Polygon 2022 by MDC-Polygon

Editorial Note:

1780 West 49th Street, Hialeah, Florida 33012, USA

Polygon is MDC Hialeah's Academic Journal. It is a multi-disciplinary online publication whose purpose is to display the academic work produced by faculty and staff. We, the editorial committee of Polygon, are pleased to publish the 2022 Spring Issue Polygon which is the fifteenth consecutive issue of Polygon. It includes seven regular research articles. We are pleased to present work from a diverse array of fields written by faculty from across the college. The editorial committee of Polygon is thankful to the Miami Dade College President Madeline Pumariega, Miami Dade College District Board of Trustees, Hialeah Campus President, Dr. Anthony Cruz, Dean of Faculty, Dr. Caridad Castro, Chairperson of Hialeah Campus Liberal Arts and Sciences, Ms. Tiffany Nicholson, Chairperson of Hialeah Campus World Languages and Communication, Professor Liliana Cobas, , Chairperson of Business, Engineering & Technology, Professor Charles Williams, III, Dean of Student Services, Dr. Nelson Magaña, Director of Hialeah Campus Administrative Services, Ms. Andrea M. Forero, Network & Multimedia Support Manager of Hialeah Campus, Mr. Leyva Alejandro, all staff and faculty of Hialeah Campus and Miami Dade College, in general, for their continued support and cooperation for the publication of Polygon. Sincerely, Editorial Board Members of Polygon: Dr. M. Shakil (Editor-in-Chief), Dr. Jaime Bestard (Editor) and Professor Victor Calderin (Editor). Advisory & Reviewer Committee of Polygon: Dr. Kelly Kennedy, Dr. Alex Gancedo, Prof. Loretta Blanchette, Prof. Rene Barientos, Dr. Melissa Lammey, Dr. Carlos Ruiz, Prof. Rodolfo Cruz, Dr. Victoria Castells, Dr. Mariana Vaillant Molina, Dr. Allison Thomas Johnson Patrons: Dr. Anthony Cruz, President, Hialeah Campus; Dr. Caridad Castro, Chairperson of Liberal Arts and Sciences; Ms. Tiffany Nicholson, Chairperson of Hialeah Campus Liberal Arts and Sciences; Professor Liliana Cobas, Chairperson of World Languages and Communication; Professor Charles Williams, III, Chairperson of Business, Engineering & Technology; Dr. Nelson Magaña, Dean of Student Services; Mr. Alexander Hernandez, Director of Learning Resources; Ms. Youdaris Mira, Chairperson of Continuing Education; Ms. Andrea M. Forero, Director of Hialeah Campus Administrative Services; Ms. Yudi Moses, Director of Admissions, Registration & Financial Aid; Mr. Jacob Shilts, Director of Student Services; Mr. Leyva Alejandro, Network & Multimedia Support Manager. Miami Dade College District Board of Trustees: Chair (Trustee) Michael Bileca; Vice Chair (Trustee) Nicole Washington; Trustee Dr. Anay Abraham; Trustee Marcell Felipe; Trustee Roberto Alonso; Trustee Ismare Monreal; Trustee Maria Bosque Blanco; Miami Dade College President Madeline Pumariega; Miami Dade College Executive Vice President and Provost Dr. Malou C. Harrison.

CONTENTS ARTICLES / AUTHOR(S)

PAGES

Jules Henri Poincare (1854-1912), the poet of the infinite - 110th anniversary of the death of the last great universalist - Adolfo L. Mendez, Ph.D., and Maria Mercedes Corredor

1 – 10

The Performance of Students Interactions While Producing Project- Based Learning Remote Strategies – Jaime Bestard, Ph.D.

11 – 19

Bootstraps Confidence Interval Estimates, with Some Uses of StatKey: An Intuitive Introduction for Elementary Statistics Students – Eric O. Hernandez, Ph.D., and M. Shakil, Ph.D.

20 – 36

1943 and 2000 - Years of Special Achievements in the History of African-American Women in Mathematical Sciences - M. Shakil, Ph.D.

37 – 53

Landau “big-O” and “small-o” symbols – A Historical Introduction, with Some Applicability - M. Shakil, Ph.D.

54 – 60

A Theoretical Study of Structures, Properties, Reaction Pathways and Biochemical Activities of 3-(2-Fluorophenyl)-1HIndazole Isomers - Subhojit Majumdar, Ph.D.

61 - 79

Use Of 3D Printers for Physics Demonstrations - Wojciech J. Walecki, Ph.D., Adrian S. Vila, And Dayron Gomez

80- 97

Previous Editions Link: https://issuu.com/mdc-polygon Disclaimer: The views and perspectives of the authors presented in their respective articles published in Polygon do not represent those of Miami Dade College. Mission of Miami Dade College As democracy’s college, Miami Dade College changes lives through accessible, high-quality teaching and learning experiences. The College embraces its responsibility to serve as an economic, cultural and civic leader for the advancement of our diverse global community. iii

Solicitation of Articles for the 2023 Issue (16th Issue) of Polygon: The editorial committee would also like to cordially invite the MDC community to submit their articles for consideration for the 2023 Issue (16th Issue) of Polygon. Guidelines for Submission POLYGON “Many Corners, Many Faces (POMM)” A premier professional refereed multi-disciplinary electronic journal of scholarly works, feature articles and papers on descriptions of Innovations at Work, higher education, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination, published by MDC Hialeah Campus Liberal Arts and Sciences Department (LAS). Editorial Board: Dr. Mohammad Shakil (Mathematics) (Editor-in-Chief) Dr. Jaime Bestard (Mathematics) Prof. Victor Calderin (English) Manuscript Submission Guidelines: Welcome from the POLYGON Editorial Team: The Department of Liberal Arts and Sciences at the Miami Dade College–Hialeah Campus and the members of editorial Committee - Dr. Mohammad Shakil, Dr. Jaime Bestard, and Professor Victor Calderin – would like to welcome you and encourage your rigorous, engaging, and thoughtful submissions of scholarly works, feature articles and papers on descriptions of Innovations at Work, higher education, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. We are pleased to have the opportunity to continue the publication of the POLYGON, which will be bi-anually during the Fall & Spring terms of each academic year. We look forward to hearing from you. General articles and research manuscripts: Potential authors are invited to submit papers for the next issues of the POLYGON. All manuscripts must be submitted electronically (via e-mail) to one of the editors at mshakil@mdc.edu, or jbestard@mdc.edu, or vcalderi@mdc.edu. This system will permit the new editors to keep the submission and review process as efficient as possible. Typing: Acceptable formats for electronic submission are MSWord, and PDF. All text, including title, headings, references, quotations, figure captions, and tables, must be typed, with 1 1/2 line spacing, and one-inch margins all around. Please employ a minimum font size of 11. Please see the attached template for the preparation of the manuscripts. Length: A manuscript, including all references, tables, and figures, should not exceed 7,800 words (or at most 20 pages). Submissions grossly exceeding this limit may not be accepted for review. Authors should keep tables and figures to a minimum and include them at the end of the text. Style: For writing and editorial style, authors must follow guidelines in the Publication Manual of the American Psychological Association (5th edition, 2001). The editors request that all text pages be numbered. You may also please refer to the attached template for the preparation of the manuscripts.

iv Abstract and keywords: All general and research manuscripts must include an abstract and a few keywords. Abstracts describing the essence of the manuscript must be 150 words or less. The keywords will be used by readers to search for your article after it is published. Book reviews: POLYGON accepts unsolicited reviews of current scholarly books on topics related to research, policy, or practice in higher education, Innovations at Work, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. Book reviews may be submitted to either themed or open-topic issues of the journal. Book review essays should not exceed 1,900 words. Please include, at the beginning of the text, city, state, publisher, and the year of the book’s publication. An abstract of 150 words or less and keywords are required for book review essays. Notice to Authors of Joint Works (articles with more than one author). This journal uses a transfer of copyright agreement that requires just one author (the Corresponding Author) to sign on behalf of all authors. Please identify the Corresponding Author for your work when submitting your manuscript for review. The Corresponding Author will be responsible for the following: ensuring that all authors are identified on the copyright agreement, and notifying the editorial office of any changes to the authorship. • securing written permission (via email) from each co-author to sign the copyright agreement on the co-author’s behalf. • warranting and indemnifying the journal owner and publisher on behalf of all coauthors. •

Although such instances are very rare, you should be aware that in the event a co-author has included content in their portion of the article that infringes the copyright of another or is otherwise in violation of any other warranty listed in the agreement, you will be the sole author indemnifying the publisher and the editor of the journal against such violation. Please contact the editorial office if you have any questions or if you prefer to use a copyright agreement for all coauthors to sign. Instructions for the Preparation of Manuscripts for the Polygon: (THE TITLE IS HERE) (12 pt, bold, 32 pt above) NAME IS HERE (11 pt16 pt above, 32 pt below) ABSTRACT is here, not exceeding 160 words. It must contain main facts of the work. (11 pt) Key words and phrases (11 pt) Introduction (11 pt, bold, 24 pt above, 12 pt below) Main Body of the Article Discussion Conclusion Acknowledgements REFERENCES (11 pt, 30 pt above, 12 pt below) [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1970.

v [2] J. Galambos and I. Simonelli, Products of Random Variables – Applications to Problems of Physics and to Arithmetical Functions, CRC Press, Boca Raton / Atlanta, 2005. [3] S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons & Fractals, 28(4) (2006), 930-937. [4] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlin. Sci. Numer. Simulat. 1(7) (2006), 15-27. (11 pt) Author’s Biographical Sketch (Optional): Dr. Y. Abu received his Master’s and Ph. D. Degrees in Mathematics from the University of Small Town, USA, in 1987, under the direction of Dr. M. Opor. Since 1989, he has been teaching at the Community College of Small Town, USA. His research interests lie in the Fractals, Solitons, Undergraduate Teaching of Mathematics, and Curriculum Development. (11 pt) Address: Department of Liberal Arts & Sciences (Mathematics Program), Community College of Small Town, P. O. Box 7777, Small Town, USA. E-mail: yabu@ccst. (11 pt)

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Jules Henri Poincare (1854-1912), the poet of the infinite 110th anniversary of the death of the last great universalist Adolfo L. Mendez, Ph.D. (*), Maria Mercedes Corredor Math and Natural Sciences Department, Miami Dade College-Padron Campus maria.corredor003@mymdc.net, amendez4@mdc.edu (*) Corresponding author ABSRACT This 2022 fulfills 110 years of the death of Jules Henri Poincare, the last universalist. This paper aims to commemorate and honor the last person in history to have a wide range of interests, knowledge, and activities in science. Poincare exerted a decisive and perpetual influence on the next generations of not only scientists but cultural heritage as well as in the growing scientific knowledge, being a reference in the scientific world. This contribution aims to give our tribute to one of the greatest minds ever illuminated. Keywords: Physics, Mathematics, Philosophy, Determinism, Chaos.

