We study the asymptotic behaviour of random walks in i.i.d. random environments on ${{\mathbb{Z}}^{d}}$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience and, in 2-dimensions, the existence of a deterministic limiting velocity.

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

This paper contains a set of results concerning paracompactness of locally nice spaces which can be proved by (variations on) the technique of “stationary sets and chaining” combined with other techniques available at the present stage of knowledge in the field. The material covered by the paper is arranged in three sections, each containing, in essence, one main result.

The main result of Section 1 says that a locally Lindelöf, submeta-Lindelöf ( = δθ-refinable) space is paracompact if and only if it is strongly collectionwise Hausdorff. Two consequences of this theorem, respectively, answer a question raised by Tall [7], and strengthen a result of Watson [9]. In the last two sections, connected spaces are dealt with. The main result of the second section can be best understood from one of its consequences which says that under 2ωl > 2ω, connected, locally Lindelöf, normal Moore spaces are metrizable.

In a recent note [2] we showed that if R is a prime ring and d ≠ 0 a derivation of R such that d(x)d(y) = d(y)d(x) for all x, y ∈ R then, if R is not a characteristic 2, R must be commutative. (If char R = 2 we showed that R must be an order in a 4-dimensional simple algebra.)

In this paper we shall consider a similar problem, namely, that of a prime ring R with involution * where d(x)d(y) = d(y)d(x) not for all x, y ∈ R but merely for symmetric elements x* = x and y* = y. Although it is clear that some results can be obtained if R is of characteristic 2, we shall only be concerned with the case char R ≠ 2. Even in this case one cannot hope to extend the result cited in the first paragraph, that is, to show that R is commutative.

Let A = (a1, a2, a3, …) be a sequence of positive integers. We let

denote the set of integers that are sums of distinct terms of A. If P(A) contains all sufficiently large integers, we say that A is complete. We shall show that certain classes of sequences that are characterized by their rate of growth are complete.

Extreme points of the unit sphere in any Orlicz space over a measure space that contains no atoms of infinite measure are characterized. In the case of a finite-valued Orlicz function and a nonatomic measure space, exposed points of the unit sphere in these spaces are characterized too. Some corollaries and examples are also given.

There are two subjects in the literature on semigroups which have recently attracted great attention: the class of completely regular semigroups (that is semigroups which are unions of their subgroups) and congruences on regular semigroups. In completely regular semigroups, the most popular subject is that of varieties, even though other aspects of them, such as structure, congruences, amalgamation, received their due attention. On the other hand, the treatment of congruences on regular semigroups became especially interesting with the emergence of the kernel-trace approach. This method proved quite successful in the case of inverse semigroups, see [6], whereas the analysis for the general regular semigroups encounters considerable difficulties, see [4].

The theory of Fourier transforms of tempered distributions as developed by Laurent Schwartz [17] is quite simple and elegant and has wide variety of applications, but there does not exist a corresponding neat and simple theory for the Hilbert transform of generalized functions (distributions) having wide applications. One of the objectives of this paper is to develop such a theory for the Hilbert transform of generalized functions and indicate its applicability to a variety of problems.

In problems of physics sometimes we need to find harmonic functions u(x, y) in the region y > 0 whose limit as y → 0+ does not exist in pointwise sense but does exist in the distributional sense. The theory of Hilbert transform of generalized functions that we are going to develop will provide an answer to the existence and uniqueness of this problem.

In a previous paper (2), one of us has derived a formula for the counting series for bicoloured graphs.2 These are graphs each of whose points has been coloured with exactly one of two colours in such a way that every two adjacent points have different colours.

In this paper we first enumerate bicoloured graphs without isolated points and connected bicoloured graphs. This leads us to corresponding problems for bicolourable graphs. Such a graph has the property that its points can be coloured with two colours so as to obtain a bicoloured graph.

In this paper, we determine when two simple generalized Cartan type $W$ Lie algebras ${{W}_{d}}(A,\,T,\,\varphi )$ are isomorphic, and discuss the relationship between the Jacobian conjecture and the generalized Cartan type $W$ Lie algebras.

Let V denote a unitary vector space with inner product (x, y). A self-adjoint linear map T: V → V is positive (positive definite) if (Tx, x) ≧ 0 ((Tx, x) ≧ 0) for all x ≠ 0 in V. We write S ≧ T(S > T) if S and T are self-adjoint and S – T ≧ 0 (S – T > 0). If U is a unitary vector space, a function f: Hom(V, V) → Hom(U, U) is monotone idf S ≧ T implies that f(S) ≧ f(T). If both U and V are taken to be the n-dimensional unitary space Cn of n-tuples of complex numbers with standard inner product, then f is a monotone matrix junction, a notion introduced for a more restrictive class of functions by Löwner (3) which has important applications in pure and applied mathematics. For orientation we refer the reader to (1), where several interesting examples of monotone and related functions are displayed in detail.

