Infinite horizon disturbance attenuation for discrete-time systems. A Popov-Yakubovich approach

It is proved that for the discrete-time linear systems with time-varying coefficients the existence of a controller which simultaneously stabilizes and provides prescribed disturbance attenuation for the resultant closed-loop system, implies the existence of global solutions to several Kalman-Szegö-Popov-Yakubovich systems. It is also proved that this fact is equivalent to the existence of the positive semidefinite stabilizing solutions to corresponding game-theoretic Riccati equations. The family of all controllers with the above mentioned properties is constructed in terms of the solutions to the cited Kalman-Szegö-Popov-Yakubovich systems. The main tool is the generalized Popov-Yakubovich theory which is essentially developed in an operator-theoretic framework.     

Milnor numbers and Euler obstruction*

Using a geometric approach, we determine the relations between the local Euler obstruction Eu_f of a holomorphic function f and several generalizations of the Milnor number for functions on singular spaces. *This work was partially supported by CNRS-CONACYT (12409) Cooperation Program. The first and third named authors partially supported by CONACYT grant G36357-E and DGPA (UNAM) grant IN 101 401.     

Studies in the field of polyethylene—V. Correlations between compressibility,crystallinity and molecular weight of high pressure natural polyethylenes

The P-V-T relations of high pressure polyethylene free of additives have been studied in a modified universal testing machine. These polymers show a rapid compression at the melting point of the crystalline phase.     

On the generators of semisimple lie algebras

It will be shown that given any element X in a simple Lie algebra $\text{Q}$ over C, there exists a Y ∈ $\text{Q}$ such that the Lie algebra generated by X and Y is $\text{Q}$. The result is extended to the real semisimple Lie algebras. In some sense the main theorem of this paper can be regarded as an extension of Morozov-Jacobson theorem concerning three dimensional simple Lie algebras (see the remark at the end of Sec. 4). A new property of a special class of regular elements, known as the cyclic elements, is given.     

Restes des formules de quadrature de Gauss et de Turán

Sans résumé Présenté par P. Turán     

Matrix Representations for Positive Noncommutative Polynomials

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ``symmetric' polynomial q(x, h) in noncommuting variables, {x<sub>1</sub>, . . . , } and {h<sub>1</sub>, . . . , } for which q(X,H) is positive semidefinite whenever are tuples of selfadjoint matrices with ||X<sub>j</sub>|| ≤ 1 but H<sub>j</sub> unconstrained. The representation we obtain is a Gram representation in the variables h where P<sub>q</sub> is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector' whose entries are polynomials in both x and h. We show that one can choose P_q such that the matrix P_q(X) is positive semidefinite for all ||X_j|| ≤ 1. The representation covers sum of square results ([Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when g_x = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically to appear IOET July 2003] when restricted to x-domains of the type ||X_j|| ≤ 1. Partially supported by NSF, DARPA and Ford Motor Co. Partially supported by NSF grant DMS-0140112 Partially supported by NSF grant DMS-0100367     

Electronic spectroscopy of the A1A' <-- X1A' system of CDF

To further investigate the Renner-Teller (RT) effect and barriers to linearity and dissociation in the simplest singlet carbene, we recorded fluorescence excitation spectra of bands involving the pure bending levels 2(n)(0) with n = 0-9 and the combination states 1(1)(0)2(n)(0) with n = 1-8 and 2(n)(0)3(1)(0) with n = 0-5 in the A(1)A'<-- X(1)A' system of CDF, in addition to some weak hot bands. The spectra were measured under jet-cooled conditions using a pulsed discharge source, and rotationally analyzed to yield precise values for the band origins and rotational constants; fluorescence lifetimes were also measured to probe for lifetime lengthening effects due to the RT interaction. The derived A state parameters are compared with previous results for CHF and with predictions of ab initio electronic structure theory. The approach to linearity in the A state is evidenced in a sharp increase in the A rotational constant with bending excitation, and a minimum in the vibrational intervals near 2(9). A fit of the vibrational intervals for the pure bending levels yields an A state barrier to linearity in good agreement both with that previously derived for CHF and ab initio predictions. From the spectra and lifetime measurements, the onset of extensive RT perturbations is found to occur at a higher energy than in CHF, consistent with the smaller A constant.