Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials'$ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial$ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya$ and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure

The LaTeX project: A case study of open-source software

This is a case study of TeX, a typesetting software that was developed by Donald E. Knuth in the late 70's. Released with an open source license, it has become a reference in scientific publishing. TeX is now used to typeset and publish much of the world's scientific literature in physics and mathematics. This case study is part of a wider effort by academics to understand the open-source phenomenon. That development model is similar to the organization of the production of knowledge in academia; there is no set organization with a hierarchy, but free collaboration that is coordinated spontaneously and winds up generating complex products that are the property of all who can understand its functioning. The case study was led by gathering qualitative data via interviews with TeX developers and quantitative data on the TeX community -- the program's code, the software that is part of the TeX distribution, the newsgroups dedicated to the software, and many other indicators of the evolution and activity in that open-source project. The case study is aimed at economists who want to develop models to understand and analyze the open-source phenomenon. It is also geared towards policy-makers who would like to encourage or regulate open- source, and towards open-source developers who wonder what are the efficient strategies to make an open-source project successful.TeX, LaTeX, case study, open source, software, innovation, organisational structure, economic history, knowledge production, knowledge diffusion.

Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids

For points in $d$ real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed $d$ by $d$ matrix over \bz. Our starting point is a given pair $(A, \mathcal D)$ with the matrix $A$ assumed expansive, and $\mathcal D$ a chosen complete digit set, i.e., in bijective correspondence with the points in \bz^d/A^T\bz^d. We give an explicit geometric representation and encoding with infinite words in letters from $\mathcal D$. We show that the attractor $X(A^T,\mathcal D)$ for an affine Iterated Function System (IFS) based on $(A,\mathcal D)$ is a set of fractions for our digital representation of points in \br^d. Moreover our positional "number representation" is spelled out in the form of an explicit IFS-encoding of a compact solenoid \sa associated with the pair $(A,\mathcal D)$. The intricate part (Theorem \ref{thenccycl}) is played by the cycles in \bz^d for the initial $(A,\mathcal D)$-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional $\mathcal D$-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces