Smooth Real-valued Function Research Articles

On the non-locality of quasiconvexity

It is shown that in the class of smooth real-valued functions on n × m matrices (n ≥ 3, m ≥ 2) there can be no “local condition” which is equivalent to quasiconvexity.

A weighted version of a rearrangement inequality

We show that if a positive absolutely continous measure causes a special relative isoperimetric inequality to hold, then Dirichlet-type integrals of sufficiently smooth real-valued functions decrease under an appropriate equimeasurable rearrangement.

Fibre singularities of functions on principal S1-bundles

Let π : E → B denote a smooth principal S 1-bundle with total space E and base space B of dimension m. In C.Z.W. Hassel Sweatman, 1993, it was noted that, as a consequence of the general results of K. Igusa, 1990, the set of smooth real-valued functions on E whose singularities on fibres are only of types A 1,…, A m + 1 is open and dense in the Whitney C ∞ topology and further results on the existence of good functions were proved there. In this article, we prove the existence of a smooth real-valued function on E with no fibre singularities of type A k for k > [ m/2] + 1.

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold and h a smooth function on M . Let ρh(x) = inf |v|=1 (Ricx(v, v)− 2Hess(h)x(v, v)). Here Ricx denotes the Ricci curvature at x and Hess(h) is the Hessian of h. Then M has finite fundamental group if ∆h − ρh < 0. Here ∆h =: ∆ + 2L∇h is the Bismut-Witten Laplacian. This leads to a quick proof of recent results on extension of Myers’ theorem to manifolds with mostly positive curvature. There is also a similar result for noncompact manifolds. An early result of Myers says a complete Riemannian manifold with Ricci curvature bounded below by a positive number is compact and has finite fundamental group. See e.g. [9]. Since then efforts have been made to get the same type of result but to allow a little bit of negativity of the curvature (see Berard and Besson[2]). Wu [12] showed that Myers’ theorem holds if the manifold is allowed to have negative curvature on a set of small diameter, while Elworthy and Rosenberg [8] considered manifolds with some negative curvature on a set of small volume, followed by recent work of Rosenberg and Yang [10]. We use a method of Bakry [1] to obtain a result given in terms of the potential kernel related to ρ(x) = inf |v|=1 Ricx(v, v), which gives a quick probabilistic proof of recent results on extensions of Myers’ theorem. Here Ricx denotes the Ricci curvature at x. Let M be a complete Riemannian manifold, and h a smooth real-valued function on it. Assume Ric−2Hess(h) is bounded from below, where Hess(h) ∗Research supported in part by NATO Collaborative Research Grants Programme 0232/87 and by SERC grant GR/H67263. 1991 Mathematical subject classification 60H30,53C21

DIFFERENTIAL FORMS ON DIFFERENTIAL SPACES

1 Differential spaces Let M be a set and let C be a family of real functions on M . The topology induced on M by the family C, i.e., the weakest topology such that all functions from the family C are continuous, will be denoted by τC. If the family C separates points of M , then (M, τC) is a Hausdorff space. Let sc(C) denote the set of all functions of the form α ◦ (f1, . . . , fk) where f1, . . . , fk ∈ C and α is a smooth real-valued function defined on an open neighborhood of f1(M)× . . .× fk(M) in R. It is easy to show ( [9]) that: C ⊂ sc(C) sc(C) = sc(sc(C)) τsc(C) = τC

The Lusternik-Schnirelmann theorem reconsidered

The celebrated theorem of Lusternik and Schnirelmann [6], as reformulated by Palais [7] or Schwartz [10], concerns the behaviour of smooth (i.e., C 1) real-valued functions on a complete Riemannian manifold. The functions are required to be bounded below (or above) and to satisfy one further condition, the Palais-Smale condition. The theorem gives a lower bound, the Lusternik-Schnirelmann category, for the number of critical points of such a function. Without the Palais-Smale condition the result is no longer true. For example the exponential function of a single real variable has no critical points, although it is bounded below and the realline has category one. The purpose of this paper is to establish, by a variant of the classical argument, versions of the Lusternik-Schnirelmann theorem which are valid without the Palais-Smale condition. They reduce to the classical theorem when the manifold is compact and the condition is necessarily satisfied, but in the noncompact case provide new information when the condition is not satisfied.

Fitting simple non-tensor-product splines to scattered noisy data on Euclidean d-space

A family of formally simple non-tensor-product splines has already proven useful for estimating smooth real-valued functions on R d from their given noisy values at finitely many ‘spots’. These splines are uniquely defined by the values they take at the ‘knots’. Choosing the knot set different from the given data spot set is a useful option (e.g., when spot design is poor, or for reasons of data reduction). Spline determination is described for this general case, together with statistical estimation of the proper degree of smoothing. Test results are presented, and use in pragmatic systems modelling and simulation is sketched.

Morse theory for codimension-one foliations

It is shown that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every smooth real-valued function on the manifold has a cusp singularity.

