Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability

In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.

Information Geometry of Reversible Markov Chains

We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels.

On some properties of normally-inflective complexes with simple inflective center in $E_3$

In the paper, we consider the special degenerate class of mormally-inflective complexes with simple inflective center in three-dimensional Euclidean space $E_3$. We prove that to construct this class of complexes one should take an arbitrary curve and draw sheaf of straight lines through each point of this curve. For arbitrary normally-inflective complex with simple inflective center we establish that such complex is fibered into two one-parametric families of congruences.

The algebraic classification of nilpotent algebras

We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.

Generation of an HBIM Library Regarding a Palace of the 19th Century in Lisbon

The growing interest within the construction industry in the preservation, rehabilitation, and conversion of heritage-value buildings has led to the implementation of Historic Building Information Modelling (HBIM) across all sectors. The rigorously accurate representation of old construction solutions for roofs, floors, and walls, and of the decorative details and finishes, commonly used in historic buildings of patrimonial value, requires specific geometric modelling processes in order to generate relevant libraries of parametric objects. This study addresses the generation of parametric families of representative architectural geometry in the context of the conversion of a building of patrimonial value. A collection of historical information and a detailed inspection in situ, recorded by means of digital images and sketches with annotative dimensions of the architectural forms, were first carried out to support the accurate representation of the building. Several families of objects were generated, namely, guillotine windows, glazed doors, ornamental stonework, staircases, and handrails, thus creating a reference library for use in further similar building projects. The study improves the potential of the BIM process for its application to buildings with distinctive architecture in the context of the preservation or conversion of heritage buildings.

On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces

A nonlinear system of differential equations describing the rotational motion of a rigid body under the action of torque of potential and circular-gyroscopic forces is considered. For this torque, the system of differential equations has three classical first integrals: the energy integral, the area integral, and the geometric integral. For the analogue of the Lagrange case, when two moments of inertia coincide and the potential depends on one angle, an additional first integral is found and integration in quadratures is performed. A number of examples is considered where parametric families of exact solutions are considered. In these examples, polynomial or analytical functions were used as a potential. In particular, we construct families of periodic and almost periodic motions, as well as families of asymptotically uniaxial rotations. We also identified movements that have limit values of opposite signs for unlimited increase and decrease of time.

On the Increasing Convex Order of Relative Spacings of Order Statistics

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions.