title: Ultraviolet

text: Medical organizations recommend that patients protect themselves from UV radiation by using sunscreen. Five sunscreen ingredients have been shown to protect mice against skin tumors. However, some sunscreen chemicals produce potentially harmful substances if they are illuminated while in contact with living cells. The amount of sunscreen that penetrates into the lower layers of the skin may be large enough to cause damage.


title: Ultraviolet

text: Sunscreen reduces the direct DNA damage that causes sunburn, by blocking UV‑B, and the usual SPF rating indicates how effectively this radiation is blocked. SPF is, therefore, also called UVB-PF, for "UV‑B protection factor". This rating, however, offers no data about important protection against UVA, which does not primarily cause sunburn but is still harmful, since it causes indirect DNA damage and is also considered carcinogenic. Several studies suggest that the absence of UV‑A filters may be the cause of the higher incidence of melanoma found in sunscreen users compared to non-users. Some sunscreen lotions contain titanium dioxide, zinc oxide, and avobenzone, which help protect against UV‑A rays.


title: Ultraviolet

text: The photochemical properties of melanin make it an excellent photoprotectant. However, sunscreen chemicals cannot dissipate the energy of the excited state as efficiently as melanin and therefore, if sunscreen ingredients penetrate into the lower layers of the skin, the amount of reactive oxygen species may be increased. The amount of sunscreen that penetrates through the stratum corneum may or may not be large enough to cause damage.


title: Ultraviolet

text: In an experiment by Hanson "et al". that was published in 2006, the amount of harmful reactive oxygen species (ROS) was measured in untreated and in sunscreen treated skin. In the first 20 minutes, the film of sunscreen had a protective effect and the number of ROS species was smaller. After 60 minutes, however, the amount of absorbed sunscreen was so high that the amount of ROS was higher in the sunscreen-treated skin than in the untreated skin. The study indicates that sunscreen must be reapplied within 2 hours in order to prevent UV light from penetrating to sunscreen-infused live skin cells.


title: Ultraviolet

text: Ultraviolet radiation can aggravate several skin conditions and diseases, including systemic lupus erythematosus, Sjögren's syndrome, Sinear Usher syndrome, rosacea, dermatomyositis, Darier's disease, Kindler–Weary syndrome and Porokeratosis.


title: Ultraviolet

text: The eye is most sensitive to damage by UV in the lower UV‑C band at 265–275 nm. Radiation of this wavelength is almost absent from sunlight but is found in welder's arc lights and other artificial sources. Exposure to these can cause "welder's flash" or "arc eye" (photokeratitis) and can lead to cataracts, pterygium and pinguecula formation. To a lesser extent, UV‑B in sunlight from 310 to 280 nm also causes photokeratitis ("snow blindness"), and the cornea, the lens, and the retina can be damaged.


title: Ultraviolet

text: Protective eyewear is beneficial to those exposed to ultraviolet radiation. Since light can reach the eyes from the sides, full-coverage eye protection is usually warranted if there is an increased risk of exposure, as in high-altitude mountaineering. Mountaineers are exposed to higher-than-ordinary levels of UV radiation, both because there is less atmospheric filtering and because of reflection from snow and ice.


title: Ultraviolet

text: Ordinary, untreated eyeglasses give some protection. Most plastic lenses give more protection than glass lenses, because, as noted above, glass is transparent to UV‑A and the common acrylic plastic used for lenses is less so. Some plastic lens materials, such as polycarbonate, inherently block most UV.


title: Ultraviolet

text: UV degradation is one form of polymer degradation that affects plastics exposed to sunlight. The problem appears as discoloration or fading, cracking, loss of strength or disintegration. The effects of attack increase with exposure time and sunlight intensity. The addition of UV absorbers inhibits the effect.


title: Ultraviolet

text: Sensitive polymers include thermoplastics and speciality fibers like aramids. UV absorption leads to chain degradation and loss of strength at sensitive points in the chain structure. Aramid rope must be shielded with a sheath of thermoplastic if it is to retain its strength.


title: Ultraviolet

text: Many pigments and dyes absorb UV and change colour, so paintings and textiles may need extra protection both from sunlight and fluorescent bulbs, two common sources of UV radiation. Window glass absorbs some harmful UV, but valuable artifacts need extra shielding. Many museums place black curtains over watercolour paintings and ancient textiles, for example. Since watercolours can have very low pigment levels, they need extra protection from UV. Various forms of picture framing glass, including acrylics (plexiglass), laminates, and coatings, offer different degrees of UV (and visible light) protection.


title: Ultraviolet

text: Because of its ability to cause chemical reactions and excite fluorescence in materials, ultraviolet radiation has a number of applications. The following table gives some uses of specific wavelength bands in the UV spectrum


title: Ultraviolet

text: Photographic film responds to ultraviolet radiation but the glass lenses of cameras usually block radiation shorter than 350 nm. Slightly yellow UV-blocking filters are often used for outdoor photography to prevent unwanted bluing and overexposure by UV rays. For photography in the near UV, special filters may be used. Photography with wavelengths shorter than 350 nm requires special quartz lenses which do not absorb the radiation.


title: Ultraviolet

text: Digital cameras sensors may have internal filters that block UV to improve color rendition accuracy. Sometimes these internal filters can be removed, or they may be absent, and an external visible-light filter prepares the camera for near-UV photography. A few cameras are designed for use in the UV.


title: Ultraviolet

text: Photography by reflected ultraviolet radiation is useful for medical, scientific, and forensic investigations, in applications as widespread as detecting bruising of skin, alterations of documents, or restoration work on paintings. Photography of the fluorescence produced by ultraviolet illumination uses visible wavelengths of light.


title: Ultraviolet

text: In ultraviolet astronomy, measurements are used to discern the chemical composition of the interstellar medium, and the temperature and composition of stars. Because the ozone layer blocks many UV frequencies from reaching telescopes on the surface of the Earth, most UV observations are made from space.


title: Ultraviolet

text: Corona discharge on electrical apparatus can be detected by its ultraviolet emissions. Corona causes degradation of electrical insulation and emission of ozone and nitrogen oxide.


title: Ultraviolet

text: EPROMs (Erasable Programmable Read-Only Memory) are erased by exposure to UV radiation. These modules have a transparent (quartz) window on the top of the chip that allows the UV radiation in.


title: Ultraviolet

text: Colorless fluorescent dyes that emit blue light under UV are added as optical brighteners to paper and fabrics. The blue light emitted by these agents counteracts yellow tints that may be present and causes the colors and whites to appear whiter or more brightly colored.


title: Ultraviolet

text: UV fluorescent dyes that glow in the primary colors are used in paints, papers, and textiles either to enhance color under daylight illumination or to provide special effects when lit with UV lamps. Blacklight paints that contain dyes that glow under UV are used in a number of art and aesthetic applications.


title: Ultraviolet

text: Amusement parks often use UV lighting to fluoresce ride artwork and backdrops. This often has the side effect of causing rider's white clothing to glow light-purple.


title: Ultraviolet

text: To help prevent counterfeiting of currency, or forgery of important documents such as driver's licenses and passports, the paper may include a UV watermark or fluorescent multicolor fibers that are visible under ultraviolet light. Postage stamps are tagged with a phosphor that glows under UV rays to permit automatic detection of the stamp and facing of the letter.


title: Ultraviolet

text: UV fluorescent dyes are used in many applications (for example, biochemistry and forensics). Some brands of pepper spray will leave an invisible chemical (UV dye) that is not easily washed off on a pepper-sprayed attacker, which would help police identify the attacker later.


title: Ultraviolet

text: In some types of nondestructive testing UV stimulates fluorescent dyes to highlight defects in a broad range of materials. These dyes may be carried into surface-breaking defects by capillary action (liquid penetrant inspection) or they may be bound to ferrite particles caught in magnetic leakage fields in ferrous materials (magnetic particle inspection).


title: Ultraviolet

text: UV is an investigative tool at the crime scene helpful in locating and identifying bodily fluids such as semen, blood, and saliva. For example, ejaculated fluids or saliva can be detected by high-power UV sources, irrespective of the structure or colour of the surface the fluid is deposited upon. UV–vis microspectroscopy is also used to analyze trace evidence, such as textile fibers and paint chips, as well as questioned documents.


title: Ultraviolet

text: Other applications include the authentication of various collectibles and art, and detecting counterfeit currency. Even materials not specially marked with UV sensitive dyes may have distinctive fluorescence under UV exposure or may fluoresce differently under short-wave versus long-wave ultraviolet.


title: Ultraviolet

text: Using multi-spectral imaging it is possible to read illegible papyrus, such as the burned papyri of the Villa of the Papyri or of Oxyrhynchus, or the Archimedes palimpsest. The technique involves taking pictures of the illegible document using different filters in the infrared or ultraviolet range, finely tuned to capture certain wavelengths of light. Thus, the optimum spectral portion can be found for distinguishing ink from paper on the papyrus surface.


title: Ultraviolet

text: Ultraviolet light helps detect organic material deposits that remain on surfaces where periodic cleaning and sanitizing may have failed. It is used in the hotel industry, manufacturing, and other industries where levels of cleanliness or contamination are inspected.


title: Ultraviolet

text: Perennial news features for many television news organizations involve an investigative reporter using a similar device to reveal unsanitary conditions in hotels, public toilets, hand rails, and such.


