Multiresolution Forecasting for Industrial Applications

The forecasting of univariate time series poses challenges in industrial applications if the seasonality varies. Typically, a non-varying seasonality of a time series is treated with a model based on Fourier theory or the aggregation of forecasts from multiple resolution levels. If the seasonality changes with time, various wavelet approaches for univariate forecasting are proposed with promising potential but without accessible software or a systematic evaluation of different wavelet models compared to state-of-the-art methods. In contrast, the advantage of the specific multiresolution forecasting proposed here is the convenience of a swiftly accessible implementation in R and Python combined with coefficient selection through evolutionary optimization which is evaluated in four different applications: scheduling of a call center, planning electricity demand, and predicting stocks and prices. The systematic benchmarking is based on out-of-sample forecasts resulting from multiple cross-validations with the error measure MASE and SMAPE for which the error distribution of each method and dataset is estimated and visualized with the mirrored density plot. The multiresolution forecasting performs equal to or better than twelve comparable state-of-the-art methods but does not require users to set parameters contrary to prior wavelet forecasting frameworks. This makes the method suitable for industrial applications.

Formation of Standing Waves Within Crystal Unitcell

We show that the historic Davisson-Germer experiment demonstrates formation of standing waves within nickel crystal unitcell. Cartesian Fourier transform cannot offer description in terms of standing waves because Cartesian Fourier theory cannot accommodate π in place of 2π. Thus, formation of standing waves within unitcell in Davisson-Germer experiment necessarily requires spherical polar coordinate description of crystal diffraction. Description in spherical polar coordinates permits to incorporate precision angles from the experiment for better convergence in structure determination calculations.

Existence and Continuous Dependence of the Local Solution of Non-Homogeneous KdV-K-S Equation in Periodic Sobolev Spaces

In this article, we prove that initial value problem associated to the non-homogeneous KdV-Kuramoto-Sivashinsky (KdV-K-S) equation in periodic Sobolev spaces has a local solution in with and the solution has continuous dependence with respect to the initial data and the non-homogeneous part of the problem. We do this in an intuitive way using Fourier theory and introducing a inspired by the work of Iorio [2] and Ayala and Romero [8]. Also, we prove the uniqueness solution of the homogeneous and non-homogeneous KdV-K-S equation, using its dissipative property, inspired by the work of Iorio [2] and Ayala and Romero [9].

Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering

To analyze and synthesize signals on networks or graphs, Fourier theory has been extended to irregular domains, leading to a so-called graph Fourier transform. Unfortunately, different from the traditional Fourier transform, each graph exhibits a different graph Fourier transform. Therefore to analyze the graph-frequency domain properties of a graph signal, the graph Fourier modes and graph frequencies must be computed for the graph under study. Although to find these graph frequencies and modes, a computationally expensive, or even prohibitive, eigendecomposition of the graph is required, there exist families of graphs that have properties that could be exploited for an approximate fast graph spectrum computation. In this work, we aim to identify these families and to provide a divide-and-conquer approach for computing an approximate spectral decomposition of the graph. Using the same decomposition, results on reducing the complexity of graph filtering are derived. These results provide an attempt to leverage the underlying topological properties of graphs in order to devise general computational models for graph signal processing.

Calculation of geometric dimensions of the levitation track

Background: The problemto reduce the metal content of the levitation track is important when we create the transport systems with magnetic suspension. Aim is develop the reasonable recommendations to choose the geometric dimensions of the levitation track. Methods: we usedthe main provisions of the electromagnetic field theory and the aspects of the Fourier theory. Results: the optimal values of the width and thickness of the levitation track are justified. Conclusion: when we choose the width of the track we should be guided by considerations related to material consumption and the appearance of the side electromagnetic forces. The values of these forces are bigger the greater the ratio of the transverse displacement of the excitation solenoid to the track width. From the point of view of electrodynamics the value of the order of several centimeters we can consider as the upper limit of the track thickness.

Modern Optics Simplified

This textbook is designed for use in a standard physics course on optics at the sophomore level. The book is an attempt to reduce the complexity of coverage found in Modem Optics to allow a student with only elementary calculus to learn the principles of optics and the modern Fourier theory of diffraction and imaging. Examples based on real optics engineering problems are contained in each chapter. Topics covered include aberrations with experimental examples, correction of chromatic aberration, explanation of coherence and the use of interference theory to design an antireflection coating, Fourier transform optics and its application to diffraction and imaging, use of gaussian wave theory, and fiber optics will make the text of interest as a textbook in Electrical and bioengineering as well as Physics. Students who take this course should have completed an introductory physics course and math courses through calculus Need for experience with differential equations is avoided and extensive use of vector theory is avoided by using a one dimensional theory of optics as often as possible. Maxwell’s equations are introduced to determine the properties of a light wave and the boundary conditions are introduced to characterize reflection and refraction. Most discussion is limited to reflection. The book provides an introduction to Fourier transforms. Many pictures, figures, diagrams are used to provide readers a good physical insight of Optics. There are some more difficult topics that could be skipped and they are indicated by boundaries in the text.