Conditions of Use:
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It is important to learn math because it makes us better people.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Author:
- Ninglu Wang
- Jessamine Bartley-Matthews
- Date Added:
- 04/09/2016
Mathematics Textbooks and Full Courses
269
affiliated resources
Conditions of Use:
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This course is a continuation of Abstract Algebra I: the student will revisit structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms. The student will also take a look at ring factorization, general lattices, and vector spaces. Later this course presents more advanced topics, such as Galois theory - one of the most important theories in algebra, but one that requires a thorough understanding of much of the content we will study beforehand. Upon successful completion of this course, students will be able to: Compute the sizes of finite groups when certain properties are known about those groups; Identify and manipulate solvable and nilpotent groups; Determine whether a polynomial ring is divisible or not and divide the polynomial (if it is divisible); Determine the basis of a vector space, change bases, and manipulate linear transformations; Define and use the Fundamental Theorem of Invertible Matrices; Use Galois theory to find general solutions of a polynomial over a field. (Mathematics 232)
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- The Saylor Foundation
- Date Added:
- 11/11/2011
Conditions of Use:
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" The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc."
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Rosales, Rodolfo R.
- Date Added:
- 01/01/2009
Conditions of Use:
Read the Fine Print
This course is oriented toward US high school students. The course is divided into 10 units of study. The first five units build the foundation of concepts, vocabulary, knowledge, and skills for success in the remainder of the course. In the final five units, we will take the plunge into the domain of inferential statistics, where we make statistical decisions based on the data that we have collected.
- Subject:
- Statistics and Probability
- Material Type:
- Full Course
- Provider:
- The Saylor Foundation
- Date Added:
- 08/28/2013
Conditions of Use:
Read the Fine Print
This course discusses how to use algebra for a variety of everyday tasks, such as calculate change without specifying how much money is to be spent on a purchase, analyzing relationships by graphing, and describing real-world situations in business, accounting, and science.
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- The Saylor Foundation
- Date Added:
- 08/28/2013
Conditions of Use:
No Strings Attached
In this course students gain proficiency in Linear Equations, Linear Inequalities, Graphing linear equations, Solving Systems of Equations, Simplifying with Polynomials, Division of Polynomials, Factoring Polynomials, Developing a Factoring Strategy, and Solving Other Algebraic Equations.
- Subject:
- Career and Technical Education
- Education
- Mathematics
- Material Type:
- Full Course
- Author:
- Boyoung Chae
- SBCTC Admin
- Date Added:
- 02/01/2018
Conditions of Use:
No Strings Attached
The College and Career Readiness Standards for Level E (High School) outline the outcomes for this course.In this course students gain proficiency in Functions, Linear Functions, Solving Quadratics, Quadratic Functions, Exponential Functions, and Logarithmic Functions.
- Subject:
- Career and Technical Education
- Education
- Mathematics
- Material Type:
- Full Course
- Author:
- Boyoung Chae
- SBCTC Admin
- Date Added:
- 02/01/2018
Conditions of Use:
Read the Fine Print
This course is oriented toward US high school students. Its structure and materials are aligned to the US Common Core Standards. Foci include: graphing, equations and inequalities.
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- The Saylor Foundation
- Date Added:
- 08/28/2013
Conditions of Use:
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This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Artin, Michael
- Date Added:
- 01/01/2011
Conditions of Use:
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" This is an introductory course in algebraic combinatorics. No prior knowledge of combinatorics is expected, but assumes a familiarity with linear algebra and finite groups. Topics were chosen to show the beauty and power of techniques in algebraic combinatorics. Rigorous mathematical proofs are expected."
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Musiker, Gregg
- Date Added:
- 01/01/2009
Conditions of Use:
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" This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry."
- Subject:
- Algebra
- Geometry
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Kedlaya, Kiran
- Date Added:
- 01/01/2009
Conditions of Use:
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In this second term of Algebraic Topology, the topics covered include fibrations, homotopy groups, the Hurewicz theorem, vector bundles, characteristic classes, cobordism, and possible further topics at the discretion of the instructor.
