Integration on supermanifolds and a generalized Cartan calculus

A suggestion by Berezin for a method of integration on supermanifolds is given a precise differential geometric meaning by assuming that a supermanifold is the total space of a fibre bundle with connection. The relevant objects for integration are identified as suitable horizontal/vertical projections of hyperforms. The latter are generalizations of differential forms having both covariant and contravariant indices. The exterior calculus of these projected hyperforms is developed, analogously to the Cartan calculus, by introducing appropriate derivations and determining their commutators, respectively anticommutators.     

New applications of the calculus of variations in the large to nonlinear elasticity

Global methods of the calculus of variations and the infinite dimensional critical point theories of Morse and Ljusternik are applied to investigate the structure of the equilibrium states of thin flexible elastic plates under general body forces. The arguments used are equally applicable to broad classes of physical systems governed by nonlinear elliptic partial differential equations.     

On Symplectic and Multisymplectic Structures and Their Discrete Versions in Lagrangian Formalism

We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively.We also explore their certain difference discrete counterparts in the relevant regularly discretized finite and infinite dimensional Lagrangian systems by means of the difference discrete variational principle with the difference being regarded as an entire grometric object and the noncommutative differential calculus on regular lattice.In order to show that in all these cases the symplectic and multisymplectic preserving properties do not necessarily depend on the relevant Euler-Lagrange equations,the Euler-Lagrange cohomological concepts and content in the configuration space are employed.     

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

In this paper an efficient computational method based on extending the sensitivity approach(SA) is proposed to find an analytic exact solution of nonlinear differential difference equations.In this manner we avoid solving the nonlinear problem directly.By extension of sensitivity approach for differential difference equations(DDEs),the nonlinear original problem is transformed into infinite linear differential difference equations,which should be solved in a recursive manner.Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained.Numerical examples are employed to show the effectiveness of the proposed approach.     

Hamiltonian systems on fibered manifolds

We offer a new geometric theory of Hamiltonian systems with an infinite number of degrees of freedom in which the Hamiltonian operators are nonlinear differential operators on fields. The Poisson bracket is carried into the vertical bracket by the mapping between functionals and Hamitonian operators which is established by a Hamiltonian structure.     

构造非线性发展方程的无穷序列复合型类孤子新解

为了构造非线性发展方程的无穷序列复合型类孤子新解, 进一步研究了G'(ξ)/G(ξ) 展开法. 首先, 给出一种函数变换, 把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题. 然后, 利用Riccati方程解的非线性叠加公式, 获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解. 在此基础上, 借助符号计算系统Mathematica, 构造了改进的(2+1)维色散水波系统和(2+1)维色散长波方程的无穷序列复合型类孤子新精确解. 关键词： G'(ξ)/G(ξ)展开法')" href="#">G'(ξ)/G(ξ)展开法 非线性叠加公式 非线性发展方程 复合型类孤子新解     

On the variational noncommutative poisson geometry

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.     

三分量磁场延拓的递推算法

提出一种三分量磁场延拓的递推算法,主要目的是解决三分量磁场的向下延拓问题.应用微分学基本原理,将具有相同水平坐标,不同垂向坐标的点上的三分量磁场,通过磁场的垂向偏导数的积分联系起来.根据磁场向量散度和旋度的性质,将磁场的垂向偏导数与水平方向的偏导数联系起来.这样,在已知一个平面上的磁场时,可以通过递推的办法逐层求取该平面以下任意平面上的三分量磁场.应用实例证明方法的快速性和准确性.     

Formal study of a chemical reaction by Grad expansion of the Boltzmann equation. II

In a previous communication the Boltzmann equations for a bimolecular chemical reaction were transformed into an infinite system of quadratic differential equations. It is shown here that the reaction rate may be calculated approximately by using the first differential equation only, which is a generalization of the phenomenological law of chemical kinetics. Then, a general method for solving the expanded Boltzmann equations by successive approximations is proposed and studied to first order.     

Stochastic differential calculus,the Moyal *-product,and noncommutative geometry

A reformulation of the Itô calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator d satisfying d² = 0 and the Leibniz rule). In this calculus, differentials do not commute with functions. The relation between both types of differential calculi is mediated by a generalized Moyal *-product. In contrast to the Itô calculus, the new framework naturally incorporates analogues of higher-order differential forms. A first step is made towards an understanding of their stochastic meaning.     

