Accurate approximate solutions to oscillatory problems with perturbing singular damping

In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form

$\text{d}{}^2\text{y}\text{d}\text{x}{}^2\text{+εμ(x)}\text{d}\text{y}\text{d}\text{x}\text{+ω}{}^2{}_{\mathrm{c}}\text{y = 0}$

, where ε is a real parameter less than unity, ω_c is a positive real constant of order unity and μ(x) is a singular function of x in the region of interest. It does not appear to be possible to find a general analytic expression for the error estimate of the approximate solution. For the case μ(x) = x^?2, however, it is shown that the approximate solution is accurate to 0(ε²), as x → 0^? from negative values, by comparing it with the numerically integrated solution. For the same case, the approximate solution is orders of magnitude more accurate than Poincaré's first-order perturbation solution, which is accurate to $\text{0(}\text{ε}{}^2\text{ln}\text{|x|}\text{|x|}\text{)}$ as x → 0^?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem.     

On recursive averaging processes and Hilbert space extensions of the contraction mapping principle

If T maps a convex domain D_T into itself, and if {ω_n} is a real sequence with range in (0, 1] then the recursive averaging process,

$\text{X}{}_{\mathrm{n+1}}\text{=(1?omega;}{}_{\mathrm{n}}\text{)}\text{X}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{+ω}{}_{\mathrm{n}}\text{T}\text{x}{}_{\mathrm{n}}\text{,}\text{x}{}_0\text{=ξ?D}{}_{\mathrm{T}}$

generates a sequence {x?_n}; with range in D_T. Under suitable conditions on D_T, T and {ω_n} the sequence {x?_n} will converge in some sense to a fixed point of T. We prove that if D_T is a closed convex subset of a complex Hilbert space H, if Tω = (1 ? ω) I + ωT is a strict contraction for some ω ? (0, 1], and if {ω_n} satisfies the conditions,

$\text{ω}{}_{\mathrm{n}}\text{→ 0}$

and

$\text{∑}\text{n=0}\text{∞}\text{ω}{}_{\mathrm{n}}\text{=∞}$

then, for arbitrary ξ ? D_T, {x?_n} converges strongly to (the unique) fixed point of T. We also prove that if D_T and {ω_n} satisfy the foregoing conditions, if T has at least one fixed point, and if Tω is non-expansive for some ω ? (0, 1], then for all ξ ? D_T, {x?_n} converges at least weakly to some fixed point of T. Finally, we apply these results to linear equations involving bounded normal operators and obtain an extension of the classical Neumann operator series.     

Parameter identification of a class of time-varying systems via orthogonal shifted legendre polynomials

A method of using orthogonal shifted Legendre polynomials for identifying the parameters of a process whose behaviour can be modelled by a linear differential equation with time-varying coefficients in the form of finite-order polynomials is presented. It is based on the repeated integration of the differential equation and the representations of $\text{∫}\text{0}\text{t}$s(τ) dτ = Ps(t) and ts(t) = Rs(t), where P and R are constant matrices and s(t) is a shifted Legendre vector whose elements are shifted Legendre polynomials. The differential input-output equation is converted into a set of overdetermined linear algebraic equations for a least squares solution. The results of simulation studies are included to illustrate the applicability of the method.     

Mathematical models of information system use

This report presents the results from a study of mathematical models relating to the usage of information systems. For each of four models, the papers developed during the study provide three types of analyses: reviews of the literature relevant to the model, analytical studies, and tests of the models with data drawn from specific operational situations. (1) The Cobb-Douglas model: x₀ = ax₁^bx₂^(1?b).This classic production model, normally interpreted as applying to the relationship between production, labor, and capital, is applied to a number of information related contexts. These include specifically the performance of libraries, both public and academic, and the use of information resources by the nation's industry. The results confirm not only the utility of the Cobb-Douglas model in evaluation of the use of information resources, but demonstrate the extent to which those resources currently are being used at significantly less than optimum levels. (2) Mixture of Poissons:

