An (r, λ)-system is an arrangement of v objects (or varieties) into subsets (or blocks) such that each variety appears in exactly r blocks and each pair of distinct varieties appears in exactly λ blocks. To avoid trivial designs, we assume that 1≤λ<r.

A geometric proof is presented that, under certain restrictions, the product of an h-cobordism with a closed manifold of Euler characteristic zero is a product cobordism. The results utilize open book decompositions and round handle decompositions of manifolds.

For the autonomous system x' = F(x,∊), the case where the second Hopf-Friedrichs condition fails is analyzed in that sufficient conditions for bifurcation of non-trivial periodic solutions to occur are given. An application to mathematical ecology is also discussed.

The purpose of this note is to prove a result which is known to hold for Fréchet spaces [1, Chapitre II, §5, Exercise 24]. M. M. Day [2, p. 37] attributes the Banach space case to H. Löwig, although the earliest version that we have been able to find is that given by G. W. Mackey in [7, Theorem 1-1]. Recently H. E. Lacey has given an elegant proof for Banach spaces [5]. It is perhaps interesting to note that the non-locally convex case can be deduced from these known results which are established by duality arguments.

It is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.

The purpose of this note is to unify Theorem 4 of G. Baumslag [2] and a result of D. M. Smirnov in [6] in a more general setting. We prove the following result.

Let γ be an increasing function on the real numbers such that γ(0) = 0 (which, by translation of axes, is no restriction) and suppose that γ(n) is a positive integer if n is a positive integer. Let γ- denote the inverse function of γ. Furthermore, let L(x) be the least integer ≥ x; let [x] be the greatest integer ≤x, and suppose that c0, c1 … is an arbitrary sequence of numbers.

Let X be an infinite connected CW-complex which is the union of an increasing sequence of subcomplexes Xr. Let cat X denote the Lusternik-Schnirelmann category of X, normalized to take the value 0 on contractible spaces. Suppose that cat Xr ≤ K (r ≥ 1). In his problem list [1], T. Ganea proved that cat Xr ≤ 2k +1 and asked (Problem 5) whether this is the best possible upper bound. The purpose of this note is to prove that cat X ≤ 2k.

The dual object T of a compact group is called a local central A(p) set if there is a constant K such that ‖X‖P < K ‖X‖1 for all irreducible characters X of G. For each γ∊Γ, Dr is an irreducible representation of G of dimension dγ. Several authors [1, 2, 3, 4] have observed that Γ is a local central Λ(p) set for p<l provided sup{dγ:γ∊Γ}>∞, and some of them [2, 3] conjectured the converse. Cecchini [1] showed that Γ is not a local central Λ(4) set if G is a compact Lie group.

In the first part of this note we give a very simple and elegant proof of the theorem on the order of the error function of the (k, r)-integers, which the authors proved earlier using elaborate calculations. We also obtain an improvement of an earlier result on the order of the same error function on the basis of the Riemann hypothesis.

We derive here simple asymptotic estimates for both the Euler and Bernoulli numbers. The derivations follow easily from known results, but I am unable to find them elsewhere in the literature. C. Jordan [1, p. 245 and p. 303] gives some related inequalities. Other properties of these two classical sets of numbers may be found in [1], [3] and [4].

In their paper [1], Buseman and Glassco state that N(C, 7, 3) = 5 has been claimed but questioned. Schouten in [2] provides a classification of the orbits of ∧3V under the action of the group of automorphisms of ∧3V induced by automorphisms of V when dim V = 7.

Let (R, *) be a ring R with an involution *, i.e., * is a map RϵR such that for all a, bϵR

The trace and norm of an element a in (R, *) are respectively

(R, *) is said to be a normal ring if for all aϵR

or equivalently,

Niven [3] gave a simple proof that π is irrational. Koksma [2] modified Niven's proof to show that er is irrational for every non-zero rational r. Dixon [1] made a similar modification to show that π is not algebraic of degree 2. In this note, we prove a general theorem which gives Niven's and Koksma's results as easy corollaries. A suitable modification in our proof also gives Dixon's result.

In this note we show that a solvable group of unipotent matrices over a skew field can be simultaneously triangularized.

It is shown that the classical Fourier transform can be extended as an algebra isomorphism onto the algebra of all complex-valued functions, which are measurable and finite a.e., under pointwise addition and multiplication. The extended Fourier transform agrees with the distributional Fourier transform on the space of all distributions which have regular transforms. It is defined on an algebra of Mikusiński-type operators in which multiplication is convolution in the subspace of integrable distributions.

Let C[0,1] be the space of all continuous real valued functions defined in [0,1] with the supremum norm

1

The subspace of C[0,1] consisting of all functions f(x) for which f(0) and f(l) are integers will be denoted by C0[0,1], Let be a sequence of real numbers satisfying:

2

A base a pseudoprime is an integer n such that

1

A Carmichael number is a composite integer n such that (1) is true for all a such that (a, n ) = l. It was shown by Carmichael [1] that, if n is a Carmichael number, then n is the product of k(>2) distinct primes P1,P2,P3, … Pk, and Pi-1|n-1(i=1, 2, 3, …, k).

A short proof is given of a biquadratic reciprocity law proved by Burde in 1969.

Let p and q be primes ≡1 (mod 4) such that (p | q) = (q | p) = 1. Then there are integers a, b, c, d with

Set