Regularized Riesz energies of submanifolds

Given a closed submanifold, or a compact regular domain, in Euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under Möbius transformations, thus giving a generalization to higher dimensions of the Möbius energy of knots.     

Some Properties of Riesz Products on the Ring of p-Adic Integers

Riesz products on the ring of p-adic integers are introduced and studied from the points of view of harmonic analysis and dynamical system relative to the shift transformation. We find necessary and sufficient conditions for a Riesz product to be invariant, quasi-invariant or quasi-Bernoulli. We study the mutual absolute continuity of two Riesz products and the almost everywhere convergence of lacunary series with respect to a Riesz product. We also compute the Hausdorff dimension and the energy dimension and prove a multifractal formalism for a given Riesz product.     

Reverse triangle inequalities for Riesz potentials and connections with polarization

We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup-norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main tool used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure.     

Maximizers for Gagliardo–Nirenberg inequalities and related non-local problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.     

星形热弹性网络系统的稳定性及Riesz基性质

研究了一类星形弹性网络系统在热效应影响以及边界反馈作用下的稳定性问题及系统相应（广义）特征向量的Riesz基性质．基于Green和Naghdi第二类热弹性理论,假设在该热弹性系统中热以有限波速传播,并且在传播过程中无能量耗散．证明了该热弹性网络系统能量渐近衰减到零．并进一步通过系统算子谱分析,讨论得出该系统算子的（广义）特征向量构成状态空间的一组Riesz基．     

Riesz basis for strongly continuous groups

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most K elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the groups form a Riesz family. This implies that if in every range of the spectral projection we construct an orthonormal basis, then the union of these bases is a Riesz basis in the Hilbert space.     

Resz means of bounded states: semi-classical asymptotic

The semi-classical asymptotic behaviour of the Riesz means of a distribution of eigenvalues is investigated, at a non-critical energy level. For Schrödinger type operators, we obtain the second term related to the periodic trajectories of the classical Hamiltonian. This oscillating term is explained for Riesz means of order one with the aim of discussing a Lieb-Thirring conjecture.     

Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces

As a generalization of the notion of Riesz seminorm, a class of seminorms on directed partially ordered vector spaces is introduced, such that (1) every seminorm in the class can be extended to a Riesz seminorm on every larger Riesz space that is majorized and (2) a seminorm on an order dense linear subspace of a Riesz space is in the class if and only if it can be extended to a Riesz seminorm on the Riesz space. The latter property yields that if a directed partially ordered vector space has an appropriate Riesz completion, then a seminorm on the space is in the class if and only if it is the restriction of a Riesz seminorm on the Riesz completion. An explicit formula for the extension is given. The class of seminorms is described by means of a notion of solid unit ball in partially ordered vector spaces. Some more properties concerning restriction and extension are studied, including extension to L- and M-seminorms.     

On the variety of Riesz spaces

Finitely generated linearly ordered Riesz spaces are described, leading to a proof that the variety of Riesz spaces is generated as a quasivariety by the Riesz space ? of real numbers. The finitely generated Riesz spaces are also described: they are the subalgebras of real-valued function spaces on root systems of finite height.     

Riesz ideals in generalized effect algebras and in their unitizations

We show that quotients of generalized effect algebras by Riesz ideals preserve some important special properties, e.g., homogeneity and hereditary Riesz decomposition properties; moreover, quotients of generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices and generalized MV-algebras with respect to Riesz ideals belong to the same class. We give a necessary and sufficient condition under which a Riesz ideal I of a generalized effect algebra P is a Riesz ideal also in the unitization E of P. We also study relations between Riesz ideals and central elements in GEAs and in their unitizations. In the last section, we demonstrate the notion of Riesz ideals by some illustrative examples. Received June 28, 2005; accepted in final form January 23, 2007.     

Riesz External Field Problems on the Hypersphere and Optimal Point Separation

We consider the minimal energy problem on the unit sphere ?? ^d in the Euclidean space ? ^d+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r ^s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d ? 2 ≤ s < d ? 1. The proof uses a maximum principle for measures supported on ?? ^d . When Q is the Riesz s-potential of a signed measure and d ? 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on ?? ^d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.     

Higher Riesz transforms and imaginary powers associated to the harmonic oscillator

Summary We consider higher order Riesz transforms for the multi-dimensional Hermite function expansions. The Riesz transforms occur to be Calderón--Zygmund operators hence their mapping properties follow by using results from a general theory. Then we investigate higher order conjugate Poisson integrals showing that at the boundary they approach appropriate Riesz transforms of a given function. Finally, we consider imaginary powers of the harmonic oscillator by using tools developed for studying Riesz transforms.     

Riesz Products, Random Walks, and the Spectrum

It is shown that to a classical Riesz product one can naturally assign a random walk; the spectrum of the shifts on the tail algebra of the random walk is defined by the measure to which the Riesz product converges. This observation is extended to general groups, which leads to some operator analogs of Riesz products. The properties of operator Riesz products are investigated.     

