# Elements of Mathematics Functions of a Real Variable: Elementary Theory The theory of generalized Taylor expansions and the Euler-MacLaurin formula are presented in the sixth chapter, and applied in the last one to the study of the Gamma function on the real line as well as on the complex plane. Although the topics of the book are mainly of an advanced undergraduate level, they are presented in the generality needed for more advanced purposes: functions allowed to take values in topological vector spaces, asymptotic expansions are treated on a filtered set equipped with a comparison scale, theorems on the dependence on parameters of differential equations are directly applicable to the study of flows of vector fields on differential manifolds, etc.

The Elements of Mathematics series is the result of this project.

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I Derivatives. First Derivative. Derivative of a vector function. Linearity of differentiation. Derivative of a product. Derivative of the inverse of a function. Derivative of a composite function. Derivative of an inverse function. Derivatives of real-valued functions.

The Mean Value Theorem.

## CounterExamples: From Elementary Calculus to the Beginnings of Analysis

Rolle's Theorem. The mean value theorem for real-valued functions. The mean value theorem for vector functions. Continuity of derivatives. Derivatives of Higher Order. Derivatives of order n. Taylor's formula. Convex Functions of a Real Variable. Definition of a convex function. Families of convex functions. Continuity and differentiability of convex functions. Criteria for convexity. Primitives and Integrals. Definition of primitives. Existence of primitives. Regulated functions. Properties of integrals. Integral formula for the remainder in Taylor's formula; primitives of higher order.

Integrals Over Non-Compact Intervals. Definition of an integral over a non-compact interval. Integrals of positive functions over a non-compact interval. Absolutely convergent integrals. Derivatives and Integrals of Functions Depending on a Parameter. Integral of a limit of functions on a compact interval. Integral of a limit of functions on a non-compact interval. Normally convergent integrals. Derivative with respect to a parameter of an integral over a compact interval. Derivative with respect to a parameter of an integral over a non-compact interval. Change of order of integration. Derivatives of the Exponential and Circular Functions.

Derivatives of the exponential functions; the number e. Derivative of logax. Derivatives of the circular functions; the number?. Inverse circular functions. The complex exponential. Properties of the function ez. The complex logarithm. Primitives of rational functions. Complex circular functions; hyperbolic functions. Expansion of the real exponential.

Expansions of the complex exponential, of cos x and sin x. The binomial expansion. Existence Theorems. The concept of a differential equation. Differential equations admitting solutions that are primitives of regulated functions. Existence of approximate solutions. Comparison of approximate solutions. Existence and uniqueness of solutions of Lipschitz and locally Lipschitz equations.

Continuity of integrals as functions of a parameter. Dependence on initial conditions. Linear Differential Equations. Existence of integrals of a linear differential equation. Linearity of the integrals of a linear differential equation. Integrating the inhomogeneous linear equation. Fundamental systems of integrals of a linear system of scalar differential equations. Adjoint equation. Infinitely divisible laws. Some ergodic theory. Mathematics and Statistics of Gambling. Probability and statistics are founded on the study of games of chance.

Nowadays, gambling in casinos, sports and the Internet is a huge business. This course addresses practical and theoretical aspects. Topics covered: mathematics of basic random phenomena physics of coin tossing and roulette, analysis of various methods of shuffling cards , odds in popular games, card counting, optimal tournament play, practical problems of random number generation. Prerequisites: Statistics and Topics in Probability: Percolation Theory. An introduction to first passage percolation and related general tools and models. Topics include early results on shape theorems and fluctuations, more modern development using hyper-contractivity, recent breakthrough regarding scaling exponents, and providing exposure to some fundamental long-standing open problems.

Course prerequisite: graduate-level probability. A topics course in combinatorics and related areas. The topic will be announced by the instructor. Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Offered every years.

This advanced course in extremal combinatorics covers several major themes in the area. These include extremal combinatorics and Ramsey theory, the graph regularity method, and algebraic methods. This is a graduate-level course on the use and analysis of Markov chains. Emphasis is placed on explicit rates of convergence for chains used in applications to physics, biology, and statistics. A variety of card shuffling processes will be studies. Central Limit and concentration. Classical functional inequalities Nash, Faber-Krahn, log-Sobolev inequalities , comparison of Dirichlet forms. Random walks and isoperimetry of amenable groups with a focus on solvable groups.

