In this paper we consider O. Bonnet III-isometry (or BIII-isometry) of surfaces in 3-dimensional Euclidean space E3 Suppose a map F: M → M* is a diffeomorphism, and F* (III*) = III, ki(m) = k*i (m*), i = 1, 2, where m ∈ M, m* ∈ M*, m* = F (m), ki and k*i are the principal curvatures of surfaces M and M* at the points m and m*, respectively, III and III* are the third fundmental forms of M and M*, respectively. In this case, we call F an O. Bonnet III-isometry from M to M*. O. Bonnet I-isometries were considered in references [1]-[5].

We distinguish three cases about BIII-surfaces, which admits a non-trivial BIII-ismetry. We obtain some geometric properties of BIII-surfaces and BIII-isometries in these three cases; see Theorems 1, 2, 3 (in Section 2). We study some special BIII-surfaces: the minimal BIII-surfaces; BIII-surfaces of revolution; and BIII-surfaces with constant Gaussian curvature; see Theorems 4, 5, 6 (in Section 3).

In this paper conditions of M-symmetry, strong, semimodularity and θ-modularity for the congruence lattice L (S) of a regular ω-semigroup S are studied. They are proved to be equivalent to modularity. Moreover it is proved that the kernel relation is a congruence on L(S) if and only if L(S) is modular, generalizing an analogous result stated by Petrich for bisimple ω-semigroups.

In this paper, two gradient properties of explicit convex functions are given.

1. General Idea. In this paper formal operators and bracketings devices are essentially the same; so are well formed formulae and correct bracketings.

A well known theorem of symbolic logic (Post languages, etc.) and universal algebra characterizes well formed formulae among (linear) strings of symbols for variables and operators in terms of a system of numerical inequalities, one of them an equality.

We give a separability criterion for the polynomials of the form

$$\begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}$$

$$\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$

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.

Let G be a finitely generated group and let R be a commutative ring, regarded as a G-module with G acting trivially. We shall determine when the cup product of two elements of H1(G, R) is zero. Our method will use the interpretation of H2(G, R) as extensions of G by R. This will give an alternative demonstration of results of Hillman and Würfel.

Throughout this paper G denotes an infinite compact connected Hausdorff Abelian group with character group X. Given a map of α X into itself, we are concerned with the set of a, ∈ G such that the function ϕa ∈ l∞ (X) defined by is a multiplier of type (þ, þ), where it can be assumed without loss of generality that 1 ≤ þ < ∞.

Let G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.

A Boolean-like ring R is a commutative ring with unity in which 2x = 0 and xy(1 + x)(1 + y) = 0 hold for all elements x, y of the ring R. It is shown in this paper that in the category of Boolean-like rings, R is injective if and only if R is a complete Boolean ring and R is projective if and only if R = {0, 1}.

Consider a k-variable normal distribution Ν (μ,Σ where mgr; = (μ1,μ2, … μk)' and Σ is diagonal matrix of unknown elements >0,i = 1,2, … k. The problem of sequential estimation of = 1 αiμi is considered. The stopping rule is shown to have some interesting limiting properties when the σi's become infinite.

In [1], it was shown that if ƒ ∈ Lp(Rn), where 1 < p < ∞, then the closed subspace of Lp (Rn) spanned by functions of the form [where a1, …, an, b1, …, bn, are real numbers; ak, ≠ 0; k = 1, …, n] coincides with the whole of Lp(Rn). In the present note, analogous results are derived for the spaces of integrable functions, essentially bounded measurable functions, bounded continuous functions, and continuous functions vanishing at infinity.

Convergent iterative sequences are constructed for the polynomials fm = z + zm, m ≧ 2, with initial point the lemniscate {z: |fm (z)| ≦1}. In the particular case m = 2 convergent iterative sequences are constructed also for f-1m, (z) with an arbitrary initial point. The method is based on a certain variational principle which allows reducing the problem to the well known situation of an analytic function mapping a simply connected domain into a proper subset of itself and possessing a fixed point in the domain.

Throughout this paper any matrix is square and of order q unless otherwise specified. I and J will represent an identity matrix and a square matrix with every element + 1 respectively; if necessary, the size will be indicated by a subscript. The Kronecker product of matrices A = (aij) and B is the block matrix whose (i, j)-th block is aijB, and is written A x B.

The purpose of this note is to introduce some “canonical” characteristic, subgroups and to observe some of their more obvious properties.

K. W. Chang generalizing a result of Lazer [3], proved in [4] the following THEOREM 1. Suppose that f:I → R+ = (0, + ∞), I = ([t0, + ∞), t0 0, is a non-decreasing function whose derivatives of orders ≧ 3 exist are continuous on ([t0, + ∞). Moreover, limt→+∞f(t) = +∞ and for some α, 1≧ α ≧ 2, and F =f-1/α then every solution x(t) of the equation tends to zero as t - + ∞.

New amicable orthogonal designs of order eight are given and they are used to construct ne orthogonal designs of order 32.

It is well known that a wide range of Special Function Theory can be realized by considering unitary representations of certain topological groups.

In this approach it is very important to determine all irreducible continuous unitary representations of the group in question.

For the group of movements this problem was initiated by Vilenkin [6]. Rather restrictive conditions were imposed in this paper and while he returned to the problem in [7], it was still not solved in full generality (among other things the representation space was assumed separable). The first complete solution appears to have been given by Thoma [5]. Here, the method was to show each irreducible continuous unitary representation equivalent to a particular representation in a space of square integrable functions.

A periodic binary array on the sequare grid is said to be sequential if and only if each row and each column of the array contains a given periodic binary sequence or some cyclic shift or reversal of this sequence. Such arrays are of interest in connection with experimental layouts. This paper extends previous results by characterizing sequential arrays on sequences of the type (1,…,1,0,…,0) and solving the problem of equivalence of such arrays (including a computation of the number of equivalence classes).

A consequence of Schreier's formula is that if G is a subgroup of the free group F of rank n < 1 and rank G ≦ n, then G = F or G is of infinite index in F. However, if S is a free sovlvable group of derived length I < 1 and H is a subgroup of S which is free solvable of the same length, then the rank of H does not exceed the rank of S. These observations led G. Baumslag to conjecture that if H is of finite index in S then H = S. In fact, we have sharper results in two directions. If H and S are free solvable of the same length, not only is H of infinite index in S, but δ1−1(S)/δ1−1(H) is torsion-free. In another direction we need not assume that S is free solvable, only that s is torsion-free and of derived length l (l > 1) and that H is not cyclic. Thus Stallings' theorem [11] that a finitely generated torsionfree group with a free subgroup of finite index is itself free has an even stronger counterpart in the variety of groups solvable of length at most l (l > 1): a torsionfree group in that variety with a non-cyclic free subgroup of finite index coincides with this subgroup. The proof relies on the following theorem: If S is a free solvable group, J is the group of automorphisms of S which induce the identity on S/S', and I is the group of inner automorphisms of S, then J/I is torsion-free. The proofs of these theorems form the bulk of the first four sections.