## Mathematics

###### CI 161. Content Area Methods and Materials in Secondary Teaching

Prerequisites: CI 152 AND CI 159 or concurrent enrollment; admission to the Single Subject Credential Program or teaching experience. Planning, delivering, and assessing content-specific instruction; academic and common core standards; identifying specific standards that require literacy strategies. (Instructional materials fee for Single Subject - Art Methods and Materials enrollees, $10)

Units: 3, Repeatable up to 999 units

Course Typically Offered: Fall, Spring

###### EHD 154B. Final Student Teaching Seminar - Mathematics

Prerequisites: Concurrent enrollment in EHD 155B. Seminar to accompany final student teaching that provides opportunities for candidates to investigate and discuss variety of topics and strategies and to reflect on issues that surface during their student teaching experience.

Units: 1

###### EHD 155B. Student Teaching in Secondary School - Math

Prerequisites: admission to student teaching, EHD 155A, CI 161 (or concurrently, depending on major departmental policy); senior or post baccalaureate standing; approval of major department including subject matter competency approval; completion of the subject matter preparation program or passing the subject matter examination(s) designated by the California Commission on Teacher Credentialing. Supervised teaching in single subject classroom; assignment is for the full day; five days per week. CR/NC grading only.

Units: 5-10, Repeatable up to 20 units

Course Typically Offered: Fall, Spring

###### MATH 2L. Quantitative Reasoning Support

Co-requisite: MATH 3 or MATH 10A or MATH 11 or MATH 45 or MATH 45H for Mathematics placement category III or IV. Support class for mathematics and quantitative reasoning courses through active learning and use of mathematical software. Review of topics in arithmetic, algebra, and geometry; develop and strengthen computational skills, mathematical concepts, problem solving, critical thinking, and study skills.

Units: 1

Course Typically Offered: Fall, Spring

###### MATH 3. College Algebra

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take MATH 3L. Equations and inequalities; rectangular coordinates; systems of equations and inequalities; polynomial, rational, exponential, and logarithmic functions and their graphs; complex numbers.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 3L. College Algebra w/ Support

Equations and inequalities; rectangular coordinates; systems of equations and inequalities; polynomial, rational, exponential, and logarithmic functions and their graphs; complex numbers.(3 lecture, 3 lab hours)

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 5. Trigonometry

Prerequisite: Mathematics placement category I or II. Can be taken concurrently with Math 3 for category standing III or IV. Concept of a function, sine and cosine functions, tables and graphs, other trigonometric functions, identities and equations. Trigonometric functions of angles, solution of triangles. (See Duplication of Courses).

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 6. Precalculus

Prerequisite: Mathematics placement category I or II. Basic algebraic properties of real numbers; linear and quadratic equations and inequalities; functions and graphs; polynomials; exponential and logarithmic functions; analytic trigonometry and functions. G. E. Foundation B4. (3 lecture, 2 activity hours)

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 10A. Structure and Concepts in Mathematics I

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take Math 10AL. Designed for prospective elementary school teachers. Development of real numbers including integers, rational and irrational numbers, computation, prime numbers and factorizations, and problem-solving strategies. Meets B4 G. E. requirement only for liberal studies majors.

Units: 3

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 10AL. Structure and Concepts in Mathematics I w/Support

Designed for prospective elementary school teachers. Development of real numbers including integers, rational and irrational numbers, computation, prime numbers and factorizations, and problem-solving strategies. Meets B4 G. E. requirement only for liberal studies majors. (3 lecture, 3 lab hours)

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 10B. Structure and Concepts in Mathematics II

Prerequisite: MATH 10A. Designed for prospective elementary school teachers. Counting methods, elementary probability and statistics. Topics in geometry to include polygons, congruence and similarity, measurement, geometric transformations, coordinate geometry, and connections between numbers and geometry with selected applications.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 11. Elementary Statistics

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take Math 11L. Illustration of statistical concepts: elementary probability models, sampling, descriptive measures, confidence intervals, testing hypotheses, chi-square, nonparametric methods, regression. It is recommended that students with credit in MATH 75 or MATH 75A and B take MATH 101.

