Related papers: Max-Plus Algebra for Complex Variables and Its App…
Discrete transforms, such as the discrete Fourier transform, are widely used in machine learning to improve model performance by extracting meaningful features. However, with numerous transforms available, selecting an appropriate one often…
In this paper, we modify and apply the recently introduced Mixed Newton Method, which is originally designed for minimizing real-valued functions of complex variables, to the minimization of real-valued functions of real variables by…
The discrete Fourier transform is approximated by summing over part of the terms with corresponding weights. The approximation reduces significantly the requirement for computer memory storage and enhances the numerical computation…
Maximization of submodular functions under various constraints is a fundamental problem that has been studied extensively. A powerful technique that has emerged and has been shown to be extremely effective for such problems is the…
Given an undirected graph $G$, we define a new object $H_G$, called the mp-chart of $G$, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the…
Integral transformations are used to estimate high order derivatives of various special functions. Applications are given to numerical integration, where estimates of high order derivatives of the integrand are needed to achieve bounds on…
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…
We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms…
Extended formulations are an important tool in polyhedral combinatorics. Many combinatorial optimization problems require an exponential number of inequalities when modeled as a linear program in the natural space of variables. However, by…
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…
We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of…
A nonlinear algebraic equation system of 5 variables is numerically solved, which is derived from the application of the Fourier transform to a differential equation system that allows modeling the behavior of the temperatures and the…
The Fourier transform of the indicator function of arbitrary polygons and polyhedra is computed for complex wavevectors. Using the divergence theorem and Stokes' theorem, closed expressions are obtained. Apparent singularities, all…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
Ultradiscretization is a limiting procedure transforming a given differential/difference equation into a ultradiscrete equation. Ultradiscrete equations are expressed by addition, subtraction and/or max. The procedure is expected to…
The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform was the motivation for the development of modern harmonic analysis. Its discrete version is also widely used in…
In this paper we consider the fundamental operations dilation and erosion of mathematical morphology. Many powerful image filtering operations are based on their combinations. We establish homomorphism between max-plus semi-ring of integers…
Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this…
We introduce a new numerical method for the computation of the inverse nonlinear Fourier transform and compare its computational complexity and accuracy to those of other methods available in the literature. For a given accuracy, the…