The main object of this paper is to investigate the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials. We first establish two relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. It can be found that many results obtained before are special cases of these two relationships. Moreover, we have a study on the sums of products of the Apostol-Bernoulli...

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials Bn(x;λ) in detail. The starting point is their Fourier series on [0, 1] which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain ...

In this paper, we use Hermite cubic finite elements to approximate the solutions of a nonlinear Euler– Bernoulli beam equation. The equation is derived from Hollomon’s generalized Hooke’s law for work hardening materials with the assumptions of the Euler–Bernoulli beam theory. The Ritz–Galerkin finite element procedure is used to form a finite dimensional nonlinear program problem, and a nonlin...

Let B n = b 1 +. .. + b n , n ≥ 1 where b 1 , b 2 ,. .. are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic order of the sums N n=1 d θ,D (B n), where d θ,D (B n) = #{d ∈ D, d ≤ n θ : d|B n } and D is a sequence of positive integers.

Let X1, X2, X3, . . . be i.i.d. random variables (Here ”i.i.d.” means ”independent and identically distributed” ), s.t. Pr[Xi = 1] = p, Pr[Xi = 0] = 1 − p. Xi is also called Bernoulli random variable. Let Sn = X1 + · · · + Xn. We will be interested in the random variable Sn which is called Binomial random variable (Sn ∼ B(n, p)). If you toss a coin for n times, and Xi = 1 represents the event t...

The design of a magnetic slit used for momentum definition and beam separation for high energy particle beams is discussed. (To be submitted to Review of Scientific Instruments) * Work supported by U. S. Atomic Energy Commission

In the framework of wireless power transmission applications, the design of planar phased arrays providing maximum beam collection efficiency (i.e., maximizing the ratio between the power radiated over a target area and the total power transmitted by the array) through a subarrayed architecture is addressed. The degrees-of-freedom of the synthesis problem at hand (i.e., the membership of the ar...

We show that the permanent of an n×n matrix with iid Bernoulli entries ±1 has probability o(1) of vanishing (or of attaining any other fixed value) in the limit n→∞.