SCALAR DIFFERENTIAL INVARIANTS, DIFFIETIES AND CHARACTERISTIC CLASSES
Publisher Summary This chapter provides a history of the theory of scalar differential invariants. Spectra of scalar differential invariants algebras are diffieties. By diffieties, objects of the category of differential equations are denoted. Diffieties are a kind of manifolds, generally infinite-dimensional, supplied with a finite-dimensional distribution called the Cartan distribution or the infinite order-contact structure. The chapter describes some geometrical structures and their scalar differential invariants and characteristic classes. It further discusses a few generalizations and variants of the preceding considerations and related problems. The chapter also discusses the topics of natural equations and special characteristic classes, characteristic classes associated with Lie pseudo-groups, Tsujishita's approach and the secondary quantizied-characteristic classes, quasi-homogeneous geometries, and nonhomogeneous geometrical structures and singularities.

title: The logic algebra for the theory of linear difierential operators
abstract: A material handling device on a sewing machine for feeding labels one at a time to a movable vacuum pickup head on the machine which moves the label to a position on the sewing machine adjacent the needle for sewing of the label onto a garment. The labels are cut from a strip wound on a reel. A reciprocable finger moves a portion of the strip along a surface and thereby unwinds the strip from the reel. On each label-feeding stroke of the finger, the strip is advanced a distance equal to the length or width of a label past a cutter and downwardly onto an elevator platform. The upper surface of the platform has at least one aperture therein which is connected to a vacuum source. The label is in a position on the platform covering the aperture and is held in position on the platform by vacuum from the vacuum source. With the label positioned and held on the platform, the cutter is actuated to cut the label from the remainder of the strip. While the cutter is cutting the label from the remainder of the strip, the elevator platform is simultaneously elevating the label to a position above the surface for subsequent pickup by the movable vacuum pickup head on the machine. As the pickup head moves into position to pick up the label, the vacuum is disconnected from the aperture in the platform so as to release its hold on the label and is connected to the pickup head. The pickup head by vacuum picks up the label and moves it from its position on the elevator platform to a position adjacent the needle for sewing of the label onto a garment. The elevator platform and cutter are lowered to their initial position. The finger then advances the strip a distance equal to the length or width of a label past the cutter and onto the platform. When the machine operator desires another label, the label is cut from the remainder of the strip, elevated, picked up, and positioned adjacent the needle and thereafter the next label is advanced past the cutter and onto the platform.

title: Symmetries and conservation laws of partial differential equations: Basic notions and results
abstract: The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.

title: From symmetries of partial differential equations towards secondary (“quantized”) calculus
abstract: Diffieties are analogues of algebraic varieties for partial differential equations. They are a kind of (generally, infinite-dimensional) manifolds supplied with an infinite-order contact structure. Secondary, or more speculatively, “quantized” calculus arises as a sort of differential calculus over filtered smooth function algebras on diffieties that respects the contact structure. This paper, written as an informal introduction and invitation to Secondary Calculus, is an account of the author's attempt to understand what should be the analogue of the Schrodinger equation for quantum field theory. So, more attention is paid to motivations than to exact constructions and formulas.

title: Geometry of nonlinear differential equations
abstract: The paper contains a survey of certain contemporary concepts and results connected with the geometric foundations of the theory of nonlinear partial differential equations. At the base of the account is situated the geometry and analysis on jet spaces, finite and infinite.

title: WHAT IS THE HAMILTONIAN FORMALISM
abstract: In this paper the basic concepts of the classical Hamiltonian formalism are translated into algebraic language. We treat the Hamiltonian formalism as a constituent part of the general theory of linear differential operators on commutative rings with identity. We take particular care in motivating the concepts we introduce. As an illustration of the theory presented here, we examine the Hamiltonian formalism in Lie algebras. We conclude by presenting a version of the orbit method in the theory of representations of Lie groups, which is a natural corollary of our view of the Hamiltonian formalism.

title: THE QUASI-ELASTIC METHOD IN CREEP STABILITY ANALYSIS
abstract: The paper presents a modification of the quasi-elastic solution technique aimed at applications to the stability analysis of viscoelastic structures. The study is concerned with two distinct instability phenomena, the bifurcation and the snapthrough buckling in the presence of creep. Respectively, two structural models are considered, a circular arch and a shallow two-bar system under sustained lateral loading. The quasi-elastic approximate solution results in the instability condition defined in terms of the load-critical time equation. The time-dependent response of the structures is studied in relation to the limited and unlimited creep behaviour of viscoelastic materials. Two numerical examples are given and accuracy of the method is discussed.

title: Brief note creep-stability analysis of viscoelastic cylindrical shells
abstract: The paper presents the creep-stability analysis of viscoelastic cylindrical shells under axial compression. The mechanical properties of the material are described by the constitutive equations of the linear viscoelastic theory in terms of convolution integral operators. The approximate analytical solution to the problem is obtained by means of a modification of the quasi-elastic method. As a result, the instability condition for the shell is formulated. It is shown that for viscoelastic materials with limited creep, there is a safe load limit below which the structure is asymptotically stable. Any load above the safe load limit leads to buckling at the corresponding critical time.