The Effect of Alkane Chain Length on the Liquid–Liquid Critical Temperatures of Oligostyrene/Linear-Alkane Mixtures

Summary. Upper critical solution temperatures (UCSTs) for liquid–liquid demixing in a set of mixtures of linear alkanes (pentane (N ₁=5) to pentacontane (N ₁=50)) with an oligostyrene (1241amu, N ₂=12) are reported. We find strong correlation between the Hildebrand solubility parameters of the alkanes and the UCST. Correlations are developed which enable predictions concerning the miscibility of mixtures of compounds with longer chains.     

Aquachlorobis[4,4,5,5‐tetramethyl‐2‐(2‐pyridyl)‐4,5‐dihydro‐1H‐imidazol‐1‐oxyl 3‐oxide‐κ2O1,N2]nickel(II) nitrate methanol disolvate monohydrate

The title complex, [NiCl(C₁₂H₁₆N₃O₂)₂(H₂O)]NO₃·2CH₄O·H₂O, was obtained from a methano­lic solution of Ni(NO₃)₂·6H₂O, 2‐pyridyl nitro­nyl nitro­xide (2‐NITpy) and (NEt₄)₂[CoCl₄]. The equatorial coordination sites of the octahedral Ni^II centre are occupied by two chelating radical ligands, with the axial positions occupied by the Cl^? and water ligands. The H₂O—Ni—Cl axis of the complex lies along a crystallographic twofold axis, so that only half the cation is present in the asymmetric unit. The Ni—Cl bond length [2.3614 (17) Å] is significantly shorter than distances typical of octahedral Ni^II centres [2.441 (5) Å]. However, with only one nitrate anion per formula unit, the oxidation state of the metal must be assigned as Ni^II. The 2‐NITpy ligands bend away from the equatorial plane, forming a hydro­phobic region around the Cl atoms. Conversely, the ligated water mol­ecule forms moderately strong hydrogen bonds with the disordered methanol solvent mol­ecules, which in turn form interactions with the water of crystallization and the disordered nitrate anion. These interactions combine to give hydro­philic regions throughout the crystal structure.     

Some non-existence results on divisible difference sets

In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:

1. character sum arguments similar to the work of Turyn [25] for ordinary difference sets,
2. involution arguments and
3. multipliers in conjunction with results on ordinary difference sets.

Among other results, we show that an abelian affine difference set of odd orders (s not a perfect square) inG can exist only if the Sylow 2-subgroup ofG is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd ordern.     

Principal part of resolvent and factorization of an increasing nonanalytic operator-function

Let a selfadjoint operator-valued functionL() be given on the interval [a,b] such thatL(a)0,L(b)0,L()0 (ab), andL() has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued functionL() can be reduced to the spectral theory of one operatorZ, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L()=M()(I–Z), where the operator-valued functionM() is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition ofL ^–1() into the sume of its principal and regular parts.     

New necessary conditions on the existence of abelian difference sets

In this paper we develop a new method to obtain identities in a group algebraGF(p)G if an abelian difference set of ordern0 (modp) exists inG. We give an explicit formula ifp ² orp ³ is the exactp-power dividingn. This generalizes the approach of Wilbrink, Arasu and the author. The proof presented here uses some knowledge about field extensions of thep-adic numbers.     

On abelian difference set codes

In this paper we determine the ranks of the incidence matrices that belong to the following types of difference sets: Twin prime power difference sets, biquadratic residues and biquadratic residues with 0. We also prove a conjecture of Assmus and Key on the code generated by the hyperovals of PG(2, q).