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On the Difference S(Z(n)) - Z(S(n)
In this paper, we prove that there exist infmitely many positive integers
n satisfying S(Z(n))> Z(S(n)) or S(Z(n)) < Z(S(n))
Differential Geometry from Differential Equations
We first show how, from the general 3rd order ODE of the form
z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on
the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s)
satisfies a special differential condition of the form, U[F]=0, the conformal
metric possesses a conformal Killing field, xi = partial with respect to s,
which in turn, allows the conformal metric to be mapped into a three
dimensional Lorentzian metric on the space (z,z',z'') or equivalently, on the
space of solutions of the original differential equation. This construction is
then generalized to the pair of differential equations, z_ss =
S(z,z_s,z_t,z_st,s,t) and z_tt = T(z,z_s,z_t,z_st,s,t), with z_s and z_t, 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,z_s,z_t,z_st,s,t). When the S and T satisfy equations
analogous to U[F]=0, namely equations of the form M[S,T]=0, the 6-space then
possesses a pair of conformal Killing fields, xi =partial with respect to s and
eta =partial with respect to t which allows, via the mapping to the four-space
of z, z_s, z_t, z_st 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.Comment: 37 pages, revised version with clarification
Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory
This article is devoted to the optimal control of state equations with memory
of the form: ?[x(t) = F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) ds), t>0, with
initial conditions x(0)=x, x(-s)=z(s), s>0.]Denoting by the
solution of the previous Cauchy problem and: where is a
class of admissible controls, we prove that is the only viscosity solution
of an Hamilton-Jacobi-Bellman equation of the form: in the sense of the
theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L.
Lions
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