# UWM Undergraduate Catalog 2017-2018

# College of Letters and Science Home Page

## MATH Mathematical Sciences

*This course information produced at 01:00 on 11/21/2017.*

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**75 Math Success Strategies. 0 cr. U.**Strategies for improving attitudes about and performance in mathematics, including study skills for mathematics and mathematical mindsets.|Fee for 1 cr assessed; credit counts toward credit load for Fin Aid and enrollment verification only. Satisfactory/Unsatisfactory only. Prereq: none.

**90 Basic Mathematics. 0 cr. U.**Arithmetic operations involving whole numbers, integers, positive and negative rational numbers; decimals, percents; ratio, proportion; radicals; descriptive statistics; units of measure; geometry; introduction to algebra.|Fee for 3 cr assessed; credits count toward credit load for Financial Aid and enrollment verification only. Not open to students eligible for math courses that carry graduation credit. Prereq: none.

**92 Mathematical Literacy for College Students I. 0 cr. U.**Introduction to numeracy, proportional reasoning, algebraic reasoning, and functions. Emphasis on developing conceptual and procedural tools that support the use of key mathematical concepts in context.|Fee for 3 cr assessed; counts as 3 cr toward cr load for Fin Aid & enrollment verification. Prereq: Placement level E or D.

**94 Foundations of Elementary Mathematics. 0 cr. U.**Arithmetic, geometry, and beginning algebra; develops mathematical reasoning, problem solving, and facility with basic mathematical objects and their relationships. Individualized instruction via adaptive learning software.|Fee for 6 cr assessed; credits count toward credit load for Financial Aid and enrollment verification only. Math 94 counts as repeat of Math 90 & 95. Prereq: none.

**95 Essentials of Algebra. 0 cr. U.**Number systems; linear equations, inequalities; exponent notation, radicals; polynomials, operations, factoring, rational expressions; coordinate geometry; linear systems; quadratic equations.|Fee for 3 cr assessed; credits count toward credit load for Financial Aid and enrollment verification only. Prereq: math placement level D.

**98 Algebraic Literacy I. 0 cr. U.**Arithmetic number systems; linear equations, inequalities; exponent notation, radicals; polynomials, operations, factoring; modeling; coordinate geometry; linear systems; quadratic equations.|Fee for 3 cr assessed; counts as 3 cr toward credit load for Fin Aid & enrollment verification only. Prereq: math placement level D.

**102 Mathematical Literacy for College Students II. 3 cr. U.**Continuation of Math 92, with an integrated approach to numeracy, proportional reasoning, algebraic reasoning, and functions.|Prereq: Grade of C or better in Math 92(P) or (C).

**103 Contemporary Applications of Mathematics. 3 cr. U.**Logical inference, probability and statistical inference, geometric growth, with selected topics such as linear programming, patterns, binary codes.|Prereq: ACT math subscore of 18 or higher, or Math Placement Level C.

**105 Intermediate Algebra. 3 cr. U.**Algebraic techniques with polynomials, rational expressions, equations and inequalities, exponential and logarithmic functions, rational exponents, systems of linear equations.|Not open for cr to students who have cr in Math 108(ER). Prereq: math placement level C.

**108 Algebraic Literacy II. 3 cr. U.**Continuation of Math 98 in polynomials, equations, and inequalities; exponential, logarithmic, and periodic functions; rational expressions and exponents; and systems of linear equations.|Not open for credit for students who have cr in Math 105(ER). Prereq: C or better Math 98.

**115 Precalculus. 4 cr. U.**Essential topics from college algebra and trigonometry for students intending to enroll in calculus.|Repeats 2 cr of Math 116 & 2 cr of Math 117. Prereq: Math Placement Level AB.

**116 College Algebra. 3 cr. U.**Function concepts. Polynomial, rational, exponential, and logarithmic functions. Systems of equations and inequalities. Matrices and determinants. Sequences and series. Analytic geometry and conic sections. Induction.|2 cr may be used to repeat 2 cr of Math 115. Prereq: math placement level B.

**117 Trigonometry. 2 cr. U.**Trigonometric functions; graphs, identities, equations, inequalities; inverse trigonometric functions; solutions of triangles with applications; complex numbers; polar coordinates.|Repeats 2 cr of Math 115. Prereq: math placement level B.

