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Abstract Algebra by Romyar Sharif

This note covers the following topics: Set theory, Group theory, Ring theory, Isomorphism theorems, Burnsides formula, Field theory and Galois theory, Module theory, Commutative algebra, Linear algebra via module theory, Homological algebra, Representation theory.

s328 Pages

Abstract Algebra in GAP by Alexander Hulpke

This book aims to give an introduction to using GAP with material appropriate for an undergraduate abstract algebra course. It does not even attempt to give an introduction to abstract algebra, there are many excellent books which do this. Topics covered includes: The GGAP user interface, Rings, Groups, Linear Algebra, Fields and Galois Theory, Number Theory.

s179 Pages

Algebraic Geometry Notes I

This note covers the following topics: Hochschild cohomology and group actions, Differential Weil Descent and Differentially Large Fields, Minimum positive entropy of complex Enriques surface automorphisms, Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces, Superstring Field Theory, Superforms and Supergeometry, Picard groups for tropical toric schemes, Complex multiplication and Brauer groups of K3 surfaces.

sNA Pages

Basic Modern Algebraic Geometry

This note covers the following topics: Functors, Isomorphic and equivalent categories, Representable functors, Some constructions in the light of representable functors, Schemes: Definition and basic properties, Properties of morphisms of schemes, general techniques and constructions.

s111 Pages

Algebraic Topology by NPTEL

This is a basic note in algebraic topology, it introduce the notion of fundamental groups, covering spaces, methods for computing fundamental groups using Seifert Van Kampen theorem and some applications such as the Brouwers fixed point theorem, Borsuk Ulam theorem, fundamental theorem of algebra.

sNA Pages

Notes On The Course Algebraic Topology

This note covers the following topics: Important examples of topological spaces, Constructions, Homotopy and homotopy equivalence, CW -complexes and homotopy, Fundamental group, Covering spaces, Higher homotopy groups, Fiber bundles, Suspension Theorem and Whitehead product, Homotopy groups of CW -complexes, Homology groups, Homology groups of CW -complexes, Homology with coefficients and cohomology groups, Cap product and the Poincare duality, Elementary obstruction theory.

s181 Pages

Methods of Applied Mathematics Notes

This note describes the following topics: Normed Linear Spaces and Banach Spaces, Hilbert Spaces, Spectral Theory and Compact Operators, Distributions, The Fourier Transform, Sobolev Spaces, Boundary Value Problems, Differential Calculus in Banach Spaces and the Calculus of Variations.

s279 Pages

Applied Finite Mathematics

This book explains the following topics: Linear Equations, Matrices, Linear Programming, Mathematics of Finance, Sets and Counting, Probability, Markov Chains, Game Theory.

sNA Pages

A Generalized Arithmetic Geometric Mean

This note explains the following topics: Classical arithmetic geometry, The Convergence Theorem, The link with the classical AGM sequence, Point counting on elliptic curves, A theta structure induced by Frobenius.

s95 Pages

Algebraic and Arithmetic Geometry

This note covers the following topics: Rational points on varieties, Heights, Arakelov Geometry, Abelian Varieties, The Brauer-Manin Obstruction, Birational Geomery, Statistics of Rational Points, Zeta functions.

s63 Pages

Introduction to School Algebra

This note covers the following topics: Symbolic Expressions, Transcription of Verbal Information into Symbolic Language, Linear Equations in One Variable, Linear Equations in Two Variables and Their Graphs, Simultaneous Linear Equations, Functions and Their Graphs, Linear functions and proportional reasoning, Linear Inequalities and Their Graphs, Exponents, Quadratic Functions and Their Graphs, The Quadratic Formula and Application.

s216 Pages

College Algebra by Avinash Sathaye

This is a set of lecture notes on introductory school algebra written for middle school teachers. Topics covered includes: Symbolic Expressions, Transcription of Verbal Information into Symbolic Language, Linear Equations in One Variable, Linear Equations in Two Variables and Their Graphs, Simultaneous Linear Equations, Functions and Their Graphs, Linear functions and proportional reasoning, Linear Inequalities and Their Graphs, Exponents, Quadratic Functions and Their Graphs, The Quadratic Formula and Applications.

s190 Pages

Mathematics Lecture Notes by NPTEL

This note covers the following topics: Numbers, functions, and sequences, Limit and continuity, Differentiation, Maxima, minima and curve sketching, Approximations, Integration, Logarithmic and exponential functions, Applications of Integration, Series of numbers and functions, Limit and continuity of scalar fields, Differentiation of scalar fields, Maxima and minima for scalar fields, Multiple Integration, Vector fields, Stokes theorem and applications.

sNA Pages

Lecture on Additive Number Theory

In Additive Number Theory we study subsets of integers and their behavior under addition. Topics covered includes:Lower Bound on Sumset, Erdos conjecture on arithmetic progressions, Szemeredi theorem, Algorithm to find Large set with 3-term AP, Condition for a set not having 3-term AP, Cardinality of set with no 3-term AP, Improved Size of A, Sum Free Sets and Prime number theorem.

