title: Jacob Bernoulli

text: Jacob collaborated with his brother on various applications of calculus. However the atmosphere of collaboration between the two brothers turned into rivalry as Johann's own mathematical genius began to mature, with both of them attacking each other in print, and posing difficult mathematical challenges to test each other's skills. By 1697, the relationship had completely broken down.


title: Jacob Bernoulli

text: Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines.


title: Jacob Bernoulli

text: By 1689 he had published important work on infinite series and published his law of large numbers in probability theory. Jacob Bernoulli published five treatises on infinite series between 1682 and 1704. The first two of these contained many results, such as the fundamental result that formula_1 diverges, which Bernoulli believed were new but they had actually been proved by Pietro Mengoli 40 years earlier and was proved by Nicole Oresme in the 14th century already. Bernoulli could not find a closed form for formula_2, but he did show that it converged to a finite limit less than 2. Euler was the first to find the limit of this series in 1737. Bernoulli also studied the exponential series which came out of examining compound interest.


title: Jacob Bernoulli

text: In May 1690 in a paper published in "Acta Eruditorum", Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent, is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. It had been studied by Huygens in 1687 and Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call separation of variables. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. In 1696 Bernoulli solved the equation, now called the Bernoulli differential equation,


title: Jacob Bernoulli

text: Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694. In 1695 he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced.


title: Jacob Bernoulli

text: Bernoulli's most original work was "Ars Conjectandi", published in Basel in 1713, eight years after his death. The work was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability. The book also covers other related subjects, including a review of combinatorics, in particular the work of van Schooten, Leibniz, and Prestet, as well as the use of Bernoulli numbers in a discussion of the exponential series. Inspired by Huygens' work, Bernoulli also gives many examples on how much one would expect to win playing various games of chance. The term Bernoulli trial resulted from this work.


title: Jacob Bernoulli

text: In the last part of the book, Bernoulli sketches many areas of mathematical probability, including probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; and the law of large numbers.


title: Jacob Bernoulli

text: Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity.


title: Jacob Bernoulli

text: In 1683 Bernoulli discovered the constant e by studying a question about compound interest which required him to find the value of the following expression (which is in fact ):


title: Jacob Bernoulli

text: One example is an account that starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25. Compounding quarterly yields $1.00×1.25 = $2.4414..., and compounding monthly yields $1.00×(1.0833...) = $2.613035...


title: Jacob Bernoulli

text: Bernoulli noticed that this sequence approaches a limit (the force of interest) for more and smaller compounding intervals. Compounding weekly yields $2.692597..., while compounding daily yields $2.714567..., just two cents more. Using as the number of compounding intervals, with interest of 100% / in each interval, the limit for large is the number that Euler later named ; with "continuous" compounding, the account value will reach $2.7182818... More generally, an account that starts at $1, and yields (1+) dollars at compound interest, will yield dollars with continuous compounding.


title: Jacob Bernoulli

text: Bernoulli wanted a logarithmic spiral and the motto "Eadem mutata resurgo" ('Although changed, I rise again the same') engraved on his tombstone. He wrote that the self-similar spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self." Bernoulli died in 1705, but an Archimedean spiral was engraved rather than a logarithmic one.


title: Advanced Media Broadcasting System

text: Advanced Media Broadcasting System (AMBS) is a broadcast media company in the Philippines. Its main offices and facilities are located at Unit 906A of the Paragon Plaza Building and at the 3rd Floor of Starmalls EDSA Shaw complex, both in Mandaluyong City, Metro Manila.


title: Advanced Media Broadcasting System

text: Founded in 1994 by the Vera family, it is owned by the Villar Group-backed Prime Asset Ventures through Planet Cable.


title: Advanced Media Broadcasting System

text: The Vera family, under the aegis of Quest Broadcasting head, attorney Jose Luis Vera, established Advanced Media Broadcasting System in 1994 with its Congressional franchise granted the following year. With the help of Jinji Buhain (niece of then-Manila Auxiliary Bishop Teodoro Buhain), AMBS bought 103.5 DWCS from the Archdiocese of Manila and changed its callsign to DWKX which gave the birth of adult contemporary station K-Lite. The station, along with four other Vera-controlled Metro Manila stations - (Jam 88.3, Wave 89.1, Magic 89.9 and 99.5 RT) and the Killerbee provincial network - formed The Radio Partners in the early 2000s before being reorganized in 2011 under the new Tiger 22 Media group.


title: Advanced Media Broadcasting System

text: In 2019, AMBS was granted a 25-year legislative franchise extension under Republic Act No. 11253, albeit without President Rodrigo Duterte's signature as the bill lapsed into law after 30 days of inaction. The Vera family and AMBS president Andrew Santiago, however, decided to sell the company to Planet Cable of real estate magnate and businessman-politician Manny Villar after its FM station (K-Lite) was in the red due the effects of the COVID-19 pandemic.


title: Advanced Media Broadcasting System

text: On January 5, 2022, the National Telecommunications Commission (NTC) awarded AMBS a provisional authority to operate for the digital Channel 16 frequency (previously assigned to ABS-CBN Corporation) for an 18-month test broadcast period; and for the analog Channel 2 frequency under a temporary assignment for simulcast purposes until analog shut-off in 2023. The construction of AMBS's own digital terrestrial television network in Metro Manila was expected to be completed in July 2022.


title: Advanced Media Broadcasting System

text: In June 2022, AMBS Manila began its test broadcast. On September 1, 2022, TV Host Willie Revillame announced that the TV station of AMBS will be named as ALLTV. Initially targeted on October 1, 2022, ALLTV made its soft launch on September 13, 2022 at 12 pm PHT, with plans of expanding it nationwide by the end of the year or even sooner; the grand launch will take place either late 2022 or early 2023. On July 15, 2022, during his contract signing with AMBS, Revillame announced that his variety show "Wowowin" would return on All TV since its final broadcast on GMA Network on February 11, 2022.


title: Advanced Media Broadcasting System

text: All TV Manila conducted a test broadcast in Metro Manila from June 27 to September 12, 2022. The station began its broadcast through soft launch on September 13, 2022.


title: Extrapolation

text: In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced so as to arrive at a (usually conjectural) knowledge of the unknown (e.g. a driver extrapolates road conditions beyond his sight while driving). The extrapolation method can be applied in the interior reconstruction problem.


title: Extrapolation

text: A sound choice of which extrapolation method to apply relies on "a priori knowledge" of the process that created the existing data points. Some experts have proposed the use of causal forces in the evaluation of extrapolation methods. Crucial questions are, for example, if the data can be assumed to be continuous, smooth, possibly periodic etc.


title: Extrapolation

text: Linear extrapolation means creating a tangent line at the end of the known data and extending it beyond that limit. Linear extrapolation will only provide good results when used to extend the graph of an approximately linear function or not too far beyond the known data.


title: Extrapolation

text: If the two data points nearest the point formula_1 to be extrapolated are formula_2 and formula_3, linear extrapolation gives the function:


title: Extrapolation

text: (which is identical to linear interpolation if formula_5). It is possible to include more than two points, and averaging the slope of the linear interpolant, by regression-like techniques, on the data points chosen to be included. This is similar to linear prediction.


title: Extrapolation

text: A polynomial curve can be created through the entire known data or just near the end (two points for linear extrapolation, three points for quadratic extrapolation, etc.). The resulting curve can then be extended beyond the end of the known data. Polynomial extrapolation is typically done by means of Lagrange interpolation or using Newton's method of finite differences to create a Newton series that fits the data. The resulting polynomial may be used to extrapolate the data.


title: Extrapolation

text: High-order polynomial extrapolation must be used with due care. For the example data set and problem in the figure above, anything above order 1 (linear extrapolation) will possibly yield unusable values; an error estimate of the extrapolated value will grow with the degree of the polynomial extrapolation. This is related to Runge's phenomenon.


title: Extrapolation

text: A conic section can be created using five points near the end of the known data. If the conic section created is an ellipse or circle, when extrapolated it will loop back and rejoin itself. An extrapolated parabola or hyperbola will not rejoin itself, but may curve back relative to the X-axis. This type of extrapolation could be done with a conic sections template (on paper) or with a computer.


title: Extrapolation

text: French curve extrapolation is a method suitable for any distribution that has a tendency to be exponential, but with accelerating or decelerating factors. This method has been used successfully in providing forecast projections of the growth of HIV/AIDS in the UK since 1987 and variant CJD in the UK for a number of years. Another study has shown that extrapolation can produce the same quality of forecasting results as more complex forecasting strategies.


title: Extrapolation

text: Can be created with 3 points of a sequence and the "moment" or "index", this type of extrapolation have 100% accuracy in predictions in a big percentage of known series database (OEIS).


title: Extrapolation

text: Typically, the quality of a particular method of extrapolation is limited by the assumptions about the function made by the method. If the method assumes the data are smooth, then a non-smooth function will be poorly extrapolated.


title: Extrapolation

text: In terms of complex time series, some experts have discovered that extrapolation is more accurate when performed through the decomposition of causal forces.


title: Extrapolation

text: Even for proper assumptions about the function, the extrapolation can diverge severely from the function. The classic example is truncated power series representations of sin("x") and related trigonometric functions. For instance, taking only data from near the "x" = 0, we may estimate that the function behaves as sin("x") ~ "x". In the neighborhood of "x" = 0, this is an excellent estimate. Away from "x" = 0 however, the extrapolation moves arbitrarily away from the "x"-axis while sin("x") remains in the interval [−1,1]. I.e., the error increases without bound.


title: Extrapolation

text: Taking more terms in the power series of sin("x") around "x" = 0 will produce better agreement over a larger interval near "x" = 0, but will produce extrapolations that eventually diverge away from the "x"-axis even faster than the linear approximation.


