We will study applications of numerical methods in Clifford algebras in ${ℝ}^4$, in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in ${ℝ}^4$. In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients can...

We survey results related to the Kreiss Matrix Theorem, especially examining extensions of this theorem to Banach space and Hilbert space. The survey includes recent and established results together with proofs of many of the interesting facts concerning the Kreiss Matrix Theorem.

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be designed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take...

The systems of an arbitrary number of linear inequalities OVer a real locally convex space have been classified in three classes, namely: consistent, weakly inconsistent and strongly inconsistent, i.e. having ordinary solutions, weak solutions or notsolutions respectively. In this paper, the third type is divided in two classes: strict-strongly and quasi-strongly inconsistent and is given a topology over a quotient space of the set of systems over finite- dimensional spaces, that yields a set of...

An algorithm is given to decompose an automorphism of a finite vector space over ℤ₂ into a product of transvections. The procedure uses partitions of the indexing set of a redundant base. With respect to tents, i.e. finite ℤ₂-representations generated by a redundant base, this is a decomposition into base changes.

Fiedler and Markham (1994) proved $${\left(\frac{\det \widehat{H}}{k}\right)}^k\ge \det H,$$ where $H={\left(H_{ij}\right)}_{i,j=1}^n$ is a positive semidefinite matrix partitioned into $n\times n$ blocks with each block $k\times k$ and $\widehat{H}={\left(\mathrm{tr}{H}_{ij}\right)}_{i,j=1}^n$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $$\det (I_n+\widehat{H})\ge \det {(I_{nk}+kH)}^{1/k}.$$

Our concern is with the group of conformal transformations of a finite-dimensional real quadratic space of signature (p,q), that is one that is isomorphic to ${ℝ}^{p,q}$, the real vector space ${ℝ}^{p+q}$, furnished with the quadratic form $x^{\left(2\right)}=x·x=-x_1^2-x_2^2-...-x_p^2+x_{p+1}^2+...+x_{p+q}^2$, and especially with a description of this group that involves Clifford algebras.

The construction of a well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition is proposed. A suitable parametrix is obtained by using a new unknown and an approximation of the transparency condition. We prove the well-posedness of the equation for any wavenumber. Finally, some numerical comparisons with well-tried method prove the efficiency of the new formulation.