The unitary operator for the free-evolution part is defined as Û0(−∞, t) ≡ T exp{−(i/~) ∫ t −∞ dt Ĥdet(t)} with T being the time-ordering, and, for the interaction part, we have defined ÛI(−∞, t) ≡ T exp{−(i/~) ∫ t −∞ dt ′Ĥ int (t ′)}. For the measurement to be linear, Ĥdet only involves linear or quadratic functions of canonical coordinates, among which their commutators are classical numbers,...

We consider the iterative resolution scheme for the Navier-Stokes equation, and focus on the second iterate, more precisely on the map from the initial data to the second iterate at a given time t. We investigate boundedness properties of this bilinear operator. This new approach yields very interesting results: a new perspective on Koch-Tataru solutions; a first step towards weak strong unique...

Projected-shear nite element methods for periodic Reissner-Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L 2-norm, H 1-norm, and the energy norm for both displacement and rotations are established and gradient superconvergence along the Gauss lines is justiied in some weak s...

We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the...

An operator formalism for bosonic system on arbitrary algebraic curves is introduced. The classical degrees of freedom are identi ed and their commutation relations are postulated. The explicit realization of the algebra formed by the elds is given in the Hilbert space equipped with a bilinear form. The construction is based on the "gaussian" representation for system on the complex sphere [Alv...

Gelfand and Dikii gave a bosonic formal variational calculus in [5, 6] and Xu gave a fermionic formal variational calculus in [13]. Combining the bosonic theory of Gelfand-Dikii and the fermionic theory, Xu gave in [14] a formal variational calculus of super-variables. Fermionic Novikov algebras are related to the Hamiltonian super-operator in terms of this theory. A fermionic Novikov algebra i...

We propose a blind source separation model for statistically independent source signals, when the mixing operator is bilinear. This model is equivalent to a linear model for separation of pairwise multiplications the source signals. We prove that if the source signals are are colored and have distinct autocorrelation functions on a given set P , then we can extract simultaneously the pairwise m...

We provide sufficient normal curvature conditions on the boundary of a domain D ⊂ R to guarantee unboundedness of the bilinear Fourier multiplier operator TD with symbol χD outside the local L 2 setting, i.e. from L1 (R) × L2 (R) → L ′ 3 (R) with P 1 pj = 1 and pj < 2 for some j. In particular, these curvature conditions are satisfied by any domain D that is locally strictly convex at a single ...

In the present paper, a linear parabolic evolution equation is considered whose bilinear form is controlled from a general Banach space. The control-to-state operator and some important properties thereof are presented. For a quadratic objective function, the gradient in the control space is derived. A-posteriori error estimators are presented for a given reduced-order model (ROM) with respect ...