**175 Mathematical Explorations for Elementary Teachers I. 3 cr. U.**Theory of arithmetic of whole numbers, fractions, and decimals. Introduction to algebra, estimation and problem-solving strategies.|Prereq: Math placement level A, AB, B, or C & regis in elementary, early childhood, or exceptional educ curric.

**176 Mathematical Explorations for Elementary Teachers II. 3 cr. U.**A continuation of Math 175 in geometry, statistics, and probability.|Prereq: grade of C or better in Math 175(P).

**194 First-Year Seminar: (Subtitled). 3 cr. U.**Specific topics are announced in the Schedule of Classes each time the class is offered.|Open only to freshmen. Students may earn cr in just one L&S First-Year Sem (course numbers 192, 193, 194). Prereq: none.

**199 Independent Study. 1-3 cr. U.**Regularly offered courses may not be taken as Independent Study.|May be retaken w/chg in topic to 9 cr max. Prereq: 2.5 gpa in all previous math courses; writ cons instr, dept chair & asst dean for SAS.

**205 Introductory Finite Mathematics. 3 cr. U.**Elements of mathematical logic, structures in sets; partitions and counting; probability theory, stochastic processes.|Prereq: math placement level B.

**211 Survey in Calculus and Analytic Geometry. 4 cr. U.**A one-semester survey with applications to business administration, economics, and non-physical sciences. Topics include coordinate systems, equations of curves, limits, differentiation, integration, applications.|May not be used as a prereq for Math 232. No cr for students with cr in Math 213, 221, or 231. Prereq: math placement level B.

**213 Calculus with Life Sciences Applications. 4 cr. U.**Limits, derivatives, graphing. Antiderivatives, the definite integral, and the fundamental theorem of calculus. Additional techniques and applications pertinent to the life sciences throughout the course. No cr for students with cr in Math 221 or 231. Prereq: math placement level A.

**221 Honors Calculus I. 5H cr. U.**Calculus of functions of one and several variables; sequences, series, differentiation, integration; introduction to differential equations; vectors and vector functions; applications.|Max of 6 cr in combination of Math 221 & 222 may count toward Honors College reqs. Prereq: math placement level A+; cons instr or Honors College dir.

**222 Honors Calculus II. 5H cr. U.**Continuation of Math 221.|Max of 6 cr in combination of Math 221 & 222 may count toward Honors College reqs. Prereq: C or better in Math 221(P) or B or better in Math 232(P); cons instr.

**231 Calculus and Analytic Geometry I. 4 cr. U.**Limits, derivatives, and graphs of algebraic, trigonometric, exponential, and logarithmic functions; antiderivatives, the definite integral, and the fundamental theorem of calculus, with applications.|No cr for students with cr in 213. Prereq: math placement level A.

**232 Calculus and Analytic Geometry II. 4 cr. U.**Continuation of Math 231. Applications of integration, techniques of integration; infinite sequences and series; parametric equations, conic sections, and polar coordinates.|Prereq: grade of C or better in Math 231(P) or grade of B or better in Math 213(P).

**233 Calculus and Analytic Geometry III. 4 cr. U.**Continuation of Math 232. Three-dimensional analytic geometry and vectors; partial derivatives; multiple integrals; vector calculus, with applications.|Counts as repeat of Math 229. Prereq: grade of C or better in Math 232(P).

**234 Linear Algebra and Differential Equations. 4 cr. U.**Elementary differential equations. Vectors; matrices; linear transformations; quadratic forms; eigenvalues; applications.|No cr for students with cr in Math 240. Prereq: grade of C or better in Math 232(P).

**240 Matrices and Applications. 3 cr. U.**Vectors, vector spaces, systems of linear equations, matrices, determinants, linear transformations, diagonalization, eigenvalues, eigenvectors; selected topics from quadratic forms, difference equations, numerical methods, and applications.|No cr for students with cr in Math 234. Prereq: Math placement level A or grade of C or better in a Math or MthStat course numbered 200 or higher.

**275 Problem Solving/Critical Thinking for Elementary Education Majors. 3 cr. U.**Course provides a strong foundation in the exploration, teaching and communication (oral and written) of mathematical concepts via problem-solving experiences and discussion.|Prereq: grade of C or better in Math 175(P) or cons instr.