Units: 3

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 11L. Elementary Statistics w/Support

Illustration of statistical concepts: elementary probability models, sampling, descriptive measures, confidence intervals, testing hypotheses, chi-square, nonparametric methods, regression. It is recommended that students with credit in MATH 75 or MATH 75A and B take MATH 101. (3 lecture, 3 lab hours)

Units: 4

GE Area: B4

###### MATH 45. What Is Mathematics?

Prerequisite: Mathematics placement category I or II. Students in Mathematics placement category III or IV must take Math 45L. Covers topics from the following areas: (I) The Mathematics of Social Choice; (II) Management Science and Optimization; (III) The Mathematics of Growth and Symmetry; and (IV) Statistics and Probability. G. E. Foundation B4.

Units: 3

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 45L. What Is Mathematics w/Support

Covers topics from the following areas: (I) The Mathematics of Social Choice; (II) Management Science and Optimization; (III) The Mathematics of Growth and Symmetry; and (IV) Statistics and Probability. G. E. Foundation B4. (3 lecture, 3 lab hours)

Units: 4

GE Area: B4

###### MATH 70. Calculus for Life Sciences

No credit if taken after MATH 75 or MATH 75A and B. Prerequisite: Mathematics placement category I or II and calculus placement according to department standards. Functions and graphs, limits, derivatives, antiderivatives, differential equations, and partial derivatives with applications in Life Sciences.

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 75. Calculus I

Prerequisites: Mathematics placement category I or II, and calculus placement according to department standards. Functions, graphs, limits, continuity, derivatives and applications, definite and indefinite integrals. (3 lecture, 2 activity hours) G.E. Foundation B4.

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 75A. Calculus with Review IA

Prerequisites: Mathematics placement category I or II, and calculus placement according to department standards. Functions, graphs, limits, continuity, derivatives, and applications, with extensive review of algebra and elementary functions. With MATH 75B, equivalent to MATH 75. G.E. Foundation B4.

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 75B. Calculus with Review IB

Prerequisite: MATH 75A. Further applications of derivatives, and definite and indefinite integrals, with extensive review of algebra and elementary functions. With MATH 75A, equivalent to MATH 75.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 76. Calculus II

Prerequisite: MATH 75 or MATH 75A and B. Techniques and applications of integration, improper integrals, conic sections, polar coordinates, infinite series. (3 lecture, 2 activity hours)

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 77. Calculus III

Prerequisite: MATH 76. Vectors, three-dimensional calculus, partial derivatives, multiple integrals, Green's Theorem, Stokes' Theorem.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 81. Elementary Differential Equations with Linear Algebra

Prerequisite: MATH 77. Introduction to ordinary linear differential equations and linear systems of differential equations; solutions by Laplace transforms. Solution of linear systems of equations; introduction to vector spaces; eigenvalues and eigenvectors. Using computer software as an exploratory tool.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 90. Directed Study

Independently arranged course of study in some limited area of mathematics either to remove a deficiency or to investigate a topic in more depth. (1-3 hours, to be arranged)

Units: 1-3

###### MATH 100. Exploring Mathematics

Prerequisite: MATH 10B. The development of mathematical reasoning, problem solving, and communication skills for effective teaching of mathematics in elementary school.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 101. Statistical Methods

Prerequisite: MATH 70 or MATH 75, or MATH 75A and B; no credit if taken after MATH 108. Application of statistical procedures to examples from biology, engineering, and social science; one- and two-sample normal theory methods; chi-square, analysis of variance, and regression; nonparametric methods. Computerized statistical packages are used.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 105. Statistical Programming and Data Analysis

Prerequisite: MATH 11 or MATH 101 or permission of instructor. Introduction to SAS and R through programming and data analysis. Topics include data access, data structure, data management and manipulation, simulations, arrays, matrices, graphics, custom functions, and standard statistical techniques in SAS/R.