title: Ultraviolet

text: UV/Vis spectroscopy is widely used as a technique in chemistry to analyze chemical structure, the most notable one being conjugated systems. UV radiation is often used to excite a given sample where the fluorescent emission is measured with a spectrofluorometer. In biological research, UV radiation is used for quantification of nucleic acids or proteins. In environmental chemistry, UV radiation could also be used to detect Contaminants of emerging concern in water samples.


title: Ultraviolet

text: In pollution control applications, ultraviolet analyzers are used to detect emissions of nitrogen oxides, sulfur compounds, mercury, and ammonia, for example in the flue gas of fossil-fired power plants. Ultraviolet radiation can detect thin sheens of spilled oil on water, either by the high reflectivity of oil films at UV wavelengths, fluorescence of compounds in oil, or by absorbing of UV created by Raman scattering in water.


title: Ultraviolet

text: In general, ultraviolet detectors use either a solid-state device, such as one based on silicon carbide or aluminium nitride, or a gas-filled tube as the sensing element. UV detectors that are sensitive to UV in any part of the spectrum respond to irradiation by sunlight and artificial light. A burning hydrogen flame, for instance, radiates strongly in the 185- to 260-nanometer range and only very weakly in the IR region, whereas a coal fire emits very weakly in the UV band yet very strongly at IR wavelengths; thus, a fire detector that operates using both UV and IR detectors is more reliable than one with a UV detector alone. Virtually all fires emit some radiation in the UVC band, whereas the Sun's radiation at this band is absorbed by the Earth's atmosphere. The result is that the UV detector is "solar blind", meaning it will not cause an alarm in response to radiation from the Sun, so it can easily be used both indoors and outdoors.


title: Ultraviolet

text: UV detectors are sensitive to most fires, including hydrocarbons, metals, sulfur, hydrogen, hydrazine, and ammonia. Arc welding, electrical arcs, lightning, X-rays used in nondestructive metal testing equipment (though this is highly unlikely), and radioactive materials can produce levels that will activate a UV detection system. The presence of UV-absorbing gases and vapors will attenuate the UV radiation from a fire, adversely affecting the ability of the detector to detect flames. Likewise, the presence of an oil mist in the air or an oil film on the detector window will have the same effect.


title: Ultraviolet

text: Ultraviolet radiation is used for very fine resolution photolithography, a procedure wherein a chemical called a photoresist is exposed to UV radiation that has passed through a mask. The exposure causes chemical reactions to occur in the photoresist. After removal of unwanted photoresist, a pattern determined by the mask remains on the sample. Steps may then be taken to "etch" away, deposit on or otherwise modify areas of the sample where no photoresist remains.


title: Ultraviolet

text: Photolithography is used in the manufacture of semiconductors, integrated circuit components, and printed circuit boards. Photolithography processes used to fabricate electronic integrated circuits presently use 193 nm UV and are experimentally using 13.5 nm UV for extreme ultraviolet lithography.


title: Ultraviolet

text: Electronic components that require clear transparency for light to exit or enter (photovoltaic panels and sensors) can be potted using acrylic resins that are cured using UV energy. The advantages are low VOC emissions and rapid curing.


title: Ultraviolet

text: Certain inks, coatings, and adhesives are formulated with photoinitiators and resins. When exposed to UV light, polymerization occurs, and so the adhesives harden or cure, usually within a few seconds. Applications include glass and plastic bonding, optical fiber coatings, the coating of flooring, UV coating and paper finishes in offset printing, dental fillings, and decorative fingernail "gels".


title: Ultraviolet

text: UV sources for UV curing applications include UV lamps, UV LEDs, and excimer flash lamps. Fast processes such as flexo or offset printing require high-intensity light focused via reflectors onto a moving substrate and medium so high-pressure Hg (mercury) or Fe (iron, doped)-based bulbs are used, energized with electric arcs or microwaves. Lower-power fluorescent lamps and LEDs can be used for static applications. Small high-pressure lamps can have light focused and transmitted to the work area via liquid-filled or fiber-optic light guides.


title: Ultraviolet

text: The impact of UV on polymers is used for modification of the (roughness and hydrophobicity) of polymer surfaces. For example, a poly(methyl methacrylate) surface can be smoothed by vacuum ultraviolet.


title: Ultraviolet

text: UV radiation is useful in preparing low-surface-energy polymers for adhesives. Polymers exposed to UV will oxidize, thus raising the surface energy of the polymer. Once the surface energy of the polymer has been raised, the bond between the adhesive and the polymer is stronger.


title: Ultraviolet

text: Using a catalytic chemical reaction from titanium dioxide and UVC exposure, oxidation of organic matter converts pathogens, pollens, and mold spores into harmless inert byproducts. However, the reaction of titanium dioxide and UVC is not a straight path. Several hundreds of reactions occur prior to the inert byproducts stage and can hinder the resulting reaction creating formaldehyde, aldehyde, and other VOC's en route to a final stage. Thus, the use of titanium dioxide and UVC requires very specific parameters for a successful outcome. The cleansing mechanism of UV is a photochemical process. Contaminants in the indoor environment are almost entirely organic carbon-based compounds, which break down when exposed to high-intensity UV at 240 to 280 nm. Short-wave ultraviolet radiation can destroy DNA in living microorganisms. UVC's effectiveness is directly related to intensity and exposure time.


title: Ultraviolet

text: UV has also been shown to reduce gaseous contaminants such as carbon monoxide and VOCs. UV lamps radiating at 184 and 254 nm can remove low concentrations of hydrocarbons and carbon monoxide if the air is recycled between the room and the lamp chamber. This arrangement prevents the introduction of ozone into the treated air. Likewise, air may be treated by passing by a single UV source operating at 184 nm and passed over iron pentaoxide to remove the ozone produced by the UV lamp.


title: Ultraviolet

text: Ultraviolet lamps are used to sterilize workspaces and tools used in biology laboratories and medical facilities. Commercially available low-pressure mercury-vapor lamps emit about 86% of their radiation at 254 nanometers (nm), with 265 nm being the peak germicidal effectiveness curve. UV at these germicidal wavelengths damage a microorganism's DNA/RNA so that it cannot reproduce, making it harmless, (even though the organism may not be killed). Since microorganisms can be shielded from ultraviolet rays in small cracks and other shaded areas, these lamps are used only as a supplement to other sterilization techniques.


title: Ultraviolet

text: UV-C LEDs are relatively new to the commercial market and are gaining in popularity. Due to their monochromatic nature (±5 nm) these LEDs can target a specific wavelength needed for disinfection. This is especially important knowing that pathogens vary in their sensitivity to specific UV wavelengths. LEDs are mercury free, instant on/off, and have unlimited cycling throughout the day.


title: Ultraviolet

text: Disinfection using UV radiation is commonly used in wastewater treatment applications and is finding an increased usage in municipal drinking water treatment. Many bottlers of spring water use UV disinfection equipment to sterilize their water. Solar water disinfection has been researched for cheaply treating contaminated water using natural sunlight. The UV-A irradiation and increased water temperature kill organisms in the water.


title: Ultraviolet

text: Ultraviolet radiation is used in several food processes to kill unwanted microorganisms. UV can be used to pasteurize fruit juices by flowing the juice over a high-intensity ultraviolet source. The effectiveness of such a process depends on the UV absorbance of the juice.


title: Ultraviolet

text: Pulsed light (PL) is a technique of killing microorganisms on surfaces using pulses of an intense broad spectrum, rich in UV-C between 200 and 280 nm. Pulsed light works with xenon flash lamps that can produce flashes several times per second. Disinfection robots use pulsed UV.


title: Ultraviolet

text: Some animals, including birds, reptiles, and insects such as bees, can see near-ultraviolet wavelengths. Many fruits, flowers, and seeds stand out more strongly from the background in ultraviolet wavelengths as compared to human color vision. Scorpions glow or take on a yellow to green color under UV illumination, thus assisting in the control of these arachnids. Many birds have patterns in their plumage that are invisible at usual wavelengths but observable in ultraviolet, and the urine and other secretions of some animals, including dogs, cats, and human beings, are much easier to spot with ultraviolet. Urine trails of rodents can be detected by pest control technicians for proper treatment of infested dwellings.


title: Ultraviolet

text: Butterflies use ultraviolet as a communication system for sex recognition and mating behavior. For example, in the "Colias eurytheme" butterfly, males rely on visual cues to locate and identify females. Instead of using chemical stimuli to find mates, males are attracted to the ultraviolet-reflecting color of female hind wings. In "Pieris napi" butterflies it was shown that females in northern Finland with less UV-radiation present in the environment possessed stronger UV signals to attract their males than those occurring further south. This suggested that it was evolutionarily more difficult to increase the UV-sensitivity of the eyes of the males than to increase the UV-signals emitted by the females.


title: Ultraviolet

text: Many insects use the ultraviolet wavelength emissions from celestial objects as references for flight navigation. A local ultraviolet emitter will normally disrupt the navigation process and will eventually attract the flying insect.


title: Ultraviolet

text: The green fluorescent protein (GFP) is often used in genetics as a marker. Many substances, such as proteins, have significant light absorption bands in the ultraviolet that are of interest in biochemistry and related fields. UV-capable spectrophotometers are common in such laboratories.


title: Ultraviolet

text: Ultraviolet traps called bug zappers are used to eliminate various small flying insects. They are attracted to the UV and are killed using an electric shock, or trapped once they come into contact with the device. Different designs of ultraviolet radiation traps are also used by entomologists for collecting nocturnal insects during faunistic survey studies.


