In a vicinity of a stationary solution we consider a real analytic system of ODE of order four, depending on a small parameter. We look for families of periodic solutions which contract to the stationary solution, when the parameter tends to zero. We apply the general methods developed in [2] for the study of complex bifurcations and in [4] for local resolutions of singularities.

A norm on ℝn is said to be permutation invariant if its value is preserved under permutation of the coordinates of a vector. The isometry group of such a norm must be closed, contain Sn and —I, and be conjugate to a subgroup of On, the orthogonal group. Motivated by this, we are interested in classifying all closed groups G such that 〈—I,Sn〉 < G < On. We use the theory of Lie groups to classify all possible infinite groups G, and use the theory of finite reflection groups to classify all possible finite groups G. In keeping with the original motivation, all groups arising are shown to be isometry groups. This completes the work of Gordon and Lewis, who studied the same problem and obtained the results for n ≥ 13.

We offer here a substantial simplification and shortening of a proof of the monotonicity of the abscissae of the points of inflection of Bessel functions of the first kind and positive order.

In [7,3.1 ] the authors show that if a space X is realcompact and locally compact, then X* is a P′-space. In this paper we show that the hypothesis of realcompactness can be weakened. We also look at other conditions on X that are sufficient to guarantee that X* is a P′-space.

It is shown that a suitable conformai change of the metric in the leaf direction of a transversally oriented Riemannian foliation on a closed manifold will make the basic component of the mean curvature harmonic. As a corollary, we deduce vanishing and finiteness theorems for Riemannian foliations without assuming the harmonicity of the basic mean curvature.

We show that every weak*-closed derivation from C[0,1] ⊂ L∞[0, 1] into L∞[0, 1] is the inverse of integration against a function in L1[0,1].

Let be the closed bidisc and T2 be its distinguished boundary. For be a slice map, that is, and Then ker Φαβ is an invariant subspace, and it is not difficult to describe ker Φαβ and In this paper, we study the set of all multipliers for an invariant subspace M such that the common zero set of M contains that of ker Φαβ.

We construct, under MAcountable, a countably compact topological subsemigroup of which is not a group, hence a counterexample for the Wallace problem. We also show that there is no p-compact counterexample for the Wallace problem, answering a question of D. Grant. Finally, we show that—in some sense—our counterexample for the Wallace problem constructed under MAcountable cannot be done in ZFC.

We provide a criterion for the class group of a complex quadratic field to have exponent at most 2. This is given in terms of the factorization of a generalized Euler-Rabinowitsch polynomial and has consequences for consecutive distinct initial prime-producing quadratic polynomials which we cite as applications.

Let R be a prime ring of characteristic not 2 and d a derivation of R. It is shown that if d2n is a derivation of R, where n is a positive integer, then d2n-1 = 0.

The defining relations for the Lie superalgebra Γ (σ1, σ2, σ3) as a contragredient algebra are discussed and a PBW type basis theorem is proved for the corresponding q-deformation.

We study the question of nonexistence of extremal functions for perturbations of some sharp inequalities such as those of Moser-Trudinger (1971) and Chang- Marshall (1985). We shall show that for each critically sharp (in a sense that will be precisely defined) inequality of the form

where is a collection of measurable functions on a finite measure space (I, μ) and O a nonnegative continuous function on [0, ∞), we have a continuous Ψ on [0, ∞) with 0 ≤ Ψ ≤ Φ, but with

not being attained even if the supremum in (1) is attained. We then apply our results to the Moser-Trudinger and Chang-Marshall inequalities. Our result is to be contrasted with the fact shown by Matheson and Pruss (1994) that if Ψ(t) = o(Φ(t) as t —> ∞ then the supremum in (2) is attained. In the present paper, we also give a converse to that fact.

Let be a representation of the finite group G over the field . If the order |G| of G is relatively prime to the characteristic of or n = 1 or 2, then it is known that the ring of invariants is Cohen-Macaulay. There are examples to show that need not be Cohen-Macaulay when |G| is divisible by the characteristic of . In all such examples is at least 4. In this note we fill the gap between these results and show that rings of invariants in three variables are always Cohen-Macaulay.

If R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

Nonnegative product linearization of the Askey-Wilson polynomials is shown for a wide range of parameters. As a corollary we obtain Rahman's result on the continuous q-Jacobi polynomials with α ≥ β > — 1 and α + β + 1 ≥ 0.

A sieve lemma is found, applicable to certain families of analytic functions on the unit disk, analogous to a large sieve lemma of P. X. Gallagher for sets of integers.

Weighted LP mean convergence of Hermite-Fejér interpolation based on the zeros of orthogonal polynomials with respect to the weight |x|2α+1(l — x2)β(α, β > — 1) is investigated. A necessary and sufficient condition for such convergence for all continuous functions is given. Meanwhile divergence of Hermite-Fejér interpolation in LP with p > 2 is obtained. This gives a possible answer to Problem 17 of P. Turân [J. Approx. Theory, 29(1980), p. 40].

AS a generalization of Morita duality, Kraemer introduced the notion of quasi-duality and showed that each left linearly compact ring has a quasi-duality. Let R be an associative ring with identity and R[[x]] the power series ring. We prove that (1) R[[x]] has a quasi-duality if and only if R has a quasi-duality; (2) R[[x]] is left linearly compact if and only if R is left linearly compact and left noetherian; and (3) R[[x]] has a Morita duality if and only if R is left noetherian and has a Morita duality induced by a bimodule RUS such that S is right noetherian.