scholarly journals An Asymptotic Formula for r-Bell Numbers with Real Arguments

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Cristina B. Corcino ◽  
Roberto B. Corcino
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Approximation ◽  
Bell Numbers

The r-Bell numbers are generalized using the concept of the Hankel contour. Some properties parallel to those of the ordinary Bell numbers are established. Moreover, an asymptotic approximation for r-Bell numbers with real arguments is obtained.

scholarly journals Multiobjective Maximization of Monotone Submodular Functions with Cardinality Constraint

2020 ◽  
pp. ijoo.2019.0041
Author(s):  
Rajan Udwani
Keyword(s):  
Asymptotic Formula ◽  
Fast Algorithm ◽  
Asymptotic Approximation ◽  
Constant Factor ◽  
The Other ◽  
Submodular Functions ◽  
Cardinality Constraint ◽  
Greedy Methods ◽  
Deterministic Formula ◽  
Single Objective

We consider the problem of multiobjective maximization of monotone submodular functions subject to cardinality constraint, often formulated as [Formula: see text]. Although it is widely known that greedy methods work well for a single objective, the problem becomes much harder with multiple objectives. In fact, it is known that when the number of objectives m grows as the cardinality k, that is, [Formula: see text], the problem is inapproximable (unless P = NP). On the other hand, when m is constant, there exists a a randomized [Formula: see text] approximation with runtime (number of queries to function oracle) the scales as [Formula: see text]. We focus on finding a fast algorithm that has (asymptotic) approximation guarantees even when m is super constant. First, through a continuous greedy based algorithm we give a [Formula: see text] approximation for [Formula: see text]. This demonstrates a steep transition from constant factor approximability to inapproximability around [Formula: see text]. Then using multiplicative-weight-updates (MWUs), we find a much faster [Formula: see text] time asymptotic [Formula: see text] approximation. Although these results are all randomized, we also give a simple deterministic [Formula: see text] approximation with runtime [Formula: see text]. Finally, we run synthetic experiments using Kronecker graphs and find that our MWU inspired heuristic outperforms existing heuristics.

scholarly journals ENCLOSED QUANTUM MECHANICAL SYSTEMS

1956 ◽  
Vol 34 (9) ◽  
pp. 914-919 ◽  
Author(s):  
T. E. Hull ◽  
R. S. Julius
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Approximation ◽  
Eigenvalue Problems ◽  
Mechanical Systems ◽  
Quantum Mechanical ◽  
Quantum Mechanical Systems

A brief description is given of the eigenvalue problems associated with enclosed quantum mechanical systems and of some attempts to deal with these problems. Another method is developed which leads to a general asymptotic formula for the eigenvalues. This formula yields a simple asymptotic approximation to the eigenvalue in each particular case, once the eigenfunction of the corresponding unrestricted system is known.

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 Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Han-Young Kim
Keyword(s):  
Bell Polynomials ◽  
Bell Numbers ◽  
Bell Number ◽  
Finite Sums

AbstractThe nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

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.

Uniform asymptotic approximation for 3D quantum scattering by a non-central potential

2016 ◽  
Vol 115 (6) ◽  
pp. 60008
Author(s):  
N. Grama ◽  
C. Grama ◽  
I. Zamfirescu
Keyword(s):  
Asymptotic Approximation ◽  
Quantum Scattering ◽  
Central Potential
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