Unitary equivalence of one-parameter groups of Toeplitz and composition operators

The composition operators on H² whose symbols are hyperbolic automorphisms of the unit disk fixing ±1 comprise a one-parameter group and the analytic Toeplitz operators coming from covering maps of annuli centered at the origin whose radii are reciprocals also form a one-parameter group. Using the eigenvectors of the composition operators and of the adjoints of the Toeplitz operators, a direct unitary equivalence is found between the restrictions to zH² of the group of Toeplitz operators and the group of adjoints of these composition operators. On the other hand, it is shown that there is not a unitary equivalence of the groups of Toeplitz operators and the adjoints of the composition operators on the whole of H², but there is a similarity between them.     

Wandering r-tuples for unitary systems

The properties of the set Wr(U) of all complete wandering r-tuples for a system U of unitary operators acting on a Hilbert space are investigated by parameterizing Wr(U) in terms of a fixed wandering r-tuple Ψ and the set of all unitary operators which locally commute with U at Ψ. The special case of greatest interest is the system 〈D,T〉 of dilation (by 2) and translation (by 1) unitary operators acting on L²(R), for which the complete wandering r-tuples are precisely the orthogonal multiwavelets with multiplicity r. We also give some examples for its application.     

套代数模中酉算子的存在性

假设套N是可接受的,则得到弱闭Alg(N)-模u中存在酉算子的充要条件是I-=I且对任意 M>0,有N->0.     

Schur Orthogonality Relations for the Finitary Infinite Symmetric Group

Let S f be the finitary infinite symmetric group. For a certain class of irreducible unitary representations of S f , a version of Schur orthogonality relations is proved. That is, we construct an invariant inner product on the matrix coefficient space of each representation and show that matrix coefficients for distinct representations are orthogonal with respect to these norms.     

A climbing girl's reflections about angles

The main research question in this paper is whether a climbing discourse can be a resource for a school-geometry discourse. The text is based on a 12-year old girl's story from an exciting climbing trip during her summer holiday. The girl uncovers some of her knowledge that had been invisible to her; she is guided to see some relations between her climbing and her understanding of angles. In the beginning, this girl believes her story does not concern angles at all. The tools for uncovering angles in her story are based on different levels of visibility and objects of the climbing discourse combined with different conceptions of space. The girl develops her consciousness about angles as natural elements in her climbing activity and she is guided to see the angle as an object of her climbing discourse.     

The theoretical research of the medium effect on the vibrational spectrum and the energy of intramolecular coordination O → Si in (benzoyloxymethyl)trifluorosilane molecule

The characteristics of (benzoyloxymethyl)trifluorosilane C₆H₅C(O)OCH₂SiF₃ containing a five-membered heterocycle closed by intramolecular coordination O → Si bond (Ia) and its most stable acyclic isomer (Ib) have been calculated by HF, MP2(Full) non-empirical methods, and DFT(B3LYP) using 6-311G(d) and 6-311 + G(2d,p) basis sets. The (C₈H₁₈, C₆H₆, (C₄H₉)₂O, CHCl₃, (CH₂)₄O, CH₂Cl₂, CH₃CN) medium effect on the energy and structural characteristics, dipole moments, and vibrational spectra of Ia and Ib isomers was calculated by the DFT(B3LYP)/6-311 + G(2d,p) method in the Onsager SCRF model approximation. The DFT(B3LYP)/6-311 + G(2d,p) calculation reasonably reproduces the medium effect on coordination energy, geometry, dipole moments, and band frequencies in the vibrational spectrum of Ia.     

An Application of Logic to Combinatorial Geometry: How Many Tetrahedra are Equidecomposable with a Cube?

It is known (see Rapp [9]) that elementary geometry with the additional quantifier “there exist uncountably many” is decidable. We show that this decidability helps in solving the following problem from combinatorial geometry: does there exist an uncountable family of pairwise non-congruent tetrahedra that are n-equidecomposable with a cube? Mathematics Subject Classification: 03B25, 03C80, 51M04, 52B05, 52B10.     

Partial geometries pg (s, t, 2) with an abelian Singer group and a characterization of the van Lint-Schrijver partial geometry

Let be a proper partial geometry pg(s,t,2), and let G be an abelian group of automorphisms of acting regularly on the points of . Then either t≡2±od s+1 or is a pg(5,5,2) isomorphic to the partial geometry of van Lint and Schrijver (Combinatorica 1 (1981), 63–73). This result is a new step towards the classification of partial geometries with an abelian Singer group and further provides an interesting characterization of the geometry of van Lint and Schrijver.The author is Postdoctoral Fellow of the Fund for Scientific Research Flanders (FWO-Vlaanderen).