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Implicit Gamma Theorems (I): Pseudoroots and Pseudospectra
No full textLet g : E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, ε)) ⊂ E, the image under g of the closed ball about x∈ E with radius ε . When g(x) expresses the solution of an equation depending on x , then the elements of g(B(x,ε )) are ε -pseudosolutions. Our aim is to investigate the size of the set g(B(x,ε )) . We derive upper and lower bounds of the following form:
g(x) + Dg (x) ( B(0, c 1 ε N))
g(B(x,ε ))
g(x) +Dg (x) ( B(0, c 2 ε ) ),
where Dg (x) denotes the derivative of g at x . We consider both the case where g is given explicitly and the case where g is given implicitly. We apply our results to the implicit function associated with the evaluation map, namely the solution map, and to the polynomial eigenvalue problem. Our results are stated in terms of an invariant γ which has been extensively used by various authors in the study of Newton's method. The main tool used here is an implicit γ theorem, which estimates the γ of an implicit function in terms of the γ of the function defining it
Effect of heat loss on flame edges in a premixed counterflow
No full textWe describe the combined influence of heat-loss and strain (characterized here by non-dimensional parameters ? and ?, respectively) on premixed flame-edges in a two-dimensional counterflow configuration. The problem is formulated as a thermo-diffusive model with a single Arrhenius reaction. In order to help classify the various flame-edge regimes, the non-adiabatic one-dimensional problem which characterizes the wings (far downstream) of the flame-edge is briefly revisited and its solutions are delimited in the ?–? plane. An analytical description of the flame-edges is then presented in the weak-strain limit ??0. This is complemented by a detailed numerical study. Several combustion regimes are found and their domains of existence are identified in the ?–? plane. These include ignition fronts, extinction fronts, solutions with propagation speeds that depend non-monotonically on the strain-rate, propagating flame tubes and stationary flame tubes. Multiplicity of solutions and hysteresis phenomena, which are partly but not exclusively associated with the one-dimensional regimes, are also identified and discussed
Uniqueness and reconstruction in magnetic resonance - electrical impedance tomography (MR - EIT)
No full textMagnetic resonance–electrical impedance tomography (MR–EIT) was first proposed in 1992. Since then various reconstruction algorithms have been suggested and applied. These algorithms use peripheral voltage measurements and internal current density measurements in different combinations. In this study the problem of MR–EIT is treated as a hyperbolic system of first-order partial differential equations, and three numerical methods are proposed for its solution. This approach is not utilized in any of the algorithms proposed earlier. The numerical solution methods are integration along equipotential surfaces (method of characteristics), integration on a Cartesian grid, and inversion of a system matrix derived by a finite difference formulation. It is shown that if some uniqueness conditions are satisfied, then using at least two injected current patterns, resistivity can be reconstructed apart from a multiplicative constant. This constant can then be identified using a single voltage measurement. The methods proposed are direct, non-iterative, and valid and feasible for 3D reconstructions. They can also be used to easily obtain slice and field-of-view images from a 3D object. 2D simulations are made to illustrate the performance of the algorithms
Mass transfer from a finite strip near an oscillating stagnation point --- implications for atherogenesis
No full textWe consider the mass transfer from a finite-length strip
near a two-dimensional, oscillating, stagnation-point flow in an incompressible, Newtonian fluid.
