By combination of least square vector machine (LSSVM) with quadratic inference func-
tions (QIF), this paper construct a new estimation method for partially linear single index panel model
with fixed effects when responses from the same cluster are correlated. Under some regular condition,
asymptotic normality of parametric estimators and convergence rate of non-parametric estimator are
derived. The ˉnite sample performances of the proposed method are investigated by Monte Carlo simu-
lation under di?erent correlation structures, and compared with penalized quadratic inference functions
method (PQIF). The proposed estimation techniques are applied to analyse the relationship between
population structure and residents’consumption rate. Our research results show that the e±ciency
of estimators are improved by the proposed method, application e?ects are good, program operation
has high speed, it is particularly suitable for analysis of linear, nonlinear relationship among economic
variables and big data.

This paper obtains the functional sample path properties of subsequence's C-R increments
for l^p-valued, 1 · p < 1, Wiener processes. By which, the functional laws of iterated logarithm
for l^p-valued Wiener processes are generalized.

BS algorithm is one of the classical algorithms for multiple change-points detection,
it may bring about too many misjudgments and a high time complexity due to the procedure of global
CUSUM statistic. On one hand, the BS algorithm is an o?-line sequential method, therefore the data
timing information is not fully utilized. On the other hand, the principle of the BS algorithm to detect
the change-points is to maximize the CUSUM statistic, which does not consider the morphological
characteristics of the statistical constituent sequence. In view of these, the paper proposes an improved
BS algorithm, named Shape-based BS algorithm, which is based on local shape recognition. Basing
on the local pattern recognition of statistic sequence not only decreases the computational complexity,
but also avoids mutual interference among change-points, and it could also promote the robustness in
discerning change points. At last, this paper uses Shape-based BS algorithm to reduce the scenarios of
electric power, and achieves satisfactory practical results.

A class of nonlinear generalized forced disturbed Klein-Gordon equation is considered by
using the homotopic mapping method. Firstly, an approximate solution of soliton to typical
nonlinear equation is solved using the method of undetermined coe±cients for the hyperbolic tangent
functions. Then, the approximate solution of soliton to nonlinear forced disturbed equation is obtained
using the homotopic mapping principle. Finally, it is point out that the approximate solution of soliton
is an analytic expression, so we can carry on analytic operation to it. But these can not obtain for the
simple simulate method.

In this paper, the local bifurcations of an enzyme catalysed system is studied. Firstly, it
shows that this system has either 1 or 2 isolated equilibria, or a singular line. The dynamical properties
of each equilibria are given. Then, when the isolated equilibria are non-hyperbolic, it exhibits that this
system may undergo transcritical bifurcation and Hopf bifurcation. By Lyapunov quantity, the order
of weak focus is proved to be 1. Finally, numerical simulations are employed to illustrate the results obtained.

Based on the fact that tumor cells not only stimulate the proliferation of immune
effector cells but also have the inhibiting effect on the growth of the cells, a tumor-immune dynamical
model is described by expressing the comprehensive effect of tumor cells on immune system with a
positive or negative action rate coefficient. By investigating the global dynamics of the model, it is
found that the saddle-node bifurcation and the bistable phenomenon may occur, which implies that
that the final state of tumor development depends on the initial state, and the corresponding threshold
conditions are obtained. And the effect of the intrinsic input of effector cells and the action rate
coefficient of tumor cells on effector cells on the dynamics of the model is analyzed. The obtained
results show that the model may have complex dynamical behaviors when the inhibition effect of
tumor cells on effector cells is strong enough.

This paper mainly studies Cauchy integral formulas for two kinds of functions and
the related problems. Firstly, the Cauchy integral formula for right hypergenic functions in Clifford
analysis is given. Then, the properties of the right hypergenic quasi-Cauchy type integral are studied.
Finally, the Cauchy integral formula for bihypergenic functions in Clifford analysis is given.

This paper solves the block problems which play a fundamental role in the optimal
control theory under the operator-theoretic framework. Speciˉc duality theory is established for 4-block
problem by computing the appropriate annihilator and preannihilator of subspaces in such optimal
problem. Existence of optimal controllers is ensured and formulas for the performance index are
derived. It is also shown that the known results about duality theories for time-varying 1-block and
2-block problems are both special cases of the presented results in this paper. Moreover, an example
is given to prove that the optimum obtained by duality theory for a compact plant is time-varying allpass.

Based on logical similarity and residual implications, the paper discusses the robust-
ness of two important reasoning methods, the ˉve implication inference method (QIP) and the similarity
inference method (FSI). The concrete results of the robustness of QIP method under four common im-
plications are given, based on the revised Kleene implication the FSI's robustness conclusion, and a
preliminary comparison of the robustness of these two inference methods.

Topological structure is one of important research contents in the field of logical algebra.
In order to describe the topological structure of negative non-involutive residuated lattices, based
on the congruences induced by normal fuzzy ideals, uniform topological spaces are established and some
of their properties are discussed. The following conclusions are proved: (1) every uniform topological
space is ˉrst-countable, zero-dimensional, disconnected, locally compact and completely regular. (2) a
uniform topological space is a T1 space i? it is a T2 space. (3) the lattice and adjoint operations in a
negative non-involutive residuated lattice are continuous under the uniform topology, which make the
negative non-involutive residuated lattice to be topological negative non-involutive residuated lattice.
Meanwhile, some necessary and sufficient conditions for the uniform topological spaces to be compact
and discrete are obtained. Finally, the relationships between algebraic isomorphism and topological
homeomorphism in topological negative non-involutive residuated lattice are discussed. The results of
this paper have a positive role to reveal internal features of negative non-involutive residuated lattices
on a topological level.

A graph is bipartite if its vertex set can be partitioned into two subsets X and Y such
that every edge has one end in X and one end in Y ; such a partition (X; Y ) is called a bipartition of the
graph. A simple bipartite graph with bipartition (X; Y ) is called a complete bipartite graph if every
vertex in X is adjacent to every vertex in Y . Furthermore, the complete bipartite graph is denoted
by K_{m;n} if |X| = m and |Y| = n. It has known that if the in-degree of each vertex in an oriented
graph of Kn;n is a or b, where a and b are two non-negative integers, then there exist two non-negative
integers s and t satisfying the equations s + t = 2n and as + bt = n^2. This paper investigates oriented
graphs of Kn;n, and proves the following results. Let s and t be two arbitrary non-negative integers.
For two non-negative integers a and b satisfying the equations s+t = 2n and as+bt = n^2, there exists
an oriented graph of Kn;n such that the in-degree of each vertex is a or b. This paper shows that the
necessary condition is also su±cient for a complete bipartite graph K_{n;n}.