s122 Pages

Calculus with Applications by Daniel Kleitman

The note is intended as a one and a half term course in calculus for students who have studied calculus in high school. It is intended to be self contained, so that it is possible to follow it without any background in calculus, for the adventurous.

sNA Pages

Calculus Lectures Notes by Gerald Hoehn

This note covers following topics of Integral and Differential Calculus: Differential Calculus: rates of change, speed, slope of a graph, minimum and maximum of functions, Derivatives measure instantaneous changes, Integral Calculus: Integrals measure the accumulation of some quantity, the total distance an object has travelled, area under a curve, volume of a region.

sNA Pages

An introduction to Category Theory

The book is aimed primarily at the beginning graduate student.It gives the de nition of this notion, goes through the various associated gadgetry such as functors, natural transformations, limits and colimits, and then explains adjunctions.

s436 Pages

Introduction To Category Theory And Categorical Logic

This note covers the following topics related to Category Theory: Categories, Functors and Natural Transformations, subcategories, Full and Faithful Functors, Equivalences, Comma Categories and Slice Categories, Yoneda Lemma, Grothendieck universes, Limits and Colimits, Adjoint Functors, Adjoint Functor Theorems, Monads, Elementary Toposes, Cartesian Closed Categories, Logic of Toposes and Sheaves.

s117 Pages

Lectures On Semiclassical Analysis

This note explains the following topics: Symplectic geometry, Fourier transform, stationary phase, Quantization of symbols, Semiclassical defect measures, Eigenvalues and eigenfunctions, Exponential estimates for eigenfunctions, symbol calculus, Quantum ergodicity and Quantizing symplectic transformations.

s211 Pages

Classical Analysis and ODEs

This note explains the following topics: linearly related sequences of difference derivatives of discrete orthogonal polynomials, identity for zeros of Bessel functions, Close-to-convexity of some special functions and their derivatives, Monotonicity properties of some Dini functions, Classification of Systems of Linear Second-Order Ordinary Differential Equations, functions of Hausdorff moment sequences, Van der Corput inequalities for Bessel functions.

sNA Pages

Enumerative Combinatorics

This book will bring enjoyment to many future generations of mathematicians and aspiring mathematicians as they are exposed to the beauties and pleasures of enumerative combinatorics. Topics covered includes: What is Enumerative Combinatorics, Sieve Methods, Partially Ordered Sets, Rational Generating Functions, Graph Theory Terminology

s725 Pages

Lecture Notes Combinatorics

This lecture note covers the following topics: What is Combinatorics, Permutations and Combinations, Inclusion-Exclusion-Principle and Mobius Inversion, Generating Functions, Partitions, Partially Ordered Sets and Designs.

s137 Pages

Commutative Algebra by Columbia University

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Basic commutative algebra will be explained in this document.

s443 Pages

A Term of Commutative Algebra

This book is a clear, concise, and efficient textbook, aimed at beginners, with a good selection of topics. Topics covered includes: Rings and Ideals, Radicals, Filtered Direct Limits, CayleyHamilton Theorem, Localization of Rings and Modules, KrullCohenSeidenberg Theory, Rings and Ideals, Direct Limits, Filtered direct limit.

s266 Pages

A Guide to Complex Variables

This book has plenty of figures, plenty of examples, copious commentary, and even in-text exercises for the students. But, since it is not a formal textbook, it does not have exercise sets. It does not have a Glossary or a Table of Notation. Topics covered includes: The Complex Plane, Complex Line Integrals, Applications of the Cauchy Theory, Isolated Singularities and Laurent Series, The Argument Principle, The Geometric Theory of Holomorphic Functions, Harmonic Functions, Infinite Series and Products, Analytic Continuation .

s185 Pages

Complex Variables by R. B. Ash and W.P. Novinger

The book is intended as a text, appropriate for use by advanced undergraduates or graduate students who have taken a course in introductory real analysis, or as it is often called, advanced calculus. No background in complex variables is assumed, thus making the text suitable for those encountering the subject for the first time. The Elementary Theory, General Cauchy Theorem , Applications of the Cauchy Theory, Families of Analytic Functions, Factorization of Analytic Functions and The Prime Number Theorem.

s220 Pages

Complex Analysis by Christian Berg

This note covers the following topics: Holomorphic functions, Contour integrals and primitives, The theorems of Cauchy, Applications of Cauchys integral formula, Argument. Logarithm, Powers, Zeros and isolated singularities, The calculus of residues, The maximum modulus principle, Mobius transformations.

s192 Pages

Complex Variables A Physical Approach

This text will illustrate and teach all facets of the subject in a lively manner that will speak to the needs of modern students. It will give them a powerful toolkit for future work in the mathematical sciences, and will also point to new directions for additional learning. Topics covered includes: The Relationship of Holomorphic and Harmonic Functions, The Cauchy Theory, Applications of the Cauchy Theory, Isolated Singularities and Laurent Series, The Argument Principle, The Geometric Theory of Holomorphic Functions, Applications That Depend on Conformal Mapping, Transform Theory.

