Some areas associated with Pythagorean triples

The set of all Pythagorean triples of positive integers are partitioned in a natural way. For each partition three triples are chosen in various ways, interpreted as points in space, and projected onto the coordinate planes. When a plane triangle is so formed its area is determined. Also, lines through two consecutive triples are obtained along with their intersections with the coordinate planes.     

Parametrization of Pythagorean triples by a single triple of polynomials

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials.     

Why 3 and 5 are always factors of primitive Pythagorean triples

Analysis of integer structure and right-end-digits can illustrate why 3 and 5 are features of primitive Pythagorean triples. The results also utilize the triangular and pentagonal numbers.     

A special matrix transformation semigroup of primitive pairs and the genealogy of Pythagorean triples

A special matrix semigroup acting on the set of primitive pairs of positive integers naturally parameterizing certainDiophantine equations (such as those defining Pythagorean triples) is constructed and studied. The Pythagorean triples are related to a trichotomy tree of solutions.     

Boundary triples and quasi boundary triples for elliptic operators

The abstract concepts of boundary triples and their recent generalizations are useful tools to parametrize the self-adjoint and maximal dissipative/maximal accumulative extensions of formally symmetric elliptic differential expressions with the help of explicit boundary conditions. In the present note the parametrizations induced by the “natural” quasi boundary triple with the Dirichlet and Neumann trace as boundary maps are compared with the parametrizations induced by a “classical” ordinary boundary triple, where a regularized Neumann trace is used for one of the boundary maps. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Markoff triples and quasifuchsian groups

We study the global behaviour of trees of Markoff triples overthe complex numbers. We relate this to the space of type-preservingrepresentations of the punctured torus group into SL(2,C). Inparticular, we explore which Markoff triples correspond to quasifuchsianrepresentations. We derive a variation of McShane's identityfor quasifuchsian groups. In the case of non-discrete representations,we attempt to relate the asymptotic behaviour of Markoff triplesto the realisability of laminations in hyperbolic 3-space. Wealso consider how some of these issues might be related formore general surfaces. 1991 Mathematics Subject Classification:57M50.     

The Pythagorean theorem and the Euclidean parallel postulate

In this note, we develop new, simple and very accurate asymptotic approximations for non-integer order derivatives of monomial functions by using the more accurate asymptotic approximations for large factorials that have recently appeared in the literature.     

Large Fredholm triples

We give several equivalent condition for Busby extensions of a given algebra to be absorbing, considerably improving our earlier results [G.A. Elliott, D. Kucerovsky, An abstract Brown-Douglas-Fillmore absorption theorem, Pacific J. Math. 198 (2001) 385-409], and establish sufficient conditions for Fredholm triples to be absorbing in a suitable sense. As an application of one of our criteria, we prove a multivariable Brown-Douglas-Fillmore type theorem.     

关于本原商高数的Miyazaki猜想

设a,b,c是适合a=2~(2r)-n~2,b=2~(r+1)n,c=2~(2r)+n~2的正整数,其中r是正整数,n是奇素数.运用初等数论方法讨论了指数Diophantine方程c~x+b~y=a~z.证明了:当2~r=n+1时,方程仅有正整数解(x,y,z)=(1,1,2);否则,方程无解。上述结果部分地证实了有关本原商高数的Miyazaki猜想。