Analogue of the degree conjecture over function fields

Under a certain assumption, similar to Manin's conjecture, we prove an upper bound on the degree of modular parametrizations of elliptic curves by Drinfeld modular curves, which is the function field analogue of the conjectured bound over the rational numbers.

     

Oligonucleotide Analogues with Integrated Bases and Backbone. Part 14

The self‐complementary tetrameric propargyl triols 8, 14, 18 , and 21 were synthesized to investigate the duplex formation of self‐complementary, ethynylene‐linked UUAA, AAUU, UAUA, and AUAU analogues with integrated bases and backbone (ONIBs). The linear synthesis is based on repetitive Sonogashira couplings and C‐desilylations (34–72% yield), starting from the monomeric propargyl alcohols 9 and 15 and the iodinated nucleosides 3, 7, 11 , and 13 . Strongly persistent intramolecular H‐bonds from the propargylic OH groups to N(3) of the adenosine units prevent the gg‐type orientation of the ethynyl groups at C(5′). As such, an orientation is required for the formation of cyclic duplexes, this H‐bond prevents the formation of duplexes connected by all four base pairs. However, the central units of the UAUA and AAUU analogues 18 and 14 associate in CDCl₃/(D₆)DMSO 10 : 1 to form a cyclic duplex characterized by reverse Hoogsteen base pairing. The UUAA tetramer 8 forms a cyclic UU homoduplex, while the AUAU tetramer 21 forms only linear associates. Duplex formation of the O‐silylated UUAA and AAUU tetramers is no longer prevented. The self‐complementary UUAA tetramer 22 forms Watson–Crick‐ and Hoogsteen‐type base‐paired cyclic duplexes more readily than the sequence‐isomeric AAUU tetramer 23 , further illustrating the sequence selectivity of duplex formation.     

The ionization of sulfurous acid in Na−Mg−Cl solutions at 25°C

The stoichiometric pK ₁ ^* and pK ₂ ^* for the ionization of sulfurous acid has been determined from emf measurements in NaCl solutions with varying concentrations of added MgCl₂ (m=0.1, 0.2 and 0.3) from I=0.5 to 6.0 molal at 25°C. These experimental results have been treated using both the ion pairing and Pitzer's specific ion-interaction models. The Pitzer parameters for the interaction of Mg²⁺ with SO₂ and HSO ₃ ^– yielded =0.085±0.004, ⁽⁰⁾ = 0.35±0.02, ⁽¹⁾ = 1.2±0.04, and C = –0.072±0.007. The Pitzer parameters ⁽⁰) = –2.8±0.4, ⁽¹⁾ = 12.9±2.9 and ⁽²⁾ = –2071±57 have been determined for the interactions of Mg²⁺ with SO ₃ ^2– . The calculated values of pK ₁ ^* and pK ₂ ^* using Pitzer's equations reproduce the measured values to within ±0.04 pK units. The ion pairing model with log K_MgSO3=2.36±0.02 and log_MgSO3 = 0.1021, reproduces the experimental values of pK ₂ ^* to ±0.01. These results demonstrate that treating the data by considering the formation of MgSO₃ yields a better fit of the experimental measurements with fewer adjustable parameters. With these derived coefficients obtained from the Pitzer equations and the ion pairing model, it is possible to make reliable estimates of the activity coefficients of HSO ₃ ^– and SO ₃ ^2– in seawater, brines and marine aerosols containing Mg²⁺ ions.     

Mechanistic aspects of the Wolff-Kishner reaction. V. The cation effect on the reaction of benzophenone hydrazone in butyl carbitol and decyl alcohol

The rate of the Wolff-Kishner reaction of benzophenone hydrazone in butyl carbitol increases as the cation of the alkoxide base is varied in the order K>Na>Li>Mg. The replacement of butyl carbitol by 1-decanol also accelerates the reaction, and an additional increase is caused by the presence of crown ether. On the basis of changes in the activation parameters, it is concluded that the reactivity of the hydrazone anion increases as the contact ion pair becomes more fully solvated and dissociated.     

Molar volumes and ion pairing of lithium halides in alcohols

The concentration dependence of the apparent molar volumes of lithium halides (and electrolytes in general) in alcohols (and solvents permitting association in general) is, in the first instance, due to changes in the degree of association and to the inherent difference between the apparent molar volumes of the ions and of the ion pairs. Previous publications on the molar volumes of electrolytes in organic solvents, disregarding altogether ion pairing, appear to be incorrect. Data from the literature for lithium chloride and lithium bromide in normal primary alcohols and several branched alcohols from C₁ to C₈ and data from our laboratory for lithium halides in 1-hexanol and 2-ethyl-1-hexanol served for the determination of φ _V and φ _E . Electrical and structural contributions to the values of these functions for the ions and for the ion pairs are discussed.     

