Algebra II is the second semester of a year-long introduction to modern algebra. The course focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.
These notes, which were created by students in a recent on-campus 18.702 Algebra II class, are offered here to supplement the materials included in OCW’s version of 18.702. They have not been checked for accuracy by the instructors of that class or by other MIT faculty members.

Algebra and Trigonometry provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra and trigonometry course. The modular approach and the richness of content ensures that the book meets the needs of a variety of courses. Algebra and Trigonometry offers a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned.

This course covers the applications of algebra to combinatorics. Topics include enumeration methods, permutations, partitions, partially ordered sets and lattices, Young tableaux, graph theory, matrix tree theorem, electrical networks, convex polytopes, and more.

This resource is a video abstract of a research paper created by Research Square on behalf of its authors. It provides a synopsis that's easy to understand, and can be used to introduce the topics it covers to students, researchers, and the general public. The video's transcript is also provided in full, with a portion provided below for preview:

"A new algorithm has successfully mapped part of the brain’s circuitry during shock therapy. For those suffering from severe depression, the approach could make for safer and more effective treatment. For brain research at large, it could lead to better ways of untangling noisy neural data to reveal real connections between different focal regions of the brain. Despite the gruesome picture painted by pop culture, modern shock therapy is a mild treatment option. In fact, over 2 million treatments are administered worldwide every year. Under general anesthesia, patients receive a small amount of current to the brain, triggering a brief seizure. The resulting changes in brain chemistry have been shown to reverse symptoms of mental health conditions like severe depression or bipolar disorder. But the procedure isn’t perfect. One of the most troubling side effects is memory loss, a result of poor targeting. To be effective and safe, induced seizures should be restricted to the pre-frontal cortex..."

The rest of the transcript, along with a link to the research itself, is available on the resource itself.

This book aims to be an accessible introduction into the design and analysis of efficient algorithms. Throughout the book we will introduce only the most basic techniques and describe the rigorous mathematical methods needed to analyze them.

The topics covered include:

The divide and conquer technique.
The use of randomization in algorithms.
The general, but typically inefficient, backtracking technique.
Dynamic programming as an efficient optimization for some backtracking algorithms.
Greedy algorithms as an optimization of other kinds of backtracking algorithms.
Hill-climbing techniques, including network flow.

The goal of the book is to show you how you can methodically apply different techniques to your own algorithms to make them more efficient. While this book mostly highlights general techniques, some well-known algorithms are also looked at in depth. This book is written so it can be read from "cover to cover" in the length of a semester, where sections marked with a * may be skipped.

This textbook is an introductory coverage of algorithms and data structures with application to graphics and geometry.

This web page contains a free electronic version of my self-published textbook Algorithms, along with other lecture notes I have written for various theoretical computer science classes at the University of Illinois, Urbana-Champaign

This course is offered to undergraduates and addresses several algorithmic challenges in computational biology. The principles of algorithmic design for biological datasets are studied and existing algorithms analyzed for application to real datasets. Topics covered include: biological sequence analysis, gene identification, regulatory motif discovery, genome assembly, genome duplication and rearrangements, evolutionary theory, clustering algorithms, and scale-free networks.

This resource is a video abstract of a research paper created by Research Square on behalf of its authors. It provides a synopsis that's easy to understand, and can be used to introduce the topics it covers to students, researchers, and the general public. The video's transcript is also provided in full, with a portion provided below for preview:

"Individuals with metabolic syndrome have a 2-fold increased risk of cardiovascular disease, but few strategies provide adequate cardiovascular risk reduction for this group. One option to lower this risk is to reduce atherogenic lipids, particularly low-density lipoprotein cholesterol. Statins are a recommended first-line therapy for this purpose, but this approach doesn’t always provide sufficient LDL-C lowering to optimally reduce cardiovascular risk. Now, researchers have shown that alirocumab, a proprotein converstase subtilisin/kexin type 9 inhibitor, or PCSK9 inhibitor, approved for LDL-C reduction, may address this need. Pooled clinical trial data from ten phase 3 clinical trials from the ODYSSEY clinical development program showed alirocumab significantly lowered LDL-C in individuals both with and without metabolic syndrome..."

The rest of the transcript, along with a link to the research itself, is available on the resource itself.

This resource is a video abstract of a research paper created by Research Square on behalf of its authors. It provides a synopsis that's easy to understand, and can be used to introduce the topics it covers to students, researchers, and the general public. The video's transcript is also provided in full, with a portion provided below for preview:

"Alirocumab, a drug that inhibits PCSK9 and helps clear the blood of atherogenic lipids, may be of benefit for individuals with type 2 diabetes who are most at risk of cardiovascular disease. According to the International Diabetes Federation, heart disease is the leading cause of death among people with type 2 diabetes. One factor that might contribute to that grim statistic is the high mixed dyslipidemia prevalence in this population. Mixed dyslipidemia is characterized by elevated triglycerides, and hence high triglyceride-rich lipoprotein, or remnant cholesterol, and high cholesterol levels; and low serum levels of high-density lipoprotein cholesterol, or HDL-C. Despite treatment with statins and other lipid-lowering therapies targeting atherogenic lipoproteins, management of mixed dyslipidemia remains challenging, especially among individuals with diabetes..."

The rest of the transcript, along with a link to the research itself, is available on the resource itself.

