In [8] we studied equivariant bifurcation problems with a symmetry group acting on parameters, from the point of view of singularity theory. We followed the now classical theory originated by Damon [5], using the ideas presented in [5, 13, 14]. We adapted general results about unfoldings, the algebraic characterization of finite determinacy, and the recognition problems, to multiparameter bifurcation problems f(x, λ)=0 with ‘diagonal’ symmetry on both the state variables and on the bifurcation parameters. More precisely, such bifurcation problems satisfy the condition f(γx, γλ)=γf(x, λ) for all γ∈Γ, where Γ is a compact Lie group

In this paper we attack the same problem from a different angle: the path formulation. This idea can be traced back to the first papers of Mather [17] and Martinet [15, 16]. It was used explicitly in Golubitsky and Schaeffer [12] (see also their earlier paper [11]) as a way of relating bifurcation problems in one state variable without symmetry to a miniversal unfolding in the sense of catastrophe theory. At that time the techniques of singularity theory were not powerful enough to handle the full power of the idea efficiently – either in theory or in computational practice. This is why the path formulation was abandoned in favour of contact equivalence with distinguished parameters, as developed in Golubitsky and Schaeffer [12]. Considerable progress has been made since then; for example Montaldi and Mond [19] use the path formulation to apply the idea of [Kscr ]V-equivalence introduced by Damon [6] to equivariant bifurcation theory. Bridges and Furter [3] studied equivariant gradient bifurcation problems using the path formulation, and defined an equivalence relation in the space of paths and their unfoldings that respects contact equivalence of the gradients. Here we describe an algebraic approach to the path formulation that has the advantage of organizing the classification of normal forms. Moreover, it minimizes the calculation involved in obtaining the normal forms (compare with the classical framework in Furter et al. [8]). The geometric approach to the path formulation using [Kscr ]V-equivalence is still open in the context of a symmetry group acting diagonally on parameters.

We give an explicit description of the construction of inverse limits in the category of locales and show a localic version of Steenrod's theorem without the Axiom of Choice; as an application we generalize the classic result by Steenrod.

We show that the non-existence of elements in the p-stem πSp of Hopf invariant one implies that: there exists no smooth map f[ratio ]M→N with only fold singularities when M is a closed n-dimensional manifold with odd Euler characteristic and N is an almost parallelizable p-dimensional manifold (n[ges ]p), provided that p≠1, 3, 7. In fact, the result itself is originally due to Kikuchi and Saeki [25, 34]. Our proof clarifies the relationship between the two problems and gives a new insight to the problem of the global singularity theory. Furthermore we generalize the above result to maps with only Morin singularities of types Ak with k[les ]3 when p≠1, 2, 3, 4, 7, 8.

Some thirty years ago Buchsbaum and Rim [1] extended the notion of multiplicity e(a1, …, an; E) for elements a1, …, an of a commutative ring R with identity and a Noetherian R-module E (≠0) with lengthR (E/[sum ]nt=1aiE) finite to give a multiplicity e((aij); E) associated with E and an m×n matrix (aij) over R satisfying a certain extended finiteness condition. One of their results states that for each of a set of m complexes depending on E, (aij) the Euler–Poincaré characteristic is a certain integer multiple of e((aij); E), at least when R is a local ring.

Some of these ideas were taken up in [4] where it is shown that when n[ges ]m−1, each of the complexes K((aij); E; t) with t∈ℤ introduced in [3] also have e((aij); E) as their Euler–Poincaré characteristic. With a slight change in viewpoint (aij) can be replaced by linear forms aj=[sum ]mi=1aijxi (j=1, …, n) of the graded polynomial ring R[x1, …, xm]; the complex K((aij); E; t) then becomes the component of degree t in a certain graded double complex

formula here

where K(a1, …, an; F) is the standard Koszul complex (see [4; section 2]). From this point of view the construction can be extended to allow the homogeneous polynomials a1, …, an to have any (possibly unequal) positive degrees [6]. The main aim of the present note is to extend similarly the results of [4] and to strengthen those results to give information on the vanishing of the multiplicity.

Let

formula here

be the complete elliptic integral of the first kind. Recently, S.-L. Qiu and M. K. Vamanamurthy have proved for all r∈(0, 1):

formula here

with r′=√1−r2. In this paper we establish that the converse inequality

formula here

holds for all r∈(0, 1). Moreover, we show that the constant factors ¼ and π/(4 log (2))−1 in (a 1) and (a 2) are best possible.

For a wide class of categories of Banach spaces, we show that the existence of an (almost) transitive element implies the existence of a separable almost transitive element. We give some applications to C0(L) spaces and abstract M-spaces. Next, we prove that Wood's conjecture on almost transitivity of the norm in C0(L) can be reduced to the case in which the one-point compactification of L is metrizable.

We construct a simple example of transitive M-space and we show the existence of almost transitive separable M-spaces that are isomorphic to C[0, 1]. These M-spaces have a rich M-structure and they are counterexamples for several questions about centralizers.