Dielectric relaxation of electrolyte solutions in acetonitrile

Complex permittivity spectra in the frequency range 0.95v (GHz)89 for acetonitrile and its solutions of LiBr, NaI, NaClO₄, and Bu₄NBr at 25°C show one Debye equation for the neat solvent whereas the superposition of a Debye process for the solute and a Cole-Cole distribution for the solvent is necessary to account for the dielectric relaxation behavior of the solutions. The reorientation of bulk acetonitrile is diffusive and only weakly coupled to viscosity. The number of solvent molecules irrotationally bound to the electrolyte is in good agreement with conventional solvation numbers for all electrolytes, when kinetic depolarization is assumed to be negligible. The solute relaxation process is dominated by the formation kinetics and reorientation of contact ion pairs. There is evidence for solvent-shared ion pairs in dilute NaClO₄ solutions.     

Raman spectral study of the constitution and equilibria of nitric acid and d-nitric acid

From relative integrated intensity measurements of the symmetric stretching vibration of nitrate ion in nitric acid solutions (both HNO ₃ /H ₂ O and DNO ₃ /D ₂ O), the mass law concentration quotients, Q were obtained as functions of concentration. By extrapolation the limiting dissociation constants were estimated to be 24.4 and 15 respectively at 25°C. It is shown that this constant refers to a process in which the ion pair H ₃ O⁺ NO ₃ ^– is in equilibrium with the dispersed, solvated H ₃ O⁺ and NO ₃ ^– ions.     

Fractional Power Series and Pairings on Drinfeld Modules

Let be an algebraically closed field containing which is complete with respect to an absolute value . We prove that under suitable constraints on the coefficients, the series converges to a surjective, open, continuous -linear homomorphism whose kernel is locally compact. We characterize the locally compact sub--vector spaces of which occur as kernels of such series, and describe the extent to which determines the series. We develop a theory of Newton polygons for these series which lets us compute the Haar measure of the set of zeros of of a given valuation, given the valuations of the coefficients. The ``adjoint' series converges everywhere if and only if does, and in this case there is a natural bilinear pairing

which exhibits as the Pontryagin dual of . Many of these results extend to non-linear fractional power series. We apply these results to construct a Drinfeld module analogue of the Weil pairing, and to describe the topological module structure of the kernel of the adjoint exponential of a Drinfeld module.

     

Effect of ion pairing on steady-state voltammetric limiting currents at microelectrodes Part I. Theoretical principles

Voltammetric studies in solutions of high resistivity are facilitated by the use of microelectrodes under steady-state conditions. Such solutions are encountered with solvents of low permittivity because of the very sparing solubility of electrolytes. Moreover, in such media the supporting electrolyte, as well as the electroactive ionic species, is usually extensively ion paired. Here we predict the limiting current that will flow in these circumstances, when a monovalent ion undergoes a one-electron transfer at a hemispherical microelectrode to form a neutral product. The ion pairing equilibria are assumed to be fast but all diffusion coefficients are treated as distinct. An analytical solution is elusive in the general case, but a simple numerical procedure allows the limiting current to be predicted for any combination of the system parameters. Several special cases are also discussed, some of which yield explicit formulae for the limiting current. In a companion paper, experimental data are compared with the theoretical predictions.     

Osmotic and activity coefficients of sodium chloride-sorbitol and potassium chloride-sorbitol solutions at 25°C

Osmotic and activity coefficients are reported for sorbitol over the range 1.8–11.1m with NaCl as the reference and 1.8–7.6m with KCl as the reference electrolyte. The osmotic coefficients over the range 0–3.7m are identical with those reported earlier by Robinson. It was found that the activity coefficients of sorbitol are uniformly about four percent lower than those of dextrose. Activity coefficients of trace quantities of NaCl in concentrated sorbitol solutions are about one-half of their value in pure NaCl solutions while the activity coefficients of trace quantities of sorbitol in concentrated NaCl solutions are about one-third of their value in pure sorbitol solutions. Potassium chloride lowers the activity coefficient of sorbitol less than does sodium chloride in solutions of similar water activity. Sorbitol lowers the activity coefficients of potassium chloride in concentrated solution but actually elevates them in dilute solutions.