Search results
Results from the WOW.Com Content Network
In combinatorial mathematics, block walking is a method useful in thinking about sums of combinations graphically as "walks" on Pascal's triangle.As the name suggests, block walking problems involve counting the number of ways an individual can walk from one corner A of a city block to another corner B of another city block given restrictions on the number of blocks the person may walk, the ...
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems.. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations.
A Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix. The example below constructs a 7 × 7 Pascal matrix, but the method works for any desired n × n Pascal matrices. The dots in the following matrices represent zero elements.
For the other two variables, one exponent increases by 1 and one exponent decreases by 1. The exponents of A are 3 and 2 (the larger being in the left term). The exponents of C are 0 and 1 (the larger being in the right term). The coefficients and larger exponents are related: 4 × 3 = 12 × 1; 4 / 12 = 1 / 3; These equations yield the ratio ...
The first five layers of Pascal's 3-simplex (Pascal's pyramid). Each face (orange grid) is Pascal's 2-simplex (Pascal's triangle). Arrows show derivation of two example terms. In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.
However, as mentioned above this only works for octant zero, that is lines starting at the origin with a slope between 0 and 1 where x increases by exactly 1 per iteration and y increases by 0 or 1. The algorithm can be extended to cover slopes between 0 and -1 by checking whether y needs to increase or decrease (i.e. dy < 0)
It can determine where to stop because, when y = x, it has reached 45°. The reason for using these angles is shown in the above picture: As x increases, it neither skips nor repeats any x value until reaching 45°. So during the while loop, x increments by 1 with each iteration, and y decrements by 1 on occasion, never exceeding 1 in one ...
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, [1] India, [2] China, Germany, and Italy.