We denote lattice join and meet by ∨ and ∧ respectively and the associated partial order by ≧. A lattice L with 0 and I is said to be orthocomplemented if it admits a dual automorphism x → x′, that is a one-one mapping of L onto itself such that which is involutive, so that for each x in L and, further, is such that for each x in L.

Recently J. Cresp and R. P. Sullivan (1975) investigated those rings R with the following properties: (*) every multiplicative subsemigroup of R is a subring. (**) every multiplicative subsemigroup of R containing 0 is a subring. For rings with (*) they obtained the following characterization.

Let R be any ring with identity, M a unital right R-module and α ≥ 0 an ordinal. Then M is a direct sum of a semisimple module and a module having Krull dimension at most α if and only if for every submodule N of M there exists a direct summand K of M such that K ⊆ N and N/K has Krull dimension at most α.

The asymptotic behaviour, for large real positive values of the independent variable t, of solutions of the ordinary differential equation is considered. (The dots denote differentiation with respect to t.) It is shown that this asymptotic behaviour can be obtained by comparison with the differential equation For both equations when a2 < 2 the solutions are unbounded oscillatory functions but when a2 > 2 the solutions are essentially monotone. The critical case a2 = 2 has not been investigated here although the present results suggest that the solutions are probably monotone. The asymptotic estimates of the solutions of equation (1) are first obtained by heuristic means commonly used by Applied Mathematicians working on singular perturbation problems. A basis for a formal expansion is obtained in terms of a recurrence relation and the first few terms are displayed explicitly. These formal expansion estimates are used to guide a rigorous analysis which is essentially an extension of the techniques used in validating solutions obtained by the Langer related equation technique. For the oscillatory case, a pair of Volterra integral equations is used to justify the asymptotic representations for an arbitrary large interval for t but not including the point at infinity.

We develop the transversal harmonic theory for a transversally symplectic flow on a manifold and establish the transversal hard Lefschetz theorem. Our main results extend the cases for a contact manifold (H. Kitahara and H. K. Pak, ‘A note on harmonic forms on a compact manifold’, Kyungpook Math. J.43 (2003), 1–10) and for an almost cosymplectic manifold (R. Ibanez, ‘Harmonic cohomology classes of almost cosymplectic manifolds’, Michigan Math. J.44 (1997), 183–199). For the point foliation these are the results obtained by Brylinski (‘A differential complex for Poisson manifold’, J. Differential Geom.28 (1988), 93–114), Haller (‘Harmonic cohomology of symplectic manifolds’, Adv. Math.180 (2003), 87–103), Mathieu (‘Harmonic cohomology classes of symplectic manifolds’, Comment. Math. Helv.70 (1995), 1–9) and Yan (‘Hodge structure on symplectic manifolds’, Adv. Math.120 (1996), 143–154).

In the preceding paper 1) one of us has proposed a definition for the “best” or most regularly growing fractional iterates of logarithmico-exponential type functions. The definition was essentially based on two observations. First, that the functional equation has exactly one solution with the property that is totally monotonic for every x > 0. Secondly, that if f(x) is a logarithmicoexponential function such that then the Abel equation has (apart from the arbitrariness of an additive constant) exactly one solution with the property that exists) and the same is true for any reasonably well-behaved function with property (3) whose manner of growth does not transcend Hardy's scale of L-functions. Thus every such function has a uniquely determined family of fractional iterates given by and these fσ(x) may be regarded as the most regularly growing iterates of f(x).

Properties of the Lebesgue function associated with interpolation at the equidistant nodes , are investigated. In particular, it is proved that the relative maxima of the Lebesgue function are strictly decreasing from the outside towards the middle of the interval [0, n], and upper and lower bounds, and an asymptotic expansion, are obtained for the smallest maximum when n is odd.

This is a sequel to the author's (Phatarfod [9]) paper in which an analogue of Wald's Fundamental Identity (F.I.) for random variables defined on a Markov chain with a finite number of states was derived. From it the sampling properties of sequential tests of simple hypotheses about the parameters occurring in the transition probabilities were obtained. In this paper we consider the case of continuous Markovian variables. We restrict our attention to the practically important case of a Normal Markov sequence X0,X1,X2,… such that the Yr being independent normal variables with mean zero and variance σ2.

