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Anderson's model for disordered systems is considered in the case where the transfer matrix elements fluctuate according to a Lorentzian distribution. It is shown that the exact ensemble averaged Green's function may be obtained if the energy level on each site depends linearly on the overlap integrals. Using this result the stability of amorphous Heisenberg ferromagnets is studied. The Weaire model of an amorphous covalent semiconductor with fluctuating matrix elements is considered. Numerical results for the density of states are given. [Russian Text Ignored]

Recently, Little demonstrated the existence of persistent states in a neural network when a certain transfer matrix has approximately degenerate maximum eigenvalues. He showed the direst analogy of the persistence of the neuronal firing patterns considered in discrete time steps, to the long range apatial order in an Ising spin crystal system. The ordered phase of the spin system occurs below the Curie point temparature. Hence in analogy, a factor representing the temparature of the neural network is assumed in Little's model for the transfer matrix. We derive his transfer matrix for a neural network and explicitly relate this temperature or “smearing” phenomenon to the experimentally observed fluctuations in the postsynaptic potentials.

This paper presents an extension to the Matrix Transfer Method which gives quadratic convergence to a natural frequency from within a range on either side. The method is described and illustrated by application to: 1 A torsional vibration problem, and 2 A lumped mass beam vibration problem.

In the high speed printer the fine quality print is obtained by means of the design on the basis of the dynamic characteristics of the aggregate motion lever for the typehead selection. This paper describes the study of these characteristics by means of the transfer matrix methods for the branch system. The results obtained are as follows : (1) The analytical method are given corresponding to the figure of the aggregate lever.(2) The method to obtain the concentrated mass model equivalent to the distributed mass system is expressed.(3) The theoretical results on the basis of the above-mentioned method agree with the experimental values well.

Exact expressions are given for the free energy of the classical Heisenberg model in one lattice dimension and two or three spin dimensions with general anisotropy; the two-dimensional result, already known, is in terms of Mathieu functions and the three-dimensional result in terms of Lame wavefunctions. The method presented solves the eigenvalue problem for an appropriate transfer matrix by relating it to solutions of the Helmholtz equation in suitable curvilinear coordinate systems. High and low temperature limits are discussed for the three-dimensional spin case and the connection given with already known results for partial anisotropy.

In one form or another, practically all exactly soluble models are one dimensional. Even the two-dimensional Ising model owes its solvability to the fact that it can be reduced to a one-dimensional problem. Rather than deal with a number of unrelated models, which have been catalogued,1 we will treat one family of models which is at the frontier of the subject at present. These models have solutions which have been recently explored and in turn promise even further solutions. In order to give some background we will first deal with the Heisenberg chain. This is connected with the two-dimensional Ising model (first solved by Onsager in 1944) which is in turn related to three other models which will be discussed, namely: Ice, F, and KDP.

This paper gives a brief description of the set of programs, elaborated recently for the analysis of three- phase voltage controlled induction motors, fed by back- to-back connected thyristor-thyristor or thyristor-diode pairs. The most interesting results of the computer analysis are also published. The set of programs is suitable for both exact calculation to check the steady-state parameters and analysis of dynamic performance. The eigenvalues of the transfer matrix are plotted in a speedtorque coordinate system. Tendency for a special subharmonic self-oscillation may be seen. Some characteristic root-loci are also represented. In addition the z-transfer function of the controlled motor, the design method of the analogue or digital control system are outlined.

A perturbed piezoelectric waveguide is constructed from a reference waveguide, which is specified by the eigenvalues and eigenfunctions, by the coordinate transformation, or other methods. In the perturbed waveguide, coupling occurs between normal modes of the reference waveguide. Coupling phenomena are analyzed by transforming Maxwell's and Newton's equations into the variational expression of the phase constant of the normal mode and then by applying the Rayleigh‐Ritz method. As an example, the couplings between purely elastic waveguide modes are discussed; and then the two‐mode approximation is applied in the case when two modes are degenerate or nearly degenerate. Mode conversion phenomena are discussed and from the viewpoint of the network theory, the mode transfer matrix relating the complex amplitudes of the two modes along the waveguide is defined and developed.

