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The Strategy of Distributed Load Balancing Based on Hybrid Scheduling | A strategy of distributed load balancing based on hybrid scheduling (LBHS) is proposed in this paper. This method is a hybrid scheduling strategy, which is based on the static allocation, and used dynamic scheduling system with the amply consideration of all the factors in the process of load balancing. This method can improve the utilization of the processor. The method is tested in the land surface disaster and ecology model. The result indicates that the strategy could improve the performance of parallel computing effectively, especially in the case that the size and data of the task is partially mastered beforehand. | A finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grunwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discrete l 2 norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis. | eng_Latn | 18,600 |
An assessment of two popular Padé-based approaches for fast frequency sweeps of time-harmonic finite element problems | Several Pade-based computational methods have been recently combined with the finite element method for the efficient solution of complex time-harmonic acoustic problems. Among these, the component-wise approach, which focuses on the fast-frequency sweep of individual degrees of freedom in the problem, is an alternative to the projection-based approaches. While the former approach allows for piecewise analytical expressions of the solution for targeted degrees of the freedom, the projection-based approaches may offer a wider range of convergence. In this work, the two approaches are compared for a range of problems varying in complexity, size and physics. This includes for instance the modelling of coupled problems with non-trivial frequency dependence such as for the modelling of sound absorbing porous materials. Conclusions are drawn in terms of computational time efficiency, implementation, and suitability of the methods depending on specific scientific problems of interest. | In this paper, a numerical method is given for solving fuzzy Fredholm integral equations of the second kind, by using Bernstein piecewise polynomial, whose coefficients determined through solving dual fuzzy linear system. Numerical examples are presented to illustrate the proposed method, whose calculations were implemented by using the Computer software MathCadV.14. | eng_Latn | 18,601 |
Optimal control of the rotation of a system of two bodies connected by an elastic rod | Abstract Optimal controls of the rotation of a mechanical system consisting of two rigid bodies, joined by an elastic rod, through a specified angle about an axis passing through the centre of mass of one of the bodies are constructed. The problem of the optimal control of the rotation of the system through a given angle with complete suppression of the oscillations of the elastic rod at the minimum of the energy functional of the control moment and the problem of time-optimality for a specified constraint on the energy functional of the control moment are solved. | In the oil production and thermal recovery in the application of oil and gas field, well of curves are regarded as the crooked canal(namely not only the canal has horizontal section but also vertical section), it instead the horizontal well before. Thus establishmented the new elbow well model, the solution of the model not only suitable for theoretical study but also easy to calculate in practical production. This article considered the problem of indeterminate percolation of spherical symmetry infinite domain described by the initial boundary value problem of the system of partial differential equation and obtained the point-source accurate solution when research on the problem of indeterminate percolation for double porosity medium. Abtained the accurate solution about mathematical model of line source from appling the point-source accurate solution, and applied the results to mathematical model of well of curves, obtained the integral expression of it’s accurate solution. | eng_Latn | 18,602 |
$L^p$-maximal regularity of degenerate delay equations with periodic conditions | Under suitable assumptions on the delay operator F, we give nec- essary and sucient conditions for the inhomogeneous abstract degenerate de- lay equations: (Mu) 0 (t) = Au(t) + Fut + f(t), (t 2 T) to have L p -maximal regularity. | Abstract This paper presents two analytical models of special multiple-state devices with repair. The failure rates are constant and the repair rates between failure states are constant, while the repair rate times between failure state and good state are arbitrarily distributed. Laplace transform of the state probabilities and steady-state availability are derived. | eng_Latn | 18,603 |
ON MODIFIED BLACK–SCHOLES EQUATION | Abstract Black–Scholes equation corresponds to the diffusion equation. It is argued that the telegraph equation is more suitable from several points of view. The corresponding modified Black–Scholes equation is proposed. The telegraph equation is also applied to spatial population dynamics. | Abstract The high frequency modes of Hamiltonian systems tend to have small amplitudes. Hence for moderately accurate integration of such problems by, say, the leapfrog method the time step tends to be limited by stability restrictions rather than accuracy restrictions. Conventional implicit symplectic methods like implicit midpoint have less severe stability restrictions but the cost of solving large nonlinear systems with dense Jacobian matrices is probably too high to make them worthwhile. To bring down the cost of implicit methods, we have designed (i) mixed implicit-explicit, and (ii) linearly implicit methods that retain the property of being symplectic. | yue_Hant | 18,604 |
A Research of Difference Schemes for Nonlinear Schr Dinger Equation | One finite difference scheme is proposed for nonlinear Schr dinger equation resolution.It is proved that the proposed scheme is convergent and stable.A numerical computing example is given to prove the theoretical analysis. | Abstract The paper is concerned with a reduced SIR model for migrant workers. By using differential inequality technique and a novel argument, we derive a set of conditions to ensure that the endemic equilibrium of the model is globally exponentially stable. The obtained results complement with some existing ones. We also use numerical simulations to demonstrate the theoretical results. | eng_Latn | 18,605 |
Acquired von Willebrand Disease in Monoclonal Gammapathies: Effectiveness of High-dose Intravenous Gamma Globulin | Acquired von Willebrand disease (AvWD) is a bleeding disorder strikingly similar to congenital von Willebrand disease in terms of clinical manifestations and laboratory findings being characterized by a prolonged bleeding time and low levels of factor VIII von Willebrand factor (FVIII/vWf) complex, but without a family history of a congenital bleeding tendency and absence of abnormal bleeding after hemostatic challenges. | In the present paper, a future cone in the Minkowski space defined in terms of the square-norm of the residual vector for an ill-posed linear system to be solved, is used to derive a nonlinear system of ordinary differential equations. Then the forward Euler scheme is used to generate an iterative algorithm. Two critical values in the critical descent tri-vector are derived, which lead to the largest convergence rate of the resultant iterative algorithm, namely the globally optimal tri-vector method (GOTVM). Some numerical examples are used to reveal the superior performance of the GOTVM than the famous methods of conjugate gradient (CGM) and generalized minimal residual (GMRES). Through the numerical tests we also set forth the rationale by assuming the tri-vector as being a better descent direction. | eng_Latn | 18,606 |
Equation of a fitted smooth spline and its analytical derivative | Interpreting spline results | Prove no existing a smooth function satisfying ... related to Morse Theory | eng_Latn | 18,607 |
Transient heat conduction in a medium with two circular cavities: Semi-analytical solution | This paper considers a transient heat conduction problem for an infinite medium with two non-overlapping circular cavities. Suddenly applied, steady Dirichlet type boundary conditions are assumed. The approach is based on superposition and the use of the general solution to the problem of a single cavity. Application of the Laplace transform results in a semi-analytical solution for the temperature in the form of a truncated Fourier series. The large-time asymptotic formulae for the solution are obtained by using the analytical solution in the Laplace domain. The method can be extended to problems with multiple cavities and inhomogeneities. | A sampling cone 3 and/or a cone gas cone 4 and/or an extraction cone 8 of a mass spectrometer are disclosed having a metallic carbide surface or coating. The metallic carbide surface or coating may comprise a transition metal carbide, a carbide alloy, or a mixed metal carbide alloy, but preferably comprises titanium carbide. The coated surface is intended to reduce adsorption of material on contact with the surface of the sampling or extraction cone. | eng_Latn | 18,608 |
Global exponential stability for coupled systems of neutral delay differential equations | In this paper, a novel class of neutral delay differential equations (NDDEs) is presented. By using the Razumikhin method and Kirchhoff's matrix tree theorem in graph theory, the global exponential stability for such NDDEs is investigated. By constructing an appropriate Lyapunov function, two different kinds of sufficient criteria which ensure the global exponential stability of NDDEs are derived in the form of Lyapunov functions and coefficients of NDDEs, respectively. A numerical example is provided to demonstrate the effectiveness of the theoretical results. | Abstract A novel method enabling to find the dependence of the solution y(x, α) on the parameter α for nonlinear boundary problems will be presented. The algorithm suggested will be illustrated on the example of simultaneous heat and mass transfer in a porous catalyst. A strategy is presented making it possible to examinate also problems having multiple solutions. This technique may successfully compete with a sequential Newton—Kantorovich method. | eng_Latn | 18,609 |
Calculate the Laplace Transform of a Function | The Laplace transform is an integral transform used in solving differential equations of constant coefficients. This transform is also extremely useful in physics and engineering. | Projection operators are defined below, given an arbitrary state | ψ ⟩ . {\displaystyle |\psi \rangle .}
| eng_Latn | 18,610 |
I'm supposed to solve $u_{xx}-3u_{xt}-4u_{tt}=0$ with initial conditions $u(x,0)=x^2$ and $u_t(x,0)=e^x$. So I factored the problem into $(u_x-4u_t)(u_x + u_t)$ and set each equal to 0 and found the 2 solutions to be $(x,t)=f(x+t/4)$ and $u(x,t)=g(x-t)$ Then I get $u(x,t)=1/2[\phi(x+t/4) + \phi(x-4)]$ + 1/2c $\int_{x-t}^{x+t/4} e^{s} ds$ =1/2[$(x+t/4)^2 + (x-t)^2$] + 1/2c[$e^{x+t/4}-e^{x-t}$] I multiply out the first part then and get a solution of: 1/2[$5/2xt-15/16t^2$]+1/2c[$e^{x+t/4}-e^{x-t}$] But the book has a solution of $x^2 + t^2/4 + 4/5[e^{x+t/4}-e^{x-t}]$ I don't understand how they get 4/5 in front of the exponentials and how $1/2[\phi(x+t/4) + \phi(x-4)]$ turns into $x^2 + t^2/4$ . Can anyone see where I went wrong? | Solve $$u_{xx}-3u_{xt}-4u_{tt}=0$$ where $u(x,0)=x^2$ and $u_t(x,0)=e^x$. My workings so far: I have factored the differential equation in the following way: $$(\delta_x-4\delta_t)(\delta_x+\delta_t)=0$$ where $\delta_x=\frac{\delta}{\delta x}$ etc. Now if we let $v$ be the solution to $(\delta_x+\delta_t)u$ then we have the following two equations: \begin{eqnarray*} (\delta_x+\delta_t)u=u_x+u_t=v\\ (\delta_x-4\delta_t)v=v_x-4v_t=0 \end{eqnarray*} Now for $v$ we simply find $$v=h(t+4x)$$ where $h$ is an arbitrary function of one variable. Now what remains is find $u$ such that $$u_x+u_t=h(t+4x)$$ I am stuck here, I thought about making a change of variables $\zeta=x+t$ and $\eta=x-t$ and thus using the product rule to show that $u_x=u_{\zeta}+u_{\eta}$ and $u_t=u_{\zeta}-u_{\eta}$ and thus $u_x+u_t=2u_{\zeta}$ and we need to solve $$u_{\zeta}=h(t+4x)$$ (I left out the factor 2 because $h$ is an arbitrary function). Do I simply integrate now and conclude $$u=f(\zeta)h(t+4x)+g(\eta)=f(x+t)h(t+4x)+g(x-t)$$ This seems wrong to me... Some help would be greatly appreciated! | If $2^p-1$ is a prime, (thus $p$ is a prime, too) then $p\mid 2^p-2=\phi(2^p-1).$ But I find $n\mid \phi(2^n-1)$ is always hold, no matter what $n$ is. Such as $4\mid \phi(2^4-1)=8.$ If we denote $a_n=\dfrac{\phi(2^n-1)}{n}$, then $a_n$ is , but how to prove it is always integer? Thanks in advance! | eng_Latn | 18,611 |
Mixed convection flow about a solid sphere embedded in a porous medium filled with a nanofluid | A steady laminar mixed convection boundary layer flow about an isothermal solid sphere embedded in a porous medium filled with a nanofluid has been studied for both cases of assisting and opposing flows. The transformed boundary layer equations were solved numerically using an implicit finite-difference scheme. Three different types of nanoparticles, namely Cu, Al2O3 and TiO2 in water-based fluid were considered. Numerical solutions were obtained for the skin friction coefficient, the velocity and temperature profiles. The features of the flow and heat transfer characteristics for various values of the nanoparticle volume fraction and the mixed convection parameters were analyzed and discussed. | In this work we consider a coupled system of m( ≥ 2) linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms with discontinuous source term. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct in magnitude. Overlapping boundary and interior layers can appear in the solution. A numerical method is constructed that involve an appropriate piecewise-uniform Shishkin mesh, which is fitted to both the boundary and interior layers. The parameter-uniform convergence of the numerical approximations is examined. | eng_Latn | 18,612 |
Uptake of gentamicin by Staphylococcus aureus possessing gentamicin-modifying enzymes: enhancement of uptake by puromycin and N,N'-dicyclohexylcarbodiimide. | Uptake of gentamicin by a gentamicin-resistant strain of Staphylococcus aureus possessing the aminoglycoside-modifying phosphotransferase enzyme APH(2") was enhanced by the protein synthesis inhibitor, puromycin or by the proton-translocating ATPase inhibitor, N,N'-dicylohexylcarbodiimide. Such enhanced uptake was inhibited by carbonyl cyanide p trifluoromethoxyphenylhydrazone or by valinomycin in the presence of potassium ions, suggesting a role for the transmembrane proton motive force in the process. The accumulated gentamicin did not cause loss of cell viability and exhibited altered chromatographic mobility compared with a control (unmodified) preparation of gentamicin. | In the present paper, the author shows that the predictor/multi-corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2-D and 3-D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG-PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM-PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi-steady solutions to the incompressible viscous flow problems: 2-D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd. | eng_Latn | 18,613 |