Spectral distance on the circle

A building block of non-commutative geometry is the observation that most of the geometric information of a compact Riemannian spin manifold M is encoded within its Dirac operator D. Especially via Connes' distance formula one is able to extract from the spectral properties of D the geodesic distance on M. In this paper we investigate the distance d encoded within a covariant Dirac operator on a trivial U(n)-fiber bundle over the circle with arbitrary connection. It turns out that the connected components of d are tori whose dimension is given by the holonomy of the connection. For n=2 we explicitly compute d on all the connected components. For n?2 we restrict to a given fiber and find that the distance is given by the trace of the module of a matrix. The latest is defined by the holonomy and the coordinate of the points under consideration. This paper extends to arbitrary n and arbitrary connection the results obtained in a previous work for U(2)-bundle with constant connection. It confirms interesting properties of the spectral distance with respect to another distance naturally associated to connection, namely the horizontal or Carnot-Carathéodory distance dH. Especially in case the connection has irrational components, the connected components for d are the closure of the connected components of dH within the Euclidean topology on the torus.     

Exponential stability of uniform Euler–Bernoulli beams with non-collocated boundary controllers

We study the stability of a robot system composed of two Euler–Bernoulli beams with non-collocated controllers. By the detailed spectral analysis, we prove that the asymptotical spectra of the system are distributed in the complex left-half plane and there is a sequence of the generalized eigenfunctions that forms a Riesz basis in the energy space. Since there exist at most finitely many spectral points of the system in the right half-plane, to obtain the exponential stability, we show that one can choose suitable feedback gains such that all eigenvalues of the system are located in the left half-plane. Hence the Riesz basis property ensures that the system is exponentially stable. Finally we give some simulation for spectra of the system.     

Spectral operators generated by damped hyperbolic equations

We announce a series of results on the spectral analysis for a class of nonselfadjoint opeators, which are the dynamics generators for the systems governed by hyperbolic equations containing dissipative terms. Two such equations are considered: the equation of nonhomogeneous damped string and the 3-dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. Nonselfadjoint boundary conditions are imposed at the ends of a finite interval or on a sphere centered at the origin respectively. Our main result is the fact the aforementioned operators are spectral in the sense of N. Dunford. The result follows from the fact that the systems of root vectors of the above operators form Riesz bases in the corresponding energy spaces. We also give asymptotics of the spectra and state the Riesz basis property results for the nonselfadjoint operator pencils associated with these operators.     

A generalization of inverse distance weighting and an equivalence relationship to noise-free Gaussian process interpolation via Riesz representation theorem

In this paper, we show the relationship between two seemingly unrelated approximation techniques. On the one hand, a certain class of Gaussian process-based interpolation methods, and on the other hand inverse distance weighting, which has been developed in the context of spatial analysis where there is often a need for interpolating from irregularly spaced data to produce a continuous surface. We develop a generalization of inverse distance weighting and show that it is equivalent to the approximation provided by the class of Gaussian process-based interpolation methods. The equivalence is established via an elegant application of Riesz representation theorem concerning the dual of a Hilbert space. It is thus demonstrated how a classical theorem in linear algebra connects two disparate domains.     

A noncompact variational problem in the theory of riesz potentials. II

We study some generalizations of the well-known problem of minimization of the Riesz energy on condensers. Under fairly general assumptions, we establish necessary and sufficient conditions for the existence of minimal measures.     

On Order Bounded Subsets of Locally Solid Riesz Spaces

In a topological Riesz space there are two types of bounded subsets: order bounded subsets and topologically bounded subsets. It is natural to ask (1) whether an order bounded subset is topologically bounded and (2) whether a topologically bounded subset is order bounded. A classical result gives a partial answer to (1) by saying that an order bounded subset of a locally solid Riesz space is topologically bounded. This paper attempts to further investigate these two questions. In particular, we show that (i) there exists a non-locally solid topological Riesz space in which every order bounded subset is topologically bounded; (ii) if a topological Riesz space is not locally solid, an order bounded subset need not be topologically bounded; (iii) a topologically bounded subset need not be order bounded even in a locally convex-solid Riesz space. Next, we show that (iv) if a locally solid Riesz space has an order bounded topological neighborhood of zero, then every topologically bounded subset is order bounded; (v) however, a locally convex-solid Riesz space may not possess an order bounded topological neighborhood of zero even if every topologically bounded subset is order bounded; (vi) a pseudometrizable locally solid Riesz space need not have an order bounded topological neighborhood of zero. In addition, we give some results about the relationship between order bounded subsets and positive homogeneous operators.     

Amarts on Riesz Spaces

The concepts of conditional expectations, martingales and stopping times were extended to the Riesz space context by Kuo, Labuschagne and Watson （Discrete time stochastic processes on Riesz spaces, Indag. Math.,15（2004）, 435-451）. Here we extend the definition of an asymptotic martingale （amart） to the Riesz spaces context, and prove that Riesz space amarts can be decomposed into the sum of a martingale and an adapted sequence convergent to zero. Consequently an amart convergence theorem is deduced.