Behavior of digital sequences through exotic numeration systems

peer reviewedMany digital functions studied in the literature, e.g., the summatory function of the base-k sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic number theory or linear algebra. In this paper we develop a method based on exotic numeration systems and we apply it on two examples motivated by the study of generalized Pascal triangles and binomial coefficients of words

The cultural challenge in mathematical cognition

In their recent paper on “Challenges in mathematical cognition”, Alcock and colleagues (Alcock et al. [2016]. Challenges in mathematical cognition: A collaboratively-derived research agenda. Journal of Numerical Cognition, 2, 20-41) defined a research agenda through 26 specific research questions. An important dimension of mathematical cognition almost completely absent from their discussion is the cultural constitution of mathematical cognition. Spanning work from a broad range of disciplines – including anthropology, archaeology, cognitive science, history of science, linguistics, philosophy, and psychology – we argue that for any research agenda on mathematical cognition the cultural dimension is indispensable, and we propose a set of exemplary research questions related to it

Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems

We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand property, i.e., we can add or remove trailing zeroes to valid representations. It is a folklore fact that the Sierpi\'{n}ski gasket is the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from the classical Pascal triangle modulo $2$. In a similar way, we describe and study the subset of $[0, 1] \times [0, 1]$ associated with the latter generalization of the Pascal triangle modulo a prime number.Comment: 30 pages; 32 figure

Pascal triangles and Sierpiński gasket extended to binomial coefficients of words

The binomial coefficient (u,v) of two finite words u and v (on a finite alphabet) is the number of times the word v appears inside the word u as a subsequence (or, as a "scattered" subword). For instance, (abbabab,ab)=4. This concept naturally extends the classical binomial coefficients of integers, and has been widely studied for about thirty years (see, for instance, Simon and Sakarovitch). In this talk, I present the research lead from October 2015 on an extension of the Pascal triangles to base-2 expansions of integers. In a first part, I define two new objects that both generalize the classical Pascal triangle and the Sierpinski gasket. In a second part, I define a new sequence extracted from the Pascal triangle in base 2 and study its regularity. In a third part, I exhibit an exact formula for the behavior of the summatory function of the latter sequence

Some generalizations of the Pascal triangle: base 2 and beyond

The binomial coefficient (u,v) of two finite words u and v (on a finite alphabet) is the number of times the word v appears inside the word u as a subsequence (or, as a "scattered" subword). For instance, (abbabab,ab)=4. This concept naturally extends the classical binomial coefficients of integers, and has been widely studied for about thirty years (see, for instance, Simon and Sakarovitch). In this talk, I present the research lead from October 2015: I give the main ideas that lead to an extension of the Pascal triangles to base-2 expansions of integers. After that, I extend the work to any Parry-Bertrand numeration system including the Fibonacci numeration system

Program and Abstracts of the Annual Meeting of the Georgia Academy of Science, 2015

The annual meeting of the Georgia Academy of Science took place March 13-14, 2015, at Georgia College & State University, Milledgeville, Georgia. The keynote speaker was Dr. Chryssa Kouveliotou of UA-Huntsville and the Marshall Space Center. Her presentation was entitled The Transient High-Energy Sky. Additional presentations were provided by members of the Academy who represented the following sections: I. Biological Sciences II Chemistry III. Earth & Atmospheric Sciences IV. Physics, Mathematics, Computer Science, Engineering & Technology V. Biomedical Sciences VI. Philosophy & History of Science VII. Science Education VIII. Anthropolog

World Cotton Germplasm Resources

Preservation of plant germplasm resources is vitally important for mankind to supply food and product security in the globalization and technological advances of the 21st century. Mankind preserved a wealth of available genetic resources of many plant species worldwide. One of the such worldwide plant germplasm resources is available for cotton, a unique natural fiber producing cash crop for mankind. Worldwide cotton germplasm collections exist in Australia, Brazil, China, India, France, Pakistan, Turkey, Russia, United States of America, and Uzbekistan. The objective of World Cotton Germplasm Resources book is to present readers with updated information on existing cotton germplasm resources, highlighting detailed inventory, description, storage conditions, characterization and utilization as well as challenges and perspectives. This book should be a comprehensive encyclopedic reading source for plant research community and students to gather important information on worldwide cotton germplasm resources