title: Extrapolation

text: This divergence is a specific property of extrapolation methods and is only circumvented when the functional forms assumed by the extrapolation method (inadvertently or intentionally due to additional information) accurately represent the nature of the function being extrapolated. For particular problems, this additional information may be available, but in the general case, it is impossible to satisfy all possible function behaviors with a workably small set of potential behavior.


title: Extrapolation

text: In complex analysis, a problem of extrapolation may be converted into an interpolation problem by the change of variable formula_6. This transform exchanges the part of the complex plane inside the unit circle with the part of the complex plane outside of the unit circle. In particular, the compactification point at infinity is mapped to the origin and vice versa. Care must be taken with this transform however, since the original function may have had "features", for example poles and other singularities, at infinity that were not evident from the sampled data.


title: Extrapolation

text: Another problem of extrapolation is loosely related to the problem of analytic continuation, where (typically) a power series representation of a function is expanded at one of its points of convergence to produce a power series with a larger radius of convergence. In effect, a set of data from a small region is used to extrapolate a function onto a larger region.


title: Extrapolation

text: Again, analytic continuation can be thwarted by function features that were not evident from the initial data.


title: Extrapolation

text: Also, one may use sequence transformations like Padé approximants and Levin-type sequence transformations as extrapolation methods that lead to a summation of power series that are divergent outside the original radius of convergence. In this case, one often obtains


title: Extrapolation

text: The extrapolated data often convolute to a kernel function. After data is extrapolated, the size of data is increased N times, here N is approximately 2–3. If this data needs to be convoluted to a known kernel function, the numerical calculations will increase Nlog(N) times even with fast Fourier transform (FFT). There exists an algorithm, it analytically calculates the contribution from the part of the extrapolated data. The calculation time can be omitted compared with the original convolution calculation. Hence with this algorithm the calculations of a convolution using the extrapolated data is nearly not increased. This is referred as the fast extrapolation. The fast extrapolation has been applied to CT image reconstruction.


title: Extrapolation

text: Extrapolation arguments are informal and unquantified arguments which assert that something is probably true beyond the range of values for which it is known to be true. For example, we believe in the reality of what we see through magnifying glasses because it agrees with what we see with the naked eye but extends beyond it; we believe in what we see through light microscopes because it agrees with what we see through magnifying glasses but extends beyond it; and similarly for electron microscopes. Such arguments are widely used in biology in extrapolating from animal studies to humans and from pilot studies to a broader population.


title: Extrapolation

text: Like slippery slope arguments, extrapolation arguments may be strong or weak depending on such factors as how far the extrapolation goes beyond the known range.


title: Explorer 1

text: Explorer 1 was the first satellite launched by the United States in 1958 and was part of the U.S. participation in the International Geophysical Year (IGY). The mission followed the first two satellites the previous year; the Soviet Union's Sputnik 1 and Sputnik 2, beginning the Cold War Space Race between the two nations.


title: Explorer 1

text: Explorer 1 was launched on 1 February 1958 at 03:47:56 GMT (or 31 January 1958 at 22:47:56 Eastern Time) atop the first Juno booster from LC-26A at the Cape Canaveral Missile Test Center of the Atlantic Missile Range (AMR), in Florida. It was the first spacecraft to detect the Van Allen radiation belt, returning data until its batteries were exhausted after nearly four months. It remained in orbit until 1970.


title: Explorer 1

text: Explorer 1 was given Satellite Catalog Number 00004 and the Harvard designation 1958 Alpha 1, the forerunner to the modern International Designator.


title: Explorer 1

text: The U.S. Earth satellite program began in 1954 as a joint U.S. Army and U.S. Navy proposal, called Project Orbiter, to put a scientific satellite into orbit during the International Geophysical Year. The proposal, using a military Redstone missile, was rejected in 1955 by the Eisenhower administration in favor of the Navy's Project Vanguard, using a booster advertised as more civilian in nature. Following the launch of the Soviet satellite Sputnik 1 on 4 October 1957, the initial Project Orbiter program was revived as the Explorer program to catch up with the Soviet Union.


title: Explorer 1

text: Explorer 1 was designed and built by the Jet Propulsion Laboratory (JPL), while a Jupiter-C rocket was modified by the Army Ballistic Missile Agency (ABMA) to accommodate a satellite payload; the resulting rocket known as the Juno I. The Jupiter-C design used for the launch had already been flight-tested in nose cone reentry tests for the Jupiter intermediate-range ballistic missile (IRBM), and was modified into Juno I. Working closely together, ABMA and JPL completed the job of modifying the Jupiter-C and building Explorer 1 in 84 days. However, before work was completed, the Soviet Union launched a second satellite, Sputnik 2, on 3 November 1957. The U.S. Navy's attempt to put the first U.S. satellite into orbit failed with the launch of the Vanguard TV-3 on 6 December 1957.


title: Explorer 1

text: Explorer 1 was designed and built by the California Institute of Technology's JPL under the direction of Dr. William H. Pickering. It was the second satellite to carry a mission payload (Sputnik 2 was the first).


title: Explorer 1

text: The total mass of the satellite was , of which were instrumentation. In comparison, the mass of the first Soviet satellite Sputnik 1 was . The instrument section at the front end of the satellite and the empty scaled-down fourth-stage rocket casing orbited as a single unit, spinning around its long axis at 750 revolutions per minute.


title: Explorer 1

text: Data from the scientific instruments was transmitted to the ground by two antennas. A 60 milliwatt transmitter fed a dipole antenna consisting of two fiberglass slot antennas in the body of the satellite operating on 108.03 MHz, and four flexible whips forming a turnstile antenna were fed by a 10 milliwatt transmitter operating on 108.00 MHz.


title: Explorer 1

text: Because of the limited space available and the requirements for low weight, the payload instrumentation was designed and built with simplicity and high reliability in mind, using germanium and silicon transistors in its electronics. A total of 20 transistors were used in Explorer 1, plus additional ones in the Army's micrometeorite amplifier. Electrical power was provided by mercury chemical batteries that made up approximately 40% of the payload weight.


title: Explorer 1

text: The external skin of the instrument section was sandblasted stainless steel with white stripes. Several other color schemes had been tested, resulting in backup articles, models, and photographs showing different configurations, including alternate white and green striping and blue stripes alternating with copper. The final coloration was determined by studies of shadow–sunlight intervals based on firing time, trajectory, orbit, and inclination.


title: Explorer 1

text: The Explorer 1 payload consisted of the Iowa Cosmic Ray Instrument without a tape data recorder which was not modified in time to make it onto the spacecraft. The real-time data received on the ground was therefore very sparse and puzzling showing normal counting rates and no counts at all. The later Explorer 3 mission, which included a tape data recorder in the payload, provided the additional data for confirmation of the earlier Explorer 1 data.


title: Explorer 1

text: The scientific instrumentation of Explorer 1 was designed and built under the direction of Dr. James Van Allen of the University of Iowa containing:


title: Explorer 1

text: After a jet stream-related delay on 28 January 1958, at 03:47:56 GMT on 1 February 1958 the Juno I rocket was launched, putting Explorer 1 into orbit with a perigee of and an apogee of having a period of 114.80 minutes, and an inclination of 33.24°. Goldstone Tracking Station could not report after 90 minutes as planned whether the launch had succeeded because the orbit was larger than expected. At about 06:30 GMT, after confirming that Explorer 1 was indeed in orbit, a news conference was held in the Great Hall at the National Academy of Sciences in Washington, D.C. to announce it to the world.


title: Explorer 1

text: The original expected lifetime of the satellite before orbital decay was three years. Mercury batteries powered the high-power transmitter for 31 days and the low-power transmitter for 105 days. Explorer 1 stopped transmission of data on 23 May 1958, when its batteries died, but remained in orbit for more than 12 years. It reentered the atmosphere over the Pacific Ocean on 31 March 1970 after more than 58,400 orbits.


title: Explorer 1

text: Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years – to address this kind of momentum-preserving energy dissipation.


title: Explorer 1

text: Sometimes the instrumentation reported the expected cosmic ray count (approximately 30 counts per second) but other times it would show a peculiar zero counts per second. The University of Iowa (under James Van Allen) observed that all of the zero counts per second reports were from an altitude of more than over South America, while passes at would show the expected level of cosmic rays. Later, after Explorer 3, it was concluded that the original Geiger counter had been overwhelmed ("saturated") by strong radiation coming from a belt of charged particles trapped in space by the Earth's magnetic field. This belt of charged particles is now known as the Van Allen radiation belt. The discovery was considered to be one of the outstanding discoveries of the International Geophysical Year.


title: Explorer 1

text: The acoustic micrometeorite detector detected 145 impacts of cosmic dust in 78,750 seconds. This calculates to an average impact rate of 8.0 impacts per second per square meter, or 29 impacts per hour per square meter, over the twelve-day period.


title: Explorer 1

text: Explorer 1 was the first of the long-running Explorers program. Four follow-up satellites of the Explorer series were launched by the Juno I launch vehicle in 1958, of these, Explorer 3 and 4 were successful, while Explorer 2 and 5 failed to reach orbit. The final flight of the Juno I booster, the satellite Beacon-1, also failed. The Juno I vehicle was replaced by the Juno II launch vehicle in 1959.


title: Explorer 1

text: A follow-up to the first mission, Explorer-1 [PRIME], was successfully launched aboard a Delta II launch vehicle in late October 2011. The PRIME was built using modern satellite construction techniques. The orbiting satellite was a backup, because the initial Explorer-1 PRIME, launched on 4 March 2011, did not reach orbit due to a launch vehicle failure.


title: Explorer 1

text: An identically constructed flight backup of Explorer 1 is on display in the Smithsonian Institution's National Air and Space Museum, Milestones of Flight Gallery in Washington, D.C. LC-26 was deactivated in 1963, and was designated for use as a museum in 1964, the Air Force Space and Missile Museum. Here too, a full-scale Explorer 1 is on display, but this one is a mockup.


