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Kepler's laws are an appropriate topic to highlight the significance of the pedal equation in physics. There are several articles that obtain Kepler's laws through conservation and gravitation laws. This can be shown more easily and ingeniously if one uses the pedal equation of an ellipse. In fact, the complete kinematics of a particle in an attractive central force field can be derived from one single pedal form. Though many articles use the pedal equation, the classical procedure (without proof) for obtaining the pedal equation is mentioned only in a few because the classical derivations can sometimes be lengthier and are not simple. In this paper, using elementary physics, we derive the pedal equation for all conic sections in a unique, short, and pedagogical way. Later, from the dynamics of a particle in the attractive central force field, we deduce the single pedal form, which elegantly describes all the possible trajectories. Also, for the purpose of completeness, we derive Kepler's laws. Key words: pin and string construction of conics, pedal equation, central force field, conservation laws, trajectories and Kepler laws. Les lois de Kepler constituent un sujet approprie pour faire ressortir la pertinence de l'equation podaire en physique. Il y a plusieurs articles qui obtiennent les lois de Kepler comme consequence de lois de conservation et de gravite. Ce resultat peut s'obtenir de facon plus facile et plus ingenieuse en utilisant l'equation podaire de l'ellipse. En fait, la cinematique complete d'une particule dans un champ de force central peut etre obtenue d'une simple formule podaire. Meme si bon nombre d'articles utilisent l'equation podaire, seulement quelques uns presentent la procedure classique pour obtenir (sans preuve) l'equation podaire. La raison est probablement que cette procedure est longue et pas simple. Notre but ici est d'utiliser des arguments physiques simples pour obtenir l'equation podaire pour toutes les sections coniques d'une facon courte, simple et pedagogique. Par la suite, a partir de la dynamique de la particule dans le champ de force central, nous deduisons la forme podaire simple qui decrit de facon elegante toutes les trajectoires possibles. Pour etre plus complets, nous obtenons ensuite les lois de Kepler. [Traduit par la Redaction] Mots-cles : construction de coniques a l'aide de punaises et de cordelettes, equation podaire, champ de force central, lois de conservation, trajectoires et lois de Kepler., 1. Introduction The accurate observations of Tycho Brahe and the discovery of the three laws by Johannes Kepler using them, can be considered the epitome of observational science. Kepler, after [...]

With the emergence and advancement of airborne laser scanning technology over the past decade, individual tree height can be easily measured over a large area of forests with a comparable degree of accuracy to conventional ground-based methods. In laser scanning based large-scale forest inventories, the need to predict diameter from remotely sensed tree height calls for a systematic evaluation of equation forms as the first step towards a well-developed approach to developing diameter-height equations. This study evaluated more than 30 height-diameter equations in the forest biometrics literature to select candidates for deriving equation forms for diameter-height equations. The evaluation was based on four criteria: (i) the height-diameter function is inversable; (ii) the inverse function is continuous and monotonically increasing over a specified working range of total tree height; (iii) diameter at breast height is equal to zero when tree height equals breast height in the inverse function; and preferably, (iv) the inverse function has an inflection point that is consistent with biological expectations. A total of 12 candidate equation forms were derived, which included five two-parameter and seven three-parameter equations. The estimation properties and predictive performance of these 12 equation forms were further evaluated and compared through repeated sampling and fitting using data from 3581 trees destructively sampled for taper measurements from Pinus radiata D. Don plantations across New South Wales, Australia. Three equation forms, including the constrained Richards, Weibull, and the combined power and exponential function, displayed superior prediction accuracy and estimation properties and so were recommended as the primary equation forms for developing diameter-height equations. The remaining equation forms were marred by either lower prediction accuracy or poorer estimation properties or both. The three recommended equation forms should only serve as basic deterministic specifications upon which other tree and stand variables should be incorporated as predictors to further improve their predictive performance. Resume: Grace a l'emergence et au progres de la technologie de balayage laser aeroporte au cours de la derniere decennie, la hauteur des arbres individuels peut etre facilement mesuree sur une grande superficie de foret avec un degre d'exactitude comparable a celui des methodes classiques au sol. Dans les inventaires forestiers a grande echelle bases sur le balayage laser, la necessite de predire le diametre a partir de la hauteur de l'arbre obtenue par teledetection exige une evaluation systematique des formes d'equation comme premiere etape d'une approche bien structuree pour produire des equations diametre-hauteur. Cette etude a evalue plus de 30 equations hauteur-diametre trouvees dans la litterature sur la biometrie forestiere pour choisir celles qui pouvaient etre utilisees pour deriver les formes d'equation des equations diametre-hauteur. L'evaluation a ete basee sur quatre criteres: (i) la fonction hauteur-diametre est inversable; (ii) la fonction inverse est continue et croissante monotone sur une plage specifiee de la hauteur totale de l'arbre; (iii) le dhp est egal a zero lorsque la hauteur l'arbre est egale a la hauteur de poitrine dans la fonction inverse; et de preference, (iv) la fonction inverse a un point d'inflexion qui est conforme aux attentes biologiques. Au total, 12 formes candidates d'equation ont ete derivees incluant cinq equations a deux parametres et sept equations a trois parametres. En outre, les proprietes d'estimation et la performance predictive de ces 12 formes d'equation ont ete evaluees et comparees par un echantillonnage et un ajustement repetes en utilisant les donnees de 3581 arbres echantillonnes de facon destructive pour mesurer le defilement dans des plantations de Pinus radiata D. Don situees un peu partout dans l'Etat de la Nouvelle-Galles du Sud, en Australie. Trois formes d'equation, incluant la fonction de Richards avec contrainte, la fonction de Weibull et la fonction exponentielle et puissance combinee, avaient une exactitude de prediction et des proprietes d'estimation superieures. Elles ont donc ete recommandees comme formes primaires d'equation pour developper les equations diametre-hauteur. Les formes d'equation restantes etaient entachees soit par une moins grande exactitude de prediction ou de mauvaises proprietes d'estimation ou les deux. Les trois formes d'equation recommandees devraient servir seulement de specifications deterministes de base sur lesquelles d'autres variables d'arbre et de peuplement devraient etre incorporees en tant que predicteurs pour ameliorer encore davantage leur performance predictive. [Traduit par la Redaction], Introduction Diameter at breast height (DBH) and total tree height from ground to tip are the two most common measurements of tree size in forest mensuration. Although DBH can be [...]

