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Admissibility for quasiregular representations of exponential solvable Lie groups

100%

Oussa V.

Colloquium Mathematicum

2013

tom 131

nr 2

241-264

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra 𝔫 of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra 𝔥. Furthermore, assume that the linear adjoint action of 𝔥 on 𝔫 is diagonalizable with non-purely imaginary eigenvalues. Let $τ = Ind_{H}^{N ⋊ H} 1$. We obtain an explicit direct integral decomposition for τ, including a description of the spectrum as a submanifold of (𝔫+𝔥)*, and a formula for the multiplicity function of the unitary irreducible representations occurring in the direct integral. Finally, we completely settle the admissibility question for τ. In fact, we show that if G = N ⋊ H is unimodular, then τ is never admissible, and if G is non-unimodular, then τ is admissible if and only if the intersection of H and the center of G is equal to the identity of the group. The motivation of this work is to contribute to the general theory of admissibility, and also to shed some light on the existence of continuous wavelets on non-commutative connected nilpotent Lie groups.

Optimal solutions for a model of tumor anti-angiogenesis with a penalty on the cost of treatment

51%

Ledzewicz U., Oussa V., Schättler H.

Applicationes Mathematicae

2009

tom 36

nr 3

295-312

The scheduling of angiogenic inhibitors to control a vascularized tumor is analyzed as an optimal control problem for a mathematical model that was developed and biologically validated by Hahnfeldt et al. [Cancer Res. 59 (1999)]. Two formulations of the problem are considered. In the first one the primary tumor volume is minimized for a given amount of angiogenic inhibitors to be administered, while a balance between tumor reduction and the total amount of angiogenic inhibitors given is minimized in the second formulation. The optimal solutions to both problems are presented and compared.

Ograniczanie wyników

1 Applicationes Mathematicae

1 Colloquium Mathematicum

2 Oussa V.

1 Ledzewicz U.

1 Schättler H.

1 2013

1 2009

Wyniki wyszukiwania

Admissibility for quasiregular representations of exponential solvable Lie groups

Optimal solutions for a model of tumor anti-angiogenesis with a penalty on the cost of treatment