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Basic Notions of Algebra

Basic Notions of Algebra

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2005
sidottu
§22. K-theory 230 A. Topological X-theory 230 Vector bundles and the functor Vec(X). Periodicity and the functors KJX). K(X) and t the infinite-dimensional linear group. The symbol of an elliptic differential operator. The index theorem. B. Algebraic K-theory 234 The group of classes of projective modules. K , K and K of a ring. K of a field and o l n 2 its relations with the Brauer group. K-theory and arithmetic. Comments on the Literature 239 References 244 Index of Names 249 Subject Index 251 Preface This book aims to present a general survey of algebra, of its basic notions and main branches. Now what language should we choose for this? In reply to the question 'What does mathematics study?', it is hardly acceptable to answer 'structures' or 'sets with specified relations'; for among the myriad conceivable structures or sets with specified relations, only a very small discrete subset is of real interest to mathematicians, and the whole point of the question is to understand the special value of this infinitesimal fraction dotted among the amorphous masses. In the same way, the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion.
Discourses on Algebra

Discourses on Algebra

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2002
nidottu
I wish that algebra would be the Cinderella ofour story. In the math­ ematics program in schools, geometry has often been the favorite daugh­ ter. The amount of geometric knowledge studied in schools is approx­ imately equal to the level achieved in ancient Greece and summarized by Euclid in his Elements (third century B. C. ). For a long time, geom­ etry was taught according to Euclid; simplified variants have recently appeared. In spite of all the changes introduced in geometry cours­ es, geometry retains the influence of Euclid and the inclination of the grandiose scientific revolution that occurred in Greece. More than once I have met a person who said, "I didn't choose math as my profession, but I'll never forget the beauty of the elegant edifice built in geometry with its strict deduction of more and more complicated propositions, all beginning from the very simplest, most obvious statements!" Unfortunately, I have never heard a similar assessment concerning al­ gebra. Algebra courses in schools comprise a strange mixture of useful rules, logical judgments, and exercises in using aids such as tables of log­ arithms and pocket calculators. Such a course is closer in spirit to the brand of mathematics developed in ancient Egypt and Babylon than to the line of development that appeared in ancient Greece and then con­ tinued from the Renaissance in western Europe. Nevertheless, algebra is just as fundamental, just as deep, and just as beautiful as geometry.
Basic Algebraic Geometry 1

Basic Algebraic Geometry 1

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2013
sidottu
Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.'' The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann--Roch theorem for curves, including a proof from first principles.Shafarevich's book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.
Basic Algebraic Geometry 2

Basic Algebraic Geometry 2

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2013
sidottu
Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.''The second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the ``Shafarevich conjecture''.The style of Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.
Basic Notions of Algebra

Basic Notions of Algebra

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2014
nidottu
§22. K-theory 230 A. Topological X-theory 230 Vector bundles and the functor Vec(X). Periodicity and the functors KJX). K(X) and t the infinite-dimensional linear group. The symbol of an elliptic differential operator. The index theorem. B. Algebraic K-theory 234 The group of classes of projective modules. K , K and K of a ring. K of a field and o l n 2 its relations with the Brauer group. K-theory and arithmetic. Comments on the Literature 239 References 244 Index of Names 249 Subject Index 251 Preface This book aims to present a general survey of algebra, of its basic notions and main branches. Now what language should we choose for this? In reply to the question 'What does mathematics study?', it is hardly acceptable to answer 'structures' or 'sets with specified relations'; for among the myriad conceivable structures or sets with specified relations, only a very small discrete subset is of real interest to mathematicians, and the whole point of the question is to understand the special value of this infinitesimal fraction dotted among the amorphous masses. In the same way, the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion.
Basic Algebraic Geometry 1

Basic Algebraic Geometry 1

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2015
nidottu
Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.'' The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann--Roch theorem for curves, including a proof from first principles.Shafarevich's book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.
Collected Mathematical Papers

