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Set-valued random differential equations in Banach space

Mariusz Michta (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider the problem of the existence of solutions of the random set-valued equation: (I) D H X t = F ( t , X t ) P . 1 , t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space K c ( E ) , of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

Seven Proofs for the Subadditivity of Expected Shortfall

Paul Embrechts, Ruodu Wang (2015)

Dependence Modeling

Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are...

Several results on set-valued possibilistic distributions

Ivan Kramosil, Milan Daniel (2015)

Kybernetika

When proposing and processing uncertainty decision-making algorithms of various kinds and purposes, we more and more often meet probability distributions ascribing non-numerical uncertainty degrees to random events. The reason is that we have to process systems of uncertainties for which the classical conditions like σ -additivity or linear ordering of values are too restrictive to define sufficiently closely the nature of uncertainty we would like to specify and process. In cases of non-numerical...

Shift inequalities of Gaussian type and norms of barycentres

F. Barthe, D. Cordero-Erausquin, M. Fradelizi (2001)

Studia Mathematica

We derive the equivalence of different forms of Gaussian type shift inequalities. This completes previous results by Bobkov. Our argument strongly relies on the Gaussian model for which we give a geometric approach in terms of norms of barycentres. Similar inequalities hold in the discrete setting; they improve the known results on the so-called isodiametral problem for the discrete cube. The study of norms of barycentres for subsets of convex bodies completes the exposition.

Shift invariant measures and simple spectrum

A. Kłopotowski, M. Nadkarni (2000)

Colloquium Mathematicae

We consider some descriptive properties of supports of shift invariant measures on ℂ ℤ under the assumption that the closed linear span (in L 2 ) of the co-ordinate functions on ℂ ℤ is all of L 2 .

Sigma-finiteness and regularity of generalized Radon measures.

J. Fernández Novoa (1990)

Collectanea Mathematica

We extend some known sigma-finiteness and regularity results for (locally finite) Radon measures to locally sigma-finite or locally moderated Radon measures of type (H), and we obtain other new ones. The main result states that the regularity and the sigma-finiteness are equivalent for alllocally moderated, diffused, Radon measures of type (H) in a T1 topological space which is either weakly metacompact or paralindelöf (resp. metalindelöf) and has a concassage of Lindelöf (resp. separable) subsets....

Simple systems are disjoint from Gaussian systems

Andrés del Junco, Mariusz Lemańczyk (1999)

Studia Mathematica

We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...

Simple topological measures and a lifting problem

Finn F. Knudsen (2008)

Fundamenta Mathematicae

We state a certain lifting conjecture and prove it in the case of a torus. From this result we are able to construct a connected dense subset of the space of intrinsic simple topological measures on the torus, consisting of push forwards of compactly supported generalized point-measures on the universal covering space. Combining this result with an observation of Johansen and Rustad, we conclude that the space of simple topological measures on a torus is connected.

Singular distributions, dimension of support, and symmetry of Fourier transform

Gady Kozma, Alexander Olevskiĭ (2013)

Annales de l’institut Fourier

We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to l 2 .We also give examples of non-symmetry...

Singular integrals and rectifiability.

Pertti Mattila (2002)

Publicacions Matemàtiques

We shall discuss singular integrals on lower dimensional subsets of Rn. A survey of this topic was given in [M4]. The first part of this paper gives a quick review of some results discussed in [M4] and a survey of some newer results and open problems. In the second part we prove some results on the Riesz kernels in Rn. As far as I know, they have not been explicitly stated and proved, but they are very closely related to some earlier results and methods.[Proceedings of the 6th International Conference...

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