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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.
This defines a probability measure on the solution of the differential equation that can be sampled. Gaussian process regression methods are based on posing the problem of solving the differential equation at hand as a Gaussian process regression problem, interpreting evaluations of the right-hand side as data on the derivative. [35]
In 2009, Adam S. Landsberg proposed the following simpler variant of the 100 prisoners problem which is based on the well-known Monty Hall problem: [13] Behind three closed doors a car, the car keys and a goat are randomly distributed. There are two players: the first player has to find the car, the second player the keys to the car.
To illustrate, the solution + = has bases with a common factor of 3, the solution + = has bases with a common factor of 7, and + = + has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively
Similar to the examples described above, we consider x, y, φ to be independent uniform random variables over the ranges 0 ≤ x ≤ a, 0 ≤ y ≤ b, − π / 2 ≤ φ ≤ π / 2 . To solve such a problem, we first compute the probability that the needle crosses no lines, and then we take its complement.
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
For instance, (x – a)(x – b) = x 2 – (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables. This was first formalized by the 16th-century French mathematician François Viète , in Viète's formulas , for the case of positive real roots.
The solution set of the equation x 2 / 4 + y 2 = 1 forms an ellipse when interpreted as a set of Cartesian coordinate pairs. Main article: Solution set The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown , that satisfies all the equations ...
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