**276 Algebraic Structures for Elementary Education Majors. 3 cr. U.**Topics for K-8 teachers. Basic patterns and rules that govern number systems, geometric transformations, and manipulation of algebraic expressions.|Counts as repeat of Math 299 w/same topic. Prereq: grade of C or better in Math 175(P) or cons instr.

**277 Geometry for Elementary Education Majors. 3 cr. U.**Topics for K-8 teachers. Geometry as measuring tool-congruence, similarity, area, volume, and coordinates; geometry as axiomatic system-definitions, conjectures, proofs, counterexamples; rigid motions, symmetry.|Prereq: grade of C or better in Math 176(P) or cons instr.

**278 Discrete Probability and Statistics for Elementary Education Majors. 3 cr. U.**Random experiments; histograms; sample spaces; equally likely outcomes for random experiments; permutations; combinations; binomial, geometric, hypergeometric distributions; expectation; conditional probabilities; max likelihood estimation & inference.|Math 278 and MthStat 278 are jointly offered; they count as repeats of one another. Prereq: grade of C or better in Math 176(P) or cons instr.

**290 Topics in Mathematics: (Subtitled). 3 cr. U.**Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered.|May be retaken w/chg in topic to 9 cr max. Prereq: satisfaction of Quantitative Literacy Part A GER.

**297 Study Abroad: (Subtitled). 1-12 cr. U.**Designed to enroll students in UWM sponsored programs before course work level, content and credits are determined and/or in specially prepared program course work.|May be retaken w/chg in topic. Prereq: acceptance for Study Abroad Prog.

**299 Ad Hoc: (Subtitled). 1-6 cr. U.**Course created expressly for offering in a specified enrollment period. Requires only dept & assoc dean approval. In exceptional circumstances, can be offered in one add'l sem.|May be retaken w/chg in topic. Prereq: none; add'l prereqs may be assigned to specific topic.

**305 Introduction to Mathematical and Computational Modeling. 3 cr. U/G.**Construction and analysis of discrete and continuous mathematical models in applied, natural, and social sciences. Elements of programming, simulations, case studies from scientific literature.|Counts as repeat of Math 690(675) w/topic Adv Math Models with Apps. Prereq: jr st; grade of C or better in Math 211(P) & one addl 200-level or higher Math or MthStat course, or grade of B or better in Math 213(P), or grade of C or better in Math 231(P), or cons instr.

**311 Theory of Interest. 4 cr. U.**Interest theory: annuities, amortization schedules, portfolio immunization. Financial derivatives: bonds, forwards, options. Topics consistent w/syllabus of SOA Financial Mathematics Exam.|Prereq: grade of C or better in Math 232(P).

**313 Linear Programming and Optimization. 3 cr. U/G.**Primal and dual formulations of linear programming problems; simplex and related methods of solution; algorithms for transportation; optimization.|Prereq: jr st; grade of C or better in Math/ElecEng 234(P) or Math 240(P); or grad st.

**315 (314) Mathematical Programming and Optimization. 3 cr. U/G.**Introduction to operations research. Network analysis; integer programming; game theory; nonlinear programming; dynamic programming.|Prereq: jr st, grade of C or better in either Math 234(P) or 240(P), & grade of C or better in either Math 211(P) or 233(P); or cons instr; or grad st.

**318 Topics in Discrete Mathematics. 3 cr. U.**Number theory topics related to cryptography; discrete structures including graphs, partial orders, Latin squares and block designs; advanced counting techniques.|Prereq: a grade of C or better in CompSci 317(P) or Math 341(P).

**320 Introduction to Differential Equations. 3 cr. U/G.**Elementary types and systems of differential equations, series solutions, numerical methods, Laplace transforms, selected applications.|No grad cr in Math Sci. Prereq: jr st, grade of C or better in both Math 232(P) & 240(P), or grade of C or better in Math/ElecEng 234(P); or grad st.

**321 Vector Analysis. 3 cr. U/G.**Topics selected from vector algebra; scalar and vector fields; line, surface, and volume integrals; theorems of Green, Gauss, and Stokes; vector differential calculus.|Prereq: jr st, grade of C or better in Math 233(P); or grad st.

