共查询到20条相似文献，搜索用时 31 毫秒
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C.C.H. Tang 《Journal of The Franklin Institute》1973,296(2):137-149
In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form , where ε is a real parameter less than unity, ωc 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 as x → 0?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem. 相似文献
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Thomas H. Johnson 《Journal of The Franklin Institute》1930,210(2):135-152
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 . The solution of this equation corresponding to the kinetic energy is where . The motion of such a particle should have the characteristics of a plane wave of frequency ν and wave-length . 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 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 . Equation (3) is solved by , where E may have a continuous set of values and represents the total energy. U1 and U2 must satisfy the equations and , where . 相似文献
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Lucas Pun 《Journal of The Franklin Institute》1973,295(3):193-216
The exact solution of the equation 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. 相似文献
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J.C. Dunn 《Journal of The Franklin Institute》1973,295(2):117-133
If T maps a convex domain DT into itself, and if {ωn} is a real sequence with range in (0, 1] then the recursive averaging process, generates a sequence {x?n}; with range in DT. Under suitable conditions on DT, T and {ωn} the sequence {x?n} will converge in some sense to a fixed point of T. We prove that if DT 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 {ωn} satisfies the conditions, and then, for arbitrary ξ ? DT, {x?n} converges strongly to (the unique) fixed point of T. We also prove that if DT and {ωn} 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 ? DT, {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. 相似文献
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W.F.G. Swann 《Journal of The Franklin Institute》1934,217(1):59-71
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 . 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 where α1, α2, etc., are constants and . 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 in P be now regarded as + 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. 相似文献
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Israel Maizlish 《Journal of The Franklin Institute》1924,197(5):667-690
Present Status of the Problem.—The scattering of X-rays is one of the outstanding problems of electromagnetic radiation which has not been solved satisfactorily. All theories (based on classical electrodynamics) presented thus far do not explain either the diminution in the scattering coefficient, or the observed asymmetry in the scattering, or both. Among such theories we may mention:J. J. Thompson's Theory.—Assuming that the scattering is done by a point electron, and making use of certain additional hypotheses, Thomson showed that the scattering coefficient of any substance is given by and that the intensity of the scattered radiation is given by where N is the number of atoms per c.c., p the number of electrons per atom, e the electronic charge, m the electronic mass, c the velocity of light, Iθ the intensity of the scattered radiation at an angle θ between the incident beam and the radius vector joining the centre of the electron and the point P distant r from the electron, and I is the intensity of the incident beam. This theory explains neither the asymmetry nor the decrease in the coefficient of scattering.Schott's Theory.—Among other things, the assumption is here made that the atom consists of coaxal rings of electron. The electrons in each ring are spaced at equal intervals and revolve with a uniform angular velocity, which, however, may be different for different rings. This theory fails to explain the observed diminution in the scattering coefficient.Debye's Theory.—In its essentials, Debye's theory has the same merits and demerits as that of Schott. Debye assumes that all the electrons in an atom are arranged in a single ring, and that they are spaced at equal intervals. This theory (and also Schott's theory) explains the asymmetry and the “excess scattering,” but is altogether unable to explain the diminution in the scattering coefficient.Modification of the Classical Theory.—The present paper presents a discussion of the possibility of modifying the classical theory (that of J. J. Thomson) so as to account for the decrease in the scattering coefficient as well as the dissymmetry. By assuming that the electron is made up of a number of parts—for simplicity, of two parts—it has been found possible to account for the diminution in the scattering coefficient without, at the same time, explaining the observed asymmetry. To accomplish both objects is what was aimed at in the combination of the present work with that of Debye. In this research the goal has not been perfection between predicted and observed results, but rather to discuss some possible modifications of the classical theory and their consequences. 相似文献
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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 = ax1bx2(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: where x0 is the usage and (nj,mj),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: 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. 相似文献
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Matrix A with characteristic polynomial Q(z) is defined positive or negative Hurwitz according to whether Q(z) or Q(-z) is a Hurwitz polynomial. Leading principle sections of the Tikhonova growth matrix have associated characteristic polynomials Pn(-z) which satisfy the recursionThat the Tikhonova growth matrix is negative Hurwitz is established through applying the Wall-Stieltjes theory of continued fraction expansions to show the Pn(-z) are Hurwitz polynomials. The Kayeya-Enestrom theorem and a procedure for refinement of the Gerschgorin estimate are used to obtain analytical bounds on spectral radii for the Tikhonova model, which provides estimates of maximal growth rates. The theory allows generalization to more complicated growth models. 相似文献
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J.L. Finck 《Journal of The Franklin Institute》1947,243(2):117-129
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 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 . 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. 相似文献
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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? with all factor matrices restricted to be nonnegative, i.e., U?0,S?0,V?0. In this paper we develop multiplicative updates for orthogonal NMTF where X≈USV? is pursued with orthogonality constraints, U?U=I, and V?V=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. 相似文献
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M.M. Cavalcanti E. Fernández-Cara A.L. Ferreira 《Journal of The Franklin Institute》2017,354(14):6405-6421
The main goal of the present paper is twofold: (i) to establish the well-posedness of a class of nonlinear degenerate parabolic equations and (ii) to investigate the related null controllability and decay rate properties. In a previous step, we consider an appropriate regularized system, where a small parameter α is involved. More precisely, the usual nonlinear term b(x)uux is replaced by b(x)zux, where and A is a Poisson–Dirichlet operator. We investigate the behavior of the null controls and their associated states as α → 0. 相似文献
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A procedure is described for determining the characteristics of adiabatic flow through a rocket nozzle with and without composition change. The method of calculation is illustrated for the expansion of pure hydrogen gas from a chamber temperature of 306° K. and a pressure of 20.42 atm. to atmospheric pressure.The study indicates that the exhaust velocity and temperature are highest for flow where complete equilibrium is reached at each temperature with respect to the reaction Flow with composition change requires a nozzle exit to nozzle throat area ratio somewhat greater than that determined for adiabatic flow without composition change for the same ratio of chamber pressure to exit pressure.The residence time in a given temperature range is computed as a function of gas temperature for the two types of flow. The results of this calculation may be used to determine the minimum required reaction rates which allow composition changes during flow through the nozzle. 相似文献
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Xiongping Dai Tingwen Huang Yu Huang Yi Luo Gang Wang Mingqing Xiao 《Journal of The Franklin Institute》2017,354(10):4126-4155
Given any finite family of real d-by-d nonsingular matrices by extending the well-known Li–Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:
we study the chaotic dynamics of the state trajectory (xn(x0, σ))n ≥ 1 with initial state governed by a switching law . Two sufficient conditions are given so that for a “large” set of switching laws σ, there exhibits the scrambled dynamics as follows: for all This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (xn(x0, σ))n ≥ 1 are extremely sensitive to the initial states . We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws. 相似文献
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