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## Light

Written by cnathael@blog.com Posted in:

Light is electromagnetic radiation, particularly radiation of a wavelength that is visible to the human eye (about 400–700 nm, or perhaps 380–750 nm[1]). In physics, the term light sometimes refers to electromagnetic radiation of any wavelength, whether visible or not.[2][3]
Four primary properties of light are:
Light, which exists in tiny "packets" called photons, exhibits properties of both waves and particles. This property is referred to as the wave–particle duality. The study of light, known as optics, is an important research area in modern physics.

## Speed of light

The speed of light in a vacuum is presently defined to be exactly 299,792,458 m/s (approximately 186,282 miles per second). This definition of the speed of light means that the metre is now defined in terms of the speed of light. Light always travels at a constant speed, even between particles of a substance through which it is shining. Photons excite the adjoining particles that in turn transfer the energy to the neighbor. This may appear to slow the beam down through its trajectory in realtime. The time lost between entry and exit accounts to the displacement of energy through the substance between each particle that is excited.
Different physicists have attempted to measure the speed of light throughout history. Galileo attempted to measure the speed of light in the seventeenth century. An early experiment to measure the speed of light was conducted by Ole Rømer, a Danish physicist, in 1676. Using a telescope, Ole observed the motions of Jupiter and one of its moons, Io. Noting discrepancies in the apparent period of Io's orbit, Rømer calculated that light takes about 22 minutes to traverse the diameter of Earth's orbit.[4] Unfortunately, its size was not known at that time. If Ole had known the diameter of the Earth's orbit, he would have calculated a speed of 227,000,000 m/s.
Another, more accurate, measurement of the speed of light was performed in Europe by Hippolyte Fizeau in 1849. Fizeau directed a beam of light at a mirror several kilometers away. A rotating cog wheel was placed in the path of the light beam as it traveled from the source, to the mirror and then returned to its origin. Fizeau found that at a certain rate of rotation, the beam would pass through one gap in the wheel on the way out and the next gap on the way back. Knowing the distance to the mirror, the number of teeth on the wheel, and the rate of rotation, Fizeau was able to calculate the speed of light as 313,000,000 m/s.
Léon Foucault used an experiment which used rotating mirrors to obtain a value of 298,000,000 m/s in 1862. Albert A. Michelson conducted experiments on the speed of light from 1877 until his death in 1931. He refined Foucault's methods in 1926 using improved rotating mirrors to measure the time it took light to make a round trip from Mt. Wilson to Mt. San Antonio in California. The precise measurements yielded a speed of 299,796,000 m/s.
Two independent teams of physicists were able to bring light to a complete standstill by passing it through a Bose-Einstein Condensate of the element rubidium, one led by Dr. Lene Vestergaard Hau of Harvard University and the Rowland Institute for Science in Cambridge, Mass., and the other by Dr. Ronald L. Walsworth and Dr. Mikhail D. Lukin of the Harvard-Smithsonian Center for Astrophysics, also in Cambridge.[citation needed]

## Electromagnetic spectrum

Electromagnetic spectrum with light highlighted
Generally, EM radiation (the designation 'radiation' excludes static electric and magnetic and near fields) is classified by wavelength into radio, microwave, infrared, the visible region we perceive as light, ultraviolet, X-rays and gamma rays.
The behavior of EM radiation depends on its wavelength. Higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. When EM radiation interacts with single atoms and molecules, its behavior depends on the amount of energy per quantum it carries.

## Refraction

Main article: Refraction
Refraction is the bending of light rays when passing from one transparent material to another. It is described by Snell's Law:
$n_1\sin\theta_1 = n_2\sin\theta_2\ .$
where θ1 is the angle between the ray and the normal in the first medium, θ2 is the angle between the ray and the normal in the second medium, and n1 and n2 are the indices of refraction, n = 1 in a vacuum and n > 1 in a transparent substance.
When a beam of light crosses the boundary between a vacuum and another medium, or between two different media, the wavelength of the light changes, but the frequency remains constant. If the beam of light is not orthogonal (or rather normal) to the boundary, the change in wavelength results in a change in the direction of the beam. This change of direction is known as refraction.
The refractive quality of lenses is frequently used to manipulate light in order to change the apparent size of images. Magnifying glasses, spectacles, contact lenses, microscopes and refracting telescopes are all examples of this manipulation.
Light refraction is the main basis of measurement for gloss. Gloss is measured using a glossmeter.

## Optics

The study of light and the interaction of light and matter is termed optics. The observation and study of optical phenomena such as rainbows and the aurora borealis offer many clues as to the nature of light as well as much enjoyment.

## Light sources

A cloud illuminated by sunlight
There are many sources of light. The most common light sources are thermal: a body at a given temperature emits a characteristic spectrum of black-body radiation. Examples include sunlight (the radiation emitted by the chromosphere of the Sun at around 6,000 K peaks in the visible region of the electromagnetic spectrum when plotted in wavelength units [1] and roughly 40% of sunlight is visible), incandescent light bulbs (which emit only around 10% of their energy as visible light and the remainder as infrared), and glowing solid particles in flames. The peak of the blackbody spectrum is in the infrared for relatively cool objects like human beings. As the temperature increases, the peak shifts to shorter wavelengths, producing first a red glow, then a white one, and finally a blue color as the peak moves out of the visible part of the spectrum and into the ultraviolet. These colors can be seen when metal is heated to "red hot" or "white hot". Blue thermal emission is not often seen. The commonly seen blue colour in a gas flame or a welder's torch is in fact due to molecular emission, notably by CH radicals (emitting a wavelength band around 425 nm).
Atoms emit and absorb light at characteristic energies. This produces "emission lines" in the spectrum of each atom. Emission can be spontaneous, as in light-emitting diodes, gas discharge lamps (such as neon lamps and neon signs, mercury-vapor lamps, etc.), and flames (light from the hot gas itself—so, for example, sodium in a gas flame emits characteristic yellow light). Emission can also be stimulated, as in a laser or a microwave maser.
Deceleration of a free charged particle, such as an electron, can produce visible radiation: cyclotron radiation, synchrotron radiation, and bremsstrahlung radiation are all examples of this. Particles moving through a medium faster than the speed of light in that medium can produce visible Cherenkov radiation.
Certain chemicals produce visible radiation by chemoluminescence. In living things, this process is called bioluminescence. For example, fireflies produce light by this means, and boats moving through water can disturb plankton which produce a glowing wake.
Certain substances produce light when they are illuminated by more energetic radiation, a process known as fluorescence. Some substances emit light slowly after excitation by more energetic radiation. This is known as phosphorescence.
Phosphorescent materials can also be excited by bombarding them with subatomic particles. Cathodoluminescence is one example of this. This mechanism is used in cathode ray tube television sets and computer monitors.
A city illuminated by light bulbs
Certain other mechanisms can produce light:
When the concept of light is intended to include very-high-energy photons (gamma rays), additional generation mechanisms include:

## Units and measures

Light is measured with two main alternative sets of units: radiometry consists of measurements of light power at all wavelengths, while photometry measures light with wavelength weighted with respect to a standardized model of human brightness perception. Photometry is useful, for example, to quantify illumination intended for human use. The SI units for both systems are summarized in the following tables.
Quantity Symbol SI unit Abbr. Notes
Radiant energy Q joule J energy
Radiant intensity I watt per steradian W·sr−1 power per unit solid angle
Radiance L watt per steradian per square metre W·sr−1·m−2 power per unit solid angle per unit projected source area.
called intensity in some other fields of study.
Irradiance E, I watt per square metre W·m−2 power incident on a surface.
sometimes confusingly called "intensity".
M watt per square metre W·m−2 power emitted from a surface.
Radiosity J or Jλ watt per square metre W·m−2 emitted plus reflected power leaving a surface
or
Lν
or
metre per hertz
W·sr−1·m−3
or
W·sr−1·m−2·Hz−1
commonly measured in W·sr−1·m−2·nm−1

or
Eν
watt per metre3
or
watt per square metre per hertz
W·m−3
or
W·m−2·Hz−1
commonly measured in W·m−2·nm−1

SI photometry units
Quantity Symbol SI unit Abbr. Notes
Luminous energy Qv lumen second lm·s units are sometimes called talbots
Luminous flux F lumen (= cd·sr) lm also called luminous power
Luminous intensity Iv candela (= lm/sr) cd an SI base unit
Luminance Lv candela per square metre cd/m2 units are sometimes called "nits"
Illuminance Ev lux (= lm/m2) lx Used for light incident on a surface
Luminous emittance Mv lux (= lm/m2) lx Used for light emitted from a surface
Luminous efficacy   lumen per watt lm/W ratio of luminous flux to radiant flux
The photometry units are different from most systems of physical units in that they take into account how the human eye responds to light. The cone cells in the human eye are of three types which respond differently across the visible spectrum, and the cumulative response peaks at a wavelength of around 555 nm. Therefore, two sources of light which produce the same intensity (W/m2) of visible light do not necessarily appear equally bright. The photometry units are designed to take this into account, and therefore are a better representation of how "bright" a light appears to be than raw intensity. They relate to raw power by a quantity called luminous efficacy, and are used for purposes like determining how to best achieve sufficient illumination for various tasks in indoor and outdoor settings. The illumination measured by a photocell sensor does not necessarily correspond to what is perceived by the human eye, and without filters which may be costly, photocells and CCDs tend to respond to some infrared, ultraviolet or both.

