A Class of k-Quantum Nonlinear Coherent States and Some of Their Properties

A class of k-quantum nonlinear coherent states, i.e., the k eigenstates of the powers B ^k (k 3) of the annihilation operator of f-oscillators, is obtained and its completeness is investigated. An alternative method to construct them is proposed. We introduce a new kind of higher-order squeezing and sub-Poissonian distribution. The quantum statistical properties of the k states are studied. The result shows that all of the eigenstates can be generated by a linear superposition of k Roy-type nonlinear coherent states. These states may form a complete Hilbert space, and the M-th order [M = (n + 1/2)k; n = 0,1, ...] squeezing effects exist in all of the k states when k is even. There is the sub-Poissonian distribution in all of the states.     

Heavy alkaline earth metal pyrazolates: synthetic pathways, structural trends, and comparison with divalent lanthanoids

Two series of heavy alkaline earth metal pyrazolates, [M(Ph(2)pz)(2)(thf)(4)] 1 a-c (Ph(2)pz=3,5-diphenylpyrazolate, M=Ca, Sr, Ba; THF=tetrahydrofuran) and [M(Ph(2)pz)(2)(dme)(n)] (M=Ca, 2 a, Sr, 2 b, n=2; M=Ba, 2 c, n=3; DME=1,2-dimethoxyethane) have been prepared by redox transmetallation/ligand exchange utilizing Hg(C(6)F(5))(2). Compounds 1 a and 2 b were also obtained by redox transmetallation with Tl(Ph(2)pz). Alternatively, direct reaction of the alkaline earth metals with 3,5-diphenylpyrazole at elevated temperatures under solventless conditions yielded compounds 1 a-c and 2 a-c upon extraction with THF or DME. By contrast, [M(Me(2)pz)(2)(Me(2)pzH)(4)] 3 a-c (M=Ca, Sr, Ba; Me(2)pzH=3,5-dimethylpyrazole) were prepared by protolysis of [M[N(SiMe(3))(2)](2)(thf)(2)] (M=Ca, Sr, Ba) with Me(2)pzH in THF and by direct metallation with Me(2)pzH in liquid NH(3)/THF. Compounds 1 a-c and 2 a-c display eta(2)-bonded pyrazolate ligands, while 3 a,b exhibit eta(1)-coordination. Complexes 1 a-c have transoid Ph(2)pz ligands and an overall coordination number of eight with a switch from mutually coplanar Ph(2)pz ligands in 1 a,b to perpendicular in 1 c. In eight coordinate 2 a,b the pyrazolate ligands are cisoid, whilst 2 c has an additional DME ligand and a metal coordination number of ten. By contrast, 3 a,b have octahedral geometry with four eta(1)-Me(2)pzH donors, which are hydrogen-bonded to the uncoordinated nitrogen atoms of the two trans Me(2)pz ligands. The application of synthetic routes initially developed for the preparation of lanthanoid pyrazolates provides detailed insight into the similarities and differences between the two groups of metals and structures of their complexes.     

Genericity of wild holomorphic functions and common hypercyclic vectors

Let T_α be the translation operator by α in the space of entire functions defined by . We prove that there is a residual set G of entire functions such that for every f∈G and every the sequence is dense in , that is, G is a residual set of common hypercyclic vectors ( functions) for the family . Also, we prove similar results for many families of operators as: multiples of differential operator, multiples of backward shift, weighted backward shifts.     

Convex sets which are intersections of closed balls

In this paper we aim to investigate different questions concerning the stability of the set of all intersections of closed balls in a normed space. We are mainly concerned with: (i) the stability of under the closure of the vector sums; (ii) the stability under the addition of balls. We prove that (i) and (ii) are different properties which have strong connections with the geometry of the space. They have interest both in finite and infinite dimension. In the former case, there is a link with linear programming theory. We also study two more stability properties related to the well-known binary intersection property. Mazur sets and Mazur spaces are introduced, as a natural family satisfying (i). We prove that every two-dimensional normed space is a Mazur space, a result which distinguishes dimension d?2 from dimension d?3. We also discuss the connections between Mazur spaces and porosity.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

Wavelet Frames on Groups and Hypergroups via Discretization of Calderón Formulas

Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we use Calderón-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in ⁿ, for an affine extension of the Heisenberg group, and on many commutative hypergroups.