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That is, x ∈ lim sup X n if and only if there exists a subsequence (X n k) of (X n) such that x ∈ X n k for all k. lim inf X n consists of elements of X which belong to X n for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ lim inf X n if and only if there exists some m > 0 such that x ∈ X n for all n > m.
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix.. This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C).
n = 1 that yield a minimax approximation or bound for the closely related Q-function: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {(a n,b n)} N n = 1 for many variations of the exponential approximations and bounds up to N = 25 have been released to open access as a comprehensive dataset. [16]
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
On one hand, the limit as n approaches infinity of a sequence {a n} is simply the limit at infinity of a function a(n) —defined on the natural numbers {n}. On the other hand, if X is the domain of a function f(x) and if the limit as n approaches infinity of f(x n) is L for every arbitrary sequence of points {x n} in X − x 0 which converges ...
The product of 1-D sinc functions readily provides a multivariate sinc function for the square Cartesian grid : sinc C (x, y) = sinc(x) sinc(y), whose Fourier transform is the indicator function of a square in the frequency space (i.e., the brick wall defined in 2-D space).
Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. Imbalanced weights may undesirably affect the matrix spectrum, leading to the need of normalization — a column/row scaling of the matrix entries ...
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...