Abstract A few important integrals involving the product of two universal associated Legendre polynomials Pl＇m＇, （x）,Pk＇n＇（x）and x2a（1-x2）-p-1,xb（1± x）-p-1and xc（1-x2）-p-1（1 ± x）axe evaluated...Abstract A few important integrals involving the product of two universal associated Legendre polynomials Pl＇m＇, （x）,Pk＇n＇（x）and x2a（1-x2）-p-1,xb（1± x）-p-1and xc（1-x2）-p-1（1 ± x）axe evaluated using the operator form of Taylor＇s theorem and an integral over a single universal associated Legendre polynomial. These integrals are more general since the quantum numbers are unequal, i.e.l＇ ≠ k＇ and m＇≠ n＇ .Their selection rules are a/so given. We also verify the correctness of those integral formulas numerically.展开更多
A class of second-order differential equations commonly arising in physics applications are considered,and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Unive...A class of second-order differential equations commonly arising in physics applications are considered,and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated Legendre Equations are examined and established. The hypergeometric solutions, presented in this work,will promote future investigations of their mathematical properties and applications to problems in theoretical physics.展开更多
Mechanotransduction refers to a physiological process by which mechanical forces, such as pressures exerted by ionized fluids on cell membranes and tissues, can trigger excitations of electrical natures that play impo...Mechanotransduction refers to a physiological process by which mechanical forces, such as pressures exerted by ionized fluids on cell membranes and tissues, can trigger excitations of electrical natures that play important role in the control of various sensory (i.e. stimuli-responsive) organs and homeostasis of living organisms. In this work, the influence of mechanotransduction processes on the generic mechanism of the action potential is investigated analytically, by considering a mathematical model that consists of two coupled nonlinear partial differential equations. One of these two equations is the Korteweg-de Vries equation governing the spatio-temporal evolution of the density difference between intracellular and extracellular fluids across the nerve membrane, and the other is Hodgkin-Huxley cable equation for the transmembrane voltage with a self-regulatory (i.e. diode-type) membrane capacitance. The self-regulatory feature here refers to the assumption that membrane capacitance varies with the difference in density of ion-carrying intracellular and extracellular fluids, thus ensuring an electromechanical feedback mechanism and consequently an effective coupling of the two nonlinear equations. The exact one-soliton solution to the density-difference equation is obtained in terms of a pulse excitation. With the help of this exact pulse solution the Hodgkin-Huxley cable equation is shown to transform, in steady state, to a linear eigenvalue problem some bound states of which can be obtained exactly. Few of such bound-state solutions are found analytically.展开更多
文摘Abstract A few important integrals involving the product of two universal associated Legendre polynomials Pl＇m＇, （x）,Pk＇n＇（x）and x2a（1-x2）-p-1,xb（1± x）-p-1and xc（1-x2）-p-1（1 ± x）axe evaluated using the operator form of Taylor＇s theorem and an integral over a single universal associated Legendre polynomial. These integrals are more general since the quantum numbers are unequal, i.e.l＇ ≠ k＇ and m＇≠ n＇ .Their selection rules are a/so given. We also verify the correctness of those integral formulas numerically.
基金Supported of Natural Sciences and Engineering Research Council of Canada under Grant No.GP249507
文摘A class of second-order differential equations commonly arising in physics applications are considered,and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated Legendre Equations are examined and established. The hypergeometric solutions, presented in this work,will promote future investigations of their mathematical properties and applications to problems in theoretical physics.
文摘Mechanotransduction refers to a physiological process by which mechanical forces, such as pressures exerted by ionized fluids on cell membranes and tissues, can trigger excitations of electrical natures that play important role in the control of various sensory (i.e. stimuli-responsive) organs and homeostasis of living organisms. In this work, the influence of mechanotransduction processes on the generic mechanism of the action potential is investigated analytically, by considering a mathematical model that consists of two coupled nonlinear partial differential equations. One of these two equations is the Korteweg-de Vries equation governing the spatio-temporal evolution of the density difference between intracellular and extracellular fluids across the nerve membrane, and the other is Hodgkin-Huxley cable equation for the transmembrane voltage with a self-regulatory (i.e. diode-type) membrane capacitance. The self-regulatory feature here refers to the assumption that membrane capacitance varies with the difference in density of ion-carrying intracellular and extracellular fluids, thus ensuring an electromechanical feedback mechanism and consequently an effective coupling of the two nonlinear equations. The exact one-soliton solution to the density-difference equation is obtained in terms of a pulse excitation. With the help of this exact pulse solution the Hodgkin-Huxley cable equation is shown to transform, in steady state, to a linear eigenvalue problem some bound states of which can be obtained exactly. Few of such bound-state solutions are found analytically.
