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Scientific theoretic and applied developments, original discoveries and inventions: - Refined theory and applied algorithms for linear systems equations (LAS) resolution called “sequential and parallel orthogonalization” (versus known basic methods of Gauss- Jordan and Cramer’s rule), improved “special” cases of square-root and conjugated gradients algorithms. - Fourier analysis refinement, the best solutions for radix 2 Fast Fourier Transform (FFT), Fermat Number Transform (FNT), new 2D real FFT and 3D FFT; - A new branch in Fourier analysis called “Fourier-splines” convolutional approximation based on finite spline functions as truncated Fourier series. - The most productional FFT-structure algorithm of array ranking (sorting), totally in-place, N*log(N) complexity. - New mathematical 3D model approximation methods based on Fourier-series and/or Fourier-splines. - The development of a project for a new codec/decodec standard based on Fourier-splines for digital 1D audio- signals, 2D images, 3D tomography etc providing correct and more efficient compression versus jpeg/mpeg for HDTV/DVD etc. - The original fast rendering 3D engine with new features (rendering without z-buffer, arbitrary polygons including non-convex and nested ones, universal transparency, textures compression with Fourier-splines). - Full set of algorithmic solutions for medical and seismic tomography primarily refined methods “Marching Cubes” and “Ray Casting”. - Cloud-to-mesh transform for regular 3D models (devices Microsoft Kinect, Apple iPhone LIDAR etc) for applications of virtual web-suiting, interior restoration and dynamic moving objects registration. - Efficient targets recognition and navigation algorithmic solution for IR-warheads, sports shooting, biological applications (i.e. DNA photos processing) etc. - Gravitational orbits and atom electrons orbitals approximation on Fourier-series basis or equivalently Fourier-splines basis (just 5 spectral Fourier components). - Generalization of classic physical laws: - gravity (I. Newton, followed a proof of galaxies stability without so called “dark matter”), - electrostatic (J.O. Coulomb), Van-der-Waals, currents (A.M. Ampere) etc based on spread masses/charges model, - final resolution of singularity problem in zero. - Stars (including the Sun) in Galaxies movement theory development as generalization of classic Newton’s solution for planets and satellites in Fourier-series based on spread masses integrals; - Planck’s law parameters fast determination algorithms based on series for different wavelengths (frequencies).