**322 Introduction to Partial Differential Equations. 3 cr. U/G.**Partial differential equations of mathematical physics, boundary value problems in heat flow, vibrations, potentials, etc. Solved by Fourier series; Bessel functions and Legendre polynomials.|Prereq: jr st, Math 320(P), & grade of C or better in Math 233(P); or grad st.

**341 Seminar: Introduction to the Language and Practice of Mathematics. 3 cr. U.**Goal: facility with mathematical language and method of conjecture, proof and counter example, with emphasis on proofs. Topics: logic, sets, functions and others.|Prereq: math placement level A.

**345 Mathematics from a Historical Perspective: (Subtitled). 3 cr. U.**Topics from the development of mathematics, such as famous problems, mathematicians, calculating devices; chronological outlines. Significant reading and writing assignments.|May be retaken w/chg in topic to 9 cr max. Add'l prereqs announced in SOC each offering. Prereq: Grade of C or better in Math 232(P); satisfaction of OWC-A.

**371 Introduction to Stochastic Models in Finance. 3 cr. U/G.**Elementary modeling of financial instruments for students in mathematics, economics, business, etc. Statistical and stochastic tools leading to the Black-Scholes model. Real data parameter fitting.|Prereq: jr st & one of the following pairs; Econ 413(431)(P) & 506(P), Bus Adm 210(P) & 350(P), Bus Adm 701(P) & 702(P), or Math 234(P) & MthStat 361(P), or cons instr; or grad st.

**381 Honors Seminar: (Subtitled). 3H cr. U.**Significant topics to illustrate to non-mathematicians the characteristic features of mathematical thought. Only H.S. algebra and geometry assumed.|May be retaken w/chg in topic to 9 cr max. Not open for cr toward a major in Math. Prereq: soph st; Honors 200(P); cons Honors College dir.

**405 Mathematical Models and Applications. 3 cr. U/G.**Modeling techniques for analysis and decision-making in social and life sciences and industry. Deterministic and stochastic modeling. Topics may vary with instructors.|Prereq: jr st; one of: grade of C or better in Math 211(P), grade of B or better in Math 213(P), or grade of C or better in Math 231(P); & grade of C or better in either Math/ElecEng 234(P) or Math 240(P); or grad st.

**413 Introduction to Numerical Analysis. 3 cr. U/G.**Root finding and solution of nonlinear systems; direct solution of linear systems; interpolation & approximation of functions; least squares; fast Fourier transform; quadrature.|Prereq: jr st, grade of C or better in Math 233(C), & grade of C or better in Math/ElecEng 234(C); or grad st.

**415 Introduction to Scientific Computing. 3 cr. U/G.**Nonlinear systems; iterative solution of linear systems; initial value problems in ordinary differential equations; boundary value problems in ordinary and partial differential equations.|Prereq: jr st, grade of C or better in Math 233(C), & grade of C or better in Math/ElecEng 234(C); or grad st.

**417 (416) Computational Linear Algebra. 3 cr. U/G.**Direct solution of linear systems; iterative solution of linear systems; least squares; eigenvalue problems.|Prereq: jr st & grade of C or better in Math/ElecEng 234(P) or Math 240(P); or grad st.

**423 (623) Complex Analysis. 3 cr. U/G.**Complex numbers; definition and properties of analytic functions of a complex variable; conformal mapping; calculus of residues; applications to mathematics and physics. See also Math 713.|Prereq: jr st; grade of C or better in Math 233(P); or grad st.

**431 Modern Algebra with Applications. 3 cr. U/G.**Groups, rings, fields, Boolean algebras with emphasis on their applications to computer science and other areas.|Does not carry grad cr in math sci. Prereq: jr st & grade of C or better in Math 232(P); or grad st.

**451 Axiomatic Geometry. 3 cr. U/G.**An axiomatic approach to Euclidean and non-Euclidean geometry (historic role of the parallel postulate and models).|Dept cons req'd for grad cr in math sci. Prereq: jr st, grade of C or better in both Math 341(P) & Math 232(C); or grad st.

**453 Transformations in Geometry. 3 cr. U/G.**Selected topics from vector geometry and geometric transformations such as the study of invariants and conics.|Recom for secondary school teachers. Dept cons req'd for grad cr in math sci. Prereq: jr st, grade of C or better in both Math 341(P) & Math 232(C); or grad st.