We quantitatively investigate the axial imaging properties of dynamic phase-contrast microscopy, with a special focus on typical combinations of tracer particles and magnifications that are used for velocimetry analysis. We show, for the first time, that a dynamic phase-contrast microscope, which is the integration of an all-optical novelty filter in a commercially available inverted microscope, can visualize three-dimensional velocity fields with a significantly reduced optical sectioning depth. The depth of field for dynamic phase-contrast microscopy is extracted from the three-dimensional response function and compared with the respective values for incoherent bright-field illumination. These results are then used to perform a depth-resolved particle image velocimetry analysis of Hagen-Poiseuille as well as electro-osmotically actuated flows in a microchannel. OCIS codes: 110.0180, 180.4315, 180.6900, 190.7070.

Classically, in open-channel networks, the flow is numerically approximated by the one-dimensional Saint Venant equations coupled with a junction model. In this study, a comparison between the one-dimensional (1D) and two-dimensional (2D) numerical simulations of subcritical flow in open-channel networks is presented and completely described allowing for a full comprehension of the modeling of water flow. For the 1D, the mathematical model used is the 1D Saint Venant equations to find the solution in branches. For junction, various models based on momentum or energy conservation have been developed to relate the flow variables at the junction. These models are of empirical nature due to certain parameters given by experimental results and moreover they often present a reduced field of validity. In contrast, for the 2D simulation, the junction is discretized into triangular cells and we simply apply the 2D Saint Venant equations, which are solved by a second-order finite-volume method. In order to give an answer to the question of luxury or necessity of the 2D approach, the 1D and 2D numerical results for steady flow are compared to existing experimental data. DOI: 10.1061/(ASCE)HY.1943-7900.0000230 CE Database subject headings: Open channels; Critical flow; Steady flow; Flow. Author keywords: Saint Venant equations; Open channels.

We study the synthesis of a slider-crank four-bar linkage whose coupler point traces a set of predefined task points. We report that there are at most 558 slider-crank four-bars in cognate pairs passing through any eight specified task points. The problem is formulated for up to eight precision points in polynomial equations. Classical elimination methods are used to reduce the formulation to a system of seven sixth-degree polynomials. A constrained homotopy technique is employed to eliminate degenerate solutions, mapping them to solutions at infinity of the augmented system, which avoids tedious post-processing. To obtain solutions to the augmented system, we propose a process based on the classical homotopy and secant homotopy methods. Two numerical examples are provided to verify the formulation and solution process. In the second example, we obtain six slider-crank linkages without a branch or an order defect, a result partially attributed to choosing design points on a fourth-degree polynomial curve. [DOI: 10.1115/1.4001878]

A fast computation algorithm for fine-pixel aerial images is presented that modifies the transmission cross coefficient approach of Hopkins as a product of two matrices. The spatial frequency of the image is calculated by the sum of diagonal and off-diagonal elements of the matrix. Let N, [N.sub.F], and M be the number of the point sources, the sampling number for fast Fourier transform, and the sampling number in the spatial frequency domain ranging twice the pupil size, respectively. The calculation time of this method is proportional to BN[[(M- 1)/2].sup.4], while that of a conventional source integration method is [2ANN.sub.F][log.sub.2] [N.sub.F], where A and B are constants and generally B < A If [N.sub.F] is sufficiently greater than M or M is small enough, which is the fine-pixel condition, this method runs faster than the source integration method. If the coherence factor is 0.9 and M [less than or equal to] 55, this method runs faster than the source integration even under the Nyquist sampling condition. OCIS codes: 110.4980, 110.5220, 110.2990.

