On the Discrete Weibull Marshall–Olkin Family of Distributions: Properties, Characterizations, and Applications

In this article, we introduce a new flexible discrete family of distributions, which accommodates wide collection of monotone failure rates. A sub-model of geometric distribution or a discrete generalization of the exponential model is proposed as a special case of the derived family. Besides, we point out a comprehensive record of some of its mathematical properties. Two distinct estimation methods for parameters estimation and two different methods for constructing confidence intervals are explored for the proposed distribution. In addition, three extensive Monte Carlo simulations studies are conducted to assess the advantages between estimation methods. Finally, the utility of the new model is embellished by dint of two real datasets.

On the Existence of Non-Spurious Solutions to Second Order Dirichlet Problem

Using the direct variational method together with the monotonicity approach we consider the existence of non-spurious solutions to the following Dirichlet problem −x¨t =ft,xt, x0 =x1 =0, where f: 0,1 × R→R is a jointly continuous and not necessarily convex function. A new approach towards deriving the discrete family of approximating problems is proposed.

Rotations in a Non-Orthogonal Frame

This paper is concerned with describing the space of matrices that describe rotations in non-orthogonal coordinates. In scenarios such as in crystallography, conformational analysis of polymers, and in the study of deployable mechanisms and rigid origami, non-orthogonal reference frames are natural. For example, non-orthogonal vectors in the direction of atomic bonds in a molecule, the lattice coordinates of a crystal, or the directions of links in a mechanism are intrinsic. In these cases it is awkward to impose an artificial orthonormal reference frame rather than choosing one that is defined by the geometry of the object being studied. With these applications in mind, we fully characterize the space of all possible non-orthogonal rotations. We find that in the 2D case, this space is a three-dimensional subset of the special linear group, SL(2, R), which is itself a three-dimensional Lie group. In the 3D case we find that the space of nonorthogonal rotations is a seven-dimensional subspace of SL(3, R), which is an eight-dimensional Lie group. In the 2D case we use the Iwasawa decomposition to fully characterize the solution. In the 3D case we parameterize this seven-dimensional space by conjugating elements of the rotation group SO(3) by elements of a discrete family of of four-parameter subgroups of GL(3, R), and using this we derive an inversion formula to extract classical orthogonal rotations from those expressed in non-orthogonal coordinates.

Fibonacci family of dynamical universality classes

Universality is a well-established central concept of equilibrium physics. However, in systems far away from equilibrium, a deeper understanding of its underlying principles is still lacking. Up to now, a few classes have been identified. Besides the diffusive universality class with dynamical exponent z=2, another prominent example is the superdiffusive Kardar−Parisi−Zhang (KPZ) class with z=3/2. It appears, e.g., in low-dimensional dynamical phenomena far from thermal equilibrium that exhibit some conservation law. Here we show that both classes are only part of an infinite discrete family of nonequilibrium universality classes. Remarkably, their dynamical exponents zα are given by ratios of neighboring Fibonacci numbers, starting with either z1=3/2 (if a KPZ mode exist) or z1=2 (if a diffusive mode is present). If neither a diffusive nor a KPZ mode is present, all dynamical modes have the Golden Mean z=(1+5)/2 as dynamical exponent. The universal scaling functions of these Fibonacci modes are asymmetric Lévy distributions that are completely fixed by the macroscopic current density relation and compressibility matrix of the system and hence accessible to experimental measurement.

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman-Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic.