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Introduction to Numerical Methods

As in adaptive integration, it is useful to devise an ode integrator that automatically finds the appropriate ∆t. The Dormand-Prince Method, which is implemented in MATLAB’s ode45.m, finds the appropriate step size by comparing the results of a fifth-order and fourth-order method. It requires six function evaluations per time step, and constructs both a fifth-order and a fourth-order method from these function evaluations

NUMERICAL METHODS

We thank the faculty and the students of various universities, Engineering colleges and others for sending their suggestions for improving this book. Based on their suggestions, we have made the follwoing changes. New problems have been added and detailed solutions for many problems are given. C-programs of frequently used numerical methods are given in the Appendix. These programs are written in a simple form and are user friendly. Modifications to these programs can be made to suit individual requirements and also to make them robust. We look forward to more suggestions from the faculty and the students. We are thankful to New Age International Limited for bringing out this Second Edition.

Т А Н Л А Н ГАН АСАРЛАР

Муқимий — XIX асрнинг иккинчи ярми ва XX асрнннг бошларида яшаган отоқлн ўзбек демократ шоириднр. Ўэбек адабиётн ва адабий тилиии ривожлантиришда Муқнмийнинг хиэматлари буюкдир. У меҳнгткашлар оммаси орасидан етншиб чиқди ва ўэ асарларида меҳнаткашларнинг фикр ва туйғуларнни, эксплуататорларга қарши қаттиқ нафратинн ҳаққоний акс аттирди. Муқимий ўэ эамонасннннг илғор кишиси эди. У ўз кўэ ўнгида содир бўлган муҳим воқиаларни эийраклик блан англаб олди.

ЕР ЮЗИ

Жаноби Ҳақ таъолий ҳазратлари бизни ердан яратмишдур. Чунки бизим туғулодурғон, ўсадурғон, тириклик қиладурғон ҳам ўлғондин сўнгра кўмиладурғон ўрнимиз шул ердур. Бас, ер бизим онамиз ва умрлик ватанимиздур. Бир кишиға ўз уйи ва маҳалласининг аҳволини билиб турмак қандай лозим бўлса, умрлик ватан аслиси бўлғон ернинг аҳволини билмак ондан ортуғроқлозимдур. Лекин бир кишининг умри ҳар қанча узун бўлса ҳам, ер юзини тамом айланиб, кўруб билолмас. Ер юзидағи инсонларни, ҳайвонларни, ҳукуматларни, шаҳарларни, тоғларни, чўлларни, денгизларни, наҳрларни ва шуларға ўхшаш керакли нарсаларнинг барчасини кўриб, билмак бир киши учун мумкин эмасдур

NUMBER THEORY

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.

SHAJARAYI XORAZMSHOHIY

O‘zbek xalqi olis o‘tmishni o‘rganish, moziyni qayta kashf etish yo‘lida muhim yutuqlarga erishdi. Ammo turg‘unlik davrida tarixshunoslik borasida ham qatag‘on etilgan sarhadlar, yolg‘on hukmronlik qilgan davrlar bor edi. Natijada, xalqimiz o‘tmishining muhim va katta bir davri tarix haqidagi tadqiqotlarda dam, adabiy asarlarda ham chinakan aksini topmadi. Bu O‘rta Osiyoning Chor Rossiyasi tomonidan zabt etilishi davridir. Arab va Chingizxon bosqinidan keyingi eng dahshatli bu bosqin tasviri, hatto, xalq tarixining nodir badiiy kashfiyoti bo‘lmshi Abdulla Oripov «O‘zbekiston» qasidasida ham aks etmagan:

МATEMATIK ANALIZ FANIDAN O‟QUV USLUBIY MAJMUA

Matematik tahlil fani funksional bog‟lanishlarni o‟rganuvchi fan bo‟lib, barcha matematik turdagi fanlar va fizika kursi uchun asos hisoblanadi. Matematik tahlil fanining tadbiqlari: yuzalarni hisoblash, jismlarning hajmlarini hisoblash, jismlarning massasini, og‟irlik markazini, inersiya momentini topish, o‟zgaruvchan kuchning jismni ko‟chirishda bajargan mexanik ishini hisoblash masalalari matematik tahlilning fizika va astronomiya yo‟nalishlarida o‟qitilishi naqadar muhim ekanligini yaqqol ko‟rsatadi

Мирий.

Мирнй XIX асрнинг иккинчи ярмидаги прогрессив адабиётиинг истеъдодли ва сермаҳсул намояндаларидан. Шоирнинг ҳаёти ва "ижодини ўрганиш бундан ўттиз етти йил аввал адабиётшунос А. Мажидий томонидан бошланган эди. У ўзининг Мирий ҳаҳидаги маҳоласида («Маориф ва ўқитғувчи», 6—7, 8-сонлар, 1927) шоирнинг ҳаёти ва ижоди ҳаҳида ҳисҳача маълумот беради. Шундан кейин Мирий ҳаҳида илмий-тадҳиқот ишлари олиб борилмади.

NUCLEAR MODELS

More than a generation of German-speaking students around the world have worked their way to an understanding and appreciation of the power and beauty of modern theoretical physics with mathematics, the most fundamental of sciences - using Walter Greiner's textbooks as their guide. The idea of developing a coherent, complete presentation of an entire field of science in a series of closely related textbooks is not a new one. Many older physicists remember with real pleasure their sense of adventure and discovery as they worked their ways through the classic series by Sommerfeld, by Planck and by Landau and Lifshitz. From the students' viewpoint, there are a great many obvious advantages to be gained through use of consistent notation, logical ordering of topics and coherence of presentation; beyond this, the complete coverage of the science provides a unique opportunity for the author to convey his personal enthusiasm and love for his subject.

