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Stability of Relative Equilibria of Point Vortices on the Sphere
No full textWe describe the linear and nonlinear stability and instability of certain configurations of point vortices on the sphere forming relative equilibria. These configurations consist of up to two rings, with and without polar vortices. Such configurations have dihedral symmetry, and the symmetry
is used both to block diagonalize the relevant matrices and to distinguish the subspaces on which their eigenvalues need to be calculated
Rationality as conformity
No full textWe address the problem of characterizing the choice processes of two like-minded yet
non-communicating agents who intend to select, from a finite set of options, the same
possible world. Hence, we call the resulting framework “Rationality-as-conformity”
Within the scope of our formalisation, in which a choice problem is defined on
non-empty subset of maps from a finite set A to the binary set {0, 1}, we introduce
and investigate three distinct logico-mathematical characterisations of Rationality-as-
conformity.
Finally, we discuss the applicability of our framework to problems such as pure
coordination games and radical interpretation which are traditionally related to “rationality”
Unusual phenomena in an extreme bushfire
Get PDFFires burning to the west of Canberra on 18th January 2003 generated a series of cumulonimbus, or pyrocumulus, clouds that may have exacerbated windspeeds as they approached suburban Canberra. Powerful whirlwinds were generated, at least one of which might have been a genuine tornado. In places, fires are reported to have emerged with an intensity that appeared unsustainable by the local vegetation. This suggests that pyrolysed gases might have been able to escape from the incomplete combustion at the fire front in sufficient concentration to ignite and burn elsewhere
Theoretical analysis of acoustic instability of a hypersonic shock layer on a porous wall
Get PDFThe possibility of controlling the laminar�turbulent transition in hypersonic shock layers by means
of porous coatings is considered. The linear stability of the shock layer to acoustic disturbances is
analyzed. A dispersion relation is derived in an analytical form and analyzed for different characteristic
values of porosity of the wall, which allows one to study the spectrum of acoustic disturbances
in the shock layer. Analytical expressions for the growth rate of instability of acoustic disturbances
are presented as functions of the reflection factor. Their structure indicates that the porous coating
effectively decreases acoustic instability of the shock layer
Rational Cherednik algebras and Hilbert schemes
Get PDFLet be the rational Cherednik algebra of type with spherical subalgebra . Then is filtered by order of differential operators, with associated graded ring
\mbox{gr} U_c = \mathbb{C} [ \mathfrak{h} \oplus \mathfrak{h}^*] where is the th symmetric group. We construct a filtered -algebra such that, under mild conditions on :
• the category -qgr of graded noetherian -modules modulo torsion is equivalent to -mod;
• the associated graded -algebra \mbox{gr}B has \mbox{gr}B-lqgr \simeq \coh \Hilb(n)
This can be regarded as saying that simultaneously gives a non-commutative deformation of and of its resolution of singularities \Hilb(n) \rightarrow \mathfrak{h} \oplus \mathfrak{h}^*. As we show elsewhere, this result is a powerful tool for studying the representation theory of and its relationship to \Hilb(n)
A generalized second order frame bundle for Fréchet manifolds
No full textWorking within the framework of Fréchet modelled infinite dimensional manifolds, we propose a generalized notion of second order frame bundle. We revise in this way the classical notion of bundles of linear frames of order
two, the direct definition and study of which is problematic due to intrinsic difficulties of the space models. However, this new structure keeps all the fundamental characteristics of a frame bundle: It is a principal Frechet bundle associated (differentially and geometrically) with the corresponding second order tangent bundle
The Effects of State Dependent and State Independent Probabilistic Updating on Boolean Network Dynamics
Get PDFWe study semi-synchronous Boolean networks with robabilistic updating schemes and various topologies (tree, loop, and random). As well as state independent probabilistic updating we investigate a state dependent scheme which allows us to control the `accuracy'
of nodes. A node is accurate at if it has been updated at , or if its state is as it would be if it had updated.
The state dependent re-evaluation probabilities are determined by the `accuracy heuristic': a stochastic equation which depends on the estimation of a distribution; we look at ways of estimating this distribution and derive variance expressions for the estimators.
Through our work on random Boolean trees we observe that (in
general) the output of a Boolean function with correlated inputs, becomes less correlated as the number of inputs is increased. We also discover that the correlation of a Boolean function's output directly affects the ability of the heuristic to achieve the node's target accuracy.
Deterministic random Boolean network dynamics are viewed in a new way, via the distribution of node output distributions (the probability a node's state is or ). This view shows the `activity' of nodes across the network. We find that as in-degree is increased the topology has less effect on the activity and the distribution of the Boolean functions dominates. We present a theoretical result to support this theory.
To understand the dynamics of probabilistically updating Boolean networks we use a numerical approximation to Flyvbjerg's frozen component.
The concept of stability in probabilistically updating Boolean networks is addressed and investigated. For the loop topology the dynamics of active loops fall into two categories: those with an odd number of inversion nodes and those with an even number. We discuss the stability of a fixed point in both cases. For the random topology we derive an annealed approximation which indicates a phase transition similar to that previously found in the deterministic networks
Sampling distributions associated with the multivariate t distribution
No full textThe known sampling distributions and simulation methods associated
with multivariate t distributions are reviewed. We believe that this
review will serve as an important reference and encourage further
research activities in the area