Almost fully decomposable infinite rank lattices over orders

Let Λ be an order over a Dedekind domain R with quotient field K. An object of , the category of R-projective Λ-modules, is said to be fully decomposable if it admits a decomposition into (finitely generated) Λ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105-128], we give a necessary and sufficient criterion for R-orders Λ in a separable K algebra A with the property that every is fully decomposable. In the present paper, we assume that is separable, but that the p-adic completion A_p is not semisimple for at least one . We show that there exists an , such that KL admits a decomposition KL=M₀⊕M₁ with finitely generated, where L∩M₁ is fully decomposable, but L itself is not fully decomposable.     

The self-similar solutions of the Tricomi-type equations

This paper analyses the properties of the family of self-similar solutions of the generalized Tricomi equation in the domain by considering initial conditions on the functions and their derivatives, posed as the Cauchy problem with homogeneous initial data. For specific values of the power k ( = 1/2 or = 3/2) and n = 1 this problem has applications in the aerodynamics of airfoils operating in transonic flows of perfect or dense gases, respectively. An integral transformation is suggested and used to represent the solutions of the Cauchy problem with homogeneous initial functions in terms of fundamental solutions of the classical wave equation (the case k = 0). Then the Cauchy problem with homogeneous initial functions for the wave equation in is solved. These results are used to derive estimates of the upper bound for solutions’ size and to obtain the asymptotics for self-similar solutions of the wave equation and of the Tricomi-type equation in the neighbourhood of their light cones.     

The evaluation subgroup of a fibre inclusion

Let be a fibration of simply connected CW complexes of finite type with classifying map . We study the evaluation subgroup Gn(E,X;j) of the fibre inclusion as an invariant of the fibre-homotopy type of ξ. For spherical fibrations, we show the evaluation subgroup may be expressed as an extension of the Gottlieb group of the fibre sphere provided the classifying map h induces the trivial map on homotopy groups. We extend this result after rationalization: We show that the decomposition G_∗(E,X;j)⊗Q=(G_∗(X)⊗Q)⊕(π_∗(B)⊗Q) is equivalent to the condition Q(h_?)=0.     

Existence of limit cycles for Liénard-type systems with p-Laplacian

The purpose of this paper is to give sufficient conditions under which an equivalent system to the equation has at least one stable limit cycle, where is the one-dimensional p-Laplacian. The main results are proved by means of phase plane analysis with the Poincaré-Bendixson theorem. Sufficient conditions are also given for the origin to be unstable and for all solutions to be bounded in the future. Jitsuro Sugie: Supported in part by Grant-in-Aid for Scientific Research 16540152     

A Class of Calogero Type Reductions of Free Motion on a Simple Lie Group

The reductions of the free geodesic motion on a non-compact simple Lie group G based on the G ₊ × G ₊ symmetry given by left- and right-multiplications for a maximal compact subgroup are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the ‘spin’ degrees of freedom are absent and we obtain the standard BC _n Sutherland model with three independent coupling constants from SU(n + 1,n) and from SU(n,n). This generalization of the Olshanetsky-Perelomov derivation of the BC _n model with two independent coupling constants from the geodesics on G/G ₊ with G = SU(n + 1,n) relies on fixing the right-handed momentum to a non-zero character of G ₊. The reductions considered permit further generalizations and work at the quantized level, too, for non-compact as well as for compact G.      

Lieb–Thirring Bounds for Complex Jacobi Matrices

We obtain various versions of classical Lieb–Thirring bounds for one- and multi-dimensional complex Jacobi matrices. Our method is based on Fan–Mirski Lemma and seems to be fairly general.      

Influence of V and Mo overlayer on magneto-optical Kerr effect in ultrathin Co films

A method for characterization of sub-nanometer thick Co/V and Co/Mo interfaces is proposed that uses magneto-optical ellipsometry. Both the polar Kerr rotation and ellipticity are fitted simultaneously to different models of interface layer. The magneto-optical data are measured for varying thicknesses of the cobalt layer and overlayer by scanning of a laser beam over the samples with two orthogonal wedges. Decrease of magneto-optical effect at both interfaces Co/V and Co/Mo were observed, which corresponds to interface layers of thicknesses ranging from one to two monoatomic layers. In the case of vanadium, the interface layer is sharper and can be explained either by reduced magnetic moment of cobalt, or by anti-parallel magnetic moment of vanadium near the Co/V interface.     

A Remark on the Linearized Stability of Positive Solutions for Systems Involving the p-Laplacian

The aim of this paper is to prove some stability result for nonlinear elliptic systems of the form where Δ_p denotes the p-Laplacian operator defined by Δ_pz = div(|∇ z|^p-2∇ z); p > 2, Ω is a bounded domain in R^N (N > 1) with smooth boundary where with h = 1 when α = 1, λ is a positive parameter and f,g are C² functin on [0,∞) × [0,∞). We prove stability and instability results of positive stationary solutions under various choices of f and g.     

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Passive linear systems τ = have their transfer function in the Schur class S . Using a parametrization of contractive block operators the transfer function is connected to the Sz.-Nagy–Foiaş characteristic function of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions and . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific Research N.W.O. (B 61-553). Received: December 22, 2006. Revised: February 6, 2007.