When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))⊂z∈C;ℜz≤−α for all t≥t0 and some α>0 is sufficient for uniform asymptotic stability of the linear time-varying system x˙=A(t)x. We prove that this class contains as a proper subset the matrix functions with the values in the special orthogonal group SO(n).

Interpolation on the special orthogonal group with high-dimensional Kuramoto model

The construction of smooth interpolation trajectories in different non-Euclidean spaces finds application in robotics, computer graphics, and many other engineering fields. This paper proposes a method for generating interpolation trajectories on the special orthogonal group SO(3), called the rotation group. Our method is based on a high-dimensional generalization of the Kuramoto model which is a well-known mathematical description of self-organization in large populations of coupled oscillators. We present the method through several simulations and visualize each simulation as trajectories on unit spheres S2. In addition, we applied our method to the specific problem of object rotation interpolation.

Chirality in Geometric Algebra

We define chirality in the context of chiral algebra. We show that it coincides with the more general chirality definition that appears in the literature, which does not require the existence of a quadratic space. Neither matrix representation of the orthogonal group nor complex numbers are used.

Approximating Correlation Matrices Using Stochastic Lie Group Methods

Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.

Generalization of Geyer’s commutation relations with respect to the orthogonal group in even dimensions

A connection between the deformed Duffin–Kemmer–Petiau (DKP) algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators $$a^{\pm }_{k}$$ a k ± and for an additional operator $$a_{0}$$ a 0 obeying para-Fermi statistics of order 2 based on the Lie algebra $${\mathfrak {s}}{\mathfrak {o}}(2M+2)$$ s o ( 2 M + 2 ) is established. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The representation for the operator $$a_{0}$$ a 0 in terms of generators of the orthogonal group SO(2M) correctly reproducing action of this operator on the state vectors of Fock space is obtained. A connection of the Geyer operator $$a_{0}^{2}$$ a 0 2 with the operator of so-called G-parity and with the CPT- operator $${\hat{\eta }}_{5}$$ η ^ 5 of the DKP-theory is established. In a para-Grassmann algebra a noncommutative, associative star product $$*$$ ∗ (the Moyal product) as a direct generalization of the star product in the algebra of Grassmann numbers is introduced. Two independent approaches to the calculation of the Moyal product $$*$$ ∗ are considered. It is shown that in calculating the matrix elements in the basis of parafermion coherent states of various operator expressions it should be taken into account constantly that we work in the so-called Ohnuki and Kamefuchi’s generalized state-vector space $${\mathfrak {U}}_{\;G}$$ U G , whose state vectors include para-Grassmann numbers $$\xi _{k}$$ ξ k in their definition, instead of the standard state-vector space $${\mathfrak {U}}$$ U (the Fock space).