All polynomials in this paper are supposed to have real coefficients. Polynomials which can be represented in the form P(X) = c ad1-x) " U + XY, with all akl > 0 or all akl < 0, (1) k+l 0 have been introduced and studied by 6.6. Lorentz i[ I]; we shall call them polynomials with positive or negative (more exactly non-negative or non-positive) coefficients, respectively, or simply Lorent...

We introduce the class of Beta-Wishart random matrices and develop a comprehensive eigenvalue theory for it. The Beta-Wishart matrices generalize the classical real, complex, and quaternion Wishart matrices to a unform and continuous model valid for any β > 0. As most distributions are expressed in terms of the hypergeometric function of a matrix argument, many new identities for this function ...

1 Review of Probability Random variables are denoted by X, Y , Z, etc. The cumulative distribution function (c.d.f.) of a random variable X is denoted by F (x) = P (X ≤ x), −∞ < x < ∞, and if the random variable is continuous then its probability density function is denoted by f(x) which is related to F (x) via f(x) = F ′(x) = d dx F (x) F (x) = ∫ x −∞ f(y)dy. The probability mass function (p.m...

This text describes the FD-NU method and its implementation for the BENCHOP-project. 1 Spatial discretizations For example, under the Black-Scholes model European option prices u satisfy the PDE ut(s, t) + 1 2 σsuss(s, t) + rsus(s, t)− ru(s, t) = 0, s > 0, t ∈ [0, T ), (1) where σ and r are the volatility and interest rate, respectively. We employ quadratically refined grids defined by si = [( ...

Let G be a graph with n vertices and m edges. The degree sequence of G is denoted by d1 ≥ d2 ≥ · · · ≥ dn. Let A(G) and D(G) = diag(di : 1 ≤ i ≤ n) be the adjacency matrix and the degree diagonal matrix of G, respectively. The Laplacian matrix of G is L(G) = D(G) − A(G). It is well known that L(G) is a symmetric, semidefinite matrix. We assume the spectrum of L(G), or the Laplacian spectrum of ...

Let the randomized query complexity of a relation for error probability ǫ be denoted by Rǫ(·). We prove that for any relation f ⊆ {0, 1} × R and Boolean function g : {0, 1} → {0, 1}, R1/3(f ◦g ) = Ω(R4/9(f) ·R1/2−1/n4(g)), where f ◦g n is the relation obtained by composing f and g. We also show that R1/3 (

We consider the problem of obtaining a Bahadur representation of sample quantiles in a certain dependence context. Before stating in what a Bahadur representation consists, let us specify some general notation. Given some random variable Y , F(·) = FY (·) is referred as the cumulative distribution function of Y , ξ(p) = ξY (p) for some 0 < p < 1 as the quantile of order p. If F(·) is absolutely...

Since the quantum searching algorithm was first proposed by Grover [1], several generalizations of the original algorithm have been developed [2]-[4]. The generalized algorithm that we will realize can be posed as follows. Let N basis states of a system constitute set D. A function F is defined as F : D → {0, 1}. The states satisfying F (x) = 1 are defined as marked states, which constitute set...

Given a commutative ring R with identity 1?0, let the set Z(R) denote of zero-divisors and Z*(R)=Z(R)?{0} be non-zero R. The zero-divisor graph R, denoted by ?(R), is simple whose vertex Z*(R) each pair vertices in are adjacent when their product 0. In this article, we find structure Laplacian spectrum graphs ?(Zn) for n=pN1qN2, where p&lt;q primes N1,N2 positive integers.

We consider the problem of obtaining a Bahadur representation of sample quantiles in a certain dependence context. Before stating in what a Bahadur representation consists, let us specify some general notation. Given some random variable Y , F(·) = FY (·) is referred as the cumulative distribution function of Y , ξ(p) = ξY (p) for some 0 < p < 1 as the quantile of order p. If F(·) is absolutely...