Nonlinear filtering theory, based upon the Fokker-Planck equation and Bayes' rule, has been around since the 1970s. This method does not require linearized dynamics and/or measurement models, and also the assumption of the Gaussian process is not necessary. However, it is extremely difficult to obtain an analytical solution of the Fokker-Planck equation with only a few exceptions, and normally the solution is obtained through numerical approaches. In this paper, this type of nonlinear filtering method is reinvestigated and applied to a relative orbit estimation problem. The time evolution of the state conditional probability density function between measurements is obtained by solving the associated Fokker-Planck equation numerically using the numerical scheme called the alternating direction implicit method. To further mitigate the computational load, an enhanced moving domain technique is employed and the probability distribution function is partitioned. DOI: 10.1061/(ASCE)AS.1943-5525.0000024 CE Database subject headings: Filters; Aerospace engineering; Orbits; Measurement; Distribution functions. Author keywords: Nonlinear filtering; Fokker-Planck equation; Bayes estimation; Aerospace control; Alternative direction implicit method.

The non-autonomous Bloch equations modelling a driven 2-level atom in the presence of an off-resonant broadband squeezed vacuum (SV) field is treated analytically. This concerns iterative solutions valid for large SV detuning parameter but for arbitrary strength of the laser field. Computational results are presented for the averaged atomic variables for various data and compared with the resonant SV field case. The iterated analytical results for nonzero SV detuning are compared with the (exact) numerical solutions of the Bloch equations, hence we have an insight about the range of other system parameters (other than the Rabi frequency) for which the iterated solutions are valid to O([10.sup.-2]) or less. The main purpose of deriving these analytical results is to calculate analytically the transient fluorescent spectrum. For an initially ground-state atom, both the SV phase and detuning parameters induce pronounced asymmetrical spectrum in the strong field case. PACS Nos: 32.50.+d, 42.50.-p, 42.50.ct Nous etudions analytiquement les equations de Bloch qui modeelisent un atome a deux niveaux soumis a un champ laser en presence d'un champ du vide (SV) comprimee a large bande et hors resonance. Ceci touche les solutions iteratives pour de grands parametres de desyntonisation du SV, mais pour une intensite arbitraire du champ laser. Les resultats calcules sont presentes pour des variables atomiques moyennes pour diffeerentes donneees et compares avec le cas du champ SV resonant. Nous comparons les resultats analytiques iteeratifs pour un SV non nul avec les solutions numeriques (exactes) des equations de Bloch, ce qui nous donne un apenju du domaine des parametres du systeme autres que la frequence de Rabi, pour lequel les solutions iteratives sont valides a une precision de 0([10.sup.-2]) ou mieux. Le but principal de ces resultats analytiques est de calculer analytiquement le spectre de fluorescence transitoire. Pour un atome initialement dans le fondamental, la phase SV et les parametres de desyntonisation peuvent tous deux conduire a un spectre fortement asymetrique en champ fort. [Traduit par la Redaction], 1. Introduction From the early days of quantum optics, theoretical studies of the interaction of radiation fields with the simplest atomic structure, namely a two-level atom, has provided valuable information [...]

Transformed Eulerian mean (TEM) equations and Eliassen--Palm (EP) flux diagnostics are presented for the general nonhydrostatic, fully compressible, deep atmosphere formulation of the primitive equations in spherical geometric coordinates. The TEM equations are applied to a general circulation model (GCM) based on these general primitive equations. It is demonstrated that a naive application in this model of the widely used approximations to the EP diagnostics, valid for the hydrostatic primitive equations using log-pressure as a vertical coordinate and presented, for example, by Andrews et al. in 1987 can lead to misleading features in these diagnostics. These features can be of the same order of magnitude as the diagnostics themselves throughout the winter stratosphere. Similar conclusions are found to hold for 'downward control' calculations. The reasons are traced to the change of vertical coordinate from geometric height to log-pressure. Implications for the modeling community, including comparison of model output with that from reanalysis products available only on pressure surfaces, are discussed. DOI: 10.1175/2010JAS3355.1

Recently, two wave models based on the stream-function theory have been derived from Hamilton's principle for gravity waves. One is the irrotational Green--Naghdi (IGN) equation and the other is the complementary mild-slope equation (CMSE). The IGN equation has been derived to describe refraction and diffraction of nonlinear gravity waves in the time domain and in water of finite but arbitrary bathymetry. The CMSE has been derived to consider the same problem in the (linear) frequency domain. In this paper, we first discuss the two models from the viewpoint of Hamilton's principle. Then the two models are applied to a resonant scattering of Stokes waves over periodic undulations, or the Bragg scattering problem. The numerical results are compared with existing numerical predictions and experimental data. It is found here that Level 3 IGN equation can describe Bragg scattering well for arbitrary bathymetry. [DOI: 10.1115/1.4000503]

