We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

In general, multiplication of operators is not essentially commutative in an algebra generated by integral-type operators and composition operators. In this paper, we characterize the essential commutativity of the integral operators and composition operators from a mixed-norm space to a Bloch-type space, and give a complete description of the universal set of integral operators. Corresponding results for boundedness and compactness are also obtained.

Every Banach space with separable second dual can be equivalently renormed to have weakly uniformly rotund dual. Under certain embedding conditions a Banach space with weakly uniformly rotund dual is reflexive.

Let A and B be closed subalgebras of Cr(K) whose direct sum is Cr(K) According to a result of Stephen Fisher the set of common zeros of elements of B is a retract of K. We give examples to show that this result is not correct.

We obtain bounds for certain functionals defined on a class of meromorphic functions in the unit disc of the complex plane with a nonzero simple pole. These bounds are sharp in a certain sense. We also discuss possible applications of this result. Finally, we generalise the result to meromorphic functions with more than one simple pole.

Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial $D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation $D(x) \equiv 0 \pmod {p}$ has a solution.

Two 2-cell embeddings i, j of a graph G into surfaces  and ′ are said to be congruent with respect to a subgroup Γ of Aut(G) if there are a homeomorphism h:  → ′ and an automorphism γ ∈ Γ such that h ∘ i = j ∘ γ. In this paper, we compute the total number of congruence classes of 2-cell embeddings of any bouquet of circles into surfaces with respect to a group consisting of graph automorphisms of a bouquet.

We introduce the Banach space of vector valued sequences lp, q(E), 1 ≤ p, q ≤ ∞, where E is a Banach space. Then we study the relation between lp, q(E) and the Schur multipliers of lpE, where E is taken to be some lr.

In this note we give a bound for the weighted star discrepancy of Faure-Niederreiter (0, m, s)-nets in prime-power base b ≥ s – 1 for the special sequence of weights given by γi = 1/i2, i ≥ 1.

We prove that the sequence {log ζ(n)}n≥2 is not holonomic, that is, does not satisfy a finite recurrence relation with polynomial coefficients. A similar result holds for L-functions. We then prove a result concerning the number of distinct prime factors of the sequence of numerators of even indexed Bernoulli numbers.

In this paper, we obtain a new result for overtwisted contact $(+1/n)$-surgery. We also give a counterexample to a conjecture by James Conway on overtwistedness of manifolds obtained by contact surgery.

Dedicated to Professor George Szekeres

Short amicable sets were introduced recently and have many applications. The construction of short amicable sets has lead to the construction of many orthogonal designs, weighing matrices and Hadamard matrices. In this paper we give some constructions for short amicable sets as well as some multiplication theorems. We also present a table of the short amicable sets known to exist and we construct some infinite families of short amicable sets and orthogonal designs.

An area-diameter relation and an area-width relation for plane lattice-point-free-convex bodies is proved. This implies relations on covering sets with respect to general lattices.

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.

As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

For a pythagorean field F with semiordering Q and associated preordering T, it is shown that the Witt ring WT (F) is isomorphic to the Witt ring W (K) whre K is a closure of F with respect to Q. For an arbitrary preordering T, it is shown how the covering number of T relates to the construction of WT (F).