Nonlinear networks and invariance

Symmetric and non-symmetric waves in the osmosis K(2, 2) equation

Emerging trends in soft set theory and related topics.

Symmetric solutions of linear matrix equations by matrix decompositions

Harmonic balance is today the most useful approach for nonlinear microwave network analysis. Usually FFT is used to convert the nonlinear elements time waveforms into the frequency domain as part of the harmonic balance process. This approach is straight forward for a single frequency. excitation, but is quite complicated and time consuming for the multi-frequency excitation case. Steer [1] proposed the Generalized Power Series, which enables for a given nonlinearity a direct calculation of the currents spectrum given the voltages spectrum. In this paper we propose a mathematical model termed Signal Space, which is similar to Steer's approach but better suited to implementation for harmonic balance. In [2] we extracted the signal space and made an approximation between the harmonic currents subspaces to enable a treatment of high nonlinearity of elements. Under describing function criteria we use the Signal Space approach to get expressions for oscillator parameters such as oscillation frequency, stability condition, injection locking bandwidth etc. Our results are compatible with Kurakawa's expressions.

Despite the fact that artificial neural networks (ANNs) have been proposed primarily as nonlinear learning systems, considerable insight into the behaviour of these networks can be gained from linear modelling techniques, for which the literature is vast (see for example [8, 137, 181]), and significantly the resultant theory is directly applicable to a powerful and special class of ANNs, i.e. linear-in-the-parameters networks, which forms the basis of this book. An obvious reason to use ANNs is their ability to approximate arbitrarily well any continuous nonlinear function, with the specific network architecture and parameter/weight adjustment algorithm determining how well learning is achieved. Of particular interest in adaptive control and estimation is the ability of algorithms to model, track and control on-line. However, it is infeasible to assume that the input signals excite the whole of the state space (a prerequisite of identification theory), and so it is necessary to consider the effects of a reduced input signal on overall functional approximation for various network architectures and associated learning laws. Here learning must be local, in that adjustable network weights or parameters should only affect the network’s output locally. In ANNs, the vast majority of supervised learning rules are based on the assumption that the nonlinear network can be locally linearised, so that it is natural to develop on-line learning algorithms with provable learning network stability and convergence conditions, and to choose ANNs which have linear-in-the-parameters with local behavioural characteristics.

The object of this study is two thin elastic isotropic rectangular plates in an infinitely long rectangular parallelepiped with an ideal fluid. The first plate is the upper base of the rectangular parallelepiped, and the second one horizontally separates ideal fluids that have different densities. The subject of the study is the normal joint plane vibrations of elastic rectangular plates and an incompressible fluid and the conditions that enable the stability of these vibrations. In the linear statement, the frequency spectrum of normal plane vibrations of two elastic isotropic plates in an infinitely long rectangular parallelepiped with an ideal incompressible fluid has been investigated. The frequency equation of joint vibrations of the plates and the ideal fluid was reduced to the form of an eighth-order determinant for arbitrary cases of fixing the contours of the plates. The case of clamped contours of the plates and the case of rebirth of the plates into membranes is analyzed. Based on analytical studies of infinite series in the transcendental frequency equation, exact stability conditions for the combined oscillations of plates and liquid were established. It has been shown that instability of oscillations of plates and liquid occurs when a heavier liquid is above a less heavy liquid. The derived stability conditions for symmetric and asymmetric oscillations of plates and liquid do not depend on the elastic parameters of the upper plate, the mass characteristics of the plates and the depths of filling liquids. The analytically obtained exact stability conditions for the combined oscillations of the plate and liquid generalize the previously obtained approximate stability conditions for this problem. The numerical calculations of the frequency equation confirmed the analytical studies of the stability conditions. The results could be used in the calculation and design of mechanical objects related to the storage and transportation of liquid cargo

Stability and bifurcations of hysteretic systems subjected to principal parametric excitation

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

Some symmetric boundary value problems and non-symmetric solutions

The laminar flow in a two-dimensional channel may produce either symmetric or non-symmetric steady solutions, depending on the value of the critical Reynolds number. It has been concluded by several researchers that the flow remains symmetric up to a certain Reynolds number depending upon the expansion ratio, while asymmetries appear at higher Reynolds numbers. The geometric expansion ratio in the current study is chosen to be 1:3, and the bifurcation phenomenon has been investigated for this sudden expansion flow. A passive control mechanism is introduced by inserting a two-dimensional plate in the flow with the aim of stabilizing the unstable symmetric flow configuration in the channel. The computations indicate that the critical Reynolds number of the symmetry-breaking bifurcation increases as the length of the plate increases.

One of the main properties of solutions of nonlinear neural networks is stability and often the direct Lyapunov method is used to study stability properties. Often the present state depends on its maximum value over a past time interval. It requires the study of a new model, so called model with supremum. Some sufficient conditions for stability of equilibrium of nonlinear neural networks with supremum, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as non-Lipschitz activation functions are studied.

The purpose of this article is to outline a nonrigorous and highly intuitive introductory approach to the sort of group representation theory used in molecular and solid state physics. We examine the real three-space spanned by orthonormal polar basis vectors {‖i〉, i=1,2,3}; first it is shown that the familiar block diagonal ’’rotation matrix’’ is simply the {‖i〉} representation of the geometric transformation operator ?φ which causes a counterclockwise rotation of any three-space polar vector by the angle φ about the ‖3〉 axis. By applying this argument to the group of symmetry operators of the ammonia molecule, it becomes self-evident that the two irreducible representations obtainable from the {‖i〉} representation of C3v correspond to subspaces of real-space invariant relative to the six operators of that group. And the basis vectors which span these invariant subspaces (and which produce the corresponding irreducible representations) have clearly defined symmetry (A1 and E) properties under the operations of C3v. This idea is further illustrated in consideration of the representation produced by the in-plane normal modes of vibration of the three hydrogen atoms in ammonia. Finally, it is shown that the approach can readily be extended into function space.

Some properties on the structure of dynamic equations and a method of systematic formulations of a fairly general class of nonlinear RLC networks and lumped parameter active networks are presented. It is shown that by a differential and integral transformation such that each range of network variables becomes a standard function space, the equation and solution are described by synthetic operators of a certain standard type. The solution is then, definitely computable. Conditions for the unique solvability, which are given by global invertibility of the synthetic operator describing the equation, are derived from the contraction mapping theorem in a Banach space. These conditions are finally expressed in terms of differential coefficients of branch characteristics.

Delay-dependent [formula omitted] filtering for complex dynamical networks with time-varying delays in nonlinear function and network couplings

We consider a finite-time stochastic stability for distributed parameter systems with free boundary condition. This mathematical model is deduced from the analysis of the tracking error of the installed pipe from the designed path in the micro-tunneling machine. After showing that the proposed mathematical system has a solution in some function spaces, we study the finite-time stability problem. The buckling phenomena of the installed pipe is simulated related to the derived sufficient conditions for the finite-time stability.