We investigate the first-order Thom–Boardman singularity sets of the dual mapping for an arbitrary (and then for a generic) smooth hypersurface in the complex projective space ℙn. Our results focus on nonemptiness, connectedness, regular stratifications and numerical invariants for these sets.

Let X be a (based) space of the homotopy type of a CW-complex. Let H(X) denote the classical (ungraded) cohomology ring Πi≥0Hi (X;Z/2). In [1] Atiyah and Hirzebruch described the group of natural ring automorphisms of H(X) (‘cohomology automorphisms’) with group operation given by composition. They showed that is isomorphic to the group of formal power series of the form with group operation given by ‘substitution’ of one power series into another. In particular the most famous ‘cohomology automorphism’, the total Steenrod Square, corresponds to x + x2.

Schauder bases in Banach spaces are studied in [5].

In ordered Banach spaces a special type of Schauder bases, the O.P. Schauder bases, are studied because then the properties of ordered spaces can be used.

We introduce a new construction in stable homotopy theory. If F and G are module spectra over a ring spectrum E, there is no well-known spectrum of E-module homomorphisms from F to G. Such a construction would not be homotopy invariant, and therefore would not serve much purpose. We show that, provided the rings and modules have A∞ structures, there is a spectrum RHomE(F, G) of derived module homomorphisms which has very pleasant properties. It is homotopy invariant, exact in each variable, and its homotopy groups form the abutment of a hypercohomology-type spectral sequence.

The purpose of this paper is to characterize the Orlicz vector-valued function spaces containing a copy or a complemented copy of l1. Pisier proved in [13] that if a Banach space E contains no copy of l1, then the space Lp(S, Σ, μ, E) does not contain it either, for 1 < p < ∞. We extend this result to the case of Orlicz vector valued function spaces, by reducing the problem to the situation considered by Pisier. Next, we pass to study the problem of embedding l1 as a complemented subspace of LΦ(E). We obtain a complete characterization when E is a Banach lattice and only partial results in case of a general Banach space. We use here in a crucial way a result of E. Saab and P. Saab concerning the embedding of l1 as a complemented subspace of C(K, E), the Banach space of all the E-valued continuous functions on the compact Hausdorff space K (see [14]). Finally, we use these results to characterize several classes of Banach spaces for which LΦ(E) has some Banach space properties, namely the reciprocal Dunford-Pettis property and Pelczyński's V property.

The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.

An n-dimensional m-component link is an oriented smooth submanifold Σn of Sn+2, where is the ordered disjoint union of m submanifolds of Sn+2, each homeomorphic to Sn. Σ is a boundary link if there is an oriented smooth submanifold Vn+1 of Sn+1, the disjoint union of the submanifolds , such that ∂Vi = Σi (i = 1,…, m). A pair (Σ, V), where Σ is a boundary link and V as above, with each Vi connected (i = 1,…, m), is called an n-dimensional special Seifert pair. In this paper, we define a notion of cobordism of special Seifert pairs and give an algebraic description of the set (group) of cobordism classes.

In this note we utilize the atomic decomposition of weighted Hardy spaces to prove weighted versions of Hardy's inequality for the Fourier transform with Muckenhoupt weight. The result extends to certain integral operators with homogeneous kernels of degree −1.

In this paper we discuss smooth and locally smooth cyclic actions on cohomology complex projective spaces. We shall show that in low dimensions the presence of a codimension-two fixed-point component forces many algebraic invariants to behave as they do in the linear examples.

The 2-variable polynomial PK of a satellite K is shown not to satisfy any formula, relating it to the polynomial of its companion and of the pattern, which is at all similar to the formulae for Alexander polynomials. Examples are given of various pairs of knots which can be distinguished by calculating P for 2-strand cables about them even though the knots themselves share the same P. Properties of a given knot such as braid index and amphicheirality, which may not be apparent from the knot's polynomial P, are shown in certain cases to be detectable from the polynomial of a 2-cable about the knot.

Write F for the finite field, , having 2m elements. Let W2(F) denote the Witt vectors of length two over F (for a definition, see [4] or [10], §10). Write F(q) for the truncated polynomial ring, F[t]/(tq).

The Landau–Kolmogorov inequality

where ‖.‖ is the ‘sup’ norm, is well known and has many interesting applications and generalizations (see [1, 4–7, 13, 16]). Its study was initiated by Landau[10] and Hadamard [8] (the case n = 2). Kolmogorov [9] succeeded in finding in explicit form the best possible constants K(n, k) = Cn, k in (1) for functions on the whole real line R. The best constants for the half line R+ are not known in explicit form except for n = 2, 3, 4, but an algorithm exists for their computation (Schoenberg and Cavaretta [15]).

We give a simple proof that a measure on a locally compact abelian group G is quasi-translation invariant with continuous translation densities if and only if it is equivalent to Haar measure on G and has a continuous positive density.

Suppose throughout that {pn} is a sequence of non-negative numbers with p0 > 0, that

and that

Let {sn} be a sequence of real numbers.

Let{N(x), x ≥ 0} be a Poisson process with intensity parameter λ > 0, and introduce

When looking for the changepoint in the Land's End data, Kendall and Kendall [7] proved for all 0 < ε1 < 1 − ε2 < 1 that

where {V(s), − ∞ < s < ∞} is an Ornstein–Uhlenbeck process with covariance function exp (−|t − s|). D. G. Kendall has posed the problem of replacing ε1 and ε2 by zero or by sequences εi(n) → 0 (n → ∞) (i = 1, 2), in (1·1). In this paper we study the latter problem and also its L2 version. The proofs will be based on the following weighted approximation of Zn.

The result known as Kuttner's theorem [2] asserts that if 0 < p < 1 and A is a Toeplitz matrix then there is a sequence which is strongly Cesàro summable with index p but which is not A summable. This theorem was extended by Maddox[3] to coregular matrices, and Thorpe [9] gave a further extension by showing that if 0 < p < 1 and X is a locally convex FK space with X ⊃ w0(p) then X ⊃ l∞. Here, w0(p) denotes the space of sequences strongly summable to 0, i.e. x∈w0(p) if and only if

and l∞ denotes the space of bounded sequences. Other proofs of Thorpe's extension and related results appear in Maddox[4, 5].

We say a real number x is normal to base r if the sequence is uniformly distributed modulo [0, 1]. Pollington[10] has proved the following result concerning the normality of numbers.

Considering a certain partition of a given compact region in the complex plane, interpolatory, convergence and quasiconformal properties of interpolants from the class of continuous piecewise functions with pieces as quadratic polynomials in z and were studied recently in [3] and [7]. In the present paper we succeed in showing that the foregoing mapping properties continue to hold for interpolants from a certain class of continuous piecewise rational polynomial functions in z and

Schweizer and Sklar, in their book Probabilistic Metric Spaces, raise the question of whether every triangle function is causal. This note provides a negative answer, by observing certain properties of a suitable partial quaternary operation on the unit interval.

The modern theory of integrable systems rests on two fundamental pillars: the classification of Lax [13] and zero-curvature equations [14, 1, 2]; and algebraic models of the classical calculus of variations [9,5] specialized to the residue calculus in modules of differential forms over rings of matrix pseudo-differential operators [9, 6]. Both these aspects of the theory are by now very well understood for integrable systems in one space dimension.