title: Ultraviolet

text: Ultraviolet radiation is helpful in the treatment of skin conditions such as psoriasis and vitiligo. Exposure to UVA, while the skin is hyper-photosensitive, by taking psoralens is an effective treatment for psoriasis. Due to the potential of psoralens to cause damage to the liver, PUVA therapy may be used only a limited number of times over a patient's lifetime.


title: Ultraviolet

text: UVB phototherapy does not require additional medications or topical preparations for the therapeutic benefit; only the exposure is needed. However, phototherapy can be effective when used in conjunction with certain topical treatments such as anthralin, coal tar, and vitamin A and D derivatives, or systemic treatments such as methotrexate and Soriatane.


title: Ultraviolet

text: Reptiles need UVB for biosynthesis of vitamin D, and other metabolic processes. Specifically cholecalciferol (vitamin D3), which is needed for basic cellular / neural functioning as well as the utilization of calcium for bone and egg production. The UVA wavelength is also visible to many reptiles and might play a significant role in their ability survive in the wild as well as in visual communication between individuals. Therefore, in a typical reptile enclosure, a fluorescent UV a/b source (at the proper strength / spectrum for the species), must be available for many captive species to survive. Simple supplementation with cholecalciferol (Vitamin D3) will not be enough as there's a complete biosynthetic pathway that is "leapfrogged" (risks of possible overdoses), the intermediate molecules and metabolites also play important functions in the animals health. Natural sunlight in the right levels is always going to be superior to artificial sources, but this might not be possible for keepers in different parts of the world.


title: Ultraviolet

text: It is a known problem that high levels of output of the UVa part of the spectrum can both cause cellular and DNA damage to sensitive parts of their bodies – especially the eyes where blindness is the result of an improper UVa/b source use and placement photokeratitis. For many keepers there must also be a provision for an adequate heat source this has resulted in the marketing of heat and light "combination" products. Keepers should be careful of these "combination" light/ heat and UVa/b generators, they typically emit high levels of UVa with lower levels of UVb that are set and difficult to control so that animals can have their needs met. A better strategy is to use individual sources of these elements and so they can be placed and controlled by the keepers for the max benefit of the animals.


title: Ultraviolet

text: The evolution of early reproductive proteins and enzymes is attributed in modern models of evolutionary theory to ultraviolet radiation. UVB causes thymine base pairs next to each other in genetic sequences to bond together into thymine dimers, a disruption in the strand that reproductive enzymes cannot copy. This leads to frameshifting during genetic replication and protein synthesis, usually killing the cell. Before formation of the UV-blocking ozone layer, when early prokaryotes approached the surface of the ocean, they almost invariably died out. The few that survived had developed enzymes that monitored the genetic material and removed thymine dimers by nucleotide excision repair enzymes. Many enzymes and proteins involved in modern mitosis and meiosis are similar to repair enzymes, and are believed to be evolved modifications of the enzymes originally used to overcome DNA damages caused by UV.


title: Ultraviolet

text: Photobiology is the scientific study of the beneficial and harmful interactions of non-ionizing radiation in living organisms, conventionally demarcated around 10 eV, the first ionization energy of oxygen. UV ranges roughly from 3 to 30 eV in energy. Hence photobiology entertains some, but not all, of the UV spectrum.


title: Gamma distribution

text: In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:


title: Gamma distribution

text: The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a formula_33 base measure) for a random variable formula_34 for which E["X"] = "kθ" = "α"/"β" is fixed and greater than zero, and E[ln("X")] = "ψ"("k") + ln("θ") = "ψ"("α") − ln("β") is fixed ("ψ" is the digamma function).


title: Gamma distribution

text: The parameterization with "k" and "θ" appears to be more common in econometrics and other applied fields, where the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution. See Hogg and Craig for an explicit motivation.


title: Gamma distribution

text: The parameterization with "α" and "β" is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (rate) parameters, such as the "λ" of an exponential distribution or a Poisson distribution – or for that matter, the "β" of the gamma distribution itself. The closely related inverse-gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.


title: Gamma distribution

text: If "k" is a positive integer, then the distribution represents an Erlang distribution; i.e., the sum of "k" independent exponentially distributed random variables, each of which has a mean of "θ".


title: Gamma distribution

text: The gamma distribution can be parameterized in terms of a shape parameter "α" = "k" and an inverse scale parameter "β" = 1/"θ", called a rate parameter. A random variable "X" that is gamma-distributed with shape "α" and rate "β" is denoted


title: Gamma distribution

text: If "α" is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion:


title: Gamma distribution

text: It can also be expressed as follows, if "k" is a positive integer (i.e., the distribution is an Erlang distribution):


title: Gamma distribution

text: The skewness of the gamma distribution only depends on its shape parameter, "k", and it is equal to formula_50


title: Gamma distribution

text: Unlike the mode and the mean, which have readily calculable formulas based on the parameters, the median does not have a closed-form equation. The median for this distribution is the value formula_52 such that


title: Gamma distribution

text: A rigorous treatment of the problem of determining an asymptotic expansion and bounds for the median of the gamma distribution was handled first by Chen and Rubin, who proved that (for formula_54)


title: Gamma distribution

text: where formula_56 is the mean and formula_57 is the median of the formula_58 distribution. For other values of the scale parameter, the mean scales to formula_59, and the median bounds and approximations would be similarly scaled by formula_30.


title: Gamma distribution

text: K. P. Choi found the first five terms in a Laurent series asymptotic approximation of the median by comparing the median to Ramanujan's formula_61 function. Berg and Pedersen found more terms:


title: Gamma distribution

text: Partial sums of these series are good approximations for high enough formula_29; they are not plotted in the figure, which is focused on the low-formula_29 region that is less well approximated.


title: Gamma distribution

text: Berg and Pedersen also proved many properties of the median, showing that it is a convex function of formula_29, and that the asymptotic behavior near formula_66 is formula_67 (where formula_68 is the Euler–Mascheroni constant), and that for all formula_69 the median is bounded by formula_70.


title: Gamma distribution

text: A closer linear upper bound, for formula_71 only, was provided in 2021 by Gaunt and Merkle, relying on the Berg and Pedersen result that the slope of formula_57 is everywhere less than 1:


title: Gamma distribution

text: which can be extended to a bound for all formula_69 by taking the max with the chord shown in the figure, since the median was proved convex.


title: Gamma distribution

text: An approximation to the median that is asymptotically accurate at high formula_29 and reasonable down to formula_78 or a bit lower follows from the Wilson–Hilferty transformation:


title: Gamma distribution

text: In 2021, Lyon proposed several closed-form approximations of the form formula_81. He conjectured closed-form values of formula_82 and formula_83 for which this approximation is an asymptotically tight upper or lower bound for all formula_69. In particular:


title: Gamma distribution

text: Lyon also derived two other lower bounds that are not closed-form expressions, including this one based on solving the integral expression substituting 1 for formula_89:


title: Gamma distribution

text: Additionally, he showed that interpolations between bounds could provide excellent approximations or tighter bounds to the median, including an approximation that is exact at formula_75 (where formula_98) and has a maximum relative error less than 0.6%. Interpolated approximations and bounds are all of the form


title: Gamma distribution

text: where formula_100 is an interpolating function running monotonically from 0 at low formula_29 to 1 at high formula_29, approximating an ideal, or exact, interpolator formula_103:


title: Gamma distribution

text: The interpolated bounds are plotted (mostly inside the yellow region) in the log–log plot shown. Even tighter bounds are available using different interpolating functions, but not usually with closed-form parameters like these.


title: Gamma distribution

text: If "X" has a Gamma("k", "θ") distribution for "i" = 1, 2, ..., "N" (i.e., all distributions have the same scale parameter "θ"), then


title: Gamma distribution

text: For the cases where the "X" are independent but have different scale parameters, see Mathai or Moschopoulos.


title: Gamma distribution

text: Indeed, we know that if "X" is an exponential r.v. with rate "λ", then "cX" is an exponential r.v. with rate "λ"/"c"; the same thing is valid with Gamma variates (and this can be checked using the moment-generating function, see, e.g.,these notes, 10.4-(ii)): multiplication by a positive constant "c" divides the rate (or, equivalently, multiplies the scale).


title: Gamma distribution

text: The gamma distribution is a two-parameter exponential family with natural parameters "k" − 1 and −1/"θ" (equivalently, "α" − 1 and −"β"), and natural statistics "X" and ln("X").


title: Gamma distribution

text: If the shape parameter "k" is held fixed, the resulting one-parameter family of distributions is a natural exponential family.


title: Gamma distribution

text: This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is ln("x").


title: Gamma distribution

text: The Kullback–Leibler divergence (KL-divergence), of Gamma("α", "β") ("true" distribution) from Gamma("α", "β") ("approximating" distribution) is given by


title: Gamma distribution

text: Written using the "k", "θ" parameterization, the KL-divergence of Gamma("k", θ) from Gamma("k", "θ") is given by


title: Gamma distribution

text: If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The compound distribution, which results from integrating out the inverse scale, has a closed-form solution known as the compound gamma distribution.


title: Gamma distribution

text: If, instead, the shape parameter is known but the mean is unknown, with the prior of the mean being given by another gamma distribution, then it results in K-distribution.


title: Gamma distribution

text: The gamma distribution formula_144 can be expressed as the product distribution of a Weibull distribution and a variant form of the stable count distribution.


title: Gamma distribution

text: Its shape parameter formula_145 can be regarded as the inverse of Lévy's stability parameter in the stable count distribution:


title: Gamma distribution

text: where formula_147 is a standard stable count distribution of shape formula_148, and formula_149 is a standard Weibull distribution of shape formula_145.


