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Differential equations are, in addition to a topic of study in mathematics, the main language in which the laws and phenomena of science are expressed. In basic terms, a differential equation is an expression that describes how a system changes from one moment of time to another, or from one point in space to another. When working with differential equations, the ultimate goal is to move from a microscopic view of relevant physics to a macroscopic view of the behavior of a system as a whole. Let’s look at a simple differential equation. Based on previous math and physics courses, you know that a car that is constantly accelerating in the x-direction obeys the equation d2x/dt2 = a, where a is the applied acceleration. This equation has two derivations with respect to time, so it is a second-order differential equation; because it has derivations with respect to only one variable (in this example, time), it is known as an ordinary differential equation, or an ODE. Let’s say that we want to solve the above ODE for the position of the car as a function of time. We can do so by using direct integration: the integration of both sides with respect to time gives us dx/dt = at + c, where c is a constant of integration. If the velocity of the car is known to be a particular value at some point in time T, we can solve for c as c = [dx/dt]t=T / aT. More simply, if the velocity is zero at time 0, then c = 0. Integrating again gives us the desired solution: x(t) = at2/2 + ct + e, where e is another constant of integration. Again, if the position of the car at t=0 is taken to be zero, then the solution for the position of the car becomes x(t) = at2/2. It is useful to note that checking the validity of a solution to an ODE is easily accomplished by substituting it back into the ODE. Unfortunately, not all differential equations are this easy to solve. Generally, an ODE is a functional relation (it would be a function, except that the “variables” are themselves functions!) between an independent variable t, a dependent function U(t), and some of its derivatives diU(t)/dti. An ODE is linear if it can be written as a functional relation in which no powers of U or its derivatives appear—otherwise, the ODE is nonlinear. For the most part, nonlinear ODEs can only be solved numerically; this course will focus on linear ODEs. This course will also introduce several other subclasses and their respective properties. However, despite centuries of study, the only practical approach to the solution of complicated ODEs that has emerged is numerical approximation. Although these numerical techniques are the subject of numerical analysis courses (see MA213: Numerical Analysis), this course will introduce you to the fundamentals behind numerical solutions. The prerequisites for this course are MA101, MA102, MA103, and MA211. Considerable motivation will be gained if PHYS101 and PHYS102 are also taken as pre- or co-requisites. This course will make use of a PDF text by Paul Dawkins of Lamar University as its principal reading material. You may wish to download this PDF at the outset of this course so that you have it on hand throughout. You can find this file by clicking here and then looking for the line that says “Here is the file you requested: Differential Equations (Math 3301).” Click on the link associated with “Differential Equations (Math 3301).”
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saylor.org Academy  ●  Online
Modern computer technology requires an understanding of both hardware and software, as the interaction between the two offers a framework for mastering the fundamentals of computing. The purpose of this course is to cultivate an understanding of modern computing technology through an in-depth study of the interface between hardware and software. In this course, you will study the history of modern computing technology before learning about modern computer architecture and a number of its important features, including instruction sets, processor arithmetic and control, the Von Neumann architecture, pipelining, memory management, storage, and other input/output topics. The course will conclude with a look at the recent switch from sequential processing to parallel processing by looking at the parallel computing models and their programming implications.
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saylor.org Academy  ●  Online
This self-contained course presents a sampling of the fields of Materials Engineering and Materials Science. This course is intended primarily for engineering students who are not planning to major in either Materials Engineering or Materials Science. We will focus primarily on the concerns of the materials engineer—the person interested in choosing materials to make a finished product. This selection is determined by compromises among material properties, ease of fabrication, and cost. In contrast, the materials scientist is concerned with understanding the relationships between material properties and the internal structure of a material—that is, atomic bonding, arrangements of atoms, grain structure, and other microscopically observable features. We leave most of these associations to advanced courses, which will use more chemistry and physics than needed for this course. The course is divided into four units: Unit 1: Ways That Materials Can Fail – What Can Go Wrong? Unit 2: Classes of Engineering Materials – What Do We Have? Unit 3: Comparison of Engineering Materials – ­Which Is Best? Unit 4: Processing of Materials – How Can We Shape It? In Unit 1, we will look at available handbook properties and laboratory test results that characterize a material’s strength or weakness to failure. We will concentrate on mechanical property failures, leaving electrical and other types of breakdown to other courses. Our concerns will be: Static, steady-state applied forces (Elastic Deformation) Ductile materials (Plastic Deformation) Brittle materials (Fast Fracture) Cyclic, vibration forces (Fatigue Failure) High temperature environments (Creep Deformation) Corrosive environments (Oxidation and Wet Corrosion) In Unit 2, we will identify four major classes of the tens of thousands of available materials: metals, polymers, ceramics, and composite materials. We will examine specific examples from each category. Unit 3 is a synthesis of the first two units. We will see the consequences of the numerical handbook values defined in Unit 1 in evaluating the materials in Unit 2. In Unit 4, we will look at how we process our materials to obtain the desired configurations for our products. Your study will include a look at casting, mechanical forming, sintering, and joining. Not all materials can be processed with all procedures.
