Point Qi Research Articles

Rearranging a sequence of points onto a line

Given a sequence of n weighted points 〈p1,p2,…,pn〉 in the plane, we consider the problem of finding a rearrangement of the points, qi for each pi, onto a line such that any two consecutive points qi and qi+1 are at distance no more than their weight difference, and the maximum distance between pi and qi over all i is minimized. We present efficient algorithms that compute optimal rearrangements for three variants of the problem either under the L1 metric or under the Euclidean metric. When the line is fully specified or partially specified by only its orientation, our algorithms take near-linear time. When we have to choose a best target line over all lines in the plane, onto which the input sequence can be rearranged with the optimal rearrangement cost, we present an O(n3log4⁡n)-time algorithm.

언택트(Untact)시대에 한국적 무용치료: 오행경혈기무의 활용방안

Due to the drastic development of culture, society and technology, modern people are pursuing psychologically, physically and mentally harmonious state for the quality of one''s personal life. Especially, the concept of health has been expanded to the system of life that elements of in the present COVID-19, which includes elements in various fields as a state of great significance in our whole life as the most important condition and core elements constituting our life. This study was conducted aiming at activating Five Elements Meridian Point Qi Dance as a new movement of Korean dance therapy composed based on the theory of meridian system and meridian point of Korean medicine in the untact era. As a converged study of dance and Korean medicine, this is a study on the utilizability of detailed qi dance program to be used at the site of dance therapy which was discovered in the course of carrying out researches on the new approach of dance therapy having views on the similarity of philosophical background between Korean dance and Korean medicine. If Five Elements Meridian Point Qi Dance is activated and become popular as health promotion program and health movement, its future value will lead to more varied and positive influence on the promotion of health and life. As we enter the untact era due to social, cultural and technical development, interest in health promotion and healthy span has increased gradually, so many systematic and scientific researches suited to the characteristics of Korean people shall be carried out continuously and settle down as promotion, program and movement of health.

오행경혈기무의 치료적 유용성

Dance is an art that assumes the form of movement and the most basic characteristics of dance is movement. As a body therapy system that circulates vitality of body and makes energy by circulating energy and blood ceaselessly through the dance movement, principle of dance therapy can be applied. Especially, “Five Elements Meridian Point Qi Dance” which is performed based on the smooth operation of energy and breath with principle of Yin-Yang and Five-Elements that combined Korean dance and Qi dance is considered valuable as a treatment program that reflects the result of Korean medicinal experiment in the domain of current dance therapy program on the basis of principle having something in common with medical principle of traditional curing movement. Oriental medicine sets a high value on the concept of Qi (氣: energy). Qi is an essential element constituting all things in the universe, which is considered a core element in the oriental philosophy that explains essence and origins of matter, existence and activity of things. Regarding human mind and brain as one, Oriental medicine takes an organic and psychosomatic correlative approach. It can be said to be contrast to atomistic and analytic system of Western medicine. Qi enables faithful body activity and maintenance of vital activity, which provides help to performance of functions such as respiratory control, muscle relaxation, body temperature control, digestion absorption control etc. As a movement using Korean dance, energy and breath, Five Elements Meridian Point Qi Dance is composed based on the principle of Yin-Yang and Five-Elements, and is a movement based on the perspective of 14 meridian systems in Korean medicine. In movements of Qi dance, balance of yin and yang is made by controlling deviation of malformation of top and bottom, front and back, inside and outside and left and right by achieving harmony in yin and yang through motion and standstill, pace, folding and unfolding, spreading and gathering of movement. In developing movements of Five Elements Meridian Point Qi Dance, this study adopted a sound (musical instrument) symbolized by movement and five yin that represents symbolic animals of Five Elements, that is to say, Blue Dragon on the left, White Tiger on the right, Vermilion Bird on the south and Black Tortoise on the north. Research in this thesis is considered valuable and significant as it prepared systematic movement and foundation that materializes its effect through organization of a practical program for the possibility of cure and treatment effect through the principle and movement composition of actual and practical dance therapy movement based on the Korean medicinal Five Elements meridian point within the category of dance therapy for movement through the principle of Five Elements meridian point on the basis of Yin-Yang and Five-Elements from the perspective of Korean medicine.