$\text{χ}{}_0\text{=}\text{∑}\text{i=0}\text{n}\text{i}\text{∑}\text{j=0}\text{p}\text{n}{}_{\mathrm{j}}\text{e}{}^{\mathrm{mj}}\text{(m}{}_{\mathrm{j}}\text{)′/i!}$

where x₀ is the usage and (n_j,m_j),j = 0 to p, are the p + 1 components of the distribution. This model of heterogeneity is applied to the usage of library materials and of thesaurus terms. In each case, both the applicability and the analytical value of the model are demonstrated. (3) Inverse effects of distance: x = a e^?md if c(d) = rdx = ad^?m if c(d) = r log(d).These two models reflect different inverse effects of distance, the choice depending upon the cost of transportation. If the cost,c(d), is linear, the usage is inverse exponential; if logarithmic, the usage is inverse power. The literature that discusses the relationship between usage of facilities and the distance from them is reviewed. The models are tested with data from the usage of the Los Angeles Public Library, both Central Library and branches, based on a survey of 3662 users. (4) Weighted entropy:

$\text{S(x}{}_1\text{,x}{}_2\text{,...,x}{}_{\mathrm{n}}\text{)= -}\text{∑}\text{i=1}\text{n}\text{r(x}{}_{\mathrm{i}}\text{P(x}{}_{\mathrm{i}}\text{)}\text{log}\text{(p(x}{}_{\mathrm{i}}\text{)).}$

This generalization of the “entropy measure of information” is designed to accommodate the effects of “relevancy”, as measured by r(x), upon the performance of information retrieval systems. The relevant literature is reviewed and the application to retrieval systems is considered.     

The representation of radiation reaction in wave mechanics

It is well known that the wave mechanical ψ equation leads to the conclusion that the centroid of the wave mechanical electron should move according to the classical electrodynamic equation of motion in which, however, the terms representing what is commonly called radiation reaction are absent. If v is the velocity of the electron, the classical rate of change of momentum is $\text{md}\text{dt}\text{{}\text{v}\text{(}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{}}$. The equation of motion including radiation reaction terms may be regarded as obtainable by replacing this quantity by one obtained by operating upon it with the operator P^?1

$\text{P={I?}\text{α}{}_1\text{kd}\text{dt}\text{+}\text{α}{}_2\text{d}\text{dt}\text{(}\text{kd}\text{dt}\text{)?·}}{}^{\mathrm{?}}$

where α₁, α₂, etc., are constants and $\text{k = (}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. The main purpose of the paper is to show that if there be any relativistically invariant ψ equation which leads to the classical equation of motion without radiation reaction terms, then by replacing the vector and scalar potentials U and ? in that equation by P(U) and P(?), a relativistically invariant equation of motion will be obtained including the radiation reaction terms, provided that the $\text{d}\text{dt}$ in P be now regarded as $\text{?}\text{?t}$ + u · grad, where u is the velocity of the wave mechanical density distribution at a point. The purpose is to use the power to produce the equation of motion as a criterion for suggesting the proper modification of the ψ equation to apply in those cases where, on the classical theory, the electron would suffer great acceleration, as in ionization by rapidly moving corpuscles.     

Perturbation solutions of the Carson–Cambi equation

The periodic differential equation (1+ε cos t)y&#x030B; + py = 0, hereby termed the Carson–Cambi equation, is the simplest second-order differential equation having a periodic coefficient associated with the second derivative. Provided |ε|<1, which is the case we examine, then the differential equation is a Hill's equation and thus possesses regions of stability and instability in the p–ε plane. Ordinary perturbation theory is employed to obtain the stable (periodic) solutions to ε³. Two-timing theory is employed to obtain solutions for values of k near the critical points k = ±$\text{1}\text{2}$, ±$\text{3}\text{2}$, ±$\text{5}\text{2}$. Three-timing is employed to extend the solution near k = ±$\text{1}\text{2}$. The solutions of the Carson–Cambi equation are compared with the solutions of the corresponding Mathieu equation.     

