On regular linear relations

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.     

On the independence of correspondences

An almost independent set-valued process on a Loeb product space is shown to be representable as the closure of a sequence of its selections which are almost independent themselves. This provides a Castaing type representation in terms of independent correspondences. Different definitions of independence for correspondences in the literature are also unified in a general setting.     

On the relative character correspondences

Let π(G,A):Irr_A(G)→Irr(C_G(A)) be the Glauberman–Isaacs correspondence, where G and A are finite groups with coprime orders and A acts on G by automorphisms. Let B be a subgroup of A. In this setting, we give some new conditions for the fixed-point subgroups C_G(A) and C_G(B) such that χπ(G,A) is an irreducible constituent of the restriction of χπ(G,B) to C_G(A) for all χIrr_A(G).     

On the properties of Lyapunov exponents of regular linear systems

We establish the sharp Baire class of the Lyapunov exponents on the space of Lyapunov regular linear systems with continuous bounded coefficients equipped with the topology of uniform or compact convergence of the coefficients on the half-line.     

On the intersection of modular correspondences

Oblatum 28-I-1992 & 29-X-1992     

On strong regular stabilizability for linear neutral type systems

The problem of strong stabilizability of linear systems of neutral type is investigated. We are interested in the case when the system has an infinite sequence of eigenvalues with vanishing real parts. This is the case when the main part of the neutral equation is not assumed to be stable in the classical sense. We discuss the notion of regular strong stabilizability and present an approach to stabilize the system by regular linear controls. The method covers the case of multivariable control and is essentially based on the idea of infinite-dimensional pole assignment proposed in [G.M. Sklyar, A.V. Rezounenko, A theorem on the strong asymptotic stability and determination of stabilizing controls, C. R. Acad. Sci. Paris Ser. I Math. 333 (8) (2001) 807-812]. Our approach is based on the recent results on the Riesz basis of invariant finite-dimensional subspaces and strong stability for neutral type systems presented in [R. Rabah, G.M. Sklyar, A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations 214 (2) (2005) 391-428].     

Sturmian maximizing measures for the piecewise-linear cosine family

Let T be the angle-doubling map on the circle $\mathbb{T}$ , and consider the 1-parameter family of piecewise-linear cosine functions $f_\theta :\mathbb{T} \to \mathbb{R}$ , defined by $f_\theta (x) = 1 - 4d_\mathbb{T} (x,\theta )$ . We identify the maximizing T-invariant measures for this family: for each ?? the f _?? -maximizing measure is unique and Sturmian (i.e. with support contained in some closed semi-circle). For rational p/q, we give an explicit formula for the set of functions in the family whose maximizing measure is the Sturmian measure of rotation number p/q. This allows us to analyse the variation with ?? of the maximum ergodic average for f _?? .     

On the equality of two plane partition correspondences

The study of column-strict plane partitions and Young tableax has spawned numerous constructive correspondences. Among these are correspondences found in the work of Bender and Knuth that send one column-strict plane partition to another, causing a specified permutation on the numbers of parts of a given size. Another correspondence, created by Schützenberger to act on standard Young tableaux and defined in an entirely different manner. has intimate connections with the Robinson-Schensted algorithm. In this paper, these correspondences are generalized to skew column-strict plane partitions and certain of their basic properties are considered. In particular, it is shown that the correspondence of Schützenberger can be considered a special case of the Bender-Knuth correspondences.     

Perturbing the boundary conditions of the generator of a cosine family

Let A be a densely defined, closed linear operator (which we shall call maximal operator) with domain D(A) on a Banach space X and consider closed linear operators L:D(A)???X and ??:D(A)???X (where ?X is another Banach space called boundary space). Putting conditions on L and ??, we show that the second order abstract Cauchy problem for the operator A _?? with A _?? u=Au and domain D(A _??):={u??D(A):Lu=??u} is well-posed and thus it generates a cosine operator function on the Banach space X.     

On the family of linear extensions of a partial order

Given a partial order P defined on a finite set X, a binary relation ?_P may be defined on X by setting x ?_Py for elements x and y in X just when more linear extensions L of P on X have xLy than yLx. A linear extension L of P on X is a linear order on X with P ? L. There exist partial orders P such that ?_P includes cycles. Thus, in a voting situation in which voters are unanimous in their preferences on the pairs in P and express all possible linearly ordered preferences on X which are consistent with P, with no two voters having the same preference order, strict simple majorities as given by ?_P can cycle.     

A Birkhoff-James cosine function for normed linear spaces

Aequationes mathematicae - The cosine function is a classical tool for measuring angles in inner product spaces, and it has various extensions to normed linear spaces. In this paper, we investigate...     

On the spectrum of cosine operator functions

In this paper we characterize the spectrum of strongly continuous cosine functions, defined in a Hilbert space, in terms of properties of the infinitesimal generator.     

More on regular linear spaces

We extend the enumeration of regular linear spaces in 1 to at most 19 points. In addition, one of the 5 missing cases in the previous list is settled. The number of regular linear spaces of type (15|2¹⁵,3³⁰) is 10,177,328. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

On the geometry of point correspondences of three conformal spaces

Point correspondences of three conformal spaces are studied on the base of G. F. Laptev??s invariant methods. We establish the basic equations and geometrical objects of the point correspondences in question. We construct invariant normalizations of the spaces, single out the basic tensors of the correspondences, establish a connection of the correspondences with the theory of multidimensional 3-webs, and find the torsion and the curvature tensors of a point correspondence. For a series of particular cases, we prove existence theorems.     

On Shimura, Shintani and Eichler-Zagier correspondences

In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.