s437 Pages

Computational Mathematics by Jose Augusto Ferreira

This lecture note explains the following topics: This Numerical Methods for ODEs, Discretizations for ODEs, The Runge-Kutta Methods, Linear Multistep Methods, Numerical Methods for PDEs, Tools of Functional Analysis, The Ritz-Galerkin Method, FDM for Time-Dependent PDES, Finite Difference Methods for Elliptic Equations, Computational Projects.

s177 Pages

Introduction to Computational Mathematics by Greg Fasshauer

This note covers the following topics: Mathematical Modeling, Eulers Method, Taylor Series, Taylor Polynomials, Floating-Point Numbers, Normalized Floating-Point Numbers , MATLAB.

s163 Pages

Notes on Infinite Sequences and Series

This book covers the following topics: Sequences, Limit Laws for Sequences, Bounded Monotonic Sequences, Infinite Series, Telescopic Series, Harmonic Series, Higher Degree Polynomial Approximations, Taylor Series and Taylor Polynomials, The Integral Test, Comparison Test for Positive-Term Series, Alternating Series and Absolute Convergence, Convergence of a Power Series and Power Series Computations.

s21 Pages

Constants And Numerical Sequences

These lectures note explains the real and complex numbers and their properties, particularly completeness; define and study limits of sequences, convergence of series, and power series.

sNA Pages

Stability criteria for nonlinear fully implicit differential algebraic systems

The goal of this note is to contribute to the qualitative theory of differential-algebraic systems by providing new asymptotic stability criteria for a class of nonlinear, fully implicit DAEs with tractability index two. Topics covered includes: State space analysis of differential-algebraic equations, Properly formulated DAEs with tractability index 2, The state space form, Index reduction via differentiation, Stability criteria for differential-algebraic systems, Asymptotic stability of periodic solutions, Lyapunovs direct method regarding DAEs.

s191 Pages

Differential algebra

This book covers the following topics: differential polynomial and their ideals, algebraic differential manifolds, structure of differential polynomials, systems of algebraic equations, constructive method, intersections of algebraic differential manifolds, Riquier's existence theorem for orthonomic system.

s189 Pages

Differential Analysis Lecture notes by Richard B. Melrose

This note covers the following topics: Measure and Integration, Hilbert spaces and operators, Distributions, Elliptic Regularity, Coordinate invariance and manifolds, Invertibility of elliptic operators, Suspended families and the resolvent, Manifolds with boundary, Electromagnetism and Monopoles.

s243 Pages

Lecture Notes In Analysis

This note covers the following topics: Introduction To PDE, Basic Tools of Analysis, Analysis of the Wave Equation in Minkowski Space, Basic Concepts in Riemannian and Lorentzian Geometry.

s235 Pages

Differentiation by Mathtutor

This note explains the following topics: Differentiation from first principles, Differentiating powers of x, Differentiating sines and cosines, Differentiating logs and exponentials, Using a table of derivatives, The quotient rule, The product rule, The chain rule, Parametric differentiation, Differentiation by taking logarithms, Implicit differentiation, Extending the table of derivatives, Tangents and normals, Maxima and minima.

sNA Pages

A Collection of Problems in Differential Calculus

This note covers the following topics: Limits and Continuity, Differentiation Rules, Applications of Differentiation, Curve Sketching, Mean Value Theorem, Antiderivatives and Differential Equations, Parametric Equations and Polar Coordinates, True Or False and Multiple Choice Problems.

s159 Pages

A First Course in Elementary Differential Equations

This note covers the following topics: Qualitative Analysis, Existence and Uniqueness of Solutions to First Order Linear IVP, Solving First Order Linear Homogeneous DE, Solving First Order Linear Non Homogeneous DE: The Method of Integrating Factor, Modeling with First Order Linear Differential Equations, Additional Applications: Mixing Problems and Cooling Problems, Separable Differential Equations, Exact Differential Equations, Substitution Techniques: Bernoulli and Ricatti Equations, Applications of First Order Nonlinear Equations, One-Dimensional Dynamics, Second Order Linear Differential Equations, The General Solution of Homogeneous Equations, Existence of Many Fundamental Sets, Second Order Linear Homogeneous Equations with Constant, Coefficients, Characteristic Equations with Repeated Roots, The Method of Undetermined Coefficients, Applications of Nonhomogeneous Second Order Linear Differential Equations.

s213 Pages

Partial Differential Equations Lectures by Joseph M. Mahaffy

This note introduces students to differential equations. Topics covered includes: Boundary value problems for heat and wave equations, eigenfunctionexpansions, Surm-Liouville theory and Fourier series, D'Alembert's solution to wave equation, characteristic, Laplace's equation, maximum principle and Bessel's functions.

sNA Pages

Differential Geometry by Rui Loja Fernandes

This note covers the following topics: Manifolds as subsets of Euclidean space, Abstract Manifolds, Tangent Space and the Differential, Embeddings and Whitneys Theorem, The de Rham Theorem, Lie Theory, Differential Forms, Fiber Bundles.

