Integral representations for hyperbolic harmonic function in the clifford algebra Cl n+1,0

Clifford algebra is introduced as a theoretical foundation for network topology expression and algorithm construction. Network nodes are coded with basis vectors in a vector space , and the edges and k‐walk routes can be expressed by 2‐blades and k‐blades, respectively, in the Clifford algebra Cl(n,0). The topologies among nodes, edges, and routes of networks can be directly calculated, and the network routes can be extended and traversed with oriented join products. The network algorithm construction processes based on Clifford algebra are instantiated by the single source shortest path algorithm. The experimental results on different scale random networks suggest that Clifford algebra is suited for network expression and relation computation. The Clifford algebra‐based shortest path algorithm is vivid and clear in geometric meaning and has great advantage on temporal and spatial complexity. Copyright © 2013 John Wiley & Sons, Ltd.

We are studying a function theory of k-hypermonogenic functions connected to k-hyperbolic harmonic functions that are harmonic with respect to the hyperbolic Riemannian metric $$ds_{k}^{2} = x_{n}^{\frac{2k}{1-n}} \left(dx_{0}^{2} + \cdots + dx_{n}^{2} \right)$$ in the upper half space \({\mathbb{R}_{+}^{n+1} = \left\{\left(x_{0}, \ldots,x_{n}\right)\,|\,x_{i} \in \mathbb{R}, x_{n} > 0\right\}}\). The function theory based on this metric is important, since in case \({k = n-1}\), the metric is the hyperbolic metric of the Poincare upper half space and Leutwiler noticed that the power function \({x^{m}\,(m \in \mathbb{N}_{0})}\), calculated using Clifford algebras, is a conjugate gradient of a hyperbolic harmonic function. We find a fundamental \({k}\)-hyperbolic harmonic function. Using this function we are able to find kernels and integral formulas for k-hypermonogenic functions. Earlier these results have been verified for hypermonogenic functions (\({k = n-1}\)) and for k-hyperbolic harmonic functions in odd dimensional spaces.

A Jacobian matrix of a general inline planar platform is studied. An inline planar platform is a manipulator with three legs, each with RPR joints, such that the revolute joints are free and align on each platform and the prismatic joints are powered. The configurations that cause the Jacobian matrix to become singular form a singularity surface that must be avoided for controllability. The Jacobian matrix is developed in the even Clifford algebra Cl+(P 2) of the projective space P 2 and its singularity surface is studied. A redundant planar platform manipulator is shown to have a block Jacobian matrix. A composite of singularity sets is developed for a redundant planar platform. A three-dimensional multi-platform manipulator is discussed.

Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space $$\mathbb {R}^{p',q'}$$ , $$n'=p'+q'=2m$$ , we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over $$\mathbb {R}^{p,q}$$ , $$n=p+q=p'+q'+1=2m+1$$ . Explicit examples are provided for dimensions $$n'=2,4,6$$ , and the resulting inverses for $$n=n'+1=3,5,7$$ . The general result for $$n=7$$ appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p, q), $$n=p+q=7$$ , only involving a single addition of multivector products in forming the determinant.

Orthonormal basis sets define isomorphisms and automorphisms in Clifford algebras. Orthonormal basis sets (ONB) are defined as sets of multivectors satisfying scalar product relations. A FORTRAN program determining ONBs is described. It is shown that any simple Clifford algebra is isomorphic to the tensor product of a Clifford algebra Cl m,m and a Clifford algebra isomorphic to ℝ, ℂ or ℍ. From the construction of matrix algebras isomorphic to Cl m,m given by the second FORTRAN program, matrix algebras with entries in ℝ, ℂ or ℍ can be used to construct isomorphisms to all simple Clifford algebras.

We develop basic properties of solutions to the Dirac-Hodge and Laplace equations in upper half space endowed with the hyperbolic metric. Solutions to the Dirac-Hodge equation are called hypermonogenic functions, while solutions to this version of Laplace's equation are called hyperbolic harmonic functions. We introduce a Borel-Pompeiu formula forC 1 functions and a Green's formula for hyperbolic harmonic functions. Using a Cauchy integral formula, we introduce Hardy spaces of solutions to the Dirac-Hodge equation. We also provide new arguments describing the conformal covariance of hypermonogenic functions and invariance of hyperbolic harmonic functions and introduce intertwining operators for the Dirac-Hodge operator and hyperbolic Laplacian.

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper, we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein–Gordon equation with a potential. With the aid of one particular solution we factorize the Klein–Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein–Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein–Gordon equation with potential. Finally, we give some examples of the application of the proposed procedure.

In this article, some properties of singular integrals with parameter are discussed. We prove a Liouville-type theorem for pseudo-monogenic functions and pseudo-harmonic functions. With the help of them, R −1 Riemann boundary value problems for pseudo-monogenic functions and pseudo-harmonic functions with values in a Clifford algebra Cl(V 3.3) are investigated in the classical sense and their explicit representation of solutions are obtained.