title: Volume of an n-ball

text: In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An -ball is a ball in an -dimensional Euclidean space. The volume of a -ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a -ball of radius is formula_1 where formula_2 is the volume of the unit -ball, the -ball of radius .


title: Volume of an n-ball

text: As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation:


title: Volume of an n-ball

text: where is Euler's gamma function. The gamma function is offset from but otherwise extends the factorial function to non-integer arguments. It satisfies if is a positive integer and if is a non-negative integer.


title: Volume of an n-ball

text: Inverting the above, the radius of an -ball of volume can be expressed recursively in terms of the radius of an - or -ball:


title: Volume of an n-ball

text: Using explicit formulas for particular values of the gamma function at the integers and half-integers gives formulas for the volume of a Euclidean ball in terms of factorials. These are:


title: Volume of an n-ball

text: The volume can also be expressed in terms of double factorials. For an odd integer , the double factorial is defined by


title: Volume of an n-ball

text: There are multiple conventions for double factorials of even integers. Under the convention in which the double factorial satisfies


title: Volume of an n-ball

text: Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume:


title: Volume of an n-ball

text: Stirling's formula for the gamma function can be used to approximate the volume when the number of dimensions is high.


title: Volume of an n-ball

text: In particular, for any fixed value of the volume tends to a limiting value of 0 as goes to infinity. For example, the volume is increasing for , achieves its maximum when , and is decreasing for .


title: Volume of an n-ball

text: Let denote the hypervolume of the -sphere of radius . The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by:


title: Volume of an n-ball

text: An important step in several proofs about volumes of -balls, and a generally useful fact besides, is that the volume of the -ball of radius is proportional to :


title: Volume of an n-ball

text: is a body (measurable set) in that space and is the body obtained by stretching in all directions by the factor then the volume of equals times the volume of . This is a direct consequence of the change of variables formula:


title: Volume of an n-ball

text: Another proof of the above relation, which avoids multi-dimensional integration, uses induction: The base case is , where the proportionality is obvious. For the inductive step, assume that proportionality is true in dimension . Note that the intersection of an "n"-ball with a hyperplane is an -ball. When the volume of the -ball is written as an integral of volumes of -balls:


title: Volume of an n-ball

text: it is possible by the inductive hypothesis to remove a factor of from the radius of the -ball to get:


title: Volume of an n-ball

text: which demonstrates the proportionality relation in dimension . By induction, the proportionality relation is true in all dimensions.


title: Volume of an n-ball

text: A proof of the recursion formula relating the volume of the -ball and an -ball can be given using the proportionality formula above and integration in cylindrical coordinates. Fix a plane through the center of the ball. Let denote the distance between a point in the plane and the center of the sphere, and let denote the azimuth. Intersecting the -ball with the -dimensional plane defined by fixing a radius and an azimuth gives an -ball of radius . The volume of the ball can therefore be written as an iterated integral of the volumes of the -balls over the possible radii and azimuths:


title: Volume of an n-ball

text: The azimuthal coordinate can be immediately integrated out. Applying the proportionality relation shows that the volume equals


title: Volume of an n-ball

text: The same technique can be used to give an inductive proof of the volume formula. The base cases of the induction are the 0-ball and the 1-ball, which can be checked directly using the facts and . The inductive step is similar to the above, but instead of applying proportionality to the volumes of the -balls, the inductive hypothesis is applied instead.


title: Volume of an n-ball

text: The proportionality relation can also be used to prove the recursion formula relating the volumes of an -ball and an -ball. As in the proof of the proportionality formula, the volume of an -ball can be written as an integral over the volumes of -balls. Instead of making a substitution, however, the proportionality relation can be applied to the volumes of the -balls in the integrand:


title: Volume of an n-ball

text: The integrand is an even function, so by symmetry the interval of integration can be restricted to . On the interval , it is possible to apply the substitution . This transforms the expression into


title: Volume of an n-ball

text: The integral is a value of a well-known special function called the beta function , and the volume in terms of the beta function is


title: Volume of an n-ball

text: The beta function can be expressed in terms of the gamma function in much the same way that factorials are related to binomial coefficients. Applying this relationship gives


title: Volume of an n-ball

text: As with the two-dimension recursive formula, the same technique can be used to give an inductive proof of the volume formula.


title: Volume of an n-ball

text: The volume of the "n"-ball formula_32 can be computed by integrating the volume element in spherical coordinates. The spherical coordinate system has a radial coordinate and angular coordinates , where the domain of each except is , and the domain of is . The spherical volume element is:


title: Volume of an n-ball

text: Each of the factors in the integrand depends on only a single variable, and therefore the iterated integral can be written as a product of integrals:


title: Volume of an n-ball

text: The integral over the radius is . The intervals of integration on the angular coordinates can, by the symmetry of the sine about , be changed to :


title: Volume of an n-ball

text: This function is both rotationally invariant and a product of functions of one variable each. Using the fact that it is a product and the formula for the Gaussian integral gives:


title: Volume of an n-ball

text: where is the -dimensional volume element. Using rotational invariance, the same integral can be computed in spherical coordinates:


title: Volume of an n-ball

text: where is an -sphere of radius (being the surface of an -ball of radius ) and is the area element (equivalently, the -dimensional volume element). The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then:


title: Volume of an n-ball

text: To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation :


title: Volume of an n-ball

text: The relations formula_48 and formula_49 and thus the volumes of "n"-balls and areas of "n"-spheres can also be derived geometrically. As noted above, because a ball of radius formula_50 is obtained from a unit ball formula_51 by rescaling all directions in formula_50 times, formula_32 is proportional to formula_54, which implies formula_55. Also, formula_56 because a ball is a union of concentric spheres and increasing radius by "ε" corresponds to a shell of thickness "ε". Thus, formula_57; equivalently, formula_48.


title: Volume of an n-ball

text: formula_49 follows from existence of a volume-preserving bijection between the unit sphere formula_60 and formula_61:


title: Volume of an n-ball

text: (formula_63 is an "n"-tuple; formula_64; we are ignoring sets of measure 0). Volume is preserved because at each point, the difference from isometry is a stretching in the "xy" plane (in formula_65 times in the direction of constant formula_66) that exactly matches the compression in the direction of the gradient of formula_67 on formula_68 (the relevant angles being equal). For formula_69, a similar argument was originally made by Archimedes in "On the Sphere and Cylinder".


title: Volume of an n-ball

text: and an ball is the set of all vectors whose norm is less than or equal to a fixed number called the radius of the ball. The case is the standard Euclidean distance function, but other values of occur in diverse contexts such as information theory, coding theory, and dimensional regularization.


title: Volume of an n-ball

text: For most values of , the surface area formula_76 of an sphere of radius (the boundary of an -ball of radius ) cannot be calculated by differentiating the volume of an ball with respect to its radius. While the volume can be expressed as an integral over the surface areas using the coarea formula, the coarea formula contains a correction factor that accounts for how the -norm varies from point to point. For and , this factor is one. However, if then the correction factor is : the surface area of an sphere of radius in is times the derivative of the volume of an ball. This can be seen most simply by applying the divergence theorem to the vector field to get


title: Volume of an n-ball

text: The volume formula can be generalized even further. For positive real numbers , define the ball with limit to be


title: Volume of an n-ball

text: Using the harmonic mean formula_83 and defining formula_84, the similarity to the volume formula for the ball becomes clear.


title: Flora MacDonald

text: Flora MacDonald (Gaelic: "Fionnghal nic Dhòmhnaill", 1722 - 5 March 1790) was a member of Clan Macdonald of Sleat, best known for helping Charles Edward Stuart evade government troops after the Battle of Culloden in April 1746. Her family generally backed the government during the 1745 Rising and MacDonald later claimed to have assisted Charles out of sympathy for his situation.


title: Flora MacDonald

text: Arrested and held in the Tower of London, she was released under a general amnesty in June 1747. She later married Allan MacDonald and the couple emigrated to North Carolina in 1773. Their support for the British government during the American War of Independence meant the loss of their American estates and they returned to Scotland, where she died in 1790.


title: Flora MacDonald

text: Flora MacDonald was born in 1722 at Milton on the island of South Uist in the Outer Hebrides, third and last child of Ranald MacDonald (d. 1723) and his second wife, Marion. Her father was a member of the minor gentry of Clan MacDonald of Clanranald, being tacksman and leaseholder of Milton and Balivanich. She had two brothers, Angus, who later inherited the Milton tack, and Ronald, who died young.


title: Flora MacDonald

text: Whilst some parts of the MacDonald clan remained Catholic, particularly in the Hebrides, Flora's family was part of South Uist's Protestant minority. Through her father's younger brother, Maighstir Alasdair MacDhòmhnaill, the Non-Juring Episcopalian Rector of Kilchoan and a Clanranald tacksman based in Moidart, she was first cousin to the Jacobite war poet Alasdair mac Mhaighstir Alasdair, who along with Sorley MacLean is considered one of the two most important figures in Scottish Gaelic literature.


title: Flora MacDonald

text: Her father died soon after her birth and in 1728 her widowed mother married Hugh MacDonald, Tacksman of Armadale, on the Isle of Skye, for Clan Macdonald of Sleat. Meanwhile, Flora MacDonald was brought up by her father's cousin, Sir Alexander MacDonald, Chief of Clan Macdonald of Sleat. Suggestions she was educated in Edinburgh cannot be confirmed.


title: Flora MacDonald

text: MacDonald was visiting Benbecula in the Outer Hebrides when Prince Charles and a small group of aides took refuge there after the Battle of Culloden in June 1746. One of his companions, Captain Conn O'Neill from County Antrim, was distantly related to MacDonald and asked for her help. MacDonald of Sleat had not joined the Rebellion and Benbecula was controlled by a pro-government militia commanded by MacDonald's step-father, Hugh MacDonald. This connection allowed her to obtain the necessary permits but she apparently hesitated, fearing the consequences for her family if they were caught. She may have been taking less of a risk than it appears; witnesses later claimed Hugh advised the Prince where to hide from his search parties.


