Abstract: The main aim of this article is to establish sampling series restoration formulae in for a class of stochastic L2-processes which correlation function possesses integral representation close to a Hankel-type transform which kernel is either Bessel function of the first and second kind Jn ;Yn respectively. The results obtained belong to the class of irregular sampling formulae and present a stochastic setting counterpart of certain older results by Zayed [25] and of recent results by Knockaert [13] for J–Bessel sampling and of currently established Y– Bessel sampling results by Jankov Maˇsirevi´c et al. [7]. The approach is twofold, we consider sampling series expansion approximation in the mean–square (or L2) sense and also in the almost sure (or with the probability 1) sense. The main derivation tools are the Piranashvili’s extension of the famous Karhunen–Cram´er theorem on the integral representation of the correlation functions and the same fashion integral expression for the initial stochastic process itself, a set of integral representation formulae for the Bessel functions of the first and second kind Jn ;Yn and various properties of Bessel and modified Bessel functions which lead to the so–called Bessel–sampling when the sampling nodes of the initial signal function coincide with a set of zeros of different cylinder functions.

Keywords: Almost sure convergence, Bessel functions of the first and second kind Jn ;Yn , correlation function, harmonizable stochastic processes, Karhunen–Cram´er–Piranashvili theorem, Karhunen processes, Kramer’s sampling theorem, mean–square convergence, sampling series expansions, sampling series truncation error upper bound, spectral representation of correlation function, spectral representation of stochastic process

Abstract: We develop and analyze a bisection type global optimization algorithm for real Lipschitz functions. The suggested method combines the branch and bound method with an always convergent solver of nonlinear equations. The computer implementation and performance are investigated in detail.

Keywords: global optimum; nonlinear equation; always convergent method; Newton method; branch and bound algorithms; Lipschitz functions

Abstract: We apply Williamson’s theorem for the diagonalization of quadratic forms by symplectic matrices to sub-Riemannian (and Riemannian) structures on the Heisenberg groups. A classification of these manifolds, under isometric Lie group automorphisms, is obtained. A (parametrized) list of equivalence class representatives is identified; a geometric characterization of this equivalence relation is provided. A corresponding classification of (drift-free) invariant optimal control problems is exhibited.

Keywords: Heisenberg group; sub-Riemannian geometry; isometry; symplectic group; invariant optimal control; cost-equivalence

Abstract: In this paper our aim is to present some monotonicity and convexity properties for the one dimensional regularization of the Coulomb potential, which has applications in the study of atoms in magnetic fields and which is in fact a particular case of the Tricomi confluent hypergeometric function. Moreover, we present some Tur´an type inequalities for the function in the question and we deduce from these inequalities some new tight upper bounds for the Mills ratio of the standard normal distribution.

Keywords: Gaussian integral; regularization of the Coulomb potential; Mills’ ratio; Tur´an type inequalities; functional inequalities; bounds; log-convexity and geometrical convexity

Abstract: We study the existence of Stackelberg equilibrium points on strategy sets which are geodesic convex in certain Riemannian manifolds by using metric projection arguments. The present results extend those obtained in Nagy [J. Global Optimization (2013)] in the Euclidean context.

Keywords: Stackelberg model; curved spaces; variational analysis

Abstract: Solving mixed integer linear programming (MILP) problems is a difficult task due to the parallel use of both integer and non-integer values. One of the most widely used solution is to solve the problem in the real space and they apply additional iteration steps (so-called cutting-plane algorithms or Gomory’s cuts) to narrow down the solution to the optimal integer solution. The ABS class of algorithms is a generalized class of algorithms which, with appropriate selection of parameters, is suitable for the solution of both integer and non-integer linear problems. Here we provide for the first time a complete ABS-based algorithm for MILP problems by adaptation of the ABS approach to Gomory’s cuttingplane algorithm. We also provide a numerical example demonstrating the working principle of our algorithm.