Collected Mathematical Papers

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2015
nidottu
This volume contains almost all mathematical papers published between 1943 and 1984 of Igor R. Shafarevich. They appear in English translations (with two exceptions, which are in French and German), some of the papers have been translated into English especially for this edition. Notes by Shafarevich at the end of the volume contain corrections and remarks on the subsequent development of the subjects considered in the papers. Igor R. Shafarevich has made a big impact on mathematics. He has worked in the fields of algebra, algebraic number theory, algebraic geometry and arithmetic algebraic geometry. His papers reflect his broad interests and include topics such as the proof of the general reciprocity law, the realization of groups as Galois groups of number fields, class field towers, algebraic surfaces (in particular K3 surfaces), elliptic curves, and finiteness results on abelian varieties, algebraic curves over number fields and lie algebras.
Basic Algebraic Geometry 2

Basic Algebraic Geometry 2

Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2016
nidottu
Shafarevich's Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, ``For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.''The second volume is in two parts: Book II is a gentle cultural introduction to scheme theory, with the first aim of putting abstract algebraic varieties on a firm foundation; a second aim is to introduce Hilbert schemes and moduli spaces, that serve as parameter spaces for other geometric constructions. Book III discusses complex manifolds and their relation with algebraic varieties, Kähler geometry and Hodge theory. The final section raises an important problem in uniformising higher dimensional varieties that has been widely studied as the ``Shafarevich conjecture''.The style of Basic Algebraic Geometry 2 and its minimal prerequisites make it to a large extent independent of Basic Algebraic Geometry 1, and accessible to beginning graduate students in mathematics and in theoretical physics.
Linear Algebra and Geometry

Linear Algebra and Geometry

Igor R. Shafarevich; Alexey O. Remizov

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2012
sidottu
This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitely generated periodic modules (similar to Jordan normal forms of linear operators). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics.
Linear Algebra and Geometry

Linear Algebra and Geometry

Igor R. Shafarevich; Alexey O. Remizov

Springer-Verlag Berlin and Heidelberg GmbH Co. K
2014
nidottu
This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitely generated periodic modules (similar to Jordan normal forms of linear operators). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics.
Geometries and Groups

Geometries and Groups

Viacheslav V. Nikulin; Igor R. Shafarevich

Springer-Verlag Berlin and Heidelberg GmbH Co. K
1987
nidottu
This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they are identical to the geometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and crystallography. The description of locally Euclidean geometries of one type shows that these geometries are themselves naturally represented as the points of a new geometry. The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions. Thus in this book we would like to introduce the reader to a theory of geometries which are different from the usual Euclidean geometry of the plane and 3-space, in terms of examples which are accessible to a concrete and intuitive study. The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries. The book does not presuppose on the part of the reader any preliminary knowledge outside the limits of a school geometry course.
Beat the Odds in Forex Trading

Beat the Odds in Forex Trading

Igor R. Toshchakov

John Wiley Sons Inc
2006
sidottu
"Beat the Odds in Forex Trading provides traders with tremendous value by disseminating the trading methods and philosophy of one of the most remarkable Forex success stories since Soros." --Alexander De Khtyar, President, Forex International Investments, Inc. Add certainty and systematization into Forex trading with this practical approach. Author and industry professional Igor Toshchakov shows how recurring market patterns--which can be recognized on a simple bar chart--can be successfully used to trade the Forex market. Written for traders at every level, this valuable resource discusses the challenges of developing a trading method, while revealing the Toshchakov's approach to the market--both from a philosophical and tactical point of view. You'll discover specific trading strategies based on recognizable market patterns, get detailed information on entry and exit points, profit targets, stop losses, risk evaluation, and much more.
Putin's 'Viper' Detachment

