The Schur Multiplier of a Pair of Groups

In this article we develop the theory of a Schur multiplier for pairs of groups. The idea of such a multiplier is implicit in the work of J.-L. Loday (1978) and others on algebraicK -theory, and in the work of Eckmann et al. (1972) and others on group homology. In contrast to their work, we focus on the general group-theoretic properties of the multiplier. These properties are systematically derived from: 1) the functoriality of the multiplier; 2) an exact homology sequence; 3) and a transfer homomorphism.     

X-ray structure of 2,6-dimethoxy-7-methylpurine

The title compound (C₈H₁₀N₄O₂) is monoclinic, with a = 7.740(2), b = 17.044(7), c = 6.992(3) Å, = 100.60(1)°, and space group P2₁/c. Two O-methyl groups are coplanar with the pyrimidine ring. Whereas, the O(6)-methyl group is directed away from the imidazole ring toward the N(1) atom, the O(2)-methyl is pointed away from the N(1) atom toward the N(3) atom. Two intermolecular hydrogen bonds H(8)···N(1) and H(711)···O(2) of 2.48(2) and 2.58(3) Å make a linear arrangement of the molecules. The conformation of the O-methyl groups explains some results of thermal rearrangement of 2,6-dialkoxy-7-methylpurines and differences in alkylations of 2,4-dialkoxypyrimidines and 2,6-dialkoxy-7-methylpurines.     

Crystal structures of four complexes produced by reaction of copper(I) cyanide with imidazoles

Four complexes of CuCN with imidazoles have been prepared by adding the ligand to a solution of CuCN in aqueous sodium thiosulfate. The imidazole ligands used were 2-methylimidazole (mim), 1,2-dimethylimidazole (dmim), 5-chloro-1-methylimidazole (clmim), and 2-phenylimidazole (phim). Complex 1, (CuCN)(mim) is monoclinic, C2/c, a = 9.565(3), b = 7.764(5), c = 8.983(8) Å, = 96.76(3)°, Z, = 4, Complex 2, (CuCN)(dmim) is monoclinic, P2₁/c, a = 8.120(2), b = 11.796(4), c = 16.375(9) Å, , = 100.87(4)°, Z = 8. Complex 3, (CuCN) (clmim) is monoclinic, C2/c, a = 24.907(4), b = 6.894(5), c = 18.259(4) Å, = 102.79(2)°, Z = 16. Complex 4, (CuCN)(phim) is orthorhombic, Pca2₁, a = 9.204(4), b = 8.125(2), c = 26.304(6) Å, Z, = 8. Complexes 1, 2, and 4 have one-dimensional chains –Cu–CN–Cu– with an imidazole bonded to each Cu. Complex 3 has a two-dimensional sheet of CuCN, again with an imidazole bonded to each Cu. In 1 the imidazole group is disordered by a two-fold rotation approximately around the N···N direction in the imidazole group. In 3 one of the imidazole groups has a disorder involving exchange of the Cl and CH₃.     

A short proof of the Approximation Conjecture for amenable groups

We give a short proof of the Approximation Conjecture with complex coefficients for amenable groups [G. Elek, The Strong Approximation Conjecture holds for amenable groups, J. Funct. Anal. 239 (1) (2006) 345–355].     

Failure of hypoellipticity for the canonical solution to $${\bar\partial_b}$$ on some nilpotent Lie groups

In this paper we show that Kohn’s solution to the \({\bar\partial_b}\) problem fails to be hypoelliptic on some class of high codimension submanifolds of \({\mathbb{C}^n}\). The examples presented here carry a Lie group structure which generalize the one-dimensional Heisenberg group.     

Covering the alternating groups by products of cycle classes

Given integers k,l?2, where either l is odd or k is even, we denote by n=n(k,l) the largest integer such that each element of An is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(An,Cl), where Cl is the set of all l-cycles in An. We prove that if k?2 and l?9 is odd and divisible by 3, then . This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368-380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87-99].     

Few-cosine spherical codes and Barnes-Wall lattices

Using Barnes-Wall lattices and 1-cocycles on finite groups of monomial matrices, we give a procedure to construct tricosine spherical codes. This was inspired by a 14-dimensional code which Ballinger, Cohn, Giansiracusa and Morris discovered in studies of the universally optimal property. Their code has 64 vectors and cosines . We construct the Optimism Code, a 4-cosine spherical code with 256 unit vectors in 16-dimensions. The cosines are . Its automorphism group has shape ²1+8⋅GL(4,2). The Optimism Code contains a subcode related to the BCGM code. The Optimism Code implies existence of a nonlinear binary code with parameters (16,256,6), a Nordstrom-Robinson code, and gives a context for determining its automorphism group, which has form .     

Pointwise bounds for L 2 eigenfunctions on locally symmetric spaces

We prove pointwise bounds for L ² eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with -rank one if the corresponding eigenvalues lie below the continuous part of the L ² spectrum. Furthermore, we use these bounds in order to obtain some results concerning the L ^p spectrum.     

Periodic decomposition of integer valued functions

We study those functions that can be written as a finite sum of periodic integer valued functions. On ℤ we give three different characterizations of these functions. For this we prove that the existence of a real valued periodic decomposition of a ℤ → ℤ function implies the existence of an integer valued periodic decomposition with the same periods. This result depends on the representation of the greatest common divisor of certain polynomials with integer coefficients as a linear combination of the given polynomials where the coefficients also belong to ℤ[x]. We give an example of an ℤ → {0, 1} function that has a bounded real valued periodic decomposition but does not have a bounded integer valued periodic decomposition with the same periods. It follows that the class of bounded ℤ → ℤ functions has the decomposition property as opposed to the class of bounded ℝ → ℤ functions. If the periods are pairwise commensurable or not prescribed, then we get more general results. Supported by OTKA grants T 43623 and K 61908.     

Effective strategy for the systematic synthesis of hydrazine derivatives

A new and efficient strategy for the systematic synthesis of hydrazine derivatives is reported. It allows the synthesis of up to tetrasubstituted hydrazine derivatives with minimal number of steps using only one protecting group or without any of them at all. Simple and readily available starting materials such as hydrazine hydrate or phenylhydrazine can be used. A variety of substrates were used to investigate scope and limitations of this strategy, additionally one full synthetic sequence was performed.