Remarks on Artin approximation with constraints

This paper is devoted to the global conservative solutions of a dissipative Camassa-Holm type equation with cubic and quartic nonlinearities. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solutions of the semilinear system. Returning to the original variables, we get the global conservative solution of the equation. Finally, we show that the peakon solutions of the equation still conserve in .

THE purpose of this paper is to describe a short method for solving arbitrary systems of linear algebraic equations, and evaluating determinants, the quantities involved being either real or complex. For proofs and for treatment of complex systems whose symmetrical coefficients are conjugate, see appendix. The cases considered are: 1. Arbitrary systems with real coefficients, which occur in obtaining stresses in structures, in solving systems of linear differential equations with constant coefficients (transient problems), etc. 2. Symmetrical systems with real coefficients, which occur with d-c networks, undamped vibration, deflections in structures, least square processes, Ritz's method, etc. 3. Symmetrical systems with complex coefficients, which occur with a-c networks, and forced vibration with dissipation. 4. Arbitrary systems with complex coefficients, which occur in certain vibration problems involving gyroscopic action. 5. Systems involving two sets of variables, which occur when the currents in a network are to be found for a variety of impressed voltages, also in the approximate solution of integral equations arising in electric field problems.

Modular kinematics is considered by several authors a valuable tool in robot simulation. The paper describes how to verify if a given set of kinematic variables that allow a modular solution of the constraint equations of a robot can form a set of independent coordinates to be used in dynamic analysis. Then, some strategies to select a topologically valid set are provided. The proposed method is applied to a planar robot example.

UDC 517.9, 533.7, 533.723 Our study concerns the existence of a global solution to the discrete kinetic equations in Sobolev spaces. We obtain a decomposition of the solution and clarify the influence of the oscillations generated by the interaction operator. We show that there exists a submanifold Mdiss of initial data for which dissipative solution exit. If initial data u 0 , v 0 , w 0 deviate from the submanifold Mdiss, then the interaction operator generates solitons, the nondissipative part of the solution. The amplitude of solitons is proportional to the distance from u 0 , v 0 , w 0 to the submanifold Mdiss, which means that the solution can stabilize as t →∞ only on compact sets of spatial variables. Bibliography :6 titles.

Propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach

The paper establishes the basic algebraic theory for the Gevrey rings. We prove the Hensel lemma, the Artin approximation theorem and the Weierstrass-Hironaka division theorem for them. We introduce a family of norms and we look at them as a family of ana