LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation ｜z〉g and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula （Huygens-Fresnel integration transformation describing optical diffraction）. The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz -- rz^＊ in the ｜Z〉g representation, while ｜z〉g in phase space is graphically denoted by an ellipse.

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders, Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation. We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure. The method can be generalized to the other fractional soliton hierarchy.

On the generalized Cauchy function and new Conjecture on its exterior singularities

This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.

OPTIMAL CONTROL OF MARKOVIAN SWITCHING SYSTEMS WITH APPLICATIONS TO PORTFOLIO DECISIONS UNDER INFLATION

This article is concerned with a class of control systems with Markovian switching, in which an It5 formula for Markov-modulated processes is derived. Moreover, an optimal control law satisfying the generalized Hamilton-Jacobi-Bellman （HJB） equation with Markovian switching is characterized. Then, through the generalized HJB equation, we study an optimal consumption and portfolio problem with the financial markets of Markovian switching and inflation. Thus, we deduce the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds. Finally, for the CRRA utility function, we explicitly give the optimal consumption and portfolio policies. Numerical examples are included to illustrate the obtained results.

Nonlocal Theory of High-Temperature Superconductivity

The Boltzmann local physical kinetics forecasts the destruction of SC regime because of the heat movement of particles. Then, the most fundamental distinction between a strange metal and a conventional metal is the absence of well-defined quasi-particles. Here, we show that the mentioned “quasi-particles” are solitons, which are formed as a result of self-organization of ionized matter. Shortcomings of the Boltzmann physical kinetics consist in the local description of the transport processes on the level of infinitely small physical volumes as elements of diagnostics. The non-local physics leads to the theory superconductivity including the high temperature diapason. The generalized non-local non-stationary London’s formula is derived.

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebrasare obtained.Two expanding integrable systems are produced with the help of the generalized zero curvature equation.One of them has complex Hamiltion structure with the help of generalized Tu formula (GTM).

WEYL TRANSFORMATIONS ON MANIFOLDS Ⅱ. Generalized Trace and Inverse Mapping Formula

The two gate formula for a diffusion X_t in R^n is considered.The formula gives an expressionof the density function of X_t in terms of the path integral of the Brownian bridge starting from xand ending on y at time t.

MOMENT GENERATING FUNCTIONS OF RANDOM VARIABLES AND ASYMPTOTIC BEHAVIOUR FOR GENERALIZED FELLER OPERATORS

Describes the representation of moment generating function for the S-lambda type random variables. Higher order asymptotic formula for generalized Feller operators; Regular n-r order moment for the random variables.

Modular Invariance and Anomaly Cancellation Formulas in Odd Dimension Ⅱ

By studying modular invariance properties of some characteristic forms, we get some generalized anomaly cancellation formulas on(4 r-1)-dimensional manifolds with no assumption that the 3 rd de-Rham cohomology of manifolds vanishes. These anomaly cancellation formulas generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds. We also generalize our previous anomaly cancellation formulas on(4 r-1)-dimensional manifolds and the Han–Yu rigidity theorem to the(a, b) case.

Generalized Integral Representations for Functions with Values in C(V_(3,3))

By using the solution to the Helmholtz equation u-λu = 0(λ≥ 0),the explicit forms of the so-called kernel functions and the higher order kernel functions are given.Then by the generalized Stokes formula,the integral representation formulas related with the Helmholtz operator for functions with values in C(V3,3) are obtained.As application of the integral representations,the maximum modulus theorem for function u which satisfies Hu = 0 is given.

Comparison principle and stability criteria for stochastic differential delay equations with Markovian switching

In the present paper we first obtain the comparison principle for the nonlinear stochastic differentialdelay equations with Markovian switching. Later, using this comparison principle, we obtain some stabilitycriteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptoticstability in the pth mean and the pth moment exponential stability of such equations. Finally, an example isgiven to illustrate the effectiveness of our results.

Exact joint laws associated with spectrally negative Levy processes and applications to insurance risk theory

We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson＇s formula is provided.

Stability of Stochastic Differential Delay Equations with Markovian Switching

The main aim of this paper is to investigate the pth moment exponential stability of stochastic differential delay equations with Markovian switching.A specific Lyapunov function is introduced to obtain the required stability,and the almost sure exponential stability for the delay equations is discussed subsequently.

The Analysis of Newly Gained Feedback Loops After Introducing the Salary-Grade-Incitement System to HR Management

In order to find some effective management policy by the feedback archetypes analysis. To the minimum archetypes generating problem of a complex system, we transform the rate variable fundamental in-tree model of SD to a diagonal-0 branch-vector matrix using the method combining graph theory and algebra. We create the branch vector matrix formula to generate minimum archetypes of the network flow diagram. We solve the problem of how to generate the minimum archetypes of the complex system archetype generating method, and create a SD model of a human resource management in an organization using this new method. We prove that there are 16 new positive feedback loops and 17 negative feedback loops of 2 to 6 order of the HR management that is based on the incitements of performance levels. At the same time, these policies restrain the performance of the organization because of the increasing of cost. By analyze the growth limited structure model, we find the policy of increasing the performance of both the employee and the organization.

ON METHOD OF SOLUTION TO ONE CLASS OF SINGULAR INTEGRAL EQUATIONS WITH SINGULARITY OF ORDER ONE

In this paper, we study one kind of singular integral equations with singularity of order one. By the generalized Plemelj formula, this class of equations are transformed into a system of linear algebraic equations. In particular, we prove the existence of solution to the equation. The general solutions and the conditions of solvability are obtained in function class H.