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Covariantization of quantized calculi over quantum groups

Seyed Ebrahim Akrami, Shervin Farzi (2020)

Mathematica Bohemica

We introduce a method for construction of a covariant differential calculus over a Hopf algebra A from a quantized calculus d a = [ D , a ] , a ∈ A , where D is a candidate for a Dirac operator for A . We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra A ∘ . We apply this method to the Dirac operator for the quantum SL ( 2 ) given by S. Majid. We find that the differential calculus obtained by our method is the...

Covering lemmas and BMO estimates for eigenfunctions on Riemannian surfaces.

Guozhen Lu (1991)

Revista Matemática Iberoamericana

The principal aim of this note is to prove a covering Lemma in R2. As an application of this covering lemma, we can prove the BMO estimates for eigenfunctions on two-dimensional Riemannian manifolds (M2, g). We will get the upper bound estimate for || log |u| ||BMO, where u is the solution to Δu + λu = 0, for λ > 1 and Δ is the Laplacian on (M2, g). A covering lemma on homogeneous spaces is also obtained in this note.

Cramér's formula for Heisenberg manifolds

Mahta Khosravi, John A. Toth (2005)

Annales de l'institut Fourier

Let R ( λ ) be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that ∫ 1 T | R ( t ) | 2 d t = c T 5 2 + O δ ( T 9 4 + δ ) , where c is a specific nonzero constant and δ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that R ( t ) = O δ ( t 3 4 + δ ) .The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the 2 n + 1 -dimensional case.

Critical configurations of planar robot arms

Giorgi Khimshiashvili, Gaiane Panina, Dirk Siersma, Alena Zhukova (2013)

Open Mathematics

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive...

Critical points of asymptotically quadratic functions

Michal Fečkan (1995)

Annales Polonici Mathematici

Existence results for critical points of asymptotically quadratic functions defined on Hilbert spaces are studied by using Morse-Conley index and pseudomonotone mappings. Applications to differential equations are given.

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