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Krull-Gabriel dimension of 1-domestic string algebras
Get PDFWe classify indecomposable pure injective modules over a wide class of 1-domestic string algebras and calculate the Krull–Gabriel dimension of these algebras
Generation of anisotropic-smoothness regularization filters for EIT
Get PDFIn the inverse conductivity problem, as in any ill-posed inverse problem, regularization techniques are necessary in order to stabilize inversion. A common way to implement regularization in electrical impedance tomography is to use Tikhonov regularization. The inverse problem is formulated as a minimization of two terms: the mismatch of the measurements against the model, and the regularization functional. Most commonly, differential operators are used as regularization functionals, leading to smooth solutions. Whenever the imaged region presents discontinuities in the conductivity distribution, such as interorgan boundaries, the smoothness prior is not consistent with the actual situation. In these cases, the reconstruction is enhanced by relaxing the smoothness constraints in the direction normal to the discontinuity. In this paper, we derive a method for generating Gaussian anisotropic regularization filters. The filters are generated on the basis of the prior structural information, allowing a better reconstruction of conductivity profiles matching these priors. When incorporating prior information into a reconstruction algorithm, the risk is of biasing the inverse solutions toward the assumed distributions. Simulations show that, with a careful selection of the regularization parameters, the reconstruction algorithm is still able to detect conductivities patterns that violate the prior information. A generalized singular-value decomposition analysis of the effects of the anisotropic filters on regularization is presented in the last sections of the paper
Once again on the supersonic flow separation near a corner
Get PDFLaminar boundary-layer separation in the supersonic flow past a corner point on a rigid body contour, also termed the compression ramp, is considered based on the viscous–inviscid interaction concept. The ‘triple-deck model’ is used to describe the interaction process. The governing equations of the interaction may be formally derived from the Navier–Stokes equations if the ramp angle [theta] is represented as [theta] = [theta]0Re[minus sign]1/4, where [theta]0 is an order-one quantity and Re is the Reynolds number, assumed large. To solve the interaction problem two numerical methods have been used. The first method employs a finite-difference approximation of the governing equations with respect to both the streamwise and wall-normal coordinates. The resulting algebraic equations are linearized using a Newton–Raphson strategy and then solved with the Thomas-matrix technique. The second method uses finite differences in the streamwise direction in combination with Chebychev collocation in the normal direction and Newton–Raphson linearization.
Our main concern is with the flow behaviour at large values of [theta]0. The calculations show that as the ramp angle [theta]0 increases, additional eddies form near the corner point inside the separation region. The behaviour of the solution does not give any indication that there exists a critical value [theta]*0 of the ramp angle [theta]0, as suggested by Smith & Khorrami (1991) who claimed that as [theta]0 approaches [theta]*0, a singularity develops near the reattachment point, preventing the continuation of the solution beyond [theta]*0. Instead we find that the numerical solution agrees with Neiland's (1970) theory of reattachment, which does not involve any restriction upon the ramp angle
Performance and analysis of saddle point preconditioner for the discrete steady-state Navier-Stokes equations
No full textWe examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers
The geometry of the classical solutions of the Garnier systems
No full textOur aim is to find a general approach to the theory of classical solutions of the Garnier system in n-variables, G_n, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system G_n. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system G_n is such that the associated Fuchsian system has l monodromy matrices equal to ±1, then it can be reduced classically to a solution of a the Garnier system with n – 1 variables G_{n – 1}. When n monodromy matrices are equal to ±1, we have classical solutions of G_n. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the Painlevé VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to ±1
Graded Manifolds and Drinfeld Doubles for Lie Bialgebroids
Get PDFWe define graded manifolds as a version of supermanifolds endowed with an extra Z-grading in the structure sheaf, called weight (not linked
with parity). Examples are ordinary supermanifolds, vector bundles, double
vector bundles (in particular, iterated constructions like TTM), etc. I give
a construction of doubles for graded QS- and graded QP-manifolds (graded
manifolds endowed with a homological vector field and a Schouten/Poisson
bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded QS-manifolds can be considered, roughly, as "generalized Lie
bialgebroids". The double for them is closely related with the analog of Drinfeld's double for Lie bialgebroids recently suggested by Roytenberg. Lie bialgebroids as a generalization of Lie bialgebras, over some base manifold, were
defined by Mackenzie and P. Xu. Graded QP-manifolds give an odd version
for all this, in particular, they contain "odd analogs" for Lie bialgebras, Manin
triples, and Drinfeld's double
Level of repair analysis and minimum cost homomorphisms of graphs
Get PDFLevel of Repair Analysis (LORA) is a prescribed procedure for defence logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, sub-assemblies, components, etc. organized into several levels of indenture and with a number of possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall life-cycle costs. For a LORA problem with two levels of indenture with three possible repair decisions, which is of interest in UK and US military and which we call LORA-BR, Barros (1998) and Barros and Riley (2001) developed certain branch-and-bound heuristics. The surprising result of this paper is that LORA-BR is, in fact, polynomial-time solvable. To obtain this result, we formulate the general LORA problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORA-BR, LORA-M, to the maximum weight independent set problem on a bipartite graph. We prove that the general LORA problem is NP-hard by using an important result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem and provide partial results. Finally, we show that our result for LORA-BR can be applied to prove that an extension of the maximum weight independent set problem on bipartite graphs is polynomial time solvable
Stability of the overshoot for Lévy processes
No full textWe give equivalences for conditions like and , where the convergence is in probability or almost sure, both as and , where is a L\'{e}vy process and and are the first exit times of out of the strip and half-plane , , respectively. We also show, using a result of Kesten, that a.s.\ as is equivalent to ``creeping'' across a level
Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero
Get PDFBifurcations are studied from a fixed point with fourfold eigenvalue zero occurring in a two degrees of freedom Hamiltonian system of second-order ordinary differential equations (ODEs) which is additionally reversible with respect to two different linear involutions. Using techniques from Catastrophe Theory we are led to a codimension 2 problem and obtain two different unfoldings of the singularity related to the hyperbolic and elliptic umbilic, respectively. The analysis of the unfolded systems is essentially concerned with the existence and properties of homoclinic and heteroclinic orbits. The studies are motivated by a problem from nonlinear optics concerning the existence of solitons in a chi^2-medium
Observable dependence of fluctuation-dissipation relations and effective temperatures
No full textWe study the nonequilibrium fluctuation-dissipation theorem (FDT) in the glass phase of Bouchaud’s trap model. We incorporate an arbitrary observable m and obtain its correlation and response functions in closed form. A limiting nonequilibrium FDT plot is approached at long times for most choices of m. In contrast to standard mean field models, however, the shape of the plot depends nontrivially on the observable, and its slope varies continuously even though there is a single scaling of relaxation times with age. Nonequilibrium FDT plots can therefore not be used to define a meaningful effective temperature Teff in this model. Consequences for the wider applicability of an FDT-derived Teff are discussed