Improving the compression rate versus L 1 error ratio in cell-average error control algorithms

Multiresolution representations of data are powerful tools in data compression. A common framework for applications is the cell-average setting. For a proper adaptation to singularities, it is interesting to develop nonlinear methods. Thus, one needs to control the stability of these representations. We introduce a generalization, depending on a parameter λ, of the classical cell-average error-control algorithms and we study the choice of the parameter to get the best relation quality vs. ratio of compression. It turns out that λ can be chosen nonlinearly and that for the L ¹ norm we can get a significant improvement over the classical error-control algorithms.     

Minimal singularities for representations of Dynkin quivers

We develop some reduction techniques for the study of singularities in orbit closures of finite dimensional modules. This enables us to classify all singularities occurring in minimal degenerations of representations of Dynkin quivers. They are all smoothly equivalent to the singularity at the zero-matrix inside thep×q-matrices of rank at most one.     

The use of phase portraits to visualize and investigate isolated singular points of complex functions

ABSTRACT

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts.     

Formation of singularities for wave equations including the nonlinear vibrating string

Under very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small-amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small-amplitude time-periodic solutions.     

Microlocal Analysis and Global Solutions of Some Hyperbolic Equations with Discontinuous Coefficients

We are concerned with analyzing hyperbolic equations with distributional coefficients. We focus on the case of coefficients with jump discontinuities considered earlier by Hurd and Sattinger in their proof of the breakdown of global distributional solutions. Within the framework of Colombeau generalized functions, however, Oberguggenberger showed the existence and uniqueness of a global solution. Within this framework we develop further a microlocal analysis to understand the propagation of singularities of such Colombeau solutions. To achieve this we introduce a refined notion of a wave-front set, extending Hörmander's definition for distributions. We show how the coefficient singularities modify the classical relation of the wave front set of the solution and the characteristic set of the operator, with a generalized notion of characteristic set.     

A family of stable nonlinear nonseparable multiresolution schemes in 2D

Multiresolution representations of data are powerful tools in data compression. For a proper adaptation to the edges, a good strategy is to consider a nonlinear approach. Thus, one needs to control the stability of these representations. In this paper, 2D multiresolution processing algorithms that ensure this stability are introduced. A prescribed accuracy is ensured by these strategies.     

Large time existence of small viscous surface waves without surface tension

The motion of a three-dimensional viscous, imcompressible fluid is governed by the Navier-Stokes equations. We study the case where the fluid is in an ocean of infinite extent and finite depth with a free surface on top. This gives rise to a nonlinear free boundary problem. The given data are the initial velocity field and the initial free surface. In general, given smooth data, the solution will develop singularities in finite time; however, the effect of viscosity and surface tension tends to prevent the ingulitrities. It was previously known that when both are present, small, appropriately smooth solutions do not develop singularities; that is, smooth solutions exist globally in time. In this paper, we show that viscosity alone will prevent the formation of singularitics, even without surface tension; i.e., small smooth data which satisfy certain natural compatibility conditions, smooth solutions exist for all time. Uniqueness of the solution for any finite time interval is also proved.     

Ten years of the analytic perturbation theory in QCD

The renormalization group method allows improving the properties of the QCD perturbative power series in the ultraviolet region. But it ultimately leads to unphysical singularities of observables in the infrared domain. The analytic perturbation theory is the next step in improving the perturbative expansions. Specifically, it involves an additional analyticity requirement based on the causality principle and implemented in the Källen-Lehmann and Jost-Lehmann representations. This approach eliminates spurious singularities of the perturbative power series and enhances the stability of the series with respect to both higher-loop corrections and the choice of the renormalization scheme. This paper is an overview of the basic stages in developing the analytic perturbation theory in QCD, including its recent applications to describing hadronic processes.     

