Variational analysis of functions of the roots of polynomials

The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies the computation of the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior of the roots, we obtain characterizations of the subderivative and regular subdifferential for these functions as well. In particular, we completely characterize the subderivative and regular subdifferential of the radius mapping (the maximum of the moduli of the roots). The abscissa and radius mappings are important for the study of continuous and discrete time linear dynamical systems. Dedicated to R. Tyrrell Rockafellar on the occasion of his 70th birthday. Terry is one of those rare individuals who combine a broad vision, deep insight, and the outstanding writing and lecturing skills crucial for engaging others in his subject. With these qualities he has won universal respect as a founding father of our discipline. We, and the broader mathematical community, owe Terry a great deal. But most of all we are personally thankful to Terry for his friendship and guidance. Research supported in part by the National Science Foundation Grant DMS-0203175. Research supported in part by the Natural Sciences and Engineering Research Council of Canada. Research supported in part by the National Science Foundation Grant DMS-0412049.     

Projection methods and approximations for ordinary differential equations

A formulation of a differential equation as projection and fixed point problem allows approximations using general piecewise functions. We prove existence and uniqueness of the approximate solution, convergence in the L₂ norm and nodal superconvergence. These results generalize those obtained earlier by Hulme for continuous piecewise polynomials and by Del four-Dubeau for discontinuous piecewise polynomials. A duality relationship for the two types of approximations is also given. This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada (Grant OGPLN-336) and by the “Ministère de l’Education du Québec” (FCAR Grant-ER-0725).     

Explicit classes of permutation polynomials of $$ \mathbb{F}_{3^{3m} } $$

Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of mathematics and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, six classes of linearized permutation polynomials and six classes of nonlinearized permutation polynomials over are presented. These polynomials have simple shapes, and they are related to planar functions. This work was supported by Australian Research Council (Grant No. DP0558773), National Natural Science Foundation of China (Grant No. 10571180) and the Research Grants Council of the Hong Kong Special Administrative Region of China (Grant No. 612405)     

Fine continuity on metric spaces

We obtain pointwise estimates for solutions of obstacle problems on metric measure spaces and prove that p-superharmonic functions are p-finely continuous. Consequently, we show that p-quasicontinuous functions are p-finely continuous at p-quasievery point. As a byproduct, we obtain the sufficiency part of the Wiener criterion in metric spaces without the assumption of linear local connectedness. The author was supported by the Swedish Research Council.     

Cohomology and Morse theory for strongly indefinite functionals

Supported in part by the Swedish Natural Science Research Council     

An index theory and existence of multiple brake orbits for star-shaped Hamiltonian systems

Supported in part by the Swedish Natural Science Research Council     

The trace to the boundary of Sobolev spaces on a snowflake

The trace to the boundary is defined for functions in a Sobolev space in a domain with fractal boundary, for instance von Koch's snowflake domain. The image and the kernel of the trace operator are characterized. Partially supported by the Swedish Natural Science Research Council     

Canonical representations of partial 2- and 3-trees

We give algorithms constructing canonical representations of partial 2-trees (series parallel graphs) and partial 3-trees. The algorithms can be implemented in log-linear space, or in linear time using quadratic space.Supported in part by a grant from the Swedish Natural Science Research Council.Research supported in part by the Office of Naval Research Contract N00014-86-K-0419.     

Homogenization of parabolic equations an alternative approach and some corrector-type results

We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type results and benefit from some interpolation properties of Sobolev spaces to identify regularity assumptions strong enough for such results to hold.This research was supported by The Swedish Research Council for the Engineering Sciences (TFR), The Swedish National Board for Industrial and Technological Development (NUTEK), and The Country of Jämtland Research Foundation     

The rodrigues type representations for a certain class of special functions

Summary An attempt is made here to present a systematic introduction to and several applications of a certain method of obtaining Rodrigues type representations for a fairly wide variety of sequences of special functions. The main results, contained in Theorems2 and3 below, are shown to apply not only to the Bessel polynomials, the classical orthogonal polynomials including, for instance, Hermite, Jacobi (and, of course, Gegenbauer, Legendre, and Tchebycheff), and Laguerre polynomials, and to their various generalizations studied in recent years, but also to such other special functions as the Bessel function and a certain class of generalized hypergeometric functions. Entrata in Redazione il 25 giugno 1977. This work was partially supported by the National Research Council of Canada under grants A-7353 and A-4027. For a preliminary report of this paper see Notices Amer. Math. Soc.,24 (1977), p. A-238, Abstract no. 77T-B43.     

Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii). The second author's work was partially supported by the fund for the promotion of research at the Technion-Israel Institute of Technology. The third author's work was partially supported by the Swedish Natural Science Research Council, and by the 1992 Wallenberg Prize from the Swedish Mathematical Society.     

Integral representation with weights I

 We describe a new approach to representation formulas for holomorphic functions, including the Cauchy-Fantappie-Leray formula, and provide a general method to generate weighted formulas. Received: 31 October 2001 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 32 A 25. The author was partially supported by the Swedish Research Council.     

On gyroscopic stabilization

The mechanisms of transition between divergence, flutter, and stability for a class of conservative gyroscopic systems with parameters are studied. Two results are obtained which state sufficient conditions for gyroscopic stabilization of conservative systems with an even dimension and a negative definite stiffness matrix. A number of examples are given to demonstrate the feasibility of the results.The work of this author was supported in part by The Danish Technical Research Council through the programme on Computer Aided Engineering Design and by The Danish Natural Science Research Council through the programme on Differential Equations.     

The numerical evaluation of cauchy principal value integrals with non-standard weight functions

The authors develop an algorithm for the numerical evaluation of the finite Hilbert transform, with respect to non-standard weight functions, by a product quadrature rule. In particular, this algorithm allows us to deal with the weight functions with algebraic and/or logarithmic singularities in the interval [−1, 1], by using the Chebyshev points as quadrature nodes. The practical application of the rule is shown to be straightforward and to yield satisfactory numerical results. Convergence theorems are also given, when the nodes are the zeros of certain classical Jacobi polynomials and the weight is defined as a generalized Ditzian-Totik weight. This work was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (first author) and by the Italian Research Council (second author).     

On reachable states in boundary control for the heat equation,and an associated moment problem

We derive a new necessary and sufficient condition for solvability of a moment problem involving real exponentials, which arises in control theory for the heat equation; this allows one to identify some novel situations for which the moment problem is solvable. Moreover, we prove a theorem in the context of boundary control for the heat equation, which allows one to construct new reachable states from known reachable states; as a corollary this implies that all polynomial functions of the space variables are reachable. Finally, we show also that trigonometric polynomials (and certain related functions involving real exponentials) are also reachable.This author wishes to thank the Mathematics Department at McGill University for its support and hospitality during the period in which this paper was completed.This work was supported by the Natural Sciences and Engineering Research Council of Canada Grant A7271.     

Random capacities and their distributions

Summary We formalize the notion of an increasing and outer continuous random process, indexed by a class of compact sets, that maps the empty set on zero. Existence and convergence theorems for distributions of such processes are proved. These results generalize or are similar to those known in the special cases of random measures, random (closed) sets and random (upper) semicontinuous functions. For the latter processes infinite divisibility under the maximum is introduced and characterized. Our result generalizes known characterizations of infinite divisibility for random sets and max-infinite divisibility for random vectors. Also discussed is the convergence in distribution of the row-vise maxima of a null-array of random semicontinuous functions.Research supported by the Swedish Natural Science Research Council     

Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise

The stochastic heat equation driven by additive noise is discretized in the spatial variables by a standard finite element method. The weak convergence of the approximate solution is investigated and the rate of weak convergence is found to be twice that of strong convergence. M. Kovács and S. Larsson supported by the Swedish Research Council (VR). Part of this work was done at Institut Mittag-Leffler. S. Larsson supported by the Swedish Foundation for Strategic Research (SSF) through GMMC, the Gothenburg Mathematical Modelling Centre.     

A class of polynomials defined by generalized Rodrigues’ formula

Summary The present paper incorporates a systematic study of linear, bilinear and bilateral generating functions, pure as well as mixed recurrence relations, and the differential equations associated with the class of polynomials {G _n ^α (x, r, p, k)|n=0, 1, 2, ... }, defined by (1.3) below, which evidently provides an elegant generalization of the various recent extensions of the classical Hermite and Laguerre polynomials. This work was supported in part by the National Research Council of Canada under Grant A7353. See also Abstract 71T-B15, Notices Amer. Math. Soc.18 (1971), p. 252. Entrata in Redazione il 16 novembre 1970.     

On the number of polynomials over GF(2) that factor into 2, 3 or 4 prime polynomials

In this paper a simple method is presented to derive formulas for the number of polynomials over GF(2) which factor into two, three, and four prime polynomials only. A table is given, summarizing the above numbers for polynomials of degree up to 127. Furthermore, the computed values are compared with an asymptotic approximation for these values.This work was supported in part by the National Swedish Board for Technical Development under grants 81-3323 and 83-4364 at the University of Lund.     

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied. This work was partially funded by the Natural Sciences and Engineering Research Council of Canada, and by the MITACS Network of Centres of Excellence.