Faced with the persisting problem of the unification of gravity with other fundamental interactions we investigate the possibility of a new paradigm, according to which the basic space of physics is a multidimensional space C associated with matter configurations. We consider general relativity in C. In spacetime, which is a 4-dimensional subspace of C, we have not only the 4-dimensional gravity, but also other interactions, just as in Kaluza-Klein theories. We then consider a finite dimensional description of extended objects in terms of the center of mass, area, and volume degrees of freedom, which altogether form a 16-dimensional manifold whose tangent space at any point is Clifford algebra Cl(1,3). The latter algebra is very promising for the unification, and it provides description of fermions.

<h2>Abstract</h2>\n<p><strong>Objectives:</strong> This research emphasis on developing solution of transverse displacement for simply supported beam subjected to thermal load using hyperbolic shear deformation theory (HYSDT). Moreover, the significant transverse displacement analysis of single, three-layer orthotropic symmetric laminated beams exposed to sinusoidal thermal loads was taken under consideration with the aim to achieve exact solution for the above condition.<strong> Methodology:</strong> In this solution, we have used hyperbolic function in reference to thickness co-ordinate by considering shear deformation theory. The governing equations, boundary conditions were <strong>developed using principle of virtual work and thus exhibiting solution by Navier’s theory. Findin g:</strong> It is observed that by considering single, triple layer transverse displacement for simply supported beam subjected to thermal loading as well as applying hyperbolic shear deformation theory (HYSDT) has predicted transverse displacement of 0.269, 0.210 and 0.562 for ply angles such as 00, 900, 00/900/00 respectively with aspect ratio 20. In accordance with this we observed uniformity among the predicted values. <strong>Novelty:</strong> In this research we have introduced an advance function such as hyperbolic shear deformation theories (HYSDT) and transverse displacement for variable lengths and thickness of the beam was calculated by using MATLAB coding. Thus, achieving precise solution.</p>\n<p><strong>Keywords</strong>: Sinusoidal load; hyperbolic shear deformation theory; governing differential equations; MATLAB coding; thermal load</p>

We give sufficient conditions which ensure that harmonic extension u = P[f] to the upper half space {(x, y) | x ? Rn, y > 0} of a function f ? Lp(Rn) satisfies estimate ?u/?y (x, y) ? C?(y)/y for every x in E ? Rn, where ? is a majorant. The conditions are expressed in terms of behaviour of the Riesz transforms Rjf of f near points in E. We briefly investigate related questions for the cases of harmonic and hyperbolic harmonic functions in the unit ball.

In a series of papers, we have developed the principles of a variant of standard gauge theories which we call’ spin Gauge Theories’, based upon Clifford algebra, and have progressed towards the formulation of a unified theory of particle and gravitational interactions. In the first paper [1], we presented a theory of the electroweak interactions of one family of leptons and quarks, introducing the concept of the ‘Frame Field’. We conjectured that the frame field was intimately involved with gravitation, and in the second paper [2], we used spin gauge theory principles and the frame field to derive the Einstein-Hilbert gravitational Lagrangian as a particle interaction; we found that the theory automatically added a’ spin gravity’ term, quadratic in the curvature, modifying the Einstein theory at short distances. In the third paper [3], we showed that the electroweak and gravitational interactions could be combined into a unified theory. The only observed particle interaction which was not included in the theory was the strong interaction; the standard’ strong’ interaction theory is generated by the SU(3) colour symmetry group of quarks. We have outlined [4] a spin gauge theory, using the Clifford algebra C4,7, which unifies the electroweak, gravitational and SU(3) interactions. In this and the accompanying paper [5], we describe in some detail how the colour group is introduced; we also report on a modification of the standard theory which is suggested to us by the structure of our unified model. We have not yet completed our work on this modification.

After general remarks upon the biomathematical tasks of quantitative description or mathematical modelling of biological systems and phenomena on different levels of organization the growth process of body length of human beings serves as illustration of the possibilities of such mathematical treatments. A short survey is given about the attempts of using differential equations for reflecting basic principles of growth processes. Further on analytical functions (as the solutions of growth differential equations) preferably may be used for an objective quantitative reproducing of measured courses of growth variables. The values of parameters contained in such expressions will be calculated by nonlinear fitting procedures. In nearly all cases we got a very high numerical precision in approximating parts of the body length growth process of man which extends from conception until the age of 18. Our attempts of a phenomenologic‐mathematical modelling of this growth process mean a subdividing of the whole process in single growth spurts with a symmetric‐sigmoidal shape of their courses, and describing the single spurt by a hyperbolic tangent function. An advantage of this kind of modelling is the fact that each of the terms is valid for the whole time interval of growth. Within the process of forming the model heuristical hints are taken into consideration arising from biological background and which would supply the possibility of interpretation and verification of the analyzed growth spurts.

In this paper we have defined some product type hyperbolic convex function and product type hyperbolic Young function and prove some results based on this. We also characterize an integral representation of the product type hyperbolic convex functions in this paper.

The object of this note is to present new expressions for the classical Mathieu series in terms of hyperbolic functions. The derivation is based on elementary arguments concerning the integral representation of the series. The results are used afterwards to prove, among others, a new relationship between the Mathieu series and its alternating companion. A recursion formula for the Mathieu series is also presented. As a byproduct, some closed-form evaluations of integrals involving hyperbolic functions are inferred.