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In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields and . The (window) nonadjacent form of τ -expansion of integers in was first investigated by Solinas. For integers in , the nonadjacent form and the window nonadjacent form of the τ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-τ expansions for integers in all Euclidean imaginary quadratic number fields.

The main problem that is solved in this paper has the following simple formulation (which is not used in its solution). The group K = Op(C) × Oq(C) acts on the space Mp, q of p × q complex matrices by (a; b) · x = axb–1, and so does its identity component K0 = SOp(C)×SOq(C). A K-orbit (or K0-orbit) in Mp,q is said to be nilpotent if its closure contains the zero matrix. The closure, , of a nilpotent K-orbit (resp. K0-orbit) in Mp,q is a union of and some nilpotent K-orbits (resp. K0-orbits) of smaller dimensions. The description of the closure of nilpotent K-orbits has been known for some time, but not so for the nilpotent K0-orbits. A conjecture describing the closure of nilpotent K0-orbits was proposed in [11] and verièd when min(p, q) ≤ 7. In this paper we prove the conjecture. The proof is based on a study of two prehomogeneous vector spaces attached to and determination of the basic relative invariants of these spaces.The above problem is equivalent to the problem of describing the closure of nilpotent orbits in the real Lie algebra so(p, q) under the adjoint action of the identity component of the real orthogonal group O(p, q).

For an integer n, let d(n) denote the ordinary divisor function. This paper studies the asymptotic behavior of the sumIt is proved in the paper that, as x → ∞,where A1 and A2 are certain constants and ε is any fixed positive real number.The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error claimed by Gafurov.

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants ofmcopies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.

In an earlier paper [10], we studied a generalized Rao bound for ordered orthogonal arrays and (T, M, S)-nets. In this paper, we extend this to a coding-theoretic approach to ordered orthogonal arrays. Using a certain association scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal arrays and linear ordered codes as well as a linear programming bound for the general case. We include some tables which compare this bound against two previously known bounds for ordered orthogonal arrays. Finally we show that, for even strength, the LP bound is always at least as strong as the generalized Rao bound.

Positive, copositive, onesided and intertwining (co-onesided) polyno- mial and spline approximations of functions f W k( 1 1) are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained. 1. Introduction and main results. We start by recalling some of the notations and definitions used throughout this paper. Let C(a b )a nd C k ( a b) be, respectively, the sets of all continuous and k-times continuously differentiable functions on (a b), and let Lp(a b), 0 p , be the set of measurable functions on (a b) such that f Lp(a b) ,w here f L p ( a b ) := b a f (x) p dx 1 p Throughout this paper L (a b) is understood as C(a b) with the usual uniform norm, to simplify the notation. We also denote by W k (a b), p 1, the set of all functions f on (a b) such that f (k 1) are absolutely continuous and f (k) Lp, and by Pn the set of all m=0 m i ( 1) m i f (x mh 2 + ih) if x mh 2 (a b), 0 otherwise. Then the m-th (usual) modulus of smoothness of f Lp(a b )i s def ined by m ( f t ( a b))p := sup 0 h t Δ m (f ( a b)) Lp (a b) We will also use the so-called -modulus, an averaged modulus of smoothness, defined for all bounded measurable functions on (a b )b y m ( f t ( a b))p := m (f t ) L p ( a b )

We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo­ geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.

The purpose of this paper is twofold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua–Marcus’ inequalities. Our results are stronger and more general than the existing ones.

The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.

This paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.We begin with an overview of these fixed point results, including concise self contained sketches of their proofs. Then we proceed with a discussion of the relations among the ten classes of self-maps with domains ⟨X, d⟩ having various topological properties that often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable-path connectedness, and d-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between these classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of ℝ with f′ ≡ 0, which constitutes also a minimal dynamical system. We finish by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.

Let R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

Let G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

Let a1, …, a9 be non-zero integers and n any integer. Suppose that a1, …, a9 ≡ n (mod 2) and (ai, aj) = 1 for 1 ≤ i ≤ j ≤ 9. In this paper we prove that (i)if aj are not all of the same sign, then the cubic equation has prime solutions satisfying (ii)if all aj are positive and , then is soluble in primes pj. These results improve our previous results with the bounds max{|aj|}14+ε and max{|aj|}43+ε in place of max{|aj|}8+ε and max{|aj|}25+ε above, respectively.

