An analysis of the Kimura 3ST model of DNA sequence evolution is given on the basis of its continuous Lie symmetries. The rate matrix commutes with a U(1) × U(1) × U(1) phase subgroup of the group GL(4) of 4 × 4 invertible complex matrices acting on a linear space spanned by the four nucleic acid base letters. The diagonal 'branching operator' representing speciation is defined, and shown to intertwine the U(1)×U(1)×U(1) action. Using the intertwining property, a general formula for the probability density on the leaves of a binary tree under the Kimura model is derived, which is shown to be equivalent to established phylogenetic spectral transform methods.The use of Markov models of stochastic change to taxonomic character distributions is part of the standard armoury of techniques for describing mutations and inferring ancestral relationships between taxa. For the simplest models, symmetries of the rate matrix under discrete group actions (Z 2 for binary types, or Z 2 × Z 2 for DNA or RNA bases in molecular applications, for example) have been used to good effect in simplifying phylogenetic analysis (for references, see below). In particular, much attention has been centred on properties of the frequently used Kimura 3ST model [1] which possesses such symmetry.A general framework for phylogenetic branching models is as follows [2]. By assumption, different taxonomic units are identified, and classified by a set of defining characteristics: for example, based on morphological features or on sequence data, say, for a particular gene or protein. To each taxon is ascribed a character probability density, and it is the task of phylogenetic reconstruction to infer ancestral relationships amongst a group of related taxa, given sample character frequencies.In this letter, we describe an approach to the analysis of symmetries of such models using continuous transformation groups. Rather than identifying the character types with

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The title compounds, LiKB(4)O(7) and LiRbB(4)O(7), are newly developed non-linear optical crystals containing two kinds of anionic groups, namely (B(3)O(8))(7-) and (B(5)O(10))(5-). The (B(3)O(8))(7-) groups form infinite spiral chains parallel to the [100] direction, which are interconnected by sharing O atoms with (B(5)O(10))(5-) groups.

We report a case of Langerhans cell histiocytosis(LCH)of the mandible that was treated by intralesional injection of corticosteroids. A 57-year-old man was referred to our clinic because of gingival swelling and tenderness around the left molar region of the mandible. Computed tomography revealed a relatively well-defined bone defect and a pathologic fracture in the left body of the mandible. An excisional biopsy was performed, and the lesion was histopathologically diagnosed as LCH. Since the patient refused surgical excision, intralesional injection of corticosteroids was performed. The lesion and the pathologic fracture became undetectable 5 months after the administration of 40mg dexamethasone. There was no recurrence after a follow-up period of 1 year 5 months. Intralesional injection of corticosteroid may be one beneficial option for the treatment of LCH of the mandible.

structure structure (solids and liquids) D 2000 10 -004 Lithium Potassium Borate and Lithium Rubidium Borate: New Non-linear Optical Crystals. -Single crystals of LiKB 4 O 7 and LiRbB 4 O 7 , newly developed non-linear optical crystals, are grown by the Czochralski method. The isotypic compounds crystallize in the orthorhombic space group P2 1 2 1 2 1 (Z = 8) and contain two kinds of anions, (B 3 O 8 ) 7− and (B 5 O 10 ) 5− . The (B 3 O 8 ) 7− groups form infinite spiral chains parallel to the [100] direction, which are interconnected by sharing O atoms with the (B 5 O 10 ) 5− groups. -(ONO, YASUHIRO; NAKAYA, MICHIKO; KAJITANI,