**489 Internship in Mathematics, Upper Division. 1-6 cr. U.**Application of advanced principles of mathematics in a business, organizational, educational, governmental, or other appropriate setting.|One cr earned for academic work based on 40 hours in internship. May be retaken to 6 cr max. Prereq: jr st; 6 cr 300-level or above in math or math stats; 2.5 gpa in the major; cons supervising faculty member.

**490 Topics in Mathematics: (Subtitled). 3 cr. U/G.**Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered. May be retaken w/chg in topic to 9 cr max.|Prereq: jr st; grade of C or better in a Math or MthStat course at the 200 level or above; or grad st.

**497 Study Abroad: (Subtitled). 1-12 cr. U/G.**Designed to enroll students in UWM sponsored programs before course work level, content and credits are determined and/or in specially prepared program course work.|May be retaken w/chg in topic. Prereq: jr st; acceptance for Study Abroad Prog.

**499 Ad Hoc: (Subtitled). 1-6 cr. U.**Course created expressly for offering in a specified enrollment period. Requires only dept & assoc dean approval. In exceptional circumstances, can be offered in one add'l sem.|May be retaken w/chg in topic. Prereq: jr st; add'l prereqs may be assigned to specific topic.

**511 Symbolic Logic. 3 cr. U/G.**First-order predicate calculus; formal properties of theoretical systems; chief results of modern mathematical logic; advanced topics such as completeness and computability.|CompSci 511, Math 511 & Philos 511 are jointly offered; they count as repeats of one another. Prereq: jr st & either Philos 212(P) or 6 cr in math at the 300-level or above.

**521 Advanced Calculus I. 3 cr. U/G.**Fundamental notions of sets and functions; limits, continuity; Riemann integral, improper integral; infinite series; uniform convergence; power series; improper integrals with a parameter.|Prereq: jr st; grades of C or better in Math 232(P) & 341(P); or grad st.

**522 Advanced Calculus II. 3 cr. U/G.**Linear functions; differentiation of functions of several variables (implicit functions, Jacobians); change of variable in multiple integrals; integrals over curves, surfaces; Green, Gauss, Stokes theorems.|Prereq: jr st, Math 521(P), grade of C or better in Math 233(P), & grade of C or better in either Math 234(P) or 240(P); or grad st.

**531 Modern Algebra. 3 cr. U/G.**Integers; groups; rings; fields; emphasis on proofs.|Prereq: jr st; grade of C or better in Math 341(P).

**535 Linear Algebra. 3 cr. U/G.**Vector spaces; linear transformations and matrices; characteristic values and vectors; canonical forms; bilinear, quadratic, and Hermitian forms; selected applications.|Prereq: jr st, grade of C or better in either Math 234(P) or 240(P), & grade of C or better in Math 341(P); or grad st.

**537 Number Theory. 3 cr. U/G.**Number theoretic functions; distribution of primes; Diophantine approximation; partitions; additive number theory; quadratic reciprocity.|Prereq: jr st, grade of C or better in both Math 232(P) & 341(P); or grad st.

**551 Elementary Topology. 3 cr. U/G.**General theory of point sets in Euclidean spaces, with emphasis on topology of two-dimensional and three-dimensional spaces; elementary notions of metric spaces; applications.|Prereq: jr st; grades of C or better in Math 233(P) & 341(P); or grad st.

**553 Differential Geometry. 3 cr. U/G.**The theory of curves and surfaces by differential methods.|Prereq: jr st, grade of C or better in all of Math 233(P), 234(P) & 341(P); or grad st.

**571 Introduction to Probability Models. 3 cr. U/G.**Probability review, Markov chains in discrete and continuous time. Random walks, branching processes, birth and death processes. Queuing theory. Applications to physical sciences, engineering, mathematics.|Prereq: jr st; grade of C or better in Math 233(P); grade of C or better in Math/ElecEng 234(P) or in both Math 240(P) & 320(P); & one calculus-based course in statistics or probability at the 300 level or above; or grad st.

