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NUMBER THEORY Structures, Examples, and Problems

One of the oldest and liveliest branches of mathematics, Number Theory, is noted for its theoretical depth and applications to other fields, including representation theory, physics, and cryptography. The forefront of Number Theory is replete with sophisticated and famous open problems; at its foundation, however, are basic, elementary ideas that can stimulate and challenge beginning students. This textbook takes a problem-solving approach to Number Theory, situating each theoretical concept within the framework of some examples or some problems for readers to solve. Starting with the essentials, the text covers divisibility, powers of integers, floor function and fractional part, digits of numbers, basic methods of proof (extremal arguments, pigeonhole principle, induction, infinite descent, inclusion-exclusion), arithmetic function, important divisibility theorems and Diophantine equations. Emphasis is also placed on the presentation of some special problems involving quadratic residues, Fermat, Mersenne, and perfect numbers, as well as famous sequences of integers such as Fibonacci, Lucas, and other important ones defined by recursive relations. By thoroughly discussing interesting examples and applications and by introducing and illustrating every key idea, by relevant problems of various levels of difficulty, the book motivates, engages and challenges the reader. The exposition proceeds incrementally, intuitively and rigorously uncovers deeper properties.

Изучаем природу родного края.

Как научить детей любить природу родного края, понимать происходящие в природе явления поможет учителю эта книга. Вней описан опыт проведения внеклассной работы с учащимися V - VII классов.

Газон ўтини ўстиришда бегона ўтларга қарши кураш чоралари

Тошкент шаҳрининг диққатга сазовор жойларида, аҳоли яшаш худудларида, истироҳат боғлари ва сайилгоҳлар атрофида яшил ҳудудлар барпо этишда адабий манбаларга асосланган ҳолда газонларнинг эстетик хусусиятлари ўрганиб чиқиш жараёнида ўсимлик учун ўта зарур бўлган иқлим шароити, ҳаво ҳарорати, тупроқ таркиби ва намлиги, шамол харакатининг йўналиши ўрганиб чиқилган. Битирув ишимни бажаришда биринчидан замонавий технологияларга асосланган махсус режа асосида газон ўтларини ўстиришни ва улардаги бегона ўтларга қарши курашни ўрганиб чиқишдан иборат.

Young People, Wellbeing and Placemaking in the Arctic

This edited book presents the results of two research projects on youth wellbeing and senses of place in the Arctic region. The contributions are united by their focus on agency. Rather than seeing youth as vulnerable and possible victims of decisions by others, they illustrate the diverse avenues that youth pursue to achieve a good life in the Arctic.

Сабзавот кўчатларини етиштириш технологиясини иқтисодий самарадорлиги

Республикамизда сабзавотларни етиштиришни янги технологияларини қўллашни ва уларни кам тарқалган турларини кўпайтириш лозим. Сабзавот маҳсулотларини кўпайтириш ва сифатини яхшилаш учун эртапишар навларни экиш ва кўчат усулидан фойдаланиш яхши самара беради.

Алишер Навоий

Навоийнинг Ҳирот ташқарисидаги боғи. Ўртада икки қаватли шийпон. Узоқда Ҳирот ва унинг қўрғони, қалъаси кўриниб туради. Шийпоннинг бир ёғи мармар ҳовуз, атрофи гулзор. Нарида чўзилиб кетган мевали дарахтлар.

ШАРҚ КЛАССИК ФИЛОЛОГИЯСИ

Ушбу тўпламга Шарқ классик филологиясининг турли қирраларига бағишланган илмий мақолалар жамланди. Китоб филологлар ва шу соҳа билан қизиқувчиларга мўлжалланган.

CELL BIOLOGY

The authors thank their families, who supported this work, and also express gratitude to their mentors, who helped to shape their views of how science should be conducted. Bill is proud to have both his longtime partner and confidante Margarete and his son Charles as advisors on the science for this edition. He would not be surprised if his daughter Irina were added to that panel for our next edition. His contributions are firstly dedicated to them. Bill also would like to thank Jonathan King, Stephen Harrison, Aaron Klug, Tony Crowther, Ron Laskey, and Uli Laemmli, who provided a diverse range of rich environments in which to learn that science at the highest level is an adventure that lasts a lifetime. Graham dedicates the book to his family, Margaret, Paul, and Lara Johnson; the Benhorins; friends Mari, Steve, and Andrew; and his partners Flower and Anna Kuo. He also thanks his mentors at the Scripps Research Institute, Arthur Olson, David Goodsell, Ron Milligan, and Ian Wilson, for developing his career. Jennifer thanks her husband Jonathan for his strong backing and her lab members for their enthusiasm for the project. Tom dedicates the book to his wife Patty, a constant source of support and inspiration for more than five decades, and his children Katie and Dan, who also provided advice on the book. He also thanks Ed Korn and the late Sus Ito for the opportunity to learn biochemistry and microscopy under their guidance, and Ed Taylor and the late Hugh Huxley, who served as role models.

