Related papers: A note on Pythagorean Triples
In this paper we determine the parity of some sequences which are related to Catalan numbers. Also we introduce a combinatorical object called, \Catalan tree", and discuss its properties.
The properties of the Pastro polynomials on the real line are studied with the help of a triplet of $q$-difference operators. The $q$-difference equation and recurrence relation these polynomials obey are shown to arise as generalized…
We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities.
Consider two circles, externally tangential,and with integer radii R1, R2; and with R1>R2.The two circles have three tangent lines in common, one of them being T1T2. If M is the midpoint of T1T2, and K the point of intersection of the lines…
We establish new pair correlation results for certain generic homogenous diagonal forms evaluated on the integers. Methods are analytic leading to explicit quantitative statements.
We introduce and study a new kind of congruent number problem on the right trapezoid.
This paper considers the problem of the valuation for integer numbers of the zeta function and of five other functions which are naturally associated to it. A relatively elementary approach is exposed, which closely connects this still…
The relationship between Salem numbers and short geodesics has been fruitful in quantitative studies of arithmetic hyperbolic orbifolds, particularly in dimensions 2 and 3. In this article, we push these connections even further. The…
We study a sequence transformation pipeline that maps certain sequences with rational generating functions to permutation-based sequence families of combinatorial significance. Many of the number triangles we encounter can be related to…
We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find…
We establish a relationship between the two important central lines of the triangle, the Euler line and the Brocard axis, in a configuration with an arbitrary rectangle and a random point. The classical Cartesian coordinate system method…
We show that the classical algebra of quaternions is a commutative $\Z_2\times\Z_2\times\Z_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.
This survey aims to collect the main results of the theory of the set-theoretical solutions to the pentagon equation obtained up to now in the literature. In particular, we present some classes of solutions and raise some questions.
The notion of polytopal map between two polytopal complexes is defined. Surprisingly, this definition is quite simple and extends naturally those of simplicial and cubical maps. It is then possible to define an induced chain map between the…
Any three basic hypergeometric series {}_{2}phi_{1} whose respective parameters (a, b, c) differ by integer powers of the base q satisfy a linear relation with coefficients which are rational functions of a, b, c, q and the variable x.…
We give details of a formerly known relation between ternary quadratic forms and quaternion orders through the even Clifford algebra. Based on this and classifications of ternary quadratic forms we give a completely explicit classification…
Power nonnegative matrices are defined as complex matrices having at least one nonnegative integer power. We exploit the possibility of deriving a Perron Frobenius-like theory for these matrices, obtaining three main results and drawing…
Lotuses are certain types of finite contractible simplicial complexes, obtained by identifying vertices of polygons subdivided by diagonals. As we explained in a previous paper, each time one resolves a complex reduced plane curve…
We describe a simple probabilistic method to cross-identify astrophysical sources from different catalogs and provide the probability that a source is associated with a source from another catalog or that it has no counterpart. When the…
We define a canonical quadratic pair on the Clifford algebra of an algebra with quadratic pair over a field. This allows us to extend to the characteristic 2 case the notion of trialitarian triples, from which we derive a characterization…