It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.

We study the global behaviour of Gaussian curvature K and normal curvature K⊥ of zero mean curvature spacelike surfaces (stationary surfaces) in a four-dimensional Lorentzian space form L4(c). In particular, we show that the only complete stationary surfaces in Minkowski space E41 with K[ges ]0 are those with K≡0≡K⊥ and we give an explicit description of them. More general results are obtained for stationary surfaces in L4(c). We also discuss applications to Willmore surfaces in both Lorentzian and Riemannian three spaces. We give new examples of complete stationary surfaces in E41 with finite total curvature.

We show that the sequence 2n+n is a universal Hilbert sequence. That is, for each polynomial P(T, Y) irreducible in ℚ(T) [Y], the polynomial P(2n+n, Y) is irreducible in ℚ[Y] for all but finitely many n. This answers a question of M. Yasumoto. Other examples, like 2n+5n, are given. They are all obtained as special cases of a more general result which is proved from classical diophantine arguments.

We give a simple inequality relating the elliptic Mahler measure of a polynomial to the traditional Mahler measure (via the length of the polynomial). These bounds are essentially sharp. We also give the corresponding result for polynomials in several variables.

We study self-similar sets with overlaps, on the line and in the plane. It is shown that there exist self-similar sets that have non-integral Hausdorff dimension equal to the similarity dimension, but with zero Hausdorff measure. In many cases the Hausdorff dimension is computed for a typical parameter value. We also explore conditions for the validity of Falconer's formula for the Hausdorff dimension of self- affine sets, and study the dimension of some fractal graphs.

For an infinite Bernoulli convolution whose support is a symmetric Cantor-type set, R. Kershner and A. Wintner [14] proved that the convolution is absolutely continuous or singular depending on whether the Lebesgue measure of the support is positive or zero respectively. Their proof employs the same construction which was developed originally by F. Hausdorff [11] in his celebrated paper, ‘Dimension und äusseres Mass’, where he introduced the notion of a fractional dimension. In this article we exploit this observation and prove a result which extends the theorem of Kershner and Wintner by considering Hausdorff instead of Lebesgue measure for the same class of infinite Bernoulli convolutions.

In this paper we prove that the conjectures of Frisch and Parisi and Arneodo et al. (called the multifractal formalism for functions) may fail for some non-homogeneous selfsimilar functions on ℝ2. In these cases, we compute the correct spectrum of singularities and we show how the multifractal formalism must be modified.

In two recent papers [1, 2], we have developed an algorithm for the solution of linear differential systems of the type

formula here

on an interval [X, ∞). Here Z is an n-component vector, ρ is a scalar factor, D is a constant diagonal matrix and R is a perturbation matrix such that R(x)=O(x−δ) as x→∞, for some δ>0.The algorithm implements a repeated transformation process by means of which (1·1) is transformed into other systems whose perturbation matrices are of successively smaller orders of magnitude as x→∞. When the perturbation reaches a prescribed accuracy, the Levinson asymptotic theorem [4, 5] provides the solution of the final system in the process, the solution also possessing that accuracy. The corresponding solution of (1·1) is then obtained by transforming back.

The algorithm has two main aspects. The first is the algebraic one of generating symbolically the matrix terms which appear in the transformation process. In [1] and [2], the algorithm was set up in such a way that this aspect is implemented in the symbolic algebra system Mathematica. The other aspect is the numerical one of deriving from the algebra the values of the solutions of (1·1), given ρ, D and R. Included here are the determination of explicit error-bounds and procedures for handling large numbers of matrix terms.

In this paper we develop these ideas in two different but not unrelated directions. First, we replace the constant matrix D by one which has entries of differing orders of magnitude as x→∞. Our methods are sufficiently indicated by two orders of magnitude, and thus we consider the system

formula here

where formula here

with constant and diagonal D˜ and D, while λ is a scalar factor such that λ(x)→∞ as x→∞. The amplification of our previous algorithm to cover this new situation is dealt with in Sections 2–4 below. At the end of Section 3, we state the Levinson asymptotic theorem in the form that we need, with the necessary accuracy incorporated. As in [1] and [2], a typical ρ(x) is xa (α>−1). However, the value α=−1 also occurs in applications, and our second extension is to include α=−1 in the scope of the algorithm. This matter is the subject of Section 5.

