1.
Sinusoid (blood vessel)
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A sinusoid is a large blood vessel that is a type of capillary similar to a fenestrated endothelium. Sinusoids are actually classified as a type of open pore capillary as opposed to continuous, fenestrated capillaries have diaphragms that cover the pores whereas open pore capillaries lack a diaphragm and just have an open pore. The open pores of endothelial cells increase their permeability. In addition, permeability is increased by large inter-cellular clefts and fewer tight junctions, the level of permeability is such as to allow small and medium-sized proteins such as albumin to readily enter and leave the blood stream. Sinusoids are found in the liver, lymphoid tissue, endocrine organs, and hematopoietic organs such as the bone marrow, sinusoids found within terminal villi of the placenta are not comparable to these because they possess a continuous endothelium and complete basal lamina. This word was first used in 1893, liver sinusoids, which help hepatocytes transport a small number of molecules to and from the blood stream
2.
Trigonometric functions
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
3.
Curve
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
4.
Oscillation
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Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. The term vibration is used to describe mechanical oscillation. Familiar examples of oscillation include a swinging pendulum and alternating current power, the simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension. Such a system may be approximated on an air table or ice surface, the system is in an equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium. However, in moving the back to the equilibrium position, it has acquired momentum which keeps it moving beyond that position. If a constant force such as gravity is added to the system, the time taken for an oscillation to occur is often referred to as the oscillatory period. All real-world oscillator systems are thermodynamically irreversible and this means there are dissipative processes such as friction or electrical resistance which continually convert some of the energy stored in the oscillator into heat in the environment. Thus, oscillations tend to decay with time there is some net source of energy into the system. The simplest description of this process can be illustrated by oscillation decay of the harmonic oscillator. In addition, a system may be subject to some external force. In this case the oscillation is said to be driven, some systems can be excited by energy transfer from the environment. This transfer typically occurs where systems are embedded in some fluid flow, at sufficiently large displacements, the stiffness of the wing dominates to provide the restoring force that enables an oscillation. The harmonic oscillator and the systems it models have a degree of freedom. More complicated systems have more degrees of freedom, for two masses and three springs. In such cases, the behavior of each variable influences that of the others and this leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks mounted on a wall will tend to synchronise. This phenomenon was first observed by Christiaan Huygens in 1665, more special cases are the coupled oscillators where energy alternates between two forms of oscillation
5.
Continuous wave
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A continuous wave or continuous waveform is an electromagnetic wave of constant amplitude and frequency, a sine wave. In mathematical analysis, it is considered to be of infinite duration, continuous wave is also the name given to an early method of radio transmission, in which a sinusoidal carrier wave is switched on and off. Information is carried in the duration of the on and off periods of the signal. Very early radio transmitters used a gap to produce radio-frequency oscillations in the transmitting antenna. The signals produced by these spark-gap transmitters consisted of strings of brief pulses of radio frequency oscillations which died out rapidly to zero. The disadvantage of damped waves was that their energy was spread over a wide band of frequencies. As a result they produced electromagnetic interference that spread over the transmissions of stations at other frequencies and this motivated efforts to produce radio frequency oscillations that decayed more slowly, had less damping. Manufacturers produced spark transmitters which generated long ringing waves with minimal damping and it was realized that the ideal radio wave for radiotelegraphic communication would be a sine wave with zero damping, a continuous wave. An unbroken continuous sine wave theoretically has no bandwidth, all its energy is concentrated at a single frequency, continuous waves could not be produced with an electric spark, but were achieved with the vacuum tube electronic oscillator, invented around 1913 by Edwin Armstrong and Alexander Meissner. Damped wave spark transmitters were replaced by continuous wave vacuum tube transmitters around 1920, what is transmitted in the extra bandwidth used by a transmitter that turns on/off more abruptly is known as key clicks. Certain types of power used in transmission may increase the effect of key clicks. The first transmitters capable of producing continuous wave, the Alexanderson alternator and vacuum tube oscillators, early radio transmitters could not be modulated to transmit speech, and so CW radio telegraphy was the only form of communication available. Continuous-wave radio was called radiotelegraphy because like the telegraph, it worked by means of a switch to transmit Morse code. However, instead of controlling the electricity in a cross-country wire and this mode is still in common use by amateur radio operators. In military communications and amateur radio, the terms CW and Morse code are used interchangeably. Morse code may be sent using direct current in wires, sound, or light, a carrier wave is keyed on and off to represent the dots and dashes of the code elements. The carriers amplitude and frequency remains constant during each code element, at the receiver, the received signal is mixed with a heterodyne signal from a BFO to change the radio frequency impulses to sound. Though most commercial traffic has now ceased operation using Morse it is popular with amateur radio operators
6.
Graph of a function
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In mathematics, the graph of a function f is the collection of all ordered pairs. If the function x is a scalar, the graph is a two-dimensional graph. If the function x is an ordered pair of real numbers, the graph is the collection of all ordered triples. Graphing on a Cartesian plane is referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the representation of the function. The concept of the graph of a function is generalized to the graph of a relation, note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers, to test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of y, if the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line y = x. In science, engineering, technology, finance, and other areas, in the simplest case one variable is plotted as a function of another, typically using rectangular axes, see Plot for details. In the modern foundation of mathematics known as set theory, a function, F = { a, if x =1, d, if x =2, c, if x =3, is. The graph of the polynomial on the real line f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, the graph of the trigonometric function f = sin cos is. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface, oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function, f = −2 Given a function f of n variables, x 1, …, x n, the normal to the graph is. This is seen by considering the graph as a set of the function g = f − z. The graph of a function is contained in a Cartesian product of sets, fibre bundles arent cartesian products, but appear to be up close. There is a notion of a graph on a fibre bundle called a section
7.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
8.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
9.
Engineering
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The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, the word engine itself is of even older origin, ultimately deriving from the Latin ingenium, meaning innate quality, especially mental power, hence a clever invention. The earliest civil engineer known by name is Imhotep, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. John Smeaton was the first self-proclaimed civil engineer and is regarded as the father of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
10.
