Planes and processes

This article presented to Combinatorics 2006 is a survey of finite projective planes and the processes used to construct them. All non-translation planes are described, fundamental processes in translation planes are defined and some of these are used to connect semi-field flocks with symplectic spreads. Hermitian ovoids are connected to extensions of derivable nets, and three types of ‘lifting’ methods are discussed. Furthermore, hyperbolic fibrations and ‘regulus-inducing’ central collineation groups are connected to flocks of quadratic cones. Finally, hyper-reguli and multiple hyper-regulus replacement are considered.     

Classical and quantum Heisenberg groups, their representations and applications

This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrödinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.     

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another.We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel.We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,…,n}.     

Mixtures of Global and Local Edgeworth Expansions and Their Applications

Edgeworth expansions which are local in one coordinate and global in the rest of the coordinates are obtained for sums of independent but not identically distributed random vectors. Expansions for conditional probabilities are deduced from these. Both lattice and continuous conditioning variables are considered. The results are then applied to derive Edgeworth expansions for bootstrap distributions, for Bayesian bootstrap distribution, and for the distributions of statistics based on samples from finite populations. This results in a unified theory of Edgeworth expansions for resampling procedures. The Bayesian bootstrap is shown to be second order correct for smooth positive “priors,” whenever the third cumulant of the “prior” is equal to the third power of its standard deviation. Similar results are established for weighted bootstrap when the weights are constructed from random variables with a lattice distribution.     

Central Regions and Dependency

The paper introduces an approach to the ordering of dependence which is based on central regions. A d-variate probability distribution is described by a nested family of sets, called central regions. Those regions are affine equivariant, compact and starshaped and concentrate about a properly defined center. They can be seen as level sets of a depth function. Special cases are Mahalanobis, zonoid, and likelihood regions. A d-variate distribution is called more dependent than another one if the volume of each central region is smaller with the first distribution. This dependence order is characterized by an inequality between determinants of certain parameter matrices if either (i) F and G are arbitrary distributions and the central regions are Mahalanobis or (ii) F and G belong to an elliptical family of distributions and the central regions are arbitrary. If the regions are zonoid regions, the dependence order implies the ordering of lift zonoid volumes. Alternatively, the dependence order is applied to the copulae of the given distributions. Generalized correlation indices are proposed which are increasing with the dependence orders.     

Thing and Object

There is a fundamental ontological difference between two kinds of entity: things and objects. Unlike things, objects are not identical to any fusion of particulars. Unlike things, objects do not have mereological parts. While things are ontologically innocent, objects are not. Objects are meaty. I defend the distinction between things and objects, and provide an account of the nature of objects.

S-continuous functions and certain weak forms of regularity and complete regularity

A topological space is called s-regular if each closed connected set and a point outside it are separated by disjoint open sets. Similarly notion of complete s-regularity is introduced; basic properties of s-regular spaces and completely s-regular spaces are studied and interrelations between them and the standard separation axioms are observed. It is shown that in the class of semilocally connected spaces s-regularity coincides with regularity and complete s-regularity coincides with complete regularity. Moreover, properties of s-continuous functions are studied and it is shown that s-regularity and completely s-regularity are preserved under certain s-continuous mappings.     

Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Rational solutions and special polynomials associated with the generalized K ₂ hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations including the Fordy-Gibbons equation. Differential-difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations Γ and −2Γ is established. Properties of the polynomials are studied. Differential-difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found.     

Independent sets and non-augmentable paths in arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs

A digraph is arc-locally in-semicomplete if for any pair of adjacent vertices x,y, every in-neighbor of x and every in-neighbor of y either are adjacent or are the same vertex. A digraph is quasi-arc-transitive if for any arc xy, every in-neighbor of x and every out-neighbor of y either are adjacent or are the same vertex. Laborde, Payan and Xuong proposed the following conjecture: Every digraph has an independent set intersecting every non-augmentable path (in particular, every longest path). In this paper, we shall prove that this conjecture is true for arc-locally in-semicomplete digraphs and quasi-arc-transitive digraphs.     

Three- and Four-Dimensional Einstein-like Manifolds and Homogeneity

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podestá and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).     

