How many normal measures can ℵ carry?

Relative to the existence of a supercompact cardinal with a measurable cardinal above it, we show that it is consistent for ?₁ to be regular and for ? to be measurable and to carry precisely τ normal measures, where τ ≥ ? is any regular cardinal. This extends the work of [2], in which the analogous result was obtained for ?_{ω +1} using the same hypotheses (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

How many graphs are unions of k‐cliques?

We study the number of n‐vertex graphs that can be written as the edge‐union of k‐vertex cliques. We obtain reasonably tight estimates for in the cases (i) k = n ? o(n) and (ii) k = o(n) but . We also show that exhibits a phase transition around . We leave open several potentially interesting cases, and raise some other questions of a similar nature. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 87–107, 2006     

How does within‐host dynamics affect population‐level dynamics? Insights from an immuno‐epidemiological model of malaria

Malaria is one of the most common mosquito‐borne diseases widespread in the tropical and subtropical regions. Few models coupling the within‐host malaria dynamics with the between‐host mosquito‐human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formulated. Applying age‐structured partial differential equations for the between‐host dynamics, we describe the asymptomatic and symptomatic infectious host population for malaria transmission. The basic reproduction numbers for the within‐host model and for the coupled system are derived, respectively. The existence and stability of the equilibria of the coupled model are analyzed. We show numerically that the within‐host model can exhibit complex dynamical behavior, possibly even chaos. In contrast, equilibria in the immuno‐epidemiological model are globally stable and their stabilities are determined by the reproduction number. Increasing the activation rate of the within‐host immune response “dampens” the sensitivity of the population level reproduction number and prevalence to the increase of the within‐host reproduction of the pathogen. From public health perspective this means that treatment in a population with higher immunity has less impact on the population‐level reproduction number and prevalence than in a population with less immunity.     

How do the positive embeddings of Banach lattices depend on the αth derivatives of K?

Let K and S be locally compact Hausdorff spaces and X be an abstract space. Suppose that T is a positive Banach lattice isomorphism from into . Then for each ordinal α the cardinalities of the αth derivatives and satisfy the following inequality Moreover, if then is a continuous image of a subset of which can be taken closed when K is compact. The first statement of this result for is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number is sharp in these vector‐valued theorems.     

How many miles to βX? II — Approximations to βX versus cofinal types of sets of metrics

Kada, Tomoyasu and Yoshinobu proved that the Stone-?ech compactification of a locally compact separable metrizable space is approximated by the collection of d-many Smirnov compactifications, where d is the dominating number. By refining the proof of this result, we will show that the collection of compatible metrics on a locally compact separable metrizable space has the same cofinal type, in the sense of Tukey relation, as the set of functions from ω to ω with respect to eventually dominating order.     

Into how many regions do <Emphasis Type="Italic">n</Emphasis> lines divide the plane if at most <Emphasis Type="Italic">n − k</Emphasis> of them are concurrent?

The number of connected components of the complement in the real projective plane to a family of n ≥2 different lines such that any point belongs to at most n − k of them is estimated. If $ n \geqslant \frac{{k^2 + k}} {2} + 3 $ n \geqslant \frac{{k^2 + k}} {2} + 3 , then the number of regions is at least (k+1)(n−k). Thus, a new proof of N. Martinov’s theorem is obtained. This theorem determines all pairs of integers (n, f) such that there is an arrangement of n lines dividing the projective plane into f regions.