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Bessel (publisher)
Bessel may refer to: * Bessel beam * Bessel ellipsoid * Bessel function in mathematics * Bessel's inequality in mathematics * Bessel's correction in statistics. * Bessel filter, a linear filter often used in audio crossover systems * Bessel Fjord, NE Greenland * Bessel Fjord, NW Greenland * Bessel (crater), a small lunar crater * Bessel transform, also known as Fourier-Bessel transform or Hankel transform * Bessel window In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the inte ..., in signal processing * Besselian date, see Epoch (astronomy)#Besselian years * , a German merchant ship in service 1928–45, latterly for the Kriegsmarine People * Friedrich Wilhelm Bessel (1784–1846), German mathematician, astronomer, and systematizer of the Bessel functions See also * Bessell {{disa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Beam
A Bessel beam is a wave whose amplitude is described by a Bessel function of the first kind. Electromagnetic, acoustic, gravitational, and matter waves can all be in the form of Bessel beams. A true Bessel beam is non-diffractive. This means that as it propagates, it does not diffract and spread out; this is in contrast to the usual behavior of light (or sound), which spreads out after being focused down to a small spot. Bessel beams are also ''self-healing'', meaning that the beam can be partially obstructed at one point, but will re-form at a point further down the beam axis. As with a plane wave, a true Bessel beam cannot be created, as it is unbounded and would require an infinite amount of energy. Reasonably good approximations can be made, however, and these are important in many optical applications because they exhibit little or no diffraction over a limited distance. Approximations to Bessel beams are made in practice either by focusing a Gaussian beam with an axicon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Ellipsoid
The Bessel ellipsoid (or Bessel 1841) is an important reference ellipsoid of geodesy. It is currently used by several countries for their national geodetic surveys, but will be replaced in the next decades by modern ellipsoids of satellite geodesy. The Bessel ellipsoid was derived in 1841 by Friedrich Wilhelm Bessel, based on several arc measurements and other data of continental geodetic networks of Europe, Russia and the British Survey of India. It is based on 10 meridian arcs and 38 precise measurements of the astronomic latitude and longitude (see also astro geodesy). The dimensions of the Earth ellipsoid axes were defined by logarithms in keeping with former calculation methods. The Bessel and GPS ellipsoids The Bessel ellipsoid fits especially well to the geoid curvature of Europe and Eurasia. Therefore it is optimal for National survey networks in these regions, despite of the fact that its axes are about 700 m shorter than that of the mean Earth ellipsoid d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel's Inequality
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let H be a Hilbert space, and suppose that e_1, e_2, ... is an orthonormal sequence in H. Then, for any x in H one has :\sum_^\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2, where ⟨·,·⟩ denotes the inner product in the Hilbert space H. If we define the infinite sum :x' = \sum_^\left\langle x,e_k\right\rangle e_k, consisting of "infinite sum" of vector resolute x in direction e_k, Bessel's inequality tells us that this series converges. One can think of it that there exists x' \in H that can be described in terms of potential basis e_1, e_2, \dots. For a complete orthonormal sequence (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel's Correction
In statistics, Bessel's correction is the use of ''n'' − 1 instead of ''n'' in the formula for the sample variance and sample standard deviation, where ''n'' is the number of observations in a sample. This method corrects the bias in the estimation of the population variance. It also partially corrects the bias in the estimation of the population standard deviation. However, the correction often increases the mean squared error in these estimations. This technique is named after Friedrich Bessel. Formulation In estimating the population variance from a sample when the population mean is unknown, the uncorrected sample variance is the ''mean'' of the squares of deviations of sample values from the sample mean (i.e. using a multiplicative factor 1/''n''). In this case, the sample variance is a biased estimator of the population variance. Multiplying the uncorrected sample variance by the factor : \frac n gives an ''unbiased'' estimator of the population variance. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Filter
