Classification of voting algorithms for N-version software

A voting algorithm in N-version software is a crucial component that evaluates the execution of each of the N versions and determines the correct result. Obviously, the result of the voting algorithm determines the outcome of the N-version software in general. Thus, the choice of the voting algorithm is a vital issue. A lot of voting algorithms were already developed and they may be selected for implementation based on the specifics of the analysis of input data. However, the voting algorithms applied in N-version software are not classified. This article presents an overview of classic and recent voting algorithms used in N-version software and the authors' classification of the voting algorithms. Moreover, the steps of the voting algorithms are presented and the distinctive features of the voting algorithms in Nversion software are defined. © Published under licence by IOP Publishing Ltd

The Assembly of Hierarchs in Krasnoyarsk: Historical and Architectural Study

The article examines the stages of construction history of the Assembly of Hierarchs in Krasnoyarsk that is a monument of architecture in the late XIX century. Archival documents and field studies of the object have been used for this research. On the basis of documentary material, the role of the first bishop of the Yenisei Diocese, Nicodemus, has been revealed in organizing and conducting activities related to site selection for the Hierarchs town church in Krasnoyarsk, with its buildings, as well as the design of interior decoration of residential buildings and chapel. A complex study design drawings of buildings in initial and final stages of formation have been presented for the first time; their architectural and planning analysis has been carried out. They allow us not only to discover creative proposals of architects, but also trace the techniques of architectural projects in the XIX century. The names of architects, who made up projects of structure and rebuilding homes for the Assembly of Hierarchs in Krasnoyarsk in the pre-revolutionary period, have been revealed. These materials given in the article make it possible to recreate an authentic architectural and planning structure of the Assembly of Hierarchs and chapel and the elements of the decor with great accuracy nowadays. The results of the study prove to have a great architectural and urban planning, historical and cultural value of the Assembly of Hierarchs in Krasnoyarsk.В статье рассмотрены этапы строительной истории Архиерейского дома в Красноярске - памятника архитектуры конца XIX века. Для исследования использованы архивные документы и натурное изучение объекта. На основе документальных материалов выявлена роль первого епископа Енисейской епархии, Никодима, в организации и проведении мероприятий, связанных с выбором участка для архиерейского подворья в Красноярске, с его застройкой, а также с оформлением внутреннего убранства жилых помещений и домовой церкви. Впервые представлены проектные чертежи комплекса исследуемых зданий начального и завершающего этапов формирования; проведен их архитектурно-планировочный анализ. Они позволяют раскрыть не только творческие предложения зодчих, но и методы составления архитектурных проектов в XIX веке. Выявлены фамилии архитекторов, составлявших в дореволюционный период проекты устройства и перестройки дома для Архиерея в Красноярске. Приведенные в статье материалы исследования позволяют с большой достоверностью воссоздать в настоящее время подлинную архитектурно-планировочную структуру Архиерейского дома и домовой церкви, элементы их декора. Результаты исследования подтверждают высокую архитектурно-градостроительную и историко-культурную ценность Архиерейского дома в Красноярске

Multiple Factorizations of Bivariate Linear Partial Differential Operators

We study the case when a bivariate Linear Partial Differential Operator (LPDO) of orders three or four has several different factorizations. We prove that a third-order bivariate LPDO has a first-order left and right factors such that their symbols are co-prime if and only if the operator has a factorization into three factors, the left one of which is exactly the initial left factor and the right one is exactly the initial right factor. We show that the condition that the symbols of the initial left and right factors are co-prime is essential, and that the analogous statement "as it is" is not true for LPDOs of order four. Then we consider completely reducible LPDOs, which are defined as an intersection of principal ideals. Such operators may also be required to have several different factorizations. Considering all possible cases, we ruled out some of them from the consideration due to the first result of the paper. The explicit formulae for the sufficient conditions for the complete reducibility of an LPDO were found also

Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2+1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.Comment: Latex, 34 page

Thermodynamic phase transitions and shock singularities

We show that under rather general assumptions on the form of the entropy function, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of the characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of non-hydrodynamic type such as the classical plasma and the ideal Bose gas are also discussed.Comment: revised version, 18 pages, 6 figure

Integrable Systems and Metrics of Constant Curvature

In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with metrics of constant curvature. Kaup-Boussinesq system has three local Hamiltonian structures and one nonlocal Hamiltonian structure associated with metric of constant curvature. Darboux theorem (reducing Hamiltonian structures to canonical form ''d/dx'' by differential substitutions and reciprocal transformations) for these Hamiltonian structures is proved

The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies

The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied.Comment: 29 pages, LaTe