We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler and Edelstein’s fixed point theorems to modular metric spaces endowed with a graph.

Let (L, λ) be a locally solid Riesz space. (L, λ) is said to have the Levi property if for every increasing λ-bounded net (xα) ⊂ L+, sup xα exists. The Levi property, appearing in literature also as weak Fatou property (Luxemburg and Zaanen), condition (B) or monotone completeness (Russian terminology), is a classical object of investigation. In this paper we are interested in some variations of the property, their mutual relationships and applications in the theory of topological Riesz spaces. In the first part of the paper we clarify the status of two problems of Aliprantis and Burkinshaw. In the second part we study ideal-injective Riesz spaces.

1. Main result. Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if "normal" is replaced by "Hermitian", "nonnegative" or "positive"? Recall that an operator T is nonnegative (resp. positive) if (Tx, X) ^ 0 (resp. (Tx, X) > 0) for any x ^ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions. If the space is finite-dimensional, then necessary and sufficient condi­tions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest(cf suc[10. h numbe] ). r An operator T is the product of positive (resp. nonnegative) operators if and only if det T > 0 (resp. det T ^ 0); in this case, 5 positive (resp. nonnegative) operators will do and 5 is the smalles(cf.t[1 ] and [13] ). Thus from now on we will only consider the infinite-dimensional space. For this case, the problems have only been slightly touched upon before. For example, in [8, Solution 144 (a) ] it was shown that the (simple) unilateral shift is not the product of finitely many normal operators; in [11] Radjavi showed that every normal operator is the product of 4 Hermitian op­erators. Other than these, there seems to be very few in the literature. In this paper, we will completely determine which operators can be expressed as such. It turns out that the classes of operators expressible as products of normal, Hermitian or nonnegative operators are identical. More precisely, we have the following

A non-Hausdorff Ascoli theorem for continuous functions was established in [6]. The present purpose is to extend this result to point-compact continuous multifunction, using Levine's generalization for closed subsets [12]. The paper is organized as follows: the object of section 2 is to establish the necessary multifunction lemmas and to introduce the notion of a Tychonoff set; section 3 generalizes to multifunction context the partial exponential law of R. H. Fox [9, p. 430], and establishes a special exponential law for multifunctions; section 4 concerns the crucial properties of even continuity for multifunctions, introduced in [8]; the main theorem of the paper is established in section 5.

All spaces considered in this paper are completely regular and T\. A continuum is a compact, connected, Hausdorff space. A continuum is hereditarily locally connected if each of its subcontinua is locally connected. The reader may consult Whyburn [5] or Kuratowski [2] for a discussion on hereditarily locally connected metric continua. Nishiura and Tymchatyn [3] recently obtained some metric characterizations of connected subsets of hereditarily locally connected metric continua. Simone [4] extended to arbitrary hereditarily locally connected continua some well-known characterizations of hereditarily locally connected metric continua. In the first section of this paper some other characterizations of hereditarily locally connected metric continua are extended to the nonmetric case. In particular, we extend Wilder's theorem to say that a continuum is hereditarily locally connected if and only if every connected subset is locally connected. In the second section of this paper there are given some uniform and some topological characterizations of connected spaces which admit a hereditarily locally connected compactification.

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

One of the attractions of finite combinatorics is its explicit constructions. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. The larger domain to be considered consists of the recursive structures. While recursive structures may be infinite they are still amenable to explicit constructions. In this paper we shall concentrate on recursive colorings of highly recursive graphs.A function f: Nk → N, where N is the set of natural numbers, is recursive if and only if there exists an algorithm (i.e., a finite computer program) which upon input of a sequence of natural numbers , after a finite number of steps, outputs . A subset of Nk is recursive provided that its characteristic function is recursive. For a more thorough definition of recursive functions and recursive relations see [10].