## Historical theories about light, in chronological order

### Hindu theories

In ancient India, the Hindu schools of Samkhya and Vaisheshika, from around the 6th–5th century BC, developed theories on light. According to the Samkhya school, light is one of the five fundamental "subtle" elements (tanmatra) out of which emerge the gross elements. The atomicity of these elements is not specifically mentioned and it appears that they were actually taken to be continuous.
On the other hand, the Vaisheshika school gives an atomic theory of the physical world on the non-atomic ground of ether, space and time. (See Indian atomism.) The basic atoms are those of earth (prthivı), water (pani), fire (agni), and air (vayu), that should not be confused with the ordinary meaning of these terms. These atoms are taken to form binary molecules that combine further to form larger molecules. Motion is defined in terms of the movement of the physical atoms and it appears that it is taken to be non-instantaneous. Light rays are taken to be a stream of high velocity of tejas (fire) atoms. The particles of light can exhibit different characteristics depending on the speed and the arrangements of the tejas atoms. Around the first century BC, the Vishnu Purana refers to sunlight as the "the seven rays of the sun".
Later in 499, Aryabhata, who proposed a heliocentric solar system of gravitation in his Aryabhatiya, wrote that the planets and the Moon do not have their own light but reflect the light of the Sun.
The Indian Buddhists, such as Dignāga in the 5th century and Dharmakirti in the 7th century, developed a type of atomism that is a philosophy about reality being composed of atomic entities that are momentary flashes of light or energy. They viewed light as being an atomic entity equivalent to energy, similar to the modern concept of photons, though they also viewed all matter as being composed of these light/energy particles.
It is written in the Rigveda that light consists of three primary colors. "Mixing the three colours, ye have produced all the objects of sight!"[5]

### Greek and Hellenistic theories

In the fifth century BC, Empedocles postulated that everything was composed of four elements; fire, air, earth and water. He believed that Aphrodite made the human eye out of the four elements and that she lit the fire in the eye which shone out from the eye making sight possible. If this were true, then one could see during the night just as well as during the day, so Empedocles postulated an interaction between rays from the eyes and rays from a source such as the sun.
In about 300 BC, Euclid wrote Optica, in which he studied the properties of light. Euclid postulated that light travelled in straight lines and he described the laws of reflection and studied them mathematically. He questioned that sight is the result of a beam from the eye, for he asks how one sees the stars immediately, if one closes one's eyes, then opens them at night. Of course if the beam from the eye travels infinitely fast this is not a problem.
In 55 BC, Lucretius, a Roman who carried on the ideas of earlier Greek atomists, wrote:
"The light & heat of the sun; these are composed of minute atoms which, when they are shoved off, lose no time in shooting right across the interspace of air in the direction imparted by the shove." – On the nature of the Universe
Despite being similar to later particle theories, Lucretius's views were not generally accepted and light was still theorized as emanating from the eye.
Ptolemy (c. 2nd century) wrote about the refraction of light in his book Optics, and developed a theory of vision whereby objects are seen by rays of light emanating from the eyes.[6]

### Optical theory

The Muslim scientist Ibn al-Haytham (965–1040), known as Alhacen or Alhazen in the West, developed a broad theory of vision based on geometry and anatomy in his 1021 Book of Optics. Al-Haytham postulated that every point on an illuminated surface radiates light rays in all directions, but that only one ray from each point can be seen: the ray that strikes the eye perpendicularly. The other rays strike at different angles and are not seen. He described the pinhole camera and invented the camera obscura, which produces an inverted image, using it as an example to support his argument.[7] This contradicted Ptolemy's theory of vision that objects are seen by rays of light emanating from the eyes. Alhacen held light rays to be streams of minute energy particles[8][not in citation given] that travelled at a finite speed.[9][10][11] He improved Ptolemy's theory of the refraction of light, and went on to discover the laws of refraction.
He also carried out the first experiments on the dispersion of light into its constituent colors. His major work Kitab al-Manazir (Book of Optics) was translated into Latin in the Middle Ages, as well his book dealing with the colors of sunset. He dealt at length with the theory of various physical phenomena like shadows, eclipses, the rainbow. He also attempted to explain binocular vision, and gave an explanation of the apparent increase in size of the sun and the moon when near the horizon, known as the moon illusion. Because of his extensive experimental research on optics, Ibn al-Haytham is considered the "father of modern optics".[12]
Ibn al-Haytham also correctly argued that we see objects because the sun's rays of light, which he believed to be streams of tiny energy particles[8] travelling in straight lines, are reflected from objects into our eyes.[9] He understood that light must travel at a large but finite velocity,[9][10][11] and that refraction is caused by the velocity being different in different substances.[9] He also studied spherical and parabolic mirrors, and understood how refraction by a lens will allow images to be focused and magnification to take place. He understood mathematically why a spherical mirror produces aberration.
Avicenna (980–1037) agreed that the speed of light is finite, as he "observed that if the perception of light is due to the emission of some sort of particles by a luminous source, the speed of light must be finite."[13] Abū Rayhān al-Bīrūnī (973–1048) also agreed that light has a finite speed, and he was the first to discover that the speed of light is much faster than the speed of sound.[14] In the late 13th and early 14th centuries, Qutb al-Din al-Shirazi (1236–1311) and his student Kamāl al-Dīn al-Fārisī (1260–1320) continued the work of Ibn al-Haytham, and they were the first to give the correct explanations for the rainbow phenomenon.[14][not in citation given]

### Physical optics

René Descartes (1596–1650) held that light was a mechanical property of the luminous body, rejecting the "forms" of Ibn al-Haytham and Whitelo as well as the "species" of Bacon, Grosseteste, and Kepler.[15] In 1637 he published a theory of the refraction of light that assumed, incorrectly, that light travelled faster in a denser medium than in a less dense medium. Descartes arrived at this conclusion by analogy with the behaviour of sound waves.[citation needed] Although Descartes was incorrect about the relative speeds, he was correct in assuming that light behaved like a wave and in concluding that refraction could be explained by the speed of light in different media.
Descartes is not he first to use the mechanical analogies but because he clearly asserts that light is only a mechanical property of the luminous body and the transmitting medium, Descartes' theory of light is regarded as start of modern physical optics.[16]

### Particle theory

Ibn al-Haytham (Alhazen, 965–1040) proposed a particle theory of light in his Book of Optics (1021). He held light rays to be streams of minute energy particles[8] that travel in straight lines at a finite speed.[9][10][11] He states in his optics that "the smallest parts of light," as he calls them, "retain only properties that can be treated by geometry and verified by experiment; they lack all sensible qualities except energy."[8] Avicenna (980–1037) also proposed that "the perception of light is due to the emission of some sort of particles by a luminous source".[13]
Pierre Gassendi (1592–1655), an atomist, proposed a particle theory of light which was published posthumously in the 1660s. Isaac Newton studied Gassendi's work at an early age, and preferred his view to Descartes' theory of the plenum. He stated in his Hypothesis of Light of 1675 that light was composed of corpuscles (particles of matter) which were emitted in all directions from a source. One of Newton's arguments against the wave nature of light was that waves were known to bend around obstacles, while light travelled only in straight lines. He did, however, explain the phenomenon of the diffraction of light (which had been observed by Francesco Grimaldi) by allowing that a light particle could create a localised wave in the aether.
Newton's theory could be used to predict the reflection of light, but could only explain refraction by incorrectly assuming that light accelerated upon entering a denser medium because the gravitational pull was greater. Newton published the final version of his theory in his Opticks of 1704. His reputation helped the particle theory of light to hold sway during the 18th century. The particle theory of light led Laplace to argue that a body could be so massive that light could not escape from it. In other words it would become what is now called a black hole. Laplace withdrew his suggestion when the wave theory of light was firmly established. A translation of his essay appears in The large scale structure of space-time, by Stephen Hawking and George F. R. Ellis.

