A Set of Almost Automorphic Functions and Applications

For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.

The Comparative Study on Two Models of Syndrome Differentiation of the Hand, Foot and Mouth Disease: An Investigation Analysis of the Signs and Symptoms on 2325 Cases

Objective To realize the characteristics of "zheng" differentiation-treatment for hand, foot and mouth disease(HFMD), a new methodology of syndrome differentiation for different stages of HFMD has been explored. Methods Total of 2 325 cases with HFMD were recorded by distributing them into exterior syndrome stage, interior syndrome stage, severe syndrome stage and recovered syndrome stage, respectively, and the main symptoms and subsidiary symptoms of different stages of HFMD have been observed. The major and minor pathogenesis of HFMD in different stages were obtained, and compared with the "2010 Guideline for the Diagnosis and Treatment of HFMD". Results It was found that the major pathogenesis of exterior stage was defined as "the invation of the wenevil to the defender of the body with the collaterals got involved ", and the minor as "qi deficiency"; in interior stage, "the fury of Gan-Yang" was the main pathogenesis, and "qi in chaos and qi deficiency" was the minor; in severe syndrome stage, "the damage of heart, liver and lung" was the main pathogenesis, and "qi in chaos" was the minor; and the pathogenesis of recovered stage was "qi-yin deficiency". Compared with the "2010 Guideline for the Diagnosis and Treatment of HFMD", it showed that "the obstruction of the fei-pi qi by the mixture of shi-re evil" and "the mixture of shi-re" in vivo was quite difficult to be explained in completely different context in the general situation; in the severe stage, the TCM clinical characteristics of syndrome differentiation might lose; in the early acute severe cases, the phenomenon that xin-yang and fei-qi almost ran out was difficult to be observed, then, the line between the severe and the acute severe became vague.Conclusions The theory of syndrome differentiation by stages of HFMD was reasonable in the actual situation of clinical description on HFMD which was expected to be further tested and widely applied in the "zheng" differentiation-treatment of HFMD in the future.

A new type of adiabatic invariants for disturbed non-conservative nonholonomic system

According to the Herglotz variational principle and differential variational principle of Herglotz type, we study the adiabatic invariants for a non-conservative nonholonomic system. Firstly, the differential equations of motion of the non-conservative nonholonomic system based upon the generalized variational principle of Herglotz type are given, and the exact invariant for the non-conservative nonholonomic system is introduced. Secondly, a new type of adiabatic invariant for the system under the action of a small perturbation is obtained. Thirdly, the inverse theorem of the adiabatic invariant is given. Finally, an example is given.

Study on the Relationship between Insulin-Resistance and Syndrome Differentiation Typing in Hypertension Patients

：Objective:To find the relationship between insulin-resistance and Syndrome Differentiation type (SDT) in hypertensive patients.Methods: Two hundred and nine patients of early stage hypertention with no complications of heart, brain or kidney were selected and classified into 4 types according to SDT, the Liver-Fire exuberant type (A), the Phlegm-Dampness abundant type (B), the Yin-Deficiency and Yang-Excess type (C) and the Deficiency of both Yin and Yang type (D). Their insulin sensitivity was examined and compared with that of 40 healthy subjects.Results:(1) Compared with the healthy subjects, all hypertensive patients had apparent insulin resistance (P<0.05).If the insulin sensitivity of healthy subjects was defined as 1.00, that of patients of type A, B, C and D were 0.54, 0.58, 0.65 and 0.80 respectively. (2) The insulin sensitivity of patients in the 4 SDT groups were compared and no significant difference was found in comparison between group A, B and C, while significant difference was found when the other three groups were compared with group D (P<0.05), the insulin sensitivity of type D was close to that of the healthy subjects. (3) The fasting blood insulin of type D was obviously lower and the insulin sensitivity of type D was obviously higher than that of the other three types as a whole (P=0.0001). (4) Multivariate regression analysis demonstrated that insulin sensitivity was closely correlated with SDT (P=0.0001). Conclusion: Insulin resistance is one of the pathological basis for SDT in hypertension.

ON THE EXISTENCE OF SOLUTIONS TO D＇－VALUED STOCHASTIC DIFFERENTIAL EQUATIONS INVOLVING EVOLUTION DRIFT

In this paper, nonstandard analysis is employed to present an existence theory of -valued stochastic differential equations involving evolution drift. And (C0, 1)-evolution systems are also defined and investigated on dual multi-Hilbertian spaces.

Pathwise Uniqueness of the Solutions toVolterra Type Stochastic DifferentialEquations in the Plane

In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.

