For positive integers n and k, with n≥2k, let , where each prime factor of u is less than k, and each prime factor of v is at least equal to k. It is shown that u<v holds with just 12 exceptions, which are determined. If , where each prime factor of U is at most equal to k, and each prime factor of V is greater than k, then U<V holds with at most finitely many exceptions, 19 of which are determined. It is conjectured that there are no others.

In this paper, we are concerned with certain generalizations of subnormal and ascendent (transfinitely subnormal) subgroups of a group. A subgroup A of a group G is called f-ascendent in G if there is a well ordered ascending complete series of subgroups of G, where for all α < λ, either Gα ⊲ Gα+1 or [Gα+1: Gα] < ∞. If such a series has finite length, A is called F-subnormal in G.

Let ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ be Dirichlet spaces with superharmonic weights induced by positive Borel measures $\unicode[STIX]{x1D707}$ on the open unit disk. Denote by $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ Möbius invariant function spaces generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$. In this paper, we investigate the relation among ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and some Möbius invariant function spaces, such as the space $BMOA$ of analytic functions on the open unit disk with boundary values of bounded mean oscillation and the Dirichlet space. Applying the relation between $BMOA$ and $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$, under the assumption that the weight function $K$ is concave, we characterize the function $K$ such that ${\mathcal{Q}}_{K}=BMOA$. We also describe inner functions in $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ spaces.

Throughout this paper only abelian l-groups will be considered and G will denote an abelian l-group. G is large in the l-group H or H is an essential extension of G if G is an l-subgroup of H and for each l-ideal L ≠ 0 of H we have L ∩ G ≠ 0. A ν-hull of G is a minimal vector lattice that contains G and is an essential extension of G. Each G admits a ν-hull (Conrad (1970)).We shall be interested in the following properties of G.

We consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all lattice of subvarieties of varieties V(A) ganarated by algebras A ε P.

If two functions of a real variable are integrable over two intervals, say of t, τ, respectively, then the product of the two functions should be integrable over the rectangular product of the two intervals of t and τ. For the Lebesgue integral, definable using non-negative functions alone, the proof is easy. For non-absolute integrals such as the Perron, Çesàro-Perron, and Marcinkiewicz-Zygmund integrals we have difficulties since the functions cannot be assumed non-negative. But the present paper gives a proof.

The usual technique for dealing with an abelian W*-algebra is to consider it, via the Gelfand theory, as the algebra of all continuous complex-valued functions on an extremally disconnected compact Hausdorff space with a separating family of normal linear functionals. An alternative approach, outlined in [2] and [10], is to develop the theory within the framework of Riesz spaces (linear vector lattices) where the order properties of the self-adjoint operators play an important and natural role. It has been known for a long time that the self-adjoint part of an abelian W*-algebra is a Dedekind complete Riesz space under the natural ordering of self-adjoint operators, but it is only relatively recently that a proof of this fact has been given that is independent of the Gelfand theory, and the interested reader may consult [2] or [10] for the details. This approach is essentially foreshadowed in [6] and provides a very satisfying introduction to the theory of commutative rings of operators. From this point of view, the spectral theorem for self-adjoint operators falls naturally into place as an easy consequence of the spectral theorem of H. Freudenthal. In this paper, the line of approach via Riesz spaces is developed further and several well known results are shown to follow as elementary consequences of the order structure of the algebra.

Random Fourier series are studied for a class of compact abelian hypergroups. The randomizing factors are assumed to be independent and uniformly subgaussian. In the presence of a natural teachnical hypothesis, an entropy condition of Dudley is shown to be sufficient for almost sure continuity. The classical results on almost sure membership in Lp, where p < ∞, are generalized to this setting. As an application, it is shown that a simple condition on the dual object implies that the de Leeuw-Kahane-Katznelson phenomenon occurs. Another application is the analogue of a classical sufficient condition for almost sure continuity. Examples illustrating the general theory are given for the hypergroup of conjugacy classes of SU(2) and for a class of compact countable hypergroups.

Let r be an integer greater than 1, and let A be a finite, nonempty set of nonzero integers. We obtain a lower bound for the number of positive squarefree integers n, up to x, for which the products ∏ p∣n(p+a) (over primes p) are perfect rth powers for all the integers a in A. Also, in the cases where A={−1} and A={+1}, we will obtain a lower bound for the number of such n with exactly r distinct prime factors.

We consider algebras for which the operation PC of pure closure of subsets satisfies the exchange property. Subsets that are independent with respect to PC are directly independent. We investigate algebras in which PC satisfies the exchange property and which are relatively free on a directly independent generating subset. Examples of such algebras include independence algebras and dinitely generated free modules over principal ideal domains.

It is shown that all the generalized Whitehead products vanish in X and all the components of XΣA have the same homotopy type when X is a T-space. It is also shown that any T-space is a G-space. The dual spaces of T-spaces are introduced and studied.

A characterization is provided here for any normal partition of the set of idempotents of a regular semigroup S. As a by-product of the method used, a new characterization of the greatest congruence on S corresponding to a given normal partition of its idempotents is obtained.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 M 10.

Let E be a vector lattice in the sense of [2]. Two elements a, b in E are said to be disjoint (written a⊥b) if ′a′^′b′ = 0. For any subset A of E, A⊥ is the set If A is a one point set {a}, then we shall use a⊥ for {a}⊥. A⊥⊥ is denned by A⊥⊥ = {A⊥}⊥. A subset B of E is said to be a polar if it is of the form A⊥ for some subset A of E. It is well known [1] that the set of all polars of E ordered by inclusion forms a Boolean algebra. We shall denote this algebra by B, and M will mean the set of all maximal ideals of B. The following properties of polars will be used in the paper.

The object of this paper is to apply, in the case n = 4, the results of Barnes and Dickson [1] concerning extreme coverings of n-dimensional Euclidean space by equal spheres whose centres form a lattice.

We prove the higher integrability of nonnegative decreasing functions, verifying a reverse inequality, and we calculate the optimal integrability exponent for these functions.

The existence of 1-factorizations of the complete multigraph λKn which cannot be decomposed into 1-factorizations with smaller λ is studied.

The group near-ring constructed from a right near-ring R and a group G is studied in the special case where the near-ring is distributively generated. In particular, results concerning homomorphisms of near-rings or of groups and the augmentation ideal are obtained which resemble closely those obtained for group rings.

A (v, k, λ) configurations is conjectured to exist for every v, k and λ satisfying λ(v − 1) = k(k − 1) and k − λ is a square if v is even, x2 = (k − λ)y2 + (−1)(v−1)/2 λz2 has a solution in integers x, y and z not all zero for v odd.

We give an explicit way of writing down a minimal set of generators for the canonical ideal of a nondegenerate curve, or of a more general smooth projective curve in a toric surface, in terms of its defining Laurent polynomial.

Let L=−Δ+V be a Schrödinger operator on ℝn where V is a nonnegative function in the space L1loc(ℝn) of locally integrable functions on ℝn. In this paper we provide an atomic decomposition for the Hardy space H1L(ℝn) associated to L in terms of the maximal function characterization. We then adapt our argument to give an atomic decomposition for the Hardy space H1L(ℝn×ℝn) on product domains.