An extended Fatou–Shishikura inequality and wandering branch continua for polynomials

Let P be a polynomial of degree d with Julia set JP. Let N˜ be the number of non-repelling cycles of P. By the famous Fatou–Shishikura inequality N˜≤d−1. The goal of the paper is to improve this bound. The new count includes wandering collections of non-(pre)critical branch continua, i.e., collections of continua or points Qi⊂JPall of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of eval(Qi)≥3 external rays. Also, we relate individual cycles, which are either non-repelling or repelling with no periodic rays landing, to individual critical points that are recurrent in a weak sense.A weak version of the inequality readsN˜+Nirr+χ+∑i(eval(Qi)−2)≤d−1 where Nirr counts repelling cycles with no periodic rays landing at points in the cycle, {Qi} form a wandering collection BC of non-(pre)critical branch continua, χ=1 if BC is non-empty, and χ=0 otherwise.

The Behnke-Stein theorem for open Riemann surfaces

Using the Riemann-Roch theorem and the set-topological part of Bishop's polyhedron lemma, we show that the usual Runge approximation theorem for compact subsets of the Riemann sphere is valid word-for-word on any compact Riemann surface X, with meromorphic functions on X playing the role of rational functions; this result is essentially equivalent to the Behnke-Stein approximation theorem. The Behnke-Stein generalization of the Runge approximation theorem [1], which is the basic tool for many existence questions on open Riemann surfaces, can be stated in several equivalent ways, for instance: Let X be a Riemann surface, and K a compact subset of X. Then, in order that every holomorphic function in a neighborhood of K be uniformly approximable on K by holomorphic functions on X, it is necessary and sufficient that X K have no connected component with compact closure in X. As is well known, the necessity part of the above theorem follows from the sufficiency part (using the theory of compact Riemann surfaces). In this note, we wish to point out how the famous special polyhedron lemma of Bishop [2], which is an elementary set-topological result, can be used (together with the Riemann-Roch theorem for compact Riemann surfaces) to give a simple proof of the following theorem. Theorem 1.1. Let X be a compact Riemann surface, and K c X a compact subset. Let Q be any subset of X K which contains (precisely) one point qi from each connected component W of X K. Then any holomorphic function on a neighborhood of K can be approximated uniformly on K by meromorphic functions on X whose poles lie in Q. Again it is well known and easy to see that Theorem 1.1 implies the sufficiency part of Behnke-Stein. The main fact needed for this implication is the following: Received by the editors March 24, 1988. 1980 Mathematics Subject Classification (1 985 Revision). Primary 30E 10, 30F10.

A Reduction of the Poincare Conjecture to Group Theoretic Conjectures

POINCARE CONJECTURE. Any simply-connected closed3 3-manifold is a 3-sphere, i.e., any homotopy 3-sphere is a 3-sphere. This paper presents a reduction of the above conjecture to the two conjectures below. Let N be an orientable closed3 surface of genus p ! 2, and let Ai, Bi, i = 1, * * *, p, be a fundamental system of N with base point o, (see No. 4)4. As it is well known, N has certain hyperbolic metrics, imposed on it by Poincare, and therefore there are geodesics on N, (see No. 3). Let now Xi and Yi be the primary simple closed geodesics on N, corresponding to Ai and Bi respectively, i = 1, * * *, p, (Lemma (12.2)). Then Xi and Yi have only one point qi in common (Nos. 3 and 12), any two of the pairs Xi, Yi different from each other, are disjoint, and if we cut N along the X and Y's, we obtain a 2-sphere with p holes having boundaries Xi YiXT1 y-1. Let X', i = 1, ... , p, be primary simple closed geodesics on N, any two of which different from each other are disjoint, and such that X' is homologous to Yi on N. Let oi be some common point of Xi and X'. Let now j be a fixed natural number > 1 and ? p. By an isotopic deformation of each one of the pairs Xi, Yi, i = 1, ... , p, on N, such that qj moves on XJ, we obtain a new fundamental system Aji, Bji, i = 1, ..., p, of N with base point oj, such that the carrier of Ajj is precisely Xi. Then

A New Foundation of Non-Euclidean, Affine, Real Projective and Euclidean Geometry

The concepts of the non-Euclidean plane geometry can be defined in terms of the notions joining and intersecting of geometric entities as follows:1 We call a system of three lines a maximal triangle if (1) no two of them intersect, (2) through any point of each of the three lines there exists exactly one line which intersects neither of the other two lines. We say that two lines L and M are parallel if there exists a line N such that L, M, N form a maximal triangle. We say the point Q lies between two points P and R if (1) P, Q, R are distinct collinear points, (2) any line through Q intersects at least one line of any maximal triangle two sides of which pass through P and R, respectively. The point Q is said to lie between three points if (1) these points are not collinear, (2) Q does not coincide with any of them nor lies between any two of them, (3) any line through Q contains at least one point between two of the three points. The points Q1, Q2, … are said to converge toward Q if for any three points the following condition holds: If Q lies between the three points, then almost all points Qi (i.e., all except at most a finite number) lie between them.