The differential analytic index in Simons–Sullivan differential K-theory

We define the Simons–Sullivan differential analytic index by translating the Freed–Lott differential analytic index via explicit ring isomorphisms between Freed–Lott differential K-theory and Simons–Sullivan differential K-theory. We prove the differential Grothendieck–Riemann–Roch theorem in Simons–Sullivan differential K-theory using a theorem of Bismut.     

What is the differential equation?

<正>An equation involving a derivative is called a differential equation.Such as,(dy)/(dx)=2x,and the function y=f(x)satisfies this equation.When we know the additional condition that y=2when x=-1,the function y=f(x)will be find exactly.The additional condition is called the initial condition.It is used to evaluate constant of integration.     

What is the differential equation？

An equation involving a derivative is called a differential equation.Such as,dy/dx=2x,and the function y=f（x）satisfies this equation.When we know the additional condition that y=2 when x=-1,the function y=f（x）will be find exactly.The additional condition is called the initial condition.It is used to evaluate constant of integration.     

评《Stochastic differential equations》

吴让泉同志的专著《Stochastic differential equations》于1985年由Pitman出版公司出版。该书除预备知识这一章外,共分三章.第一章的主要部分介绍了Emeyy关于半鞅随机微分方程解的存在性及唯一性结果(1978年),此外介绍了作者与毛学荣对这一结果的一个推广.第二章详细讨论了线性随机微分方程解的表示,部分结果是经典的(例如见 Arnold:“SDE”,John Wiley＆Sons,1974),部分结果(随机 Liouville公式)是作者新近的工作(但Jacod在“Sem.Prob.XVI,1982”中讨论了非连续半鞅的线性随机微分方程,所得结果更一     

Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations

In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward–backward stochastic differential equations (FBSDEs) is established, which can be regarded as an extension of the Feynman–Kac formula to the non-Markovian framework.     

On Beurling’s boundary differential relation

We prove that for an arbitrary positive continuous function Φ on the complex plane ? there exists an injective disc algebra function φ and n ∈ ? such that ξ ? φ(ξ ⁿ ) solves Beurling’s boundary differential relation |f′(ξ)| = Φ(f(ξ)) on ?Δ. Moreover, if the growth of Φ is sublinear, the existence of univalent solutions of Beurling’s boundary differential relation is shown.     

Stochastic differential equations—some new ideas

In this paper we present a general method to study stochastic equations for a broader class of driving noises. We explain the main principles of this approach in the case of stochastic differential equations driven by a Wiener process. As a result we construct strong solutions of Itô equations with discontinuous and even functional coefficients. We point out that our construction of solutions does not rely on a pathwise uniqueness argument. Further we find that solutions of a larger class of Itô diffusions actually live in a Fréchet space, which is substantially smaller than the Meyer–Watanabe test function space.     

Ax–Schanuel for linear differential equations

We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby (The theory of exponential differential equations, 2006, Sel Math 15(3):445–486, 2009) and Crampin (Reducts of differentially closed fields to fields with a relation for exponentiation, 2006) we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.     

Hyers–Ulam stability of hypergeometric differential equations

Aequationes mathematicae - In the present paper by applying the series method we prove the Hyers–Ulam stability of the homogeneous hypergeometric differential equation in a subclass of...     

Affine differential geometry of surfaces in ℝ4

Employing the method of moving frames, i.e. Cartan's algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM ² in ⁴ relative to the action of the general affine group on ⁴. The invariants found include a normal bundle, a quadratic form onM ² with values in the normal bundle, a symmetric connection onM ² and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM ², obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined.     

RESEARCH ANNOUNCEMENTS A Cone Pseudo—differential Calculus

The calculus of pseudo-differential operators on singular spaces and theconcept of ellipti-city in operator algebras on manifolds with singularitieshave become an enormous challenge for analysists. The so-called cone algebras(with discrete and continuous asymptotics) are investigated by manymathematicians, especially by B. W. Schulze, who developed and enrichedcone and edge pseudo-differential calculus, see Schulze［4-7］, Rempel and Schulze ［2, 3］. In this note,we construct a cone pseudo-differentialcalculus for operators which respect conormal asymptotics of a prescribedasymptotic type.     

Hyers-Ulam-Rassias stability of κ-Caputo fractional differential equations

The paper is connected with the existence of solutions and Hyers-Ulam stability for a class of nonlinear fractional differential equations with κ-Caputo fractional derivative in boundary value problems. The existence and uniqueness results are obtained by utilizing the Banach fixed point theorem and Leray-Schauder nonlinear alternative theorem. In addition, two sufficient conditions to guarantee the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of boundary value problems of fractional differential equations are also presented. Finally, theoretical results are illustrated by two numerical examples.     

On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation

(a(t)|x^′|^αsgnx^′)^′+b(t)|x|^βsgnx=0${(a(t){\left|x^{\prime }\right|}^{\alpha }\mathrm{sgn}{x}^{\prime })}^{\prime }+b(t){|x|}^{\beta }\mathrm{sgn}x=0$

in the super-linear case α<β$\alpha <\beta$. Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) =0 in Ω, u=0 on Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u~3,u~2(u-p),u~2(u~2 -p),     

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.     

Examining students’ variational reasoning in differential equations

In this study, we explored how a sample of eight students used variational reasoning while discussing ordinary differential equations (DEs). Our analysis of variational reasoning draws on the literature with regard to student thinking about derivatives and rate, students’ covariational reasoning, and different multivariational structures that can exist between multiple variables. First, we found that while students can think of “derivative” as a variable in and of itself and also unpack derivative as a rate of change between two variables, the students were often able to think of “derivative” in these two ways simultaneously in the same explanation. Second, we found that students made significant usage of covariational reasoning to imagine relationships between pairs of variables in a DE, and that mental actions pertaining to recognizing dependence/independence were especially important. Third, the students also conceptualized relationships between multiple variables in a DE that matched different multivariational structures. Fourth, importantly, we identified a type of variational reasoning, which we call “feedback variation”, that may be unique to DEs because of the recursive relationship between a function’s value and its own rate of change.