### Wave theory

In the 1660s, Robert Hooke published a wave theory of light. Christiaan Huygens worked out his own wave theory of light in 1678, and published it in his Treatise on light in 1690. He proposed that light was emitted in all directions as a series of waves in a medium called the Luminiferous ether. As waves are not affected by gravity, it was assumed that they slowed down upon entering a denser medium.
Thomas Young's sketch of the two-slit experiment showing the diffraction of light. Young's experiments supported the theory that light consists of waves.
The wave theory predicted that light waves could interfere with each other like sound waves (as noted around 1800 by Thomas Young), and that light could be polarized, if it were a transverse wave. Young showed by means of a diffraction experiment that light behaved as waves. He also proposed that different colors were caused by different wavelengths of light, and explained color vision in terms of three-colored receptors in the eye.
Another supporter of the wave theory was Leonhard Euler. He argued in Nova theoria lucis et colorum (1746) that diffraction could more easily be explained by a wave theory.
Later, Augustin-Jean Fresnel independently worked out his own wave theory of light, and presented it to the Académie des Sciences in 1817. Simeon Denis Poisson added to Fresnel's mathematical work to produce a convincing argument in favour of the wave theory, helping to overturn Newton's corpuscular theory. By the year 1821, Fresnel was able to show via mathematical methods that polarization could be explained only by the wave theory of light and only if light was entirely transverse, with no longitudinal vibration whatsoever.
The weakness of the wave theory was that light waves, like sound waves, would need a medium for transmission. A hypothetical substance called the luminiferous aether was proposed, but its existence was cast into strong doubt in the late nineteenth century by the Michelson-Morley experiment.
Newton's corpuscular theory implied that light would travel faster in a denser medium, while the wave theory of Huygens and others implied the opposite. At that time, the speed of light could not be measured accurately enough to decide which theory was correct. The first to make a sufficiently accurate measurement was Léon Foucault, in 1850.[17] His result supported the wave theory, and the classical particle theory was finally abandoned.

### Electromagnetic theory

A linearly-polarized light wave frozen in time and showing the two oscillating components of light; an electric field and a magnetic field perpendicular to each other and to the direction of motion (a transverse wave).
In 1845, Michael Faraday discovered that the plane of polarization of linearly polarized light is rotated when the light rays travel along the magnetic field direction in the presence of a transparent dielectric, an effect now known as Faraday rotation.[18] This was the first evidence that light was related to electromagnetism. In 1846 he speculated that light might be some form of disturbance propagating along magnetic field lines.[19] Faraday proposed in 1847 that light was a high-frequency electromagnetic vibration, which could propagate even in the absence of a medium such as the ether.
Faraday's work inspired James Clerk Maxwell to study electromagnetic radiation and light. Maxwell discovered that self-propagating electromagnetic waves would travel through space at a constant speed, which happened to be equal to the previously measured speed of light. From this, Maxwell concluded that light was a form of electromagnetic radiation: he first stated this result in 1862 in On Physical Lines of Force. In 1873, he published A Treatise on Electricity and Magnetism, which contained a full mathematical description of the behaviour of electric and magnetic fields, still known as Maxwell's equations. Soon after, Heinrich Hertz confirmed Maxwell's theory experimentally by generating and detecting radio waves in the laboratory, and demonstrating that these waves behaved exactly like visible light, exhibiting properties such as reflection, refraction, diffraction, and interference. Maxwell's theory and Hertz's experiments led directly to the development of modern radio, radar, television, electromagnetic imaging, and wireless communications.

### The special theory of relativity

The wave theory was wildly successful in explaining nearly all optical and electromagnetic phenomena, and was a great triumph of nineteenth century physics. By the late nineteenth century, however, a handful of experimental anomalies remained that could not be explained by or were in direct conflict with the wave theory. One of these anomalies involved a controversy over the speed of light. The constant speed of light predicted by Maxwell's equations and confirmed by the Michelson-Morley experiment contradicted the mechanical laws of motion that had been unchallenged since the time of Galileo, which stated that all speeds were relative to the speed of the observer. In 1905, Albert Einstein resolved this paradox by revising the Galilean model of space and time to account for the constancy of the speed of light. Einstein formulated his ideas in his special theory of relativity, which advanced humankind's understanding of space and time. Einstein also demonstrated a previously unknown fundamental equivalence between energy and mass with his famous equation
$E = mc^2 \,$
where E is energy, m is, depending on the context, the rest mass or the relativistic mass, and c is the speed of light in a vacuum.

### Particle theory revisited

Another experimental anomaly was the photoelectric effect, by which light striking a metal surface ejected electrons from the surface, causing an electric current to flow across an applied voltage. Experimental measurements demonstrated that the energy of individual ejected electrons was proportional to the frequency, rather than the intensity, of the light. Furthermore, below a certain minimum frequency, which depended on the particular metal, no current would flow regardless of the intensity. These observations appeared to contradict the wave theory, and for years physicists tried in vain to find an explanation. In 1905, Einstein solved this puzzle as well, this time by resurrecting the particle theory of light to explain the observed effect. Because of the preponderance of evidence in favor of the wave theory, however, Einstein's ideas were met initially by great skepticism among established physicists. But eventually Einstein's explanation of the photoelectric effect would triumph, and it ultimately formed the basis for wave–particle duality and much of quantum mechanics.

### Quantum theory

A third anomaly that arose in the late 19th century involved a contradiction between the wave theory of light and measurements of the electromagnetic spectrum emitted by thermal radiators, or so-called black bodies. Physicists struggled with this problem, which later became known as the ultraviolet catastrophe, unsuccessfully for many years. In 1900, Max Planck developed a new theory of black-body radiation that explained the observed spectrum correctly. Planck's theory was based on the idea that black bodies emit light (and other electromagnetic radiation) only as discrete bundles or packets of energy. These packets were called quanta, and the particle of light was given the name photon, to correspond with other particles being described around this time, such as the electron and proton. A photon has an energy, E, proportional to its frequency, f, by
$E = hf = \frac{hc}{\lambda} \,\!$
where h is Planck's constant, λ is the wavelength and c is the speed of light. Likewise, the momentum p of a photon is also proportional to its frequency and inversely proportional to its wavelength:
$p = { E \over c } = { hf \over c } = { h \over \lambda }.$
As it originally stood, this theory did not explain the simultaneous wave- and particle-like natures of light, though Planck would later work on theories that did. In 1918, Planck received the Nobel Prize in Physics for his part in the founding of quantum theory.

### Wave–particle duality

The modern theory that explains the nature of light includes the notion of wave–particle duality, described by Albert Einstein in the early 1900s, based on his study of the photoelectric effect and Planck's results. Einstein asserted that the energy of a photon is proportional to its frequency. More generally, the theory states that everything has both a particle nature and a wave nature, and various experiments can be done to bring out one or the other. The particle nature is more easily discerned if an object has a large mass, and it was not until a bold proposition by Louis de Broglie in 1924 that the scientific community realized that electrons also exhibited wave–particle duality. The wave nature of electrons was experimentally demonstrated by Davisson and Germer in 1927. Einstein received the Nobel Prize in 1921 for his work with the wave–particle duality on photons (especially explaining the photoelectric effect thereby), and de Broglie followed in 1929 for his extension to other particles.

### Quantum electrodynamics

The quantum mechanical theory of light and electromagnetic radiation continued to evolve through the 1920s and 1930's, and culminated with the development during the 1940s of the theory of quantum electrodynamics, or QED. This so-called quantum field theory is among the most comprehensive and experimentally successful theories ever formulated to explain a set of natural phenomena. QED was developed primarily by physicists Richard Feynman, Freeman Dyson, Julian Schwinger, and Shin-Ichiro Tomonaga. Feynman, Schwinger, and Tomonaga shared the 1965 Nobel Prize in Physics for their contributions.

## Light pressure

Light pushes on objects in its path, just as the wind would do. This pressure is most easily explainable in particle theory: photons hit and transfer their momentum. Light pressure can cause asteroids to spin faster,[20] acting on their irregular shapes as on the vanes of a windmill. The possibility to make solar sails that would accelerate spaceships in space is also under investigation.[21][22]
Although the motion of the Crookes radiometer was originally attributed to light pressure, this interpretation is incorrect; the characteristic Crookes rotation is the result of a partial vacuum.[23] This should not be confused with the Nichols radiometer, in which the motion is directly caused by light pressure.[24]

## Spirituality

An intricate display for the feast of St. Thomas at Kallara Pazhayapalli in Kottayam, Kerala, India dramatically illustrates the importance of light in religion.
The sensory perception of light plays a central role in spirituality (vision, enlightenment, darshan, Tabor Light). The presence of light as opposed to its absence (darkness) is a common metaphor of good and evil, knowledge and ignorance, and similar concepts. This idea is prevalent in both Eastern and Western spirituality.