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Behrens, Mark
- Date Added:
- 01/01/2006
Conditions of Use:
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Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Wehrheim, Katrin
- Date Added:
- 01/01/2010
Conditions of Use:
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Continues 18.100, in the direction of manifolds and global analysis. Differentiable maps, inverse and implicit function theorems, n-dimensional Riemann integral, change of variables in multiple integrals, manifolds, differential forms, n-dimensional version of Stokes' theorem. 18.901 helpful but not required.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Guillemin, Victor
- Date Added:
- 01/01/2005
Conditions of Use:
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Applied Calculus instructs students in the differential and integral calculus of elementary functions with an emphasis on applications to business, social and life science. Different from a traditional calculus course for engineering, science and math majors, this course does not use trigonometry, nor does it focus on mathematical proofs as an instructional method.
- Subject:
- Calculus
- Material Type:
- Full Course
- Textbook
- Provider:
- Lumen Learning
- Provider Set:
- Candela Courseware
- Author:
- Dale Hoffman
- David Lippman
- Shana Calaway
- Date Added:
- 03/31/2016
Conditions of Use:
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Laszlo Tisza was Professor of Physics Emeritus at MIT, where he began teaching in 1941. This online publication is a reproduction the original lecture notes for the course "Applied Geometric Algebra" taught by Professor Tisza in the Spring of 1976. Over the last 100 years, the mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group theory and many others. These sophisticated tools provide powerful machinery for describing the physical world, however, their physical interpretation is often not intuitive. These course notes represent Prof. Tisza's attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics. Prof. Tisza revisits many elementary problems with an advanced treatment in order to help develop the geometrical intuition for the algebraic machinery that may carry over to more advanced problems. The lecture notes came to MIT OpenCourseWare by way of Samuel Gasster, '77 (Course 18), who had taken the course and kept a copy of the lecture notes for his own reference. He dedicated dozens of hours of his own time to convert the typewritten notes into LaTeX files and then publication-ready PDFs. You can read about his motivation for wanting to see these notes published in his Preface below. Professor Tisza kindly gave his permission to make these notes available on MIT OpenCourseWare.
- Subject:
- Algebra
- Physics
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Tisza, L
- Date Added:
- 01/01/2009
Conditions of Use:
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Focuses on modeling, quantification, and analysis of uncertainty by teaching random variables, simple random processes and their probability distributions, Markov processes, limit theorems, elements of statistical inference, and decision making under uncertainty. This course extends the discrete probability learned in the discrete math class. It focuses on actual applications, and places little emphasis on proofs. A problem set based on identifying tumors using MRI (Magnetic Resonance Imaging) is done using Matlab.
- Subject:
- Statistics and Probability
- Material Type:
- Full Course
- Homework/Assignment
- Lecture
- Lecture Notes
- Reading
- Provider:
- ArsDigita University
- Provider Set:
- ArsDigita University
- Author:
- Rajeev Surati
- Tina Kapur
- Date Added:
- 02/16/2011
I designed the course for graduate students who use statistics in their research, plan to use statistics, or need to interpret statistical analyses performed by others. The primary audience are graduate students in the environmental sciences, but the course should benefit just about anyone who is in graduate school in the natural sciences. The course is not designed for those who want a simple overview of statistics; well learn by analyzing real data. This course or equivalent is required for UMB Biology and EEOS Ph.D. students. It is a recommended course for several of the intercampus graduate school of marine science program options.
- Subject:
- Statistics and Probability
- Material Type:
- Full Course
- Provider:
- UMass Boston
- Provider Set:
- UMass Boston OpenCourseWare
- Author:
- Eugene Gallagher
- Date Added:
- 10/14/2015
Conditions of Use:
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This course is an arithmetic course intended for college students, covering whole numbers, fractions, decimals, percents, ratios and proportions, geometry, measurement, statistics, and integers using an integrated geometry and statistics approach. The course uses the late integers modelintegers are only introduced at the end of the course.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Textbook
- Provider:
- Lumen Learning
- Provider Set:
- Candela Courseware
- Author:
- David Lippman
- Date Added:
- 03/31/2016
Conditions of Use:
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The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Stanley, Richard
- Date Added:
- 01/01/2003
Conditions of Use:
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The Art of the Probable" addresses the history of scientific ideas, in particular the emergence and development of mathematical probability. But it is neither meant to be a history of the exact sciences per se nor an annex to, say, the Course 6 curriculum in probability and statistics. Rather, our objective is to focus on the formal, thematic, and rhetorical features that imaginative literature shares with texts in the history of probability. These shared issues include (but are not limited to): the attempt to quantify or otherwise explain the presence of chance, risk, and contingency in everyday life; the deduction of causes for phenomena that are knowable only in their effects; and, above all, the question of what it means to think and act rationally in an uncertain world. Our course therefore aims to broaden students’ appreciation for and understanding of how literature interacts with--both reflecting upon and contributing to--the scientific understanding of the world. We are just as centrally committed to encouraging students to regard imaginative literature as a unique contribution to knowledge in its own right, and to see literary works of art as objects that demand and richly repay close critical analysis. It is our hope that the course will serve students well if they elect to pursue further work in Literature or other discipline in SHASS, and also enrich or complement their understanding of probability and statistics in other scientific and engineering subjects they elect to take.
