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We introduce a new flexible modified alpha power (MAP) family of distributions by adding two parameters to a baseline model. Some of its mathematical properties are addressed. We show empirically that the new family is a good competitor to the Beta-F and Kumaraswamy-F classes, which have been widely applied in several areas. A new extension of the exponential distribution, called the modified alpha power exponential (MAPE) distribution, is defined by applying the MAP transformation to the exponential distribution. Some properties and maximum likelihood estimates are provided for this distribution. We analyze three real datasets to compare the flexibility of the MAPE distribution to the exponential, Weibull, Marshall–Olkin exponential and alpha power exponential distributions.

In analyzing bivariate data sets, data with common observations are frequently encountered and, in this case, existing absolutely continuous bivariate distributions are not applicable. Only a few models, such as the bivariate distribution proposed by Marshall and Olkin (J Am Stat Assoc 62(317):30–44, 1967), have been developed to model such data sets and the choice of models to fit data sets having common observations is very limited. In this paper, three general classes of bivariate distributions for modeling data with common observations are developed. To develop the bivariate distributions, we employ a probability model in reliability. Considering a system with two components, it is assumed that, when the first failure of the components occurs, with some probability, it immediately causes the failure of the remaining component, and, with complementary probability, the residual lifetime of the remaining component is shortened according to some stochastic order. It will be shown that, by specifying the underlying distributions contained in the joint distribution, numerous families of bivariate distributions can be generated. Therefore, this work provides substantially increased flexibility in modeling data sets with common observations. The developed models are fitted to two real-life data sets and it is shown that these models outperform the existing models in terms of fitting performance and their performances are satisfactory.

This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so-called skew-normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew-elliptical families, and some forms of semiparametric formulations, extending the relevance of the field much beyond the original theme of 'skewness'. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.

We study two-parametric families of spatial orbits given in the analytic form f(x,y,z)=c1, g(x,y,z)=c2 (c1, c2 = const.) which are produced by three-dimensional potentials V=V(x,y,z) inside a material concentration. These potentials must verify two linear partial differential equations (PDEs) which are the basic equations of the 3D Inverse Problem of Newtonian Dynamics and the well-known Poisson’s equation. A suitable class of potentials for this case is the axisymmetric potentials V=B(x2+y2,z) which have applications in astrophysical problems. For the given density function ρ=ρ(x,y,z), ρ=ρ0=const., or, ρ=ρ(z) and a pre-assigned family of orbits, three-dimensional potentials producing this family of orbits are found in each case. We focus our interest on the cored, logarithmic potentials and another one of fourth degree describing elliptical galaxies. The two-parametric families of straight lines in 3D space are also considered.

An analytical approximate method for calculating multidimensional integrals of analytic functions is proposed, in which the integrand is approximated by a power series. This approach transforms the original system of nonlinear equations with integral components into a system of equations with a polynomial left-hand side. To solve equations of this class, an analytical method based on abstract power series is developed. A recurrent procedure is developed for the analytical solution of this class of nonlinear equations.

One of the main problems of astrophysics is to determine the mean field potential of galaxies. The potentials describe the motion in the central region of the galaxy. We can obtain families of star orbits in a galaxy from the astronomical observations. In the present paper, using the tools of the 3D Inverse problem of Dynamics, we study three-dimensional potentials of the form V=A(x2+py2+qz2), (A is an arbitrary function and p,q=const.), which are compatible with a preassigned two-parametric family of spatial regular curves f(x,y,z)=c1, g(x,y,z)=c2 (c1,c2=const). We establish three differential conditions which are fulfilled by the “slope functions” α(x,y,z), β(x,y,z) corresponding to the above two-parametric family of orbits. If these conditions are satisfied, then we can find such a potential by quadratures. We offer pertinent examples of potentials which are mainly used in Galactic Dynamics, e.g. axisymmetric potentials, the potential of a homogeneous sphere, potentials coming from power law density profile and potentials which represent the central region of a perturbed triaxial galaxy. Special cases are also examined. Finally, potentials producing families of straight lines are taken into account. Two-dimensional potentials is a special category and it is studied separately.

A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.

We study three-dimensional potentials of the form V=U(xp+yp+zp), where U is an arbitrary function of C2-class, and p∈Z, which produces a preassigned two-parametric family of spatial regular orbits given in the solved form f(x,y,z) = c1, g(x,y,z) = c2 (c1, c2 = const). These potentials have to satisfy two linear PDEs, which are the basic equations of the 3D inverse problem of Newtonian dynamics. The functions f and g can be represented uniquely by the ”slope functions” α(x,y,z) and β(x,y,z). The orbital functions α(x,y,z) and β(x,y,z) have to satisfy three differential conditions according to the theory of the inverse problem. If these conditions are satisfied, then we can find such a potential analytically. We offer pertinent examples of potentials that are mainly used in physical problems. The values obtained for p lead to cases of well-known potentials, such as the Newtonian, cored, logarithmic, polynomial and quadratic ones. New families of orbits produced by the 3D harmonic oscillator are found. Pertinent examples are given and cover all cases. Two-dimensional potentials belong to a special category of potentials and are studied separately. The families of straight lines in 3D space are also examined.

Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and up-to-date account of the subject. Robust optimization is designed to meet some major challenges associated with uncertainty-affected optimization problems: to operate under lack of full information on the nature of uncertainty; to model the problem in a form that can be solved efficiently; and to provide guarantees about the performance of the solution. The book starts with a relatively simple treatment of uncertain linear programming, proceeding with a deep analysis of the interconnections between the construction of appropriate uncertainty sets and the classical chance constraints (probabilistic) approach. It then develops the robust optimization theory for uncertain conic quadratic and semidefinite optimization problems and dynamic (multistage) problems. The theory is supported by numerous examples and computational illustrations. An essential book for anyone working on optimization and decision making under uncertainty,Robust Optimizationalso makes an ideal graduate textbook on the subject.