Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

AMS (MOS). Mathematics Subject Classification. 58F05, 58F27, 58F30.Generalizing the degenerate KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth sub-manifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold will persist; b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a sub-isoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.The first author is partially supported by NSFC grant 19971042, National 973 key Project: Nonlinearity in China and the outstanding youth project of Ministry of Education of China. The second author is partially supported by NSF grant DMS9803581. This work is partially done when the second and third authors were visiting the National University of Singapore

FACTORS AFFECTING CONSUMER PREFERENCES FOR MAJOR FOOD MARKETS IN TAIWAN

The Food Industry Research and Development Institute conducted a nationwide survey of food consumption in Taiwan in 1999. A sample of 1200 consumers responded to a questionnaire, which asked whether there was a use/visit experience in the past year for each of six types of food markets: traditional vegetable market, supermarket, hypermarket, consumer cooperative, chain convenience store, and grocery store. Questions also asked which types of food markets consumers used most frequently for the purchase of fresh foods, planned purchases or occasional purchases of processed foods, and what factors influenced consumers to choose their most frequently used market. Nearly 84 percent of surveyed consumers have used/visited traditional vegetable markets in the past year, while 47 percent choose traditional markets as their most frequented marketplace for buying fresh food in Taiwan. Some 81 percent of surveyed consumers have used or visited a supermarket in past year, while 17.7 percent and 29.0 percent choose supermarkets as the most frequented marketplace for planned and occasional purchases of processed foods, respectively. We apply stepwise logistic regression to identify significant sociodemographic factors (such as gender, age, and others) which influence the choice of each of the six major types of food markets and to identify the promotional factors which positively or negatively influence consumers' preferences for the most often used market. Female consumers have almost four times higher odds of frequently using traditional vegetable markets in the past year. The unmarried, divorced/widowed, and/or those who had higher family monthly incomes demonstrate lower odds of frequently using traditional markets. For purchases of fresh foods, female respondents, older consumers, and/or those living in northern Taiwan have higher odds of choosing traditional vegetable markets as the most frequently used market, while occupation "chief," and those with higher levels of education have lower odds of choosing traditional markets. Price level influences consumers positively in choosing traditional vegetable markets, while products with mark registrations, such as GMP or CAS, negatively influence this choice. For planned purchases of processed foods, older consumers are more likely than others to choose traditional vegetable markets, and price level is also a positive influence for choosing these markets. For occasional purchases of processed foods, manual laborers and/or those living in northern Taiwan have higher odds of choosing traditional vegetable markets, while unmarried respondents have lower odds. Price level and products with nutrition labeling are positive promotional factors influencing consumers to choose traditional vegetable markets. For the other five market types, sociodemographic factors influence consumers' choices of markets differently in each case, but marketing factors also influence choices of which type of marketplace to use most frequently.Consumer/Household Economics,

Measurable Concurrence of Mixed States

We show that bipartite concurrence for rank-2 mixed states of qubits is written by an observable which can be exactly and directly measurable in experiment by local projective measurements, provided that four copies of the composite quantum system are available. In addition, for a tripartite quantum pure state of qubits, the 3-tangle is also shown to be measurable only by projective measurements on the reduced density matrices of a pair of qubits conditioned on four copies of the state.Comment: 3 page