Units: 3

Course Typically Offered: Fall

###### MATH 106. Applied Linear Statistical Models

Prerequisites: MATH 75; MATH 11 or MATH 101 or permission of instructor. Topics include simple linear regression, parameter inference, interval estimation, prediction, diagnostics and remedial measures, multiple linear regression, model selection and validation, generalized linear models, ridge regression, LASSO.

Units: 3

Course Typically Offered: Spring

###### MATH 107. Mathematical Statistics

Prerequisite: MATH 77 (may be taken concurrently). Probability theory; discrete and continuous distributions; random variables; conditional distributions; multivariate distributions; limit theorems; maximum likelihood methods.

Units: 3

Course Typically Offered: Fall

###### MATH 108. Advanced Mathematical Statistics

Prerequisite: MATH 107. Statistical inferences; sufficiency; optimal hypothesis tests; inferences from normal theory, nonparametric statistics; elementary decision theory; Bayesian statistics.

Units: 3

Course Typically Offered: Spring

###### MATH 109. Applied Probability

Prerequisite: MATH 107. Introduction to stochastic processes and their applications in science and industry. Markov chains, queues, stationary time series.

Units: 3

Course Typically Offered: Spring

###### MATH 110. Symbolic Logic

(Similar to PHIL 145; consult department.) Prerequisite: MATH 75 or MATH 75A and B. An informal treatment of the theory of logical inference, statement calculus, truth-tables, predicate calculus, interpretations applications.

Units: 3

Course Typically Offered: Spring

###### MATH 111. Transition to Advanced Mathematics

Prerequisite: MATH 76. Introduction to the language and problems of mathematics. Use of LaTeX as a typesetting tool. Topics include set theory, symbolic logic, types of proofs, and mathematical induction. Emphasis on students constructing, explaining, and justifying mathematical arguments through active learning.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 114. Discrete Structures

Prerequisite: MATH 111. Counting techniques, matrix algebra, graphs, trees and networks, recurrence relations and generating functions, applied modern algebra.

Units: 3

Course Typically Offered: Fall

###### MATH 116. Theory of Numbers

Prerequisite: MATH 111. Divisibility theory in the integers, primes and their distribution, congruence theory, Diophantine equations, number theoretic functions, primitive roots, indices, the quadratic reciprocity law.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 118. Graph Theory

Prerequisite: MATH 111. Trees, connectivity, Euler and Hamilton paths, matchings, chromatic problems, planar graphs, independence, directed graphs, networks.

Units: 3

Course Typically Offered: Spring - even

###### MATH 121. Numerical Analysis I

Prerequisites: MATH 77 and either CSCI 40 or ECE 71. Computer arithmetic, solutions of equations using iterative techniques, interpolation, numerical differentiation, quadrature, and numerical ordinary differential equations. Use of numerical software libraries.

Units: 3

Course Typically Offered: Spring

###### MATH 122. Numerical Analysis II

Prerequisites: MATH 121, MATH 152. Systems of linear equations, Gaussian elimination with pivoting, matrix inversion, determinant of a matrix, SVD, LU and Cholesky factorization of a matrix, iterative techniques, orthogonal matrix, QR factorization, Gram-Schmidt and Householder methods, approximating eigenvalues, systems of nonlinear equations, steepest descent techniques, Newton's method, and rational approximation.

Units: 3

Course Typically Offered: Fall

###### MATH 123. Mathematical Modeling

Prerequisite: MATH 77. Application of mathematical techniques to solve selected problems in areas such as ecology, biology, economics, finance, social sciences, life sciences, physical sciences and engineering. The emphasis will be on the building of mathematical models and on interpreting the solutions of these models in terms of real-life applications.

Units: 3

Course Typically Offered: Spring - odd

###### MATH 128. Applied Complex Analysis

Prerequisite: MATH 77. Analytic functions of a complex variable, contour integration, series, singularities of analytic functions, the residue theorems, conformal mappings; emphasis on engineering and physics applications.

