problem state - ορισμός. Τι είναι το problem state
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Τι (ποιος) είναι problem state - ορισμός

CONSTRUCT CIRCLES THAT ARE TANGENT TO THREE GIVEN CIRCLES IN A PLANE
Apollonius' problem; Problem of apollonius; Apollonius problem; Appolonius' problem; Apollonius's problem; Four coins problem
  • Figure 13: A symmetrical Apollonian gasket, also called the Leibniz packing, after its inventor [[Gottfried Leibniz]].
  • Figure 2: Four complementary pairs of solutions to Apollonius's problem; the given circles are black.
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  • ''r''<sub>''outer''</sub> + ''r''<sub>''inner''</sub>}} of the inner and outer radii, while twice its center distance ''d''<sub>''s''</sub> equals their difference.
  • ''r''<sub>''outer''</sub> − ''r''<sub>''inner''</sub>}} of the inner and outer radii, while twice its center distance ''d''<sub>''s''</sub> equals their sum.
  • The set of points with a constant ratio of distances ''d''<sub>1</sub>/''d''<sub>2</sub> to two fixed points is a circle.
  • ''r''<sub>2</sub> + ''r''<sub>''s''</sub>}}, respectively, so their difference is independent of ''r''<sub>''s''</sub>.
  • Figure 11: An Apollonius problem with no solutions. A solution circle (pink) must cross the dashed given circle (black) to touch both of the other given circles (also black).
  • radical center]] (orange).
  • Figure 9: The two tangent lines of the two tangent points of a given circle intersect on the [[radical axis]] ''R'' (red line) of the two solution circles (pink). The three points of intersection on ''R'' are the poles of the lines connecting the blue tangent points in each given circle (black).
  • Figure 6: A conjugate pair of solutions to Apollonius's problem (pink circles), with given circles in black.
  • Figure 1: A solution (in purple) to Apollonius's problem. The given circles are shown in black.
  • Figure 4: Tangency between circles is preserved if their radii are changed by equal amounts. A pink solution circle must shrink or swell with an internally tangent circle (black circle on the right), while externally tangent circles (two black circles on left) do the opposite.
  • Figure 12: The two solutions (red) to Apollonius' problem with mutually tangent given circles (black), labeled by their curvatures.
  • Figure 5: Inversion in a circle. The point ''P''<nowiki>'</nowiki> is the inverse of point ''P'' with respect to the circle.

problem state      
IBM jargon for user mode, the opposite of "supervisor state". On IBM System 360, 370 and 390 mainframes {privileged instructions} may only be executed in "supervisor state". Application programs request the operating system to perform these operations by using the Supervisor Call (SVC) instruction. (1995-02-15)
Problem statement         
CONCISE DESCRIPTION OF THE ISSUES THAT NEED TO BE ADDRESSED BY A PROBLEM SOLVING TEAM
Problem Statement
A problem statement is a concise description of an issue to be addressed or a condition to be improved upon. It identifies the gap between the current (problem) state and desired (goal) state of a process or product.
Knapsack problem         
  • multiple constrained problem]] could consider both the weight and volume of the boxes. <br />(Solution: if any number of each box is available, then three yellow boxes and three grey boxes; if only the shown boxes are available, then all except for the green box.)
  • A demonstration of the dynamic programming approach.
PROBLEM IN COMBINATORIAL OPTIMIZATION
0/1 knapsack problem; 0-1 knapsack problem; Unbounded knapsack problem; Unbounded Knapsack Problem; Binary knapsack problem; Napsack problem; Backpack problem; 0-1 Knapsack problem; Integer knapsack problem; Knapsack Problem; Algorithms for solving knapsack problems; Methods for solving knapsack problems; Approximation algorithms for the knapsack problem; Bounded knapsack problem; Multiple knapsack problem; Rucksack problem; Computational complexity of the knapsack problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.

Βικιπαίδεια

Problem of Apollonius

In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 BC – c. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts).

In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions. François Viète found such a solution by exploiting limiting cases: any of the three given circles can be shrunk to zero radius (a point) or expanded to infinite radius (a line). Viète's approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent to finding a position from the differences of its distances to three known points. This has applications in navigation and positioning systems such as LORAN.

Later mathematicians introduced algebraic methods, which transform a geometric problem into algebraic equations. These methods were simplified by exploiting symmetries inherent in the problem of Apollonius: for instance solution circles generically occur in pairs, with one solution enclosing the given circles that the other excludes (Figure 2). Joseph Diaz Gergonne used this symmetry to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of the given circles. These developments provide a geometrical setting for algebraic methods (using Lie sphere geometry) and a classification of solutions according to 33 essentially different configurations of the given circles.

Apollonius' problem has stimulated much further work. Generalizations to three dimensions—constructing a sphere tangent to four given spheres—and beyond have been studied. The configuration of three mutually tangent circles has received particular attention. René Descartes gave a formula relating the radii of the solution circles and the given circles, now known as Descartes' theorem. Solving Apollonius' problem iteratively in this case leads to the Apollonian gasket, which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy–Littlewood circle method.

Παραδείγματα από το σώμα κειμένου για problem state
1. On fears of proliferation, Pallone said India should not be considered as a problem state.
2. "Reality in Florida, we have a hurricane problem," state Emergency Management Director Craig Fugate said Wednesday.
3. Although India has never been a signatory of the nuclear Non–Proliferation Treaty, it should not be considered as a problem state with regard to non–proliferation issues.
4. We need Central help as 23 of the 38 districts are infested with the problem," state Deputy Chief Minister Sushil Kumar Modi said on Sunday.
5. "However, this isn‘t Russia‘s problem." State–owned Gazprom, the world‘s biggest gas producer and Russia‘s largest company, is building a $10.5 billion link between Germany, its biggest customer, and fields in Western Siberia.