1,2,3, some inductive real sequences and a beautiful algebraic pattern

By rewriting the relation 1+2=3 as √12+√22=√32 , a right triangle is looked at. Some geometrical observations in connection with plane parqueting lead to an inductive sequence of right triangles with √12+√22=√32 as initial one converging to the segment [0,1] of the real line. The sequence of their hypotenuses forms a sequence of real numbers which initiates some beautiful algebraic patterns. They are determined through some recurrence relations which are proper for being evaluated with computer algebra

A pair of rational double sequences

Double sequences appear in a natural way in cases of iteratively given sequences if the iteration allows to determine besides the successors from the predecessors also the predecessors from their followers. A particular pair of double sequences is considered which appears in a parqueting-reflection process of the complex plane. While one end of each sequence is a natural number sequence, the other consists of rational numbers. The natural numbers sequences are not yet listed in OEIS Wiki. Complex versions from the double sequences are provided

Biharmonic Green functions

The harmonic Green and Neumann function and a particular Robin function are used to construct bi-harmonic Green, Neumann and particular Robin functions. Moreover hybrid bi-harmonic Green functions are given. They all are constructed via a convolution of the mentioned harmonic particular fundamental solutions. In case of the unit disc they are explicitly expressed. Besides these 9 bi-harmonic Green functions there is another bi-harmonic Green function in explicit form for the unit disc not defined by convolution. Related boundary value problems are not all well posed. In case they are, the unique solutions are given. For the other cases solvability conditions are determined and the unique solutions found. There are all together 12 Dirichlet kind boundary value problems for the inhomogeneous bi-harmonic equation treated. The investigation is restricted to the two dimensional case and complex notation is used

Boundary value problems in complex analysis II

This is the continuation of an investigation of basic boundary value problems for first order complex model partial deferential equations. Model second order equations are the Poisson and the inhomogeneous Bitsadze equations. Deferent kinds of boundary conditions are posed as combinations of the Schwarz, the Dirichlet, and the Neumann conditions. Solvability conditions and the solutions are given in explicit form for the unit disc. Exemplarily the inhomogeneous polyanalytic equation is investigated as a model equation of arbitrary order

Estimates of solutions to linear elliptic systems and equations

Whenever nonlinear problems have to be solved through approximation methods by solving related linear problems a priori estimates are very useful. In the following this kind of estimates are presented for a variety of equations related to generalized first order Beltrami systems in the plane and for second order elliptic equations in $ℝ^m$. Different types of boundary value problems are considered. For Beltrami systems these are the Riemann-Hilbert, the Riemann and the Poincaré problem, while for elliptic equations the Dirichlet problem as well as entire solutions are involved