Acceleration of convergence of some infinite sequences {An} whose asymptotic expansions involve fractional powers of n via the ${\tilde {d}}^{(m)}$ transformation

2020 ◽  
Vol 85 (4) ◽  
pp. 1409-1445
Author(s):  
Avram Sidi
Keyword(s):  
Asymptotic Expansions ◽  
Acceleration Of Convergence ◽  
Fractional Powers ◽  
Infinite Sequences

Unfolding the high orders of asymptotic expansions with coalescing saddles: singularity theory, crossover and duality

1993 ◽  
Vol 443 (1917) ◽  
pp. 107-126 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Singularity Theory ◽  
Exact Formula ◽  
Large Parameter ◽  
Fractional Powers ◽  
Varying Parameters ◽  
Distant Cluster ◽  
Contour Integrals ◽  
Uniform Approximations ◽  
Uniform Asymptotic Expansions

We study the leading behaviour of the late coefficients (high orders r ) of asymptotic expansions in a large parameter k , for contour integrals involving a cluster of coalescing saddles, and thereby establish the form of the divergence of the expansions. The two principal cases are: 'saddle-to-cluster’, where the integral is through a simple saddle and its expansion diverges because of a distant cluster; and 'cluster-to-saddle', where the integral is through a cluster and its expansion diverges because of a distant simple saddle. In both, the large- r coefficients are dominated by the 'factorial divided by power' familiar in asymptotics, but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parameters A = { A 1 , A 2 ....} in the integrand. The 'crossover' between different forms is described by a series of canonical integrals, built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence. The arguments of these integrals involve not only the A but also fractional powers of r , which by a curious duality replace the powers of the original large parameter k which occur in uniform approximations involving these integrals. A by-product of the cluster-to-saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions.

Productly linearly independent sequences

2015 ◽  
Vol 52 (3) ◽  
pp. 350-370
Author(s):  
Jaroslav Hančl ◽  
Katarína Korčeková ◽  
Lukáš Novotný
Keyword(s):  
Rational Numbers ◽  
Infinite Products ◽  
Linearly Independent ◽  
New Concepts ◽  
Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

Non-repetitive sequences

1970 ◽  
Vol 68 (2) ◽  
pp. 267-274 ◽  
Author(s):  
P. A. B. Pleasants
Keyword(s):  
Repetitive Sequences ◽  
Finite Set ◽  
Infinite Sequences

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

scholarly journals Asymptotic Expansions and Strategies in the Online Increasing Subsequence Problem

2021 ◽  
Vol 174 (1) ◽  
Author(s):  
Amirlan Seksenbayev
Keyword(s):  
Dynamic Programming ◽  
Asymptotic Expansions ◽  
Dual Problem ◽  
Infinite Length ◽  
Expected Time ◽  
Selection Strategies ◽  
Dynamic Programming Equation ◽  
Optimal Values ◽  
Design Selection ◽  
Selection Of

AbstractWe study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size $n$ n . In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length $k$ k from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with $O(1)$ O ( 1 ) remainder in the first problem and $O(k)$ O ( k ) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible.

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