Putin's 'Viper' Detachment

Igor R Salikov

PEN SWORD BOOKS LTD
2026
sidottu
With the collapse of the USSR in 1991, the Soviet armed forces were largely disbanded and some of their personnel were distributed among the newly independent republics. With few options, the author joined the Redut organisation which consisted of Russian mercenaries operating abroad. In the years that followed, the rise of Valdimir Putin after he became President led to a strong nationalist sentiment developing in the Kremlin. Two former Soviet republics, initially Chechnya and then Georgia, were the first to be subjected to a growing Russian aggression. Special sabotage teams and assault battalions were created, each based on the Redut group, and the author, who had acquired the call sign ‘Viper’, soon found himself once again in Russian pay. This time, though, he was leading a special forces group in operations in Donetsk in the eastern part of Ukraine. In the autumn of 2021, ‘Viper’ was appointed to command the intelligence unit of the Redut group as part of the GRU. He was given three secret agents. These were Ukrainian special forces officers who had defected to Russia. They were handed a secret mission – to attack the headquarters in Kyiv of the Ukrainian Counter-Intelligence and Security Service, the SBU, and destroy all its files ahead of Putin’s full-scale invasion of the country. Putin's ‘Viper’ Detachment is the first full account by ‘Viper’ himself of this secret operation and how his unit had to fight its way out of Ukraine after the attempt on the SBU headquarters failed. A tense, full-throttle, struggle ensued, which ultimately led the battle-hardened author refusing to fight or kill anymore. He pulled his platoon out of the Ukraine and into Belarus, pursued by Putin’s counter-intelligence officers. Eventually, ‘Viper’ managed to escape Russia and went to the International Criminal Court in Hague to testify about war crimes committed by Putin's government. This is the first time the world will read about this secret operation recounted by the man who led it. In this book, ‘Viper’ reveals true nature of the hidden war in Ukraine.
The Professor's Daughter: A Fictional Memoir

The Professor's Daughter: A Fictional Memoir

R. Igor Gamow

Big Bang Productions, LLC
2010
nidottu
"The Professor's Daughter" is a fictional memoir about an eccentric professor who's been a non-conformist his entire career. After his wife's death, he tries to re-establish his relationship with his estranged daughter, Athena. She wants to know about her father's secret life that he lived apart from both her and her mother. Reluctantly, the professor tells Athena about the other women in his life through stories filled with passion, humor and irreverence. In writing what he calls a "fictional memoir", Gamow knows that there is always a risk that friends and family might take offense. However, his aim is to explore the mysteries of romance and have a good-natured laugh at the wild chemistry created when mixing men, women and love. "The Professor's Daughter" is a fictionalized memoir in which the life and loves of a recently widowed professor are recounted to his estranged daughter. Drawn home by the death of her mother, Athena must confront her father and the memories of the night years ago that pushed them apart. In the telling of "his side of the story," the professor tells of the many loves in his life, and expounds on his unique perspective on romantic love. Gamow's extensive experience in film is evident in this narrative, which is filled with vibrant characters, crisp dialogue, and a well developed sense of scene. A highly entertaining read, with a flare for the risqu ." - Stephanie Walker, Literary Editor, Boulder, Colorado "I am both shocked and enthralled after reading The Professor's Daughter Like a number of his heroes: Watson and Crick, Einstein, and Charles Darwin, the controversial Professor Gamow has turned heads and ruffled feathers with a surprising new publication. I have known him over the past twelve years as a teacher, mentor, research partner, and friend. Nothing could have prepared me for this revealing "memoir," which sheds an interesting light on a man with an extremely divisive history. I have long been inspired by Professor Gamow's quixotic idealism, but this book will cause many to question whether or not he has gone too far. I highly recommend this read as it places an interesting twist on decades of rumors, hearse, and speculation." - Aaron M. Shupp, MD Candidate, University of Colorado School of Medicine "In typical Gamowian fashion, this book entertains while stretching the imagination with both humor and innovation." - Gino Segre, Professor of Physics and Astronomy at the University of Pennsylvania and Author of "Faust in Copenhagen" "Gamow paints a picture that will leave the reader questioning just how much is real. The dialogue is crisp and believable, making the story an easy read..." "The change in point of view is smooth and easy to follow. Gamow leads the reader to an almost instant empathy with Athena. Like her, the reader is waiting for her father to prove himself. The plot moves smoothly from the time she hears of her mother's death, goes home and reconnects with those who influenced her childhood; and as her father returns to tell his story. The unexpected ending leaves the reader wondering." Pat Avery ForeWord Clarion Reviews
The Story of Pi

The Story of Pi

R. Igor Gamow

Independently Published
2019
nidottu
TWO WORLDS, TWO STORIES. WHAT IS THE CONNECTION? On the sea-swept coast of Ireland a disillusioned physician finds an abandoned baby and decides to raise her as his own daughter. Discontented with traditional means of education, he creates a new and fantastical method to teach the girl, the likes of which the world has never seen...In a fire tower perched high atop a tall West Virginian mountain, a National Park Service ranger stands vigil over the valley below and the mysterious school nestled within it. Will a tragedy he witnesses on his watch be a catalyst for healing or harm?