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The AP exam for European History is divided into two sections, comprising 55 multiple-choice questions (with four answer choices), three short-answer questions, and two essay responses (one thematic Long Essay Question (LEQ) and one Document Based Question (DBQ)). [3]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
The College Board announced the development of AP World History: Ancient, which focuses exclusively on earlier periods, including prehistory. [2] Students in the United States usually take the course in their sophomore year of high school, although they are not generally required to do so, as some take it in senior and freshman year.
The test is offered by the College Board. Approximately 2,900 colleges and universities will grant college credits for each test. Both U.S. and international schools grant CLEP credit. Most of the tests are 90 minutes long. As of 2023, they cost $90 each; they will cost $93 in the 2023–2024 school year. [2]
Another board member, Fields medalist Alain Connes, hoped that the publicity around the unsolved problems would help to combat the "wrong idea" among the public that mathematics would be "overtaken by computers". [5] Some mathematicians have been more critical.
Beginning with the May 2011 AP Exam administration, the College Board changed the scoring method of AP Exams. [28] [29] Total scores on the multiple-choice section are now based on the number of questions answered correctly. Points are no longer deducted for incorrect answers and, as was the case before, no points are awarded for unanswered ...
LEQ := λm.λn.ISZERO (SUB m n), and since m = n, if LEQ m n and LEQ n m, it is straightforward to build a predicate for numerical equality. The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" expressions in lambda calculus. For example, the predecessor function can be defined as:
The first major result was that of John von Neumann in 1933, [4] giving an affirmative answer for compact groups. The locally compact abelian group case was solved in 1934 by Lev Pontryagin . The final resolution, at least in the interpretation of what Hilbert meant given above, came with the work of Andrew Gleason , Deane Montgomery and Leo ...