j To learn more, view ourPrivacy Policy. 1 = The action functional n ) 1 still occur in the Lagrangian, and a system of equations in n coordinates still has to be solved.[3]. t For eg. WebThe definition of Partial Differential Equations (PDE) is a differential equation that has many unknown functions along with their partial derivatives. {\displaystyle M} 0 ) Below are a few examples of linear differential equations. t = This defines a family of solutions of the PDE; so, we can choose (x, y, u) = x + uy, Example 2. / These equations are usually combined into a single equation. along the real axis or imaginary axis; in either case it should give the same result. u 0 = . Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if {G, H} = 0, then G is conserved and the symplectomorphisms are symmetry transformations. Suppose that In this work, we present our developments in the context of solving two main classes of problems: data-driven solution ) z = Repeated roots. {\displaystyle (u_{1},\dots ,u_{N})} WebFor vector partial differential equations, the basis functions may take values in . This system of equations first appeared in the work of Jean le Rond d'Alembert. Thanks to all authors for creating a page that has been read 2,415,027 times. For example, the equation below is one that we will discuss how to solve in this article. It is also customary to define a Hugoniot function:[17]. = This is done using a straightforward generalization of the Wirtinger derivative, where the function in question is required to have the (partial) Wirtinger derivative with respect to each complex variable vanish. We will take u to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the velocity vector of the fluid at each point of the plane is equal to the gradient of u, defined by. Neil S. Trudinger was born in Ballarat, Australia in 1942. n when the limit is taken at the origin. D {\displaystyle {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}. {\displaystyle \left\{{\begin{aligned}{D\mathbf {u} \over Dt}&=-\nabla w+{\frac {1}{\mathrm {Fr} }}{\hat {\mathbf {g} }}\\\nabla \cdot \mathbf {u} &=0\end{aligned}}\right.}. t The Hamiltonian vector field induces a Hamiltonian flow on the manifold. q r Two real and distinct roots. 2 {\displaystyle E=\gamma mc^{2}} to explain a circle there is a general equation: (x h). , and I denotes the identity matrix. = q or equivalently in mechanical variables, as: This parameter is always real according to the second law of thermodynamics. {\displaystyle p(x)\neq 0,\ q(x)\neq 0.} The Poisson bracket has the following properties: This is called Liouville's theorem. n Moreover, because the composition of a conformal transformation with another conformal transformation is also conformal, the composition of a solution of the CauchyRiemann equations with a conformal map must itself solve the CauchyRiemann equations. = q D y , f F ) i This effectively reduces the problem from n coordinates to (n 1) coordinates: this is the basis of symplectic reduction in geometry. m WebIn mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms.They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincar conjecture and the Calabi conjecture.They are difficult to + denotes the outer product. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. As a natural generalization of the fractional Schrdinger equation, the variable-order fractional Schrdinger equation has been exploited to study fractional quantum phenomena:[55]. Using differential equations Radioactive decay is calculated. WebThe CahnHilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. , , + There are CauchyRiemann equations, appropriately generalized, in the theory of several complex variables. {\displaystyle \nabla _{F}} M x is the right eigenvector of the matrix s 2, (at = 2, D2 = 1/2m for a particle of mass m), and the operator (2)/2 is the 3-dimensional fractional quantum Riesz derivative defined by. = p {\displaystyle t} and = We see that indeed the CauchyRiemann equations are satisfied, Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but important theoretical results can be derived from them, because coordinates and momenta are independent variables with nearly symmetric roles. ( , d It is a set of nonlinear differential equations that approximates the electrical characteristics of excitable cells such as neurons and muscle cells.It is a continuous-time dynamical system.. Alan ) + Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities ) , However, we already mentioned that for a thermodynamic fluid the equation for the total energy density is equivalent to the conservation equation: Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as: t L , Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. solving Laplace's equation in spherical coordinates. subscripts label the N-dimensional space components, and 0 The function 1 For the history of the subject, see the thesis (in French): Stphane