sNA Pages

Differential Geometry in Toposes

This note explains the following topics: From KockLawvere axiom to microlinear spaces, Vector bundles,Connections, Affine space, Differential forms, Axiomatic structure of the real line, Coordinates and formal manifolds, Riemannian structure, Well-adapted topos models.

s93 Pages

Differential Topology of Fiber Bundles

This note explains the following topics: The concept of a fiber bundle, Morphisms of Bundles, Vector Bundles, Principal Bundles, Bundles and Cocycles, Cohomology of Lie Algebras, Smooth G-valued Functions, Connections on Principal Bundles, Curvature and Perspectives.

s146 Pages

Introduction to Differential Topology by Uwe Kaiser

This book gives a deeper account of basic ideas of differential topology than usual in introductory texts. Also many more examples of manifolds like matrix groups and Grassmannians are worked out in detail. Topics covered includes: Continuity, compactness and connectedness, Smooth manifolds and maps, Regular values and Sards theorem, Manifolds with boundary and orientations, Smooth homotopy and vector bundles, Intersection numbers, vector fields and Euler characteristic.

s110 Pages

Lecture Notes for College Discrete Mathematics

This note explains the following topics: Sets, Sums and products, The Euclidean algorithm, Numeral systems, Counting, Proof techniques, Pascal's triangle, Recurrence sequences.

s203 Pages

Applied Discrete Mathematics William Shoaff

This note explains the following topics: positional and modular number systems, relations and their graphs, discrete functions, set theory, propositional and predicate logic, sequences, summations, mathematical induction and proofs by contradiction.

sNA Pages

Elliptic Functions An Elementary Text Book for Students of Mathematics

This note explains the following topics: Elliptic Integrals, Elliptic Functions, Periodicity of the Functions, Landens Transformation, Complete Functions, Development of Elliptic Functions into Factors, Elliptic Integrals of the Second Order, Numerical Calculations.

s147 Pages

Elliptic Curve Cryptography by Christian Wuthrich

Elliptic-curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.

s98 Pages

Fourier Analysis and Related Topics

Aim of this note is to provide mathematical tools used in applications, and a certain theoretical background that would make other parts of mathematical analysis accessible to the student of physical science. Topics covered includes: Power series and trigonometric series, Fourier integrals, Pointwise convergence of Fourier series, Summability of Fourier series, Periodic distributions and Fourier series, Metric, normed and inner product spaces, Orthogonal expansions and Fourier series, Classical orthogonal systems and series, Eigenvalue problems related to differential equations, Fourier transformation of well-behaved functions, Fourier transformation of tempered distributions, General distributions and Laplace transforms.

s341 Pages

Topics in Fourier Analysis

This note is an overview of some basic notions is given, especially with an eye towards somewhat fractal examples, such as infinite products of cyclic groups, p-adic numbers, and solenoids. Topics covered includes: Fourier series, Topological groups, Commutative groups, The Fourier transform, Banach algebras, p-Adic numbers, r-Adic integers and solenoids, Compactifications and Completeness.

s182 Pages

A tale of two fractals

This book is devoted to a phenomenon of fractal sets, or simply fractals. Topics covered includes: Sierpinski gasket, Harmonic functions on Sierpinski gasket, Applications of generalized numerical systems, Apollonian Gasket, Arithmetic properties of Apollonian gaskets, Geometric and group-theoretic approach.

s134 Pages

Lectures on fractal geometry and dynamics

Goal of this course note is primarily to develop the foundations of geometric measure theory, and covers in detail a variety of classical subjects. A secondary goal is to demonstrate some applications and interactions with dynamics and metric number theory.

s96 Pages

Fractional Calculus Integral and Differential Equations of Fractional Order

This note covers the following topics: Introduction To Fractional Calculus, Fractional Integral Equations, Fractional Differential Equations and The Mittag-leffler Type Functions.

s56 Pages

Fractional Calculus Definitions and Applications

The first chapter explains definition of fractional calculus. The second and third chapters, look at the Riemann-Liouville definitions of the fractional integral and derivative. The fourth chapter looks at some fractional differential equations with an emphasis on the Laplace transform of the fractional integral and derivative. The last chapter describes application problemsa mortgage problem and a decay-growth problem.

s61 Pages

Functional Analysis by ETH Zurich

This note covers the following topics: Principles of Functional Analysis, The Weak and Weak Topologies, Fredholm Theory, Spectral Theory, Unbounded Operators, Semigroups of Operators.

s427 Pages

Functional analysis and its applications

Functional analysis plays an important role in the applied sciences as well as in mathematics itself. These notes are intended to familiarize the student with the basic concepts, principles andmethods of functional analysis and its applications, and they are intended for senior undergraduate or beginning graduate students. Topics covered includes: Normed and Banach spaces, Continuous maps, Differentiation, Geometry of inner product spaces , Compact operators and Approximation of compact operators.

s92 Pages

Geometric Algebra and its Application to Mathematical Physics

This thesis is an investigation into the properties and applications of Cliffords geometric algebra. Topics covered includes: Grassmann Algebra and Berezin Calculus, Lie Groups and Spin Groups, Spinor Algebra, Point-particle Lagrangians, Field Theory, Gravity as a Gauge Theory.

