All things in the data age can be described by data record. In data analysis, the problem
of matrix completion is to supplement some missing data. This kind of problem has been studied to a
certain extent. For instance, the desired results are achieved by solving the nuclear norm regularized
least squares problem. In this paper, starting from the duality of the problem, the alternating direction
method of multipliers (ADMM) is used to solve the problem. Under some assumptions, the global
convergence of the inexact dual ADMM (dADMM) are discussed. In the numerical experiments, by
comparing it with the primal ADMM (pADMM) to demonstrate the superiority of the algorithm.

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.

This paper is concerned with the oscillatory behavior of third-order nonlinear delay
dynamic equations on time scales. By using Riccati transformation and inequality techniques, the
Leighton-type, Philos-type and Kamenev-type oscillation criteria for a class of delay dynamic equations
on time scales are obtained. Our results improve and generalize the corresponding results in the existing
literatures, and give an example to verify the validity of the obtained results.

For functional clustering, similarity measure is one of the major approaches. However,
most researches measure the similarity of functional data from a single perspective, using either a
numerical distance approach or a curve shape approach. This paper proposes a new similarity measure
based on extreme point bias compensation. This new measure gives consideration to the numerical
distance and curve shape simultaneously. And the empirical results show the validity of the new
measure. Further, a multifunction clustering analysis method, the function entropy weight method, is
developed, which enriches the functional clustering analysis methods.

In this paper, new exact solutions of some nonlinear equations in mathematical physics
are constructed by using the symbolic computation software Maple. Firstly, the two-soliton solution for
an integrable nonlocal discrete mKdV equation is obtained via the Hirota’s bilinear method, and the
asymptotic behavior is analyzed. A kind of explicit expression for the N-soliton solution also is given.
Secondly, abundant families of travelling wave solutions of the multicomponent Klein-Gordon system
and long wave-short wave system are obtained directly by means of the Jacobi elliptic functions. When
the modulus m → 1, those solutions degenerate as the corresponding hyperbolic function solutions
including the bell-type soliton solution.

A new method for solving nonlinear Schr?odinger equation with wave operator is
proposed. Fristly, the characteristics of the equilibrium solution of the transformed di?erential equation
are analyzed in the differential dynamical system. Secondly, the envelope periodic solutions in different
forms are obtained by combining Lame equation, a new Lame function and Jacobi elliptic function
expansion method. Finally, under extreme conditions, new solitary wave solutions and other forms of
solutions are presented.

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.

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.

A digraph D is called quasi-transitive if, for every triple x, y, z of distinct vertices of
D such that xy and yz are arcs of D, there is at least one are between x and z. Seymour’s second
neighbourhood conjecture asserts that every oriented graph D has a vertex v such that d+D(x) 6 d++D (x),
where an oriented graph is a digraph with no cycle of length two. A vertex that satisfies Seymour’s
second neighbourhood conjecture is called a Seymour vertex. Fisher proved that Seymour’s second
neighbourhood conjecture restricted to tournaments is true, where any tournament contains at least
one Seymour vertex. Havet and Thomass′e proved that a tournament T with no vertex of out-degree
zero has at least two Seymour vertices. Observe that quasi-transitive oriented graphs is a superclass of
tournaments. In this paper, Seymour’s second neighbourhood conjecture on quasi-transitive oriented
graphs is the core problem. Notice the relationship between Seymour vertices of a quasi-transitive
oriented graph and an extended tournament. It is proved that the conjecture is true for quasi-transitive
oriented graphs. Furthermore, every quasi-transitive oriented graph has at least a Seymour vertex and
every quasi-transitive oriented graph with no vertex of out-degree zero has at least two Seymour vertices.

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 presents a modified DBSCAN clustering algorithm with adaptive parameter, and applies it to find potential information clusters of related fund accounts in the stock market.
The algorithm overcome the shortages that parameter ε in the traditional DBSCAN algorithm is oversensitive, and it cannot perform well on multi-densities data sets. Moreover, based on the characteristics
of real data, a new distance is defined to describe the similarity between two funds, which also makes
the modified algorithm better for solving practical problem. Finally, the effectiveness of the modified
algorithm is verified by numerical experiments based on simulated data and real data.

