The Cauchy problem for the nonlinear Schrödinger equation in H1

We consider the initial value problem for the nonlinear Schrödinger equation in H¹(Rⁿ). We establish local existence and uniqueness for a wide class of subcritical nonlinearities. The proofs make use of a truncation argument, space-time integrability properties of the linear equation, anda priori estimates derived from the conservation of energy. In particular, we do not need any differentiability property of the nonlinearity with respect to x.Research supported by NSF grants DMS 8201639 and DMS 8703096.     

A dynamic solution in N-person cooperative game theory

We formulate a cooperative game as an extended form game in which each player in turn proposes payoffs to a coalition over M steps. Payoffs at time t are discounted by a penalty function f(t). If all players in a coalition agree to their payoffs, they receive them. Under a convergence hypothesis verified by computer for three players in many cases, we compute the payoffs resulting from a coalition pattern and give necessary conditions for particular patterns. The resulting solution is related to the Nash bargaining solution and the competitive solution.     

The fermat equation over quadratic fields

Kummer's method of proof is applied to the Fermat equation over quadratic fields. The concept of an m-regular prime, p, is introduced and it is shown that for certain values of m, the Fermat equation with exponent p has no nontrivial solutions over the field Q(√m).     

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.

Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor is bounded by for some exponent and some positive constant . The -exponent of tractability is defined as the smallest power of in these bounds. It is shown by using Monte Carlo quadrature that the -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

     

Pluronic L61 accelerates flip-flop and transbilayer doxorubicin permeation

It has recently been found that Pluronics (block copolymers of ethylene oxide, EO, and propylene oxide, PO) favor the permeability and accumulation of anthracycline antibiotics, for example doxorubicin (Dox), in tumor cells. In an effort to understand these results, the interaction of EO(2)/PO(32)/EO(2) (Pluronic L61) with unilamellar egg yolk vesicles (80-100 nm in diameter) was examined. A partition coefficient K(p)=[Pl](membrane)/[Pl](water)=45 was determined. This corresponds to adsorption of about 20 polymer molecules to the surface of each vesicle in a 20 microM polymer solution. Despite this rather weak adsorption, Pluronic has a substantial effect upon the transmembrane permeation rate of Dox and upon the phospholipid flip-flop rate within the bilayers. Thus, the Dox permeation rate increases threefold and the flip-flop rate increases sixfold in 20 microM Pluronic. The two rates increase linearly with the amount of adsorbed polymer. The obvious ability of Pluronics to increase the mobility of membrane components may have important biomedical consequences.     

Reactions of the perfluoroalkyltriazapentadiene Ph2N3C2(C3F7)2H with acids,bases, and water

Treatment of the bis(perfluoroalkyl)triazapentadiene PhN=C(C(3)F(7))-N=C(C(3)F(7))-NHPh, 2, with bases affords salts of the conjugate base. Alkylation of the Na(+) salt with CH(3)I yields PhN=C(C(3)F(7))-N=C(C(3)F(7))-NMePh. The crystal structure of [Bu(4)N][Ph(2)N(3)C(2)(C(3)F(7))(2)] demonstrates a twisted, zigzag geometry for the anion in the solid state, but in solution, it is conformationally unstable. Both compounds are stable in aqueous methanol, but hydrolysis occurs under acidic conditions. Protonation with CF(3)SO(3)H acid occurs at the sp(2) nitrogen to give [PhN-(H) HC(C(3)F(7))-C(C(3)F(7))-NHPh][CF(3)SO(3)]. Heating 2 in CF(3)SO(3)H produces 2,4-bis(heptafluoropropyl)-1,3-quinazoline.