Keywords: linear programming; ABS methods; mixed integer problem; cutting-plane methods; Gomory’s cuts

Abstract: The aim of this paper is the numerical simulation of anisotropic mean curvature of graphs in the context of relative geometry, developed in [1]. We extend results in [4] to our problem; we prove an existence theorem and energy equality. The numerical scheme is based on the method of lines where the spatial derivatives are approximated by finite differences [2]. We then solve the resulting ODE system by means of the adaptive Runge-Kutta-Merson method. To show the stability of the scheme we prove the discrete version of the energy equality. Finally, we show experimental order of convergence and results of numerical experiments with various anisotropy settings.

Keywords: anisotropy; mean curvature; Finsler geometry; method of lines; FDM

Abstract: A graph is regular if all its vertices have the same degree. Otherwise a graph is irregular. To measure how irregular a graph is, several graph topological indices were proposed including: the Collatz-Sinogowitz index [8], the variance of the vertex degrees [7], the irregularity of a graph [4], and recently proposed the total irregularity of a graph [1]. Here, we compare the above mentioned irregularity measures for bidegreed graphs.

Keywords: topological graph indices; complete split graph; 2-walk linear graph

Abstract: In this paper, the finite-time stability (FTS) of linear continuous time-delay systems is studied. By using suitable Lyapunov-like function and Jensen’s and Coppel’s inequality, a FTS condition is derived as a set of algebraic inequalities. The comparison of this method with some previous one is done and it has been showed that the numerical computation is reduced.

Keywords: time delay systems; finite-time stability; continuous systems; Jensen’s integral inequality; Coppel’s inequality

Abstract: New vector description of kinetic pressures on shaft bearings of a rigid body nonlinear dynamics with coupled rotations around no intersecting axes is first main result presented in this paper. Mass moment vectors and vector rotators coupled for pole and oriented axes, defined by K. Hedrih in 1991, are used for obtaining vector expressions for kinetic pressures on the shaft bearings of a rigid body dynamics with coupled rotations around no intersecting axes. A complete analysis of obtained vector expressions for kinetic pressures on shaft bearings give us a series of the kinematical vectors rotators around both directions determined by axes of the rigid body coupled rotations around no intersecting axes. As an example of defined dynamics, we take into consideration a heavy gyro-rotordisk with one degree of freedom and coupled rotations when one component of rotation is programmed by constant angular velocity. For this system with nonlinear dynamics, series of graphical presentation transformations in realizations with changes of eccentricity and angle of inclination (skew position) of heavy rigid body-disk in relation to self rotation axis are presented, as well as in realization with changing orthogonal distance between axes of coupled rotations. Angular velocity of kinetic pressures components in vector form are expressed by using angular acceleration and angular velocity of component coupled rotations of gyrorotor-disk.

Keywords: coupled rotation; no intersecting axes; deviational mass moment vector; rotator; kinetic pressures; kinetic pressure components; nonlinear dynamics; gyrorotordisk; eccentricity; angle of inclination, deviation kinetic couple; fixed point; graphical presentations; three parameter analysis

Abstract: In this paper, we present a calculation methodology of the testing duration of the products’ reliability, using the Weibull distribution, which allows the estimation of the mean duration of a censored and/or complete test, as well as of the confidence intervals for this duration. By using these values we can improve the adequate planning and allocation of material and human resources for the specific testing activities. The proposed methodology and the results’ accuracy were verified using the Monte Carlo data simulation method.

Keywords: reliability; test plan; Weibull distribution; Monte Carlo simulation

Abstract: One crisis which human beings will probably face in the upcoming decades is the water crisis. The crisis in arid and semi-arid regions covering a large part of Iran would be much more severe. Thus, using novel methods of water collection such as construction of underground dam is so important. Decision making and selection of an appropriate option in construction of such dams is one basic challenge. The major issue in construction of such dams is selecting an appropriate location. Selecting the best location for building underground dams is a challenge due to involvement of a wide range of influential factors. In this paper, analytic hierarchy process (AHP), one of multi-criteria decision making (MCDM) techniques in fuzzy environment is applied to select the optimal alternative for construction of an underground dam in a case study. Results show that using AHP in the fuzzy environment improves decision making through considering more important factors in decision making.

Keywords: underground dam; multi-criteria decision making; fuzzy theory; AHP