2 Henri Jules Poincare: A great universalist. “To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.” H. Poincare. Unlike Albert Einstein, Henri Poincare was not a cultural icon, he was a mathematician practically unknown in his day. Before Einstein, Poincare had the correct equations for the Lorentz transformations, anticipating Einstein in the basic foundations of the Theory of Special Relativity. However, although some of his work was fundamental in many of the German wiseman’s achievements, Poincare never achieved the popular world fame of Einstein. During his life, his sphere of influence did not include much beyond his academic colleagues even though he posed a puzzle that remained unsolved for almost a century. According to his biographers, what dominates his physical appearance is an expression of constant distraction. When speaking to him, one had a feeling that the man was not following or understanding what was said. Very few were able to follow his thoughts and he had no students. But after his unexpected death on July 17th, 1912; he was immortalized as the last universalist due to his enormous contribution and influence, not only in Mathematics but in Physics, Astronomy, Philosophy, and arts. Poincare contributed to a wide range of pure and practical mathematics topics, including celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity, and physical cosmology. Becoming a universalist for him was simply the consequence of letting his personality run wild. He got a drop of interest in a subject that he nourished and fed with knowledge until he became a connoisseur in the topic and made a difference. Additionally, Poincare read and wrote widely, beginning with popular science writings and progressing to more advanced texts. By the end of his life, he had written more than 16 books for the lay public which is 3 times more than Einstein. Altogether he wrote nearly five hundred research papers, along with many pamphlets and lecture notes. One of the great paradoxes in the history of science is the fact that Poincare received dozens of nominations for the Nobel Prize, but unfairly he was never awarded. He was the scientist that started doubting everything. In a book he wrote in 1904 on definitions in Mathematics he explains: “It is not enough to doubt everything, one must know why one doubts” (Poincare, 1904). This appears as his motto for all the other breakthroughs he accomplished in science, a philosophical point of view of mathematics, science, and logical thinking. The impact on science is that Poincare opened the doors for a new vision of natural sciences and Philosophy: concepts such as space,

3 determinism, intuition, among others, were revisited by and after him, with a fundamental impact on Geometry, Theory of Relativity, Quantum Mechanics, Non-linear Dynamics, all of them, ingredients of the new great scientific revolution. In recognition of his diversity of knowledge, as well as its originality and depth of thought, he was a member of more than thirty prestigious international scientific societies. When he was admitted to the Académie Françoise - a mainly literary organization - he was recognized as a scientific and cultural personality "whose reputation is established as an axiom" (Masson, 1909). In addition to his dozens of published volumes, which cover topics in number theory, calculus, general theory of functions, topology (of which he is one of its founders), differential calculus, electricity, thermodynamics, and astronomy, his prolific work includes remarkable volumes in philosophy of science, in which Poincare gave his original vision of science, in what constitutes the deepest and most rigorous analysis of science carried out until him. Summarizing the breadth and depth of his thought, another greatest physicist contemporary of Poincare, referred to his colleague as “… his extraordinary power of abstract construction is equilibrated by constant care for reality; he is realist in Mathematics as in Physics. The tree of his thought branched to infinity, is solidly attached to the soil by deep roots…” (Langevin, 1913). The depth, and creativity of Poincare's scientific-philosophical thought, as well as his original vision of Nature, has been compared to that of paradigmatic geniuses such as Aristotle, Descartes, or Leibnitz. Such a vast range of knowledge and interest in science has not existed in a single mind since Poincare. But, such is the impact of Poincare's work, that it is not reduced only to the scientificphilosophical field: contemporary art is also a debtor of the last universalist. Artists like Maurits Escher, Marcel Duchamp, and Pablo Picasso, who establish contemporary cultural patterns, were greatly influenced by Poincare's works by rejecting logic and common sense and expressing nonsense, and irrationality. Escher's vision of impossible objects from an n-dimensional perspective, conceptual art, and the Dada movement by Duchamp, as well as Picasso's revolutionary cubism, reflect Poincare's conception of space and objects. These artists were particularly struck by Poincare’s advice on how to view a fourth dimension, considered essential for the early-20th-century avant-garde art movement that revolutionized European painting and sculpture.

1. N-body problem when n = 3. Determinism in crisis. “...small differences in the initial conditions may produce very great ones in the final phenomena.

4 Predictions become impossible, and we have the fortuitous phenomenon.” H. Poincare.

In the past 20th century, three great revolutions in science marked a turn in the vision of how Nature works. Two of these three great revolutions in the 20th century, the General Theory of Relativity and Quantum Mechanics, removed two-pillar statements of Newtonian physics, the illusion of absolute space and time, as well as the dream of controllable measurement processes. The third revolution in the 20th century removed Newton’s conception of a deterministic world that is completely predictable: Chaos Theory. By the end of the 17th century, Isaac Newton replaced the old Aristotelian picture of the physical universe with the foundation of Classical Mechanics. The Newtonian worldview legacy is as simple as it is great: every event can be predicted, within a reasonable margin of error, according to physical laws, which only need to be discovered, combined appropriately, and applied accurately to determine what the future motions of objects will be. The Newtonian deterministic world states that if the precise location and velocity of every atom in the Universe are known, their past and future values for any given time are entailed; they can be calculated from the laws of Newton’s classical mechanics. The Universe is totally predictable. But after Newton’s works, the stability of the orbit of the Moon around Earth was one of the early subjects about orbit stabilities under a gravitational field with perturbations from other massive bodies like the Sun. The Swiss mathematician and astronomer Leonard Euler, one of the greatest mathematicians in history, found a periodic solution for that problem but considering that the Earth and the Sun orbited each other in circular orbits and that the Moon was massless. The problem was revisited by Poincare. The fundamental problem was to predict the motions (that is, the orbits) of three bodies mutually attracted by gravity, given their initial positions and velocities. In 1890, Poincare found that Newton’s solutions are useless: after a short time, the orbits become so tangled that it is not possible to predict the position of each body. What a challenge! Newton’s determinism seems to fall in crisis. What Poincare discovered was that a general solution to that problem is essentially impossible due to the proper dynamics of such a system of gravitationally linked bodies, which is highly sensitive to initial conditions. Poincare was the first to articulate this sensitivity in the study of the so-called three-body problem. In his works on the three-body problem, Poincare

5 became the first scientist to discover what we call today a chaotic system which laid the foundations of modern Chaos Theory. For that reason, although Chaos Theory is a relatively new discipline, Henri Poincare is considered the father of the discipline. 2. Philosophy of science. Intuition instead of formalism and logic. “Mathematics should be felt rather than formulated.” H. Poincare. Poincare’s contribution to the Philosophy of Science is initially focused on the nature of Mathematics. Before Poincare, Mathematicians were generally divided into formalists and logistics. As a follower of the German philosopher Immanuel Kant, the greatest philosopher of Modern Philosophy, Poincare identified himself as an intuitionist of Mathematics. His work was based on the understanding of the pure forms of Kant’s transcendental intuition with the inventiveness of mathematicians. Contrary to most mathematicians, for Poincare, Mathematics was more intuition and invention than discoveries. For him, intuition rises prior to all experiences, and an experiment can only be an opportunity to use it. As a Kantian, for Poincare, the mind comes armed with a priori intuitions, which provide the form of all experience, and just these a priori intuitions supply the non-empirical content of mathematics that is not provided by some external reality, making the mathematical knowledge as an intuitive subject matter. What is intuition in Poincare’s philosophy is not limited to mathematicians. Philosophers that followed him, emphasized the role that prior intuition plays in the process of knowledge before any primary empirical experience. Philosophers have struggled to comprehend Poincare’s self-critical way of thought, which refuses to capture the entire world in a single concept. In his words: “The soul of a true philosopher is a battlefield; this is not a peaceful monarchy where there is room for one master only”. Poincare’s epistemology is resumed as the struggle for him was not understanding or “knowing” a subject, it was explaining how he knew it. He spent the majority of his time trying to understand what he knew to be true and making productive use of it. This is how he decided what should be proven and what topic was worthy of his endeavors.

3. Poincare and us. Poincare’s legacy at MDC. “The scientist does not study nature because it is useful; he studies it because he delights in it, and

6 he delights in it because it is beautiful.” H. Poincare. The figure of Henri Poincare is, in general, far from the curriculum of subjects in higher education, despite his caliber as a scientific personality who has left an imprint on science and culture. That is why, considering ourselves indebted to his legacy, we consider it appropriate to bring his work and personality to the college, so that it serves as an inspiration to the generations of students we are training as future professionals. During the Fall 2020 Term, was created the Poincare Group for the Studies on Nonlinear Dynamical Systems. Honoring Poincare's legacy, the group includes undergraduate students from Miami Dade College-Padron Campus, Math and Natural Sciences Department. As part of the undergraduate academic training of natural science and engineering students, the group has the fundamental intention of motivating and promoting the study of deterministic systems in Nature, from physical to biological and social systems. The group works under the MDC’s projects “Searching for determinism in solar activity indices” and “Detecting chaotic behavior in natural phenomena” with some significant results to date, with the students being the protagonists of the activities and contributions carried out, which are summarized, mainly, in participation in two scientific meetings (Florida Undergraduate Research Conference, FURC 2021 and MDC Summer Research Institute Symposium 2021), one scientific publication (POLYGON, issue 2021), series of Lectures on Chaos and Nonlinear Dynamics, and series of Debates on Poincare’s work and legacy, among others. At present, there are two scientific research works in progress.

4. Conclusion… but there are more. “Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.” -H. Poincare The originality in Poincare’s vision of Nature and his effect on modern science is massive although goes practically unnoticed. Better known are his contributions to topology - the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures, but precisely because of the intricacy of his contribution. The so-called Poincare Conjecture, enunciated by him in 1904, has a particular impact when we ask ourselves what the shape of the Universe is like, it was not proven and therefore validated as a theorem until 2002 (Perelman, 2002). His indifferent personality in the face of fame and his humility were notorious and constant

7 throughout his prolific life. He did not claim glory for himself despite being the man who changed the course of science. “The subliminal self is in no way inferior to the conscious self… It knows how to choose, guess… It knows how to guess better than the conscious self since it succeeds where the latter has failed. In a word, is not the subliminal self superior to the conscious self? (Poincare, 1904). The most shocking thing about his story is that despite being the giant that he was, he does not enjoy the admiration of the general public that he deserves. People do not recognize his name as they do with Einstein or Newton, even being Poincare who opened the doors to the Theory of Relativity or the Theory of Chaos. He was a prolific, honest, humble, and humorous writer who deserves our greatest admiration and respect. May this be a modest contribution to preserve the memory and recognize the legacy of one of the most original thinkers y one of the most prolific scientists that have ever existed: the last great universalist.

REFERENCES [1] F. Masson: Henri Poincare and the French Academy. Popular Science Monthly. 75 (1909), 267273. [2] G. Perelman: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002). [3] H. Poincare: The Foundations of Science: Science and Hypothesis, the value of science and method. University Press of America (1904), 553 pp. [4] P. Langevin: L’oeuvre d’Henri Poincare: Le Physicien. Revue de Metaphysique et de Morale. 21 (1913), 675-718.

8 Figures: Figure 1: Henri Poincare in his personal office. Photo taken near his death.

9 Figure 2: Henri Poincare and Marie Curie (sitting at the center of the picture) are absorbed in their work while the others (H. Lorentz, W. Wien, M. Planck, de Broglie, E. Rutherford, A. Einstein and P. Langevin, among others) pose for a historic photo. 1911 Solvay Conference.

10 Figure 3. Relativity, lithograph print by M. Escher in 1953. This work is influenced by the concepts about the space of Poincare.

11 The Performance of Students Interactions While Producing Project- Based Learning Remote Strategies Jaime Bestard, Ph.D. Professor, Mathematics and Statistics MDC- Hialeah Campus, 1780 W 49th St, Suite 2334, Hialeah, FL 33012 Email: jbestard@mdc.edu ABSRACT The increasing proportion of dual enrollment students, younger and still taking their general education main courses in the K-12 System, makes the situation of engagement and motivation in remote instruction more compelling, coming from a very detailed rubric based in point by point evaluation rooted system, to the college education rooted in the quality of the discussions and production of arguments, such students find the early higher education challenging when it comes to excel in quality and the qualitative assessment of their learning needs and progress is object of re- considerations, with innovative procedures, that is the case of the Problem -Based Learning driven courses like STA2023 STATISTICAL METHODS, have been exercised during the last two long pandemic years, and experimented using the remote instructional techniques, with results that point to highly significant differences in performance to the team work of two students class settings.

Key Words: Pedagogic, Student Performance, High Impact Practices, Project, Based Learning.

12 Introduction The increasing proportion of dual enrollment students, younger and still taking their general education main courses in the K-12 System, makes this situation of engagement and motivation more compelling, coming from a very detailed rubric based in point by point evaluation rooted system, such students find the early higher education challenging when it comes to excel in quality and the qualitative assessment of their learning needs and progress is object of re- considerations, with innovative procedures, that is the case of the Problem -Based Learning driven courses like STA2023 STATISTICAL METHODS. The theme of engagement has sparked as an outstanding challenge in the remote instructional modalities that populated more than 40 % of the courses at Miami Dade College nowadays. In the discipline of Mathematics, which has several courses considered difficult by many, it is quite important to keep the engagement of students in a class at a very high level. Such courses that integrate quantitative reasoning, across the curriculum, like STA2023 Statistical Methods, setting the basis for consistent employability and professional success in every field, the use of the High Impact Pedagogic Practices is not falling behind, and such courses cannot take the luxury of falling behind in a population where diversity and minorities are the greater proportion, including the first time in college students.