**575 High School Mathematics from an Advanced Viewpoint. 3 cr. U/G.**Number systems; algebra of polynomials; theory of equations; functions; modeling; geometric measurement; geometric transformations; connections between advanced mathematics and high school topics.|Counts as repeat of Math 690(675) w/similar topic. Prereq: jr st, either Math 451(P) or 453(P), & either Math 431(P) or 531(P); or cons instr; or grad st.

**581 Introduction to the Theory of Chaotic Dynamical Systems. 3 cr. U/G.**Iterated mappings, one parameter families, attracting and repelling periodic orbits, topological transitivity, Sarkovski's theorem, chaos, bifurcation theory, period doubling route to chaos, horseshoe maps, attractors.|Prereq: jr st & Math 521(P), 529(P) or 621(P), or cons instr; or grad st.

**591 Undergraduate Seminar: (Subtitled). 1 cr. U.**Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered.|May be retaken w/chg in topic to 4 cr max. Prereq: jr st; cons instr.

**599 Capstone Experience. 1 cr. U.**Student writes a paper under supervision of an advisor on an approved topic not covered in the student's regular course work.|May be retaken w/chg in topic to 2 cr max. Prereq: sr st; cons instr.

**601 Advanced Engineering Mathematics I. 3 cr. U/G.**Sequences and series, elementary complex analysis; Fourier series; linear and nonlinear ordinary differential equations; matrix theory, elementary functional analysis; elementary solution of partial differential equations.|Prereq: jr st; grade of C or better in both of Math 233(P) and Math/ElecEng 234(P); 3 cr Math at 300-level or above; or cons instr; or grad st.

**602 Advanced Engineering Mathematics II. 3 cr. U/G.**Continuation of Math 601. Partial differential equations, Fourier and Laplace transforms, convolutions, special functions, mathematical modeling.|Prereq: jr st; Math 601(P).

**615 Numerical Solution of Partial Differential Equations. 3 cr. U/G.**Finite difference solution of elliptic boundary value problems and of evolution problems; solution of hyperbolic conservation laws; finite volume methods; finite element methods.|Prereq: jr st; Math 413(P), 415(414)(P), or 417(416)(P); Math 322(P) or 602(P); or cons instr.

**617 Optimization. 3 cr. U/G.**Unconstrained and constrained optimization: linear, nonlinear, and dynamic programming; barrier, penalty, and Lagrangian methods; Karush-Kuhn-Tucker theory, quadratic, and sequential quadratic programming; evolutionary algorithms.|Prereq: jr st; Math 321(P) or 602(P); or grad st or cons instr.

**621 Introduction to Analysis I. 3 cr. U/G.**Topology of Euclidean space; continuity; differentiation of real and vector-valued functions; Riemann-Stieltjes integration.|Prereq: jr st: grades of C or better in Math 233(P), 341(P), & either 234(P) or 240(P); cons dept advisor; or grad st.

**622 Introduction to Analysis II. 3 cr. U/G.**Continues Math 621. Sequences and series of functions; uniform convergence; power series; functions of several variables; inverse and implicit function theorems; differential forms; Stokes' theorem.|Prereq: jr st; Math 621(P) or cons instr; or grad st.

**631 Modern Algebra I. 3 cr. U/G.**Group theory, including normal subgroups, quotients, permutation groups, Sylow's theorems, Abelian groups; field theory; linear algebra over general fields.|Prereq: jr st; grade of C or better in Math 341(P) & either Math 234(P) or 240(P); cons dept advisor; or grad st.

**632 Modern Algebra II. 3 cr. U/G.**Continuation of Math 631. Ring theory, including ideals, quotient rings, Euclidean rings, polynomial rings, unique factorization; modules, including vector spaces, linear transformations, canonical forms; bilinear forms.|Prereq: jr st; Math 631(P) or cons instr; or grad st.

**690 (675) Topics in Mathematics: (Subtitled). 3 cr. U/G.**Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered.|May be retaken w/chg in topic to 9 cr max. Prereq: jr st, at least one U/G Math or MthStat course; cons instr; or grad st.

**699 Independent Reading. 1-3 cr. U.**See Advanced Independent Study. For further information, consult dept chair.|May be retaken w/chg in topic. Prereq: jr st; 2.5 gpa, & writ cons instr, dept chair, & asst dean for SAS.

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