On the existence and uniqueness of solutions in nonlinear complementarity theory

A complementarity problem is said to be globally uniquely solvable (GUS) if it has a unique solution, and this property will not change, even if any constant term is added to the mapping generating the problem.A characterization of the GUS property which generalizes a basic theorem in linear complementarity theory is given. Known sufficient conditions given by Cottle, Karamardian, and Moré for the nonlinear case are also shown to be generalized. In particular, several open questions concerning Cottle's condition are settled and a new proof is given for the sufficiency of this condition.A simple characterization for the two-dimensional case and a necessary condition for then-dimensional case are also given.The research described in this paper was carried out while N. Megiddo was visiting Tokyo Institute of Technology under a Fellowship of the Japan Society for the Promotion of Science.     

Computation of all solutions to a system of polynomial equations

This paper proposes a homotopy continuation method for approximating all solutions to a system of polynomial equations in several complex variables. The method is based on piecewise linear approximation and complementarity theory. It utilizes a skilful artificial map and two copies of the triangulationJ ₃ with continuous refinement of grid size to increase the computational efficiency and to avoid the necessity of determining the grid size a priori. Some computational results are also reported.     

Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property

We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let be a compact metric space.

First we show the following. If is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map with , then is topologically mixing. If and are commuting expansive onto continuous maps with POTP and if is topologically transitive with period , then for some dividing , , where the , , are the basic sets of with such that all have period , and the dynamical systems are a factor of each other, and in particular they are conjugate if is a homeomorphism.

Then we prove an extension of a basic result in symbolic dynamics. Using this and many techniques in symbolic dynamics, we prove the following. If is a topologically transitive, positively expansive onto continuous map having POTP, and is a positively expansive onto continuous map with , then has POTP. If is a topologically transitive, expansive homeomorphism having POTP, and is a positively expansive onto continuous map with , then has POTP and is constant-to-one.

Further we define `essentially LR endomorphisms' for systems of expansive onto continuous maps of compact metric spaces, and prove that if is an expansive homeomorphism with canonical coordinates and is an essentially LR automorphism of , then has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.

     

D-Fructose-6-phosphate aldolase-catalyzed one-pot synthesis of iminocyclitols

A one-pot chemoenzymatic method for the synthesis of a variety of new iminocyclitols from readily available, non-phosphorylated donor substrates has been developed. The method utilizes the recently discovered fructose-6-phosphate aldolase (FSA), which is functionally distinct from known aldolases in its tolerance of different donor substrates as well as acceptor substrates. Kinetic studies were performed with dihydroxyacetone (DHA), the presumed endogenous substrate for FSA, as well as hydroxy acetone (HA) and 1-hydroxy-2-butanone (HB) as donor substrates, in each case using glyceraldehyde-3-phosphate as acceptor substrate. Remarkably, FSA used the three donor substrates with equal efficiency, with kcat/KMvalues of 33, 75, and 20 M-1 s-1, respectively. This level of donor substrate tolerance is unprecedented for an aldolase. Furthermore, DHA, HA, and HB were accepted as donors in FSA-catalyzed aldol reactions with a variety of azido- and Cbz-amino aldehyde acceptors. The broad substrate tolerance of FSA and the ability to circumvent the need for phosphorylated substrates allowed for one-pot synthesis of a number of known and novel iminocyclitols in good yields, and in a very concise fashion. New iminocyclitols were assayed as inhibitors against a panel of glycosidases. Compounds 15 and 16 were specific alpha-mannosidase inhibitors, and 24 and 26 were potent and selective inhibitors of beta-N-acetylglucosaminidases in the submicromolar range. Facile access to these compounds makes them attractive core structures for further inhibitor optimization.     

Controlling 3(10)-helix and alpha-helix of short peptides in the solid state

L-Leu hexapeptide containing alpha-aminoisobutyric acid (Aib) forms a right-handed (P) 3(10)-helix, whereas that containing cyclic alpha,alpha-disubstituted amino acid Ac(5)c(dOM) assumes a right-handed (P) alpha-helix in the solid state.     

The d (3)Pi(g)-c (3)Sigma(u) (+) band system of C2

A two-dimensional fluorescence (excitation/emission) spectrum of C2 produced in an acetylene discharge was used to identify and separate emission bands from the d (3)Pi(g)<--c (3)Sigma(u) (+) and d (3)Pi(g)<--a (3)Pi(u) excitations. Rotationally resolved excitation spectra of the (4<--1), (5<--1), (5<--2), and (7<--3) bands in the d (3)Pi(g)<--c (3)Sigma(u) (+) system of C2 were observed by laser-induced fluorescence spectroscopy. The molecular constants of each vibrational level, determined from rotational analysis, were used to calculate the spectroscopic constants of the c (3)Sigma(u) (+) state. The principal molecular constants for the c (3)Sigma(u) (+) state are B(e)=1.9319(19) cm(-1), alpha(e)=0.018 55(69) cm(-1), omega(e)=2061.9 cm(-1), omega(e)x(e)=14.84 cm(-1), and T(0)(c-a)=8662.925(3) cm(-1). We report also the first experimental observations of dispersed fluorescence from the d (3)Pi(g) state to the c (3)Sigma(u) (+) state, namely, d (3)Pi(g)(v=3)-->c (3)Sigma(u) (+)(v=0,1).     

An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems Over Symmetric Cones

This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let and be a finite dimensional real vector space and a symmetric cone embedded in ; examples of and include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a pair of the N-dimensional Euclidean space and N-dimensional second-order cones, and a pair of the space of m × m real symmetric (or complex Hermitian) matrices and the cone of their positive semidefinite matrices. Sums of squares relaxations are further extended to a polynomial optimization problem over , i.e., a minimization of a real valued polynomial a(x) in the n-dimensional real variable vector x over a compact feasible region , where b(x) denotes an - valued polynomial in x. It is shown under a certain moderate assumption on the -valued polynomial b(x) that optimal values of a sequence of sums of squares relaxations of the problem, which are converted into a sequence of semidefinite programs when they are numerically solved, converge to the optimal value of the problem. Research supported by Grant-in-Aid for Scientific Research on Priority Areas 16016234.