ON LEVIN TYPE COMPARISON THEOREMS FOR CERTAIN SECOND ORDER DIFFERENTIAL EQUATIONS

In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison theorems.

INTEGRABLE TYPES OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION SETS OF HIGHER ORDERS

Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publications both at home and abroad in recent years Based on these papers and in virtue of Leibniz formula,and transformation set technique,this paper puts forth the solution to nonlinear ordinary differential equation set of higher-orders, moveover,its integrability is proven.The results obtained are the generalization of those in the references.

On Some Bending Problems of Prismatic Shell with the Thickness Vanishing at Infinity

The present work is devoted to the bending problems of prismatic shell with the thickness vanishing at infinity as an exponential function. The bending equation in the zero approximation of Vekua＇s hierarchical models is considered. The problem is reduced to the Dirichlet boundary value problem for elliptic type partial differential equations on half-plane. The solution of the problem under consideration is constructed in the integral form.

H_(2)/H_(∞) Control for Stochastic Jump-Diffusion Systems with Markovian Switching

In this paper, a stochastic H2/H∞ control problem is investigated for Poisson jumpdiffusion systems with Markovian switching, which are driven by a Brownian motion and a Poisson random measure with the system parameters modulated by a continuous-time finite-state Markov chain.A stochastic jump bounded real lemma is proved, which reveals that the norm of the perturbation operator below a given threshold is equivalent to the existence of a global solution to a parameterized system of Riccati type differential equations. This result enables the authors to obtain sufficient and necessary conditions for the existence of H2/H∞ control in terms of two sets of interconnected systems of Riccati type differential equations.

Darcy-Forchheimer mangetized flow based on differential type nanoliquid capturing Ohmic dissipation effects

Hydromagnetic nanoliquid establish an extraordinary category of nanoliquids that unveil both liquid and magnetic attributes.The interest in the utilization of hydromagnetic nanoliquids as a heat transporting medium stem from a likelihood of regulating its flow along with heat transportation process subjected to an externally imposed magnetic field.This analysis reports the hydromagnetic nanoliquid impact on differential type(second-grade)liquid from a convectively heated extending surface.The well-known Darcy-Forchheimer aspect capturing porosity characteristics is introduced for nonlinear analysis.Robin conditions elaborating heat-mass transportation effect are considered.In addition,Ohmic dissipation and suction/injection aspects are also a part of this research.Mathematical analysis is done by implementing the basic relations of fluid mechanics.The modeled physical problem is simplified through order analysis.The resulting systems(partial differential expressions)are rendered to the ordinary ones by utilizing the apposite variables.Convergent solutions are constructed employing homotopy algorithm.Pictorial and numeric result are addressed comprehensively to elaborate the nature of sundry parameters against physical quantities.The velocity profile is suppressed with increasing Hartmann number(magnetic parameter)whereas it is enhanced with increment in material parameter(second-grade).With the elevation in thermophoresis parameter,temperature and concentration of nanoparticles are accelerated.

Normal forms of linear second order partial differential equations on the plane

The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane.We discuss the results obtained in the last decades and some problems,which are important both for the development of this theory and the applications.The reduction theorem,which was used to obtain many of recent results in the theory,is included in the paper in the parametric form together with proof.There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.

Harnack Differential Inequalities for the Parabolic Equation u_t= LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F>0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.

OSCILLATORY BEHAVIOR OF EVEN ORDER MIXED NEUTRAL DIFFERENTIAL EQUATIONS

In this paper,we shall consider a class of even order mixed neutral differential equations. Some new oscillations criteria of the solutions are obtained.

THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE 0F PERIODIC SOLUTIONS OF A TYPE OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence of periodic solutions of a type of neutral functional equation system are obtained,and at the same time, we present a method with formula shows how to find the periodicsolutions.

On a Class of Second Order Partial Differential Operators of Non-principal Type

§1.IntroductionThis paper deals with linear partial differential operators with real principalsymbol.Let P(x,D)be such an operator of mth order with C~∞ coefficients definedin an open subset Ω of R^n and p_m(x,ξ)be its principal symbol.According to thedefinition given by Duistermaat and Hmander(see[1]),P(x,D)is called ofprincipal type at x^0 ∈Ω if for any ξ∈R^n\0 satisfying p_m(x^0,ξ)=0,x=x^0 is not theprojection in Ω of the bicharacteristic strip of P(x,D)through(x^0,ξ).Under thiscondition,they proved that there exists a neighborhood U of x^0,U,such that forany real number s,