Methods The Project -Based Learning (PBL), as a High Impact Pedagogic practice requires the use of authentic cases in class, this is a challenge for the instructional personnel that must re-invent themselves with updating such themes and topics for the exercises in the course and in the projects. When the society entered in the confinement because the pandemic in the Spring of 2020, there were some early approaches to the application of the PBL in the then “in person” typical instructional modality, seminars, exchanges with the national champions of the modality including institutions that fully apply the modality for engineering majors and literature review, the early applications took some preference from the side of the students that did have other choices, indeed more convenient for them, while moving faster to complete their requirements for graduation. The change in the landscape of the hiring screening procedures in the corporate and public sectors, using the soft skills more than the cold records to hire professionals, made it attractive, in that time, to develop such skills in our courses, the implementation started by STA2023Statistical Methods, obviously to make the students reflect on the authentic data characteristics findings, from the descriptive

13 and also from the inferential perspective and the collaboration with funding from a Title V Hialeah Campus grant permitted to acquire the updated version installed campus- wide of the statistical package MINITAB, widely used in the pharmacological industry as in other STEM employers. The course moved into a progressive instruction, integrating the MDC Learning Outcome 1 “ Communication in listening speaking and writing “ to the Learning Outcome 2” Use of Quantitative Reasoning “, to the Learning Outcome 3” Critical Thinking and Logical Reasoning”( also phrased as the discipline/program learning outcomes) and to the Learning Outcome 4” Allocating Information Literacy” that permitted to assess the performance of students gradually and progressively with written progressive essays that described from a simple data analysis, using the STEM and LEAF PLOT, the BOXPLOT, HISTOGRAM and DESCRIPTORS PANEL for one data set in class and discussing the corresponding Exploratory Data Analysis with measures of center, variability, shape of the distribution and presence of outliers and evolution in time respective paragraphs, to the more extensive essays that include the use of inferential procedures like the Anderson-Darling Normality Test and the most elementary non parametric procedures like Chi-Square Goodness of Fit tests to validate the sample the students collected in their authentic data bases. In the Spring 2020 the confinement due to the Pandemic started and with it, the remote instructional needs with the evolution from the synchronic and a-synchronic modalities to the current MDC-LIVE instructional modality, that has reached about 41 % of the current course offering college – wide. The strategy that evolved from the in person to the remote a-synchronous to synchronous and then to MDC-LIVE deliveries transit as fast as the situation worsened with the pandemic, and the development of the High Impact Pedagogic Strategies for those remote modalities populated the landscape of the best pedagogical practices. The scaffolding process that is imperative to introduce in the practice of the project-based learning is conducive to also practice a transparent informative and instructive delivery of the main goals as the corresponding theoretical and practical background the students need to put together with the teamwork and the accountability that needs to be constantly in revision for the good conduction of the integrative process in the course. All those components activate the system of feedback and formative assessment the instructional personnel need to develop to control the correct execution and guidance in balance with the creativity that the students must develop. In order to illustrate the application of the Project – Based Learning and the scaffolding process

14 that early in the course produces the integration students need to trigger further achievement of major goals, there is an example of exercises where the STA2023 students navigate by enhancing their perspective of what is the progressive improvement in the written description of data sets, during the first three weeks of classes the students follow a common exercise related to describe a common sample of the age of students in the class, where they participate in the construction of the stem and leaf plot, the determination of the five numbers summary, the mode and the construction of the box plot mostly during class time, and presenting the written essay, very elementary at the beginning where they phrase the five main characteristics of data: Central tendency, Variability , Shape of the distribution and presence of outliers using the very basic fences criterion. The second phase incorporates a sample the students take on their own individually and repeat the exercise incorporating other elements that support the descriptive statistics of the sample like the descriptors panel with the mean the variance, the coefficient of variation and the z-score. In this phase the students submit an essay that incorporates such elements together the previous ones, producing the deeper description. At this point with about two weeks of the course passed the selection of the best individual discussions of cases brings the constitution of teams that will produce the third phase comparing their individual selected sample versus the initial common sample or simply comparing two cases optional for the class and the instructor, team formation start the important inclusion of some early inferential procedures to document the validation of the sample and the use of collateral variables that may support such validation with the use of normality tests and the chi-square Goodness of fit test and the Chi-square independence test, obviously without technology that would be almost impossible, but there is the early XXI century statistics class supportive materials and software, that may evolve in a further artificial intelligence applications to produce multivariable instruction. The submission of this third low stakes assignment incorporates a first module assessment of evidence based learning, developing several soft skills that will definitely shape a better professional performance and long lasting better employability, teaching how to produce and conduct teamwork technical discussions and opening the door to the course project where the teams select the theme from a given set of problems involving global initiatives and current social and natural sciences problems. With the selection of the theme and topic the teams start over again the already practiced method in a large-scale data collection, validation of the sample and describing in full. It is important to notice the implementation of the sessions of the project while the course advances in content, and that the accountability of the members of the team is guaranteed as per the continuous progressive checkups that

15 ensure the consistency in the development of the participative and collaborative skills. The projects include some collateral variables which analysis include some social and civic awareness while commenting the validation of the sample. Observe the corresponding possibility to assess communication skills, obviously quantitative reasoning, and critical thinking as allocation of information literacy. The assessment was formative in terms of the % of content delivered in the course producing the linear regression. The case for Project Based Learning and Undergraduate Research as High Impact Pedagogic Practices is also used in MAC1105 College Algebra and MAC2233 Business Calculus, using trends from the current events like the pandemic where the students can verify in a real scenario of authentic data the main properties of the functions and the interpretation of the applications of derivatives by graphically determine the slope of tangents and secants to the functions, and their asymptotes. This hands-on work engages students in evidence – based learning practices that are very effective and motivating.

Results The data collected was related to the STA2023 sections taught during the term Spring 2019 (sections 5581, 8840, 5583); in the term Summer 2020 (section 2506) and term Fall 2020 (sections 5562, 5565, 10077) The design of the experiment was proceeded by a pilot over the summer term 2018, when the details and organization were tested in an instructional environment, some adjustments were necessary and updates from the state of the art in the institutions that have experience on the strategies The pre- experiment set the margin of error (5%), the significance level (0.05), and the estimation of the necessary sample size (n=168 students) to accomplish such statistical requirements, also this pilot permitted the optimization of the experimental design by the cohorts and control groups (sections 5581 and 5562) and two choices of Project Based Learning: Individual (section 8840) and Group (sections 5583, 2506, 5565, 10077) The MDC indicators was collected for the sections involved in the study, as presented in Chart 1:

Chart 1: Pass Rate, Success Quotient and Retention Rate per section R TERM Section PR SQ R N students 2193(Spring 2020) (Control)5581 50 58 86 21 (PBL indiv.)8840 52 68 76 29 (PBL team)5583 61 69 89 17 2195(Summer 2020) (PBL team)2506 76 87 88 17 2197(Fall 2020) (Control)5562 50 60 83 23 (PBL team)5565 71 85 84 30 (PBL team)10077 73 80 91 11 The experiment consisted in using three groups of actions: a) the cohort with PBL in teams of two students, b) the cohort of individual PBL, and c) the control unit with traditional format. The individual results of the students in the projects were recorded per section in each course and term to compare the individual effectiveness of the instructional pedagogy, as shown in Chart 2:

Grades A B C F W

5581 2/ 9.09% 4/18.18% 5/22.72% 8/36.36% 3/13.63%

Chart 2: individual results per students in sections Sections 5583 8840 2506 5562 5/3.61% 4/23.53% 4/22.22% 2/8.33% 6/20.68% 1/5.88% 5/27.27% 5/20.83% 4/13.79% 6/3.53% 4/22.22% 5/20.83% 7/24.14% 5/29.41% 2/11.11% 8/32.66% 7/24.14% 1/5.88% 2/11.11% 4/16.33%

5565 9/30.00% 8/26.62% 5/16.66% 4/13.32% 4/13.32%

10077 7/63.63% 1/9.09% 0/0.00% 2/18.18% 1/9.09%

The level of guidance and intervention was set up as: a) the one that used the PBL techniques of minimal orientation, forming teams of two students to address the topic, b) the one that conducted individual PBL techniques to address the topic, and c) the traditional method of conducting the course with the lectures and exercises, all according to competencies. It is important to declare that the PBL choices did accomplished the competencies of the course as well, and that the indicators of the analysis involved only the result in the course work and not the final grade which correlation may be object of an independent analysis.

Analysis As part of the analysis, correlation regression tests were conducted to document the potential

17 existence of association between the Pass rate and the proportion of content delivery as shown in chart 3 and to model such association. Group a)EXP 1 a)EXP 2 c)Ctrl Groups

Chart 3: Correlation -regression results Cohort/ size PR (%) r/ p-value R sq PBL Individual/ 57 87.1 0.84/ 0.023 0.54 PBL Team/ 76 88.7 0.86/ 0.001 0.49 Traditional/ 35 84 0.89/0.003 0.43

Significance *a) vs. c) ** b) vs. c) * a), b) vs. c)

Legend: PR-Pass Rate, r coefficient of correlation, R sq Coefficient of Determination, linear regression PR vs % content delivered; * p-value< 0.05 significant; ** p-value< 0.01 highly significant As observed in the chart, there is a strong and significant correlation in each treatment with respect to the % of content delivered, including the traditional delivery, when there is a team project, during the semester, as shown in the greatest value of the slope in the regression model, explaining a better and more effectiveness per unit of course progression, when the students receive the PBL pedagogical practices than in the traditional content delivery. The results of the analysis of variance for the experimental and control units, showing significant differences (p-value<0.001) in at least one of the choices as previously pointed for the cohort of PBL in team (observe side by side box plots in Figure 1) All the groups show a strong correlation with the progression of the semester, but it is remarkable that the simple form increase at a faster rate.

Boxplot of 5581, 5583, ... 5

Data

1 5581

5583

8840

2506

5562

5565

10077

FIGURE 1: Individual performances in the experimental and control units Observe the higher performance of the sections involving the performance of the sections 5565 and 10077 that were under PBL Teams, still not significant differences at 5% significance level when it comes to the instructional delivery of competencies.

Concluding Remarks a) As per the study, it is observable that the student success indexes of PASS RATE and SUCCESS QUOTIENT have a significant increase when applying the PBL Teams of two students, while the RETENTION RATE experiences some decay in those courses provided the level of accountability demanded to and by the team. b) The Performance is highly significantly higher when assessed progressively and formative across the content delivery in the conditions of remote team PBL instructional strategy compared to the traditional lecturing method. c) The Performance is significantly higher when assessed progressively and formative across the content delivery in the conditions of remote individual PBL instructional strategy compared to the traditional

19 lecturing method. d) The Performance is highly significantly higher when assessed progressively and formative across the content delivery in the conditions of both remote team and individual PBL instructional strategy compared to the traditional lecturing method.

Acknowledgements First, the author is thankful to the reviewer for his valuable and constructive suggestions which considerably improved the presentation of the paper. The author would also like to thank the Editorial Committee of Polygon for accepting this paper for publication in Polygon.

BIBLIOGRAPHIC REFERENCES [1] “Project Based Learning in the First Year”; Woobe & Stoddard, AACU, 2019 [2] “Getting Started With Team Based Learning”; Sibley, Ostafichuk, et All, Stylus, 2019 [3] “Creating Wicked Students, Designing Courses for a Complex World”; Hanstedr, Stylus, 2018

20 Bootstraps Confidence Interval Estimates, with Some Uses of StatKey: An Intuitive Introduction for Elementary Statistics Students Eric O. Hernandez, Ph.D. Department of Mathematics Miami Dade College, Kendall Campus Miami, Florida, USA, E-mail: eherna13@mdc.edu M. Shakil, Ph.D. Department of Mathematics Miami Dade College, Hialeah Campus Hialeah, Florida, USA, E-mail: mshakil@mdc.edu ABSTRACT In this paper, we present an intuitive introduction to bootstrap methods for producing better confidence interval estimate of population parameters. We present both theory and examples to show how this is achieved. Some uses of the recently developed software, StatKey, for the bootstrap confidence interval estimates and randomization tests, have also been presented. Keywords:Bootstraps; Confidence Intervals; Estimates; Mean; Normal Distribution; Percentiles. Mathematics Subject Classification (MSC2020): 62-01; 62-02; 62-04; 62-08; 62F10; 62F40.

