
Author 
W. B. Vasantha Kandasamy, Florentin Smarandache 
ISBN10 
9781599731797 
Year 
2012 
Pages 
150 
Language 
en 
Publisher 
Infinite Study 
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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 = n1.

Author 
Lynne Vanne 
ISBN10 
9781304637130 
Year 
201311 
Pages 
40 
Language 
en 
Publisher 
Lulu.com 
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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

Author 
Corneliu Lala 
ISBN10 
9789731991795 
Year 
20150218 
Pages 
442 
Language 
en 
Publisher 
INFAROM Publishing 
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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., MultiState), POWERBALL (U.S.A., MultiState), EUROMILLIONS (Europe, MultiCountry), EUROJACKPOT (Europe, MultiCountry), 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.  Multisystem 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.

Author 
Thomas Taylor 
ISBN10 
0787314064 
Year 
20060601 
Pages 

Language 
en 
Publisher 
Health Research Books 
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Author 
John Stillwell 
ISBN10 
9783319015774 
Year 
20131016 
Pages 
244 
Language 
en 
Publisher 
Springer Science & Business Media 
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While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the settheoretic aspects of analysis, this text makes the best of two worlds: it combines a downtoearth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

Author 
Ian Stewart 
ISBN10 
9780191016486 
Year 
20150312 
Pages 
432 
Language 
en 
Publisher 
OUP Oxford 
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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 firstyear 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 epsilondelta 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.

Author 
Donileen R. Loseke 
ISBN10 
9781412997201 
Year 
20120301 
Pages 
194 
Language 
en 
Publisher 
SAGE 
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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.

Author 
Andreas Philippou 
ISBN10 
902772234X 
Year 
19860430 
Pages 
304 
Language 
en 
Publisher 
Springer Science & Business Media 
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It isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. O. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Oulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and largescale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Author 
T.W. Cusick 
ISBN10 
9780080541846 
Year 
19980420 
Pages 
430 
Language 
en 
Publisher 
Elsevier 
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This book is almost entirely concerned with stream ciphers, concentrating on a particular mathematical model for such ciphers which are called additive natural stream ciphers. These ciphers use a natural sequence generator to produce a periodic keystream. Full definitions of these concepts are given in Chapter 2. This book focuses on keystream sequences which can be analysed using number theory. It turns out that a great deal of information can be deducted about the cryptographic properties of many classes of sequences by applying the terminology and theorems of number theory. These connections can be explicitly made by describing three kinds of bridges between stream ciphering problems and number theory problems. A detailed summary of these ideas is given in the introductory Chapter 1. Many results in the book are new, and over seventy percent of these results described in this book are based on recent research results.

Author 
David Newsome Susan A. Moore Ross Kingston Dowling 
ISBN10 
1845412761 
Year 
20011128 
Pages 
340 
Language 
en 
Publisher 
Channel View Publications 
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The book covers all facets of tourism in natural areas. The book is underpinned by a strong foundation of environmental understanding. It then describes the range of impacts, which occur when tourism takes place in the natural environment and illustrates how managers can plan, develop and appropriately manage tourism developments in natural areas. Finally, the book addresses ongoing management concerns such as monitoring environmental change and the need to introduce appropriate management strategies.

Author 
Paul Callaghan 
ISBN10 
9783540432876 
Year 
20020220 
Pages 
248 
Language 
en 
Publisher 
Springer Science & Business Media 
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Author 
Talal Ghannam 
ISBN10 
9781456463694 
Year 
20110104 
Pages 
214 
Language 
en 
Publisher 
Talal Ghannam 
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(Please note that this is the 1st edition. a much larger 2nd edition has already been published.)What is it that relates all these different things together?The subatomic particles and the geometrical patterns of the Vedic square.The procession of the equinox; the Mayan dooms day; the Hindu's Brahma cycle; and the destruction of Atlantis. The hydrogen atom and the golden section. Fibonacci numbers, alchemy and consciousness.Nikola Tesla, the octave scale and Pythagoras' music of the spheres.Plants' phyllotaxis, Solfeggio frequencies and the planets of our solar system. It is: Numbers, or more precisely, their digital root.In this book the author examines the amazing world of numbers, particularly those which have intrigued and fascinated ancient and modern mathematicians alike. However, he does it from a very novel point of view; by implementing the digital root operation, in which the individual digits of any of these numbers are summed up until a single digit is left over. The author will show that when applying this simple operation to magical numbers, and to many other groups of numbers, an amazing world of hidden interconnections; repetition cycles; numerical symmetries; and geometrical patterns emerges. Especially so when the geometrical (the circle) and the numerical aspects of the digital root world are combined together. It is in this circular/numerical world where numbers, individually or collectively, exist in their most basic, never the less, perfect and symmetrical states, and where the basic nine numbers are differentiated into triplet groups of intriguing properties and presence.Along the way, the author will also show how these new revelations of the digital root world are in fact corroborating and strengthening the numerological and mystical qualities that have been claimed to numbers by philosophers and mystics throughout the ages. This book will take us on a numerical and spiritual journey: starting from prime and figurate numbers; to Fibonacci sequence and the golden section; to alchemy and the Mayan calendar; to finally conclude with the mysterious 4th dimension. In all, this book presents a very unique and intriguing perspective of the numerical world, which will forever alter our perception not just toward numbers, but also toward the whole universe at large.

Author 
John P. Mayberry 
ISBN10 
0521770343 
Year 
2000 
Pages 
424 
Language 
en 
Publisher 
Cambridge University Press 
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This 2001 book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

Author 
Leo Corry 
ISBN10 
9780191007064 
Year 
20150827 
Pages 
368 
Language 
en 
Publisher 
OUP Oxford 
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The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be? In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century. Focusing on both foundational debates and practical use numbers, and showing how the story of numbers is intimately linked to that of the idea of equation, this book provides a valuable insight to numbers for undergraduate students, teachers, engineers, professional mathematicians, and anyone with an interest in the history of mathematics.