It is an old problem to find how a collection of congruent plane figures should be arranged without overlapping to cover the largest possible fraction of the plane or some region of the plane. If similar figures of arbitrary different sizes are permitted, Vitali's theorem ([7] p. 109) guarantees that packings which cover almost all points are possible. It is natural to study the diameters of figures used in such a packing and we will investigate this for the case of a closed disk packed with smaller open disks.

In 1955, J. Suranyi and P. Turán [8] introduced the nomenclature (0, 2)-interpolation for the problem of finding polynomials of degree ≤ 2n-1 whose values and second derivatives are prescribed in certain given nodes. In a series of papers ([2], [3], [8]) Professor Turán and his associates discussed the problems of existence, uniqueness, explicit representation and convergence of such interpolatory 2 polynomials when the nodes are the zeros of (1-x2) P′n- 1(x), Pn- 1(x) being the Legendre polynomial of degree n - 1.

Let f(t) be integrable L(-π, π) and periodic with period 2 π and let

1.1

be its Fourier series. The series

1.2

obtained by term by term differentiation of the series (1.1) is called the derived Fourier series of f.

The purpose of this note is to give a precise formulation of the correspondence principle between classical and quantum mechanics, for the case of momentum operators. (See e. g. I. Segal, Journ. Math. Phys. (1960), 475).

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation

1.1

using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

It could perhaps be reasonably maintained that for most students of calculus the definite integral is in fact the Cauchy Integral. That is to say, where F is a primitive of f.

If done heuristically, the discussion of the Riemann integral serves two purposes. Firstly, it indicates that every continuous function has a primitive.

Holsztyński [1] called a map f : X → Y f r om a space X into a space Y ‘universal for all maps of X into Y f if for all maps g : X → Y there exists a point x ∊ X such that f(x) = g(x) , i. e., if f has a coincidence with all maps from X into Y. As the word ‘universal’ is already widely used with different meanings, we prefer the more precise term ‘coincidence producing’ for these maps. Such maps must clearly be surjective.

The pk+2 - integral of James [2] is strong enough to integrate a trigonometric series of the form

1.1

which is summable (C, k) in [0,2π], provided an extra condition holds involving the conjugate series

1.2

Considering series with coefficients o(n), Taylor [5] constructed an integral (the AP-integral) which successfully integrates series of the form (1. 1) which are Abel summable provided an extra condition holds involving the Abel means of the conjugate series (1.2).

Compact complex homogeneous manifolds have been studied in great detail by Borel, Goto, Remmert and Wang (cf., (13)): it was shown that every compact, connected complex homogeneous manifold M is a holomorphic fiber bundle over a projective algebraic homogeneous manifold B with a connected, complex parallelizable fiber F.

In 1940, I. Niven [2] proved that the gaussian integer z = x + iy is the sum of two squares of gaussian integers if, and only if, y is even and not both of 1/2x and 1/2y are rational odd integers. In this note we calculate the total number g2(z) of representations of z in this form.

1

where a, b, c, d are rational integers, if and only if

2

It is well known that Sobolev′s Lemma on the continuity of functions possessing L2 distributional derivatives of sufficiently high order is a simple consequence of elementary properties of the Fourier transform in L2 (e.g. [1, p. 174]). (In fact this statement remains true if 2 is replaced by p, 1 ≤ p ≤ 2). In this note we show that imbedding theorems of the type Wm, p ⊂Lq can also be obtained using Fourier transforms and an elementary lemma which reduces the cases p > 2 to the case p = 2. The simplicity of this approach is obtained at the expense of a slight loss of generality in the imbedding theorem.

Let d denote a fixed integer > 1 and let GF(q) denote the finite field of q = pn elements. We consider q fixed ≥ A(d), where A(d) is a (large) constant depending only on d. Let

1

where each aiεGF(q). Let nr(r = 2, 3, …, d) denote the number of solutions in GF(q) of

for which x1, x2, …, xr are all different.

A filter is a set theoretical concept and as such, its structure is independent of any topology which can be put on the given space. However, an O-filter, whose counterpart in the theory of nets is the O-nets of Robertson and Franklin [2], is defined with respect to the topology on the given space. The purpose of this paper is to give necessary and sufficient conditions for every O-filter to be an ultrafilter and for every Cauchy filter to be an O-filter.

As is well known, every real function is the pointwise (uniform) limit of a sequence of functions with a finite (countable) range of values. Monna [5] and Kvačko [4] suggested some extensions of this theorem to functions with values in a separable metric space. In the present note we give some further generalizations, with an emphasis on uniform approximations which have many applications in the generalized theory of measure and integration. In particular, we consider measurable abstract functions (mappings).

An orientation of a graph G is an assignment of a unique direction to each line of G. The result is called an oriented graph. Two orientations of a graph are regarded as equivalent if the resulting oriented graphs are isomorphic as directed graphs. For example, the graph C3 consisting of a cycle of length 3 (a triangle) shown in Figure 1(a), has exactly two orientations D1 and D2; see Figure 1(b) and (c).

In [6] J. Rainwater obtained the following theorem.

Theorem. Let N be a normed linear space, {xn} a bounded sequence of elements in N and X ∊ N. for each extreme point f of the unit ball of N✶, then {xn} converges weakly to x.

Now let X be a compact Hausdorff space and H a linear subspace of C(X) (all real-valued continuous functions on X ) which separates the points of X and contains the constant functions. If x∊X, then MX(H) denotes the set of positive linear functionals μ on C(X) such that μ(h) = h(x) for all h in H.

The function was defined by G.N. Watson, [9, (i)]in 1931 by the integral relation∗

+ another term with μ and v interchanged;

If S is a semigroup, then an S-set AS is a set A together with a representation of S by mappings of A into itself. In this article, the theory of injective envelopes is carried from R-modules to S-sets. These results are known to hold in every Grothendieck category, but the category EnsS of (right) S-sets is not even additive.

Let M be a metric space with metric ρ which has the following properties.

1. M is finitely compact.

2. There exists a geodesic in M joining each two points of M.

3. For all a, b∈M, ρ(a, b) is equal to the length of a geodesic joining a and b.

Let p denote a prime number and let κp denote the finite field of p elements. Let f(x) ∊ κp[x] be of fixed degree d ≥ 2. We suppose that p is also fixed, large compared with d, say, p ≥ p0(d). By V(f) we denote the number of distinct values of f(x), x ∊ κp. We call f maximal if V(f) = p and quasi-maximal if V(f) = p + O(1). Clearly a maximal polynomial is quasi-maximal but it is not known under what conditions the converse holds.