Signal processing
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According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. Oppenheim and Schafer further state that the digitalization or digital refinement of techniques can be found in the digital control systems of the 1940s and 1950s. Feature extraction, such as understanding and speech recognition. Quality improvement, such as reduction, image enhancement. Including audio compression, image compression, and video compression and this involves linear electronic circuits as well as non-linear ones. The former are, for instance, passive filters, active filters, additive mixers, integrators, non-linear circuits include compandors, multiplicators, voltage-controlled filters, voltage-controlled oscillators and phase-locked loops. Continuous-time signal processing is for signals that vary with the change of continuous domain, the methods of signal processing include time domain, frequency domain, and complex frequency domain. This technology was a predecessor of digital processing, and is still used in advanced processing of gigahertz signals. Digital signal processing is the processing of digitized discrete-time sampled signals, processing is done by general-purpose computers or by digital circuits such as ASICs, field-programmable gate arrays or specialized digital signal processors. Typical arithmetical operations include fixed-point and floating-point, real-valued and complex-valued, other typical operations supported by the hardware are circular buffers and look-up tables. Examples of algorithms are the Fast Fourier transform, finite impulse response filter, Infinite impulse response filter, nonlinear signal processing involves the analysis and processing of signals produced from nonlinear systems and can be in the time, frequency, or spatio-temporal domains. Nonlinear systems can produce complex behaviors including bifurcations, chaos, harmonics
11.
Amplitude
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The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the extreme values. In older texts the phase is called the amplitude. Peak-to-peak amplitude is the change between peak and trough, with appropriate circuitry, peak-to-peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an oscilloscope. Peak-to-peak is a measurement on an oscilloscope, the peaks of the waveform being easily identified and measured against the graticule. This remains a common way of specifying amplitude, but sometimes other measures of amplitude are more appropriate. In audio system measurements, telecommunications and other areas where the measurand is a signal that swings above and below a value but is not sinusoidal. If the reference is zero, this is the absolute value of the signal, if the reference is a mean value. Semi-amplitude means half the peak-to-peak amplitude, some scientists use amplitude or peak amplitude to mean semi-amplitude, that is, half the peak-to-peak amplitude. It is the most widely used measure of orbital wobble in astronomy, the RMS of the AC waveform. For complicated waveforms, especially non-repeating signals like noise, the RMS amplitude is used because it is both unambiguous and has physical significance. For example, the power transmitted by an acoustic or electromagnetic wave or by an electrical signal is proportional to the square of the RMS amplitude. For alternating current electric power, the practice is to specify RMS values of a sinusoidal waveform. One property of root mean square voltages and currents is that they produce the same heating effect as direct current in a given resistance, the peak-to-peak value is used, for example, when choosing rectifiers for power supplies, or when estimating the maximum voltage that insulation must withstand. Some common voltmeters are calibrated for RMS amplitude, but respond to the value of a rectified waveform. Many digital voltmeters and all moving coil meters are in this category, the RMS calibration is only correct for a sine wave input since the ratio between peak, average and RMS values is dependent on waveform. If the wave shape being measured is greatly different from a sine wave, true RMS-responding meters were used in radio frequency measurements, where instruments measured the heating effect in a resistor to measure current. The advent of microprocessor controlled meters capable of calculating RMS by sampling the waveform has made true RMS measurement commonplace
12.
Frequency
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Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as frequency, which emphasizes the contrast to spatial frequency. The period is the duration of time of one cycle in a repeating event, for example, if a newborn babys heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as vibrations, audio signals, radio waves. For cyclical processes, such as rotation, oscillations, or waves, in physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by the Greek letter ν or ν. For a simple motion, the relation between the frequency and the period T is given by f =1 T. The SI unit of frequency is the hertz, named after the German physicist Heinrich Hertz, a previous name for this unit was cycles per second. The SI unit for period is the second, a traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. As a matter of convenience, longer and slower waves, such as ocean surface waves, short and fast waves, like audio and radio, are usually described by their frequency instead of period. Spatial frequency is analogous to temporal frequency, but the axis is replaced by one or more spatial displacement axes. Y = sin = sin d θ d x = k Wavenumber, in the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has a relationship to the wavelength. Even in dispersive media, the frequency f of a wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave. In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes, f = c λ. When waves from a monochrome source travel from one medium to another, their remains the same—only their wavelength. For example, if 71 events occur within 15 seconds the frequency is, the latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an error in the calculated frequency of Δf = 1/, or a fractional error of Δf / f = 1/ where Tm is the timing interval. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small, an older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope
13.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
14.
Radian
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The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
15.
Phase (waves)
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Phase is the position of a point in time on a waveform cycle. A complete cycle is defined as the interval required for the waveform to return to its initial value. The graphic to the right shows how one cycle constitutes 360° of phase, the graphic also shows how phase is sometimes expressed in radians, where one radian of phase equals approximately 57. 3°. Phase can also be an expression of relative displacement between two corresponding features of two waveforms having the same frequency, in sinusoidal functions or in waves phase has two different, but closely related, meanings. One is the angle of a sinusoidal function at its origin and is sometimes called phase offset or phase difference. Another usage is the fraction of the cycle that has elapsed relative to the origin. Phase shift is any change that occurs in the phase of one quantity and this symbol, φ is sometimes referred to as a phase shift or phase offset because it represents a shift from zero phase. For infinitely long sinusoids, a change in φ is the same as a shift in time, if x is delayed by 14 of its cycle, it becomes, x = A ⋅ cos = A ⋅ cos whose phase is now φ − π2. It has been shifted by π2 radians, Phase difference is the difference, expressed in degrees or time, between two waves having the same frequency and referenced to the same point in time. Two oscillators that have the frequency and no phase difference are said to be in phase. Two oscillators that have the frequency and different phases have a phase difference. The amount by which such oscillators are out of phase with each other can be expressed in degrees from 0° to 360°, if the phase difference is 180 degrees, then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then interference will occur. It is common for waves of electromagnetic, acoustic or other energy to become superposed in their transmission medium, when that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes, time is sometimes used to express position within the cycle of an oscillation. A phase difference is analogous to two athletes running around a track at the same speed and direction but starting at different positions on the track. They pass a point at different instants in time, but the time difference between them is a constant - same for every pass since they are at the same speed and in the same direction. If they were at different speeds, the difference is undefined
16.
