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On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

A. Pajor, L. Pastur (2009)

Studia Mathematica

We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix H ( 0 ) and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of H ( 0 ) and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...

On the lower semicontinuous quasiconvex envelope for unbounded integrands (I)

Marcus Wagner (2009)

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K n m instead of the whole space n m as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous...

On the measurability of sets of pairs of intersecting nonisotropic straight lines of type beta in the simply isotropic space

Adrijan Varbanov Borisov, Margarita Georgieva Spirova (2009)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

The measurable sets of pairs of intersecting non-isotropic straight lines of type β and the corresponding densities with respect to the group of general similitudes and some its subgroups are described. Also some Crofton-type formulas are presented.

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