**Decomposition** is the process by which organic substances are broken down into a more simple organic matter. The process is a part of the nutrient cycle and is essential for recycling the finite matter that occupies physical space in the biosphere. Bodies of living organisms begin to decompose shortly after death. Animals, such as worms, also help decompose the organic materials. Organisms that do this are known as decomposers. Although no two organisms decompose in the same way, they all undergo the same sequential stages of decomposition. The science which studies decomposition is generally referred to as *taphonomy* from the Greek word *taphos*, meaning tomb.

One can differentiate **abiotic** from **biotic decomposition** (biodegradation). The former means "degradation of a substance by chemical or physical processes, e.g., hydrolysis.^{[1]} The latter means "the metabolic breakdown of materials into simpler components by living organisms",^{[2]} typically by microorganisms.

Decomposition begins at the moment of death, caused by two factors: 1.) autolysis, the breaking down of tissues by the body's own internal chemicals and enzymes, and 2.) putrefaction, the breakdown of tissues by bacteria. These processes release compounds such as cadaverine and putrescine, that are the chief source of the unmistakably putrid odor of decaying animal tissue.

Prime decomposers are bacteria or fungi, though larger scavengers also play an important role in decomposition if the body is accessible to insects, mites and other animals. The most important arthropods that are involved in the process include carrion beetles, mites,^{[3]}^{[4]} the flesh-flies (Sarcophagidae) and blow-flies (Calliphoridae), such as the green-bottle fly seen in the summer. In North America, the most important non-insect animals that are typically involved in the process include mammal and bird scavengers, such as coyotes, dogs, wolves, foxes, rats, crows and vultures. Some of these scavengers also remove and scatter bones, which they ingest at a later time. Aquatic and marine environments have break-down agents that include bacteria, fish, crustaceans, fly larvae ^{[5]} and other carrion scavengers.

Five general stages are used to describe the process of decomposition in vertebrate animals: fresh, bloat, active decay, advanced decay, and dry/remains.^{[6]} The general stages of decomposition are coupled with two stages of chemical decomposition: autolysis and putrefaction.^{[7]} These two stages contribute to the chemical process of decomposition, which breaks down the main components of the body. With death the microbiome of the living organism collapses and is followed by the necrobiome that undergoes predictable changes over time.

Among those animals that have a heart, the "fresh" stage begins immediately after the heart stops beating. From the moment of death, the body begins cooling or warming to match the temperature of the ambient environment, during a stage called algor mortis.^{[8]} Shortly after death, within three to six hours, the muscular tissues become rigid and incapable of relaxing, during a stage called rigor mortis. Since blood is no longer being pumped through the body, gravity causes it to drain to the dependent portions of the body, creating an overall bluish-purple discolouration termed livor mortis or, more commonly, lividity.

Once the heart stops, the blood can no longer supply oxygen or remove carbon dioxide from the tissues. The resulting decrease in pH and other chemical changes causes cells to lose their structural integrity, bringing about the release of cellular enzymes capable of initiating the breakdown of surrounding cells and tissues. This process is known as autolysis.

Visible changes caused by decomposition are limited during the fresh stage, although autolysis may cause blisters to appear at the surface of the skin.^{[9]}

The small amount of oxygen remaining in the body is quickly depleted by cellular metabolism and aerobic microbes naturally present in respiratory and gastrointestinal tracts, creating an ideal environment for the proliferation of anaerobic organisms. These multiply, consuming the body's carbohydrates, lipids, and proteins, to produce a variety of substances including propionic acid, lactic acid, methane, hydrogen sulfide, and ammonia. The process of microbial proliferation within a body is referred to as putrefaction and leads to the second stage of decomposition, known as bloat.^{[10]}

Blowflies and flesh flies are the first carrion insects to arrive, and they seek a suitable oviposition site.^{[6]}

The bloat stage provides the first clear visual sign that microbial proliferation is underway. In this stage, anaerobic metabolism takes place, leading to the accumulation of gases, such as hydrogen sulfide, carbon dioxide, methane, and nitrogen. The accumulation of gases within the bodily cavity causes the distention of the abdomen and gives a cadaver its overall bloated appearance.^{[11]} The gases produced also cause natural liquids and liquefying tissues to become frothy.^{[8]} As the pressure of the gases within the body increases, fluids are forced to escape from natural orifices, such as the nose, mouth, and anus, and enter the surrounding environment. The buildup of pressure combined with the loss of integrity of the skin may also cause the body to rupture.^{[11]}