Fourier analysis
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In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics, for example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis, in mathematics, the term Fourier analysis often refers to the study of both operations. The decomposition process itself is called a Fourier transformation and its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations. Each transform used for analysis has an inverse transform that can be used for synthesis. This wide applicability stems from many useful properties of the transforms, The transforms are linear operators and, the exponential functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. Therefore, the behavior of a linear time-invariant system can be analyzed at each frequency independently, the discrete version of the Fourier transform can be evaluated quickly on computers using Fast Fourier Transform algorithms. In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum, the FT method is used to decode the measured signals and record the wavelength data. Fourier transformation is also useful as a representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation of small pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, a large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. One function is transformed into another, and the operation is reversible, evaluating this quantity for all values of f produces the frequency-domain function. Then s can be represented as a recombination of complex exponentials of all frequencies, s = ∫ − ∞ ∞ S ⋅ e 2 i π f t d f. The complex number, S, conveys both amplitude and phase of frequency f, see Fourier series for more information, including the historical development. The DTFT is the dual of the time-domain Fourier series. Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function and that is a cornerstone in the foundation of digital signal processing. Furthermore, under idealized conditions one can theoretically recover S and s exactly
17.
Wavelength
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In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency. Wavelength is commonly designated by the Greek letter lambda, the concept can also be applied to periodic waves of non-sinusoidal shape. The term wavelength is also applied to modulated waves. Wavelength depends on the medium that a wave travels through, examples of wave-like phenomena are sound waves, light, water waves and periodic electrical signals in a conductor. A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric, water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary, wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle near the waters surface moves in a circle of the same diameter as the wave height. The range of wavelengths or frequencies for wave phenomena is called a spectrum, the name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, in a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the speed is the speed of light. Thus the wavelength of a 100 MHz electromagnetic wave is about, the wavelength of visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm. For sound waves in air, the speed of sound is 343 m/s, the wavelengths of sound frequencies audible to the human ear are thus between approximately 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light, a standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed, the stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Consequently, wavelength, period, and wave velocity are related just as for a traveling wave, for example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. In that case, the k, the magnitude of k, is still in the same relationship with wavelength as shown above
18.
Plane wave
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In the physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes. The solutions in x → of n → c ⋅ x → − t = c o n s t. comprise the plane with normal vector n →, thus, the points of equal field value of A always form a plane in space. This plane then shifts with time t, along the direction of propagation n → with velocity c, the term is often used to mean the special case of a monochromatic, homogeneous plane wave. A monochromatic plane wave is one in which the amplitude is a function of x and t. A homogeneous plane wave is one in which the planes of constant phase are perpendicular to the direction of propagation n →, however, many waves are approximately plane waves in a localized region of space. For example, a source such as an antenna produces a field that is approximately a plane wave far from the antenna in its far-field region. Two functions that meet the criteria of having a constant frequency. One of the simplest ways to use such a sinusoid involves defining it along the direction of the x-axis, the equation below, which is illustrated toward the right, uses the cosine function to represent a harmonic and homogeneous plane wave travelling in the positive x direction. A = A o cos In the above equation, A is the magnitude or disturbance of the wave at a point in space. An example would be to let A represent the variation of air pressure relative to the norm in the case of a sound wave, a o is the amplitude of the wave which is the peak magnitude of the oscillation. K is the wave number or more specifically the angular wave number and equals 2π/λ. K has the units of radians per unit distance and is a measure of how rapidly the disturbance changes over a distance at a particular point in time. X is a point along the x-axis, Y and z are not part of the equation because the waves magnitude and phase are the same at every point on any given y-z plane. This equation defines what that magnitude and phase are, ω is the wave’s angular frequency which equals 2π/T, where T is the period of the wave. ω has the units of radians per unit time and is a measure of how rapidly the disturbance changes over a length of time at a particular point in space. T is a point in time φ is the phase shift of the wave and has the units of radians. Note that a phase shift, at a given moment of time. A phase shift of 2π radians shifts it exactly one wavelength, other formalisms which directly use the wave’s wavelength λ, period T, frequency f and velocity c are below
19.
Dot product
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In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. Sometimes it is called inner product in the context of Euclidean space, algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them, the dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance, the equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. In such a presentation, the notions of length and angles are not primitive, so the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry. For instance, in space, the dot product of vectors and is. In Euclidean space, a Euclidean vector is an object that possesses both a magnitude and a direction. A vector can be pictured as an arrow and its magnitude is its length, and its direction is the direction that the arrow points. The magnitude of a vector a is denoted by ∥ a ∥, the dot product of two Euclidean vectors a and b is defined by a ⋅ b = ∥ a ∥ ∥ b ∥ cos , where θ is the angle between a and b. In particular, if a and b are orthogonal, then the angle between them is 90° and a ⋅ b =0. The scalar projection of a Euclidean vector a in the direction of a Euclidean vector b is given by a b = ∥ a ∥ cos θ, where θ is the angle between a and b. In terms of the definition of the dot product, this can be rewritten a b = a ⋅ b ^. The dot product is thus characterized geometrically by a ⋅ b = a b ∥ b ∥ = b a ∥ a ∥. The dot product, defined in this manner, is homogeneous under scaling in each variable and it also satisfies a distributive law, meaning that a ⋅ = a ⋅ b + a ⋅ c. These properties may be summarized by saying that the dot product is a bilinear form, moreover, this bilinear form is positive definite, which means that a ⋅ a is never negative and is zero if and only if a =0. En are the basis vectors in Rn, then we may write a = = ∑ i a i e i b = = ∑ i b i e i. The vectors ei are a basis, which means that they have unit length and are at right angles to each other. Hence since these vectors have unit length e i ⋅ e i =1 and since they form right angles with each other, thus in general we can say that, e i ⋅ e j = δ i j
20.
Wave
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In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium. Frequency refers to the addition of time, wave motion transfers energy from one point to another, which displace particles of the transmission medium–that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations, around almost fixed locations, there are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed, restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors, when the molecules collide, they also bounce away from each other. This keeps the molecules from continuing to travel in the direction of the wave, the second main type, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields generated by charged particles. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this varies depending on the type of wave. Further, the behavior of particles in quantum mechanics are described by waves, in addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields. While mechanical waves can be transverse and longitudinal, all electromagnetic waves are transverse in free space. A single, all-encompassing definition for the wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value, however, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon as a results in a blurred line. The term wave is often understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium and it may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity and this difference in origin introduces certain wave characteristics particular to the properties of the medium involved
21.