ҲУКМДОРГА ЎГИТЛАР

Хожа Убайдуллоҳ Аҳрорнинг муриди ва издоши, нақшбандия тариқатининг атоқли шайхларидан бири Мавлоно Муҳаммад Қози XV аср охири ва XVI аср бошларида яшаб фаолият кўрсатган. Муҳаммад Қозининг асосий асари машҳур «Силсилат ул-орифин»дан ташқари, давлатни бошқариш юзасидан йўл-йўриқ ва насиҳатларни ўз ичига олган яна бир рисоласи XVI асрнинг нодир ёзма манбаларидан бўлган «Тарихи Рашидий» асари орқали бизгача етиб келган. Рисола Заҳириддин Муҳаммад Бобурнинг холаваччаси ва Муҳаммад Қозининг шогирди, яъни «Тарихи Рашидий» асарининг муаллифи Мирзо Муҳаммад Ҳайдарга бағишланган бўлиб, унинг саволларига жавоб тариқасида ёзилган

Notes on Euclidean Geometry

This book is a compilation and distillation of my notes, as participant and later as instructor, from the Math Olympiad Program (MOP), the annual summer program to prepare U.S. high school students for the International Mathematical Olympiad (IMO). As such, it has an overt and a covert mission. The overt mission is to assist students in preparing for the USA Mathematical Olympiad (USAMO) and the IMO, as American students have historically fared poorly on problems in Euclidean geometry. The covert, and perhaps more important, mission is to introduce said students (and anyone else who happens to read this) to a lineage of mathematics stretching from ancient times to the present; hence the inclusion of advanced topics in inversive and projective geometry which may segue into the study of complex analysis, algebraic geometry, or the like.

ХИМИЯ БИОЛОГИЧЕСКИ АКТИВНЫХ ВЕЩЕСТВ

Электронный курс лекций посвящен структурно-функциональным особенностям и химическим свойствам основных классов биологически активных веществ (белков, углеводов, липидов, витаминов, антибиотиков и др.). Описаны методы химического синтеза и структурного анализа пере- численных классов соединений, их свойства и воздействие на биологические системы, а также распространение в природе.

ҲОЗИРГИ ЎЗБЕК ТИ Л И Д А С И Ф А Т

Узбек тили сифат категориясининг илмпй таҳлиляга бағишланган мазкур монографияда сифатнииг умумий характеристикаси, унинг семантик турлари, С1руктураси, даражалари илмий асосда ўрганилади. Китоб тилшуносларга, олий ўкув юртларинипг филология факультетларининг ўқитувчи ва талабаларига. аспирантларига мўлжалланган.

ФЕЪЛ СТИЛИСТИКАСИ

Тил инсон томонидан яратилган ва инсонга «ҳадя этилган» буюк инъомдир. Тил бор экан, алоқа бор, тарих бор, маданият ва адабиёт бор. Ўзбек адабий тилининг такомили ва равнақи ўзбек бадиий адабиётининг юксалиши билан чамбарчас боғлиқ. Кейинги даврларда ўзбек бадиий адабиёти жуда катта одимлар билан олға кетди, кўп миллатли адабиётимиз даврасидан муносиб ўрин олди. Шу қатори ўзбек бадиий прозаси кейинги йилларда ҳам сон, ҳам сифат жиҳатидан юксалди. Айниқса ҳикоя, роман жанрларининг ўсиши, сатирик, лирик, тарихий асарларнинг кўплаб яратилиши, ўзбек тили бисотидаги ноёб жавоҳирларни кўрсатиш тил бойлигини намойиш этишда катта омил бўлиб хизмат этди. Шунинг билан бирга ёзувчиларимиз халқ тилидаги битмас-туганмас бойликларни адабий тил билан йўғириб, бадиий тил хазинасини яна ҳам кенгайтирдилар. Бу ҳодисаларни қайд этиш унга илмий ёндошиш, таҳлил этиш эса тилшуносларнинг асосий вазифаларидандир.

PHILOSOPHY OF LITERATURE

Literature is a discipline which poses its own characteristic philosophical questions. Literary theorists have been primarily engaged with discussions of the nature of literature, while ‘analytic’ philosophers have addressed literary problems within the framework of aesthetics, or in a manner which is accessible only to a philosophical audience. The present book redresses the balance by examining issues in the philosophy of literature from an analytic standpoint accessible to both students of literature and students of philosophy. The book discusses definitions of literature, the distinction between oral and written literature and the identity of literary works. The author offers a wideranging discussion of the nature of fiction, in which both irony and non-literary fictions are analysed. An examination of our emotional involvement with fictional characters and events, followed by a discussion of the concept of imagination as an essential factor in our apprehension of literary works. Various theories of metaphor are then considered, and postmodernist theories of authorship discussed. Issues about truth and morality in literature are also raised. Finally, the book asks whether literary appraisals are objective or subjective and proposes a qualified subjective view

CONTEST PROBLEM BOOK

The New York City Interscholastic Mathematics League is the oldest and largest local organization of its type in the United States. Records exist of competitions given by the League since the early 1920s, and there are indications that these contests were the continuation of an even earlier tradition. The contests have contributed to the unique position that New York City holds in mathematics education today. In national and international contests, New York City high school students have shown a high degree of talent, interest, and achievement. Many of these students have gone on to make their careers in mathematics, and some of them have contributed problems that appear in the present book. Thus the New York City IML contests have been instrumental in developing a mathematical problem-solving community in New York City, which in turn has made its contribution to the field on a national level.