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The thick flame asymptotic limit and Damköhler's hypothesis
No full textWe derive analytical expressions for the burning rate of a flame propagating in a prescribed steady parallel flow whose scale is much smaller than the laminar flame thickness.In this specific context, the asymptotic results can be viewed as an analytical test of Damköhler's hypothesis relating to the influence of the small scales in the flow on the flame; the increase in the effective diffusion processes is described. The results are not restricted to the adiabaticequidiffusional case, which is treated first, but address also the influence of non-unit Lewis numbers and volumetric heat losses. In particular, it is shown that non-unit Lewis numbereffects become insignificant in the asymptotic limit considered. It is also shown that the dependence of the effective propagation speed on the flow is the same as in the adiabatic equidiffusional case, provided it is scaled with the speed of the planar non-adiabatic flame
Krylov subspace iterative techniques: on the detection of brain activity with electrical impedance tomography
Get PDFn this paper, we review some numerical techniques based on the linear Krylov subspace iteration that can be used for the efficient calculation of the forward and the inverse electrical impedance tomography problems. Exploring their computational advantages in solving large-scale systems of equations, we specifically address their implementation in reconstructing localized impedance changes occurring within the human brain. If the conductivity of the head tissues is assumed to be real, the preconditioned conjugate gradients (PCGs) algorithm can be used to calculate efficiently the approximate forward solution to a given error tolerance. The performance and the regularizing properties of the PCG iteration for solving ill-conditioned systems of equations (PCGNs) is then explored, and a suitable preconditioning matrix is suggested in order to enhance its convergence rate. For image reconstruction, the nonlinear inverse problem is considered. Based on the Gauss-Newton method for solving nonlinear problems we have developed two algorithms that implement the PCGN iteration to calculate the linear step solution. Using an anatomically detailed model of the human head and a specific scalp electrode arrangement, images of a simulated impedance change inside brain's white matter have been reconstructed
Group theoretic conditions for existence of robust relative homoclinic trajectories
Get PDFWe consider robust relative homoclinic trajectories (RHTs) for G-equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these results we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries
Free Lie algebras and formal power series
No full textLet G be a group and K a field. If V is a graded KG-module of the form V=V1plus sign in circleV2plus sign in circlecdots, three dots, centered , where each Vn is finite dimensional, then the free Lie algebra L(V) acquires the structure of a graded KG-module, L(V)=L1(V)plus sign in circleL2(V)plus sign in circlecdots, three dots, centered . The isomorphism types of V and L(V) may be described by the power series ∑ngt-or-equal, slanted1[Vn]tn and ∑ngt-or-equal, slanted1[Ln(V)]tn with coefficients from the Green ring. The main object of study is the function on power series which maps ∑[Vn]tn to ∑[Ln(V)]tn for every graded KG-module V. Closed formulae are given in certain cases, and these are closely related to character formulae of Brandt and others
On odd Laplace operators
No full textAbstract We consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an 'orbit space' of volume forms. This includes earlier results for the odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on M is partitioned into orbits by the action of a natural groupoid whose arrows correspond to the solutions of the quantum Batalin–Vilkovisky equations. We compare this situation with that of Riemannian and even Poisson manifolds. In particular, we show that the square of an odd Laplace operator is a Poisson vector field defining an analog of Weinstein's 'modular class'
About a homoclinic pitchfork bifurcation in reversible systems with additional Z2-symmetry
Get PDFThis paper studies bifurcations from a homoclinic orbit to a degenerate fixed point. We consider reversible Z2-symmetric systems of ordinary differential equations (ODEs) and assume the existence of a symmetric homoclinic orbit to a fixed point which itself undergoes a pitchfork bifurcation. We are interested in bifurcations from the primary homoclinic orbit in an unfolding of the degenerate situation.
The studies are motivated by numerical investigations on a model-system of second-order ODEs. There one finds a similar behaviour in the local and the global bifurcation. While locally two new fixed points are created, numerical computations show that at the same time two homoclinic orbits to these fixed points bifurcate from the primary orbit. We call the global scenario a reversible homoclinic pitchfork bifurcation.
An analysis of this homoclinic bifurcation is performed in a general frame. Depending on the sign of a higher order coefficient in the normal form we distinguish two cases of the local pitchfork bifurcation: the eye case (which is the one encountered in the model-system) and the figure-eight case. Adopting Lin's method to the non-hyperbolic situation the bifurcation of one-homoclinic orbits to the local centre manifold of the fixed point is investigated. Rigorous existence results for homoclinic orbits to fixed points and periodic orbits are derived. The global bifurcation picture is found to depend crucially on the local bifurcation
Reduced C*-algebra of the p-adic group GL(n) II
Get PDFThe reduced C*-algebra of the p-adic group GL(n) admits a Bernstein decomposition. We give a minimal refinement of this decomposition, and provide structure theorems for the reduced Iwahori-Hecke C*-algebra and the reduced spherical C*-algebra. This leads to a very explicit description of the tempered dual of GL(n) in terms of Bernstein parameters and extended quotients. We also prove that Plancherel measure (on the tempered dual of a reductive p-adic group) is
rotation-invariant
Computing Nearest Covariance and Correlation Matrices
Get PDFWe look at two matrix nearness problems posed by a finance
company, where nearness is measured in the Frobenius norm.
Correlation and covariance matrices are computed from sampled stock data with missing entries by a technique that produces matrices that are not positive semidefinite. In the first problem we find the nearest correlation matrix that is positive semidefinite and preserves any correlations known to be exact. In the second problem we investigate how the missing elements in the data should be chosen in order to generate the nearest covariance matrix to the indefinite matrix from the completed set of data. We show how the former problem can be solved using an alternating projections algorithm and how the latter problem can be investigated using a multi-directional search optimization method
Imperfect homoclinic bifurcations
Get PDFExperimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system