Method of reducing the threshold of the high-precision digital closed-loop fiber optic gyroscope | A novel method of reducing the threshold of the high precision digital closed-loop fiber optic gyroscope(DCFOG) is studied.The reason leading to the threshold increasing is proposed due to the gyroscope dead band which is mainly caused by the electronic cross-coupling interference in the modulation and demodulation circuit.The digital closed-loop transfer model for fiber-optic gyroscope is modified and the electronic cross-coupling interference from the feedback channel to the output signal of the detector in the forward channel is added.This model is also used to realize a real-time dynamic simulation for the dead band.The simulation results are basically consistent with experimental results.Unlike the random modulation method,this study inhibits the dead zone mainly by optimizing the electronic circuit and improving the circuit electromagnetic compatibility.We propose a new method to test the fiber optic gyro threshold.The results show that the gyroscope threshold is comparable to the noise level. | For soft soils time-dependent consolidation processes are highly non-linear and require adistinct analysis of the overlying mechanisms. In the present study thus an analytical, rigoroussolution for one-dimensional consolidation of a clay deposit with permeable top andimpermeable bottom under haversine cyclic loading with restperiod is introduced. The solutionwas achieved using Fourier harmonic analysis for the periodic function representingthe rate of imposition of excess pore water pressure. Comparing the derived solution to asolution by Razouki & Schanz (2011) using an implicite finite difference technique andavailable solutions in literature, advantages are illuminated. Additionally, the presented,analytical solution is used to validate a numerical solution of the same conditional problemvia finite element method. | eng_Latn | 18,614 |
HOMOTOPY METHOD FOR INVERSING TWO PARAMETERS OF 2-D WAVE EQUATION IN POROUS MEDIA | The homotopy method is used to inverse material parameters of porous media. According to that the computed response should fit to the measured one, the parameter inversion problem of porous media is reduced to a problem of nonlinear operator equations zero in this paper. The homotopy method with large scope convergence is used to find the solution of inversion problem. The homotopy method is used to inverse the parameters of 2-D wave equations in porous media which has an analytical solution given by Paul in 1976. Numerical simulations indicate that the homotopy method is effective and robust. | e18068Background: One method of therapy deintensification in locally advanced (LA) HPVOPC is to reduce total radiation dose during concomitant chemoradiation (CRT) in a sequential therapy plan. We ... | yue_Hant | 18,615 |
EXPERIMENTAL STUDY ON THE PERFORMANCE OF REGENERATION OF LIQUID DESICCANT AIR-CONDITIONING SYSTEM | In this paper, on the basis of the mechanism of liquid desiccant dehumidification and regeneration, an experimental equipment of liquid desiccant air-conditioning system is set up, the regenerator is a counter-flow packed tower and it is made of stainless steel to withstand the corrosive effect of the liquid desiccant. In experiments, calcium choride is applied as the liquid desiccant and the process of regeneration is studied. The results show that the mass flow rate and humidity of the air as well as the temperation and mass flow rate of the diluted solution can affect the heat and mass transfer of regeneration. | The differential equations of the continuous time LQ control problem are discretized, then the mixed-energy condensation algorithm is established for the continuous time and linear equality constraint LQ control problem, the above algorithm can be used to solve Riccati equation with the linear constraint effectively. An example is given. | yue_Hant | 18,616 |
Viscous flow behaviour of fluorozirconate glasses | Abstract The isothermal viscous flow properties of two fluorozirconate glass compositions have been studied by parallel plate rheometry at temperatures approaching the fibre-drawing viscosity region. An unusual flow behaviour has been found, in which sharp end points occur where deformation ceases under load. This behaviour has been characterised in terms of a threshold yield stress with hysteresis effects. The materials are prone to crystallisation, but no obvious direct link has been found between such effects and the flow behaviour, which may thus be intrinsic. | This work describes a high order accurate discontinuous finite element method for the numerical solution of the equations governing compressible inviscid flows. Our investigation has focused on the problem of correctly prescribing the boundary conditions along curved boundaries. “Ale show by numerical testing that, in the presence of curved boundaries, a high order approximation of the solution requires a corresponding high-order approximation of the geometry of the domain. Numerical solutions of transonic flows are presented which illustrate the versatility and the accuracy of the proposed method. | eng_Latn | 18,617 |
An integrated optical parallel adder as a first step towards light speed data processing | Integrated optical circuits with nanophotonic devices have attracted significant attention due to its low power dissipation and light-speed operation. With light interference and resonance phenomena, the nanophotonic device works as a voltage-controlled optical pass-gate like a pass-transistor. This paper first introduces a concept of the optical pass-gate logic, and then proposes a parallel adder circuit based on the optical pass-gate logic. Experimental results obtained with an optoelectronic circuit simulator show advantages of our optical parallel adder circuit over a traditional CMOS-based parallel adder circuit. | Abstract In many applications, such as atmospheric chemistry, large systems of ordinary differential equations (ODEs) with both stiff and nonstiff parts have to be solved numerically. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linear stability of popular second order methods like extrapolated BDF, Crank-Nicolson leap-frog and a particular class of Adams methods. We present results for problems with decoupled eigenvalues and comment on some specific CFL restrictions associated with advection terms. | eng_Latn | 18,618 |
Numerical Analysis of Free Surface Shock Waves around Bow by Modified MAC-Method | Computational schemes and numerical stability conditions of a modified version of Marker and Cell method are studied and a new computing program is developed using second-order upstream differencing representations of the momentum conservation equations and a SOR iterative method for solving a Poisson equation for the pressure, which is applicable to 3-D wave making problems of steadily advancing floating bodies in deep water. Computed results are given for nonlinear bow-waves of wedge models. | In the oil production and thermal recovery in the application of oil and gas field, well of curves are regarded as the crooked canal(namely not only the canal has horizontal section but also vertical section), it instead the horizontal well before. Thus establishmented the new elbow well model, the solution of the model not only suitable for theoretical study but also easy to calculate in practical production. This article considered the problem of indeterminate percolation of spherical symmetry infinite domain described by the initial boundary value problem of the system of partial differential equation and obtained the point-source accurate solution when research on the problem of indeterminate percolation for double porosity medium. Abtained the accurate solution about mathematical model of line source from appling the point-source accurate solution, and applied the results to mathematical model of well of curves, obtained the integral expression of it’s accurate solution. | eng_Latn | 18,619 |
Evaluation of the anesthetic efficacy of alfaxalone in oscar fish (Astronotus ocellatus) | OBJECTIVE To evaluate effects of alfaxalone on heart rate (HR), opercular rate (OpR), results of blood gas analysis, and responses to noxious stimuli in oscar fish (Astronotus ocellatus). ANIMALS 6 healthy subadult oscar fish. PROCEDURES Each fish was immersed in water containing 5 mg of alfaxalone/L. Water temperature was maintained at 25.1°C, and water quality was appropriate for this species. The HR, OpR, response to noxious stimuli, and positioning in the tank were evaluated, and blood samples for blood gas analysis were collected before (baseline), during, and after anesthesia. RESULTS Immersion anesthesia of oscar fish with alfaxalone (5 mg/L) was sufficient for collection of diagnostic samples in all fish. Mean ± SD induction time was 11 ± 3.8 minutes (minimum, 5 minutes; maximum, 15 minutes), and mean recovery time was 37.5 ± 13.7 minutes (minimum, 20 minutes; maximum, 55 minutes). There was a significant difference in OpR over time, with respiratory rates significantly decreasing between baseline... | Many questions in natural science and engineering can be transformed into nonlinear equations and solved.Aimed at the problems of the classical algorithms for solving nonlinear equations,such as high sensitivity to the initial guess of the solution,poor convergence reliability and can't get all solutions,etc.,the artificial fish-swarm algorithm(AFSA)was put forward to solve nonlinear equations.The numerical examples in linkage synthesis and approximate synthesis show that this algorithm has the characteristic of high precision and fast convergence speed,and resolves the difficulty in getting all solutions. | eng_Latn | 18,620 |
A PDE Model for Imatinib-Treated Chronic Myelogenous Leukemia | We derive a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia (CML). This model is a continuous extension of the agent-based CML model of Roeder et al. (Nat. Med. 12(10), 1181-1184, 2006) and of its recent formulation as a system of difference equations (Kim et al. in Bull. Math. Biol. 70(3), 728-744, 2008). The new model is formulated as a system of partial differential equations that describe various stages of differentiation and maturation of normal hematopoietic cells and of leukemic cells. An imatinib treatment is also incorporated into the model. The simulations of the new PDE model are shown to qualitatively agree with the results that were obtained with the discrete-time (difference equation and agent-based) models. At the same time, for a quantitative agreement, it is necessary to adjust the values of certain parameters, such as the rates of imatinib-induced inhibition and degradation. | Abstract In many applications, such as atmospheric chemistry, large systems of ordinary differential equations (ODEs) with both stiff and nonstiff parts have to be solved numerically. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linear stability of popular second order methods like extrapolated BDF, Crank-Nicolson leap-frog and a particular class of Adams methods. We present results for problems with decoupled eigenvalues and comment on some specific CFL restrictions associated with advection terms. | kor_Hang | 18,621 |
Lyapunov stability and its application to systems of ordinary differential equations | An outline and a brief introduction to some of the concepts and implications of Lyapunov stability theory are presented. Various aspects of the theory are illustrated by the inclusion of eight examples, including the Cartesian coordinate equations of the two-body problem, linear and nonlinear (Van der Pol's equation) oscillatory systems, and the linearized Kustaanheimo-Stiefel element equations for the unperturbed two-body problem. | Abstract A review is undertaken of the various forms that have been obtained for the recurrence relation from which the eigenvalues of Laplace's tidal equation may be obtained. Such forms are shown to be analytically consistent and are discussed in relation to their subsequent numerical evaluation. By determining eigenvalues of equivalent depth for a given frequency of oscillation instead of the other way around the problem becomes a straightforward one of matrix diagonalization. If solutions are based on normalized Legendre functions the matrix is symmetric. A method of evaluating the related wind functions is described. | eng_Latn | 18,622 |
Determining the parameters of a stratified piecewise constant medium for the unknown shape of an impulse source | For a hyperbolic wave equation with some parameter λ, we consider the problem of finding the piecewise constant wave propagation speed and a series of parameters in the conjugation condition. Moreover, the shape is assumed unknown of the impulse point source that excites the oscillation process. We prove that, under certain assumptions on the structure of the medium, its sought parameters are determined uniquely from the displacements of points of the boundary given for two different values of λ. We give an algorithm for solving the problem. | Abstract This paper presents two analytical models of special multiple-state devices with repair. The failure rates are constant and the repair rates between failure states are constant, while the repair rate times between failure state and good state are arbitrarily distributed. Laplace transform of the state probabilities and steady-state availability are derived. | eng_Latn | 18,623 |
Numerical study on small-scale longitudinal heat conduction in cross-wavy primary surface heat exchanger | Abstract The longitudinal heat conduction is usually neglected during the analysis of cross-wavy (CW) primary heat exchanger. In this paper, an anisotropic heat transfer numerical model is proposed to analyze the local thermal field of primary surface plate affected by the small-scale longitudinal heat conduction. Moreover, a detached numerical model is used for simplifying grid and improving computational stability. It shows that the temperature distribution is non-uniform in the anisotropic heat transfer model. The small-scale longitudinal heat conduction makes the temperature distribution of plate become more uniform and enhances the heat exchanger performance. The temperature difference has little effect on the contribution of longitudinal heat conduction, while the contribution of longitudinal heat conduction increases with the increase of the inlet temperature and Reynolds number. | In this paper the physical meaning of a nonlinear partial differential equation (nPDE) of the fourth order relating to wave theory is deduced to the first time. The equation under consideration belongs to a class of less studied nPDEs and an explicit physical meaning is not known until now. This paper however bridges the gap between some known results and a concrete application concerning wave theory. We show how one can derive locally supercritical solitary waves as well as locally and nonlocally forced supercritical waves and analytical solutions are given explicitly. ::: ::: Keywords: Nonlinear partial differential equations, evolution equations, supercritical solitary waves, locally supercritical waves, non-locally supercritical waves. | eng_Latn | 18,624 |
Numerical Integration Method to Determine Periodic Solutions of Nonlinear Systems | A new method to obtain periodic solution of nonlinear systems is proposed by combining secant method with numerical integration. This method can be applied to nonlinear systems with discontinuous characteristics without special treatment. To check the validity of the proposed method, we applied this method to nonlinear systems with discontinuous characteristics, such as piecewise linear spring system, system with Coulomb’s friction and impact damper system. Comparing the results by the proposed method with the results by simple numerical integration or experimental results, the validity of the proposed method is confirmed.Copyright © 2005 by ASME | identical to the Schuler-period found in terrestrial inertial navigation systems (INS), and in the Schuler Pendulum model. On the other hand, the existence of the Schuler-period in INS is well known to engineers and technicians performing INS work, but the fact that this period is identical to the period of LEO satellites is barely known in the INS community, if at all, let alone its existence in the Hill-Clohessy-Wiltshire equations. In this paper we examine the four phenomena; namely, Schuler pendulum, INS Schuler oscillations, LEO satellite orbital period, and the period of the relative motion between two adjacent LEO satellites. In particular we show why the LEO satellite’s orbital period is identical to the well-known INS Schuler period. We also show that if the INS error equations are generalized to a non-terrestrial case, a generalized form of the Schuler oscillation exists, which takes the form of the Hill-Clohessy-Wiltshire equations for satellite relative motion. | eng_Latn | 18,625 |