title: Gamma distribution

text: Finding the maximum with respect to "θ" by taking the derivative and setting it equal to zero yields the maximum likelihood estimator of the "θ" parameter, which equals the sample mean formula_153 divided by the shape parameter "k":


title: Gamma distribution

text: where formula_115 is the digamma function and formula_158 is the sample mean of ln("x"). There is no closed-form solution for "k". The function is numerically very well behaved, so if a numerical solution is desired, it can be found using, for example, Newton's method. An initial value of "k" can be found either using the method of moments, or using the approximation


title: Gamma distribution

text: which is within 1.5% of the correct value. An explicit form for the Newton–Raphson update of this initial guess is:


title: Gamma distribution

text: Consistent closed-form estimators of "k" and "θ" exists that are derived from the likelihood of the generalized gamma distribution.


title: Gamma distribution

text: Using the sample mean of "x", the sample mean of ln("x"), and the sample mean of the product "x"·ln("x") simplifies the expressions to:


title: Gamma distribution

text: These estimators are not strictly maximum likelihood estimators, but are instead referred to as mixed type log-moment estimators. They have however similar efficiency as the maximum likelihood estimators.


title: Gamma distribution

text: Although these estimators are consistent, they have a small bias. A bias-corrected variant of the estimator for the scale "θ" is


title: Gamma distribution

text: With known "k" and unknown "θ", the posterior density function for theta (using the standard scale-invariant prior for "θ") is


title: Gamma distribution

text: Integration with respect to "θ" can be carried out using a change of variables, revealing that 1/"θ" is gamma-distributed with parameters "α" = "Nk", "β" = "y".


title: Gamma distribution

text: In Bayesian inference, the gamma distribution is the conjugate prior to many likelihood distributions: the Poisson, exponential, normal (with known mean), Pareto, gamma with known shape "σ", inverse gamma with known shape parameter, and Gompertz with known scale parameter.


title: Gamma distribution

text: Consider a sequence of events, with the waiting time for each event being an exponential distribution with rate formula_177. Then the waiting time for the formula_178-th event to occur is the gamma distribution with integer shape formula_179. This construction of the gamma distribution allows it to model a wide variety of phenomena where several sub-events, each taking time with exponential distribution, must happen in sequence for a major event to occur. Examples include the waiting time of cell-division events, number of compensatory mutations for a given mutation, waiting time until a repair is necessary for a hydraulic system, and so on.


title: Gamma distribution

text: The gamma distribution has been used to model the size of insurance claims and rainfalls. This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process – much like the exponential distribution generates a Poisson process.


title: Gamma distribution

text: The gamma distribution is also used to model errors in multi-level Poisson regression models because a mixture of Poisson distributions with gamma-distributed rates has a known closed form distribution, called negative binomial.


title: Gamma distribution

text: In wireless communication, the gamma distribution is used to model the multi-path fading of signal power; see also Rayleigh distribution and Rician distribution.


title: Gamma distribution

text: In oncology, the age distribution of cancer incidence often follows the gamma distribution, wherein the shape and scale parameters predict, respectively, the number of driver events and the time interval between them.


title: Gamma distribution

text: In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals.


title: Gamma distribution

text: In bacterial gene expression, the copy number of a constitutively expressed protein often follows the gamma distribution, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of protein molecules produced by a single mRNA during its lifetime.


title: Gamma distribution

text: In genomics, the gamma distribution was applied in peak calling step (i.e., in recognition of signal) in ChIP-chip and ChIP-seq data analysis.


title: Gamma distribution

text: In Bayesian statistics, the gamma distribution is widely used as a conjugate prior. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution. It is also the conjugate prior for the exponential distribution.


title: Gamma distribution

text: Given the scaling property above, it is enough to generate gamma variables with "θ" = 1, as we can later convert to any value of "β" with a simple division.


title: Gamma distribution

text: Suppose we wish to generate random variables from Gamma("n" + "δ", 1), where n is a non-negative integer and 0 < "δ" < 1. Using the fact that a Gamma(1, 1) distribution is the same as an Exp(1) distribution, and noting the method of generating exponential variables, we conclude that if "U" is uniformly distributed on (0, 1], then −ln("U") is distributed Gamma(1, 1) (i.e. inverse transform sampling). Now, using the ""α"-addition" property of gamma distribution, we expand this result:


title: Gamma distribution

text: where "U" are all uniformly distributed on (0, 1] and independent. All that is left now is to generate a variable distributed as Gamma("δ", 1) for 0 < "δ" < 1 and apply the ""α"-addition" property once more. This is the most difficult part.


title: Gamma distribution

text: Random generation of gamma variates is discussed in detail by Devroye, noting that none are uniformly fast for all shape parameters. For small values of the shape parameter, the algorithms are often not valid. For arbitrary values of the shape parameter, one can apply the Ahrens and Dieter modified acceptance-rejection method Algorithm GD (shape "k" ≥ 1), or transformation method when 0 < "k" < 1. Also see Cheng and Feast Algorithm GKM 3 or Marsaglia's squeeze method.


title: Gamma distribution

text: where formula_188 is the integer part of "k", "ξ" is generated via the algorithm above with "δ" = {"k"} (the fractional part of "k") and the "U" are all independent.


title: Gamma distribution

text: While the above approach is technically correct, Devroye notes that it is linear in the value of "k" and generally is not a good choice. Instead, he recommends using either rejection-based or table-based methods, depending on context.


title: Gamma distribution

text: For example, Marsaglia's simple transformation-rejection method relying on one normal variate "X" and one uniform variate "U":


title: Gamma distribution

text: With formula_195 generates a gamma distributed random number in time that is approximately constant with "k". The acceptance rate does depend on "k", with an acceptance rate of 0.95, 0.98, and 0.99 for k=1, 2, and 4. For "k" < 1, one can use formula_196 to boost "k" to be usable with this method.


title: Derivative

text: In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.


title: Derivative

text: The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.


title: Derivative

text: Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.


title: Derivative

text: The process of finding a derivative is called differentiation. The reverse process is called "antidifferentiation". The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.


title: Derivative

text: A function of a real variable is "differentiable" at a point of its domain, if its domain contains an open interval containing , and the limit


title: Derivative

text: exists. This means that, for every positive real number formula_2 (even very small), there exists a positive real number formula_3 such that, for every such that formula_4 and formula_5 then formula_6 is defined, and


title: Derivative

text: If the function is differentiable at , that is if the limit exists, then this limit is called the "derivative" of at , and denoted formula_8 (read as " prime of ") or formula_9 (read as "the derivative of with respect to at ", " by at ", or " over at "); see , below.


title: Derivative

text: If is differentiable at , then must also be continuous at . As an example, choose a point and let be the step function that returns the value 1 for all less than , and returns a different value 10 for all greater than or equal to . cannot have a derivative at . If is negative, then is on the low part of the step, so the secant line from to is very steep, and as tends to zero the slope tends to infinity. If is positive, then is on the high part of the step, so the secant line from to has slope zero. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist.


title: Derivative

text: However, even if a function is continuous at a point, it may not be differentiable there. For example, the absolute value function given by is continuous at , but it is not differentiable there. If is positive, then the slope of the secant line from 0 to is one, whereas if is negative, then the slope of the secant line from 0 to is negative one. This can be seen graphically as a "kink" or a "cusp" in the graph at . Even a function with a smooth graph is not differentiable at a point where its tangent is vertical: For instance, the function given by is not differentiable at .


title: Derivative

text: In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative.


title: Derivative

text: Most functions that occur in practice have derivatives at all points or at almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, for example if the function is a monotone function or a Lipschitz function, this is true. However, in 1872 Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point is a meager set in the space of all continuous functions. Informally, this means that hardly any random continuous functions have a derivative at even one point.


title: Derivative

text: Let be a function that has a derivative at every point in its domain. We can then define a function that maps every point to the value of the derivative of at . This function is written and is called the "derivative function" or the "derivative of" .


title: Derivative

text: Sometimes has a derivative at most, but not all, points of its domain. The function whose value at equals whenever is defined and elsewhere is undefined is also called the derivative of . It is still a function, but its domain may be smaller than the domain of .


title: Derivative

text: Using this idea, differentiation becomes a function of functions: The derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain and whose range is a set of functions. If we denote this operator by , then is the function . Since is a function, it can be evaluated at a point . By the definition of the derivative function, .


title: Derivative

text: For comparison, consider the doubling function given by ; is a real-valued function of a real number, meaning that it takes numbers as inputs and has numbers as outputs:


title: Derivative

text: Because the output of is a function, the output of can be evaluated at a point. For instance, when is applied to the square function, , outputs the doubling function , which we named . This output function can then be evaluated to get , , and so on.


title: Derivative

text: Let be a differentiable function, and let be its derivative. The derivative of (if it has one) is written and is called the "second derivative of ". Similarly, the derivative of the second derivative, if it exists, is written and is called the "third derivative of ". Continuing this process, one can define, if it exists, the th derivative as the derivative of the th derivative. These repeated derivatives are called "higher-order derivatives". The th derivative is also called the derivative of order and denoted .


title: Derivative

text: If represents the position of an object at time , then the higher-order derivatives of have specific interpretations in physics. The first derivative of is the object's velocity. The second derivative of is the acceleration. The third derivative of is the jerk. And finally, the fourth through sixth derivatives of are snap, crackle, and pop; most applicable to astrophysics.


