Gaussian graph model which studies the relationships between the independent random variables is one of the topic of machine learning. In this paper, a method is proposed to link the estimation of separate graphical models through a hierarchical penalty. By studying the highdimensional statistical properties of the new model, the asymptotic consistency and sparsistency of the proposed parameter estimation are proved.

Through compound distributions, a parameter λ is introduced to expand a family of distributions . It has been shown that ageing property DFR is transmitted from one random variable to the new one. In addition, the closure of this model under different stochastic orders viz. likelihood ratio order, shifted likelihood ratio orders and shifted hazard rate orders is discussed. Also, this method is applied to exponential distribution and Log-normal distribution. Finally, application of the extended distribution to a data set representing the remission times of bladder cancer patients is given and its goodness-of-fit is demonstrated.

Let $X_1,\cdots ,X_h$ be independent $ (N,d_1,\alpha _1 ),\cdots ,(N,d_h,\alpha _h)$-stable processes respectively, where $\alpha _1, \cdots , \alpha _h $ may be different real numbers in $(0,2]$. The corresponding multi-parameter process with stable components is defined as the following $N$-parameter random field on $\mathbf{R}^d\left(d= \sum\limits_{i=1}^{h}d_{i}\right)$, $X(t)= (X_1(t),X_2(t),\cdots,X_h(t)),$ $\forall t \in \mathbf{R}_+^N$. Under the condition that $N>d,$ it is proved that there is a (jointly continuous) local time for $X(t)$. The result is peculiar to the process $X(t)$, because there is no local time under any condition for the one-parameter process with stable components.

The paper discusses a perturbed Markov-modulated dual risk model with constant dividend barrier, in which, the gain arrivals, gains sizes and expenses are influenced by a Markov process. A system of integro-differential equations for the expected total discounted dividend payments until ruin is derived. Moreover, in the two-state model, explicit results are obtained when both claim amounts are exponentially distributed and mixed exponentially distributed. Finally, some numerical examples are given.

In this paper, some new exact tests and confidence intervals of variance components in panel data model with four random effects are developed by using generalized $p$ value and generalized confidence interval. Invariance of these tests and confidence intervals under scale transformation is also discussed. It is showed that the generalized $p$ value is feasible and effective to resolve the hypothesis testing problems with nuisance parameters. Simulation shows that the test of generalized $p$ value is powerful with its testing size approximating nominal level.

Using the method of composite expansion and the theory of differential inequality, a class of singularly perturbed boundary value problems for fourth order equations are studied. The formal asymptotic solutions of the problems are constructed by the method of composite expansion. Then the existence of solutions for the original problems and the uniform validity of the formal asymptotic solutions are proved by the theory of differential inequality. At last, an example is given to illustrate the significance of the obtained result.

This paper investigates the impulse control of functional differential equations with the general form of $\dot{x}(t)=f(t, x(t), x(t-r))$. By employing comparison theorems, it proves that if $f$ is continuous, the boundness or attractiveness of the solutions can be obtained by impulse control, and if $f$ satisfies the weak Lipschitz Conditions, the equations can be stablized, asymptotically stablized or exponentially stablized by impulse control, and the algorithm of impulse control is provided.

By using the fixed-point theorem on cone, the authors consider the multiple positive solutions for $m$- point fractional boundary value problem with p-Laplacian operator in infinite interval, sufficient conditions for the existence of at least three positive solutions are obtained. Moreover, an example is given to show the effectiveness of the works.

In this paper, the existence of single and multiple positive solutions of the Neumann boundary value problem $$ \left\{\begin{array}{ll} \ u''(t)+m^2(t)u(t)=f(t,u(t)), t\in(0,1),\\[2ex] u'(0)=0, u'(1)=0 \ \end{array} \right. $$ are studied. By using Dancer's global bifurcation theorem, the optimal sufficient conditions for the existence of single and multiple positive solutions of the above mentioned problem concerning the first eigenvalue of the relevant linear problem are obtained.

For a quadratic algebra $A=k\langle X\rangle/(\Re)$, when the set $\Re$ of relations satisfies certain symmetric condition, $A$ is an Artin-Schelter-regular PBW algebra. Furthermore, $A$ is a binomial skew polynomial ring, with respect to an enumeration of $X$.

In this paper, a higher order uniform convergence of the SIPG method for 1-d singularly perturbed problem of convection-diffusion type is analyzed on Shishkin mesh. A uniform error estimate of $\mathcal{O}((N^{-1}\ln N)^k)$ is obtained in energy norm, if $k$-th ($k\geq 1$) piecewise polynomial is used and the total number of element is $N$. The numerical experiments verify the theoretical results.

Using the results of crossing number of complete bipartite graph $cr(K_{5,n}) =Z(5,n)$ by Kleitman D J, the crossing numbers of the join graph of three specific 5-vertex graphs $G_{12},G_{15},G_{18}$ with the paths $P_{n}$ are given. In addition, an update to description of join graph of 5-vertex graphs with the paths is given too, whose crossing numbers are known.