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Calculus: Early Transcendentals 8th Edition

$$a_n = \sin{(\frac{n\pi}{2})}$$
Our first term is $a_1 = 1$. The sequence resembles the relationship of $\sin(x)$ at evaluations of $x = n\frac{\pi}{2}$ as $n = 1, 2, 3, 4, \cdots.$ Hence $$a_n = \sin{(\frac{n\pi}{2})}$$

format: plain


text:
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## Hybrid Logic (Stanford Encyclopedia of Philosophy)

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## Prolegomena of hypergroup theory.(English)Zbl 0785.20032

Supplemento alla Rivista di Matematica Pura ed Applicata. Tricesimo: Aviani Editore. 215 p. (1993).
As recalled by the author in the introduction of his book, hypergroups (multigroups) were introduced by F. Marty in 1934 and independently by H. S. Wall in 1937 as a natural generalization of groups. Since then the investigation into hypergroup theory knew no respite and developed in the large setting of hyperstructures (multialgebras). Corsini’s book is an introduction to hypergroup theory. In fact this is an enlarged new edition of the Italian version dating from 1986.
Each section begins with brief historical comments and references and is illustrated with interesting examples. The first section (“Introduction”) contains the fundamental definitions and results on hypergroups. The second section (“Homomorphisms”) deals with homomorphisms of various types and correlated equivalences. The results of the third section “Complete parts” are presented in the context of the general theory of hypergroups and also in connection of the heart’s structure (the heart of a hypergroup $$H$$ is the intersection of all the subhypergroups of $$H$$ which are complete parts). Sections four, “Subhypergroups”, and five, “Regular hypergroups”, deal with some special types of subhypergroups (closed, normal and invertible, conjugable). The hypergroups of type $$U$$ and those of type $$C$$ are also briefly presented. In the following four sections, namely “Join spaces, canonical hypergroups and their generalizations”, “$$m$$-complete hypergroups”, “Cogroups”, “Cyclic hypergroups”, the most important results on some remarkable classes of hypergroups such as $$m$$-complete hypergroups, 1-hypergrups, cogroups, cyclic hypergroups, are given. The section “Associativity semihypergroups and hypergroups” presents associativity semihypergroups and hypergroups as well as (partial) $$\tau$$-groupoids and constructions of partial $$\tau$$-groupoids. The Cartesian product of semihypergroups is also the object of this section. In the next section constructions of hypergroups and semihypergroups are given. In connection with the hypergroup theory the notion of hyperring, hypermodule, vectorial hyperspace and the principal results concerning them are presented in the section “Hyperrings, hypermodules and vectorial hyperspaces”.
The book (which closes with “Index of symbols”, “Analytical index” and with the exhaustive bibliography of 384 titles) is self-contained and the organization of the material aims successfully at making proofs as clear as possible. Corsini’s book not only gathers a large number of results on hypergroup theory, it also implicitly proposes a great amount of research work.

### MSC:

 20N20 Hypergroups 20-02 Research exposition (monographs, survey articles) pertaining to group theory


text:
# How to find the row median of columns having same name in data.table object in R?

R ProgrammingServer Side ProgrammingProgramming

To find the row median of columns having same name in data.table object in R, we can follow the below steps −

• First of all, create a data.table with some columns having same name.

• Then, use tapply along with colnames and median function to find the row median of columns having same name.

## Example

#### Create the data.table

Let’s create a data.table as shown below −

library(data.table)
DT<-
data.table(x=rpois(25,5),y=rpois(25,10),x=rpois(25,5),y=rpois(25,2),x=rpois(25,4),y=rpo is(25,10),check.names=FALSE)
DT

## Output

On executing, the above script generates the below output(this output will vary on your system due to randomization) −

     x  y  x y x  y
1:   4  15 6 1 6  9
2:   4  11 3 3 4  9
3:   1  6  8 2 2 12
4:   6  9  4 2 6 12
5:   8 10  1 1 2  9
6:   6  9  7 4 2  4
7:   8  8  3 2 1  8
8:   2 10  7 2 5 16
9:   7  5  8 1 5 9
10:  5 11  9 5 7 10
11:  3  4  1 5 6  6
12: 10 10  6 3 6 13
13:  7  5  4 2 3  8
14:  4  9  0 2 5 14
15:  6 11  8 2 4 10
16:  6  8  3 3 7  7
17:  7  8  4 2 1  8
18:  8  9  7 4 3 15
19:  4 16  5 1 3  7
20:  7 15  4 1 3 12
21:  6 10  5 1 1  9
22:  4 11  3 0 2  9
23:  4 12  4 3 4 17
24:  4  8  2 1 5  6
25:  5  9  1 0 2  5
x  y  x y x  y

Find the row median of columns having same name

Using tapply along with colnames and median function to find the row median of columns having same name in data.table object DT −

library(data.table)
DT<-
data.table(x=rpois(25,5),y=rpois(25,10),x=rpois(25,5),y=rpois(25,2),x=rpois(25,4),y=rpo is(25,10),check.names=FALSE)
t(apply(DT,1, function(x) tapply(x,colnames(DT),median)))

## Output

      x y
[1,]  6 9
[2,]  4 9
[3,]  2 6
[4,]  6 9
[5,]  2 9
[6,]  6 4
[7,]  3 8
[8,]  5 10
[9,]  7 5
[10,] 7 10
[11,] 3 5
[12,] 6 10
[13,] 4 5
[14,] 4 9
[15,] 6 10
[16,] 6 7
[17,] 4 8
[18,] 7 9
[19,] 4 7
[20,] 4 12
[21,] 5 9
[22,] 3 9
[23,] 4 12
[24,] 4 6
[25,] 2 5
Published on 10-Nov-2021 06:29:47


text:
# Average speed that you observe for the button as it falls

1. Feb 13, 2005

You are at an airport watching people go by on a "people mover" conveyer belt that moves at a constant speed of 0.60m/s. As a person goes by, they drop a button from a height of 1.2m. What is the average speed that you observe for the button as it falls to the ground?

well i have:
x=1.2m
v=0.60 m/s
a = 0

so i'll use this formula:

V^2=v(0)^2 +2a(x-x(0))

well a is 0 so the right side cancels to zero.

and v(0) = 0.60

so that means the answer is 0.60m/s, but it's incorrectly. the answer should be 2.5m/s. can someone tell me what i'm doing wrong?

2. Feb 13, 2005

### vsage

Where's the movement in the y direction?

3. Feb 13, 2005

$$v_{avg} = \frac {\int _0 ^T [v_x^2(t) + v_y^2(t)]^{1/2}dt }{T}$$


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# Jean is arranging paint brushes used by artists for an impressionist's

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Jean is arranging paint brushes used by artists for an impressionist's  [#permalink]

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01 Jul 2018, 02:14
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Jean is arranging paint brushes used by artists for an impressionist's exhibit. He has 4 from Van Gogh, 5 from Monet, 3 from Manet, 4 from Degas, and 2 from Toulouse-Lautrec to choose from. All brushes from the same artist are considered identical for ordering. Jean wants to group all the brushes from Van Gogh together. How many ways can he arrange the paint brushes by the artist that used them?
A) 5!
B) 15!
C) $$\frac{15!}{4!}$$
D) $$\frac{11!}{5!3!4!2!}$$
E) $$\frac{15!}{5!3!4!2!}$$

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Re: Jean is arranging paint brushes used by artists for an impressionist's  [#permalink]

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01 Jul 2018, 08:09
If we don't care of order, that group of Van Gogh brushes can be counted as 1 item
So, in this case we have:
Van Gogh = 1
Monet = 5
Manet = 3
Degas = 4
Toulouse-Lautrec = 2

Total ways to arrange 15 items = (1 + 5 + 3 + 4 + 2)! = 15!
As order doesn't matter we should multiply by factorial of number of each repeating group

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Re: Jean is arranging paint brushes used by artists for an impressionist's  [#permalink]

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01 Jul 2018, 12:22

This is a permutation question and the order is important.
Van Gogh brushes is considered as 1 entity.
So we have Van Gogh = 1,Monet = 5,Manet = 3,Degas = 4 and Toulouse-Lautrec = 2
So N = (1 + 5 + 3 + 4 + 2) = 15

Total arrangements = n!/(p!+q!+...) so soln is : 15!/(5!*3!*4!*2!)
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Re: Jean is arranging paint brushes used by artists for an impressionist's  [#permalink]

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01 Jul 2018, 12:52
souvonik2k wrote:
Jean is arranging paint brushes used by artists for an impressionist's exhibit. He has 4 from Van Gogh, 5 from Monet, 3 from Manet, 4 from Degas, and 2 from Toulouse-Lautrec to choose from. All brushes from the same artist are considered identical for ordering. Jean wants to group all the brushes from Van Gogh together. How many ways can he arrange the paint brushes by the artist that used them?
A) 5!
B) 15!
C) $$\frac{15!}{4!}$$
D) $$\frac{11!}{5!3!4!2!}$$
E) $$\frac{15!}{5!3!4!2!}$$

Consider the 4 Van Gogh brushes as 1 unit, since they need to be grouped together.

