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Stochastic and Deterministic Analysis of SIS Household Epidemics
Get PDFWe analyse SIS epidemics amongst populations partitioned into
households. The analysis considers both the stochastic and
deterministic model and unlike previous analysis, we consider
general infectious period distributions. For the deterministic
model, we prove the existence of an endemic equilibrium for the
epidemic if and only if the threshold parameter, .
Furthermore, by utilising Markov Chains we show that the total
number of infectives converges to the endemic equilibrium as time
. For the stochastic model, we prove a law
of large numbers result for the convergence of the mean number of
infectives per household in the stochastic model to the
deterministic limit. This is followed by the derivation of a
Gaussian limit process for the fluctuations of the stochastic
model
A short review on Landsberg spaces
Get PDFThis short review is concerned with real finite-dimensional Finsler manifolds
(M,F) with Finsler structures F:TM-->[0,infty) that
satisfy the Landsberg conditions. In particular this includes the case of
Berwald manifolds since their Chern connections on the pullback of TM are fibre-independent.
The aim is to provide an annotated collection of references to geometric
results that seem important in the study of Landsberg spaces and to suggest some
areas for further work in this context
A Narrowband Level Set Method Applied to EIT in Brain for Cryosurgery Monitoring
Get PDFIn this paper we investigate the feasibility of applying a novel level set reconstruction technique to electrical imaging of the human brain. We focus particularly on the potential application of Electrical Impedance Tomography (EIT) to cryosurgery monitoring. In this application, cancerous tissue is treated by a local freezing technique using a small needle-like cryosurgery probe. The interface between frozen and non-frozen tissue can be expected to have a relatively high contrast in conductivity and we treat the inverse problem of locating and monitoring this interface during the treatment. A level set method is used as a powerful and flexible tool for tracking the propagating interfaces during the monitoring process. For calculating sensitivities and the Jacobian when deforming the interfaces we employ an adjoint formula rather than a direct differentiation technique. Particulary we are using a narrowband technique for this procedure. This combination of an adjoint technique and a narrowband technique for calculating Jacobians results in a computationally efficient and extremely fast method for solving the inverse problem. Moreover, due to the reduced number of unknowns in each step of the narrowband approach compared to a pixel- or voxel-based technique, our reconstruction scheme tends to be much more stable. We demonstrate that our new method also outperforms its pixel-/ voxel-based counterparts in terms of image quality in this application
A Review on the Numerical Solution of the 1D Euler Equations
Get PDFThis paper presents a review on the numerical solution of the Euler equations. Three different high resolution versions of the following schemes are considered: Roe's scheme, the HLLE scheme and the AUSM+ scheme. We present a variety of test cases, each designed to test the robustness of each scheme and compare the results to determine which scheme was the most accurate
Reconstructing projective schemes from Serre subcategories
Get PDFGiven a positively graded commutative coherent ring which is finitely generated as an -algebra, a bijection between the tensor Serre subcategories of and the set of all
subsets of the form with quasi-compact open complement for all is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of
isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces
({\rm Proj} A,\cc O_{{\rm Proj} A})\simeq ({\rm spec}({\rm qgr} A),{\cal O}_{{\rm qgr} A}),
where is a ringed space associated to the lattice of tensor Serre subcategories of
A Schur-Newton Method for the Matrix p'th Root and its Inverse
Get PDFNewton's method for the inverse matrix th root,
, has the attraction that it
involves only matrix multiplication.
We show that if the starting matrix is for then
the iteration converges quadratically to
if the eigenvalues of lie in
a wedge-shaped convex set containing the disc
.
We derive an optimal choice of for the case where has real, positive
eigenvalues.
An application is described to roots of transition matrices from Markov models,
in which for certain problems the convergence condition is satisfied with
.
Although the basic Newton iteration is numerically unstable,
a coupled version is stable and a simple modification of it provides a new
coupled iteration for the matrix th root.
For general matrices we develop a hybrid algorithm that computes a Schur
decomposition, takes square roots of the upper (quasi)triangular factor,
and applies the coupled Newton iteration to a matrix for which fast convergence
is guaranteed.
The new algorithm can be used to compute either
or , and
for large that are not highly composite
it is more efficient than the method of Smith based entirely on the Schur
decomposition
The reopening of a collapsed fluid-filled elastic tube
No full textWe present an experimental study of the reopening mechanics of a collapsed liquid-filled elastic tube. The experiment is a simple mechanical model of pulmonary airway reopening and aims to assess the robustness of existing theoretical models. A metre-long horizontal elastic tube of inner radius Ri=4.88 ± 0.14mm is filled with silicone oil and is carefully collapsed mechanically. The injection of nitrogen at a constant flow rate results in the steady propagation of an air finger, after the decay of initial transients. This behaviour is observed over the realizable range of the capillary numbers Ca, which measures the ratio of viscous and capillary forces. With increasing Ca, the transition region between the collapsed and reopened sections of the tube shortens, and the height of the tube behind the bubble tip increases. We also find that air fingers can propagate in partially reopened tubes, in which the transmural pressure is negative far behind the finger tip.
The effect of viscosity on the reopening dynamics was explored by performing experiments using three different grades of silicone oil, with kinematic viscosities of 1000cS, 200cS and 100cS. A direct comparison between the experimental pressure dependence on Ca and numerical simulations of the zero-gravity three-dimensional airway-reopening model of Hazel & Heil (Trans. ASME: J. Biomech. Engng, vol. 128, 2006, p. 473) highlights some significant differences. Within the experimental parameter range, gravity profoundly influences the reopening mechanics in several ways. The reopening tube is supported by a rigid base, which induces an asymmetry about the horizontal mid-plane of the collapsed tube, resulting in distinct phases of reopening as Ca increases. In addition, buoyancy forces act on the air finger, which is observed to propagate near the top of the cross-section of the tube, leaving a thicker fluid-lining below. In the limit of small Ca, the height of the reopened tube increases significantly with viscosity. Experimental evidence suggests that this increase in viscosity leads to significant changes in the film configuration behind the propagating finger, caused by the increased contribution of buoyancy forces. The altered film configuration changes the mechanical load on the tube walls and, hence, the shape of the reopened tube
The Decomposition of Lie Powers
No full textLet be a group, a field of prime characteristic and a finite-dimensional -module. Let denote the free Lie algebra on regarded as an -submodule of the free associative algebra (or tensor algebra) . For each positive integer , let and be the th homogeneous components of and , respectively. Here is called the th Lie power of . Our main result is that there are submodules , , ... of such that, for all , is a direct summand of and, whenever and is not divisible by , the module is the direct sum of , , ..., . Thus every Lie power is a direct sum of Lie powers of -power degree. The approach builds on an analysis of as a bimodule for and the Solomon descent algebra