Attracting orbits in Newton’s method

It is well known that the dynamical system generated by Newton’s Method applied to a real polynomial with all of its roots real has no periodic attractors other than the fixed points at the roots of the polynomial. This paper studies the effect on Newton’s Method of roots of a polynomial "going complex". More generally, we consider Newton’s Method for smooth real-valued functions of the form f μ ( x ) = g ( x ) + μ {f_\mu }(x) = g(x) + \mu , μ \mu a parameter. If μ 0 {\mu _0} is a point of discontinuity of the map μ → \mu \to (the number of roots of f μ {f_\mu } ), then, in the presence of certain nondegeneracy conditions, we show that there are values of μ \mu near μ 0 {\mu _0} for which the Newton function of f μ {f_\mu } has nontrivial periodic attractors.

Two-Metric Projection Methods for Constrained Optimization

This paper is concerned with the problem min $\{ f(x)\mid x \in X\} $ where X is a convex subset of a linear space H, and f is a smooth real-valued function on H. We propose the class of methods $x_{k + 1} = P(x_k - \alpha _k g_k )$, where P denotes projection on X with respect to a Hilbert space norm $\| \cdot \|$ , $g_k $ denotes the Frechet derivative of f at $x_k $ with respect to another Hilbert space norm $\| \cdot \|_k $ on H, and $\alpha _k $ is a positive scalar stepsize. We thus remove an important restriction in the original proposal of Goldstein [1] and Levitin and Poljak [2], where the norms $\| \cdot \|$ and $\| \cdot \|_k $ must be the same. It is therefore possible to match the norm $\| \cdot \|$ with the structure of X so that the projection operation is simplified while at the same time reserving the option to choose $\| \cdot \|_k $on the basis of approximations to the Hessian of f so as to attain a typically superlinear rate of convergence. The resulting methods are particularly attractive for large-scale problems with specially structured constraint sets such as optimal control and nonlinear multi-commodity network flow problems. The latter class of problems is discussed in some detail.

Applications of catastrophe theory to the physical sciences

Catastrophe Theory was introduced by René Thom in the late 1960's, as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings. Thom envisaged a very general approach to topological changes in the solutions to parametrized systems of equations (such as differential and difference equations), and in particular discussed the special case of ‘elementary’ catastrophe theory: singularities of smooth real-valued functions. Popular expositions have tended to overemphasize this special case, but it remains the major source of ideas and methods. Here we survey the applications of Catastrophe Theory to the physical sciences (physics, chemistry, engineering, fluid mechanics, etc.). For brevity we confine attention to an area lying between ‘elementary’ and ‘general’ Catastrophe Theory, usually known as Singularity Theory. This is the theory of singularities of smooth vector-valued functions, which mathematically is a straightforward (though non-trivial) generalization of the real-valued case. In the last few years it has developed into a powerful and useful technique in several areas of theoretical physics, notably optics and bifurcation theory. Equivariant Catastrophe Theory, taking account of symmetry, is likely to prove especially interesting.

Local characterization of phase diagrams

A classification of the local topological type of phase diagrams for fluid mixtures is obtained. The classification derives from ideas of Thom and results of Arnol'd in the theory of singularities of smooth real-valued functions, together with the assumption that the state of the system corresponds to a minimum of a smooth function on a space of internal variables which can have arbitrarily high dimension. Griffith's notions of elementary and compound entities emerge in a natural manner. The methods used are phenomenological. Some attention is given to the question of why these seem to work so well and how they might be rigorously established.

Universal unfoldings in Banach spaces: reduction and stability

This article is concerned with the bifurcation of critical points of a smooth real-valued function on a Banach space. There are two somewhat different aspects considered. The first, referred to as reduction, is a refinement of the Liapunov-Schmidt procedure by incorporating techniques from the theory of determinate germs and their universal unfoldings. A definite reduction procedure is given, whereby a germ or unfolding given in a Banach space may be replaced by a germ or unfolding in a finite-dimensional space in such a way that the geometry of the bifurcation set and the overlying critical point set is preserved. The aim is to provide a practicable tool, not an exhaustive theoretical discussion. How practicable it is may be seen from another paper (Magnus and Poston(7)) in which it is applied to a problem in elasticity; indeed, the present paper in part was originally the ‘machinery’ section of that paper. The second aspect, stability, is of more theoretical interest. The well known structural stability of universal unfoldings (see (5)) is extended to infinite-dimensions. Only a local theory is given.

Variable domain second order evolution equations

Well-posedness criteria are given abstract evolution equations of the form for with Cauchy data Here A(t), P(t), N(t) are linear operators on a Hilbert space with N(t) bounded. These are the kinds of equations in [J.Functional Analysis 4 (1969), 50-70], but there it was assumed that each A(t) was self-adjoint and that Dom (A(t)) did not depend on t. These restrictions are relaxed in the present paper. As an application, the following simple but nontrivial mixed hyperbolic problem is solved: Here α,β,γ,δ,∊ are smooth real-valued functions on with β positive; a,b, are smooth real-valued functions on and f 1,f 2 are smooth complex-valued functions on [0,1] with .