Upper bounds on a two-term exponential sum*

We obtain upper bounds for mixed exponential sums of the type where p^m is a prime power with m⩾ 2 and X is a multiplicative character (mod p^m). If X is primitive or p⫮(a, b) then we obtain |S(χ,f,p ^m)| ⩽2np ^{2/3 m}. If X is of conductor p and p⫮( a, b) then we get the stronger bound |S(χ,f,p ^m)|⩽np ^m/2. This paper is dedicated to Prof. Wang Yuan on the occasion of his 70th birthday.     

Small solutions of the congruencea1x12 +a2x22 +a3x32 +a4x42 ≡c(modp)

For any integersa ₁,a ₂,a ₃,a ₄ andc witha ₁ a ₂ a ₃ a ₄≢0(modp), this paper shows that there exists a solutionX=(x ₁,x ₂,x ₃,x ₄) ∈Z ⁴ of the congruencea ₁ x ₁ ² +a ₂ x ₂ ² +a ₃ x ₃ ² +a ₄ x ₄ ² ≡c(modp) such that

The interaction of an osmium-carbon triple bond with copper(I), silver(I) and gold(I) to give mixed dimetallocyclopropene species and the structures of Os(AgCl) (CR) Cl(CO) (PPh3)2

The osmium carbyne complex, Os(CR)Cl(CO)(PPh₃)₂, (R  p-tolyl) reacts with Group I halides to form the mixed dimetallocyclopropene species, $\text{Os(Cul)(C}$R)Cl(CO)(PPh₃)₂, $\text{Os(AgCl)(C}$R)Cl(CO)(PPh₃)₂, $\text{Os(AuCl)(C}$R)Cl(CO)(PPh₃)₂, and [$\text{Os[Ag(OClO}{}_3\text{)](C}$R)Cl(CO)(MeCN)(PPh₃)₂] ClO₄ X-ray crystal structure determination of $\text{Os(AgCl)(C}$R)Cl(CO)(PPh₃)₂ confirms the presence of a three-membered ring and the structure can be viewed as the “acetylene-like” interaction of an osmium—carbon triple bond with AgCl. In acid solution AgCl is precipitated and an alkylidene complex results from proton addition to the carbyne ligand.     

An improved Mordell type bound for exponential sums

For a sparse polynomial , with and , we show that

thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.

     

Gaps between integers with the same prime factors

We give numerical and theoretical evidence in support of the conjecture of Dressler that between any two positive integers having the same prime factors there is a prime. In particular, it is shown that the abc conjecture implies that the gap between two consecutive such numbers is , and it is shown that this lower bound is best possible. Dressler's conjecture is verified for values of and up to .

     

New structural types and different oxidation levels in the family of Mn6-oxime single-molecule magnets

The synthesis and magnetic properties of three new members of a family of salicyaldoxime based [Mn6] single-molecule magnets possessing new structural types, core topologies and Mn oxidation state distributions are reported. The isostructural complexes [MnIII6O2(R-sao)6(X)2(EtOH)6] (R = Et, X = Br (1); R = Me, X = I (2)) exhibit single-molecule magnet behaviour with spin Hamiltonian parameters S = 12, g = 1.98 and D = -0.36 cm(-1) in both cases. The hexametallic cluster [MnIII4MnIV2O2(OMe)(4-)(Et-sao)6(MeOH)2].MeOH (3.MeOH) possesses a planar rod-like topology and a mixed valent [MnIV4MnIII2] core, which is unprecedented in this family of [Mn6] SMMs.     

Solid supported chemical syntheses of both components of the lantibiotic lacticin 3147

Lantibiotics are antimicrobial peptides produced by bacteria. Some are employed for food preservation, whereas others have therapeutic potential due to their activity against organisms resistant to current antibiotics. They are ribosomally synthesized and posttranslationally modified by dehydration of serine and threonine residues followed by attack of thiols of cysteines to form monosulfide lanthionine and methyllanthionine rings, respectively. Chemical synthesis of peptide analogues is a powerful method to verify stereochemistry and access structure-activity relationships. However, solid supported synthesis of lantibiotics has been difficult due to problems in generating lanthionines and methyllanthionines with orthogonal protection and good stereochemical control. We report the solid-phase syntheses of both peptides of a two-component lantibiotic, lacticin 3147. Both successive and interlocking ring systems were synthesized on-resin, thereby providing a general methodology for this family of natural products.     

Magnetism and disorder effects on muon spin rotation measurements of the magnetic penetration depth in iron-arsenic superconductors

It is shown that attempts to accurately deduce the magnetic penetration depth λ of overdoped BaFe(1.82)Co(0.18)As? single crystals by transverse-field muon spin rotation (TF μSR) are thwarted by field-induced magnetic order and strong vortex-lattice disorder. We explain how substantial deviations from the magnetic field distribution of a nearly perfect vortex lattice by one or both of these factors is also significant for other iron-arsenic superconductors, and this introduces considerable uncertainty in the values of λ obtained by TF μSR.     

The connection between Hall effect and magnetism

Experimental measurements of Hall effect on amorphous metals and, in particular, alloys containing magnetic components, often exhibit a simple relationship between Hall coefficientR _H and magnetic susceptibility. The scattering times, however, do not seem to affect either sign or magnitude of the Hall coefficient. This is in strong contrast to conventional theories of the Hall effect in materials. The present paper demonstrates that a theoretical foundation can be found for the experimental observations.