Hence we have Total # of brushes = 1 + 5 + 3 + 4 + 2 = 15

So we have total # of ways to arrange the brushes = 15!/(5!3!4!2!)

Thanks,
GyM
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Re: Jean is arranging paint brushes used by artists for an impressionist's  [#permalink]

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01 Jul 2018, 16:31
4 van Gogh is "1"
VmmmmmMMMddddtt
Formula:
(1+5+3+4+2)! / (1!5!3!4!2!)
Re: Jean is arranging paint brushes used by artists for an impressionist's &nbs [#permalink] 01 Jul 2018, 16:31
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# Group action with unique word

This must be known or easy for some of you, but here goes:

Suppose $$f_0,f_1:[n]\to [n]$$ are invertible functions, where $$[n]=\{0,\dots,n-1\}$$ is a set of $$n$$ elements. For a word $$w=w_1\dots w_m\in\{0,1\}^m$$ we define $$f_w=f_{w_m}\circ f_{w_{m-1}}\circ\dots\circ f_{w_1}$$ (or make it the opposite order, if you prefer). Suppose $$c, d$$ and $$m$$ are such that there is exactly one word $$w\in\{0,1\}^m$$ with $$f_w(c)=d$$. Does it follow that $$n\ge m+1$$?

I've checked that it does follow for all $$m\le 11$$ (eleven).

## 1 Answer

Fix $$c\in [n]$$. Let $$\mathcal R_m(c)$$ be $$\{f_w(c)\colon |w|=m\}$$ and $$r_m(c)=|\mathcal R_m(c)|$$. We define $$\mathcal R_0(c)$$ to be $$\{c\}$$.

Claim: Let $$m\ge 0$$. If $$r_m(c)=r_{m+1}(c)$$, then for all $$M>m$$ and for all $$d\in\mathcal R_M(c)$$, there are two $$w$$'s in $$\{0,1\}^M$$ with $$f_w(c)=d$$.

Proof: Let $$S=\mathcal R_m(c)$$. Then $$\mathcal R_{m+1}(c)=f_0(S)\cup f_1(S)$$. By invertibility if the $$f_i$$, $$f_0(S)$$ and $$f_1(S)$$ each have cardinality $$r_m(c)=r_{m+1}(c)$$, and their union also has cardinality $$r_{m+1}(c)$$. It follows that $$f_0(S)=f_1(S)$$. Now let $$w\in \{0,1\}^M$$ and let $$f_w(c)=d$$. Let $$w$$ be the concatenation of $$u$$ (of length $$m+1$$) and $$v$$ of length $$M-(m+1)\ge 0$$. By the above, there exists a $$u'$$ also of length $$m+1$$ with the opposite $$(m+1)$$st symbol such that $$f_{u'}(c)=f_{u}(c)$$. Now $$f_v\circ f_{u'}(c)=f_v\circ f_u(c)$$. $$\square$$

It follows that if $$c$$ and $$d$$ are such that there is a unique $$w$$ of length $$m$$ with $$f_w(c)=d$$, then $$r_{j+1}(c)>r_j(c)$$ for each $$j, so that $$n\ge r_m(c)\ge m+r_0(c)=m+1$$.

I should comment that this proof somewhat resembles that of the Morse-Hedlund theorem in symbolic dynamics.

• Yes: I used $|f_i(S)|=|S|$. Also there are easy counter-examples if you don’t have invertibility. – Anthony Quas Sep 27 at 15:10
• I edited the answer to make the role of invertibility explicit. – Anthony Quas Sep 27 at 15:17
• Hey Bjørn, it would feed my ego to be a co-author on a logic paper, but I don’t think this is much of a contribution. – Anthony Quas Sep 27 at 15:54
• The standard practice in this situation would be to acknowledge Anthony Quas's help in your paper, and possibly cite this question using a weblink. – Derek Holt Sep 27 at 16:18
• I agree with @DerekHolt. That would be just fine. – Anthony Quas Sep 27 at 16:26


text:
# For intro to analysis. Please answer true or false and give justification for your answer. Thanks!...

###### Question:

For intro to analysis. Please answer true or false and give justification for your answer. Thanks!

Mark the answers as ”TRUE” or "FALSE” on the front sheet. 1. Let f: Rd + R be continuous. Then the set {x e Rd : f(x) = 5} is closed. 2. Let f: Rd → R be continuous. Then the set {x € Rd: f(x) < 5} is open. 3. The rank of a p xq matrix is equal to min(p, q). 4. There exists a one-to-one linear map L : R5 → R4 5. There exists a surjective linear map L: R4 → R5

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# Inverse Variation

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In this section, we learn about inverse variation and inverse variation as a power. In an inverse variation problem, we say that y varies inversely with (as) x if there is a constant value k such that: y = k/x. The problems we see in this section are basically the same as in the last section with direct variation. We are given an opening scenario that allows us to find k. We are then told to find y when x is a certain value. We begin by writing the generic formula: y = k/x. We substitute the given values and solve for the constant of variation k. We then rewrite the equation with the known value of k and x. Once this is done, we can solve for the unknown y. When we see inverse variation as a power, we use the same technique.
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text:
# Wheat and chessboard problem

(Redirected from Second half of the chessboard)
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By the time that the fifth square is reached on the chessboard, the board contains a total of 31 grains of wheat

The wheat and chessboard problem (sometimes expressed in terms of rice grains) is a mathematical problem expressed in textual form as:

If a chessboard were to have wheat placed upon each square such that one grain were placed on the first square, two on the second, four on the third, and so on (doubling the number of grains on each subsequent square), how many grains of wheat would be on the chessboard at the finish?

The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8 + ... and so forth for the 64 squares. The total number of grains equals 18,446,744,073,709,551,615 (the 64th Mersenne number), much higher than what most intuitively expect.

The exercise of working through this problem may be used to explain and demonstrate exponents and the quick growth of exponential and geometric sequences. It can also be used to illustrate sigma notation. When expressed as exponents, the geometric series is: 20 + 21 + 22  + 23 + ... and so forth, up to 263. The base of each exponentiation, "2", expresses the doubling at each square, while the exponents represent the position of each square (0 for the first square, 1 for the second, etc.).

## Solutions

The simple, brute-force solution is to just manually double and add each step of the series:

${\displaystyle T_{64}=1+2+4+\cdots +9,223,372,036,854,775,808=18,446,744,073,709,551,615}$
where ${\displaystyle T_{64}}$ is the total number of grains.

The series may be expressed using exponents:

${\displaystyle T_{64}=2^{0}+2^{1}+2^{2}+\cdots +2^{63}}$

and, represented with capital-sigma notation as:

${\displaystyle \sum _{i=0}^{63}2^{i}.\,}$

It can also be solved much more easily using:

${\displaystyle T_{64}=2^{64}-1.\,}$

A proof of which is:

${\displaystyle s=2^{0}+2^{1}+2^{2}+\cdots +2^{63}.}$

Multiply each side by 2:

${\displaystyle 2s=2^{1}+2^{2}+2^{3}+\cdots +2^{63}+2^{64}.}$

Subtract original series from each side:

${\displaystyle 2s-s=-2^{0}+2^{64}}$
${\displaystyle \therefore s=2^{64}-1.\,}$

## Origin and story

The wheat and chess problem appears in different stories about the invention of chess. One of them includes the geometric progression problem. The story is first known to have been recorded in 1256 by Ibn Khallikan.[1] Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Minister) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager prize for a brilliant invention, only to have court treasurers report the unexpectedly huge number of wheat grains would outstrip the ruler's resources. Versions differ as to whether the inventor becomes a high-ranking advisor or is executed.[2]

Macdonnell also investigates the earlier development of the theme.[3]

[According to al-Masudi's early history of India], shatranj, or chess was invented under an Indian king, who expressed his preference for this game over backgammon. [...] The Indians, he adds, also calculated an arithmetical progression with the squares of the chessboard. [...] The early fondness of the Indians for enormous calculations is well known to students of their mathematics, and is exemplified in the writings of the great astronomer Āryabaṭha (born 476 A.D.). [...] An additional argument for the Indian origin of this calculation is supplied by the Arabic name for the square of the chessboard, (بيت, "beit"), 'house'. [...] For this has doubtless a historical connection with its Indian designation koṣṭhāgāra, 'store-house', 'granary' [...].