Dugowson. We first integrate, We then take the partial derivative of our result with respect to, If our differential equation is not exact, then there are certain instances where we can find an integrating factor that makes it exact. u e If n=i{\displaystyle n_{\pm }=\alpha \pm \beta i} are the roots to the characteristic equation, then we obtain a complex function as our solution. ^ = z = . a After integrating it becomes y^3/3 -3y^2/2=x+c. p For z along the real line, Mathematical Methods for the Physical Sciences Two Semester Course. t Source terms like tanx{\displaystyle \tan x} and xn{\displaystyle x^{-n}} warrant the use of variation of parameters to find the particular solution. q z t + u q Integrate both sides. Enter the email address you signed up with and we'll email you a reset link. ) along which partial differential equations (PDEs) degenerate into ordinary differential equations (ODEs). , , H The existence of partial derivatives satisfying the CauchyRiemann equations there doesn't ensure complex differentiability: u and v must be real differentiable, which is a stronger condition than the existence of the partial derivatives, but in general, weaker than continuous differentiability. The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). 1 Also, the CauchyRiemann equations imply that the dot product The Hamiltonian can induce a symplectic structure on a smooth even-dimensional manifold M2n in several equivalent ways, the best known being the following:[6]. = v const F i {\displaystyle {\frac {\partial }{\partial t}}{\begin{pmatrix}\rho \\\mathbf {j} \\0\end{pmatrix}}+\nabla \cdot {\begin{pmatrix}\mathbf {j} \\{\frac {1}{\rho }}\,\mathbf {j} \otimes \mathbf {j} +p\mathbf {I} \\{\frac {\mathbf {j} }{\rho }}\end{pmatrix}}={\begin{pmatrix}0\\\mathbf {f} \\0\end{pmatrix}}}. The respective differential equation on s Communications in Mathematical Physics. {\displaystyle {\mathcal {L}}} b . y Partial Differential Equations Sheet II: D15b.pdf (224.6KB) D15b.ps (377.9KB) Thu 16 Oct 2014: D15c: Partial Differential Equations Sheet III: D15c.pdf (160.9KB) D15c.ps (297.5KB) Wed 26 Nov 2014: D15d: Partial Differential Equations Sheet IV: D15d.pdf (215.3KB) D15d.ps (382.2KB) Fri 28 Nov 2014: D15e: Additional can be measured experimentally whereas canonical momentum 1 0 See also Geodesics as Hamiltonian flows. g {\displaystyle {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} 1 = t ) In Clifford algebra the complex number 1 Here, homogeneous does not refer to homogeneous functions, but the fact that the equation is set to 0. P f If the symplectic manifold has dimension 2n and there are n functionally independent conserved quantities Gi which are in involution (i.e., {Gi, Gj} = 0), then the Hamiltonian is Liouville integrable. 0 j degrees of freedom, the Lagrangian mechanics defines the energy function, The inverse of the Legendre transform of We must then use reduction of order to find the second linearly independent solution. The repeated roots case will have to wait until the section on reduction of order.Two real and distinct roots. Variation of parameters is a more general method of solving inhomogeneous differential equations, particularly when the source term does not contain a finitely many number of linearly independent derivatives. The journal is intended to be accessible to a broad spectrum of researchers into numerical approximation of PDEs p p(x)=0. As a closed nondegenerate symplectic 2-form . A u Viewed as conjugate harmonic functions, the CauchyRiemann equations are a simple example of a Bcklund transform. + = If not, then the solution can be left in integral form. 1 It is used to represent many types of phenomenons like sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation, and quantum mechanics. The complex-valued function Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. Biography of Neil S. Trudinger. The total differential of the Lagrangian is: The term in parentheses on the left-hand side is just the Hamiltonian p E D Equations are considered to have infinite solutions. t r where {\displaystyle r} Differential equations relate a function with one or more of its derivatives. Conditions required of holomorphic (complex differentiable) functions, "CauchyRiemann" redirects here. D (canonical or symplectic coordinates) in which the symplectic form becomes: (In algebraic terms, one would say that the d t WebHistory. {\displaystyle p(x)=0.} ( We can then simply integrate to obtain our answer. p s , and . x t are isomorphic). . const Differential equations relate a function with one or more of its derivatives. p(x)0,q(x)0. + + p . Now consider the potential values of m ) The Hamiltonian satisfies, In phase space coordinates t b The only forces acting on the mass are the reaction from the sphere and gravity. where meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy. Webdevoted to this topic.1,2,3,4,5 CFD provides numerical approximation to the equations that govern fluid motion. v the above equality can be written as. There are 12 references cited in this article, which can be found at the bottom of the page. The index in the fractional Schrdinger equation is the Lvy index, 1 < 2. F The structure of this differential equation is such that each term is multiplied by a power term whose degree is equal to the order of the derivative. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not causal, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). , The polynomial being set to 0 is deemed the characteristic equation. D x Ryan Ph.d. P A R T A Ordinary Differential Equations (ODEs) Chap. i I [14] For n = 2, this system is equivalent to the standard CauchyRiemann equations of complex variables, and the solutions are holomorphic functions. in this case is a vector, and x Ordinary Differential Equations, 1923.But Bateman is more of a compendium of results, and in this. wikiHow is where trusted research and expert knowledge come together. m {\displaystyle C^{\infty }(M,\mathbb {R} )} In the following we show a very simple example of this solution procedure. , u corresponding to the eigenvalue D Repeated roots to the homogeneous differential equation with constant coefficients. {\displaystyle p} With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stressenergy tensor, and energy and momentum were likewise unified into a single concept, the energymomentum vector. u e d Solutions to the NavierStokes equations are used in many practical applications. Vect WebThe (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields as they occur in classical physics such as mechanical waves (e.g. 2 t v {\displaystyle q} H , The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called "the Hamiltonian mechanics" of the Hamiltonian system. the Hugoniot curve, whose shape strongly depends on the type of material considered. t Indeed, following Rudin,[5] suppose f is a complex function defined in an open set C. Then, writing z = x + iy for every z , one can also regard as an open subset of R2, and f as a function of two real variables x and y, which maps R2 to C. We consider the CauchyRiemann equations at z = z0. N In regions where the state vector y varies smoothly, the equations in conservative form can be put in quasilinear form: where called conservative methods.[1]. z j ) V n On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains: and by defining the specific total enthalpy: one arrives to the CroccoVazsonyi form[15] (Crocco, 1937) of the Euler momentum equation: In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form: by defining the specific total Gibbs free energy: From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow. ) = {\displaystyle \mathbf {p} } s In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows.[7]. Select a standard coordinate system (, ) on . For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. 2 WebBasic assumptions. In this article, we show the techniques required to solve certain types of ordinary differential equations whose solutions can be written out in terms of elementary functions polynomials, exponentials, logarithms, and trigonometric functions and their inverses. be the distance from the center of curvature of the streamline, then the second equation is written as follows: where {\displaystyle {\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}} The + , the other in terms of ( ) Fluid dynamics has a wide range of applications, including calculating d We will see in the next section on how to solve the corresponding. To create this article, 35 people, some anonymous, worked to edit and improve it over time. This has the advantage that kinetic momentum n Conversely, if f: C C is a function which is differentiable when regarded as a function on R2, then f is complex differentiable if and only if the CauchyRiemann equations hold. z It is possible to construct a continuous function satisfying the CauchyRiemann equations at a point, but which is not analytic at the point (e.g., f(z) = z5/|z|4). R y Lamb in his famous classical book Hydrodynamics (1895), still in print, used this identity to change the convective term of the flow velocity in rotational form:[13]. Because this is a second-order equation, we know that this is, A useful way to check if two solutions are linearly independent is by way of the, In terms of linear algebra, the solution set of this differential equation spans a vector space with dimension equal to the order of the differential equation. , = 1 1 {\displaystyle \nabla f=0} There are many ways to choose these n solutions, but we are certain that there cannot be more than n of them. We choose as right eigenvector: The other two eigenvectors can be found with analogous procedure as: Finally it becomes apparent that the real parameter a previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. x I 0 is the number density of the material. Because this differential equation is a linear equation, the general solution consists of a linear combination of the individual solutions. Learn more A differential equation is an equation that relates a function with one or more of its derivatives. q u Now, consider dds (x + uy) = 1y dds(x + u) x + uy2 dyds , = x + uy x + uy = 0. satisfies the CauchyRiemann equations. m y n j = 0 is the specific entropy, q {\displaystyle f:\Omega \to \mathbb {R} ^{n}} here is considered a constant (polytropic process), and can be shown to correspond to the heat capacity ratio. 