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Finite-tight sets

100%

Florescu L.

Open Mathematics

2007

tom 5

nr 4

619-638

We introduce two notions of tightness for a set of measurable functions - the finite-tightness and the Jordan finite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune-Valadier’s theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness locates the great growths of a set of measurable mappings on a finite family of sets of small measure. In the Euclidean case, the Jordan finite-tight sets form a subclass of finite-tight sets for which the finite family of sets of small measure is composed by d-dimensional intervals. The main result affirms that each tight set H ⊆ W 1,1 for which the set of the gradients ∇H is a Jordan finite-tight set is relatively compact in measure. This result offers very good conditions to use fiber product lemma for obtaining a relaxed lower semicontinuity condition.

Results in Q-measure

63%

Jisha C. R.

Mathematica Applicanda

2022

tom Vol. 50, no. 2

183--215

This paper introduces the notion of a generalized measure for a sequence of functions with oscillation and concentration effects. This measure is constructed by averaging the sequence of Borel measurable functions using singular or regular perturbations. In this way, the generalized limits of such sequences are conceptualized by enlarging the space of functions to measure spaces. It is a modification of the Young measure. This modified measure was termed a Q-measure. It can be difficult to determine the Young measure for a broad function. The Q-measure can be easily calculated for particular functions. This is one of the advantages of this study. As an application of the measure, we can define another weaker type of Monotone convergence theorem, the Lebesgue-dominated convergent theorem. A notion of average for underlying sequences to define the Q-measure is given, as also its application in signal analysis and atmospheric sciences.

W tym artykule autor wprowadza nową miarę, którą nazywa miarą Q, reprezentującą słabą∗ granicę barycentrum ciągu funkcji borelowskich. Omawia niektóre wyniki zwi¡zane z tą miarą, co jest pomocne przy wyznaczaniu miary Q dla poszczególnych typów funkcji. Ponadto omówiono zastosowanie koncepcji średniej w analizie sygnałów i naukach o atmosferze.

Q-functional applications

63%

Jisha C. R.

Mathematica Applicanda

2022

tom Vol. 50, no. 2

217--233

The Q-measure indicate a weak∗ limit of the barycenter of a sequence of Borel measurable functions. In this paper, we will look only at Q-functional. Q-functional is defined by Q- measure, it is useful in the field of optimization. Computational results for Q-functional are presented and compared with Young functional. The obtained analytical results demonstrate relative error in Q-functional is lesser compared to Young functional.

Miara Q wyznacza słabą∗ granicę barycentrum ciągu funkcji borelowskich. W tym artykule przyjrzymy się tylko funkcjonałom Q. Q-funkcjonalność jest definiowana przez miarę Q i jest przydatna w zastosowaniu do zadań optymalizacji. Przedstawiono wyniki obliczeń dla funkcjonału Q i porównano je z funkcjonałem Younga. Otrzymane wyniki analityczne pokazują, że błąd względny w funkcjonale Q jest mniejszy w porównaniu z funkcjonałem Younga.

Three boundary value problems for second order differential inclusions in Banach spaces

51%

Azzam D. , Castaing C. , Thibault L.

Control and Cybernetics

2002

tom Vol. 31, no 3

659-693

The paper studies, in the context of Banach spaces, the problem of three boundary conditions for both second order differential inclusions and second order ordinary differential equations. The results are obtained in several new settings of Sobolev-type spaces involving Bochner and Pettis integrals. Some classes of second order multivalued evolution equations associated with m-accretive operators are also considered. Applications to some control problems are provided with the help of narrow convergence for Young measures.