This exercise can be used to demonstrate how quickly exponential sequences grow, as well as to introduce exponents, zero power, capital-sigma notation and geometric series. Updated for modern times using pennies and the hypothetical question, "Would you rather have a million dollars or the sum of a penny doubled every day for a month?", the formula has been used to explain compounded interest.[4] [5]

## Second half of the chessboard

An illustration of Ray Kurzweil's second half of the chessboard principle. The letters are abbreviations for the SI metric prefixes.

In technology strategy, the "second half of the chessboard" is a phrase, coined by Ray Kurzweil,[6] in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy. While the number of grains on the first half of the chessboard is large, the amount on the second half is vastly (232 > 4 billion times) larger.

The number of grains of wheat on the first half of the chessboard is 1 + 2 + 4 + 8 + ... + 2,147,483,648, for a total of 4,294,967,295 (232 − 1) grains, or about 279 tonnes of wheat (assuming 65 mg as the mass of one grain of wheat).[7]

The number of grains of wheat on the second half of the chessboard is 232 + 233 + 234 + ... + 263, for a total of 264 − 232 grains. This is equal to the square of the number of grains on the first half of the board, plus itself. The first square of the second half alone contains more grains than the entire first half. On the 64th square of the chessboard alone there would be 263 = 9,223,372,036,854,775,808 grains, more than two billion times as many as on the first half of the chessboard.

On the entire chessboard there would be 264 − 1 = 18,446,744,073,709,551,615 grains of wheat, weighing about 1,199,000,000,000 metric tons. This is about 1,645 times the global production of wheat in 2014 (729,000,000 metric tons).[8]

## Use

Carl Sagan titled the second chapter of his final book The Persian Chessboard and wrote that when referring to bacteria, "Exponentials can't go on forever, because they will gobble up everything."[9] Similarly, The Limits to Growth uses the story to present suggested consequences of exponential growth: "Exponential growth never can go on very long in a finite space with finite resources."[10]

## References

1. ^ Clifford A. Pickover (2009), The Math Book: From Pythagoras to the 57th Dimension, New York : Sterling. ISBN 9781402757969. p. 102
2. ^ Tahan, Malba (1993). The Man Who Counted: A Collection of Mathematical Adventures. New York: W.W. Norton & Co. pp. 113–115. ISBN 0393309347. Retrieved 2015-04-05.
3. ^ Macdonell, A. A. (2011-03-15). "Art. XIII.—The Origin and Early History of Chess". Journal of the Royal Asiatic Society of Great Britain & Ireland. 30 (01): 117–141. doi:10.1017/S0035869X00146246. Retrieved 2015-04-05.
4. ^ https://www.bloomberg.com/news/videos/b/92966fc7-c54d-4405-8fa6-cbefd05bbd6f
5. ^ http://mathforum.org/dr.math/faq/faq.doubling.pennies.html
6. ^ Kurzweil, Ray (1999). The Age of Spiritual Machines: When Computers Exceed Human Intelligence. New York: Penguin. p. 37. ISBN 0-670-88217-8. Retrieved 2015-04-06.
7. ^ "Encyclopedia Britannica: Grain, unit of weight". 29 April 2004. Retrieved 2 March 2017.
8. ^ "FAOSTAT". faostat3.fao.org. Retrieved 2 March 2017.
9. ^ Sagan, Carl (1997). Billions and Billions: Thoughts On Life And Death At the Brink Of The Millennium. New York: Ballantine Books. p. 17. ISBN 0-345-37918-7.
10. ^ Meadows, Donella H., Dennis L. Meadows, Jørgen Randers, and William W. Behrens III (1972). The Limits to Growth, p. 21, at Google Books. New York: University Books. ISBN 0-87663-165-0. Retrieved 2015-04-05.


text:
randRange(4, 6) randRange(75, 98, LENGTH) (function() { var sum = 0; jQuery.each(SCORES, function( i, elem ) { sum += elem; }); return sum; })()

In his(1) course(1) class, person(1) took LENGTH plural(exam(1)). His(1) scores were toSentence(SCORES).

What was his(1) average score on the plural(exam(1))?

SUM / LENGTH

The average is the sum of his(1) scores divided by the number of scores.

There are LENGTH scores and their sum is SCORES.join(" + ") = SUM.

His(1) average score is SUM \div LENGTH = SUM / LENGTH.

randRange(3, 7) randRange(77, 88) randRange(OLD_AVG + 2, 99)

On the first COUNT plural(exam(1)) of his(1) course(1) class, person(1) got an average score of OLD_AVG.

What does he(1) need on the next exam(1) to have an overall average of NEW_AVG?

NEW_AVG + COUNT * (NEW_AVG - OLD_AVG)

Let his(1) score on the next exam(1) be x.

The sum of all of his(1) scores is then COUNT \cdot OLD_AVG + x.

The same sum must also be equal to COUNT + 1 \cdot NEW_AVG.

Solve: x = COUNT + 1 \cdot NEW_AVG - COUNT \cdot OLD_AVG = (COUNT + 1) * NEW_AVG - COUNT * OLD_AVG.

randRange(3, 6) randRange(2, COUNT - 2) randRange(77, 88) (OLD_AVG * COUNT + 100 * REMAINING) / (COUNT + REMAINING)

person(1) has taken COUNT plural(exam(1)) and his(1) average score so far is OLD_AVG.

If he(1) gets 100, a perfect score, on the remaining REMAINING plural(exam(1)), what will his(1) new average be?

NEW_AVG

If he(1) gets 100 on the remaining plural(exam(1)), the sum of his(1) scores will be COUNT \cdot OLD_AVG + REMAINING \cdot 100 = COUNT * OLD_AVG + 100 * REMAINING.

His(1) overall average will then be COUNT * OLD_AVG + 100 * REMAINING \div COUNT + REMAINING = NEW_AVG.


text:
Tuesday, June 14, 2011

D0 rejects CDF's claim on top-antitop mass difference, too

First: major LHC detectors reach one inverse femtobarn!

By 21:10 Prague Summer Time tonight, the CMS and ATLAS at CERN have collected 1/fb of data each: see viXra blog.

This was the original plan for the whole year 2011 - and it's reached in mid June! The figure includes 47/pb from 2010, but 47/pb is collected in 36 hours in average these days, so on Wednesday, the 2011 run will probably reach a femtobarn, too. About 95% of the data have been recorded, so in a few more days, one also surpasses 1 recorded femtobarn of the data.

The LHC could collect 4/fb by the end of 2011 and, because of expected luminosity increases, 15/fb by the end of 2012 which should be enough to discover the Higgs boson even at the most inaccessible places. In fact, if the Higgs boson isn't in the data that have already been collected by now, then it's probably lighter than 135 GeV. It follows that the vacuum of the Standard Model is unstable and needs new, SUSY-like particles to be saved.

Now, let's look at the opposite side of the Atlantic

A Japanese translation of the text below is available.

In March, Tommaso Dorigo hyped a preprint by the CDF Collaboration,
Measurement of the mass difference between top and antitop quarks
As far as I remember, I hadn't discussed that paper on this blog because I considered it and still consider it a very bad, offensive piece of work. At any rate, they claimed that the top quark was much lighter than the antitop antiquark:
M(top) - M(antitop) = -3.3 +- 1.4(stat) +- 1.0(syst)
Using this lousy measurement, they wrote a paper phrased in such a way that they offered a 2-sigma evidence that the CPT "hypothesis" (which, of course, implies that antiparticles have to have exactly the same masses as particles) was wrong.

CDF, a pretty detector that has produced lots of rubbish lately (and maybe it's the people behind it who did it)

This is just offensive. It's the kind of shoddy research with the most sensationalist claims supported by the weakest possible (and, independently of that, flawed) evidence that belongs to the climate "science" but surely not to particle physics.