A synthesis technique is presented, in the form of a computer algorithm, for linear, minimum-phase, multivariable plants with defined ranges of parameter uncertainty, to satisfy precisely specifications of the 'basically non-interacting' type. These are that the elements of the system transfer matrix T(s) = [ταβ (s)] satisfy k i l ω ≤ τ ii j ω ≤ k i 2 ω , τ ij j ω ≤ k ij ω i ≠ j , over the range of parameter uncertainty. The design is optimum in the sense of achieving this with loop transfer elements which decrease in magnitude at the fastest possible rate vs w . This is important for the reduction of sensor noise effects, whose typical large amplification tends to saturate plant elements.

The question of achieving a variety of specific design goals (e.g., arbitrary pole placement, decoupling, and exact model matching) for linear multivariable systems subjected to unknown step disturbances on the entire internal state and output is constructively resolved via a synthesis procedure which simultaneously combines integral feedforward compensation with asymptotic state estimation. The technique relies on the ability to express the desired closed-loop transfer matrix of the compensated system as the product of the open-loop transfer matrix and any proper rational transfer matrix, assuming the given system has no zeros at the origin and a stable closed-loop design is sought. The method employs some recent results involving the differential operator approach to multivariable system analysis and synthesis, and the treatment is confined almost exclusively to the frequency domain.

Exact renormalization-group recursion relations are studied for a spin-1 Ising model in one dimension. In this model, critical lines are marked by a double degeneracy of the largest eigenvalue of the transfer matrix, and the tricritical point by a triple degeneracy. This is directly related to the tricritical fixed point being more unstable than the critical fixed points.

A restricted cell model for hard, parallel squares is presented both in a discrete and a continuous version. The model is solved exactly by means of a transfer matrix method and the thermodynamic properties are calculated. Some correlation functions are also obtained, which show that the long range order decays at least as fast as 1/r 2.

Previous calculations have given accurate first-order focal properties for the two-tube electrostatic lens at discrete voltage ratios. For computer optimization, calculations involving systems of two-tube lenses, one must be able to calculate the focal properties continuously over some arbitrary range of voltage ratios. Hence the data must be displayed in a continuous manner, and a method of interpolation is needed which yields functions having a high degree of smoothness. Special care must be taken to describe the lens behavior correctly near zero strength or for the voltage ratio approaching unity. A satisfactory solution to this problem has been achieved using cubic splines. The resulting functions of the focal properties are continuous and have continuous first and second derivatives. The total beam behavior, and hence the system design, is determined by the transfer matrix which is obtained from the focal properties. To achieve sufficient accuracy in the lens calculations over the entire range of required focal properties, the region near zero lens strength had to be treated separately.

The toroidal condenser is studied as an electrostatic mirror for motion in the vicinity of its symmetry plane. The elements of the transfer matrix for the in- and off-plane optical properties are given explicitly to second order. The basic quantities entering the matrix are shown to be obtainable from one quadrature and the solutions of two differential equations which can be solved in succession. No fringing field or space charge effects are considered.

The potential energy expression and the (14 by 14) stiffness matrix of a straight thin-walled beam element of open asymmetric cross section, subjected to initial axial force, initial bending moments, and initial bimoment, are derived. The transformation matrix relating the forces and displacements (including bimoment and warping parameter) at the adjacent end cross section of two elements meeting at an angle is deduced as the limiting case of a transfer matrix of a curved beam. To cope with asymmetric cross sections, some element displacements and forces are referred to the shear center and others to the cross-sectional centroid and the matrix for transformation from shear center to centroid is set up. The incremental larger-displacement analysis is formulated using the Eulerian coordinate approach with updating of the local coordinate systems at each load increment. The deformed beams are imagined to be composed of straight elements. Results of lateral post-buckling analysis of various beams are presented.

