The expected values of invariant polynomials with matrix argument of elliptical distributions

ThisresearchissupportedbytheChineseAcademyofSciences.1.IntroductionTherearemailypublicatiollsonmatrixvariatedistributions(ormatrixdistributionforsimplicity),inparticular,abollttheirexpectedvaluesofzonalpolynomialsoftheirquadraticforms(of.[12],[151,[171and…     

The Jung Theorem in metric spaces of curvature bounded above

The classical Jung Theorem states in essence that the diameter of a compact set in satisfies where is the circumradius of . The theorem was extended recently to the hyperbolic and the spherical -spaces. Here, the estimate above is extended to a class of metric spaces of curvature introduced by A. D. Alexandrov. The class includes the Riemannian spaces. The extended estimate is of the form where is a positive integer suitably defined for the set and its circumcenter. It can be that is not unique or does not exist. In the latter case, no estimate is derived. In case of a Riemannian -dimensional space, an integer always exists and satisfies . Then . In case of , one has .

     

Hypercomplex structures on four-dimensional Lie groups

The purpose of this paper is to classify invariant hypercomplex structures on a -dimensional real Lie group . It is shown that the -dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group of the quaternions, the multiplicative group of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, and , respectively, and the semidirect product . We show that the spaces and possess an of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian -manifolds are determined.

     

Oscillating singularities on cantor sets: A grand-canonical multifractal formalism

The singular behavior of functions is generally characterized by their Hölder exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then elaborate on a grand-canonical multifractal formalism that describes statistically the fluctuations of both the Hölder and the oscillation exponents. We prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities.     

The Al-Zn-Ga Phase Diagram Part II

The first part of this paper presented five experimental isopletic cuts in the Al-Zn-Ga ternary phase diagram. On these cuts, two isobaric ternary invariant reactions were determined and a significant retrograde miscibility of Ga in a α′_SS solid solution was observed. In the second part, the two isobaric invariant reactions are studied more precisely. In particular, the composition of the invariant phases are given and the Ga miscibility in the α_SS ternary solid solution is studied. Isothermal sections are established. The results confirm the existence of a vanishing point in the liquidus area, conjugated with a ternary critical point at about 290°C. A general perspective shape of the equilibria in the diagram is proposed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Invariant strings and pattern-recognizing properties of one-dimensional cellular automata

A cellular automaton is a discrete dynamical system whose evolution is governed by a deterministic rule involving local interactions. It is shown that given an arbitrary string of values and an arbitrary neighborhood size (representing the range of interaction), a simple procedure can be used to find the rules of that neighborhood size under which the string is invariant. The set of nearestneighbor rules for which invariant strings exist is completely specified, as is the set of strings invariant under each such rule. For any automaton rule, an associated filtering rule is defined for which the only attractors are spatial sequences consisting of concatenations of invariant strings. A result is provided defining the rule of minimum neighborhood size for which an arbitrarily chosen string is the unique invariant string. The applications of filtering rules to pattern recognition problems are discussed.     

The goodness of ergodic adiabatic invariants

For a slowly time-dependent Hamiltonian system exhibiting chaotic motion that ergodically covers the energy surface, the phase space volume enclosed inside this surface is an adiabatic invariant. In this paper we examine, both numerically and theoretically, how the error in this ergodic adiabatic invariant scales with the slowness of the time variation of the Hamiltonian. It is found that under certain circumstances, the error is diffusive and scales likeT ^–1/2, whereT is the characteristic time over which the Hamiltonian changes. On the other hand, for other cases (where motion in the Hamiltonian has a long-time 1/t tail in a certain correlation function), the error scales like [T ^–1 ln(T)]^1/2. Both of these scalings are verified by numerical experiments. In the situation where invariant tori exist amid chaos, the motion may not be fully ergodic on the entire energy surface. The ergodic adiabatic invariant may still be useful in this case and the circumstances under which this is so are investigated numerically (in particular, the islands have to be small enough).     

Spectrum and eigenfunctions of the Frobenius-Perron operator of the tent map

The spectrum and eigenfunctions of the Frobenius-Perron operator induced by the tent map are discussed in detail. Special attention is paid to the case where the critical point of the map lies on an aperiodic trajectory and the differences from maps with a periodic critical trajectory are stressed. It is shown that the relevant eigenvalues of the spectrum are not sensitive to the aperiodicity of the critical trajectory. All other parts of the spectrum and all eigenfunctions in particular are changed drastically if the critical trajectory becomes aperiodic. The intimate connection between the point spectrum and the kneading invariant is established and the critical slowing down as well as the band splitting are a consequence of its properties. The structure of the infinite sequence of null spaces and its implications on the spectrum of the operator are discussed. It is shown that any initial distributionP(0,x) of bounded variation can be projected uniquely onto the eigenfunctions of the relevant eigenvalues and that the time dependence ofP(n, x) is determined by this expansion up to an errorO( ⁿ). From this the stationary and the asymptotic behavior of the correlation function x(n) x can be derived exactly.     

多复变数Bergman空间的支配集

本文利用伪双曲度量球对单位球上的Bergman空间的支配集给出完整刻画.证明方法是将Luecking在单位圆盘上的三个重要引理推广到单位球上,从而刻画单位球上的Bergman空间的支配集.     

第三类超Cartan域上的比较定理

本文给出了第三类超Cartan域上不变Kalher度量下的全纯截曲率的表达式.利用其Bergman度量的完备性,构造了一个不比Bergman度量小的完备的不变Kalher度量,证明了在此Kalher度量下的全纯截曲率有一个负上界,从而证明了第三类超Cartan域的Bergman度量与Kobayashi度量的比较定理.