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saylor.org Academy  ●  Online
In this course, we will study the history of Eastern (Orthodox) Christian art. The course begins with an overview of the emergence of Christianity in the Late Antique period and the formation of the Christian visual language that grew out of the Classical tradition. The course then follows the development of Christian art after the fall of the Roman Empire and the emergence of a “new Rome” in the East: the Byzantine Empire. A series of reading assignments paired with lectures and virtual tours will introduce you to important works of Early Christian and Byzantine art and will also give you an understanding of the central debates of Early Christian and Byzantine art historical scholarship. By the time you finish the course, you will be able to identify the most important artworks from this period and understand how their appearances relate to the social, political, and religious environment in which they were produced. You will also be able to trace the ways in which Early Christian and Byzantine art changed over time and identify some of the proposed reasons for these changes. Further, you will gain an understanding of the unique position of the Early Christian and Byzantine worlds and the works of art and architecture produced in these cultures. An understanding of Early Christian and Byzantine Art will round out your art historical education, enhancing your knowledge of art produced by other cultures and in different historical periods.
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saylor.org Academy  ●  Online
This course will introduce you to the major topics, problems, and methods of philosophy and surveys the writings of a number of major historical figures in the field. Philosophy can be – and has been – defined in many different ways by many different thinkers. In a scholarly sense, philosophy is the study of the history of human thought. It requires familiarity with great ideas understood through the various major thinkers in world history. In its most general sense, philosophy is simply the investigation of life’s “big questions.” We will explore such fundamental questions in several of the core areas of philosophy, including metaphysics, epistemology, political philosophy, ethics, and the philosophy of religion. With the help of commentaries and discussions from a number of contemporary philosophers, we will read and reflect on texts by major Western and non-Western thinkers including Lao Tzu, Buddha, Confucius, Plato, Aristotle, Saint Anselm, René Descartes, Blaise Pascal, John Locke, Immanuel Kant, Thomas Hobbes, John Stuart Mill, Friedrich Nietzsche, Karl Marx, and Bertrand Russell. This course aims to not only familiarize you with philosophers and problems but to also improve your ability to think critically about the issues, develop your own ideas about them, and express these ideas clearly and persuasively in writing. Unit 1 introduces philosophy as a discipline and provides a sense of its subject matter and methodology. Unit 2 addresses topics in metaphysics and epistemology – traditionally the “core” areas of philosophy. Units 2, 3, and 4 cover moral, political, and religious philosophy, respectively. Each unit presents selections from a set of philosophers whose works are traditionally compared on the same themes in order to set up contrasting approaches and opinions.
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saylor.org Academy  ●  Online
How does the brain function? How does it interact with the body in order to control and mediate behaviors and actions? Though psychologists have long studied these questions, the workings of the brain remain, in large part, a mystery. In this course, we will explore the field of psychology devoted to the pursuit of these questions: neuropsychology or the study of the structure and function of the brain as it relates to psychological processes. We will study significant findings in the field, noting that technological improvements have often enabled substantial advancements in field research. You may, for example, take MRIs or PET scans – devices used to diagnose medical problems – for granted, but these have only relatively recently enabled researchers to study the brain in greater detail. While a formal background in biology is not required for this course, you will find that neuropsychology relies heavily on the discipline. In fact, psychologists and biologists have often explored similar issues, though typically from vastly different perspectives. Accordingly, you may find supplemental biology materials useful if you are entirely unfamiliar with the brain and the nervous system. This course will begin with a brief history of neuropsychology. We will then study the nervous system and the structure of the brain, identifying its different lobes and cortices, before concluding with a discussion of how the brain provides us with higher functioning abilities (i.e., learning, remembering, and communicating).
