In this note we consider some problems related to the following question: What is the smallest integer m(n) for which there exists a family Fn of sets A1, A2,…, Am(n) with the following properties, (i) each member of Fn has n elements and (ii) if S is a set which meets each member of Fn, then S contains at least one member of Fn?

Erdős and Hajnal [1] observed that

and that m(l) = 1, m(2) = 3, m(3) = 7.

This paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

In this paper, we discuss the solvability of the p-k-Hessian inequality $\sigma _{k}^{\frac 1k} ( \lambda ( D_{i} (|Du|^{p-2}$ $ D_{j}u ) ) ) \geq f(u)$ on the entire space $\mathbb {R}^{n}$ and provide a necessary and sufficient condition, which can be regarded as a generalized Keller–Osserman condition. Furthermore, we obtain the optimal regularity of solution.

In 1936 A. M. Turing published his analysis of the notion of effective computability. Very roughly speaking, its object was to distinguish between numbers defined by existential statements and those defined by purely constructive ones. Guided by his schematizing of what goes on when a person calculates using paper and pencil, Turing introduced the concept of an A-machine, eventually to be called a Turing machine. Such a machine consists of a box and a linear tape divided into squares, indefinitely long in both directions. The tape passes through the box and exactly one square of it is always within the box. The system changes only at certain discrete times: t = 0, 1, …. Each square can carry exactly one of the finite set of symbols: s1, s2, …, sn, one of which may be the blank.

In [l], Moser derives a recurrence relation and studies the limiting behaviour of the Expectations En of the following game. “A real number is drawn at random from [0,1]. We may either keep the number selected, or reject it and draw again. We can then either keep the second number chosen or reject it, and draw again, and so on. Suppose we have at most n choices. What stopping rule gives the largest En and how can we estimate En?”

It is well known that ℳn, the multiplicative group n consisting of n-rowed square matrices with integer entries and determinant equal to ±1 can be generated by:

It is also known that U4 can be generated by U1, U2, and U3 (cf.[2] p.85).

However, by a construction which is much simpler than the one just mentioned for U4, it is possible to generate U3 by just U2 and U4. Since U2 and U4 affect only the first

This note is concerned with some new integral inequalities which are extensions of the results in [2]. The method by which these results are obtained is due to D. C. Benson [1]. Throughout the present note we shall assume 1<p<∞ and f(x) a non-negative measurable function.

A new oscillation criterion is given for the delay differential equation ${x}'\,(t)\,+\,p(t)x\,(t\,-\,\text{ }\!\!\tau\!\!\text{ (t)})\,=\,0$, where $p,\,\text{ }\!\!\tau\!\!\text{ }\,\in \,\text{C}\,\text{( }\!\![\!\!\text{ 0,}\,\infty ),\,\text{ }\!\![\!\!\text{ 0,}\,\infty )\text{)}$ and the function $T$ defined by $T(t)\,=\,t-\,\text{ }\!\!\tau\!\!\text{ }\,\text{(t),}\,\text{t}\ge \,\text{0}$ is increasing and such that ${{\lim }_{t\to \infty }}\,T(t)\,=\,\infty $. This criterion concerns the case where $\lim \,{{\inf }_{t\to \infty }}\int _{T(t)}^{t}p(s)ds\le \frac{1}{e}$.

It is shown that no pair of orthogonal Steiner triple systems of order 9 exists.

We give a classification result for "extreme-monic" polynomials in several variables having measure 1. The result implies a recent several-variable generalization, by D. W. Boyd, of Kronecker's classical theorem (that all zeros of a monic integral polynomial, with non-zero constant term and measure 1, are roots of unity).

It is proved that a Fréchet space is quasinormable if and only if every null sequence in the strong dual converges equicontinuously to the origin. This answers positively a question raised by Valdivia. As a consequence a positive answer to a problem of Jarchow on Fréchet Schwartz spaces is obtained.

Several authors have investigated "rings of quotients" of a given ring R . (See, for example, Johnson [7], Johnson and Wong [8], Utumi [11], Findlay and Lambek [5], Lambek [9], and Bourbaki [2].)

This paper presents variants of the Holder inequality for integrals of functions (as well as for sums of real numbers) and its inverses. In these contexts, all possible transliterations and some extensions to more than two functions are also mentioned.

We strengthen certain results concerning actions of $\left( \mathbb{C},\,+ \right)$ on ${{\mathbb{C}}^{3}}$ and embeddings of ${{\mathbb{C}}^{2}}$ in ${{\mathbb{C}}^{3}}$, and show that these results are in fact valid over any field of characteristic zero.

A family of sets {Aα} is said by Miller [3] to have property B if there exists a set S which meets all the sets Aα and contains none of them. Property B has been extensively studied in several recent papers (see the references in [2] and the last chapter of P. Erdös and A. Hajnal, On chromatic number of graphs and set systems, Acta. Math. Acad. Sci. Hung. 17 (1966) 61–99).

The homogeneous Besov-Lipschitz spaces, usually defined by difference operators or Fourier transform, are studied in terms of mean oscillation, and several equivalent characterisations are given.

We show that, in a complete metric space, every selfmap that is a “weak directional contraction” admits a fixed point.

A family of sets is said to possess property if there exists a set such that and for every We consider the following question raised by P. Erdös |1|: let n and N be positive integers, n ≥ 2 and N ≥ 2n - 1 and let S be a set of N elements; what is the least integer (provided such an integer exists), for which there exists a family of subsets of S satisfying

1. (a) |F| = n for each

1. (b)

1. (c) does not have property

1. (d) if and then has property

A new proof is given for the weak amenability of the group algebras of locally compact groups.

Let C(X, Y) denote the set of continuous functions from a metric space X to a metric space Y. Viewing elements of C(X, Y) as closed subsets of X × Y, we say {fn} converges topologically to f if Li fn = Lsfn = f. If X is connected, then topological convergence in C(X,R) does not imply pointwise convergence, but if X is locally connected and Y is locally compact, then topological convergence in C(X, Y) is equivalent to uniform convergence on compact subsets of X. Pathological aspects of topological convergence for seemingly nice spaces are also presented, along with a positive Baire category result.