10E9solution to the elliptic Painlev equation

OF THE DISSERTATION On some nonlocal elliptic and parabolic equations by Tianling Jin Dissertation Director: YanYan Li We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem involving nonlocal conformally invariant operators. Regularity properties for solutions of some degenerate elliptic equations as well as a Liouville type theorem are established, and used in our blow up analysis. We also introduce a fractional Yamabe flow and show that on the conformal spheres (Sn, [gSn ]) it converges to the standard sphere up to a Mobius diffeomorphism. These arguments can be applied to obtain extinction profiles of solutions of some fractional porous medium equations, which are further used to improve a Sobolev inequality via a quantitative estimate of the remainder term.

In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area means finding the majority of points on the elliptic curve and minimizing the time to compute the solution in contrast with the necessary time to compute the inverse solution. We can compute the positive solution of PDE (partial differential equation) like oscillations of f(s)/s around the principal eigenvalue λ1 of -Δ in H 0 1 (Ω).Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves.

The Elliptic Gamma Function and SL(3, [formula omitted])⋉[formula omitted

We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms. The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented, thus arriving at a compact formulation. In general, a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case. The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations. The convergence of the scheme has been established using the matrix analysis, and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix. The difference scheme has been applied to test convection diffusion equation, steady state Burger’s equation, ocean model and a semi-linear elliptic equation. The computational results confirm the theoretical order and accuracy of the method.

Exact Jacobian elliptic function solutions and hyperbolic function solutions for Sawada–Kotere equation with variable coefficient

In this paper we consider binary cubic diophantine equations of every form and shape, solely subjected to the requirement that they represent elliptic curves defined over Q. How to deal with standard Weierstras equations is well understood, but comparatively little is known about elliptic equations of different type. We give a detailed analysis of the general situation and subsequently apply the elliptic logarithm method to a variety of unusual elliptic equations, notably to some equations directly related to Krawtchouk polynomials. 1991 Mathematics subject classification: 11D25, 11G05, 11Y50, 12D10

Film hole irrigation is a relatively low cost and high efficiency irrigation method, which can significantly improve the efficiency of agricultural water use. In order to establish the quantitative model of film hole irrigation between cumulative infiltration and the wetting body and the irrigation volume model of crops, the infiltration process and wetting body characteristics of four different soils (Xi’an silt loam, silt, silt loam and loam) were studied in laboratory experiments and numerical HYDRUS simulation experiments. The relationship between cumulative infiltration and wetting body radius was established using a mathematical method, and a crop irrigation volume model was proposed based on the root distribution and the required water content of different crops. The experimental results showed that the shape of the wetting body of film hole irrigation is approximately half of the rotating ellipsoid, and the curve shape of the wetting front can be expressed using an elliptic equation. From the center of the film hole to the surface of the wetting front, the soil water content of the wetting body gradually decreases, and the change rate of water content gradually increases, reaching its maximum value near the wetting front. Furthermore, the distribution of water content in the wetting body can be accurately expressed using an elliptic curve equation. The cumulative infiltration of film hole irrigation is proportional to the third power of the equivalent radius of the wetting body, and the equivalent radius is equal to the geometric mean of the horizontal and vertical migration distances of the wetting front. In addition, based on the distribution of crop roots and the demand of crop roots on soil water content, the irrigation model of crops was established. This study provides a theoretical basis for the calculation of the irrigation volume for film hole irrigation under the condition of experiment, and has a guiding significance for the field experiment and application of film hole irrigation in different crops in future.

For the transverse electric polarization case (TE) we present a treatment of the optical reflectivity and transmissivity of a slab whose dielectric coefficient is a real valued function of the light intensity. If this function is numerically integrable with respect to the light intensity, our treatment can serve as an algorithm for a numerical solution of the nonlinear wave equation. If the dielectric function is proportional to the intensity, an analytical solution of the cubic wave equation is given for the electric field strength and for the phase of the field in terms of Weierstrass' elliptic functions and first elliptic theta functions, respectively. Evaluating this solution by means of a computer algebra system yields the reflectivity, transmissivity and phase dependency on the incident field intensity and on parameters characteristic for the problem. Certain combinations of the parameters lead to bistable and multivalued behavior. The solution found is used to determine the relative extrema of the reflectivity and the critical values of the thickness and of the incident intensity. The results are a generalization of linear optics results. Application of the analysis to the cubic-quintic wave equation yields the general analytic solution which is used to detemine the reflectivity of a semi-infinite nonlinear medium.

We deal with an ant colony based routing model for wireless multi‐hop networks. Our model adopts an elliptic curve equation, which is beneficial to design pheromone dynamics for load balancing and packet delivery robustness. Due to the attribute of an elliptic curve equation, our model prevents the over‐utilization of a specific node, distinctively from conventional ant colony based schemes. Numerical simulations exhibit the characteristics of our model with respect to various parameters.

On complex soliton solutions, complex elliptic solutions and complex rational function solutions for the Sasa-Satsuma model equation with variable coefficients

Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations

We introduce the notion of a “good" solution of a fully nonlinear uniformly elliptic equation. It is proven that “good" solutions are equivalent to $L^p$-viscosity solutions of such equations. The main contribution of the paper is an explicit construction of elliptic equations with strong solutions that approximate any given fully nonlinear uniformly elliptic equation and its $L^p$-viscosity solution. The results also extend some results about “good" solutions of linear equations.

We report a new 9-point compact discretization of order two in y- and order four in x-directions, based on cubic spline approximation, for the solution of two-dimensional nonlinear elliptic partial differential equations of the form $$ A\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial x^{2} }} + B\left( {x,y} \right)\frac{{\partial^{2} u}}{{\partial y^{2} }} = f\left( {x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y}} \right),\left( {x,y} \right) \in\Omega $$ defined in the domain \( \Omega = \left\{ {\left( {x,y} \right):0 0 \) and \( B\left( {x,y} \right) > 0 \) in \( \Omega \). The corresponding Dirichlet boundary conditions are prescribed by $$ u\left( {x,y} \right) = \psi \left( {x,y} \right),\quad \left( {x,y} \right) \in \partial\Omega $$ The main spline relations are presented and incorporated into solution procedures for elliptic partial differential equations. Available numerical methods based on cubic spline approximations for the numerical solution of nonlinear elliptic equations are of second-order accurate. Although 9-point finite difference approximations of order four accurate for the solution of nonlinear elliptic differential equations are discussed in the past, but these methods require five evaluations of the function f. In this piece of work, using the same number of grid points and three evaluations of the function f, we have derived a new stable cubic spline method of order 2 in y- and order 4 in x-directions for the solution of nonlinear elliptic equation. However, for a fixed parameter (Δy/Δx 2), the proposed method behaves like a fourth order method. The accuracy of the proposed method is exhibited from the computed results. The proposed method is applicable to Poisson’s equation and two-dimensional Navier-Stokes’ equations of motion in polar coordinates, which is main highlight of the work. The convergence analysis of the proposed cubic spline approximation for the nonlinear elliptic equation is discussed and we have shown under appropriate conditions the proposed method converges. Some physical examples and their numerical results are provided to justify the advantages of the proposed method.

The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This book, reissued with a new Foreword, presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.

We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.