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.

This paper is concerned with the convergence of Rayleigh-Ritz approximations to the solution of an elliptic boundary value problem. Although the work arose in connection with the aerofoil problem (and it is to this problem that the results obtained are immediately applied), the methods here employed are suitable for use on the wider class of problem mentioned above.

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

We consider continuous Gaussian stochastic process indexed by a compact subset of a vector space over a local field. Under suitable conditions we obtain an asymptotic expression for the probability that such a process will exceed a high level. An important component in the proof of these results is a theorem of independent interest concerning the amount of ‘time’ which the process spends at high levels.

Let $X=GC$ be a group, where C is a cyclic group and G is either a generalized quaternion group or a dihedral group such that $C\cap G=1$. In this paper, X is characterized and, moreover, a complete classification for $X$ is given, provided that G is a generalized quaternion group and C is core-free.

Let $A_{{\it\alpha}}^{p}$ be the weighted Bergman space of the unit ball in ${\mathcal{C}}^{n}$, $n\geq 2$. Recently, Miao studied products of two Toeplitz operators defined on $A_{{\it\alpha}}^{p}$. He proved a necessary condition and a sufficient condition for boundedness of such products in terms of the Berezin transform. We modify the Berezin transform and improve his sufficient condition for products of Toeplitz operators. We also investigate products of two Hankel operators defined on $A_{{\it\alpha}}^{p}$, and products of the Hankel operator and the Toeplitz operator. In particular, in both cases, we prove sufficient conditions for boundedness of the products.

Let X0, X1…Xn,… be a stationary Gaussian process. We give sufficient conditions for the expected number of real zeros of the polynomial Qn (z) = Σnj =o X jzj to be (2/ π)log n as n tends to infinity.

Suppose Xi, i = 1,…,n are indepedent and identically distributed with E/X1/r < ∞, r = 1,2,…. If Cov (( − μ)r, S2) = 0 for r = 1, 2,…, where μ = EX1, S2 = , and , then we show X1 ~ N (μ, σ2), where σ2 = Var(X1). This covariance zero condition charaterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of and S2 using semi invariants. More generally, if Cov = 0 for r = 1,…, k, then E((X1 − μ)/σ)r+2 = EZr+2 for r = 1,… k, where Z ~ N(0, 1). Conversely Corr may be arbitrarily close to unity in absolute value, but for unimodal X1, Corr2( < 15/16, and this bound is the best possible.

A one-dimensional tiling is a bi-infinite string on a finite alphabet, and its tiling semigroup is an inverse semigroup whose elements are marked finite substrings of the tiling. We characterize the structure of these semigroups in the periodic case, in which the tiling is obtained by repetition of a fixed primitive word.

In this paper, we obtain the projective cover of the orbit space X/G in terms of the orbit space of the projective space of X, when X is a Tychonoff G-space and G is a finite discrete group. An example shows that finiteness of G is needed.

We give a transcendence measure of special values of functions satisfying certain functional equations. This improves an earlier result of Galochkin, and gives a better upper bound of the type for such a number as an S-number in the classification of transcendental numbers by Mahler.

In the dual object of an infinite compact, connected group, every infinite Sidon set contains an infinite subset on which full interpolation can be performed using very small classes of measures (discrete measures on arbitrarily small sets or nonnegative discrete measures). In particular, the Figà-Talamanca–Rider subset of an infinite product of compact, connected, simple Lie groups has these kinds of interpolation. This substantially improves previous interpolation results.

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).

Let g1, g2, …, gn be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of real zeros of the random algebraic equations Σnk=1 Kσ gk(ω)tk = C, where C is a constant independent of t and not necessarily zero. This average is (1/π) log n, when n is large and σ is non-negative.

An approach to p-adic interpolation via divided differences is used to give alternative proofs of results of van der Poorten on p-adic exponential polynomials and to derive a p-adic analogue of Turan's first main theorem on sums of powers.

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).

If {Xn: 1 ≦ n < ∞} are independent, identically distributed random variables having E(X1) = 0 and Var(X1) = 1, the most elementary form of the central limit theorem implies that P(n-½Sn≧ zn) → 0 as n → ∞, where Sn = Σnk=1 X,k, for all sequences {zn:1 ≧ n gt; ∞} for which zn → ∞. The probability P(n-½ Sn ≧ zn) is called a “large deviation probability”, and the rate at which it converges to 0 has been the subject of much study. The objective of the present article is to complement earlier results by describing its asymptotic behavior when n-½zn → ∞ as n → ∞, in the case of absolutely continuous random variables having moment-generating functions.

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.

We are concerned with existence results for nonlinear scalar Neumann boundary value problems u″ + g(x, u) = 0, u′(0) = u′(π) = 0 where g(x, u) satisfies Carathéodory conditions and is (possibly) unbounded. On the one hand we only assume that the function (sgn u)g(x, u) is bounded either from above or from below in some function space, and we impose conditions which relate the asymptotic behavior of the function (for¦u¦large) with the first two eigenvalues of the corresponding linear problem (here G(x, u) = is the potential generated by g). On the other hand we consider cases where the function (sgn u)g(x, u) is unbounded. The potential G(x, u) is not necessarily required to satisfy a convexity condition. Our method of proof is variational, we make use of the Saddle Point Theorem.

Let L = HK be a semidirect product of a normal locally finite π′-group H by a locally finite π′-group K, where π, is a set of primes. Suppose CK(H) = 1 and L is Sylow π-sparse (which in the countable case just says that the Sylow π-subgroups of L are conjugate). This paper completes the characterization of those groups which can occur as K—this had previously been obtained under the assumption that L is locally soluble. The answer is the same—essentially that the groups occurring are those having a subgroup of finite index which is a subdirect product of so-called “pinched” groups.