s244 Pages

Applications of Geometric Algebra in Computer Vision

There are four main areas discussed in this guide is: Geometric Algebra, Projective Geometry, Multiple View Tensors and 3DReconstruction.

s174 Pages

Introduction to Geometric Topology

The aim of this book is to introduce hyperbolic geometry and its applications to two- and three-manifolds topology. Topics covered includes: Hyperbolic geometry, Hyperbolic space, Hyperbolic manifolds, Thick-thin decomposition, The sphere at infinity, Surfaces, Teichmuller space, Topology of three-manifolds, Seifert manifolds, Constructions of three-manifolds, Three-manifolds, Mostow rigidity theorem, Hyperbolic Dehn filling.

s448 Pages

Surgery and Geometric Topology

This book covers the following topics: Cohomology and Euler Characteristics Of Coxeter Groups, Completions Of Stratified Ends, The Braid Structure Of Mapping Class Groups, Controlled Topological Equivalence Of Maps in The Theory Of Stratified Spaces and Approximate Fibrations, The Asymptotic Method In The Novikov Conjecture, N Exponentially Nash G Manifolds and Vector Bundles, Controlled Algebra and Topology.

s162 Pages

Euclidean Geometry by Rich Cochrane and Andrew McGettigan

This is a great mathematics book cover the following topics: Equilateral Triangle, Perpendicular Bisector, Angle Bisector, Angle Made by Lines, The Regular Hexagon, Addition and Subtraction of Lengths, Addition and Subtraction of Angles, Perpendicular Lines, Parallel Lines and Angles, Constructing Parallel Lines, Squares and Other Parallelograms, Division of a Line Segment into Several Parts, Thales' Theorem, Making Sense of Area, The Idea of a Tiling, Euclidean and Related Tilings, Islamic Tilings.

s102 Pages

Topics in Geometry Dirac Geometry Lecture Notes

This is an introductory note in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry.

sNA Pages

Graph Theory Lecture Notes by NPTEL

The intension of this note is to introduce the subject of graph theory to computer science students in a thorough way. This note will cover all elementary concepts such as coloring, covering, hamiltonicity, planarity, connectivity and so on, it will also introduce the students to some advanced concepts.

sNA Pages

Structural Graph Theory Lecture Notes

This note covers the following topics: Immersion and embedding of 2-regular digraphs, Flows in bidirected graphs, Average degree of graph powers, Classical graph properties and graph parameters and their definability in SOL, Algebraic and model-theoretic methods in constraint satisfaction, Coloring random and planted graphs: thresholds, structure of solutions and algorithmic hardness.

s123 Pages

Notes on Group Theory

This note covers the following topics: Notation for sets and functions, Basic group theory, The Symmetric Group, Group actions, Linear groups, Affine Groups, Projective Groups, Finite linear groups, Abelian Groups, Sylow Theorems and Applications, Solvable and nilpotent groups, p-groups, a second look, Presentations of Groups, Building new groups from old.

s126 Pages

Geometric Group Theory Preliminary Version Under revision

The goal of this book is to present several central topics in geometric group theory, primarily related to the large scale geometry of infinite groups and spaces on which such groups act, and to illustrate them with fundamental theorems such as Gromovs Theorem on groups of polynomial growth. Topics covered includes: Geometry and Topology, Metric spaces, Differential geometry, Hyperbolic Space, Groups and their actions, Median spaces and spaces with measured walls, Finitely generated and finitely presented groups, Coarse geometry, Coarse topology, Geometric aspects of solvable groups, Gromovs Theorem, Amenability and paradoxical decomposition, Proof of Stallings Theorem using harmonic functions.

s817 Pages

Lecture Notes on Introduction to Harmonic Analysis

This note explains the following topics: The Fourier Transform and Tempered Distributions, Interpolation of Operators, The Maximal Function and Calderon-Zygmund Decomposition, Singular Integrals, Riesz Transforms and Spherical Harmonics, The Littlewood-Paley g-function and Multipliers, Sobolev Spaces.

s217 Pages

Harmonic Analysis Lecture Notes

This textbook presents more than any professor can cover in class. The first part of the note emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. Topics covered includes: Fourier series, Fourier coefficients, Fourier integrals,Fourier transforms, Hilbert and Riesz transforms, Fourier series and integrals, Band limited functions, Band limited functions, Periodization and Poisson summation.

s176 Pages

Workbook In Higher Algebra

This note explains the following topics: Group Theory, Sylows Theorem, Field and Galois Theory, Elementary Factorization Theory, Dedekind Domains, Module Theory, Ring Structure Theory, Tensor Products.

s194 Pages

Higher algebra, a sequel to elementary algebra for schools

This book is intended as a sequel to our Elementary Algebra for Schools. The first few chapters are devoted to a fuller discussion of Ratio, Proportion, Variation, and the Progressions, and then introduced theorems with examples.