Let Bn denote the open unit ball in Cn. We write V to denote Lebesgue volume measure on Bn normalized so that V(Bn)=1. Fix −1<γ<∞ and let Vγ denote the measure given by dVγ(z)=cγ (1−[mid ]z[mid ]2)γdV(z), for z∈Bn, where cγ=Γ(n+γ+1)/ (n!Γ(γ+1)); then Vγ(Bn)=1. The weighted Bergman space A2,γ(Bn) is the space of all analytic functions in L2(Bn, dVγ). This is a closed linear subspace of L2(Bn, dVγ). Let Pγ denote the orthogonal projection of L2(Bn, dVγ) onto A2,γ(Bn). For a function f∈L∞(Bn) the Toeplitz operator Tf is defined on A2,γ(Bn) by Tfh=Pγ(fh), for h∈A2,γ(Bn). It is clear that Tf is bounded on A2,γ(Bn) with ∥Tf∥[les ]∥f∥∞. In this paper we will consider the question for which f∈L∞(Bn) the operator Tf is compact on A2,γ(Bn). Although a complete answer has been given by the author and D. Zheng (see the next section), the condition for compactness is somewhat unnatural. In this article we will give a more natural description for compactness of Toeplitz operators with sufficiently nice symbols. We will describe compactness in terms of behaviour of the so-called Berezin transform of the symbol, which has been useful in characterizing compactness of Toeplitz operators with positive symbols (see [5, 9]). Before we can define this Berezin transform we need to introduce more notation.

If each of E and F is a real Banach space, H a compact Hausdorff space, C(H, E) the Banach space (sup norm ∥·∥∞) of continuous E-valued functions defined on H, L: C(H, E)→F a continuous linear transformation (=operator) with representing measure m, [sum ] the σ-algebra of Borel subsets of H and m˜(A) the semivariation of m on A∈[sum ], then m maps [sum ] into B(E, F**), the Banach space of all operators from E into F** (= the bidual of F), ∥L∥=m˜(H) and L(f)=∫ fdm. The reader may consult [9] or [6] for a detailed discussion of the Riesz representation theorem in this setting.

This paper deals with an initial value problem of a neutral functional-differential equation with variable time delay. The effect of the neutral term on the existence and uniqueness of solutions is investigated. A comparison theorem is given and is applied to a functional Riccati equation. The asymptotic behaviour of solutions of a linear perturbation of the generalized pantograph equation is studied.

An orientation-preserving, recurrent homeomorphism of the two-sphere which is not the identity is shown to admit exactly two fixed points. A recurrent homeomorphism of a compact surface with negative Euler characteristic is periodic.

For n[ges ]2, let S(Rn) be the space of Schwartz class functions. The dual space of S(Rn) is denoted by S′(Rn) and is called the tempered distributions. For f∈S(Rn), define fˆ(w)=∫Rne−iw·xf(x)dx and for T∈S′(Rn), define Tˆ∈S′(Rn) by the formula 〈Tˆ, f〉=〈T, fˆ〉 for all f∈S(Rn). For 1[les ]p[les ]∞, let FLp(Rn)= {T∈S′(Rn)[ratio ] Tˆ∈Lp(Rn)}. Let β be a multi-index of nonnegative integers, β=(β1, β2, …, βn−1) with [mid ]β[mid ]= β1+β2+…+βn−1. For a nonnegative integer k and bounded Borel measures σβ on a compact subset K of Rn, [mid ]β[mid ][les ]k, we define T∈S′(Rn) by the formula

formula here

If T is given by (1), then we know that supp(T)⊂K and the order of T is at most k. In particular, when k=0, we say that T is a bounded measure on K. Denote EK(Rn)= {T∈S′(Rn)[ratio ]T is a bounded measure on K} and DK(Rn)= {T∈S′(Rn)[ratio ] supp(T)∪K}. If in addition we assume some connectness property of K, then one can show ([5, th. 2·3·10]) that if T∈DK(Rn), then T does have the representation (1) for some k[ges ]0. The question is when we can say that k=0. From the Hausdorff–Young inequality and the proof of theorem 7·6·6 in [5], one sees that if the interior of K (in Rn) is not empty, then DK(Rn) ∩FLp(Rn) ⊂EK(Rn) if and only if 1[les ]p[les ]2. Much remains to be understood if the interior of K is empty.

In this paper, we seek to prove a similar result for compact subsets of hypersurfaces of Rn, motivated from the differential equations with constant coefficients.

We consider the time evolution of an unbounded interface between a viscous fluid and a displacing fluid of negligible viscosity in a Hele-Shaw geometry. Using a conformal mapping transformation z=F(ζ, t) from the lower half ζ-plane to the flow domain Ω(t) in the z-plane, the dynamics is formulated in terms of an integro-differential equation for F on the real ζ axis. This formulation is similar to earlier results for radial and channel geometries. Using a certain explicit solution it is shown that the zero surface tension problem is ill-posed in the sense of Hadamard. Furthermore, certain earlier formal asymptotic results for small but non-zero surface tension [47], are rigorously established. This is achieved by relating the similarity reduction of the Harry–Dym equation to the Painlevé II equation, and by studying this equation in the complex plane. In particular, it is shown that there exists a unique solution of Painlevé II equation which satisfies the appropriate far-field algebraic decay condition in a certain sector Ŝ of the complex plane and which in Ŝ is free of singularities. This solution asymptotes to certain elliptic functions in other sectors of the complex plane and contains a set of denumerably infinite number of singularities that approach the boundary of Ŝ from the outside at large distances from the origin (in the similarity variable). This is consistent with earlier numerical results.