title: Flora MacDonald

text: Passes were issued allowing passage to the mainland for MacDonald, a boat's crew of six men and two personal servants, including Charles disguised as an Irish maid called Betty Burke. On 27 June, they landed near Sir Alexander's house at Monkstadt, near Kilbride, Skye. In his absence, his wife Lady Margaret arranged lodging with her steward, MacDonald of Kingsburgh, who told Charles to remove his disguise, as it simply made him more conspicuous. The next day, Charles was taken from Portree to the island of Raasay; MacDonald remained on Skye and they never met again.


title: Flora MacDonald

text: Two weeks later, the boatmen were detained and confessed; MacDonald and Kingsburgh were arrested and taken to the Tower of London. After Lady Margaret interceded on her behalf with the chief Scottish legal officer, Duncan Forbes of Culloden, she was allowed to live outside the Tower under the supervision of a "King's Messenger" and released after the June 1747 Act of Indemnity. Aristocratic sympathisers collected over £1,500 for her, one of the contributors being Frederick, Prince of Wales, heir to the throne; she allegedly told him she helped Charles out of charity and would have done the same for him.


title: Flora MacDonald

text: On 6 November 1750, at the age of 28, she married Allan MacDonald, a captain in the British Army and Kingsburgh's eldest son. The couple first lived at Flodigarry on Skye and inherited the family estate in 1772 after Kingsburgh died. The writer and Jacobite sympathiser Samuel Johnson met her in 1773 during his visit to the island, and later described her as "a woman of soft features, gentle manners, kind soul and elegant presence". He was also author of the inscription on her memorial at Kilmuir: "a name that will be mentioned in history, and if courage and fidelity be virtues, mentioned with honour".


title: Flora MacDonald

text: During the 1756–1763 Seven Years' War, Allan MacDonald had served with some distinction in the 114th and 62nd Regiments of Foot but proved to be a poor businessman. After quarrelling with his landlord over debts and rent, he and Flora emigrated in 1774 to Anson County, North Carolina, and settled on a plantation they named "Killegray". When the American Revolutionary War began in 1775, Allan raised the Anson Battalion of the Loyalist North Carolina Militia, a total of around 1,000 men, including their sons Alexander and James.


title: Flora MacDonald

text: Tradition records that as the Anson battalion assembled in Cross Creek on 15 February 1776, Flora "addressed them in their own Gaelic tongue and excited them to the highest pitch of warlike enthusiasm", a tradition known among the Scottish clans as a ""brosnachadh-catha"" or "incitement to battle." They then set off for the coast to link up with some 2,000 British reinforcements commanded by General Henry Clinton, who in reality had only just sailed from Cork in Ireland. Early on the morning of 17 February, they were ambushed at Moore's Creek Bridge by Patriot militia led by Richard Caswell and along with his troops, Allan MacDonald was taken prisoner.


title: Flora MacDonald

text: After the battle, Flora was interrogated by North Carolina's Committee of Safety, before which she exhibited "spirited behaviour." In April 1777, all Loyalist-owned property was confiscated by the North Carolina Provincial Congress and Flora was evicted from Killegray, with the loss of all her possessions. After 18 months in captivity, Allan was released as part of a prisoner exchange in September 1777 and posted to Fort Edward, Nova Scotia as commander of the 84th Regiment of Foot. Here he was joined by Flora in August 1778.


title: Flora MacDonald

text: After a harsh winter in Halifax, Nova Scotia, in September 1779 MacDonald took passage for London in the "Dunmore," a British privateer; during the voyage, she broke her arm and ill-health delayed her return to Scotland until spring 1780. She spent the next few years living with various family members, including Dunvegan, home of her son-in-law Major General Alexander MacLeod, the largest landowner in Skye after the MacDonalds. The compensation received for the loss of their property in North Carolina was insufficient to allow them to resettle in Nova Scotia and Allan returned to Scotland in 1784. Kingsburgh was now occupied by Flora's half-sister and her husband, and Allan instead took up farming in nearby Penduin.


title: Flora MacDonald

text: According to historian J.P. MacLean, in later life Flora often said that she first served the House of Stuart and then the House of Hanover and was worsted in the cause of each. She died in 1790 at the age of 68 and was buried in Kilmuir Cemetery, followed by her husband in September 1792. They had seven surviving children, two daughters and five sons, two of whom were lost at sea in 1781 and 1782; a third son John made his fortune in India, enabling his parents to spend their last years in some comfort.


title: Flora MacDonald

text: Traditional portrayals of the escape focus on Charles, MacDonald being relegated to a secondary role and receiving less credit than is her due. She rarely spoke of the episode and her last contact with Charles was when they parted ways at Portree; it appears at least one motive was that his presence endangered her family.


title: Flora MacDonald

text: The Victorians created a Scottish cultural identity expressed through tartans, the late 19th-century inventions of Burns Suppers and Highland Games, and the co-option of romantic icons like Mary, Queen of Scots and Bonnie Prince Charlie. In 1878, MacDonald took her place in this pantheon with the publication of an alleged "Autobiography"; ghost-written by her granddaughter Flora Frances Wylde, it contains so many mistakes that it could not have been written by her. Many of these errors were repeated by Charles Ewald in his 1886 book "The Life and Times of Prince Charles Edward" which remains the basis for many popular perspectives on her life and motivations.


title: Flora MacDonald

text: In 1884, Sir Harold Boulton wrote an adaptation of an existing melody which he named "The Skye Boat Song". This was soon followed by the first performance of the Scottish highland dance "Flora MacDonald's Fancy", while a bronze statue was erected at Inverness Castle in 1896, with her dog Flossie by her side. The Flora MacDonald Academy, formerly Flora Macdonald College, in Red Springs, North Carolina is named after her and two of her children are interred on the campus; until 2009, it was also the site of the Flora Macdonald Highland Games.


title: Flora MacDonald

text: MacDonald was painted several times by Scottish portrait artist Allan Ramsay (1713–1784), most of which have now survived. The one used in this article was done after her release from the Tower in 1749–1750; in 2015, a previously unrecorded painting, allegedly also by Ramsay, was discovered in Florida.


title: Evolution of cetaceans

text: The evolution of cetaceans is thought to have begun in the Indian subcontinent from even-toed ungulates 50 million years ago (mya) and to have proceeded over a period of at least 15 million years. Cetaceans are fully aquatic marine mammals belonging to the order Artiodactyla and branched off from other artiodactyls around 50 mya. Cetaceans are thought to have evolved during the Eocene (56-34 mya), the second epoch of the present-extending Cenozoic Era. Molecular and morphological analyses suggest Cetacea share a relatively recent closest common ancestor with hippopotami and that they are sister groups. Being mammals, they surface to breathe air; they have 5 finger bones (even-toed) in their fins; they nurse their young; and, despite their fully aquatic life style, they retain many skeletal features from their terrestrial ancestors. Research conducted in the late 1970s in Pakistan revealed several stages in the transition of cetaceans from land to sea.


title: Evolution of cetaceans

text: The two modern parvorders of cetaceans – Mysticeti (baleen whales) and Odontoceti (toothed whales) – are thought to have separated from each other around 28–33 mya in a second cetacean radiation, the first occurring with the archaeocetes. The adaptation of animal echolocation in toothed whales distinguishes them from fully aquatic archaeocetes and early baleen whales. The presence of baleen in baleen whales occurred gradually, with earlier varieties having very little baleen, and their size is linked to baleen dependence (and subsequent increase in filter feeding).


title: Evolution of cetaceans

text: The aquatic lifestyle of cetaceans first began in the Indian subcontinent from even-toed ungulates 50 million years ago, over a period of at least 15 million years, but a jawbone discovered in Antarctica may reduce this to 5 million years. Archaeoceti is an extinct parvorder of Cetacea containing ancient whales. The traditional hypothesis of cetacean evolution, first proposed by Van Valen in 1966, was that whales were related to the mesonychids, an extinct order of carnivorous ungulates (hoofed animals) that resembled wolves with hooves and were a sister group of the artiodactyls (even-toed ungulates). This hypothesis was proposed due to similarities between the unusual triangular teeth of the mesonychids and those of early whales. However, molecular phylogeny data indicates that whales are very closely related to the artiodactyls, with hippopotamuses as their closest living relative. Because of this observation, cetaceans and hippopotamuses are placed in the same suborder, Whippomorpha. Cetartiodactyla (formed from the words Cetacea and Artiodactyla) is a proposed name for an order that includes both cetaceans and artiodactyls. However, the earliest anthracotheres, the ancestors of hippos, do not appear in the fossil record until the Middle Eocene, millions of years after "Pakicetus", whereas the first known whale ancestor appeared during the Early Eocene; this difference in timing implies that the two groups diverged well before the Eocene. Molecular analysis identifies artiodactyls as being very closely related to cetaceans, so mesonychids are probably an offshoot from Artiodactyla, and cetaceans did not derive directly from mesonychids, but the two groups may share a common ancestor.


title: Evolution of cetaceans

text: The molecular data are supported by the discovery of "Pakicetus", the earliest archaeocete. The skeletons of "Pakicetus" show that whales did not derive directly from mesonychids. Instead, they are artiodactyls that began to take to the water soon after artiodactyls split from mesonychids. Archaeocetes retained aspects of their mesonychid ancestry (such as the triangular teeth) which modern artiodactyls, and modern whales, have lost. The earliest ancestors of all hoofed mammals were probably at least partly carnivorous or scavengers, and today's artiodactyls and perissodactyls became herbivores later in their evolution. Whales, however, retained their carnivorous diet because prey was more available and they needed higher caloric content in order to live as marine endotherms (warm-blooded). Mesonychids also became specialized carnivores, but this was likely a disadvantage because large prey was uncommon. This may be why they were out-competed by better-adapted animals like the hyaenodontids and later Carnivora.