Intestinal anaerobic bacteria transform haemoglobin into sulfhemoglobin and other colored pigments. The associated gases which accumulate within the body at this time aid in the transport of sulfhemoglobin throughout the body via the circulatory and lymphatic systems, giving the body an overall marbled appearance.^{[12]}

If insects have access, maggots hatch and begin to feed on the body's tissues.^{[6]} Maggot activity, typically confined to natural orifices, and masses under the skin, causes the skin to slip, and hair to detach from the skin.^{[8]} Maggot feeding, and the accumulation of gases within the body, eventually leads to post-mortem skin ruptures which will then further allow purging of gases and fluids into the surrounding environment.^{[10]} Ruptures in the skin allow oxygen to re-enter the body and provide more surface area for the development of fly larvae and the activity of aerobic microorganisms.^{[11]} The purging of gases and fluids results in the strong distinctive odors associated with decay.^{[6]}

Active decay is characterized by the period of greatest mass loss. This loss occurs as a result of both the voracious feeding of maggots and the purging of decomposition fluids into the surrounding environment.^{[11]} The purged fluids accumulate around the body and create a cadaver decomposition island (CDI). Liquefaction of tissues and disintegration become apparent during this time and strong odors persist.^{[6]} The end of active decay is signaled by the migration of maggots away from the body to pupate.^{[10]}

Decomposition is largely inhibited during advanced decay due to the loss of readily available cadaveric material.^{[11]} Insect activity is also reduced during this stage.^{[8]} When the carcass is located on soil, the area surrounding it will show evidence of vegetation death.^{[11]} The CDI surrounding the carcass will display an increase in soil carbon and nutrients, such as phosphorus, potassium, calcium, and magnesium;^{[10]} changes in pH; and a significant increase in soil nitrogen.^{[13]}

During the dry/remains stage, the resurgence of plant growth around the CDI may occur and is a sign that the nutrients present in the surrounding soil have not yet returned to their normal levels.^{[11]} All that remains of the cadaver at this stage is dry skin, cartilage, and bones,^{[6]} which will become dry and bleached if exposed to the elements.^{[8]} If all soft tissue is removed from the cadaver, it is referred to as completely skeletonized, but if only portions of the bones are exposed, it is referred to as partially skeletonised.^{[14]}

A dead body that has been exposed to the open elements, such as water and air, will decompose more quickly and attract much more insect activity than a body that is buried or confined in special protective gear or artifacts. This is due, in part, to the limited number of insects that can penetrate a coffin and the lower temperatures under soil.

The rate and manner of decomposition in an animal body is strongly affected by several factors. In roughly descending degrees of importance, they are:

- Temperature;
- The availability of oxygen;
- Prior embalming;
- Cause of death;
- Burial, depth of burial, and soil type;
- Access by scavengers;
- Trauma, including wounds and crushing blows;
- Humidity, or wetness;
- Rainfall;
- Body size and weight;
- Clothing;
- The surface on which the body rests;
- Foods/objects inside the specimen's digestive tract (bacon compared to lettuce).

The speed at which decomposition occurs varies greatly. Factors such as temperature, humidity, and the season of death all determine how fast a fresh body will skeletonize or mummify. A basic guide for the effect of environment on decomposition is given as Casper's Law (or Ratio): if all other factors are equal, then, when there is free access of air a body decomposes twice as fast than if immersed in water and eight times faster than if buried in earth. Ultimately, the rate of bacterial decomposition acting on the tissue will depend upon the temperature of the surroundings. Colder temperatures decrease the rate of decomposition while warmer temperatures increase it. A dry body will not decompose efficiently. Moisture helps the growth of microorganisms that decompose the organic matter, but too much moisture could lead to anaerobic conditions slowing down the decomposition process.^{[15]}

The most important variable is a body's accessibility to insects, particularly flies. On the surface in tropical areas, invertebrates alone can easily reduce a fully fleshed corpse to clean bones in under two weeks. The skeleton itself is not permanent; acids in soils can reduce it to unrecognizable components. This is one reason given for the lack of human remains found in the wreckage of the *Titanic*, even in parts of the ship considered inaccessible to scavengers. Freshly skeletonized bone is often called "green" bone and has a characteristic greasy feel. Under certain conditions (normally cool, damp soil), bodies may undergo saponification and develop a waxy substance called adipocere, caused by the action of soil chemicals on the body's proteins and fats. The formation of adipocere slows decomposition by inhibiting the bacteria that cause putrefaction.

