# No 4 (2015)

### The Usefulness of Cooperation in Two-Person Games with Quadratic Pay off Functions on the Rectangle

Aboubacar M.

#### Abstract

In this paper we study an important question for the game theory of two players, about essentiality of such games. The investigation is carried out in a particular case, in the class of games with quadratic payoff functions on a rectangle. The essentiality in two players’ games means that by joining the two players in union, both players can get positive additions to guaranteed payoff. The essentiality of two players’ games has not always occurred. Thus, the joining of the two players in union, in general, may be useful and sometimes (in case of absence of essentiality) useless. In applications, for example in analysis of the economic activities of firms or countries, the question of usefulness of the union acquires a lot of interest. In the general game theory, the question of essentiality of games is given a little attention at the moment. Apparently, this is due to the difficulty of this problem in the general case. Note that the games with quadratic payoff functions are frequently used in game theory for modeling different kinds of processes being investigated, for example, in mathematical economics.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):5-9
5-9

### Sufficient Conditions of Solvability of a Functional Differential Equation with Orthotropic Contractions in Weighted Spaces

Tasevich A.L.

#### Abstract

In this paper the solvability of a functional-differential equation is studied in the scale of Kondrat’ev weighted spaces. The equation is considered in the real plane, it has constant coefficients and transformations of arguments of required function, and this transformation consists in the contraction of one argument and the expansion of another. These transformations are called by orthotropic contractions here. The considered problem boils down to an invertibility of a difference operator on the real line with variable smooth coefficients stabilized in the infinity. Sufficient conditions of the invertibility of the difference operator and the initial functional differential operator were obtained in algebraic form. It is well-known that the properties of functional differential equations are largely defined by the structure of point orbits under the action of a group generated by transformations attended in the equation. The orbits of isotropic contractions are situated on the rays passing through the origin and condense near the origin”- a fixed point of the operator. In case of contraction of one argument and expansion of another the orbits are situated on curves having a form of hyperbolas. Herewith the origin is a fixed point of the operator as before. Therefore it is natural to assume that problems with orthotropic contractions differ in their properties and analysis methods from problems with isotropic contractions.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):10-17
10-17

### Simplified Sea Level Estimation Based on Logistic Equation and Paleoclimate Data

Sorokin L.V.

#### Abstract

During the pre-industrial period the variation of Global Surface Air Temperature on 1 ºC change Global Mean Sea Level on 24.79 m (SL < 0) and in the modern time - 7.5 m (SL > 0). For the global warming period the rising rate of the Global Mean Sea Level will slowing down in 3.3 times simultaneously increases the Global Surface Air Temperature rate in 3.3 times. In the present time the Global Surface Air Temperature is 0.8 ºC higher then pre-industrial level that will lead to the sea-level rise on 6 m. The logistic equation for the current climate conditions (global warming on 0.8 ºC) has provide us a solution that the sea level within next 95 years will rise up to +5 m and will stabilized at +6 m level within the relaxation time of 150 years. The EU Strategy on adaptation to climate change assume that the global warming must be limited to below 2 ºC above pre-industrial level, that within some ages will lead to the sea-level rise on 15 m (15 times grater than for the A1B scenario, with the Sea-level rise up to 1 m). So in the future conditions the EU Strategy on adaptation to climate change do not assume rapid climate changes that will significantly increase the economic losses from Sea Level Rise.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):18-29
18-29

### Research of Acceleration Calculations in Solving Scientific Problems on the Heterogeneous Cluster HybriLIT

Alexandrov E.I., Belyakov D.V., Matveyev M.A., Podgainy D.V., Streltsova O.I., Torosyan S.G., Zemlyanaya E.V., Zrelov P.V., Zuev M.I.

#### Abstract

The paper presents some test results of the heterogeneous computing cluster HybriLIT put into operation at the Laboratory information technologies of the Joint Institute for Nuclear Research. The structure of the cluster includes computational nodes with NVIDIA graphical accelerators and Intel Xeon Phi coprocessors. The necessity of integration of such a computational platform in the JINR Multifunctional Information and Computing Complex is determined by a global tendency to use hybrid computing architectures for carrying out massive-parallel computations in applied scientific problems solution. Test of the cluster aimed at: first of all, test of the efficiency of hardware and software settings that include operational system, resource manager, file system, compilers, and, secondly, test of the efficiency of using different architectures for the solution of particular applied problems in order to provide user guides on specialized libraries. For realization of the cluster test, an approach that includes test computations by means of standard program packages such as Linpack and program complexes established in LIT has been developed. The presented results show that the use of hybrid computing architectures allow accelerate the solution of applied scientific problems, and heterogeneous computing cluster HybriLIT is an effective means of accomplishing this aim.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):30-37
30-37

### Model of Hydrogen Atom Quantum Measurements on Rigged Hilbert Spaces

Zorin A.V.

#### Abstract

The measurement procedure makes the isolated (closed) quantum system to be the open one. The operators of observables of rather simple explicit form are converted into pseudo-differential operators of more complex form. The author has proposed the method of establishing consistency between the theoretical data of conventional quantum mechanics of (isolated) quantum objects and experimental data on the measured values of the observables of corresponding open quantum objects. In this paper, the proposed correspondence is used for the construction of rigged Hilbert spaces, in which the operators of measured the observables of hydrogen-like atom admit spectral decomposition.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):38-45
38-45

### Modulated Characteristics of Nanosize p-i-n SOI Units

Masalsky N.V.