Wind wave
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In fluid dynamics, wind waves, or wind-generated waves, are surface waves that occur on the free surface of bodies of water. They result from the wind blowing over an area of fluid surface, Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft high, when directly generated and affected by local winds, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells, more generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago, wind waves in the ocean are called ocean surface waves. Wind waves have an amount of randomness, subsequent waves differ in height, duration. The key statistics of wind waves in evolving sea states can be predicted with wind wave models, although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves. The great majority of large breakers seen at a result from distant winds. Water depth All of these work together to determine the size of wind waves. Further exposure to that wind could only cause a dissipation of energy due to the breaking of wave tops. Waves in an area typically have a range of heights. For weather reporting and for analysis of wind wave statistics. This figure represents an average height of the highest one-third of the waves in a time period. The significant wave height is also the value a trained observer would estimate from visual observation of a sea state, given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm. Wave formation on a flat water surface by wind is started by a random distribution of normal pressure of turbulent wind flow over the water. This pressure fluctuation produces normal and tangential stresses in the surface water and it is assumed that, The water is originally at rest. There is a distribution of normal pressure to the water surface from the turbulent wind. Correlations between air and water motions are neglected, the second mechanism involves wind shear forces on the water surface
22.
Sound
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In physics, sound is a vibration that propagates as a typically audible mechanical wave of pressure and displacement, through a transmission medium such as air or water. In physiology and psychology, sound is the reception of such waves, humans can hear sound waves with frequencies between about 20 Hz and 20 kHz. Sound above 20 kHz is ultrasound and below 20 Hz is infrasound, other animals have different hearing ranges. Acoustics is the science that deals with the study of mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound. A scientist who works in the field of acoustics is an acoustician, an audio engineer, on the other hand, is concerned with the recording, manipulation, mixing, and reproduction of sound. Auditory sensation evoked by the oscillation described in, sound can propagate through a medium such as air, water and solids as longitudinal waves and also as a transverse wave in solids. The sound waves are generated by a source, such as the vibrating diaphragm of a stereo speaker. The sound source creates vibrations in the surrounding medium, as the source continues to vibrate the medium, the vibrations propagate away from the source at the speed of sound, thus forming the sound wave. At a fixed distance from the source, the pressure, velocity, at an instant in time, the pressure, velocity, and displacement vary in space. Note that the particles of the medium do not travel with the sound wave and this is intuitively obvious for a solid, and the same is true for liquids and gases. During propagation, waves can be reflected, refracted, or attenuated by the medium, the behavior of sound propagation is generally affected by three things, A complex relationship between the density and pressure of the medium. This relationship, affected by temperature, determines the speed of sound within the medium, if the medium is moving, this movement may increase or decrease the absolute speed of the sound wave depending on the direction of the movement. For example, sound moving through wind will have its speed of propagation increased by the speed of the if the sound and wind are moving in the same direction. If the sound and wind are moving in opposite directions, the speed of the wave will be decreased by the speed of the wind. Medium viscosity determines the rate at which sound is attenuated, for many media, such as air or water, attenuation due to viscosity is negligible. When sound is moving through a medium that does not have constant physical properties, the mechanical vibrations that can be interpreted as sound can travel through all forms of matter, gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium, sound cannot travel through a vacuum. Sound is transmitted through gases, plasma, and liquids as longitudinal waves and it requires a medium to propagate
23.
Light
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Light is electromagnetic radiation within a certain portion of the electromagnetic spectrum. The word usually refers to light, which is visible to the human eye and is responsible for the sense of sight. Visible light is defined as having wavelengths in the range of 400–700 nanometres, or 4.00 × 10−7 to 7.00 × 10−7 m. This wavelength means a range of roughly 430–750 terahertz. The main source of light on Earth is the Sun, sunlight provides the energy that green plants use to create sugars mostly in the form of starches, which release energy into the living things that digest them. This process of photosynthesis provides virtually all the used by living things. Historically, another important source of light for humans has been fire, with the development of electric lights and power systems, electric lighting has effectively replaced firelight. Some species of animals generate their own light, a process called bioluminescence, for example, fireflies use light to locate mates, and vampire squids use it to hide themselves from prey. Visible light, as all types of electromagnetic radiation, is experimentally found to always move at this speed in a vacuum. In physics, the term sometimes refers to electromagnetic radiation of any wavelength. In this sense, gamma rays, X-rays, microwaves and radio waves are also light, like all types of light, visible light is emitted and absorbed in tiny packets called photons and exhibits properties of both waves and particles. This property is referred to as the wave–particle duality, the study of light, known as optics, is an important research area in modern physics. Generally, EM radiation, or EMR, is classified by wavelength into radio, microwave, infrared, the behavior of EMR depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths, when EMR interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries. There exist animals that are sensitive to various types of infrared, infrared sensing in snakes depends on a kind of natural thermal imaging, in which tiny packets of cellular water are raised in temperature by the infrared radiation. EMR in this range causes molecular vibration and heating effects, which is how these animals detect it, above the range of visible light, ultraviolet light becomes invisible to humans, mostly because it is absorbed by the cornea below 360 nanometers and the internal lens below 400. Furthermore, the rods and cones located in the retina of the eye cannot detect the very short ultraviolet wavelengths and are in fact damaged by ultraviolet. Many animals with eyes that do not require lenses are able to detect ultraviolet, by quantum photon-absorption mechanisms, various sources define visible light as narrowly as 420 to 680 to as broadly as 380 to 800 nm
24.
Cosine
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In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides, trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, more precise definitions are detailed below. Trigonometric functions have a range of uses including computing unknown lengths. In this use, trigonometric functions are used, for instance, in navigation, engineering, a common use in elementary physics is resolving a vector into Cartesian coordinates. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another and that is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides and it is these ratios that the trigonometric functions express. To define the functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows, The hypotenuse is the side opposite the right angle, the hypotenuse is always the longest side of a right-angled triangle. The opposite side is the side opposite to the angle we are interested in, in this side a. The adjacent side is the side having both the angles of interest, in this case side b, in ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation. The following definitions apply to angles in this 0° – 90° range and they can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case sin A = opposite hypotenuse = a h and this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angle A, since all such triangles are similar. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, in our case cos A = adjacent hypotenuse = b h
25.