Units: 3

Course Typically Offered: Fall

###### MATH 133. Number Theory for Liberal Studies

Prerequisite: MATH 10B or permission of instructor. The historical development of the concept of number and arithmetic algorithms. The magnitude of numbers. Basic number theory. Special numbers and sequences. Number patterns. Modular arithmetic.

Units: 3

Course Typically Offered: Fall

###### MATH 134. Geometry for Liberal Studies

Prerequisite: MATH 10B or permission of instructor. The use of computer technology to study and explore concepts in Euclidean geometry. Topics include, but are not restricted to, properties of polygons, tilings, and polyhedra.

Units: 3

Course Typically Offered: Spring

###### MATH 137. Exploring Statistics

Prerequisite: MATH 10B or permission of instructor. Descriptive and inferential statistics with a focus on applications to mathematics education. Use of technology and activities for student discovery and understanding of data organization, collection, analysis and inference.

Units: 3

Course Typically Offered: Fall

###### MATH 138. Exploring Algebra

Prerequisite: MATH 10B or permission of instructor. Designed for prospective school teachers who wish to develop a deeper conceptual understanding of algebraic themes and ideas needed to become competent and effective mathematics teachers.

Units: 3

Course Typically Offered: Spring

###### MATH 139. Advanced Algebra for Middle School Teachers

Prerequisite: MATH 6 or MATH 138. Basic structures of modern algebra from a middle school mathematics curriculum perspective. Algebraic structures, polynomial equations, and elementary linear algebra.

Units: 4

Course Typically Offered: Fall

###### MATH 143. History of Mathematics

Prerequisite: MATH 75 or MATH 75A and 75B. History of the development of mathematical concepts in algebra, geometry, number theory, analytical geometry, and calculus from ancient times through modern times. Theorems with historical significance will be studied as they relate to the development of modern mathematics.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 145. Problem Solving

Prerequisite: MATH 111; EHD 50 (may be enrolled concurrently). A study of formulation of problems into mathematical form; analysis of methods of attack such as specialization, generalization, analogy, induction, recursion, etc. applied to a variety of non-routine problems. Topics will be handled through student presentation.

Units: 3

Course Typically Offered: Fall

###### MATH 149S. Capstone Mathematics for Teachers

Prerequisites: MATH 151; MATH 161; MATH 171 (MATH 161 and 171 may be taken concurrently). Secondary school mathematics from an advanced viewpoint. This course builds on students' work in upper division mathematics to deepen their understanding of the mathematics taught in secondary schools. In collaboration with local in-service teachers and university faculty, students will actively explore topics in number theory, algebra, analysis, geometry, and apply their content knowledge in a service-learning context.

Units: 4

Course Typically Offered: Spring

###### MATH 151. Principles of Algebra

Prerequisite: MATH 111. Equivalence relations; groups, cyclic groups, normal sub-groups, and factor groups; rings, ideals, and factor rings; integral domains and polynomial rings; fields and field extensions.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 152. Linear Algebra

Prerequisite: MATH 77. Vector spaces, linear transformations, matrices, determinants, eigenvalues and eigenvectors, linear functions, inner-product spaces, bilinear forms, quadratic forms, orthogonal and unitary transformations, selected applications.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 161. Principles of Geometry

Prerequisite: MATH 111. The classical elliptic, parabolic, and hyperbolic geometries developed on a framework of incidence, order and separation, congruence; coordinatization. Theory of parallels for parabolic and hyperbolic geometries. Selected topics of modern Euclidean geometry.

Units: 3

Course Typically Offered: Spring

###### MATH 165. Differential Geometry

Prerequisite: MATH 77 and MATH 111. Study of geometry in Euclidean space by means of calculus, including theory of curves and surfaces, curvature, theory of surfaces, and intrinsic geometry on a surface.

