An alternative point of view is to consider the functional (varrho mapsto int phi,mathrm{d}varrho +int phi ^c,mathrm{d}nuu ) for fixed (phi ), in which case the first variation is of course (phi ); then it is easy to see that the first variation of the supremum may be obtained (in case of uniqueness) just by selecting the optimal (phi ).
Warning: this definition is not equivalent to true convexity along the geodesic, since we only compare intermediate instants t to 0 and 1, and not to other intermediate instants; however, in case of uniqueness of geodesics, or if we required the same condition to be true for all geodesics, then we would recover the same condition.
In case of uniqueness, the reconstruction of a patient's genome would be possible.
The idea consists of using two steps: in the first step the optimal values of the ℓ1 and ℓ∞ errors are calculated, and in the second step, in case of non-uniqueness of solutions, one of the multiple solutions is selected according to a different criterion.
In the case of uniqueness, both limits coincide with this solution.
We reprove in an extremely simple way the classical theorem that time periodic dissipative systems imply the existence of harmonic periodic solutions, in the case of uniqueness.
In the trivial case of uniqueness, (T_{omega}colon mathbb {R}to mathbb {R}) must be strictly increasing (otherwise, we get a contradiction) by which no purely subharmonic (i.e. those with (n>1)) nω-periodic solutions can exist.
We also provide some new insights toward the uniqueness of gait in case of foot motion.
whatever is (nu ) (this is typically true when (f'(0)=-infty ), as it is the case for the entropy functional, and more generally in case of linear or fast diffusion), which provides uniqueness up to additive constants of the potential.
The new flexible model may be written: (4) where, in case of K covariates, b1 t) = 0 and β1 k = 0, k = 1,..., K for uniqueness.
Thanks to these theories, several results concerning attractors in the case of equations without uniqueness have been obtained recently for differential inclusion [5, 6], parabolic equations [8 10], the phase-field equation [11], the wave equation [12], the three-dimensional Navier-Stokes equation [3, 13], etc.