- Subject:
- Arts and Humanities
- Literature
- Philosophy
- Religious Studies
- Statistics and Probability
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Jackson, Noel
- Kibel, Alvin
- Raman, Shankar
- Date Added:
- 01/01/2008
This course is a review of basic mathematics skills. Here's what's covered:
-fundamental numeral operations of addition, subtraction, multiplication
division of whole numbers, fractions, and decimals
-ratio and proportion
-percent
-systems of measurement
-an introduction to geometry
NOTE: Open Campus courses are non-credit reviews and tutorials and cannot be used to satisfy requirements in any curriculum at BPCC. (Basic Mathematics Course by Bossier Parish Community College is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Based on a work at http://bpcc.edu/opencampus/index.html.)
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- Bossier Parish Community College
- Author:
- Michelle Barnickel
- Date Added:
- 10/11/2017
Conditions of Use:
No Strings Attached
Topics include signed numbers, decimal numbers, exponential notation, scientific notation, solving and graphing linear equations, an introduction to polynomials, and systems of linear equations and their graphs. Geometrical topics include lines and angles, closed curves and convex polygons, triangles and similarities, and symmetry and proportion in nature and art. Students may complete this course during the first three weeks of the semester by passing the MyMathLab modules. Students will then be eligible to take either MAT 099 Intermediate Algebra, MAT 114-Quantitative Reasoning or MAT 120-Intro to Statistics the following semester. This course does not satisfy degree requirements. Students may complete this course during the first three weeks of the semester by passing the MyOpenMath Acceleration assignments.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- Roxbury Community College
- Author:
- Valerie Dietel-Brenneman
- Date Added:
- 05/15/2019
BPCC Open Campus - Math 097: Basic Mathematics is a review of basic mathematics skills. Here's what's covered: -fundamental numeral operations of addition, subtraction, multiplication division of whole numbers, fractions, and decimals -ratio and proportion -percent -systems of measurement -an introduction to geometry NOTE: Open Campus courses are non-credit reviews and tutorials and cannot be used to satisfy requirements in any curriculum at BPCC.
- Subject:
- Education
- Mathematics
- Material Type:
- Assessment
- Full Course
- Lecture
- Lesson
- Unit of Study
- Author:
- Allison Martin
- Date Added:
- 10/11/2017
Conditions of Use:
No Strings Attached
This course is a continuation of MAT087, Basic Mathematics. Topics include signed numbers, decimal numbers, exponential notation, scientific notation, solving and graphing linear equations, an introduction to polynomials, and systems of linear equations and their graphs. Geometrical topics include lines and angles, closed curves and convex polygons, triangles and similarities, and symmetry and proportion in nature and art. Students may complete this course during the first three weeks of the semester by passing the MyMathLab modules. Students will then be eligible to take either MAT 099 Intermediate Algebra, MAT 114-Quantitative Reasoning or MAT 120-Intro to Statistics the following semester. This course does not satisfy degree requirements.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- Roxbury Community College
- Author:
- John McColgan
- Date Added:
- 05/15/2019
Conditions of Use:
No Strings Attached
This course is a continuation of MAT087, Basic Mathematics. Topics include signed numbers, decimal numbers, exponential notation, scientific notation, solving and graphing linear equations, an introduction to polynomials, and systems of linear equations and their graphs. Geometrical topics include lines and angles, closed curves and convex polygons, triangles and similarities, and symmetry and proportion in nature and art. Students may complete this course during the first three weeks of the semester by passing the MyMathLab modules. Students will then be eligible to take either MAT 099 Intermediate Algebra, MAT 114-Quantitative Reasoning or MAT 120-Intro to Statistics the following semester. This course does not satisfy degree requirements.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- Northern Essex Community College
- Author:
- Jim Sullivan
- Date Added:
- 05/15/2019
Conditions of Use:
Read the Fine Print
This course is also intended to provide the student with a strong foundation for intermediate algebra and beyond. Upon successful completion of this course, you will be able to: simplify and solve linear equations and expressions including problems with absolute values and applications; solve linear inequalities; find equations of lines; and solve application problems; add, subtract, multiply, and divide various types of polynomials; factor polynomials, and simplify square roots; evaluate, simplify, multiply, divide, add, and subtract rational expressions, and solve basic applications of rational expressions. This free course may be completed online at any time. It has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. (Mathematics 001)
