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The optimum of the linear cost function is where the red line intersects the polygon. The red line is a level set of the cost function, and the arrow indicates the direction in which we are optimizing. A closed feasible region of a problem with three variables is a convex polyhedron.
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1 , and x = 8, y = 9, z = 2 . There is a unique plane in three-dimensional space which passes through the three points with these coordinates , and this plane is the set of all points whose coordinates are solutions of the equation.
A set of 20 points in a 10 × 10 grid, with no three points in a line. The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line.
For example, given a = f(x) = a 0 x 0 + a 1 x 1 + ··· and b = g(x) = b 0 x 0 + b 1 x 1 + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba ...
The "nine dots" puzzle. The puzzle asks to link all nine dots using four straight lines or fewer, without lifting the pen. The nine dots puzzle is a mathematical puzzle whose task is to connect nine squarely arranged points with a pen by four (or fewer) straight lines without lifting the pen or retracing any lines.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
Each curve in this example is a locus defined as the conchoid of the point P and the line l.In this example, P is 8 cm from l. In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.
Including 0, the set has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice of base giving a logarithmic unit) gives an isomorphism with the log semiring (with 0 corresponding to ), and its units (the finite numbers, excluding ) correspond to the positive real numbers.