We use the properties of subharmonic functions to prove the following results, First, for any lattice system with finite-range forces there is a gap in the spectrum of the transfer matrix, which persists in the thermodynamic limit, if the fugacityz lies in a regionE of the complex plane that contains the origin and is free of zeros of the grand partition function (with periodic boundary conditions) as the thermodynamic limit is approached. Secondly, if the transfer matrix is symmetric (for example, with nearest and next-nearest neighbor interactions in two dimensions), and if infinite-volume Ursell functions exist that are independent of the order in which the various sides of the periodicity box tend to infinity, then these Ursell functions decay exponentially with distance for all positivez inE. (For the Ising ferromagnet with two-body interactions, exponential decay holds forz inE even if the range of interaction is not restricted to one lattice spacing). Thirdly, if the interaction potential decays moreslowly than any decaying exponential, then so do all the infinite-volume Ursell functions, for almost all sufficiently small fugacities in the case of general lattice systems, and for all real magnetic fields in the case of Ising ferromagnets.

Although the XY model cannot have a transfer matrix in the sense of an Ising model, it is shown that the partition function is as if there were one. The partition function for the closed anisotropic XY chain with N sites and arbitrary external field is shown to take the form lambda 0N+ sigma k=1infinity gk lambda kN where the gk are integers. This means that there could be a transfer matrix of infinite size, the lambda k being its eigenvalues and the gk being the corresponding degeneracies.

The paper deals with the classical problem of axi-symmetric transmission of low amplitude waves through a circular pipe containing a viscous liquid. Exact governing equations are identified and solved, the radial as well as the axial component of the velocity being considered. Attention is drawn to certain fallacies underlying the conventional approach. The parameters required in the formulation of the transfer matrix for a pipe have been evaluated. In order to evaluate the response at the terminal point of a branched system for a sinusoidal input at one of the ends, a general algorithm has been developed.

The transfer matrix elements of real crossed electric and magnetic fields with circular main ion path are given in a second order radial and in a first order axial approximation. Fringing field and curved boundary effects are included in the same approximations, for normal entry and exit. The results can be used for converging, diverging or dispersive, superimposed or pure magnetic and electric fields, as well as for Wien filters (infinite main path radius).

For special mass spectrometers 1 ) consisting of three sector fields the ion optical properties and derived in a second order approximation with the use of the matrix method 2,3 ). All fringing field effects are included in the calculations. The field arrangements are chosen in such a manner that stigmatic focusing is achieved on the image surface. Similar to the parabola spectrograph the instruments produce an energy dispersion in the direction of the mass lines. The transfer matrix product of the whole field combination is derived using (27×27)-matrices which combine the radial and axial transfer matrices of the single fields 4,5 ). In this product also a matrix is used which allows the connection of the first two fields which have a common mean plane of deflection. Another matrix renders the connection of the second and the third field the mean planes of deflection of which are vertical to each other. Numerical results are given for the geometrical dimensions and the image aberrations of one example of these field arrangements.

This paper generalizes the well known Tsypkin method for use in prediction of self or forced oscillations in multivariable relay control systems. The method is valid for all systems having linear plants whose transfer matrix elements have at least two more poles than zeros. The relays may have hysteresis and dead band and need not be identical. It is shown that certain multivariable relay control systems can be represented by a related set of single variable control systems for stability analysis purposes. This result permits Tsypkin's stability criteria for single variable systems to be used to determine the stability of these systems. The methods developed are illustrated by an example and the theoretically predicted behaviour is confirmed by analogue simulation.

The time constant of a continuous stirred-tank reactor and the transfer function of a simulated second-order system with two random inputs were determined. In the first case the experimental result agreed within 11% of the known value; in the second case the results agreed with those predicted from a theoretical analysis. To determine the transfer matrix of a fifth-order system an impractical amount of medium speed computer time was required; the results were not considered satisfactory. However the length of calculation compares favorably with other methods of total identification for processes which have extraneous noise associated with them. Because of this the application of the present technique in adaptive control systems appears to be favorable.

The problem of diagonalizing the transfer matrix for the two dimensional Ising model with all boundary spins equal to +1 is solved by use of the spinor method. This provides a simple proof that the spontaneous magnetization is actually given by the well known formula for the long range order with torodial boundary conditions, and this means that the critical temperature is precisely that temperature above which the state is unique and below which it is non unique. An expression for the magnetization at finite distance from the boundary is also given, and a simple derivation of the formula for the surface tension between two coexisting phases is presented. Finally the relation between the degeneracy of the spectrum and the phase transition is discussed.

We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization $M(H,T)$. The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of $M(H,T)$ converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for $H=O$, which reproduces exactly the critical-point behavior of the full Ising model.

The transfer matrix method is known to be well suited for a complete analysis of a lumped as well as distributed element, one-dimensional, linear dynamical system with a marked chain topology. However, general subroutines of the type available for classical matrix methods are not available in the current literature on transfer matrix methods. In the present article, general expressions for various aspects of analysis-viz., natural frequency equation, modal vectors, forced response and filter performance—have been evaluated in terms of a single parameter, referred to as velocity ratio. Subprograms have been developed for use with the transfer matrix method for the evaluation of velocity ratio and related parameters. It is shown that a given system, branched or straight-through, can be completely analysed in terms of these basic subprograms, on a stored program digital computer. It is observed that the transfer matrix method with the velocity ratio approach has certain advantages over the existing general matrix methods in the analysis of one-dimensional systems.

There exist several multivariable control systems with the characteristic of having equal inputs and equal outputs; equality of the outputs is usually achieved by means of high performance components in each of the control system loops. Such a high performance is required in order to reduce the output sensitivity to disturbances and parametric variations within prescribed amounts. In practice we try to make the system transfer matrix a diagonal matrix.

The isochronous cyclotron considered here consists of a ring of radial magnet sectors separated by straight sections. The general features of such cyclotrons are discussed, and a complete analysis is presented of their orbit properties using the hard-edge approximation. This analysis proceeds through the transfer matrix technique, which is simplified by the symmetry of the magnet structure; the only complication arises from the inconstancy of the magnetic field index resulting from the isochronism condition. Computed orbit properties are presented and discussed for a wide variety of possible geometries; these properties include the extreme values of the beam widths, as well as the focusing frequencies, as a function of the energy. Problems associated with certain resonances are also considered. The results indicate that isochronous cyclotrons of this type should be quite capable of producing proton beams at energies up to about 500 MeV.

In linear accelerators with drift tubes the bunching of particles around the synchronous one is obtained at the expense of focusing. Nevertheless, it is possible to ensure simultaneous radial and axial stabilities if, in the accelerating gaps, we abandon the cylindrical symmetry. If the potential function in a gap between two drift-tubes has quadrupolar symmetry, it is possible, with certain assumptionsabout the gap geometry, to expand this function as a double Fourier series. Thus we are able to calculate the transfer matrix from one drift tube to the next. Formulae giving axial and radial R. F. impulses, obtained in the thin-lens approximation for the quadrupole symmetry planes are easily applicable. The effect of the R.F. is due to "circular" and "quadrupolar" impulses; the former is independent of the meridian plane, whereas the sign of tne latter changes from one quadrupolar symmetry plane to the other. Tne radial and axial motions of the particles cease to be independent. The chosen potential functiorn enables us to calculate the second order transfer elements containing the coupling terms between the two degrees of freedom.

