As an improvement of the combinatorial realization of totally positive matrices via
the essential positive weightings of certain planar network by S. Fomin and A. Zelevinsky [7], in
this paper, we give a test method of positive definite matrices via the planar networks and the
so-called mixing-type sub-cluster algebras respectively, introduced here originally.
This work firstly gives a combinatorial realization of all matrices through planar network, and
then sets up a test method for positive definite matrices by LDU-decompositions and the horizontal weightings of all lines in their planar networks. On the other hand, mainly the relationship is built between positive definite matrices and mixing-type sub-cluster algebras.

The single machine parallel-batch scheduling with deteriorating jobs and rejection
is considered in this paper. A job is either rejected, in which a rejection penalty should be paid,
or accepted and processed on the machine. Each job’s processing time is an increasing linear
function of its starting time. The machine can process any number of jobs simultaneously as a
batch. The processing time of a batch is equal to the largest processing time of the jobs in the
batch. The objectives are to minimize the makespan and the total weighted completion time,
respectively, under the condition that the total rejection penalty cannot exceed a given upper
bound Q. We show that both problems are NP-complete and present dynamic programming
algorithms and fully polynomial time approximation schemes (F P T ASs) for the considered
problems.

This paper concerns the minimization problem of L2 norm of curl of vector fields
prescribed full trace on the boundary of a multiconnected bounded domain. The existence of
the minimizers in H1 are shown by orthogonal decompositions of vector function spaces and a
constructed auxiliary variational problem. And the H2 estimate of the type II divergence-free
part of the minimizers is established by div-curl-gradient type estimates of vector fields.

We explain and prove some lemmas of the approximate coupling and we give some
details of the Matlab implementation of this method. A particular invertible SDEs is used to
show the convergence result for this method for general d, which will give an order one error bounds.

Let {X, Xn, n ≥ 1} be a sequence of identically distributed pairwise negative quadrant dependent (PNQD) random variables and {an, n ≥ 1} be a sequence of positive constants with an = f(n) and f(θk)/f(θk?1) ≥ β for all large positive integers k, where 1 < θ ≤ β and
f(x) > 0 (x ≥ 1) is a non-decreasing function on [b, +∞) for some b ≥ 1. In this paper, we obtain the strong law of large numbers and complete convergence for the sequence {X, Xn, n ≥ 1},which are equivalent to the general moment condition ∑∞n=1 P(|X| > an) < ∞. Our results extend and improve the related known works in Baum and Katz [1], Chen at al. [3], and Sung[14].

In this work, we consider the inverse nodal problem for the Sturm-Liouville problem
with a weight and the jump condition at the middle point. It is shown that the dense nodes of the
eigenfunctions can uniquely determine the potential on the whole interval and some parameters.

In this paper, we consider testing the hypothesis concerning the means of two independent semicontinuous distributions whose observations are zero-inflated, characterized by
a sizable number of zeros and positive observations from a continuous distribution. The continuous parts of the two semicontinuous distributions are assumed to follow a density ratio model.
A new two-part test is developed for this kind of data. The proposed test takes the sum of
one test for equality of proportions of zero values and one conditional test for the continuous
distribution. The test is proved to follow a χ
2 distribution with two degrees of freedom. Simulation studies show that the proposed test controls the type I error rates at the desired level,
and is competitive to, and most of the time more powerful than two popular tests. A real data
example from a dietary intervention study is used to illustrate the usefulness of the proposed test.

This paper provides a selective review of the recent developments on econometric/statistical modeling in quantile treatment effects under both selection on observables and on
unobservables. First, we discuss identification, estimation and inference of quantile treatment
effects under the framework of selection on observables. Then, we consider the case where the
treatment variable is endogenous or self-selected, for which an instrumental variable method
provides a powerful tool to tackle this problem. Finally, some extensions are discussed to the
data-rich environments, to the regression discontinuity design, and some other approaches to
identify quantile treatment effects are also discussed. In particular, some future research works
in this area are addressed.

We deal with the boundedness of solutions to a class of fully parabolic quasilinear
repulsion chemotaxis systems
{
ut = ? · (?(u)?u) + ? · (ψ(u)?v), (x, t) ∈ ? × (0, T),
vt = ?v ? v + u, (x, t) ∈ ? × (0, T),
under homogeneous Neumann boundary conditions in a smooth bounded domain ? ? R
N (N ≥3), where 0 < ψ(u) ≤ K(u + 1)α, K1(s + 1)m ≤ ?(s) ≤ K2(s + 1)m with α, K, K1, K2 > 0 and
m ∈ R. It is shown that if α ? m < 4N+2 , then for any sufficiently smooth initial data, the
classical solutions to the system are uniformly-in-time bounded. This extends the known result
for the corresponding model with linear diffusion.