Null controllability of some nonlinear degenerate 1D parabolic equations     

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)uu_x is replaced by b(x)zu_x, where $z={(\text{Id}.?{\alpha }^{2}A)}^{?1}u$ 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   

The representation of radiation reaction in wave mechanics     

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 $\text{md}\text{dt}\text{{}\text{v}\text{(}\text{I}\text{?}\text{v}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{}}$. 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 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.  相似文献   

Eigenvalue localization for the tikhonova model of crystal growth     

Charlie H. Cooke  Philip W. Smith 《Journal of The Franklin Institute》1983,316(2):127-133

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 P_n(-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 P_n(-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   

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   

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   

Accurate approximate solutions to oscillatory problems with perturbing singular damping     

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, 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.  相似文献   

Angular region: a measure of underdamped behavior     

F.M. Reza 《Journal of The Franklin Institute》1982,314(3):191-202

The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil 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: A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system, When a comparison between positive operators A and B is available, say B²=KA, then The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.  相似文献   

The scattering of X-rays     

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, I_θ 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   

The reflection of hydrogen atoms from lithium fluoride     

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 $\text{mv}{}^2\text{2}$ is where . 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 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   

A fuzzy model of document retrieval systems     

Valiollah Tahani 《Information processing & management》1976,12(3):177-187

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

W.F.G. Swann 《Journal of The Franklin Institute》1934,218(2):173-241

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.  相似文献   

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

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   

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   

Stabilization of carrier-frequency servomechanisms. II. Design formulae for proportional-derivative networks     

Andrew Sobczyk 《Journal of The Franklin Institute》1948,246(2):95-121

In an alternating current servomechanism, the error is proportional to the modulation envelope of a modulated-carrier error signal. It is shown in part I that for stability and fidelity of the servo, it is highly desirable that the effect of the controller includes a proportional-derivative action on the modulation envelope. This action may be obtained with various forms of RC networks, including the parallel “T,” bridge “T,” and Wien Bridge forms.This part contains detailed design procedures and tables of values for the various types of proportional-derivative networks. Several forms of parallel “T” networks arise from the fact that there are five independent time constants in the network, while in order to realize the desired transfer characteristic it is necessary to impose only four conditions. It is indicated how the remaining degree of freedom may be used to obtain the most suitable input and output impedances for the source and load impedances with which the parallel “T” is to be used. The derivations for the parallel “T” formulae are given in an Appendix.Tolerance requirements on the components of parallel “T” and bridge “T” networks are derived. If ±1 per cent components are used at 60 cycles, the resonant frequency will lie between 56.4 and 63.6 cycles, and the notch width (rejection band width) will be within ±0.99 cps. of the correct value. In order to guarantee that the phase shift at 60 cycles is within ±10°, the percentage deviation of each part must be less than ($\text{9.0}\text{T}{}_{\mathrm{d}}\text{ω}{}_0$), where ω₀ is the carrier angular frequency, T_d the derivative time constant.  相似文献   

The concept of resilience with respect to indicating instruments     

Frederick J. Schlink 《Journal of The Franklin Institute》1919,187(2):147-169

This paper deals particularly with those instruments of the index-and-scale and value-controlling types of the class of non-integrating instruments, as distinguished from integrating instruments and those used for comparison purposes strictly. Independent corroboration is adduced regarding the general characteristics of the hysteresis phenomena discussed in the author's earlier paper on the variance of measuring instruments.It is shown that the area of the hysteresis loop obtained on complete cyclic calibration is a measure of the energy dissipated in the operation of the instrument and that the smallness of the area of this loop, which may be used as a factor in a function exactly analogous to the resiliency in the case of other quasielastic bodies, is a measure of the excellence of the instrument as regards the reproducibility or invariance of its indications, so far as concerns mechanical sources of variation. The specific properties of the hysteresis loop are set down analytically and the physical nature and causes of the imperfect resilience of instruments are discussed in detail.Since the motion of an index or pointer through a displacement implies the existence of a motive force to bring about the deflection, in the presence of an equivalent reactive or restoring force opposing it, the essentials of a work diagram apparently always exist in the results of a properly perforined cyclic calibration. Indications are given of the methods to be followed in the process of reducing the results of the calibration to such form that the loop obtained correctly represents energy dissipation during a cycle, typifying an integral of the form when B_y and D_y are the projections of the extremities B and D of the loop on the, axis of deflection or reading.The restrictions which surround the carrying out of cyclic calibrations in a manner calculated to obtain results of the character required are discussed, including the requirements of slow, aperiodic change of the variables, unreversed between the chosen extreme turning points, and accomplished in the absence of jarring or vibration. Attention is directed to the necessity of accustoming the instrument to the particular cycle over which it is to be calibrated, to the end of regularizing its performance.Both the form and area of the hysteresis loop should be observed, in order to arrive at regional as well as aggregate effects of the causes producing the variance. The amount and scope of the effects of vibratory treatment in modifying and diminishing the causes of lag are indicated.The possible causes of the lag known to exist in. the case of instruments using a surface of discontinuity between fluids as the indicating element are discussed, and it is shown that known phenomena perhaps hitherto unrecognized in their relation to instrument design and calibration may account for the variancy noted. Detailed experimental consideration of these factors is hoped for.Comparison of instrument performances on the basis of the hysteresis loss requires equivalence of the ranges of operation, or possibly reduction of the results on the basis of information not vet available in a form, capable of general application.The resiliency determination opens up a field for type-testing and selection of instruments on a basis quite discrete from that pertaining to the ordinary methods of calibration, in that the former permits selection between types of instruments rather than between individuals, making clearly discriminable the differences between the characteristics of given operating principles, designs of mechanical details, or qualities of workmanship. The more general methods of diminishing variance educible from the foregoing considerations are indicated.  相似文献   

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

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   

Statistical description of the size properties of non uniform particulate substances     

Theodore Hatch  Sarah P. Choate 《Journal of The Franklin Institute》1929,207(3):369-387

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Analysis of the particle-size distribution of the grains producing a photographic image     

R.P. Loveland  H.M. Menihan  A.P.H. Trivelli 《Journal of The Franklin Institute》1948,246(6):459-481

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An econometric property of the g-index     

L. Egghe 《Information processing & management》2009

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The Gray image of a class of constacyclic codes over polynomial residue rings     

《Journal of The Franklin Institute》2014,351(12):5467-5479

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