In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the...In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the property of the equidistant conjugate points of a triangle is generalized to high dimensional simplex.展开更多
Recently we have proposed anew method combininginterior and exterior approaches to solve linear programming problems. This method uses an interior point, and from there connected to the vertex of the so called station...Recently we have proposed anew method combininginterior and exterior approaches to solve linear programming problems. This method uses an interior point, and from there connected to the vertex of the so called station cone which is also a solution of the dual problem. This allows us to determine the entering vector and the new station cone. Here in this paper, we present a new modified algorithm for the case, when at each iteration we determine a new interior point. The new building interior point moves toward the optimal vertex. Thanks to the shortened from both inside and outside, the new version allows to find quicker the optimal solution. The computational experiments show that the number of iterations of the new modified algorithm is significantly smaller than that of the second phase of the dual simplex method.展开更多
In this paper we present a new method combining interior and exterior approaches to solve linear programming problems. With the assumption that a feasible interior solution to the input system is known, this algorithm...In this paper we present a new method combining interior and exterior approaches to solve linear programming problems. With the assumption that a feasible interior solution to the input system is known, this algorithm uses it and appropriate constraints of the system to construct a sequence of the so called station cones whose vertices tend very fast to the solution to be found. The computational experiments show that the number of iterations of the new algorithm is significantly smaller than that of the second phase of the simplex method. Additionally, when the number of variables and constraints of the problem increase, the number of iterations of the new algorithm increase in a slower manner than that of the simplex method.展开更多
An orthoscheme or Pythagorean simplex is a solid in n-dimensional Euclidean space whose faces are right triangles.In 1956,Hadwiger asked whether an ndimensional general(not necessarily Pythagorean)simplex can always b...An orthoscheme or Pythagorean simplex is a solid in n-dimensional Euclidean space whose faces are right triangles.In 1956,Hadwiger asked whether an ndimensional general(not necessarily Pythagorean)simplex can always be decomposed into a finite number of Pythagorean simplexes.Tschirpke proved in 1994 that this division is always possible in 5D space.Coxeter proved that a 3D Pythagorean simplex can be split into three smaller ones.In a 2024 paper,I generalized Coxeter’s trisection to prove that the dissection of an n-dimensional Pythagorean simplex into n pieces of the same type is possible if each leg of the original solid is equal to the unit distance.In the present paper,I extend this proof to an n-dimensional Pythagorean simplex with legs of arbitrary measure.This means the proof of the Hadwiger conjecture in the special case of a Pythagorean simplex.展开更多
基金Supported by the Technological Project of Jiangxi Province Education Department(GJJ 08389)
文摘In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the property of the equidistant conjugate points of a triangle is generalized to high dimensional simplex.
文摘Recently we have proposed anew method combininginterior and exterior approaches to solve linear programming problems. This method uses an interior point, and from there connected to the vertex of the so called station cone which is also a solution of the dual problem. This allows us to determine the entering vector and the new station cone. Here in this paper, we present a new modified algorithm for the case, when at each iteration we determine a new interior point. The new building interior point moves toward the optimal vertex. Thanks to the shortened from both inside and outside, the new version allows to find quicker the optimal solution. The computational experiments show that the number of iterations of the new modified algorithm is significantly smaller than that of the second phase of the dual simplex method.
文摘In this paper we present a new method combining interior and exterior approaches to solve linear programming problems. With the assumption that a feasible interior solution to the input system is known, this algorithm uses it and appropriate constraints of the system to construct a sequence of the so called station cones whose vertices tend very fast to the solution to be found. The computational experiments show that the number of iterations of the new algorithm is significantly smaller than that of the second phase of the simplex method. Additionally, when the number of variables and constraints of the problem increase, the number of iterations of the new algorithm increase in a slower manner than that of the simplex method.
文摘An orthoscheme or Pythagorean simplex is a solid in n-dimensional Euclidean space whose faces are right triangles.In 1956,Hadwiger asked whether an ndimensional general(not necessarily Pythagorean)simplex can always be decomposed into a finite number of Pythagorean simplexes.Tschirpke proved in 1994 that this division is always possible in 5D space.Coxeter proved that a 3D Pythagorean simplex can be split into three smaller ones.In a 2024 paper,I generalized Coxeter’s trisection to prove that the dissection of an n-dimensional Pythagorean simplex into n pieces of the same type is possible if each leg of the original solid is equal to the unit distance.In the present paper,I extend this proof to an n-dimensional Pythagorean simplex with legs of arbitrary measure.This means the proof of the Hadwiger conjecture in the special case of a Pythagorean simplex.
