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An example of a 3 × 3 × 3 magic cube. In this example, no slice is a magic square. In this case, the cube is classed as a simple magic cube.. In mathematics, a magic cube is the 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an n × n × n pattern such that the sums of the numbers on each row, on each column, on each pillar and on each of the four ...
Packing squares in a square: Optimal solutions have been proven for n from 1-10, 14-16, 22-25, 33-36, 62-64, 79-81, 98-100, and any square integer. The wasted space is asymptotically O(a 3/5). Packing squares in a circle: Good solutions are known for n ≤ 35. The optimal packing of 10 squares in a square
In ()-(), L1-norm ‖ ‖ returns the sum of the absolute entries of its argument and L2-norm ‖ ‖ returns the sum of the squared entries of its argument.If one substitutes ‖ ‖ in by the Frobenius/L2-norm ‖ ‖, then the problem becomes standard PCA and it is solved by the matrix that contains the dominant singular vectors of (i.e., the singular vectors that correspond to the highest ...
A solution is optimal if the sequence of moves is as short as possible. The highest value of this, among all initial configurations, is known as God's number, [3] or, more formally, the minimax value. [4] God's algorithm, then, for a given puzzle, is an algorithm that solves the puzzle and produces only optimal solutions.
This set of words contains many expressions that turn out to denote equal values in every lattice. For example, if a is some element of X , then a ∨ 1 = 1 and a ∧ 1 = a . The word problem for free bounded lattices is the problem of determining which of these elements of W ( X ) denote the same element in the free bounded lattice FX , and ...
Even order Chebyshev filters implemented with passive elements, typically inductors, capacitors, and transmission lines, with terminations of equal value on each side cannot be implemented with the traditional Chebyshev transfer function without the use of coupled coils, which may not be desirable or feasible, particularly at the higher ...
A fractional set cover is an assignment of a fraction (a number in [0,1]) to each set in , such that for each element x in the universe, the sum of fractions of sets that contain x is at least 1. The goal is to find a fractional set cover in which the sum of fractions is as small as possible.
Any point in the solution set can be obtained by first choosing a value for z, and then computing the corresponding values for x and y. Each free variable gives the solution space one degree of freedom , the number of which is equal to the dimension of the solution set.