In this paper we introduce the concept of risk measure and give a detailed discussion of the characters of redundant risk and interchangeable risk. Additionally, we give a full explanation for the financial meanings of the risk allocation principles. We show the different properties between covariance risk allocation function and relative risk allocation function and prove that the commonly used risk budgeting model is a special form of risk allocation function. Finally, we try to reveal the equivalent relationship for each pair of risk allocation functions.

In this paper,applying the special structure of nonnegative Hamiltonian operators, the distribution of point spectrum of a class of nonnegative Hamiltonian operators is obtained.It is proved that imaginary axis is a subset of its continuous spectrum and resolvent set, by the above result, necessary and sufficient condition of invertibility for a class of nonnegative Hamiltonian operators is obtained.

By means of randomization, the concept of randomized truth degree and randomized logic pseudo-metric of formulas in R₀ three-valued propositional logic are introduced. The concept of randomized logic metric space is also introduced and it is proved that the new built randomized concepts are extensions of the corresponding concepts in quantified logic.

This paper introduced a new class L^*_(m) of heavy-tailed distribution functions, some properties about it are studied. Under the assumption that the distributions of the summands of a (non-standard) random walk {S_n}_n≥ 0 belong to L^*_(m), we obtain the asymptotic estimate for the distribution of maximum of {S_n}_n≥ 0. Application in risk theory is investigated.

In this paper, mathematical problems with nonlinear complementarity constraints are considered. By means of Fischer-Burmeister function, the nonlinear complementarity condition is transformed into a nonsmooth equation. Then, during the iteration, a corresponding smooth system approximates the nonsmooth equation. The smooth optimization is solve by SQP algorithm for standard constrained optimization. Global convergence of the algorithm is established under appropriate assumptions.

Under weaker condition, in this paper, we study the existence of positive solutions for the singular boundary value problem
x''' + a(t)F(t,x)=0, 0<t<1,
x(0)=x''(0)=x(1)=0
where a(t), F(t, x) may be singular at t=0, t=1 and x=0.

Exponential family nonlinear models, or generalized nonlinear models are generalized from generalized linear models and nonlinear regression. For separable and continous exponential family models (such as normal, gamma, and inverse Gaussian models), this paper discusses the tests for departures from nominal dispersion in the framework of longitudinal nonlinear models with varying dispersion and/or random effects. The score test statistics are constructed and their asymptotic distributions and local powers are derived. The properties of test statistics are investigated through Monte Carlo simulations. We illustrate our test methods by data concerning the yield of coastal Bermuda grass.

A time series model in random environment domain is presented, and the probability property of the model is discussed by utilizing the theory of stochastic stability on Markov Chains. Sufficient condition for adjoint non-recurence of the model is given.

The paper proposes a simple and convenient questionnaire design technique of sensitive questions survey.We discuss the estimator and variance of the proportion for the sensitive attribute. The result showed that under same protection of privacy , the efficient for the questionnaire design technique and traditional strategies have their own advantages and disadvantages by the different parameters.

Based on the moment of inertia of the samples on the surface of a unit sphere, the basic characterzation of d-dimentional samples with a uniform distribution on the surface of a unit sphere is proved, the strong consistency and asymptotic multinormality of the centroid and the eigenvalue estimators of the covariance matrix of the unit sphere uniform distribution are obtained, the asymptotic Chi squared statistic for testing uniformity of a given sample on the surface of a unit sphere is suggested. The consistency of the goodness-of-fit tests is proved and the empirical power of tests is made.

By using the cone theory and the fixed point index, we will study the existence of positive solutions of a class of boundary value problem with dependence on the first order derivative.

A k-edge coloring of a graph G is a mapping φ from E(G) to the color set {1,2,...,k} such that any two adjacent edges have different colors. The chromatic index χ'(G) is the smallest integer k such that G admits a k-edge coloring. In this paper, we prove that every graph G with maximum degree 6 which is embeddable in a surface of nonegative Euler characteristic has χ'(G)=6 if either G has no 4-cycle with a chord, or G has no 5- and 6-cycle with a chord.

The asymptotic synchronization in a lattice of x_i-coupled nonidentical Lorenz equations is considered, the external coupling matrix is an n× n irreducible symmetric real matrix having zero row sums and nonpositive off-diagonal elements. The uniform bounded dissipativeness of the coupled Lorenz systems is discussed by Lyapunov stability theory. Under this condition, applying Cauchy-Schwarz inequality to prove that asymptotic synchronization occurs for the coupled Lorenz systems with x-component coupling provided the coupling coefficient is sufficiently large. That is, the difference between any two components of a solution is bounded by the quantity O(ε) as t→∞, where ε is the maximal deviation of parameters of nonidentical Lorenz Equations.

In this artcle, a class model of impulsive competition system with infinitely distributed delays and feedback controls is considered. By using the continuation theorem of coincidence degree theory, homotopy invariance property and Lyapunov's approach, some sufficient conditions ensuring the existence and stability of positive periodic solutions of the system are obtained. The results improve and extend some recent works.

A necessary and sufficient condition is proved for a generalized steepest descent approximation to converge to the zeros of quasi-accretive locally bounded operators defined on proper subsets of uniformly smooth real Banach space. Related results deal with the strongly convergence of the scheme to a solution of equations involving φ-strongly quasi-accretive operators. These results extend and unify corresponding ones by Xu and Roach, Xu Zhongben and Jiang Yaolin, Chidume, Xu, Zhang and Roach, Zhou and others.

This paper is concerned with the existence and global exponential stability of periodic solutions for a nonlinear periodic system,arising from the description of the states of neurons in delayed Cohen-Grossberg type. We consider non-decreasing activations which may also have jump discontinuities in order to model the ideal situation where the gain of the neuron amplifiers is very high and tends to infinity. Under suitable assumptions on the interconnection matrices, we deduce some sufficient conditions ensuring existence as well as global exponential stability of periodic solution, the presented condition concerns the theory of M-matrices and is easy to check. Furthermore, due to the possible discontinuities of the activations functions, we introduce a suitable notation of limit to study the convergence of the output of the delayed neural networks.an numerical example are given to illustrate the theoretical results.

Based on the embedding theorem of fuzzy sets, we consider a sort of parameter family for the distribution of imprecise data and propose the corresponding point estimation approach. With the help of the multivariate distribution function, the estimates of the parameters for the distribution of LR-fuzzy data are obtained.

In this paper, a partially linear EV functional relationship model under longitudinal data is considered. By using a non-parametric kernel weight function and generalized least square method, the estimators of unknown parameter β, the variance of errors σ² as well as the unknown function g(.) are constructed. The asymptotic normality of $\widehat\beta$ and $\widehat\sigma^2$ is obtained. At the same time, the convergence rate of \widehat g(.) is investigated. The results obtained in this paper are good as the corresponding ones for independent samples.