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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

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The four conic sections arise when a plane cuts through a double cone in different ways. If the plane cuts through parallel to the side of the cone (case 1), a parabola results (to be specific, the parabola is the shape of the planar graph that is formed by the set of points of intersection of the plane and the cone). If the plane is perpendicular to the cone's axis of symmetry (case 2, lower plane), a circle results. If the plane cuts through at some angle between these two cases (case 2, upper plane) — that is, if the angle between the plane and the axis of symmetry is larger than that between the side of the cone and the axis, but smaller than a right angle — an ellipse results. If the plane is parallel to the axis of symmetry (case 3), or makes a smaller positive angle with the axis than the side of the cone does (not shown), a hyperbola results. In all of these cases, if the plane passes through the point at which the two cones meet (the vertex), a degenerate conic results. First studied by the ancient Greeks in the 4th century BCE, conic sections were still considered advanced mathematics by the time Euclid (fl. c. 300 BCE) created his Elements, and so do not appear in that famous work. Euclid did write a work on conics, but it was lost after Apollonius of Perga (d. c. 190 BCE) collected the same information and added many new results in his Conics. Other important results on conics were discovered by the medieval Persian mathematician Omar Khayyám (d. 1131 CE), who used conic sections to solve algebraic equations.

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Knot theory is the branch of topology that studies mathematical knots, which are defined as embeddings of a circle S1 in 3-dimensional Euclidean space, R3. This is basically equivalent to a conventional knotted string with the ends of the string joined together to prevent it from becoming undone. Two mathematical knots are considered equivalent if one can be transformed into the other via continuous deformations (known as ambient isotopies); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways, but the most common method is by planar diagrams (known as knot projections or knot diagrams). Given a method of description, a knot will have many descriptions, e.g., many diagrams, representing it. A fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot.

Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals. Classical knot theory, as initiated by Max Dehn, J. W. Alexander, and others, is primarily concerned with the knot group and invariants from homology theory such as the Alexander polynomial.

The discovery of the Jones polynomial by Vaughan Jones in 1984, and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools as quantum groups and Floer homology. (Full article...)

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