It is known that the Williamson construction for Hadamard matrices can be generalized to constructions using sums of tensor products. This paper describes a specific construction using real monomial representations of Clifford algebras, and its connection with graphs of amicability and anti-amicability. It is proven that this construction works for all such representations where the order of th...

We consider a straightforward extension of the 4-dimensional spacetime M4 to the space of extended events associated with strings/branes, corresponding to points, lines, areas, 3-volumes, and 4-volumes in M4. All those objects can be elegantly represented by the Clifford numbers X ≡ xγA ≡ x γa1...ar , r = 0, 1, 2, 3, 4. This leads to the concept of the so-called Clifford space C, a 16-dimension...

We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if −1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, t...

We review a rough scheme of quantum mechanics using the Clifford algebra. Following the steps previously published in a paper by another author [31], we demonstrate that quantum interference arises in a Clifford algebraic formulation of quantum mechanics. In 1932 J. von Neumann showed that projection operators and, in particular, quantum density matrices can be interpreted as logical statements...

, 773 (2006); 314 Science et al. Clifford B. Saper, Body Temperature BIOMEDICINE: Enhanced: Life, the Universe, and www.sciencemag.org (this information is current as of February 7, 2007 ): The following resources related to this article are available online at http://www.sciencemag.org/cgi/content/full/314/5800/773 version of this article at: including high-resolution figures, can be found in ...

In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained complex matrices corresponding to basis of quaternions. Also, investigated features matrices. get Pauli Then, have shown that produced these is isomorphic Clifford Cl(E_αβ^3) space E_αβ^3.
 Finally, studied relations among symplecti...

Mathias Soeken, D. Michael Miller, and Rolf Drechsler Institute of Computer Science, University of Bremen, Germany Department of Computer Science, University of Victoria, BC, Canada The Pauli matrices are a set of three 2×2 complex Hermitian, unitary matrices. In this article, we investigate the relationships between certain roots of the Pauli matrices and how gates implementing those roots are...

In this paper we give some general results on the non-splitextension group $overline{G}_{n} = 2^{2n}{^{cdot}}Sp(2n,2), ngeq2.$ We then focus on the group $overline{G}_{4} =2^{8}{^{cdot}}Sp(8,2).$ We construct $overline{G}_{4}$ as apermutation group acting on 512 points. The conjugacy classes aredetermined using the coset analysis technique. Then we determine theinertia factor groups and Fischer...