Entropy, harmonic functions, and Poisson boundary following Kaimanovich-Vershik theory.

Introduction to Stochastic Differential Equations. Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: or equivalent and differential equations. Introduction to mathematical models of complex static and dynamic stochastic systems that undergo sudden regime change in response to small changes in parameters. Examples from materials science phase transitions , power grid models, financial and banking systems.

Special emphasis on mean field models and their large deviations, including computational issues. Dynamic network models of financial systems and their stability. Topics in Financial Math: Market microstructure and trading algorithms. Introduction to market microstructure theory, including optimal limit order and market trading models. Random matrix theory covariance models and their application to portfolio theory. Statistical arbitrage algorithms. Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk.

Term structure models and interest rate derivatives. Optimal stopping and American options. Computation and Simulation in Finance. Monte Carlo, finite difference, tree, and transform methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Functions of Several Complex Variables. Holomorphic functions in several variables, Hartogs phenomenon, d-bar complex, Cousin problem. Domains of holomorphy. Plurisubharmonic functions and pseudo-convexity.

Stein manifolds.

Algebra 16 - Real-Valued Functions of a Real Variable

Levi problem and its solution. Riemann surfaces and holomorphic maps, algebraic curves, maps to projective spaces. Calculus on Riemann surfaces. Elliptic functions and integrals. Riemann-Hurwitz formula. Riemann-Roch theorem, Abel-Jacobi map. Uniformization theorem. Hyperbolic surfaces. Suitable for advanced undergraduates. Topics of contemporary interest in algebraic geometry.

Topics in number theory: L-functions. The Riemann Zeta function and Dirichlet L-functions, zero-free regions and vertical distribution of the zeros, primes in arithmetic progressions, the class number problem, Hecke L-functions and Tate's thesis, Artin L-functions and the Chebotarev density theorem, Modular forms and Maass forms.

Topics may include 1 subadditive and multiplicative ergodic theorems, 2 notions of mixing, weak mixing, spectral theory, 3 metric and topological entropy of dynamical systems, 4 measures of maximal entropy. Topics of contemporary interest in number theory. The theory of linear and nonlinear partial differential equations, beginning with linear theory involving use of Fourier transform and Sobolev spaces. Topics: Schauder and L2 estimates for elliptic and parabolic equations; De Giorgi-Nash-Moser theory for elliptic equations; nonlinear equations such as the minimal surface equation, geometric flow problems, and nonlinear hyperbolic equations.

Symplectic Geometry and Topology. Linear symplectic geometry and linear Hamiltonian systems. Symplectic manifolds and their Lagrangian submanifolds, local properties. Symplectic geometry and mechanics. Contact geometry and contact manifolds. Relations between symplectic and contact manifolds.

## Éléments de mathématique - Wikipedia

Hamiltonian systems with symmetries. Momentum map and its properties. Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Applications include Fourier imaging the theory of diffraction, computed tomography, and magnetic resonance imaging and the theory of compressive sensing. Algebraic Combinatorics and Symmetric Functions.

Symmetric function theory unifies large parts of combinatorics. Theorems about permutations, partitions, and graphs now follow in a unified way. Topics: The usual bases monomial, elementary, complete, and power sums. Schur functions. Representation theory of the symmetric group. Littlewood-Richardson rule, quasi-symmetric functions, combinatorial Hopf algebras, introduction to Macdonald polynomials.

Throughout, emphasis is placed on applications e. Prerequisite: A and B, or equivalent. Crystal Bases: Representations and Combinatorics. Crystal Bases are combinatorial analogs of representation theorynof Lie groups. We will explore different aspects of thesenanalogies and develop rigorous purely combinatorial foundations. Geometry and Topology of Complex Manifolds. Complex manifolds, Kahler manifolds, curvature, Hodge theory, Lefschetz theorem, Kahler-Einstein equation, Hermitian-Einstein equations, deformation of complex structures.

