Local boundary dilatation of quasiconformal maps in the disk

In this paper we partly give an affirmative answer to a problem proposed by F. Gardiner and N. Lakic by studying the gluing of quasiconformal maps.

     

Kobayashi＇s and Teichmiiller＇s Metrics on the Teichmiiller Space of Symmetric Circle Homeomorphisms

We give a direct proof of a result of Earle, Gardiner and Lakic, that is, Kobayashi’s metric and Teichmüller’s metric coincide with each other on the Teichmüller space of symmetric circle homeomorphisms.     

A note on Hamilton sequences for extremal Beltrami coefficients

F. P. Gardiner gave a sufficient condition for a sequence to be a Hamilton sequence for an extremal Beltrami coefficient. In this note, we shall consider the converse problem, proving that the condition is not necessary.

     

Some inequalities for the Poincaré metric of plane domains

In this paper, the Poincaré (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain In particular, another proof of a recent result of Gardiner and Lakic [7] is given with explicit constant. This and some other constants in this paper involve particular values of complete elliptic integrals and related special functions. A concrete estimate for the hyperbolic distance near a boundary point is also given, from which refinements of Littlewood’s theorem are derived.This research was carried out during the first-named author’s visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS.     

Unification of extremal length geometry on Teichmüller space via intersection number

In this paper, we give a framework for the study of the extremal length geometry of Teichmüller space after S. Kerckhoff, F. Gardiner and H. Masur. There is a natural compactification using extremal length geometry introduced by Gardiner and Masur. The compactification is realized in a certain projective space. We develop the extremal length geometry in the cone which is defined as the inverse image of the compactification via the quotient mapping. The compactification is identified with a subset of the cone by taking an appropriate lift. The cone contains canonically the space of measured foliations in the boundary. We first extend the geometric intersection number on the space of measured foliations to the cone, and observe that the restriction of the intersection number to Teichmüller space is represented by an explicit formula in terms of the Gromov product with respect to the Teichmüller distance. From this observation, we deduce that the Gromov product extends continuously to the compactification. As an application, we obtain an alternative approach to a characterization of the isometry group of Teichmüller space. We also obtain a new realization of Teichmüller space, a hyperboloid model of Teichmüller space with respect to the Teichmüller distance.     

Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space II

In this paper, we study the asymptotic behavior of Teichmüller geodesic rays in the Gardiner–Masur compactification. We will observe that any Teichmüller geodesic ray converges in the Gardiner–Masur compactification. Therefore, we get a mapping from the space of projective measured foliations to the Gardiner–Masur boundary by assigning the limits of associated Teichmüller rays. We will show that this mapping is injective but is neither surjective nor continuous. We also discuss the set of points where this mapping is bicontinuous.     

The behavior of invariants of finite degree under interdependent transformations of links

A new approach to the definition of the notion of finite-degree invariants of oriented links is described. It is proved that using new transformations, which are much more general than usual, actually leads to the same theory of such invariants. Applying these general transformations we also prove that the invariants of finite degree are polynomials in the gleams if the Hopf projection of the link is fixed. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 141–147. Translated by N. Yu. Netsvetaev.     

HYBRID ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS

ＨＹＢＲＩＤＡＬＧＥＢＲＡＩＣＭＵＬＴＩＬＥＶＥＬＰＲＥＣＯＮＤＩＴＩＯＮＩＮＧＭＥＴＨＯＤＳ￥ＢａｉＺｈｏｎｇｚｈｉ（白中治）（ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，复旦大学，邮编：２００４３３）Ａｂｓｔｒａｃｔ：Ａｃｌａｓｓｏｆｈｙｂｒｉｄａｌｇｅｂｒ...     

Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions

We revisit the classical theory of multidimensional scalar conservation laws. We reformulate the notion of the classical Kruzkov entropy solutions and study some new properties as well as the well-posedness of the initial value problem with inhomogeneous fluxes and general initial data. We also consider Dirichlet boundary value problems. We put forward a new and transparent definition for solutions and give a simple proof for their well-posedness in domains with smooth boundaries. Finally, we introduce the notion of saturated solutions and show that it is well-posed.     

Abstract Volterra operators

We propose a new general definition of Volterra operators. Several types of evolutionary operators, including Volterra ones in the sense of A.N. Tikhonov, satisfy this definition. For equations with generalized Volterra operators we introduce the notions of local, global, and maximally extended solutions. For solutions to nonlinear equations we formulate the existence, uniqueness, and extendability conditions. The theorems proved in this paper imply both known and new results on the solvability of concrete equations. We adduce an example of the application of obtained results to the study of the Cauchy problem for functional differential equations.     