Let π be a set of primes and let G be a π-separable group (all groups considered are finite). Two subsets Xπ(G) and Bπ(G) of the set Irr(G) of irreducible characters of G play an important role in the character theory of π-separable groups and particularly solvable groups. If p is prime and π is the set of all other primes, then the Bπ characters of G give a natural one-to-one lift of the Brauer characters of G into Irr(G). More generally, they have been used to define Brauer characters for sets of primes.

The π-special characters of G (i.e., Xπ(G)) restrict irreducibly and in a one-to-one fashion to a Hall-π-subgroup of G. If an irreducible character χ is quasi-primitive, it factors uniquely as a product of a π-special character an a π′-special character. This is a particularly useful tool in solvable groups.

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.

The purpose of this note is to prove an improved version of Jones' Aposyndetic Decomposition Theorem. Corollaries to the new theorem re-emphasize the importance of understanding aposyndetic, homogeneous continua.

The proof is a synthesis of results about homogeneous continua with results from an unexpected source: completely regular mappings. Completely regular mappings occur naturally and often in the study of homogeneous continua, which is a surprising and pleasing phenomenon, since these mappings were invented for quite another purpose [1]. The author believes that these maps are likely to provide even more new information about homogeneous continua.

A continuum is a compact, connected, nonvoid metric space. A curve is a one-dimensional continuum. A continuum M is homogeneous if for each pair of points p and q belonging to M, there exists a homeomorphism h: M → M such that h(p) = q.

Correspondence between torsion-free connections with nilpotent skew-symmetric curvature operator and IP Riemann extensions is shown. Some consequences are derived in the study of four-dimensional IP metrics and locally homogeneous affine surfaces.

If a quasigroup (L, .) has finite order n, then there are n2 principal loop-isotopes. Some of these n2 loops may be isomorphic, and the main purpose of this paper is to obtain theorems that describe the isomorphism classes. Using these results and a computer, we have determined all the loops of order 6. These are listed (using the Fisher and Yates (2) designations) at the end of the paper.

This paper consists of two parts. The first contains an outline of the theorems and principal results and the second (§§2-6) gives proofs of the theorems and additional details. The theorems concern properties of Darboux continuous functions and functions having generalized Darboux properties. The corresponding results are shown to have interesting applications to the theory of non-absolutely convergent integrals.

A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that ${{\ell }^{1}},{{c}_{0}}$ and ${{\ell }^{\infty }}$ are the only symmetric sequence spaces with the Dunford- Pettis property, and that in the class of symmetric spaces on $(0,\,\alpha ),\,0\,<\,\alpha \,\le \,\infty$, the only spaces with the Dunford-Pettis property are ${{\text{L}}^{1}},{{\text{L}}^{\infty }},{{\text{L}}^{1}}\cap {{\text{L}}^{\infty }},{{\text{L}}^{1}}+{{\text{L}}^{\infty }},{{({{\text{L}}^{\infty }})}^{\text{o}}}$ and ${{({{\text{L}}^{1}}+{{\text{L}}^{\infty }})}^{\text{o}}}$, where ${{\text{X}}^{\text{o}}}$ denotes the norm closure of ${{\text{L}}^{1}}\cap {{\text{L}}^{\infty }}$ in $X$. It is also proved that all Banach dual spaces of ${{\text{L}}^{1}}\cap {{\text{L}}^{\infty }}$ and ${{\text{L}}^{1}}+{{\text{L}}^{\infty }}$ have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces ${{({{\text{L}}^{1}}+{{\text{L}}^{\infty }})}^{\text{o}}}$ and ${{({{\text{L}}^{\infty }})}^{\text{o}}}$ have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Köthe-Bochner spaces.

For a functor G:𝒜→𝔛 and a class 𝔇 of small categories containing the terminal category 1 we form the extension

and call G right 𝔇-pro-adjoint if and only if Pro (𝔇, G) is right adjoint. Here Pro (𝔇, 𝒜) is the completion of 𝒜with respect to 𝔇; it coincides with the usual pro-category of 𝒜 in case 𝔇 = directed sets. For this 𝔇 a full embedding Gis dense in the sense of Mardešić [11] if and only if it is right 𝔇-pro-adjoint in the above sense; this has been proved recently by Stramaccia [15]. The most important example is the embedding of the homotopy category of pointed CW-complexes into the homotopy category of pointed topological spaces (cf. [2]).