The language of jets. Thom transversality theorem. Holonomic approximation theorem. Applications: immersion theory and its generaliazations.

Differential relations and Gromov's h-principle for open manifolds. Applications to symplectic geometry. Mappings with simple singularities and their applications. Method of convex integration. Topics in Partial Differential Equations. Covers a list of topics in mathematical physics. The specific topics may vary from year to year, depending on the instructor's discretion. Background in graduate level probability theory and analysis is desirable. The purpose of this course is to show beautiful surprises and instructive paradoxes in a maximal diversity of fluid phenomena, and to understand them with minimal models.

Some deep currents will develop across multiple lectures. Evolution Equations in Differential Geometry. The theory of surfaces and 3-manifolds. Curves on surfaces, the classification of diffeomorphisms of surfaces, and Teichmuller space. The mapping class group and the braid group. Knot theory, including knot invariants.

Decomposition of 3-manifolds: triangulations, Heegaard splittings, Dehn surgery. Loop theorem, sphere theorem, incompressible surfaces. Geometric structures, particularly hyperbolic structures on surfaces and 3-manifolds. May be repeated for credit up to 6 total units. Homotopy groups, fibrations, spectral sequences, simplicial methods, Dold-Thom theorem, models for loop spaces, homotopy limits and colimits, stable homotopy theory.

Possible topics: principal bundles, vector bundles, classifying spaces. Connections on bundles, curvature. Topology of gauge groups and gauge equivalence classes of connections.

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• Characteristic classes and K-theory, including Bott periodicity, algebraic K-theory, and indices of elliptic operators. Spectral sequences of Atiyah-Hirzebruch, Serre, and Adams. Cobordism theory, Pontryagin-Thom theorem, calculation of unoriented and complex cobordism. Incompressible surfaces, irreducible manifolds, prime decomposition, Morse theory, Heegaard diagrams, Heegaard splittings, the Thurston norm, sutured manifold theory, Heegaard Floer homology, sutured Floer homology. Only for mathematics graduate students.

Advanced Topics in Convex Optimization. Modern developments in convex optimization: semidefinite programming; novel and efficient first-order algorithms for smooth and nonsmooth convex optimization. Emphasis on numerical methods suitable for large scale problems arising in science and engineering. Applied mathematics through toys and magic. This course is a series of case-studies in doing applied mathematics on surprising phenomena we notice in daily life. Almost every class will show demos of these phenomena toys and magic and suggest open projects. In each class I will try to make the discussion self-contained and to give everybody something to take home, regardless of the background.

Research seminar for graduate students working in logic and formal philosophy. Presentations on contemporary topics by seminar participants and outside visitors. Maybe be repeated for credit. This course provides unified coverage of linear algebra and multivariable differential calculus. It discusses applications connecting the material to many quantitative fields. Linear algebra in large dimensions underlies the scientific, data-driven, and computational tasks of the 21st century. The linear algebra portion of the course includes orthogonality, linear independence, matrix algebra, and eigenvalues as well as ubiquitious applications: least squares, linear regression, Markov chains relevant to population dynamics, molecular chemistry, and PageRank , singular value decomposition essential in image compression, topic modeling, and data-intensive work in the natural sciences , and more.

The multivariable calculus material includes unconstrained optimization via gradients and Hessians used for energy minimization in physics and chemistry , constrained optimization via Lagrange multipliers, crucial in economics , gradient descent and the multivariable Chain Rule which underlie many machine learning algorithms, such as backpropagation , and Newton's method a crucial part of how GPS works.

The course emphasizes computations alongside an intuitive understanding of key ideas, making students well-prepared for further study of mathematics and its applications to other fields. The widespread use of computers makes it more important, not less, for users of math to understand concepts: in all scientific fields, novel users of quantitative tools in the future will be those who understand ideas and how they fit with applications and examples. Prerequisite: 21, 42, or the math placement diagnostic offered through the Math Department website in order to register for this course.

