Discrete analysis on a radial lattice

A discrete model for analytic functions is constructed using lattice points of the complex plane arranged in radial form. The discrete analytic functions are defined as solutions of a finite-difference approximation to the polar Cauchy-Riemann equations. The resulting discrete powerz⁽ⁿ⁾ (an analogue of zⁿ) has a simple algebraic form (a direct analogue of ?ⁿ exp{inθ}) and has some surprising properties. For example every discrete polynomial ∑₀^ma_nz⁽ⁿ⁾ has a factorization in terms of the zeros of its classical counterpart ∑₀^ma_nzⁿ every discrete entire function has a power series representation ∑ a_nz⁽ⁿ⁾.     

On the greatest prime factor of with effective constants

Let denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of exceeding in which the constants are effectively computable. As a result we prove that it is possible to calculate a value such that for every x_0$"> there is a with the greatest prime factor of exceeding . The novelty of our approach is the avoidance of any appeal to Siegel's Theorem on primes in arithmetic progression.

     

Hydroamination of Dihapto-Coordinated Benzene and Diene Complexes of Tungsten: Fundamental Studies and the Synthesis of γ-Lycorane

Reactions are described for complexes of the form WTp(NO)(PMe₃)(η²-arene) and various amines, where the arene is benzene or benzene with an electron-withdrawing substituent (CF₃, SO₂Ph, SO₂Me). The arene complex is first protonated to form an η²-arenium species, which then selectively adds the amine. The resulting η²-5-amino-1,3-cyclohexadiene complexes can then be subjected to the same sequence with a second nucleophile to form 3-aminocyclohexene complexes, where up to three stereocenters originate from the arene carbons. Alternatively, 1,3-cyclohexadiene complexes containing an ester group at the 5 position (also prepared from an arene) can be treated with acid followed by an amine to form trisubstituted 3-aminocyclohexenes. When the amine is primary, ring closure can occur to form a cis-fused bicyclic γ-lactam. Highly functionalized cyclohexenes can be liberated from the tungsten through oxidative decomplexation. The potential utility of this methodology is demonstrated in the synthesis of the alkaloid γ-lycorane. An enantioenriched synthesis of a lactam precursor to γ-lycorane is also described. This compound is prepared from an enantioenriched version of the tungsten benzene complex. Regio- and stereochemical assignments for the reported compounds are supported by detailed 2D-NMR analysis and 13 molecular structure determinations (SC-XRD).     

A structurally characterized nitrous oxide complex of vanadium

Nitrous oxide (N(2)O), a widespread greenhouse gas, is a thermodynamically potent and environmentally green oxidant that is an attractive target for activation by metal centers. However, N(2)O remains underutilized owing to its high kinetic stability, and the poor ligand properties of this molecule have made well-characterized metal-N(2)O complexes a rarity. We now report a vanadium-pyrrolide system that reversibly binds N(2)O at room temperature and provide the first single-crystal X-ray structure of such a complex. Further characterization by vibrational spectroscopy and DFT calculations strongly favor assignment as a linear, N-bound metal-N(2)O complex.     

Well-defined vanadium organoazide complexes and their conversion to terminal vanadium imides: structural snapshots and evidence for a nitrene capture mechanism

We report the synthesis and structural characterization of a family of well-defined organoazide complexes supported by a three-fold-symmetric pyrrolide scaffold and their conversion to the corresponding terminal imido congeners. Kinetic measurements on a series of structurally homologous but electronically distinct vanadium organoazide complexes reveal that the azido-to-imido transformations proceed via a process that is first-order in the metal complex and has a positive entropy of activation. Further studies suggest that these reactions may involve metal-mediated generation and capture of nitrene fragments.     

Diophantine inequalities with mixed powers

It is shown that if λ₁,…, λ₆ are nonzero real numbers, not all of the same sign, such that $\text{λ}{}_1\text{λ}{}_2$ is irrational, then the values taken by λ₁x₁² + λ₂x₂² + λ₃x₃³ + λ₄x₄³ + λ₅x₅⁵ + λ₆x₆⁵ for integral x₁,…, x₆ are everywhere dense on the real line. Similar results are proved with other combinations in place of the two fifth powers.     

Single value rating methods

A variety of sound and partition interactions have been examined and the numerical changes produced in the various sound ratings are compared with the partition ratings. Some recommendations are made regarding the use of single value ratings for partition performance.