Erdös, Harary and Tutte have defined the chromatic number of the plane to be the least number of sets partitioning the plane such that no set contains two points at unit distance apart. By analogy, the chromatic number χ(S1), of the sphere, Sr of radius r is defined to be the least number of sets partitioning the surface of Sr such that no set contains two points at unit chordal distance apart. In this paper it is proven that and that this bound is best possible since .

A chain C in a distributive lattice L is called strongly maximal in L if and only if for any homomorphism φ of L onto a distributive lattice K, the chain (Cφ)0 is maximal in K, where (Cφ)0 = Cφ if 0 ∉ K, and (Cφ)0 = Cφ ∪ {0}, otherwise. Gratzer (1971, Theorem 28) states that if B is a generalized Boolean lattice R-generated by L and C is a chain in L, then C R-generates B if and only if C is strongly maximal in L. In this note (Theorem 4.6), we prove the following assertion, which is not far removed from Gratzer's statement: let B be a generalized Boolean lattice R-generated by L and C be a chain in L. If 0 ∈ L, then C generates B if and only if C is strongly maximal in L. If 0 ∉ L, then C generates B if and only if C is strongly maximal in L and [C)L = L. In Section 5 (Example 5.1) a counterexample to Gratzer's statement is provided.

An entire function f(z) is of bounded index if there exists a non-negative integer N such that . The least such integer N is called the index of f (se Lepson (1966)).

With respect to a given family of functions F, a function is said to be F-convex, if it is supported, at each point, by some member of F. For particular choices of F one obtains the convex functions and the generalized convex functions in the sense of Beckenbach. F-convex functions are characterized and studied, retaining some essential results of classical convexity.

Let X be a real Banach space and let K be a bounded closed convex subset of X. We prove that the set of strongly exposing functions K^ of K is a (norm) dense G8 in X* if and only if for any bounded closed convex subset C such that K⊄C, there exists a point x in K which is a strongly exposed point of conv (C ∪ K). As an application, we show that if X* is weakly compact generated, then for any weakly compact subset K in X, the set K^ is a dense G8 in X*.

Let p be a prime ≡ 1 (mod 4) and put v = p(p + 1)/2. It is proved in this paper that there exist four symmetric circulant matrices A, B, C, D of order υ such that where Iv is the identity matrix of order υ. This result is used to construct Hadamard matrices of order 4υ that are of the type originally prescribed by Williamson.

Duchon (1967, 1969) and Duchon and Kluvánek (1967) considered the problems of the existence of countably additive tensor products for vector measures. Duchon and Kluvánek (1967) showed that a countably additive product with respect to the inductive tensor topology always exists while Kluvánek (1970) presented an example which showed that this was not the case for the projective tensor topology. Kluvánek considered yet another tensor product topology which is stronger than the inductive topology but weaker than the projective tensor topology and showed that a countably additive product for two vector measures always exists for this particular tensor topology, see Kluvánek (1973). He has conjectured that this topology is the strongest tensor topology (given by a cross norm) which always admits products for any two arbitrary vector measures. In this note we use an example of Kluvánek (1974) to show that this conjecture is indeed true when one of the factors in the tensor product is l2 and the other factor is metrizable. The construction used also clarifies a conjecture made by Swartz (to appear) concerning products of Hilbert space valued measures.

Throughout this note, let E, F and G be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

It is proved that there exists no compatible tight Riesz order on a complemented modular lattice. An example is provided of a complemented lattice with a compatible tight Riesz order.

A continuous real-linear operator A = A0 + i1A1 + i2A2 + i3A3 on a quaternionic Hilbert space is called sesquihermitian if the linear operators Av are Hermitian; this condition is independent of the choice of quaternion basis (i1,i2,i3). The joint spectral distribution of the Av provides a functional calculus for sesquihermitian operators and real-valued C∞-functions on . This calculus is independent of the quaternion basis and extends naturally to quaternion-valued functions to give a continuous quaternion-linear mapping from the algebra of these functions to that of sesquihermitian operators. The mapping is not, in general, multiplicative unless the Av commute, in which case it agrees with that for several commuting operators on complex Hilbert space.

Schur's matrix Mn is ordinarily defined to be the n by n matrix (εjk), 0 ≦ j, k < n, where ε = exp (2 πi/n). This matrix occurs in a variety of areas including number theory, statistics, coding theory and combinatorics. In this paper, we investigate Pn, the permanent of Mn, which is define by where π ranges over all n! permutations on {0,1, …, n – 1}. Pn occurs, for example, in the study of circulants. Specifically, let Xn denote the n by n circulant matrix (xi, j) with xi, j = xi, j, where the subscript is reduced modulo n. The determinant of Xn is a homogeneous polynomial of degree n in the xi and can be written as Then Pn = A (1,1, … 1). Typical of the results established in this note are: (i) P2n = 0 for all n, (ii) Pp ≡ p ! (mod p3) for p a prime >3. (iii) If pa divides n then divides Pn. Also, a table of values of Pn is given for 1 ≦ n ≦ 23.

An Hadamard matrix H is a square matrix of order n all of whose entries are ± 1 such thatThere are matrices of order 1 and 2and for all other Hadamard matrices the order n is a multiple of 4, n = 4m. It is a reasonable conjecture that Hadamard matrices exist for every order which is a multiple of 4 and the lowest order in doubt is 268. With every Hadamard matrix H4m a symmetric design D exists with

Let A be a complex sequentially complete, locally convex (not necessarily commutative) topological algebra with the defining family {pα}α∈D of seminorms in which (*): for each sequence xn → 0 there exists xm ∈ {xn} such that xmk → 0 as k → ∞. Then each multiplicative linear functional on a Fréchet algebra satisfying the above condition (*) is Continuous.

These results answer open questions (1) and (2) (Mem. Amer. Math. Soc. 11, 1953) in the affirmative for Fréchet algebras in which (*) holds. It is also shown that a positive linear functional on such algebras with identity and continuous involution is continuous, thus partially generalizing Shah's result (1959).

Let Γ be a graph with finite vertex set V. Γ is homogeneous if whenever U1, U2 ⊆ V are such that the vertex subgraphs (U1), (U2) are isomorphic, then every isomorphism from (U1) to (U2) extends to an automorphism of Γ; homogeneous graphs were studied by Sheehan (1974) and were classified by the author. Γ is locally homogeneous if whenever U ⊆ V, then every automorphism of (U) extends to an automorphism of Γ. We prove that every locally homogeneous graph is homogeneous.

This note is concerned with a certain property of point processes. We prove that if N1, N2 and N3 are three independent point processes, then the bivariate point process (N1 + N3+ N2 + N3) uniquely determines the point processes N1, N2 and N3.

Throughout, R will denote an associative ring with center Z. For elements x, y of R and k a positive integer, we define inductively [x, y]0 = x, [x, y] = [x, y]1 = xy − yx, [x, y, y, hellip, y]k = [[x, y, y, hellip, y]k−1, y]. A ring R is said to satisfy the k-th Engel condition if [x, y, y, hellip, y]k = 0. By an integral domain we mean a nonzero ring without nontrivial zero divisors.

The associative and the commutative laws are characterized by preservation under the construction of powers and addition of a (new) unit element. This is used to generate the varieties defined by the two laws from two element groupoids.

Let S be a closed bounded convex set in d-dimensional Euclidean space Ed. The width w(S) of S is the minimum distance between supporting hyperplanes of S, and L(S) is the number of integral lattice points in the interior of S.