1. Lyapunov Exponents for Differential Equations 2. Abstract Theory of Lyapunov Exponents 3. Regularity of Lyapunov Exponents Associated with Differential Equations 4. Lyapunov Stability Theory 5. The Oseledets Decomposition 6. Dynamical Systems with Nonzero Lyapunov Exponents. Multiplicative Ergodic Theorem 7. Nonuniform Hyperbolicity. Regular Sets 8. Examples of Nonuniformly Hyperbolic System...

In this paper we prove that there exists an absolute constant c > 0 such that for every natural number k, every non-bipartite 2-connected graph with average degree at least ck contains k cycles with consecutive odd lengths. This implies the existence of the absolute constant d > 0 that every non-bipartite 2-connected graph with minimum degree at least dk contains cycles of all lengths modulo k,...

Fundamental notions concerning absolute structure and absolute configuration, and their determination from single-crystal diffraction measurements, are presented and reviewed. A glossary of terms with definitions useful in this field is provided. For absolute structure and its determination, the separate but interacting influences of the structure and the inversion-distinguishing power of an X-...

The problems of absolute stability and existence of forced oscillations for systems with sector restricted nonlinearities are considered. An overview of the results for systems described by ordinary differential equations with constant and periodic coefficients is presented. A result is obtained on exponential absolute stability for systems described by ordinary differential equations. Next, a ...

A considerable body of experimental and theoretical work claims the existence of negative absolute temperatures in spin systems and ultra-cold quantum gases. Here, we clarify that such findings can be attributed to the use of a popular yet inconsistent entropy definition, which violates fundamental thermodynamic relations and fails to produce sensible results for simple analytically tractable c...

The presence of the equator and poles has a profound effect on the critical dynamonumberDf for the onset of dynamo waves and their subsequent nonlinear evolution. The instability threshold is associated with the onset of absolute instability, which is necessary for the existence of a globally unstablemode. With increasingD thewaves become very rapidly nonlinear and the region of activity extend...

Experimental measurements of acoustic transmission through a solid-solid two-dimensional binary-composite medium constituted of a triangular array of parallel circular steel cylinders in an epoxy matrix are reported. Attention is restricted to propagation of elastic waves perpendicular to the cylinders. Measured transmitted spectra demonstrate the existence of absolute stop bands, i.e., band ga...

15 3. Stable and unstable filtrations 17 3.1. Existence and subfoliation 17 3.2. Absolute continuity 19 4. Central Foliations 21 4.1. Normal hyperbolicity 21 4.2. Integrability of the central foliation and dynamical coherence 23 4.3. Smoothness of central leaves via normal hyperbolicity 25 4.4. Robustness of the central foliation 26 5. Intermediate Foliations 27 5.1. Nonintegrability of interme...

For delay Lur’e control systems with multiple non-linearities and time-varying uncertainties, necessary and sufficient conditions for the existence of Lyapunov functional in the extended Lur’e form with negative definite derivative to guarantee robust absolute stability are derived by solving a set of linear matrix inequalities(LMIs). Moreover, some new delay-dependent absolute stability criter...