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Software crack illustration. Software cracking (known as "breaking" mostly in the 1980s [1]) is an act of removing copy protection from a software. [2] Copy protection can be removed by applying a specific crack. A crack can mean any tool that enables breaking software protection, a stolen product key, or guessed password. Cracking software ...
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...
In mathematics, Frullani integrals are a specific type of improper integral named after the Italian mathematician Giuliano Frullani.The integrals are of the form ()where is a function defined for all non-negative real numbers that has a limit at , which we denote by ().
and the problem is, given the continuous kernel function and the function , to find the function .. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely (,) = (), and the limits of integration are ±∞, then the right hand side of the equation can be rewritten as a convolution of the functions and and therefore ...
The contour integral of a complex function: is a generalization of the integral for real-valued functions. For continuous functions in the complex plane , the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter.
Consider the following second-order problem, ′ + + = () =, where = {,, <is the Heaviside step function.The Laplace transform is defined by, = {()} = ().Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,
When we recently wrote about the toughest math problems that have been solved, we mentioned one of the greatest achievements in 20th-century math: the solution to Fermat’s Last Theorem. Sir ...
For example, suppose we want to find the integral ∫ 0 ∞ x 2 e − 3 x d x . {\displaystyle \int _{0}^{\infty }x^{2}e^{-3x}\,dx.} Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it.