A set oriented definition of finite-time Lyapunov exponents and coherent sets

The problem of phase space transport, which is of interest from both the theoretical and practical point of view, has been investigated extensively using geometric and probabilistic methods. Two important tools to study this problem that have emerged in recent years are finite-time Lyapunov exponents (FTLE) and the Perron–Frobenius operator. The FTLE measures the averaged local stretching around reference trajectories. Regions with high stretching are used to identify phase space transport barriers. One probabilistic method is to consider the spectrum of the Perron–Frobenius operator of the flow to identify almost-invariant densities. These almost-invariant densities are used to identify almost invariant sets. In this paper, a set-oriented definition of the FTLE is proposed which is applicable to phase space sets of finite size and reduces to the usual definition of FTLE in the limit of infinitesimal phase space elements. This definition offers a straightforward connection between the evolution of probability densities and finite-time stretching experienced by phase space curves. This definition also addresses some concerns with the standard computation of the FTLE. For the case of autonomous and periodic vector fields we provide a simplified method to calculate the set-oriented FTLE using the Perron–Frobenius operator. Based on the new definition of the FTLE we propose a simple definition of finite-time coherent sets applicable to vector fields of general time-dependence, which are the analogues of almost-invariant sets in autonomous and time-periodic vector fields. The coherent sets we identify will necessarily be separated from one another by ridges of high FTLE, providing a link between the framework of coherent sets and that of codimension one Lagrangian coherent structures. Our identification of coherent sets is applied to three examples.     

Strict (ε, δ)-Subdifferentials and Extremality Conditions

A new abstract definition of extremality is introduced extending traditional extremality notions in optimization problems. The set of points satisfying the new definition includes points not necessarily optimal in the usual sense but nevertheless having some extremal properties. Necessary and sufficient extremality conditions are derived. Contrary to usual notions of extremality the new one is stable relative to small deformations of the data.     

Combinatorially homogeneous graphs

Gardiner classified ultrahomogeneous graphs and posed the problem of defining “combinatorial homogeneity”. Later, Ronse proved that homogeneous graphs are ultrahomogeneous by classifying such graphs. In this paper, we give a direct proof that (suitably defined) combinatorially homogeneous graphs are ultrahomogeneous. Also, we clasify combinatorially C-homogeneous graphs.     

Local projection stabilization for advection–diffusion–reaction problems: One-level vs. two-level approach

Local projection stabilization (LPS) of finite element methods is a new technique for the numerical solution of transport-dominated problems. The main aim of this paper is a critical discussion and comparison of the one- and two-level approaches to LPS for the linear advection–diffusion–reaction problem. Moreover, the paper contains several other novel contributions to the theory of LPS. In particular, we derive an error estimate showing not only the usual error dependence on the mesh width but also on the polynomial degree of the finite element space. Based on this error estimate, we propose a definition of the stabilization parameter depending on the data of the solved problem. Unlike other papers on LPS methods, we observe that the consistency error may deteriorate the convergence order. Finally, we explain the relation between the LPS method and residual-based stabilization techniques for simplicial finite elements.     

Addendum to: Comparison between different types of abstract integrals in riesz spaces

In [3] we did not give explicitly the definition of measurability for realvalued functions, with respect to finitely additive measures with values in a Dedekind complete Riesz space. We note that, in [3], all involved functions are intended to be measurable. We now report the definition of measurability, which we gave in [2] (Definition 3.2). Lavoro svolto nell’ ambito dello G.N.A.F.A. del C.N.R.     

Information and system modelling

In this paper, we first give a clear mathematical definition of information. Then based on this definition of information we consider two routes of system modelling. One route is with stochastic information and the other route is with deterministic information. The route with stochastic information gives the usual information theory where information is carried by random variables or stochastic processes. With this route of stochastic information we can derive quantum mechanics. Then our new feature is the route with deterministic information. We show that with deterministic information we can establish deterministic quantum systems (which are quantum systems with no probability interpretation). From these deterministic quantum systems we can derive the three laws of thermodynamics and resolve the paradox between the second law of thermodynamics and the evolution phenomena of the world. We resolve this paradox by clarifying the relation between Shannon information entropy, Boltzmann entropy and the entropy for the second law. This clarification also solves the negative entropy problem of Schroedinger. These deterministic quantum systems which are established with deterministic information can be regarded as solutions to the the debate between Bohr and Einstein and the measurement problem of quantum mechanics because of their deterministic nature and their quantum structure.     

Equazioni ellittiche del secondo ordine di tipo non variazionale in aperti non limitati

Summary In this paper we study the Dirichlet problem, for second order, linear elliptic partial differential equations with discontinuous coefficients in unbounded domains. We obtain some results about existence and uniqueness of the solution in W²().

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Dynamics in dumbbell domains III. Continuity of attractors

In this paper we conclude the analysis started in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597] and continued in [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in Lp and H¹ norms.     

Asymptotic behaviour of linear Dirichlet parabolic problems with variable operators depending on time in varying domains

We study the asymptotic behaviour of the solutions of linear parabolic Dirichlet problems when the coefficients and the domains where the problems are posed vary simultaneously. In the limit problem it appear the H-limit of the operators, and as it is usual in the homogenization of Dirichlet problems, a new term of order zero. We also obtain a corrector result.