In this paper, the apparent resistivity, being a concept commonly used in the resistivity well-logging, is deeply analyzed. Some properties which should be satisfied by the apparent resistivity, as well as the rationality of this concept are explored by means of theoretical consideration with related numerical examples.

In this paper, the existence and stability of a steady state are investigated. The existence theory and consisitency conditions of the solution are gotten.Furthermore, the instability of the steady state are proved by using the variational characterization of eigenvalues.

In this paper, some differential inequalities are established for a class of second order nonlinear differential equations, and some zero point comparison results are derived by using these inequalities. The results generalize some classical Sturm Comparison Theorems.

A class of nonlinear singularly perturbed initial-boundary value problem with ultra parabolic equation is considered. Firstly, the comparison theorem is introduced for corresponding problem. Then the formally asymptotic solution is constructed by using the singular perturbation method. Finally, using the comparison theorem, the existence and its asymptotic behavior of generalized solution for the problem are proved.

Based on new non-semisimple matrix Lie algebras, the general method of constructing the nonlinear bi-integrable couplings of soliton hierarchy is introduced. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. As an application, the nonlinear bi-integrable couplings of Broer-Kaup-Kupershmidt hierarchy and their Hamiltonian structures are given. Finally, some errors exist in reference are pointed out, and a set of new formulae using the theory of source are set up, also the nonlinear bi-integrable couplings of Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is derived based on the new formulae.

Boundary value problems $A(y)\Psi_f^{*+}(y)+B(y)\Psi_f^{*-}(y)=G(y)L(\Psi_f^{*+}(y),\Psi_f^{*-}(y))$ for hypergenic functions in Clifford analysis are mainly discussed. Firstly, the related properties of hypergenic quasi Cauchy integral are considered; secondly, using Schauder fixed point principle the existence of the solution to the nonlinear boundary value problem is proved; finally, the existence and uniqueness of the solution to the linear boundary value problem are proved by virtue of the principle of contraction mapping.

Firstly, the $A_n(R)$-valued higher order Teodorescu operator $T$ in $\mathbf{R}^n$ is defined and its properties in $L^\gamma$ space are discussed. Secondly, its norm is estimated and a modified higher order Teodorescu operator $T^*$ is introduced. And then, that the operator $T^*$ has a unique fixed point by the Banach’s contract mapping principle is proved. Finally, that the Mann iterative sequence strongly converges to the fixed point of $T^*$ is proved and an iterative sequence of the solution of a singular integral equation is given.

This paper proved a class of improved Hardy-type inequalities on anistropic Heisenberg group by some subtle skills, which extend some recent results of improved Hardy-type inequalities on Heisenberg group.

This paper investigates an online hierarchical scheduling problem on three parallel identical machines. Each job has a unit processing time and a hierarchy 1 or 2. Each machine has also a hierarchy 1 or 2. The job with a hierarchy 1 can only be processed on the machine with hierarchy 1 and the job with a hierarchy 2 can be processed on any one of three machines. The goal is to minimize the total completion time of all jobs. For the case where one machine with hierarchy 1 and two machines with hierarchy 2, an optimal online algorithm with competitive ratio of 17/14 is presented. For the case where two machines with hierarchy 1 and one machine with hierarchy 2, an optimal online algorithm with competitive ratio of 43/36 is presented.

With the help of rearrangement, this paper finds one class of functions in $H^p(\mathbf{R}^n)(0<p\leq 1)$ whose $H^p$-norms and $L^p$-norms are equivalent. More precisely, for each $f\in L^p(\mathbf{R}^n)$, there exists a function $g\in H^p(\mathbf{R}^n)$ satisfying $d_f=d_g$ and $\Vert g\Vert_{L^p}\leq\Vert g\Vert_{H^p}\leq C_p\Vert g\Vert_{L^p}$. Moreover, this paper also introduces a method to construct an $L^\infty$-atom of $H^p(\mathbf{R}^n)$.

In the paper, an implicit monotone projection iterative scheme with errors is designed in Hilbert spaces. The iterative sequence is proved to be strongly convergent to the common zero of the sum of infinite family of $m$-accretive mappings and inversely strongly accretive mappings, which extends the previous corresponding studies from the finite cases to infinite ones. Secondly, by using splitting methods, one kind of $p$-Laplacian-like parabolic systems is converted to operator equations. The existence of the non-trivial solution of the p-Laplacian-like parabolic systems is obtained and the relationship between the non-trivial solution and the common zero of the sum of infinite $m$-accretive mappings and inversely strongly accretive mappings is being set-up. Finally, the iterative approximate sequence of the non-trivial solution of the $p$-Laplacian-like parabolic systems is constructed. The work done in this paper extends and complements some previous corresponding work.

This paper considers the problem of variable selection in high-dimensional longitudinal linear regression models with monotone missing patterns. A new variable selection procedure is proposed based on the smooth-threshold inverse probability weighted generalized estimating equation. The proposed procedure avoids the convex optimization problem without using penalty functions. Besides, the proposed method can automatically eliminate inactive predictors by setting the corresponding parameters to be zero, and simultaneously estimate the nonzero regression coefficients. Under some regularity conditions, the variable selection procedure is proved to have Oracle property. Finally, some simulation studies are conducted to examine the finite sample property of the proposed variable selection procedure.