21 Introduction Efron (1979) introduced the Bootstrap method for assessing the errors in a statistical estimation problem. It spread like brush fire in statistical sciences within a couple of decades. We attempt first to explain the idea behind the method and the purpose of it at a rather rudimentary level. The primary task of a statistician is to summarize a sample-based study and generalize the finding to the parent population in a scientific manner. A technical term for a sample summary number is (sample) statistic. Some basic sample statistics are sample mean, sample median, sample standard deviation etc. Of course, a summary statistic like the sample mean will fluctuate from sample to sample and a statistician would like to know the magnitude of these fluctuations around the corresponding population parameter in an overall sense. This is then used in assessing Margin of Errors. The entire picture of all possible values of a sample statistic presented in the form of a probability distribution is called a sampling distribution. There is a plenty of theoretical knowledge of sampling distributions, which can be found in any textbooks of mathematical statistics. A general intuitive method applicable to just about any kind of sample statistic that keeps the user away from the technical tedium has got its own special appeal. Bootstrap is such a method. To understand bootstrap, suppose it were possible to draw repeated samples (of the same size) from the population of interest, a large number of times. Then, one would get a fairly good idea about the sampling distribution of a particular statistic from the collection of its values arising from these repeated samples. But, that does not make sense as it would be too expensive and defeat the purpose of a sample study. The purpose of a sample study is to gather information cheaply in a timely fashion. The idea behind bootstrap is to use the data of a sample study at hand as a “surrogate population”, for the purpose of approximating the sampling distribution of a statistic; i.e. to resample (with replacement) from the sample data at hand and create a large number of “phantom samples” known as bootstrap samples. The sample summary is then computed on each of the bootstrap samples (usually a few thousand). A histogram of the set of these computed values is referred to as the bootstrap distribution of the statistic. In bootstrap’s most elementary application, one produces a large number of “copies” of a sample statistic, computed from these phantom bootstrap samples. Then, a small percentage, say 100(a / 2) % (usually a = 0.05), is trimmed off from the lower as well as from the upper end of these numbers. The range of remaining 100(1-a) % values is declared as the confidence limits of the corresponding unknown population summary number of interests, with level of confidence 100(1-a) %. The above method is referred to as bootstrap percentile method. For the Bootstrap method, the interested readers are also referred to Efron (1981, 1982, 1987, 1993, 1994), Efron and Tibshirani (1993), and DiCiccio and Efron (1996), among others.

22 Bootstraps Confidence Intervals There are many different methods for estimating CI from a bootstrapped distribution. These methods use the bootstrap distribution in different ways to arrive at CI. In what follows, these are described as given below: The normal interval method only uses the bootstrap distribution to get an estimate of the standard error (SE), which it then uses in the more traditional CI formula. The remaining methods actually derive the estimate entirely from the bootstrapped distribution. The percentile interval, and studentized interval all conclude with the same step of identifying the percentiles corresponding to the desired CI as the upper and lower bounds (e.g., 2.5% and 97.5% for a 95% CI). These estimates differ however in adjustments made to the bootstrap distribution before this step. The percentile interval makes no adjustments, the studentized interval converts the distribution to studentized statistics, correcting each statistic by its associated standard error. Finally, the basic interval method corrects the distribution for bias and then identifies the lower and upper bounds that capture the desired CI level using a slightly more complex formula. DiCiccio and Efron (1996) recommend that at least 2000 replications are used when conducting bootstrap resampling. The idea behind this method is that the resamples can be viewed as thousands of potential samples from the population. Together, the estimates from the resamples represent the possible range of the estimate in the population. A robust empirical CI can then be estimated from the bootstrap distribution. The selection of an appropriate bootstrapping method generally can be determined based on responses of your research questions. These questions target the underlying assumptions of the methods in order to select one that best fits the data. In general, these assumptions are tested by examining whether we have sufficient information for the method (i.e., standard error) and examining the bootstrap distribution. Remember the bootstrap distribution theoretically reflects the range of possible estimates in the population, thus we can use this information to review for potential problems that can introduce bias into our estimate.

Applications and Simulations Study Confidence intervals are important aspects of inferential methods that rely on an approximated sampling distribution. Confidence intervals use data from a sample to estimate a population parameter. The decision of using a confidence interval depends on the research question. If we want to estimate a population parameter, we use a confidence interval. Two Approaches to Inference: We have the following two main approaches to inference:

23 (I) Traditional Inference: a. Assume some distribution (e.g. normal or t) to describe the behavior of sample statistics. b. Estimate parameters for that distribution from sample statistics. c. Calculate the desired quantities from the theoretical distribution. (II) Simulation: a. Generate many samples (by computer) to show the behavior of sample statistics. b. Calculate the desired quantities from the simulation distribution. Illustration: In what follows, we illustrate the above-mentioned approaches through some examples. Traditional Inference: It is are described below by using an example of a random sample of 25 Mustangs cars. Example: What is the average price of a used Mustang car? We have selected a random sample of n = 25 Mustangs (as given in Figure 1) from the website of autotrader.com. We record the price (in $1,000’s) for each car (as given in Figure 2).

Figure 1: Sample of Mustangs (Source: autotrader.com)

MustangPrice

Dot Plot

25 Price

Figure 2: Price (in $1,000’s) for each Mustang car

25 As computed, our best estimate for the average price of used Mustangs $15,98, as given below:

𝑛𝑛 = 25

𝑥𝑥̅ = 15.98 𝑠𝑠 = 11.1

But how accurate is the above estimate?

Goal: Find an interval that is likely to contain the mean price for all Mustangs Confidence Interval Key idea: How much do we expect the mean price to vary when we take samples of 25 cars at a time? Solution: We provide it below step by step for the above-mentioned example of the random sample of 25 Mustangs cars. 1. Check conditions: n = 25 2. Which formula? CI for a mean:

3. Calculate summary stats:

4. df = 25 – 1 = 24 5. To find t*, since for

using the t-Table (Table 1), the following critical value for t* is obtained: t*=2.064.

Table 1: t-Distribution Critical Values

6. Plugging the above values, we obtain the 95 % CI as follows:

or,

15.98 ± 2.064 ∙ 11.11� √25

15.98 ± 4.59 = (11.39, 20.57)

7. Interpret in context: “We are 95% confident that the mean price of all used Mustang cars at this site is between $11,390 and $20,570.”

Sampling Distribution of Sample Means: As we stated above, our answer to the key idea “How much do we expect the mean price to vary when we take samples of 25 cars at a time” appears to be fine, but, in fact, statistically, the process is not very helpful at building understanding of a CI. Therefore, the following question arises: Question: “Can we arrive at the same answer in a way that also builds understanding”?

27 Answer: The answer to the above question is in affirmative, that is, “yes”, as explained below:

Key Concept: How much do sample statistics vary? If we take samples of 25 Mustangs at a time, what sort of distribution should we expect to see for 𝑥𝑥̅ ′𝑠𝑠?

For this, we need Sampling Distribution of Sample Means, 𝑥𝑥̅ ′𝑠𝑠, which is described as given below. Producing a Sampling Distribution: For this, possible following are the traditional approaches. 1. Know the value of the parameter and distribution of the population. 2. Take thousands of samples from the population. 3. Rely on theoretical approximations.

Remarks: (1) and (2) are not practical in real situations, whereas (3) is difficult for introductory students Bootstrap: First, we need the following key concept: Key Concept: How much do sample statistics vary? How can we figure out how much sample statistics vary when we only have ONE sample? Answer: The answer to the above question is “Bootstrap”, as given in the Flow-Chart (Figure 3)

Figure 3: A Flow-Chart for Bootstrap

28 Producing a Bootstrap Distribution: How do we get a CI from the bootstrap distribution? It is described as follows:

Method 1: Standard Error •

Find the standard error (SE) as the standard deviation of the bootstrap statistics.

•

Find an interval with

•

𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ± 2 ⋅ 𝑆𝑆𝑆𝑆

as illustrated in Figure 4.

Figure 4: Standard Error (SE) of the Bootstrap Statistics

29 Method 2: Percentile Interval •

For a 95% interval, find the endpoints that cut off 2.5% of the bootstrap means from each tail, leaving 95% in the middle.

StatKey Software for Bootstrap Confidence Interval Estimates: It is important to note that recently a statistical software, known as StatKey, has been developed by the Lock family for the bootstrap confidence interval estimates and randomization tests; (see Table 2). The StatKey is a free online technology designed to help introductory students understand and easily implement bootstrap intervals and randomization tests. The interested readers are also referred to Lock et al. (2014) and Lock et al (2021), among others. Please also refer to: www.lock5stat.com/StatKey Table 2: StatKey (Source: lock5stat.com/statkey)

30 Bootstrap Distribution for Mustang Price Means: To illustrate the Bootstrap confidence interval estimates, we consider the Bootstrap distribution for Mustang price means by using the statistical software, StatKey, adopted from Lock et al (2021). Method 1: Standard Error: Using StatKey, it is illustrated in Figure 5.

Figure 5: (Method 1: Standard Error) Bootstrap Distribution for Mustang Price Means

31 Method 2: Percentile Interval: Example 1: Using StatKey, it is illustrated in Figure 6, for 95% CI via Percentiles.

Figure 6: (95 % Percentile Interval) Bootstrap Distribution for Mustang Price Means

32 Example 2: Using StatKey, it is illustrated in Figure 7, for 99% CI via Percentiles.

Figure 7: (99 % Percentile Interval) Bootstrap Distribution for Mustang Price Means

Analysis of Results and Discussions We have the following analysis of our Bootstrap Confidence Intervals results in Section 3.

Bootstrap Confidence Intervals Analysis: Method 1 (Standard Error: Statistic ± 2 SE): i.

Great preparation for moving to traditional methods.

Method 2 (Percentiles): i.

Great at building understanding of confidence level.

ii.

Same process works for different parameters

Transition to Traditional Formulas: It is illustrated in Figure 8.

Figure 8: Transition to Traditional Formulas Golden Rule of Bootstraps: It is illustrated in Figure 9.

Figure 9: Golden Rule of Bootstraps

34 Advantages of Choosing Bootstraps Confidence Interval Estimates: In what follows, we present the advantages and disadvantages of the above methods, with reference to the various examples as described above, that is, choosing a method, traditional or Bootstraps. Our “Final Thoughts” about the bootstraps confidence interval estimates are given below. Final Thoughts: a. Pros: •

Illustrates directly how statistics vary from sample to sample

•

Follows naturally from sampling/statistics

•

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ± 2 ∗ 𝑆𝑆𝑆𝑆

•

generalizes easily to traditional formulas

•

Same process can be applied to lots of statistics

•

Can connect to tests, but doesn’t require tests

b. Cons: •

Requires software.

•

Tedious to demonstrate “by hand”.

•

Doesn’t always “work”.

Concluding Remarks For assessing the errors in a statistical estimation problem, Efron (1979) introduced 43 years ago, a nonparametric method, called “bootstrap”, without the use of the powerful statistical packages and technology that has been evolved. However, as pointed out by Efron (1979), bootstrap method is remarkably easy to implement on computer. It provides several advantages over the traditional parametric approach, and is easy to describe and applies to arbitrarily complicated situations. Moreover, in Efron’s bootstrap method, distribution assumptions, such as normality, are never made. As further pointed out by Efron (1979), the jackknife is a linear approximation method for the bootstrap. In this paper, an intuitive introduction to bootstrap methods for producing better confidence interval estimate of population

35 parameters is presented. Some uses of the recently developed software, StatKey, developed by the Lock family, for the bootstrap confidence interval estimates and randomization tests, have also been presented. We hope that this paper will be useful to researchers in various fields of applied research. Moreover, it is hoped that this paper will help us in developing a lesson plan to show how the teaching of basic ideas of bootstrap methods for producing better confidence interval estimate of population parameters can help the students to gain the knowledge and insights from some real-world data. Also, such studies are very important in view of the facts that there is a great emphasis on health literacy and building healthy communities. It is hoped that by implementing the techniques discussed in this paper in preparing lesson plans will help us to develop in our students the quantitative analytic skills to evaluate and process numerical data, which is one of the Gen Ed Outcomes of Miami Dade College.

Acknowledgements First, the authors are thankful to the reviewer for his valuable and constructive suggestions which considerably improved the presentation of the paper. The authors would also like to thank the Editorial Committee of Polygon for accepting this paper for publication in Polygon. Further, the authors would like to acknowledge their sincere indebtedness and thanks to the works of various authors and resources on the subject which they consulted during the preparation of this research project. Finally, last but not the least, the authors would also like to express thanks to Miami Dade College for providing them an opportunity to serve as a mathematics faculty in the college at its Kendall and Hialeah Campuses, without which it was impossible to conduct their research and complete this paper.

