Exact controllability and stabilizability of the Korteweg-de Vries equation

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation

 

on the interval , with periodic boundary conditions

 

where the distributed control is restricted so that the ``volume' of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control is allowed to act on the whole spatial domain , it is shown that the system is globally exactly controllable, i.e., for given and functions , with the same ``volume', one can alway find a control so that the system (i)--(ii) has a solution satisfying

If the control is allowed to act on only a small subset of the domain , then the same result still holds if the initial and terminal states, and , have small ``amplitude' in a certain sense. In the case of closed loop control, the distributed control is assumed to be generated by a linear feedback law conserving the ``volume' while monotonically reducing . The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.

     

The effect of random restrictions on formula size

We consider the formula size complexity of boolean functions over the base consisting of ∧, ∨, and ¬ gates. In 1961 Subbotovskaya proved that, for any boolean function on n variables, setting all but a randomly chosen ?n variables to randomly chosen constants, reduces the expected complexity of the induced function by at least a factor. This fact was used by Subbotovskaya to derive a lower bound of Ω(n^1.5) for the formula size of the parity function, and more recently by Andreev who exhibited at lower bound of ΩC(n^2.5/log^O(1)(n)) for a function in P. We present the first improvement of Subbotovskaya's result, showing that the expected formual complexity of a function restricted as above is at most an O(?^γ) franction of its original complexity, for This allows us to give an improved lower bound of Ω(n^2.556…) for Andreev's function. At the time of discovery, this was the best formula lower bound known for any function in NP. © 1993 John Wiley & Sons, Inc.     

Spin density distributions in radical anions of heterocyclic amine N-oxides

The unrestricted Hartree-Fock (UHF) method of Snyder and Amos is used to calculate, in the -electron approximation, the spin density distributions in radical anions of heterocyclic amine N-oxides. The computed spin densities are observed to be in good agreement with the experimental values. The computed spin density distribution of the radical anion of pyridine N-oxide is consistent with the greater susceptibility of pyridine N-oxide relative to pyridine to electrophilic nitration. Also, the calculations are consistent with the lower basicity of the N-oxides relative to the parent bases.     

The application of C-N.M.R. spectroscopy to the determination of the stereochemistry of Diels-Alder adducts

Long-range (³J) ¹³C-H coupling is a reliable probe to evaluate the stereochemistry of cycloaddition products. The method is best applied to carbonyl containing adducts but not restricted to them. Several structures have been revised and new ones evaluated.     

An abundance of nodulation factors

In this issue of Chemistry & Biology, Morón et al. [1] report that Rhizobium tropici CIAT899 produces different Nod factors in response to flavonoid induction under differing environmental conditions. This unanticipated environmental dependence has implications for altering or potentially improving the host-bacteria interaction in bean nodulation.     

Trypsin activity assay in substrate-specific one- and two-dimensional gels: a powerful method to separate and characterize novel proteases in active form in biological samples

To separate and identify the proteases, a substrate-specific, sensitive assay in sodium dodecyl sulfate (SDS)-polyacrylamide gels after two-dimensional (2-D) electrophoresis has been developed. This method allows simultaneous determination of protease cleavage specificity, molecular weight, isoelectric point, and if necessary, amino acid sequencing. After isoelectric focusing in immobilized pH gradient (IPG) strips (pH 6-11) (first dimension), trypsin was electrophoresed in 12% SDS polyacrylamide gels (second dimension) copolymerized with Boc-Gln-Ala-Arg-MCA (4-methyl-coumaryl-7-amide). The gels were washed in cold 2.5% Triton X-100 and water, and incubated in assay buffer (6.3 mM Bicine, 100 mM NaCl). Trypsin cleavage of the peptide-MCA generated fluorescent 7-amino-4-methyl-coumarin. In 1-D gels, as low as 500 pg trypsin could be detected and trypsin band volumes correlated linearly with the amounts of trypsin (R(2) = 0.999). In 2-D gels, the lowest amount of trypsin detected was 1 ng. The linear regression of spot volume and loading amount was still good (R(2) = 0.974). To optimize renaturation conditions, 5x5 min washes with 2.5% Triton X-100 and water, respectively, gave the strongest band volume. For fluorescence development, an assay buffer at pH 9 was the best; incubation at 37 degrees C for 30 min was sufficient. The method has application for identifying novel proteases as it does not rely on antibodies.     

Does alpha-fluorination affect the structural trans-influence and kinetic trans-effect of an alkyl ligand? Molecular structures of Pd(TMEDA)(CH(3))(R(F)) and a kinetic study of the trans to cis isomerization of Pt(TMEDA)(CH(3))(2)I(R(F)) [R(F) = CF(2)CF(3), CFHCF(3), CH(2)CF(3)

Reaction of Pd(TMEDA)(CH(3))(2) [TMEDA = tetramethylethylenediamine] with fluoroalkyl iodides R(F)I affords a series of square planar Pd(II) complexes Pd(TMEDA)(CH(3))(R(F)) [R(F) = CF(2)CF(3) (9), CFHCF(3) (10), CH(2)CF(3) (11)], presumably by oxidative addition followed by reductive elimination of CH(3)I. The solid-state structures of each compound have been determined by single crystal X-ray diffraction studies, allowing the effect of increasing alpha-fluorination on the structural trans-influence of alkyl ligands to be examined. In these compounds there is no significant difference observed in the trans-influence of the three fluorinated alkyl ligands toward the trans-N atom, although a significant cis-influence on the neighboring methyl ligand is apparent. Oxidative addition of the same series of fluoroalkyl ligands to the corresponding Pt(TMEDA)(CH(3))(2) affords octahedral Pt(IV) complexes trans-Pt(TMEDA)(CH(3))(2)(R(F))I [R(F) = CF(2)CF(3) (12), CFHCF(3) (13), CH(2)CF(3) (14)] as the kinetic products. In each case, subsequent isomerization to the corresponding all cis-isomers is observed; in the case of 13, the stereocenter at the alpha-carbon results in two diastereomeric cis-isomers, which are formed at different rates. The molecular structures of 13 and its more stable all cis-isomer 16b have been crystallographically determined. Kinetic studies of the trans-cis isomerization reactions show the mechanism to involve a polar transition state, presumably involving iodide dissociation, followed by rearrangement of the cation, and iodide recombination. High dielectric solvents increase the rate, but solvent coordinating ability has no effect. Dissolved salts (LiI, LiOTf) show normal accelerative salt effects, with no inhibition in the case of added iodide, consistent with the formation of an intimate ion pair intermediate. The kinetic parameters show that the trans-effects of fluoroalkyl ligands in these compounds follow the order expected from the relative sigma-donor properties of the ligands, with CF(2)CF(3) < CFHCF(3) < CH(2)CF(3).