In extremely dry or cold conditions, the normal process of decomposition is halted – by either lack of moisture or temperature controls on bacterial and enzymatic action – causing the body to be preserved as a mummy. Frozen mummies commonly restart the decomposition process when thawed (see Ötzi the Iceman), whilst heat-desiccated mummies remain so unless exposed to moisture.

The bodies of newborns who never ingested food are an important exception to the normal process of decomposition. They lack the internal microbial flora that produce much of decomposition and quite commonly mummify if kept in even moderately dry conditions.

Aerobic decomposition takes place in the presence of oxygen. This is most common to occur in nature. Living organisms that use oxygen to survive feed on the body. Anaerobic decomposition takes place in the absence of oxygen. This could be place where the body is buried in organic material and oxygen can not reach it. This process of putrefaction has a bad odor accompanied by it due to the hydrogen sulfide and organic matter containing sulfur.^{[15]}

Embalming is the practice of delaying decomposition of human and animal remains. Embalming slows decomposition somewhat, but does not forestall it indefinitely. Embalmers typically pay great attention to parts of the body seen by mourners, such as the face and hands. The chemicals used in embalming repel most insects, and slow down bacterial putrefaction by either killing existing bacteria in or on the body themselves or by "fixing" cellular proteins, which means that they cannot act as a nutrient source for subsequent bacterial infections. In sufficiently dry environments, an embalmed body may end up mummified and it is not uncommon for bodies to remain preserved to a viewable extent after decades. Notable viewable embalmed bodies include those of:

- Eva Perón of Argentina, whose body was injected with paraffin was kept perfectly preserved for many years, and still is as far as is known (her body is no longer on public display).
- Vladimir Lenin of the Soviet Union, whose body was kept submerged in a special tank of fluid for decades and is on public display in Lenin's Mausoleum.
- Other Communist leaders with pronounced cults of personality such as Mao Zedong, Kim Il-sung, Ho Chi Minh, Kim Jong-il and most recently Hugo Chávez have also had their cadavers preserved in the fashion of Lenin's preservation and are now displayed in their respective mausoleums.

- Pope John XXIII, whose preserved body can be viewed in St. Peter's Basilica.
- Padre Pio, whose body was injected with formalin prior to burial in a dry vault from which he was later removed and placed on public display at the San Giovanni Rotondo.

A body buried in a sufficiently dry environment may be well preserved for decades. This was observed in the case for murdered civil rights activist Medgar Evers, who was found to be almost perfectly preserved over 30 years after his death, permitting an accurate autopsy when the case of his murder was re-opened in the 1990s.^{[16]}

Bodies submerged in a peat bog may become naturally "embalmed", arresting decomposition and resulting in a preserved specimen known as a bog body. The time for an embalmed body to be reduced to a skeleton varies greatly. Even when a body is decomposed, embalming treatment can still be achieved (the arterial system decays more slowly) but would not restore a natural appearance without extensive reconstruction and cosmetic work, and is largely used to control the foul odors due to decomposition.

An animal can be preserved almost perfectly, for millions of years in a resin such as amber.

There are some examples where bodies have been inexplicably preserved (with no human intervention) for decades or centuries and appear almost the same as when they died. In some religious groups, this is known as incorruptibility. It is not known whether or for how long a body can stay free of decay without artificial preservation.^{[17]}

Various sciences study the decomposition of bodies under the general rubric of forensic science because the usual motive for such studies is to determine the time and cause of death for legal purposes:

- Forensic taphonomy specifically studies the processes of decomposition in order to apply the biological and chemical principles to forensic cases in order to determine post-mortem interval (PMI), post-burial interval as well as to locate clandestine graves.
- Forensic pathology studies the clues to the cause of death found in the corpse as a medical phenomenon.
- Forensic entomology studies the insects and other vermin found in corpses; the sequence in which they appear, the kinds of insects, and where they are found in their life cycle are clues that can shed light on the time of death, the length of a corpse's exposure, and whether the corpse was moved.
^{[18]}^{[19]} - Forensic anthropology is the medico-legal branch of physical anthropology that studies skeletons and human remains, usually to seek clues as to the identity, age, sex, height and ethnicity of their former owner.
^{[20]}^{[21]}

