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Topic Review
Predictive Maintenance of Ball Bearing Systems
In the era of Industry 4.0 and beyond, ball bearings remain an important part of industrial systems. The failure of ball bearings can lead to plant downtime, inefficient operations, and significant maintenance expenses.
  • 105
  • 01 Feb 2024
Topic Review
Petr–Douglas–Neumann Theorem
In geometry, the Petr–Douglas–Neumann theorem (or the PDN-theorem) is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908. The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940 and also by B H Neumann (1909–2002) in 1941. The naming of the theorem as Petr–Douglas–Neumann theorem, or as the PDN-theorem for short, is due to Stephen B Gray. This theorem has also been called Douglas's theorem, the Douglas–Neumann theorem, the Napoleon–Douglas–Neumann theorem and Petr's theorem. The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals.
  • 315
  • 28 Nov 2022
Topic Review Peer Reviewed
Pandemic Equation and COVID-19 Evolution
The Pandemic Equation describes multiple pandemic waves and has been applied to describe the COVID-19 pandemic. Using the generalized approaches of solid-state physics, we derive the Pandemic Equation, which accounts for the effects of pandemic mitigation measures and multiple pandemic waves. The Pandemic Equation uses slow and fast time scales for “curve flattening” and describing vaccination and mitigation measures and the Scaled Fermi–Dirac distribution functions for describing transitions between pandemic waves. The Pandemic Equation parameters extracted from the pandemic curves can be used for comparing different scenarios of the pandemic evolution and for extrapolating the pandemic evolution curves for the periods of time on the order of the instantaneous Pandemic Equation characteristic time constant. The parameter extraction for multiple locations could also allow for uncertainty quantification for such pandemic evolution predictions.
  • 450
  • 19 Apr 2024
Topic Review
Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order
A DDE is a single-variable differential equation, usually called time, in which the derivative of the solution at a certain time is given in terms of the values of the solution at earlier times. Moreover, if the highest-order derivative of the solution appears both with and without delay, then the DDE is called of the neutral type. The neutral DDEs have many interesting applications in various branches of applied science, as these equations appear in the modeling of many technological phenomena. The problem of studying the oscillatory and nonoscillatory properties of DDEs has been a very active area of research in the past few decades.
  • 585
  • 19 Jul 2022
Topic Review
Oscillation of Solutions for Fractional Difference Equations
Oscillation is one of the important branches in applied mathematics and can be induced or destroyed by the introduction of nonlinearity, delay, or a stochastic term. The oscillation of differential and difference equations contributes to many realistic applications, such as torsional oscillations, the oscillation of heart beats, sinusoidal oscillation, voltage-controlled neuron models, and harmonic oscillation with damping. 
  • 598
  • 29 Apr 2022
Topic Review Video
Object 3d Reconstruction
The present paper summarized the existing methods of 3D reconstruction of objects by the Shape-From-Focus (SFF) method. This is a method for recovering depth from an image series of the same object taken with different focus settings, referred to as a multifocal image.
  • 474
  • 08 Oct 2021
Topic Review
Non-Standard Calculus
In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to infinitesimal calculus. It provides a rigorous justification that were previously considered merely heuristic. Nonrigourous calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless. Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."
  • 323
  • 27 Oct 2022
Topic Review
Moser–De Bruijn Sequence
In number theory, the Moser–De Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4, or equivalently the numbers whose binary representations are nonzero only in even positions. These numbers grow in proportion to the square numbers, and are the squares for a modified form of arithmetic without carrying. No two doubled sequence members differ by a square, and every non-negative integer has a unique representation as the sum of a sequence member and a doubled sequence member. This decomposition into sums can be used to define a bijection between the integers and pairs of integers, to define coordinates for the Z-order curve, and to construct inverse pairs of transcendental numbers with simple decimal representations. A simple recurrence relation allows values of the Moser–De Bruijn sequence to be calculated from earlier values, and can be used to prove that the Moser–De Bruijn sequence is a 2-regular sequence.
  • 303
  • 10 Oct 2022
Topic Review
Mersenne Conjectures
In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one.
  • 735
  • 10 Oct 2022
Topic Review Peer Reviewed
Mechanics and Mathematics in Ancient Greece
This entry presents an overview on how mechanics in Greece was linked to geometry. In ancient Greece, mechanics was about lifting heavy bodies, and mathematics almost coincided with geometry. Mathematics interconnected with mechanics at least from the 5th century BCE and became dominant in the Hellenistic period. The contributions by thinkers such as Aristotle, Euclid, and Archytas on fundamental problems such as that of the lever are sketched. This entry can be the starting point for a deeper investigation on the connections of the two disciplines through the ages until our present day.
  • 1.7K
  • 13 Apr 2022
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