Effective JSJ decompositions

We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

We give a description of the model-theoretic relation of forking independence in terms of the notion of JSJ decompositions in non-abelian free groups

A description of the general solution of given bounded periodicity exponent is obtained for an arbitrary system of equations in a free group. On the basis of this result an algorithm is constructed that computes the rank of coefficient-free systems of equations. Bibliography: 7 titles.

Given a finite irredundant system of equations to be solved over the free group, one has four non-negative integers: the number of equations, the number of indeterminants, the rank of the system and the Abelian rank of the system. We show which four-tuples can actually occur.

The contemporary advancement in smart technology has led to a greater business prospective. Smart system like cyber-physical system (CPS) has become prevalent in many area of applications. Challenges in integration of CPS with group of network has physical components to sense the state of changes in the real world is paving way towards the introduction of enhanced security mechanism. In this paper, we propose a security framework for group of network is a pseudo-free to communicate with the cyber physical system to assure high security. Pseudo-freeness is a very strong assumption of security used in cryptography. Group is said to be pseudo-free group, if an adversary who is allowed to see group description, is unable to solve non-trivial equations. Non-trivial equations are the equations that have no solution in a free group. The notion of pseudo-freeness was introduced with few open problems whether such groups exists in cryptosystem or not. The conjecture that the RSA group is pseudo-free has been settled for case of static adversary who is given only the group description and its goal is to come up with an equation and solution. Later on the notion of pseudo-freeness has been extended to adaptive adversary for case of univariate equations. In case of adaptive pseudo-free groups, the adversary learns solutions of some non-trivial equations before producing a new equation. Here we make extension to the notion of pseudo-free groups from univariate to multivariate equations and system of equations in face of adaptive adversary. We define and prove the security of adaptive pseudo-free groups for case of multivariate equations and system of equations. According to the results, this framework can be used as a system design for a group to have strong security with cyber physical system.

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.

This paper presents an algorithm based on the spectral Laguerre method for approximation of time derivatives as applied to a problem of seismic wave propagation in porous media with energy dissipation. The initial system of equations is written as a first-order hyperbolic system in terms of velocities, stresses, and pore pressure. To numerically solve the problem, a combination of an analytical Laguerre transform and a finite-difference method is used. The method proposed in the paper is an analog of a well-known spectral method based on the Fourier transform. However, unlike the Fourier transform, the integral Laguerre transform with respect to time reduces the initial problem to a system of equations in which the expansion parameter is present only in the right-hand side of the equations as a recurrence relation. As compared to the finite-difference method, with an analytical transform in the spectral method it is possible to reduce the original problem to a system of differential equations having only derivatives with respect to the spatial coordinates. This allows using the known stable difference scheme for the recurrence solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Because of the presence of a second longitudinal wave with low velocity, the use of difference schemes in all the coordinates to obtain stable solutions requires a small step consistent both with respect to time and space, which inevitably increases the execution time.

Equations in free groups have become prominent recently in connection with the solution to the well-known Tarski conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P -space. The special case of one-variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial-time algorithm for finding solution sets. One-variable equations are the only general subclass of equations in free groups for which such results are known. We improve on previous attempts to use cancellation arguments by employing a new method of reduction motivated by techniques from formal language theory. Our paper is self-contained; we assume only knowedge of basic facts about free groups.

Inelastic analysis of flexible bars using simplified nonlinear equivalent systems

Numerical modeling of multi-element airfoil's aerodynamics employs the Reynolds averaged Navier-Stokes equations of incompressible environment, which are related via a single-parametric differential turbulence model by Spalart-Allmaras. The system of initial equations has been recorded with respect to an arbitrary curvilinear coordinate system. The pressure and velocity fields have been aligned by using an artificial compressibility method modified for calculating the nonstationary problems. The system of initial equations has been integrated numerically by applying a control volume method. The counter-flow Rogers-Kwak approximation has been used for convection flows, based on the Roe scheme of third-order accuracy. The turbulence models, in order to approximate the convection components, employed a TVD scheme with the third-order flow limiter ISNAS. The paper reports results from calculating a turbulent flow around a multi-element airfoil in a wide range of the angles of attack. The result of the current research is the performed analysis of the flow field around a multi-element airfoil, pressure coefficients, the lifting force, as well as the drag force. Physical features in a flow structure at flowing around the multi-element airfoil 30P30N have been identified. In the investigated range of the angles of attack, flowing around a airfoil in the takeoff and landing configuration is stationary in nature except for the regions where the flow is detached from sharp edges, such as the slat's inside part and a region in the tail part of the main profile. There are recirculation currents within these regions. With an increase in the angle of attack, dimensions of the detachable zone at the slat's inner surface decrease while remaining almost unchanged in the tail part of the main profile. At the top surface of the main profile there forms a jet of air, due to the acceleration of the flow between the slat and the leading edge of the main profile. The existence of a gap between the main profile and the flap leads to the interference of jet currents at the upper surface of the slat. It has been shown that the takeoff and landing configuration demonstrates the higher values of the lifting force coefficient than the cruise configuration, especially at large angles of attack. The calculation results agree well with the data by other authors.

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form $$W(x,x')=0$$ (which is called here the equivalent system of the 3rd type). It is important that the function $$W(x,x')$$ can be built a priori. Once $$W(x,x')$$ is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition $$W(x,x')=0$$ ; (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.

Based on a model of a double-velocity and two-temperature medium the authors constructed a system of equations that describes plane or axisymmetric turbulent flow of a gas suspension in a channel with permeable walls. The system of equations of motion and heat transfer reduces to a system of ordinary differential equations, whose integration is much less difficult than solution of the initial system. The authors obtained the distribution of the velocity and local characteristics of turbulence in the channel with injection with allowance for the inverse effect of a condensed phase.

A system of equations for the electron and lattice temperatures around and along the path of a 700-MeV heavy (uranium) ion in nickel (one-layer material) is solved numerically in the axially symmetric cylindrical coordinate system under the assumption of temperature-dependent specific heat and thermal conductivity. The obtained dependences of the lattice temperature on the radius (distance from the ion path) and depth suggest that the ionization energy loss of a 700-MeV uranium ion in nickel is sufficient to melt the material. A comparative analysis with the linear model is performed and the maximum radius and depth of the region where the target material can melt is estimated. Then, the initial system of equations is solved for the region around and along the path of a 710-MeV heavy (bismuth 209Bi) ion in the two-layer material Ni(2 μm)-W with constant thermophysical parameters. The obtained dependences of the lattice temperature on the radius and depth show that the ionization energy loss of a 710-MeV bismuth ion in this two-layer material is sufficient for melting. The maximum radius and depth of the regions in the target material where phase transitions may occur are estimated.

We obtain exact plane-symmetric solutions of the spinor field equations with a nonlinear term that is an arbitrary functions of the invariant\(S = \bar \psi \psi \) and with the self-gravitational field taken into account. Conditions are formulated for which the initial system of Einstein's equation and the spinor field equations with a power-law nonlinearity have regular solutions with localized (negative) spinor field energy density: so-called soliton-like solutions. Exact solutions of the spinor field equations are also obtained in flat space—time in this case and it is shown that the initial system of equations does not have soliton-like solutions. Hence the self-gravitational field plays a crucial (regularizing) in soliton-like solutions of the nonlinear spinor field equations.

An algorithm for solving a class of systems of equations in a free group is constructed, and the structure of the general solution is studied.