The optimal dividend payment problem in an insurance company whose surplus follows the classical Cram′er-Lundberg process with cluster claims is considered. A Hawkes process is
introduced so that the occurrence of a claim in the risky asset price triggers more sequent jumps.
Using dynamic programming principle and viscosity solution theory, it shows that the optimal value
function is a viscosity solution of the associated Hamilton-Jacobi-Bellman(HJB) equation. The optimal value function can be characterized as the smallest viscosity supersolution of the HJB equation.
Finally, some numerical results are exhibited and a barrier line strategy is introduced.

In this paper, a class of quasilinear elliptic equations is considered in $\mathbf{R}^{N}$. By virtue of critical point theory for nonsmooth functionals, the multiple weak solutions of the equations are obtained.

By relaxing the hypothesis that the in°uence of independent variable in parametric
spatial lag model is a linear or nonlinear function of some known form, a nonparametric spatial lag
model with random independent variables is considered. The GMM estimation method of the model
is constructed, the asymptotic properties of the estimators are derived and the small sample perfor-
mances of the estimates are investigated by Monte Carlo simulation.In addition, the estimation methods
proposed are applied to estimate the growth rate of TFP of China's provinces and municipalities.

Bayesian analysis for joint mean and variance models is studied in this paper, in
which Gibbs sampler and Metropolis-Hastings algorithm are used to calculate Bayesian estimations of
unknown parameters and Bayesian case deletion diagnostic. Simulation studies and a real example are
used to illustrate the proposed methodology.

According to the denseness and the closedness of range, the point spectrum of a
bounded linear operator is split into four disjoint parts, i.e., four classes of point spectra. For 3 × 3
upper triangular operator matrices, the possible point spectra SD,E,Fσp,i(MD,E,F )(i = 1, 2, 3, 4) are
given by means of the analysis method and block operator technique.

The deep learning method for pulmonary nodule detection is generally divided into
two stages: candidate nodule detection and false positive nodule elimination. Based on the two-stage
method, an incremental learning acceleration scheme is proposed that integrates new data to improve
the accuracy of the system. The training model of historical data screens new data and selects the
data with poor performance as an input for the continuous training of the two-stage model. The above
methods are tested on LUNA16 and TIANCHI17 two classic data sets. Using only half of the new
ones, the new model can achieve the same e?ect as the traditional two-stage method.

The topic of the asymptotic distribution of the Wilcoxon two-sample statistic under
an associated sample is investigated in this paper. Using Hoeffding decomposition, it is shown that the
asymptotic distribution of the Wilcoxon two-sample statistic under an associated sample is the normal
distribution, which generalizes the existing asymptotic result of the Wilcoxon two-sample statistic under
a negatively associated sample.

For two-stage series repairable system, a change-point control chart is used to monitor
the system to reveal state of the system, and then the corresponding maintenance policy is carried
out. Firstly, a change-point control chart is given to monitor the multistage system. Secondly, the
possible scenarios and the probability of occurrence of each scenario are analyzed, assuming that
assignable causes may occur in both two-stage, which follow general distribution; and the probability
of the corresponding maintenance policy is obtained based on the scenarios. Thirdly, a procedure
for calculating average revenue is presented according to renew-reward theory. Then, a numerical
example is given to illustrate the application of the proposed integrated model. The results show
that the integrated model outperforms the maintenance model in terms of average revenue earning.
Finally, using fractional factor design, a sensitivity analysis is conducted to develop insights into input
parameters that in°uence the integration e?orts.

In this paper, the asymptotic properties of $\widehat{\alpha}$ are considered, which is a least squares estimator for the parameter $\alpha$ of a weighted fractional bridge $$X_0=0,~~ \text{d}X_t=-\alpha\frac{X_t}{T-t}\text{d}t +\text{d}B^{a,b}_t, \ \ 0\leq t0, T>0$. The estimator has various convergence depending on the value of $\alpha$ as $t\rightarrow T.$ The rate of corresponding convergence is proved as well.