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat Group delay and phase delay, group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems. The filter's name is a reference to German mathematician Friedrich Bessel (1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949. The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot (signal), overshoot, the Bessel filter has a small amount of overshoot, but still much less than other common frequency-domain filters, such as Butterworth filters. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Fjord
Bessel Fjord is a fjord in northeastern Greenland. Administratively it belongs to the NE Greenland National Park area. History The area around the mouth of this fjord was referred to as "Bessel Bay" at the time of the 1869–70 Second German North Polar Expedition led by Carl Koldewey. It was named after German astronomer and director of the Königsberg university observatory Friedrich Wilhelm Bessel (1784–1846). Later, the 1906–08 Denmark expedition applied the name to the fjord itself. This fjord marked the northern border of Erik the Red's Land in 1932–1933.Spencer Apollonio, ''Lands That Hold One Spellbound: A Story of East Greenland,'' 2008 Geography Bessel Fjord stretches north of Norlund Land from west to east for about 53 km. Troms Island lies in its mouth in the Greenland Sea. This fjord marks the border between King Frederick VIII Land to the north and King Christian X Land to the south. It is located in Erik the Red's Land, in the Greenland Caledonites. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Fjord, NW Greenland
Bessel Fjord ( da, Bessels Fjord) is a fjord in northwestern Greenland. Administratively it belongs to the Avannaata municipality. Knud Rasmussen described the fjord entrance in the following terms: Geography Bessel Fjord stretches roughly from north to south for about 60 km. It is a long and narrow fjord lined with high mountains rising steeply from the shore. Hannah Island (Greenland), Hannah Island, a small island, lies in the area of its mouth by the Kennedy Channel, Cape Bryan is on the western side of the mouth and Cape Maynard on the northeastern.''Prostar Sailing Directions 2005 Greenland and Iceland Enroute,'' p. 93 This fjord is located northeast of Washington Land, at the northern end of Daugaard-Jensen Land. The Petermann Peninsula forms its eastern shore. There are large ice caps on both landmasses flanking the fjord. Bibliography *H.P. Trettin (ed.), ''Geology of the Innuitian Orogen and Arctic Platform of Canada and Greenland''. Geological Survey of Canada ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel (crater)
Bessel is a small Lunar craters, lunar impact crater that is located in the southern half of the Mare Serenitatis. The crater was named after the German astronomer Friedrich Wilhelm Bessel in 1935. Gazetteer of Planetary Nomenclature, International Astronomical Union (IAU) Working Group for Planetary System Nomenclature (WGPSN) Despite its small size, this is the largest crater to lie entirely within the Lunar mare, mare. It lies to the north-northeast of the crater Menelaus (crater), Menelaus. This crater is circular and bowl-shaped with a rim that has a higher albedo than the floor or the surrounding mare. The outer rim is not significantly worn, and there are no features of note on the interior, apart from some slumping of material from the inner walls to the floor. Bessel is not of sufficient size to have developed the wiktionary:terrace, terr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hankel Transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor ''k'' along the ''r'' axis. The necessary coefficient of each Bessel function in the sum, as a function of the scaling factor ''k'' constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval. Definition The Hankel transform of order \nu of a function ''f''(''r'') is given by : F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrmr, where J_\nu is the Bessel function of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Window
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epoch (astronomy)
In astronomy, an epoch or reference epoch is a moment in time used as a reference point for some time-varying astronomical quantity. It is useful for the celestial coordinates or orbital elements of a celestial body, as they are subject to perturbations and vary with time. These time-varying astronomical quantities might include, for example, the mean longitude or mean anomaly of a body, the node of its orbit relative to a reference plane, the direction of the apogee or aphelion of its orbit, or the size of the major axis of its orbit. The main use of astronomical quantities specified in this way is to calculate other relevant parameters of motion, in order to predict future positions and velocities. The applied tools of the disciplines of celestial mechanics or its subfield orbital mechanics (for predicting orbital paths and positions for bodies in motion under the gravitational effects of other bodies) can be used to generate an ephemeris, a table of values giving the posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
