Stable Mappings Research Articles

Phi-stable operators and inner approximation-solvability

We extend, by applying a theorem of Petryshyn (1970), the approximation-solvability of the nonlinear functional equations involving strongly stable Hilbert space mappings to the case of strongly ϕ \phi -stable mappings—a new and rather general class of mappings. These mappings constitute a generalization of monotone mappings. Finally, we upgrade the obtained results to the case of Banach space mappings.

DI-Domains as prime information systems

This paper introduces prime information systems. Prime information systems determine dI-domains and stable approximable mappings give stable functions, and vice versa. The notion of rigid embedding is captured by a subsystem relation. Under this relation, prime information systems form a complete partial order (cpo). Constructions like lifting, sum, product, and function space are proposed which induce continuous functions on the cpo. In this way recursive prime information systems can be defined using fixed point theory.

DI-Domains as Information Systems*

This paper introduces stable information systems. Stable information systems determine DI-domains and stable approximable mappings give stable functions, and vice versa. The notation of rigid embedding is captured by a subsystem relation. Under this relation, stable information systems form a cpo. Constructions like lifting, sum, product and function space are proposed which induce continuous functions on the cpo. In this way recursive stable information systems can be defined using fixed point theory

Stable mappings of discriminant varieties

Smooth mappings defined on discriminant varieties of -versal unfoldings of isolated singularities arise in many interesting geometrical contexts, for example when classifying outlines of smooth surfaces in ℝ3 and their duals, or wave-front evolution [1, 2, 5]. In three previous papers we have classified various stable mappings on discriminants. When the isolated singularity is weighted homogeneous the discriminant is not a local smooth product, and this makes the classification of stable germs considerably easier than in general. Moreover, discriminants arising from weighted homogeneous singularities predominate in low dimensions, so such classifications are very useful for applications.

Stable mappings of 3-manifolds into the plane

Stable mappings of 3-manifolds into the plane

On the Number of Cusps of Stable Perturbations of a Plane-to-Plane Singularity

Let $f:(R^{2}, O)\rightarrow(R^{2},0)$ be a smooth map germ. By the theorem of Whitney [Wh], $f$ can be approximated (in the semi-local sense) by a $C^{\infty}$ stable mapping. In other words, $f$ is a ”degeneration” of neighboring stable mappings, which we call stable perturbations of $f$. Then it will be natural to expect that stable perturbations have several properties in common reflecting the structure of the ”generating” map-germ $f$. In this paper we concentrate on investigating the number of cusps of stable perturbations of a generic plane-to-plane singularity. For instance, we observe that the number $\kappa(\tilde{f})$ modulo 2 of cusps of a stable perturbation $\tilde{f}$ of a generic map-germ $f:(R^{2}, O)\rightarrow(R^{2},0)$ is a topological invariant of $f$ (Theorem 2.4). In fact $\kappa(\tilde{f})$ mod 2 is determined by the number of branches of the locus of critical points of $f$ and the mapping degree of $f$ (Theorem 2.1). Thus if two generic map-germs $f,$ $ g:(R^{2},0)\rightarrow$

Time Dependent Nonlinear Oscillations with Many Periodic Solutions

We investigate the local structure of nonlinear operators defining time dependent nonlinear oscillations. We explicitly describe the local structure of such operators using as models stable mappings between finite-dimensional spaces. This structure theorem allows us to prove the existence of many periodic solutions for such nonlinear oscillations. We determine the maximum number of such periodic solutions occurring near equilibrium. As one consequence we deduce that there exist oscillations with polynomial nonlinear terms of degree $2n$ for which there exist $(n + 1)^2 $ periodic solutions.

Generating families for images of Lagrangian submanifolds and open swallowtails

SummaryIn this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL2(ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.

M-T topologically stable mappings are uniformly stable.

M-T topologically stable mappings are uniformly stable.

Detecting Cohomologically Stable Mappings

Let $f$ be a cohomologically stable mapping defined from a compactum $X$ to the $(n + 1)[ - {\text {cell}}{I^{n + 1}}$, let $\pi :{I^{n + 1}} \to {I^n}$ be the projection, and let $A = {I^n} \times \{ 1\}$ and $B = {I^n} \times \{ - 1\}$ be opposite faces of ${I^{n + 1}}$. If $S$ is a separator or a continuum-wise separator of ${f^{ - 1}}(A)$ and ${f^{ - 1}}(B)$ in $X$, then $\pi f |S$ is cohomologically stable. This result is used to extend certain computations of cohomological dimension that are due to Walsh, who considered only the special case of the identity mapping on ${I^{n + 1}}$.

Detecting cohomologically stable mappings

Let f f be a cohomologically stable mapping defined from a compactum X X to the ( n + 1 ) [ − cell I n + 1 (n + 1)[ - {\text {cell}}{I^{n + 1}} , let π : I n + 1 → I n \pi :{I^{n + 1}} \to {I^n} be the projection, and let A = I n × { 1 } A = {I^n} \times \{ 1\} and B = I n × { − 1 } B = {I^n} \times \{ - 1\} be opposite faces of I n + 1 {I^{n + 1}} . If S S is a separator or a continuum-wise separator of f − 1 ( A ) {f^{ - 1}}(A) and f − 1 ( B ) {f^{ - 1}}(B) in X X , then π f | S \pi f |S is cohomologically stable. This result is used to extend certain computations of cohomological dimension that are due to Walsh, who considered only the special case of the identity mapping on I n + 1 {I^{n + 1}} .

