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Topic Review
Sound Source Localization and Detection Methods
Many acoustic detection and localization methods have been developed. However, all of the methods require capturing the audio signal. Therefore, any method’s essential element and requirement is using an acoustic sensor. In addition to converting sound waves into an electrical signal, they also perform other functions, such as: reducing ambient noise, or capturing sounds with frequencies beyond the hearing range of the human ear. Classic methods have stood the test of time and are still widely used due to their simplicity, reliability, and effectiveness. There are three main mathematical methods for determining the sound source. These include triangulation, trilateration, and multilateration
  • 160
  • 28 Dec 2023
Biography
Slavik Jablan
After a long and brave battle with a serious illness, our dear friend and colleague Slavik Jablan passed away on 26 February 2015. The world is deprived of a remarkable mathematician, a great artist, a wonderful man and a dear friend. He made significant contributions to many areas of mathematics: geometry, group theory, mathematical crystallography, the theory of symmetry, antisymmetry, color
  • 758
  • 26 Sep 2022
Topic Review
Sine
In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). For an angle [math]\displaystyle{ x }[/math], the sine function is denoted simply as [math]\displaystyle{ \sin x }[/math]. More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series, or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin. The word "sine" (Latin "sinus") comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.
  • 1.8K
  • 11 Oct 2022
Topic Review
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions f(z1, z2, ... zn) on the space Cn of n-tuples of complex numbers. As in complex analysis, which is the case n = 1 but of a distinct character, these are not just any functions: they are supposed to be holomorphic or complex analytic, so that locally speaking they are power series in the variables zi. Equivalently, as it turns out, they are locally uniform limits of polynomials; or local solutions to the n-dimensional Cauchy–Riemann equations.
  • 340
  • 27 Oct 2022
Topic Review
Role of Visual Tools in Understanding Mathematical Culture
The term mathematical culture’ cannot be naturally defined; we will understand it in the same way as ‘good mathematics’ is understood by, i.e., good ways of solving mathematical issues, good mathematical techniques, good mathematical applications and cultivating of mathematical insight, creativity and beauty of mathematics. Cultivation of a mathematical culture means teaching how to see the roots of mathematics in reality (in nature, in society, but also in mathematics itself), getting to know the world of mathematical concepts, understanding this world and being able to apply it in a cultivated and correct way when solving various problems.
  • 170
  • 25 Jul 2023
Topic Review
Right Triangle
A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side a may be identified as the side adjacent to angle B and opposed to (or opposite) angle A, while side b is the side adjacent to angle A and opposed to angle B. If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.
  • 702
  • 17 Oct 2022
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.
  • 78
  • 01 Feb 2024
Biography
Pierre Deligne
Pierre René, Viscount Deligne (French: [dəliɲ]; born 3 October 1944) is a Belgian mathematician. He is known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre
  • 391
  • 08 Dec 2022
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.
  • 297
  • 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.
  • 340
  • 19 Apr 2024
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