title: Derivative

text: A function need not have a derivative (for example, if it is not continuous). Similarly, even if does have a derivative, it may not have a second derivative. For example, let


title: Derivative

text: On the real line, every polynomial function is infinitely differentiable. By standard differentiation rules, if a polynomial of degree is differentiated times, then it becomes a constant function. All of its subsequent derivatives are identically zero. In particular, they exist, so polynomials are smooth functions.


title: Derivative

text: The derivatives of a function at a point provide polynomial approximations to that function near . For example, if is twice differentiable, then


title: Derivative

text: If is infinitely differentiable, then this is the beginning of the Taylor series for evaluated at around .


title: Derivative

text: A point where the second derivative of a function changes sign is called an "inflection point". At an inflection point, the second derivative may be zero, as in the case of the inflection point of the function given by formula_18, or it may fail to exist, as in the case of the inflection point of the function given by formula_19. At an inflection point, a function switches from being a convex function to being a concave function or vice versa.


title: Derivative

text: The symbols formula_20, formula_21, and formula_22 were introduced by Gottfried Wilhelm Leibniz in 1675. It is still commonly used when the equation is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by


title: Derivative

text: and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation


title: Derivative

text: for the "n"th derivative of formula_25. These are abbreviations for multiple applications of the derivative operator. For example,


title: Derivative

text: With Leibniz's notation, we can write the derivative of formula_27 at the point formula_28 in two different ways:


title: Derivative

text: Leibniz's notation allows one to specify the variable for differentiation (in the denominator), which is relevant in partial differentiation. It also can be used to write the chain rule as


title: Derivative

text: Sometimes referred to as "prime notation", one of the most common modern notations for differentiation is due to Joseph-Louis Lagrange and uses the prime mark, so that the derivative of a function formula_31 is denoted formula_32. Similarly, the second and third derivatives are denoted


title: Derivative

text: To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses:


title: Derivative

text: The latter notation generalizes to yield the notation formula_37 for the "n"th derivative of formula_31 – this notation is most useful when we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can become cumbersome.


title: Derivative

text: Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If formula_39, then


title: Derivative

text: denote, respectively, the first and second derivatives of formula_27. This notation is used exclusively for derivatives with respect to time or arc length. It is typically used in differential equations in physics and differential geometry. The dot notation, however, becomes unmanageable for high-order derivatives (order 4 or more) and cannot deal with multiple independent variables.


title: Derivative

text: Euler's notation uses a differential operator formula_43, which is applied to a function formula_31 to give the first derivative formula_45. The "n"th derivative is denoted formula_46.


title: Derivative

text: If is a dependent variable, then often the subscript "x" is attached to the "D" to clarify the independent variable "x".


title: Derivative

text: although this subscript is often omitted when the variable "x" is understood, for instance when this is the only independent variable present in the expression.


title: Derivative

text: The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using "rules" for obtaining derivatives of more complicated functions from simpler ones.


title: Derivative

text: Here are the rules for the derivatives of the most common basic functions, where "a" is a real number.


title: Derivative

text: Here are some of the most basic rules for deducing the derivative of a compound function from derivatives of basic functions.


title: Derivative

text: Here the second term was computed using the chain rule and third using the product rule. The known derivatives of the elementary functions "x", "x", sin("x"), ln("x") and , as well as the constant 7, were also used.


title: Derivative

text: Relative to a hyperreal extension of the real numbers, the derivative of a real function at a real point can be defined as the shadow of the quotient for infinitesimal , where . Here the natural extension of to the hyperreals is still denoted . Here the derivative is said to exist if the shadow is independent of the infinitesimal chosen.


title: Derivative

text: A vector-valued function y of a real variable sends real numbers to vectors in some vector space R. A vector-valued function can be split up into its coordinate functions , meaning that . This includes, for example, parametric curves in R or R. The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y("t") is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,


title: Derivative

text: if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of y exists for every value of "t", then y′ is another vector-valued function.


title: Derivative

text: If is the standard basis for R, then y("t") can also be written as . If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y("t") must be


title: Derivative

text: This generalization is useful, for example, if y("t") is the position vector of a particle at time "t"; then the derivative y′("t") is the velocity vector of the particle at time "t".


title: Derivative

text: In other words, every value of "x" chooses a function, denoted "f", which is a function of one real number. That is,


title: Derivative

text: In this expression, "a" is a "constant", not a "variable", so "f" is a function of only one real variable. Consequently, the definition of the derivative for a function of one variable applies:


title: Derivative

text: The above procedure can be performed for any choice of "a". Assembling the derivatives together into a function gives a function that describes the variation of "f" in the "y" direction:


title: Derivative

text: This is the partial derivative of "f" with respect to "y". Here ∂ is a rounded "d" called the partial derivative symbol. To distinguish it from the letter "d", ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".


title: Derivative

text: In general, the partial derivative of a function in the direction "x" at the point ("a", ..., "a") is defined to be:


title: Derivative

text: In the above difference quotient, all the variables except "x" are held fixed. That choice of fixed values determines a function of one variable


title: Derivative

text: In other words, the different choices of "a" index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.


title: Derivative

text: This is fundamental for the study of the functions of several real variables. Let be such a real-valued function. If all partial derivatives of are defined at the point , these partial derivatives define the vector


title: Derivative

text: which is called the gradient of at . If is differentiable at every point in some domain, then the gradient is a vector-valued function that maps the point to the vector . Consequently, the gradient determines a vector field.


title: Derivative

text: If "f" is a real-valued function on R, then the partial derivatives of "f" measure its variation in the direction of the coordinate axes. For example, if "f" is a function of "x" and "y", then its partial derivatives measure the variation in "f" in the "x" direction and the "y" direction. They do not, however, directly measure the variation of "f" in any other direction, such as along the diagonal line . These are measured using directional derivatives. Choose a vector


title: Derivative

text: In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector. Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. To see how this works, suppose that where u is a unit vector in the direction of v. Substitute into the difference quotient. The difference quotient becomes:


title: Derivative

text: This is "λ" times the difference quotient for the directional derivative of "f" with respect to u. Furthermore, taking the limit as "h" tends to zero is the same as taking the limit as "k" tends to zero because "h" and "k" are multiples of each other. Therefore, . Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.


title: Derivative

text: If all the partial derivatives of "f" exist and are continuous at x, then they determine the directional derivative of "f" in the direction v by the formula:


title: Derivative

text: This is a consequence of the definition of the total derivative. It follows that the directional derivative is linear in v, meaning that .


title: Derivative

text: The same definition also works when "f" is a function with values in R. The above definition is applied to each component of the vectors. In this case, the directional derivative is a vector in R.


title: Derivative

text: When "f" is a function from an open subset of R to R, then the directional derivative of "f" in a chosen direction is the best linear approximation to "f" at that point and in that direction. But when , no single directional derivative can give a complete picture of the behavior of "f". The total derivative gives a complete picture by considering all directions at once. That is, for any vector v starting at a, the linear approximation formula holds:


title: Derivative

text: Just like the single-variable derivative, is chosen so that the error in this approximation is as small as possible.


title: Derivative

text: If "n" and "m" are both one, then the derivative is a number and the expression is the product of two numbers. But in higher dimensions, it is impossible for to be a number. If it were a number, then would be a vector in R while the other terms would be vectors in R, and therefore the formula would not make sense. For the linear approximation formula to make sense, must be a function that sends vectors in R to vectors in R, and must denote this function evaluated at v.


title: Derivative

text: To determine what kind of function it is, notice that the linear approximation formula can be rewritten as


title: Derivative

text: Notice that if we choose another vector w, then this approximate equation determines another approximate equation by substituting w for v. It determines a third approximate equation by substituting both w for v and for a. By subtracting these two new equations, we get


title: Derivative

text: If we assume that v is small and that the derivative varies continuously in a, then is approximately equal to , and therefore the right-hand side is approximately zero. The left-hand side can be rewritten in a different way using the linear approximation formula with substituted for v. The linear approximation formula implies:


title: Derivative

text: This suggests that is a linear transformation from the vector space R to the vector space R. In fact, it is possible to make this a precise derivation by measuring the error in the approximations. Assume that the error in these linear approximation formula is bounded by a constant times ||v||, where the constant is independent of v but depends continuously on a. Then, after adding an appropriate error term, all of the above approximate equalities can be rephrased as inequalities. In particular, is a linear transformation up to a small error term. In the limit as v and w tend to zero, it must therefore be a linear transformation. Since we define the total derivative by taking a limit as v goes to zero, must be a linear transformation.


title: Derivative

text: In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors. In particular, the numerator and denominator of the difference quotient are not even in the same vector space: The numerator lies in the codomain R while the denominator lies in the domain R. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator and denominator. To make precise the idea that is the best linear approximation, it is necessary to adapt a different formula for the one-variable derivative in which these problems disappear. If , then the usual definition of the derivative may be manipulated to show that the derivative of "f" at "a" is the unique number such that


title: Derivative

text: because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero. This last formula can be adapted to the many-variable situation by replacing the absolute values with norms.


title: Derivative

text: The definition of the total derivative of "f" at a, therefore, is that it is the unique linear transformation such that


title: Derivative

text: Here h is a vector in R, so the norm in the denominator is the standard length on R. However, "f"′(a)h is a vector in R, and the norm in the numerator is the standard length on R. If "v" is a vector starting at "a", then is called the pushforward of v by "f" and is sometimes written .


