On Modified Second Paine–de Hoog–Anderssen Boundary Value Problem

This article deals with a special case of the Sturm–Liouville boundary value problem (BVP), an eigenvalue problem characterized by the Sturm–Liouville differential operator with unknown spectra and the associated eigenfunctions. By examining the BVP in the Schrödinger form, we are interested in the problem where the corresponding invariant function takes the form of a reciprocal quadratic form. We call this BVP the modified second Paine–de Hoog–Anderssen (PdHA) problem. We estimate the lowest-order eigenvalue without solving the eigenvalue problem but by utilizing the localized landscape and effective potential functions instead. While for particular combinations of parameter values that the spectrum estimates exhibit a poor quality, the outcomes are generally acceptable although they overestimate the numerical computations. Qualitatively, the eigenvalue estimate is strikingly excellent, and the proposal can be adopted to other BVPs.

Anisotropy Properties of Turbulence in Flow Over Seepage Bed

The present study analyses the Reynolds stress anisotropy in the non-uniform sediment beds under the condition of no seepage and downward seepage flow. The results show the estimation of the deviation measure from the isotropic turbulence in view of Reynolds stress tensor for turbulent flow in the presence of seepage through the channel bed. The investigation presents the Lumley triangle for flow turbulence, Eigen values, and the invariant functions for the whole flow depth subjected to no seepage and seepage beds. The longitudinal profile of anisotropy tensor within the near-bed zone for seepage flow provides the higher anisotropic stream than those of no seepage flow, while the remaining (transverse and vertical) profiles of anisotropy tensor in the vicinity of bed for seepage flows provides lower anisotropic stream. The anisotropic invariant maps show the near bed anisotropy inclining to be a two-component isotropy subjected to no seepage and seepage flow. With the increase in vertical distance from bed surface that is close to the water surface, the data sets of anisotropic invariant maps for no seepage and seepage flows show a trend of one-component isotropy, while it has an affinity to develop a three-component isotropy in the vicinity of mid zone of the flow depth. Invariant function data sets present a well two-component isotropy in the near bed region of flow and a quasi-three component isotropy in the outer region of flow for seepage flows as compared to no seepage flow.

Invariants for a Dynamical System with Strong Random Perturbations

In this chapter we consider the invariant method for stochastic system with strong perturbations, and its application to many different tasks related to dynamical systems with invariants. This theory allows constructing the mathematical model (deterministic and stochastic) of actual process if it has invariant functions. These models have a kind of jump-diffusion equations system (stochastic differential Itô equations with a Wiener and a Poisson paths). We show that an invariant function (with probability 1) for stochastic dynamical system under strong perturbations exists. We consider a programmed control with Prob. 1 for stochastic dynamical systems – PSP1. We study the construction of stochastic models with invariant function based on deterministic model with invariant one and show the results of numerical simulation. The concept of a first integral for stochastic differential equation Itô introduce by V. Doobko, and the generalized Itô – Wentzell formula for jump-diffusion function proved us, play the key role for this research.

Reynolds stress anisotropy in flow over two-dimensional rigid dunes

Characteristics of turbulence anisotropy in flow over two-dimensional rigid dunes are analysed. The Reynolds stress anisotropy is envisaged from the perspective of the stress ellipsoid shape. The spatial evolutions of the anisotropic invariant map (AIM), anisotropic invariant function, eigenvalues of the scaled Reynolds stress tensor and eccentricities of the stress ellipsoid are investigated at various streamwise distances along the vertical. The data plots reveal that the oblate spheroid axisymmetric turbulence appears near the top of the crest, whereas the prolate spheroid axisymmetric turbulence dominates near the free surface. At the dune trough, the axisymmetric contraction to the oblate spheroid diminishes, as the vertical distance below the crest increases. At the reattachment point and one-third of the stoss-side, the oblate spheroid axisymmetric turbulence formed below the crest appears to be more contracted, as the vertical distance increases. The AIMs suggest that the turbulence anisotropy up to edge of the boundary layer follows a looping pattern. As the streamwise distance increases, the turbulence anisotropy at the edge of the boundary layer approaches the plane-strain limit up to two-thirds of the stoss-side, intersecting the plane-strain limit at the top of the crest and thereafter moving towards the oblate spheroid axisymmetric turbulence.

Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.

Polymer Dynamics Through Group Invariance of SL(2R) - Type in a Fractal Paradigm

We analyze polymer dynamics in a fractal paradigm. Then, it is shown that polymer dynamics in the form of Schrödinger - type regimes imply synchronization processes of the polymers� structural units, through joint invariant function of two simultaneous isomorphic groups of SL(2R) - type, as solutions of Stoka equations. In this context, period doubling, damped oscillations, self - modulation and chaotic regimes emerge as natural behaviors in the polymer dynamics. The present model can also be applied to a large class of materials, such as biomaterials, biocomposites and other advanced materials.