1-Hyperreflexivity and complete hyperreflexivity

The subspaces and subalgebras of B(H) which are hyperreflexive with constant 1 are completely classified. It is shown that there are 1-hyperreflexive subspaces for which the complete hyperreflexivity constant is strictly greater than 1. The constants for CT⊗B(H) are analyzed in detail.     

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph $C^{?}$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.     

On hereditarily and super R-Hopfian and L-co-Hopfian abelian groups

An Abelian group is said to be R-Hopfian [L-co-Hopfian] if every surjective [injective] endomorphism has a right [left] inverse. An Abelian group G is said to be hereditarily R-Hopfian [hereditarily L-co-Hopfian] if each subgroup of G is R-Hopfian [L-co-Hopfian]; similarly G is super R-Hopfian [super L-co-Hopfian] if each homomorphic image of G is R-Hopfian [L-co-Hopfian]. The various classes of hereditarily and super R-Hopfian and L-co-Hopfian groups are studied and necessary conditions for groups to have these properties are derived; in several, but not all, cases, su?cient conditions are also obtained.     

Codes from lattice and related graphs, and permutation decoding

Codes of length n² and dimension 2n−1 or 2n−2 over the field F_p, for any prime p, that can be obtained from designs associated with the complete bipartite graph K_n,n and its line graph, the lattice graph, are examined. The parameters of the codes for all primes are obtained and PD-sets are found for full permutation decoding for all integers n≥3.     

Adjoints and Monads Related to Compact Lattices and Compact Lawson Idempotent Semimodules

For compact Hausdorff Lawson lattices L that are idempotent semirings with lower semicontinuous multiplications, free compact Hausdorff Lawson L-idempotent semimodules are constructed over compacta and over compact Hausdorff Lawson lower semilattices. It is shown that these free objects are spaces of L-valued regular measures. Related left adjoint functors and monads are described.     

Subtotally positive and Monge matrices

A real matrix is called k-subtotally positive if the determinants of all its submatrices of order at most k are positive. We show that for an m × n matrix, only mn inequalities determine such class for every k, 1 ? k ? min(m,n). Spectral properties of square k-subtotally positive matrices are studied. Finally, completion problems for 2-subtotally positive matrices and their additive counterpart, the anti-Monge matrices, are investigated. Since totally positive matrices are 2-subtotally positive as well, the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices.     

On torsionless and dense acts

Torsionless acts over a monoid S are investigated, in particular torsionless factor acts of 2 -free and 1 -free acts. Monoids over which free or projective acts are torsionless and vice versa are characterized. Some necessary conditions for torsionless acts to be principally weakly flat, weakly flat or strongly flat are given. First results on dense acts are mentioned and several examples, mostly on the basis of cofree acts, are presented to illustrate these concepts. August 15, 2000     

Left SF-Rings and Regular Rings

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.     

Near-Rings and Partitioned Groups

Over a commutative ring R with identity, free modules M with 2 distinguished submodules are studied. The category Rep₂R of such objects M have the obvious morphisms between them, which are homomorphisms between .R-modules preserving the distinguished submodules. The endo-morphisms for each M constitute a subalgebra End_RM of the algebra End_RM and the readability of λ-generated R-algebras A as End_RM is considered for cardinals λ. Despite the fact that 4 is the minimal number of distinguished submodules for realizing any algebra over a field il, we are able to prove a similar result in Rep₂R for many rings R including R = Z and algebras which are cotorsion-free. Several examples illustrate the boarder line of our main result. The main theorem is applied for constructing Butler groups in [11]     

Lie algebras and recurrence relations III:q-analogs and quantized algebras

q-Analogs of the basic structures discussed in Lie Algebras and Recurrence Relations I are presented. Theq-Heisenberg-Weyl (qHW) and qsl(2) algebras are discussed in detail. Presently it is known that such structures are very closely tied in with the theory of quantum groups. Among other topics, coherent state representations and their interpretations asq-identities forq-Hermite and Al-Salam-Chihara (q-Meixner) polynomials are discussed. A discussion of Clebsch-Gordan coefficients for a qsu(2)-type algebra is presented.