The classical Abraham-Lorentz formula for the radiation reaction on a point charge suffers from two notorious defects: runaways and preacceleration. Recently, several authors have advocated as an alternative the Landau-Lifshitz formula, which has neither fault. The latter formula is often presented as an approximation to Abraham-Lorentz, raising the delicate question of how an approximation can be considered more accurate than the original. For a spherical shell of finite size, the equation for the radiation reaction is noncontroversial. We begin there, obtain the Abraham-Lorentz and Landau-Lifshitz expressions as limiting cases, and undertake some numerical studies to determine which is superior. [c] 2010 American Association of Physics Teachers. [DOI: 10.1119/1.3269900]

This paper describes the derivation and numerical solution of a lubrication equation appropriate for high Knudsen number flows and certain types of striated rough surfaces. The derivation begins with the compressible form of the lubrication equation together with the nonlinear series form of the Poiseuille flow reported by Fukui and Kaneko (1990, 'A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems,' ASME J. Tribol., 112, pp. 78-83.). A multiple-scale analysis is performed on the lubrication equation for a finite-width time-dependent bearing and is limited to either stationary-transverse or longitudinal striated surface roughness of very. short length scale. The rough surface averaging that takes place within the multiple-scale analysis includes a fully coupled treatment of the Poiseuille flow. What results is an especially nonlinear lubrication equation with averaged surface roughness effects that is appropriate for high Knudsen number analysis. A rotational transformation is also introduced to provide the roughness averaged lubrication equation in a form that allows analysis of the skewed orientation of a recording head slider with roughness defined relative to the direction of disk motion but with the lubrication equation conveniently expressed in the coordinate system of the slider. A factored-implicit numerical algorithm is described that provides the solution of the roughness averaged lubrication equation. Even though the lubrication equation is highly nonlinear, the numerical scheme is crafted to be fully second-order, time-accurate, and noniterative for tracking the solution in time either to an asymptotic steady-state or in response to a dynamic event. Numerical solutions of several simple geometry bearings are presented that utilize parameters that are typical of the slider-disk interface of current hard disk drives. It is anticipated that the primary benefit of this work may be the ability to accurately and efficiently include the influence of discrete disk data tracks in the air bearing design of very low clearance recording head sliders. [DOI: 10.1115/1.4001023] Keywords: data storage, high Knudsen number, rough surface, air bearing interface, lubrication equation

Equations for Rayleigh scattering in a mixture of gases are derived and compared to frequent approximations in the literature. The traditional Rayleigh scattering equation as modified by King for scatter from a pure gas is correct, whereas another version sometimes appearing in modern literature is erroneous. Use of a mixture's refractive index, which is equivalent to assuming the isotropic molecular polarizabilities of the component gases are identical, is an approximation. Another common approximation is using only number-density weighting of the King factors. Approximation errors can be large when the major components of a mixture have disparate optical properties. Fortunately, the errors for Earth's air are much smaller and comparable to errors from other sources. OCIS codes: 290.5870, 290.1310, 010.1320, 280.3640.

We present a method for orientation-selective edge detection and enhancement based on the irradiance transport equation. The proposed technique distinguishes the sign of the derivative of the intensity pattern along an arbitrarily selected direction. The method is based on the capacity of liquid-crystal displays to generate simultaneously a contrast reverted replica of the image displayed on it. When both images (the original one and its replica) are imagined across a slightly defocused plane, one obtains an image with enhanced first derivatives. Unlike most Fourier methods, the proposed technique works well with a low-coherence light source, and it does not require precise alignment. The proposed method does not involve numerical processing, and thus it could be potentially useful for processing large images in real-time applications. Validation experiments are presented. [c] 2010 Optical Society of America OCIS codes: 100.2980, 100.2810.

In this paper, unsteady water quality modeling and the associated sensitivity equations are solved for water distribution systems. A new solution algorithm is proposed, designed for slow varying velocity and based on a time splitting method to separate and solve efficiently each phenomenon such as advection and chemical reaction. This numerical approach allows simultaneous solution of both the direct problem and the sensitivity equations. Special attention is given to the treatment of advection, which is handled with a total variation diminishing scheme. The general model presented in this study permits global sensitivity analysis of the system to be performed and its efficiency is illustrated on two pipe networks. The importance of the sensitivity analysis is shown as part of the calibration process on a real network. DOI: 10.1061/(ASCE)HY.1943-7900.0000138 CE Database subject headings: Water distribution systems: Sensitivity analysis: Water quality: Hydraulic models. Author keywords: Water distribution systems: Sensitivity analysis; Water quality; Models: Algorithm.