The University of Tennessee Anthropological Research Facility (better known as the Body Farm) in Knoxville, Tennessee has a number of bodies laid out in various situations in a fenced-in plot near the medical center. Scientists at the Body Farm study how the human body decays in various circumstances to gain a better understanding of decomposition.

Decomposition of plant matter occurs in many stages. It begins with leaching by water; the most easily lost and soluble carbon compounds are liberated in this process. Another early process is physical breakup or fragmentation of the plant material into smaller bits which have greater surface area for microbial colonization and attack. In smaller dead plants, this process is largely carried out by the soil invertebrate fauna, whereas in the larger plants, primarily parasitic life-forms such as insects and fungi play a major breakdown role and are not assisted by numerous detritivore species.

Following this, the plant detritus (consisting of cellulose, hemicellulose, microbial products, and lignin) undergoes

chemical alteration by microbes. Different types of compounds decompose at different rates. This is dependent on their chemical structure.

For instance, lignin is a component of wood, which is relatively resistant to decomposition and can in fact only be decomposed by certain fungi, such as the black-rot fungi. Wood decomposition is a complex process involving fungi which transport nutrients to the nutritionally scarce wood from outside environment.^{[22]} Because of this nutritional enrichment the fauna of saproxylic insects may develop^{[23]} and in turn affect dead wood, contributing to wood decomposition and nutrient cycling in the forest floor.^{[23]} Lignin is one such remaining product of decomposing plants with a very complex chemical structure causing the rate of microbial breakdown to slow. Warmth increases the speed of plant decay, by the same amount regardless of the composition of the plant^{[24]}

In most grassland ecosystems, natural damage from fire, insects that feed on decaying matter, termites, grazing mammals, and the physical movement of animals through the grass are the primary agents of breakdown and nutrient cycling, while bacteria and fungi play the main roles in further decomposition.

The chemical aspects of plant decomposition always involve the release of carbon dioxide. In fact, decomposition contributes over 90 percent of carbon dioxide released each year.^{[24]}

The decomposition of food, either plant or animal, called *spoilage* in this context, is an important field of study within food science. Food decomposition can be slowed down by conservation. The spoilage of meat occurs, if the meat is untreated, in a matter of hours or days and results in the meat becoming unappetizing, poisonous or infectious. Spoilage is caused by the practically unavoidable infection and subsequent decomposition of meat by bacteria and fungi, which are borne by the animal itself, by the people handling the meat, and by their implements. Meat can be kept edible for a much longer time – though not indefinitely – if proper hygiene is observed during production and processing, and if appropriate food safety, food preservation and food storage procedures are applied.

Spoilage of food is attributed to contamination from microorganisms such as bacteria, molds, and yeasts, along with natural decay of the food.^{[25]} These decomposition bacteria reproduce at rapid rates under conditions of moisture and preferred temperatures. When the proper conditions are lacking the bacteria may form spores which lurk until suitable conditions arise to continue reproduction.^{[25]}

The rate of decomposition is governed by three sets of factors—the physical environment (temperature, moisture and soil properties), the quantity and quality of the dead material available to decomposers, and the nature of the microbial community itself.^{[26]}

Decomposition rates are low under very wet or very dry conditions. Decomposition rates are highest in wet, moist conditions with adequate levels of oxygen. Wet soils tend to become deficient in oxygen (this is especially true in wetlands), which slows microbial growth. In dry soils, decomposition slows as well, but bacteria continue to grow (albeit at a slower rate) even after soils become too dry to support plant growth. When the rains return and soils become wet, the osmotic gradient between the bacterial cells and the soil water causes the cells to gain water quickly. Under these conditions, many bacterial cells burst, releasing a pulse of nutrients.^{[26]} Decomposition rates also tend to be slower in acidic soils.^{[26]} Soils which are rich in clay minerals tend to have lower decomposition rates, and thus, higher levels of organic matter.^{[26]} The smaller particles of clay result in a larger surface area that can hold water. The higher the water content of a soil, the lower the oxygen content^{[27]} and consequently, the lower the rate of decomposition. Clay minerals also bind particles of organic material to their surface, making them less accessible to microbes.^{[26]} Soil disturbance like tilling increases decomposition by increasing the amount of oxygen in the soil and by exposing new organic matter to soil microbes.^{[26]}

