Calculus of Functions of One Variable I. Limits, derivatives, and integrals of functions of one variable. Honors version available Requisites: Calculus of the elementary transcendental functions, techniques of integration, indeterminate forms, Taylor's formula, infinite series. Calculus of Functions of Several Variables.
Vector algebra, solid analytic geometry, partial derivatives, multiple integrals. Limits, derivatives, and integrals of functions of one variable, motivated by and applied to discrete-time dynamical systems used to model various biological processes. Techniques of integration, indeterminate forms, Taylor's series; introduction to linear algebra, motivated by and applied to ordinary differential equations; systems of ordinary differential equations used to model various biological processes. Permission of the instructor.
Elective topics in mathematics. This course has variable content and may be taken multiple times for credit. Undergraduate Seminar in Mathematics. A seminar on a chosen topic in mathematics in which the students participate more actively than in usual courses. May be repeated for credit. Directed Exploration in Mathematics. By permission of the director of undergraduate studies. Experimentation or deeper investigation under the supervision of a faculty member of topics in mathematics that may be, but need not be, connected with an existing course.
No one may receive more than seven semester hours of credit for this course. May be repeated for credit; may be repeated in the same term for different topics; 7 total credits. Revisiting Real Numbers and Algebra. Central to teaching precollege mathematics is the need for an in-depth understanding of real numbers and algebra. This course explores this content, emphasizing problem solving and mathematical reasoning.
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This course serves as a transition from computational to more theoretical mathematics. Topics are from the foundations of mathematics: First Course in Differential Equations. Introductory ordinary differential equations, first- and second-order differential equations with applications, higher-order linear equations, systems of first-order linear equations introducing linear algebra as needed. First Course in Differential Equations Laboratory.
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Course is computational laboratory component designed to help students visualize ODE solutions in Matlab. Emphasis is on differential equations motivated by applied sciences. Some applied linear algebra will appear as needed for computation and modeling purposes. Undergraduate Reading and Research in Mathematics. Permission of the director of undergraduate studies. This course is intended mainly for students working on honors projects.
No one may receive more than three semester hours credit for this course. Mathematical Methods in Biostatistics. Special mathematical techniques in the theory and methods of biostatistics as related to the life sciences and public health. Includes brief review of calculus, selected topics from intermediate calculus, and introductory matrix theory for applications in biostatistics. Teaching and Learning Mathematics. Study of how people learn and understand mathematics, based on research in mathematics, mathematics education, psychology, and cognitive science.
This course is designed to prepare undergraduate mathematics majors to become excellent high school mathematics teachers. It involves field work in both the high school and college environments. An investigation of various ways elementary concepts in mathematics can be developed. Applications of the mathematics developed will be considered. An examination of high school mathematics from an advanced perspective, including number systems and the behavior of functions and equations. Designed primarily for prospective or practicing high school teachers.
A general survey of the history of mathematics with emphasis on elementary mathematics. Some special problems will be treated in depth. The real numbers, continuity and differentiability of functions of one variable, infinite series, integration. Functions of several variables, the derivative as a linear transformation, inverse and implicit function theorems, multiple integration.
Geometry of Principal Sheaves (Hardcover, 2005 ed.)
Functions of a Complex Variable with Applications. The algebra of complex numbers, elementary functions and their mapping properties, complex limits, power series, analytic functions, contour integrals, Cauchy's theorem and formulae, Laurent series and residue calculus, elementary conformal mapping and boundary value problems, Poisson integral formula for the disk and the half plane. Linear differential equations, power series solutions, Laplace transforms, numerical methods.
Mathematical Methods for the Physical Sciences I. Theory and applications of Laplace transform, Fourier series and transform, Sturm-Liouville problems. Students will be expected to do some numerical calculations on either a programmable calculator or a computer. This course has an optional computer laboratory component: Students will need a CCI-compatible computing device. Introduction to boundary value problems for the diffusion, Laplace and wave partial differential equations.
Bessel functions and Legendre functions. Introduction to complex variables including the calculus of residues.
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Elementary Theory of Numbers. Divisibility, Euclidean algorithm, congruences, residue classes, Euler's function, primitive roots, Chinese remainder theorem, quadratic residues, number-theoretic functions, Farey and continued fractions, Gaussian integers. Elements of Modern Algebra. Binary operations, groups, subgroups, cosets, quotient groups, rings, polynomials.
Introduction to mathematical theory of probability covering random variables; moments; binomial, Poisson, normal and related distributions; generating functions; sums and sequences of random variables; and statistical applications. Linear Algebra for Applications. Algebra of matrices with applications: Counting selections, binomial identities, inclusion-exclusion, recurrences, Catalan numbers.
Selected topics from algorithmic and structural combinatorics, or from applications to physics and cryptography. Introduction to topics in topology, particularly surface topology, including classification of compact surfaces, Euler characteristic, orientability, vector fields on surfaces, tessellations, and fundamental group. Euclidean and Non-Euclidean Geometries.
Critical study of basic notions and models of Euclidean and non-Euclidean geometries: Mathematical and Computational Models in Biology. This course introduces analytical, computational, and statistical techniques, such as discrete models, numerical integration of ordinary differential equations, and likelihood functions, to explore various fields of biology.
Mathematical and Computational Models in Biology Laboratory. This lab introduces analytical, computational, and statistical techniques, such as discrete models, numerical integration of ordinary differential equations, and likelihood functions, to explore various fields of biology. Topics will vary and may include iteration of maps, orbits, periodic points, attractors, symbolic dynamics, bifurcations, fractal sets, chaotic systems, systems arising from differential equations, iterated function systems, and applications.
Mathematical Modeling in the Life Sciences. Requires some knowledge of computer programming. Model validation and numerical simulations using ordinary, partial, stochastic, and delay differential equations. Springer Shop Bolero Ozon.
MATHEMATICS (MATH) < University of North Carolina at Chapel Hill
Geometry of Principal Sheaves. The book provides a detailed introduction to the theory of connections on principal sheaves in the framework of Abstract Differential Geometry ADG. This point of view complies with the demand of contemporary physics to cope with non-smooth models of physical phenomena and spaces with singularities. Starting with a brief survey of the required sheaf theory and cohomology, the exposition then moves on to differential triads the abstraction of smooth manifolds and Lie sheaves of groups the abstraction of Lie groups.
Based on sheaf-theoretic methods and sheaf - homology, the presentGeometry of Principal Sheaves embodies the classical theory of connections on principal and vector bundles, and connections on vector sheaves, thus paving the way towards a uni? We elaborate on the aforementioned brief description in the sequel.
However, the theory of smooth manifolds is inadequate to cope, for - stance, with spaces like orbifolds, spaces with corners, or other spaces with more complicated singularities. This is a rather unfortunate situation, since one cannot apply the powerful methods of di? The ix x Preface same inadequacy manifests in physics, where many geometrical models of physical phenomena are non-smooth. Review This Product No reviews yet - be the first to create one!