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Showing papers on "Constant (mathematics) published in 1975"


Journal ArticleDOI
TL;DR: In this article, the authors show that the processes both of preparing for and of responding to a disoriented test form consist of the mental rotation of an image, and that both sorts of mental rotation are carried out at essentially the same constant rate.

471 citations


Journal ArticleDOI
V. L. Highland1
TL;DR: In this article, the experimentalist's familiar formula for multiple scattering is investigated in terms of the more exact theory, and a new value for the constant is suggested: Es=17.5 MeV.

425 citations


Journal ArticleDOI
TL;DR: In this article, an approximate expression for the force between unequal diffuse double layers at constant charge was derived using a method based on the "compression" approach developed previously for the case of similar double layers.

417 citations


Journal ArticleDOI
TL;DR: In this paper, the authors constructed the quantization of a classical mechanics whose phase space is a classical complex symmetric space, and established the important qualitative differences between the quantisation of such mechanics and the quantification of ordinary mechanics with plane phase spaces.
Abstract: By means of the method described in the author's paper Quantization (Math. USSR Izv. 8 (1974), 1109-1165), we construct the quantization of a classical mechanics whose phase space is a classical complex symmetric space. We establish the important qualitative differences between the quantization of such mechanics and the quantization of ordinary mechanics with plane phase spaces: for all the spaces considered, except for the sphere, Planck's constant is bounded above. Moreover, in the compact case Planck's constant takes on only discrete values.Bibliography: 17 items.

335 citations


Journal ArticleDOI
TL;DR: In this article, a class of mean annual, zonally averaged energy-balance climate models of the Budyko-Sellers type are studied by a spectral (expansion in Legendre polynomials) method.
Abstract: A class of mean annual, zonally averaged energy-balance climate models of the Budyko-Sellers type are studied by a spectral (expansion in Legendre polynomials) method. Models with constant thermal diffusion coefficient can be solved exactly, The solution is approached by a rapidly converging sequence with each succeeding approximant taking into account information from ever smaller space and time scales. The first two modes represent a good approximation to the exact solution as well as to the present climate. The two-mode approximation to a number of more general models are shown to be either formally or approximately equivalent to the same truncation in the constant diffusion case. In particular, the transport parameterization used by Budyko is precisely equivalent to the two-mode truncation of thermal diffusion. Details of the dynamics do not influence the first two modes which fortunately seem adequate for the study of global climate change. Estimated ice age temperatures and ice line latitud...

321 citations


Journal ArticleDOI
TL;DR: In this paper, a simple theory based on a hard sphere model is used to calculate the self-diffusion constant and the shear viscosity of liquid carbon tetrachloride.
Abstract: A simple theory based on a hard sphere model is used to calculate the self−diffusion constant and the shear viscosity of liquid carbon tetrachloride. The theory accounts for the coupling between translational and rotational motions of molecules. This feature distinguishes the hard sphere theory presented herein with that given recently by other workers. It is shown that the coupling has a large (factor of 2) effect on the diffusion constant, and thus the coupling should not be neglected even for molecules as spherical as carbon tetrachloride. The approximations that are needed to arrive at a hard sphere theory for transport coefficients are discussed in detail. This analysis reveals the strengths and limitations of such a theory.

295 citations


Journal ArticleDOI
TL;DR: In this paper, the Sieder-Tate equation for liquids and the two equations of Petukhov for gases and liquids are combined to calculate heat transfer coefficients to variable property liquid metals.

238 citations


Journal ArticleDOI
TL;DR: In this article, a model is presented whereby a system of two interacting electrical double layers can minimise its electrical free energy at all distances of separation, where the surface charge on each particle is described in terms of surface site equilibria which maintain constant chemical potential of potential determining ions.
Abstract: A model is presented whereby a system of two interacting electrical double layers can minimise its electrical free energy at all distances of separation. The surface charge on each particle is described in terms of surface site equilibria which maintain constant chemical potential of potential determining ions. The ability of the system to react (i.e. buffer) to changes in the interparticle medium during approach, depends on the surface site dissociation constants, the point of zero charge of the surface and the ionic strength. It is shown that essentially perfect regulation, i.e. essentially infinite buffer capacity, is observed with systems such as AgI for which the potential of the single double layer is given by the Nernst equation; for such a system the usual assumption of interaction at constant potential is sensibly correct. For systems where the activity of potential determining ions is set at a value far removed from a surface dissociation constant the interaction is well approximated by constant charge interaction. The interaction with regulation involves solving a set of transcendental equations for self consistent values of surface charge and potential at all separations for any given set of bulk parameters. It is a general treatment that replaces constant charge and constant potential assumptions and is applicable to oxide colloids and amphoteric biosurfaces in particular.

