Calcul différentiel et intégral adapté aux substitutions par Volterra

In 1887 Volterra was in search of a general vision for analysis. His most famous work led him to the definition of functionals, or more precisely to develop a differential and integral calculus for “functions that depend on other functions” or “line functions.”However, Volterra's efforts to define a general context for certain analytical problems would also lead him to extend the notions of derivation and integration to substitutions—matrices whose coefficients are functions—which have an important role in the study of differential linear equations.In a memoir entitled Sui fondamenti della teoria delle equazioni differenziali lineari Volterra establishes a differential and integral calculus for substitutions. This work, which allows one to think of linear differential equations through two operations on substitutions—derivation and integration, also makes it possible to analyse the progression strategy implemented by the Italian mathematician in his search for a generalized analysis from the beginning of his career.We are examining the selection and reorganization processes that have enabled Volterra to transpose a well-established theory for ordinary functions to a framework adapted to substitutions. We thus reveal a dynamic of progress towards generality, and explore the elements on which his thoughts are based.Far from being an anecdotal, this text, which does not solve any conjecture, allows us to see a coherence in Volterra's way of progressing, and clarifies his role in the search for an analysis which would gradually become the functional analysis of the 20th century.     

变量代换法在解微分方程中的应用

归纳总结利用变量代换求解不同类型常微分方程的方法与技巧,并借助两个实例的多种解法加以说明.     

Symmetries of compatibility conditions for systems of differential equations

When symmetries of differential equations are applied, various types of associated systems of equations appear. Compatibility conditions of the associated systems expressed in the form of differential equations inherit Lie symmetries of the initial equations. Invariant solutions to compatibility systems are known as orbits of partially invariant and generic solutions involved in the Lie group foliation of differential equations and so on. In some cases Bäcklund transformations and differential substitutions connecting quotient equations for compatibility conditions and initial systems naturally arise. Besides, Ovsiannikov's orbit method for finding partially invariant solutions is essentially based on such symmetries.     

微分中值定理的应用及推广

介绍应用微分中值定理时,构造所需辅助函数的两种有效方法：观察法和解微分方程法。并通过变量代换法化无限为有限,将罗尔定理的应用推广到无限区间上。     

Some exact solutions of the non-linear Dirac-Hamilton system

We obtain a family of substitutions which reduce a system of multi-dimensional Dirac—Hamilton differential equations to ordinary differential equations. We construct large classes of exact solutions to this system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 610–616, May, 1990.     

Differential substitutions of the miura transformation type

The existence of a nontrivial integral and a sufficiently large set of symmetries with respect to one of the characteristics of a hyperbolic equation implies the existence of nontrivial integrals and symmetries with respect to the other characteristic. We classify differential substitutions of the form ω=P(x, u, u _x ) relating families of evolution equations that depend on an arbitrary function. Among such substitutions, there are infinitely many pairwise-nonequivalent ones. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 3, pp. 336–348, September, 1998.     

“Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom

We construct solutions of analogues of a time-dependent Schrödinger equation corresponding to an isomonodromic polynomial Hamiltonian of a Garnier system with two degrees of freedom. The solutions are determined by solutions of linear differential equations whose compatibility condition is the given Garnier system. With explicit substitutions, these solutions reduce to solutions of the Belavin–Polyakov–Zamolodchikov equations with four times and two spatial variables.     

Hyperbolic equations with third-order symmetries

We present a complete list of nonlinear one-field hyperbolic equations with integrable third-order x and y symmetries. The list includes both equations of the sine-Gordon type and equations linearizable by differential substitutions.     

First-order differential substitutions for equations integrable on \mathbb{S}^n

We determine necessary conditions under which integrable vector evolution equations of third order admit Miura-type transformations. For equations integrable on the n-dimensional sphere, we obtain first-order differential substitutions.     

Integrable vector isotropic equations admitting differential substitutions of first order

A symmetry classification of integrable vector evolution equations of third order admitting Miura-type transformations is presented. We obtain the Bäcklund autotransformation for the new equation as well as differential substitutions relating the solutions of some integrable isotropic equations.     

