Parallel scheduling of recursively defined arrays

A new method of automatic generation of concurrent programs which constructs arrays defined by sets of recursive equations is described. It is assumed that the time of computation of an array element is a linear combination of its indices, and integer programming is used to seek a succession of hyperplanes along which array elements can be computed concurrently. The method can be used to schedule equations involving variable length dependency vectors and mutually recursive arrays. Portions of the work reported here have been implemented in the PS automatic program generation system

Capstone Mathematics and Technology: A Collection of Mathematical Technology Enhanced Activities for Students and Teachers

The purpose of this project is to provide an introduction to how technology can be used in the mathematical classroom to enhance students\u27 learning of mathematics, while at the same time leading students to a richer and deeper understanding of those mathematical concepts. The topics were selected based on their relevance to the Utah State Core Curriculum for middle and secondary mathematics courses. It was intended that each lesson plan would challenge a preservice mathematics educator to build relationships between different areas of mathematics and/or to create deeper understandings of specific mathematical concepts. At the same time many of the lesson plans can be used at the high school level to teach mathematical ideas. The ideas arc not too complex for the secondary level, but their extensions that will hopefully inspire a preservice mathematics teacher to search for deeper understandings. It is hoped that these lessons will promote a desire in the students who work out the activities to create their own lesson plans, plans which relate activities, mathematical topics and technologies together for deeper understanding

Zero-sum triangles for involutory, idempotent, nilpotent and unipotent matrices

In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for the recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent and unipotent properties, especially nilpotent and unipotent matrices of index 2. With the zero-sum rule we also give the conditions for the special matrices and the generic methods for the generation of those special matrices. Some of the generated integer triangles have been found by other methods, but most of them are newly discovered, and many combinatorial identities can be found with them. The results may also interest the economists for trading analysis and simulation

SOME NOTIONS ON LEAST COMMON MULTIPLES

There is the well known result that n! divides the product of any set of n consecutive numbers. Using this idea we define Smarandache LCM Ratio Sequence ofthe rth kind as SLRS(r)

Parallel scheduling of recursively defined arrays

AbstractThis paper describes a new method of automatic generation of concurrent programs which construct arrays defined by sets of recursive equations. We assume that the time of computation of an array element is a linear combination of its indices, and we use integer programming to seek a succession of hyperplanes along which array elements can be computed concurrently. The method can be used to schedule equations involving variable length dependency vectors and mutually recursive arrays. Portions of the work reported here have been implemented in the PS automatic program generation system

Chemical Graphs. XL.1 Three Relations Between the Fibonacci Sequence and the Numbers of Kekule Structures for Non-branched cata-Condensed Polycyclic Aromatic Hydrocarbons

Fo·r benze.notd or non-benzenoid ca:ta1fusenes having a non- ibranched string 01f cata-co.ndensed rings, the numbers K of Kekule structures (perfect matching·s) can be expressed vi<t the recurrence relationship (1); as a coa.-ollary when each annelated .segment has exactly two ring.s, the numbers O\u27f Kelm.le structures form the Fiibonacci sequence. Coro.nary 2 presents a second re.lationshi:p with Fiibonacci numbers. Algebraic expressions for the number of Kekule struc- 1tures in non-brainched cata.fusenes 1n terms of hexago.n numbers iJn each linearly condensed segment can be obtained. The numbers of terms in .such .a,,lgebraic expressivns lead to a new numerical triangle (Table I) which is related to Pascal\u27s triangle, and which pwvides a third link with the F ~bonacci numbers expressed either by relation (7) or by the equivalent relation (10)

An Alternate Approach to Alternating Sums: A Method to DIE for

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