Sensitivity optimization problem for uncertain discrete SISO system

We consider an optimal stabilization problem for an uncertain system described as a linear time-invariant plant closed by an uncertain feedback which is a conic nonlinearity. We apply the so-called S-procedure losslessness theorem by Yakubovich to reduce the original problem to a parametrized H-infinity optimization problem. We prove that the minimal cost in this parameterized optimization is a concave function of the parameter. This fact makes applicable the standard methods of the H-infinity control theory and the convex analysis in developing optimal linear feedback design algorithms for the original problem.This work was supported by the Swedish Academy of Science and by the Göran Gustafsson Foundation. The author is grateful to the anonymous referees for many useful remarks.     

Quadrature domains for harmonic functions

This paper establishes a conjecture of Gustafsson, Sakai andShapiro by showing that any quadrature domain (for harmonicfunctions) with respect to a signed measure is also a quadraturedomain with respect to a positive measure.     

Self-Couplings and the concentration function

The concentration function of a random variable may be estimated using couplings of the variable with itself.Supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.     

Coupling and Poisson approximation

We give an overview of the Stein-Chen method for establishing Poisson approximations of various random variables. Couplings of certain variables are used to gives explicit bounds for the total variation distance between the distribution of a random variable and a Poisson variable. Some applications are given. In some cases, explicit couplings may be used to obtain good estimates; in other applications it suffices to show the existence of couplings with certain monotonicity properties.Supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.     

Flagf-vectors and thecd-index

Partially supported by the Goran Gustafsson Foundation at the Royal Institute of Technology (Kungl. Tekniska H?gskolan), Stockholm, and by NSF grants #DMS-8401376 and #DMS-9206374     

Exponential convergence toward equilibrium for homogeneous Fokker–Planck-type equations

We consider homogeneous solutions of the Vlasov–Fokker–Planck equation in plasma theory proving that they reach the equilibrium with a time exponential rate in various norms. By Csiszar–Kullback inequality, strong L¹-convergence is a consequence of the ‘sharp’ exponential decay of relative entropy and relative Fisher information. To prove exponential strong decay in Sobolev spaces H^k, k ⩾ 0, we take into account the smoothing effect of the Fokker–Planck kernel. Finally, we prove that in a metric for probability distributions recently introduced in [9] and studied in [4, 14] the decay towards equilibrium is exponential at a rate depending on the number of moments bounded initially. Uniform bounds on the solution in various norms are then combined, by interpolation inequalities, with the convergence in this weak metric, to recover the optimal rate of decay in Sobolev spaces. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons, Ltd.     

Some Bialgebroids Constructed by Kadison and Connes–Moscovici are Isomorphic

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.      

On the geometry of subsets of positive reach

We prove that sets of positive reach in Riemannian manifolds and more generally, almost convex subsets in spaces with an upper curvature bound have an upper curvature bound with respect to the inner metric.Mathematics Subject Classification (2000): 53C20     

Cohomology theories of Hopf bimodules and cup-product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by Gerstenhaber and Schack, and by Ospel. We prove, when all the spaces involved are finite dimensional, that they are all equal to the Ext functor on the module category of an associative algebra X associated to A, as described by Cibils and Rosso. We also give an expression for a cup-product in the cohomology defined by Ospel, and prove that it corresponds to the Yoneda product of extensions.     

Neighborhood Complexes of Stable Kneser Graphs

It is shown that the neighborhood complexes of a family of vertex critical subgraphs of Kneser graphs—the stable Kneser graphs introduced by L. Schrijver—are spheres up to homotopy. Furthermore, it is shown that the neighborhood complexes of a subclass of the stable Kneser graphs contain the boundaries of associahedra (simplicial complexes encoding triangulations of a polygon) as a strong deformation retract.* The first author was partially supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. The second author was supported by the graduate school Algorithmische Diskrete Mathematik, which is funded by the Deutsche Forschungsgemeinschaft, grant GRK 219/3. The DAAD partially supported a stay at KTH, Stockholm, in December 1998, where this work was done: DAAD program AZ 313/S-PPP     

A symmetry result for the p-laplacian equation via the moving planes method

Thsi paper is concerned with a positive solution u of the non–homogeneous p–Laplacian equation in an open, bounded, connected subset Ω of Rⁿ with C² boundary. We assume that u verifies overdetermined boundary conditions and we prove that of us has only one critial point Ω thenΩ is a ball and u is radially symmetric; to prove this result we use the moving planes method introduced by J.Serrien. Using the same technique we also prove that the result is stable in the following sense: the boundary of Ω tends to the boundary of a sphere as the diameter of the critical set u tends to 0.     

Spectral Deviations for the Damped Wave Equation

We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the “entropy” that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction.     

Cohomology Theories of Hopf Bimodules and Cup-Product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.     

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.     

A Poincaré–Birkhoff–Witt criterion for Koszul operads

The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to prove that an operad is Koszul. We define the notion of Poincaré–Birkhoff–Witt basis in the context of operads. Then we show that an operad having a Poincaré–Birkhoff–Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincaré–Birkhoff–Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie, Poisson) have a Poincaré–Birkhoff–Witt basis. We also prove by our methods that new operads are Koszul.     

On adaptive wavelet estimation of a quadratic functional from a deconvolution problem

We observe a stochastic process where a convolution product of an unknown function and a known function is corrupted by Gaussian noise. We wish to estimate the squared \mathbbL²{\mathbb{L}^2} -norm of the unknown function from the observations. To reach this goal, we develop adaptive estimators based on wavelet and thresholding. We prove that they achieve (near) optimal rates of convergence under the mean squared error over a wide range of smoothness classes.     

Discretization results for the Huff and Pareto-Huff competitive location models on networks

In this paper we prove that there always exists a finite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto-Huff model, we prove that there is always an optimal solution at, or an ε-optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.     

The exact packing measure of Lévy trees

We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize the Aldous continuum random tree which corresponds to the Brownian case. In this paper, we prove that Lévy trees always have an exact packing measure: we explicitly compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.     

Power algebras and generalized quotient algebras

The notion of a good quotient relation has been introduced as an attempt to generalize the notion of a quotient algebra to relations on an algebra which are not necessarily congruences. In order to make it possible to prove generalized versions of 'power isomorphism theorems', the more restrictive notions of very good, Hoare good and Smyth good relation have been introduced. In this paper we describe the relationships between Hoare good, Smyth good and very good relations. As a consequence, we prove that every structure preserving relation on an algebra is very good. Received September 25, 1998; accepted in final form January 14, 1999.     

A weak countable choice principle

A weak choice principle is introduced that is implied by both countable choice and the law of excluded middle. This principle suffices to prove that metric independence is the same as linear independence in an arbitrary normed space over a locally compact field, and to prove the fundamental theorem of algebra.