1 ) Well, equations are used in 3 fields of mathematics and they are: Equations are used in geometry to describe geometric shapes. 1.1 Basic Concepts. + Various types of finite element methods AEM. t In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. In the following we list some very simple equations of state and the corresponding influence on Euler equations. [1], The Euler equations can be applied to incompressible or compressible flow. WebGet 247 customer support help when you place a homework help service order with us. to be a conformal mapping (that is, angle-preserving) is that. {\displaystyle {\boldsymbol {q}}:[a,b]\to M} In fact the tensor A is always diagonalizable. {\displaystyle \mathbf {F} } The integration introduces an arbitrary constant on both sides, but we may consolidate them on the right side. If one expands the material derivative the equations above are: Coming back to the incompressible case, it now becomes apparent that the incompressible constraint typical of the former cases actually is a particular form valid for incompressible flows of the energy equation, and not of the mass equation. By differentiating the CauchyRiemann equations a second time, one shows that u solves Laplace's equation: The function v also satisfies the Laplace equation, by a similar analysis. Then the Euler momentum equation in the steady incompressible case becomes: The convenience of defining the total head for an inviscid liquid flow is now apparent: That is, the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant. P ^ p ) {\displaystyle \mathbf {y} } e The function H is known as "the Hamiltonian" or "the energy function." d {\displaystyle y.} 2 u u Hamiltonian systems can be generalized in various ways. A J We wish to find a function (x,y),{\displaystyle \varphi (x,y),} called the potential function, such that ddx=0. The original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. Algorithms are used as specifications for performing calculations and data processing.More advanced algorithms can perform automated deductions (referred t This means that, in complex analysis, a function that is complex-differentiable in a whole domain (holomorphic) is the same as an analytic function. Characteristic equation. / , {\displaystyle e} The hypothesis that f obey the CauchyRiemann equations throughout the domain is essential. Considering the first equation, variable must be changed from density to specific volume. Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate ) If F and G are smooth functions on M then the smooth function 2(IdG, IdF) is properly defined; it is called a Poisson bracket of functions F and G and is denoted {F, G}. q ( 2 In most applications, the functions represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between them. t In the case where the cometric is degenerate at every point q of the configuration space manifold Q, so that the rank of the cometric is less than the dimension of the manifold Q, one has a sub-Riemannian manifold. X This equation tells us that y{\displaystyle y} and its derivatives are all proportional to each other. D For an ideal polytropic gas the fundamental equation of state is:[19]. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. This is in fact a special case of a more general result on the regularity of solutions of hypoelliptic partial differential equations. If a differential equation has only one independent variable then it is called an ordinary differential equation. An equation is a statement in which the values of the mathematical expressions are equal. Holomorphy is the property of a complex function of being differentiable at every point of an open and connected subset of C (this is called a domain in C). d , which can be calculated in the following way: Grouping by i Combining the characteristic and compatibility equations, dxds = y + u, (2.11), dyds = y, (2.12), duds = x y (2.13). q (i) Equations of First Order/ Linear Partial Differential Equations, (ii) Linear Equations of Second Order Partial Differential Equations. and the smooth Lagrangian Holomorphic functions are analytic and vice versa. wikiHow is a wiki, similar to Wikipedia, which means that many of our articles are co-written by multiple authors. turns ( The LiouvilleArnold theorem says that, locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism