CARTESIAN CLOSEDNESS AND THE MCNEILLE COMPLETION OF AN INITIALLY STRUCTURED CATEGORY

Abstract

An adaptation of a theorem by Herrlich [5] shows that every initially structured category A can be fully embedded in a topological category A_C, which is, in fact, a MacNeille completion of A. It is then shown that A is Cartesian closed if and only if A_C is.

Also developed is the notion of a Cartesian closed initially structured (CCIS) hull of a category. The theory of the CCIS hull is analogous to that of the Cartesian closed topological (CCT) hull. It is proved that a category has a CCT hull-if and only if it has a CCIS hull; and this allows the list of conditions equivalent to the existence of a CCT hull to be supplemented.

Examples are given, drawn mainly from the various categories of binary relations.     

EMBEDDING C0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract

We show that the normed space of μ-measurable Pettis integrable functions on a probability space with values in a Banach space X contains a copy of the sequence space c₀ if and only if X contains a copy of c₀. In this case, if the probability μ has infinite range, a copy of c₀ consisting of μ-measurable functions can be found, such that it is complemented in the bigger space of all weakly μ-measurable Pettis integrable functions.     

ON THE EXISTENCE AND NON-EXISTENCE OF COMPLEMENTARY RADICAL AND SEMISIMPLE CLASSES

ABSTRACT

In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.     

THE NUMBER AND SIZE OF ORBITS OF SOME PERMUTATION GROUPS

Abstract

It is shown that Aut ?, the group of homeomorphisms of the rational numbers with the usual topology, has 2 ^N_o orbits on the power set P(?). We call S ? ? a moiety if S and its complement in ? are infinite. It is shown that the orbit of any moiety S under Aut ? has cardinality 2^N_o while the orbit of S under Aut(?, ≤), the group of order preserving automorphisms of ?, has cardinality N_o if and only if S is a finite union of disjoint rational intervals with rational endpoints.     

CONTINUITY OF THE ONE-SIDED BEST UNIFORM APPROXIMATION OPERATOR

Abstract

The purpose of this paper is to relate the continuity and selection properties of the one-sided best uniform approximation operator to similar properties of the metric projection. Let M be a closed subspace of C(T) which contains constants. Then the one-sided best uniform approximation operator is Hausdorff continuous (resp. Lipschitz continuous) on C(T) if and only if the metric projection P_M is Haudorff continuous (resp. Lipschitz continuous) on C(T). Also, the metric projection P_M admits a continuous (resp. Lipschitz continuous) selection if and only if the one-sided best uniform approximation operator admits a continuous (resp. Lipschitz continuous) selection.     

A MODEL OF THE ENZOOTIOLOGY OF LYME DISEASE IN THE ATLANTIC NORTHEAST OF THE UNITED STATES

A mathematical model is presented for the dynamics of the rate of infection of the Lyme disease vector tick Ixodes dammini (Acari: Ixodidae) by the spirochete Borrelia burgdorferi, in the Atlantic Northeast of the United States. According to this model, moderate reductions in the abundance of white-tailed deer Odocoileus virginianus may either decrease or increase the spirochete infection rate in ticks, provided the deer are not reservoir hosts for Lyme disease. Expressions for the basic reproductive rate of the disease are computed analytically for special cases, and it is shown that as the basic reproductive rate increases, a proportional reduction in the tick population produces a smaller proportional reduction in the infection rate, so that vector control is less effective far above the threshold. The model also shows that control of the mouse reservoir hosts Peromyscus leucopus could reduce the infection rate if the survivorship of juvenile stages of ticks were reduced as a consequence. If the survivorship of juvenile stages does not decline as the rodent population is reduced, then rodent reduction can increase the spirochete infection rate in the ticks.     

CONSISTENCY OF THE FIELD EQUATIONS IN THE SCALE COVARIANT THEORY OF GRAVITATION

ABSTRACT

In the theory of general relativity, it is well known that the integrability conditions for Einstein's field equations are the conservation equations. The latter guarantee that if the constraint equations G^oa = kT^oa hold on an initial surface, and if G^av = kT^av at all times, then the constraint equations will hold at all earlier and later times. This ensures the consistency of the field equations and underlies the initial value approach to their solution. We investigate the corresponding result in the scale co-variant theory of gravitation.     

ON THE SEPARATION PROPERTY OF SYMMETRIC ORDINARY SECOND-ORDER DIFFERENTIAL EXPRESSIONS

A 1-variable calculus type argument is used to show that, for a function f : R ² → R, if for all (a, b) ? R ² we have that f o c is smooth for every smooth curve c : R → R ² nonsingular except at 0 and with c(0) = (a, b), then f is smooth. This strengthens Boman's theorem. In fact, we use an even more special collection of smooth curves to prove Boman's theorem. It is shown using a related special collection of smooth curves how the upper half cone can be viewed largely as a model for polar coordinates. Our proof here shows how the use of Frölicher spaces can reduce questions in several dimensions to those of one real variable.     