THE CAUCHY–SCHWARZ MASTER CLASS

This lively, problem-oriented text is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy–Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered — either by one of history’s famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of H¨older, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.

TI{E USSR OLYMPIAD PROBLEM BOOK

THIS BOOK CONTAIDES 320 unconventional problems in algebra, arithmetic, elementary number theory, and trigonometry. Most of these problems first appeared in competitive examinations sponsored by the School Mathematical Society of the Moscow State University and in the Mathematical Olympiads held in Moscow. The book is designed for students having a mathematical hackground at the high school level; very many of the problems are within reach of seventh and eighth grade students of outstanding ability. Solutions are given for all the problems. The solutions for the more difficult problems are especially detailed. The third (Russian) edition differs from the second chiefly in the elimination of errors detected in the second edition. Therefore, the preface to the second edition is retained,

Касбий таълим жараёнида педагогик лойиҳалаш ва педагогик технологияларни қўллаш

Малакавий битирув ишининг мақсади: Касбий таълим жараёнида педагогик лойиҳалаш ва педагогик технологияларни қўллаш асосида касбҳунар коллежи таълим мазмунини янгилаш орқали малакали мутахассисларни тайёрлаш.

GEOMETRY

Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor the author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Грамматика

Сборник упражнений охватывает все разделы английской грамматики. Упражнения основаны на несложной лексике. Они содержат достаточно материала для запоминания основных грамматических форм и выработки навыков их применения. Дополнен теоретическим материалом и словарем

The Pythagorean Theorem

The Pythagorean Theorem has been with us for over 4000 years and has never ceased to yield its bounty to mathematicians, scientists, and engineers. Amateurs love it in that most new proofs are discovered by amateurs. Without the Pythagorean Theorem, none of the following is possible: radio, cell phone, television, internet, flight, pistons, cyclic motion of all sorts, surveying and associated infrastructure development, and interstellar measurement. The Pythagorean Theorem, Crown Jewel of Mathematics chronologically traces the Pythagorean Theorem from a conjectured beginning, Consider the Squares (Chapter 1), through 4000 years of Pythagorean proofs, Four Thousand Years of Discovery (Chapter 2), from all major proof categories, 20 proofs in total. Chapter 3, Diamonds of the Same Mind, presents several mathematical results closely allied to the Pythagorean Theorem along with some major Pythagorean “spin-offs” such as Trigonometry. Chapter 4, Pearls of Fun and Wonder, is a potpourri of classic puzzles, amusements, and applications. An Epilogue, The Crown and the Jewels, summarizes the importance of the Pythagorean Theorem and suggests paths for further exploration. Four appendices service the reader: A] Greek Alphabet, B] Mathematical Symbols, C] Geometric Foundations, and D] References. For the reader who may need a review of elementary geometric concepts before engaging this book, Appendix C is highly recommended. A Topical Index completes the book.

Касб ҳунар коллежи ўқувчиларида маънавий сифатларни шакллантиришнинг самарали омиллари

Битирув малакавий ишининг мақсади-ўрта махсус таълим тизимидаги маънавий-маърифий фаолият мазмуни, шакл ва методлари орқали ўқувчи ёшларда маънавий сифатларни шаклланишини ўрганиш.

Рубоиёт

Умар Хайёмни дунёга машҳур қилган унинг рубоийларидир. Умар Хайём шоир бўлмаган, лекин ҳаёт ҳақида баъзи мулоҳазаларини 4қаторлик шеър қилиб қоғозга тушириб қўяверган. XVIII асрга кадар унинг рубоийлари ҳақида тадкикотлар олиб борилмаган. У хакда дастлаб Оксфорд университети профессори Томас Гайд, ундан сўнг Фон Гомер Биргестел ва Меме Николослар тадкикот олиб боришган. 1859 йили инглиз шоири ва таржимони Эдвард ФитзЖералд Умар Хайёмнинг 70 рубоийсини таржима қилиб эълон қилгандан сўнг, бу шеърлар жуда машҳур бўлиб кетган ва инглиз тилидан жаҳоннинг бошқа тилларига ҳам таржима қилинган.