Variational Theory of Motion of Curved, Twisted and Extensible Elastic Rods

By assimilating biological systems, both structural and functional, into multifractal objects, their behavior can be described in the framework of the scale relativity theory, in any of its forms (standard form in Nottale’s sense and/or the form of the multifractal theory of motion). By operating in the context of the multifractal theory of motion, based on multifractalization through non-Markovian stochastic processes, the main results of Nottale’s theory can be generalized (specific momentum conservation laws, both at differentiable and non-differentiable resolution scales, specific momentum conservation law associated with the differentiable–non-differentiable scale transition, etc.). In such a context, all results are explicated through analyzing biological processes, such as acute arterial occlusions as scale transitions. Thus, we show through a biophysical multifractal model that the blocking of the lumen of a healthy artery can happen as a result of the “stopping effect” associated with the differentiable-non-differentiable scale transition. We consider that blood entities move on continuous but non-differentiable (multifractal) curves. We determine the biophysical parameters that characterize the blood flow as a Bingham-type rheological fluid through a normal arterial structure assimilated with a horizontal “pipe” with circular symmetry. Our model has been validated based on experimental clinical data.

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Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

Journal of Mathematical Chemistry ◽

10.1007/s10910-021-01282-y ◽

2021 ◽

Author(s):

Mariusz Pawlak ◽

Marcin Stachowiak

Keyword(s):

Spherical Harmonics ◽

Interaction Potential ◽

Chem Phys ◽

Basis Set ◽

Matrix Elements ◽

Trigonometric Functions ◽

Variational Theory ◽

Molecular Collision ◽

The Matrix ◽

Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

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Fermat Metrics

Symmetry ◽

10.3390/sym13081422 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1422

Author(s):

Antonio Masiello

Keyword(s):

Variational Methods ◽

Morse Theory ◽

Sagnac Effect ◽

Variational Theory ◽

Global Hyperbolicity ◽

Fermat Principle ◽

Multiple Image ◽

Light Rays ◽

Stationary Spacetimes

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

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