On Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions with periodic coefficients and Eisenstein series

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of \({SL_2(\mathbb{Z})}\) and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given.     

Distribution of averages of Ramanujan sums

The average value of a certain normalization of Ramanujan sums is determined in terms of Bernoulli numbers and odd values of the Riemann zeta function. The distribution of values and limiting behavior of such a normalization are then studied along subsets of Beurling type integers with positive density and sequences of moduli with constraints on the number of distinct prime factors.     

Block‐brush copolymers via ROMP and sequential double click reaction strategy

We report an efficient way, sequential double click reactions, for the preparation of brush copolymers with AB block‐brush architectures containing polyoxanorbornene (poly (ONB)) backbone and poly(ε‐caprolactone) (PCL), poly(methyl methacrylate) (PMMA) or poly(tert‐butyl acrylate) (PtBA) side chains: poly(ONB‐g‐PMMA)‐b‐poly(ONB‐g‐PCL) and poly(ONB‐g‐PtBA)‐b‐poly(ONB‐g‐PCL). The living ROMP of ONB affords the synthesis of well‐defined poly(ONB‐anthracene)₂₀‐b‐poly (ONB‐azide)₅ block copolymer with anthryl and azide pendant groups. Subsequently, well‐defined linear alkyne end‐functionalized PCL (PCL‐alkyne), maleimide end‐functionalized PMMA (PMMA‐MI) and PtBA‐MI were introduced onto the block copolymer via sequential azide‐alkyne and Diels‐Alder click reactions, thus yielding block‐brush copolymers. The molecular weight of block‐brush copolymers was measured via triple detection GPC (TD‐GPC) introducing the experimentally calculated dn/dc values to the software. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Immobilization of invertase in conducting copolymers of 3-methylthienyl methacrylate

Immobilization of invertase in conducting copolymer matrices of 3-methylthienyl methacrylate with pyrrole and thiophene was achieved by constant potential electrolysis using sodium dodecyl sulfate (SDS) as the supporting electrolyte. Polythiophene (PTh) was also used in entrapment process for comparison. Kinetic parameters, Michaelis-Menten constant, K(m), and the maximum reaction rate, V(max), were investigated. Operational stability and temperature optimization of the enzyme electrodes were also examined.     

Square-free criteria for polynomials using no derivatives

We provide some square-free criteria for primitive polynomials over unique factorization domains, which do not make use of derivatives or discriminants. Using some ideas of Ostrowski we establish nonvanishing conditions for determinants of matrices with polynomial entries and deduce square-free criteria for polynomials in several variables.

     

Heterotrimetallic Mixed‐Valent Molecular Precursors Containing Periodic Table Neighbors: Assignment of Metal Positions and Oxidation States

A known trinuclear structure was used to design the heterobimetallic mixed‐valent, mixed‐ligand molecule [Co^II(hfac)₃?Na?Co^III(acac)₃] ( 1 ). This was used as a template structure to develop heterotrimetallic molecules [Co^II(hfac)₃?Na?Fe^III(acac)₃] ( 2 ) and [Ni^II(hfac)₃?Na?Co^III(acac)₃] ( 3 ) via isovalent site‐specific substitution at either of the cobalt positions. Diffraction methods, synchrotron resonant diffraction, and multiple‐wavelength anomalous diffraction were applied beyond simple structural investigation to provide an unambiguous assignment of the positions and oxidation states for the periodic table neighbors in the heterometallic assemblies. Molecules of 2 and 3 are true heterotrimetallic rather than a statistical mixture of two heterobimetallic counterparts. Trinuclear platform 1 exhibits flexibility in accommodating a variety of di‐ and trivalent metals, which can be further utilized in the design of molecular precursors for the NaMM′O₄ functional oxide materials.     

Diophantine approximation with mild divisibility constraints

Given an irrational number α and a sequence B of coprime positive integers with the sum of inverses convergent, we investigate the problem of finding small values of nα, with n B-free.     

Medium and structural effects on the ionization constants of some pyrazole carboxylic acid derivatives

Abstract  Stoichiometric ionization constants of some pyrazole carboxylic acids [4-benzoyl-1-(3-nitrophenyl)-5-phenyl-1H-pyrazole-3-carboxylic acid, 4-benzoyl-1-(4-nitrophenyl)-5-phenyl-1H-pyrazole-3-carboxylic acid, 4-(ethoxycarbonyl)-1,5-diphenyl-1H-pyrazole-3-carboxylic acid, 4-(ethoxycarbonyl)-1-(3-nitrophenyl)-5-phenyl-1H-pyrazole-3-carboxylic acid, 4-(ethoxycarbonyl)-1-(4-nitrophenyl)-5-phenyl-1H-pyrazole-3-carboxylic acid] were determined in ethanol–water mixtures of 50, 60, 70% ethanol (v/v) by a potentiometric titration method. Titrations were performed in an ionic strength of 0.10 M NaCl at 25.0 ± 0.1 °C using an Orion 960 automatic titrator under a nitrogen atmosphere. Using the potentiometric titration data, ionization constants were calculated in three different ways. The effects of structure and solvent on the acidity of pyrazole carboxylic acids are also discussed. Graphical abstract        

Iron(III) complex of a tridentate <Emphasis Type="Italic">NNN</Emphasis> type ligand: synthesis,X-ray structure,thermal analysis and voltammetric studies

A mononuclear iron(III) complex with 2,6-bis(3,4,5-trimethyl-N-pyrazolyl)pyridine (btmpp) has been synthesized, and characterized by X-ray crystallography, IR, UV–Vis and TGA–DTA methods. The Fe(III/II) couple was characterized by way of cyclic voltammetry using DMF solvent, where the complex was observed to have an irreversible behaviour.