REFERENCES [1] Efron, B. (1979). Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics, 7(1), 1–26. http://www.jstor.org/stable/2958830 [2] Efron, B. (1981). Nonparametric Estimates of Standard Error: The Jackknife, the Bootstrap and Other Methods. Biometrika, 68(3), 589–599. https://doi.org/10.2307/2335441 [3] Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF Regional Conference Series in Applied Mathematics, Monograph 38, SIAM, Philadelphia, USA. http://dx.doi.org/10.1137/1.9781611970319

36 [4] Efron, B. (1987). Better Bootstrap Confidence Intervals. Journal of the American Statistical Association, 82(397), 171–185. https://doi.org/10.2307/2289144 [5] Efron, B. (1993). Bayes and Likelihood Calculations from Confidence Intervals. Biometrika, 80(1), 3–26. https://doi.org/10.2307/2336754 [6] Efron, B. (1994). Missing Data, Imputation, and the Bootstrap. Journal of the American Statistical Association, 89, 463-475. [7] Efron, B., and Tibshirani, R. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York, USA. [8] DiCiccio, T. J., and Efron, B. (1996). Bootstrap Confidence Intervals. Statistical Science, 11(3), 189– 212. http://www.jstor.org/stable/2246110 [9] Lock Morgan, K., Lock, R. H., Lock, P. F., Lock, E. F., and Lock, D. F. (2014). STATKEY: ONLINE TOOLS FOR BOOTSTRAP INTERVALS AND RANDOMIZATION TESTS. Invited Paper, the Ninth International Conference on Teaching Statistics (ICOTS 9), Flagstaff, AZ, USA. [10]

Lock, R. H., Lock, P. F., Lock Morgan, K., Lock, E. F., Lock, and D. F. (2021). Statistics:

Unlocking the power of data (3rd edition). John Wiley & Sons, Inc., Hoboken, NJ, USA. [11]

autotrader.com

[12]

lock5stat.com/statkey

37 1943 and 2000 - Years of Special Achievements in the History of African-American Women in Mathematical Sciences M. Shakil, Ph.D. Dept of Math, Liberal Arts & Sciences Miami-Dade College, Hialeah Campus, Fl., USA E-mail: mshakil@mdc.edu ABSTRACT In this paper, the years 1943 and 2000, which are regarded as the years of special achievements in the history of African-American women in mathematical sciences, are presented. Keywords: African-Americans, women, mathematical sciences. AMS Subject Classifications: 01A05; 01A07; 01A70; 01A85; 01A90; 00A15

38 INTRODUCTION African-American mathematicians have contributed in both large and small ways that is overlooked when chronicling the history of science and mathematics. By describing the scientific history of African-American men and women within mathematical sciences, we can see how the efforts of individuals have advanced human understanding in the world around us. The study of African cultural influences in mathematical sciences is challenged by the pragmatic difficulties of racist historical omissions on several theoretical fronts. Afro-Americans have had an uphill journey in all spheres of education, for ages. This has its roots in the social backdrop in which they have had to live. However there have been many scientists, astronomers, mathematicians and professionals in every other scientific fields as well, who were able to triumph over the repressions and racial discriminations to reach the level of excellence that was required of them to compete with their white counterparts. The accomplishments of these stalwarts of science and mathematics give lie to all those who undermine and underrate the abilities of the Afro-American scholars. The contributions of the African-American scholars and their abilities in the fields of science and mathematics are enormous. The African-American women are also not far behind, and their accomplishments in the field of mathematical sciences are no less than men’s, which are remarkable and noteworthy. History bears testimony to their achievements. For details on these, see, for example, Kenshaft (1981,1987), Donaldson (1989), Dean (1996), Dean et al. (1999), Williams (2008a, b) and Shakil (2010), among others. As pointed above, the purpose of this paper is to highlight the achievements of African-American women in the field of mathematical sciences, with special reference to the years 1943 and 2000. The organization of this paper is as follows. Section 2 contains the achievements of African-Americans in the field of mathematical sciences. Section 3 contains some highlights of early women in mathematics. In Section 4, we present the years 1943 and 2000, which are regarded as the years of special achievements in the history of African-American women in mathematical sciences. The concluding remarks are presented in section 5.

ACHIEVEMENTS OF AFRICAN-AMERICANS IN MATHEMATICAL SCIENCES, 1925 - 2004 In this section, we present the achievements of African-Americans in the field of mathematical sciences, which can be divided into four different periods beginning from 18th century to the present. These periods can be further classified and indexed by the year as provided in the following Tree

39 Diagram (Figure 1). For details, see Williams (2008a, b) and Shakil (2010), among others.

Figure 1: A TREE DIAGRAM

STATISTICS OF AFRICAN-AMERICAN Ph.D.’s IN MATHEMATICS (1925 – 2004) The statistics on the numbers of African-Americans receiving Ph.D.’s in the field of mathematical sciences during the period 1925-2004 have been presented in the following graph (Figure 2).

# African-Americans Ph.D.'s in the Mathematical Sciences 1925 - 2004 Total: 392

Ph.D.'s in Mathematics (%)

60.00% 50.00% 40.00% # African-Americans Ph.D.'s in the Mathematical Sciences

30.00% 20.00% 10.00% 0.00% 1925 - 1944 1945 - 1964 1965 - 1984 1985 - 2004 YEAR

Figure 2: STATISTICS OF AFRICAN-AMERICAN Ph.D.’s IN MATHEMATICS (1925 – 2004 •

Per our list as prepared above, a total of 390 African-Americans had received a Ph.D. in mathematics during the period 1925 – 2004. For the sake of our statistical analysis, we have divided this period into four different sub-periods: 1925 – 1944, 1945 – 1964, 1965 – 1984, and 1985 – 2004.

•

It is interesting to note that the maximum number of African-Americans receiving Ph.D.’s in mathematics was during the period 1985 – 2004. According to our analysis, the maximum number of African-Americans received their Ph.D.’s in mathematics in the years 1988 and 2001, 16 in each year. The maximum number of African-American Ph.D.’s in mathematics has been earned by the students of the University of Michigan, the total being 16 (during the period 1938 – 1984).

41 HIGHLIGHTS ON THE ACHIEVEMENTS OF AFRICAN-AMERICANS IN MATHEMATICAL SCIENCES In the following paragraph, achievements of African-Americans in the field of mathematical sciences are highlighted. •

For the interest of the reads, the statistics on the numbers of U.S. Citizens receiving Ph.D.’s in the field of mathematical sciences during the period 1994-2003 have been presented in the following Table 1. Table 1: Numbers of U.S. Citizens receiving Ph.D.’s in Mathematics (1994 – 2003)

(Source: AMS-MAA Annual Survey Reports 1998 – 2004 for PhD's in the mathematical science at US Universities) •

The following Table 2 presents the “Ratio of African-American Men and Women Ph.D.’s in Mathematical Sciences.

42 Table 2: RATIO OF AFRICAN-AMERICAN MEN & WOMEN Ph.D.’s IN MATHEMATICAL SCIENCES

(Source: AMS-MAA Annual Survey Reports 1998 – 2004 for PhD's in the mathematical science at US Universities) •

It is interesting to note that, during the period 1925 - 1947, 12 African-Americans earned a Ph.D. in Mathematics.

•

1925: Elbert Frank Cox was the first African-American to earn a Ph.D. in Mathematics in 1925 from Cornell University. There were 28 Ph.D.'s awarded in the United States that year.

Figure 3: Dr. Elbert Frank Cox •

Furthermore, half of all African-Americans who had earned their Ph.D.'s in Mathematics, by the time of the year 1945, were students of the University of Michigan.

•

During the period 1943 - 1969, thirteen African-American women earned a Ph.D. in Mathematics.

•

In 1929, Dr. Dudley Woodard was the first African-American to publish a research paper in mathematical sciences in an accredited mathematics journal, entitled, “On two dimensional analysis situs with special reference to the Jordan Curve Theorem,”Fundamenta Mathematicae, 13 (1929), 121-145. For details, see Williams (2008a, b) and Shakil (2013), among others.

Dr. Dudley Weldon Woodard (1881-1965) B.S., Wilberforce University, 1903; B.S. and M.S. University of Chicago, 1906 and 1907; Ph.D., University of Pennsylvania, 1928. Ph.D. Thesis: “On Two-Dimensional Analysis Situs with Special Reference to the Jordan Curve Theorem”; Advisor Professor John R. Kline, a renowned topologist. This portrait taken from the 1927 issue of the Bison, the Howard University yearbook, when Dr. Woodard was Dean of the College of Arts and Sciences. Photograph courtesy of Moorland-Spingarn Research Center, Howard University Archives, Washington, D.C. (Source: Williams, Scott W. (2008b) “Mathematicians of the African Diaspora,” www.math.buffalo.edu/mad/index.html) Figure 4: Dr. Dudley Weldon Woodard

•

The first African-American publication in a top research journal was Dr. William W. S. Claytor's Topological Immersian of Peanian Continua in a Spherical Surface, Annals of Mathematics, 35 (1934), 809-835.

•

Dr. Gloria Ford Gilmer is considered to be the first African-American woman to publish the first two (non-Ph.D.-thesis) mathematics research papers, jointly with another African-American, Dr. Luna I. Mishoe, in the year 1956, entitled: o

“On the limit of the coefficients of the eigenfunction series associated with a certain non-selfadjoint differential system”, Proc. Amer. Math. Soc. 7 (1956), 260-266.

“On the uniform convergence of a certain eigenfunction series”, Pacific J. Math. 6 (1956), 271-278.

•

In 1964, Dr. David Blackwell became the first African-American mathematician to Chair a department, Department of Statistics, at one of the major universities, University of California at Berkeley.

•

In 1965, Dr. David Blackwell became the first African-American named to The National Academy of Sciences.

•

In 1974, Dr. J. Ernest Wilkins, Jr., became the President of the American Nuclear Society.

•

In 1979, Dr. David Blackwell won the von Neumann Theory Prize of the Operations Research Society of America.

•

It is also interesting to know that one of the most important landmarks and rarest achievements in the field of mathematical sciences was when three African-American Women, Drs. Tasha Innis, Kimberly Weems, and Sherry Scott, received the Ph.D. in mathematics, in the same year 2000, from the same university, University of Maryland, College Park, Maryland.

SOME HIGHLIGHTS ON WOMEN IN MATHEMATICS It appears from historical records that before the twentieth century not many women had accomplished much in mathematics. However, the following are the some of the early women mathematicians who have despite many barriers contributed enormously in the field of mathematical sciences.

46 TEN EARLY WOMEN MATHEMATICIANS In the following paragraph, achievements of some early women mathematicians are highlighted. For details, see for example, http://womenshistory.about.com/b/2005/05/03/women-mathematicians.htm, among others. Hypatia of Alexandria: (born c. 350–370; died 415 AD) - Greek - philosopher, astronomer, mathematician. HYPATIA Born c. 350–370 (Alexandria) Died 415 (Alexandria)

This fictional portrait of Hypatia by Jules Maurice Gaspard, originally the illustration for Elbert Hubbard's 1908 fictional biography, has now become, by far, the most iconic and widely reproduced image of her. (Source: https://en.wikipedia.org/wiki/Hypatia) Figure 5: Hypatia of Alexandria •

Elena Cornaro Piscopia: (1646-1684) - Italian (Venice) - mathematician, theologian.

•

Maria Agnesi: (1718-1799) - Italian (Milan) – mathematician

•

Sophie Germain: (1776-1830) - French – mathematician.

•

Mary Fairfax Somerville: (1780-1872) - Scottish and British - mathematician - known as the

47 "Queen of Nineteenth Century Science." •

Ada Lovelace (Augusta Byron, Countess of Lovelace): (1815-1852) - British - mathematician The daughter of Byron, the poet.

•

Charlotte Angas Scott: (1848-1931) - English, American - mathematician, educator.

•

Sofia Kovalevskaya: (1850-1891) - Russian – mathematician.

•

Alicia Stott: (1860-1940) - English – mathematician.

•

Amalie Emmy Noether: (1882-1935) - German, Jewish, American - mathematician - Called by Albert Einstein "the most significant creative mathematical genius thus far produced since the higher education of women began."

•

It will be interesting to note that the first American Woman to earn a Ph.D. in Mathematics was Dr. Winifred Edgerton Merrill from Columbia University in 1886.

Dr. Winifred Edgerton Merrill (1882 – 1951) (Source: https://en.wikipedia.org/wiki/Winifred_Edgerton_Merrill) Figure 6: Dr. Winifred Edgerton Merrill

•

In the early half of the twentieth century many African-American women obtained a Master’s Degree in Mathematics.

•

It was not until 1943, 18 years after the first African-American Dr. Elbert Frank Cox earned a Ph.D. in Mathematics from Cornell University in 1925 that an African-American woman (Dr. Euphemia Lofton Haynes) reached that level.