Periodic boundary value problems of fourth order impulsive differential equations | This paper considers the existence of solutions for periodic boundary value problems of fourth order impulsive differential equations. By using the method of upper and lower solutions together with the monotone iterative technique, new existence results of coupled solutions and uniqueness of problems are obtained. | For general centered self‐similar unsteady flows separated by a shock wave from an unsteady ambient medium, the profile distributions of physical variables are, in general, obtainable by purely numerical means only. However, one integral of motion based on mass and energy conservation relations is found to exist universally, except in the singular case where the ambient density decay exponent falls on the geometry expansion index. This work formally derives and presents this integral in closed form. | eng_Latn | 18,626 |
Interaction between material length scale and imperfection size for localisation phenomena in viscoplastic media | For the description of localised shear modes viscoplasticity theory is used to regularise the initial value problem. The viscous term in the constitutive equation has been shown to implicitly introduce a material length scale in the model. A relation between the material length scale and the width of the shear band in a perfect bar has been derived. The influence of imperfections on the viscoplastic solution has also been studied and the length scale set by the size of the imperfect zone has been related to the material length scale. The features of viscoplastic soluions in which these two length scale effects are present, have been studied for one- and two-dimensional problems. | In this paper, we consider the second order wave equation discretized in space by summation-by-parts-simultaneous approximation term (SBP-SAT) technique. Special emphasis is placed on the accuracy analysis of the treatment of the Dirichlet boundary condition and of the grid interface condition. The result shows that a boundary or grid interface closure with truncation error $\mathcal{O}(h^p)$ converges of order $p + 2$ if the penalty parameters are chosen carefully. We show that stability does not automatically yield a gain of two orders in convergence rate. The accuracy analysis is verified by numerical experiments. | eng_Latn | 18,627 |
Natural convection of non-Newtonian fluids through permeable axisymmetric and two-dimensional bodies in a porous medium | Abstract The present work concerns the natural convection of non-Newtonian power-law fluids with or without yield stress over the permeable two-dimensional or axisymmetric bodies of arbitrary shape in a fluid-saturated porous medium. Using the fourth-order Runge–Kutta scheme method and shooting method we obtain the local non-similarity solutions. The parameters that include the dimensionless yield stress Ω , permeable constant c and power index n are studied, and the heat flux and the wall temperature are taken into consideration as variables. The local non-similarity solutions are found to be in excellent agreement with the exact solution. It is found that the results depend strongly on the values of the yield stress parameter, the wall temperature distributions, the lateral mass flux rate, and the heat flux at the boundary. | Abstract The paper studies the existence, uniqueness, regularity and the approximation of solutions to the reaction–diffusion equation endowed with a general nonlinear regular potential and Cauchy–Neumann boundary conditions. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation. | eng_Latn | 18,628 |
Structure preserving numerical methods for the Vlasov equation | To preserve a number of physically relevant invariants is a major concern when considering long time integration of the Vlasov equation. In the present work we consider the semi-Lagrangian discontinuous Galerkin method for the Vlasov-Poisson system. We discuss the performance of this method and compare it to cubic spline interpolation, where appropriate. In addition, numerical simulations for the two-stream instability are shown. | This work revisits the constant stepsize stochastic approximation algorithm for tracking a slowly moving target and obtains a bound for the tracking error that is valid for all time, using the Alekseev non-linear variation of constants formula. | eng_Latn | 18,629 |
Three dimensional finite difference numerical simulation of elastic wave propagation under borehole conditions | The improved Higdon absorbing boundary condition was derived in this paper. Various borehole and formation conditions were simulated using the fourth order accuracy in space and second order accuracy in time centered finite difference method. The numerical simulate result of orthorhombic medium formation and elliptic borehole conditions are presented. The wave field of monopole and dipole source are simulated and analysed, which agree well with the theory of elastic wave propagation.It indicates that with the improved Higdon absorbing boundary condition , the three dimensional program on acoustic log can be used to simulate the wave field of complex borehole conditions. | In this paper verification methods for Fredholm integral equations are considered. By these methods the numerical approximation is computed together with mathematically guaranteed error bounds of high quality. The foundations for such numerics is described and then applied to Fredholm integral equations. We conclude by some numerical examples demonstrating the effectiveness of such new numerical schemes. | eng_Latn | 18,630 |
Greenhouse Gas Emissions and Urban Congestion: Incorporation of Carbon Dioxide Emissions and Associated Fuel Consumption into Texas A&M Transportation Institute Urban Mobility Report | The Texas A&M Transportation Institute's Urban Mobility Report (UMR) is acknowledged to be the most authoritative source of information about traffic congestion and its possible solutions. As policy makers from the local to national levels devise strategies to reduce greenhouse gas (GHG) emissions, the level of interest in the environmental impact of urban congestion has increased. To this end, the researchers developed and applied a methodology to determine carbon dioxide (CO2) emissions caused by congestion for inclusion in the UMR. The methodology also estimated fuel consumption on the basis of the CO2 emissions estimates. The researchers developed a five-step methodology with data from three primary data sources: (a) FHWA's Highway Performance Monitoring System, (b) INRIX traffic speed data, and (c) the U. S. Environmental Protection Agency's Motor Vehicle Emissions Simulator model. Results were intuitive and reasonable when emission rates (pounds of CO2 per mile) were compared with the emissions inve... | In the present paper, the author shows that the predictor/multi-corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2-D and 3-D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG-PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM-PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi-steady solutions to the incompressible viscous flow problems: 2-D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd. | eng_Latn | 18,631 |
Arc length based WENO scheme for Hamilton-Jacobi Equations | In this article, novel smoothness indicators are presented for calculating the nonlinear weights of weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and gives a high-resolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme. | We define in this article an approach to tackle the problem of the computation of the general solution of word equations. Parametric transformation, Nielsen's transformation and Rouen's transformation allowing to collect in one transformation some unbounded sequences of elementary transformations are given. | eng_Latn | 18,632 |
ON THE NUMERICAL METHODS FOR DISCONTINUITIES AND INTERFACES | Discontinuous solutions or interfaces are common in nature, for examples, shock waves or material interfaces. However, their numerical computation is difficult by the feature of discontinuities. In this paper, we summarize the numerical approaches for discontinuities and interfaces appearing mostly in the system of hyperbolic conservation laws, and explain various numerical methods for them. We explain two numerical approaches to handle discontinuities in the solution: shock capturing and shock tracking, and illustrate their underlying algorithms and mathematical problems. The front tracking method is explained in details and the level set method is outlined briefly. The several applications of front tracking are illustrated, and the research issues in this field are discussed. | There are three questions in the first exploration of mathematical model system of relative dosage.First,three-section model is irrational and should be replaced by first-section. Second,The same medicine in the different models,there are no difference with endpoint of usual dose.Third,The different medicines have no difference with different or same usual dose.Sowe come up with new mathematical model system and illuminate the rational and applicability. | yue_Hant | 18,633 |
Explosive solutions of parabolic stochastic partial differential equations with L$\acute{e}$vy noise | In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a L$\acute{\mbox{e}}$vy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that the positive solutions will blow up in finite time in mean $L^{p}$-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrated the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by L$\acute{\mbox{e}}$vy noise has a global solution. | Many physical phenomena can be described by biexponential curve. To study the physical phenomena, there is a need to obtain their mathematical expressions. Generally, the expressions are acquired based on the fitting of experimental data, and the general method to fit exponential curve is nonlinear least squares (NLLS). In the NLLS method, it is crucial to choose good enough starting values for the parameters, otherwise, the fitting might fail to converge. In this paper, one new method is proposed to solve the problem, which utilizes the Fourier transform to transform the non-linear exponential fitting model into a linear one. It makes the estimation of the parameters independent of the initial values and easy to converge. The fitting result shows that the proposed method can be used to fit the curve expressed by difference of double exponentials. | eng_Latn | 18,634 |
Differentiate and Analyze the Treatment of Chronic Obstructive Pulmonary Disease | In recent years,accompany with unceasingly deepen of clinic and basic research of chronic obstructive pulmonary disease(COPD),there is some deeply understanding on COPD.Refer to the current situation of prevention and cure and clinical research of COPD,by to intensively differentiate and analyze the familiar problem of COPD,to further research some opinion of treatment of COPD. | The American option pricing problem can be described mathematically bythe well-known Black-Scholes equation with a free boundary. It is difficult to solvethis problem since the free boundary is unknown.To accomplish that, we make use of the Implicit Finite-difference Method and Penalty Method which means to add a continuous penalty term with a small parameter 2 into the Black-Scholes equation,to transform the original free boundary problem into a nonlinear partial differential equation with a fixed boundary. | eng_Latn | 18,635 |
Second Order Uniformly Convergent Numerical Method for a Coupled System of Singularly Perturbed Reaction-Diffusion Problems with Discontinuous Source Term | In this work we consider a coupled system of m( ≥ 2) linear singularly perturbed equations of reaction-diffusion type coupled in the reaction terms with discontinuous source term. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct in magnitude. Overlapping boundary and interior layers can appear in the solution. A numerical method is constructed that involve an appropriate piecewise-uniform Shishkin mesh, which is fitted to both the boundary and interior layers. The parameter-uniform convergence of the numerical approximations is examined. | In this paper, we develop a methodology for finite time rotor angle stability analysis using the theory of normal hyperbolic surfaces. The proposed method would bring new insights to the existing techniques, which are based on asymptotic analysis. For the finite time analysis we have adopted the Theory of normally hyperbolic surfaces. We have connected the repulsion rates of the normally hyperbolic surfaces, to the finite time stability. Also, we have characterized the region of stability over finite time window. The parallels have been drawn with the existing tools for asymptotic analysis. Also, we have proposed a model free method for online stability monitoring. | eng_Latn | 18,636 |
Elastoplastic model for unsaturated soil with incorporation of the soil-water characteristic curve | An elastoplastic model is proposed in this paper that incorporates the soil-water characteristic curve (SWCC) for obtaining soil parameters of unsaturated soil. The SWCC is shown to govern the rate of change in the soil parameters for the elastoplastic model with respect to matric suction. A series of isotropic consolidation tests under different matric suctions and tests for obtaining SWCC were carried out on statically compacted kaolin specimens. Nanyang Technological University (NTU) mini suction probes were installed along the height of the specimen to measure pore-water pressures during isotropic consolidation and SWCC tests. The results of isotropic consolidation tests demonstrate the strong influence of matric suction on compressibility and stiffness of the soil specimens. The experiments were also simulated using the proposed elastoplastic model and SWCC of the compacted kaolin. The simulated results agree closely with the experimental results. In addition, the proposed elastoplastic model was als... | Abstract The problem of interaction between an axisymmetrically loaded thin circular plate and a supporting elastic medium is reduced to that of solving an integral equation for the unknown normal contact pressure. The supporting medium is an isotropic elastic layer of constant thickness lying, with or without friction, on a semi-infinite isotropic elastic base or on a rigid base. For the solution of the resulting integral equation an effective numerical procedure is employed and some numerical results are presented. | eng_Latn | 18,637 |
Numerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials | In this paper, a numerical method is given for solving fuzzy Fredholm integral equations of the second kind, by using Bernstein piecewise polynomial, whose coefficients determined through solving dual fuzzy linear system. Numerical examples are presented to illustrate the proposed method, whose calculations were implemented by using the Computer software MathCadV.14. | Abstract Boundary value problems involving continuous flow reactors have been considered in which tubular and well-stirred tank reactors have been considered together with an axial dispersion model for the tubular reactor. This formulation does away with the customary but non-physical discontinuity in the state of the feed stream at the inlet to the stirred tank reactor. The problems, restricted to isothermal reactors entertaining first order reaction systems, have been solved by means of an elegant formalism in tune with the general theory of self-adjoint operators in abstract Hilber space and consistent with the elementary treatment. | eng_Latn | 18,638 |
Fontaine’s Forgotten Method for Inexact Differential Equations | SummaryIn 1739 Alexis-Claude Clairaut published the modern integrating factor method of solving inexact ordinary differential equations (ODEs). He was motivated by a 1738 Alexis Fontaine paper with a different method which requires solving a difficult partial differential equation (PDE). Here we revisit Fontaine's method, examine his modest attempt to solve the PDE, and utilize a different technique to give (we believe) the first family of ODEs solvable by Fontaine's method with no obvious solution using the modern technique. | Abstract A one-phase Stefan problem with latent heat a power function of position is investigated. The second kind of boundary condition is involved, and the surface heat flux is considered as a corresponding power function of time. The problem can be viewed as a special case of the shoreline movement problem under the conditions of nonlinear variation of ocean depth and a surface flux that varies as a power of time. An exact solution is constructed using the similarity transformation technique. Theoretical proof for the existence and the uniqueness of the exact solution is conducted. Solutions for some special cases presented in the literature are recovered. In the end, computational examples of the exact solution are presented, and the results can be used to verify the accuracy of general numerical phase change algorithms. | eng_Latn | 18,639 |
Limiting Profiles for Solutions of Differential-Delay Equations | This expository paper discusses the ”limiting profile” as e approaches zero of the graph of x e , where x e is a periodic solution either of ex′ e (t) = f(x e (t), x e (t - 1)) or of ex′ e (t) = f(x e (t), x e (t - r)), r = r(x e (t)), and f and r are given functions. A variety of theorems from the literature are summarized and proofs of some of the simpler results are sketched. The paper is divided into four sections. The first section gives some background material on fixed point theory, degree theory and the fixed point index. The second section discusses the equation ex′(t) = f(x(t), x(t - 1)) and treats the family of examples f(x, y) = -x -μy + y3 , μ > 1. The final two sections give a sampling of theorems concerning ex′(t) = f(x(t), x(t - r)), r = r(x(t)), and attempt to give some indication of the techniques involved in the analysis. | A novel 2- and 3-D space spectral domain analysis (SSDA) simulator for microstrip and CPW transmission line discontinuities is described. The characteristic impedance, effective dielectric constant, and S-parameters of different type of transmission lines and discontinuities shapes can be computed. The topology of structure and the computed results are displayed using a graphical interface. The simulator has been implemented on Sun Sparcstations. > | eng_Latn | 18,640 |
On the solution of high order stable time integration methods | Evolution equations arise in many important practical problems. They are frequently stiff, i.e. involves fast, mostly exponentially, decreasing and/or oscillating components. To handle such problems, one must use proper forms of implicit numerical time-integration methods. In this paper, we consider two methods of high order of accuracy, one for parabolic problems and the other for hyperbolic type of problems. For parabolic problems, it is shown how the solution rapidly approaches the stationary solution. It is also shown how the arising quadratic polynomial algebraic systems can be solved efficiently by iteration and use of a proper preconditioner. | In this paper, a numerical method is given for solving fuzzy Fredholm integral equations of the second kind, by using Bernstein piecewise polynomial, whose coefficients determined through solving dual fuzzy linear system. Numerical examples are presented to illustrate the proposed method, whose calculations were implemented by using the Computer software MathCadV.14. | eng_Latn | 18,641 |
Jacob's ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^4$ | The elementary geometric properties of Jacob's ladders of the second order lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral of $|\zeta(1/2+it)|^4$. These formulae cannot be obtained by methods of Balasubramanian, Heath-Brown and Ivic. | A mathematical model of the dynamic behavior of a heterogeneous liquid-liquid reaction system in a plug flow reactor is formulated. Numerical investigation of the oscillatory instability of the reactor is carried out. The region of oscillatory regimes in a parametric space is examined for the Semenov parameter — the temperature at the entrance to the reactor. A hard birth of low-frequency high-amplitude oscillations is detected. The death of these oscillations and occurrence of high-frequency low-amplitude oscillations is analyzed. A model of this phenomenon based on the classical theory of stability of systems described by systems of ordinary differential equations is proposed. | eng_Latn | 18,642 |
Synthetic routes to nanomaterials containing anthracyclines: noncovalent systems | Chemotherapy still constitutes a basic treatment for various types of cancer. Anthracyclines are effective antineoplastic drugs that are widely used in clinical practice. Unfortunately, they are characterized by high systemic toxicity and lack of tumour selectivity. A promising way to enhance treatment effectiveness and reduce toxicity is the synthesis of systems containing anthracyclines either in the form of complexes for the encapsulation of active drugs or their covalent conjugates with inert carriers. In this respect nanotechnology offers an extensive spectrum of possible solutions. In this review, we discuss recent advances in the development of anthracycline prodrugs based on nanocarriers such as copolymers, lipids, DNA, and inorganic systems. The review focuses on the chemical architecture of the noncovalent nanocarrier–drug systems. | The novel contribution of this paper relies in the proposal of a fully implicit numerical method designed for nonlinear degenerate parabolic equations, in its convergence/stability analysis, and in the study of the related computational cost. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures: in particular we investigate the mutual benefit of combining in various ways suitable preconditioners with V-cycle algorithms. Numerical experiments in one and two spatial dimensions for the validation of our multi-facet analysis complement this contribution. | eng_Latn | 18,643 |
Modelling uncertainties in the stationary seepage problem | This paper considers the uncertainties in the stationary seepage problem. Initially, this investigation considers the uncertainties in the results of a finite element method analysis of a stationary seepage problem, where the properties of the porous medium can be considered as random variables. Random field theory is introduced to describe the uncertainty in the hydraulic properties of a porous media. The method of updating of stochastic random fields, under the condition that some of their sample realisations are observed and known is outlined. 1 Introduction In the modelling of seepage problems in porous media many kinds of simplifications are used which can introduce uncertainty into the results of the analysis, as illustrated by the following examples; (i) hydraulic characteristics can be considered a | The paper deals with the stability of the semi-implicit Milstein method for stochastic differential e-quations with time delay. By studying the difference equation, which is the outcome of applying the numerical method to a linear test equation, conditions under which the method is MS-stable and GMS-stable are determined. Moreover, some numerical experiments are given. | eng_Latn | 18,644 |
On streamline diffusion arising in Galerkin FEM with predictor/multi‐corrector time integration | In the present paper, the author shows that the predictor/multi-corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2-D and 3-D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG-PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM-PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi-steady solutions to the incompressible viscous flow problems: 2-D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd. | We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak H{\"o}rmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process. | eng_Latn | 18,645 |
PERTURBED PERIODIC SOLUTION FOR BOUSSINESQ EQUATION | We consider the solution of the good Boussinesq equation Utt-Uxx+Uxxxx=(U2)xx,-∞ 0, the difference between the true solution u(x,t;ɛ) and the N-th partial sum of the asymptotic series is bounded by φN+1 multiplied by a constant depending on T and N, for all −∞ < x < ∞, 0 ⩽ |ɛ|t and 0⩽|ɛ| ⩽ ɛ0. | The paper presents a method of the evaluation of capacitance of paraboloidal and spherical bowls using the method of moments based on the pulse function as the basis function and point matching for testing. The analysis is carried out by dividing the surface into curvilinear rectangular and triangular subsections. Numerical data on capacitance are presented. | yue_Hant | 18,646 |
Fractional calculus description of DMTA transient in long-memory materials | Abstract Fractional calculus descriptions of polymer viscoelasticity are becoming increasingly popular, because they allow a concise description of non-Debye relaxation and memory of strain history using a small number of parameters. However, use of fractional calculus to this end is frequently restricted to description of dynamic behaviour, such as in dynamic mechanical thermal analysis (DMTA), where the dependence of the complex modulus on frequency can be expressed algebraically in closed form. However, this approach is only valid in the steady state. The problem of the approach to steady state and of the effect of the slowly-decaying transient on DMTA measurements is addressed here. | A method for the algebraic approximation of attractors recently developed by Foias and Temam is adapted for application to autonomous galloping oscillators. We compare results obtained by algebraic approximation on the one hand and numerical integration on the other. We conclude with an assessment of the limitations of the method as applied to our systems. | eng_Latn | 18,647 |
Some new integrable systems of two-component fifth order equations | In this work, we develop some fifth-order integrable coupled systems of weight 0 and 1 which possess seventh-order symmetry. We establish four new systems, where in some cases, related recursion operator and bi-Hamiltonian formulations are given. We also investigate the integrability of the developed systems. | Abstract A novel method enabling to find the dependence of the solution y(x, α) on the parameter α for nonlinear boundary problems will be presented. The algorithm suggested will be illustrated on the example of simultaneous heat and mass transfer in a porous catalyst. A strategy is presented making it possible to examinate also problems having multiple solutions. This technique may successfully compete with a sequential Newton—Kantorovich method. | eng_Latn | 18,648 |
Three higher-dimensional Virasoro integrable models: Multiple soliton solutions | In this work, we study three higher-dimensional Virasoro integrable models, namely the (3+1)-dimensional Nizhnik-Novikov-Veselov equation, the (3+1)-dimensional breaking soliton equation, and a (3+1)-dimensional extended breaking soliton equation. The three equations are among the Virasoro integrable models. We use the simplified form of the Hirota’s method to establish multiple soliton solutions for each equation. We determine the constraint conditions between the coefficients of the spatial variables to guarantee the existence of the multiple soliton solutions for each model. | In this paper, a numerical method is given for solving fuzzy Fredholm integral equations of the second kind, by using Bernstein piecewise polynomial, whose coefficients determined through solving dual fuzzy linear system. Numerical examples are presented to illustrate the proposed method, whose calculations were implemented by using the Computer software MathCadV.14. | eng_Latn | 18,649 |
CRE Method for Solving mKdV Equation and New Interactions Between Solitons and Cnoidal Periodic Waves | In nonlinear physics, the modified Korteweg de-Vries (mKdV) as one of the important equation of nonlinear partial differential equations, its various solutions have been found by many methods. In this paper, the CRE method is presented for constructing new exact solutions. In addition to the new solutions of the mKdV equation, the consistent Riccati expansion (CRE) method can unearth other equations. | A kinetic energy budget over the Indian region is computed for the period 4–9 July 1973, when a twin monsoon depression-one in the Bay of Bengal and another in the Arabian sea were the dominant synoptic features. The generation term caused by the cross-contour flow is a dominant source to the kinetic energy. The dissipation term is computed as a residual and is a major sink for the kinetic energy. The horizontal flux divergence is also a sink term but is much smaller in magnitude than other major source and sink terms. From the results it may be inferred that the generation term is the most important for the maintenance of monsoon disturbances. | eng_Latn | 18,650 |
Advancing Dynamic Evolutionary Optimization Using In-Memory Database Technology | This paper reports on IMDEA (In-Memory database Dynamic Evolutionary Algorithm), an approach to dynamic evolutionary optimization exploiting in-memory database (IMDB) technology to expedite the search process subject to change events arising at runtime. The implemented system benefits from optimization knowledge persisted on an IMDB serving as associative memory to better guide the optimizer through changing environments. For this, specific strategies for knowledge processing, extraction and injection are developed and evaluated. Moreover, prediction methods are embedded and empirical studies outline to which extent these methods are able to anticipate forthcoming dynamic change events by evaluating historical records of previous changes and other optimization knowledge managed by the IMDB. | Abstract In many applications, such as atmospheric chemistry, large systems of ordinary differential equations (ODEs) with both stiff and nonstiff parts have to be solved numerically. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linear stability of popular second order methods like extrapolated BDF, Crank-Nicolson leap-frog and a particular class of Adams methods. We present results for problems with decoupled eigenvalues and comment on some specific CFL restrictions associated with advection terms. | kor_Hang | 18,651 |
Parameter Estimations for the Burgers Equation | Nonhysteretic infiltration in nonswelling soil is modelled by the Burgers equation under appropriate physical conditions. For this nonlinear partial differential equation a modal scheme for estimating parameters such as soil water diffusivity and conductivity are introduced, and numerical experiments are performed. | We define in this article an approach to tackle the problem of the computation of the general solution of word equations. Parametric transformation, Nielsen's transformation and Rouen's transformation allowing to collect in one transformation some unbounded sequences of elementary transformations are given. | eng_Latn | 18,652 |
