[tag=IN] [lemma=linear] [□]?
[tag=IN] [lemma=linear] [□=□]?
[tag=IN] [lemma=linear] [□ | □]?
While remarkably successful for linear PDE 's , Fourier methods provide little information for nonlinear PDE 's , and many interesting PDE 's are nonlinear . Chen et al at PSB'99 present a number of linear differential equation models , including both mRNA and protein levels . Here strlen takes time proportional to the length of the string , and it gets called on every iteration of the loop , leading to quadractic running time , instead of linear . This talk is concerned with a feedback control synthesis problem for linear systems to meet design specifications given in terms of multiple frequency domain inequalities in ( semi ) finite ranges . Contracts normally include seven courses beyond linear algebra , balanced so as to represent analytic , axiomatic , and applied mathematics . Review of basic ordinary differential equations ; systems of linear differential equations ; nonlinear systems ; phase portraits ; linearization and stability ; conservative systems ; Lyapunov functions ; cycles and long term behavior of solutions ; series solutions near ordinary and regular singular points ; Bessel functions . It is somewhat distressing that the theory of linear algebras must again and again be developed from the beginning , for the fundamental concepts of this branch of mathematics crop up everywhere in mathematics and physics , and a knowledge of them should be as widely disseminated as the elements of differential calculus . The implementation is at least as complete as other existing clp ( R ) implementations : It solves linear equations over rational or real valued variables , covers the lazy treatment of nonlinear equations , features a decision algorithm for linear inequalities that detects implied equations , removes redundancies , performs projections ( quantifier elimination ) , allows for linear dis-equations , and provides for linear optimization . Space is thus converted into a new morphic language ' , which shifts the focus of linear architectural plans on the solids in a structure ( the walls ) to the voids ( rooms or open spaces such as atria ) . Monte Carlo simulations are performed which support the analytical predictions in the weak feedback limit and provide results beyond linear perturbation theory . OreModules is a Maple implementation of algorithms which compute free resolutions , extension modules , left / right / generalized inverses and other algebraic objects for linear systems of differential equations ( ODE , PDE ) , differential time-delay systems , discrete systems etc.