The reflection of hydrogen atoms from lithium fluoride

The paper describes the phenomena associated with the reflection of a sharply defined beam of hydrogen atoms from a crystal of LiF. Of primary interest is the fact that the atoms show interference effects in agreement with the wave mechanics theory and plane grating diffraction patterns are photographed. Evidence of the thermal agitation of the surface ions is obtained from the diffuse reflection with surrounds the specular beam.The Schrödinger wave equation for the motion of a free particle of mass m is

$\text{▽}{}^2\text{ψ ?}\text{4πmi}\text{h}\text{?ψ}\text{?t}\text{= 0}\text{(I)}$

. The solution of this equation corresponding to the kinetic energy $\text{mv}{}^2\text{2}$ is

where

$\text{v }\text{mv}{}^2\text{2}\text{and}\text{σ}\text{mv}\text{h}$

. The motion of such a particle should have the characteristics of a plane wave of frequency ν and wave-length $\text{λ =}\text{1}\text{σ}$. The experiments of various investigators¹ have shown the validity of the wave theory of the motion of the free electron and have given values of the wave-length in agreement with the theory.The free motion of atoms, ions and molecules should likewise have wave characteristics. In the case of the hydrogen atom, as the simplest example, the complete wave equation may be written in the form

$\text{I}\text{m}\text{▽}{}^2\text{x,y,zψ +}\text{I}\text{μ}\text{▽}{}^2\text{η,μζψ ?}\text{8π}{}^2\text{μ?ψ}\text{mh}{}^2\text{η}{}^2\text{+ μ}{}^2\text{+ ζ}{}^2$

$\text{?}\text{4πi}\text{h}\text{?ψ}\text{?t}\text{= 0,}\text{(3)}$

where x, y, z, are the coördinates of the center of mass of the atom and ξ, η, ζ the coördinates of the electron with respect to the center of mass. If m_? and m₊ are the masses of electron and proton, m and μ have the significance

$\text{m = m}{}_{\mathrm{?}}\text{+ m}{}_{\mathrm{+}}\text{and}\text{I}\text{μ}\text{=}\text{I}\text{m}{}_{\mathrm{?}}\text{+}\text{I}\text{m}{}_{\mathrm{+}}$

. Equation (3) is solved by

$\text{ψ = U}{}_1\text{(x,y,z) U}{}_2\text{(η, ν ζ) ?}{}^{\mathrm{<mtext>2\pi iEt</mtext><mtext>h</mtext>}}$

, where E may have a continuous set of values and represents the total energy. U₁ and U₂ must satisfy the equations

$\text{▽}{}_1{}^2\text{U}{}_1\text{+}\text{8π}{}^2\text{mβU}{}_1\text{h}{}^2\text{= 0,}\text{(4)}$

and

$\text{▽}{}_2{}^2\text{U}{}_2\text{+}\text{8π}{}^2\text{μ}\text{h}{}^2\text{(α ?}\text{μ?}\text{m}\text{η}{}^2\text{+ ν}{}^2\text{+ ζ}{}^2\text{)}\text{U}{}_2\text{= 0}\text{(5)}$

, where

$\text{α + β + E}$

.     

Angular region: a measure of underdamped behavior

The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil

$\text{P(s)=s}{}^2\text{I+sB+A,}$

where the operator A depends on the dissipative and reactive elements of the system, while B depends solely on the reactive elements. The operator P(s) for every applied stimulus vector signal x must satisfy:

$\text{(Bx,x)}{}^2\text{<4(Ax,x).}$

A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system,

$\text{|}\text{tan}\text{φ|?}\text{[4(Ax,x)?(Bx,x)}{}^2\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(Bx,x)}\text{.}$

When a comparison between positive operators A and B is available, say B²=KA, then

$\text{|}\text{tan}\text{φ|?}\text{√(4?K}{}^2\text{)}\text{K}\text{.}$

The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.     

An equation of state for a system with a single type of transformation

Based on theory of a previous paper, the writer has developed an equation of state for a system with a single type of transformation. This equation is of the form

$\text{h=A+Bv+Cp+Dpv?T(E+Fv+Gp+Hpv)}$

where h = ε + pv is the total heat, p the pressure, v the specific volume, T the temperature, and p, v, T are considered independent variables. A, B, C, etc., are constants for the system. The latent eat at constant (p, T) is given by

$\text{λp,T=(v}{}_2\text{?v}{}_1\text{)(}\text{?h}\text{?v}\text{)P,T= (v}{}_2\text{?v}{}_1\text{)[(B?TF)+p(D?TH)]}$

. These equations are checked with data on saturated and superheated ammonia, and the agreement is good to within a few tenths of a per cent. Also, checks with data on saturated and superheated steam show agreement within several per cent.     