This resource is a video abstract of a research paper created by Research Square on behalf of its authors. It provides a synopsis that's easy to understand, and can be used to introduce the topics it covers to students, researchers, and the general public. The video's transcript is also provided in full, with a portion provided below for preview:

"A recent analysis suggests that ixazomib, an oral proteasome inhibitor (or PI), is tolerable and enhances therapeutic responses in patients with multiple myeloma. Ixazomib is approved for use in combination with lenalidomide and dexamethasone for patients who have already received 1 or more prior therapy. The promising results obtained for non-transplant patients taking ixazomib alone, if their disease has responded to primary induction therapy, point to a new possible treatment option for multiple myeloma. Multiple myeloma is a form of cancer that develops in bone marrow. Here, the body normally generates white blood cells that help fight off infection. But in multiple myeloma, malignant cells gradually crowd out these disease-fighting cells, compromising the body’s immune response, while also damaging the bones. The malignant cells also secrete large amounts of a non-functional protein which leads to kidney failure and other harms..."

The rest of the transcript, along with a link to the research itself, is available on the resource itself.

Poster showing Uncle Sam with a truck full of citizens holding Liberty Bonds. The truck bears a sign, "To Berlin." United Cigars.

What constitutes a recession? Two quarters of negative real GDP growth? Well, most of the time that “rule of thumb” definition works, but not always. The NBER Business Cycle Dating Committee has a more nuanced way of determining what a recession is. This essay discusses where recessions come from, how they are determined, and how they end.

The lab will start off with some conversation starters having to do with numbers, each prompt has a quantitative aspect involved. During the main activity lab students will get to learn each other's experiences in other countries, focusing on country similarities.

The presidential campaign of 1836 viewed as a card game by a satirist in sympathy with the Whigs. Opposing candidates Martin Van Buren (Democrat) and William Henry Harrison (Whig) face each other across a card table. Behind Van Buren stands his vice-presidential running mate Richard M. Johnson. Behind Harrison is incumbent President Andrew Jackson, who smokes a clay pipe and stands on tip-toes to spy on Harrison's hand. With his left hand he signals to Van Buren. Jackson: "What a h---ll of a hand old Harrison's got. I'm afraid Martin and Dick Johnson will go off with a flea in their ear." Johnson: "The old general is making signs that Harrison has the two highest trump cards and low. Martin he'll catch your Jack and then the jig's up! You'd better beg." Van Buren: "I ask one." Harrison: "Take it! now look out for your Jack!" On the wall above the table is a painting of the Battle of the Thames, one of Harrison's celebrated military victories a well as the occasion on which Johnson is reported to have slain the Indian chief Tecumseh. The print is probably by Robinson draughtsman Edward W. Clay, judging from its similarity to his "Grand Match Between the Kinderhook Poney ..." (no. 1836-14) and other signed work of the period.|Entered . . . 1836 by H.R. Robinson.|Published May 1836 by the proprietor H.R. Robinson, 48 Cortlandt St. N. York.|Title appears as it is written on the item.|Weitenkampf, p. 44.|Forms part of: American cartoon print filing series (Library of Congress)|Published in: American political prints, 1766-1876 / Bernard F. Reilly. Boston : G.K. Hall, 1991, entry 1836-11.

Graphic Novel and Student Discussion Guide on the central ideas of John Stewart Mills work on the necessity of free speech in a democratic society.

The major figures in American national politics in 1838 are gently satirized, each characterized as riding a favorite issue or "hobbyhorse." At the lead (far left) is President Martin Van Buren, riding a horse "Sub-Treasury," which he calls his "Old Hickory nag." The artist refers to Van Buren's independent treasury program, a system whereby federal funds were to be administered by revenue-collecting agencies or local "sub-treasuries" rather than by a national bank. The Independent Treasury Bill was perceived as an outgrowth of predecessor Jackson's anti-Bank program. Another hobbyhorse, "United States Bank" (center), is shared by Whig senators Henry Clay and Daniel Webster, leaders of congressional opposition to Jackson and Van Buren's respective fiscal agendas. Clay says, "Either you or I must get off Dan, for this horse wont carry double!" Webster responds, "Dash my Whig if I get off Hal!" Directly behind Van Buren Democratic Senator Thomas Hart Benton rides a horse "Specie Currency," an allusion to Benton's championing of hard money economics. Benton was identified with administration efforts to curb the use of currency in favor of "specie" or coin, and to increase the ratio of gold to silver in circulation. He says, "My Golden Poney carries more weight than any of them!" Behind Clay and Webster is South Carolina senator John C. Calhoun, advocate of state's rights and the driver of Southern nullification of the "Tariff of Abominations." On the right are William Henry Harrison, in military uniform and riding an "Anti-Masonic" hobby, and Massachusetts Congressman John Quincy Adams on his "Abolition" mount. Harrison's horse is named after the party which supported his 1836 bid for the Presidency. When he says, ". . . unless there is another Morgan abduction, I'm afraid he'll [the horse] lose his wind!" he alludes to the suspicious 1826 death of William Morgan (purportedly at the hands of Masons) which fueled considerable anti-Masonic sentiment in the United States. Adams laments, "This horse, instead of being my Topaz, is my Ebony." |Entd . . . 1838 by H.R. Robinson.|Printed & publd. by H.R. Robinson, 52 Cortlandt St. N.Y.|Signed with monogram: C (Edward Williams Clay).|The print was registered for copyright on March 16, 1838.|Title appears as it is written on the item.|Blaisdell and Selz, no. 16.|Davison, no. 104.|Weitenkampf, p. 53.|Forms part of: American cartoon print filing series (Library of Congress)|Published in: American political prints, 1766-1876 / Bernard F. Reilly. Boston : G.K. Hall, 1991, entry 1838-1.

This book is intended for those who have already learned to read Arabic but are still struggling with the basics of Arabic grammar. The book moves from the very simple to the very complex, and includes examples and exercises in each chapter to reinforce learning. The key to the exercises is found in the back of the book. The book is difficult to find, as it was published at the Defense Language Institute. A free non-searchable PDF version is available through this site, and a searchable PDF is also available for purchase.