The paper aims to investigate curvature inheritance (CI) symmetry in M-projectively flat spacetimes. It is shown that the CI spacetime a conformal motion. We have proved M-projective tensor follows property along vector field ξ, when admits conditions of both and motion or ξ. Also, we derived some results for with perfect fluid following Einstein equations (EFEs) cosmological term admitting an ...

We study the distribution of the exponential functional I(ξ, η) = ∫∞ 0 exp(ξt−)dηt, where ξ and η are independent Lévy processes. In the general setting, using the theory of Markov processes and Schwartz distributions, we prove that the law of this exponential functional satisfies an integral equation, which generalizes Proposition 2.1 in [9]. In the special case when η is a Brownian motion wit...

and Applied Analysis 3 Proof. Let ξ be fixed. If φ t is in S, then its diffraction Fresnel transform certainly exists. Moreover, differentiating the right-hand side of 2.3 with respect to ξ, under the integral sign, ktimes, yields a sum of polynomials, pk t ξ , say of combinations of t and ξ. That is, ∣ ∣ ∣ ∣ ∣ d dtk Fd ( φ ) ξ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ pk t ξ φ t exp ( i ( α1t 2 − 2tξ α2ξ ) 2γ1 )∣ ∣...

Let {p θ , θ ∈ Θ} be a family of probability densities with respect to µ where Θ is an open subinterval (finite, infinite or semi-infinite) of the real line. The Fisher Information for this family is defined by: I(θ) := E θ ∂ ∂θ log p θ (X) 2 Note that the densities p θ can be on arbitrary spaces; the only condition for defining I(θ) is that θ is real. The function ∂ ∂θ log p θ (X) is called th...

and Applied Analysis 3 aR (X, Y) ξ + b [S (Y, ξ)X − S (X, ξ) Y +g (Y, ξ) QX − g (X, ξ) QY] − τ n [ a n − 1 + 2b] [g (Y, ξ)X − g (X, ξ) Y] = 0. (24) Here, taking into account of (16), we have [η (Y)X − η (X)Y] [a (α 2 − ρ) + b (n − 1) (α 2 − ρ) − τ n ( a n − 1 + 2b)] + b [η (Y]QX − η (X)QY] = 0. (25) Let Y = ξ be in (25); then also by using (18) we obtain [−X − η (X) ξ] [a (α 2 − ρ) − τ n ( a n ...

1 Overview A typical result on large deviations of sums with random terms states that if ξ t are independent scalar random variables with zero means and such that " ξ t has as light tails as a Gaussian N (0, 4σ 2 t) random variable " , specifically, E exp{ξ 2 t /σ 2 t } ≤ O(1), (1) and S N = N t=1 ξ t , then Prob |S N | > t σ 2 1 + ... + σ 2 N ≤ O(1) exp{−O(1)t 2 } (2) (from now on, all O(1)'s ...

Let ξ be a real-valued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that (under a finite moment assumption) the probability distribution nσn(Mn(ξ)) 2 is universal in the sen...

We consider the nonlinear Sturm-Liouville problem −(p(x)u′(x))′ + q(x)u(x) = f(x, u(x)) + h(x), in (0, π), c00u(0) + c01u ′(0) = 0, c10u(π) + c11u ′(π) = 0, where: p ∈ C1[0, π], q ∈ C0[0, π], with p(x) > 0 for all x ∈ [0, π]; ci0 + ci1 > 0, i = 0, 1; h ∈ L2(0, π). We suppose that f : [0, π] × R → R is continuous and there exist increasing functions ζl, ζu : [0,∞)→ R, and positive constants A, B...