This paper characterizes the Banach space E for the sequential continuity and the continuity on bounded sets of the composition map m: C(S, E)wk x K{E,F)wk —> C(S,F)wk. Here, K(E,F) denotes the Banach space of compact linear operators on the Banach space E to the Banach space F with the usual operator norm, and for any Banach space E, Ewk denote the Banach space E with the weak topology. Also we denote by C(S, E) the Banach space of E valued continuous functions on a nonvoid compact Hausdorff space S with sup norm.

In this paper we present a new approach to classical Karamata's results concerning the Hardy-Littlewood tauberian theorem.

Let G be a group with identity e, let λ be a normal supernilpotent radical in the category of associative rings and let λref be the reflected radical in the category of G-graded rings. Then for A a G-graded ring, λref(A) is the largest graded ideal of A whose intersection with Ae is λ (Ae). For λ = B, the prime radical, we compare Bref(A) to BG(A) = B(A)G, the largest graded ideal in B(A).

A classical theorem of Marcinkiewicz states that a function is Perron integrable iff it has one continuous major and one continuous minor function. Using an elaboration of this remarkable theorem three applications are made; to obtain a new proof of a recent characterization of the Perron integral, to proofs of some theorems on interchange of limits and integration and to extend classical existence theorems for ordinary differential equations.

In this note we shall give a simple and more direct proof of the Kuiper- Kuo Theorem. Also, we shall simplify Kuiper's proof of the Morse Lemma.

We define an arithmetical function, f(n), which gives a lower bound for the rank of elliptic curves, y2 = x3 + nx, n square-free. Thus, if f(n) is unbounded for square-free values of n, then there are elliptic curves of arbitrarily large rank. We show that f(n) is unbounded as n ranges over all integers.

Mahler's measure of a monic polynomial is equal to the product of modules of its roots which lie outside the unit circle. By classical theorem of Kronecker it is strictly greater than 1 for any polynomial that is not a product of cyclotomic factors. In this case a number of lower bounds of the measure, depending either on the degree of the polynomial or on the number of its non-zero coefficients, has been found. Here is given an improvement of the bound of the latter type previously found by the author, A. Schinzel and W. Lawton.

Computations of the Iwasawa λ -invariant for imaginary quadratic fields showed a discrepancy in the proportion of even and odd traces of certain integers from these imaginary quadratic fields. This paper shows that such a discrepancy is in some sense to be expected and that the proportion of even and odd traces of principal generators of powers of prime ideals in imaginary quadratic fields is related to the 3-primary component of the class group.

We consider the density in C[a, b] of generalized polynomials of the form The main point of this note is that total positivity of K(x, t) has little relationship to density: There is a symmetric, analytic, totally positive (in fact ETP (∞)) kernel K for which these generalized polynomials are not dense.

A Banach space X is said to have the Dunford-Pettis Property if every weakly compact linear operator T: X —> Y, where Y is any Banach space, is completely continuous (that is, T maps weakly convergent sequences to strongly convergent ones). In this paper, we prove that if A is a nest algebra of operators on a separable, infinite dimensional Hilbert space, then A fails to have the Dunford-Pettis Property. We also investigate a certain algebra associated to A, analogous to a construction used by Bourgain and others in connection with the Dunford-Pettis Property for function algebras. We show that this algebra must lie between A and the quasi-triangular algebra A + K and we give examples to show that either extreme or something in between is possible. Finally, we consider the algebra of analytic Toeplitz operators and give a result for the corresponding associated algebra which is analogous to a result of Cima, Jansen, and Yale for H∞.

Let E and F be real subspaces of Cn which contain no non trivial complex subspace of Cn and intersecting only at the origin. We prove in a special case that every continuous function on E ∪ F can be asymptotically approximated on E ∪ F by an entire function.

We prove that every continuous function on the product of certain curves can be asymptotically approximated by entire functions.

Let G be a finite group, let k be a field and let R be a finite dimensional fully G-graded k-algebra. Also let L be a completely reducible R-module and let P be a projective cover of R. We give necessary and sufficient conditions for P|R1 to be a projective cover of L|R1 in Mod (R1). In particular, this happens if and only if L is R1-projective. Some consequences in finite group representation theory are deduced.

We find conditions under which the numerator of the zeta-function of the curve y2+y = xd over Fp, where d — 2g +1 is a prime, d ≠ p, is irreducible over Q. This leads to the generalized Mersenne problem of "almost primality" of the number of points on the jacobian of such a curve over an extension of Fp, which has application to public key cryptography.

Let S be a smooth surface contained as an ample divisor in a smooth complex projective threefold X, which is a P1 -bundle, and assume that induces OP1 (1) on the fibres of X. The following fact is proven. The restriction to S of the bundle projection of X is exactly the reduction morphism of the pair provided that this one is not a conic bundle. The proof is very simple and does not involve any consideration on the nefness of the adjoint bundle Some applications of the proof are given.

We consider two generalizations R0w and R0 of the usual symmetry axiom for topological spaces to arbitrary closure spaces and convergence spaces. It is known that the two properties coincide on Top and define a non-simple subcategory. We show that R0W defines a simple subcategory of closure spaces and R0 a non-simple one. The last negative result follows from the stronger statement that every epireflective subcategory of R0 Conv containing all T1 regular topological spaces is not simple. Similar theorems are shown for the topological categories Fil and Mer.

Nehari's theorem for norms of bounded Hankel operators is revisited. Using it, the absolute values of Toeplitz operators are studied. This gives a theorem of Widom and Devinatz for invertible Toeplitz operators.

In [2], Ladas and Sficas made two conjectures about the asymptotic behavior of solutions of some neutral differential equations. In this paper we confirm that these conjectures are indeed correct.

If A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.

A complete answer is given to the question: Under what circumstances is the product of three harmonic homologies in PG(2, F) again a harmonic homology ? This is the natural question to ask in seeking a generalization to projective geometry of the Three Reflection Theorem of metric geometry. It is found that apart from two familiar special cases, and with one curious exception, the necessary and sufficient conditions on the harmonic homologies produce exactly the Three Reflection Theorem.