s593 Pages

An Episodic History of Mathematics

The purpose of this book, is to acquaint the student with mathematical language and mathematical life by means of a number of historically important mathematical vignettes. This book will also serve to help the prospective school teacher to become inured in some of the important ideas of mathematicsboth classical and modern.

s483 Pages

A Short Account of the History of Mathematics

This is the classic resource on the history of math providing a deeper understanding of the subject and how it has impacted our culture, all in one essential volume. The subject-matter of this book is a historical summary of the development of mathematics, illustrated by the lives and discoveries of those to whom the progress of the science is mainly due. It may serve as an introduction to more elaborate works on the subject, but primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know.

s466 Pages

Categories and Homological Algebra

This course note introduces the reader to the language of categories and to present the basic notions of homological algebra, first from an elementary point of view, with the notion of derived functors, next with a more sophisticated approach, with the introduction of triangulated and derived categories.

s125 Pages

Homological Algebra I (PDF 85P)

Covered topics are: General manipulations of complexes, More on Koszul complexes, General manipulations applied to projective resolutions, Tor, Regular rings, review of Krull dimension, Regular sequences and Tor, CohenMacaulay rings and modules, Injective and divisible modules, Injective resolutions, A definition of Ext using injective resolutions, Duality and injective hulls, Gorenstein rings, Bass numbers.

s85 Pages

Measure and Integration

This graduate-level lecture note covers Lebesgue's integration theory with applications to analysis, including an introduction to convolution and the Fourier transform.

sNA Pages

Guide to Integration

This note covers the following topics: Elementary Integrals, Substitution, Trigonometric integrals, Integration by parts, Trigonometric substitutions, Partial Fractions.

s96 Pages

K Theory Notes

This note covers the following topics: Vector Bundles and Bott Periodicity, K-theory Represented by Fredholm Operators, Representations of Compact Lie Groups, Equivariant K-theory.

s97 Pages

K theory For Operator Algebras

This note will develop the K-theory of Banach algebras, the theory of extensions of C algebras, and the operator K-theory of Kasparov from scratch to its most advanced aspects. Topics covered includes: Survey of Topological K-Theory, Operator K-Theory, Preliminaries, K-theory Of Crossed Products, Theory Of Extensions, Kasparovs Kk-theory.

s314 Pages

Introduction to Lie Groups by Alistair Savage

This note focus on the so-called matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of background knowledge. Topics covered includes: Matrix Lie groups, Topology of Lie groups, Maximal tori and centres, Lie algebras and the exponential map, Covering groups.

s111 Pages

Lie Groups Representation Theory and Symmetric Spaces

This note covers the following topics: Fundamentals of Lie Groups, A Potpourri of Examples, Basic Structure Theorems, Complex Semisimple Lie algebras, Representation Theory, Symmetric Spaces.

s178 Pages

A Brief Introduction to Linear Algebra

This note covers the following topics: Linear Algebra, Matrix Algebra, Homogeneous Systems and Vector Subspaces, Basic Notions, Determinants and Eigenvalues, Diagonalization, The Exponential of a Matrix, Applications,Real Symmetric Matrices, Classification of Conics and Quadrics, Conics and the Method of Lagrange Multipliers, Normal Modes.

sNA Pages

Introduction to Applied Linear Algebra

This book is meant to provide an introduction to vectors, matrices, and least squares methods, basic topics in applied linear algebra. Our goal is to give the beginning student, with little or no prior exposure to linear algebra, a good grounding in the basic ideas, as well as an appreciation for how they are used in many applications, including data fitting, machine learning and artificial intelligence, tomography, image processing, finance, and automatic control systems. Topics covered includes: Vectors, Norm and distance, Clustering, Matrices, Linear equations, Matrix multiplication, Linear dynamical systems, Least squares, Multi-objective least squares, Constrained least squares.

s473 Pages

Differentiable manifolds

The purpose of these notes is to introduce and study differentiable manifolds. Topics covered includes: Manifolds in Euclidean space, Abstract manifolds, The tangent space, Topological properties of manifolds, Vector fields and Lie algebras, Tensors, Differential forms and Integration.

s132 Pages

Lectures on the Geometry of Manifolds

This book introduces the reader to the concept of smooth manifold through abstract definitions and, more importantly, through many we believe relevant examples. Topics covered includes: Manifolds, Natural Constructions on Manifolds, Calculus on Manifolds, Riemannian Geometry, Elements of the Calculus of Variations, The Fundamental group and Covering Spaces, Cohomology, Characteristic classes, Classical Integral Geometry, Elliptic Equations on Manifolds and Dirac Operators.

s570 Pages

Introduction to Mathematical Analysis I

Goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels.

sNA Pages

Mathematical Analysis Volume I by Elias Zakon

This text is an outgrowth of lectures given at the University of Windsor, Canada. Topics covered includes: Set Theory, Real Numbers. Fields, Vector Spaces, Metric Spaces, Function Limits and Continuity, Differentiation and Anti differentiation.

s365 Pages

Infinite Series Notes (PDF 22P)

This note covers the following topics related to Infinite Series: Definitions and basic examples, Positive series, Series with mixed signs and Power series

s22 Pages

Notes on Infinite Series (PDF 61P)