We investigate the ability of a ID fully nonlinear Boussinesq model including breaking to reproduce surf zone waves in terms of wave height and nonlinear intraphase properties such as asymmetry and skewness. An alternative approach for wave-breaking parameterization including roller effects through diffusive-type terms on both, the mass conservation and momentum equations is developed and validated on regular wave and solitary wave experiments as an attempt to improve wave height and left-right asymmetry estimates. The new approach is able to reproduce wave height decay, and intraphase nonlinear properties within the entire surf zone of spilling breakers without requiring temporal evolution of model parameters. CE Database subject headings: Surf zones; Breaking waves; Boussinesq equations; Wave propagation. Author keywords: Surf zone waves; Wave-breaking; Boussinesq-type equations; Nonlinear wave propagation. DOI: 10.1061/(ASCE)WW.1943-5460.0000022

The propagation of a dispersive shock wave is studied in a quintic-derivative nonlinear Schrodinger (Q-DNLS) equation, which may describe, Por example, the wave propagation on a discrete electrical transmission line. II is shown that a physical system described by a Q-DNLS equation without a dissipative term may support the propagation of shock waves. The influence of the derivative nonlinearity terms on the shock is analyzed. Using the round exact shock solutions of the Q-DNLS equation as the initial input signal, we investigate numerically the spatiotemporal stability of the shock signal in the network. PACS Nos: 42.65.Tg, 42.25.Bs, 84.40.Az, 02.60.Cb Nous etudions la propagation d'une onde de choc dispersive a l'aide d'une equation differentielle quintique non lineaire de Schrodinger (equation Q-DNLS), qui peut decrire, par exemple. la propagation d'onde sur une ligne de transmission electrique discrete. Nous montrons qu'un systeme physique decrit par une equation Q-DNLS sans terme dissipatif peut permettre la propagation d'onde de choc. Nous etudions l'influence des termes non lineaires en derivee sur l'onde de choc. Utilisant comme signal de depart, la solution exacte pour l'onde de choc obtenue de l'equation Q-DNLS, nous etudions la stabilite spatiotemporelle du signal de choc dans la ligne. [Traduit par la Redaction], 1. Introduction Since the pioneering works by Hirota and Suzuky [l] on electrical transmission lines simulating the Toda lattice, a growing interest has been devoted to the use of the [...]

In this paper, some complete monotonicity properties of functions related to the gamma function and Barnes G-function are proved. Keywords Gamma function, double gamma function, Barnes G-function, complete monotonicity, asymptotic expansion., [section] 1. Introduction A function f is said to be completely monotonic over (a, b), where -[infinity] [less than or equal to] a < b [less than or equal to] [...]

For any positive integer n, the Pseudo-Smarandache function Z(n) is defined as Z(n) = min {m : m [member of] N, n | m (m+1)/2}. And the F. Smarandache function SL(n) is defined as the smallest positive integer k such that n [absolute value of [1, 2, ..., k]], where [1, 2, ..., k] denotes the least common multiple of 1, 2, ..., k. In this paper, we use the elementary methods and congruences to study the solvability of the equation Z(n) + SL(n) = n, and give its all positive integer solutions. Keywords Pseudo-Smarandache function, F. Smarandache LCM function, equation, solution, congruences., [section] 1. Introduction and results For any positive integer n [greater than or equal to] 1, the Pseudo-Smarandache function Z(n) is defined as the smallest positive integer m such that [...]

Let [t.sup.(e)] (n) denote the number of e-squarefree e-divisor of n. The aim of the present paper is to establish an asymptotic formula for the mean value of the function ([t.sup.(e)])r, where r [greater than or equal to] 1 is a fixed integer. Keywords E-squarefree e-divisor function, the generalized divisor function, convolution method., [section] 1. Introduction An integer d = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called an e-divisor of n =[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > 1 if [b.sub.i]|[a.sub.i] for [...]

Let n > 1 be an integer, [??](n) denote the number of regular integers m(mod n) such that 1 [less than or equal to] m [less than or equal to] n. In this paper we shall investigate the mean value of the function log [??](n) by the convolution method. Keywords Regular integers (mod n), mean value, convolution method, Perron's formula., [section] 1. Introduction and main results Let n > 1 be an integer. Consider the integers m for which there exists an integer x such that [m.sup.2]x = m(mod n). [...]

The homothetic motion in 2-dimensional Euclidean space [E.sup.2], the relation between the velocities of this motion and geometric results for the pole curves were studied in ref [1]. In this paper a canonical relative system of any plane with respect to other planes are given. Therefore in a homothetic motion E/E', Euler-Savary formula giving the relation between the curvature of trajectory curves drawn in the fixed plane E' by the points of the moving plane E is obtained. In the special case of homothetic scale h identically equal to 1, we get the Euler-Savary formula which was given by Muller [5]. Finally some geometrical results are reached using Euler-Savary formula. Keywords Euler-Savary formula, homothetic motion, kinematics., § 1. Introduction This study deals with instantaneous geometric plane kinematics. Which is the study, for a certain instant during a continuous motion, of the differential geometric properties? So if [...]