title: Evolution of cetaceans

text: "Indohyus" was a small chevrotain-like animal that lived about 48 million years ago in what is now Kashmir. It belongs to the artiodactyl family Raoellidae, which is believed to be the closest sister group of Cetacea. "Indohyus" is identified as an artiodactyl because it has two trochlea hinges, a trait unique to artiodactyls. Approximately the size of a raccoon or domestic cat, this omnivorous creature shared some traits of modern whales, most notably the involucrum, a bone growth pattern which is the diagnostic characteristic of any cetacean; this is not found in any other species. It also showed signs of adaptations to aquatic life, including dense limb bones that reduce buoyancy so that they could stay underwater, which are similar to the adaptations found in modern aquatic mammals such as the hippopotamus. This suggests a similar survival strategy to the African mousedeer or water chevrotain which, when threatened by a bird of prey, dives into water and hides beneath the surface for up to four minutes.


title: Evolution of cetaceans

text: The first fossils of the Indohyus were unearthed by Indian geologist A. Ranga Rao. He discovered a few teeth and a jawbone amongst rocks that he had collected. After his death, his widow Leelavathi Rao donated the rocks to professor Hans Theweissan. His technician accidentally broke open a couple of the donated rocks and discovered additional Indohyus fossils.


title: Evolution of cetaceans

text: The pakicetids were digitigrade hoofed mammals that are thought to be the earliest known cetaceans, with "Indohyus" being the closest sister group. They lived in the early Eocene, around 50 million years ago. Their fossils were first discovered in North Pakistan in 1979, located at a river not far from the shores of the former Tethys Sea. After the initial discovery, more fossils were found, mainly in the early Eocene fluvial deposits in northern Pakistan and northwestern India. Based on this discovery, pakicetids most likely lived in an arid environment with ephemeral streams and moderately developed floodplains millions of years ago. By using stable oxygen isotopes analysis, they were shown to drink fresh water, implying that they lived around freshwater bodies. Their diet probably included land animals that approached water for drinking or some freshwater aquatic organisms that lived in the river. The elongated cervical vertebrae and the four, fused sacral vertebrae are consistent with artiodactyls, making "Pakicetus" one of the earliest fossils to be recovered from the period following the Cetacea/Artiodactyla divergence event.


title: Evolution of cetaceans

text: Pakicetids are classified as cetaceans mainly due to the structure of the auditory bulla (ear bone), which is formed only from the ectotympanic bone. The shape of the ear region in pakicetids is highly unusual and the skull is cetacean-like, although a blowhole is still absent at this stage. The jawbone of pakicetids also lacks the enlarged space (mandibular foramen) that is filled with fat or oil, which is used in receiving underwater sound in modern cetaceans. They have dorsal orbits (eye sockets facing up), which are similar to crocodiles. This eye placement helps submerged predators observe potential prey above the water. According to a 2009 study, the teeth of pakicetids also resemble the teeth of fossil whales, being less like a dog's incisors, and having serrated triangular teeth, which is another link to more modern cetaceans.


title: Evolution of cetaceans

text: It was initially thought that the ears of pakicetids were adapted for underwater hearing, but, as would be expected from the anatomy of the rest of this creature, the ears of pakicetids are specialized for hearing on land. However, pakicetids were able to listen underwater by using enhanced bone conduction, rather than depending on the tympanic membrane like other land mammals. This method of hearing did not give directional hearing underwater.


title: Evolution of cetaceans

text: Pakicetids have long thin legs, with relatively short hands and feet which suggest that they were poor swimmers. To compensate for that, their bones are unusually thick (osteosclerotic), which is probably an adaptation to make the animal heavier to counteract the buoyancy of the water. According to a 2001 morphological analysis by Thewissen et al., pakicetids display no aquatic skeletal adaptation; instead they display adaptations for running and jumping. Hence pakicetids were most likely aquatic waders.


title: Evolution of cetaceans

text: "Ambulocetus", which lived about 49 million years ago, was discovered in Pakistan in 1994. They were vaguely crocodile-like mammals, possessing large brevirostrine jaws. In the Eocene, ambulocetids inhabited the bays and estuaries of the Tethys Sea in northern Pakistan. The fossils of ambulocetids are always found in near-shore shallow marine deposits associated with abundant marine plant fossils and littoral mollusks. Although they are found only in marine deposits, their oxygen isotope values indicate that they consumed a range of water with different degrees of salinity, with some specimens having no evidence of sea water consumption and others that did not ingest fresh water at the time when their teeth were fossilized. It is clear that ambulocetids tolerated a wide range of salt concentrations. Hence, ambulocetids represent a transition phase of cetacean ancestors between fresh water and marine habitat.


title: Evolution of cetaceans

text: The mandibular foramen in ambulocetids had increased in size, which indicates that a fat pad was likely to be housed in the lower jaw. In modern toothed whales, this fat pad in the mandibular foramen extends posteriorly to the middle ear. This allows sounds to be received in the lower jaw, and then transmitted through the fat pad to the middle ear. Similar to pakicetids, the orbits of ambulocetids are on the top of the skull, but they face more laterally than in pakicetids.


title: Evolution of cetaceans

text: Ambulocetids had relatively long limbs with particularly strong hind legs, and they retained a tail with no sign of a fluke. The hindlimb structure of Ambulocetids shows that their ability to engage in terrestrial locomotion was significantly limited compared to that of contemporary terrestrial mammals, and likely did not come to land at all. The skeletal structures of the knee and ankle indicates that the motion of the hindlimbs was restricted into one plane. This suggests that, on land, propulsion of the hindlimbs was powered by the extension of dorsal muscles. They probably swam by pelvic paddling (a way of swimming which mainly utilizes their hind limbs to generate propulsion in water) and caudal undulation (a way of swimming which uses the undulations of the vertebral column to generate force for movements), as otters, seals and modern cetaceans do. This is an intermediate stage in the evolution of cetacean locomotion, as modern cetaceans swim by caudal oscillation (a way of swimming similar to caudal undulation, but is more energy efficient).


title: Evolution of cetaceans

text: Recent studies suggest that ambulocetids were fully aquatic like modern cetaceans, possessing a similar thoracic morphology and being unable to support their weight on land. This suggests that complete abandonment of the land evolved much earlier among cetaceans than previously thought.


title: Evolution of cetaceans

text: Remingtonocetids lived in the Middle-Eocene in South Asia, about 49 to 43 million years ago. Compared to family Pakicetidae and Ambulocetidae, Remingtonocetidae was a diverse family found in north and central Pakistan and western India. Remingtonocetids were also found in shallow marine deposits, but they were obviously more aquatic than ambulocetidae. This is demonstrated by the recovery of their fossils from a variety of coastal marine environments, including near-shore and lagoonal deposits. According to stable oxygen isotopes analysis, most remingtonocetids did not ingest fresh water, and had hence lost their dependency on fresh water relatively soon after their origin.


title: Evolution of cetaceans

text: The orbits of remingtonocetids faced laterally and were small. This suggests that vision was not an important sense for them. The nasal opening, which eventually becomes the blowhole in modern cetaceans, was located near the tip of the snout. The position of the nasal opening had remained unchanged since pakicetids. One of the notable features in remingtonocetids is that the semicircular canals, which are important for balancing in land mammals, had decreased in size. This reduction in size had closely accompanied the cetacean radiation into marine environments. According to a 2002 study done by Spoor et al., this modification of the semicircular canal system may represent a crucial ‘point of no return’ event in early cetacean evolution, which excluded a prolonged semi-aquatic phase.


title: Evolution of cetaceans

text: Compared to ambulocetids, remingtonocetids had relatively short limbs. Based on their skeletal remains, remingtonocetids were probably amphibious cetaceans that were well adapted to swimming, and likely to swim by caudal undulation only.


title: Evolution of cetaceans

text: The protocetids form a diverse and heterogeneous group known from Asia, Europe, Africa, and North America. They lived in the Eocene, approximately 48 to 35 million years ago. The fossil remains of protocetids were uncovered from coastal and lagoonal facies in South Asia; unlike previous cetacean families, their fossils uncovered from Africa and North America also include open marine forms. They were probably amphibious, but more aquatic compared to remingtonocetids. Protocetids were the first cetaceans to leave the Indian subcontinent and disperse to all shallow subtropical oceans of the world.


title: Evolution of cetaceans

text: There were many genera among the family Protocetidae. There were different degrees of aquatic adaptations in this group, with some able to support their weight on land, and others not. The discovery of the southeastern Pacific "Peregocetus" indicates they crossed the Atlantic and achieved circumstances-equatorial distribution by 40 mya. Their amphibious nature is supported by the discovery of a pregnant "Maiacetus", in which the fossilised fetus was positioned for a head-first delivery, suggesting that "Maiacetus" gave birth on land. If they gave birth in the water, the fetus would be positioned for a tail-first delivery to avoid drowning during birth.


title: Evolution of cetaceans

text: Unlike remingtonocetids and ambulocetids, protocetids have large orbits which are oriented laterally. Increasingly lateral-facing eyes might be used to observe underwater prey, and are similar to the eyes of modern cetaceans. Furthermore, the nasal openings were large and were halfway up the snout. The great variety of teeth suggests diverse feeding modes in protocetids.


title: Evolution of cetaceans

text: In both remingtonocetids and protocetids, the size of the mandibular foramen had increased. The large mandibular foramen indicates that the mandibular fat pad was present. However the air-filled sinuses that are present in modern cetaceans, which function to isolate the ear acoustically to enable better underwater hearing, were still not present. The external auditory meatus (ear canal), which is absent in modern cetaceans, was also present. Hence, the method of sound transmission that were present in them combines aspects of pakicetids and modern odontocetes (toothed whales). At this intermediate stage of hearing development, the transmission of airborne sound was poor due to the modifications of the ear for underwater hearing while directional underwater hearing was also poor compared to modern cetaceans.