1943 AND 2000 - YEARS OF SPECIAL ACHIEVEMENTS IN THE HISTORY OF AFRICAN-AMERICAN WOMEN IN MATHEMATICAL SCIENCES The achievements of African-American women in mathematics are no less than other mathematicians. For a chronology of African-American women who have excelled and contributed to the knowledge and advancement of mathematical sciences during the period 1943 – 2001, the interested readers are referred to Williams (2008a, b). In what follows, we present the years 1943 and 2000, which are regarded as the years of special achievements in the history of African-American women in mathematical sciences.

1943 – FIRST LANDMARK OF SPECIAL ACHIEVEMENT IN THE HISTORY OF AFRICAN-AMERICAN WOMEN IN MATHEMATICAL SCIENCES A notable development of African-American women in mathematical sciences began with Euphemia Lofton Haynes, who earned a Ph.D. in Mathematics from Catholic University of America in 1943, with a dissertation, supervised by Dr. Aubrey Edward Landry, entitled, “The Determination of Sets of Independent Conditions Characterizing Certain Special Cases of Symmetric Correspondences”. Dr. Haynes was born on September 11, 1890, in Washington, D.C., United States. She was the first child and only daughter of William S. Lofton, a dentist and financier, and Lavinia Day Lofton, a kindergarten teacher. Dr. Euphemia Lofton Haynes’s other alma mater were: University of the District of Columbia, Smith College, and University of Chicago. Dr. Euphemia Lofton Haynes was a professor of mathematics at the University of the District of Columbia. She also chaired the Division of Mathematics and Business Education at the University of the District of Columbia, which she had created and dedicated it to the training of African American teachers. Pope John XXIII awarded her the Papal decoration of honor, Pro Ecclesia et Pontifice, in 1959. She was named a Fellow of the American Association for the Advancement of Science in 1998. Dr. Haynes passed away on July 25, 1980 in her

49 hometown, Washington, D.C.

(Source: Williams, Scott W. (2008b) “Mathematicians of the African Diaspora,” www.math.buffalo.edu/mad/index.html) Figure 7: Dr. Euphemia Lofton Haynes

2000 - ANOTHER LANDMARK OF SPECIAL ACHIEVEMENT IN THE HISTORY OF AFRICAN-AMERICAN WOMEN IN MATHEMATICAL SCIENCES One of the most important landmarks and special achievements in the history of mathematical sciences was when three African-American women, Tasha Innis, Kimberly Weems, and Sherry Scott, received their doctorate degrees in mathematics in the same year 2000 from the same university, University of Maryland, College Park, Maryland.

Figure 8: Dr. Tasha Innis, Dr. Kimberly Weems, Dr. Sherry Scott at the time of their graduation in 2000, University of Maryland, College Park, Maryland

Figure 9: Dr. Tasha R. Inniss

Figure 10: Dr. Sherry Scott Joseph

Figure 11: Dr. Kimberly Weems

52 CONCLUDING REMARKS The purpose of this paper was to highlight the achievements of African-American women in the field of mathematical sciences, with special reference to the years 1943 and 2000. It is evident that these women mathematicians remain as source of inspiration to us to excel in mathematics and other fields of knowledge, and achieve our goals. The achievements of these women mathematicians, despite the difficulties they had to overcome, stand as a beacon for us. It is hoped that by implementing the discussions on the achievements of African-American women in the field of mathematical sciences of this paper in preparing lesson plans will help us to develop in our students the following Gen Ed Outcomes of Miami Dade College: a) Demonstrate knowledge of diverse cultures, including global and historical perspectives. b) Demonstrate an appreciation for aesthetics and creative activities.

ACKNOWLEDGMENTS First, the author is thankful to the reviewer for his valuable and constructive suggestions which considerably improved the presentation of the paper. The author would also like to thank the Editorial Committee of Polygon for accepting this paper for publication in Polygon. Further, the author would like to acknowledge his sincere indebtedness and thanks to the works of various authors and resources on the subject which he has consulted during the preparation of this research project. Special mention must be made of Dr. Scott W. Williams, Professor of Mathematics, The State University of New York at Buffalo, whose works I have liberally consulted, particularly his website “Mathematicians of the African Diaspora,” www.math.buffalo.edu/mad/index.html. Finally, last but not the least, the author would also like to express his thanks to Miami Dade College for providing him an opportunity to serve as a mathematics faculty in the college at its Hialeah Campus, without which it was impossible to conduct my research and complete this paper.

53 REFERENCES [1] Dean, Nathaniel, editor (1996), “African Americans in Mathematics,” DIMACS 34, American Mathematical Society.

[2] Dean, N., McZeal, C., Williams, P., Editors (1999), “African Americans in Mathematics II,” Contemporary Math., 252, American Mathematical Society [3] Donaldson, James A. (1989), “Black Americans in Mathematics,” in A Century of Mathematics in America, Part III, HISTORY OF MATHEMATICS, Volume 3, American Mathematical Society, 449-469. [4] Kenshaft, Patricia C. (1981), “Black Women in Mathematics in the United States,” American Mathematical Monthly, 592-604. [5] Kenshaft, Patricia C. (1987), “Black Men and Women in Mathematical Research,” Journal of Black Studies, 19:2, 170-90. [6] SHAKIL, M. (2010). AFRICAN-AMERICANS IN MATHEMATICAL SCIENCES - A CHRONOLOGICAL INTRODUCTION. Polygon Spring 2010 Vol. 4, 27-42. 4. 27-42. [7] SHAKIL, M.. (2013). Dr. Dudley Weldon Woodard, the First African-American Mathematician to Publish a Research Paper in an International Accredited Mathematics Journal – A Historical Introduction. Polygon 2013. 54-77. [8] Williams, Scott W. (1999), Black Research Mathematicians, African Americans in Mathematics II, Contemporary Math. 252, AMS, 165-168. [9] Williams, Scott W. (2008a) “A Modern History of Blacks in Mathematics,” www.math.buffalo.edu/mad/madhist.html [10]

Williams, Scott W. (2008b) “Mathematicians of the African Diaspora,”

www.math.buffalo.edu/mad/index.html

54 Landau “big-O” and “small-o” symbols A Historical Introduction, with Some Applicability M. Shakil, Ph.D. Dept of Math, Liberal Arts & Sciences Miami-Dade College, Hialeah Campus, Fl., USA E-mail: mshakil@mdc.edu ABSTRACT In this paper, a historical introduction and some applicability of Landau “big-O” and “small-o” symbols have been presented.

Keywords: Landau, Big-O, small-o. AMS Subject Classifications: 01A05; 01A55; 01A60; 01A85; 00A05; 00A06

55 Introduction “Big-O” and “small-o” notations are known as Landau symbols. These have been used in mathematics for more than a century. They play important roles in mathematics, statistics and computer science, specially to describe the limiting behavior of a function when the argument tends towards a particular value or infinity. In mathematics, these are used in calculus, mathematical analysis, numerical analysis, statistics, etc., particularly, in the analysis of the growth of functions, and also to describe the asymptotic behavior of functions. In computer science, these are used in the analysis of algorithms and to describe the behavior of complexity functions for large n. For details on of Landau “big-O” and “small-o” symbols and their applicability, the interested readers are referred to Olmsted (1961), Hardy and Wright (1979), Hammerlin and Hoffmann (1991), Rosen (1994), and Bishop et al. (2007), among others. The organization of this paper is as follows. Section 2 contains some historical background. Section 3 contains definitions and some basic properties of Landau “O” and “o” symbols. In Section 4, we present some applicability of Landau symbols. The concluding remarks are presented in section 5.

Some Historical Background The German mathematician Paul Gustav Heinrich Bachmann (1837-1920) first introduced “bigO” notation in 1894 in his book “Analytische Zahlentheorie” on number theory. It is a member of a family of notations invented by Bachmann (1894), and later by Landau (1909), and others. The “big-O” notation is also known as the Bachmann–Landau notation or the asymptotic notation. The letter O was chosen by Bachmann to stand for the German word, “Ordnung”, meaning the “order of approximation”. Paul Bachmann was one of the most important mathematicians of 20th century. He studied mathematics, at the University of Berlin and later at Gottingen, where he attended lectures presented by Dirichlet, the famous number theorist. He received his doctorate under the German number theorist Kummer in 1862 on group theory. Bachmann made essential contributions in the field of number theory. He had interest in music also, and played piano. The “big-O” and “small-o” notations are sometimes called Landau symbols after the German mathematician Edmund Landau (1877-1938), who was also one of the greatest mathematicians of 20th century and who used this notation throughout his work. Landau earned his doctorate in 1899 under the direction of the famous mathematician Frobenius. Landau’s main contributions to mathematics were in the field of analytic number theory and mathematical analysis, including complex analysis.

56 1. Definitions and Some Properties of Landau “O” and “o” Symbols In this section, we present the definitions and some basic properties of Landau “O” and “o” symbols. (i)

Definitions: Consider two functions f, g: D → R (field of real numbers), D ⊂ R, where g(x) ≠ 0 for x ∈ D.



f is said to be of the order “big-O” with respect to g as x → x0, if ∃ some 𝑓𝑓(𝑥𝑥)

constant C > 0 and a 𝛿𝛿 > 0, such that � 𝑔𝑔(𝑥𝑥)� ≤ 𝐶𝐶 ∀ 𝑥𝑥 ∈ 𝐷𝐷 with 𝑥𝑥 ≠ 𝑥𝑥0 and 

|𝑥𝑥 − 𝑥𝑥0 | < 𝛿𝛿 . It is represented as f (x) = O(g(x)) as x → x0.

f is said to be of the order “small-o” with respect to g as x → x0, if ∀ constant

𝑓𝑓(𝑥𝑥) C > 0,  ∃ 𝛿𝛿 > 0 such that � 𝑔𝑔(𝑥𝑥)� ≤ 𝐶𝐶 ∀ 𝑥𝑥 ∈ 𝐷𝐷 with 𝑥𝑥 ≠ 𝑥𝑥0 and |𝑥𝑥 − 𝑥𝑥0 | <

(ii)

𝛿𝛿 . It is represented as f (x) = o(g(x)) as x → x0.

Some Basic Properties: As x → x0, we have the following basic properties of Landau “O” and “o” Symbols: 

f (x) = O(f(x)).



f (x) = o(g(x)) ⇒ f (x) = O(g(x)).

   • 

f (x) = K ∙ O(g(x)) for some K ∈ R ⇒ f (x) = O(g(x)).

f (x) = O(g1(x)) and g1 (x) = O(g2(x)) ⇒ f (x) = O(g2(x)). f1 (x) = O(g1(x)) and f2 (x) = O(g2(x))

f1 (x) ∙ f2 (x) = O(g1(x) ∙ g2(x)).

f (x) = O(g1(x) ∙ g2(x)) ⇒ f (x) = g1(x) ∙ O(g2(x)).

Remark. The analogs of properties (iii) – (vi) also hold for the symbol “o.’

2. Some Applicability of Landau “O” and “o” Symbols In section, some applicability of Landau “O” and “o” symbols are presented.

57 3. Some Examples from Mathematics In what follows, some examples from mathematics provided here. (i)

Let f(x) = anxn + an-1xn-1 + … + a1x + a0 be a polynomial of degree n, where n is a positive integer and a0, a1, …, an-1, an are real numbers. Then f(x) is O(xn).

(ii) (iii) (iv)

The sum of the first n positive integers ∑𝑛𝑛𝑘𝑘=1 𝑘𝑘 is O(n2). log n! is O(n log n).

Let f: [0, 1] → R be a function with f(0) = 0. If f is continuous or continuously differentiable on the interval [0, 1], then f(x) = o(1) and f(x) = O(x) as x → 0,

respectively. (v)

Let (𝑎𝑎𝜇𝜇 ) be a sequence of real numbers, and suppose a constant K exists such that

�𝑎𝑎𝜇𝜇 + 1 −

𝑎𝑎𝜇𝜇 �

≤ K ∀ 𝜇𝜇 ∈ 𝑁𝑁. Then 𝑎𝑎𝜇𝜇 = 𝑂𝑂(𝜇𝜇) as 𝜇𝜇 → ∞.

4. Some Examples from Computer Science

In what follows, some examples from computer science are provided here. For this, we need some basic definitions, which are given below: (i)

DEFINITION OF ALGORITHM: An algorithm is a definite procedure for solving a problem using a finite number of steps.