The problem is investigated using a combination of asymptotic and numerical methods. The aim of the study is to
determine the effect of the location of the strip, relative to the time-averaged position of the stagnation point,
on the mass transfer into the fluid. The study is motivated
by the problem of mass transfer from an injured region of the arterial wall into the blood, a process that may be of considerable importance in atherogenesis. For physiologically realistic parameter values,
we find that the fluid flow is quasi-steady, but the
mass transfer exhibits genuine time-dependence and
a high-frequency asymptotic solution provides an accurate
prediction of the time-average mass transfer. In this regime, there is a significant reduction in mass transfer when the centre of the strip is
located at the point of zero time-averaged wall shear rate, or equivalently wall shear stress, which
may serve to explain, at least partially, the correlation between arterial disease and regions of low wall shear stress
Vortex dynamics on cylinders
No full textPoint vortices on a cylinder (periodic strip) are studied geometrically, using local integrals of motion. The Hamiltonian formalism is developed, a non-existence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging
Phytoplankton biomass and residual nitrate in the pelagic ecosystem
No full textWe develop and analyse a simple, two-compartment (chlorophyll and nitrate) model of the surface mixed layer of the ocean. The mixed-layer depth is modulated intermittently to simulate the effects of storms. The optical properties of the water column are linked to changes in the chlorophyll biomass. The model can be treated analytically. Mathematical bounds are found for the autotrophic biomass and the residual nitrate in terms of the intensity and frequency of storms and the bio-optical properties of the phytoplankton. The results are discussed in the context of the high-nutrient, low-chlorophyll regimes, where unconsumed nitrate is a persistent occurrence
When maximising entropy gives the rational closure
No full textWe introduce a generalization of epsilon-probability functions
and a new correspondence with rational consequence relations. The
rational consequence relations corresponding to the generalized epsilon-probability
functions of maximum
entropy are investigated and it is shown that
in this case the analogous notion of `maximum entropy closure' equals the
rational closure, thus
reconfirming the rational closure of a conditional knowledge base as
the simplest, least prejudiced, rational consequence relation
satisfying that knowledge base
Model theory and modules
Get PDFThe model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the notation R-Mod denotes the category of left R-modules. By Ab we mean the category of abelian groups. In Part 1 we introduce the general concepts and in Part 2 we discuss these in more specific contexts. References within the text, as well as those in the bibliography, are neither complete nor comprehensive but are intended to lead the reader to a variety of source
Early stage kinetics in a unified model of shear-induced demixing and mechanical shear banding instabilities
No full textWe present a unified model of shear-induced demixing and "mechanical" shear banding instabilities in polymeric and surfactant solutions, by combining a simple flow instability with a two-fluid approach to concentration fluctuations. Within this model, we calculate the "spinodal" limit of stability of initially homogeneous shear states to demixing/banding, and predict the selected length and time scales at which inhomogeneity first emerges after a shear start-up "quench" into the unstable region, finding qualitative agreement with experiment. Our analysis is the counterpart, for this driven phase transition, of the Cahn-Hilliard calculation for unsheared fluid-fluid demixing
Steady finite-Reynolds-number flows in three-dimensional collapsible tubes
Get PDFA fully coupled finite-element method is used to investigate the steady flow of a viscous fluid through a thin-walled elastic tube mounted between two rigid tubes. The steady three-dimensional Navier–Stokes equations are solved simultaneously with the equations of geometrically nonlinear Kirchhoff–Love shell theory. If the transmural (internal minus external) pressure acting on the tube is sufficiently negative then the tube buckles non-axisymmetrically and the subsequent large deformations lead to a strong interaction between the fluid and solid mechanics. The main effect of fluid inertia on the macroscopic behaviour of the system is due to the Bernoulli effect, which induces an additional local pressure drop when the tube buckles and its cross-sectional area is reduced. Thus, the tube collapses more strongly than it would in the absence of fluid inertia. Typical tube shapes and flow fields are presented. In strongly collapsed tubes, at finite values of the Reynolds number, two ’jets‘ develop downstream of the region of strongest collapse and persist for considerable axial distances. For sufficiently high values of the Reynolds number, these jets impact upon the sidewalls and spread azimuthally. The consequent azimuthal transport of momentum dramatically changes the axial velocity profiles, which become approximately \uTheta-shaped when the flow enters the rigid downstream pipe. Further convection of momentum causes the development of a ring-shaped velocity profile before the ultimate return to a parabolic profile far downstream