Existence and uniqueness of a class of double pyramidal central configurations in six-body problems

Under the necessary conditions for a double pyramidal central configuration with a diamond base to exist in the real number space, the existence and uniqueness of such configurations were studied by employing combinedly the algebraic method and numerical calculation. It is found that there exists a planar curl triangle region G in a square Q such that any point in G and given by the ratio of the two diagonal lengths of the diamond base and the ratio of one diagonal length of the base to the height of the double pyramid configuration determines a unique double pyramid central configuration, while all points in Q-G have no referance to any central configuration.

Existence and uniqueness for a class of double pyramidal central configurations with a concave pentagonal base

Based on some necessary conditions for double pyramidal central configurations with a concave pentagonal base, for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with a concave pentagonal base in 7-body problems are proved and the range of the ratio between radius and half-height is obtained, within which the 7 bodies involved form a central configuration or form uniquely a central configuration.

Existence and uniqueness for a class of nine-bodies central configurations

On some necessary conditions for double pyramidal central configurations with concave heptagon for any given ratio of masses, the existence and uniqueness of a class of double pyramidal central configurations with concave heptagon base for nine-body problems is proved in this paper, and the range of the ratio cr of the circularity radius of the heptagon to the half-height of the double pyramidal central configuration involved in this class configurations is obtained, which is in the interval （√3/3,1.099 600 679）, and the configuration involved in the bodies with any σ∈ （√3/3, 1.099 600 679） can form a central configuration which is a uniquely central configuration is proved.

Comparison of the Pattern of Weight Training on Muscle Strength and Endurance in Karate Men

Some studies performed weight training for improvement physical fitness in martial arts. The purpose of this study was to compare weight training with two loading patterns （double pyramid and reverse step） on muscular strength and muscular endurance in karate men. The subjects who have experienced six years in karate are participated voluntarily. These subjects have been divided randomly into three groups： double pyramid group, reverse step group and control group. Characteristics of subject： double pyramid group （age： 20.55 ± 2.62 years, weight： 73.45 ± 11.17 kg and height： 176.86 ±5.04 cm）, reverse step group （age： 20.50 ± 3.92 years, weight： 75.35 ± 11.24 kg and height： 178.90 ± 9.24 cm） and control group （age： 21.36 ±4.65 year ,weight： 75.77 ± 6.07 kg and height： 178.64± 4.67 cm）. All subjects performed weight training during eight weeks. Data were statistically analyzed by text dependent, analysis of variance and tukey test with a 0.05 significant level. The result shows that a program of weight training double-pyramid method significantly increases muscular strength and muscular endurance, while a period of weight training reverses step method only significantly increase muscle endurance but no change in muscular strength is observed. The coaches are recommended to use a double pyramid method to improve muscular strength and muscular endurance training with regard to training load for preparing their athletes.