Frontiers Of 4d And 5d Transition Metal Oxides | Thank you for downloading frontiers of 4d and 5d transition metal oxides. As you may know, people have look numerous times for their chosen books like this frontiers of 4d and 5d transition metal oxides, but end up in malicious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they juggled with some infectious virus inside their laptop. frontiers of 4d and 5d transition metal oxides is available in our book collection an online access to it is set as public so you can get it instantly. Our digital library saves in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the frontiers of 4d and 5d transition metal oxides is universally compatible with any devices to read. | By differential operator,this paper study the method for solving a class of the fourth order linear differential equations and the stability of its solution,make the extension to results of paper[2]. | yue_Hant | 18,653 |
Oscillation phenomena in gravity-driven drainage in coarse porous media | Correlations between rainfall amount and water content in coarse soils were studied. Situations were registered in which the intensity of the gravity-driven flow in the soil exhibited irregular oscillations which could not be explained within the scope of the standard Richards' theory. The problem was analyzed by means of a set of laboratory experiments. It seems that the observed global (macroscopic) oscillations can be qualitatively explained as a result of correlated behavior of the local (microscopic) flows. The search for a satisfactory quantitative description of those effects promises to be a field of useful and interesting research activities. | In this paper, we consider the second order wave equation discretized in space by summation-by-parts-simultaneous approximation term (SBP-SAT) technique. Special emphasis is placed on the accuracy analysis of the treatment of the Dirichlet boundary condition and of the grid interface condition. The result shows that a boundary or grid interface closure with truncation error $\mathcal{O}(h^p)$ converges of order $p + 2$ if the penalty parameters are chosen carefully. We show that stability does not automatically yield a gain of two orders in convergence rate. The accuracy analysis is verified by numerical experiments. | eng_Latn | 18,654 |
The synthesis, testing and use of 5-fluoro-alpha-D-galactosyl fluoride to trap an intermediate on green coffee bean alpha-galactosidase and identify the catalytic nucleophile. | 5-Fluoro-alpha-D-galactopyranosyl fluoride was synthesized and its interaction with the active site of an alpha-galactosidase from green coffee bean (Coffea arabica), a retaining glycosidase, characterized kinetically and structurally. The compound behaves as an apparently tight binding (Ki = 600 nM) competitive inhibitor, achieving this high affinity through reaction as a slow substrate that accumulates a high steady-state concentration of the glycosyl-enzyme intermediate, as evidenced by ESiMS. Proteolysis of the trapped enzyme coupled with HPLC/MS analysis allowed the localization of a labeled peptide that was subsequently sequenced. Comparison of this sequence information to that of other members of the same glycosidase family revealed the active site nucleophile to be Asp145 within the sequence LKYDNCNNN. The importance of this residue to catalysis has been confirmed by mutagenesis studies. | Abstract The method of Galerkin is used to obtain a semidiscrete formulation of the time-dependent multigroup diffusion equa-tions which are then integrated in time by the θ-method to obtain a system of algebraic equations. For given initial and boundary conditions, successive estimates of the neutron flux for each energy group and the precursor concentrations for each delayed neutron group are obtained. Since the precursor concentrations are not necessarily continuous across an interface, the global basis functions are treated in a different form than the corresponding ones for the neutron flux. A computer code is implemented using quadratic Lagrange basis functions. Numerical results for a one-dimensional subcritical transient problem show that this method is as accurate as the usual finite difference schemes but considerably more efficient. | eng_Latn | 18,655 |
Numerical Solution of the American Option Pricing by the Penalty Method | The American option pricing problem can be described mathematically bythe well-known Black-Scholes equation with a free boundary. It is difficult to solvethis problem since the free boundary is unknown.To accomplish that, we make use of the Implicit Finite-difference Method and Penalty Method which means to add a continuous penalty term with a small parameter 2 into the Black-Scholes equation,to transform the original free boundary problem into a nonlinear partial differential equation with a fixed boundary. | Abstract We extend to a particular class of nonlinear difference equations the classical method of equivalent linearization. We show that the method can be used to obtain an approximation to the periodic solutions of these equations. In particular, we can determine the parameters of the limit cycles and limit points. Three examples illustrating the method are presented. | eng_Latn | 18,656 |
Species delineation and hybrid identification using diagnostic nuclear markers for Plectropomus leopardus and Plectropomus maculatus | We thank King Abdullah and University of Science and Technology (KAUST) Bioscience Core Laboratory for laboratory support. We acknowledge Jean-Francois Flot and Tane Sinclair-Taylor for helpful discussions and assistance with figures. Financial support was provided by KAUST baseline research funds to M. L. B. We also thank anonymous reviewers for their constructive comments. | In the present paper, the author shows that the predictor/multi-corrector (PMC) time integration for the advection–diffusion equations induces numerical diffusivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin finite element method (GFEM). The transient 2-D and 3-D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind/Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG-PMC turned out to be overly diffusive due to the additional PMC streamline diffusion, while the solutions of the GFEM-PMC were comparatively accurate without significant damping and phase error. A similar tendency was seen also in the quasi-steady solutions to the incompressible viscous flow problems: 2-D driven cavity flow and natural convection in a square cavity. Copyright © 2002 John Wiley & Sons, Ltd. | eng_Latn | 18,657 |
Calculation of critical points from cubic two‐constant equations of state | A computational modification of the Heidemann-Khalil method for calculating the critical temperatures and pressures for general phase-equilibrium problems greatly reduced the computing time for simple two-constant cubic equations of state. For systems where the unlike binary interaction parameters can be derived from the pure-component parameters using the geometric mean values, a further simplification lowers the computing times to a few milliseconds, regardless of the number of components. | Second order Cauchy Pompeiu formulas are given in the case of one and of several complex variables, for Douglis-algebra-valued functions of one complex variable and for Clifford-algebra-valued functions of several real variables. The kernels of the integral operators attained provide fundamental solutions for the respective differential operators. | eng_Latn | 18,658 |
Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs | Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some case studies are discussed. | Details of the application of the Feynman graph summing method to the ..pi..-d system are given. (SDF) | eng_Latn | 18,659 |
One-Dimensional Fully Developed Turbulent Flow through Coarse Porous Medium | A fully developed turbulent regime is considered as a specific case of non-Darcy flow, and an analytical approach has been developed to determine normal depth, water surface profile, and seepage discharge of the flow through coarse porous medium in steady condition. The results of a laboratory rock drain with length, height, and width of 6.4, 0.8, and 0.8 m, respectively, and longitudinal slope of 0.04 were compared with the analytical solution developed in this study, and the results showed a good agreement between analytical and experimental data. To see the compatibility of the solution, a Darcy-based form of the solution (Pavlovsky's method) and a flow analysis of buried streams (FABS) model are compared with the proposed solution and experimental data. Compared with Pavlovsky's solution and FABS model, the results showed a satisfactory agreement with experimental records from water surface profiles through rock drain. DOI: 10.1061/(ASCE)HE.1943-5584.0000937. © 2014 American Society of Civil Engineers. | Abstract Previous expressions for thermo- and diffusiophoresis in the free molecule regime are reviewed, compared, and simplified where possible. A simple first order derivation is then presented for the force exerted on a particle, and hence for its resulting velocity, when both phoretic effects are present in a multicomponent gas mixture. The results are obtained in a form which facilitates their application to practical problems. After rearrangement, the equation for particle velocity is shown to be essentially identical to previous expressions, except for small second order effects. An example of application is given. | eng_Latn | 18,660 |
Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method | A meshless local boundary integral equation (LBIE) method is proposed to solve the unsteady two-dimensional Schrodinger equation. The method is based on the LBIE with moving least-squares (MLS) approximation. For the MLS approximation, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. A time-stepping method is employed to deal with the time derivative. An efficient method for dealing with singular domain integrations that appear in the discretized equations is presented. Finally, numerical results are considered for some examples to demonstrate the accuracy, efficiency and high rate of convergence of this method. Copyright © 2008 John Wiley & Sons, Ltd. | Abstract The high frequency modes of Hamiltonian systems tend to have small amplitudes. Hence for moderately accurate integration of such problems by, say, the leapfrog method the time step tends to be limited by stability restrictions rather than accuracy restrictions. Conventional implicit symplectic methods like implicit midpoint have less severe stability restrictions but the cost of solving large nonlinear systems with dense Jacobian matrices is probably too high to make them worthwhile. To bring down the cost of implicit methods, we have designed (i) mixed implicit-explicit, and (ii) linearly implicit methods that retain the property of being symplectic. | eng_Latn | 18,661 |
Comparison between the Stanford-Binet: L-M and the Stanford-Binet: Fourth Edition with a Group of Gifted Children. | Abstract Comparisons were made between scores on form L-M and the Fourth Edition of the Stanford-Binet Intelligence Scale for 32 students classified as gifted by the public schools. The L-M was administered by psychological examiners in the school system and the Fourth Edition was administered by the authors. Significant differences were found between the L-M IQ and the Composite Score of the Fourth Edition and between the L-M IQ and the four area scores of the Fourth Edition. The implications of these findings are discussed. | Abstract The development of hp -version discontinuous Galerkin methods for hyperbolic conservation laws is presented in this work. A priori error estimates are derived for a model class of linear hyperbolic conservation laws. These estimates are obtained using a new mesh-dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation. The results generalize and extend previous results on mesh-dependent norms to hp -version discontinuous Galerkin methods. A posteriori error estimates which provide bounds on the actual error are also developed in this work. Numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution. | eng_Latn | 18,662 |
Auxiliary equation method for the mKdV equation with variable coefficients | Abstract By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients. As a result, new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation. | Active methylene compounds were carboxylated by employing the reagent system, diphenylcarbodiimide and potassium carbonate, in dimethyl sulfoxide at room temperature and atmospheric pressure. The reaction proceeded even in the absence of carbon dioxide, but the carboxylation easily proceeded in carbon dioxide atmosphere. | eng_Latn | 18,663 |
An analysis of the fractional step method | The fractional step method for solving the incompressible Navier-Stokes equations in primitive variables is analyzed as a block LU decomposition. In this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved. In addition, it is shown that poor temporal accuracy (first-order) is not due to boundary conditions, but due to the method itself. A generalized block LU decomposition that overcomes this difficulty is presented, allowing arbitrarily high temporal order of accuracy. The generalized decomposition is shown to be useful for a wide range of problems including steady problems. Technical issues, such as stability and the appropriate pressure update scheme, are also addressed. Numerical simulations of the steady, incompressible Navier-Stokes equations in a square domain confirm the theoretical results. 18 refs., 4 figs. | Study classical general mathematics model bag fetch ball,arrange in an order,put ball not to enter case analytical method of problem,utilize the these problems analytical method to solve some classical and general probability and calculate the problem. | eng_Latn | 18,664 |
Digital signal propagation in dispersive media | In this article, the propagation of digital and analog signals through media which, in general, are both dissipative and dispersive is modeled using the one-dimensional telegraph equation. Input signals are represented using impulsive, Heaviside unit step, Gaussian, rectangular pulse, and both unmodulated and modulated sinusoidal pulse type boundary data. Applications to coaxial transmission lines and freshwater signal propagation, for both digital and analog signals, are included. The analysis presented here supports the finding that digital transmission in dispersive media is generally superior to that of analog. The boundary data (input signals) give rise to solutions of the telegraph equation which contain propagating discontinuities. It is shown that the magnitudes of these discontinuities, as a function of distance, can be found without the need of solving the governing equation. Thus, for digital signals in particular, signal strength at a given distance from the input source can be easily determin... | In this paper, aimed at the neutron transport equations of eigenvalue problem under 2-D cylindrical geometry on unstructured grid, the discrete scheme of Sn discrete ordinate and discontinuous finite is built, and the parallel computation for the scheme is realized on MPI systems. Numerical experiments indicate that the designed parallel algorithm can reach perfect speedup, it has good practicality and scalability. | eng_Latn | 18,665 |
