The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean ...The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>=d<sup>2</sup> with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d<sup>2</sup> is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>+d<sup>2</sup>=e<sup>2</sup> and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d<sup>2</sup>, respectively, e<sup>2</sup>.展开更多
Shutting down a link for the purposes of a scheduled routine maintenance does cause the forwarding path to change. If these changes are not done in a required order will cause not only transient micro loops but also a...Shutting down a link for the purposes of a scheduled routine maintenance does cause the forwarding path to change. If these changes are not done in a required order will cause not only transient micro loops but also an overload in some links. Currently, some ISP operators use a graceful link shutdown procedure by first setting up the Interior Gateway Protocol (IGP) link metric to MAX_METRIC -1 and then shutdown the link. In this paper, we present a Pythagorean Triple Metric Sequence as a method to use to shutdown a link during such network operations. Conducting a link shutdown of any desired link for maintenance purpose is a very delicate duty that requires extreme care to prevent transient loops during such topological changes. We thus wish to demonstrate that there exists a Pythagorean Triple Metric Sequence for any given link that can be used to shutdown a link during the routine maintenance by ISPs.展开更多
Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposi...Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposite side b and modding out by π-γ. The only Hr for which the structure is known is Hπ/2, which is free abelian. We prove that for generalγ, Hr has an element of order two iff 2(1- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x3-3x2+1=0 has a rational solution 0<x<1. This shows that the set of values ofγ for which Hr has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in Hr. Finally, we give some examples of higher order torsion elements in Hr.展开更多
文摘The Pythagorean triples (a, b | c) of planar geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup> with integers (a, b, c) are generalized to 3D-Pythagorean quadruples (a, b, c | d) of spatial geometry which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>=d<sup>2</sup> with integers (a, b, c, d). Rules for a parametrization of the numbers (a, b, c, d) are derived and a list of all possible nonequivalent cases without common divisors up to d<sup>2</sup> is established. The 3D-Pythagorean quadruples are then generalized to 4D-Pythagorean quintuples (a, b, c, d | e) which satisfy the equation a<sup>2</sup>+b<sup>2</sup>+c<sup>2</sup>+d<sup>2</sup>=e<sup>2</sup> and a parametrization is derived. Relations to the 4-square identity are discussed which leads also to the N-dimensional case. The initial 3D- and 4D-Pythagorean numbers are explicitly calculated up to d<sup>2</sup>, respectively, e<sup>2</sup>.
文摘Shutting down a link for the purposes of a scheduled routine maintenance does cause the forwarding path to change. If these changes are not done in a required order will cause not only transient micro loops but also an overload in some links. Currently, some ISP operators use a graceful link shutdown procedure by first setting up the Interior Gateway Protocol (IGP) link metric to MAX_METRIC -1 and then shutdown the link. In this paper, we present a Pythagorean Triple Metric Sequence as a method to use to shutdown a link during such network operations. Conducting a link shutdown of any desired link for maintenance purpose is a very delicate duty that requires extreme care to prevent transient loops during such topological changes. We thus wish to demonstrate that there exists a Pythagorean Triple Metric Sequence for any given link that can be used to shutdown a link during the routine maintenance by ISPs.
文摘Let 0<γ<π be a fixed pythagorean angle. We study the abelian group Hr of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in Hr is defined by adding the angles β opposite side b and modding out by π-γ. The only Hr for which the structure is known is Hπ/2, which is free abelian. We prove that for generalγ, Hr has an element of order two iff 2(1- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x3-3x2+1=0 has a rational solution 0<x<1. This shows that the set of values ofγ for which Hr has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z2 or Z3 in Hr. Finally, we give some examples of higher order torsion elements in Hr.