A bicovariant differential algebra of a quantum group

A bicovariant differential algebra of four basic objects (coordinate functions, differential 1-forms, Lie derivatives and inner derivations) within a differential calculus on a quantum group is shown to be produced by a direct application of the cross-product construction to the Woronowicz differential complex, whose Hopf algebra properties account for the bicovariance of the algebra. A correspondence with classical differential calculus, including Cartan identity, and some other useful relations are considered. An explicit construction of a bicovariant differential algebra on GL_q(N) is given and its (co)module properties are discussed.     

Geometry of quantum principal bundles I

A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential forms on the base manifold with an appropriate differential calculus on the structure quantum group. Relations between the calculus on the group and the calculus on the bundle are investigated. A concept of (pseudo)tensoriality is formulated. The formalism of connections is developed. In particular, operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. Generalizations of the first Structure Equation and of the Bianchi identity are found. Illustrative examples are presented.     

Differential calculus on Jordan algebras and Jordan modules

Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules.

     

On algebraic models of dynamical systems

We describe a universal algebraic model which, being read appropriately, yields (periodic and infinite) discrete dynamical systems, as well as their continuous limits, which cover all differential scalar Lax systems. For this model we give: Two different constructions of an infinity of integrals; modified equations; deformations; infinitesimal automorphisms. The basic tools are supplied by symbolic calculus and the abstract Hamiltonian formalism.     

Analysis of a time fractional wave-like equation with the homotopy analysis method

The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when ?f=?g=−1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus.     

On the Singularity Structure of WKB Solution of the Boosted Whittaker Equation: its Relevance to Resurgent Functions with Essential Singularities

Inspired by the symbol calculus of linear differential operators of infinite order applied to the Borel transformed WKB solutions of simple-pole type equation [Kamimoto et al. (RIMS Kôkyûroku Bessatsu B 52:127–146, 2014)], which is summarized in Section 1, we introduce in Section 2 the space of simple resurgent functions depending on a parameter with an infra-exponential type growth order, and then we define the assigning operator \({\mathscr{A}}\) which acts on the space and produces resurgent functions with essential singularities. In Section 3, we apply the operator \({\mathscr{A}}\) to the Borel transforms of the Voros coefficient and its exponentiation for the Whittaker equation with a large parameter so that we may find the Borel transforms of the Voros coefficient and its exponentiation for the boosted Whittaker equation with a large parameter. In Section 4, we use these results to find the explicit form of the alien derivatives of the Borel transformed WKB solutions of the boosted Whittaker equation with a large parameter. The results in this paper manifest the importance of resurgent functions with essential singularities in developing the exact WKB analysis, the WKB analysis based on the resurgent function theory. It is also worth emphasizing that the concrete form of essential singularities we encounter is expressed by the linear differential operators of infinite order.     

A survey on the stability of fractional differential equations

Recently, fractional calculus has attracted much attention since it plays an important role in many fields of science and engineering. Especially, the study on stability of fractional differential equations appears to be very important. In this paper, a brief overview on the recent stability results of fractional differential equations and the analytical methods used are provided. These equations include linear fractional differential equations, nonlinear fractional differential equations, fractional differential equations with time-delay. Some conclusions for stability are similar to that of classical integer-order differential equations. However, not all of the stability conditions are parallel to the corresponding classical integer-order differential equations because of non-locality and weak singularities of fractional calculus. Some results and remarks are also included.     

Dirichlet forms and white noise analysis

We use the white noise calculus as a framework for the introduction of Dirichlet forms in infinite dimensions. In particular energy forms associated with positive generalized white noise functionals are considered and we prove criteria for their closability. If the forms are closable, we show that their closures are Markovian (in the sense of Fukushima).     

Effect of viscous dissipation on natural convection flow between vertical parallel plates with time-periodic boundary conditions

This article investigates the natural convection flow of viscous incompressible fluid in a channel formed by two infinite vertical parallel plates. Fully developed laminar flow is considered in a vertical channel with steady-periodic temperature regime on the boundaries. The effect of internal heating by viscous dissipation is taken into consideration. Separating the velocity and temperature fields into steady and periodic parts, the resulting second order ordinary differential equations are solved to obtain the expressions for velocity, and temperature. The amplitudes and phases of temperature and velocity are also obtained as well as the rate of heat transfer and the skin friction on the plates. In presence of viscous dissipation, fluids of relatively small Prandtl number has higher temperature than the channel plates and as such, heat is being transferred from the fluid to the plate.