1. Introduction. In this paper we give several general constructions for lattice packings of spheres in real n-dimensional space Rn and complex space Cn. These lead to denser lattice packings than any previously known in R36, R64, R80, …, R128, …. A sequence of lattices is constructed in Rn for n = 24m ≦ 98328 (where m is an integer) for which the density Δ satisfies log2 Δ ≈ – (1.25 …)n, and another sequence in Rn for n = 2m (m any integer) withThe latter appear to be the densest lattices known in very high dimensional space. (See, however, the Remark at the end of this paper.) In dimensions around 216 the best lattices found are about 2131000 times as dense as any previously known.Minkowski proved in 1905 (see [20] and Eq. (23) below) that lattices exist with log2 Δ > –n as n → ∞, but no infinite family of lattices with this density has yet been constructed.

The purpose of automata theory is to study and classify those properties of words that may be defined by a finite structure, say a finite automaton or a finite monoid. It seems natural to consider the same problem for infinite words. This amounts to studying the asymptotic behaviour of finite automata. As is well-known, this breaks the equivalence between determinism and non-determinism of finite automata. The study of the infinite behaviour of finite automata is based on a deep theorem due to B-&-uuml;chi and Mc Naughton: the recognizable sets of infinite words are the finite boolean combinations of deterministic ones (i.e. recognized by deterministic automata). The aim of this paper is to build an analogous theorem for two-sided infinite sequences. We define a biinfinite word as the equivalence class under the shift of a two-sided infinite sequence. The recognizable sets of biinfinite words are defined in a natural way and one is led to a two-sided notion of determinism. This notion seems to be new and justifies the consideration of biinfinite words. The main result of this paper is the extension to biinfinite words of the theorem of B-&-uuml;chi and Mc Naughton: the recognizable sets of biinfinite words are the finite boolean combinations of deterministic ones (Theorem 3.1). There exist three available proofs of B-&-uuml;chi-Mc Naughton's theorem. The original one by Mc Naughton [4] is hard to read. The proof given by Eilenberg in his book [2] has been constructed by Sch-&-uuml;tzenberger and Eilenberg from Mc Naughton's proof; it is similar to that of Rabin [5]. Finally, Sch-&-uuml;tzenberger gave a further proof in [6], which makes the argument more direct by using the methods of the theory of finite monoids. The proof of our main result follows closely Sch-&-uuml;tzenberger's method. This method allows to reduce the two-sided case to the one-sided case, although this seems very difficult to obtain directly. In the first section, we briefly recall the theory of the one-sided infinite behaviour of finite automata. In particular, we give Sch-&-uuml;tzenberger's proof of B-&-uuml;chi-Mc Naughton's theorem. The elements of this proof are used in the proof of our main result. In the second section we define the notions of biinfinite word, biautomaton and deterministic biautomaton. The last section contains the proof of our main result.

This paper is a continuation of [5]. As stated there, the problem is to explicitly decompose the space L2 = L2(G(F)\G(A)) into simpler invariant subspaces, and to deal with the associated continuous spectrum in case G is a connected reductive algebraic group defined over a global function field. In that paper the solution was begun by studying Eisenstein series associated to cusp forms on Levi components of parabolic subgroups; these Eisenstein series and the associated intertwining operators were shown to be rational functions satisfying functional equations. To go further it is necessary to consider more general Eisenstein series and intertwining operators, and to show that they have similar properties. Such Eisenstein series arise from the cuspidal ones by a residue taking process, which is detailed in a disguised form suitable for induction in the first part of this paper: the main result is a preliminary form of the spectral decomposition.

In an earlier paper [6] we considered the application of Lagrangian methods to the enumeration of plane rooted trees with given colour partition. We obtained an expression which generalised Tutte’s result [9], and a correspondence, which, when specialised, gives the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte Theorem [1]. A corollary of these results is a one-to-one correspondence [4], between trees and generalised derangements, for which no combinatorial description has yet been found.In this paper we extend these methods to the enumeration of rooted labelled trees to demonstrate how another pair of well-known and apparently unrelated theorems may be obtained as the result of a single enumerative approach. In particular, we show that a generalisation of Good's result [3], also considered by Knuth [7], and the matrix tree theorem [8] have a common origin in a single system of functional equations, and that they correspond to different coefficients in the power series solution.

