Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

Injective resolutions of Noetherian rings and cogenerators

We give new construction of injective resolutions of complexes and bimodules. Applying this construction to an injective resolution of a Noetherian ring, we construct a -embedding cogenerator for the category of modules of projective dimension . Moreover, for a Noetherian projective -algebra , we show that satisfies the Auslander condition if and only if the flat dimension of every -module is equal to or larger than the one of the injective hull .

     

An improvement of discrete Tardos fingerprinting codes

It has been proven that the code lengths of Tardos’s collusion-secure fingerprinting codes are of theoretically minimal order with respect to the number of adversarial users (pirates). However, the code lengths can be further reduced as some preceding studies have revealed. In this article we improve a recent discrete variant of Tardos’s codes, and give a security proof of our codes under an assumption weaker than the original Marking Assumption. Our analysis shows that our codes have significantly shorter lengths than Tardos’s codes. For example, when c = 8, our code length is about 4.94% of Tardos’s code in a practical setting and about 4.62% in a certain limit case. Our code lengths for large c are asymptotically about 5.35% of Tardos’s codes. A part of this work was presented at 17th Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-17), Bangalore, India, December 16–20, 2007.     

Theoretical studies on magnetic interactions between Cu(II) ions in hydroxypyridone nucleobases

The origin of a ferromagnetic interaction between Cu²⁺ ions in the Cu²⁺–DNA system which reported by Tanaka et al. is examined by using DFT calculations. In order to consider effects of an entanglement and a dis-entanglement of the double helix chain, three types of structural disorders i.e. distance, rotation angle and discrepancy in XY-plane, are considered in the model dimer structure. All calculated results show that J_ab values are weak anti-ferromagnetic couplings. Boltzmann distribution simulation indicates that the high spin (HS) species exist 21% at 1.5 K by thermal excitation within the model structure.     

Molecular distortion effect on ff-emission in a Pr(III) complex with 4,7-diphenyl-1,10-phenanthroline.

The electronic and structural behaviour of a Pr(III) complex with 4,7-diphenyl-1,10-phenanthroline, [Pr(bathophen)(2)(NO(3))(3)], is investigated with respect to the effect of configuration changes on the Pr(III) centre. [Pr(bathophen)(2)(NO(3))(3)] luminesces from the excited states of the ligand and the metal ion. The fluorescence, ff-emission ((1)D(2)-->(3)H(4)), and phosphorescence bands appear at 394, 608.2 and 482 nm, respectively, in the solid state. In acetonitrile, the complex also shows multiple emissions. From the time-resolved emission and the lifetime measurements, the excitation energy-transfer in [Pr(bathophen)(2)(NO(3))(3)] is clarified, that is, the upper excited triplet level of the ligand acts as an energy donor, while the (1)D(2) levels of Pr(III) is the acceptor. Additionally, the emission phenomena of the complex can be modified by molecular distortion, particularly by rotation of the phenyl groups in the ligand.