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High Quality Content by WIKIPEDIA articles! A solved game is a game whose outcome can be mathematically predicted, regardless of the moves played. Games which haven't been mathematically solved in this sense are said to be "unsolved". Solving a game generally involves assuming both players will always make the best move possible (that is, the move which, in the long run, will ensure they either win or draw that game). This article focuses on two-player games that have been solved.

In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all internal angles equal to 4?/5 (144°). Its Schlafli symbol is {10}.

High Quality Content by WIKIPEDIA articles! The Quadrature of the Parabola is a treatise on geometry, written by Archimedes in the 3rd century B.C. Written as a letter to his friend Dositheus, the work presents 24 propositions regarding parabolas, culminating in a proof that the area of a parabolic segment (the region enclosed by a parabola and a line) is 4/3 that of a certain inscribed triangle. The proof uses the method of exhaustion. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. This represents the most sophisticated use of the method of exhaustion in ancient mathematics, and remained unsurpassed until the development of integral calculus in the 17th century.

High Quality Content by WIKIPEDIA articles! In abstract algebra, the split-quaternions or coquaternions are elements of an associative algebra introduced by James Cockle in 1849 under the latter name. They are also known as para-quaternions (particularly in recent literature on para-quaternionic geometry) or hyperbolic quaternions, although historically the latter term has a different meaning. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain zero divisors, nilpotent elements, and nontrivial idempotents.

Kirk Trigsted, Randall Gallaher, Kevin Bodden

Normal 0 false false false Intermediate Algebra by Trigsted, Gallaher, and Bodden is the first online, completely “clickable” Intermediate Algebra text to take full advantage of MyMathLab’s features and benefits. Like many instructors, Kirk Trigsted experienced marked improvement in student learning when teaching with MyMathLab. After observing that most students started their assignments by going directly to the exercises in MyMathLab without first consulting their textbooks, he was inspired to design an eText and course to better meet the needs and expectations of students. The coupling of a true eText with a fully designed course structure within MyMathLab provides a seamless learning experience for readers. Most importantly, the Trigsted MyMathLab course drives students into the book, before starting their homework. Foundations; Equations and Inequalities in One Variable; Graphs and Functions; Systems of Linear Equations and Inequalities; Polynomial Expressions and...

Robert Goldblatt

This is an introduction to nonstandard analysis based on a course of lectures given several times by the author. It is suitable for use as a text at the beginning graduate or upper undergraduate level, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions; a source of new ideas, objects and proofs; and a wellspring of powerful new principles of reasoning (transfer, overflow, saturation, enlargement, hyperfinite approximation etc.). The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective, emphasizing the role of the transfer principle as a working tool of mathematical practice. It then sets out the theory of enlargements of fragments of the mathematical...

High Quality Content by WIKIPEDIA articles! In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis) that lies on the same plane. Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid, times the figure's area (Pappus's second centroid Theorem). Rotating a curve A representative disk is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length w) around some axis (located r units away), so that a cylindrical volume of ??r2w units is enclosed.

William L. Briggs, Lyle Cochran

Drawing on their decades of teaching experience, William Briggs and Lyle Cochran have created a calculus text that carries the teacher’s voice beyond the classroom. That voice–evident in the narrative, the figures, and the questions interspersed in the narrative–is a master teacher leading readers to deeper levels of understanding. The authors appeal to readers’ geometric intuition to introduce fundamental concepts and lay the foundation for the more rigorous development that follows. Comprehensive exercise sets have received praise for their creativity, quality, and scope. Functions; Limits; Derivatives; Applications of the Derivative; Integration; Applications of Integration; Integration Techniques; Sequences and Infinite Series; Power Series; Parametric and Polar Curves. For all readers interested in single variable and multivariable calculus for mathematics, engineering, and science.

High Quality Content by WIKIPEDIA articles! In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry. Technically, a scheme is a topological space together with commutative rings for all its open sets, which arises from "glueing together" spectra (spaces of prime ideals) of commutative rings.

High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! Zhi-Wei Sun is a Chinese mathematician, working primarily on number theory, combinatorics, and group theory. Currently he works as a professor in Nanjing University.Born in Huai'an, Jiangsu, Sun and his twin brother Sun Zhihong proved a theorem about what are now known as the Wall-Sun-Sun primes that guided the search for counterexamples to Fermat's last theorem. In 2003, he presented a unified approach to three famous topics of Paul Erd?s in combinatorial number theory: covering systems, restricted sumsets, and zero-sum problems or EGZ Theorem.