The quality and quantity of the material available to decomposers is another major factor that influences the rate of decomposition. Substances like sugars and amino acids decompose readily and are considered labile. Cellulose and hemicellulose, which are broken down more slowly, are "moderately labile". Compounds which are more resistant to decay, like lignin or cutin, are considered recalcitrant.^{[26]} Litter with a higher proportion of labile compounds decomposes much more rapidly than does litter with a higher proportion of recalcitrant material. Consequently, dead animals decompose more rapidly than dead leaves, which themselves decompose more rapidly than fallen branches.^{[26]} As organic material in the soil ages, its quality decreases. The more labile compounds decompose quickly, leaving an increasing proportion of recalcitrant material. Microbial cell walls also contain recalcitrant materials like chitin, and these also accumulate as the microbes die, further reducing the quality of older soil organic matter.^{[26]}

- Cadaverine
- Chemical decomposition
- Ecosystem
- Humus
- Leachate
- Microbiology of decomposition
- Peat (turf)
- Putrescine
- Staling

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- Media related to Decomposition at Wikimedia Commons
- 1Lecture.com – Food decomposition (a Flash animation)

Preceded by Death |
Stages of human developmentDecomposition |
Succeeded by Skeletonization |

Biodegradation is the breakdown of organic matter by microorganisms, such as bacteria, fungi.

CalcinationThe IUPAC defines calcination as "heating to high temperatures in air or oxygen". However, calcination is also used to mean a thermal treatment process in the absence or limited supply of air or oxygen applied to ores and other solid materials to bring about a thermal decomposition. A calciner is a steel cylinder that rotates inside a heated furnace and performs indirect high-temperature processing (550–1150 °C, or 1000–2100 °F) within a controlled atmosphere.

Chemical decompositionChemical decomposition, analysis or breakdown is the separation of a single chemical compound into its two or more elemental parts or to simpler compounds. Chemical decomposition is usually regarded and defined as the exact opposite of chemical synthesis. In short, the chemical reaction in which two or more products are formed from a single reactant is called a decomposition reaction.It is of mainly 3 types 1.electrolytic decomposition 2.photo decomposition 3.thermal decomposition.

The details of a decomposition process are not always well defined but some of the process is understood; much energy is needed to break bonds. Since all decomposition reactions break apart the bonds holding it together in order to produce into its simpler basic parts, the reactions would require some form of this energy in varying degrees. Because of this fundamental rule, it is known that most of these reactions are endothermic although exceptions do exist.

The stability of a chemical compound is eventually limited when exposed to extreme environmental conditions such as heat, radiation, humidity, or the acidity of a solvent. Because of this chemical decomposition is often an undesired chemical reaction. However chemical decomposition is being used in a growing number of ways.

For example this method is employed for several analytical techniques, notably mass spectrometry, traditional gravimetric analysis, and thermogravimetric analysis. Additionally decomposition reactions are used today for a number of other reasons in the production of a wide variety of products. One of these is the explosive breakdown reaction of sodium azide [(NaN3)2] into nitrogen gas (N2) and sodium (Na). It is this process which powers the life-saving airbags present in virtually all of today's automobiles.Of the six known basic decomposition reactions, this discussion will focus on what are referred to as the 'three broad types' and considered to be the most common. These three are the thermal, electrolytic, and catalytic decomposition reactions.

Cholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃ-/) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g. Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

Decomposition of time seriesThe decomposition of time series is a statistical task that deconstructs a time series into several components, each representing one of the underlying categories of patterns. There are two principal types of decomposition, which are outlined below.

