Form Invariance and Conserved Quantity for Non-holonomic Systems with Variable Mass and Unilateral Constraints

The paper studies the form invariance and a type of non-Noether conserved quantity called Mei conserved quantity for non-holonomic systems with variable mass and unilateral constraints.Acoording to the invariance of the form of differential equations of motion under infinitesimal transformations,this paper gives the definition and criterion of the form invariance for non-holonomic systems with variable mass and unilateral constraints.The condition under which a form invariance can lead to Mei conservation quantity and the form of the conservation quantity are deduced.An example is given to illustrate the application of the results.     

First Digits’ Shannon Entropy

Related to the letters of an alphabet, entropy means the average number of binary digits required for the transmission of one character. Checking tables of statistical data, one finds that, in the first position of the numbers, the digits 1 to 9 occur with different frequencies. Correspondingly, from these probabilities, a value for the Shannon entropy H can be determined as well. Although in many cases, the Newcomb–Benford Law applies, distributions have been found where the 1 in the first position occurs up to more than 40 times as frequently as the 9. In this case, the probability of the occurrence of a particular first digit can be derived from a power function with a negative exponent p > 1. While the entropy of the first digits following an NB distribution amounts to H = 2.88, for other data distributions (diameters of craters on Venus or the weight of fragments of crushed minerals), entropy values of 2.76 and 2.04 bits per digit have been found.     

Geometric control of hybrid systems

In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support functions.     

Shape invariance in phase space

Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the implications of the supersymmetric quantum mechanics and shape invariance techniques to the phase space formalism. We show that shape invariance induces a new set of relations between the Wigner functions of the system, that allows for their direct calculation, once we know one of them. The simple harmonic oscillator and the Morse potential are solved as examples.     

Parameter Invariance in Field Theory and the Hamiltonian Formalism

The deparametrization problem for parameter‐invariant Lagrangian densities defined over J¹(N, F), is solved in terms of a projection onto a suitable jet bundle. The Hamilton‐Cartan formalism for such Lagrangians is then introduced and the pre‐symplectic structure of such variational problems is proved to be projectable through the aforementioned projection. Specific examples with physical meaning are also analyzed. 1998 PACS codes. 02.20.Tw Infinite‐dimensional Lie groups, 02.30.Wd Calculus of variations and optimal control, 02.40.Ky Riemannian geometries, 02.40.Ma Global differential geometry, 02.40.Vh Global analysis and analysis on manifolds, 04.20.Fy Canonical formalism, Lagrangians, and variational principles, 11.10.Ef Lagrangian and Hamiltonian approach, 11.10.Kk Field theories in dimensions other than four, 11.25.Sq Nonperturbative techniques; string field theory. 1991 Mathematics Subject Classification. Primary: 58E30 Variational principles; Secondary: 53B20 Local Riemannian geometry, 58A20 Jets, 58E12 Applications to minimal surfaces (problems in two independent variables), 58G35 Invariance and symmetry properties, 81S10 Geometric quantization, symplectic methods, 83E30 String and superstring theories.     

Scoring indices,top-truncated preferences,and splitting invariance

We present a new characterization of the class of weight-based scoring indices for ranking problems with top-truncated preferences. The main novel axiom is Splitting Invariance: if an alternative is split up into a number of distinct yet unranked alternatives, then the total score of these alternatives should increase by the score of the original alternative, and the scores of the other alternatives should not change.