Head start (positioning)
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In positioning, a head start is a start in advance of the starting position of others in competition, or simply toward the finish line or desired outcome. Depending on the situation, a start may be inherent, obtained by special privilege, earned through ones accomplishments. While not guaranteeing success, a head start will increase such chances, in competitive sports, such as a race, a head start refers to a start ahead of other competitors, allowing a shorter distance to the finish line. The idea of a start may seem unfair. But in some cases, a start is an advantage that may be earned by one more of the competitors. Also, adults who are racing against children may provide children with a start, knowing the children are slower. In modern pentathlon, the leader after the first four events takes a head start in the 3,000 metre cross country event based on his lead over his opponent. The first competitor to the line wins. In baseball, base runners who are attempting to steal bases or advance more bases in the event of a hit may attempt to get a start off their base before the pitcher throws his pitch. In traffic, at a red light, many motorists will start to inch up in anticipation of the light turning green and this will give them a head start on other motorists on the road. A head start could be a start in an attempt to achieve some goal. In such cases, the start is usually earned by working harder or by using more efficient means of reaching that point
26.
Harmonics
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A harmonic is any member of the harmonic series, the divergent infinite series. Every term of the series after the first is the mean of the neighboring terms. The phrase harmonic mean likewise derives from music, the term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves, a harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the frequency, the sum of harmonics is also periodic at that frequency. On strings, harmonics that are bowed have a glassy, pure tone, harmonics may also be called overtones, partials or upper partials. In some music contexts, the harmonic, overtone and partial are used fairly interchangeably. Most acoustic instruments emit complex tones containing many individual partials, rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials. Oscillators that produce harmonic partials behave somewhat like one-dimensional resonators, and are long and thin. Wind instruments whose air column is open at one end, such as trumpets and clarinets. However they only produce partials matching the odd harmonics, at least in theory, the reality of acoustic instruments is such that none of them behaves as perfectly as the somewhat simplified theoretical models would predict. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials, antique singing bowls are known for producing multiple harmonic partials or multiphonics. An overtone is any partial higher than the lowest partial in a compound tone, the relative strengths and frequency relationships of the component partials determine the timbre of an instrument. This chart demonstrates how the three types of names are counted, In many musical instruments, it is possible to play the upper harmonics without the note being present. In a simple case this has the effect of making the note go up in pitch by an octave, in some cases it also changes the timbre of the note. This is part of the method of obtaining higher notes in wind instruments. The extended technique of playing multiphonics also produces harmonics, on string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch
27.
Timbre
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The physical characteristics of sound that determine the perception of timbre include spectrum and envelope. Singers and instrumental musicians can change the timbre of the music they are singing/playing by using different singing or playing techniques, for example, a violinist can use different bowing styles or play on different parts of the string to obtain different timbres. On electric guitar and electric piano, performers can change the timbre using effects units, in simple terms, timbre is what makes a particular musical sound have a different sound from another, even when they have the same pitch and loudness. For instance, it is the difference in sound between a guitar and a playing the same note at the same volume. Experienced musicians are able to distinguish between different instruments of the type based on their varied timbres, even if those instruments are playing notes at the same pitch. Tone quality and tone color are synonyms for timbre, as well as the attributed to a single instrument. However, the texture can also refer to the type of music, such as multiple. The sound of an instrument may be described with words such as bright, dark, warm, harsh. There are also colors of noise, such as pink and white, in visual representations of sound, timbre corresponds to the shape of the image. The Acoustical Society of America Acoustical Terminology definition 12, Timbre has been called. the psychoacousticians multidimensional waste-basket category for everything that cannot be labeled pitch or loudness. Many commentators have attempted to decompose timbre into component attributes, the richness of a sound or note a musical instrument produces is sometimes described in terms of a sum of a number of distinct frequencies. The lowest frequency is called the frequency, and the pitch it produces is used to name the note. The dominant frequency is the frequency that is most heard, for example, the dominant frequency for the transverse flute is double the fundamental frequency. Other significant frequencies are called overtones of the frequency, which may include harmonics. Harmonics are whole number multiples of the frequency, such as ×2, ×3, ×4. There are also sometimes subharmonics at whole number divisions of the fundamental frequency, most instruments produce harmonic sounds, but many instruments produce partials and inharmonic tones, such as cymbals and other indefinite-pitched instruments. When the tuning note in an orchestra or concert band is played, each instrument in the orchestra or concert band produces a different combination of these frequencies, as well as harmonics and overtones. The sound waves of the different frequencies overlap and combine, the timbre of a sound is also greatly affected by the following aspects of its envelope, attack time and characteristics, decay, sustain, release and transients
28.
Musical note
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In music, the term note has three primary meanings, A sign used in musical notation to represent the relative duration and pitch of a sound, A pitched sound itself. Notes are the blocks of much written music, discretizations of musical phenomena that facilitate performance, comprehension. In the former case, one note to refer to a specific musical event, in the latter. Two notes with fundamental frequencies in an equal to any integer power of two are perceived as very similar. Because of that, all notes with these kinds of relations can be grouped under the pitch class. However, within the English-speaking and Dutch-speaking world, pitch classes are represented by the first seven letters of the Latin alphabet. A few European countries, including Germany, adopt an almost identical notation, the eighth note, or octave, is given the same name as the first, but has double its frequency. The name octave is also used to indicate the span between a note and another with double frequency, for example, the now-standard tuning pitch for most Western music,440 Hz, is named a′ or A4. There are two systems to define each note and octave, the Helmholtz pitch notation and the scientific pitch notation. Letter names are modified by the accidentals, a sharp ♯ raises a note by a semitone or half-step, and a flat ♭ lowers it by the same amount. In modern tuning a half step has a ratio of 12√2. The accidentals are written after the name, so, for example, F♯ represents F-sharp, B♭ is B-flat. Additional accidentals are the double-sharp, raising the frequency by two semitones, and double-flat, lowering it by that amount, in musical notation, accidentals are placed before the note symbols. Systematic alterations to the seven lettered pitches in the scale can be indicated by placing the symbols in the key signature, explicitly noted accidentals can be used to override this effect for the remainder of a bar. A special accidental, the natural symbol ♮, is used to indicate an unmodified pitch, effects of key signature and local accidentals do not accumulate. If the key signature indicates G♯, a flat before a G makes it G♭, though often this type of rare accidental is expressed as a natural. Likewise, a sharp sign on a key signature with a single sharp ♯ indicates only a double sharp. Assuming enharmonicity, many accidentals will create equivalences between pitches that are written differently, for instance, raising the note B to B♯ is equal to the note C
29.
