Abstract
In recent years the umbral calculus has emerged from the shadows to provide an elegant correspondence framework that automatically gives systematic solutions of ubiquitous difference equations --- discretized versions of the differential cornerstones appearing in most areas of physics and engineering --- as maps of well-known continuous functions. This correspondence deftly sidesteps the use of more traditional methods to solve these difference equations. The umbral framework is discussed and illustrated here, with special attention given to umbral counterparts of the Airy, Kummer, and Whittaker equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de Vries, and Toda systems.
Highlights
INTRODUCTIONThrough appropriate umbral deformation of the continuum solutions, the corresponding discrete difference equations may be automatically solved
Robust theoretical arguments have established an anticipation of a fundamental minimum measurable length in Nature, of order LPlanck ≡ hGN /c3 = 1.6162 × 10−35 m, the corresponding mass and time being and LPlanck /c = 5.3911 ×1M0P−la4n4cks.≡Thehecs/sGenNce=o2f .s1u7c6h5a×rg1u0m−e8nktgs is the following
In a system or process characterized by energy E, no lengths smaller than L can be measured, where L is the larger of either the Schwarzschild horizon radius of the system (∼ E) or, for energies smaller than the Planck mass, the Compton wavelength of the aggregate process (∼ 1/E)
Summary
Through appropriate umbral deformation of the continuum solutions, the corresponding discrete difference equations may be automatically solved. In (21) the linearity of the umbral deformation functional was exploited, together with the fact that the umbral image of an exponential is an exponential, albeit with interesting modifications, to discretize well-behaved functions occurring in solutions of physical differential equations through their Fourier expansion. This discrete shadowing of the Fourier representation functional should be of utility in inferring wave disturbance propagation in discrete spacetime lattices. We pay particular attention to the umbral counterparts of the Airy, Kummer, and Whittaker equations, and their solutions, and to the umbral maps of solitons for the Sine-Gordon, Korteweg–de Vries, and Toda systems
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