The Method of Capillary Gas Chromatography Used to Measure Naphthalene Contents in Air at Work | The method of capillary gas chromatography used to measure naphthalene contents in air at work is employed. The method has higher sensitivity, precision, and accuracy and is easy to practise. The precision is CV=3.5%～5.9%; accuracy rate of discovered is 95.8%～98.1%; minimum value tested out is 2×10-4μg. | Abstract The paper studies the existence, uniqueness, regularity and the approximation of solutions to the reaction–diffusion equation endowed with a general nonlinear regular potential and Cauchy–Neumann boundary conditions. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation. | eng_Latn | 18,666 |
Technical Note Numerical evaluation of the Graetz series | Abstract The Graetz problem describes the temperature (or concentration) field in fully-developed laminar flow in a circular tube where the wall temperature (or concentration) profile is a step-function. A great deal of analysis has been performed on this fundamental problem: a detailed survey of the relevant abundant literature may be found in [1] . The infinite series solution of the problem, referred to as the Graetz series, is | A new method of calculating the greatest common divisor of several integers by using the row elementary operation is given.An application of this method is also given. | eng_Latn | 18,667 |
Constrained Control of Positive Discrete-Time Systems With Delays | This brief addresses the control problem of linear time-invariant discrete-time systems with delays. The control is under positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. The contribution lies in three aspects. First, a necessary and sufficient condition is established for the existence of such controllers for discrete-time delayed systems. Second, a sufficient condition is provided under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Third, sufficient conditions are proposed for discrete-time delayed systems with uncertainties, whether or not bounded control is considered. All the results are formulated as linear programming problems, hence easy to be verified. And the controllers are explicitly constructed if existent. | The aim of this paper is to solve numerically a class of problems on conservation laws, modelled by hyperbolic partial differential equations. In this paper, primary focus is over the idea of fuzzy logic-based operators for the simulation of problems related to hyperbolic conservation laws. Present approach considers a novel computational procedure which relies on using some operators from fuzzy logic to reconstruct several higher-order numerical methods known as the flux-limited methods. Further optimization of the flux limiters is discussed. The approach ensures better convergence of the approximation and preserves the basic properties of the solution of the problem under consideration. The new limiters are further applied to several real-life problems like the advection problem to demonstrate that the optimized schemes ensure better results. Simulation results are included wherever required. | eng_Latn | 18,668 |
Implicit High-Order Compact Differencing Methods: Study of Convergence and Stability | Compact differencing can deliver high-order accuracy using only a limited span of stencils, but incurring a costly matrix in version. Hence, use of a stable implicit time discretizaton becomes favorable in order to offset the computation cost by allowing a large time step. A practical way to reduce the burden of inverting a large matrix from multidimensional problems is to split the implicit operator into a series of smaller operators. Undesirable consequences can surface, such as (1) loss of stability, and/or (2) loss of accuracy. Here, we propose a consistent implicit compact method and study the stability and accuracy of steady and unsteady solutions. | The methods of false color composite band choice,samples size,and samples methods,influence classification precisions,were introduced.This article is very important reference to advance classification precision. | eng_Latn | 18,669 |
Make the solution of KMnO4 IN 1M? | How do you make 0.1 normal solution of kmno4? | How do you make 0.1 normal solution of kmno4? | eng_Latn | 18,670 |
Formal exact operator solutions to nonlinear differential equations | The compact explicit expressions for formal exact operator solutions to Cauchy problem for sufficiently general systems of nonlinear differential equations (ODEs and PDEs) in the form of chronological operator exponents are given. The variant of exact solutions in the form of ordinary (without chronologization) operator exponents are proposed. | We consider the third-order nonlocal boundary value problem u ′′′ (t) = f(t,u(t)), a.e. in (0,1), u(0) = 0, u ′ (�) = 0, u ′′ (1) = �[u ′′ ], | eng_Latn | 18,671 |
I'm learning about astrodynamics on my own and I was wondering why the $r$ is cubed in the vector notation for of Newton's Law of Universal Gravitation: $$\vec{F}_g=\frac{Gm_1m_2}{|\vec{r}|^3}\vec{r}$$ I am familiar with Newton's Law of Universal Gravitation of the form: $$F_g=\frac{Gm_1m_2}{r^2}$$ Is there something obvious I'm missing? | I am reading an older physics book that my professor gave me. It is going over Coulomb's law and Gauss' theorem. However, the book gives both equations with an $r^3$, not $r^2$, in the denominator. Can somebody please explain why it is given as r^3? An image is attached for reference. Also for equation 1-24, can somebody please explain how the middle side is equal to the right side with the del operator? | Using the method of characteristics, find a solution to Burgers' equation \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g & \text{on } \mathbb{R} \times\{t=0\} \end{cases} with the initial conditions $$g(x)=\begin{cases} 0 & \text{if }x < 0 \\ 1 & \text{if }0 \le x \le 1 \\ 0 & \text{if }x > 1 \end{cases}$$ First, I realized that the equation $u_t+\left(\frac{u^2}{2} \right)_x =0$ is equivalent to this form: $$u_t+uu_x =0$$ Then should I generally follow the method of solution as outlined in the answer of ? Note that this is not a duplicate question of that page. Rather I want to know if that page can be used for my problem, even though my ICs are different. By the way, the solution printed in my book (PDE Evans, 2nd edition, page 142) is $$u(x,t) = \begin{cases} 0 & \text{if } x < 0 \\ \frac xt & \text{if } 0 < x < t \\ 1 & \text{if } t < x < 1 + \frac t2 \\ 0 & \text{if } x > 1 + \frac t2 \tag{$0 \le t \le 2$} \end{cases}$$ | eng_Latn | 18,672 |
I am trying to program a $n$-body problem simulation. To calculate the position and velocity after a time-step I want to split it in multiple 2 body problems. Now I am stuck, trying to find the velocity and position of two bodies (ignoring all the others) after one time-step with given mass, start position and velocity. I am able to calculate the force between the two, but I am unable to solve the differential equation $$ \ddot{\vec{r_{12}}(t)} = G M \frac{\vec{r_{12}(t)}}{|\vec{r_{12}(t)}|^3} \\ M := m_1 + m_2 \\ \vec{r_{12}(t)} := \vec{r_1}(t) - \vec{r_2}(t) $$ to calculate the position relative to each other. What is $\vec{r}(t)$ and how can I calculate it? And is there another and maybe faster way to get the positions and velocities? Edit: As @Sofia pointed out, this was not clear: $\vec{r_1}$ and $\vec{r_2}$ are the location vectors of the two masses. | Two particles with initial positions and velocities $r_1,v_1$ and $r_2,v_2$ are interacting by the inverse square law (with G=1), so that $$ {d^2r_1\over dt^2} = - { m_2(r_1-r_2)\over |r_1-r_2|^3} $$ $$ {d^2r_2\over dt^2} = - { m_1(r_2-r_1)\over |r_1-r_2|^3} $$ (the inverse square law along the line of separation). What is the complete solution of these differential equations? What is the position of the two objects as a function of time? After reading a lot on Wikipedia, I've come to the definition of center of mass and relative coordinates: $$R(t) ~=~ \frac{m_1 r_1 + m_2 r_2}{m_1+m_2}$$ $$\ddot{r}(t) r(t)^2 ~=~ (m_1+m_2)G$$ Where $R$ is the center of mass, and $r$ is the displacement between the particles... Is this correct? How do I proceed to solve the differential equation? | The exercise is: Prove that $-(-v)=v$ for every $v \in V$ Proof Suppose $v \in V$ and $V$ is a vector space. Then $-(-v) \in V$ as result of the scalar multiplication property and $-(-v)=-(-1\cdot v)=-1 \cdot(-1\cdot v) =(-1 \cdot-1)\cdot v = 1 \cdot v = v $ The desired result $-(-v)=v$ holds. The solution manual gives the proof: Proof I just wanted to make sure that the way I did the proof isn't missing anything?Thanks | eng_Latn | 18,673 |
I apologise, since this question does already exist. But I cannot for the life of me understand how it is solved. Could anyone provide me with a simplified version of the solution? | I am asked to solve $$u_x+u_y=1$$ If is was homogeneous i.e., $u_x+u_y$ the answer would be $u(x,y)=f(y-x)$ where $f$ is an arbitrary function. I have found the following set of solutions: $$u(x,y)=\lambda x +(1-\lambda)y$$ where $\lambda$ is an arbitrary constant(real or imaginary). I just have no idea what method other then trial and error would have lead me here. Any ideas? Thanks! | The entire site is blank right now. The header and footer are shown, but no questions. | eng_Latn | 18,674 |
Determine if $x(t)$ is stable, where $x(t)$ is a solution to the differential equation: $x'''+x''+4x'+4x = e^t(H(t) - H(t-3))$ and x(t) is defined as: $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})\times H(t-3)$ How do I approach this? Do I use eigenvalues? If so, I get 2i, - 2i and 1, is that correct and what do these values imply? Any help is greatly appreciated! | Determine wether the differential equation is stable or not for the following conditions: $x'''(0) = x''(0) = x'(0) = x(0) =0$ $\frac{d^3x(t)}{dt^3}+\frac{d^2x(t)}{dt^2}+4\frac{dx(t)}{dt}+4x(t) = e^t(H(t)-H(t-3))$ Now, I got that the solution to this differential equation reduces x(t) by use of Laplace Transforms, however, how do I determine whether or not it is stable? $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})*H(t-3)$ Now I first thought I would look at it graphically and I used Matlab software to get a plot as shown below: And from the plot above it clearly appears to diverge (though this is not a proof). Anyone have an idea as to an elegant proof to show the solution is unstable? Anyone help would be greatly appreciated! | Where to find cross section data for $e^{-}$ + $p$ $\longrightarrow$ $p$ + $e^{-}$ ? does not include it. | eng_Latn | 18,675 |
$(2x^2+1)y''-4xy'+4y=0$ $y_1(x) = x$ is given and I have to find $y_2(x)$ and solve equation. thanks. | I need help solving this past exam question, the professor posted exam questions with out solutions to last year's exam, I tried reduction of order, however that seemed to complicate things. And then I tried Euler-Cauchy method however I realized that won't work due to the $(2x^2 + 1)y''$. Given $y_1=x$ is a solution $$(2x^2 + 1)y'' − 4xy' + 4y = 0$$ Find the second solution $y_2$ for equation. | The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$ of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail method. In case more solutions are existing or not how to check? What are the solutions of $x^4 - 2y^2 = 1$? | eng_Latn | 18,676 |
I need help to solve initial-value quasilinear problem: $xu_y - yu_x = u$, $u(x,0)=h(x)$ Here what I did: The initial curve $\Gamma: <x=s, y-0, z=h(s)> $ The characteristic equation $\frac{dx}{dt}=-y$, $\frac{dy}{dt}=x$, $\frac{dz}{dt}=z$.. So $x=-yt + s$, $y=xt$ and $z=h(s)e^t$. Solve for $s$ I get $s=\frac{x^2 + y^2}{x}$ then solution for this problem is $u=h(\frac{x^2 + y^2}{x})e^{y/x}$. Is it right? I am not sure because the answer from text book, PDE 4th Edition by Fritz John, is $u=h(\sqrt{x^2+y^2})e^{arctan(y/x)}$ Please help. Thanks | I am trying to find the equation of characteristic curves and solution of the partial differential equation $$x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=u$$ $u(x,0)=\sin(\frac{\pi}{4}x)$ . What I did: $$\frac{dx}{-y}=\frac{dy}{x}=\frac{du}{u}$$ with From first two equalities $$\frac{dx}{-y}=\frac{dy}{x}$$ by solving we get $x^2+y^2=c_1$ for some constant $c_1$. Now I am stuck with the third and rest of the equalities to come up with a suitable manipulation to solve the problem. How can I do this? Any help would be great. Thanks. | Let $F$ be a field and $x\in F$. If $x=-x$ and $1\neq -1$, then $$0=\frac{x+x}{1+1}=\frac{1+1}{1+1}x=x.$$ This means that the statement in the title is true if and only if $1\neq -1$. But how do we know that $1\neq -1$ for an arbitary field? | eng_Latn | 18,677 |
Is $$\dfrac{x^2 + 2x}{x}$$ a polynomial? | Is $\frac{x^2+x}{x+1}$ a polynomial? Fist question can be: on which field/ring or etc? In basic, let's take over $\mathbb R$. Actually, it is $x$ if $x \ne -1$. Can we say again it is a polynomial, or a near-polynomial or what? Can we say it is a polynomial over $\mathbb R - \{-1\}$? But there is structure here for $\mathbb R-\{-1\}$. This question can be academically, not sure, but I want to ask it in the level of primary school, then academically. | I am able to solve simple differential equations like : $$\dfrac{dy}{dx} = 3x^2 + 2x$$ We simply bring $dx$ to other site and integrate. But how do we find solutions of differential equations like : $$\frac{d^2x}{dt^2} + \dfrac{kx}{m} = 0$$ ? We have been told the solution is $x(t) = A\cos ( \omega t + \phi_o)$ where $\omega\ =\sqrt{\dfrac{k}{m}}$ but how do we actually find it? | gle_Latn | 18,678 |
Working on this. Have to find a solution. Very new to DE: $$\begin{cases} 2yx'(y)-1=x^2(y) \\ x(y=1)=1 \\ \end{cases} $$ Writing this as: \begin{align} 2y \frac{dx}{dy}-1&=x^2y\\ \frac{1}{x^2}2y\,dx-1&=y\,dy\\ \frac{1}{x^2}\,dx &= 1+y\,dy\\ \int \frac{1}{x^2}\,dx &= \int 1+y\,dy\\ -\frac{1}{x}+C &= \frac{y^2}{2}+y+C \end{align} Is this correct? Feel like I'm just doing algebra, but don't really understand if it's correct. Would love feedback. | Really stuck with this, \begin{cases} 2tx'(t)-1=x^2t \\ x(t=1)=1 \\ \end{cases} I don't understand the variables. What does it mean that $x$ differentiated have $t$ as a function value, like this: $x'(t)$? Don't get it :( | The new Top-Bar does not show reputation changes from Area 51. | yue_Hant | 18,679 |
I got this (C#) : Random RNG = new Random(); decimal divab50 = RNG.Next(50,100); decimal divbl50 = RNG.Next(6,50); decimal decreturn = divab50 / divbl50; Console.WriteLine(decreturn); How can I round the decreturn var to two decimals? I've tried Math.Round and String.Format they don't seem to work for vars generated in RNG. I think. I'm new at c# just started | I want to do this using the Math.Round function | I see an equation like this: $$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$ and solve it by "separating variables" like this: $$y\textrm{d}y = e^x\textrm{d}x$$ $$\int y\textrm{d}y = \int e^x\textrm{d}x$$ $$y^2/2 = e^x + c$$ What am I doing when I solve an equation this way? Because $\textrm{d}y/\textrm{d}x$ actually means $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$ they are not really separate entities I can multiply around algebraically. I can check the solution when I'm done this procedure, and I've never run into problems with it. Nonetheless, what is the justification behind it? What I thought of to do in this particular case is write $$\int y \frac{\textrm{d}y}{\textrm{d}x}\textrm{d}x = \int e^x\textrm{d}x$$ $$\int \frac{\textrm{d}}{\textrm{d}x}(y^2/2)\textrm{d}x = e^x + c$$ then by the fundamental theorem of calculus $$y^2/2 = e^x + c$$ Is this correct? Will such a procedure work every time I can find a way to separate variables? | eng_Latn | 18,680 |