- Subject:
- Algebra
- Material Type:
- Full Course
- Provider:
- The Saylor Foundation
- Date Added:
- 04/16/2012
In this beginning algebra course, you'll learn about
fundamental operations on real numbers
exponents
solving linear equations and inequalities
applications
functions
graphing linear equations
slope
systems of linear equations
- Subject:
- Algebra
- Material Type:
- Full Course
- Lecture
- Lesson
- Module
- Provider:
- Bossier Parish Community College
- Author:
- Gail Hendrix
- Date Added:
- 10/16/2017
Conditions of Use:
Read the Fine Print
This course covers a range of algebraic topics: Setting up and solving linear equations, graphing, finding linear relations, solving systems of equations, working with polynomials, factoring, working with rational and radical expressions, solving rational and radical equations, solving quadratic equations, and working with functions. More importantly, this course is intended to provide you with a solid foundation for the rest of your math courses. As such, emphasis will be placed on mathematical reasoning, not just memorizing procedures and formulas. There is enough content in this course to cover both beginning and intermediate college-level algebra.
- Subject:
- Algebra
- Material Type:
- Full Course
- Textbook
- Provider:
- Lumen Learning
- Provider Set:
- Candela Courseware
- Author:
- Tyler Wallace
- Date Added:
- 03/31/2016
Conditions of Use:
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Biomedical research today is not only rigorous, innovative and insightful, it also has to be organized and reproducible. With more capacity to create and store data, there is the challenge of making data discoverable, understandable, and reusable. Many funding agencies and journal publishers are requiring publication of relevant data to promote open science and reproducibility of research.
In order to meet to these requirements and evolving trends, researchers and information professionals will need the data management and curation knowledge and skills to support the access, reuse and preservation of data.
This course is designed to address present and future data management needs.
- Subject:
- Health, Medicine and Nursing
- Information Science
- Measurement and Data
- Material Type:
- Full Course
- Provider:
- Harvard University
- Author:
- Elaine Martin
- Julie Goldman
- Date Added:
- 01/05/2018
Conditions of Use:
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This course covers sensing and measurement for quantitative molecular/cell/tissue analysis, in terms of genetic, biochemical, and biophysical properties. Methods include light and fluorescence microscopies; electro-mechanical probes such as atomic force microscopy, laser and magnetic traps, and MEMS devices; and the application of statistics, probability and noise analysis to experimental data.
- Subject:
- Electronic Technology
- Biology
- Statistics and Probability
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- So, Peter
- Date Added:
- 01/01/2006
Conditions of Use:
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This course provides a broad understanding of the application of biostatistics in a regulatory context. Reviews the relevant regulations and guidance documents. Includes topics such as basic study design, target population, comparison groups, and endpoints. Addresses analysis issues with emphasis on the regulatory aspects, including issues of missing data and informative censoring. Discusses safety monitoring, interim analysis and early termination of trials with a focus on regulatory implications.
- Subject:
- Health, Medicine and Nursing
- Statistics and Probability
- Material Type:
- Full Course
- Lecture Notes
- Reading
- Syllabus
- Provider:
- Johns Hopkins Bloomberg School of Public Health
- Provider Set:
- JHSPH OpenCourseWare
- Author:
- Mary Foulkes
- Simon Day
- Date Added:
- 09/15/2008
Conditions of Use:
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" BLOSSOMS stands for Blended Learning Science or Math Studies. It is a project sponsored by MIT LINC (Learning International Networks Consortium) a consortium of educators from around the world who are interested in using distance and e-Learning technologies to help their respective countries increase access to quality education for a larger percentage of the population.BLOSSOMS Online"
- Subject:
- Mathematics
- Material Type:
- Full Course
- Provider:
- MIT
- Provider Set:
- MIT OpenCourseWare
- Author:
- Larson, Richard C.