(ii)

DEFINITION OF COMPLEXITY: Let A be an algorithm for solving a problem P with n ∈ N input data, where N is the set of non-negative integers. The mapping TA : N → N from the number of input data to the number of basic operations carried out by the algorithm is called the complexity of A.

(iii)

TWO FORMS OF COMPLEXITY: There are two forms of complexity as given below: i. Time Complexity: An analysis of the time required to solve a problem of a particular size involves the time complexity of the algorithm. ii. Space or Memory Complexity: An analysis of the computer memory required to solve a problem involves the space complexity of the algorithm.

58 (iv)

Examples of the Time Complexity of Some Algorithms: These are given below: i. The algorithm for finding the maximum of a set of n elements has time complexity O(n), measured in terms of the number of comparisons used. ii. The time complexity of the linear search algorithm requires at most O(n) comparisons. iii. The time complexity of the binary search algorithm requires at most O (log n) comparisons. iv. The average-case performance of the linear search algorithm, assuming that the element x is in the list, requires at most O(n) comparisons.

5. Some Examples from Probability In what follows, some examples from probability are provided here. For details, the interested readers are referred to Bishop et al. (2007), among others. (i)

Let (𝑋𝑋𝑛𝑛 ) be a set of random variables and (𝑎𝑎𝑛𝑛 ) be a corresponding set of constants, both

indexed by n, which need not be discrete. Then, the set of values (𝑋𝑋𝑛𝑛 /𝑎𝑎𝑛𝑛 ) converges to

zero in probability as n approaches an appropriate limit, can be expressed by using small

(ii)

o notation as follows: 𝑋𝑋𝑛𝑛 = 𝑜𝑜𝑝𝑝 (𝑎𝑎𝑛𝑛 ).

If for any ε > 0, there exists a finite M > 0 and a finite N > 0 such that P(|𝑋𝑋𝑋𝑋/𝑎𝑎𝑎𝑎|>𝑀𝑀) <

ε, ∀ 𝑛𝑛>𝑁𝑁 , then we say that the set of values (𝑋𝑋𝑋𝑋/𝑎𝑎𝑎𝑎) is stochastically bounded. This is expressed by using big O notation as follows: 𝑋𝑋𝑋𝑋=𝑂𝑂𝑂𝑂(𝑎𝑎𝑎𝑎) as 𝑛𝑛→∞.

Concluding Remarks Thus, the Landau Symbols “O” and “o” play very important and significant roles in mathematics and computer science. The Landau Symbols are very useful in the study of the analysis of the growth of functions, the asymptotic analysis, the analysis of algorithms and the behavior of complexity functions for large n. It is hoped that by implementing the discussions on the Landau Symbols “O” and “o” of this paper in preparing lesson plans will help us to develop in our students the following Gen Ed Outcomes of

59 Miami Dade College: a. Solve problems using critical and creative thinking and scientific reasoning. b. Demonstrate knowledge of diverse cultures, including global and historical perspectives. c. Demonstrate an appreciation for aesthetics and creative activities.

ACKNOWLEDGMENTS First, the author is thankful to the reviewer for his valuable and constructive suggestions which considerably improved the presentation of the paper. The author would also like to thank the Editorial Committee of Polygon for accepting this paper for publication in Polygon. Further, the author would like to acknowledge his sincere indebtedness and thanks to the works of various authors and resources on the subject which he has consulted during the preparation of this research project. Finally, last but not the least, the author would also like to express his thanks to Miami Dade College for providing him an opportunity to serve as a mathematics faculty in the college at its Hialeah Campus, without which it was impossible to conduct my research and complete this paper.

REFERENCES [1] Bachmann, Paul (1894). Analytische Zahlentheorie [Analytic Number Theory] (in German). Vol. 2. Leipzig: Teubner. [2] Landau, Edmund (1909). Handbuch der Lehre von der Verteilung der Primzahlen [Handbook on the theory of the distribution of the primes] (in German). Leipzig: B. G. Teubner. p. 883. [3] Olmsted, J. M. H. (1961). Advanced Calculus, Prentice-Hall, Inc., New Jersey, USA. [4] Hardy, G. H., and Wright, E. M. (1979). An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, England, UK. [5] Hammerlin, G., and Hoffmann, K-H. (1991): Numerical Mathematics, Springer-Verlag, Inc., New York, USA [6] Rosen, K. H. (1994). Discrete Mathematics & its applications, McGraw-Hill, Inc., New York

60 [7] Bishop, Y. M., Fienberg S. E., and Holland, P. W. (2007). Discrete multivariate analysis, Springer, New York, USA

61 A Theoretical Study of Structures, Properties, Reaction Pathways and Biochemical Activities of 3-(2-Fluorophenyl)-1H-Indazole Isomers Subhojit Majumdar, Ph.D. Department of Chemistry, Liberal Arts and Sciences Miami Dade College, Hialeah Campus, FL 33012, USA Email: smajumda@mdc.edu ABSTRACT Computational chemistry and theoretical chemical analyses are important tools for chemistry research today. With the constant improvement of technologies and methodologies, computational results can yield very similar data obtained from physical experiments. Being an alternative to physical experiments have its advantages as they can be cost effective and can be conducted at a chemical and hazards free safe environment. Since there no biproducts formed, this will help us to achieve the results that we want while protecting the environment, health, and well-beings of all. Many chemically relevant information on structures, properties, spectra, reaction pathways, bioactivities etc. can be achieved for any desired molecule with the use of proper theoretical methods. Keywords: Heterocyclic Chemistry, Density Functional Theory, Frontier Molecular Orbitals, Potential Energy Surface, Chemical Descriptors, Docking, Intermolecular Interactions

62 Introduction Heterocyclic compounds containing nitrogen are important building blocks for many biologically active molecules including natural products and commercially available drugs. A variety of nitrogen sources, including amines, amides, hydrazones, pyrimidines, and isocyanides have been disclosed for the synthesis of diverse bioactive and pharmacologically interesting N-containing heterocycles 1. World's largest selling drugs are nitrogen containing heterocycles 2. It is because of their ubiquitous nature which is key scaffolds of many biological molecules and pharmaceutical products. Thus, many chemists from all over the world were interested and started synthesizing different methods for the synthesis of these heterocycles. One of these which have biological, agricultural and an industrial application is Indazole 3. Indazole and its derivatives possess several activities such as anti-inflammatory, anti-tumor, anti-HIV, and anti-platelet. As medicinally important scaffolds, they have attracted considerable attention from chemists 4. Indazoles are one of the most important classes of nitrogen-containing heterocyclic compounds bearing a [4.3.0] bicyclic ring structure made up of a pyrazole ring and a benzene ring fused together. There have been many studies done on several derivatives of indazole 5. Given the usefulness and potential, we chose to investigate a derivative of indazole. F

F N N

N H N

2H -indazole

1H -indazole

N N H

3-(2-fluorophenyl)-1H -indazole (1H-FPI)

H N

(S)-3-(2-fluorophenyl)7aH -indazole (A-cisoid)

Figure 1. Indazole tautomers and 2-flourine phenyl derivatives. Indazole exists in two tautomeric forms. 1H indazole is thermodynamically more stable than 2H indazole and due to this factor, 1H indazole exists at a higher percentage compared to 2H sibling (Figure 1). In this study, we used 2-fluorophenyl derivative of 1H-indazole and explored the possibilities of synthesizing other isomers from it.

Computational Methods For the theoretical analysis, density functional theory (DFT) was used for computational calculation. Density Functional Theory (DFT) attempts to solve for the energy and geometry of a system by finding its exact electron density (ρ). If one knows the exact electron density of a system, one can

63 determine the energy and other properties of the system precisely. ρ is also much simpler to calculate than the total wavefunction used in the orbital approximation used in Hartree-Fock methods. Computational calculations were performed with B3LYP/3-21+G basic sets methods. B3LYP represents Becke’s threeparameter hybrid functional method 6 with Lee–Yang–Parr’s correlation functional (LYP) 7.

Results and Discussions 1.1. Analysis of 3-(2-Fluorophenyl)-1H-Indazole (1H-FPI) Structure of 1H-FPI what's optimized using density functional theory 3-21G+ and the minimum energy structure was located the calculations were run in gas phase with solvent present (figure 2). The conditions were set to standard temperature and pressure. While analyzing the structure the molecule, the structure was planar with C1 symmetry. F17 on the fluorophenyl group was pointing away from H23 connected to N25 atom in pyrazole moiety.

Figure 2. Optimized structure of 1H-FPI (left), and plot of change in energy during optimization process to achieve stable minima (right). FT-IR spectra of 1H-FPI indicates stretching frequencies for aromatic C=C at around 1500-1636 -1

cm . Out of plane bending for arene CH appears at 1177 and 1088. In plane arene CH bending peaks appear at 1185, 121215 and 1220. Intense peak for N-H bond appears at 3628 cm-1 along with CH stretching ranging from 3185 to 3297 cm-1 (figure 3).

Figure 3. Calculated FT-IR spectra of 1H-FPI calculated with DFT B3LYP/3-21G+ basis set. MEP mapping for 1H-FPI shows that hydrogen connected to N25 shows intense blue color indicating acidic nature (figure 4). Some of the electron densities are centered around C6-N24 as well as on F17. Frontier molecular orbital theory shows that in HOMO-1, there are some electron densities on N24 as well as the benzene ring of the indole moiety.

Figure 4. MEP mapping calculated with DFT B3LYP/3-21G+ basis set (left) and calculated DOS spectrum of 1H-FPI. Further study of molecular orbital (MO) theory shows different electronic energy levels. We looked closely into the two frontier molecular orbitals i.e., highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) which are located at 55th and 56th energy levels. Analysis of density of States (DOS) shows that there is significant orbital contribution for all four MOs. HOMO shows electron density on N25 as well as between N24-C6 which is due to pi orbital interactions. Molecular orbitals in LUMO are concentrated around N24 and C6-C11. The energy difference between HOMO and LUMO is 4.4930 eV (figure 5).

Figure 5. Partial frontier MO diagram (left) and selected molecular orbitals of 1H-FPI. 1.2. Analysis of (S)-3-(2-fluorophenyl)-7H-indazole (A-cisoid) The structure of A-cisoid was optimized using DFT and 3-21G+ basic sets and energy minima was located. The calculations were conducted in gas phase, i.e., no solvent was assigned. The temperature was set to 298.15K and the barometric pressure was set to 1 atm. In this current study, we explored (S)-3-(2-fluorophenyl)-7aH-indazole (A-cisoid). A-cisoid was one of many calculated isomers which is not a planer molecule as the hydrogen attached to C7 of the indazole moiety which is sp3 hybridized (Figure 6). C3 of indazole moiety is connected to phenyl group. Florine atom was attached to the second carbon of the phenyl substituent. Florine was chosen to introduce electronegativity as well as asymmetry to the benzene substituent. Figure 6 (left) shows the optimized structure of Frequency and geometry optimization was achieved in 49 steps and the change in energy during optimization cycle is shown on the right side of the same figure. Unlike Indazole, A-cisoid shows puckered geometry with 2-fluorophenyl group being out of plane compared to pyrazole ring. The dihedral angle ∠C1-C6-C11-C13 was measured at 34.60. Florine being a bigger atom with high electron density creates steric interaction with H10 forcing F17 pushed out of plane. C22 was designated as the bridged carbon with an axial hydrogen (H23). H23 and F17 pointed at the same side making the structure cisoid.

Figure 6. Optimized structure of A-cisoid (left), and plot of change in energy during optimization process to achieve stable minima (right).

Figure 7. FT-IR spectra of A-cisoid calculated with DFT B3LYP/3-21G+ basis set. Infra-Red data and spectrum was obtained as part of the calculation output (figure 7). Relatively intense peak at 799 cm-1 was due to out of plane bending of the arene hydrogens. Peaks at 1511 and 1329 cm-1 shows various in plane bending of arene hydrogens. H23 showed intense and distinctive peaks at 1141 cm-1 due to in plane bending and at 2912 cm-1 due to C22-H23 stretching. Molecular electrostatic potential map (MEP) derived from the optimized structure of A-cisoid structure indicated that there is significantly higher electron density (red color) around nitrogen and hydrogens bear the positive charges (Figure 8). It is important to point out that fluorine does not show much electron density despite of being the most electronegative atom in the periodic table. Due to being in conjugation with the phenyl group, fluorine disperses its electron density (lone pairs) through resonance.

Figure 8. MEP mapping calculated with DFT B3LYP/3-21G+ basis set (left) and calculated DOS spectrum of A-cisoid.