Units: 3

Course Typically Offered: Fall

###### MATH 171. Intermediate Mathematical Analysis I

Prerequisite: MATH 111. Natural and rational numbers, real numbers as a complete ordered field, its usual topology, sequences and series of real numbers, functions of a real variable, limits, continuity, uniform continuity, differentiability, generalized mean value theorem, Riemann integrals, and power series.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 172. Intermediate Mathematical Analysis II

Prerequisite: MATH 77 and MATH 171. Pointwise and uniform convergence of sequences and series of functions, convergence of sequences in higher dimensions, continuity and differentiability of functions of several variables. The inverse and implicit function theorems; topics in integration theory in higher dimensions.

Units: 4

Course Typically Offered: Spring

###### MATH 181. Differential Equations

Prerequisite: MATH 81. Definition and classification of differential equations; general, particular, and singular solutions; existence theorems; theory and technique of solving certain differential equations: phase plane analysis, elementary stability theory; applications.

Units: 3

Course Typically Offered: Fall

###### MATH 182. Partial Differential Equations

Prerequisites: MATH 81 or MATH 123. Classical methods for solving partial differential equations including separation of variables, Green's functions, the Riemann-Volterra method and Cauchy's problem for elliptic, parabolic, and hyperbolic equations; applications to theoretical physics.

Units: 3

Course Typically Offered: Spring

###### MATH 190. Independent Study

See Academic Placement. Approved for RP grading.

Units: 1-3, Repeatable up to 6 units

Course Typically Offered: Fall, Spring

###### MATH 191T. Proseminar

Prerequisites: Permission of instructor. Presentation of advanced topics in mathemathics in the field of the student's interest.

Units: 1-3, Repeatable up to 9 units

###### MATH 191T. Mathematical Software and Programming

Prerequisite: Math 76 and CSCI 40 or ECE 71. Recommended to be taken concurrently with Math 77 or Math 111. Introduction to programming usage in mathematics and statistics including computer algebra software, interpreted object-oriented high-level programming language, one programming language related to statistics, database management, optimization, and./or cloud computing. (Offered Fall 2020)

Units: 3

###### MATH 192. Undergraduate Mathematics Seminar

Prerequisite: MATH 76 or consent of the instructor. Presentations on various topics in mathematics. The course is intended for STEM students with a strong interest in mathematics. It is an upper division elective course.

Units: 1, Repeatable up to 3 units

Course Typically Offered: Fall

###### MATH 198. Senior Project

Prerequisites: Senior standing or permission of instructor; MATH 151, MATH 171, and MATH 152. Independent investigation and presentation of an advanced topic in mathematics. Satisfies the senior major requirement for the B.A. in Mathematics.

Units: 3

###### MATH 200. Research Methods in Mathematics Education

This course covers quantitative and qualitative methods of researching how people think and learn about mathematics, and how research informs the teaching of mathematics. Content includes research design, use of statistical analyses, and critical examination of research in mathematics education.

Units: 3

###### MATH 201. Cognition in Mathematics

This course explores theories and empirical studies which examine the development of students' and teachers' knowledge and practices in mathematics. A central theme of the course is the examination of research on the use of technology in the teaching of mathematics.

Units: 3

###### MATH 202. Fundamental Concepts of Mathematics

Prerequisites: MATH 151, MATH 161 and MATH 171. Fundamental notions regarding number theory, number systems, algebra of number fields; functions.

Units: 3

###### MATH 216T. Topics in Number Theory

Prerequisite: MATH 116. An investigation of topics having either historical or current research interest in the field of number theory. (Formerly MATH 216)

Units: 3, Repeatable up to 6 units

###### MATH 220. Coding Theory

Prerequisites: MATH 151 and MATH 152. Basic concepts in coding theory, properties of linear and on-linear codes, standard decoding algorithms, cyclic codes, BCH-codes.

Units: 3

###### MATH 223. Applied Operator Theory

Prerequisite: graduate standing or permission of instructor. Fundamentals of abstract spaces and spectral theory of operators with applications. Resolvent set and spectrum of a linear operator. Bounded and unbounded linear operators. Compact operators and the Fredholm alternative. Symmetry and self-adjointness.

