Asymptotic formulas for sequence factorial of arithmetic progression

This note provides asymptotic formulas for approximating the sequence factorial of members of a finite arithmetic progression by using Stirling, Burnside and other more accurate asymptotic formulas for large factorials that have appeared in the literature.     

The Pythagorean theorem and the Euclidean parallel postulate

In this note, we develop new, simple and very accurate asymptotic approximations for non-integer order derivatives of monomial functions by using the more accurate asymptotic approximations for large factorials that have recently appeared in the literature.     

Multiple prime covers of the riemann sphere

A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C _q of prime order q such that X/C _q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.     

Reconstructing pluricanonical embeddings of general k-gonal curves via odd theta-characteristics

Abstract Here we investigate r-canonical embeddings of general k-gonal curves of genus g from the point of view of Caporaso–Sernesi’s reconstruction procedure via odd theta-characteristics. Keywords: Theta-characteristic, general k-gonal curve, trigonal curve, pluricanonical embedding, Hilbert scheme Mathematics Subject Classification (2000): 14H50, 14N05     

Real line bundles on k-gonal real curves

For a generalk-gonal complex curve of genusg its variety of special line bundlesL with deg(L) =d andh ⁰(L) >r is known to contain an irreducible component of the expected dimension ρ_g (d, r) provided that the Brill-Noether number ρ_g (d, r) is non-negative andr ≤ k - 2. It is the purpose of this note to transfer this result of Brill-Noether type to the case ofk-gonal real curves, for real line bundles.     

On the Asymptotic Expansion of Hadamard Products

We consider functions f and g which are holomoxphic on closed sectors in C where they admit an asymptotic representation at ∞ in the form of power series in z^-1. We give a simple geometrical condition under which the Hadamard product f*g of f and g porsesses again an asymp totic expansion at ∞. It turns out that the asymptotic expansion of f*g is essentially the formal Hsdamd product of the asymptotic expansions of f and g. Our result yields a slight generaliestion of a well known theorem of W. B. Ford.     

Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, 3j Symbols, and Character Localization

In this paper, we employ a technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index J) of generic Wigner matrix elements D^J_MM￠(g){D^{J}_{MM'}(g)} . We use this result to derive asymptotic formulae for the character χ ^J (g) of an SU(2) group element and for Wigner’s 3j symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for χ ^J (g) is in fact exact. The result hints at a “Duistermaat-Heckman like” localization property for discrete sums.     

Linear series on a general k-gonal curve

This paper is a contribution towards a Brill-Noether theory for the moduli space of smooth &-gonal curves of genusg. Specifically, we prove the existence of certain special divisors on a generalk-gonal curveC of genusg, and we detect an irreducible component of the “expected” dimension in the varietyW ^r _d (C), (r ≤k — 2) of special divisors ofC. The latter induces a new proof of the existence theorem for special divisors on a smooth curve.     

Approximation Numbers of Identity Operators between Symmetric Sequence Spaces

We prove asymptotic formulas for the behavior of approximation quantities of identity operators between symmetric sequence spaces. These formulas extend recent results of Defant, Masty o, and Michels for identities l_pⁿF_n with an n-dimensional symmetric normed space F_n with p-concavity conditions on F_n and 1p2. We consider the general case of identities E_nF_n with weak assumptions on the asymptotic behavior of the fundamental sequences of the n-dimensional symmetric spaces E_n and F_n. We give applications to Lorentz and Orlicz sequence spaces, again considerably generalizing results of Pietsch, Defant, Masty o, and Michels.     

On gonality of Riemann surfaces

A compact Riemann surface X is called a (p, n)-gonal surface if there exists a group of automorphisms C of X (called a (p, n)-gonal group) of prime order p such that the orbit space X/C has genus n. We derive some basic properties of (p, n)-gonal surfaces considered as generalizations of hyperelliptic surfaces and also examine certain properties which do not generalize. In particular, we find a condition which guarantees all (p, n)-gonal groups are conjugate in the full automorphism group of a (p, n)-gonal surface, and we find an upper bound for the size of the corresponding conjugacy class. Furthermore we give an upper bound for the number of conjugacy classes of (p, n)-gonal groups of a (p, n)-gonal surface in the general case. We finish by analyzing certain properties of quasiplatonic (p, n)-gonal surfaces. An open problem and two conjectures are formulated in the paper.     

Asymptotic Limits for a Two-Dimensional Recursion

In this paper we consider the general two-dimensional recursion g(n + 1, r + 1) = g(n + 1, r) + bg(n, r) + g(n, r + 1) with boundary conditions g(n, 0) = g(0, r) = 1 We develop various exact and asymptotic formulas for the g(n, r).     

On q-n-gonal Klein Surfaces

We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism φ of prime order n with quotient space X/（φ） of algebraic genus q. These Klein surfaces axe called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the （admissible） values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given： q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.     

On the uniqueness of (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">h</Emphasis>)-gonal automorphisms of Riemann surfaces

Let X be a compact Riemann surface of genus g ≥  2. A cyclic subgroup of prime order p of Aut(X) is called properly (p, h)-gonal if it has a fixed point and the quotient surface has genus h. We show that if p > 6h + 6, then a properly (p, h)-gonal subgroup of Aut(X) is unique. We also discuss some related results.     

An asymptotic approach for a semi‐Markovian inventory model of type (s,S)

In this study, we constructed a stochastic process (X(t)) that expresses a semi‐Markovian inventory model of type (s, S) and it is shown that this process is ergodic under some weak conditions. Moreover, we obtained exact and asymptotic expressions for the n^th order moments (n = 1,2,3, … ) of ergodic distribution of the process X(t), as S ? s → ∞ . Finally, we tested how close the obtained approximation formulas are to the exact expressions. Copyright © 2012 John Wiley & Sons, Ltd.     

ASYMPTOTIC APPROXIMATION BYBERNSTEIN-DURRMEYER OPERATORS AND THEIR DERIVATIVES

The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators M_n. We derive the complete asymptotic expansion of the operators M_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of M_n,     

Asymptotic approximation by de la Vallée Poussin means and their derivatives

The concern of this paper is the study of local approximation properties of the de la Vallée Poussin means V_n. We derive the complete asymptotic expansion of the operators V_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly.     

Asymptotic approximation by Bernstein-Durrmeyer operators and their derivatives

The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators M_n. We derive the complete asymptotic expansion of the operators M_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of M_n,     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

On cyclic <Emphasis Type="Italic">p</Emphasis>-gonal Riemann surfaces with several <Emphasis Type="Italic">p</Emphasis>-gonal morphisms

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.