Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 4

Search results for: combinatorics

4 A Generalization of Planar Pascal’s Triangle to Polynomial Expansion and Connection with Sierpinski Patterns

Authors: Wajdi Mohamed Ratemi


The very well-known stacked sets of numbers referred to as Pascal’s triangle present the coefficients of the binomial expansion of the form (x+y)n. This paper presents an approach (the Staircase Horizontal Vertical, SHV-method) to the generalization of planar Pascal’s triangle for polynomial expansion of the form (x+y+z+w+r+⋯)n. The presented generalization of Pascal’s triangle is different from other generalizations of Pascal’s triangles given in the literature. The coefficients of the generalized Pascal’s triangles, presented in this work, are generated by inspection, using embedded Pascal’s triangles. The coefficients of I-variables expansion are generated by horizontally laying out the Pascal’s elements of (I-1) variables expansion, in a staircase manner, and multiplying them with the relevant columns of vertically laid out classical Pascal’s elements, hence avoiding factorial calculations for generating the coefficients of the polynomial expansion. Furthermore, the classical Pascal’s triangle has some pattern built into it regarding its odd and even numbers. Such pattern is known as the Sierpinski’s triangle. In this study, a presentation of Sierpinski-like patterns of the generalized Pascal’s triangles is given. Applications related to those coefficients of the binomial expansion (Pascal’s triangle), or polynomial expansion (generalized Pascal’s triangles) can be in areas of combinatorics, and probabilities.

Keywords: pascal’s triangle, generalized pascal’s triangle, polynomial expansion, sierpinski’s triangle, combinatorics, probabilities

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3 Development of a Psychometric Testing Instrument Using Algorithms and Combinatorics to Yield Coupled Parameters and Multiple Geometric Arrays in Large Information Grids

Authors: Laith F. Gulli, Nicole M. Mallory


The undertaking to develop a psychometric instrument is monumental. Understanding the relationship between variables and events is important in structural and exploratory design of psychometric instruments. Considering this, we describe a method used to group, pair and combine multiple Philosophical Assumption statements that assisted in development of a 13 item psychometric screening instrument. We abbreviated our Philosophical Assumptions (PA)s and added parameters, which were then condensed and mathematically modeled in a specific process. This model produced clusters of combinatorics which was utilized in design and development for 1) information retrieval and categorization 2) item development and 3) estimation of interactions among variables and likelihood of events. The psychometric screening instrument measured Knowledge, Assessment (education) and Beliefs (KAB) of New Addictions Research (NAR), which we called KABNAR. We obtained an overall internal consistency for the seven Likert belief items as measured by Cronbach’s α of .81 in the final study of 40 Clinicians, calculated by SPSS 14.0.1 for Windows. We constructed the instrument to begin with demographic items (degree/addictions certifications) for identification of target populations that practiced within Outpatient Substance Abuse Counseling (OSAC) settings. We then devised education items, beliefs items (seven items) and a modifiable “barrier from learning” item that consisted of six “choose any” choices. We also conceptualized a close relationship between identifying various degrees and certifications held by Outpatient Substance Abuse Therapists (OSAT) (the demographics domain) and all aspects of their education related to EB-NAR (past and present education and desired future training). We placed a descriptive (PA)1tx in both demographic and education domains to trace relationships of therapist education within these two domains. The two perceptions domains B1/b1 and B2/b2 represented different but interrelated perceptions from the therapist perspective. The belief items measured therapist perceptions concerning EB-NAR and therapist perceptions using EB-NAR during the beginning of outpatient addictions counseling. The (PA)s were written in simple words and descriptively accurate and concise. We then devised a list of parameters and appropriately matched them to each PA and devised descriptive parametric (PA)s in a domain categorized information grid. Descriptive parametric (PA)s were reduced to simple mathematical symbols. This made it easy to utilize parametric (PA)s into algorithms, combinatorics and clusters to develop larger information grids. By using matching combinatorics we took paired demographic and education domains with a subscript of 1 and matched them to the column with each B domain with subscript 1. Our algorithmic matching formed larger information grids with organized clusters in columns and rows. We repeated the process using different demographic, education and belief domains and devised multiple information grids with different parametric clusters and geometric arrays. We found benefit combining clusters by different geometric arrays, which enabled us to trace parametric variables and concepts. We were able to understand potential differences between dependent and independent variables and trace relationships of maximum likelihoods.

Keywords: psychometric, parametric, domains, grids, therapists

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2 An Optimized Approach to Generate the Possible States of Football Tournaments Final Table

Authors: Mouslem Damkhi


This paper focuses on possible states of a football tournament final table according to the number of participating teams. Each team holds a position in the table with which it is possible to determine the highest and lowest points for that team. This paper proposes an optimized search space based on the minimum and maximum number of points which can be gained by each team to produce and enumerate the possible states for a football tournament final table. The proposed search space minimizes producing the invalid states which cannot occur during a football tournament. The generated states are filtered by a validity checking algorithm which seeks to reach a tournament graph based on a generated state. Thus, the algorithm provides a way to determine which team’s wins, draws and loses values guarantee a particular table position. The paper also presents and discusses the experimental results of the approach on the tournaments with up to eight teams. Comparing with a blind search algorithm, our proposed approach reduces generating the invalid states up to 99.99%, which results in a considerable optimization in term of the execution time.

Keywords: combinatorics, enumeration, graph, tournament

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1 Geometric Intuition and Formalism in Passing from Indivisibles to Infinitesimals: Pascal and Leibniz

Authors: Remus Titiriga


The paper focuses on Pascal's indivisibles evolving to Leibniz's infinitesimals. It starts with parallel developments by the two savants in Combinatorics (triangular numbers for Pascal and harmonic triangles for Leibniz) and their implication in determining the sum of mathematical series. It follows with a focus on the geometrical contributions of Pascal. He considered the cycloid and other mechanical curves the epitome of geometric comprehensibility in a series of challenging problems he posed to the mathematical world. Pascal provided the solutions in 1658, in a volume published under the pseudonym of Dettonville, using indivisibles and ratios between curved and straight lines. In the third part, the research follows the impact of this volume on Leibniz as the initial impetus for the elaboration of modern calculus as an algorithmic method disjoint of geometrical intuition. Then paper analyses the further steps and proves that Leibniz's developments relate to his philosophical frame (the search for a characteristic Universalis, the consideration of principle of continuity or the rule of sufficient reason) different from Pascal's and impacting mathematical problems and their solutions. At this stage in Leibniz's evolution, the infinitesimals replaced the indivisibles proper. The last part of the paper starts with speculation around "What if?". Could Pascal, if he lived more, accomplish the same feat? The document uses Pascal's reconstructed philosophical frame to formulate a positive answer. It also proposes to teach calculus with indivisibles and infinitesimals mimicking Pascal and Leibniz's achievements.

Keywords: indivisibles, infinitesimals, characteristic triangle, the principle of continuity

Procedia PDF Downloads 57