The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

The aim of the present note is to describe the possible products when taking all the nonzero elements of a finite ring in some sequence. Compared with the analogous situation for finite groups, where the set of products of all elements has been shown in [2] to be a whole coset of the derived group, for rings the set of the above mentioned products will be proved either to be as large as possible or to consist of one or two elements only.

Let R be a complete intersection ring, and let M and N be R-modules. It is shown that the vanishing of ExtiR(M, N) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most n–1. We also estimate the complete intersection dimension of M*, the dual of M, in terms of vanishing of cohomology modules, ExtiR(M,N).

A well-known theorem of P. Hall says that if a group $G$ contains a normal nilpotent subgroup $N$ such that $G/N'$ is nilpotent then $G$ is nilpotent. We give a similar sufficient condition for a group $G$ to be an extension of a group of finite exponent by a nilpotent group.Supported by CNPq-Brazil.

Let {a, b} designate the Pythagorean ratio (a2 - b2)/2ab between the sides of a rational right angled triangle. This paper studies the circumstances in which Pythagorean ratios can occupy consecutive places in an arithmetic progression. Part I deals with sets of three such ratios, while Part II discusses sets of four ratios.

Let G be a finite abelian group of rank m, M an oriented compact connected surface, and F(G, M) the set of all orientation preserving free G-actions on M. Two actions φ1, φ2εF(G, M) are equivalent if there exists an orientation preserving homeomorphism h of M such that

hφ1(f) for all f ε G.

If R is a local (Noetherian) ring, it is well known that R is regular if and only if its completion is regular. It is the purpose of this note to show that a similar result is true for Noether lattices.

In this note, we give a new proof of the fact that an affine semiprime algebra $R$ of Gelfand-Kirillov dimension 1 satisfies a polynomial identity. Our proof uses only the growth properties of the algebra and yields an explicit upper bound for the pi degree of $R$.

For a given (d−1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h0(Γ),h1(Γ),. . .,hd(Γ)) and set h−1(Γ)=0. The known Swartz equality implies that if Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi(Δ)+(d−i+1)hi−1(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25(1) (1996), 137–148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18(3) (2004/05), 647–661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.

We characterize real hypersurfaces of type (A) and ruled real hypersurfaces in a complex projective space in terms of two φ-invariances of their shape operators, and give geometric meanings of these real hypersurfaces by observing their some geodesics.

In this paper we present a new discreteness criterion for a non-elementary subgroup G of SL(2, ℂ) containing elliptic elements by using a loxodromic (resp. an elliptic) transformation as a test map that need not be in G.

Let G be a group. The rth term LrG of the lower central series of G is the subgroup generated by the r-fold commutators

where , and for r > 1,

Let(π_{λ}, ℋ_{λ}) be a unitary highest weight representation of the connected Lie group G and ℊ its Lie algebra. Thenℊ contains an invariant closed convex cone W_{\rm{max}} such that, for each X∈W_{\rm{max}}^0, the selfadjoint operatori·dπ_{λ}(X) is bounded from above. We show that for each suchX , the space ℋ_{λ}^{∞} of smooth vectors for the action of G on ℋ_{λ} coincides with the set𝒟^{∞}(dπ_{λ}(X)) of smooth vectors for the generally unbounded operator dπ_{λ}(X).

We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such that

This extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].

A generalization is given of a theorem of F. Brafman [1] on the equivalence of generating relations for a certain sequence of functions. The main result, contained in Theorem 2 below, may be applied to several special functions including the classical orthogonal polynomials such as Hermite, Jacobi (and, of course, Legendre and ultraspherical), and Laguerre polynomials.

Since Azumaya introduced the notion of A-injectivity in 1974, several generalizations have been investigated by a number of authors. We introduce some more generalizations and discuss their connection to the previous ones.

An MS-algebra is an algebra (L; ∨, ∧, ∘, 0, 1) of type (2, 2, 1, 0, 0) such that (L; ∨, ∧, 0, 1) is a distributive lattice with smallest element 0 and greatest element 1, and x ↦ x∘ is a unary operation such that l∘ = 0, x ≤ x∘∘ for all x ∈ L, and (x ∧ y)∘ = x∘ ∨ y∘ for all x, y ∈ L. These algebras belong to the class of Ockham algebras introduced by Berman [3]; see also [2,10,15]. A double MS-algebra is an algebra (L, ∨, ∧, ∘, +, 0, 1) of type (2, 2, 1, 1, 0, 0) such that (L, ∘) and (Ld, +) are MS-algebras, where Ld denotes the dual of L, and the operations ∘, + are linked by the identities x∘+ = x∘∘ and x+∘ = x++. We refer to [5, 6, 7, 8] for the basic properties of MS-algebras and double MS-algebras. Concerning the latter, the properties x∘∘∘ = x∘, x+++ = x+, and x∘ ≤ x+will be used frequently. The class of double MS-algebras is congruencedistributive and consequently the results of [13] can be applied. As to general results in lattice theory and universal algebra, the reader may consult [1, 9, 12].

We characterize in terms of Darboux transformations the spaces in the Segal–Wilson rational Grassmannian, which lead to commutative rings of differential operators having coefficients which are rational functions of ex. The resulting subgrassmannian is parametrized in terms of trigonometric Calogero–Moser matrices.

We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.

Banach spaces whose duals possess the Radon-Nikodym property have been studied extensively in the past (cf. [5]). It has been shown recently in [4] that a C*-algebra is scattered if and only if its Banach dual possesses the Radon-Nikodym property. This result extends the well-known result of Pełczynski and Semandini [8] that a compact Hausdorff space Ωis dispersed if and only if C(Ω)* has the Radon-Nikodym property. The purpose of this note is to give a transparent proof of a more general result for Jordan algebras which unifies the aforementioned results. We prove that the dual of a JB-algebra A possesses the Radon-Nikodym property if and only if the state space of A is the cr-convex hull of its pure states. We also consider the projective tensor products of the duals of JB-algebras in this context.