A new method for sector optimum partition of airspace is proposed by dividing the fright altitude into several layers according to the distribution characteristics of the controller＇s workloads in an airspace. On the...A new method for sector optimum partition of airspace is proposed by dividing the fright altitude into several layers according to the distribution characteristics of the controller＇s workloads in an airspace. On the basis of the original distribution of the waypoints at each level of altitude, the sweel5 line algorithm of Voronoi diagram is used to divide them into certain polygons （ elements）, and the controller＇s workloads are calculated in each Voronoi polygon. Then by the rule about balance of controller＇s workload and by adding conditions of control handover or coordination for the sector, a mathematical model for the controller＇s workload based sector optimization is built. By the model, the Voronoi polygons are optimally partitioned. As a result, a 3D sector optimum partition of the whole airspace is formed by combining the sector optimum partitions at every layer. The actual airspace partition for Xiamen Airport has proved the reasonability and effectiveness of the 3D sector optimum partition of airspace proposed.展开更多
We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in...We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.展开更多
A recursive method based on successive computations of perimeters of inscribed regular polygons for estimating π is formulated by employing the Pythagorean theorem alone without resorting to any trigonometric calcula...A recursive method based on successive computations of perimeters of inscribed regular polygons for estimating π is formulated by employing the Pythagorean theorem alone without resorting to any trigonometric calculations. The approach is classical but the formulation of coupled recursion relations is new. Further, use of infinite series for computing π is explored by an improved version of Leibniz’s series expansion. Finally, some remarks with reference to π are made on a relatively recently rediscovered Sumerian tablet depicting geometric figures.展开更多
Hilbert’s Tenth Problem(HTP)asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by...Hilbert’s Tenth Problem(HTP)asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by Matiyasevich in 1970.In this paper we obtain some further results on HTP over Z.We prove that there is no algorithm to determine for any P(z1,...,z9)∈Z[z1,...,z9]whether the equation P(z1,...,z9)=0 has integral solutions with z9≥0.Consequently,there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z1,...,z11)=0(with integer coefficients)in 11 unknowns has integral solutions,which provides the best record on the original HTP over Z.We also prove that there is no algorithm to test for any P(z1,...,z17)∈Z[z1,...,z17]whether P(z12,...,z172)=0 has integral solutions,and that there is a polynomial Q(z1,...,z20)∈Z[z1,...,z20]such that{Q(z12,...,z202):z1,...,z20∈Z}∩{0,1,2,...}coincides with the set of all primes.展开更多
基金The National Natural Science Foundation of China (No.60472117)
文摘A new method for sector optimum partition of airspace is proposed by dividing the fright altitude into several layers according to the distribution characteristics of the controller＇s workloads in an airspace. On the basis of the original distribution of the waypoints at each level of altitude, the sweel5 line algorithm of Voronoi diagram is used to divide them into certain polygons （ elements）, and the controller＇s workloads are calculated in each Voronoi polygon. Then by the rule about balance of controller＇s workload and by adding conditions of control handover or coordination for the sector, a mathematical model for the controller＇s workload based sector optimization is built. By the model, the Voronoi polygons are optimally partitioned. As a result, a 3D sector optimum partition of the whole airspace is formed by combining the sector optimum partitions at every layer. The actual airspace partition for Xiamen Airport has proved the reasonability and effectiveness of the 3D sector optimum partition of airspace proposed.
文摘We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.
文摘A recursive method based on successive computations of perimeters of inscribed regular polygons for estimating π is formulated by employing the Pythagorean theorem alone without resorting to any trigonometric calculations. The approach is classical but the formulation of coupled recursion relations is new. Further, use of infinite series for computing π is explored by an improved version of Leibniz’s series expansion. Finally, some remarks with reference to π are made on a relatively recently rediscovered Sumerian tablet depicting geometric figures.
基金supported by National Natural Science Foundation of China(Grant No.11971222)。
文摘Hilbert’s Tenth Problem(HTP)asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring Z of integers.This was finally solved negatively by Matiyasevich in 1970.In this paper we obtain some further results on HTP over Z.We prove that there is no algorithm to determine for any P(z1,...,z9)∈Z[z1,...,z9]whether the equation P(z1,...,z9)=0 has integral solutions with z9≥0.Consequently,there is no algorithm to test whether an arbitrary polynomial Diophantine equation P(z1,...,z11)=0(with integer coefficients)in 11 unknowns has integral solutions,which provides the best record on the original HTP over Z.We also prove that there is no algorithm to test for any P(z1,...,z17)∈Z[z1,...,z17]whether P(z12,...,z172)=0 has integral solutions,and that there is a polynomial Q(z1,...,z20)∈Z[z1,...,z20]such that{Q(z12,...,z202):z1,...,z20∈Z}∩{0,1,2,...}coincides with the set of all primes.