Topological properties of real simple germs, curves, and the nice dimensions n > p

Infinitesimally stable germs play an important role both as germs from which global C∞-stable mappings are constructed and as germs representing versal unfoldings of (C∞ or holomorphic) germs. Because of the presence of moduli, the C∞ (or analytic) classification of these germs is insufficient and the topological classification of these germs must be understood as well. Here we consider the classification of such germs in the region where no moduli occur. This region is important for several reasons. Most importantly, it contains the infinitesimally stable germs occurring in the nice dimensions. These are the dimensions in which globally infinitesimally stable mappings are dense among the proper C∞-mappings. In (4), it was proved that the topological and C∞-classifications agree for infinitesimally stable germs f: n, 0 → p, 0 in the nice dimensions, n ≤ p. This was then used in (5) to characterize those topologically stable germs which are C∞-stable.

Denaturation: an example of a catastrophe II. Two-state transitions

In this paper statistical arguments are used within the framework of catastrophe theory to provide a quantitative description of protein transconformation reactions. For a protein-denaturant system, the accessible states (native and denatured) are characterized in terms of approximating Gaussian distributions. The possible transformations between the states of the system upon varying the experimental constraints are studied by assuming that the system is governed by a potential function (in the sense of Thom). The transitions observed to occur are modeled using the theory of singularities of stable mappings. Existing experimental data is used to construct model governing potential functions for proteins undergoing two-state transistions. The examples treated are ribonuclease subject to ( T, pH), ichthyocol collagen subject to ( T, CaCl 2), and lysozyme subject to ( T, pH). Once the potential function has been identified for a given protein-denaturant system it may be used to predict: (i) transition curves for values of the constraints intermediate between those reported in the literature; (ii) transition curves for experiments in which the constraints are varied in an arbitrary manner, e.g., variable pH studies under isothermal conditions; (iii) estimates of difference in thermodynamic potential (i.e., free energy) between the states involved in a transition.

Equivalence of stable mappings between two-dimensional manifolds

Equivalence of stable mappings between two-dimensional manifolds

ON SMOOTH MAPPINGS OF THE CIRCLE INTO ITSELF

In this article is constructed the set , open and everywhere dense in , of -stable mappings. is totally disconnected and is topologically conjugate to the topological Markov chain with a finite number of states; for we have and topologically conjugate to . For there exists a hyperbolic structure on .Diagrams: 1Bibliography: 9 items

ON THE PROBLEM OF CLASSIFICATION OF POLYNOMIAL ENDOMORPHISMS OF THE PLANE

The paper is a continuation of the author's paper [5] (Math. USSR Sbornik 6 (1968), 97-114).§1 concerns the iterations of a polynomial of degree on a singular set . It is assumed that the critical points of lie either in the domains of attraction of finite attracting cycles or at infinity. The theorems of [5] (Theorem 1 concerning the topological isomorphism of the transformation and of a shift on the space of one-sided -ary sequences with a finite number of identifications; Theorem 2: ) are generalized for the case of a disconnected .In §2 the author investigates the iterations of on the entire plane . He shows (Theorem 3) that the dynamical systems and are topologically isomorphic for sufficiently small in the case of polynomials satisfying one of the hypotheses of §1 and a certain coarse condition of nonconjugacy of the iterations of distinct critical points.Hypothesis: the set of structurally stable mappings investigated in the paper is everywhere dense in the space of coefficients.Nine figures; bibliography of eight titles.

Stability of C ∞ Mappings: II. Infinitesimal Stability Implies Stability

The purpose of this paper is to prove that a proper, infinitesimally stable mapping of one finite dimensional Co manifold into another is stable. This result is stated as Theorem 1 of ? 3. The method we use to show this result proves somewhat more; two things which it proves are stated as Theorems 2 and 3 in ? 3. The interest of this is the following. Stable mappings play an important role in the theory of singularities of differentiable mappings (cf. [8] and [9]). Therefore it is desirable to have methods for proving that particular mappings are stable. The result of this paper is a step towards such methods, since it is generally easier to prove that a mapping f is infinitesimally stable than it is to prove that f is stable. This paper is organized as follows. In ? 1, we set down our notation and terminology. All of this is quite standard except that manifolds have corners. However it seemed desirable to be explicit, since the precise meaning of the terminology varies from author to author. In ? 2, we introduce the topology W = WO, on the space of Co mappings of one manifold into another. The main results (? 3) are formulated in terms of this topology. In general, composition is not continuous with respect to the topology W. A major part of ? 2 is devoted to proving that various restrictions of the composition mapping are continuous. In ? 3, we state the main results. Note that Theorem 3 trivially implies Theorems 1 and 2. We will refer to the previous paper [5] in this series as I. In ? 4, we apply the results of I. Proposition 1 is essentially a restatement of the division theorem in terms of topologies introduced in ? 2. Proposition 2, on the other hand, introduces something new, in that it enlarges the set of functions that we can divide by. In ? 5, we interpret a result of Seeley in terms of the topologies that we