Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j} together with the Lie-algebraic representation of raising and lowering operators acting on the lattice h\BZ^n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous improvements throughout the paper was don

(Discrete) Almansi Type Decompositions: An umbral calculus framework based on osp(1∣2)\mathfrak{osp}(1|2) symmetries

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials \BR[\underline{x}] shall be described in terms of the generators of the Weyl-Heisenberg algebra. The extension of \BR[\underline{x}] to the algebra of Clifford-valued polynomials $\mathcal{P}$ gives rise to an algebra of Clifford-valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra $\mathfrak{osp}(1|2)$. This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of Almansi decomposition in Clifford analysis (c.f. \cite{Ryan90,MR02,MAGU}) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces $\ker (D')^k$. We will discuss afterwards how the symmetries of \mathfrak{sl}_2(\BR) (even part of $\mathfrak{osp}(1|2)$) are ubiquitous on the recent approach of \textsc{Render} (c.f. \cite{Render08}), showing that they can be interpreted in terms of the method of separation of variables for the Hamiltonian operator in quantum mechanics.Comment: Improved version of the Technical Report arXiv:0901.4691v1; accepted for publication @ Math. Meth. Appl. Sci http://www.mat.uc.pt/preprints/ps/p1054.pdf (Preliminary Report December 2010

New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention

Solutions For The Klein–gordon And Dirac Equations On The Lattice Based On Chebyshev Polynomials

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein–Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a well-adapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein–Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed. © 2015, Springer Basel.10237939913/07590-8, FAPESP, Fundação de Amparo à Pesquisa do Estado de São PauloFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Becher, P., Joos, H., The Dirac–Kähler equation and fermions on the lattice (1982) Z. Phys. C, 15, pp. 343-365Bohm, D., Space, time, and the quantum theory understood in terms of discrete structural process (1965) Proceedings of the International Conference on Elementary Particles, Kyoto, pp. 252-287Borici, A., Creutz fermions on an orthogonal lattice (2008) Phys. Rev. D, 78 (7), p. 074504Borštnik, N.M., Nielsen, H.B., Dirac-Kähler approach connected to quantum mechanics in Grassmann space (2000) Phys. Rev. D, 62 (4), p. 044010Cerejeiras, P., Kähler, U., Ku, M., Sommen, F., Discrete Hardy Spaces (2014) J. Fourier Anal. Appl., 20 (4), pp. 715-750Chan, Y.-S., Fannjiang, A.C., Paulino, G.H., Integral equations with hypersingular kernels-theory and applications to fracture mechanics (2003) Int. J. Eng. Sci., 41 (7), pp. 683-720Chelkak, D., Smirnov, S., Universality in the 2D Ising model and conformal invariance of fermionic observables (2012) Invent. Math., 189 (3), pp. 515-580Cole, E.A.B., Transition from a continuous to a discrete space-time scheme (1970) Il Nuovo Cimento A, 66 (4), pp. 645-656Constales, D., Faustino, N., Kraußhar, R.S., Fock spaces. Landau operators and the time-harmonic Maxwell equations (2011) J. Phys. A Math. Theor., 44 (13), p. 135303Creutz, M., Local chiral fermions The XXVI International Symposium on Lattice Field Theory (2008). arXiv, 808, p. 0014. , arXiv:0808.0014da Rocha, R., Vaz, J., Jr., Extended Grassmann and Clifford algebras (2006) Adv. Appl. Clifford Algebr., 16 (2), pp. 103-125da Veiga, P.A.F., O’Carroll, M., Schor, R., Excitation spectrum and staggering transformations in lattice quantum models (2002) Phys. Rev. E, 66 (2), p. 027108Dimakis, A., Müller-Hoissen, F., Discrete differential calculus: graphs, topologies, and gauge theory (1994) J. Math. Phys., 35 (12), pp. 6703-6735Faustino, N., Kähler, U., Sommen, F., Discrete Dirac operators in Clifford analysis (2007) Adv. Appl. Clifford Algebr., 17 (3), pp. 451-467Faustino, N., (2009) Discrete Clifford analysis, Dissertation, , http://hdl.handle.net/10773/2942, Ria Repositório Institucional, Universidade de AveiroFaustino, N., Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle (2014) Appl. Math. Comput., 247, pp. 607-622Friedan, D., A proof of the Nielsen–Ninomiya theorem (1982) Commun. Math. Phys, 85 (4), pp. 481-490Froyen, S., Brillouin-zone integration by Fourier quadrature: special points for superlattice and supercell calculations (1989) Phys. Rev. B, 39 (5), pp. 3168-3172Gürlebeck, K., Hommel, A., On finite difference Dirac operators and their fundamental solutions (2001) Adv. Appl. Clifford Algebr., 11 (2), pp. 89-106Kanamori, I., Kawamoto, N., Dirac–Kaehler fermion from Clifford product with noncommutative differential form on a lattice (2004) Int. J. Mod. Phys. A, 19 (5), pp. 695-736Kogut, J., Susskind, L., Hamiltonian formulation of Wilson’s lattice gauge theories (1975) Phys. Rev. D, 11 (2), pp. 395-408Mercat, C., Discrete Riemann surfaces and the Ising model (2001) Commun. Math. Phys., 218 (1), pp. 177-216Monaco, R.L., de Oliveira, E.C., A new approach for the Jeffreys–Wentzel–Kramers–Brillouin theory (1994) J. Math. Phys., 35 (12), pp. 6371-6378Montvay, I., Münster, G., (1994) Quantum Fields on a Lattice, , Cambridge University Press, CambridgeNielsen, H.B., Ninomiya, M., A no-go theorem for regularizing chiral fermions (1981) Phys. Lett. B, 105 (2), pp. 219-223Rabin, J., Homology theory of lattice fermion doubling (1982) Nucl. Phys. B, 201 (2), pp. 315-332Rodrigues, W.A., Jr., de Oliveira, E.C., (2007) The Many Faces of Maxwell, Dirac and Einstein Equations: A Clifford Bundle Approach, , 722, Springer, HeidelbergVaz, J., Jr., Clifford-like calculus over lattices (1997) Adv. Appl. Clifford Algebr., 7 (1), pp. 37-70Vaz, J., (1997) Clifford algebras and Witten’s monopole equations In: Apanasov, B., Rodrigues, U. (eds). Geometry, topology and physics: interfaces in computer science and operations research, 2nd edn, vol 2, pp, 277-300. , Walter de Gruyter & Co., BerlinWilson, W.K., Confinement of quarks (1974) Phys. Rev. D, 10 (8), pp. 2445-245