Parallel prefix sums algorithms are one of the simplest and most useful building blocks for constructing parallel algorithms. A generic implementation is valuable because of the wide range of applications for this method. This thesis presents a generic C++ implementation of parallel prefix sums. The implementation applies two separate parallel prefix sums algorithms: a recursive doubling (RD) algorithm and a binary-tree based (BT) algorithm. This implementation shows how common communication patterns can be separated from the concrete parallel prefix sums algorithms and thus simplify the work of parallel programming. For each algorithm, the implementation uses two different synchronization options: barrier synchronization and point-to-point synchronization. These synchronization options lead to different communication patterns in the algorithms, which are represented by dependency graphs between tasks. The performance results show that point-to-point synchronization performs better than barrier synchronization as the number of processors increases. As part of the applications for parallel prefix sums, parallel radix sort and four parallel tree applications are built on top of the implementation. These applications are also fundamental parallel algorithms and they represent typical usage of parallel prefix sums in numeric computation and graph applications. The building of such applications become straighforward given this generic implementation of parallel prefix sums.

Every (convex) star polygon with n vertices can be associated with a permutation on {1,2, . . . , n } . It is known that the sum of interior angles of the polygon is solely determined by £». In this thesis, we give an exact formula to calculate the sum of interior angles in term of £». We make use of this formula to derive a recurrence relation concerning the number of star polygons having a particular value of sums of interior angles.

Σκοπός της παρούσης εργασίας είναι η παρουσίαση διαφόρων μεθόδων υπολογισμού των ροπών κατανομής των ριζών των ορθογωνίων πολυωνύμων, ισοδύναμα των αθροισμάτων Newton των ριζών, δηλαδή με, ενός πολυωνύμου βαθμού και των αθροισμάτων Stieltjes που είναι αθροίσματα της μορφής, όπου και είναι οι ρίζες μιας λύσης μιας ομογενούς διαφορικής εξίσωσης δεύτερης τάξης. Κάποια από αυτά τα αθροίσματα βρίσκουν εφαρμογή στο να φράσουμε τις ρίζες κάποιων ειδικών συναρτήσεων, ενώ άλλα χρησιμοποιούνται για την μελέτη της ασυμπτωτικής κατανομής των ριζών των ορθογωνίων πολυωνύμων και για τη μελέτη της μονοτονίας των ριζών. Παρουσιάζουμε δύο μεθόδους υπολογισμού των ροπών της κατανομής των ριζών των ορθογωνίων πολυώνυμων. Στην πρώτη μέθοδο υπολογίζουμε τα αθροίσματα Netwon των ριζών χρησιμοποιώντας τις ιδιοτιμές ενός τριδιαγώνιου πίνακα, ενώ στη δεύτερη μέθοδο ο υπολογισμός των αθροισμάτων Netwon γίνεται μέσω των συντελεστών των διαφορικών εξισώσεων που ικανοποιούν τα πολυώνυμα. Θα πρέπει εδώ να τονίσουμε ότι η δεύτερη μέθοδος μας επιτρέπει να υπολογίσουμε αριθμητικά τα αθροίσματα Netwon. / We present some methods for computing the distirbution of the roots of orthogonal polynomials, equivantly of Newton sums, and we sketch their use in determining some properties of second order ordinary differential equations.

Thesis (Ph.D.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/06/08). Includes bibliographical references.

abstract: ABSTRACT This thesis attempts to answer two questions based upon the historical observation that 1^2 +2^2 +· · ·+24^2 = 70^2. The first question considers changing the starting number of the left hand side of the equation from 1 to any perfect square in the range 1 to 10000. On this question, I attempt to determine which perfect square to end the left hand side of the equation with so that the right hand side of the equation is a perfect square. Mathematically, Question #1 can be written as follows: Given a positive integer r with 1 less than or equal to r less than or equal to 100, find all nontrivial solutions (N,M), if any, of r^2+(r+1)^2+···+N^2 =M^2 with N,M elements of Z+. The second question considers changing the number of terms on the left hand side of the equation to any fixed whole number in the range 1 to 100. On this question, I attempt to determine which perfect square to start the left hand side of the equation with so that the right hand side of the equation is a perfect square. Mathematically, Question #2 can be written as follows: Given a positive integer r with 1 less than or equal to r less than or equal to 100, find all solutions (u, v), if any, of u^2 +(u+1)^2 +(u+2)^2 +···+(u+r-1)^2 =v^2 with u,v elements of Z+. The two questions addressed by this thesis have been on the minds of many mathematicians for over 100 years. As a result of their efforts to obtain answers to these questions, a lot of mathematics has been developed. This research was done to organize that mathematics into one easily accessible place. My findings on Question #1 can hopefully be used by future mathematicians in order to completely answer Question #1. In addition, my findings on Question #2 can hopefully be used by future mathematicians as they attempt to answer Question #2 for values of r greater than 100. / Dissertation/Thesis / M.A. Mathematics 2010

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 31-32).

We consider the greatest prime factors of integers of certain form.

We consider the greatest prime factors of integers of certain form.

Thesis (Doctoral)--Rheinische Friedrich-Wilhelms-Universtät Bonn, 1990. / Includes bibliographical references.

The problem we study in this thesis asks under which conditions an r-factorization of Kₘʰ can be embedded into an s-factorization of Kₙʰ. This problem is a generalization of a problem posed by Peter Cameron which asks under which conditions a 1-factorization of Kₘʰ can be embedded into a 1-factorization of Kₙʰ. This was solved by Häggkvist and Hellgren. We study sufficient conditions in the case where s = h and m divides n. To that end, we take inspiration from a paper by Amin Bahmanian and Mike Newman and simplify the problem to the construction of an "acceptable" partition. We introduce the notion of irreducible sums and link them to the main obstacles in constructing acceptable partitions before providing different methods for circumventing these obstacles. Finally, we discuss a series of open problems related to this case.