Report on the work of the Bartol Research Foundation, 1933–1934

Using the velocity analyzer of Zartman with improved technique the combined velocity spectrum of Bi atoms and Bi₂ molecules was obtained at 827°, 851°, 875°, 899°, 922°, 947° C. From the spectral distribution curves the relative abundance of Bi atoms and Bi₂ molecules in the beams at the above temperatures could be determined to 1 per cent. The vapor pressure curve of Bi was obtained experimentally by the method of effusion and the values so obtained were combined with the degree of dissociation of the vapor as computed from the beams to give the heat of dissociation. The heat of dissociation was computed from the data, assuming the pressure to be given by the temperature of the crucible T_c. In calculating the heat of dissociation, the equilibrium temperature was taken as that of the slit chamber T_s which was 24° above T_c. The results of these calculations plotted with log₁₀K_p as ordinates against $\text{1}\text{T}{}_{\mathrm{s}}$ give a straight line whose slope yields the value of the heat of dissociation as 77,100±1200 calories. The curves for the distribution of velocities observed and computed on the assumption of a given ratio of Bi atoms to Bi₂ molecules in the beam were compared in an attempt to test the law of distribution of velocities. On the high velocity side agreement in two curves was obtained within the limits of experimental accuracy. On the low velocity side important deviations were noted of such a sort that the observed curves below a velocity $\text{α}\text{2}$, (α is the most probable velocity) gave more molecules than the theory demanded. Other deviations were observed on some of the runs taken with a fourth slit in which a deficiency of molecules was observed between velocities of .75α and $\text{α}\text{2}$. This deviation was probably due to a warping of the fourth slit carriage due to heat. The nature of the variation at velocities less than $\text{α}\text{2}$ indicated the presence of molecules of greater mass than Bi₂ in the beam and at the lower temperatures a distinct peak corresponding to Bi₈ molecules was observed which were present to less than 2 per cent. The vapor pressure curve for Bi was determined by least square reduction of the observed points to be given by $\text{log}{}_{10}\text{P = ? 52.23 ×}\text{195.26}\text{T}\text{+ 8.56}$ between 1100° and 1220° abs. It lies very close to the extrapolated curve given in the International Critical Tables.     

An experimental investigation of forced vibrations

The solution of the differential equation y″ + 2Ry′ + n²y = E cos pt is written in a new form which clearly exhibits many important facts thus far overlooked by theoretical and experimental investigators. Writing s = n ? p, and Δn = n ? √n² ? R², it is found: (a) When s ≠ Δn, there are “beats,” and the first “beat” maximum is greater than any later maximum while the first “beat” minimum is less than any later “beat” minimum. The “beat” frequency is $\text{(s ?}\text{Δ}\text{n)}\text{2π}$. (b) When n² ? p² = R², there are no “beats,” and the resultant amplitude grows monotonically from zero to the amplitude of the forced vibration, (c) At resonance, when n = p, we still have maxima which occur with a frequency $\text{Δ}\text{n}\text{2π}$ in a damped system. (d) The absence of “beats” is neither a sufficient nor a necessary condition for resonance in a damped system.In the experimental investigation the upper extremity of a simple pendulum was moved in simple harmonic motion and photographic records obtained of the motion of the pendulum bob. Different degrees of damping were used, ranging from very small to critical.The experimental results are in excellent agreement with theory.     

Orthogonal nonnegative matrix tri-factorization for co-clustering: Multiplicative updates on Stiefel manifolds

Matrix factorization-based methods become popular in dyadic data analysis, where a fundamental problem, for example, is to perform document clustering or co-clustering words and documents given a term-document matrix. Nonnegative matrix tri-factorization (NMTF) emerges as a promising tool for co-clustering, seeking a 3-factor decomposition X≈USV^?$\mathit{X}\approx {\mathit{USV}}^{?}$ with all factor matrices restricted to be nonnegative, i.e., U?0,S?0,V?0.$\mathit{U}?0\text{,}\mathit{S}?0\text{,}\mathit{V}?0\text{.}$ In this paper we develop multiplicative updates for orthogonal NMTF where X≈USV^?$\mathit{X}\approx {\mathit{USV}}^{?}$ is pursued with orthogonality constraints, U^?U=I,${\mathit{U}}^{?}\mathit{U}=\mathit{I}\text{,}$ and V^?V=I${\mathit{V}}^{?}\mathit{V}=\mathit{I}$, exploiting true gradients on Stiefel manifolds. Experiments on various document data sets demonstrate that our method works well for document clustering and is useful in revealing polysemous words via co-clustering words and documents.     

