Counting matroids in minor-closed classes

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without $M(K_4)$-minor which asymptoticaly matches the best known lower bound on the number of all matroids, due to Knuth.Comment: 13 pages, 3 figure

On the number of matroids

We consider the problem of determining $m_n$, the number of matroids on $n$ elements. The best known lower bound on $m_n$ is due to Knuth (1974) who showed that $\log \log m_n$ is at least $n-3/2\log n-1$. On the other hand, Piff (1973) showed that $\log\log m_n\leq n-\log n+\log\log n +O(1)$, and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on $\log\log m_n$ that is within an additive $1+o(1)$ term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.Comment: Final version, 17 page

An entropy argument for counting matroids

We show how a direct application of Shearers' Lemma gives an almost optimum bound on the number of matroids on $n$ elements.Comment: Short note, 4 page

Scaling-up vaccine production: implementation aspects of a biomass growth observer and controller

Abstract This study considers two aspects of the implementation of a biomass growth observer and specific growth rate controller in scale-up from small- to pilot-scale bioreactors towards a feasible bulk production process for whole-cell vaccine against whooping cough. The first is the calculation of the oxygen uptake rate, the starting point for online monitoring and control of biomass growth, taking into account the dynamics in the gas-phase. Mixing effects and delays are caused by amongst others the headspace and tubing to the analyzer. These gas phase dynamics are modelled using knowledge of the system in order to reconstruct oxygen consumption. The second aspect is to evaluate performance of the monitoring and control system with the required modifications of the oxygen consumption calculation on pilot-scale. In pilot-scale fed-batch cultivation good monitoring and control performance is obtained enabling a doubled concentration of bulk vaccine compared to standard batch productio

Theory of resistor networks: The two-point resistance

The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for two-point resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a Moebius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.Comment: 30 pages, 5 figures now included; typos in Example 1 correcte

Infinite Networks of Identical Capacitors

The capacitance between the origin and any other lattice site in an infinite square lattice of identical capacitors is studied. The method is generalized to infinite Simple Cubic (SC) lattice. We make use of the superposition principle and the symmetry of the infinite gridComment: 16 pages, 5 figures, 2 table

The exact evaluation of the corner-to-corner resistance of an M x N resistor network: Asymptotic expansion

We study the corner-to-corner resistance of an M x N resistor network with resistors r and s in the two spatial directions, and obtain an asymptotic expansion of its exact expression for large M and N. For M = N, r = s =1, our result is R_{NxN} = (4/pi) log N + 0.077318 + 0.266070/N^2 - 0.534779/N^4 + O(1/N^6).Comment: 12 pages, re-arranged section

Fc receptor-mediated immunity against Bordetella pertussis

The relevance of specific Abs for the induction of cellular effector functions against Bordetella pertussis was studied. IgG-opsonized B. pertussis was efficiently phagocytosed by human polymorphonuclear leukocytes (PMN). This process was mediated by the PMN IgG receptors, FcγRIIa (CD32) and FcγRIIIb (CD16), working synergistically. Furthermore, these FcγR triggered efficient PMN respiratory burst activity and mediated transfer of B. pertussis to lysosomal compartments, ultimately resulting in reduced bacterial viability. Bacteria opsonized with IgA triggered similar PMN activation via FcαR (CD89). Simultaneous engagement of FcαRI and FcγR by B. pertussis resulted in increased phagocytosis rates, compared with responses induced by either isotype alone. These data provide new insights into host immune mechanisms against B. pertussis and document a crucial role for Ig-FcR interactions in immunity to this human pathogen.Centro de Investigación y Desarrollo en Fermentaciones Industriale

Fc receptor-mediated immunity against Bordetella pertussis

The relevance of specific Abs for the induction of cellular effector functions against Bordetella pertussis was studied. IgG-opsonized B. pertussis was efficiently phagocytosed by human polymorphonuclear leukocytes (PMN). This process was mediated by the PMN IgG receptors, FcγRIIa (CD32) and FcγRIIIb (CD16), working synergistically. Furthermore, these FcγR triggered efficient PMN respiratory burst activity and mediated transfer of B. pertussis to lysosomal compartments, ultimately resulting in reduced bacterial viability. Bacteria opsonized with IgA triggered similar PMN activation via FcαR (CD89). Simultaneous engagement of FcαRI and FcγR by B. pertussis resulted in increased phagocytosis rates, compared with responses induced by either isotype alone. These data provide new insights into host immune mechanisms against B. pertussis and document a crucial role for Ig-FcR interactions in immunity to this human pathogen.Centro de Investigación y Desarrollo en Fermentaciones Industriale