Iterated sums of arithmetic progressions

Using generating functions we obtain a closed form expression involving two binomial coefficients for the iterated or k-fold summation of an arbitrary arithmetic progression of real numbers. As a contrast we obtain the same closed form expression using an elementary method based on an examination of Pascal's triangle. Some combinatorial interpretations of the iterated sums are also provided

If Archimedes would have known functions

These are notes and slides from a Pecha-Kucha talk given on March 6, 2013. The presentation tinkered with the question whether calculus on graphs could have emerged by the time of Archimedes, if the concept of a function would have been available 2300 years ago. The text first attempts to boil down discrete single and multivariable calculus to one page each, then presents the slides with additional remarks and finally includes 40 "calculus problems" in a discrete or so-called 'quantum calculus' setting. We also added some sample Mathematica code, gave a short overview over the emergence of the function concept in calculus and included comments on the development of calculus textbooks over time.Comment: 31 pages, 36 figure

Analogues Between Leibniz\u27s Harmonic Triangle and Pascal\u27s Arithmetic Triangle

This paper will discuss the analogues between Leibniz\u27s Harmonic Triangle and Pascal\u27s Arithmetic Triangle by utilizing mathematical proving techniques like partial sums, committees, telescoping, mathematical induction and applying George Polya\u27s perspective. The topics presented in this paper will show that Pascal\u27s triangle and Leibniz\u27s triangle both have hockey stick type patterns, patterns of sums within shapes, and have the natural numbers, triangular numbers, tetrahedral numbers, and pentatope numbers hidden within. In addition, this paper will show how Pascal\u27s Arithmetic Triangle can be used to construct Leibniz\u27s Harmonic Triangle and show how both triangles relate to combinatorics and arithmetic through the coefficients of the binomial expansion. Furthermore, combinatorics plays an increasingly important role in mathematics, starting when students enter high school and continuing on into their college years. Students become familiar with the traditional arguments based on classical arithmetic and algebra, however, methods of combinatorics can vary greatly. In combinatorics, perhaps the most important concept revolves around the coefficients of the binomial expansion, studying and proving their properties, and conveying a greater insight into mathematics

A Gemoetrical Approach to Computing Expected Cycle Times for Class-Based Storage Layouts in AS/RS

An exact, geometry-based analytical model is presented that can be used to compute the expected cycle time for a storage/retrieval (S/R) machine, executing single-commands, dual-commands, or both, in a rack structure that has been laid out in pre-specified storage zones for classes of goods.The rack may be either square-in-time or non-square-in-time.The approach is intuitively appealing, and it does not assume any certain layout shape, such as traditional "L-shaped" class layouts.The model has been coded in Turbo Pascal, and can be used by designers as a tool for quickly evaluating alternative layout configurations with respect to expected S/R cycle time in an AS/RS, and thereby the throughput of an automated warehouse over time.This model has been successfully applied in a major manufacturing plant in Europe to evaluate reconfigurations of their rack storage layouts over the past five years.Automated storage and retrieval systems;AS/RS;class-based storage.

Generalizations of Pascal\u27s Triangle: A Construction Based Approach

The study of this paper is based on current generalizations of Pascal\u27s Triangle, both the expansion of the polynomial of one variable and the multivariate case. Our goal is to establish relationships between these generalizations, and to use the properties of the generalizations to create a new type of generalization for the multivariate case that can be represented in the third dimension. In the first part of this paper we look at Pascal\u27s original Triangle with properties and classical applications. We then look at contemporary extensions of the triangle to coefficient arrays for polynomials of two forms. The first of a general polynomial in one variable with terms of each power and coefficients of one, the second the sum of an arbitrary number of terms typical to a multinomial expansion. We look at construction of the resulting objects, properties and applications. We then relate the two objects together through substitution and observe a general process in which to do so. In the second part of the paper I observe an application of the current generalizations to the classical problem The Gambler\u27s Game of Points to games of alternative point structures. The paper culminates with a generalization I have made for a particular case of the second equation, moving the current four dimensional generalization into the third dimension for observation and study. We see the relationships of this generalization to those from our overview in part one, and develop the main theorem of study from the construction of its arrangement. From this theorem we are able to derive several interesting combinatorial identities from our construction