Temperature distribution in toroidal electrical coils of rectangular cross section

The essential content of a recent paper by the present writer comprises a comprehensive discussion of the physical bases underlying derivation of formulas for calculating the temperature distribution T, maximum temperature T_m and average temperature T_a in a toroidal electrical coil of rectangular cross section, internally generated heat and change of wire resistance with temperature being taken into account. Illustratively, the solution for the boundary value condition of constant surface temperature and uniform equivalent thermal conductivity was obtained.For the most part, however, problems that arise in practice are not encompassed in the comparatively simple boundary conditions of constant temperature. Experiment shows that in general the boundary condition is $\text{T ? T′ = ?}\text{K?}\text{?n}$; whereof n denotes the outward drawn normal to the coil surface, $\text{K = (}\text{k}{}_{\mathrm{n}}\text{h}\text{)}$ the ratio of the equivalent thermal conductivity in the direction of n to the emissivity of the boundary surface, and T and T′ are the corresponding temperatures in the coil surface and the immediately adjacent ambient medium. Again, it frequently ensues in practice that the thermal conductivity is substantially different in the directions of the two principal axes of the cross section.In the present paper formulas for T, T_m, and T_a are obtained for electrical coils of ratio of external to internal radius greater than (roughly) two whereof (i) the thermal conductivity is different in the directions of the two principal axes of the cross section, (ii) K is different on but constant over each of the four faces of the coil, and (iii) no restriction is made as to T′ except that over each face it be expressible in a generalized Fourier series. Determination of T is posed as a boundary-value problem in the mathematical theory of heat; the formal solution of T effected by expansions in orthogonal functions; and T_m and T_a then determined through use of their known relationships with T. The resulting formulas are in the form of rapidly-converging singly-infinite trigonometric-hyperbolic series. Illustrative of application of these general formulas, the maximum temperatures in a coil of given dimensions subject to two different sets of surface conditions are calculated and found to be in excellent agreement with the known measured values.The just-mentioned formulas encompass practically all cases encountered in practice except those coils which do not satisfy the restriction as to ratio of radii. For these latter formulas for T, T_m, and T_a are obtained pursuant to conditions of (i) equivalent thermal conductivity different in the directions of the two principal axes of the cross section, (ii) K, and likewise T′, different on but constant over each of the four faces of the coil. These formulas are in the form of rapidly-converging singly-infinite trigonometric- Bessel function (of zero order) series: Illustratively, the maximum temperature in a coil of given dimensions is calculated and found to be in excellent agreement with the known measured value.     

A useful identity in circuit theory

A useful identity expressing the derivative of an unknown variable x_k of X with respect to an entry in the coefficient matrix of a linear system AX = B is presented. If the derivatives of x_k with respect to each entry of A or their combinations are required, then the identity avoids the repeated solution of the linear equations, and may result in a symbolic solution provided A^-1 is known. A method to measure the sensitivity in an n-port linear system is proposed, and its relationships to Kron's method of tearing and Branin's formulae are also discussed.     

Initial conditioned solutions of a second-order nonlinear conservative differential equation with a periodically varying coefficient

The exact solution of the equation

$\text{d}{}^2\text{x}\text{dt}{}^2\text{+dx+d′f(wt)x}{}^3\text{=0,}$

where d, d' and w are positive constants, and ?(wt) is a rectangular periodic function of time is discussed. The equation describes approximately the transversal movement of a particle in an alternating gradient accelerator. The exact solution is obtained in the form of a composite recurrent relation containing five particular solutions. Each of these solutions corresponds to a specific well-defined area of the phase plane of the initial conditions. The dynamical behaviour and the stability of the movement are examined analytically.