Mathematics of CLIFFORD - A Maple package for Clifford and Grassmann algebras

CLIFFORD performs various computations in Grassmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: 'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.Comment: 24 pages, update contains new material included in published versio

Idempotents of Clifford Algebras

A classification of idempotents in Clifford algebras C(p,q) is presented. It is shown that using isomorphisms between Clifford algebras C(p,q) and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one sided ideals in Clifford algebras. Some low dimensional examples are discussed

Finding Octonionic Eigenvectors Using Mathematica

The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The nonassociativity of the octonions makes computations tricky, and all of these results were first obtained via brute force (but exact) Mathematica computations. Some of them, such as the computation of real eigenvalues, have subsequently been implemented more elegantly; others have not. We describe here the use of Mathematica in analyzing this problem, and in particular its use in proving a generalized orthogonality property for which no other proof is known.Comment: LaTeX2e, 22 pages, 8 PS figures (uses included PS prolog; needs elsart.cls and one of epsffig, epsf, graphicx

Clifford geometric parameterization of inequivalent vacua

We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras --as Clifford algebras-- by different filtrations resp. induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C^*-algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non-statistical, non-definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U(2)-symmetry producing a gap-equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov-Valatin transformations is given.Comment: Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.

Characterization Of Membrane Associated Mucins In Ocular Surface Disease

Mucins on the ocular surface are found in the tear film and are attached to corneal and conjunctival epithelial cells on the eye. The bulbar conjunctiva of the ocular surface can be divided into four anatomical regions: temporal, superior, nasal, and inferior. The palpebral conjunctiva is the epithelial layer of the inner surfaces of the upper and lower eyelids. In the tears, mucins provide lubrication of the ocular surface through formation of a hydrophilic gel. The primary mucin in the tear film is MUC5AC which is secreted by goblet cells that are located in varying densities within the bulbar conjunctiva. On the apical surface of the eye, membrane associated mucins (MAMs) form a protective barrier known as the glycocalyx. The highly O-glycosylated MAMs in the glycocalyx create a hydrophilic surface that attracts the tear film. The MAMs identified on the human ocular surface in the superficial cell layers, represented by “MUC” followed by a number representing order of discovery, are MUC1, MUC4, and MUC16. These mucins are expressed and secreted by the corneal and conjunctival epithelial cells. Galectin-3, a β-galactoside binding lectin, recognizes the carbohydrate galactose found on MUC1 and MUC16 and colocalizes with these MAMs in the glycocalyx. Galectin-3 is an essential component of the glycocalyx as without it, barrier function is impeded. In dry eye disease, inflammation is a core mechanism that can have negative consequences on the ocular surface. Chronic inflammation can lead to damage to the epithelial cells and tear film instability resulting in poor ocular surface hydration. Reduction of goblet cells and reduced MUC5AC are potential contributing factors to dry eye disease. The glycocalyx and glycosylation of the MAMs may also be negatively impacted such that the glycocalyx becomes disrupted. The primary purpose of this research was to investigate expression of MAMs in the regions of the bulbar conjunctiva and the palpebral conjunctiva of the upper eyelid. In addition, the secondary goal of this research was to develop an affinity assay for in vivo use on human tear samples that would enable researchers to evaluate the affinity of the interaction of MUC16 and galectin-3