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In this article we consider the role of modeling in some aspects of the conservation of African wildlife. The first study is concerned with the endangered black rhino. Animals are being translocated from high density areas to new sites in an attempt to build up the South African population as rapidly as possible. We investigate the efficacy of different translocation strategies. Next we discuss a spatial stochastic metapopulation model used to test management strategies to enhance the survival likelihood of the rare samango monkey. The value of the model is to encourage the use of corridor policies even though there may be little apparent observable benefit in a manager's lifetime. Finally, we look at some commercial aspects of exploiting wildlife on a sustainable basis. By increasing the profitability of wildlife enterprises the incentive to conserve is increased. We look at improving the financial returns from game ranches.

(2004). Optimising the global/local equity split. Investment Analysts Journal: Vol. 33, No. 60, pp. 39-47.

In this continuation of our multiseries on Grothendieck's ‘Resume', we look at the projective and injective properties of general tensor norms. We also introduce the concepts of left and right injectivity as well as left and right projectivity. Mathematics Subject Classification (2000): 47B10, 46B28, 46E15, 46E30 Key words: Injective and projective tensor norm, left injective, right injective, left projective, right projective. Quaestiones Mathematicae 26(2003), 479–499

In this continuation of our multiseries on Grothendieck's 'Resume', we look at the special place of C(K)-spaces and L 1-spaces in the general metric theory of tensor products.

In this part of our exposition and expansion of Grothendieck's work on the metric theory of tensor products we look at the behaviour of the injective and surjective tensor norms in the case where one of the component spaces is a classical sequence space as exposed in Grothendieck's “ Sur certaines classes de suites dans les espaces de Banach, et le theorem de Dvoretzky-Rogers ”. Mathematics Subject Classification (2000) : 46B28, 46B45, 47B07. Key words: Weakly p -summable sequences; absolutely p -summable sequences; projective tensor product; injective tensor product; compact operators. Quaestiones Mathematicae 25 (2002), 95-118

In this continuation of our exposition and expansion of Grothendieck's ‘Resume,' we broach the subject of bilinear forms and operators of type α, where α is a tensor norm, and follow that up with the closely related subject of α-nuclearity. Mathematics Subject Classification (2000) : 46B28, 46B07, 46B10. Key words: α-forms;α-integral operators; (metric) accessibility; α-nuclear forms (operators). Quaestiones Mathematicae 25 (2002), 73-94

This paper presents the first of a multi-part series of papers on the metric theory of tensor products according to Grothendieck's “Resume de la theorie metrique des produits tensoriels topologiques” It contains the basics on tensor norms: a discussion of the special character of the injective and the projective norms, fundamental operations on tensor norms, extension of tensor norms to spaces of infinite dimensions. Mathematics Subject Classification (2000): 46B28, 46B07, 46B04. Key words: Reasonable crossnorms; projective norm; injective norm; integral bilinear form; tensor norm; (metric) accessibility. Quaestiones Mathematicae 25 (2002), 37-72

Samango monkeys inhabit the highly fragmented forests of the midlands of Kwazulu-Natal, South Africa. Not all forest patches are occupied and the survival of monkey populations in the face of further fragmentation provides the motivation for a theoretical consideration of metapopulation persistence under different management policy conditions. The primary aims of the model are to examine the influence of connectivity on metapopulation persistence and also to provide a temporal scale for metapopulation persistence in the case of this species. We develop a stochastic, spatial population model, in which we model edge and core troops in a patch separately, and take into account the effect of demographic and environmental stochasticity, as well as the impact of natural catastrophes. The effects of different levels of connectivity between forest patches are investigated, and we allow for a variable number of patches in the metapopulation. We compare the management options of reintroduction of troops and of providing corridor links between patches in a metapopulation. The model shows that neither management option has any meaningful effect on short-term (< 200 years) survival. In the long-term, however, corridors significantly improve metapopulation persistence. In the absence of interventive management to improve connectivity, samango monkey metapopulations (typically 8 forest patches) show poor persistence probabilities, less than 0.75 to 800 years and less than 0.34 to 1500 years. In contrast these survival probabilities are 0.9 to 800 years and 0.8 to 1500 years when connectivity is provided. The creation of corridors are of the utmost importance for the long term survival of the species.

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].

A density-dependent (Allee effect based) model is used to examine the population dynamics and recovery potential of low-density samango monkey ( Cercopithecus mitis ) populations. Life history parameters are well known for this species from a number of studies. These data are incorporated in a model which shows that below a density of 0.3–0.4 of carrying capacity, free-ranging, age-structured populations are at risk of local extinction. Such low-density populations cannot tolerate more than a further 30–35% reduction in size (distributed evenly across the age/sex classes) due to stochastic catastrophic events, without proceeding to extinction within 50 years. The results of this model suggest that low-density samango monkey populations are especially demographically vulnerable to natural catastrophic events and environmental variability.

Both classical and filter-theoretic characterisations of z-, C*-, and C-embedding have long been known in the literature. Localised versions of these embeddings deal with the extendibility of a single function from a subspace to the space itself and have been characterised in classical (non filter-theoretic) terms. In this paper we obtain some filter-theoretic characterisations of these localisations.

