The exact probability distribution of the first digit of integer powers up to an arbitrary but fixed number of digits is derived. Based on its asymptotic distribution, it is shown that it approaches Benford's law very closely for sufficiently high powers

In 1983, Bayliss, Goldstein, and Türkei [2] proposed a preconditioner based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [11]...

Given integer-valued and more general real-valued wagers, Feller(1968), Ethier and Khoshnevisan(2002) have established upper and lower bounds on the probability of ruin, which often turn out to be very close to each other. However, the exact calculation of these bounds depends on the unique...

We define the notion of power stable ideal in the polynomial ring R[X] over an integral domain and prove that any radical ideal in [], where is a Dedekind domain, is power stable. We then give an example of a primary ideal which is not power stable

DNA sequences exhibit significant intra-genomic fluctuations with respect to a model of conditional uniformity under which (1) the expected frequency of each dinuceotide (base step) is proportional to the product of the marginal (base) frequencies and (2) the observed number follows a Poisson...

We construct artificial boundary conditions (ABCs) for solving the diffusion equation in a 2D convex computational domain with a piecewise smooth artificial boundary. We start from the domain decomposition, representing the plane R R as a union of an interior (computational) and exterior domains...

In this paper we study the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions in Banach spaces.We derive conditions under which the integral solutions exit

We show a lower bound for the boundary crossing probability [0,1] : h(z) + with a Brownian bridge, a trend function and a boundary function. By that we get also a lower bound for the boundary crossing probability P{∃z ∈ [0,1] : h(z) + Bo(z) u(z)}. It turns out that the bound improves the...