Square wave
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A square wave is a non-sinusoidal periodic waveform, in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. The transition between minimum to maximum is instantaneous for a square wave, this is not realizable in physical systems. Square waves are encountered in electronics and signal processing. Its stochastic counterpart is a two-state trajectory, a similar but not necessarily symmetric wave, with arbitrary durations at minimum and maximum, is called a pulse wave. The ratio of the period to the total period of any rectangular wave is called the duty cycle. A true square wave has a 50% duty cycle and this is the basis of pulse width modulation. Square waves are encountered in digital switching circuits and are naturally generated by binary logic devices. They are used as timing references or clock signals, because their fast transitions are suitable for triggering synchronous logic circuits at precisely determined intervals, to avoid this problem in very sensitive circuits such as precision analog-to-digital converters, sine waves are used instead of square waves as timing references. In musical terms, they are described as sounding hollow. Additionally, the effect used on electric guitars clips the outermost regions of the waveform. Simple two-level Rademacher functions are square waves, any periodic function can substitute the sinusoid in this definition. Note, y = atan sin x + acot sin x A square wave can be expressed as the case of an infinite series of sinusoidal waves. The ideal square wave contains only components of odd-integer harmonic frequencies, Sawtooth waves and real-world signals contain all integer harmonics. A curiosity of the convergence of the Fourier series representation of the wave is the Gibbs phenomenon. Ringing artifacts in non-ideal square waves can be shown to be related to this phenomenon, the Gibbs phenomenon can be prevented by the use of σ-approximation, which uses the Lanczos sigma factors to help the sequence converge more smoothly. An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting and this is impossible to achieve in physical systems, as it would require infinite bandwidth. Square waves in physical systems have only finite bandwidth and often exhibit ringing effects similar to those of the Gibbs phenomenon or ripple effects similar to those of the σ-approximation. For a reasonable approximation to the shape, at least the fundamental and third harmonic need to be present
30.
Triangle wave
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A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function, like a square wave, the triangle wave contains only odd harmonics, demonstrating odd symmetry. However, the higher harmonics roll off much faster than in a square wave
31.
Sawtooth wave
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The sawtooth wave is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a saw with a zero rake angle. The convention is that a sawtooth wave ramps upward and then sharply drops, however, in a reverse sawtooth wave, the wave ramps downward and then sharply rises. It can also be considered the case of an asymmetric triangle wave. The piecewise linear function x = t − ⌊ t ⌋ = t − floor or x = t based on the function of time t is an example of a sawtooth wave with period 1. A more general form, in the range −1 to 1, a sawtooth can be constructed using additive synthesis. The infinite Fourier series x r e v e r s e s a w t o o t h =2 A π ∑ k =1 ∞ k sin k converges to a sawtooth wave. A conventional sawtooth can be constructed using x s a w t o o t h = A2 − A π ∑ k =1 ∞ k sin k where A is amplitude. Note, cot y = -tan x In digital synthesis, these series are only summed over k such that the highest harmonic and this summation can generally be more efficiently calculated with a fast Fourier transform. An audio demonstration of a sawtooth played at 440 Hz and 880 Hz and 1760 Hz is available below, both bandlimited and aliased tones are presented. Sawtooth waves are known for their use in music, the sawtooth and square waves are among the most common waveforms used to create sounds with subtractive analog and virtual analog music synthesizers. The sawtooth wave is the form of the vertical and horizontal deflection signals used to generate a raster on CRT-based television or monitor screens, oscilloscopes also use a sawtooth wave for their horizontal deflection, though they typically use electrostatic deflection. On the waves ramp, the field produced by the deflection yoke drags the electron beam across the face of the CRT. On the waves cliff, the magnetic field collapses, causing the electron beam to return to its resting position as quickly as possible. Frequency is 15.734 kHz on NTSC,15.625 kHz for PAL, the vertical deflection system operates the same way as the horizontal, though at a much lower frequency. The ramp portion of the wave must appear as a straight line, if otherwise, it indicates that the voltage isnt increasing linearly, and therefore that the magnetic field produced by the deflection yoke is not linear. As a result, the beam will accelerate during the non-linear portions. This would result in a television image squished in the direction of the non-linearity, extreme cases will show marked brightness increases, since the electron beam spends more time on that side of the picture
32.
Joseph Fourier
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The Fourier transform and Fouriers law are also named in his honour. Fourier is also credited with the discovery of the greenhouse effect. Fourier was born at Auxerre, the son of a tailor and he was orphaned at age nine. Fourier was recommended to the Bishop of Auxerre, and through this introduction, the commissions in the scientific corps of the army were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics. He took a prominent part in his own district in promoting the French Revolution and he was imprisoned briefly during the Terror but in 1795 was appointed to the École Normale, and subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique. Fourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, as scientific adviser, cut off from France by the English fleet, he organized the workshops on which the French army had to rely for their munitions of war. He also contributed several papers to the Egyptian Institute which Napoleon founded at Cairo. After the British victories and the capitulation of the French under General Menou in 1801, in 1801, Napoleon appointed Fourier Prefect of the Department of Isère in Grenoble, where he oversaw road construction and other projects. However, Fourier had previously returned home from the Napoleon expedition to Egypt to resume his academic post as professor at École Polytechnique when Napoleon decided otherwise in his remark. The Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place, hence being faithful to Napoleon, he took the office of Prefect. It was while at Grenoble that he began to experiment on the propagation of heat and he presented his paper On the Propagation of Heat in Solid Bodies to the Paris Institute on December 21,1807. He also contributed to the monumental Description de lÉgypte, Fourier moved to England in 1816. Later, he returned to France, and in 1822 succeeded Jean Baptiste Joseph Delambre as Permanent Secretary of the French Academy of Sciences, in 1830, he was elected a foreign member of the Royal Swedish Academy of Sciences. In 1830, his health began to take its toll, Fourier had already experienced, in Egypt and Grenoble. At Paris, it was impossible to be mistaken with respect to the cause of the frequent suffocations which he experienced. A fall, however, which he sustained on the 4th of May 1830, while descending a flight of stairs, shortly after this event, he died in his bed on 16 May 1830. His name is one of the 72 names inscribed on the Eiffel Tower, a bronze statue was erected in Auxerre in 1849, but it was melted down for armaments during World War II. Joseph Fourier University in Grenoble is named after him and this book was translated, with editorial corrections, into English 56 years later by Freeman
33.