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saylor.org Academy  ●  Online
This course introduces major theories of economic development and to place them in a historical context. In his contributory introduction “Economic Growth, Economic Development and Human Development” in The Development Economics Reader (2008), edited by Giorgio Secondi and published by Routledge, Secondi defines economic development as the “branch of economics that studies relatively poor countries.” In the same book, Mahbub ul Haq, writing under the title “The Human Development Paradigm,” suggests that the “basic purpose of development is to enlarge people’s choices,” which is in line with the views expressed by the Nobel laureate Amartya Sen. Whether development is simply studying poor countries or expanding people’s choices in poor countries, one of the essential requirements is that there must be a means for making the choices available. This means that economic development must include growth, but growth can take place without economic development. Without economic growth, the choices cannot be expanded. At the same time, however, economic growth can take place and people’s choices can still be limited. Therefore, economic development requires economic growth, but the inverse is not required. Indeed, Pearce defines economic development as “the process of improving the standards of living and well-being of the population of developing countries by raising per capita income” (1992). In essence, theories of economic development attempt to explain the process that less developed countries (LDCs) go through to become developed countries (DCs). Further, because economic development is more than economic growth and involves changes in all aspects of the society, both social and political, the discipline tends to be interdisciplinary, drawing from other social sciences such as sociology and geography. In this course, in addition to discussing the theories of development, we will also try to explain how the theories are applied and how successful they have been in explaining the development patterns of various countries. The oft cited example of Ghana and Malaysia might be instructive at this point. Both countries gained independence from the British in 1957 (Ghana a few months earlier in March and Malaysia in August). At the time, both countries were roughly at the same level of development. If you fast forward to the 2000s, Malaysia’s per capita GDP is five times that of Ghana’s per capita GDP; it is $16,200 in Malaysia and $3,100 in Ghana. The literacy rate in Malaysia is 88.7% and 67.3% in Ghana. Malaysia has 11 times more physicians per 1,000 people than Ghana, and life expectancy is 74.04 in Malaysia but 61.45 in Ghana. There are many questions which the development economist wants the answers to, but mainly this boils down to: What accounts for the vast differences in the many measures of human development indices? We study economic development to learn from the Malaysians so that we can offer useful advice to the Ghanaians. The units in this course are stacked as building blocks; each successive unit depends on a thorough understanding of the previous unit. The course begins with some of the stylized facts of the countries classified as developing countries. You will find that they are not all alike. Some countries, such as the oil rich ones, may have a very high per capita GDP. You will learn the definitions of major terms and concepts. The sections that follow the introductory unit outline the major theories of economic development, tracing their development throughout history as the dialogue on development economics has progressed. Finally, the course will present a number of development successes and failures and will prompt you to draw your own conclusions on the validity of certain theories based on case studies.
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saylor.org Academy  ●  Online
This course is an introduction to linear algebra. It has been argued that linear algebra constitutes half of all mathematics. Whether or not everyone would agree with that, it is certainly true that practically every modern technology relies on linear algebra to simplify the computations required for Internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. Linear algebra can be viewed either as the study of linear equations or as the study of vectors. It is tied to analytic geometry; practically speaking, this means that almost every fact you will learn in this course has a picture associated with it. Learning to connect the facts with their geometric interpretation will be very useful for you. The book which is used in the course focuses both on the theoretical aspects as well as the applied aspects of linear algebra. As a result, you will be able to learn the geometric interpretations of many of the algebraic concepts in this subject. Additionally, you will learn some standard techniques in numerical linear algebra, which allow you to deal with matrices that might show up in applications. Toward the end, the more abstract notions of vector spaces and linear transformations on vector spaces will be introduced. In college algebra, one becomes familiar with the equation of a line in two-dimensional space: y = mx+b. Lines can be generalized to planes and “hyperplanes” in many-dimensional space; these objects are all described by linear relations. Linear transformations are ways of rotating, dilating, or otherwise modifying the underlying space so that these linear objects are not deformed. Linear algebra, then, is the theory and practice of analyzing linear relations and their behavior under linear transformations. According to the second interpretation listed above, linear algebra focuses on vectors, which are mathematical objects in many-dimensional space characterized by magnitude and direction. You can also think of them as a string of coordinates. Each string may represent the state of all the stocks traded in the DOW, the position of a satellite, or some other piece of data with multiple components. Linear transformations change the magnitude and direction of vectors—they transform the coordinates without changing their fundamental relationships with one another. Linear transformations are often written in a compact and easily-readable way by using matrices. Linear algebra may at first seem dry and difficult to visualize, but it is one of the most useful subjects you can learn if you wish to become a business-person, a physicist, a computer-programmer, an engineer, or a mathematician. Remember, the prerequisite of this course is one variable calculus course and a reasonable background in college algebra.