This note covers the following topics related to Infinite Series: Cauchy Root Test, Comparison Test, Dalembert (or Cauchy) Ratio Test, Alternating Series, Absolute And Conditional Convergence, Improvement Of Convergence, Rearrangement Of Double Series, Taylors Expansion, Power Series, Indeterminate Forms, Binomial Theorem, Mathematical Induction, Operations On Series Expansions Of Functions, Bernoulli Numbers, Euler-maclaurin Integration Formula and Dirichlet Series.

s61 Pages

Infinite Series Notes (PDF 22P)

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sNA Pages

Notes on Infinite Series (PDF 61P)

Currently this section contains no detailed description for the page, will update this page soon.

sNA Pages

Modern Elementary Geometry

This note covers the following topics: The classical theorem of Ceva, Ceva, Menelaus and Selftransversality, The general transversality theorem, The theorems of Hoehn and Pratt-Kasapi, Circular products of ratios involving circles, Circle transversality theorems, A basic lemma and some applications, Affinely Regular Polygons, Linear transformations; smoothing vectors, Affine-Regular Components, The general Napoleon's Theorem, The iteration of smoothing operations.

s164 Pages

Introductory modern geometry of point, ray, and circle

This book explains all the fundamental concepts in modern geometry.

s164 Pages

Single and Multivariable Calculus

This note explains the following topics: Analytic Geometry, Instantaneous Rate of Change: The Derivative, Rules for Finding Derivatives, Transcendental Functions, Curve Sketching, Applications of the Derivative, Integration, Techniques of Integration, Applications of Integration, Polar Coordinates, Parametric Equations, Sequences and Series, Vector Functions, Partial Differentiation, Multiple Integration, Vector Calculus, Differential Equations.

sNA Pages

Multivariable Calculus Notes

This note covers the following topics: Vectors and the geometry of space, Directional derivatives, gradients, tangent planes, introduction to integration, Integration over non-rectangular regions, Integration in polar coordinates, applications of multiple integrals, surface area, Triple integration, Spherical coordinates, The Fundamental Theorem of Calculus for line integrals, Green's Theorem, Divergence and curl, Surface integrals of scalar functions, Tangent planes, introduction to flux, Surface integrals of vector fields, The Divergence Theorem.

sNA Pages

Number Theory Lecture Notes by Andrew Sutherland

This note in number theory explains standard topics in algebraic and analytic number theory. Topics covered includes: Absolute values and discrete valuations, Localization and Dedekind domains, ideal class groups, factorization of ideals, Etale algebras, norm and trace, Ideal norms and the Dedekind-Kummer thoerem, Galois extensions, Frobenius elements, Complete fields and valuation rings, Local fields and Hensel's lemmas , Extensions of complete DVRs, Totally ramified extensions and Krasner's lemma , Dirichlet's unit theorem, Riemann's zeta function and the prime number theorem, The functional equation , Dirichlet L-functions and primes in arithmetic progressions, The analytic class number formula, The Kronecker-Weber theorem, Class field theory, The main theorems of global class field theory, Tate cohomology, profinite groups, infinite Galois theory, Local class field theory, Global class field theory and the Chebotarev density theorem.

sNA Pages

Analytic Number Theory Lecture Notes by Andreas Strombergsson

This note covers the following topics: Primes in Arithmetic Progressions, Infinite products, Partial summation and Dirichlet series, Dirichlet characters, L(1, x) and class numbers, The distribution of the primes, The prime number theorem, The functional equation, The prime number theorem for Arithmetic Progressions, Siegels Theorem, The Polya-Vinogradov Inequality, Sums of three primes, The Large Sieve, Bombieris Theorem.

s295 Pages

Numerical Complex Analysis

This note covers the following topics: Fourier Analysis, Least Squares, Normwise Convergence, The Discrete Fourier Transform, The Fast Fourier Transform, Taylor Series, Contour integration, Laurent series, Chebyshev series, Signal smoothing and root finding, Differentiation and integration, Spectral methods, Ultraspherical spectral methods, Functional analysis, Spectrum, Infinite-dimensional linear algebra, Linear partial differential equations, Laplace's equation, RiemannHilbert problems, Matrix-valued RiemannHilbert.

sNA Pages

Introduction to Numerical Analysis by Doron Levy

This lecture note covers the following topics: Methods for Solving Nonlinear Problems, Interpolation, Approximations, Numerical Differentiation and Numerical Integration.

s127 Pages

Theory of Probability by Prof. Scott Sheffield

This note covers topics such as sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales.

sNA Pages

Probability Theory and Statistics Lecture notes

The aim of the notes is to combine the mathematical and theoretical underpinning of statistics and statistical data analysis with computational methodology and practical applications. Topics covered includes: Notion of probabilities, Probability Theory, Statistical models and inference, Mean and Variance, Sets, Combinatorics, Limits and infinite sums, Integration.

