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The proofs of Theorem 2.2 of K. J. Dykema and M. Rørdam, Purely infinite simple C*-algebras arising from free product constructions, Canad. J. Math. 50 (1998), 323–341 and of Theorem 3.1 of K J. Dykema, Purely infinite simple C*-algebras arising from free product constructions, II,Math. Scand. 90 (2002), 73–86 are corrected.

It is shown that an old direct argument of Erdős and Heilbronn may be elaborated to yield a result of the current inverse type.

One result concerning the series representation for the continuous Jacobi transform in Walter and Zayed [1] is improved, the same thought also can be applied to the related results in [1].

R. A. Brualdi [1] presents a construction yielding matrices whose Birkhoff representation consists of the maximum number of permutation matrices and having 0(2n2) line sum. In this note a counterexample to such a construction is given. Furthermore, a new construction is presented, yielding matrices with lower line sums.

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 Banach convolution algebras l1(ω) and their continuous counterparts L1(ℝ+, ω) are much studied, because (when the submultiplicative weight function ω is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of “nice” weights ω, the only closed ideals they have are the obvious, or “standard”, ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in l1(ω). His proof was successfully exported to the continuous case L1(ℝ+, ω) by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in l1(ω) and L1(ℝ+, ω). The new proof is based on the idea of a “nonstandard dual pair” which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in L1(ℝ+, ω) containing functions whose supports extend all the way down to zero in ℝ+, thereby solving what has become a notorious problem in the area.

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).

Habib Muzaffar of Carleton University has pointed out to the authors that in their paper [A] only the resultfollows from the prime ideal theorem with remainder for ideal classes, and not the stronger resultstated in Lemma 5.2. This necessitates changes in Sections 5 and 6 of [A]. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of [A], each and every occurrence of o(s − 1) should be replaced by o(1).Sections 5 and 6 of [A] have been rewritten to incorporate the abovementioned correction and are given below. They should replace the original Sections 5 and 6 of [A].

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.

Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$ , where $a_n, b_n\in \{-1,1\}$ . The associated Borel probability measure $$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$ is called a Moran Sierpinski-type measure. In this paper, we show that $\mu _{M, \{D_n\}}$ is a spectral measure if and only if $3\mid \rho _i$ for each $i=1, 2$ . The special case is the Sierpinski-type measure with $a_n=b_n=1$ for all $n\in {\mathbb N}$ , which is proved by Dai et al. [Appl. Comput. Harmon. Anal. (2020), https://doi.org/10.1016/j.acha.2019.12.001].

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 starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ! F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals overthe group of F-adele points of G ,o f cusp forms on the group ofE-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to "periodic" automorphic forms, are related to analogous bi-period distributions associated to "periodic" au- tomorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the "fundamental lemma", which conjectures that the unit elements of the Hecke algebras on GandH havematching orbitalintegrals. Evenin stating this conjecture, oneneeds to intro- duce a "transfer factor". A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for SL(2). The geometric side of the trace formula for a test function f ! on the group of adele points of a reductive group G over a number field F ,i s as um of orbital integrals off ! parametrized by rational conjugacy classes, in G(F). It is obtained on integrating over the diagonal x = y the kernel Kf ! (x, y )o f ac onvolution operatorr(f ! ). Each such orbital integral can be expressed as an average of weighted sums of such orbital integrals over the stable conjugacy class, which is the set of rational points in the conjugacy class under the points of the group over the algebraic closure. Each such weighted sum is conjecturally related to a stable (a sum where all coefficients are equal to 1) such sum on an endoscopic group H of the group G.T his process of stabilization has been introduced by Langlands to establishliftingofautomorphicandadmissiblerepresentationsfromtheendoscopicgroups H to the original group G. The purpose of this paper is to develop an analogue in the context of the symmetric space G(E)/G(F), where E/F is a quadratic number field extension. Integrating the kernel Kf ! (x, y )o f the convolution operatorr(f ! ) for the test function f ! on the group of E- adele points of the group G over two independent variables x and y in the subgroup of F-adele points of G, we obtain a sum of bi-orbital integrals of f ! over rational bi-conjugacy classes. We introduce a notion of stable bi-conjugacy, and stabilize the geometric side of the bi-period summation formula. Thus we express the weighted sums in the stable bi

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 )

The purpose of this paper is to address conjectures raised in [2]. We show that one of the conjectures is false and we advance the proof of another by proving it for an infinite set of cases. Furthermore, we give hard evidence as to why the conjecture is true and show what remains to be done to complete the proof. Finally, we prove a conjecture given by S. Louboutin, for Mathematical Reviews, in his discussion of the aforementioned paper.

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.