There are many concepts which arise naturally in a discussion of injectivity in an equational class; for example, weak injective algebras, absolute subretracts, essential extensions, the congruence extension property, and the amalgamation property (see [3; 9; 17; 18]). It has already been demonstrated in several papers, notably [9; 17; 26; 27; 28], that the study of these concepts is greatly enriched by the assumption that the algebras under consideration have distributive congruence lattices. In this work attention is focused on weak injective algebras (Section 2) and the congruence extension property (Section 3). In Section 1 our terminology is introduced, Jonsson's lemma and its immediate corollaries are stated, and a diagramatic interpretation of Jonsson's lemma is given ; this Jonsson Diagram is the basis of all of our proofs. Our aim in Sections 2 and 3 is to reduce considerations of weak injectivity and congruence extension to the subdirectly irreducible algebras. For example, we prove (Theorem 2.2) that if K is a congruence distributive equational class whose subdirectly irreducible members form an axiomatic class, then a subdirectly irreducible member of K is a weak injective in K provided it is a weak injective "within" the class of subdirectly irreducible algebras. This result is then applied to prove (Theorem 2.5) that if K is a congruence-distributive equational class generated by a finite simple algebra A, then the weak injectives in K are precisely the Boolean extensions of A by complete Boolean algebras. The main result of Section 3 (Theorem 3.3) states that if K is a congruence-distributive equational class whose subdirectly irreducible algebras form an axiomatic class, then K satisfies the congruence extension property if and only if the subdirectly irreducible members of K satisfy the congruence extension property. The paper closes with a discussion, in Section 4, of some applications of the results.

The isomorphism problem for modular group algebras of finite p-groups appears to be still far from a solution (see [7] for a survey of the existing results). It is therefore of interest to investigate the problem for special classes of groups.The groups we consider here are the p-groups of maximal class, which were extensively studied by Blackburn [1]. In this paper we solve the modular isomorphism problem for such groups of order not larger than pp+1, having an abelian maximal subgroup, for odd primes p.What we in fact do is to generalize methods used by Passman [5] to solve the isomorphism problem for groups of order p4. In Passman's paper the case of groups of maximal class is actually the most difficult one.

Throughout this paper, R will be a commutative integral domain with identity and x an indeterminate. If ƒ ∈ R[x], let CR(ƒ) denote the ideal of R generated by the coefficients of ƒ. Define SR = {ƒ ∈ R[x]: cR(ƒ) = R} and UR = {ƒ ∈ R(x): cR(ƒ)– 1 = R}. For a,b ∈ R, write . When no confusion may result, we will write c(ƒ), S, U, and (a:b). It follows that both S and U are multiplicatively closed sets in R[x] [7, Proposition 33.1], [17, Theorem F], and that R[x]s ⊆ R[x]U.The ring R[x]s, denoted by R(x), has been the object of study of several authors (see for example [1], [2], [3], [12]). An especially interesting paper concerning R(x) is that of Arnold's [3], where he, among other things, characterizes when R(x) is a Priifer domain. We shall make special use of his results in our work.

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.

In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions. Remarkably, the number of such trees which are planted is a multiple of the number of spanning arbores­ cences on a graph with the edge partition as its adjacency matrix. This result is used (Section 4) to obtain the number of rooted plane i£-chromatic trees with fixed chromatic partition, a number given by Tutte [5] for planted trees in the case K = 2. Finally, a new combinatorial correspon­ dence between two sets of trees is given (Section 5) which yields the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte (BEST) theorem as a special case. We use a familiar decomposition of rooted trees. Associated with this is a system of functional equations which may be solved by a specialisa­ tion, given in Section 2, of the Lagrange theorem in many variables. The specialisation accounts for the persistence, in combinatorial enumeration, of determinants of matrices with row and column constraints (see, for example, [6]). Throughout this paper we use the notation: the number of non-root vertices of colour i is nt = ]C^i/

In this paper we investigate the structure of certain spaces of operator valued measures and the Lp spaces they generate. The work is motivated by our earlier paper [1] in which we studied the Lp spaces generated by matrix valued measures. The present results can thus be regarded as a generalization of this “finite dimensional” situation.

