16,029 research outputs found
Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations
Asymptotic stability and boundedness have been two of most popular topics in the study of stochastic functional differential equations (SFDEs) (see e.g. Appleby and Reynolds (2008), Appleby and Rodkina (2009), Basin and Rodkina (2008), Khasminskii (1980), Mao (1995), Mao (1997), Mao (2007), Rodkina and Basin (2007), Shu, Lam, and Xu (2009), Yang, Gao, Lam, and Shi (2009), Yuan and Lygeros (2005) and Yuan and Lygeros (2006)). In general, the existing results on asymptotic stability and boundedness of SFDEs require (i) the coefficients of the SFDEs obey the local Lipschitz condition and the linear growth condition; (ii) the diffusion operator of the SFDEs acting on a C2,1-function be bounded by a polynomial with the same order as the C2,1-function. However, there are many SFDEs which do not obey the linear growth condition. Moreover, for such highly nonlinear SFDEs, the diffusion operator acting on a C2,1-function is generally bounded by a polynomial with a higher order than the C2,1-function. Hence the existing criteria on stability and boundedness for SFDEs are not applicable andwesee the necessity to develop new criteria. Our main aim in this paper is to establish new criteria where the linear growth condition is no longer needed while the up-bound for the diffusion operator may take a much more general form
Efficient Genomic Interval Queries Using Augmented Range Trees
Efficient large-scale annotation of genomic intervals is essential for
personal genome interpretation in the realm of precision medicine. There are 13
possible relations between two intervals according to Allen's interval algebra.
Conventional interval trees are routinely used to identify the genomic
intervals satisfying a coarse relation with a query interval, but cannot
support efficient query for more refined relations such as all Allen's
relations. We design and implement a novel approach to address this unmet need.
Through rewriting Allen's interval relations, we transform an interval query to
a range query, then adapt and utilize the range trees for querying. We
implement two types of range trees: a basic 2-dimensional range tree (2D-RT)
and an augmented range tree with fractional cascading (RTFC) and compare them
with the conventional interval tree (IT). Theoretical analysis shows that RTFC
can achieve the best time complexity for interval queries regarding all Allen's
relations among the three trees. We also perform comparative experiments on the
efficiency of RTFC, 2D-RT and IT in querying noncoding element annotations in a
large collection of personal genomes. Our experimental results show that 2D-RT
is more efficient than IT for interval queries regarding most of Allen's
relations, RTFC is even more efficient than 2D-RT. The results demonstrate that
RTFC is an efficient data structure for querying large-scale datasets regarding
Allen's relations between genomic intervals, such as those required by
interpreting genome-wide variation in large populations.Comment: 4 figures, 4 table
Stochastic population dynamics under regime switching II
This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69-84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population
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