Today, the CDF's competitors at D0 have released their paper on the very same question:
Direct measurement of the mass difference between top and antitop quarks
There is no sign of the mass difference, of course. In the previous D0 paper, the very same mass difference was +3.8 +- 3.4 (stat) +- 1.2 (syst) - yes, the mass difference was going in the opposite direction than the CDF claim! The newest D0 paper says it is
M(top) - M(antitop) = +0.8 +- 1.8(stat) +- 0.5(syst)
Again, it's the opposite sign than the CDF claim but the mass difference is as zero as you could hope. The accuracy of the measurement remains disappointing but the preposterous idea suggested by the CDF paper that the experiments would begin to uncover a huge, multi-GeV mass difference between the top and the antitop has clearly been debunked.

Something is really wrong with the CDF detector and/or the CDF Collaboration and/or their methods.

Newsy Science: this is actually a fair and clear popular report on the Wjj 150 GeV events

In March, I had some exchanges with an experimenter on Dorigo's blog. The first thing I wrote was:
I think that the interpretation chosen by this [CDF] paper is shameful sensationalism. Paying attention to 2-sigma deviations is bad enough, but using 2-sigma deviations to "disprove" one of the most important principles of the discipline is a really bad taste.

Whoever had the idea to interpret their inability to measure the masses of tops and antitops more accurately in this far-reaching way should be ashamed.
Sean agreed with me. Andrea Giammanco, an experimenter who is not even a CDF member, tried to defend their work:
Hi Lubos,

I suspect you didn't actually read the paper.
I didn't find a single sentence of the paper where they claim that this result "disproves" anything. If I missed it, please point me to the exact line.

The most bold sentence that I found is that it deviates at 2 sigma level from the CPT-symmetry expectation. What is your criticism, exactly? That they should have written "it agrees at 2 sigma level with the CPT-symmetry expectation"? Whomever is able to read until that point in the paper also knows that the two statements are the same.

I must also add that I can't understand the criticism in some of the comments above for looking for an effect that "cannot" be there.

I find healthy to look for deviations even when you have no reason whatsoever to expect a deviation; I understand in general the criticism that, with finite resources, these must be prioritized, but this is a particularly uncontroversial case because data are anyway available as a by-product of all the rest of the Tevatron program, and performing this study doesn't require any other cost than a few months of salary of just two people (*).

There are several instructive historical examples of important effects that could have been observed before, because the data for their discovery were already available or easy to produce, but none of the "owners" of the data had considered to look for that particular effect (or, even worse, had actually stumbled into a 2-3 sigma deviation and paid no attention, remodelling the background or inflating the systematic uncertainty to take it into account and therefore unwittingly hiding it under the carpet.)

(*) their number and identity can be seen here:
http://www-cdf.fnal.gov/physics/new/top/2010/mass/TMT_massdiff_p28_public/
So I gave him a longer explanation:
Dear Andrea, what I find offensive is their arrogant suggestion - included to the very title, and much of the abstract as well as the paper - that they have measured a nonzero difference between top and antitop masses.

To deduce what are the masses of the particles from their measurements, you effectively have to use some quantum field theory at one level or another. Quantum field theory implies that the mass difference is zero. So the most accurate measurement is to do nothing and to conclude that the difference is exactly zero.

In this sense, the most outrageous sentence about that paper is the last sentence, one that the measurement of the top-antitop mass difference - which is a whopping 3.3 GeV according to the CDF paper - is the "most accurate" measurement of the quantity.

This is just bullshit.

The most accurate measurement of the mass difference, done with a clever definition of the mass, is 0.000000 +- 0.000000 GeV, and neither they nor you have presented any real evidence that this is not the case. The CDF result is a striking sign of the immense inaccuracy with which they measure the mass of the heaviest quark, so it's really disingenious for them to claim that they've just produced the most accurate result.

They have produced one of the most *inaccurate* results for this quantity ever written down.

It is healthy to look for whatever effects but it is totally unhealthy to publish papers claiming to have measured effects that almost certainly don't exist, without having any real evidence for such claims.

CDF should have verified that they don't have any 5-sigma deviation, and because they don't, they should use the obvious identity mass(top) = mass(antitop) as a method to make their other measurements more accurate. In other words, they should have used the top-antitop average mass, 172.5 GeV, as the actual input to use for other measurements. They failed to do all such things which really indicates that most of the mass measurements depending on the matter-antimatter difference are likely to have similar 3.4-GeV-like errors, too.

They have surely nothing to boast about in this context, so it is irritating that they apparently do.

Cheers
LM
Obviously, they just don't feel that way. They produced the worst (largest) bullshit number ever describing the top-antitop difference - over 3.3 GeV even though it is obviously 0.000 GeV and many others have come really close to it - and they don't see any problem about claiming that the 3.3 GeV result for the mass difference is the "most accurate measurement". Holy crap.

Obviously, the conversations were completely unproductive. Maybe it's because the experimenters don't actually understand the laws of physics. Maybe they're being honest but they just don't know that to suggest that they have measured a nonzero mass difference of a particle and its antiparticle because of a 2-sigma bump is just idiotic. It's like claiming that the rectangular triangle with legs 3 and 4 has hypotenuse of 5.05 plus minus 0.02 meters which deviates from the Pythagorean "hypothesis" by 2 sigma.

Hopefully, they at least understand the experiments, so they should be able to understand that the D0 has just confirmed that I have always been right and that the CDF paper had actually made the most inaccurate measurement of the m(top)-m(antitop) mass difference ever, it should be ashamed of that paper, and it had absolutely no moral right to suggest far-reaching interpretations of their shoddy measurement.

Claims about the CPT violation (or, equivalently, about the nonzero matter-antimatter mass difference) are extraordinary claims and if you're a serious person, you simply shouldn't make them based on some illusions or 2-sigma signals.

I think that the CPT-symmetry is exact, a direct consequence of the Lorentz symmetry and analyticity. Just Wick-rotate your scattering problem to the Euclidean spacetime, rotate the "tz" plane by 180 degrees, and Wick-rotate back to the Minkowski space. You will get the CPT image of the original process. Because the analytic continuation and the "tz" rotation - which is a kind of the Lorentz boost - were the only operations and they're symmetries, it follows that CPT has to hold.

(You may be confused why the reversal of the "tz" plane gives you CPT rather than PT. While the obvious geometric transformation induced by this operation is PT, the particles also start to move backwards in time after this PT, and a particle moving backwards in time has to be interpreted as an antiparticle. So the full transformation is CPT, not PT, and one has to be sensitive to properly derive that the non-geometric "C" factor is included there as well.)

But even if you had doubts whether the CPT symmetry would be exact forever, it's just exact according to all measurements that had been done so far. So a claim that it doesn't hold - and that it's even violated by 2 percent by ordinary top quarks - is just an extraordinary claim and 2-sigma deviations are surely not good enough to claim to support the claim.

1. Dear Lubos,

I am watching in disbelief what’s going on with dubious claims promoted by CDF…as I am also watching in disbelief how millions of dollars are awarded to Verlinde for his misguided research on entropic gravity …truly pathetic!

2. It is amazing, too. But the Spinoza prize is an internal Dutch affair. Maybe the Dutch science hasn't produced anything better recently than this piece of quite manifestly incorrect pop-science entropic nonsense.


text:
# stringi-search-regex

0th

Percentile

##### Regular Expressions in stringi

A regular expression is a pattern describing, possibly in a very abstract way, a part of text. Thanks to many regex functions in stringi, regular expressions may be a very powerful tool in your hand to perform string searching, substring extraction, string splitting, etc., tasks.

##### Details

All stri_*_regex functions in stringi use the ICU regex engine. Its settings may be tuned up (for example to perform case-insensitive search), see the stri_opts_regex function for more details.

Regular expression patterns in ICU are quite similar in form and behavior to Perl's regexes. Their implementation is loosely inspired by JDK 1.4 java.util.regex. ICU Regular Expressions conform to the Unicode Technical Standard #18 (see References section) and its features are summarized in the ICU User Guide (see below). A good general introduction to regexes is (Friedl, 2002). Some general topics are also covered in the Rmanual, see regex.

##### Regex Functions in stringi

Note that if a given regex pattern is empty, then all functions in stringi give NA in result and generate a warning. On a syntax error, a quite informative failure message is shown.

If you would like to search for a fixed pattern, refer to stringi-search-coll or stringi-search-fixed. This allows to do a locale-aware text lookup, or a very fast exact-byte search, respectively.