Time series
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A time series is a series of data points indexed in time order. Most commonly, a series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data, examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. Time series are very frequently plotted via line charts, Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values, Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in there is no natural ordering of the observations. Time series analysis is also distinct from data analysis where the observations typically relate to geographical locations. A stochastic model for a series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. Methods for time series analysis may be divided into two classes, frequency-domain methods and time-domain methods, the former include spectral analysis and wavelet analysis, the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the stationary stochastic process has a certain structure which can be described using a small number of parameters. In these approaches, the task is to estimate the parameters of the model describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure, Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate. A time series is one type of Panel data, Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel. A data set may exhibit characteristics of both data and time series data. One way to tell is to ask what makes one data record unique from the other records, if the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a data field and an additional identifier which is unrelated to time. If the differentiation lies on the identifier, then the data set is a cross-sectional data set candidate
34.
Superposition principle
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So that if input A produces response X and input B produces response Y then input produces response. The homogeneity and additivity properties together are called the superposition principle, a linear function is one that satisfies the properties of superposition. It is defined as F = F + F Additivity F = a F Homogeneity for scalar a and this principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a system where the input stimulus is the load on the beam. Because physical systems are only approximately linear, the superposition principle is only an approximation of the true physical behaviour. The superposition principle applies to any system, including algebraic equations, linear differential equations. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. By writing a very general stimulus as the superposition of stimuli of a specific, simple form, for example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the principle, each of these sinusoids can be analyzed separately. According to the principle, the response to the original stimulus is the sum of all the individual sinusoidal responses. Fourier analysis is common for waves. For example, in theory, ordinary light is described as a superposition of plane waves. As long as the principle holds, the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves. Waves are usually described by variations in some parameter space and time—for example, height in a water wave, pressure in a sound wave. The value of this parameter is called the amplitude of the wave, in any system with waves, the waveform at a given time is a function of the sources and initial conditions of the system. In many cases, the equation describing the wave is linear, when this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side, with regard to wave superposition, Richard Feynman wrote, No-one has ever been able to define the difference between interference and diffraction satisfactorily
35.
Standing wave
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In physics, a standing wave – also known as a stationary wave – is a wave in a medium in which each point on the axis of the wave has an associated constant amplitude. The locations at which the amplitude is minimum are called nodes, standing waves were first noticed by Michael Faraday in 1831. Faraday observed standing waves on the surface of a liquid in a vibrating container, franz Melde coined the term standing wave around 1860 and demonstrated the phenomenon in his classic experiment with vibrating strings. For waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy, as an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges. Such waves are often exploited by glider pilots, standing waves and hydraulic jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom. Many standing river waves are popular river surfing breaks, the effect is a series of nodes and anti-nodes at fixed points along the transmission line. The failure of the line to power at the standing wave frequency will usually result in attenuation distortion. In practice, losses in the line and other components mean that a perfect reflection. The result is a standing wave, which is a superposition of a standing wave. The degree to which the wave resembles either a standing wave or a pure traveling wave is measured by the standing wave ratio. Another example is standing waves in the ocean formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, in one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. The reflected wave has to have the amplitude and frequency as the incoming wave. At locations which are multiples of a quarter wavelength x =. −3 λ2, − λ, − λ2,0, λ2, λ,3 λ2, called the nodes, the amplitude is always zero, whereas at locations which are odd multiples of a quarter wavelength x =. −5 λ4, −3 λ4, − λ4, λ4,3 λ4,5 λ4, called the anti-nodes, the amplitude is maximum, with a value of twice the amplitude of the original waves. The distance between two nodes or anti-nodes is λ/2. Standing waves can occur in two- or three-dimensional resonators
36.
Tension (physics)
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Tension is the opposite of compression. At the atomic level, when atoms or molecules are pulled apart each other and gain potential energy with a restoring force still existing. Each end of a string or rod under such tension will pull on the object it is attached to, to restore the string/rod to its relaxed length. In physics, tension, as a force, as an action-reaction pair of forces. The ends of a string or other object transmitting tension will exert forces on the objects to which the string or rod is connected and these forces due to tension are also called passive forces. Tension in a string is a scalar quantity. A string or rope is often idealized as one dimension, having length but being massless with zero cross section. If there are no bends in the string, as occur with vibrations or pulleys, then tension is a constant along the string, by Newtons Third Law, these are the same forces exerted on the ends of the string by the objects to which the ends are attached. If the string curves around one or more pulleys, it still have constant tension along its length in the idealized situation that the pulleys are massless and frictionless. A vibrating string vibrates with a set of frequencies that depend on the strings tension and these frequencies can be derived from Newtons laws of motion. Each microscopic segment of the string pulls on and is pulled upon by its neighboring segments, tension = τ where x is the position along the string. If the string has curvature, then the two pulls on a segment by its two neighbors will not add to zero, and there will be a net force on that segment of the string, causing an acceleration. With solutions that include the various harmonics on a stringed instrument, tension is also used to describe the force exerted by the ends of a three-dimensional, continuous material such as a rod or truss member. Such a rod elongates under tension, a system is in equilibrium when the sum of all forces is zero. ∑ F → =0 For example, consider a system consisting of an object that is being lowered vertically by a string with tension, T, at a constant velocity. ∑ F → = T → + m g → =0 A system has a net force when a force is exerted on it. Acceleration and net force always exist together, ∑ F → ≠0 For example, consider the same system as above but suppose the object is now being lowered with an increasing velocity downwards therefore there exists a net force somewhere in the system. In this case, negative acceleration would indicate that | m g | > | T |
37.
Crest (physics)
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A crest is the point on a wave with the maximum value or upward displacement within a cycle. A crest is a point on the wave where the displacement of the medium is at a maximum, a trough is the opposite of a crest, so the minimum or lowest point in a cycle. When in antiphase – 180° out of phase – the result is destructive interference, superposition principle Wave Kinsman, Blair, Wind Waves, Their Generation and Propagation on the Ocean Surface, Dover Publications, ISBN 0-486-49511-6,704 pages
38.
