Synthesis, crystal structure and magnetic property of a new nickel selenite chloride: Ni5(SeO3)4Cl2

The new nickel selenite chloride, Ni₅(SeO₃)₄Cl₂, was obtained by high-temperature solid state reaction of NiCl₂, Ni₂O₃ and SeO₂ in a 1:2:4 molar ratio at 700 °C in an evacuated quartz tube. Its structure was established by single-crystal X-ray diffraction. Ni₅(SeO₃)₄Cl₂ crystallizes in the triclinic system, space group P-1 (No. 2) with cell parameters of a=8.076(2), b=9.288(2), c=9.376(2) Å, α=101.97(3), β=105.60(3), γ=91.83(3)° and Z=2. All nickel(II) ions in Ni₅(SeO₃)₄Cl₂ are octahedrally coordinated by selenite oxygens or/and chloride anions (([Ni(1)O₅Cl], [Ni(2)O₄Cl₂], [Ni(3)O₅Cl], [Ni(4)O₆] and [Ni(5)O₄Cl]). The structure of the title compound features a condensed three-dimensional (3D) network built by Ni(II) ions interconnected by SeO₃²⁻ anions as well as Cl⁻ anions. Magnetic property measurements show strong antiferromagnetic interaction between nickel(II) ions.     

Bruck nets,codes, and characters of loops

Numerous computational examples suggest that if _{k-1 k} are (k- 1)- and k-nets of order n, then rank _{p k} - rank _{p k-1} n - k + 1 for any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that if p ^e n, then rank _{p 3} 3n - 2 - e, where equality holds if and only if the loop G coordinatizing ₃ has a normal subloop K such that G/K is an elementary abelian group of order p ^e . Furthermore if n is squarefree, then rank _p = 3n - 3 for every prime p ¦ n, if and only if ₃ is cyclic (i.e., ₃ is coordinated by a cyclic group of order n).The validity of our conjectured lower bound would imply that any projective plane of squarefree order, or of order n 2 mod 4, is in fact desarguesian of prime order.     

Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations

Summary This paper is a continuation of our previous work [10] on projection methods. We study first existing higher order projection schemes in the semidiscretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.This work is partially supported by NSF grant MS-8802596.     

Rare numbers

Suppose thatX ₁,X ₂,... is a sequence of iid random variables taking values inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofn ^–(1–) Y _n for [0, 1].     

Gap-Free Set Partitions

A set partition is called “gap-free” if its block sizes form an interval. In other words, there is at least one block of each size between the smallest and largest block sizes. Let B(n) and G(n), respectively, denote the number of partitions and the number of gap-free partitions of the set [n]. We prove that      

Reactions of (di)manganese carbonyl ions with 1,4,7-trimethyl-1,4,7-triazacyclononane (Me3TACN) in the gas phase

The gas-phase reactions of a series of (di)manganese carbonyl positive ions with 1,4,7-trimethyl-1,4,7-triazacyclononane (Me(3)TACN) have been examined with the aid of Fourier transform ion cyclotron resonance (FTICR) mass spectrometry. The monomanganese carbonyl ions, [Mn(CO)(n)](+) (n = 2-5), react predominantly by ligand exchange and to a minor extent by electron transfer with the formation of the radical cation of Me(3)TACN. For the [Mn(CO)(n)](+) (n = 2-4) ions, the ligand exchange results in the exclusive formation of a [Mn(Me(3)TACN)](+) complex, whereas small amounts of [Mn(CO)(Me(3)TACN)](+) ions are also generated in the reactions of the [Mn(CO)(5)](+) ion. The [Mn(2)(CO)(n)](+) ions (n = 2, 4 and 5) react also by competing electron transfer and ligand exchange. The reaction of the [Mn(2)(CO)(2)](+) and [Mn(2)(CO)(4)](+) ions is associated with cleavage of the Mn--Mn bond as evidenced by the pronounced formation of [Mn(Me(3)TACN)](+) ions. For [Mn(2)(CO)(5)](+), the ligand exchange leads mainly to the formation of [Mn(2)(CO)(n)(Me(3)TACN)](+) (n = 1-3) ions. These primary product ions react subsequently by the incorporation of a second Me(3)TACN molecule to afford [Mn(2)(CO)(Me(3)TACN)(2)](+) and [Mn(2)(CO)(2)(Me(3)TACN)(2)](+) ions. Both of these latter species incorporate an oxygen molecule with formation of ions with the assigned composition of [Mn(2)(O(2))(CO)(Me(3)TACN)(2)](+) and [Mn(2)(O(2))(CO)(2)(Me(3)TACN)(2)](+).