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Ezra T Newman - One of the best experts on this subject based on the ideXlab platform.

  • Differential geometry from Differential Equations
    Communications in Mathematical Physics, 2001
    Co-Authors: Simonetta Frittelli, Carlos N Kozameh, Ezra T Newman
    Abstract:

    We first show how, from the general 3rd order ODE of the form \(\), one can construct a natural Lorentzian conformal metric on the four-dimensional space \(\). When the function \(\) satisfies a special Differential condition the conformal metric possesses a conformal Killing field, \(\), which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space \(\)) or equivalently, on the space of solutions of the Original Differential Equation. This construction is then generalized to the pair of Differential Equations, zss=S(z,zs,zt,zst,s,t) and ztt=T(z,zs,zt,zst,s,t), with zs and zt the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,zs,zt,zst,s,t). When the S and T satisfy Differential conditions analogous to those of the 3rd order ode, the 6-space then possesses a pair of conformal Killing fields, \(\) and \(\) which allows, via the mapping to the four-space of (z,zs,zt,zst) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein Equations.

Simonetta Frittelli - One of the best experts on this subject based on the ideXlab platform.

  • Differential geometry from Differential Equations
    Communications in Mathematical Physics, 2001
    Co-Authors: Simonetta Frittelli, Carlos N Kozameh, Ezra T Newman
    Abstract:

    We first show how, from the general 3rd order ODE of the form \(\), one can construct a natural Lorentzian conformal metric on the four-dimensional space \(\). When the function \(\) satisfies a special Differential condition the conformal metric possesses a conformal Killing field, \(\), which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space \(\)) or equivalently, on the space of solutions of the Original Differential Equation. This construction is then generalized to the pair of Differential Equations, zss=S(z,zs,zt,zst,s,t) and ztt=T(z,zs,zt,zst,s,t), with zs and zt the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,zs,zt,zst,s,t). When the S and T satisfy Differential conditions analogous to those of the 3rd order ode, the 6-space then possesses a pair of conformal Killing fields, \(\) and \(\) which allows, via the mapping to the four-space of (z,zs,zt,zst) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein Equations.

Carlos N Kozameh - One of the best experts on this subject based on the ideXlab platform.

  • Differential geometry from Differential Equations
    Communications in Mathematical Physics, 2001
    Co-Authors: Simonetta Frittelli, Carlos N Kozameh, Ezra T Newman
    Abstract:

    We first show how, from the general 3rd order ODE of the form \(\), one can construct a natural Lorentzian conformal metric on the four-dimensional space \(\). When the function \(\) satisfies a special Differential condition the conformal metric possesses a conformal Killing field, \(\), which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space \(\)) or equivalently, on the space of solutions of the Original Differential Equation. This construction is then generalized to the pair of Differential Equations, zss=S(z,zs,zt,zst,s,t) and ztt=T(z,zs,zt,zst,s,t), with zs and zt the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,zs,zt,zst,s,t). When the S and T satisfy Differential conditions analogous to those of the 3rd order ode, the 6-space then possesses a pair of conformal Killing fields, \(\) and \(\) which allows, via the mapping to the four-space of (z,zs,zt,zst) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein Equations.

Jaegers, Peter James - One of the best experts on this subject based on the ideXlab platform.

  • Lie group invariant finite-difference schemes for the neutron diffusion Equation
    2026
    Co-Authors: Jaegers, Peter James
    Abstract:

    Finite difference techniques are used to solve a variety of Differential Equations. For the neutron diffusion Equation, the typical local truncation error for standard finite difference approximation is on the order of the mesh spacing squared. To improve the accuracy of the finite difference approximation of the diffusion Equation, the invariance properties of the Original Differential Equation have been incorporated into the finite difference Equations. Using the concept of an invariant difference operator, the invariant difference approximations of the multi-group neutron diffusion Equation were determined in one-dimensional slab and two-dimensional Cartesian coordinates, for multiple region problems. These invariant difference Equations were defined to lie upon a cell edged mesh as opposed to the standard difference Equations, which lie upon a cell centered mesh. Results for a variety of source approximations showed that the invariant difference Equations were able to determine the eigenvalue with greater accuracy, for a given mesh spacing, than the standard difference approximation. The local truncation errors for these invariant difference schemes were found to be highly dependent upon the source approximation used, and the type of source distribution played a greater role in determining the accuracy of the invariant difference scheme than the local truncation error.U of I OnlyETDs are only available to UIUC Users without author permissio

Y N Reddy - One of the best experts on this subject based on the ideXlab platform.