In a paper published in 1975 [1, § 3], D. S. Mitrinovič and P. M. Vasič used the so-called “centroid method” to obtain two new inequalities which are complementary to (the discrete version of) Jensen's inequality for convex functions. In this paper we shall present a very general version of such inequalities using the same geometric ideas used in [1] but not using the centroid method itself. At the same time we shall extend the domain of the inequalities (even in the discrete case), and clarify the value of the constant (λ) appearing in the inequality. We give applications of the theorems to some general means and also to the classical means. Our first result is given asTHEOREM 1. Let v be a nonnegative measure on a σ-algebra of subsets of a set D and let q, f be real v-measurable functions on D such that q(x) > 0, –∞ < x1 ≦ f(x) ≦ x2 < ∞ for all x ∊ D and ∫D qdv = 1.

In a previous paper [13] one of us considered Brauer groups Br(C) and class groups Cl(C) attached to certain monoidal categories C of divisorial R-lattices. That paper showed that the groups arising for a suitable pair of categories C1 ⊆ C2 could be related by a tidy exact sequenceIt was shown that this exact sequence specializes to a number of exact sequences which had formerly been handled separately. At the same time the conventional setting of noetherian normal domains was replaced by that of Krull domains, thus generalizing previous results while also simplifying the proofs. This work was carried out in an affine setting, and one aim of the present paper is to carry these results over to Krull schemes. This will enable us to recover the non-affine version of an exact sequence obtained by Auslander [1, p. 261], as well as to introduce a new, non-affine version of a different sequence derived by the same author [2, Theorem 1].

Several years ago, C. Long wrote two papers ([3], [4]) that related to F(n) the number of ordered factorizations of n. The second of these papers [4] was devoted entirely to a discussion of conjectured formula for F(n).

In this paper we will give brief proofs of two results on the undecidability of a first-order theory using a construction which we call a modified Boolean power. Modified Boolean powers were introduced by Burris in late 1978, and the first results were announced in [2]. Subsequently we succeeded in using this construction to prove the results in this paper, namely Ershov's theorem that every variety of groups containing a finite non-abelian group has an undecidable theory, and Zamjatin's theorem that a variety of rings with unity which is not generated by finitely many finite fields has an undecidable theory. Later McKenzie further modified the construction mentioned above, and combined it with a variant of one of Zamjatin's constructions to prove the sweeping main result of [3]. The proofs given here have the advantage (over the original proofs) that they use a single construction.

Throughout this paper rings are understood to be commutative with unity. In this paper we prove the general approximation theorem for valuations whose infinite ideals have large Jacobson radicals. We give an example in which it is shown that approximation theorems for Manis valuations do not hold in the general case. Also we prove that every valuation pair (Rv, Pv) of a total quotient ring T(R) whose infinite ideal has large Jacobson radical is a Prüfer valuation pair.

Conditions for a subgroup, F, of PSL(2, R) to be discrete have been investigated by a number of authors. Jørgensen's inequality [5] gives an elegant necessary condition for discreteness for subgroups of PSL(2, C). Purzitsky, Rosenberger, Matelski, Knapp, and Van Vleck, among others [12, 13, 14, 9, 16, 17, 18, 19, 20, 7, 21] studied two generator discrete subgroups of PSL(2, R) in a long series of papers. The complete classification of two generator subgroups was surprisingly complicated and elusive. The most complete result appears in [20].In this paper we use the results of [20] to prove that a nonelementary subgroup F of PSL(2, R) is discrete if and only if every non-elementary subgroup, G, generated by two hyperbolics is discrete (Theorem 5.2) and that F contains no elliptics if and only if each such G is free (Theorem 5.1). Thus, we produce necessary and sufficient conditions for a non-elementary subgroup F of PSL(2, R) to be a discrete group without elliptic elements (Theorem 6.1) or a discrete group containing only hyperbolic elements (Theorem 7.1).

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains. For example, in 1953 Whyburn showed that every map is the restriction of a compact map [17]. Similarly, Krolevec proved in 1967 that each locally perfect map can be extended to a perfect map [12] and in an as yet unpublished paper, Dickman obtained the same result for arbitrary maps [4]. In this paper we show that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. These results are then used to obtain analogues and improvements of recent theorems of Arhangel'skiï, Ĉoban, Hodel, and Proizvolov.

The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.

This paper is a continuation of two papers [4], [5] and brings out the solution of the ballot problem in its general form.In [5], Narayana has considered a generalised occupancy problem which can be viewed as a problem in compositions of integers. In what follows, we use the definitions of [6].