An integer a is called regular (mod n) if there is an integer x such that [a.sup.2]x [equivalent to] a(mod n). Let [??](n) denote the number of regular integers a(mod n) such that 1 [less than or equal to] a [less than or equal to] n, [phi](n) is the Euler function. In this paper we investigate the mean value of the function [([??](n)/[phi](n)).sup.r], where r [greater than or equal to] 1 is a fixed integer. Keywords Regular integers (mod n), Euler's function, average order, convolution method, Euler product., [section] 1. Introduction Let n > 1 be an integer. Consider the integers a for which there exists an integer x such that [a.sup.2]x = a(mod n). Properties of these [...]

In this paper we will establish some new methods for solving third and fourth degree equations. Keywords Cardano method, Ferrari method, equation., [section] 1. Introduction In this paper, using new transformations, we will introduce new elementary methods for solving third and fourth degree equations, which essentially differ from Cardano's method and Ferrari's [...]

Researchers developed several mathematical models for simulating furrow irrigation using the Saint-Venant equations. Most of these irrigation models use numerical techniques to solve these equations, which in general, require extensive programming and computational skills. Moreover, several of these models consider uniform soil and use empirical equations for modeling infiltration. In this article, a physically based furrow irrigation model was presented for simulating flow in irrigated furrows under both uniform and layered soils. The model consisted of an overland flow and an infiltration module that are modeled using analytical solution of the zero-inertia and the Green and Ampt lone-dimensional and two-dimensional infiltration equations) equations, respectively. Furthermore, the infiltration was also modeled using the Kostiakov-Lewis infiltration equation. The model considered all possible furrow shapes and included graphical user interface. The developed model was evaluated using the field data and the model performance was discussed in the second part of the article. DOI: 10.1061/(ASCE)IR.1943-4774.0000086 CE Database subject headings: Furrow irrigation; Hydraulic models; Mathematical models.

We show how the Proca equations can be deduced from first principles in a way that is similar to the derivations of the Dirac relativistic electron equation and Maxwell equations. [DOI: 10.1119/1.3137042]

An alternative derivation of the first-order relativistic contribution to perihelic precession is presented. Orbital motion in the Schwarzschild geometry is considered in the Keplerian limit, and the orbit equation is derived for approximate elliptical motion. The method of solution makes use of coordinate transformations and the correspondence principle rather than the standard perturbative approach. The form of the resulting orbit equation is similar to that derived from Newtonian mechanics and includes first-order corrections to Kepler's orbits due to general relativity. The associated relativistic contribution to perihelic precession agrees with established first-order results. The reduced radius for the circular orbit is in agreement to first-order with that calculated from the Schwarzschild effective potential. The method of solution is understandable by undergraduate students. [DOI: 10.1119/1.3159611]

The formulation of the dynamic equations of motion proposed by Udwadia-Kalaba is discussed from the point of view of numerical efficiency. Since this formulation requires the computation of a pseudoinverse matrix, it has investigated the influence of the method of pseudoinverse computation on the dynamic simulation of an overconstrained linkage. Finally, it has been proposed that a new dynamic equation which explicitly computes accelerations also in the case of mathematical models with rank deficient Jacobian and mass matrices. DOI: 10.1061/(ASCE)0893-1321 (2009)22:4(365) CE Database subject headings: Matrix methods; Computation; Simulation; Aerospace engineering.

Let R be a 2-torsion-free prime ring and J anonzero Jordan ideal and a subring of R. For a ([sigma], [tau])-derivation d : R [right arrow] R, we prove the following results: (1) If d is a ([sigma], [sigma]]) derivation which acts as a homomorphism or as an anti-homomorphism on J, then either d = 0 on R or J [subset or equal to] Z(R); (2) If F is a generalized ([sigma], [sigma]])-derivation which acts as a homomorphism or as an anti- homomorphism on J, then either d = 0 on R or J [subset or equal to] Z(R); (3) If d is a ([sigma], [tau])-derivation which acts as a homomorphism on J, then d = 0 on R. Keywords Fuzzy Prime rings, derivations, ([sigma], [tau])-derivations, generalized derivations, generalized ([sigma], [tau])-derivations., [sections]1. Introduction Throughout the present paper, R will denote an associvative ring with center Z(R). We will write for all x,y [member of] R, [x, y] = xy - yx [...]

The main purpose of this paper it to studied the mean value properties of the Smarandache Superior m-th power part sequence ssmp(n) and the Smarandache Inferior m-th power part sequence simp(n), and give several interesting asymptotic formula for them. Keywords Smarandache Superior m-th power part sequence, Smarandache Inferior m-th power part sequences, mean value, asymptotic formula., [section]1. Introduction and Results For any positive integer n, the Smarandache Superior m-th power part sequence SSMP(n) is defined as the smallest m-th power greater than or equal to n. [...]

