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.

Quantile autoregressive (QAR) models, commonly adopted in varying-coefficients time
series modelling, has been shown its popularity in both theoretical and empirical studies. Equipped
with autoregressive structure, QAR models sometimes involve extra information in the data collection
process which is known as dependent auxiliary information. An empirical likelihood approach is used
to construct the quantile estimates. The asymptotic normality of the estimates is established conditionally on the lagged values of the response. Under the framework of empirical likelihood method with
dependent auxiliary information, the Wald test statistics are developed for testing the linear restriction
on the parameters. Both the simulation and the empirical study results indicate that the proposed
method yields more efficiency than the traditional one. Therefore, the results for general constant
coefficients QAR model under independent and identically distributed assumptions could be extended
to a class of varying coefficients QAR model with dependent structure.

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.

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.

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.

The controllability of a class of hierarchical age-structured population system is studied, in which the state equation is described by a nonlinear partial integro-differential equation with
a boundary condition of global feedback. The approximate controllability of the system is established
by means of frozen coefficients, a result for linear system and the Kakutani fixed point theorem of
multi-valued mappings. The conclusion shows that the population state can be adjusted by individuals
migration process.

In this paper, a temperature field model is constructed by using non-Fourier heat
conduction law, i.e. a class of singularly perturbed hyperbolic equations with small parameters in
an unbounded domain. The singularly perturbed two-parameter hyperbolic nonlinear equations with
discontinuous coefficients are obtained when the heat conduction coefficients jump due to sharp temperature changes. By using the singularly perturbed biparametric expansion method, the asymptotic
solution of the problem is obtained. Firstly, the expansion of the problem is obtained by using the
singularly perturbed method, and the existence and uniqueness of the internal and external solutions
are obtained. The position expression of the jump of the thermal conductivity coefficient is determined
by the method of separating variables, and the slit method is used to connect the slits of the two sides of
the jump position of the thermal conductivity coefficient, thus the asymptotic expansion of the solution
is obtained. Secondly, the uniform validity of the asymptotic solution is obtained by estimating the
residual term, and the distribution of the complete temperature field is obtained.

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.

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.

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.

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.