title: Derivative

text: If the total derivative exists at a, then all the partial derivatives and directional derivatives of "f" exist at a, and for all v, is the directional derivative of "f" in the direction v. If we write "f" using coordinate functions, so that , then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of "f" at a:


title: Derivative

text: The existence of the total derivative "f"′(a) is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on a.


title: Derivative

text: The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if "f" is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative "f"′("x"). This 1×1 matrix satisfies the property that is approximately zero, in other words that


title: Derivative

text: Up to changing variables, this is the statement that the function formula_100 is the best linear approximation to "f" at "a".


title: Derivative

text: The total derivative of a function does not give another function in the same way as the one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target.


title: Derivative

text: The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a jet, cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors. It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the jet bundle. The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the "k"th order jet of a function and its partial derivatives of order less than or equal to "k".


title: Derivative

text: By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to R. The "k"th order total derivative may be interpreted as a map


title: Derivative

text: which takes a point x in R and assigns to it an element of the space of "k"-linear maps from R to R – the "best" (in a certain precise sense) "k"-linear approximation to "f" at that point. By precomposing it with the diagonal map Δ, , a generalized Taylor series may be begun as


title: Derivative

text: where f(a) is identified with a constant function, are the components of the vector , and and are the components of and as linear transformations.


title: Derivative

text: The concept of a derivative can be extended to many other settings. The common thread is that the derivative of a function at a point serves as a linear approximation of the function at that point.


title: Derivative

text: Calculus, known in its early history as "infinitesimal calculus", is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently discovered calculus in the mid-17th century. However, each inventor claimed the other stole his work in a bitter dispute that continued until the end of their lives.


title: Zyklon B

text: Zyklon B (; translated Cyclone B) was the trade name of a cyanide-based pesticide invented in Germany in the early 1920s. It consisted of hydrogen cyanide (prussic acid), as well as a cautionary eye irritant and one of several adsorbents such as diatomaceous earth. The product is notorious for its use by Nazi Germany during the Holocaust to murder approximately 1.1 million people in gas chambers installed at Auschwitz-Birkenau, Majdanek, and other extermination camps.


title: Zyklon B

text: Hydrogen cyanide, a poisonous gas that interferes with cellular respiration, was first used as a pesticide in California in the 1880s. Research at Degesch of Germany led to the development of Zyklon (later known as Zyklon A), a pesticide that released hydrogen cyanide upon exposure to water and heat. It was banned after World War I, when Germany used a similar product as a chemical weapon. Degussa purchased Degesch in 1922. Their team of chemists, which included and Bruno Tesch, devised a method of packaging hydrogen cyanide in sealed canisters along with a cautionary eye irritant and one of several adsorbents such as diatomaceous earth. The new product was also named Zyklon, but it became known as Zyklon B to distinguish it from the earlier version. Uses included delousing clothing and fumigating ships, warehouses, and trains.


title: Zyklon B

text: The Nazis started using Zyklon B in extermination camps in early 1942 to murder prisoners during the Holocaust. Tesch was executed in 1946 for knowingly selling the product to the SS for use on humans. Hydrogen cyanide is now rarely used as a pesticide but still has industrial applications. Firms in several countries continue to produce Zyklon B under alternative brand names, including Detia-Degesch, the successor to Degesch, who renamed the product Cyanosil in 1974.


title: Zyklon B

text: Hydrogen cyanide is a poisonous gas that interferes with cellular respiration. Cyanide prevents the cell from producing adenosine triphosphate (ATP) by binding to one of the proteins involved in the electron transport chain. This protein, cytochrome c oxidase, contains several subunits and has ligands containing iron groups. The cyanide component of Zyklon B can bind at one of these iron groups, heme a3, forming a more stabilized compound through metal-to-ligand pi bonding. As a result of the formation of this new iron–cyanide complex, the electrons that would situate themselves on the heme a3 group can no longer do so. Instead, these electrons destabilize the compound; thus, the heme group no longer accepts them. Consequently, electron transport is halted, and cells can no longer produce the energy needed to synthesize ATP. Death occurs in a human being weighing within two minutes of inhaling 70 mg of hydrogen cyanide.


title: Zyklon B

text: Hydrogen cyanide, discovered in the late 18th century, was used in the 1880s for the fumigation of citrus trees in California. Its use spread to other countries for the fumigation of silos, goods wagons, ships, and mills. Its light weight and rapid dispersal meant its application had to take place under tents or in enclosed areas. Research by Fritz Haber of the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry led to the founding in 1919 of Deutsche Gesellschaft für Schädlingsbekämpfung mbH (Degesch), a state-controlled consortium formed to investigate military use of the chemical. Chemists at Degesch added a cautionary eye irritant to a less volatile cyanide compound which reacted with water in the presence of heat to become hydrogen cyanide. The new product was marketed as the pesticide Zyklon (cyclone). As a similar formula had been used as a weapon by the Germans during World War I, Zyklon was soon banned.


title: Zyklon B

text: Deutsche Gold- und Silber-Scheideanstalt (German Gold and Silver Refinery; Degussa) became sole owners of Degesch in 1922. There, beginning in 1922, , Bruno Tesch, and others worked on packaging hydrogen cyanide in sealed canisters along with a cautionary eye irritant and adsorbent stabilizers such as diatomaceous earth. The new product was also labelled as Zyklon, but it became known as Zyklon B to distinguish it from the earlier version. Heerdt was named the inventor of Zyklon B in the Degesch patent application (number DE 438818) dated 20 June 1922. The Deutsches Patent- und Markenamt awarded the patent on 27 December 1926. Beginning in the 1920s, Zyklon B was used at U.S. Customs facilities along the Mexican border to fumigate the clothing of border crossers.


title: Zyklon B

text: In 1930, Degussa ceded 42.5 percent ownership of Degesch to IG Farben and 15 percent to Th. Goldschmidt AG, in exchange for the right to market pesticide products of those two companies through Degesch. Degussa retained managerial control.


title: Zyklon B

text: While Degesch owned the rights to the brand name Zyklon and the patent on the packaging system, the chemical formula was owned by Degussa. Schlempe GmbH, which was 52 percent owned by Degussa, owned the rights to a process to extract hydrogen cyanide from waste products of sugar beet processing. This process was performed under license by two companies, Dessauer Werke and Kaliwerke Kolin, who also combined the resulting hydrogen cyanide with stabilizer from IG Farben and a cautionary agent from Schering AG to form the final product, which was packaged using equipment, labels, and canisters provided by Degesch. The finished goods were sent to Degesch, who forwarded the product to two companies that acted as distributors: Heerdt-Linger GmbH (Heli) of Frankfurt and Tesch & Stabenow (Testa) of Hamburg. Their territory was split along the Elbe river, with Heli handling clients to the west and south, and Testa those to the east. Degesch owned 51 percent of the shares of Heli, and until 1942 owned 55 percent of Testa.


title: Zyklon B

text: Prior to World War II Degesch derived most of its Zyklon B profits from overseas sales, particularly in the United States, where it was produced under license by Roessler & Hasslacher prior to 1931 and by American Cyanamid from 1931 to 1943. From 1929, the United States Public Health Service used Zyklon B to fumigate freight trains and clothes of Mexican immigrants entering the United States. Uses in Germany included delousing clothing (often using a portable sealed chamber invented by Degesch in the 1930s) and fumigating ships, warehouses, and trains. By 1943, sales of Zyklon B accounted for 65 percent of Degesch's sales revenue and 70 percent of its gross profits.


title: Zyklon B

text: In early 1942, the Nazis began using Zyklon B as the preferred killing tool in extermination camps during the Holocaust. They used it to murder roughly 1.1 million people in gas chambers at Auschwitz-Birkenau, Majdanek, and elsewhere. Most of the victims were Jews, and by far the majority of murders using this method took place at Auschwitz. Distributor Heli supplied Zyklon B to Mauthausen, Dachau, and Buchenwald, and Testa supplied it to Auschwitz and Majdanek; camps also occasionally bought it directly from the manufacturers. Some 56 tonnes of the 729 tonnes sold in Germany in 1942–44 were sold to concentration camps, amounting to about 8 percent of domestic sales. Auschwitz received 23.8 tonnes, of which 6 tonnes were used for fumigation. The remainder was used in the gas chambers or lost to spoilage (the product had a stated shelf life of only three months). Testa conducted fumigations for the Wehrmacht and supplied them with Zyklon B. They also offered courses to the SS in the safe handling and use of the material for fumigation purposes. In April 1941, the German agriculture and interior ministries designated the SS as an authorized applier of the chemical, which meant they were able to use it without any further training or governmental oversight.


title: Zyklon B

text: Rudolf Höss, commandant of Auschwitz, said that the use of Zyklon-B to murder prisoners came about on the initiative of one of his subordinates, SS-"Hauptsturmführer" (captain) Karl Fritzsch, who had used it to murder some Russian POWs in late August 1941 in the basement of Block 11 in the main camp. They repeated the experiment on more Russian POWs in September, with Höss watching. Block 11 proved unsuitable, as the basement was difficult to air out afterwards and the crematorium (Crematorium I, which operated until July 1942) was some distance away. The site of the murders was moved to Crematorium I, where more than 700 victims could be murdered at once. By the middle of 1942, the operation was moved to Auschwitz II–Birkenau, a nearby satellite camp that had been under construction since October 1941.


