Roots of Unity in Orders

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite rings. Our techniques, which are taken from commutative algebra, may have further potential in the context of cryptology and computer algebra.     

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any $k\ge 2$, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.     

Chemical Composition of Flowers of Gossypium hirsutum

Chemistry of Natural Compounds -     

Constituents of Ferula ferulioides

Chemistry of Natural Compounds -     

A PYRIDYL-TRIAZINE COMPLEX OF Co(II): SPECTROSCOPIC,STRUCTURAL, AND DOCKING STUDIES

Journal of Structural Chemistry - A new cobalt(II) complex of 5-phenyl-3-(pyridin-2-yl)-1,2,4-triazine (PPTA), [Co(PPTA)2(NO3)2] (1), is prepared and identified by the elemental analysis, FTIR...     

Synthesis and Optoelectronic Properties of a Quinoxalino-Phenanthrophenazine (QPP) Extended Tribenzotriquinacene (TBTQ)

A six-step synthesis towards a tribenzotriquinacene (TBTQ) bearing three quinoxalinophenanthrophenazine (QPP) units is presented. The optoelectronic properties are investigated and the effect of the three-dimensional arrangement of the individual QPP planes is examined using optical spectroscopy, electrochemical analysis and quantum-chemical calculations.     

SYNTHESES,CRYSTAL STRUCTURES,AND ANTIMICROBIAL ACTIVITIES OF ZINC COMPLEXES DERIVED FROM 2-AMINO-N′- (PYRIDIN-2-YLMETHYLENE)BENZOHYDRAZIDE

Journal of Structural Chemistry - New zinc complexes, [ZnBr2(HL)] (1), [ZnBr(HL)(NCS)]·0.5H2O (2), [Zn(HL)I2] (3), and [ZnL2] (4), where L is the monoanionic form of...     

Long-Chain Alkane Dehydrogenation over Hierarchically Porous Ti-Doped Pt–Sn–K/TiO2–Al2O3 Catalysts

Kinetics and Catalysis - Hierarchically porous γ-Al2O3, TiO2–Al2O3 composite supports, and Pt–Sn–K/Al2O3 and Pt–Sn–K/TiO2–Al2O3 catalysts were prepared...     

Comparison of Yield,Chemical Composition and Antimicrobial Activity of Distichochlamys citrea Rhizome Essential Oils Obtained by Different Extraction Methods

Moscow University Chemistry Bulletin - Hydrodistillation (HD), steam distillation (SD) and microwave-assisted hydrodistillation (MAHD) extraction methods were assessed for their effects on yield,...     

Chemical Constituents of Artemisia sieversiana

Chemistry of Natural Compounds -