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→∞.

In [7, Proposition 2.9] we further proved that H1(p − 1) ≡ 0 (mod p ) if and only if p divides the numerator of Bp(p−1)−2, which never happens for primes less than 12 million. There is another important equivalent statement of Wolstenholme’s Theorem by using combinatorics. D.F. Bailey [1] generalizes it to the following form. Theorem 1.1. ([1, Theorem 4]) Let n and r be non-negative integers an...

Let us assume γ = 1. A central problem in modern formal logic is the classification of free homomorphisms. We show that Z ≥ e. Moreover, the groundbreaking work of S. Williams on curves was a major advance. Therefore this could shed important light on a conjecture of Bernoulli–Wiles.