Here we give examples and classifications of varieties with strange behaviour for the enumeration of contacts (answering a question raised by Fulton, Kleiman, MacPherson). Then we give upper and lower bounds (in terms of the degree) for the non-zero ranks of a projective variety.

This paper is concerned with local compactness of the minimal primal ideal space of a C*-algebra, a sufficient condition is given. The property in question has bad hereditary properties as is shown by examples.

In the present article we generalize Theorem 2.3 of [6] in the case of JV algebras without a unit element and we obtain as a consequence that the multiplicativity of the involution ((xy)* = y*x*) in the definition of a JH*-algebra is redundant (see [3]). We end this paper with a theorem on unital JH*-algebra which is a nonassociative extension of the main result in [4].

Let 5 be a monoid. A right S-system A is called strongly flat if the functor A ⊗ — (from the category of left S-systems into the category of sets) preserves pullbacksand equalizers. (This concept arises in B. Stenström, Math. Nachr. 48(1971), 315-334 under the name weak flatness). The main result of the present paper is a proof that for A to be strongly flat it is in fact sufficient that A ⊗ — preserve only pullbacks. The approach taken is to develop an "interpolation" condition for pullback-preservation, and then to show its equivalence to Stenström's conditions for strong flatness.

It is shown that a two-sided continuous ring with ascending chain condition on left annihilators is quasi-Frobenius.

A sufficient condition for the boundedness of a multiplier from a Sobolev space of index t > 1 / 4 to one of opposite index — t is obtained. The condition relates the indices of the Sobolev spaces to which the multiplier belongs to the pairs of Sobolev spaces between which the multiplier is bounded. The result is applied to homogeneous multipliers and a description of these multipliers in this setting is presesented. Extensions to higher dimensions are indicated.

Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

We present examples showing that certain results on the invariance of link polynomials under generalized mutation are the best possible. They show, moreover, that this generalized mutation cannot be effected by a sequence of ordinary mutations. One of the examples also shows that the reduced Jones polynomial can be a more sensitive invariant than the Jones polynomial itself.

Let A(D) be the usual disc algebra, and let A+ be the subalgebra of A(D) consisting in those f € A(D) which have an absolutely convergent series of Taylor coefficients. Set m1 = {f € A(D) | f(1) = 0}, m = m1 ∩ A+, let G be the function and K = A+ ∩ I1.

We show that the quotient algebras m1/I1 and m/K are similar in the sense of J. Esterle.

We prove that a bijective transformation on the set of circles in the real inversive plane which preserves pairs of circles a fixed inversive distance ρ > 0 apart must be induced by a Möbius transformation.

A basic double series is expressed in terms of two 5ϕ4 series which extends Bailey's transformation of an 8ϕ7 series into two 4ϕ3 's. From this formula we derive some quadratic transformations; one of them is a new q-analogue of a transformation due to Whipple. Product formulas as well as Gasper-Rahman's q-Clausen formula are also given as special cases.

A class of weakly coupled systems of semilinear elliptic partial differential equations is considered in an exterior domain in ℝN, N > 3. Necessary and sufficient conditions are given for the existence of a positive solution (componentwise) with the asymptotic decay u(x) = O(|x|2-N) as |x| —> ∞. Additional results concern the existence and structure of positive solutions u with finite energy in a neighbourhood of infinity.

Consider a collection of topological spheres in Euclidean space whose intersections are essentially topological spheres. We find a bound for the number of components of the complement of their union and discuss conditions for the bound to be achieved. This is used to give a necessary condition for independence of these sets. A related conjecture of Griinbaum on compact convex sets is discussed.

We deal here with the existence of half-way nonzero divisors for complete intersections and with related properties of their Hilbert functions.

Piecewise-linear (nonambient) isotopy of classical links may be regarded as link theory modulo knot theory. This note considers an adaptation of new (and old) polynomial link invariants to this theory, obtained simply by dividing a link's polynomial by the polynomials of the individual components. The resulting rational functions are effective in distinguishing isotopy classes of links, and in demonstrating that certain links are essentially knotted in the sense that every link in its isotopy class has a knotted component. We also establish geometric criteria for essential knotting of links.

On the set F of complex-valued arithmetic functions we construct an infinite family of convolutions, that is, binary operations ψ of the form

so that (F, +, ψ) is a commutative ring, for which the unity is unbounded. Here + denotes pointwise addition.

It is proved that for every Hausdorff space ℝ and for every Hausdorff (regular or Moore) space X, there exists a Hausdorff (regular or Moore, respectively) space S containing X as a closed subspace and having the following properties:

• la) Every continuous map of S into ℝ is constant.

• b) For every point x of S and every open neighbourhood U of x there exists an open neighbourhood V of x, V ⊆ U such that every continuous map of V into ℝ is constant.

• 2) Every continuous map f of S into S (f ≠ identity on S) is constant.

We present first examples of complete transversely affine foliations on compact manifolds with leaves whose closures are not submanifolds. Moreover, we prove that under some additional assumptions the closures of leaves form a singular foliation.

As a generalization of Kiang and Tan's proximately nonexpansive semigroups, the notion of a proximately uniformly Lipschitzian semigroup is introduced and an existence theorem of common fixed points for such a semigroup is proved in a Banach space whose characteristic of convexity is less than one.

Using LePotier's vanishing theorem, we establish a lower bound on the rank of nontrivial free differential complex in terms of the dimension of the support for its cohomology. Our bound specializes to the one predicted by the syzygy theorem of Evans and Griffith.