title: Zyklon B

text: The first gas chamber at Auschwitz II–Birkenau was the "red house" (called Bunker 1 by SS staff), a brick cottage converted to a gassing facility by tearing out the inside and bricking up the windows. It was operational by March 1942. A second brick cottage, called the "white house" or Bunker 2, was converted some weeks later. According to Höss, Bunker 1 held 800 victims and Bunker 2 held 1,200 victims. These structures were in use for mass-murder until early 1943. At that point, the Nazis decided to greatly increase the gassing capacity of Birkenau. Crematorium II was originally designed as a mortuary with morgues in the basement and ground-level incinerators; they converted it into a killing factory by installing gas-tight doors, vents for the Zyklon B to be dropped into the chamber, and ventilation equipment to remove the gas afterwards. Crematorium III was built using the same design. Crematoria IV and V, designed from the beginning as gassing centers, were also constructed that spring. By June 1943, all four crematoria were operational. Most of the victims were murdered using these four structures.


title: Zyklon B

text: The Nazis began shipping large numbers of Jews from all over Europe to Auschwitz in the middle of 1942. Those who were not selected for work crews were immediately gassed. Those selected to die generally comprised about three-quarters of the total and included almost all children, women with small children, all the elderly, and all those who appeared on brief and superficial inspection by an SS doctor not to be completely fit. The victims were told that they were to undergo delousing and a shower. They were stripped of their belongings and herded into the gas chamber.


title: Zyklon B

text: A special SS bureau known as the Hygienic Institute delivered the Zyklon B to the crematoria by ambulance. The actual delivery of the gas to the victims was always handled by the SS, on the order of the supervising SS doctor. After the doors were shut, SS men dropped Zyklon B pellets through vents in the roof or holes in the side of the chamber. The victims were dead within 20 minutes. Johann Kremer, an SS doctor who oversaw gassings, testified that the "shouting and screaming of the victims could be heard through the opening and it was clear that they fought for their lives".


title: Zyklon B

text: "Sonderkommandos" (special work crews forced to work at the gas chambers) wearing gas masks then dragged the bodies from the chamber. The victims' glasses, artificial limbs, jewelry, and hair were removed, and any dental work was extracted so the gold could be melted down. If the gas chamber was crowded, which they typically were, the corpses were found half-squatting, their skin discolored pink with red and green spots, with some foaming at the mouth or bleeding from their ears. The corpses were burned in the nearby incinerators, and the ashes were buried, thrown in the river, or used as fertilizer. With the Soviet Red Army approaching through Poland, the last mass gassing at Auschwitz took place on 30 October 1944. In November 1944, "Reichsführer-SS" Heinrich Himmler, head of the SS, ordered gassing operations to cease throughout the Reich.


title: Zyklon B

text: After World War II ended in 1945, Bruno Tesch and Karl Weinbacher of Tesch & Stabenow were tried in a British military court and executed for knowingly providing Zyklon B to the SS for use on humans. Gerhard Peters, who served as principal operating officer of Degesch and Heli and also held posts in the Nazi government, served two years and eight months in prison as an accessory before being released due to amendments to the penal code.


title: Zyklon B

text: Use of hydrogen cyanide as a pesticide or cleaner has been banned or restricted in some countries. Most hydrogen cyanide is used in industrial processes, made by companies in Germany, Japan, the Netherlands and the US. Degesch resumed production of Zyklon B after the war. The product was sold as Cyanosil in Germany and Zyklon in other countries. It was still produced as of 2008. Degussa sold Degesch to Detia-Freyberg GmbH in 1986. The company is now called Detia-Degesch. Up until around 2015, a fumigation product similar to Zyklon B was in production by Lučební závody Draslovka of the Czech Republic, under the trade name Uragan D2. "Uragan" means "hurricane" or "cyclone" in Czech.


title: Zyklon B

text: Subsequent use of the word "Zyklon" in trade names has prompted angry reactions in English-speaking countries. The name "Zyklon" on portable roller coasters made since 1965 by Pinfari provoked protests among Jewish groups in the U.S. in 1993 and 1999. In 2002, British sportswear and football equipment supplier Umbro issued an apology and stopped using the name "Zyklon", which had appeared since 1999 on the box for one of its trainers, after receiving complaints from the Simon Wiesenthal Center and the Beth Shalom Holocaust Centre. Also in 2002, Siemens withdrew its application for an American trademark of the word "Zyklon", which their subsidiary BSH Bosch und Siemens Hausgeräte had proposed to use for a new line of home appliances in the United States. (The firm was already using the name in Germany for one of their vacuum cleaners.) Protests were lodged by the Simon Wiesenthal Center after the trademark application was reported to BBC News Online by one of their readers. French company IPC's product names used "Cyclone" for degreasers and suffix "B" for biodegradable: "Cyclone B" was renamed ("green cap") in 2013 after protests from Jewish groups. A rabbi said the name was "horrible ignorance at best, and a Guinness record in evil and cynicism if the company did know the history of the name of its product."


title: Zyklon B

text: Holocaust deniers claim that Zyklon B gas was not used in the gas chambers, relying for evidence on the discredited research of Fred A. Leuchter, who found low levels of Prussian blue in samples of the gas chamber walls and ceilings. Leuchter attributed its presence to general delousing of the buildings. Leuchter's negative control, a sample of gasket material taken from a different camp building, had no cyanide residue. In 1999, James Roth, the chemist who had analyzed Leuchter's samples, stated that the test was flawed because the material that was sent for testing included large chunks, and the chemical would only be within 10 microns of the surface. The surface that had been exposed to the chemical was not identified, and the large size of the specimens meant that any chemical present was diluted by an undeterminable amount. In 1994, the Institute for Forensic Research in Kraków re-examined Leuchter's claim, stating that formation of Prussian blue by exposure of bricks to cyanide is not a highly probable reaction. Using microdiffusion techniques, they tested 22 samples from the gas chambers and delousing chambers (as positive controls) and living quarters (as negative controls). They found cyanide residue in both the delousing chambers and the gas chambers but none in the living quarters.


title: Yeast

text: Yeasts are eukaryotic, single-celled microorganisms classified as members of the fungus kingdom. The first yeast originated hundreds of millions of years ago, and at least 1,500 species are currently recognized. They are estimated to constitute 1% of all described fungal species.


title: Yeast

text: Yeasts are unicellular organisms that evolved from multicellular ancestors, with some species having the ability to develop multicellular characteristics by forming strings of connected budding cells known as pseudohyphae or false hyphae. Yeast sizes vary greatly, depending on species and environment, typically measuring 3–4 µm in diameter, although some yeasts can grow to 40 µm in size. Most yeasts reproduce asexually by mitosis, and many do so by the asymmetric division process known as budding. With their single-celled growth habit, yeasts can be contrasted with molds, which grow hyphae. Fungal species that can take both forms (depending on temperature or other conditions) are called dimorphic fungi.


title: Yeast

text: The yeast species "Saccharomyces cerevisiae" converts carbohydrates to carbon dioxide and alcohols through the process of fermentation. The products of this reaction have been used in baking and the production of alcoholic beverages for thousands of years. "S. cerevisiae" is also an important model organism in modern cell biology research, and is one of the most thoroughly studied eukaryotic microorganisms. Researchers have cultured it in order to understand the biology of the eukaryotic cell and ultimately human biology in great detail. Other species of yeasts, such as "Candida albicans", are opportunistic pathogens and can cause infections in humans. Yeasts have recently been used to generate electricity in microbial fuel cells and to produce ethanol for the biofuel industry.


title: Yeast

text: Yeasts do not form a single taxonomic or phylogenetic grouping. The term "yeast" is often taken as a synonym for "Saccharomyces cerevisiae", but the phylogenetic diversity of yeasts is shown by their placement in two separate phyla: the Ascomycota and the Basidiomycota. The budding yeasts or "true yeasts" are classified in the order Saccharomycetales, within the phylum Ascomycota.


title: Yeast

text: The word "yeast" comes from Old English "gist", "gyst", and from the Indo-European root "yes-", meaning "boil", "foam", or "bubble". Yeast microbes are probably one of the earliest domesticated organisms. Archaeologists digging in Egyptian ruins found early grinding stones and baking chambers for yeast-raised bread, as well as drawings of 4,000-year-old bakeries and breweries. Vessels studied from several archaeological sites in Israel (dating to around 5,000, 3,000 and 2,500 years ago), which were believed to have contained alcoholic beverages (beer and mead), were found to contain yeast colonies that had survived over the millennia, providing the first direct biological evidence of yeast use in early cultures. In 1680, Dutch naturalist Anton van Leeuwenhoek first microscopically observed yeast, but at the time did not consider them to be living organisms, but rather globular structures as researchers were doubtful whether yeasts were algae or fungi. Theodor Schwann recognized them as fungi in 1837.


title: Yeast

text: In 1857, French microbiologist Louis Pasteur showed that by bubbling oxygen into the yeast broth, cell growth could be increased, but fermentation was inhibited – an observation later called the "Pasteur effect". In the paper ""Mémoire sur la fermentation alcoolique,"" Pasteur proved that alcoholic fermentation was conducted by living yeasts and not by a chemical catalyst.