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saylor.org Academy  ●  Online
The study of public policy is intended to offer every citizen an understanding of the various and vast roles played by the different branches of the U.S. federal government as well as by state, county, and local governments in various areas of contemporary American life. It is also a field that focuses on the priorities of American society as portrayed in the public policy choices that elected representatives make on the part of citizens and the size of different interest groups that advocate on behalf of particular policy goals. This course looks at the process of making public policy from beginning to end and in a wide array of particular policy areas that are of importance to contemporary American society. Moreover, because the process of public policymaking is best explored by examining particular instances of public debate over a wide array of specific policy areas, this course will adopt a case study approach to explore particular topics. Unit 1 will introduce this case study approach as well various actors involved in the making of American public policy and the process of setting the public policy agenda. Unit 2 explores the process of public policy formulation be examining a variety of case studies, including energy and fuel economy policy. Unit 3 examines the implementation of public policies once they have been agreed upon, the allocating of funding to pay for these projects through the budgetary process, and the evaluation of these projects to determine their effectiveness in achieving the goals they were established to advance. Unit 4 explores various areas of American economic policy, while Unit 5 looks at several topics of interest in the field of national security policy. Unit 6 examines various issues in contemporary American public health policy, while Unit 7 explores the public policy responses of the U.S. government to a number of environmental concerns. Unit 8 looks at several topics of interest within the broader field of education policy, while Unit 9 focuses on aspects of public policy that impact rural communities. Unit 10 concludes this course by exploring several areas of ongoing debate within American social policy, including immigration reform, civil rights legislation, and the criminal justice system.
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saylor.org Academy  ●  Online
In this course, you will learn about the marketing process and examine the range of marketing decisions that an organization must make in order to sell its products and services. You will also learn how to think like a marketer, discovering that the focus of marketing has always been on the consumer. You will begin to ask, “Who is the consumer of goods and services?” What does the consumer need? What does the consumer want? Marketing is an understanding of how to communicate with the consumer, and is characterized by four activities: Creating products and services that serve consumers Communicating a clear value proposition Delivering products and services in a way that optimizes value Exchanging, or trading, value for those offerings Many people incorrectly believe that marketing and advertising are one in the same. In reality, advertising is just one of many tools used in marketing, which is the process by which firms determine which products to offer, how to price those products, and to whom they should be made available. We will also explore various ways in which marketing departments and independent agencies answer these questions – whether through research, analysis, or even trial-and-error. Once a company identifies its customer and product, marketers must then determine the best way to capture the customer’s attention. Capturing the customer’s attention may entail undercutting competitors on price, aggressively marketing a product with promotions and advertising (as with “As Seen on TV” ads), or specifically targeting ideal customers. The strategy a marketing firm chooses for a particular product is vital to the success of the product. The idea that “great products sell themselves” is simply not true. By the end of this course, you will be familiar with the art and science of marketing a product. This course provides students the opportunity to earn actual college credit. It has been reviewed and recommended for 3 credit hours by The National College Credit Recommendation Service (NCCRS). While credit is not guaranteed at all schools, we have partnered with a number of schools who have expressed their willingness to accept transfer of credits earned through Saylor. You can read more about our NCCRS program here.
