Goo Ishikawa

CLASSIFICATION OF GENERIC INTEGRAL DIAGRAMS AND FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

An adhesive composition for plastic-metal foil composite packaging materials which comprises (A) a polyester glycol having a molecular weight of 3,000 to 100,000, a polyester-polyurethane polyol having a molecular weight of 3,000 to 100,000 or a mixture thereof, (B) a silane coupling agent and (C) an organic polyisocyanate, the molar equivalent ratio (NCO/H) of isocyanate group (NCO) to active hydrogen (H) thereof being within the range of 1 to 10. The adhesive composition provides a satisfactory packaging material which does not have the problem of delamination between the metal foil and plastic film even on hot-water sterilization as it contains foods in sealed relation and which ensures long shelf lives and an improved tolerance for foodstuffs.

Local invariants of singular surfaces in an almost complex four-manifold

In this paper we define two local invariants, the local self-intersection index and the Maslov index, for singular surfaces in an almost complex four-manifold and prove formulae involving these invariants, which generalize formulae of Lai and Givental.

Transversalities for Lagrange singularities of isotropic mappings of corank one

0. Introduction. The Lagrange singularity theory connects the Lagrange classification of Lagrange immersions with the classification of families of functions, that is, generating families. (The theory of Hormander, Arnol’d, Zakalyukin, . . ., [AGV]). Though there exist detailed studies on singular Lagrange varieties (e.g. [A1], [A2], [DP], [G1], [G2], [J], [Z]), the nice connection between Lagrange singularities and generating families seems to break, in particular, when we try to classify generic Lagrange non-immersions or generic isotropic mappings. Our attempt is then to study singularities of isotropic mappings in the framework of singularity theory of differential mappings (Thom-Mather theory: e.g. [T], [M2]), and to classify their generic singularities under the Lagrange equivalence. In the singularity theory, the transversality theorem is a powerful tool to grasp generic conditions for differentiable mappings, with respect to partial derivatives (see, for instance, [M1], [M2]). In general, if we fix a space of mappings, a generic condition on a mapping should be described by the transversality of the jet section of the mapping to a stratification naturally defined in a jet space of sufficiently higher order. Then, “the transversality theorem” claims the density of the subspace of mappings satisfying the transversality condition, relatively to an appropriate topology on the space of mappings. As a rule, the validity of “the theorem”, however, depends on the space of mappings. Then, the purpose of this paper is to formulate explicitly and to prove the transversality theorem for the space of isotropic mappings of corank at most one, endowed with

Solution surfaces of Monge–Ampère equations

Abstract In this paper we examine the singularities of solution surfaces of Monge–Ampere equations and study their global and local effects on the solutions for certain kinds of equations in the framework of contact geometry. In particular, as a byproduct, we give a simple proof to the classical Hartman–Nirenbergs theorem by using the notion of projective duality and provide a new example of compact developable hypersurfaces in the real projective space R P 4

Singularities of tangent surfaces to generic space curves

We give the complete solution to the local diffeomorphism classification problem of generic singularities which appear in tangent surfaces, in as wider situations as possible. We interpret tangent geodesics as tangent lines whenever a (semi-)Riemannian metric, or, more generally, an affine connection is given in an ambient space of arbitrary dimension. Then, given an immersed curve, we define the tangent surface as the ruled surface by tangent geodesics to the curve. We apply the characterization of frontal singularities found by Kokubu, Rossman, Saji, Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first author related to the procedure of openings of singularities.

Classification of phase singularities for complex scalar waves and their bifurcations

Motivated by the importance and universal character of phase singularities which were clarified recently, we study the local structure of equi-phase loci near the dislocation locus of complex valued planar and spatial waves, from the viewpoint of singularity theory of differentiable mappings, initiated by Whitney and Thom. The classification of phase singularities is reduced to the classification of planar curves by radial transformations due to the theory of du Plessis, Gaffney and Wilson. Then fold singularities are classified into hyperbolic and elliptic singularities. We show that the elliptic singularities are never realized by any Helmholtz waves, while the hyperbolic singularities are realized in fact. Moreover, the classification and realizability of Whitneys cusp, as well as its bifurcation problem, are considered in order to explain the three point bifurcation of phase singularities. In this paper, we treat the dislocation of linear waves mainly, developing the basic and universal method, the method of jets and transversality, which is applicable also to nonlinear waves.

Monge-Ampere Systems with Lagrangian Pairs

The classes of Monge-Amp\`ere systems, decomposable and bi-decomposable Monge-Amp\`ere systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of Lagrangian pairs in contact structures. We show that the Lagrangian pair is uniquely determined by such a bi-decomposable system up to the order, if the number of independent variables

Duality on Geodesics of Cartan Distributions and Sub-Riemannian Pseudo-Product Structures

\geq 3

Openings of differentiable map-germs and unfoldings

. We remark that, in the case of three variables, each bi-decomposable system is generated by a non-degenerate three-form in the sense of Hitchin. It is shown that several classes of homogeneous Monge-Amp\`ere systems with Lagrangian pairs arise naturally in various geometries. Moreover we establish the upper bounds on the symmetry dimensions of decomposable and bi-decomposable Monge-Amp\`ere systems respectively in terms of the geometric structure and we show that these estimates are sharp (Proposition 4.2 and Theorem 5.3).

Symplectic Bifurcations of Plane Curves and Isotropic Liftings

Abstract Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown in (15), that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.