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In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a Boolean sum of the adjacency matrix of the original directed graph and its matrix transpose, where the zero and one entries of are treated as logical, rather than numerical, values, as in the following example:
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is triangular. This is a stronger property than being Hessenberg . Just as in the complex case, a family of commuting real matrices { A i } may be simultaneously brought to quasi-triangular form by an orthogonal matrix.
The interpolant takes the form of a weighted sum of radial basis functions. [1] [2] RBF interpolation is a mesh-free method, meaning the nodes (points in the domain) need not lie on a structured grid, and does not require the formation of a mesh. It is often spectrally accurate [3] and stable for large numbers of nodes even in high dimensions.
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables.
The behavior of the logistic map is shown in Cobweb plot form. The animation shows the change in behavior as the parameter (r in the figure) is increased from 1 to 4, starting from an initial value of 0.2.) The logistic map is a discrete dynamical system defined by the quadratic difference equation:
Thus every shear matrix has an inverse, and the inverse is simply a shear matrix with the shear element negated, representing a shear transformation in the opposite direction. In fact, this is part of an easily derived more general result: if S is a shear matrix with shear element λ, then S n is a shear matrix whose shear element is simply nλ.
This can also be seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it. Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, the ...