A collection of examples of first order linear neutral differential delay equations having a nonoscillatory solution with lim sup = oo and lim inf = 0 at oo is given. This disproves a recent conjecture about the asymptotic behavior of solutions to such equations. In a paper in 1986, Grammatikopoulos, Grove and Ladas [3] proved some asymptotic properties of nonoscillatory solutions of the first order linear differential delay equation (1) -[v(0 + py(t - r)] + qy(t - a) = 0 where # ^ 0, /?, r and a are real constants. The asymptotic behavior of solutions of ( 1 ) in several cases involving various sign conditions on q, r, p and/? — 1 was left unresolved in [3], but two conjectures covering these unresolved cases were given in that paper. Before stating these conjectures, we observe that y satisfies (1) if and only if — y satisfies (1). Thus we can without loss of generality assume that a nonoscillatory solution of (1) is eventually positive, i.e., is positive on [>o, oo) for some real number to. CONJECTURE 1. Suppose p < 0 and qr < 0. Then linv^ y(t) = oo or lim

Let $\{M_{n}\}_{n=1}^{\infty }$ be a sequence of expanding matrices with $M_{n}=\operatorname{diag}(p_{n},q_{n})$, and let $\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$ be a sequence of digit sets with ${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$, where $p_{n}$, $q_{n}$, $a_{n}$ and $b_{n}$ are positive integers for all $n\geqslant 1$. If $\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}, then the infinite convolution $\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set $\unicode[STIX]{x1D6EC}$ such that $\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$.

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.

A theorem of Burgess and Stephenson asserts that in an exchange ring with central idempotents, every maximal left ideal is also a right ideal. The proof uses sheaf-theoretic techniques. In this paper, we give a short elementary proof of this important theorem.

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.

The preservation principle of local theta correspondences of reductive dual pairs over a p-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams and Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

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.

In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings {T1}t≥0, i.e., a family such that T0(x) = x, Ts+t = Ts(Tt(x)), where the domain is a metric space (M; d). In particular, we prove that under suitable conditions the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings.

Let X be a compact Hausdorff space. In this paper, we give an example to show that there is u ∊⊗ C(X) Mn with det(u(x)) = 1 for all x ∊ X and u ~h 1 such that the C* exponential length of u (denoted by ~h cel(u)) cannot be controlled by π. Moreover, in simple inductive limit C*-algebras, similar examples also exist.

In a previous paper the authors developed an intersection theory for subspaces of rational functions on an algebraic variety X over k = ℂ. In this short note, we first extend this intersection theory to an arbitrary algebraically closed ground field k. Secondly we give an isomorphism between the group of Cartier b-divisors on the birational class of X and the Grothendieck group of the semigroup of subspaces of rational functions on X. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative point of view on Cartier b-divisors and their intersection theory.

In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset that is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras.

Čebyšev centers have been studied extensively. In this paper an alternate concept of center for infinite bounded point sets is introduced. Some of the results in this paper for this new type of center are similar to previous results for Čebyšev centers.

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.

Let be an infinite regular cardinal. A poset L is called an -lattice if and only if for all XL satisfying 0 < |X| < m, ∧ X and ∨ X exist.This paper is a part of a sequence of papers, [5], [6], [7], [8], developing the theory of -lattices. For a survey of some of these results, see [9].The -lattice is described in [6]; γ denotes the zero and γ′ the unit of . In particular, formulas for -joins and meets are given. (We repeat the essentials of this description in Section 4.)In [6] we proved the theorem stated below. Our proof was based on characterization of (the free -lattice on P) due to [1]; as a result, our proof was very computational.

One of the original applications of catastrophe theory envisaged by Thom was that of discussing the local structure of the focal set for a (generic) smooth submanifold M ⊆ Rn + 1. Thom conjectured that for a generic M there would be only finitely many local topological models, a result proved by Looijenga in [4]. The objective of this paper is to extend Looijenga's result from the smooth category to the algebraic category (in a sense explained below), at least in the case when M has codimension 1.Looijenga worked with the compactified family of distance-squared functions on M (defined below), thus including the family of height functions on M whose corresponding catastrophe theory yields the local structure of the focal set at infinity. For the family of height functions the appropriate genericity theorem in the smooth category was extended to the algebraic case in [1], so that the present paper can be viewed as a natural continuation of the first author's work in this direction.

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.

A finite set of generators for the isotopy classes of selfhomeomorphisms of closed surfaces was given by Lickorish in three papers [2; 3; 4]. In [5] the group of isotopy classes for a particular, well-known cube with holes was presented. There the structure was “tight” enough to allow the computation of the relators as well as the generators. In this paper we give a finite set of generators for the group of isotopy classes of self-homeomorphisms on the solid double torus, the cube with two handles. Let us remark that the group of isotopy classes for the solid torus is well-known.

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.