The exact solutions of the Duffin-Kemmer-Petiau (DKP) oscillator for spin-0 particles have been studied in noncommutative phase space. The results show that due to the noncommutative effect, the energy spectrum of the DKP oscillator for spin-0 particles is no longer degenerate. In addition, we obtain the nonrelativistic limit of the energy spectrum. PACS Nos: 02.40.Gh, 03.65.-w, 03.65.Ge, 03.65.Pm Nous avons etudie sur un espace de phase non commutatif les solutions exactes de l'oscillateur Duffin-Kemmer-Petiau (DKP) pour des particules de spin zero. Les resultats montrent qu'a cause de la non commutativite, le spectre d'energie de particules massives de spin nul dans le potentiel d'oscillateur de Dirac n'est pas degenere. Nous obtenons aussi la limite non relativiste du spectre d'energie. [Traduit par la Redaction], 1. Introduction It is well known that the Duffin-Kemmer-Petiau (DKP) equation is a first-order relativistic wave equation for spin 0 and 1 bosons [1-3], which describes some problems in nuclear [...]

The Klein-Gordon equation in the presence of a spatially one-dimensional Hulthen potential is solved exactly and the scattering solutions are obtained in terms of hypergeometric functions. The transmission coefficient is derived by the matching conditions on the wave functions and the conditions for the existence of transmission resonances are investigated. It is shown how the zero-reflection condition depends on the shape of the potential. PACS Nos: 03.65.Nk, 03.65.Pm Nous solutionnons exactement l'equation de Klein-Gordon en 1+1 D avec un potentiel a une dimension spatiale de Hulthen et nous obtenons les solutions de diffusion sous la forme de fonctions hypergeometriques. Le coefficient de transmission est obtenu en mariant les conditions sur les fonctions d'onde et nous etudions les conditions d'existence d'une transmission resonante. Nous montrons que la condition de reflexion nulle depend de la forme du potentiel. [Traduit par la Redaction], The study of low-momentum scattering in the Schrodinger equation in one-dimensional even potentials shows that, as momentum goes to zero, the reflection coefficient goes to unity unless the potential V(x) [...]

By using Prandtl's mixing length theory, to model two-dimensional Reynolds stress equations, the pumping performance of a labyrinth screw pump (LSP) is studied and several key parameters are empirically determined. As a result, two innovative concepts, a cell head coefficient [K.sub.f] and a pump total head coefficient [K.sub.b], are proposed. A simple empirical equation quantifying the effects of the main geometric parameters of the threads' on the pump performance is obtained and compared with Golubiev's experimental results (1965, 'Studies on Seal for Rotating Shafts of High-Pressure Pumps,' Wear, 8, pp. 270-288; 1981, Labyrinth-Screw Pumps and Seals for Corrosive Media, 2nd ed., Mashinostroenie, Moscow, pp. 34-49). Both theoretical study and Golubiev's results indicate that with an increase in screw lead, [K.sub.f] increases while [K.sub.b] decreases. [K.sub.f] is inversely proportional to power of screw-sleeve relative diametrical clearance, and the power exponent varies with different shapes of thread. Finally, [K.sub.f] decreases with an increase in the relative depth of the thread groove over a wide range. Furthermore, some empirical relations between [K.sub.f] and screw lead, the screw-sleeve relative diametrical clearance and the relative depth of thread groove are fitted, respectively, based on the derived relation between [K.sub.f] and thread geometric parameters and Golubiev's experimental data, which would provide a theoretical basis for LSP design. [DOI: 10.1115/1.3129128] Keywords: labyrinth screw pump (LSP), pumping performance, cell head coefficient, pump total head coefficient

In this paper, we propose a novel explicit equation for friction factor, which is valid for both smooth and rough wall turbulent flows in pipes and channels. The form of the proposed equation is based on a new logarithmic velocity profile and the model constants are obtained by using the experimental data available in the literature. The proposed equation gives the friction factor explicitly as a function of Reynolds number and relative roughness. The results indicate that the present model gives a very good prediction of the friction factor and can reproduce the Colebrook equation and its Moody plot. Therefore, the new approximation for the friction factor provides a rational, accurate, and practically useful method over the entire range of the Moody chart in terms of Reynolds number and relative roughness. [DOI: 10.1115/1.3129132] Keywords: friction factor, turbulent flows, boundary layers, mathematical modeling

Given generalized reflection matrices R, S, i.e., R * = R, [R.sup.2] = I, S * = S, [S.sup.2] = I, a complex matrix A is said to be generalized reflexive (or anti-reflexive), if RAS = A (or RAS = -A). In this paper, Smarandache iterative algorithm is proposed to solve matrix equation AXB + CYD = F with generalized reflexive matrix dual (X,Y). For any initial iterative matrix pair ([X.sub.1],[Y.sub.1]), we show that a solution of this equation can be obtained within finite iteration steps in the absence of roundoff errors. Some numerical examples illustrate the feasibility and efficiency of this algorithm. Keywords Matrix equation, generalized reflexive matrices, Smarandache iterative algorithm, least-norm solution, optimal approximation., §1. Introduction Let [C.sup.m x n.sub.k] be the set of all m x n complex matrices with rank k, and U[C.sup.nxn] the set of all unitary matrices in [C.sup.n x [...]