Units: 3

###### MATH 228. Functions of a Complex Variable

Prerequisite: MATH 128. Representation theorems of Weierstrass and Mittag-Leffler, normal families, conformal mapping and Riemann mapping theorem, analytic continuation, Dirichlet problem.

Units: 3

###### MATH 232. Mathematical Models with Technology

Prerequisite: graduate standing in mathematics or permission of instructor. A technology-assisted study of the mathematics used to model phenomena in statistics, natural science, and engineering.

Units: 3

###### MATH 250. Perspectives in Algebra

Prerequisite: graduate standing in mathematics or permission of instructor. Study of advanced topics in algebra, providing a higher perspective to concepts in the high school curriculum. Topics selected from, but not limited to, groups, rings, fields, and vector spaces.

Units: 3

###### MATH 251. Abstract Algebra I

Prerequisite: MATH 151 or permission of instructor. Semi-direct products of groups; isomorphism theorems. Group actions; Sylow theorems; classification of groups of small order; finitely generated Abelian groups. Rings and ideals; quotient rings; domains (ED, PID, UFD); polynomial rings.

Units: 3

###### MATH 252. Abstract Algebra II

Prerequisite: MATH 251. Field extensions; automorphisms of fields; Galois theory. Additional topics to be chosen from (1) modules, (2) linear and multilinear algebra and (3) representation theory.

Units: 3

###### MATH 260. Perspectives in Geometry

Prerequisite: MATH 151 and MATH 152 or permission of instructor. Geometry from a transformations point of view. Projective geometry: theorems of Ceva, Menelaus, Desargues, and Pappus; conics; coordinatization. Transformations of the plane (Euclidean and projective); tessellations; wallpaper groups. Further topics to be selected from Incidence Geometry, Differential Geometry, or Algebraic Geometry.

Units: 3

###### MATH 263. Point Set Topology

Prerequisite: MATH 172. Basic concepts of point set topology, set theory, topological spaces, continuous functions; connectivity, compactness and separation properties of spaces. Topics selected from function spaces, metrization, dimension theory.

Units: 3

###### MATH 270. Perspectives in Analysis

Prerequisite: graduate standing in mathematics or permission of instructor. An overview of the development of mathematical analysis, both real and complex. Emphasizes interrelation of the various areas of study , the use of technology, and relevance to the high school mathematics curriculum.

Units: 3

###### MATH 271. Real Analysis

Prerequisite: MATH 172. Lebesgue's measure and integration theory on the real line. Limit theorems and types of convergence. Lp spaces. Differentiation and integration.

Units: 3

###### MATH 272. Functional Analysis

Prerequisite: MATH 271 or permission of instructor. Elements of the theory of abstract spaces. The three fundamental principles of linear functional analysis (Hahn-Banach Theorem, Uniform Boundedness Principle, and Open Mapping Theorem) and their implications. Duality and reflexivity of normed vector spaces, geometry of Hilbert spaces. (Formerly MATH 291T)

Units: 3

###### MATH 290. Independent Study

See Academic Placement. Approved for RP grading.

Units: 1-3, Repeatable up to 6 units

###### MATH 291T. Seminar

Prerequisite: graduate standing. Presentation of current mathematical research in field of student's interest.

Units: 1-3, Repeatable up to 6 units

###### MATH 298. Research Project in Mathematics

Prerequisite: graduate standing. Independent investigation of advanced character as the culminating requirement for the master's degree. Approved for RP grading.

Units: 3

###### MATH 298C. Project Continuation

Pre-requisite: Project MATH 298. For continuous enrollment while completing the project. May enroll twice with department approval. Additional enrollments must be approved by the Dean of Graduate Studies.

Units: 0

###### MATH 299. Thesis in Mathematics

Prerequisite: See Criteria for Thesis and Project. Preparation, completion, and submission of an acceptable thesis for the master's degree. Approved for RP grading.

Units: 3