230 citations


Journal ArticleDOI
TL;DR: In this article, a simple radiative balance climate model is presented which includes the ice feedback mechanism, zonal averaging, constant homogeneous cloudiness, and ordinary diffusive thermal heat transfer.
Abstract: A simple radiative balance climate model is presented which includes the ice feedback mechanism, zonal averaging, constant homogeneous cloudiness, and ordinary diffusive thermal heat transfer. The simplest version of the model with only one free parameter is solved explicitly in terms of hypergeometric functions and is used to study ice sheet latitude as a function of solar constant. A multiple branch structure of this function is found and discussed along with comparison to earlier results. A stability analysis about the equilibrium solutions shows that the present climate as well as an ice-covered earth are stable while an intermediate solution is unstable for small perturbations away from equilibrium.

204 citations


Journal ArticleDOI
TL;DR: This paper presents a technique for determinating time-varying feedback gains of linear systems with quadratic performance criteria by developing an operational matrix for solving state equations and solving the piecewise constant gains problem.
Abstract: This paper presents a technique for determinating time-varying feedback gains of linear systems with quadratic performance criteria. The gains are approximated by the piecewise constants which axe naturally determined by Walsh functions. After introducing Walsh functions in the beginning we develop an operational matrix for solving state equations. Then using the operational matrix we solve the piecewise constant gains problem.

147 citations


Book ChapterDOI
Jane Cullum1, W. E. Donath1, P. Wolfe1
01 Jan 1975
TL;DR: In this paper, the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix is derived and a convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the original matrix is held constant.
Abstract: Properties of the sum of the q algebraically largest eigenvalues of any real symmetric matrix as a function of the diagonal entries of the matrix are derived Such a sum is convex but not necessarily everywhere differentiable A convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the matrix is held constant An implementation of this procedure is described and numerical results are included

Journal ArticleDOI
TL;DR: From the perspective developed here, the relative fitnesses of alternative evolutionary “strategies” determine the trajectory of the Michaelis constant over evolutionary time.

Journal ArticleDOI
TL;DR: In this article, the first-order spin-rotation coupling constant (γ) and the second-order centrifugal spin-orbit constant (A D = 2 A J ) in 2 Π states of OD were analyzed by the direct two-state fit approach.

Journal ArticleDOI
TL;DR: This paper is concerned with the determination of suboptimal feedback laws for the linear systems with quadratic performance criteria and the Walsh functions approach to the solution of piecewise constant gains of optimal control is concentrated.
Abstract: This paper is concerned with the determination of suboptimal feedback laws for the linear systems with quadratic performance criteria. The time-varying gains are approximated by the piecewise constant gains which are naturally determined by using Walsh functions. An increase of the number of intervals of Walsh functions enables us to approximate the true optimal control more closely ; and a decrease of the number of intervals makes the implementation easier. Therefore the proposed method is simple in theory and flexible in practice. The beginning part of the paper, being tutorial in nature, is on Walsh functions, the middle part developes an operational method for solving state equations and the final part is concentrated on the Walsh functions approach to the solution of piecewise constant gains of optimal control.

01 Nov 1975
TL;DR: In this paper, a curve fitting impact method is presented and discussed, where a mathematical model (usually a polynomial) is fitted to the test data so that the sum of squares of the deviations of the data points from the fitted curve is minimum.
Abstract: Quality assurance programs such as those related to nuclear pressure vessel materials usually require impact property measurements taken over a range of temperatures. These property measurements frequently include impact energy, fracture toughness, lateral expansion, or other quantities exhibiting the typical ductile-brittle transition. It is often difficult, however, to develop a suitable relationship between impact test data and temperature. Impact properties tend to vary in a sigmoidal fashion, starting at a fairly constant low value, increasing sharply over a short temperature range, and leveling off at a fairly constant higher value. The best approach to this task is through regression analysis. In this technique a mathematical model (usually a polynomial) is fitted to the test data so that the sum of squares of the deviations of the data points from the fitted curve is minimum. There are standard and relatively simple methods for fitting such equations which can handle transformations of the data before analysis. A curve fitting impact method is presented and discussed.

Journal ArticleDOI
01 Jul 1975-Nature
TL;DR: In this paper, a simple, statistical, geometric model of multicomponent random disk packing, which ignores the gaps that occur in real packings, is compared with experimental packings.
Abstract: A simple, statistical, geometric model of multicomponent random disk packing, which ignores the gaps that occur in real packings, is compared with experimental packings. The model gives a good qualitative description of the changes that occur in packing structures as the proportions of the disks in a mixture changes, and considers the different types of contact that occur between disks. The average coordinance for disk packings is constant for all mixture proportions.

