About the sharpness of the stability estimates in the Kreiss matrix theorem

One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices under consideration. This so-called resolvent condition is known to imply, for all , the upper bounds and . Here is the spectral norm, is the constant occurring in the resolvent condition, and the order of is equal to .

It is a long-standing problem whether these upper bounds can be sharpened, for all fixed 1$">, to bounds in which the right-hand members grow much slower than linearly with and with , respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each 0$">, there are fixed values 0, K>1$"> and a sequence of matrices , satisfying the resolvent condition, such that for .

The result proved in this paper is also relevant to matrices whose -pseudospectra lie at a distance not exceeding from the unit disk for all 0$">.

     

On the evolution governed by the infinity Laplacian

We investigate the basic properties of the degenerate and singular evolution equation which is a parabolic version of the increasingly popular infinity Laplace equation. We prove existence and uniqueness results for both Dirichlet and Cauchy problems, establish interior and boundary Lipschitz estimates and a Harnack inequality, and also provide numerous explicit solutions. The first author is partially supported by the ESF program ``Global and Geometric Aspects of Nonlinear Partial Differential Equations'     

On the shellability of the order complex of the subgroup lattice of a finite group

We show that the order complex of the subgroup lattice of a finite group is nonpure shellable if and only if is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.

     

Interpolation spaces between the Lipschitz class and the space of continuous functions

It is shown that the complex interpolation spaces and do not coincide with or and also that the couple is not a Calderón couple. Similar results are also obtained for the couples and when .

     

Curve-straightening and the Palais-Smale condition

This paper considers the negative gradient trajectories associated with the modified total squared curvature functional . The focus is on the limiting behavior as tends to zero from the positive side. It is shown that when spaces of curves exist in which some trajectories converge and others diverge. In one instance the collection of critical points splits into two subsets. As tends to zero the critical curves in the first subset tend to the critical points present when . Meanwhile, all the critical points in the second subset have lengths that tend to infinity. It is shown that this is the only way the Palais-Smale condition fails in the present context. The behavior of the second class of critical points supports the view that some of the trajectories are `dragged' all the way to `infinity'. When the curves are rescaled to have constant length the Euler figure eight emerges as a `critical point at infinity'. It is discovered that a reflectional symmetry need not be preserved along the trajectories. There are examples where the length of the curves along the same trajectory is not a monotone function of the flow-time. It is shown how to determine the elliptic modulus of the critical curves in all the standard cases. The modulus must satisfy when the space is limited to curves of fixed length and the endpoints are separated by the vector .

     

On the comparison of the spaces

The notion of -variation and the space arise in the study of regularity properties of solutions to perturbed conservation laws. In this article we show that this notion is equivalent to variation in the regular sense, and therefore the space is the same as the space in the sense of Cesari-Tonelli. We also point out some connection between the space and the Favard classes for translation semigroups.

     

Extensions of the Kantorovich Inequality and the Bauer-Fike Inequality

This paper proves a Kantorovich-type inequality on the matrix of the type $\frac{1}{2}(Q^H_1 AQ_1 Q^H_1A^{-1} Q_1+Q^H_1A^{-1}Q_1Q^H_1AQ_1)$, where $A$ is an $n\times n$ positive definite Hermitian matrix and $Q_1$ is an $n\times m$ matrix with rank $(Q_1)=m$. The result is applied to get an extension of the Bauer-Fike inequality on condition numbers of similarities that block diagonalized matrices.     

Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

Complex specializations of the reduced Gassner representation of the pure braid group

We will give a necessary and sufficient condition for the specialization of the reduced Gassner representation to be irreducible. It will be shown that for , is irreducible if and only if .

     

On the characterization of the kernel of the geodesic X-ray transform

Let be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space . We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in .

     

A characterization of the Clifford torus

In this paper, we prove that an -dimensional closed minimal hypersurface with Ricci curvature of a unit sphere is isometric to a Clifford torus if , where is the squared norm of the second fundamental form of .

     

Regularity of the Solution of the Prandtl Equation

Mathematical Notes - Solvability and regularity of the solution of the Dirichlet problem for the Prandtl equation $$\frac{u(x)}{p(x)}-\frac{1}{2\pi}\int_{-1}^1\frac{u'(t)}{t-x}\,dt=f(x)$$ is...     

convergence of the reconstruction formula for the potential function

It is known that the potential function of the Sturm-Liouville problem can be reconstructed from the nodal data by a pointwise limit. We show that this convergence is in fact .

     

On the eigenvalue ratio for vibrating strings

For vibrating strings with concave densities or symmetric single-barrier densities, the ratio of the first two eigenvalues is minimized when the density is constant; while, for vibrating strings with symmetric single-well densities, the ratio is maximized when the density is constant.

     

Stability of the local spectrum

We give some conditions implying the equality of local spectra

where is a (bounded linear) operator on a complex Banach space and is defined by means of a local functional calculus. Moreover, we give conditions implying the stability of the local spectrum for the holomorphic and the meromorphic functional calculi.

     

Orthogonality of the range and the kernel of some elementary operators

We prove the orthogonality of the range and the kernel of an important class of elementary operators with respect to the unitarily invariant norms associated with norm ideals of operators. This class consists of those mappings , , where is the algebra of all bounded Hilbert space operators, and , , , are normal operators, such that , and . Also we establish that this class is, in a certain sense, the widest class for which such an orthogonality result is valid. Some other related results are also given.

     

A variant of the diamond principle for combinatorial ideals

We use a variant of the diamond principle to show many ideals on are not -saturated if is large. For instance, the -indescribable ideal is not -saturated if is almost ineffable.

     

Open colorings, the continuum and the second uncountable cardinal

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or ``open coloring axioms.' In particular it will be shown that the conjunction of two well-known axioms, and , implies that the size of the continuum is .

     

On the exact Hausdorff dimension of the set of Liouville numbers

Let denote the set of Liouville numbers. For a dimension function h, we write () for the h-dimensional Hausdorff measure of . In this paper we locate the exact cut-point at which the Hausdorff measure of drops from infinity to zero. Namely, if h is a dimension function that increases faster than any power function near 0, then ()=, and if h is a dimension function that increases slower than some power function near 0, then ()=0. This answers a question asked by R. D. Mauldin.Mathematics Subject Classification (2000): 28A80