Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We provide approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has gone through two major rounds of revision. In particular, a section on the performance of our algorithm on application-motivated problems has been added and a more comprehensive literature review is presente

Infrared spectra of C2H4 dimer and trimer

Spectra of ethylene dimers and trimers are studied in the nu11 and (for the dimer) nu9 fundamental band regions of C2H4 (~2990 and 3100 cm-1) using a tunable optical parametric oscillator source to probe a pulsed supersonic slit jet expansion. The deuterated trimer has been observed previously, but this represents the first rotationally resolved spectrum of (C2H4)3. The results support the previously determined cross-shaped (D2d) dimer and barrel-shaped (C3h or C3) trimer structures. However, the dimer spectrum in the nu9 fundamental region of C2H4 is apparently very perturbed and a previous rotational analysis is not well verified.Comment: 21 pages, 4 figure

Biodiversity of benthic invertebrates in Aras River

Benthic invertebrate species and their change was studied in Aras River during a hydro- biological research on the middle and terminal parts of Aras River that spanned the years 1995-1996 and 2005-2006. We found 91 species of benthic invertebrates of which 85 species were identified during 1995-1996 and 49 species during 2005-2006. The highest rate of biodiversity was seen in molluscs with 19 species and chironomid larvae with 17 species. Forty-two species had wide distribution and the remaining occurred only in special habitats. The biomass of invertebrates reduced from the upper reach of the river to the middle and lower reaches because of the changes in river bed from soil to sand. It is concluded that the formation of different habitats in different sections of the Aras River has a crucial role in the change observed in biodiversity of the benthic invertebrates

An experimental investigation of the chopping of helicopter main rotor tip vortices by the tail rotor

The chopping of helicopter main rotor tip vortices by the tail rotor was experimentally investigated. This is a problem of blade vortex interaction (BVI) at normal incidence where the vortex is generally parallel to the rotor axis. The experiment used a model rotor and an isolated vortex and was designed to isolate BVI noise from other types of rotor noise. Tip Mach number, radical BVI station, and free stream velocity were varied. Fluctuating blade pressures, farfield sound pressure level and directivity, velocity field of the incident vortex, and blade vortex interaction angles were measured. Blade vortex interaction was found to produce impulsive noise which radiates primarily ahead of the blade. For interaction away from the blade tip, the results demonstrate the dipole character of BVI radiation. For BVI close to the tip, three dimensional relief effect reduces the intensity of the interaction, despite larger BVI angle and higher local Mach number. Furthermore, in this case, the radiation patern is more complex due to diffraction at and pressure communication around the tip

Three new infrared bands of the He-OCS complex

Three new infrared bands of the weakly-bound He-OCS complex are studied, using tunable lasers to probe a pulsed supersonic slit jet expansion. They correspond to the (0400) <-- (0000), (1001)<-- (0000), and (0401) <-- (0000) transitions of OCS at 2105, 2918, and 2937 cm-1, respectively. The latter band is about 7900 times weaker than the previously studied OCS nu1 fundamental. Vibrational shifts relative to the free OCS monomer are found to be additive. Since carbonyl sulfide has previously been shown to be a valuable probe of superfluid quantum solvation effects in helium clusters and droplets, the present results could be useful for future studies of vibrational effects in such systems.Comment: 16 pages, 1 figure, 4 table

Oro-Dental Health Status and Salivary Characteristics in Children with Chronic Renal Failure

Children suffering from decreased renal function may demand unique considerations regarding special oral and dental conditions they are encountered to. It is mentioned that renal function deterioration may affect the hard or soft tissues of the mouth. Having knowledge about the high prevalence of dental defects, calculus, gingival hyperplasia, modified salivary composition and tissue responses to the dental plaque may aid the physician and the dentist to help nurture the patient with chronic renal failure through the crisis, with an aesthetically satisfying and functioning dentition

Dimension Reduction via Colour Refinement

Colour refinement is a basic algorithmic routine for graph isomorphism testing, appearing as a subroutine in almost all practical isomorphism solvers. It partitions the vertices of a graph into "colour classes" in such a way that all vertices in the same colour class have the same number of neighbours in every colour class. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and Ullman (Disc. Math., 1994) and Godsil (Lin. Alg. and its App., 1997) established a tight correspondence between colour refinement and fractional isomorphisms of graphs, which are solutions to the LP relaxation of a natural ILP formulation of graph isomorphism. We introduce a version of colour refinement for matrices and extend existing quasilinear algorithms for computing the colour classes. Then we generalise the correspondence between colour refinement and fractional automorphisms and develop a theory of fractional automorphisms and isomorphisms of matrices. We apply our results to reduce the dimensions of systems of linear equations and linear programs. Specifically, we show that any given LP L can efficiently be transformed into a (potentially) smaller LP L' whose number of variables and constraints is the number of colour classes of the colour refinement algorithm, applied to a matrix associated with the LP. The transformation is such that we can easily (by a linear mapping) map both feasible and optimal solutions back and forth between the two LPs. We demonstrate empirically that colour refinement can indeed greatly reduce the cost of solving linear programs

A Complete Characterization of the Gap between Convexity and SOS-Convexity

Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the definition of convexity, its first order characterization, and its second order characterization, are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming whereas deciding convexity is NP-hard. If we denote the set of convex and sos-convex polynomials in $n$ variables of degree $d$ with $\tilde{C}_{n,d}$ and $\tilde{\Sigma C}_{n,d}$ respectively, then our main contribution is to prove that $\tilde{C}_{n,d}=\tilde{\Sigma C}_{n,d}$ if and only if $n=1$ or $d=2$ or $(n,d)=(2,4)$. We also present a complete characterization for forms (homogeneous polynomials) except for the case $(n,d)=(3,4)$ which is joint work with G. Blekherman and is to be published elsewhere. Our result states that the set $C_{n,d}$ of convex forms in $n$ variables of degree $d$ equals the set $\Sigma C_{n,d}$ of sos-convex forms if and only if $n=2$ or $d=2$ or $(n,d)=(3,4)$. To prove these results, we present in particular explicit examples of polynomials in $\tilde{C}_{2,6}\setminus\tilde{\Sigma C}_{2,6}$ and $\tilde{C}_{3,4}\setminus\tilde{\Sigma C}_{3,4}$ and forms in $C_{3,6}\setminus\Sigma C_{3,6}$ and $C_{4,4}\setminus\Sigma C_{4,4}$, and a general procedure for constructing forms in $C_{n,d+2}\setminus\Sigma C_{n,d+2}$ from nonnegative but not sos forms in $n$ variables and degree $d$. Although for disparate reasons, the remarkable outcome is that convex polynomials (resp. forms) are sos-convex exactly in cases where nonnegative polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for computer assisted proofs of the paper added to arXi