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saylor.org Academy  ●  Online
Numerical analysis is the study of the methods used to solve problems involving continuous variables. It is a highly applied branch of mathematics and computer science, wherein abstract ideas and theories become the quantities describing things we can actually touch and see. The real number line is an abstraction where many interesting and useful ideas live, but to actually realize these ideas, we are forced to employ approximations of the real numbers. For example, consider marking a ruler at \sqrt{2}. We know that \sqrt{2} \approx 1.4142, but if we put the mark there, we know we are in error for there is an infinite sequence of nonzero digits following the 2. Even more: a number doesn’t have any width, yet any mark we make would have a width, and in that width lives an infinite number of real numbers. You may ask yourself: isn’t it sufficient to represent \sqrt{2} with 1.414? This is the kind of question that this course will explore. We have been trying to answer such questions for over 2,000 years (it is said that people have given their lives for the idea of \sqrt{2}, and they certainly wouldn’t think 1.414 sufficient). Modern computers can perform billions of arithmetic operations per second and trying to predict the path of a tropical storm can require many trillions of operations. How do we carry out such simulations and how do our approximations affect the result? The answer to the first question is certainly colored by the second! Numerical analysis is a broad and growing discipline with many open questions. This course is designed to be a first look at the discipline. Over the course of this semester, we will survey some of the basic problems and methods needed to simulate the solutions of ordinary differential equations. We will build the methods ourselves, starting with computer arithmetic, so that you will understand all of the pieces and how they fit together in state of the art algorithms. Along the way, we will write programs to solve equations, plot curves, integrate functions, and solve initial value problems. At the end of some chapters we will suggest – in a section called “Of Things Not Covered” – some topics that would have been included if we had more time or other avenues to explore if you are interested in the topics presented in the unit. The prerequisites for taking this course are MA211: Linear Algebra, MA221: Differential Equations, and either MA302/CS101: Introduction to Computer Science or a background in some programming language. Programming ideas will be illustrated in pseudocode and implemented in the open-source high-level computing environment. Numerical Analysis is the field of mathematics that applies numerical approximations in order to solve mathematical problems of continuous variables.In most cases, numerical analysis does not have the goal of finding exact answers to complex problems, as most mathematical problems cannot be solved through the application of a finite number of elementary mathematical operations.Rather, numerical analysis focuses on the development and study of algorithms that will quickly obtain approximate solutions.By analyzing these algorithms, we can evaluate their errors and stability and in turn decide when it is safe to use a particular numerical algorithm. The first known application of the methods of numerical analysis appears on Babylonian tablet YBC 7289, which is roughly dated between 1700-2000 BC.(Evidence suggests that the writer was a mathematics student.)The tablet features an incised square whose sides have a length of 0.5 units and a diagonal line that connects opposite corners of the square.The diagonal line is labeled 0:42 25 35 (in sexagesimal notation), which tells us that the Babylonians thought that the square root of 2 is 1.41421296 (in decimal notation).(The actual square root of 2 is 1.41421356… to 9 decimal places.)The Babylonian value is in error by roughly 7 parts in 100,000—an accuracy that could not have been obtained by direct measurement.As the square root of 2 is an irrational number, it cannot be directly calculated.Although not known for sure, it is likely that the value for the square root of two was originally calculated by Heron’s method, a simple version of the Newton-Raphson method for finding successively better approximations to the roots of a function. This course will focus on the applications of the methods of numerical analysis.We will cover enough of the mathematical background to allow you to intelligently discuss the convergence, error, and stability properties of numerical analysis algorithms, but will place emphasis on solving certain classes of problems that often arise in scientific or engineering contexts.These include approximating functions, finding roots of polynomial and other nonlinear functions, solving systems of linear equations, finding eigenvalues and eigenfunctions, optimizing constrained multi-dimensional functions, evaluating integrals, and solving ordinary and partial differential equations. The prerequisites for taking MA213 are MA211 (Linear Algebra), MA221 (Differential Equations), and either MA302/CS101 (Introduction to Computer Science) or a solid background in JAVA programming.While not necessary, treating MA222 (Partial Differential Equations) as a co-requisite is advised.Many problems and methods will be presented and used within an open-source Java-based computing environment.
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saylor.org Academy  ●  Online
The study of United States intelligence and national security operations is an analysis of how the various branches of government work together and, as a check upon each other, how they work to protect and promote American interests at home and abroad. The purpose of this course is to provide you with an overview of national security policy analysis and the United States intelligence community. As you progress through this course, you will learn about strategic thought and strategy formulation, develop the ability to assess national security issues and threats, and cultivate an understanding of the political and military institutions involved in the formulation and execution of national security policy through diplomacy, intelligence operations, and military force. This course will examine problems and issues regarding United States national security policy. A large section of the course will deal with the major actors and institutions involved in making and creating national security policy and the intelligence community. National security is the most critical role of your government, without which, all other policies could not be created. You will begin this course with an overview of national security interests in unit 1. In units 2 – 4, you will learn about the roles and powers possessed by each actor in the United States national security process, including responsibilities of the president, the executive branch, Congress, the military, and intelligence agencies. In unit 5, you will review the policymaking process and will consider policy analysis. In units 6 – 9, you will study specific types of national security issues and strategies that the government has used to solve these problems. Some problems include the threat of nuclear, chemical, and biological warfare; the impact of regional, sectarian, and tribal conflicts on national security interests; the threat of terrorism; and the impact of economic strife and scarce resources.
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