given the 2 PDE $$ \Delta u-au_{tt}+u_{t}=0$$ and $$ \Delta u + Du*Df=0 $$ here $ \Delta u $ is the Laplacian $ Du= grau$ and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$ my doubt is what term should i include to get the linear part of the equations $ u_{t} $ and $Du*Df $ from the Euler Lagrange equations in $ R^{n} $ thanks | given the 2 PDE $$ \Delta u-au_{tt}+u_{t}=0$$ and $$ \Delta u + Du*Df=0 $$ here $ \Delta u $ is the Laplacian $ Du= grad(u) $ is the gradient and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$ my doubt is what term should i include to get the linear part of the equations $ u_{t} $ and $Du*Df $ from the Euler Lagrange equations in $ R^{n} $ thanks | given the 2 PDE $$ \Delta u-au_{tt}+u_{t}=0$$ and $$ \Delta u + Du*Df=0 $$ here $ \Delta u $ is the Laplacian $ Du= grad(u) $ is the gradient and * means scalar product $u_{t} = \frac{\partial u}{\partial t}$ my doubt is what term should i include to get the linear part of the equations $ u_{t} $ and $Du*Df $ from the Euler Lagrange equations in $ R^{n} $ thanks | eng_Latn | 18,681 |
The equation $$TdS=dU+PdV$$ is a combination of $$dQ=dU+PdV$$ and $$TdS=dQ.$$ But in some process that is very irreversible, $TdS>dQ$, should we write $$TdS>dU+PdV$$ instead of the first equation? I guess the equation is different for reversible and irreversible process. | Combining the first and second law of thermodynamics we can get the following equation $$TdS=dU-P_{ext}dV$$ First question: Is this equation applicable for irreversible processes such that that $dS≠\dfrac{dQ}{T}$? Second question:If the system temperature $T_{sys}$ is smaller than the surrounding temperature $T_{sur}$, which temperature should we put in the equation? I have this question because sometimes people use $T_{sur}$ instead of $T_{sys}$ (e.g. , ) but the equation is supposed to describe changes in the system. | Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation arises out of following problem : A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed $u$. Without any delay, the dog starts with running with constant speed $v>u$ to catch the cat. Initially, $v$ is perpendicular to $u$ and $L$ is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of $v, u$ and $L$. See my solution below : Let initially dog be at $D$ and cat at $C$ and after time $dt$ they are at $D'$ and $C'$ respectively. Dog velocity is always pointing towards cat. Let $DA = dy, \;AD' = dx$ Let $CC'=udt,\;DD' = vdt$ as interval is very small so $DD'$ can be taken straight line. Also we have $\frac{DA}{DC}= \frac{AD'}{ CC'}$ using triangle property. $\frac{dy}{L}= \frac{dx}{udt}\\ dx = \frac{dy.udt}{L}$ $\sqrt{(dx)^2 + (dy)^2} = DD' = vdt \\ \sqrt{(\frac{dy.udt}{L})^2 + (dy)^2} = vdt $ Here $t$ varies from $0-T$, and $y$ varies from $0-L$. Now how to proceed? | eng_Latn | 18,682 |
So I have tried many methods including Laplace transform and differentiation but the p and q seems to be unsolvable, I would like to know how does the form change from the first equation, then substitude the Y = e^t to get the second equation. also my second question is why is how can i solve for p and q if i dont have any initial conditions? | If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation: $$z^2w'' + \alpha zw' + \beta w = 0$$ where $w$ is a function of $z$ and $\alpha, \beta$ are constants. How would the change of variables $t =\ln(z)$ transform this equation? | Example sentence: I love when your dog just lets you sit there to pet them. You don’t necessarily know if they are enjoying it, but they love you enough to just sit there with you for a bit. Is this correct? We assume the words "you" and "your" refer to the speaker of the sentence, and not to the listener, as second-person usually does. But it also refers to dog owners in general. I have always been curious about this. | eng_Latn | 18,683 |
This equation $(x^3+xy^2−y)dx+(y^3+x^2y+x)dy=0 $ is Integration factor found by inspection. The answer on the book is $2\arctan(\frac{y}{x})=c-x^2-y^2$ | $(x^3+xy^2-y)dx+(y^3+x^2y+x)dy=0$ I tried to find on Wolfram Alpha but it showed only the solution not in step-by-step, I know that the factor is $\frac{1}{x^2+y^2}$ but how to find it? | I was doing some practice problems that my professor had sent us and I have not been able to figure out one of them. The given equation is: $-y^2dx +x^2dy = 0$ He then asks us to verify that: $ u(x, y) = \frac{1}{(x-y)^2}$ is an integrating factor. I multiplied through to get: $\frac{-y^2}{(x-y)^2}dx + \frac{x^2}{(x-y)^2}dy = 0$ However, the partial derivatives of these do not equal each other so I am a bit confused... | eng_Latn | 18,684 |
I was studying about solving expressions of the form $\sqrt{6+\sqrt{6+\sqrt {6+\cdots}}}$ I know how to solve this, but how can this equation have a constant variable if it is continued indefinitely? I know upto limits and basic derivatives. Thanks | (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12) For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x > 0$. Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining $$x_{n+1} = \sqrt{c+x_n}$$ Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation. Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$. This is being repurposed in an effort to cut down on duplicates, see here: . and here: . | Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation arises out of following problem : A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed $u$. Without any delay, the dog starts with running with constant speed $v>u$ to catch the cat. Initially, $v$ is perpendicular to $u$ and $L$ is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of $v, u$ and $L$. See my solution below : Let initially dog be at $D$ and cat at $C$ and after time $dt$ they are at $D'$ and $C'$ respectively. Dog velocity is always pointing towards cat. Let $DA = dy, \;AD' = dx$ Let $CC'=udt,\;DD' = vdt$ as interval is very small so $DD'$ can be taken straight line. Also we have $\frac{DA}{DC}= \frac{AD'}{ CC'}$ using triangle property. $\frac{dy}{L}= \frac{dx}{udt}\\ dx = \frac{dy.udt}{L}$ $\sqrt{(dx)^2 + (dy)^2} = DD' = vdt \\ \sqrt{(\frac{dy.udt}{L})^2 + (dy)^2} = vdt $ Here $t$ varies from $0-T$, and $y$ varies from $0-L$. Now how to proceed? | eng_Latn | 18,685 |
THE PROBLEM Consider a pendulum with an hollow spheric bob of negligible weight filled with 1.1 kg of fine sand and hanging from an inextensible wire of negligible mass and $l= 30 m$ long. At the point which is diametrically opposite to the point where it is hung to the wire, the sphere has a small hole from which the sand can come out with a constant flow of 1 g/s. In the rest position the hole is at a distance $h = 1 m$ from the horizontal floor. At time $t = 0$ the sphere is placed at a distance $d = 30 cm$ from its rest position, the hole is opened and the sphere is let go (see the drawing for a better understanding). Neglecting any friction and referring to a Cartesian axis that runs along the floor in the direction of the motion of the sphere and whose origin is at the point closest to the rest position of the sphere, compute: the position where the sand falls as a function of the time $t$; the maximum distance at which the sand is deposited; the points where the greatest amount of sand is accumulated; the distribution of the mass of sand on the floor. Assume that the sand remains exactly at the point where it touches the floor. HINT Some useful trigonometric formulas: $$cos(arctan(x))=\frac{1}{\sqrt{1+x^2}}\qquad\qquad \sin(arctan(x))=\frac{x}{\sqrt{1+x^2}}$$ (WORKING ON) THE SOLUTION I suppose that the first thing to notice is that the little-angle approximation can be used since the starting angle $\vartheta_{\text{MAX}}$ (at $t=0$) is easily computed as $$\vartheta_{\text{MAX}}=arcsin(d/l)=arcsin(0.01)\simeq 0.01000017$$ and it's very small. Hence $sin(\vartheta)$ can be approximated with $\vartheta$ and $cos(\vartheta)$ with $1$. In this approximation, it is well known that the pendulum period is $$T = 2\pi\sqrt{\frac{l^*}{g}}$$ Now, strictly speaking, $l^*$ should be the distance between the point in which the wire is hung and the center of mass of the bob. At $t=0$ this means $l^* = l+R$, where $R$ is the radius of the hollow sphere. With the sand flowing away, the center of mass will shift downwards, increasing $l^*$ and hence increasing the time period only to be back at its initial position $l+R$ when there's no sand left inside the bob. However, since the problem does not give the value for $R$, I assumed that a point-like approximation for the bob could be used, hence using $l^* = l$ regardless of how much sand is left inside the sphere. I guessed this is what who wrote the problem was asking for since the period so computed is $T \simeq 11s$: a number matching too well the $1.1 g/s$ of the constant flow of sand to be a coincidence (giving the sphere emptied in $100$ periods). Assuming all of the above is right, I went on to write the equations of motion for the flowing sand: \begin{array}{ll} x(t) &= x_s + v_x t = l\,sin(\vartheta(t)) + l \dot{\vartheta(t)} t \simeq l\,\vartheta(t) + l \dot{\vartheta(t)} t\\ y(t) &= y_s + v_y t - \frac{g}{2}t^2= h + l(1-cos(\vartheta)) + l \dot{\vartheta(t)} sin(\vartheta(t)) t - \frac{g}{2}t^2 \simeq h + l \dot{\vartheta(t)} \vartheta(t) t - \frac{g}{2}t^2 \end{array} Then I'd like to set $y(t)=0$ in order to solve for $t$ and substitute inside the equation for $x(t)$...but I don't know how to proceed. Plus, how should the given HINT be useful? Any help to solve this would be extremely appreciated! EDIT (why this is not a duplicate) This question is not a duplicate because I'm not asking of how the period of pendulum varies with the flowing of the sand but, assuming it does not vary (an approximation), I'm asking about the position where the sand falls as a function of the time $t$; the maximum distance at which the sand is deposited; the points where the greatest amount of sand is accumulated; the distribution of the mass of sand on the floor. | How do you find time period as a function of time for a simple pendulum that is in the form of a hollow sphere that is filled with mercury and there is a hole in the bottom through which the mercury is constantly falling at a fixed rate? I tried creating a function for the time period to check how it varies with the mass of mercury, but I had too many variables. EDIT: This question was a concept based multiple choice question in my homework book. The option were time period becomes erratic, time period increases, time period decreases, time period first increases then decreases. | I have this HW where I have to calculate the $74$th derivative of $f(x)=\ln(1+x)\arctan(x)$ at $x=0$. And it made me think, maybe I can say (about $\arctan(x)$ at $x=0$) that there is no limit for the second derivative, therefore, there are no derivatives of degree grater then $2$. Am I right? | eng_Latn | 18,686 |
$$ \begin{align}x^2&=x+x+\cdots\ (\text{$x$ times})\\ \implies \dfrac{\mathrm d}{\mathrm dx}(x^2)&=\dfrac{\mathrm d}{\mathrm dx}(x)+\dfrac{\mathrm d}{\mathrm dx}(x)+\cdots \ (\text{$x$ times})\\ \implies 2x&=1+ 1+\cdots \ (\text{$x$ times})\\ \implies 2x&= x \\ \implies 2&=1 \end{align} $$ Which is obviously wrong. So, where am I wrong? | Consider the following: $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$ Therefore, $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$ Take the derivative of lhs and rhs and we get: $\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}} = 2x$ Which simplifies to: $x = 2x$ and hence $1 = 2$. Clearly something is wrong but I am unable pinpoint my mistake. | I am able to solve simple differential equations like : $$\dfrac{dy}{dx} = 3x^2 + 2x$$ We simply bring $dx$ to other site and integrate. But how do we find solutions of differential equations like : $$\frac{d^2x}{dt^2} + \dfrac{kx}{m} = 0$$ ? We have been told the solution is $x(t) = A\cos ( \omega t + \phi_o)$ where $\omega\ =\sqrt{\dfrac{k}{m}}$ but how do we actually find it? | eng_Latn | 18,687 |
$$x = y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^2$$ I think it can be converted into a Clairaut. form but not able to do it | I've recently been learning about differential equations, and my teacher has been giving some particularly difficult examples out for those of us who finish early. He gave us the following differential equation: $$x=y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^{2}$$ With the hint "Differentiate with respect to $y$, then let $\frac{dy}{dx}=p$". I've solved it after the lesson and would like someone to check whether or not I've done so correctly. Following the hint, $$\frac{dx}{dy}=\frac{1}{p}=p+y\frac{dp}{dy}-2p\frac{dp}{dy}$$ Which rearranges to $$p-\frac{1}{p}+y\frac{dp}{dy}=2p\frac{dp}{dy}$$ Multiplying through by $\frac{dy}{dp}$, $$\left(p-\frac{1}{p}\right)\frac{dy}{dp}+y=2p$$ Rearranging into the form $\frac{dy}{dp}+f(p)y=g(p)$, and writing $p-\frac{1}{p}=\frac{p^{2}-1}{p}$ $$\frac{dy}{dp}+\frac{p}{p^{2}-1}y=\frac{2p^{2}}{p^{2}-1}$$ Our integrating factor is $$e^{\int\frac{p}{p^{2}-1}dp}=e^{\frac{1}{2}\ln(p^{2}-1)}=\sqrt{p^{2}-1}$$ So we have $$\sqrt{p^{2}-1}\frac{dy}{dp}+\frac{p}{\sqrt{p^{2}-1}}y=\frac{2p^{2}}{\sqrt{p^{2}-1}}$$ The left hand side is, by design, $\frac{d}{dp}(y\sqrt{p^{2}-1})$, and the right hand side can be integrated as follows: $$\int \frac{2p^{2}}{\sqrt{p^{2}-1}}dp=\int\frac{2(p^{2}-1)+2}{\sqrt{p^{2}-1}}dp=\int 2\sqrt{p^{2}-1}dp+\int \frac{2}{\sqrt{p^{2}-1}} dp$$ These can both be solved using the substitution $p=\cosh(u)$, and gives $$y\sqrt{p^{2}-1}=p\sqrt{p^{2}+1}-\cosh^{-1}(p)+\alpha$$ Where $\alpha$ is an arbitrary constant. Dividing, we finally have $$y=p+\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}$$ Here I was stuck for a long time, until I realised that we also have the equation we began with: $x=py-p^{2}$, which we can solve as a quadratic in $p$ to also get $$p=\frac{y\pm \sqrt{y^{2}-4x}}{2} \implies p^{2}=\frac{y^{2}+y^{2}-4x \pm 2y\sqrt{y^{2}-4x}}{4}$$ Thus, $$y=\frac{y\pm \sqrt{y^{2}-4x}}{2}+\frac{\cosh^{-1}\left(\frac{y\pm \sqrt{y^{2}-4x}}{2}\right)+\alpha}{\sqrt{\frac{y^{2}-2x \pm y\sqrt{y^{2}-4x}}{2}-1}}$$ This is as good as I can do - we have a relation between $y$ and $x$ - am I done? Is this the correct answer? Is there an easier way? Thank you for your time. Following the suggestion by @Valentin $$\frac{dy}{dp}=1+\frac{\frac{1}{\sqrt{p^{2}-1}}\sqrt{p^{2}-1}-p\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}=1+p\frac{1-\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}$$ Hence, $$dx=\frac{dy}{p}=dp\left(\frac{1}{p}+\frac{1-\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}\right)$$ | Using the method of characteristics, find a solution to Burgers' equation \begin{cases} u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\ \qquad \qquad \, \, u=g & \text{on } \mathbb{R} \times\{t=0\} \end{cases} with the initial conditions $$g(x)=\begin{cases} 0 & \text{if }x < 0 \\ 1 & \text{if }0 \le x \le 1 \\ 0 & \text{if }x > 1 \end{cases}$$ First, I realized that the equation $u_t+\left(\frac{u^2}{2} \right)_x =0$ is equivalent to this form: $$u_t+uu_x =0$$ Then should I generally follow the method of solution as outlined in the answer of ? Note that this is not a duplicate question of that page. Rather I want to know if that page can be used for my problem, even though my ICs are different. By the way, the solution printed in my book (PDE Evans, 2nd edition, page 142) is $$u(x,t) = \begin{cases} 0 & \text{if } x < 0 \\ \frac xt & \text{if } 0 < x < t \\ 1 & \text{if } t < x < 1 + \frac t2 \\ 0 & \text{if } x > 1 + \frac t2 \tag{$0 \le t \le 2$} \end{cases}$$ | eng_Latn | 18,688 |
In all my practice I have simply taken as given that $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}$$ When does this not hold? | I've taken multivariate calculus and am wondering if I can see a specific function where the order of taking the partial derivative matters. I've been told that there are some exceptions where $ \dfrac{\partial ^2 f}{\partial x \partial y} \ne \dfrac{\partial ^2 f}{\partial y \partial x} $, so I'm curious to see what this looks like. EDIT: And why would this true? | I am trying to find the equation of characteristic curves and solution of the partial differential equation $$x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=u$$ $u(x,0)=\sin(\frac{\pi}{4}x)$ . What I did: $$\frac{dx}{-y}=\frac{dy}{x}=\frac{du}{u}$$ with From first two equalities $$\frac{dx}{-y}=\frac{dy}{x}$$ by solving we get $x^2+y^2=c_1$ for some constant $c_1$. Now I am stuck with the third and rest of the equalities to come up with a suitable manipulation to solve the problem. How can I do this? Any help would be great. Thanks. | eng_Latn | 18,689 |
