Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics provides diverse tools to characterize data objects. In this work, we study a specific tool known as Euler characteristic (EC). The EC general, low-dimensional, and interpretabl...

Two-dimensional reactive Navier-Stokes equations are solved using a simple implicit Beam-Warming finite difference scheme. Comparisons of the detonation wave solutions of reactive Euler equations and reactive Navier-Stokes equations show that physical diffusion is important at high resolution when the numerical diffusion becomes negligible. Hence, for accurate detonation wave solutions it is ne...

We study the integrals of type I(a) = ∫ On ∏ u aij ij du, depending on a matrix a ∈Mp×q(N), whose exact computation is an open problem. Our results are as follows: (1) an extension of the “elementary expansion” formula from the case a ∈M2×q(2N) to the general case a ∈Mp×q(N), (2) the construction of the “best algebraic normalization” of I(a), in the case a ∈M2×q(N), (3) an explicit formula for ...

In this article we present a feedback control law which gives exponential convergence of a nonholonomic underwater vehicle to a constant desired connguration. This is achieved using a piecewise smooth feedback control law which is based on previous work on the control of nonholonomic mobile robots in the plane. The kinematic model of the underwater vehicle is given in SE(3) by homogeneous trans...

Several applications the tracking of attitude and position of a body based on velocity data. It is tempting to use direction cosine matrices (DCM), for example, to track attitude based on angular velocity data, and to integrate the linear velocity data separately in a suitable frame. In this chapter we make the case for using bivectors as the attitude tracking method of choice since several fea...

Some spectral properties of the transition matrix of an oriented graph indicate the preconditioning of Euler-Richardson (ER) iterative scheme as a good way to compute efficiently the vertexrank vector associated with such graph. We choose the preconditioner from an algebra U of matrices, thereby obtaining an ERU method, and we observe that ERU can outperform ER in terms of rate of convergence. ...

Wherever anisotropic behaviour in physical measurements or models is encountered matrices provide adequate means to describe this anisotropy. Prominent examples are the di usion tensor magnetic resonance imaging in medical imaging or the stress tensor in civil engineering. As most measured data these matrix-valued data are also polluted by noise and require restoration. The restoration of scala...

One of the simplest and, at the same time, most important invariants of a topological space is the Euler characteristic. A generalization of the notion of the Euler characteristic to the equivariant setting, that is, to spaces with an action of a group (say, finite) is far from unique. An equivariant analogue of the Euler characteristic can be defined as an element of the ring of representation...

Nonlocally related systems for the Euler and Lagrange systems of twodimensional dynamical nonlinear elasticity are constructed. Using the continuity equation, i.e., conservation of mass of the Euler system to represent the density and Eulerian velocity components as the curl of a potential vector, one obtains the Euler potential system that is nonlocally related to the Euler system. It is shown...

Recently the new q-Euler numbers and polynomials related to Frobenius-Euler numbers and polynomials are constructed by Kim (see[3]). In this paper, we study the generalized q-Euler numbers and polynomials attached to χ related to the new q-Euler numbers and polynomials which is constructed in [3]. Finally, we will derive some interesting congruence on the generalized q-Euler numbers and polynom...