Analysis on generalized Clifford algebras

In this article, we study the analysis related to generalized Clifford algebras Cn(a), where a is a non-zero vector. If {e1, . . ., en} is an orthonormal basis, the multiplication is defined by relations (Equation presented) for aj = ej · a. The case a = 0 corresponds to the classical Clifford algebra. We define the Dirac operator as usual by D = Σj ej∂xj and define regular functions as its null solution. We first study the algebraic properties of the algebra. Then we prove the basic formulas for the Dirac operator and study the properties of regular functions.Peer reviewe

Hypermonogenic Functions of Two Vector Variables

In this paper we introduce the modified Dirac operators and , where is differentiable function, and is the Clifford algebra generated by the basis vectors of . We look for solutions of the system , where the first and third variables are invariant under rotations. These functions are called -hypermonogenic functions. We discuss about axially symmetric functions with respect to the symmetric group . Some examples of axially symmetric -hypermonogenic functions generated by homogeneous functions and hypergeometric functions are presented.Peer reviewe

Homogeneous (α, k) -polynomial solutions of the fractional riesz system in hyperbolic space

In this paper we study the fractional analogous of the Laplace-Beltrami equation and the Riesz system studied previously by H. Leutwiler , in \BR^3. In both cases we replace the integer derivatives by Caputo fractional derivatives of order $0 <\alpha <1$. We characterize the space of solutions of the fractional Laplace-Beltrami equation, and we calculate its dimension. We establish relations between the solutions of the fractional Laplace-Beltrami equation and the solutions of the fractional Riesz system. Some examples of the polynomial solutions will be presented. Moreover, the behaviour of the obtained results when $\alpha=1$ is presented, and a final remark about the consideration of Riemann-Liouville fractional derivatives instead of Caputo fractional derivatives is made

Understanding the interactions of cellulose fibres and deep eutectic solvent of choline chloride and urea

This is the author accepted manuscript. The final version is available from Springer Verlag via the DOI in this record.A deep eutectic solvent composed of choline chloride (ChCl) and urea has been recently introduced as a promising cellulose compatible medium that enables e.g. fibre spinning. This paper clarifies the influence of such a solvent system on the structure and chemical composition of the cellulosic pulp fibres. Special emphasis was placed on the probable alterations of the chemical composition due to the dissolution of the fibre components and/or due to the chemical derivatisation taking place during the DES treatment. Possible changes in fibre morphology were studied with atomic force microscopy and scanning electron microscopy. Chemical compositions of pulp fibres were determined from the carbohydrate content, and by analysing the elemental content. Detailed structural characterisation of the fibres was carried out using spectroscopic methods; namely X-Ray Photoelectron Spectroscopy, solid state Nuclear Magnetic Resonance and Raman Spectroscopy. No changes with respect to fibre morphology were revealed and negligible changes in the carbohydrate composition were noted. The most significant change was related to the nitrogen content of the pulp after the DES treatment. Comprehensive examination using spectroscopic methods revealed that the nitrogen originated from strongly bound ChCl residuals that could not be removed with a mild ethanol washing procedure. According to Raman spectroscopic data and methylene blue adsorption tests, the cationic groups of ChCl seems to be attached to the anionic groups of pulp by electrostatic forces. These findings will facilitate the efficient utilisation of DES as a cellulose compatible medium without significantly affecting the native fibre structure.The authors acknowledge the Finnish Funding Agency for Innovation (TEKES) for funding the work via Design Driven Value Chains in the World of Cellulose 2.0 project. The Academy of Finland (Project ID 300367) is acknowledged for enabling the research mobility of T.T. to the University of Exeter, UK. Unto Tapper (VTT) is thanked for the SEM imaging, Atte Mikkelson, Ritva Heinonen and Marita Ikonen (VTT) for the chemical analysis and Robertus Nugroho (Aalto University) for the AFM imaging

Adhesion of perfume-filled microcapsules to model fabric surfaces

The retention and adhesion of melamine formaldehyde (MF) microcapsules on a model fabric surface in aqueous solution were investigated using a customised flow chamber technique and atomic force microscopy (AFM). A cellulose film was employed as a model fabric surface. Modification of the cellulose with chitosan was found to increase the retention and adhesion of microcapsules on the model fabric surface. The AFM force–displacement data reveal that bridging forces resulting from the extension of cellulose chains dominate the adhesion between the microcapsule and the unmodified cellulose film, whereas electrostatic attraction helps the microcapsules adhere to the chitosan-modified cellulose film. The correlation between results obtained using these two complementary techniques suggests that the flow chamber device can be potentially used for rapid screening of the effect of chemical modification on the adhesion of microparticles to surfaces, reducing the time required to achieve an optimal formulation