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This paper is dedicated to providing an introduction into multidimensional integer trigonometry. We start with an exposition of integer trigonometry in two dimensions, which was introduced in 2008, and use this to generalise these integer…
Transcendental functions, such as exponentials and logarithms, appear in a broad array of computational domains: from simulations in curvilinear coordinates, to interpolation, to machine learning. Unfortunately they are typically expensive…
We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for…
Sympathetic cooling of trapped ions has been established as a powerful technique for manipulation of non-laser-coolable ions (Raizen1992,Waki1992,Bowe1999,Barrett2003). For molecular ions, it promises vastly enhanced spectroscopic…
Randomized iterative algorithms have attracted much attention in recent years because they can approximately solve large-scale linear systems of equations without accessing the entire coefficient matrix. In this paper, we propose two novel…
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…
Various properties of the general two-center two-electron integral over the explicitly correlated exponential function are analyzed for the potential use in high precision calculations for diatomic molecules. A compact one dimensional…
The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional…
Correlation is a common technique for the detection of shifts. Its generalization to the multidimensional geometric correlation in Clifford algebras has been proven a useful tool for color image processing, because it additionally contains…
Polar codes are the first class of error correcting codes that provably achieve the channel capacity at infinite code length. They were selected for use in the fifth generation of cellular mobile communications (5G). In practical scenarios…
In this paper, we improved the performance of the contrast source inversion (CSI) method by incorporating a so-called cross-correlated cost functional, which interrelates the state error and the data error in the measurement domain. The…
The J-orthogonal matrix, also referred to as the hyperbolic orthogonal matrix, is a class of special orthogonal matrix in hyperbolic space, notable for its advantageous properties. These matrices are integral to optimization under…
A coarse grid correction (CGC) approach is proposed to enhance the efficiency of the matrix exponential and $\varphi$ matrix function evaluations. The approach is intended for iterative methods computing the matrix-vector products with…
Polar codes were introduced in 2009 by Arikan as the first efficient encoding and decoding scheme that is capacity achieving for symmetric binary-input memoryless channels. Recently, this code family was extended by replacing the…
Hyperbolic rotation is commonly used to effectively model knowledge graphs and their inherent hierarchies. However, existing hyperbolic rotation models rely on logarithmic and exponential mappings for feature transformation. These models…
Geometric optimization problems are at the core of many applications in geometry processing. The choice of a representation fitting an optimization problem can considerably simplify solving the problem. We consider the Nonlinear…
The radial polynomials of the 2D (circular) and 3D (spherical) Zernike functions are tabulated as powers of the radial distance. The reciprocal tabulation of powers of the radial distance in series of radial polynomials is also given, based…
The circular cartogram, also known as the Dorling map, is a widely used tool for visualizing statistical data. It represents regions as circles with their areas in proportion to the statistical values and requires circle displacement to…
Nowadays, a lot of old floor plans exist in printed form or are stored as scanned raster images. Slight rotations or shifts may occur during scanning. Bringing floor plans of this form into a machine readable form to enable further use,…
A modified Green operator is proposed as an improvement of Fourier-based numerical schemes commonly used for computing the electrical or thermal response of heterogeneous media. Contrary to other methods, the number of iterations necessary…