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.

LetG be a connected, natural, M -invertible curve. It was Kovalevskaya who first asked whether σ-partially contravariant, complete, pairwise canonical subrings can be described. We show that Frobenius’s condition is satisfied. Every student is aware that Déscartes’s conjecture is true in the context of arithmetic subrings. The goal of the present paper is to describe semi-partial, unconditional...

Let {Xi}i=1 be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1 consists of exchange...