title: Yeast

text: By the late 18th century two yeast strains used in brewing had been identified: "Saccharomyces cerevisiae" (top-fermenting yeast) and "S. carlsbergensis" (bottom-fermenting yeast). "S. cerevisiae" has been sold commercially by the Dutch for bread-making since 1780; while, around 1800, the Germans started producing "S. cerevisiae" in the form of cream. In 1825, a method was developed to remove the liquid so the yeast could be prepared as solid blocks. The industrial production of yeast blocks was enhanced by the introduction of the filter press in 1867. In 1872, Baron Max de Springer developed a manufacturing process to create granulated yeast, a technique that was used until the first World War. In the United States, naturally occurring airborne yeasts were used almost exclusively until commercial yeast was marketed at the Centennial Exposition in 1876 in Philadelphia, where Charles L. Fleischmann exhibited the product and a process to use it, as well as serving the resultant baked bread.


title: Yeast

text: The mechanical refrigerator (first patented in the 1850s in Europe) liberated brewers and winemakers from seasonal constraints for the first time and allowed them to exit cellars and other earthen environments. For John Molson, who made his livelihood in Montreal prior to the development of the fridge, the brewing season lasted from September through to May. The same seasonal restrictions formerly governed the distiller's art.


title: Yeast

text: Yeasts are chemoorganotrophs, as they use organic compounds as a source of energy and do not require sunlight to grow. Carbon is obtained mostly from hexose sugars, such as glucose and fructose, or disaccharides such as sucrose and maltose. Some species can metabolize pentose sugars such as ribose, alcohols, and organic acids. Yeast species either require oxygen for aerobic cellular respiration (obligate aerobes) or are anaerobic, but also have aerobic methods of energy production (facultative anaerobes). Unlike bacteria, no known yeast species grow only anaerobically (obligate anaerobes). Most yeasts grow best in a neutral or slightly acidic pH environment.


title: Yeast

text: Yeasts vary in regard to the temperature range in which they grow best. For example, "Leucosporidium frigidum" grows at , "Saccharomyces telluris" at , and "Candida slooffi" at . The cells can survive freezing under certain conditions, with viability decreasing over time.


title: Yeast

text: In general, yeasts are grown in the laboratory on solid growth media or in liquid broths. Common media used for the cultivation of yeasts include potato dextrose agar or potato dextrose broth, Wallerstein Laboratories nutrient agar, yeast peptone dextrose agar, and yeast mould agar or broth. Home brewers who cultivate yeast frequently use dried malt extract and agar as a solid growth medium. The fungicide cycloheximide is sometimes added to yeast growth media to inhibit the growth of "Saccharomyces" yeasts and select for wild/indigenous yeast species. This will change the yeast process.


title: Yeast

text: The appearance of a white, thready yeast, commonly known as kahm yeast, is often a byproduct of the lactofermentation (or pickling) of certain vegetables. It is usually the result of exposure to air. Although harmless, it can give pickled vegetables a bad flavor and must be removed regularly during fermentation.


title: Yeast

text: Yeasts are very common in the environment, and are often isolated from sugar-rich materials. Examples include naturally occurring yeasts on the skins of fruits and berries (such as grapes, apples, or peaches), and exudates from plants (such as plant saps or cacti). Some yeasts are found in association with soil and insects. Yeasts from the soil and from the skins of fruits and berries have been shown to dominate fungal succession during fruit decay. The ecological function and biodiversity of yeasts are relatively unknown compared to those of other microorganisms. Yeasts, including "Candida albicans", "Rhodotorula rubra", "Torulopsis" and "Trichosporon cutaneum", have been found living in between people's toes as part of their skin flora. Yeasts are also present in the gut flora of mammals and some insects and even deep-sea environments host an array of yeasts.


title: Yeast

text: An Indian study of seven bee species and nine plant species found 45 species from 16 genera colonize the nectaries of flowers and honey stomachs of bees. Most were members of the genus "Candida"; the most common species in honey stomachs was "Dekkera intermedia" and in flower nectaries, "Candida blankii". Yeast colonising nectaries of the stinking hellebore have been found to raise the temperature of the flower, which may aid in attracting pollinators by increasing the evaporation of volatile organic compounds. A black yeast has been recorded as a partner in a complex relationship between ants, their mutualistic fungus, a fungal parasite of the fungus and a bacterium that kills the parasite. The yeast has a negative effect on the bacteria that normally produce antibiotics to kill the parasite, so may affect the ants' health by allowing the parasite to spread.


title: Yeast

text: Certain strains of some species of yeasts produce proteins called yeast killer toxins that allow them to eliminate competing strains. (See main article on killer yeast.) This can cause problems for winemaking but could potentially also be used to advantage by using killer toxin-producing strains to make the wine. Yeast killer toxins may also have medical applications in treating yeast infections (see "Pathogenic yeasts" section below).


title: Yeast

text: Marine yeasts, defined as the yeasts that are isolated from marine environments, are able to grow better on a medium prepared using seawater rather than freshwater. The first marine yeasts were isolated by Bernhard Fischer in 1894 from the Atlantic Ocean, and those were identified as "Torula" sp. and "Mycoderma" sp. Following this discovery, various other marine yeasts have been isolated from around the world from different sources, including seawater, seaweeds, marine fish and mammals. Among these isolates, some marine yeasts originated from terrestrial habitats (grouped as facultative marine yeast), which were brought to and survived in marine environments. The other marine yeasts were grouped as obligate or indigenous marine yeasts, which are confined to marine habitats. However, no sufficient evidence has been found to explain the indispensability of seawater for obligate marine yeasts. It has been reported that marine yeasts are able to produce many bioactive substances, such as amino acids, glucans, glutathione, toxins, enzymes, phytase, and vitamins with potential applications in the food, pharmaceutical, cosmetic, and chemical industries as well as for marine culture and environmental protection. Marine yeast was successfully used to produce bioethanol using seawater-based media which will potentially reduce the water footprint of bioethanol.


title: Yeast

text: Yeasts, like all fungi, may have asexual and sexual reproductive cycles. The most common mode of vegetative growth in yeast is asexual reproduction by budding, where a small bud (also known as a bleb or daughter cell) is formed on the parent cell. The nucleus of the parent cell splits into a daughter nucleus and migrates into the daughter cell. The bud then continues to grow until it separates from the parent cell, forming a new cell. The daughter cell produced during the budding process is generally smaller than the mother cell. Some yeasts, including "Schizosaccharomyces pombe", reproduce by fission instead of budding, and thereby creating two identically sized daughter cells.


title: Yeast

text: In general, under high-stress conditions such as nutrient starvation, haploid cells will die; under the same conditions, however, diploid cells can undergo sporulation, entering sexual reproduction (meiosis) and producing a variety of haploid spores, which can go on to mate (conjugate), reforming the diploid.


title: Yeast

text: The haploid fission yeast "Schizosaccharomyces pombe" is a facultative sexual microorganism that can undergo mating when nutrients are limiting. Exposure of "S. pombe" to hydrogen peroxide, an agent that causes oxidative stress leading to oxidative DNA damage, strongly induces mating and the formation of meiotic spores. The budding yeast "Saccharomyces cerevisiae" reproduces by mitosis as diploid cells when nutrients are abundant, but when starved, this yeast undergoes meiosis to form haploid spores. Haploid cells may then reproduce asexually by mitosis. Katz Ezov et al. presented evidence that in natural "S. cerevisiae" populations clonal reproduction and selfing (in the form of intratetrad mating) predominate. In nature, mating of haploid cells to form diploid cells is most often between members of the same clonal population and out-crossing is uncommon. Analysis of the ancestry of natural "S. cerevisiae" strains led to the conclusion that out-crossing occurs only about once every 50,000 cell divisions. These observations suggest that the possible long-term benefits of outcrossing (e.g. generation of diversity) are likely to be insufficient for generally maintaining sex from one generation to the next. Rather, a short-term benefit, such as recombinational repair during meiosis, may be the key to the maintenance of sex in "S. cerevisiae".


title: Yeast

text: Some pucciniomycete yeasts, in particular species of "Sporidiobolus" and "Sporobolomyces", produce aerially dispersed, asexual ballistoconidia.


title: Yeast

text: The useful physiological properties of yeast have led to their use in the field of biotechnology. Fermentation of sugars by yeast is the oldest and largest application of this technology. Many types of yeasts are used for making many foods: baker's yeast in bread production, brewer's yeast in beer fermentation, and yeast in wine fermentation and for xylitol production. So-called red rice yeast is actually a mold, "Monascus purpureus". Yeasts include some of the most widely used model organisms for genetics and cell biology.


title: Yeast

text: Alcoholic beverages are defined as beverages that contain ethanol (CHOH). This ethanol is almost always produced by fermentation – the metabolism of carbohydrates by certain species of yeasts under anaerobic or low-oxygen conditions. Beverages such as mead, wine, beer, or distilled spirits all use yeast at some stage of their production. A distilled beverage is a beverage containing ethanol that has been purified by distillation. Carbohydrate-containing plant material is fermented by yeast, producing a dilute solution of ethanol in the process. Spirits such as whiskey and rum are prepared by distilling these dilute solutions of ethanol. Components other than ethanol are collected in the condensate, including water, esters, and other alcohols, which (in addition to that provided by the oak in which it may be aged) account for the flavour of the beverage.


title: Yeast

text: Brewing yeasts may be classed as "top-cropping" (or "top-fermenting") and "bottom-cropping" (or "bottom-fermenting"). Top-cropping yeasts are so called because they form a foam at the top of the wort during fermentation. An example of a top-cropping yeast is "Saccharomyces cerevisiae", sometimes called an "ale yeast". Bottom-cropping yeasts are typically used to produce lager-type beers, though they can also produce ale-type beers. These yeasts ferment well at low temperatures. An example of bottom-cropping yeast is "Saccharomyces pastorianus", formerly known as "S. carlsbergensis".