into a new Hamiltonian with the conserved quantities Gi as coordinates; the new coordinates are called action-angle coordinates. N Now consider the molar heat capacity associated to a process x: according to the first law of thermodynamics: Now inverting the equation for temperature T(e) we deduce that for an ideal polytropic gas the isochoric heat capacity is a constant: and similarly for an ideal polytropic gas the isobaric heat capacity results constant: This brings to two important relations between heat capacities: the constant gamma actually represents the heat capacity ratio in the ideal polytropic gas: and one also arrives to the Meyer's relation: The specific energy is then, by inverting the relation T(e): The specific enthalpy results by substitution of the latter and of the ideal gas law: From this equation one can derive the equation for pressure by its thermodynamic definition: By inverting it one arrives to the mechanical equation of state: Then for an ideal gas the compressible Euler equations can be simply expressed in the mechanical or primitive variables specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. WebOverview Phase space coordinates (p,q) and Hamiltonian H. Let (,) be a mechanical system with the configuration space and the smooth Lagrangian . + v ) Sorry, preview is currently unavailable. n The equation for an orientation-preserving mapping d2ydx2+p(x)dydx+q(x)y=0. L = which satisfies the CauchyRiemann equations everywhere, but fails to be continuous at z=0. WebIn continuum mechanics and thermodynamics, a control volume (CV) is a mathematical abstraction employed in the process of creating mathematical models of physical processes. i Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. In quantum mechanics, the wave function will also undergo a local U(1) group transformation[5] during the Gauge Transformation, which implies that all physical results must be invariant under local U(1) transformations. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory. in joules) through multiplication with the Boltzmann constant. is differentiable at any point z in the complex plane. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. + j 2 p ^ , yielding: One may now equate these two expressions for , where , obeying Lagrange's equations: Rearranging and writing in terms of the on-shell : Since these calculations are off-shell, one can equate the respective coefficients of u WebAn invariant set is a set that evolves to itself under the dynamics. {\displaystyle {\frac {\mathrm {d} \varphi }{\mathrm {d} x}}=0.}. {\displaystyle \left\{{\begin{aligned}{Dv \over Dt}&=v\nabla \cdot \mathbf {u} \\[1.2ex]{\frac {D\mathbf {u} }{Dt}}&=v\nabla p+\mathbf {g} \\[1.2ex]{Dp \over Dt}&=-\gamma p\nabla \cdot \mathbf {u} \end{aligned}}\right.}. , The quantities This implies that the gradient of u must point along the t The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. e {\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x}}={\frac {g(x,y)}{h(x,y)}}.} Therefore, we will put forth an ansatz an educated guess on what the solution will be. n = is the specific volume, The first equation is nonlinear because of the sine term. Reduction of order applies if we know a solution y1(x){\displaystyle y_{1}(x)} to this equation, whether found by chance or given in a problem. ) i This equation tells us that an exponential function multiplied by a polynomial must equal 0. , def In the equation, X is the independent variable. By expanding the material derivative, the equations become: In fact for a flow with uniform density The solution of the initial value problem in terms of characteristic variables is finally very simple. S.D. N d b q(x)=0. d u Spherical coordinates are used to describe the position of the mass in terms of (r, , ), where r is fixed, r = l. Hamilton's equations can be derived by a calculation with the Lagrangian ( v V i - Example, Formula, Solved Examples, and FAQs, Line Graphs - Definition, Solved Examples and Practice Problems, Cauchys Mean Value Theorem: Introduction, History and Solved Examples. At the points where t + The following dimensionless variables are thus obtained: Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at apix): { However, these equations are even harder to find applications of in the sciences, and integrating factors, though. const {\displaystyle {\mathcal {H}}} t cannot. i The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field. In this example, the time derivative of q is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum. { {\displaystyle (N+2)N} i Conversely, if f : C C is a function which is differentiable when regarded as a function on R 2, then f is complex differentiable if and only if the CauchyRiemann equations hold. {\displaystyle \lambda _{i}} {\displaystyle q^{i}} q Check a table of integrals to verify. 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