I have that convergence requires that the following be true $$ \forall\epsilon>0\exists N: n\ge N:|a_n-a|<\epsilon $$ for some sequence $\{a_n\}_{n\in\mathbb{N}}$ that converges to $a$ for $n\to\infty$. I want to use the reverse of this requirement. I do not know how to express it but I am guessing $$ \exists\epsilon>0\forall N:n\ge N:|a_n-a|\ge\epsilon $$ it is expressed like the above. Can you help me out? | My task is to write a precise mathematical statement that "the sequence $(a_n)$ does not converge to a number $\mathscr l$" So, I have my definition of a convergent sequence: "$\forall\varepsilon>0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|<\varepsilon$ $\forall n \in \Bbb N$ with $n>N$" Would the correct negation of this be "$\forall\varepsilon>0$ $\exists N\in\Bbb R$ such that $|x_n -\mathscr l|>\varepsilon$ $\forall n \in \Bbb N$ with $n>N$"? It doesn't seem that this is the answer as the next part of my task is to prove that a sequence is divergent using my formed proof, but it'd be difficult to do since it's a general proof of divergence and not just a proof that $(a_n)$ doesn't converge a specific number $\mathscr l$ Perhaps I should find a prove that $(a_n)$ tends to $\pm\infty$? This is more simple but it does not include monotone sequences such as $x_n:=(-1)^n$. Can someone assist me with this task? All comments and answers are appreciated. | How do I solve the following second order partial differential equation? $4u_{xx}+5u_{xy}+u_{yy}+u_x+u_y=2$ I have classified the equation to be hyperbolic and changed variables to obtain the canonical form as $u_{\epsilon\nu}={1\over{3u_\nu}}-{8\over9}$ which I believe is correct but I am struggling to find the general solution? A step by step solution would be appreciated. | eng_Latn | 18,690 |
If $f\left( x \right) = x\cos \frac{1}{x},x \ge 1$. Then which of the following is the correct option for the domain $x \ge 1$ (A) $f(x+2)-f(x)<2$ (B) $f(x+2)-f(x)>2$ My approach is as follow $T\left( x \right) = f\left( {x + 2} \right) - f\left( x \right) = \left( {x + 2} \right)\cos \frac{1}{{x + 2}} - x\cos \frac{1}{x}$ $T\left( x \right) = x\left( {\cos \frac{1}{{x + 2}} - \cos \frac{1}{x}} \right) + 2\cos \frac{1}{{x + 2}} \Rightarrow T\left( x \right) = x\left( {2\sin \left( {\frac{{\frac{1}{{x + 2}} + \frac{1}{x}}}{2}} \right)\sin \left( {\frac{{\frac{1}{x} - \frac{1}{{x + 2}}}}{2}} \right)} \right) + 2\cos \frac{1}{{x + 2}}$ $T\left( x \right) = x\left( {2\sin \left( {\frac{{2x + 2}}{{2x\left( {x + 2} \right)}}} \right)\sin \left( {\frac{2}{{2x\left( {x + 2} \right)}}} \right)} \right) + 2\cos \frac{1}{{x + 2}} \Rightarrow T\left( x \right) = x\left( {2\sin \left( {\frac{{x + 1}}{{x\left( {x + 2} \right)}}} \right)\sin \left( {\frac{1}{{x\left( {x + 2} \right)}}} \right)} \right) + 2\cos \frac{1}{{x + 2}}$ $T\left( x \right) = 2\left( {\left( x{\sin \left( {\frac{{x + 1}}{{x\left( {x + 2} \right)}}} \right)\sin \left( {\frac{1}{{x\left( {x + 2} \right)}}} \right)} \right) + \cos \frac{1}{{x + 2}}} \right)$ From here onward how do I proceed | I have this function $$f(x) = x\cos(1/x)$$ defined for all $x \geq 1$. How do I prove that $$f(x+2)-f(x)>2$$ for all $x\geq 1$? I found $$f'(x)= \cos(1/x) + \frac{\sin(1/x)}{x}$$ and $$f''(x)=-\frac{\cos(1/x)}{x^3}.$$ How do i use this information? | I think I have found the final answer $\left(4 \left[\frac{2t \cosh(4t) - \sinh(4t)}{t^2}\right]\right)$ but need to verify it. | yue_Hant | 18,691 |
$$x^2+y^2=\frac{dy}{dx}$$ Originally, this is a question I asked on Quora. Maybe this question has been solved but I was wondering if it can be done by integrating factor? $$x^2=\frac{dy}{dx}-y^2$$ Using method of integrating factor $$\text{I.F}= e^{\int-ydx}$$ $$\text{I.F}= e^{-yx}$$ Can it be done in this way? | $$y'=x^2+y^2$$ I know, that this is a kind of Riccati equation, but is it possible to solve it with only simple methods? Thank you | I am having trouble finding the derivative of the following: $$y = x^x+x^3+3^x+3^3$$ $$\frac{dy}{dx}= x \times x^{(x-1)}+3x^2+3^x\ln3+0$$ I think the $x^x$ part is wrong. Any help would be appreciated. | eng_Latn | 18,692 |
In a few of the kinematic equations there is a $2$ or a $0.5$ coefficient. Why is this? For example the kinematic equation for distance is: $$\text{previous velocity} * \text{time} + \frac{1}{2} * \text{acceleration} * \text{time}^2$$ But why the $\frac{1}{2}$? If I use the equation for acceleration to get to that equation I don't have a $\frac{1}{2}$? Here: $$\frac{\Delta v}{t} = a | \cdot t$$ $$v_{new} - v_{old} = a \cdot t | + v_{old}$$ $$\frac{s}{t} = a \cdot t + v_{old} | \cdot t$$ $$s = a \cdot t^2 + v_{old} \cdot t$$ Are my calculations wrong? If so, could someone please show me where I went wrong or explain to me how the $\frac{1}{2}$ comes into play? | I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? | Let $F$ be a field and $x\in F$. If $x=-x$ and $1\neq -1$, then $$0=\frac{x+x}{1+1}=\frac{1+1}{1+1}x=x.$$ This means that the statement in the title is true if and only if $1\neq -1$. But how do we know that $1\neq -1$ for an arbitary field? | eng_Latn | 18,693 |
If $m$ varies then find the range of $c$ for which the line $y=mx + c$ touches the parabola $y^2 = 8(x+2)$ . My Attempt: Put the value $y = mx + c$ in the parabola equation and then done $\Delta = 0$ or $\Delta >0$ I am getting $16/(m^2 + 8m) >0$ But in this, how do I neglect $m$? | if m varies then find the range of c for which the line $y=mx + c$ touches the parabola $y^2 = 8(x+2)$ . I tried Put the value $y = mx + c in$ the parabola equation and then done $D = 0$ or $D>0 $ I am getting $16/(m^2 + 8m) >0 $ But in this , how do I neglect m | Been trying to this line to parse for a good 20mins, but all attempts have failed: \beta = $-$0.84, t = $-$2.09, P = 0.04 Could anyone tell me why I'm getting a Missing $ inserted error here? | eng_Latn | 18,694 |
How can I solve the following second order differential equation $$ \frac{d^2\theta \:}{dt^2\:}\:=-\frac{g\theta \:\:}{L} $$ with initial conditions $$θ(0) = θ_0 , v(0) = v_0 $$ I know that I can rewrite the equation as $$ \frac{d^2\theta \:}{dt^2\:}\:+\frac{g\theta \:\:}{L} = 0 $$ , find characteristic polynomial and solve the equation. I am just confused on how to solve this problem because of all the constants. Thanks for help in advance | The differential equation $$ {d^2\alpha \over dt^2} + {g \over L} \cdot \alpha = 0 $$ describes a 1-dimensional mathematical pendulum, where $\alpha $ is the angle, $ g = 9.82 $, and $ L = 0.2 $ is the lentgh of the string. What is the position of the pendulum after 1 second if the velocity at time equals zero is $ 0 $ m/s and the angle at time equals zero is $ {\pi \over 60} $ radians. To my understanding this is a homogenous equation, ergo $$ y'' + Cy = 0$$ And with the help of the characteristic equation I get the complex roots $ Ci \, $ and $ -Ci $. This is a farily simple differential equation, but I have a feeling my calculations so far are wrong (mainly because I didn't expect to see complex roots for this equation). | Let $F$ be a field and $x\in F$. If $x=-x$ and $1\neq -1$, then $$0=\frac{x+x}{1+1}=\frac{1+1}{1+1}x=x.$$ This means that the statement in the title is true if and only if $1\neq -1$. But how do we know that $1\neq -1$ for an arbitary field? | eng_Latn | 18,695 |
Two particles of same mass in a 2D frame collide with known initial (i) velocities. I would like to know the final (f) velocities of them after the collision. As in any other collision, momentum is conserved after the collision. Writing in components: $$ v_{x,1}^{i} + v_{x,2}^{i} = v_{x,1}^{f} + v_{x,2}^{f} $$ $$ v_{y,1}^{i} + v_{y,2}^{i} = v_{y,1}^{f} + v_{y,2}^{f} $$ The total energy (not the mechanical) is also conserved. K accounts for the thermal energy. $$ v_{x,1}^{2,i} + v_{y,1}^{2,i} + v_{x,2}^{2,i} + v_{y,2}^{2,i}= v_{x,1}^{2,f} + v_{y,1}^{2,f} + v_{x,2}^{2,f} + v_{y,2}^{2,f} + 2 \cdot K/m $$ I obtain with this 3 equations for 4 unknown quantities, the x and y components of the velocities of particle 1 and particle 2. How could this be solved?. What information should I add? I can only think of modelling the collision with a potential of some kind. | I am trying to calculate the final velocities of two equal mass 2-dimensional circles after an elastic collision. I have tried to figure it out using formulas I know from high school physics, but nothing seems to work. My known variables are: initial x and y velocities for each circle x and y distances between the centres of the circles at the time of the collision The variables I would like to solve for are: final x and y velocities for each circle Here are diagrams showing all the variables: | Find the general solution of $xy''-(1+x)y'+y=x^2$ knowing that the homogeneous equation has the following solution: $e^{ax}$, where $a$ is a parameter you have to find. I have found that $a=1$ or $a=1/x$. | eng_Latn | 18,696 |
Disclaimer - I'm not a mathematician, I'm a dirty physicist. My work often involves performing calculus on various things without thinking about what I'm doing too much (I leave the proof of various identities etc for the pure mathematicians to worry about). However I've often noticed that mathematicians get a little upset when I do tricks such as treating the differential $\frac{\text{d}y}{\text{d}x}$ as if it were a fraction. The simplest example I can think of is how I think about the chain rule: $$\frac{\text{d}y}{\text{d}x} = \frac{\text{d}y}{\text{d}u}\frac{\text{d}u}{\text{d}x}$$ In my head, I imagine the $\text{d}u$ terms cancelling, which is why this works. Indeed, this is how I'll explain to others how to use the chain rule when asked about it. My question is the following: Is is dangerous to think about differentials in this way? After all, one of the very first examples of calculus I've ever seen (back when I was a baby in high-school) was the derivative of $y=x^2$ calculated from first principles in the following way: \begin{align} y&=x^2\\ y+\delta y&=(x+\delta x)^2\\ y+\delta y&=x^2+2x\delta x+\delta x^2\\ \require{cancel}\cancel{x^2}+\delta y&=\cancel{x^2}+2x\delta x+\delta x^2\\ \delta y&=2x\delta x+\delta x^2\\ \frac{\delta y}{\delta x}&=2x+\delta x\\ \text{Now let }&\delta x\rightarrow0\text{ leaving}\\ \frac{\delta y}{\delta x}&=2x\\ \frac{\text{d}y}{\text{d}x}&=2x \end{align} And to me, this is just treating $\frac{\delta y}{\delta x}$ as a fraction. I know that technically you're doing $0/0$ if you think about it, but are there any examples where treating $\text{d}y/\text{d}x$ is really inappropriate? | In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x)dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential). Then if we rearrange we get $dy/dx$ which could be seen as a ratio. I wonder if the author says this because $dx$ is an independent variable, and $dy$ is a dependent variable, for $dy/dx$ to be a ratio both variables need to be independent.. maybe? | Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation arises out of following problem : A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed $u$. Without any delay, the dog starts with running with constant speed $v>u$ to catch the cat. Initially, $v$ is perpendicular to $u$ and $L$ is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of $v, u$ and $L$. See my solution below : Let initially dog be at $D$ and cat at $C$ and after time $dt$ they are at $D'$ and $C'$ respectively. Dog velocity is always pointing towards cat. Let $DA = dy, \;AD' = dx$ Let $CC'=udt,\;DD' = vdt$ as interval is very small so $DD'$ can be taken straight line. Also we have $\frac{DA}{DC}= \frac{AD'}{ CC'}$ using triangle property. $\frac{dy}{L}= \frac{dx}{udt}\\ dx = \frac{dy.udt}{L}$ $\sqrt{(dx)^2 + (dy)^2} = DD' = vdt \\ \sqrt{(\frac{dy.udt}{L})^2 + (dy)^2} = vdt $ Here $t$ varies from $0-T$, and $y$ varies from $0-L$. Now how to proceed? | eng_Latn | 18,697 |
I know that if we have a lagrangian such that $$ L'=L+\frac{d}{dt}(f(q,t))$$ then the equation of motion will be the same for $L$ or $L'$. But I would like to know if there is a proof of the opposite, ie : If $L$ and $L'$ describe the same motion, then $$ L'=L+\frac{d}{dt}(f(q,t))$$ I don't know how to prove it? | So I was reading this: and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any. So, if I have a Lagrangian and ADD an arbitrary function of $\dot{q}$, $q$ and $t$ in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative? EDIT Ok, I changed my question a little bit: Question: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? (This was used in Qmechanic's answer of this other question: , equation 14.) Also, why people only talk about things that change the Lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau&Lifshitz's mechanics book only consider this kind of change in the Lagrangian? | Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation arises out of following problem : A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed $u$. Without any delay, the dog starts with running with constant speed $v>u$ to catch the cat. Initially, $v$ is perpendicular to $u$ and $L$ is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of $v, u$ and $L$. See my solution below : Let initially dog be at $D$ and cat at $C$ and after time $dt$ they are at $D'$ and $C'$ respectively. Dog velocity is always pointing towards cat. Let $DA = dy, \;AD' = dx$ Let $CC'=udt,\;DD' = vdt$ as interval is very small so $DD'$ can be taken straight line. Also we have $\frac{DA}{DC}= \frac{AD'}{ CC'}$ using triangle property. $\frac{dy}{L}= \frac{dx}{udt}\\ dx = \frac{dy.udt}{L}$ $\sqrt{(dx)^2 + (dy)^2} = DD' = vdt \\ \sqrt{(\frac{dy.udt}{L})^2 + (dy)^2} = vdt $ Here $t$ varies from $0-T$, and $y$ varies from $0-L$. Now how to proceed? | eng_Latn | 18,698 |
$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}.$$ I don't understand partial derivative by "function" (e.g. $q_j$). $q$ can be displacement. Then $\dot{q}$ is velocity. Both can be represented in terms of $t$. So by eliminating $t$, $q$ can be represented in terms of $\dot{q}$ vice versa. Hence, all $q$ terms in $L$ can be replaced with $\dot{q}$ terms making RHS of the above equation $0$. What's wrong with this? | In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat them as independent variables? | Consider this equation : $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$ where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$. Now how to proceed ? This equation arises out of following problem : A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed $u$. Without any delay, the dog starts with running with constant speed $v>u$ to catch the cat. Initially, $v$ is perpendicular to $u$ and $L$ is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of $v, u$ and $L$. See my solution below : Let initially dog be at $D$ and cat at $C$ and after time $dt$ they are at $D'$ and $C'$ respectively. Dog velocity is always pointing towards cat. Let $DA = dy, \;AD' = dx$ Let $CC'=udt,\;DD' = vdt$ as interval is very small so $DD'$ can be taken straight line. Also we have $\frac{DA}{DC}= \frac{AD'}{ CC'}$ using triangle property. $\frac{dy}{L}= \frac{dx}{udt}\\ dx = \frac{dy.udt}{L}$ $\sqrt{(dx)^2 + (dy)^2} = DD' = vdt \\ \sqrt{(\frac{dy.udt}{L})^2 + (dy)^2} = vdt $ Here $t$ varies from $0-T$, and $y$ varies from $0-L$. Now how to proceed? | eng_Latn | 18,699 |
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