- Date Added:
- 02/16/2011
Conditions of Use:
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This course provides an introduction to applied concepts in Calculus that are relevant to the managerial, life, and social sciences. Students should have a firm grasp of the concept of functions to succeed in this course. Topics covered include derivatives of basic functions and how they can be used to optimize quantities such as profit and revenues, as well as integrals of basic functions and how they can be used to describe the total change in a quantity over time.
- Subject:
- Calculus
- Material Type:
- Full Course
- Textbook
- Provider:
- Lumen Learning
- Provider Set:
- Candela Courseware
- Date Added:
- 03/31/2016
Conditions of Use:
No Strings Attached
MATH&148 is a calculus course for business students. It is designed for students who want a brief course in calculus. Topics include differential and integral calculus of elementary functions. Problems emphasize business and social science applications. Translating words into mathematics and solving word problems are emphasized over algebra. Applications are mainly business oriented (e.g. cost, revenue, and profit). Mathematical theory and complex algebraic manipulations are not mainstays of this course, which is designed to be less rigorous than the calculus sequence for scientists and engineers. Topics are presented according to the rule of four: geometrically, numerically, analytically, and verbally. That is, symbolic manipulation must be balanced with graphical interpretation, numerical examples, and writing. Trigonometry is not part of the course.
- Subject:
- Calculus
- Material Type:
- Assessment
- Full Course
- Reading
- Syllabus
- Textbook
- Provider:
- Washington State Board for Community & Technical Colleges
- Provider Set:
- Open Course Library
- Date Added:
- 05/03/2013
Conditions of Use:
Read the Fine Print
Introductory survey of quantitative methods (QM), or the application of statistics in the workplace. Examines techniques for gathering, analyzing, and interpreting data in any number of fieldsĺÎĺ from anthropology to hedge fund management.
- Subject:
- Business and Communication
- Management
- Statistics and Probability
- Material Type:
- Activity/Lab
- Full Course
- Homework/Assignment
- Reading
- Syllabus
- Provider:
- The Saylor Foundation
- Date Added:
- 11/10/2011
Conditions of Use:
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Full course of Algebra 1 is presented online by Georgia Virtual Learning. Audio, video, text, games and activities are included to engage ninth grade students in learning.
- Subject:
- Algebra
- Functions
- Material Type:
- Activity/Lab
- Full Course
- Reading
- Provider:
- Georgia Department of Education
- Provider Set:
- Georgia Virtual Learning
- Date Added:
- 08/02/2013
Conditions of Use:
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This Geometry Concept Collection is a rigorous presentation of high school geometry. It is fully correlated with the Common Core State Standards.
- Subject:
- Geometry
- Material Type:
- Full Course
- Homework/Assignment
- Interactive
- Lecture Notes
- Lesson Plan
- Textbook
- Unit of Study
- Provider:
- CK-12 Foundation
- Author:
- Kaitlyn Spong
- Date Added:
- 08/01/2014
Conditions of Use:
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In CK-12 Middle School Math Concepts – Grade 8, the learning content is divided into concepts. Each concept is complete and whole providing focused learning on an indicated objective. Theme-based concepts provide students with experiences that integrate the content of each concept. Students are given opportunities to practice the skills of each concept through real-world situations, examples, guided practice and explore more practice. There are also video links provided to give students an audio/visual way of connecting with the content.
- Subject:
- Mathematics
- Material Type:
- Full Course
- Lesson Plan
- Textbook
- Unit of Study
- Provider:
- CK-12 Foundation
- Provider Set:
- CK-12 FlexBook
- Date Added:
- 03/28/2016
Conditions of Use:
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Published in 1991 by Wellesley-Cambridge Press, the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications.
In addition to the Textbook, there is also an online Instructor's Manual and a student Study Guide. Prof. Strang has also developed a related series of videos, Highlights of Calculus, on the basic ideas of calculus.
- Subject:
- Mathematics
- Calculus
- Material Type:
- Full Course
- Provider:
- MIT
- Author:
- Gilbert Strang
- Date Added:
- 01/01/1991