The Catalan-Larcombe-French sequence {Pn}n≥0 arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence {P2 n – Pn–1Pn+1}n≥1 are strictly log-con...

We say that a real number α is in Log space if its n-th bit (n ∈ Z) can be computed in O(log |n|) space. We show that π = 3.14159 · · · is in Log space. This implies that π is in the complexity class SC. The latter result has been conjectured by Lipton, and widely assumed to be true from the work of Bailey, Borwein and Plouffe (BBP) and others. Our result extends to other constants such as log ...

We prove that the maximum degree ∆n of a random series-parallel graph with n vertices satisfies ∆n/ log n → c in probability, and E∆n ∼ c log n for a computable constant c > 0. The same result holds for outerplanar graphs.

We consider the problem of sorting elements on a series of stacks, introduced by Tarjan and Knuth. We improve the asymptotic lower bound for the number of stacks necessary to sort n elements to 0.561 log 2 n+O(1). This is the first significant improvement since the previous lower bound, 1 2 log 2 n+O(1), was established by Knuth in 1972.

Results of diversity and community ecology studies strongly depend on sampling depth. Completely surveyed communities follow log-normal distribution, whereas power law functions best describe incompletely censused communities. It is arguable whether the statistics behind those theories can be applied to voluminous next generation sequencing data in microbiology by treating individual DNA sequen...

ABSTRACT A series of mono-alkylcarboxylic acid derivatives of tetraphenylporphyrin have been prepared. All the porphyrins were completely characterized by use of mass, 1H NMR, UV-visible, and fluorescence spectroscopy. Experimental log P were determined by use of reversed-phase thin-layer chromatography with use of log PRekker. These porphyrins are potential photosensitizers in photodynamic the...

This paper deals with the problem of broadcasting in minimum time in the telephone and message-passing models. Approximation algorithms are developed for arbitrary graphs, as well as for several restricted graph classes. In particular, an O(p n)-additive approximation algorithm is given for broadcasting in general graphs, and an O(log log n= log n) (multiplicative) ratio approximation is given ...

This paper discusses the option pricing problems using statistical series expansion for the price process of an underlying asset. We derive the Edgeworth expansion for the stock log return via extracting dynamics structure of time series. Using this result, we investigate influences of the non-Gaussianity and the dependency of log return processes for option pricing. Numerical studies show some...

a linear combination of the Poisson-distributed variables, fn. We rewrite the log-likelihood ratio in terms of sn ≡ (A/F0) τν ( e − 1 ) e−n/(τν). Most calculations in this paper rely on taking series approximations in sn and performing the summations to leading order. This approach is valid as long as A < F0, since τν ( e − 1 ) e−n/(τν) is bounded above by 1 for n ≥ 1, and the interval of conve...