title: Evolution of cetaceans

text: Some protocetids had short, wide fore- and hindlimbs that were likely to have been used in swimming, but the limbs gave a slow and cumbersome locomotion on land. It is possible that some protocetids had flukes. However, it is clear that they were adapted even further to an aquatic life-style. In "Rodhocetus", for example, the sacrum (a bone that, in land-mammals, is a fusion of five vertebrae that connects the pelvis with the rest of the vertebral column) was divided into loose vertebrae. However, the pelvis was still connected to one of the sacral vertebrae. The ungulate ancestry of these archaeocetes is still underlined by characteristics like the presence of hooves at the ends of the toes in "Rodhocetus".


title: Evolution of cetaceans

text: The foot structure of "Rodhocetus" shows that protocetids were predominantly aquatic. A 2001 study done by Gingerich et al. hypothesized that "Rodhocetus" locomoted in the oceanic environment similarly to how ambulocetids pelvic paddling, which was supplemented by caudal undulation. Terrestrial locomotion of Rodhocetus was very limited due to their hindlimb structure. It is thought that they moved in a way similar to how eared seals move on land, by rotating their hind flippers forward and underneath their body.


title: Evolution of cetaceans

text: Basilosaurids and dorudontines lived together in the late Eocene around 41 to 33.9 million years ago, and are the oldest known obligate aquatic cetaceans. They were fully recognizable whales which lived entirely in the ocean. This is supported by their fossils usually found in deposits indicative of fully marine environments, lacking any freshwater influx. They were probably distributed throughout the tropical and subtropical seas of the world. Basilosaurids are commonly found in association with dorudontines, and were closely related to one another. The fossilised stomach contents in one basilosaurid indicates that it ate fish.


title: Evolution of cetaceans

text: Although they look very much like modern cetaceans, basilosaurids lacked the 'melon organ' that allows toothed whales to use echolocation. They had small brains; this suggests they were solitary and did not have the complex social structures of some modern cetaceans. The mandibular foramen of basilosaurids covered the entire depth of the lower jaw as in modern cetaceans. Their orbits faced laterally, and the nasal opening had moved even higher up the snout, closer to the position of the blowhole in modern cetaceans. Furthermore, their ear structures were functionally modern, with the insertion of air-filled sinuses between ear and skull. Unlike modern cetaceans, basilosaurids retained a large external auditory meatus.


title: Evolution of cetaceans

text: Both basilosaurids have skeletons that are immediately recognizable as cetaceans. A basilosaurid was as big as the larger modern whales, with genera like "Basilosaurus" reaching lengths of up to long; dorudontines were smaller, with genera like "Dorudon" reaching about long. The large size of basilosaurids is due to the extreme elongation of their lumbar vertebrae. They had a tail fluke, but their body proportions suggest that they swam by caudal undulation and that the fluke was not used for propulsion. In contrast, dorudontines had a shorter but powerful vertebral column. They too had a fluke and, unlike basilosaurids, they probably swam similarly to modern cetaceans, by using caudal oscillations. The forelimbs of basilosaurids were probably flipper-shaped, and the external hind limbs were tiny and were certainly not involved in locomotion. Their fingers, however, retained the mobile joints of their ambulocetid relatives. The two tiny but well-formed hind legs of basilosaurids were probably used as claspers when mating. The pelvic bones associated with these hind limbs were not connected to the vertebral column as they were in protocetids. Essentially, any sacral vertebrae can no longer be clearly distinguished from the other vertebrae.


title: Evolution of cetaceans

text: Both basilosaurids and dorudontines are relatively closely related to modern cetaceans, which belong to parvorders Odontoceti and Mysticeti. However, according to a 1994 study done by Fordyce and Barnes, the large size and elongated vertebral body of basilosaurids preclude them from being ancestral to extant forms. As for dorudontines, there are some species within the family that do not have elongated vertebral bodies, which might be the immediate ancestors of Odontoceti and Mysticeti. The other basilosaurids became extinct.


title: Evolution of cetaceans

text: All modern baleen whales or mysticetes are filter-feeders which have baleen in place of teeth, though the exact means by which baleen is used differs among species (gulp-feeding within balaenopterids, skim-feeding within balaenids, and bottom plowing within eschrichtiids). The first members of both groups appeared during the middle Miocene. Filter feeding is very beneficial as it allows baleen whales to efficiently gain huge energy resources, which makes the large body size in modern varieties possible. The development of filter feeding may have been a result of worldwide environmental change and physical changes in the oceans. A large-scale change in ocean current and temperature could have contributed to the radiation of modern mysticetes. The earlier varieties of baleen whales, or "archaeomysticetes", such as "Janjucetus" and "Mammalodon" had very little baleen and relied mainly on their teeth.


title: Evolution of cetaceans

text: There is also evidence of a genetic component of the evolution of toothless whales. Multiple mutations have been identified in genes related to the production of enamel in modern baleen whales. These are primarily insertion/deletion mutations that result in premature stop codons. It is hypothesized that these mutations occurred in cetaceans already possessing preliminary baleen structures, leading to the pseudogenization of a "genetic toolkit" for enamel production. Recent research has also indicated that the development of baleen and the loss of enamel-capped teeth both occurred once, and both occurred on the mysticete stem branch.


title: Evolution of cetaceans

text: Generally it is speculated the four modern mysticete families have separate origins among the cetotheres. Modern baleen whales, Balaenopteridae (rorquals and humpback whale, "Megaptera novaengliae"), Balaenidae (right whales), Eschrichtiidae (gray whale, "Eschrictius robustus"), and Neobalaenidae (pygmy right whale, "Caperea marginata") all have derived characteristics presently unknown in any cetothere and vice versa (such as a sagittal crest). Mysticetes are also known for their gigantism, as baleen whales are among the largest organisms to ever have lived; they reach lengths greater than 20 m and weigh more than 100,000 kg. This gigantism is directly related to their feeding mechanism - mysticete size has been found to be dependent on the amount of baleen a mysticete can use to filter its prey. Additionally, size is a positively selected trait that gives mysticetes a boost in fitness. Mysticete populations will therefore slowly become even more gigantic as whales with larger amounts of baleen are selected.


title: Evolution of cetaceans

text: The adaptation of echolocation occurred when toothed whales (Odontoceti) split apart from baleen whales, and distinguishes modern toothed whales from fully aquatic archaeocetes. This happened around 34 million years ago in a second cetacean radiation. Modern toothed whales do not rely on their sense of sight, but rather on their sonar to hunt prey. Echolocation also allowed toothed whales to dive deeper in search of food, with light no longer necessary for navigation, which opened up new food sources. Toothed whales echolocate by creating a series of clicks emitted at various frequencies. Sound pulses are emitted, reflected off objects, and retrieved through the lower jaw. Skulls of "Squalodon" show evidence for the first hypothesized appearance of echolocation. "Squalodon" lived from the early to middle Oligocene to the middle Miocene, around 33–14 million years ago. "Squalodon" featured several commonalities with modern toothed whales: the cranium was well compressed (to make room for the melon, a part of the nose), the rostrum telescoped outward into a beak, a characteristic of the modern toothed whales that gave "Squalodon" an appearance similar to them. However, it is thought unlikely that squalodontids are direct ancestors of modern toothed whales.


title: Evolution of cetaceans

text: The first oceanic dolphins such as kentriodonts, evolved in the late Oligocene and diversified greatly during the mid-Miocene. The first fossil cetaceans near shallow seas (where porpoises inhabit) were found around the North Pacific; species like "Semirostrum" were found along California (in what were then estuaries). These animals spread to the European coasts and Southern Hemisphere only much later, during the Pliocene. The earliest known ancestor of arctic whales is "Denebola brachycephala" from the late Miocene around 9–10 million years ago. A single fossil from Baja California indicates the family once inhabited warmer waters.


title: Evolution of cetaceans

text: Ancient sperm whales differ from modern sperm whales in tooth count and the shape of the face and jaws. For example, "Scaldicetus" had a tapered rostrum. Genera from the Oligocene and Miocene had teeth in their upper jaws. These anatomical differences suggest that these ancient species may not have necessarily been deep-sea squid hunters like the modern sperm whale, but that some genera mainly ate fish. Contrary to modern sperm whales, most ancient sperm whales were built to hunt whales. "Livyatan" had a short and wide rostrum measuring across, which gave the whale the ability to inflict major damage on large struggling prey, such as other early whales. Species like these are collectively known as killer sperm whales.


title: Evolution of cetaceans

text: Beaked whales consist of over 20 genera. Earlier variety were probably preyed upon by killer sperm whales and large sharks such as megalodon. In 2008, a large number of fossil ziphiids were discovered off the coast of South Africa, confirming the remaining ziphiid species might just be a remnant of a higher diversity that has since gone extinct. After studying numerous fossil skulls, researchers discovered the absence of functional maxillary teeth in all South African ziphiids, which is evidence that suction feeding had already developed in several beaked whale lineages during the Miocene. Extinct ziphiids also had robust skulls, suggesting that tusks were used for male-male interactions.


title: Evolution of cetaceans

text: Modern cetaceans have internal, rudimentary hind limbs, such as reduced femurs, fibulas, and tibias, and a pelvic girdle. "Indohyus" has a thickened ectotympanic internal lip of the ear bone. This feature compares directly to that of modern cetaceans. Another similar feature was the composition of the teeth, which contained mostly calcium phosphate which is needed for eating and drinking by aquatic animals, though, unlike modern day toothed whales, they had a heterodont (more than one tooth morphology) dentition as opposed to a homodont (one tooth morphology present) dentition. Although they somewhat resembled a wolf, the fossils of pakicetids showed the eye sockets were much closer to the top of their head than that of other terrestrial mammals, but similar to the structure of the eyes in cetaceans. Their transition from land to water led to reshaping of the skull and food processing equipment because the eating habits were changing. The change in position of the eyes and limb bones is associated with the pakicetids becoming waders. The ambulocetids also began to develop long snouts, which is seen in current cetaceans. Their limbs (and hypothesized movement) were very similar to otters.


