systems. In constructing the asymptotic expansions of slow invariant manifolds, it is commonly assumed that a limiting algebraic equation allows one to find a slow surface explicitly. This is not always possible due to the fact that the degenerate equation for this surface (small parameter equal to zero) is either a high degree polynomial or transcendental. In many problems, however, the slow surface can be described in a parametric form. In this case, the slow invariant manifold can be found in parametric form using asymptotic expansions. If this is not possible, it is necessary to use an implicit presentation of the slow surface and obtain asymptotic representations for the slow invariant manifold in an implicit form. The results of development of the mathematical theory of these approaches and the applications of this theory to some examples related to modelling combustion processes, including those in sprays, are presented.

Asymptotic representations of solutions with slowly varying derivatives

Second-order differential equations with power and exponential nonlinearities on the right hand side play an important role in the development of a qualitative theory of differential equations. The authors of most works devoted to the establishment of asymptotic representations of solutions investigate equations with power and with regularly varying nonlinearities. Recently, the consideration of differential equations with exponential and a wider class than exponential functions - rapidly varying functions - has begun. In this paper, the asymptotic representations of solutions with slowly varying first-order derivatives of some new class of second-order differential equations with rapidly and regularly varying nonlinearities are established.

Asymptotic Representations of Rapidly Varying Solutions of Essentially Nonlinear Second-Order Differential Equations

We establish asymptotic representations for rapidly varying solutions of essentially nonlinear secondorder differential equations.

Asymptotic Behavior of Slowly Varying Solutions of Second-Order Ordinary Binomial Differential Equations with Rapidly Varying Nonlinearity

We establish new conditions for the existence of slowly varying solutions of a second-order differential binomial nonautonomous equation with rapidly varying nonlinearity. We also determine the asymptotic representations for these solutions and their first derivatives as t ↑ ω, ω ≤ + ∞.