Journal ArticleDOI
TL;DR: It is shown that for a sufficiently small adaptation constant, the mean error in the estimator weights converges to a finite limit, generally nonzero.
Abstract: In recent years adaptive linear estimation based upon the gradient-following algorithm has been proposed in a wide range of applications. However, little analysis on the convergence of the estimation has appeared when the elements of the data sequence are dependent. This paper presents such an analysis under the assumptions of stationarity and M -dependence (all data sets separated by more than a constant M are statistically independent). It is shown that for a sufficiently small adaptation constant, the mean error in the estimator weights converges to a finite limit, generally nonzero. In addition, hounds on the norm of the mean weight-deviation and on the mean norm-square of the weight-deviation are found and shown to converge to asymptotic bounds, which can be made arbitrarily small by decreasing the adaptation constant and increasing the data block length over which gradient estimates are made.

Journal ArticleDOI
TL;DR: The requirement of nonsingularity will be removed for single-input systems, and another similar control will be given which stabilizes a multi-input system even when the system has a singular matrix.
Abstract: A feedback gain will be discussed which can easily be computed and stabilizes a discrete constant linear system. The results extend the work of Kleinman [1]. In particular, the requirement of nonsingularity will be removed for single-input systems, and another similar control will be given which stabilizes a multi-input system even when the system has a singular matrix.

Journal ArticleDOI
TL;DR: This work considers the problem of placing records in a 2-dimensional storage array so that expected distance between consecutive references is minimized and a simple placement heuristic which uses only relative frequency of access for different records is shown to be within an additive constant of optimal when distance is measured by the Euclidean metric.
Abstract: We consider the problem of placing records in a 2-dimensional storage array so that expected distance between consecutive references is minimized. A simple placement heuristic which uses only relative frequency of access for different records is shown to be within an additive constant of optimal when distance is measured by the Euclidean metric. For the rectilinear and maximum metrics, we show that there is no such heuristic. For the special case in which all access probabilities are equal, however, heuristics within an additive constant of optimal do exist, and their implementation requires solution of differential equations for which we give numerical solutions.

Journal ArticleDOI
TL;DR: An algorithm is presented for solving nonlinear ordinary differential equations that generates a broad class of implicit linear differentiation formulas that is more efficient as to the number of arithmetic operations than Brayton's algorithm, as well as more general and systematic.
Abstract: An algorithm is presented for solving nonlinear ordinary differential equations that generates a broad class of implicit linear differentiation formulas. Specific interest is concerned in one type of formula which for constant timestep reduces to the well-known Gear formulas. Although these formulas are mathematically fully equivalent to the BDF formulas as presented by Brayton et al. [1], their construction is quite different. Instead of using previously calculated function values, we employ predictions extrapolated from these values to set up and evaluate the differentiation formula. A recursive relation for these predictions is derived in order to simplify their calculation in the next timestep. As predictions are needed for order and error control, our algorithm appears to be more efficient as to the number of arithmetic operations than Brayton's algorithm, as well as more general and systematic. Moreover, a change of the order can be accomplished without extra work. Due to the available predictions, an interpolation to determine function values at intermediate time instants, as for instance required in plotting procedures, can be performed in a fast way.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the higher dimensional analogs of the following classical characteristics of compact planar sets: transfinite diameter, Cebysev constant, and capacity.
Abstract: This article considers the higher dimensional analogs of the following classical characteristics of compact planar sets: transfinite diameter, Cebysev constant, and capacity.An affirmative solution is given to the problem, posed by F. Leja in 1957, of whether for the ordinary limit of the sequence defining transfinite diameter exists. The concept of -capacity is introduced, and it is compared with transfinite diameter and another Cebysev constant .For an arbitrary compact set in an analog is considered of a classical theorem of Polya estimating the sequence of Hankel determinants constructed from the coefficients in the power series expansion of an analytic function in a neighborhood of infinity. The estimate comes from the transfinite diameter of the singular set of the function.Bibliography: 10 items.

Journal ArticleDOI
TL;DR: In this article, the Kramer's problem for variable collision frequency was solved, and it was found that the velocity defect in the Knudsen layer is quite sensitive to the velocity dependence of the collision frequency.
Abstract: Since the BGK model is based on the assumption of constant collision frequency, and since this model has been found inadequate in describing some experimental data, the numerical study of a variable collision frequency model proposed earlier by Cercignani and Loyalka and Ferziger is described. Specifically, the Kramer’s problem for this model is solved, and it is found that the ’’velocity defect’’ in the Knudsen layer is quite sensitive to the velocity dependence of the collision frequency. In fact, for the hard sphere collision frequency, the present results agree reasonably well with the recent experimental data of Reynolds, Smolderen, and Wendt.

