Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.

Consider a Vn with metric tensor gαβ. It is well known that the only tensor containing derivatives of gαβno higher than the second order, linear in the second order derivatives, and having vanishing covariant divergence is the Einstein tensor

(1)

Suppose that a system of ennuple vectors hiαand their inverses hαi is set up at each point of the Vn.

It's well known (see Endo [1]) that for a commutative ring A, if A is semihereditary then w.gl. dim. A ≤ 1. It seems worth recording the noncommutative version of this.

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

Here an example of a countable, nowhere first countable Hausdorff space is given.

A new concept—that of pseudoconsistency—which seems to be particularly appropriate for the estimation of a quantile is introduced. It is shown without any conditions whatsoever on the underlying distributionthat the sample quantile is strongly pseudoconsistent for the corresponding population quantile. The asymptotic distribution of the sample quantiles and order statistics is derived when the underlying distribution is discrete.

The purpose of this paper is to determine the conditions under which the maximal ideals of the ring C*(X)—the bounded real-valued continuous functions on a completely regular Hausdorff space X—are closed under pointwise convergence of sequences. Whereas the maximal ideals of C*(X) are closed under pointwise convergence of nets if and only if X is compact, it is shown that a necessary and sufficient condition for their pointwise sequential closure is that X be pseudocompact (i.e. that all real-valued continuous functions of X be bounded).

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write

(1.1)

If

(1.2)

as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,

(1.3)

The following square functional equation

(1)

was considered (for example [l]-[9]) previously.

Put

1

For the Appell function F4 defined by [3, p. 224]

2

Saxena [5, p. 216] proved a reduction formula in the form

3

where

It may be of interest to point out that formula (3) is not correct. Indeed we first establish the following general result.

McCarthy [4] generalized a necessary and sufficient condition for an arithmetic function to be totally multiplicative to the incidence algebra on a partially ordered set. Several equivalent conditions for an arithmetic function to be totally multiplicative are known [1], [2]. In this paper we generalize several of these (and some apparently new ones) to the regular convolution rings of Narkiewicz [5]. We also investigate the prime factorization of arithmetic functions in a certain subring of some of these regular convolution rings.

We use the notations of [2], particularly for operators on classes of structures; in addition, P*r(K) (respectively, Pr(K)) denotes the class of all reduced products of families of structures in K (those with respect to a proper dual ideal, respectively).

Professor Eliopoulous studied almost tangent structure on manifolds M2n in [1], [2], [3]. An almost tangent structure F is a field of class C∞ of linear operations on M2n such that at each point x in M2n, Fx maps the complexified tangent space into itself and that Fx is of rank n everywhere and satisfies F2=0.

In this note we state and prove the following

Any equiaffinity acting on the points of an n-dimensional vector space (n ≥2) leaves invariant the members of a one parameter family of hypersurfaces defined by polynomials p(xl…,xn)=c of degree m ≤n.

The theorem, restricted to the real plane, appears to have been discovered almost simultaneously by Coxeter [4] and Komissaruk [5]. The former paper presents an elegant geometric argument, showing that the result follows from the converse of Pascal's theorem. The present approach is more closely related to that of [5], in which the transformations are reduced to a canonical form.

The nonsolvable R-groups as defined by Walter [3] are groups of orders (q3+l)q3(q — 1), q = 32n+1, n ≥ 0. These are the groups of Ree type discussed by Ward [4] together with the Ree group R(3) of order 28.27.2. The R-group with parameter q has a doubly transitive representation of degree q3+1 but in this note we will prove that it cannot contain a sharply doubly transitive subset.

In this note we outline a proof of the Calderon extension theorem by a technique similar to that for the Whitney extension theorem. For classical proofs, see Calderon [2] and Morrey [4]. See also Palais [6, p. 170]. Our purpose is thus to give a more unified proof of the theorem in the various cases. In addition, the proof applies to the Holder spaces Ck+α, which was used in [3], and applies to regions satisfying the "cone condition" of Calderon.

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.

The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].

In this note we present a very simple proof of the upcrossings inequality (see [6], and the note at the end) for martingale sequences—one of the basic results in the theory of martingales—which does not make use of the notion of optional random variable, as is done in the usual proofs of the inequality.

Let H be a Hilbert space with inner product 〈,). A well-known theorem of von Neumann states that, if S is a symmetric operator in H, then S has a selfadjoint extension in H if and only if S has equal deficiency indices. This result was extended by Naimark, who proved that, even if the deficiency indices of S are unequal, there always exists a Hilbert space H1 such that H ⊆ H1 and S has a selfadjoint extension in H1.