Detailed analytic study of the compact pairwise model for SIS epidemic propagation on networks

The global behaviour of the compact pairwise approximation of SIS epidemic propagation on networks is studied. It is shown that the system can be reduced to two equations enabling us to carry out a detailed study of the dynamic properties of the solutions. It is proved that transcritical bifurcation occurs in the system at $\tau = \tau _c = \frac{\gamma n}{\langle n^{2}\rangle-n}$, where $\tau$ and $\gamma$ are infection and recovery rates, respectively, $n$ is the average degree of the network and $\langle n^{2}\rangle$ is the second moment of the degree distribution. For subcritical values of $\tau$ the disease-free steady state is stable, while for supercritical values a unique stable endemic equilibrium appears. We also prove that for subcritical values of $\tau$ the disease-free steady state is globally stable under certain assumptions on the graph that cover a wide class of networks

A common optical algorithm for the evaluation of specular spin polarized neutron and M\"ossbauer reflectivities

Using the general approach of Lax for multiple scattering of waves a 2x2 covariant expression for the reflectivity of polarized slow neutrons of a magnetic layer structure of arbitrary complexity is given including polarization effects of the external magnetic field. The present formalism is identical to the earlier published one for the (nuclear) resonant X-ray (Mossbauer) reflectivity and properly takes the effect of the external magnetic field of arbitrary direction on the neutron beam into account. The form of the reflectivity matrix allows for an efficient numerical calculation.Comment: 4 pages, no figures, PNCMI2000 - Proceeding of the Third International Workshop on Polarized Neutron

Variational occupation numbers to a M\"uller-type pair-density

Based on a parametric point-wise decomposition, a kind of isospectral deformation, of the exact one-particle probability density of an externally confined, analytically solvable interacting two-particle model system we introduce the associated parametric ($p$) one-matrix and apply it in the conventional M\"uller-type partitioning of the pair-density. Using the Schr\"odinger Hamiltonian of the correlated system, the corresponding approximate ground-state energy $E_p$ is then calculated. The optimization-search performed on $E_p$ with such restricted informations has a robust performance and results in the exact ($ex$) ground-state energy for the correlated model system $E_p=E_{ex}$.Comment: 11 pages, 1 figur

Cascade Product of Permutation Groups

We define the cascade product of permutation groups as an external product, an explicit construction of substructures of the iterated wreath product that are much smaller than the full wreath product. This construction is essential for computational implementations of algebraic hierarchical decompositions of finite automata. We show how direct, semidirect, and wreath products and group extensions can all be expressed as cascade products, and analyse examples of groups that can be constructed isomorphically by this generic extension giving them a hierarchically coordinatized form.Comment: 12 pages, 4 figures, related software package SgpDec http://sgpdec.sf.net, v4: tree action diagrams adde

Computational Understanding and Manipulation of Symmetries

For natural and artificial systems with some symmetry structure, computational understanding and manipulation can be achieved without learning by exploiting the algebraic structure. Here we describe this algebraic coordinatization method and apply it to permutation puzzles. Coordinatization yields a structural understanding, not just solutions for the puzzles.Comment: 14 pages, 5 figures, v2 major revision of computational example

Compact Notation for Finite Transformations

We describe a new notation for finite transformations. This compact notation extends the orbit-cycle notation for permutations and builds upon existing notations. It gives insight into the structure of transformations and reduces the length of expressions without increasing the number of types of symbols.Comment: 7 pages, 4 figures, compact notation implemented in SgpDec http://gap-packages.github.io/sgpdec

Green's J{\mathcal J}-classes and Subduction Classes in Finite Transformation Semigroups

We establish the connection between Green's ${\mathcal J}$-classes and the subduction equivalence classes defined on the image sets of an action of a semigroup. The construction of the skeleton order (on subduction equivalence classes) is shown to depend in a functorial way on transformation semigroups and surjective morphisms, and to factor through the $\leq_{\mathcal L}$-order and $\leq_{\mathcal J}$-order on the semigroup and through the inclusion order on image sets. For right regular representations, the correspondence between the ${\mathcal J}$-class order and the skeleton is one of isomorphism.Comment: 8 page

Subgroup Chains and Lagrange Coordinatizations of Finite Permutation Groups

We give a general constructive proof for hierarchical coordinatizations (Lagrange Decompositions) of permutation groups. The generalization originates from the investigation of how the subgroup chains of finite permutation groups yield different coordinate systems. The study is motivated by the practical needs and the verification of an existing computational implementation. Large scale machine calculated examples are also presented.Comment: 10 pages, 1 figur

Computational Holonomy Decomposition of Transformation Semigroups

We present an understandable, efficient, and streamlined proof of the Holonomy Decomposition for finite transformation semigroups and automata. This constructive proof closely follows the existing computational implementation. Its novelty lies in the strict separation of several different ideas appearing in the holonomy method. The steps of the proof and the constructions are illustrated with computed examples.Comment: 16 pages, 4 figures, final version will be published elsewher

ReactionKinetics---A Mathematica Package with Applications I. Requirements for a Reaction Kinetics Package

Requirements are formulated for a reaction kinetics package to be useful for an as wide as possible circle of users and illustrated with examples using ReactionKinetics, a Mathematica based package.Comment: 9 pages, 5 figure