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In mathematical analysis, the staircase paradox is a pathological example showing that limits of curves do not necessarily preserve their length. [1] It consists of a sequence of "staircase" polygonal chains in a unit square , formed from horizontal and vertical line segments of decreasing length, so that these staircases converge uniformly to ...
Also known as the "Up and Down Test" or "the staircase method", a Bruceton analysis relies upon two parameters: first stimulus and step size. A stimulus is provided to the sample, and the results noted.
By taking conjugates, the number p k (n) of partitions of n into exactly k parts is equal to the number of partitions of n in which the largest part has size k. The function p k (n) satisfies the recurrence p k (n) = p k (n − k) + p k−1 (n − 1) with initial values p 0 (0) = 1 and p k (n) = 0 if n ≤ 0 or k ≤ 0 and n and k are not both ...
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
C n is the number of ways to tile a stairstep shape of height n with n rectangles. Cutting across the anti-diagonal and looking at only the edges gives full binary trees. The following figure illustrates the case n = 4: C n is the number of ways to form a "mountain range" with n upstrokes and n downstrokes that all stay above a horizontal line ...
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3); the special case for n = 4 is known as a tesseract.It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.
In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by the construction of a dimensionless quantity, in the general framework of dimensional analysis and in particular applications such as fluid mechanics.
The vertical and horizontal lines are reflected off or refracted through in the following sequence: the line containing the segment corresponding to the coefficient of x n−1, then of x n−2 etc. Choosing θ so that the path lands on the terminus, −tan(θ) is a root of this polynomial. For every real zero of the polynomial, there will be ...