##### References

Regular expressions -- ICU User Guide, http://userguide.icu-project.org/strings/regexp

J.E.F. Friedl, Mastering Regular Expressions, O'Reilly, 2002

Unicode Regular Expressions -- Unicode Technical Standard #18, http://www.unicode.org/reports/tr18/

Unicode Regular Expressions -- Regex tutorial, http://www.regular-expressions.info/unicode.html

Other search_regex: stri_opts_regex; stringi-search
Other stringi_general_topics: stringi-arguments; stringi-encoding; stringi-locale; stringi-search-boundaries; stringi-search-charclass; stringi-search-coll; stringi-search-fixed; stringi-search; stringi, stringi-package


text:
Change the chapter
Question
Find the range of visible wavelengths of light in crown glass.
$250\textrm{ nm} \le \lambda_n \le 500\textrm{ nm}$
Solution Video


text:
# Modelling and solving temporal reasoning as propositional satisfiability

@article{Pham2008ModellingAS,
title={Modelling and solving temporal reasoning as propositional satisfiability},
author={Duc Nghia Pham and John Thornton and A. Sattar},
journal={Artif. Intell.},
year={2008},
volume={172},
pages={1752-1782}
}
• Published 2008
• Mathematics, Computer Science
• Artif. Intell.
Representing and reasoning about time dependent information is a key research issue in many areas of computer science and artificial intelligence. One of the best known and widely used formalisms for representing interval-based qualitative temporal information is Allen's interval algebra (IA). The fundamental reasoning task in IA is to find a scenario that is consistent with the given information. This problem is in general NP-complete. In this paper, we investigate how an interval-based… Expand
A Generalised Approach for Encoding and Reasoning with Qualitative Theories in Answer Set Programming
• Computer Science
• Theory and Practice of Logic Programming
• 2020
Experimental results show that the proposed encodings outperform one of the two reasoners, but fall behind the other, an acceptable trade-off given the added benefits of handling any type of reasoning as well as the interpretability of logic programs. Expand
Improved Algorithms for Allen's Interval Algebra: a Dynamic Programming Approach
• Computer Science
• IJCAI
• 2021
The constraint satisfaction problem (CSP) is an important framework in artificial intelligence used to model e.g. qualitative reasoning problems such as Allen’s interval algebra (A). There is strongExpand
Qualitative Spatial and Temporal Reasoning with Answer Set Programming
• Jason Jingshi Li
• Computer Science
• 2012 IEEE 24th International Conference on Tools with Artificial Intelligence
• 2012
Three novel encodings which translate problems in qualitative spatial and temporal reasoning into logic programs for answer set programming solvers are introduced, each reflecting a different type of modeling abstraction. Expand
Decomposition and tractability in qualitative spatial and temporal reasoning
• Computer Science
• Artif. Intell.
• 2013
A sufficient condition under which consistency can be decided in polynomial time for networks of bounded treewidth in QSTR is identified, and this condition is satisfied by a range of calculi including the Interval Algebra, Rectangle Al algebra, Block Algebra), RCC8, and RCC5. Expand
SAT vs. Search for Qualitative Temporal Reasoning
A SAT encoding of IA is presented that simulates the use of tractable subsets in native solvers and shows that the refutation of any inconsistent IA network can always be done by SAT (via the authors' new encoding) as efficiently as by native search. Expand
Satisfiability Checking in Łukasiewicz Logic as Finite Constraint Satisfaction
• Mathematics, Computer Science
• Journal of Automated Reasoning
• 2011
An efficient, polynomial-time algorithm is proposed for establishing a strong upper bound on the required number of truth degrees, allowing us to check the satisfiability of sets of formulas in $\mathcal{L}_{\infty}$, and more generally, sets of fuzzy clauses over Łukasiewicz logic formulas, by solving a small number of constraint satisfaction problems. Expand
Theory propagation and reification
• Computer Science
• Sci. Comput. Program.
• 2015
It is demonstrated how reification provides a natural way for orchestrating reification in the propositional part of the problem with theory propagation in the constraint component of SMT, allowing an elegant implementation of DPLL(T) solvers in Prolog. Expand
A Divide-and-Conquer Approach for Solving Interval Algebra Networks
• Mathematics, Computer Science
• IJCAI
• 2009
A divide-and-conquer method that recursively partitions a given problem into smaller sub-problems in deciding consistency is introduced and a key theoretical property of a qualitative calculus is identified that ensures the soundness and completeness of this method. Expand
Efficient automated planning with new formulations
• Computer Science
• 2011
This work strongly suggests a shifting trend of planning from STRIPS to SAS+, and shows the power of formulating planning problems as Satisfiability, and developed a new SAS+ based SAT encoding scheme (SASE) for planning. Expand
A SAT Approach for Maximizing Satisfiability in Qualitative Spatial and Temporal Constraint Networks
• Computer Science
• KR
• 2016
To efficiently solve the MAX-QCN problem, this paper introduces and study two families of encodings of the partial maximum satisfiability problem (PMAX-SAT), based on a forbidden covering with regard to the composition table of the considered qualitative calculus. Expand

#### References

SHOWING 1-10 OF 88 REFERENCES
Reasoning About Qualitative Temporal Information
• P. V. Beek
• Computer Science, Mathematics
• Artif. Intell.
• 1992
This paper defines computationally efficient procedures for solving two related reasoning tasks that arise in interval and point algebras: Given (possibly indefinite) knowledge of the relationships between some intervals or points, find one or more scenarios that are consistent with the information provided, and find all the feasible relations between every pair of interval or points. Expand
Encoding Plans in Propositional Logic
• Computer Science
• KR
• 1996
A novel encoding based on the theory of causal planning, that exploits the notion of “lifting” from the theorem-proving community is introduced, which strictly dominates the others in terms of asymptotic complexity. Expand
Exact and approximate reasoning about temporal relations 1
• Computer Science
• Comput. Intell.
• 1990
This paper addresses a fundamental reasoning task that arises in applications of the algebra: Given (possibly indefinite) knowledge about the relationships between intervals, find all feasible relationships between two intervals, called the minimal labels problem. Expand
Complexity and algorithms for reasoning about time: a graph-theoretic approach
• Mathematics, Computer Science
• JACM
• 1993
This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics and shows that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NP-complete. Expand
A Local Search Approach to Modelling and Solving Interval Algebra Problems
• Computer Science
• J. Log. Comput.
• 2004
This paper looks at modelling and solving problems formulated using Allen's interval algebra and proposes a new constraint weighting algorithm derived from DLM and shows that local search outperforms existing consistency-enforcing algorithms on those problems that the existing techniques find most difficult. Expand
Efficient solution techniques for disjunctive temporal reasoning problems
• Computer Science
• Artif. Intell.
• 2003
A new algorithm is presented, called Epilitis, which integrates strategies for efficient DTP solving from the previous literature, including conflictdirected backjumping, removal of subsumed variables, and semantic branching, and further adds no-good recording as a central technique and can achieve a nearly two order-of-magnitude speed-up over previously published algorithms on benchmark problems in the DTP literature. Expand
Solving hard qualitative temporal reasoning problems: Evaluating the efficiency of using the ORD-Horn class
• B. Nebel
• Computer Science, Mathematics
• Constraints
• 2004
It is proved that Ladkin and Reinefeld's algorithm is complete when using the ORD-Horn class, and evidence is given that combining search methods orthogonally can dramatically improve the performance of the backtracking algorithm. Expand
Towards an Efficient SAT Encoding for Temporal Reasoning
• Mathematics, Computer Science
• CP
• 2006
Encoding IA into SAT produces better results than existing approaches, and the phase transition region maps directly from the IA encoding to each SAT encoding, but, surprisingly, the location of the hard region varies according to the encoding scheme. Expand
Temporal Constraint Networks
• Computer Science, Mathematics
• Artif. Intell.
• 1989
It is shown that the STP, which subsumes the major part of Vilain and Kautz's point algebra, can be solved in polynomial time and the applicability of path consistency algorithms as preprocessing of temporal problems is studied, to demonstrate their termination and bound their complexities. Expand
The design and experimental analysis of algorithms for temporal reasoning
• Computer Science
• 1996
This paper discusses the design and empirical analysis of algorithms for a temporal reasoning system based on Allen's influential interval-based framework for representing temporal information, and develops techniques that can result in up to a ten-fold speedup over an already highly optimized implementation. Expand


text: A spirolateral is a continuous path drawn by repeating a sequence of line segments of lengths $a_1, a_2, a_3, ... a_n$ with a given angle of turn between each line segment and the next one. (Alternatively the path can be considered as a repeated sequence of 'bound' vectors: $\overrightarrow {P_1P}_2, \overrightarrow{P_2P}_3,... \overrightarrow{P_n P}_{n+1}$, each vector starting at the endpoint of the previous vector.)