title: Evolution of cetaceans

text: Limblessness in cetaceans does not represent a regression of fully formed limbs nor the absence of limb bud initiation, but rather arrest of limb bud development. Limb buds develop normally in cetacean embryos. Limb buds progress to the condensation phase of early skeletogenesis, where nerves grow into the limb bud and the apical ectodermal ridge (AER), a structure that ensures proper limb development, appears functional. Occasionally, the genes that code for longer extremities cause a modern whale to develop miniature legs (atavism).


title: Evolution of cetaceans

text: "Pakicetus" had a pelvic bone most similar to that of terrestrial mammals. In later species, such as "Basilosaurus", the pelvic bone, no longer attached to the vertebrae and the ilium, was reduced. Certain genes are believed to be responsible for the changes that occurred to the cetacean pelvic structure, such as BMP7, PBX1, PBX2, PRRX1, and PRRX2. The pelvic girdle in modern cetaceans were once thought to be vestigial structures that served no purpose at all. The pelvic girdle in male cetaceans is different in size compared to females, and the size is thought to be a result of sexual dimorphism. The pelvic bones of modern male cetaceans are more massive, longer, and larger than those of females. Due to the sexual dimorphism displayed, they were most likely involved in supporting male genitalia that remain hidden behind abdominal walls until sexual reproduction occurs.


title: Evolution of cetaceans

text: Early archaeocetes such as "Pakicetus" had the nasal openings at the end of the snout, but in later species such as "Rodhocetus", the openings had begun to drift toward the top of the skull. This is known as nasal drift. The nostrils of modern cetaceans have become modified into blowholes that allow them to break to the surface, inhale, and submerge with convenience. The ears began to move inward as well, and, in the case of "Basilosaurus", the middle ears began to receive vibrations from the lower jaw. Today's modern toothed whales use their melon organ, a pad of fat, for echolocation.


title: Evolution of cetaceans

text: There are three major radiation events that mark diversification and speciation in the evolutionary history of Cetacea. The first occurred around the middle Eocene (40 Mya) when these early cetaceans abandoned riverine and shallow coastal habitats, setting the scene for Protocetidae – the first fully marine cetacean. With the oceans and its nutrients at their disposal, rapidly diversifying protocetids were also responsible for the first major geographic expansion, dispersing throughout North Africa, Europe, and North America. The second of three major radiation events occurred near the start of the Oligocene (~34 Mya) when Neoceti diverged from Basilosauridae. This radiation event concurrently occurs with the breakup of Gondwana and the opening of the Southern Ocean, wildly changing ocean ecosystems, productivity, and temperature gradients. The timing of this second radiation event is not coincidental, as the following diversification of cetaceans was likely due to new ecological opportunities the change in oceans gave them. The final major radiation event, occurring throughout the middle Miocene and into the Pliocene (12 Mya to 2 Mya), was not due to a specific event but is associated with widespread generic expansion of odontocetes and mysticetes. Some modern genera of cetaceans began to emerge, including Balaenoptera, a genus of rorquals that includes the blue whale. Delphinidae, ocean dolphins, also arose during this radiation event in the late Miocene.


title: Evolution of cetaceans

text: Culture is group-specific behavior transferred by social learning. Tool use to aid with foraging is one example. Whether or not a dolphin uses a tool affects its eating behavior, which causes differences in diet. Also, using a tool allows a new niche and new prey to open up for that particular dolphin. Due to these differences, fitness levels change within the dolphins of a population, which further causes evolution to occur in the long run. Culture and social networks have played a large role in the evolution of modern cetaceans, as concluded in studies showing dolphins preferring mates with the same socially learned behaviors, and humpback whales using songs between breeding areas. For dolphins particularly, the largest non-genetic effects on their evolution are due to culture and social structure.


title: Evolution of cetaceans

text: Based on a 2014 study, the population of Indo-Pacific bottlenose dolphins ("Tursiops" sp.) around Shark Bay of Western Australia can be divided into spongers and nonspongers. Spongers put sea sponges on their snouts as protection against abrasions from sharp objects, stingray barbs, or toxic organisms. The sponges also help the dolphins target fish without swim bladders, since echolocation cannot detect these fish easily against a complex background. Spongers also specifically forage in deep channels, but nonspongers are found foraging in both deep and shallow channels. This foraging behavior is mainly passed on from mother to child. Therefore, since this is a group behavior being passed down by social learning, this tool use is considered a cultural trait.


title: Evolution of cetaceans

text: Researchers in a 2014 study in Shark Bay found the fatty acid analyses between the West and East Gulf populations to differ, which is due to the two areas having different food sources. However, when comparing data from within the West Gulf, the spongers vs. the nonspongers in the deep channels had very different fatty acid results even though they are in the same habitat. Nonspongers from deep and shallow channels had similar data. This suggests that sponging was the cause of the different data and not the deep vs. shallow channels. Sponging opened up a new niche for the dolphins and allowed them access to new prey, which caused long-term dietary changes. By producing different food sources within a population, there is less intrapopulation competition for resources, showing character displacement. As a result, the carrying capacity increases since the entire population does not depend on one food source. The fitness levels within the population also change, thus allowing this culture to evolve.


title: Evolution of cetaceans

text: Social structure forms groups with individuals that interact with one another, and this allows for cultural traits to emerge, exchange, and evolve. This relationship is especially seen in the bottlenose dolphin populations in southwestern Australia, which have been known to beg for food from fishermen. This begging behavior was spread through the population due to individual (dolphins spending time around boats) and social (dolphins spending time with other dolphins who express begging behavior) learning.


title: Evolution of cetaceans

text: Culture can, however, impact social structure by causing behavior matching and assortive mating. Individuals within a certain culture are more likely to mate with individuals using the same behaviors rather than a random individual, thus influencing social groups and structure. For example, the spongers of Shark Bay preferentially stick with other spongers. Also, some bottlenose dolphins in Moreton Bay, Australia followed prawn trawlers to feed on their debris, while other dolphins in the same population did not. The dolphins preferentially associated with individuals with same behavior even though they all lived in the same habitat. Later on, prawn trawlers were no longer present, and the dolphins integrated into one social network after a couple of years.


title: Evolution of cetaceans

text: Social networks can still affect and cause evolution on their own by impending fitness differences on individuals. According to a 2012 study, male calves had a lower survival rate if they had stronger bonds with juvenile males. However, when other age and sex classes were tested, their survival rate did not significantly change. This suggests that juvenile males impose a social stress on their younger counterparts. In fact, it has been documented that juvenile males commonly perform acts of aggression, dominance, and intimidation against the male calves. According to a 2010 study, certain populations of Shark Bay dolphins had varying levels of fitness and calf success. This is either due to social learning (whether or not the mother passed on her knowledge of reproductive ability to the calves), or due to the strong association between mother dolphins in the population; by sticking in a group, an individual mother does not need to be as vigilant all the time for predators.


title: Evolution of cetaceans

text: Genetic studies conducted on Clymene dolphins ("Stenella clymene") focused on their natural histories, and the results show that the origin of the species was actually an outcome of hybrid speciation. Hybridization between spinner dolphins ("Stenella longirostris") and striped dolphins ("Stenella coeruleoalba") in the North Atlantic was caused by constant habitat sharing of the two species. Relationships between these three species had been speculated according to notable resemblances between anatomies of the Clymene and the spinner dolphins, resulting in the former being regarded as subspecies of the latter until 1981, and the possibility of the Clymene dolphin as a hybrid between the spinner and the striped dolphins have come to question based on anatomical and behavioral similarities between these two species.


title: Evolution of cetaceans

text: Genome sequences done in 2013 revealed that the Yangtze river dolphin, or "baiji" ("Lipotes vexillifer"), lacks single nucleotide polymorphisms in their genome. After reconstructing the history of the baiji genome for this dolphin species, researchers found that the major decrease in genetic diversity occurred most likely due to a bottleneck event during the last deglaciation event. During this time period, sea levels were rising while global temperatures were increasing. Other historical climate events can be correlated and matched with the genome history of the Yangtze river dolphin as well. This shows how global and local climate change can drastically affect a genome, leading to changes in fitness, survival, and evolution of a species.


title: Evolution of cetaceans

text: The European population of common dolphins ("Delphinus delphis") in the Mediterranean have differentiated into two types: eastern and western. According to a 2012 study, this seems to be due to a recent bottleneck as well, which drastically decreased the size of the eastern Mediterranean population. Also, the lack of population structure between the western and eastern regions seems contradictory of the distinct population structures between other regions of dolphins. Even though the dolphins in the Mediterranean area had no physical barrier between their regions, they still differentiated into two types due to ecology and biology. Therefore, the differences between the eastern and western dolphins most likely stems from highly specialized niche choice rather than just physical barriers. Through this, environment plays a large role in the differentiation and evolution of this dolphin species.


title: Evolution of cetaceans

text: The divergence and speciation within bottlenose dolphins has been largely due to climate and environmental changes over history. According to research, the divisions within the genus correlate with periods of rapid climate change. For example, the changing temperatures could cause the coast landscape to change, niches to empty up, and opportunities for separation to appear. In the Northeast Atlantic, specifically, genetic evidence suggests that the bottlenose dolphins have differentiated into coastal and pelagic types. Divergence seems most likely due to a founding event where a large group separated. Following this event, the separate groups adapted accordingly and formed their own niche specializations and social structures. These differences caused the two groups to diverge and to remain separated.


