I present a new algorithm for computing binomial coefficients modulo 2N. The proposed method has an O(N3·Multiplication(N)+N4) preprocessing time, after which a binomial coefficient C(P, Q) with 0≤Q≤P≤2N-1 can be computed modulo 2N in O(N2·log(N)·Multiplication(N)) time. Multiplication(N) denotes the time complexity of multiplying two N-bit numbers, which can range from O(N2) to O(N·log(N)·log(...

We solve the Kato problem for divergence form elliptic operators whose heat kernels satisfy a pointwise Gaussian upper bound. More precisely, given the Gaussian hypothesis, we establish that the domain of the square root of a complex uniformly elliptic operator L = −div(A∇) with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate ‖ √ Lf‖2 ∼ ‖∇f‖2...

We consider sums involving the product of reciprocal binomial coefficient and polynomial terms and develop some double integral identities. In particular cases it is possible to express the sums in closed form, give some general results, recover some known results in Coffey and produce new identities.

We consider sums involving the product of reciprocal binomial coefficient and polynomial terms and develop some double integral identities. In particular cases it is possible to express the sums in closed form, give some general results, recover some known results in Coffey [8] and produce new identities.

By employing the Gaussian decomposition method, the analytical formulas of the Gaussian-beam Z-scan traces have been derived for an optically thin material exhibiting both refractive and absorptive parts of third-order nonlinearity, with Gaussian or hyperbolic secant squared laser pulses of femtosecond duration. The formulas have been verified experimentally with femtosecond-pulsed Z scans on a...

An estimation problem of the mean µ of an inverse Gaussian distribution IG(µ, C µ) with known coefficient of variation c is treated as a decision problem with entropy loss function. A class of Bayes estimators is constructed, and shown to include MRSE estimator as its closure. Two important members of this class can easily be computed using continued fractions

While it has long been known that many models of high energy scattering give cross sections which rise as ln s, the determination of the coefficient of this term is rarely given. We show that in gaussian and exponential eikonal models an exact expression for the cross section can be obtained, which demonstrates the ln s asyptoptic behaviour and determines its coefficient. The coefficient is uni...

While it has long been known that many models of high energy scattering give cross sections which rise as ln s, the determination of the coefficient of this term is rarely given. We show that in gaussian and exponential eikonal models an exact expression for the cross section can be obtained, which demonstrates the ln s asyptoptic behaviour and determines its coefficient. The coefficient is uni...

The Poisson regression is a popular approach in modeling count data. However, many situations often the variance of data greater than mean (over-dispersed data) and generalized or mixed models such as gamma (negative binomial), inverse Gaussian, lognormal, Lindley have been proposed alternatives to for describing over-dispersed In some situations, source over-dispersion large percentage zeros d...