“…The polynomials and factoring, binomial theorem, and rational expressions are also elaborated. …”
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“…Real numbers, inequalities and intervalsFunction, domain and rangeBasic coordinate geometryPolar coordinatesMathematical inductionBinomial theoremCombination of functionsSymmetry in functions and graphsInverse functionsComplex numbers; real and imaginary formsGeometry of complex analysisModulus-argument form of a complex numberRoots of complex numbersLimitsOne-sided limitsDerivativesLeibniz's formulaDifferentialsDifferentiation of inverse trigonometric functionsImplicit differentiationParametrically defined curves and parametric differentiationThe exponential functionThe logarithmic functionHy…”
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“…Preface to the Fifth EditionPreface to the Fourth EditionPreface to the Third EditionPreface to the Second EditionPreface to the First EditionAuthorGreek LettersElementary Algebra and GeometryFundamental Properties (Real Numbers)ExponentsFractional ExponentsIrrational ExponentsLogarithmsFactorialsBinomial TheoremFactors and ExpansionProgressionComplex NumbersPolar FormPermutationsCombinationsAlgebraic EquationsGeometryPythagorean TheoremDeterminants, Matrices, and Linear Systems of EquationsDeterminantsEvaluation by CofactorsProperties of DeterminantsMatricesOperationsPropertiesTransposeIdenti…”
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“…Topics include logic, set theory, relations and functions, vectors and matrices, graph theory, counting and binomial theorem, probability, Boolean algebra, and linear programming and …”
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“…Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell–Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.…”
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“…Beginning with an exploration of permutations and combinations that culminates in the Binomial Theorem, the text goes on to guide the study of ordinary and exponential generating functions. …”
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“…With lower virulence the chance of a lesion following inoculation of virus is still described by the binomial theorem, but the actual distribution is primarily of susceptible cells not of viral particles. …”
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“…IntroductionDiophantine EquationsModular ArithmeticPrimes and the Distribution of PrimesCryptographyDivisibilityDivisibilityEuclid's Theorem Euclid's Original Proof The Sieve of Eratosthenes The Division Algorithm The Greatest Common Divisor The Euclidean Algorithm Other BasesLinear Diophantine EquationsThe Postage Stamp Problem Fermat and Mersenne Numbers Chapter Highlights Problems Unique FactorizationPreliminary Results The Fundamental Theorem of Arithmetic Euclid and the Fundamental Theorem of ArithmeticChapter Highlights Problems Applications of Unique Factorization A Puzzle Irrationality Proofs The Rational Root Theorem Pythagorean Triples Differences of Squares Prime Factorization of Factorials The Riemann Zeta Function Chapter Highlights Problems CongruencesDefinitions and Examples Modular Exponentiation Divisibility TestsLinear Congruences The Chinese Remainder TheoremFractions mod m Fermat's Theorem Euler's Theorem Wilson's Theorem Queens on a Chessboard Chapter Highlights Problems Cryptographic ApplicationsIntroduction Shift and Affine Ciphers Secret Sharing RSA Chapter Highlights Problems Polynomial Congruences Polynomials Mod Primes Solutions Modulo Prime PowersComposite Moduli Chapter Highlights Problems Order and Primitive RootsOrders of Elements Primitive Roots DecimalsCard Shuffling The Discrete Log Problem Existence of Primitive Roots Chapter Highlights Problems More Cryptographic Applications Diffie-Hellman Key Exchange Coin Flipping over the Telephone Mental Poker The ElGamal Public Key Cryptosystem Digital Signatures Chapter Highlights Problems Quadratic Reciprocity Squares and Square Roots Mod Primes Computing Square Roots Mod p Quadratic Equations The Jacobi Symbol Proof of Quadratic Reciprocity Chapter Highlights Problems Primality and Factorization Trial Division and Fermat Factorization Primality Testing Factorization Coin Flipping over the Telephone Chapter Highlights Problems Geometry of NumbersVolumes and Minkowski's Theorem Sums of Two Squares Sums of Four Squares Pell's Equation Chapter Highlights Problems Arithmetic FunctionsPerfect Numbers Multiplicative Functions Chapter Highlights Problems Continued Fractions Rational Approximations; Pell's Equation Basic TheoryRational Numbers Periodic Continued Fractions Square Roots of Integers Some Irrational Numbers Chapter Highlights Problems Gaussian Integers Complex Arithmetic Gaussian Irreducibles The Division Algorithm Unique Factorization Applications Chapter Highlights Problems Algebraic IntegersQuadratic Fields and Algebraic IntegersUnits Z[√-2] Z[√3] Non-unique Factorization Chapter Highlights Problems Analytic MethodsΣ1/p Diverges Bertrand's Postulate Chebyshev's Approximate Prime Number Theorem Chapter Highlights Problems Epilogue: Fermat's Last Theorem Introduction Elliptic Curves Modularity Supplementary Topics Geometric SeriesMathematical InductionPascal's Triangle and the Binomial TheoremFibonacci NumbersProblems Answers and Hints for Odd-Numbered ExercisesIndex…”
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