Over the past 30 years the black rhinoceros (Diceros bicornis) populations in Africa have dwindled dramatically. To enhance the survival prospects of the species, a national conservation strategy has been developed in South Africa. Its main goal is to formulate and implement policies to increase the southern African rhino population as rapidly as possible. This involves translocating animals from areas where the population is approaching the ecological carrying capacity and establishing new viable populations in other suitable reserves. A non-linear differential equation model for a population of black rhino was developed. The model is used with a combination of analytical and numerical techniques to investigate a number of issues relating to the translocation of rhino from well-stocked, high-density areas to low-density areas with small herds or no herds. Firstly, the model is used to determine the maximum sustainable yield from the well-stocked reserves. The model is then applied to a newly established population to determine optimal import policies. Finally, the model is extended to include both an established exporting population and a new understocked importing population. Simulations are performed to give an indication of the number and age of animals which should be translocated to maximise the growth rate of the total rhino population in southern Africa.

In this paper we present an elementary proof of a general duality result for precompact sets which can be considered as a far-reaching generalization of a well-known result of Grothendieck on precompactness in dual systems. It is then shown that a number of known results can be deduced from it, amongst others a general form of the Arzela-Ascoli theorem and Grothendieck's duality theorem itself.

This chapter discusses about the Riesz theorem. The use of the Riesz theorem that is used in tandem with the Hahn–Banach theorem is examined in the chapter. One of the first applications of the Riesz theorem is in characterizing weakly convergent sequences in metric spaces. The partnership of the Riesz Theorem with various forms of the Hahn–Banach theorem is the most powerful in all abstract analysis. The most basic form of the Hahn–Banach theorem extending bounded linear functional from a subspace of a Banach space to the whole space without changing the norm is analyzed in the chapter. The Banach's version of the extension theorem is applied wherein a linear functional that is dominated by a positively homogeneous, subadditive functional is extended to the whole space with domination remaining. One remarkable consequence of Choquet's theorem is the characterization of weakly null sequences in Banach spaces. It is shown that between Hilbert spaces, the 2-summing operators and Hilbert–Schmidt operators are precisely the same, while the trace-class coincides with the nuclear operators.

“(BY ∗∗ , weak). To see that (BX∗∗ , weak∗) × BY is υ-measurable, we call on Choquet’s theory of capacities and K-analytic sets. More specifically, since BY is Polish it is a Kσδ set; it follows that (BX∗∗ , weak ∗) × BY is a Kσδ subset of K , its Cech-Stone compactification. As such it is universally “f -capacitable”, where f ranges over all the regular capacities defined on K; in particular, it is universally measurable with respect to the regular Borel measures on K [cf. G. Choquet, Lectures on Analysis, Vol. I, W. A. Benjamin, Inc., 1969, especially pp. 141–156]. Moreover

A study is made of the norm wp (1 ≤ p ≤ ∞) on the tensor product of two Banach spaces E and F. It is shown that wp is a tensor norm, and a representation is deduced for the elements in the completion\(E\tilde \otimes _{w_p } F\) of E ⊗ F equipped with wp. Finally it is shown that the wp-nuclear operators in the sense of Grothendieck [3] coincide with those operators factoring compactly throughp (if 1 ≤ p ≤ ∞) or Co (if p=∞), with related equalities concerning the idea1 norms.

The main theme of this paper is a study in some detail of Banach ideals of continuous linear operators between Banach spaces factoring compactly through lp (1≤p

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.

Sheep farmers in the Cape Midlands region of South Africa frequently sustain stock losses through predation by caracal lynx. Further losses are incurred because of hyrax competing with the sheep for pasture. As hyrax constitute the principal food of lynx, culling either hyrax or lynx has complex feedback effects. A system dynamics model of this system comprising sheep, hyrax, lynx, pasture, and revenue sectors has been formulated. The model is used to simulate the system under different culling strategies. By comparing stock positions and farming revenue generated over a number of years under different culling strategies, an optimal culling policy is determined using an optimization procedure. A parameter sensitivity analysis reveals that the policy recommendations are relatively insensitive to parameter uncertainty.

The purpose of this paper is to investigate pairwise almost compact bitopological spaces. These spaces satisfy a bitopological compactness criterion which is strictly weaker than pairwise C-compactness and is independent of other well-known bitopological compactness notions. Pairwise continuous maps from such spaces to pairwise Hausdorff spaces are pairwise almost closed, the property is invariant under suitably continuous maps, is inherited by regularly closed subspaces and may be characterized in terms of certain covers as well as the adherent convergence of certain open filter bases. Some new natural bitopological separation axioms are introduced and in conjunction with pairwise almost compactness yield interesting results, including a sufficient condition for the bitopological complete separation of disjoint regularly closed sets by semi-continuous functions.

After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l ∞ ,τ(l ∞ ,l 1)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indE a, a∈A, ofseparable Banach spacesE a, and every Mackey null sequence inE is a null sequence in someE a. IfE is ultrabornological, thenE can be represented as lim indE a,a∈A, allE a separable Banach spaces, such that every fast null sequence inE is a null sequence in someE a.

Many are the delights and treasures to be found in Grothendieck’s Ž . famous or ought we say infamous? ‘‘Resume’’ 8 : his wondrous inequal ity, beautiful lifting theorems for continuous vector-valued functions, stunning characterizations of Hilbert spaces; the list goes on and on, with patience and grit the main necessities for the curious. One of our favorites has to do with series in L-spaces. Here’s what it Ž . 1 Ž . 2 says: if f is an unconditionally summable sequence in L , then f l n n n n 1 L . This is a crisp improvement of a theorem of Orlicz 16 which asserts 2 that Ý f and is the objective of this largely expository note. 1 n n To be sure, the L-space in question can be any L-space but for sake of 1Ž . clarity we will deal only with L 0, 1 .