Mathematical models of information system use     

Robert M. Hayes  Harold Borko 《Information processing & management》1983,19(3):173-186

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: 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: 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   

On the existence and the magnitude of electronic charges     

Alfred Landé 《Journal of The Franklin Institute》1940,229(6):767-774

In order to quantize Dirac's classical point electron¹ we supplement Einstein's classical equation (E/c)² ? p² = b2 with a reciprocal classical equation (CΔt)² ? (Δr)² = a² where b = mc and a is Dirac's signal radius. Δt is the time saved by a light signal in various states of motion of the electron, and a/c is the rest time saved. Our former efforts² of obtaining an integral equation for the probability amplitude have been rectified by Born.³ There is no solution of the integral equation, however, unless advanced and retarded phases are introduced simultaneously, along with Dirac's advanced and retarded potentials. We have obtained a transcendental equation for the eigen-value μ = αγ where α is the Sommerfeld fine-structure constant, and γ is the numerical factor in Dirac's signal radius a = γe²/mc². The smallest eigen-value is μ = 0.0299.That is, ab = hγ = h/210.  相似文献   

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

J.C. Dunn 《Journal of The Franklin Institute》1973,295(2):117-133

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, 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, and 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.  相似文献   

Theory of nonlinear systems     

Y.H. Ku 《Journal of The Franklin Institute》1983,315(1):1-26

This paper gives a general review of the Theory of Nonlinear Systems. In 1960, the author presented a paper “Theory of Nonlinear Control” at the First IFAC Congress at Moscow. Professor Norbert Wiener, who attended this Congress, drew attention to his work on the synthesis and analysis of nonlinear systems in terms of Hermitian polynomials in the Laguerre coefficients of the past of the input.Wiener's original idea was to use white noise as a probe on any nonlinear system. Applying this input to a Laguerre network gives u₁, u₂,…, u_s, and then to a Hermite polynomial generator gives V(α)'s. Applying the same input to the actual nonlinear system gives output c(t). Putting c(t) and V(α)'s through a product averaging device, we get $\text{c(t)V(α)}\text{=}\text{A}{}_{\mathrm{\alpha }}\text{2л}{}^{\mathrm{<mtext>s</mtext><mtext>2</mtext>}}$, where the upper bar denotes time average and A_α's can be considered as characteristic coefficients of the nonlinear system. A desired output z(itt) may replace c(itt) to get a new set of A_α's.The Volterra functional method suggested by Wiener in 1942 has been greatlydeveloped from 1955 to the present. The method involves a multi-dimensional convolution integral with multi- dimensional kernels. The associated multi-dimensional transforms are given by Y.H. Ku and A.A. Wolf (J. Franklin Inst., Vol. 281, pp. 9–26, 1966). Wiener extended the Volterra functionals by forming an orthogonal set of functionals known as G-functionals, using Gaussian white noise as input. Volterra kernels and Wiener kernels can be correlated and form the characteristic functions of nonlinear systems.From an extension of the linear system to the nonlinear system, the input-output crosscorrelation φ_xy can be shown to be equal to the convolution of system impulse response h₁ with the autocorrelation φ_xx. Using the white noise as input, where its power density spectrum is a constant, say, A, the crosscorrelation is given by φ_xy(σ) = Ah₁(σ), while the autocorrelation is φ_xx(τ) = Au(τ). This extension forms the basis of an optimum method for nonlinear system identification. Measurement of kernels can be made through proper circuitry.Parallel to the Volterra series and the Wiener series, another series based on Taylor-Cauchy transforms developed since 1959 are given for comparison. The Taylor-Cauchy transform method can be applied in the analysis of simultaneous nonlinear systems. It is noted that the Volterra functional method and the Taylor-Cauchy transform method give identical final results.A selected Bibliography is appended not only to include other aspects of nonlinear system theory but also to show the wide application of nonlinear system characterization and identification to problems in biology, ecology, physiology, cybernetics, control theory, socio- economic systems, etc.  相似文献   

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

Chyi Hwang  Tong-Yi Guo 《Journal of The Franklin Institute》1984,318(1):56-69

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.  相似文献   

On the compression of search trees     

Francisco Claude  Patrick K. Nicholson  Diego Seco 《Information processing & management》2014

Let X=_x1,_x2,…,_xn$X=x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_n$ be a sequence of non-decreasing integer values. Storing a compressed representation of X that supports access and search is a problem that occurs in many domains. The most common solution to this problem uses a linear list and encodes the differences between consecutive values with encodings that favor small numbers. This solution includes additional information (i.e. samples) to support efficient searching on the encoded values. We introduce a completely different alternative that achieves compression by encoding the differences in a search tree. Our proposal has many applications, such as the representation of posting lists, geographic data, sparse bitmaps, and compressed suffix arrays, to name just a few. The structure is practical and we provide an experimental evaluation to show that it is competitive with the existing techniques.  相似文献   

An econometric property of the g-index     

L. Egghe 《Information processing & management》2009

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Stability criteria for certain classes of nonlinear difference equations     

Sheldon P. Gordon 《Journal of The Franklin Institute》1973,295(4):293-315

The general mth order difference equation X(n+m)+a₁X(n+m?1)+…+a_mX(n) = F[n,X(n),…,X(n+m?1)] is considered. The stability properties of its solutions are studied using the discrete form of Liapunov's direct method. A quadratic form is selected as a possible Liapunov function V(n,X) and a scheme is developed for determining appropriate conditions on this function to insure that its total difference ΔV(n,X) is negative semi-definite or negative definite with respect to the difference equation. The approach is applied to the fourth-order difference equation in full detail to illustrate the method for determining the conditions which imply either uniform stability or uniform asymptotic stability and specific results are obtained. Several comments on, and extensions of, the work done by Puri and Drake for the cases m = 2 and m = 3 are presented.The results of the present approach in the homogeneous case where F[n,X(n),…,X(n+m?1)] = 0 are compared with the usual Schur-Cohn criteria and are shown to be at least as good.  相似文献   

Stability and dissipative analysis for a class of stochastic system with time-delay     

Hao Zhang  Huaicheng Yan 《Journal of The Franklin Institute》2010,347(5):882-3355

This paper deals with the stability and dissipative problem of a class of stochastic hybrid system. The system under study involves Markovian jump, impulsive effects and time delay, which are often encountered in practice and are the sources of instability. Our attention is focused on analysis of whether the stochastic hybrid system with time-delay is stochastically asymptotically stable and strictly (Q, S, R) dissipative. By introducing an extra artificial time instance, the equivalent system is obtained and the sufficient conditions are derived by using linear matrix inequality (LMI) techniques. The main results of this paper unify the existing results on H_∞ control.  相似文献