#### Abstract

Optical waveguide devices - modulators and switches are one of the most demanded photon devices on whom further possibility of increase of keyword parameters of integrated optical circuits and telecommunication systems depends. Search of new development approaches of necessary element basis on the basis of combination of planar technology of integrated optics and the thin-film technology “silicon on an insulator” is necessary for increase of their competitiveness. For silicon devices methods of electrooptical modulation are based on effect of dispersion of the free carriers which density depends on the level of applied voltage. In operation the perspectives of application of p-i-n of electrooptical modulators executed on the basis of the “silicon on an insulator” technology are discussed. In comparison with other class of modulators on the basis of lithium niobate, the researched devices potentially have a row of advantages, such as the low level of power consumption (within 1 W), nanosized miniaturization, high throughput (modulation frequency band over 100 GHz). The researched nanophoton devices are hi-tech and expensive. Their simulation at the physical layer is important and as for understanding of the physical processes proceeding in them and for optimization of their parameters for the purpose of achievement of optimum characteristics. On the basis of computer simulation in this operation possibility of optimization of parameters of nanosized electrooptical p-i-n of the waveguide modulators which potentially will allow to realize highly effective optical modulation is researched.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):46-55
46-55

### Spinor Model of the Transition of the Universe from the Slow Expansion to the Accelerating One

Kovalchukov N.A., Shikin G.N., Yuschenko L.P.

#### Abstract

According to the astronomical data our Universe is expanding with acceleration in the present stage. One of the possible explanation of such evolution is that in the present epoch about 70% of the total mass of the Universe is represented by dark energy(cosmic vacuum). Under simulating of the matter of the Universe we use the state equation of the form P =Wɛ, where P is pressure, ɛ is energy density. Under such state equation W = -1 corresponds to dark energy, that leads us to the state equation of the ideal ﬂuid with negative pressure: P= -ɛ < 0. This case corresponds to the cosmological constant. The transition from the slow expansion to the accelerating one means that at some stage of the evolution acceleration a(t) changes it’s sign from minus to plus. In this case the parameter W changes from values W > -1 to the value W = -1. It is notable that the case - 1 < W < -1 3 corresponds to the quintessence, and W < -1 - to the phantom matter. In this work we consider the transition of the Universe from the slow expansion to the accelerating one on the assumption that the Universe is ﬁlled by spinor matter.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):56-60
56-60

### Magnetic Islands and Confinement of Charged Particlesin Toroidal Magnetic Systems

Podturova O.I.

#### Abstract

Results of the study of charged particle trajectories in a toroidal magnetic configuration like tokamak with “magnetic islands” are presented. Particle trajectories are calculated using the three-dimensional numerical code based on numerical integration of exact equations of motion. Magnetic configuration is described analytically with use of an universal approach in terms of Hamiltonian function. This approach generalizes the traditional streaming presentation of magnetic field and guaranties the solenoidality condition being satisfied exactly. To describe magnetic configuration with islands, iterative procedure is used; the initial (basic) magnetic configuration is chosen to have nested magnetic surfaces. Small helical perturbation splits the rational surface resonant with the perturbation helicity. In the vicinity of such surface the chain of magnetic islands is formed. Our study of the motion of charged particles showed that the existence of the magnetic island has no qualitative impact on the Poincare cross-section of the trapped particle trajectory for having a standard “banana” form, typical for basic tokamak configuration. As for passing particle, which moves predominantly along the magnetic field line, its trace of the trajectory has an island structure in the poloidal cross-section. It is shown that both trapped and passing particles in their motion can cross the separatrix between magnetic island and nested magnetic surfaces; the crossing can happen in any point of separatrix. Thus, it is shown that the magnetic island has no “barrier” properties and isn’t capable to provide the improved confinement of charged particles.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):61-73
61-73

### Searches for a dineutron in the nd → p( nn) charge-exhcange process at enegry T n = 0.5∇⋅2.0 GeV

Shindin R.A., Guriev D.K., Livanov A.N., Yudin I.P.

#### Abstract

Our measurements of diﬀerential cross-sections of both np ‎→‎ pn elastic and nd ‎→‎ p(nn) quasielastic charge-exchange reactions under 0∘ over the energy range Tn = 0.5∇⋅2.0 GeVallowed to deﬁne the ration Rdp between them and it separates two Flip and Non-Flip parts of the free np ‎→‎ pn process using the Dean theorem. The main idea of this theorem calculations is based on the Pauli principle for two slow neutrons in the ﬁnal state that allowed to express the quantitative diﬀerence between the elastic and quasielastic np-reactions. However they should also diﬀer in their quality because the deuteron is converted to the nn-pair and the nd-interaction becomes non absolutely elastic. In the ﬁrst view we can take into account the binding energy ɛ ≈ 2.3 MeV but this simple correction can not explain all features which are observed in the momentum spectrum of the secondaries protons. We need to suppose that two slow neutrons in the ﬁnal state form the intermediate system which has own distribution of internal Fermi momentum and it cannot be described by the suitable Hulthen expression or using other approaches for deuteron. Also it cannot be explaned by the hypoteisis of quasi particle dineutron with mass about of 2 mn.
Discrete and Continuous Models and Applied Computational Science. 2015;(4):74-81
74-81

### Svedeniya ob avtorakh

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Discrete and Continuous Models and Applied Computational Science. 2015;(4):82-83
82-83

### Guidelines for Authors

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Discrete and Continuous Models and Applied Computational Science. 2015;(4):84-85
84-85