Abstract:
The joint normality of two random vectors is obtained based on normal conditional with linear regression and constant covariance matrix of each vector given the value of the other without assuming the existence of the joint density. This result is applied to a characterization of matrix variate normal distribution.

Abstract:
Consider the max-stable process $\eta(t) = \max_{i\in\mathbb N} U_i \rm{e}^{\langle X_i, t\rangle - \kappa(t)}$, $t\in\mathbb{R}^d$, where $\{U_i, i\in\mathbb{N}\}$ are points of the Poisson process with intensity $u^{-2}\rm{d} u$ on $(0,\infty)$, $X_i$, $i\in\mathbb{N}$, are independent copies of a random $d$-variate vector $X$ (that are independent of the Poisson process), and $\kappa: \mathbb{R}^d \to \mathbb{R}$ is a function. We show that the process $\eta$ is stationary if and only if $X$ has multivariate normal distribution and $\kappa(t)-\kappa(0)$ is the cumulant generating function of $X$. In this case, $\eta$ is a max-stable process introduced by R. L. Smith.

We introduce a new class of the slash distribution using the epsilon half normal distribution. The newly defined model extends the slashed half normal distribution and has more kurtosis than the ordinary half normal distribution. We study the characterization and properties including moments and some measures based on moments of this distribution. A simulation is conducted to investigate asymptotically the bias properties of the estimators for the parameters. We illustrate its use on a real data set by using maximum likelihood estimation.

Abstract:
We demonstrate a new technique for characterizing two-photon quantum states based on joint temporal correlation measurements using time resolved single photon detection by femtosecond upconversion. We measure for the first time the joint temporal density of a two-photon entangled state, showing clearly the time anti-correlation of the coincident-frequency entangled photon pair generated by ultrafast spontaneous parametric down-conversion under extended phase-matching conditions. The new technique enables us to manipulate the frequency entanglement by varying the down-conversion pump bandwidth to produce a nearly unentangled two-photon state that is expected to yield a heralded single-photon state with a purity of 0.88. The time-domain correlation technique complements existing frequency-domain measurement methods for a more complete characterization of photonic entanglement in quantum information processing.

Abstract:
The joint distribution of two off-diagonal Wishart matrix elements was useful in recent work on geometric probability [Finch 2010]. Not finding such formulas in the literature, we report these here.

Abstract:
The two-dimensional normal grain growth has been simulated with Monte Carlo method. With a newly modified algorithm, the attained time exponent of grain growth n equals 0.49±0.01,very close to the theoretical value 0.5. By simulating the complete process of normal grain growth, the grain size distribution is found to be initially a gamma distribution, then varies continuously and slowly with time, finally approaches the function proposed by Hillert in 1965 at the quasi steady grain growth stage. The so-called "self-similarity" of the grain size distribution is discussed according to the new simulation results.

Abstract:
We compute the two-particle quantities relevant for superconducting correlations in the two-dimensional Hubbard model within the dynamical cluster approximation. In the normal state we identify the parameter regime in density, interaction, and second-nearest-neighbor hopping strength that maximizes the $d_{x^2-y^2}$ superconducting transition temperature. We find in all cases that the optimal transition temperature occurs at intermediate coupling strength, and is suppressed at strong and weak interaction strengths. Similarly, superconducting fluctuations are strongest at intermediate doping and suppressed towards large doping and half-filling. We find a change in sign of the vertex contributions to $d_{xy}$ superconductivity from repulsive near half filling to attractive at large doping. $p$-wave superconductivity is not found at the parameters we study, and $s$-wave contributions are always repulsive. For negative second-nearest-neighbor hopping the optimal transition temperature shifts towards the electron-doped side in opposition to the van Hove singularity which moves towards hole doping. We surmise that an increase of the local interaction of the electron-doped compounds would increase $T_c$.

Abstract:
In this paper, using Marshall-Olkin transformation, a new class of Extended Power Log-normal distribution which includes the Power Log-normal and Log-normal distributions as special cases is introduced. Its characterization and statistical properties are studied. A real survival dataset is analyzed and the results show that the proposed model is flexible and appropriate.

Abstract:
We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for probability measures on $\mathbb{R}^+$. Finally, we describe the density of the free multiplicative analogue of the normal distributions as an example and prove unimodality and some symmetry properties of these measures.