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On the isoperimetry of graphs with many ends

Christophe Pittet (1998)

Colloquium Mathematicae

Let X be a connected graph with uniformly bounded degree. We show that if there is a radius r such that, by removing from X any ball of radius r, we get at least three unbounded connected components, then X satisfies a strong isoperimetric inequality. In particular, the non-reduced l 2 -cohomology of X coincides with the reduced l 2 -cohomology of X and is of uncountable dimension. (Those facts are well known when X is the Cayley graph of a finitely generated group with infinitely many ends.)

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in n i , i ≤ m, then for every F in the Grassmannian G N , n , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, π F ( μ μ ) , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

On the lattice of polynomials with integer coefficients: the covering radius in L p ( 0 , 1 )

Wojciech Banaszczyk, Artur Lipnicki (2015)

Annales Polonici Mathematici

The paper deals with the approximation by polynomials with integer coefficients in L p ( 0 , 1 ) , 1 ≤ p ≤ ∞. Let P n , r be the space of polynomials of degree ≤ n which are divisible by the polynomial x r ( 1 - x ) r , r ≥ 0, and let P n , r P n , r be the set of polynomials with integer coefficients. Let μ ( P n , r ; L p ) be the maximal distance of elements of P n , r from P n , r in L p ( 0 , 1 ) . We give rather precise quantitative estimates of μ ( P n , r ; L ) for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of μ ( P n , r ; L p ) for p ≠ 2. It follows that μ ( P n , r ; L p ) n - 2 r - 2 / p as n → ∞. The results partially...

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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