SMYTH COMPLETENESS IN TERMS OF NETS: THE GENERAL CASE

Abstract

Smyth completeness is the appropriate notion of completeness for quasi-uniform spaces carrying an additional topology to serve as domains of computation [2, 3]. The goal of this paper is to provide a better understanding of Smyth completeness by giving a characterization in terms of nets. We develop the notion of computational Cauchy net and an appropriate notion of strong convergence to get the result that a space is Smyth complete if and only if every computational Cauchy net strongly converges. As we are dealing with typically non-symmetric spaces, this is not an instance of the classical net-filter translation in general topology.     

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT

In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P ε P} and H(p) = {A} if B ? A with B ε N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.     

On planarity of compact, locally connected, metric spaces

Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅ or K _3;3. The “thumbtack space” consisting of a disc plus an arc attaching just at the centre of the disc shows the assumption of 2-connectedness cannot be dropped. In this work, we introduce “generalized thumbtacks” and show that a compact, locally connected metric space is homeomorphic to a subset of the sphere if and only if it does not contain K ₅, K _3;3, or any generalized thumbtack, or the disjoint union of a sphere and a point.     

ON THE STABILITY OF A NON-LINEAR AGE-DEPENDENT POPULATION MODEL

Abstract

A simple age-dependent population model is considered, and a conjecture by Gurtin and MacCamy that Hopf bifurcation into stable orbits can not take place is shown to be correct under certain conditions.     

THE EDGE-CHROHATIC NUMBER OF A CIRCULANT

ABSTRACT

The circulant graph C_p > a₁,…, a_k < with 0 > a₁ >…> a_k >(pt1)/2 has p vertices labeled 0,1,…,p-1 and x and y are adjacent if and only if x—y = ± a_i (mod p) for some i. We prove the following regarding the chromatic index of a circulant: if d = gcd (a₁,…, a_k, p), then x' (C_p > a_l,…,a_k) = Δ(C p > a₁,…,a_k) if and only if p/d is even.     

On residual finiteness of direct products of algebraic systems

It is well known that if two algebraic structures A and B are residually finite then so is their direct product. Here we discuss the converse of this statement. It is of course true if A and B contain idempotents, which covers the case of groups, rings, etc. We prove that the converse also holds for semigroups even though they need not have idempotents. We also exhibit three examples which show that the converse does not hold in general.      

MECHANISMS OF THE ICOSAHEDRAL COMPOUND OF TEN TETRAHEDRA

Ten regular tetrahedra can be arranged in such a way that their vertices are coincident with the vertices of a regular dodecahedron and that two tetrahedra meet at each vertex of the dodecahedron. If the resultant structure is considered as a bar-and-joint structure, there will be 60 bars, lying along the edges of the tetrahedra, and 20 joints at the vertices of the dodecahedra; six bars meet at each joint. Although the structure more than satisfies Maxwell's rule, it is known to admit finite mechanisms. Recently, a new method for detecting symmetric finite mechanisms in symmetric bar-and-node structures has been developed. The method only requires a count of the number of bars, and the number of nodes, that are left unmoved by each of the symmetry operations allowable for the structure. This paper will describe the application of this method to the structure described above. The structure has icosahedral symmetry, I _h , and the analysis confirms the existence of the mechanisms with C _3v and C _5v symmetry that have previously been detected using ad-hoc methods. This revised version was published online in August 2006 with corrections to the Cover Date.     

INSTANCES AND RAMIFICATIONS OF THE SEMI-ADJOINT SITUATION I. PRESTABLE REFLECTION AND SYMMETRIC UNADS

Abstract

It is well known that there is a one to one correspondence between idempotent monads in a category and reflective subcategories. In this paper it is examined what replaces the reflective subcategory if the idempotent monad is replaced (a) by a monad and (b) by a symmetric unad. It is shown that in case (a) one obtains the weakly reflective subcategory of objects injective relative to the functor part of the monad. In case (b) one obtains a proto-reflection and it is shown that (for complete categories) the associated orthogonal subcategory is reflective if and only if there exists a free monad associated to the unad.     

ON THE SPECTRAL RADIUS OF IRREDUCIBLE AND WEAKLY IRREDUCIBLE OPERATORS IN BANACH LATPICES

Abstract

If T is an operator on a Banach lattice E we call T weakly irreducible if E contains no non-trivial T-invariant bands. We prove that if E is order complete and if the weakly irreducible operator T > 0 is in (E′_oo ? E)^⊥⊥ then T has positive spectral radéus. Prom this follows that Jentesch's theorem holds in arbitrary Banach function spaces.

If [Ttilde] denotes the restriction of T′ to E′_oo, 0 ? T an order continuous operator, then T is weakly irreducible if and only if [Ttilde]: E′_oo→E′_oo is weakly irreducible.

Finally we show that the majorizing, irreducible operator T ≥ 0, has positive spectral radius if either Tⁿ is weakly compact or E has property (P) or T is strongly majorizing.