On the Structure of Hv0(D) and hv0(D)

Weighted spaces of harmonic and holomorphic functions on the unit disc are discussed. It is shown that these spaces are always subspaces of c₀. Moreover, for many weights, it is shown that the weighted space of holomorphic functions has a basis.     

Higher Lefschetz Traces and Spherical Euler Characteristics

Higher analogs of the Euler characteristic and Lefschetz number are introduced. It is shown that they possess a variety of properties generalizing known features of those classical invariants. Applications are then given. In particular, it is shown that the higher Euler characteristics are obstructions to homotopy properties such as the TNCZ condition, and to a manifold being homologically Kähler.

     

Duality triads of higher rank: Further properties and some examples

It is shown that duality triads of higher rank are closely related to orthogonal matrix polynomials on the real line. Furthermore, some examples of duality triads of higher rank are discussed. In particular, it is shown that the generalized Stirling numbers of rank r give rise to a duality triad of rank r.     

The Generalized Cauchy-Riemann-Fueter Equation and Handedness

A generalization of the Cauchy-Riemann condition in complex analysis is described for complex numbers, quaternions and complex quaternions. The generalization called here generalized Cauchy-Riemann-Fueter analycity encompasses not just the left and right-handed versions of quaternion analysis but also generates other variants for complex quaternions. These multiple variants are shown to satisfy an analogue of Cauchy's Theorem and to have similarities with the generalized Cauchy-Riemann conditions that define monogenic functions on R ^{n + 1}; they are also similar to Fueter-type operators and the Moisil-Theodoresco operator. The multiple variants are shown to have an interpretation that unifies analycity into a single definition. Thus left and right-handedness in quaternions are shown to be two sides of the same concept, and likewise for complex quaternions. This is then shown to have possible physical interpretation for example in understanding the nature of chirality and the 'arrow of time'.     

Comparative exponential dichotomies and column diagonal dominance

It is shown that the dichotomy obtained by Lazer for diagonally dominant linear systems is a weaker condition than that of exponential dichotomy but that the two conditions are equivalent in the case of bounded coefficients. Thus Fink's result that exponential decay of all solutions implies exponential dichotomy does not extend to the case of mixed growth and decay. It is also shown that column dominant systems of mixed sign admit exponential dichotomies when the coefficients are bounded.     

Some Results on Green''s Relations for Monoids Presented by Monadic String-Rewriting Systems

For finite monadic string-rewriting systems that are confluent and for finite special string-rewriting systems that are λ -confluent it is shown that the R -class and the L -class of a string can be characterized by a regular set of irreducible strings. It follows that for these systems the relation D is decidable. Actually it is shown that this relation is decidable in polynomial time.     

Some results on Green’s relations for monoids presented by monadic string-rewriting systems

For finite monadic string-rewriting systems that are confluent and for finite special string-rewriting systems that are λ -confluent it is shown that the R -class and the L -class of a string can be characterized by a regular set of irreducible strings. It follows that for these systems the relation D is decidable. Actually it is shown that this relation is decidable in polynomial time.     

Regularp-groups and words giving rise to commutative group operations

Groups with words which give rise to commutative operations are considered. Finitep-groups with this property are shown to be regular. It is also shown that finitep-groups of class less thanp, among other groups, have this property.     

Quasi-symmetric 2, 3, 4-designs

Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.     

The exterior derivative as a Killing vector field

Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms. Partially supported by DGICYT grants #PB91-0324, and SAB94-0311; CONACyT grant #3189-E9307.     

The complete tunnel axiom

The complete tunnel axiom (abbreviated CTA) is that there is a continuous function from the Stone-Čech remainder ω¹ of the integers onto a linearly ordered space, such that every point- inverse has empty interior. The concept of a complete tunnel through a space is defined and its presence in ω¹ is shown equivalent to the CTA. Various modifications of this concept are defined and some are shown equivalent to statements about ω¹ and about compactifications of the Mrówka-Isbell space Ψ. The strongest of these statements is shown to follow from CH, but the weakest is negated by the Proper Forcing Axiom (PFA).     

Two Proof‐Theoretic Remarks on EA + ECT

In this note two propositions about the epistemic formalization of Church's Thesis (ECT) are proved. First it is shown that all arithmetical sentences deducible in Shapiro's system EA of Epistemic Arithmetic from ECT are derivable from Peano Arithmetic PA + uniform reflection for PA. Second it is shown that the system EA + ECT has the epistemic disjunction property and the epistemic numerical existence property for arithmetical formulas.     

Some topological properties of support points of convex sets

It is shown, as a corollary to Trojanski’s renorming theorem, that the set of support points (of certain closed convex sets) is connected. Some connectedness properties are established for support functionals, and it is also shown that the support points form an F_σ.     

Boundedness of certain automorphism groups of an open manifold

It is shown that certain diffeomorphism or homeomorphism groups with no restriction on support of an open manifold (being the interior of a compact manifold) are bounded. It follows that these groups are uniformly perfect. In order to characterize the boundedness several conditions on automorphism groups of an open manifold are introduced. In particular, it is shown that the commutator length diameter of the automorphism group D(M){\mathcal{D}(M)} of a portable manifold M is estimated by 4.     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

Properties (u) and (V*) of Pelczynski in symmetric spaces of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of τ-measurable operators are respectively equivalent to properties (u) and (V ^*) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These results are non-commutative analogues of well-known characterisations in the setting of Banach lattices.     

Random sets of finite perimeter

An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of L¹ functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of particles with finite perimeter are introduced and it is shown that their union sets are random sets with locally finite perimeter.     

Product structures in motivic cohomology and higher Chow groups

It is shown that the product structures of motivic cohomology groups and of higher Chow groups are compatible under the comparison isomorphism of Voevodsky (2002) [11]. This extends the result of Weibel (1999) [14], where he used the comparison isomorphism which assumed that the base field admits resolution of singularities.The mod n motivic cohomology groups and product structures in motivic homotopy theory are defined, and it is shown that the product structures are compatible under the comparison isomorphisms.     

Exact bounds on average granular pressures in a general bin model

An exact theory of averaged static bin pressures that takes into account the general variations of the pressure ratio, the bulk density, the grain-wall friction coefficient, the cross-sectional area, and the perimeter of the bin, with respect to the depth of the stored grain in a bin is presented.Exact solutions are then presented for the general differential inequalities involved. It is shown that the essential features of Janssen's classical results are retained. However, it is shown that smaller bounds than his lower bounds on bin pressures are quite possible contrary to the presently accepted notion of bin pressures. For a particular case an estimate on an improved lower bound on the overburden pressure is obtained and is shown to be more than 10% lower than that predicted by Janssen's formula.     

A Combinatorial Study of Multiplexes and Ordinary Polytopes

Bisztriczky defines a multiplex as a generalization of a simplex, and an ordinary polytope as a generalization of a cyclic polytope. This paper presents results concerning the combinatorics of multiplexes and ordinary polytopes. The flag vector of the multiplex is computed, and shown to equal the flag vector of a many-folded pyramid over a polygon. Multiplexes, but not other ordinary polytopes, are shown to be elementary. It is shown that all complete subgraphs of the graph of a multiplex determine faces of the multiplex. The toric h -vectors of the ordinary five-dimensional polytopes are given. Graphs of ordinary polytopes are studied. Their chromatic numbers and diameters are computed, and they are shown to be Hamiltonian. Received December 8, 2000, and in revised form June 21, 2001. Online publication November 2, 2001.