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Chvátal's art gallery theorem, named after Václav Chvátal, gives an upper bound on the minimal number of guards. It states: It states: "To guard a simple polygon with n {\displaystyle n} vertices, ⌊ n / 3 ⌋ {\displaystyle \left\lfloor n/3\right\rfloor } guards are always sufficient and sometimes necessary."
The result, x 2, is a "better" approximation to the system's solution than x 1 and x 0. If exact arithmetic were to be used in this example instead of limited-precision, then the exact solution would theoretically have been reached after n = 2 iterations ( n being the order of the system).
In computational fluid dynamics, the MacCormack method (/məˈkɔːrmæk ˈmɛθəd/) is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations.
An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique solving (2) and, therefore, P1. This solution is a-priori only a member of H 0 1 ( 0 , 1 ) {\displaystyle H_{0}^{1}(0,1)} , but using elliptic regularity, will be smooth if f {\displaystyle f} is.
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition).
Using low-degree polynomials over a finite field of size , it is possible to extend the definition of Reed–Muller codes to alphabets of size .Let and be positive integers, where should be thought of as larger than .
The elements of an arithmetico-geometric sequence () are the products of the elements of an arithmetic progression (in blue) with initial value and common difference , = + (), with the corresponding elements of a geometric progression (in green) with initial value and common ratio , =, so that [4]
A plot of the smoothstep(x) and smootherstep(x) functions, using 0 as the left edge and 1 as the right edgeSmoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics, [1] [2] video game engines, [3] and machine learning.