Discretization of implicit ODEs for singular root-finding problems

This paper addresses the use of dynamical system theory to tackle singular root-finding problems. The use of continuous-time methods leads to implicit differential systems when applied to singular nonlinear equations. The analysis is based on a taxonomy of singularities and uses previous stability results proved in the context of quasilinear implicit ODEs. The proposed approach provides a framework for the systematic formulation of quadratically convergent iterations to singular roots. The scope of the work includes also the introduction of discrete-time analysis techniques for singular problems which are based on continuous-time stability and numerical stability. Some numerical experiments illustrate the applicability of the proposed techniques.     

Students’ geometric thinking with cube representations: Assessment framework and empirical evidence

While representations of 3D shapes are used in the teaching of geometry in lower secondary school, it is known that such representations can provide difficulties for students. In order to assess students’ thinking about 3D shapes, we constructed an assessment framework based on existing research studies and data from G7-9 students (aged 12–15). We then applied our framework to assess students’ geometric thinking in lessons. We report two cases of qualitative findings from a classroom experiment in which Grade 7 students (aged 12–13) tackled a problem in 3D geometry that was, for them, quite challenging. We found that students who failed to answer given problems did not mentally manipulate representations effectively, while others could mentally manipulate representations and reason about them in order to reach correct solutions. We conclude with the proposition that this finding shows the framework can be used by teachers in instruction to assess their students’ 3D geometric thinking.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

On the Solution of the Stokes Equation on Regions with Corners

The detailed behavior of solutions to Stokes equations on regions with corners has been historically difficult to characterize. The solutions to Stokes equations on regions with corners are known to develop singularities in the vicinity of corners; in particular, the solutions are known to have infinite oscillations along almost every ray that meet at the corner. While the nature of singularities for the differential equation have been analyzed in great detail, very little is known about the nature of singularities for the corresponding integral equations. In this paper, we observe that, when the Stokes equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of the form , where t is the distance from the corner and the parameters μ_j, β_j are real, and are determined via an explicit formula depending on the angle at the corner. In addition to being analytically perspicuous, these representations lend themselves to the construction of highly accurate and efficient numerical discretizations, significantly reducing the number of degrees of freedom required for the solution of the corresponding integral equations. The results are illustrated by several numerical examples. © 2020 Wiley Periodicals LLC     

Qualitative behaviour of incompressible two-phase flows with phase transitions: The isothermal case

A thermodynamically consistent model for incompressible two-phase flows with phase transitions is considered mathematically. The model is based on first principles, i.e., balance of mass, momentum and energy. In the isothermal case, this problem is analysed to obtain local well-posedness, stability of non-degenerate equilibria, and global existence and convergence to equilibria of solutions which do not develop singularities.     

A Lipschitz metric for the Hunter–Saxton equation

We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.     

Variation homological diagrams,¶and corner singularities

We describe the general homological framework (the variation arrays and variation homological diagrams) in which can be studied hypersurface isolated singularities as well as boundary singularities and corner singularities from the point of view of duality. We then show that any corner singularity is extension, in a sense which is defined, of the corner singularities of less dimension on which it is built. This framework is also used to rewrite Thom–Sebastiani type properties for isolated singularities and to establish them for boundary singularities. Received: 27 June 2000 / Revised version: 18 October 2000     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Path-following energy optimization in unilateral contact problems

Path-following (load incrementation) methods are studied in this paper for elastostatic analysis problems with unilateral contact relations in the framework of a large displacement theory by means of the parametric optimization techniques. Finite element discretization yields sparse polynomial optimization problems with equality and inequality constraints. For such sparse problems generically appearing singularities along the path of solutions are completely classified. Perturbations involving only a minimal number of parameters are shown to be sufficient to guarantee these generic situations. This clarifies stability and uniqueness questions for the solution along the examined path.     

Asymptotic representations of solutions of essentially nonlinear cyclic systems of ordinary differential equations

We obtain asymptotic representations for a class of solutions of cyclic systems of ordinary differential equations with properly varying singularities.