Numerical calculations are carried out on the two‐dimensional square Ising model for the structure of the interface between two phases. The method consists of stabilizing the interface in a finite system, calculating interface properties (magnetization profile and surface tension), and then extrapolating to the limit of infinite sized systems where the stabilizing forces may be removed. From the numerical work we make the following tentative conclusions: (1) For temperatures below the critical temperature, the limit of infinite size for fixed magnetic field H (H 〉0) followed by the limit H→O+ does not give rise to an asymptotic degeneracy of eigen values of the transfer matrix. (2) The structure of the interface is sensitive to ``infinitesimal'' forces stabilizing the interface. (3) The surface tension is independent of these same ``infinitesimal'' forces, always being given by the Onsager expression. The first conclusion above implies that an Ising model of infinite size stabilized by an arbitrarily smal...

Because of a possible strong influence of the secondary gamma-radiation on the total dose it is necessary to perform coupled neutron and gamma-ray calculations. The secondary gamma-rays produced by neutron interactions become sources in the gamma-ray groups through a group-to-group transfer matrix. Neutron and gamma-ray dose and fluxes as well as the total neutron and gamma-ray dose in two different serpentine concretes were calculated using a 40 group coupled set (22 neutron and 18 gamma-ray), which describes well the oxygen total cross section valley near 2.37 MeV. The results of discrete ordinate calculations for 250 cm concrete slab thickness show that the secondary gamma-ray dose becomes very important for lower energies and larger slab thicknesses.

The Courant-Snyder parametrization for one-dimensional linear motion in periodic systems is generalized to two-dimensional coupled linear motions. The 10-parameter 4×4 symplectic transfer matrix across a period is expressed in two normal-mode invariant phase-advances, four normal-mode periodic "amplitude" functions, and four periodic functions which reflect the strength and structure of the coupling. These parametric functions are also given by differential equations containing the periodic force coefficients appearing in the Hamiltonian. Two bilinear invariants can be constructed. With given horizontal and vertical aperture limitations, these invariants give the four-dimensional phase-space acceptance volume and the propagation of the horizontal and vertical emittances of the beam.

After reviewing the transfer matrix method introduced by Schmidt, and by Hori and Asahi for the problem of the vibration of a disordered linear chain, we discuss its extension by Yamamoto, Okada, and the present author to more complex systems. Classifying the frequency regions by the 's-value', which is defined as half a number of eigenvalues with absolute value 1 of the transfer matrix, we study the general aspects of normal modes in each region. When s=l, a 'phase shift' is shown to be a very useful quantity for expressing end effects on the frequency and intensity of the infrared absorption of a chain molecule. As an application we discuss the analysis of the experimental data on the rocking-twisting modes of normal paraffins, which indicates the practical importance of our theory.

An analytical method of the elastic and the inelastic lateral buckling of restrained beams is presented using the transfer matrix method. The field transfer matrix is derived under the assumption that each element is in a uniform bending moment and it is presented that the various restraints against the out of plane deformations at the supports and in the spans can be considered in the analysis using the point transfer matrix. The treatments of the boundary conditions and the effects of the locations of loadings are given. Using this method, the numerical examples are presented for simply supported beams and two-span continuous beams.

A first‐order difference equation is derived in the ratio of successive partition functions along the lattice for one‐dimensional lattice systems with a nonconstant 2 × 2 transfer matrix. In the case of DNA, the distribution functions of these ratios are easy to generate. Numerical values for the expected free energy, obtained using unsophisticated computer techniques, are reported. They agree with those given by Lehman et al. to within 2% computational accuracy. The random Ising model is also considered and the DNA generating equations shown to apply in a special case.

The transfer matrices of linear bilateral and unilateral networks are distinguished by the vanishing of the determinant of the matrix, in the case of the latter, and the non-existence of the matrix of the reversed four-pole. Illustrative examples are given. The voltage gain, current gain, input impedance and output impedance of a four-pole are deduced in terms of the transfer matrix elements. Expressions are developed for the transfer matrices of the four-poles resulting from combining pairs of four-poles, which may be either bilateral or unilateral, in parallel - parallel, in series - parallel and in series - series connection. These expressions are used to obtain the transfer matrices of some passive and active networks, including feedback amplifiers, to illustrate the possibilities of the technique.