Fourier transform
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The Fourier transform decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform is called the frequency domain representation of the original signal, the term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, linear operations performed in one domain have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the domain corresponds to multiplication by the frequency. Also, convolution in the domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, after performing the desired operations, transformation of the result can be made back to the time domain. Functions that are localized in the domain have Fourier transforms that are spread out across the frequency domain and vice versa. The Fourier transform of a Gaussian function is another Gaussian function, Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can also be generalized to functions of variables on Euclidean space. In general, functions to which Fourier methods are applicable are complex-valued, the latter is routinely employed to handle periodic functions. The fast Fourier transform is an algorithm for computing the DFT, the Fourier transform of the function f is traditionally denoted by adding a circumflex, f ^. There are several conventions for defining the Fourier transform of an integrable function f, ℝ → ℂ. Here we will use the definition, f ^ = ∫ − ∞ ∞ f e −2 π i x ξ d x. When the independent variable x represents time, the transform variable ξ represents frequency. Under suitable conditions, f is determined by f ^ via the inverse transform, f = ∫ − ∞ ∞ f ^ e 2 π i ξ x d ξ, the functions f and f ^ often are referred to as a Fourier integral pair or Fourier transform pair. For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions, the Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. Many other characterizations of the Fourier transform exist, for example, one uses the Stone–von Neumann theorem, the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group. In 1822, Joseph Fourier showed that some functions could be written as an sum of harmonics
39.
Harmonic series (music)
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A harmonic series is the sequence of sounds where the base frequency of each sound is an integer multiple of the lowest base frequency. Pitched musical instruments are based on an approximate harmonic oscillator such as a string or a column of air. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves, interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency. The musical timbre of a tone from such an instrument is determined by the relative strengths of each harmonic. A complex tone can be described as a combination of many simple periodic waves or partials, each with its own frequency of vibration, amplitude, a partial is any of the sine waves of which a complex tone is composed. A harmonic is any member of the series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The fundamental is also considered a harmonic because it is 1 times itself, a harmonic partial is any real partial component of a complex tone that matches an ideal harmonic. An inharmonic partial is any partial that does not match an ideal harmonic, Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial. Unpitched, or indefinite-pitched instruments, such as cymbals, gongs, or tam-tams make sounds that are rich in inharmonic partials, an overtone is any partial except the lowest partial. The term overtone does not imply harmonicity or inharmonicity and has no special meaning other than to exclude the fundamental. It is the relative strengths of the different overtones that gives an instrument its particular timbre, some electronic instruments, such as synthesizers, can play a pure frequency with no overtones. Synthesizers can also combine pure frequencies into more complex tones, such as to other instruments. Certain flutes and ocarinas are very nearly without overtones, in most pitched musical instruments, the fundamental is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency waves occur with varying prominence and give each instrument its characteristic tone quality, the fact that a string is fixed at each end means that the longest allowed wavelength on the string is twice the length of the string. Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, theoretically, these shorter wavelengths correspond to vibrations at frequencies that are 2,3,4,5,6, etc. times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies, the harmonic series is an arithmetic series. In terms of frequency, the difference between consecutive harmonics is therefore constant and equal to the fundamental, but because human ears respond to sound nonlinearly, higher harmonics are perceived as closer together than lower ones
40.
Helmholtz equation
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In mathematics, the Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation ∇2 A + k 2 A =0 where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude. The Helmholtz equation often arises in the study of problems involving partial differential equations in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, for example, consider the wave equation u =0. Separation of variables begins by assuming that the function u is in fact separable. Substituting this form into the equation, and then simplifying, we obtain the following equation. Notice the expression on the left-hand side depends only on r, as a result, this equation is valid in the general case if and only if both sides of the equation are equal to a constant value. Rearranging the first equation, we obtain the Helmholtz equation, ∇2 A + k 2 A = A =0 and we now have Helmholtzs equation for the spatial variable r and a second-order ordinary differential equation in time. The solution in time will be a combination of sine and cosine functions, with angular frequency of ω. Alternatively, integral transforms, such as the Laplace or Fourier transform, are used to transform a hyperbolic PDE into a form of the Helmholtz equation. Because of its relationship to the equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology. The solution to the spatial Helmholtz equation A =0 can be obtained for simple geometries using separation of variables, the two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieus differential equation, the solvable shapes all correspond to shapes whose dynamical billiard table is integrable, that is, not chaotic. When the motion on a billiard table is chaotic, then no closed form solutions to the Helmholtz equation are known. The study of systems is known as quantum chaos, as the Helmholtz equation. An interesting situation happens with a shape where about half of the solutions are integrable, a simple shape where this happens is with the regular hexagon. Another simple shape where this happens is with an L shape made by reflecting a square down, if the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form A r r +1 r A r +1 r 2 A θ θ + k 2 A =0 and we may impose the boundary condition that A vanish if r = a, thus A =0. The method of separation of variables leads to solutions of the form A = R Θ
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Instantaneous phase
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Instantaneous phase and instantaneous frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase of a function s, is the real-valued function, ϕ = arg . And for a function s, it is determined from the functions analytic representation, sa. When φ is constrained to its value, either the interval. Otherwise it is called unwrapped phase, which is a function of argument t. Unless otherwise indicated, the form should be inferred. S = A cos , where ω >0, S a = A e j, ϕ = ω t + θ. In this simple example, the constant θ is also commonly referred to as phase or phase offset. φ is a function of time, θ is not, in the next example, we also see that the phase offset of a real-valued sinusoid is ambiguous unless a reference is specified. S = A sin = A cos , where ω >0, S a = A e j, ϕ = ω t − π /2. In both examples the local maxima of s correspond to φ = 2πN for integer values of N and this has applications in the field of computer vision. Instantaneous angular frequency is defined as, ω = d ϕ d t, if φ is wrapped, discontinuities in φ will result in dirac delta impulses in f. This instantaneous frequency, ω, can be derived directly from the real and imaginary parts of sa, instead of the complex arg without concern of phase unwrapping. ϕ = arg = atan2 +2 m 1 π = arctan + m 2 π 2m1π, discontinuities can then be removed by adding 2π whenever Δφ ≤ −π, and subtracting 2π whenever Δφ > π. That allows φ to accumulate without limit and produces an unwrapped instantaneous phase, an equivalent formulation that replaces the modulo 2π operation with a complex multiplication is, ϕ = ϕ + arg , where the asterisk denotes complex conjugate. The discrete-time instantaneous frequency is simply the advancement of phase for that sample ω = arg , a vector-average phase can be obtained as the arg of the sum of the complex numbers without concern about wrap-around. Analytic signal Frequency modulation Cohen, Leon