Integral Calculus of Several Variables. Iterated integrals, line and surface integrals, vector analysis with applications to vector potentials and conservative vector fields, physical interpretations. Divergence theorem and the theorems of Green, Gauss, and Stokes. Prerequisite: 51 or equivalents. Ordinary Differential Equations with Linear Algebra. Ordinary differential equations and initial value problems, systems of linear differential equations with constant coefficients, applications of second-order equations to oscillations, matrix exponentials, Laplace transforms, stability of non-linear systems and phase plane analysis, numerical methods.

Modern Mathematics: Continuous Methods.

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This is the first part of a theoretical i. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space as local graphs, integration on Euclidean space, and many examples. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject.

Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor. Modern Mathematics: Discrete Methods. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. Students should have an interest in a theoretical approach to the subject.

A proof-based introduction to manifolds and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space with many examples , differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. This is the second part of a proof-based sequence in discrete mathematics.

This course covers topics in elementary number theory, group theory, and discrete Fourier analysis. For example, we'll discuss the basic examples of abelian groups arising from congruences in elementary number theory, as well as the non-abelian symmetric group of permutations. Prerequisites: 61DM or 61CM. A proof-based course on ordinary differential equations and other applications of derivatives. Topics include the inverse and implicit function theorems, implicitly-defined submanifolds of Euclidean space, linear systems of differential equations and necessary tools from linear algebra, stability and asymptotic properties of solutions to linear systems, existence and uniqueness theorems for nonlinear differential equations, behavior of solutions near an equilibrium point, and Sturm-Liouville theory.

Third part of a proof-based sequence in discrete mathematics. The first half of the quarter gives a fast-paced coverage of probability and random processes with an intensive use of generating functions. Strategy and mathematical theories of the game of Go, with guest appearance by a professional Go player. Proof Positive: Principles of Mathematics. What is a mathematical proof, and where do proofs come from?

Students will become comfortable with fundamental techniques of mathematical proof through practice with interesting and accessible examples from many areas of math. Students will additionally hone their communication skills and develop their ability to formulate and answer precise mathematical questions. Topics include direct proof, proof by contrapositive, proof by contradiction, many applications of mathematical induction, constructing good definitions, and useful writing habits. To be considered for enrollment, please email masonr stanford.

Capillary Surfaces: Explored and Unexplored Territory. Preference to sophomores. Capillary surfaces: the interfaces between fluids that are adjacent to each other and do not mix. Recently discovered phenomena, predicted mathematically and subsequently confirmed by experiments, some done in space shuttles.

Interested students may participate in ongoing investigations with affinity between mathematics and physics. How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why.

Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the seminar develops develops some of the main results in that material systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. Mathematics of Knots, Braids, Links, and Tangles. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants.

The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics. Stanford , California Bulletin ExploreDegrees Search bulletin Search. Math Discovery Lab. Applied Matrix Theory. Functions of a Complex Variable. Graph Theory. Applied Group Theory.

Linear Algebra and Matrix Theory. Functions of a Real Variable. Complex Analysis. Mathematics of Computation. Groups and Rings. Galois Theory. Partial Differential Equations. Stochastic Processes. Celestial Mechanics. Differential Geometry. Algebraic Geometry. Analysis on Manifolds. Differential Topology. Algebraic Topology. Elementary Theory of Numbers. Algebraic Number Theory. Analytic Number Theory.

Discrete Probabilistic Methods. Set Theory. Fundamental Concepts of Analysis. Elementary Functional Analysis. Polya Problem Solving Seminar. Senior Honors Thesis. Practical Training. Reading Topics. Real Analysis. Continuation of B. Modern Algebra I. Modern Algebra II. Lie Theory. Complex Differential Geometry. Calculus, ACE. Mathematical Methods of Imaging.

Theory of Probability I. Theory of Probability II. Theory of Probability III. Topics in Combinatorics. Elements of Mathematics Functions of a Real Variable: Elementary Theory Elements of Mathematics Functions of a Real Variable: Elementary Theory Elements of Mathematics Functions of a Real Variable: Elementary Theory Elements of Mathematics Functions of a Real Variable: Elementary Theory Elements of Mathematics Functions of a Real Variable: Elementary Theory Elements of Mathematics Functions of a Real Variable: Elementary Theory Elements of Mathematics Functions of a Real Variable: Elementary Theory

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