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Background theory
Published in Michael de Podesta, Understanding the Properties of Matter, 2020
This chapter is divided into four sections: Matter: Here we outline our assumptions about what constitutes the matter of the world.The electromagnetic field: Here we look at the general properties of all the known fields, and focus on some detailed properties of the electromagnetic field, which is by far the most important from our point of view.Classical and quantum mechanics: Having looked at the components of the world, we will look at some of the tools we use to understand how the components interact.Thermodynamics and statistical mechanics: Finally we look at techniques for calculating the properties of large numbers of particles.
Algebraic Geometry
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
In this chapter, we will focus on hyperelliptic curves of genus 2. They have been an object of much mathematical interest since the eighteenth century and there exists continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryptography, etc. The study of hyperelliptic curves is articulated well in Cohen and Frey (2006), Frey and Shaska (2019), Simpson (2019), and Silverman (1986), and that is where much of this material is derived from.
Direct Current (dc) Electronics
Published in Dale R. Patrick, Stephen W. Fardo, Electricity and Electronics Fundamentals, 2020
Dale R. Patrick, Stephen W. Fardo
Scientific notation simplifies multiplying and dividing large numbers of small decimals. For example: 4800×0.000045×800×0.0058=(4.8×103)×(4.5×10−5)×(8×102)×5.8×10−3)=(4.8×4.5×8×5.8)×(103−5+2−3)=1002.24×10−3=1.0022495,000÷0.0008=9.5×1048×104=9.5×104(−4)8=9.5×108=1.1875×108=118,750,000
Frequency-Dependent Discrete Implicit Monte Carlo Scheme for the Radiative Transfer Equation
Published in Nuclear Science and Engineering, 2023
Elad Steinberg, Shay I. Heizler
For a more complete description of the DIMC algorithms, the algorithm of photon creation, the single photon propagation, and the algorithm of material particle merging, see Ref. 31. The only difference in the frequency-dependent algorithm is that when a new radiation photon is emitted or when there is an effective scattering event, the photon’s new frequency is sampled in the same manner as in the emission case in the frequency-dependent IMC and ISMC. The propagation is determined via the frequency dependency of the opacity. We note that as in the classic IMC, the Monte Carlo particles that represent the photons are not photons per se, but rather “packets of energy,” that represent large numbers of photons (calculating the real number of photons is unrealistic). As a consequence, there is not a direct relation between the frequency of the energy packet and its energy, which may represent many low or (high)-energy photons.
A comprehensive review of froth surface monitoring as an aid for grade and recovery prediction of flotation process. Part B: Texture and dynamic features
Published in Energy Sources, Part A: Recovery, Utilization, and Environmental Effects, 2023
Fardis Nakhaei, Mehdi Irannajad, Sima Mohammadnejad
Over the last few years, a novel branch of machine learning known as deep learning has developed the state of the art technology in computer vision. Convolutional neural networks (CNNs) have been particularly successful at tasks such as image classification and object detection. CNNs are on the basis of a large number of convolutional layers achieved from convolution of image using many small size kernels. These large numbers of kernels extract a vast number of characteristics that are hidden in the image. However, the use of such features is not straightforward and needs activation functions and pooling. After feature extraction, they will be connected to the outputs using a fully coupled NN (Karimpouli and Tahmasebi 2019). At first, Montes-Atenasa et al. (2016) successfully predicted bubble size and bubble rate in froth flotation-like slurry columns from computational fluid dynamics (CFD) data by applying deep neural networks (DNN) with relatively low error.
Stability reliability of a cutting slope in Laohuzui Hydropower Station in Tibet of China
Published in Geomatics, Natural Hazards and Risk, 2019
Hongkai Dong, Fei Ye, Wenxi Fu
The Monte Carlo method is based on the law of large numbers in mathematics. Theoretically, the more trial runs are used in an analysis, the more accurate the solution will be. However, how many trials are required in a probabilistic analysis? According to the studies by Harr (1987), the number of Monte Carlo trials increases geometrically with the level of confidence and the number of variables, and the empirical equation is given by where N is the number of Monte Carlo trials; a is the desired level of confidence (0–100%), expressed in decimal form; Ka/2 is the normal standard deviate corresponding to the level of confidence, expressed as , and which can be obtained by inquiring the normal distribution table; and n is the number of random variables.