Composition algebras of the second kind

The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e² = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 428–447, July–August, 2007.     

Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space $\text{H}$ the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from $\text{H}$^q to $\text{H}$^p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for $\text{H}$^q is the “inflation” of a spectral measure for $\text{H}$. In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫(in0, 2π]e^?iθE(dθ) is a unitary operator on $\text{H}$ and if T is a bounded linear operator from $\text{H}$^q to $\text{H}$^q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process (Uⁿx)_?∞^∞ ?$\text{H}$^q (for each x = (xⁱ)_i^j ∈ $\text{H}$^q, Uⁿx = (Uⁿxⁱ)_i=1^q), then T is a spectral integral, ∫(_0,2π)]Φ(θ) E(dθ), and the Banach norm of T, |T|_B = ess sup |Φ(θ)|_B.     

Fock Space and the Poisson Process

Using the Wiener–Poisson isomorphism, we show that if(F_t)_{0 t 1} is a real, bounded, predictable process adaptedto the filtration of a compensated Poisson process (X_t)_{0 t 1}, and if is the operator corresponding to multiplication by , then for any regular self-adjoint quantum semimartingale , the essentially self-adjoint quantumsemimartingale satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process toa measure space (M, M, µ) as an isometry I: L² (M, M,µ) L²(, F, P) and give a new construction of the generalisedWiener–Poisson isomorphism W_I: F₊ (L²(M)) L² (, F, P)using exponential vectors. Using C^*-algebra theory, given anymeasure space we construct a canonical generalised Poisson process.Unlike other constructions, we make no a priori use of Poissonmeasures. 2000 Mathematics Subject Classification 60G20, 60G35,46L53, 81S25.     

On potential isomorphism and non-structure

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game. The research of the first and second author was partially supported by Academy of Finland grant 40734 Mathematics Subject Classification (2000): 03C55, 03C45     

关于几乎等距嵌入的一个注记

本文将指出：对于足够小的ε＞ ０,不存在从无穷维“抽象 ＬＰ”空 间（０＜ ｐ＜∞）到空间（ｃ）的ε－等距线性算子．     

21阶亚循环群的弱3-DCI性

本文证明21阶亚循环群G不是弱3-DCI群,否定了Babai和Frankl的一个猜测.此外还决定了群G的所有3元生成子集的CI-性.     

On the isometric isomorphism of probabilistic metric spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｒｅａｒｅｔｗｏｋｉｎｄｓｏｆｉｓｏｍｅｔｒｉｃｉｓｏｍｏｒｐｈｉｓｍｉｎｐｒｏｂａｂｉｌｉｓｔｉｃｍｅｔｒｉｃｓｐａｃｅｔｈｅｏｒｙ (ｓｅｅ[1 -9] ) .ＴｈｅｆｉｒｓｔｉｓｔｈａｔａＰＭｓｐａｃｅ (Ｅ ,Ｆ)ｉｓｉｓｏｍｅｔｒｉｃａｌｌｙｉｓｏｍｏｒｐｈｉｃｔｏａｎｏｔｈｅｒＰＭｓｐａｃｅ(Ｅ′,Ｆ′) ,ａｎｄｔｈｅｓｅｃｏｎｄｉｓｔｈａｔａＰＭｓｐａｃｅｓ(Ｅ ,Ｆ)ｉｓｉｓｏｍｅｔｒｉｃａｌｌｙｉｓｏｍｏｒｐｈｉｃｔｏａｇｅｎｅｒａｔｉｎｇｓｐａｃｅｏｆｑｕａｓｉ＿ｍｅｔ…     

Isomorphous rare‐earth bis[bis(2,6‐diisopropylphenyl)phosphate] complexes and their self‐assembly into two‐dimensional frameworks by intramolecular hydrogen bonds

The crystal structures of rare‐earth diaryl‐ or dialkylphosphate derivatives are poorly explored. Crystals of bis[bis(2,6‐diisopropylphenyl)phosphato‐κO ]chloridotetrakis(methanol‐κO )neodymium methanol disolvate, [Nd(C₂₄H₃₄O₄P)Cl(CH₄O)₄]·2CH₃OH, (1), and of the lutetium, [Lu(C₂₄H₃₄O₄P)Cl(CH₄O)₄]·2CH₃OH, (2), and yttrium, [Y(C₂₄H₃₄O₄P)Cl(CH₄O)₄]·2CH₃OH, (3), analogues have been obtained by reactions between lithium bis(2,6‐diisopropylphenyl)phosphate and LnCl₃(H₂O)₆ (in a 2:1 ratio) in methanol. Compounds (1)–(3) crystallize in the C 2/c space group. Their crystal structures are isomorphous. The molecule possesses C ₂ symmetry with a twofold crystallographic axis passing through the Ln and Cl atoms. The bis(2,6‐diisopropylphenyl)phosphate ligands all display a κ¹O‐monodentate coordination mode. The coordination polyhedron for the metal atom [coordination number (CN) = 7] is a distorted pentagonal bipyramid. Each [Ln{O₂P(O‐2,6‐ⁱPr₂C₆H₃)₂}₂Cl(CH₃OH)₄] molecular unit exhibits two intramolecular O—H…O hydrogen bonds, forming six‐membered rings, and two intramolecular O—H…Cl interactions, forming four‐membered rings. Intermolecular O—H…O hydrogen bonds connect each unit via four noncoordinating methanol molecules with four other units, forming a two‐dimensional hydrogen‐bond network. Crystals of bis[bis(2,6‐diisopropylphenyl)phosphato‐κO ]tetrakis(methanol‐κO )(nitrato‐κ²O ,O ′)neodymium methanol disolvate, [Nd(C₂₄H₃₄O₄P)(NO₃)(CH₄O)₄]·2CH₃OH, (4), have been obtained in an analogous manner from NdCl₃(H₂O)₆. Compound (4) also crystalizes in the C 2/c space group. Its crystal structure is similar to those of (1)–(3). The κ²O ,O ′‐bidentate nitrate anion is disordered over a twofold axis, being located nearly on it. Half of the molecule is crystallographically unique (CN_Nd = 8). Unlike (1)–(3), complex (4) exhibits disorder of all three methanol molecules, one isopropyl group of the phosphate ligand and the NO₃⁻ ligand. The structure of (4) displays intra‐ and intermolecular O—H…O hydrogen bonds similar to those in (1)–(3). Compounds (1)–(4) represent the first reported mononuclear bis[bis(diaryl/dialkyl)phosphate] rare‐earth complexes.