On symmetry classification of third-order evolutionary systems of divergent type

A symmetry classification is presented for integrable two-field third-order evolutionary systems of divergent type. The list contains thirteen exactly integrable systems. For eleven of them, differential substitutions that relate the systems with the known systems by Drinfeld-Sokolov, Ito, and Hirota-Satsuma are found. The two remaining systems seem to be new. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 141–161, 2006.     

Differential transformations of parabolic second-order operators in the plane

Darboux’s classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form Lu = (D _x ² + a(x, y)D _x + b(x, y)D _y + c(x, y))u = 0. We prove a general theorem that provides a way to determine admissible differential substitutions for such parabolic equations. It turns out that higher order transforming operators can always be represented as a composition of first-order operators that define a series of consecutive transformations. The existence of inverse transformations imposes some differential constrains on the coefficients of the initial operator. We show that these constraints may imply famous integrable equations, in particular, the Boussinesq equation.     

Hyperbolic Equations Admitting Differential Substitutions

We show that the first n–1 Laplace invariants of a scalar hyperbolic equation obtained from an equation of the same form under a differential substitution of the nth order have a zeroth order with respect to one of the characteristics. It follows that all Laplace invariants of an equation admitting substitutions of an arbitrarily high order must have a zeroth order. Three special cases of such equations are considered: those admitting autosubstitutions, those obtained from a linear equation by a differential substitution, and those with solutions depending simultaneously on both an arbitrary function of x and an arbitrary function of y.     

Topological conjugacy of constant length substitution dynamical systems

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.     

两个符号的等长代换子系统的混沌性态

本文研究了由两个符号的等长代换生成的子系统的混沌性态．通过分类研究,我们得到：（1）非奇异代换系统不是Li－Yorke混沌的,作为特例,Morse极小系统不是混沌的;（2）奇异代换子系统都是Li－Yorke混沌的．     

The spectrum of dynamical systems arising from substitutions of constant length

Probability Theory and Related Fields - Minimal flows and dynamical systems arising from substitutions are considered. In the case of substitutions of constant length the trace relation of the flow...     

Enumeration of diagonally colored Young diagrams

This paper is concerned with the study of diffraction intensities of a relevant class of binary Pisot substitutions via exponential sums. Arithmetic properties of algebraic integers are used to give a new and constructive proof of the fact that there are no diffraction intensities outside the Fourier module of the underlying cut and project schemes. The results are then applied in the context of random substitutions.     

Bounds on the number of steps of WDS required for checking the positivity of integral forms

Examples show that many integral forms can be efficiently verified to be positive through a special form of variable substitutions, i.e., weighted difference substitutions (WDS), but it was unknown how many steps of substitutions are needed, or furthermore, whether this method is adapted for checking positivity of all integral forms. In this paper, we give upper bounds of step numbers of WDS required in checking whether an integral form is positive or nonnegative, thus deducing that the positivity of integral forms can be completely verified through the WDS method.     

Invertible substitutions and Sturmian sequences

It is known on the one hand that a Sturmian sequence can be generated geometrically by the intersections of a straight line with the unit grid in the plane, and on the other hand that fixed points of invertible substitutions are Sturmian. We give a new characterization of invertible substitutions, which allows to determine the straight line which generates the fixed point of a given invertible substitution.     

THE SCENERY FLOW FOR GEOMETRIC STRUCTURES ON THE TORUS： THE LINEAR SETTING

50. IntroductionWe begin by recalling some wellknown relationshiPs. First, ther is the one-to-one corre-spondence between closed orbits of the g6odesic fiow on the modular surfaCe and conjugacyclasses of hyperbolic toral automorphisms. (This can be seen directly from the definitions(see Remaxk 1.3 in 51 below).) Secondly one knows that it is possible to code this geodesicflow using coatinued fractions and via circle rotations (cf [9, 42, 2, 7J). Thirdly, there is astrong relation between hyp…