ar X iv : m at h / 06 01 65 8 v 2 [ m at h . O C ] 2 3 Ju n 20 06 FurtherResults onStrictLyapunovFunctions for RapidlyTime - VaryingNonlinear Systems ⋆

ar X iv : m at h . O C / 0 60 16 58 v 2 2 3 Ju n 20 06 FurtherResults onStrictLyapunovFunctions for RapidlyTime - VaryingNonlinear Systems ⋆

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform globa...

2 6 Ja n 20 06 FurtherResults onStrictLyapunovFunctions for RapidlyTime - VaryingNonlinear Systems ⋆

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform globa...

ar X iv : m at h / 06 06 67 1 v 1 [ m at h . A C ] 2 7 Ju n 20 06 PRÜFER ⋆ – MULTIPLICATION DOMAINS AND ⋆ – COHERENCE

ar X iv : m at h / 06 06 33 9 v 1 [ m at h . SP ] 1 4 Ju n 20 06 Eigenfunction expansions associated with 1 d periodic differential operators of order 2 n

We prove an explicit formula for the spectral expansions in L(R) generated by selfadjoint differential operators (−1) d dx2n + n−1

ar X iv : m at h / 06 06 31 2 v 1 [ m at h . A C ] 1 3 Ju n 20 06 ASYMPTOTIC BEHAVIOR OF MULTIGRADED REGULARITY

Let S be a standard N k-graded polynomial ring over a field K, let I be a multigraded homogeneous ideal of S, and let M be a finitely generated Z k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I n M is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.