Let N be a finite set of n elements. A collection ﹛S1, S2, … , Sm﹜ of subsets of N is called a determining collection if an arbitrary subset T of N is uniquely determined by the cardinalities of the intersections Si ⋂ T, 1 ≤ i ≤ m. The purpose of this paper is to study the minimum value D(n) of m for which a determining collection of m subsets exists.This problem can be expressed as a coin-weighing problem (1; 7).In a recent paper Cantor (1) showed that D(n) = O(n/log log n), thus proving a conjecture of N. J. Fine (3) that D(n) = o(n). More recently Erdös and Rényi (2), Söderberg and Shapiro (7), Berlekamp, Mills, and Leo Moser have independently found proofs that D(n) = O(n/log n).

In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.

The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.

In this paper I use techniques developed by Mirsky and Perfect (5) to generalize the extremely close relationship between transversal theory and the theory of matroids or independence structures. I extend in two directions a fundamental theorem of Rado (8) and use the techniques of Mirsky and Perfect to obtain easy proofs of known and unknown results about systems of representatives with repetition.2. Basic concepts. In this section I review the results used subsequently. Throughout the paper, S will denote a finite set and A will denote the collection of subsets of S, {Ai i ∈ I}, where I is a finite index set. |K| will denote the cardinality of a set K and I use the notation

1. Introduction. In one of his papers on limits for the characteristic roots of a Matrix, Brauer (1) has stated a theorem, which connects the roots of a given square matrix A, with those of a matrix A* derived from A by a certain process. The proof of this theorem involves a continuity argument and in a recent paper on the construction of stochastic matrices Hazel Perfect (5) has given a proof which avoids considerations of continuity.

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

In this paper, we continue the study of integral extensions begun in [7]. Whereas in the previous paper, we dealt exclusively with the extension A[x]/(α(x)), α(x) a monic polynomial over A, we now deal with arbitrary integral extensions. Applications of the results presented herein will be made in subsequent papers.To simplify our presentation, we make the following conventions. By an algebra, we will always mean a commutative complex algebra with an identity element, usually denoted by e

This paper is a continuation of the earlier papers (1, 5) in which the author studied matrices with entries from the algebra C() of all continuous, complex-valued functions on an extremely disconnected, compact Hausdorff space . (Such spaces are sometimes called Stonian, after M. H. Stone, who first considered them in (8). They arise naturally as maximal ideal spaces of abelian W*-algebras.) In this note, three theorems are proved.

This paper continues the study of continuous selections begun in (13; 14; 15) and the expository paper (12). The purpose of these papers, which is described in detail in the introduction to (13), can be summarized here as follows. If X and Y are topological spaces, and ϕ a function (called a carrier) from X to the space 2Y of non-empty subsets of F, then a selection for ϕ is a continuous f: X → Y such that f(x) ∈ ϕ(x) for every x ∈ X. For reasons which are explained in (13), we restrict our attention to carriers which are lower semi-continuous (l.s.c), in the sense that, whenever U is open in Y, then is open in X. Our purpose in these papers is to find conditions for the existence and extendability of selections.The principal purpose of this paper is to generalize the following result, which is half of the principal theorem (Theorem 3.2˝) of (13) (and is repeated as Theorem I of (12)).

In this paper we give some new characterizations of the projection property in Archimedean Riesz spaces. Our approach primarily explores the interrelationships between such things as the band structure or the prime ideal structure of an Archimedean vector lattice and corresponding structures of its Dedekind completion. Our results show that, in general, there is a ‘strong“ relationship if and only if the original vector lattice has the projection property. The main result of this paper is Theorem 2.6 which both summarizes and extends all of the results we obtain prior to it.

It is the purpose of this paper to establish a closer connection between the logarithmic capacity of sets and double trigonometric series. In (9), closed sets of logarithmic capacity zero were established as sets of uniqueness for a particular class of double trigonometric series under circular (C, 1) summability. By slightly changing this class of series but still maintaining closed sets of logarithmic capacity zero as sets of uniqueness, it is shown in this paper that closed sets of positive logarithmic capacity form sets of multiplicity.