无限维空间中的实零点定理

在本文中，我们建立了无限维空间中的实零点定理，同时从仿射空间的拓扑结构和域的序结构两个方面，分别刻划了适合无限维实零点定理的序域．此外，本文有例子表明，对任意的基数α，确实存在适合α维实零点定理的序域．     

A tropical nullstellensatz

We suggest a version of Nullstellensatz over the tropical semiring, the real numbers equipped with operations of maximum and summation.

     

Maximally differential ideals in regular local rings

It is shown that ifA is a regular local ring andI is a maximally differential ideal inA, thenI is generated by anA-sequence.     

Hilbert’s Nullstellensatz for analytic trigonometric polynomials

This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if ${\left\{f_j\right\}}_{j=1}^{n\ge 2}$ are analytic trigonometric polynomials without common zero in the finite complex plane $?$ then there are analytic trigonometric polynomials ${\left\{g_j\right\}}_{j=1}^{n\ge 2}$ obeying ${\sum }_{j=1}^{n\ge 2}{f}_j{g}_j=1$ in $?$, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on $?$.     

Bounds for solutions of a differential inequality

This work compares the solutions of an th order differential inequality plus boundary conditions with the solution of the related differential equation with boundary conditions. The differential operator is assumed to be disconjugate. It is proved that under suitable conditions the ratio of these solutions is monotone. The solution of the inequality can be replaced by the corresponding Green's function.

     

Computation of differential Chow forms for ordinary prime differential ideals

In this paper, we propose algorithms for computing differential Chow forms for ordinary prime differential ideals which are given by characteristic sets. The algorithms are based on an optimal bound for the order of a prime differential ideal in terms of a characteristic set under an arbitrary ranking, which shows the Jacobi bound conjecture holds in this case. Apart from the order bound, we also give a degree bound for the differential Chow form. In addition, for a prime differential ideal given by a characteristic set under an orderly ranking, a much simpler algorithm is given to compute its differential Chow form. The computational complexity of the algorithms is single exponential in terms of the Jacobi number, the maximal degree of the differential polynomials in a characteristic set, and the number of variables.     

A new algorithm for calculating two-dimensional differential transform of nonlinear functions

A new algorithm for calculating the two-dimensional differential transform of nonlinear functions is developed in this paper. This new technique is illustrated by studying suitable forms of nonlinearity. Three strongly nonlinear partial differential equations are then solved by differential transform method to demonstrate the validity and applicability of the proposed algorithm. The present framework offers a computationally easier approach to compute the transformed function for all forms of nonlinearity. This gives the technique much wider applicability.     

A note on fuzzy differential equations

We study the effect of the forcing term to the solution of a fuzzy differential equation.     

A differential graded approach to the silting theorem

A silting theorem was established by Buan and Zhou as a generalisation of the classical tilting theorem of Brenner and Butler. In this paper, we give an alternative proof of the theorem by using differential graded algebras.     

Arithmetic partial differential equations, II

We continue the study of arithmetic partial differential equations initiated in [7] by classifying “arithmetic convection equations” on modular curves, and by describing their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions (Mori, 1995 [13], Buium, 2003 [4]) of the same modular forms; in this sense, our arithmetic convection equations can be seen as “unifying” the two types of expansions. The theory can be generalized to one of “arithmetic heat equations” on modular curves, but we prove that they do not carry “arithmetic wave equations.” Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.     

Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations

In this paper, we state and prove a new formula expressing explicitly the derivatives of shifted Chebyshev polynomials of any degree and for any fractional-order in terms of shifted Chebyshev polynomials themselves. We develop also a direct solution technique for solving the linear multi-order fractional differential equations (FDEs) with constant coefficients using a spectral tau method. The spatial approximation with its fractional-order derivatives (described in the Caputo sense) are based on shifted Chebyshev polynomials T_L,n(x) with x ∈ (0, L), L > 0 and n is the polynomial degree. We presented a shifted Chebyshev collocation method with shifted Chebyshev–Gauss points used as collocation nodes for solving nonlinear multi-order fractional initial value problems. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques and to compare with the existing results.     

Triviality of differential Galois cohomology of linear differential algebraic groups

For a partial differential field K, we show that the triviality of the first differential Galois cohomology of every linear differential algebraic group over K is equivalent to K being algebraically, Picard–Vessiot, and linearly differentially closed. This cohomological triviality condition is also known to be equivalent to the uniqueness up to an isomorphism of a Picard–Vessiot extension of a linear differential equation with parameters.     

Statistical linear differential games: A treatment of incomplete information

A zero-sum, two-player linear differential game of fixed duration is considered in the case when the information is incomplete but a statistical structure gives both players the possibility tospy the value of an unknown parameter in the payoff. Considerations of topological vector spaces and functional analysis allow one to demonstrate, via a classical Sion's theorem, sufficient conditions for the existence of a value.The author is indebted to Professor J. Fichefet for his helpful remarks and indications.     

Magnus integrators for solving linear-quadratic differential games

We consider Magnus integrators to solve linear-quadratic N$N$-player differential games. These problems require to solve, backward in time, non-autonomous matrix Riccati differential equations which are coupled with the linear differential equations for the dynamic state of the game, to be integrated forward in time. We analyze different Magnus integrators which can provide either analytical or numerical approximations to the equations. They can be considered as time-averaging methods and frequently are used as exponential integrators. We show that they preserve some of the most relevant qualitative properties of the solution for the matrix Riccati differential equations as well as for the remaining equations. The analytical approximations allow us to study the problem in terms of the parameters involved. Some numerical examples are also considered which show that exponential methods are, in general, superior to standard methods.     

Banach空间上泛函微分方程的渐近性

§ 1　IntroductionIn the last few years,invariant sets and attractors of (functional) differentialequations have been extensively discussed and various interesting results on the invariantsets and attractors,and estimates on the basin of attraction have been reported(see,forinstance,[1 ,2 ,4,6,1 0 ,1 3 ,1 5,1 8] ) .However,not much hasbeen developed in the directionof giving criteria on the existence of invariant sets and attractors for the functionaldifferential equations even though there ar…     

Composition constants for raising the orders of unconventional schemes for ordinary differential equations

Many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general-purpose software designed to solve numerically a wide range of differential equations. If those properties are to be exploited fully for the sake of better numerical stability, accuracy and/or speed, the differential equations may have to be solved by unconventional methods. This short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods, called reflexive.

     

Local exactness in a class of differential complexes

The article studies the local exactness at level in the differential complex defined by commuting, linearly independent real-analytic complex vector fields in independent variables. Locally the system admits a first integral , i.e., a complex function such that and . The germs of the ``level sets' of , the sets , are invariants of the structure. It is proved that the vanishing of the (reduced) singular homology, in dimension , of these level sets is sufficient for local exactness at the level . The condition was already known to be necessary.