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Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HMk), then Y ·X and Y/X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.

Let F be a closed subset of ℝn and let P(x) denote the metric projection (closest point mapping) of x ∈ ℝn onto F in lp-norm. A classical result of Asplund states that P is (Fréchet) differentiable almost everywhere (a.e.) in ℝn in the Euclidean case p = 2. We consider the case 2 < p < ∞ and prove that the ith component Pi(x) of P(x) is differentiable a.e. if Pi(x) 6= xi and satisfies Hölder condition of order 1/(p−1) if Pi(x) = xi.

We develop a potential theory for a Riesz type kernel in a homogeneous space and characterize the compact sets K with capacity zero as the sets K for which every continous function f on K is the restriction to K of a continuous potential U^σf_k of an absolutely continuous measure σ _f supported in an arbitrarily small neighbourhood of K. The measure σ _f can be choosen as a suitable restriction of a single measure σ that only depends on the set K and the kernel k.

Mathematical theorems and their proofs are often described as being beautyful, natural or explanatory. This paper treats the last case  and discusses explanatory versus non explanatory proofs using  a convergence theorem for positive series due to Kummer, called Kummer's test. This example was studied by Pringsheim already in the beginning in the last century and has recently been discussed by Hafner and Mancuso in relation to Steiner's criteria for an explanatory proof. We provide a mathematical analysis of Kummer's test and give a proof that we claim is explanatory in this sense. Besides this we answer a number of questions that are frequently asked about the test.

We study classes of multivariate (bivariate) random variables with hyperbolically completely monotone densities (MVHCM) and prove that these classes are closed with respect to the Laplace transform. We then use this result to define new classes of bivariate generalized gamma convolutions (BVGGC_L) that contain Bondesson's class BVGGC in the strong sense.

Everyn nonconstant polynomial with complex coefficients has at least one root in the field of complex numbers

A set E in a space X is called a polar set in X, relative to a kernel k(x; y), if thereis a nonnegative measure in X such that the potential Uk(x) = ∞ precisely when x ∈ E. Polarsets have been characterized in various classical cases as G-sets (countable intersections of opensets) with capacity zero. We characterize polar sets in a homogeneous space (X; d; ) for severalclasses of kernels k(x; y), among them the Riesz -kernels and logarithmic Riesz kernels. The latercase seems to be new even in Rⁿ.

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces L^p(R^d) and multi-parameter Morrey spaces L^p λ(R^d), where R^d = Rⁿ¹ × Rⁿ² ×···× R^nk , λ = (λ₁,...,λ_k) and 0 < λi ≤ n_i, 1 ≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.

Gram-Schmidts algoritm ger en parvis ortogonal mängd från en linjärt oberoende mängd i ett allmännt vektorrum med inre produkt. Vi ger en algoritm som konstruerar en mängd där vektorerna parvis har en godtycklig på förhand given vinkel, i alla fall där detta är möjligt.

The Gram-Schmidt algorithm produces a pairwise ortogonal set of vectors from a linearly independent set in a general vector space with an inner product. We give an algorithm that produces vectors that have pairwise the same angle, in all cases where this is possible.

A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ _n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.