Inverse problems on star-type graphs: differential operators of different orders on different edges

We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.     

A canonical form for nonderogatory matrices under unitary similarity

A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms for

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nonderogatory complex matrices up to unitary similarity, and

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pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues.

The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles.     

Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric,skew-symmetric,or Hermitian forms

Tridiagonal canonical forms of square matrices under congruence or *congruence, pairs of symmetric or skew-symmetric matrices under congruence, and pairs of Hermitian matrices under *congruence are given over an algebraically closed field of characteristic different from 2.     

Littlewood–Paley Theory and Function Spaces with Alocp Weights

Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted Besov–Lipschitz and Triebel–Lizorkin spaces on ℝⁿ with weights that are locally in A_p but may grow or decrease exponentially at infinity are investigated. Square–function characterizations of the weighted L^p and Hardy spaces with the above class of weights are also obtained. A certain local variant of the Calderón reproducing formula is constructed and widely used in the proofs.     

On Lelong-Bremermann Lemma

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma:

Let be a continuous plurisubharmonic function on a Stein manifold of dimension Then there exists an integer , natural numbers , and analytic mappings such that the sequence of functions

converges to uniformly on each compact subset of .

In the case when is a domain in the complex plane, it is shown that one can take in the theorem above (Section 3); on the other hand, for -circular plurisubharmonic functions in the statement of this theorem is true with (Section 4). The last section contains some remarks and open questions.

     

A large closed queueing network with autonomous service and bottleneck

This paper studies the queue-length process in a closed Jackson-type queueing network with the large number N of homogeneous customers by methods of the theory of martingales and by the up- and down-crossing method. The network considered here consists of a central node (hub), being an infinite-server queueing system with exponentially distributed service times, and k single-server satellite stations (nodes) with generally distributed service times with rates depending on the value N. The service mechanism of these k satellite stations is autonomous, i.e., every satellite server j serves the customers only at random instants that form a strictly stationary and ergodic sequence of random variables. Assuming that the first k-1 satellite stations operate in light usage regime the paper considers the cases where the kth satellite station is a bottleneck node. The approach of the paper is based both on development of the method from the paper by Kogan and Liptser [16], where a Markovian version of this model has been studied, and on development of the up- and down-crossing method. This revised version was published online in June 2006 with corrections to the Cover Date.     

Invariant ordering on the simply connected covering of the Shilov boundary of a symmetric domain

The Shilov boundary of a symmetric domain D = G/K of tube type has the form G/P, where P is a maximal parabolic subgroup of the group G. We prove that the simply connected covering of the Shilov boundary possesses a unique (up to inversion) invariant ordering, which is induced by the continuous invariant ordering on the simply connected covering of G and can readily be described in terms of the corresponding Jordan algebra.     

Formation of x-ray beams with the aid of a tapered microcapillary

The possibility of obtaining x-ray microbeams with a prescribed intensity using a system containing a x-ray tube with a transmission thin-film anode and a tapered glass microcapillary is investigated. Zh. Tekh. Fiz. 68, 90–93 (May 1998)