Let G be a group. With each element a in G we associate the mappings ρ(a) and λ(a) of G into itself defined as follows, The product of mappings is defined as usual. Let P(G) and ∧(G) denote respectively the semigroups generated by the set of all ρ's and λ's. These semigroups will be called the commutation semigrowps of G.

Let 𝒦 be the class of all right R-modules that are kernels of nonzero homomorphisms φ:E1→E2 for some pair of indecomposable injective right R-modules E1,E2. In a previous paper, we completely characterized when two direct sums A1⊕⋯⊕An and B1⊕⋯⊕Bm of finitely many modules Ai and Bj in 𝒦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many Ai and Bj in 𝒦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 𝒦 with the class 𝒰 of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).

Let X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

We prove that the lattice points on the circles x2 + y2 = n are well distributed for most circles containing lattice points.

Motivated by a problem of characterising a family of Cayley graphs, we study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\mathsf{Aut}(G)$. It is shown that such groups correspond to complete multipartite graphs which are normal edge-transitive Cayley graphs.

The structure of semigroups with atomistic congruence lattices (that is, each congruence is the supremum of the atoms it contains) is studied. For the weakly reductive case the problem of describing the structure of such semigroups is solved up to simple and congruence free semigroups, respectively. As applications, all commutative, finite, completely semisimple semigroups, respectively, with atomistic congruence lattices are described.

Abstract: An orthogonality relation is an abstract relation on a group having properties similar to the relation on a l-group given by x ⊥ y if ¦x¦ ∧¦y│ = 0. A group G with an orthogonality relation ⊥ is isomorphically represented as a subgroups of the group Γ of continuous global sections of a sheaf of groups. If the stalks of the sheaf are torsion-free and G has and element 1 satisfying 1 ⊥ = (0) then Γ can be ordered so that is is an l-group and x ⊥ y if and only if │x¦ ∧ ¦y¦ = 0 in Γ. An l-group G is complemented if for all x, y ∈ G there is an a ∈ x⊥ ⊕ x⊥⊥ with y ∈ a⊥⊥: equivalent conditions are given for G to be complemented.

When a disc rotates in a fluid at rest, fluid near the disc acquires azimuthal momentum because of the viscous torque of the disc and outwards radial momentum under the action of centrifugal forces. The resultant flow is essentially a swirling jet. Away from the disc continuity requires the existence of an axial flow towards the disc to compensate for the fluid which has been thrown outwards. If the disc is finite there is a discontinuity in the boundary conditions at the edge of the disc where the no-slip condition is suddenly replaced by a condition of zero stress in the plane of the disc. The flow discharged from the edge of the disc is essentially a wake embedded in a swirling radial jet. It appears that no investigation of this wake has yet been made.

Let Cb be the admissible functorial quasi-uniformity on the completely regular bispaces which is spanned by the upper quasi-uniformity on the real line. Answering a question posed by B. Banaschewski and G. C. L. Brümmer in the affirmative we show that CbX is transitive for every strongly zero-dimensional bispace X.

Hypercentrally embedded subgroups of finite groups can be characterized in terms of permutability as those subgroups which permute with all pronormal subgroups of the group. Despite that, in general, hypercentrally embedded subgroups do not permute with the intersection of pronormal subgroups, in this paper we prove that they permute with certain relevant types of subgroups which can be described as intersections of pronormal subgroups. We prove that hypercentrally embedded subgroups permute with subgroups of prefrattini type, which are intersections of maximal subgroups, and with F-normalizers, for a saturated formation F. In the soluble universe, F-normalizers can be described as intersection of some pronormal subgroups of the group.

This paper is taken from the author's doctoral dissertation at New York University; the work was supported by the National Science Foundation, Grant G19674. The author would like to thank Professor Gilbert Baumslag for suggesting the questions treated in this paper, and for his very helpful guidance.

For a locally compact group G, the von Neumann kernel, n(G), is the intersection of the kernels of the finite dimensional (continuous) unitary representations of G. In this paper we calculate n(G) explicitly for a general connected locally compact group and for certain classes of non-connected groups.