Maslov's complex germ and the asymptotic formula for the Gibbs canonical distribution

1998 ◽  
Vol 64 (4) ◽  
pp. 537-550
Author(s):  
O. Yu. Shvedov
Keyword(s):  
Asymptotic Formula ◽  
Complex Germ ◽  
Canonical Distribution ◽  
Gibbs Canonical Distribution

Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

2007 ◽  
Vol 44 (02) ◽  
pp. 285-294 ◽  
Author(s):  
Qihe Tang
Keyword(s):  
Asymptotic Formula ◽  
Continuous Time ◽  
Heavy Tails ◽  
Tail Behavior ◽  
Renewal Model ◽  
Discounted Aggregate Claims ◽  
Aggregate Claims ◽  
Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

Approximation properties of Szász type operators involving Charlier polynomials

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 479-487
Author(s):  
Didem Arı
Keyword(s):  
Asymptotic Formula ◽  
Weighted Space ◽  
Approximation Properties ◽  
Charlier Polynomials

In this paper, we give some approximation properties of Sz?sz type operators involving Charlier polynomials in the polynomial weighted space and we give the quantitative Voronovskaya-type asymptotic formula.

scholarly journals On the least common multiple of random q-integers

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Probabilistic Model ◽  
Expected Value ◽  
Random Set ◽  
Common Multiple ◽  
Least Common Multiple

AbstractFor every positive integer n and for every $$\alpha \in [0, 1]$$ α ∈ [ 0 , 1 ] , let $${\mathcal {B}}(n, \alpha )$$ B ( n , α ) denote the probabilistic model in which a random set $${\mathcal {A}} \subseteq \{1, \ldots , n\}$$ A ⊆ { 1 , … , n } is constructed by picking independently each element of $$\{1, \ldots , n\}$$ { 1 , … , n } with probability $$\alpha $$ α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of $${\mathcal {A}}$$ A .Let q be an indeterminate and let $$[k]_q := 1 + q + q^2 + \cdots + q^{k-1} \in {\mathbb {Z}}[q]$$ [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of $$X := \deg {\text {lcm}}\!\big ([{\mathcal {A}}]_q\big )$$ X : = deg lcm ( [ A ] q ) , where $$[{\mathcal {A}}]_q := \big \{[k]_q : k \in {\mathcal {A}}\big \}$$ [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al.

scholarly journals On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

scholarly journals On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Symmetric Square ◽  
The Mean ◽  
Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

XLIX.—Generalizations of a Problem of Pillai

1949 ◽  
Vol 62 (4) ◽  
pp. 460-469
Author(s):  
L. Mirsky
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Main Concern ◽  
Positive Integers ◽  
Infinite Set

I. Throughout this paper k1, …, k3 will denote s ≥ I fixed distinct positive integers. Some years ago Pillai (1936) found an asymptotic formula, with error term O(x/log x), for the number of positive integers n ≤ x such that n + k1, …, n + k3 are all square-free. I recently considered (Mirsky, 1947) the corresponding problem for r-free integers (i.e. integers not divisible by the rth power of any prime), and was able, in particular, to reduce the error term in Pillai's formula.Our present object is to discuss various generalizations and extensions of Pillai's problem. In all investigations below we shall be concerned with a set A of integers. This is any given, finite or infinite, set of integers greater than 1 and subject to certain additional restrictions which will be stated later. The elements of A will be called a-numbers, and the letter a will be reserved for them. A number which is not divisible by any a-number will be called A-free, and our main concern will be with the study of A-free numbers. Their additive properties have recently been investigated elsewhere (Mirsky, 1948), and some estimates obtained in that investigation will be quoted in the present paper.

scholarly journals The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

Asymptotic formula for solutions to elliptic equations near the Lipschitz boundary

2005 ◽  
Vol 184 (2) ◽  
pp. 185-213 ◽  
Author(s):  
Vladimir Kozlov ◽  
Vladimir Maz’ya
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Equations ◽  
Lipschitz Boundary
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