s294 Pages

A Basic Course in Real Analysis

This note explains the following topics: Rational Numbers and Rational Cuts, Irrational numbers, Dedekind's Theorem, Cantor's Theory of Irrational Numbers, Equivalence of Dedekind and Cantor's Theory, Finite, Infinite, Countable and Uncountable Sets of Real Numbers, Types of Sets with Examples, Metric Space, Various properties of open set, closure of a set.

sNA Pages

Real Analysis Notes by Prof. Sizwe Mabizela

This note explains the following topics: Logic and Methods of Proof, Sets and Functions , Real Numbers and their Properties, Limits and Continuity, Riemann Integration, Introduction to Metric Spaces.

s120 Pages

Lecture Notes Riemannian Geometry By Andreas Strombergsson

This note explains the following topics: Manifolds, Tangent spaces and the tangent bundle, Riemannian manifolds, Geodesics, The fundamental group. The theorem of Seifert-van Kampen, Vector bundles, The Yang-Mills functional, Curvature of Riemannian manifolds, Jacobi Fields, Conjugate points.

s241 Pages

An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity

This book covers the following topics: Differentiable Manifolds, Differential Forms, Riemannian Manifolds, Curvature, Geometric Mechanics, Relativity.

s272 Pages

Foundations of Module and Ring Theory

On the one hand this book intends to provide an introduction to module theory and the related part of ring theory. Topics covered includes: Elementary properties of rings, Module categories, Modules characterized by the Hom-functor, Notions derived from simple modules, Finiteness conditions in modules, Dual finiteness conditions, Pure sequences and derived notions, Relations between functors and Functor rings.

s616 Pages

Ring Theory by wikibook

This wikibook explains ring theory. Topics covered includes: Rings, Properties of rings, Integral domains and Fields, Subrings, Idempotent and Nilpotent elements, Characteristic of a ring, Ideals in a ring, Simple ring, Homomorphisms, Principal Ideal Domains, Euclidean domains, Polynomial rings, Unique Factorization domain, Extension fields.

sNA Pages

Set Theory Some Basics And A Glimpse Of Some Advanced Techniques

Goal of these notes is to introduce both some of the basic tools in the foundations of mathematics and gesture toward some interesting philosophical problems that arise out of them. Topics covered includes: Axioms and representations, Backbones and problems, advanced set theory.

s91 Pages

Lectures On Set Theory

This note covers the following topics: Logic, Elementary Set Theory, Generic Sets And Forcing, Infinite Combinatorics, Pcf, Continuum Cardinals.

s602 Pages

Stochastic Calculus Lecture Notes

This note covers the following topics: Discrete probability, Forward and Backward Equations for Markov chains, Martingales and stopping times, Continuous probability, Integrals involving Brownian motion, The Ito integral with respect to Brownian motion, Path space measures and change of measure.

sNA Pages

A Constructive Formalization of the Fundamental Theorem of Calculus (PDF 19P)

This note contains Basic Coq Notation, The Real Numbers, Sequences and Series, Continuous Functions, theorems on Differentiation , theorems on Integration, Transcendental Functions

s19 Pages

Topology Notes by Franz Rothe

This note explains the following topics: Metric spaces, Topological spaces, Limit Points, Accumulation Points, Continuity, Products, The Kuratowski Closure Operator, Dense Sets and Baire Spaces, The Cantor Set and the Devils Staircase, The relative topology, Connectedness, Pathwise connected spaces, The Hilbert curve, Compact spaces, Compact sets in metric spaces, The Bolzano-Weierstrass property.

s126 Pages

Notes on String Topology

String topology is the study of algebraic and differential topological properties of spaces of paths and loops in manifolds. Topics covered includes: Intersection theory in loop spaces, The cacti operad, String topology as field theory, A Morse theoretic viewpoint, Brane topology.

s95 Pages

Trigonometry Notes by Brooke Quinlan

This note covers the following topics: Angles and Their Measure, Right Triangle Trigonometry , Computing the Values of Trigonometric Functions of Acute Angles, Trigonometric Functions of General Angles, Graphs of the Sine and Cosine Functions, Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions, Phase Shifts, The Inverse Trigonometric Functions, Trigonometric Identities, Sum and Difference Formulas, Double-angle and Half-angle Formulas, Trigonometric Equation, Applications Involving Right Triangle, Area of a Triangle.

s125 Pages

Trigonometry Lecture Notes And Exercises by Daniel Raies

This note provides an introduction to trigonometry, an introduction to vectors, and the operations on functions. Topics covered includes: New functions from old functions, Trigonometry in circles and triangles, trigonometric functions, vectors.

s400 Pages

Notes on Vector Calculus

This note covers the following topics: Subsets of Euclidean space, vector fields, and continuity, Differentiation in higher dimensions, Tangent spaces, normals and extrema, Multiple Integrals, Line Integrals, Greens Theorem in the Plane, DIV, GRAD, and CURL, Change of Variables, Parametrizations, Surface Integrals, The Theorems of Stokes and Gauss.

s100 Pages

Lectures on Vector Calculus

This note explains the following topics: Vector Algebra and Index notation, Coordinate system, Integration, Integral Theorems, Permutations and Determinants.

s98 Pages