Journal ArticleDOI
TL;DR: For 981 undergraduate classes, mean student ratings of instruction decrease with increasing class size as discussed by the authors, and this relationship remains strong when other variables known or believed to influence ratings are held constant.
Abstract: For 981 undergraduate classes, mean student ratings of instruction decrease with increasing class size This relationship remains strong when other variables known or believed to influence ratings are held constant

Journal ArticleDOI
TL;DR: In this article, the sound speed and density profiles are partitioned in layers such that in each layer the square of the index of refraction can be approximated by a straight line and the density by a constant.
Abstract: In a stratified ocean model, in which the sound speed and density become constant as the depth coordinate becomes infinite, the pressure field can be represented as a finite sum of modes plus an integral superposition of modes. This later contribution is given by an integration around a branch cut. In this analysis the sound‐speed and density profiles are partitioned in layers such that in each layer the square of the index of refraction can be approximated by a straight line and the density by a constant. The profile is terminated in a high‐speed isovelocity half space. Using this profile, it is possible to express the depth‐dependent portion of the pressure field in terms of Airy functions. Evaluation of the modal and branch line contributions shows that the branch integral can contribute significantly to the pressure field over a range equal to at least one water depth and in some cases to many water depths.Subject Classification: 30.20, 30.25.

Patent
21 Jan 1975
TL;DR: In this article, the authors proposed methods and arrangements in which certain parameters, such as arc time and a short-circuit current delay time are controlled or maintained constant, so that always a constant bead size is assured owing to a constant power, after which said bead of constant size is first partly introduced into the molten pool and subsequently the short circuit current for separation is applied.
Abstract: In short circuit arc welding there are many parameters which influence the automatically controlled welding procedure, but which frequently may affect the quality of the weld. In some circumstances the welding bead which is formed does not flow in to the molten pool of the workpiece, but disintegrates into many small drops around the weld owing to an excessive short-circuit current. The invention proposes methods and arrangements in which certain parameters, such as arc time and a short-circuit current delay time are controlled or maintained constant, so that always a constant bead size is assured owing to a constant power, after which said bead of constant size is first partly introduced into the molten pool and subsequently the short-circuit current for separation is applied.

Journal ArticleDOI
TL;DR: The average number of levels that a new element moves up when inserted into a heap is investigated and it is shown that this average is bounded by a constant and its asymptotic behaviour is discussed.
Abstract: The average number of levels that a new element moves up when inserted into a heap is investigated. Two probabilistic models under which such an average might be computed are proposed. A `Lemma of Conservation of Ignorance' is formulated and used in the derivation of an exact formula for the average in one of these models. It is shown that this average is bounded by a constant and its asymptotic behaviour is discussed. Numerical data for the second model are also provided and analyzed.

Journal ArticleDOI
01 Feb 1975
TL;DR: In this article, the minimal constants for infinite A(2n) sets in discrete noncommutative groups were computed and an alternate proof of Leinert's theorem on A(o) sets was obtained.
Abstract: We compute the minimal constants for infinite A(2n) sets in discrete noncommutative groups and as a consequence we obtain an alternate proof of Leinert's theorem on A(o) sets.

Journal ArticleDOI
R Ramji Rao1
TL;DR: In this paper, the second Gruneisen constant of a number of cubic and uniaxial solids has been evaluated at the room temperature from thermodynamic data using a modified formulation of Basset et al.
Abstract: The second Gruneisen constant of a number of cubic and uniaxial solids has been evaluated at the room temperature from thermodynamic data using a modified formulation of Basset et al. A method to calculate the Anderson-Gruneisen parameter of a solid from its pressure derivatives data is described.

Journal Article
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) of the agreement with the Scuola Normale Superiore di Pisa are defined.
Abstract: © Scuola Normale Superiore, Pisa, 1975, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Journal ArticleDOI
TL;DR: A many-state Markov model has been developed for the purpose of providing various performance criteria for computer software and it is demonstrated that the numerical solution is superior to the so-called exact solution.
Abstract: A many-state Markov model has been developed for the purpose of providing various performance criteria for computer software. The software system under consideration is assumed to be fairly large, of the order of 105 words of code, so that statistical deductions become meaningful, and is assumed to initially contain an unknown number of unknown bugs. The model provides estimates and predictions of the most probable number of errors that will have been corrected at a given time t in the operation of this software package based on preliminary modeling of the error occurrence rate l as well as the error correction policy m. The model also provides predictions for the availability A(t) and for the reliability R(t) of the system. The differential equations corresponding to the Markov model are solved for the case when l and m are constant using an exact (closed-form) solution. The numerical solution is also obtained for this case for verification and demonstrative purposes. The more interesting and important case, from an applications point of view, is that when l and m are not constant, but rather functions of the state of debugging achieved. This case is solved numerically only, since the exact solution is cumbersome. It is also demonstrated that the numerical solution is superior to the so-called exact solution. Finally, some extensions and modifications of the basic Markov model are briefly discussed.