title: Evolution of cetaceans

text: Two endemic, distinctive types of short-finned pilot whale, "Tappanaga" (or "Shiogondou") the larger, northern type and "Magondou" the smaller, southern type, can be found along the Japanese archipelago where distributions of these two types mostly do not overlap by the oceanic front border around the easternmost point of Honshu. It is thought that the local extinction of long-finned pilot whales in the North Pacific in the 12th century could have triggered the appearance of "Tappanaga", causing short-finned pilot whales to colonize the colder ranges of the long-finned variant. Whales with similar characteristics to the "Tappanaga" can be found along Vancouver Island and northern USA coasts as well.


title: Sinner Man

text: "Sinner Man" or "Sinnerman" is an African American traditional spiritual song that has been recorded by a number of performers and has been incorporated in many other media and arts. The lyrics describe a sinner attempting to hide from divine justice on Judgment Day. It was recorded in the 1950s by Les Baxter, the Swan Silvertones, the Weavers and others, before Nina Simone recorded an extended version in 1965.


title: Sinner Man

text: The earliest recording of the song to bear the title "Sinner Man" was by the Les Baxter Orchestra in 1956, as the B-side of the Capitol Records single "Tango of the Drums". The lead vocal was by folk singer Will Holt, who shared the credit for writing the song with Baxter. However, the song clearly bears a close resemblance, in both melody and lyrics, to "On the Judgement Day", which was recorded by gospel group The Sensational Nightingales in 1954 and released the following year on the Peacock label. The writing of The Sensational Nightingales' song was credited to two of the group's singers, Julius Cheeks and Ernest James. Some of the lyrics in "Sinner Man", including "The rock cried out, 'No hiding place'", appear to derive from those in the spiritual, "No Hiding Place Down Here", recorded in 1928 by the Old South Quartette.


title: Sinner Man

text: A version of "Sinner Man" released in 1956, by Swedish-American folk singer William Clauson, credited Baxter, Holt, Cheeks and James as co-writers. Another gospel group, the Swan Silvertones, released their version of the song in 1957 on the Vee-Jay label, and folk singer Guy Carawan issued a version in 1958. Carawan wrote that he had learned the song in 1956 from Bob Gibson. Pete Seeger also refers to Bob Gibson as the one who 'taught us' the song, during his Live concert at Mandel Hall, Chicago, in 1957. Most modern recorded versions derive from the 1956 recording by Les Baxter. Further changes and additions were codified in 1959 by the folk music group the Weavers. The Weavers' performance of the song appears on their compilation albums "Gospel" and "Reunion at Carnegie Hall Part 2".


title: Sinner Man

text: "Sinnerman" (spelled as one word) is one of Nina Simone's most famous songs. She recorded her definitive 10-minute-plus version on her 1965 album "Pastel Blues", on which the credit is simply given as "Arranged by Nina Simone" . Simone learned the lyrics of the song in her childhood when it was used at revival meetings by her mother, a Methodist minister, to help people confess their sins. In the early days of her career during the early sixties, when she was heavily involved in the Greenwich Village scene, Simone often used the long piece to end her live performances. An earlier version of the song exists, recorded live at The Village Gate, but was not used on the 1962 Colpix album "Nina at the Village Gate". It was added as a bonus track to the 2005 CD release.


title: Sinner Man

text: Simone's version of "Sinnerman" has been sampled by Kanye West for the Talib Kweli song "Get By" (2003), by Timbaland for the song "Oh Timbaland" (2007), and by Felix da Housecat for Verve Record's "Verve Remixed" series ("Verve Remixed 2" (2003), "Verve Remixed Plus" (2005)). French rapper Abd al-Malik sampled Simone's version for the title track of his 2006 album "Gibraltar". The 2018 Hozier track "Nina Cried Power" and the 2020 Celeste track "Stop This Flame" also sample the song.


title: Sinner Man

text: Nina Simone's version was used as a soundtrack to the art gallery theft scene in the 1999 film "The Thomas Crown Affair". Her version is also played during the end credits of David Lynch's 2006 experimental film "Inland Empire", and was featured in the video game "Marc Ecko's Getting Up".


title: Sinner Man

text: It was first recorded by the group at Studio One in Kingston, Jamaica in early 1966; Peter Tosh and Bunny Wailer shared lead vocals.


title: Sinner Man

text: A different version entitled "Downpressor" was recorded by Peter Tosh & The Wailers in 1970 ("downpressor" meaning "oppressor" in Rastafarian vocabulary). The song featured Tosh on lead vocals and Bunny Wailer on background. An instrumental version was also released. Songwriting credit for this version is sometimes given to Peter Tosh.


title: Sinner Man

text: Another version was recorded in 1971 as "Oppressor Man". This version was billed as a Peter Tosh solo single, and the B Side of the single featured an instrumental version. It was one of the rarer songs from the period before being included, along with its version, on JAD's Black Dignity compilation in 2004.


title: Sinner Man

text: Tosh recorded the song again as "Downpresser Man" for his 1977 solo album "Equal Rights" and released a live recording of the song in a medley with "Equal Rights" on his 1983 album "Captured Live".


title: Sinner Man

text: The band His Name Is Alive released a version of the song on their 1997 EP "Nice Day" (as "Oh Sinner Man").


title: Sinner Man

text: The song has also been covered by 16 Horsepower on their album "Folklore" (2002); by Irish singer Sinéad O'Connor on her 2005 album "Throw Down Your Arms"; and by the ska-punk band Goldfinger.


title: Sinner Man

text: On the Netflix show, "Lucifer", Lucifer (played by Tom Ellis) plays the piano and sings the song for the crowd at his nightclub in the sixth episode of the first season.


title: Sinner Man

text: In the HBO show, "Lovecraft Country", a cover by Alice Smith of "Sinnerman" is played over the closing credits for each episode.


title: Sinner Man

text: In the HBO show "The Righteous Gemstones", a cover of "Sinnerman" by The Travelers Three (1962) is played over the closing credits of season 2 episode 3.


title: Apple IIGS

text: The Apple IIGS (styled as II), the fifth and most powerful of the Apple II family, is a 16-bit personal computer produced by Apple Computer. While featuring the Macintosh look and feel, and resolution and color similar to the Amiga and Atari ST, it remains compatible with earlier Apple II models. The "GS" in the name stands for "Graphics and Sound," referring to its enhanced multimedia hardware, especially its state-of-the-art audio.


title: Apple IIGS

text: The microcomputer is a radical departure from any previous Apple II, with a 16-bit 65C816 microprocessor, direct access to megabytes of random-access memory (RAM), and bundled mouse. It is the first computer from Apple with a color graphical user interface (color was introduced on the Macintosh II six months later) and Apple Desktop Bus interface for keyboards, mice, and other input devices. It is the first personal computer with a wavetable synthesis chip, using technology from Ensoniq.


title: Apple IIGS

text: The IIGS set forth a promising future and evolutionary advancement of the Apple II line, but Apple increasingly focused on the Macintosh platform. Apple ceased IIGS production in December 1992.


title: Apple IIGS

text: The Apple IIGS made significant improvements over the Apple IIe and Apple IIc. It emulates its predecessors via a custom chip called the Mega II and uses the then-new WDC 65C816 16-bit microprocessor. The processor runs at , which is faster than the 8-bit processors used in the earlier Apple II models. The 65C816 allows the IIGS to address considerably more RAM.


title: Apple IIGS

text: The clock was a deliberate decision to limit the IIGS's performance to less than that of the Macintosh. This decision had a critical effect on the IIGS's success; the original 65C816 processor used in the IIGS was certified to run at up to . Faster versions of the 65C816 processor were readily available, with speeds of between 5 and 14 MHz, but Apple kept the machine at 2.8 MHz throughout its production run.


title: Apple IIGS

text: Its graphical capabilities are the best of the Apple II series, with higher resolution video modes and more color. These include a 640×200-pixel mode with 2-bit color and a 320×200 mode with 4-bit color, both of which can select 4 or 16 colors (respectively) at a time from a palette of 4,096 colors. By changing the palette on each scanline, it is possible to display up to 256 colors or more per screen. With clever programming, it is possible to make the IIGS display as many as 3,200 colors at once.


title: Apple IIGS

text: Audio is generated by a built-in Ensoniq 5503 digital synthesizer chip, which has its own dedicated RAM and 32 channels of sound. These channels can be paired to produce 15 voices in stereo.


title: Apple IIGS

text: The IIGS supports both 5.25-inch and 3.5-inch floppy disks and has seven general-purpose expansion slots compatible with those on the Apple II, II+, and IIe. It also has a memory expansion slot for up 8 MB of RAM. The IIGS has ports for external floppy disk drives, two serial ports for devices such as printers and modems (which can also be used to connect to a LocalTalk network), an Apple Desktop Bus port to connect the keyboard and mouse, and composite and RGB video ports.


title: Apple IIGS

text: A real-time clock is maintained by a built-in battery (a non-replaceable 3.6-volt lithium battery; removable in a later-revision motherboard).


title: Apple IIGS

text: The IIGS also supports booting from an AppleShare server, via the AppleTalk protocol, over LocalTalk cabling. This was over a decade before NetBoot offered the same capability to computers running Mac OS 8 and beyond.


title: Apple IIGS

text: In addition to supporting all graphics modes of previous Apple II models (40 and 80 columns text, Low and Double-Low, High and Double-High resolution) the Apple IIGS's Video Graphics Chip (VGC) introduced a new graphic mode called "Super-High Resolution". This new mode offers an increased screen resolution and a vastly wider color palette, with none of the limitations of earlier Apple II graphic modes (such as color bleeding and fringing). Super-High-Resolution supports 200 lines, in either 320 or 640 pixels horizontally. Both modes use a 12-bit palette for a total of 4,096 possible colors, with up to 256 colors (or more) on screen, though not all colors can appear onscreen at the same time.


title: Apple IIGS

text: Each scan line on the screen can independently select either 320- or 640-line mode, fill mode (320-mode only), and any of the 16 palettes, allowing graphics modes to be mixed on the screen. This is most often seen in graphics programs where the menu bar is constantly in 640-pixel resolution and the working area's mode can be changed depending on the user's needs.