Figure 9. Partial MO diagram (left) and selected molecular orbitals of A-cisoid. Understanding the location of the electron density is critical to predict active sites of the molecule as well as reaction pathways. Based on the MEP map, it is evident that the nitrogen(s) in the pyrazole ring will act as electrophilic center and H23 can act as electrophile in a proton transfer reaction which we will explore later in this article. Among other major contributor of atomic orbitals, HOMO involved 2s orbitals of C6, C4, C1 and C5 atoms as well as 2px orbitals on C6 and C12, 2py on C1, C5, C6 and 2pz orbital on C6. Furthermore, 2pz orbital contribution from N25 is evident based on the molecular orbitals (Figure 9). This finding supports major electron density around N25 as showed in MEP earlier (Figure 8). LUMO was center of special interest as it is the only MO out of all four as mentioned which showed 1s orbital contribution from H23 which further solidifies its electrophilic nature as discussed earlier.

68 1.3. Isomerization Reactions of A-cisoid

Scheme 1. Calculated isomerization pathways of A-cisoid using DFT/3-21G+ basic sets Based on the investigation, an attempt was made to create a potential Energy Surface (PES) of several reaction profiles and locate the plausible transition states and intermediates which lead to several products based on frontier molecular orbital analysis. The saddle points called the transition states are the highest energy points between two consecutive minima. Reactive intermediates are the metastable states detectable by trapping reactions which gives valuable information on the direction of the reaction pathways. Computational chemistry proves itself to be a valuable tool in studying such reaction pathways and constructing PESs for the whole reaction path. It was constructed by calculating the relative energies of all steps of the reactions. Thermochemical and Physical properties of all the molecules resulted from this investigation was explored. First, we investigated the path it takes to convert over to 3-(2-fluorophenyl)-1H-indazole (1HFPI). In this section we explore theoretical proton transfer of H1 to C7 in the benzene ring as well as ring opening and closing mechanism involving the fluorophenyl substituent. Based on the calculation, we concluded that the proton transfer from While analyzing frontier orbitals of A-transoid, it is evident that the MO residue on N25 due to 2py contribution interacts with 1s of H23 (Figure 10) while the C22-N25 bond continues to weaken (from HOMO-1 of A-transoid) and break eventually resulting in INT-1 (Figure 11). We found one higher energy saddle point as the transition state of this process confirmed by one imaginary frequency generated by TS-7 at the higher energy minima at an optimized state.

Figure 10. Path 1: Conversion of A-cisoid to 1H-FPI via transition state (TS-6). The MEP mapping of INT-1 (Figure 12) shows that N25 is not electron dense (Mulliken charge is -0.341e) and most electron density resides on F17. 2-fluorophenyl group is ∠C1-C6-C11-C13 dihedral angle is around 64.5 degrees which cuts of any possibility of resonance resulting in localized charge on fluorine solely due to its high electronegativity. Upon following a ring closing mechanism to recreate the indazole ring involving 2-fluorophenyl group showed higher energy excited state (Figure 13). When ring recreation was conducted using INT-1 (Path 5), N25 attached to C12 producing (S)-4-fluoro-3-phenyl7aH-indazole (4H-FPI). Due to lack of electron availability, activation energy was calculated to be 36.7 kcal/mol which is rather high. This suggests that INT-1 and INT-2 should be very stable species. We found one higher energy saddle point as the transition state of this process confirmed by one imaginary frequency generated by TS-3 at the optimized state.

Figure 11. Path 3: Frontier MO interaction of A-transoid (left) showing HOMO and LUMO interaction (Orange dotted lines) to achieve the transition state, TS-7 (Top right) and INT-1 (bottom right).

Figure 12. MEP mapping calculated with DFT B3LYP/3-21G+ basis set (left) and calculated DOS spectrum of INT-1. While analyzing HOMO and LUMO for both INT-1, it is evident that both frontier orbitals are centered around the N25-N24-C6. HOMO shows contribution from electron cloud in 2pz orbital at N25. This can be seen in the MEP mapping of INT-1 (figure 12). Pi bonding orbitals from N24-C6 also contribute to HOMO. Orbital contribution in LUMO mainly comes from pi* antibonding MO involving 2py of N25 and N24. Electron density map for INT-2 shows slightly higher electron density on N25 (Figure 14). Mullikan charge on N25 was calculated to be -0.493e. This finding quite the opposite of what we found in INT-1. The phenyl group containing fluorine had better alignment with C6-N24-N25 plane (dihedral angle = 44.50) which allowed electron density flow from F17 to N25. This also explains higher stability of INT-2 over INT-1. Energy gap between HOMO and LUMO for INT-2 was calculated around 3.5534 eV. This value is slightly higher than that of INT-1 which is calculated to me 3.5171 eV.

Figure 13. Path 5: Conversion of INT-1 to 4H-FPI via transition state (TS-3).

Figure 14. MEP mapping calculated with DFT B3LYP/3-21G+ basis set (left) and calculated DOS spectrum of INT-2.

Figure 15. Path 7: Conversion of INT-2 to 7H-FPI via transition state (TS-5). An attempt was made to locate the transition state (TS-4) as shown in scheme 2. As there is no reaction involved and INT-1 to INT-2 conversion requires simple bond rotation, this part of the process was not described. While the mechanism was carried out using INT-2, the cyclization process (Path 7) involved a bond formation between N25 and C13 forming (R)-7a-fluoro-3-phenyl-7aH-indazole (7-FPI). This process yielded tetrahedral geometry around C13 pushing the F17 to move out of the indazole plane (Figure 15). We found one higher energy saddle point as the transition state of this process confirmed by one imaginary frequency generated by TS-5 at the higher energy minima at an optimized state. HOMO of TS-5 shows that there is accumulation of electron density between N25 and C12 which further confirms the bond formation. 1.4. Summary of the potential energy surface of the isomerization of 1H-FPI In scheme 2, we showed the potential energy surface diagram of isomerization of 1H-FPI. Acisoid showed a clear intention to convert over to A-cisoid with relative enthalpy of +49.3 kcal/mol

72 indicating an endothermic process. Conversion of A-cisoid to 1H-FPI will be thermodynamically more favorable.

Scheme 2. Schematic energy profile diagram of potential energy surface for isomerization reaction pathways calculated in kcal/mol unit using DFT/3-21G+ basic sets. Numbers in the parentheses represents the corrected enthalpy and without parentheses represents corrected free energy relative to A-cisoid. While pulling our focus on A-cisoid which became the starting point for other exploration, it became clear that reverse path 1 showed higher activation energy (Ea) of 12.7kcal/mol. A relatively lower energy pathway (path 2) was found. A-cisoid was calculated to be in equilibrium with A-transoid form with Ea of 9.4 kcal/mol. In a kinetically controlled reaction, path 2 can be a favorable way for A-cisoid to convert over to A-transoid. This is essentially a biaryl bond rotation around C6-C11. Both A-cisoid (Path 4) and A-transoid (Path 3) structures converged over to a stable intermediate (INT-1). This reaction was carried out by breaking C22-N25 bond and opening the pyrazole ring as part of indazole moiety. Atransoid showed slightly favorable path to produce INT-1 due to smaller Ea (8.7 kcal/mol) compared to Acisoid which showed to have Ea equal to 9.6 kcal/mol. INT-1 itself was a stable species with ∠C6-N24N25 angle being 1800. It was 22.7 kcal/mol more stable compared to A-cisoid. The phenyl groups on both sides were found to be out of plane as they were in both A-cisoid and A-transoid form. INT-1 could easily be converted over to INT-2 (Path 6) following an equilibrium pathway which involved rotation around C6-C11 bond. Activation energy for this process was calculated to be 6.3 kcal/mol. F17 was further away from N25 in INT-1, and they came closer when INT-1 was converted over to INT-2. This change in positioning, 4-FPI and 7-FPI were formed from their respective intermediates. While comparing path 5

73 and path 7, we found that path 5 had relatively lower activation energy (36.7kcal/mol). Path 7 showed activation energy of 44.5 kcal/mol. 4-FPI turned out to be 4.4 kcal/mol more stable compared to 7-FPI. Thus, between 4-FPI and 7-FPI, 4-FPI was relatively more thermodynamically as well as kinetically controlled product. A-cisoid and A-transoid are conformers. Same can be said for INT-1 and INT-2. Conformers are essentially same molecules with different positioning of atoms due to bond rotation. Both conformer pairs were calculated to be in equilibrium with low activation energy. Excluding the higher energy conformers and enantiomers, we concluded that there were five structural isomers namely 1H-FPI, A-cisoid, INT-2, 4-FPI, and 7-FPI in this current study. 1.5. Quantum chemical descriptors of 1H-FPI and its isomers Based on frontier molecular orbital analysis done using density functional theory, we can derive valuable chemical properties of the isomers. Properties such as ionization potential energy (IE), electron affinity (EA), Chemical hardness (η), chemical softness (σ), electronegativity (EN), chemical potential (μ) nucleophilicity (ω), and electrophilicity (ε). These parameters are used by chemists in a regular basis to understand chemical and physical natures of molecules (Table 1). Energies of HOMO and LUMO of molecules play a key role in predicting these quantities. We have already shown such molecular orbitals for 1H-FPI (figure 5). We have shown MOs for the other four isomers in figure 13 and 14. Based on the diagrams of all five isomers, it is evident that most electron density is localized around the nitrogen which act as nucleophilic centers. MEP mapping also supports this remark. This was the key behind finding different reaction paths involving nitrogen, especially N25. Table 1. Chemical parameters of the isomers of A-cisoid calculated using DFT/3-21G+ in gas phase at 298.15K. Isomers

EHOMO (eV)

ELUMO (eV)

ΔE

A-cisoid

-6.28

-2.87

6.28 2.87 3.41 1.71 0.59 4.58 -4.58 6.14 0.16

1H-FPI

-6.16

-1.66

6.16 1.66 4.49 2.25 0.45 3.91 -3.91 3.41 0.29

INT-2

-5.73

-2.18

5.73 2.18 3.55 1.78 0.56 3.95 -3.95 4.40 0.23

4-FPI

-6.32

-3.13

6.32 3.13 3.20 1.60 0.63 4.72 -4.72 6.98 0.14

7-FPI

-6.63

-3.77

6.63 3.77 2.85 1.43 0.70 5.20 -5.20 9.48 0.11

74 1.6. Study of Molecular Docking Prediction of activity spectra for substances (PASS) is an online tool which offers useful information on how small molecules can interact with bulky biomolecules such as proteins and enzymes etc. PASS predicts different types of activities based on the structure of a compound. The concept of the biological activity spectrum was introduced to describe the properties of biologically active substances. In this study, all molecular docking calculations were performed on Auto Dock-Vina software 8. Table 2. PASS prediction of bioactivity spectrum: Probability to be active (Pa) or inactive (Pi). Isomers

Activity

1H-FPI

0,840

0,005

Signal transduction pathways inhibitor

A-cisoid

0,748

0,001

Calcitonin gene-related peptide 1 receptor antagonist

INT-2

0,784

0,004

Arylmalonate decarboxylase inhibitor

4-FPI

0,802

0,003

Antimitotic

0,995

0,002

Tumor necrosis factor alpha release inhibitor

0,994

0,003

Antiarthritic

7-FPI

Based on the five isomers of 1H-FPI, we conducted a comprehensive search, and the PASS prediction results were obtained as shown in table 2. Based on many probabilities to be active (Pa) as predicted by PASS shown in table 2, 1H-FPI and 7-FPI were chosen due to high probability of activities. 1.7. Interaction of 1H-FPI with Spleen Tyrosine Kinase (PDB DOI: 4XG7) For the binding analysis of 1H-FPI, we selected Spleen tyrosine kinase (SYK) which is a nonreceptor tyrosine kinase that mediate signaling by receptors of the adaptive immune response 9. PDB file of the crystal structure of SKY was obtained from RCSB website. Optimized structure of 1-FPI was used in this experiment. AutoDock, Pymol, and Discovery Studios were the tools used for docking and visualization. Out of nine modes analyzed, only the first mode showed strong affinity towards SKY (table 3).

75 Table 3. A list of several modes of Binding Affinities of 1H-FPI with spleen tyrosine kinase.

Mode

Affinity

Distance from best mode

Kcal/mol

rmsd i.b.

umsd u.b.

-7.0

0.00

-6.8

2.407

3.509

Table 4. List of intermolecular interactions of 1H-FPI with spleen tyrosine kinase in mode 1. From Interactions