On the domain dependence of solutions to the compressible Navier–Stokes equations of a barotropic fluid

We prove a general compactness result for the solution set of the compressible Navier–Stokes equations with respect to the variation of the underlying spatial domain. Among various corollaries, we then prove a general existence theorem for the system in question with no restrictions on smoothness of the spatial domain. Copyright © 2002 John Wiley & Sons, Ltd.     

The matrix equation AX-YB=C and related problems

The main result of the paper is a theorem, using which a new proof of Roth’s theorem is obtained, a new solvability criterion for the matrix equation AX-YB = C is proved, a formula for a particular solution of the latter is derived, and the least of the orders of square nonsingular matrices containing a given rectangular matrix as a submatrix is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 15–23.     

Isometries of the Dirichlet space among the composition operators

We show that every composition operator which is an isometry of the Dirichlet space is induced by a univalent full map of the disk into itself that fixes the origin. This is an analogue of the Hardy space result for inner functions due to Nordgren. The proof relies on the Stone-Weierstrass theorem and the Riesz representation theorem.

     

Scaled-oscillation and regularity

We give sufficient conditions for Sobolev and Lipschitz functions in terms of their lower scaled-oscillation. The sharpness of these conditions is shown by examples. Our examples also show that a Stepanov-type differentiability theorem does not hold under the boundedness assumption of the lower scaled-oscillation.

     

Sufficiency in Quantum Statistical Inference

This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. As an application we discuss exponential families. Our factorization theorem also implies two further important results, previously known only in finite Hilbert space dimension, but proved here in generality: the Koashi-Imoto theorem on maps leaving a family of states invariant, and the characterization of the general form of states in the equality case of strong subadditivity. Supported by the EU Research Training Network Quantum Probability with Applications to Physics, Information Theory and Biology and Center of Excellence SAS Physics of Information I/2/2005. Supported by the Hungarian grant OTKA T032662     

New examples of Osserman metrics with nondiagonalizable Jacobi operators

Explicit examples of Osserman 4-manifolds with exactly two distinct eigenvalues of the Jacobi operators, α and β=4α≠0, are given. The former has multiplicity two and is a double root of the minimal polynomial of the Jacobi operators.     

On an Application of Newton's Method to Nonlinear Operators with w-Conditioned Second Derivative

We present a new approach to study the convergence of Newton's method in Banach spaces, which relax the conditions appearing in the usual studies. The approach is based on the bound required for the second derivative of the operator involved. An application to a nonlinear integral equation is presented.     

On a conjecture of Cox and Katz

This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces. Received: 8 February 2001; in final form: 17 September 2001 / Published online: 1 February 2002