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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.
Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically. [23] Analytic geometry allows the study of curves unrelated to circles and lines.
It means that the length of an optimal solution in HTM ≤ the length of an optimal solution in QTM. The maximal number of face turns needed to solve any instance of the Rubik's Cube is 20, [2] and the maximal number of quarter turns is 26. [3] These numbers are also the diameters of the corresponding Cayley graphs of the Rubik's Cube group. In ...
Solving an equation corresponds to adding and removing objects on both sides in such a way that the sides stay in balance until the only object remaining on one side is the object of unknown mass. [145] Word problems are another tool to show how algebra is applied to real-life situations. For example, students may be presented with a situation ...
Solving the equation by separation of variables means seeking a solution of the form of a product of spatial and temporal parts [18] (,) = (), where () is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and () is a function of time only.
The solution, known as the fold-and-cut theorem, states that any shape with straight sides can be obtained. A practical problem is how to fold a map so that it may be manipulated with minimal effort or movements. The Miura fold is a solution to the problem, and several others have been proposed. [43]
Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers. [52]
In a liquid mixture, the fugacity of each component is equal to that of a vapor component in equilibrium with the liquid. In an ideal solution, the fugacities obey the Lewis-Randall rule: =, where x i is the mole fraction in the liquid and f ∗ i is the fugacity of the pure liquid phase. This is a good approximation when the component ...
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