Exploring The Extension Of Natural Operations On Intervals Matrices And Complex Numbers

Author: W. B. Vasantha Kandasamy, Florentin Smarandache
Publisher: Infinite Study
ISBN: 9781599731797
Size: 17.19 MB
Format: PDF, Docs
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In this book we explore the possibility of extending the natural operations on reals to intervals and matrices. The extension to intervals makes us define a natural class of intervals in which we accept [a, b], a greater than b. Further, we introduce a complex modulo integer in Z_n (n, a positive integer) and denote it by iF with iF^2 = n-1.

Mr Natural S Christmas Carol Fake Book Strictly By The Numbers

Author: Lynne Vanne
Publisher: Lulu.com
ISBN: 9781304637130
Size: 18.38 MB
Format: PDF, Mobi
View: 72

19 of your favorite Christmas Carols - Strictly by the Numbers - Includes melody numbers, chord numbers and lyrics for each song Tips on how to make your own arrangement Songs are playable in any key on any instrument Includes Mr. Natural's Major Scale Intervalometer for piano/keyboards

Worldwide Lottery Games In Naturally Optimized Systems Pick 5

Author: Corneliu Lala
Publisher: INFAROM Publishing
ISBN: 9789731991795
Size: 12.15 MB
Format: PDF, ePub, Docs
View: 62

Worldwide Lottery Games In Naturally Optimized Systems Pick 5 has been developed on the authors’ initiative, based on the belief that nothing in the Universe happens by chance because there are laws governing everything, and by increasing our knowledge, we can triumph over mere chance. This is a book for players and/or groups of players (syndicates) who want to play to more than seventy lottery games from at least fifty countries over five continents, as: MEGA MILLIONS (U.S.A., Multi-State), POWERBALL (U.S.A., Multi-State), EUROMILLIONS (Europe, Multi-Country), EUROJACKPOT (Europe, Multi-Country), SUPER LOTTO (China), SIKKIM THUNDERBALL (India), THUNDERBALL (U.K.), LOTO (France) SANS TOPU (Turkey), EL GORDO (Spain), GOSLOTO 5 iz 36 (Russia), MINI LOTO (Japan), QUINA (Brazil) and so on. Worldwide Lottery Games In Naturally Optimized Systems Pick 5 contains 122 systems built with the help of original mathematical models. It is an original book comprising 51 simple variants systems, 38 pivoted variants systems and 33 combined variants systems for which the number of played numbers has values between 9 and 67 inclusive. Each system has its main characteristics, the winnings index and the unfolding on variants. The categories and the winnings will certainly be within the limits of one of the situations specified in the winnings index of the system used. All the playing systems selected in the present book are originally and naturally optimized, because the main parameters of the component combinatorial structures have optimal values. Therefore, 90 playing systems are at a level of absolute performance, which means with a smaller number of combinations, of the same category, it is not possible to get higher winnings indexes. The other 32 playing systems are at the highest level of current performance. More, all the playing systems are highly balanced. Experience in recent years has confirmed the value of the naturally optimized systems included in the book, through the numerous and substantial wins obtained by their help in many countries. Using the naturally optimized systems is a smart strategy for playing the lottery. FIVE ADVANTAGES OF NATURALLY OPTIMIZED SYSTEMS - Harmony with nature They are generated in harmony with nature, as they are based on the principles of balance, symmetry and proportion – fundamental principles of the creation, known everywhere in the world for thousands of years, which ensures their durability and high quality. - Optimum performance They are at a level of absolute performance, or at the highest level of current performance, as the main parameters of their combinatorial structures have optimum values. - Multi-system compatibility They are useful both to players and to groups of players worldwide, for lottery games of 5, 6 and 7 numbers in simple variant. - Guaranteed wins They help the lottery players to obtain guaranteed wins, strictly according to the previsions of the winnings indexes. - Accessibility They are easily accessible, can be understood by any player and their use does not require great effort, the action being made as simple as possible.

The Foundations Of Mathematics

Author: Ian Stewart
Publisher: OUP Oxford
ISBN: 9780191016486
Size: 17.19 MB
Format: PDF, ePub, Docs
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The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

Methodological Thinking

Author: Donileen R. Loseke
Publisher: SAGE
ISBN: 9781412997201
Size: 10.61 MB
Format: PDF, Mobi
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Thinking Methodologically: Basic Principles of Social Research Design focuses on the underlying logic of social research and encourages students to understand research methods as a way of thinking. In this book author Donileeen Loseke provides an overview of the basic principles of social research, including the foundations of research (data, concepts, theory), the characteristics of research questions, the importance of literature reviews, measurement (conceptualization and operationalization), data generation techniques (experiments, surveys, interviews, observation, document analysis) and sampling. Relationships among these components of research are stressed, and the repeated, explicit lesson throughout these pages is that it is not possible to argue that one or another form of research is “better” than any other and that good researchers understand the differences among—and appreciate the capabilities of—different tools.