The calculation of steaming current in oil pipes is involved in solving the charge conservation equation. It is a nonlinear equation without accurate analytical solution under turbulent-flow condition. Based on some reasonable simplifications, an approximate analytical expression for the equation and a due formula for streaming current are derived. It is proved with experimental results that the formula can accurately predict the streaming current, thus providing a method that is able to reliably and conveniently compute streaming electrification in oil pipe. Index Terms--Oil pipe, streaming current, charge conservation equation, analytical solution, turbulent flow.

Quantum decoherence is a proposed mechanism for the emergence of classical physics from the quantum mechanics. It has been developed extensively in recent years, but is sufficiently technically complicated to discourage widespread understanding. In this paper we provide a gentle introduction to quantum decoherence. We introduce state operators and their density matrix representations to describe composite systems, such as an experiment and its environment. We illustrate how the loss of information about a subsystem can cause a quantum system to appear classical. We first analyze a discrete example of phase randomization, then a Bell state, and finally a continuous system. In the final case we provide an accessible derivation of a major early result of decoherence theory, the master equation of quantum Brownian motion. We conclude by applying the master equation to the decoherence of a simple harmonic oscillator, with results reminiscent of our earlier discrete examples.

Two-dimensional liquid chromatography (2DLC) is a very powerful way to greatly increase the resolving power and overall peak capacity of liquid chromatography. The traditional 'product rule' for peak capacity usually overestimates the true resolving power due to neglect of the often quite severe under-sampling effect and thus provides poor guidance for optimizing the separation and biases comparisons to optimized one-dimensional gradient liquid chromatography. Here we derive a simple yet accurate equation for the effective two-dimensional peak capacity that incorporates a correction for under-sampling of the first dimension. The results show that not only is the speed of the second dimension separation important for reducing the overall analysis time, but it plays a vital role in determining the overall peak capacity when the first dimension is under-sampled. A surprising subsidiary finding is that for relatively short 2DLC separations (much less than a couple of hours), the first dimension peak capacity is far less important than is commonly believed and need not be highly optimized, for example, through use of long columns or very small particles.

A quartz crystal microbalance (QCM) consists of a resonator, which measures the resonance frequency of the quartz slab. When coupled with a network analyzer or coupled with impulse excitation technology, QCM gives additional impedance or dissipation information, respectively. This report provides a set of equations that bring the QCM community a convergence of the dissipation and impedance analysis. Equations derived from the complex frequency shift were applied to quantitatively analyze the dissipation data of polymer brushes obtained from QCMD. The obtained viscoelastic properties of polymer brushes were then compared with those obtained by the Voigt model method. We believe that these equations will be useful in quantitative studies of interfacial phenomena accompanied with mass or viscoelasticity changes.

We show how to obtain the first multipole contributions to the electromagnetic radiation emitted by an arbitrary localized source directly from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field. This procedure avoids the unnecessary calculation of the electromagnetic potentials. [DOI: 10.1119/1.2990666]

For any positive integer n, we define the arithmetical function F(n) as F(1) = 0. If n > 1 and n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] be the prime power factorization of n, then F(n) = alpl + [[alpha].sub.2][p.sub.2] + ... + [[alpha].sub.k] [p.sub.k]. Let S(n) be the Smarandache function. The main purpose of this paper is using the elementary method and the prime distribution theory to study the mean value properties of [(F(n) - S(n)).sup.2], and give a sharper asymptotic formula for it. Keywords Additive function, Smarandache function, Mean square value, Elementary method, Asymptotic formula., [section]1. Introduction and result Let [Florin] (n) be an arithmetical function, we call [Florin] (n) as an additive function, if for any positive integers m, n with (m, n) = [...]

A modified equilibrium distribution function for the Bhatnagar--Gross--Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity--acousfic and entropy--acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Harten's first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier--Stokes equations is required.

Bertrand's theorem is formulated as the solution of an inverse problem for classical one-dimensional motion. We show that the solutions of this problem, if suitably restricted, can be obtained by solving an elementary equation. This approach provides a compact and elegant proof of Bertrand's theorem.

A new equation consisting mole fractions of sequences and glass transition temperatures of periodic copolymers which could provide useful information on the bond rotation flexibility is described.

Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

Abstract The main purpose of this (lamer is using the elementary methods to study the solutions of an equation involving the Smarandache-type multiplicative function, and give its all positive integer [...]

Abstract In this paper, we use the elementary method to study the solutions of an equation involving the Smarandache dual function [[bar.s].sub.k] (n), and give its all solutions. Keywords Smarandache [...]

Abstract: The inversion and convolution formulas are established for an intergral transform on [-1, 1]. Analytic properties of the images of functions defined in [-1, 0] and [0, 1] under such a transform can be applied to the solution of convolution type equations. Bibliography: 12 titles. Article History: Registration Date: 30/12/2005 Article note: Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 67--93.

Abstract In this paper we generalize the abc-theorem for n-polynomials over [??][x] in which [??] is an algebraically closed field of characteristic zero. This generalization is obtained by considering the [...]