EcosystemAn ecosystem is a community of living organisms in conjunction with the nonliving components of their environment, interacting as a system. These biotic and abiotic components are linked together through nutrient cycles and energy flows. Energy enters the system through photosynthesis and is incorporated into plant tissue. By feeding on plants and on one-another, animals play an important role in the movement of matter and energy through the system. They also influence the quantity of plant and microbial biomass present. By breaking down dead organic matter, decomposers release carbon back to the atmosphere and facilitate nutrient cycling by converting nutrients stored in dead biomass back to a form that can be readily used by plants and other microbes.Ecosystems are controlled by external and internal factors. External factors such as climate, the parent material which forms the soil and topography, control the overall structure an ecosystem, but are not themselves influenced by the ecosystem.Ecosystems are dynamic entities—they are subject to periodic disturbances and are in the process of recovering from some past disturbance. Ecosystems in similar environments that are located in different parts of the world can end up doing things very differently simply because they have different pools of species present. Internal factors not only control ecosystem processes but are also controlled by them and are often subject to feedback loops.Resource inputs are generally controlled by external processes like climate and parent material. Resource availability within the ecosystem is controlled by internal factors like decomposition, root competition or shading. Although humans operate within ecosystems, their cumulative effects are large enough to influence external factors like climate.Biodiversity affects ecosystem functioning, as do the processes of disturbance and succession. Ecosystems provide a variety of goods and services upon which people depend.

Even and odd functionsIn mathematics, **even functions** and **odd functions** are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function is an even function if is an even integer, and it is an odd function if is an odd integer.

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these integers are further restricted to prime numbers, the process is called prime factorization.

When the numbers are sufficiently large, no efficient, non-quantum integer factorization algorithm is known. An effort by several researchers, concluded in 2009, to factor a 232-digit number (RSA-768) utilizing hundreds of machines took two years and the researchers estimated that a 1024-bit RSA modulus would take about a thousand times as long. However, it has not been proven that no efficient algorithm exists. The presumed difficulty of this problem is at the heart of widely used algorithms in cryptography such as RSA. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.

Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, e.g., to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically.

Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure.

LU decompositionIn numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. LU decomposition was introduced by mathematician Tadeusz Banachiewicz in 1938.

Matrix decompositionIn the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

Nitrogen cycleThe nitrogen cycle is the biogeochemical cycle by which nitrogen is converted into multiple chemical forms as it circulates among atmosphere, terrestrial, and marine ecosystems. The conversion of nitrogen can be carried out through both biological and physical processes. Important processes in the nitrogen cycle include fixation, ammonification, nitrification, and denitrification. The majority of Earth's atmosphere (78%) is atmosphere nitrogen, making it the largest source of nitrogen. However, atmospheric nitrogen has limited availability for biological use, leading to a scarcity of usable nitrogen in many types of ecosystems.

The nitrogen cycle is of particular interest to ecologists because nitrogen availability can affect the rate of key ecosystem processes, including primary production and decomposition. Human activities such as fossil fuel combustion, use of artificial nitrogen fertilizers, and release of nitrogen in wastewater have dramatically altered the global nitrogen cycle.

Partial fraction decompositionIn algebra, the **partial fraction decomposition** or **partial fraction expansion** of a rational function (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, inverse Laplace transforms. The concept was discovered in 1702 by both Johann Bernoulli and Gottfried Leibniz independently.

In symbols, one can use *partial fraction expansion* to change a rational fraction in the form

where *f* and *g* are polynomials, into an expression of the form

where:

As factorization of polynomials may be difficult, a coarser decomposition is often preferred, which consists of replacing factorization by square-free factorization. This amounts to replace "irreducible" by "square-free" in the preceding description of the outcome.

Principal component analysis**Principal component analysis** (**PCA**) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables (entities each of which takes on various numerical values) into a set of values of linearly uncorrelated variables called **principal components**. If there are observations with variables, then the number of distinct principal components is . This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components. The resulting vectors (each being a linear combination of the variables and containing *n* observations) are an uncorrelated orthogonal basis set. PCA is sensitive to the relative scaling of the original variables.

PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of **X** (Golub and Van Loan, 1983), eigenvalue decomposition (EVD) of **X**^{T}**X** in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's *Principal Component Analysis*), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics.

PCA is mostly used as a tool in exploratory data analysis and for making predictive models. It is often used to visualize genetic distance and relatedness between populations. PCA can be done by eigenvalue decomposition of a data covariance (or correlation) matrix or singular value decomposition of a data matrix, usually after a normalization step of the initial data. The normalization of each attribute consists of *mean centering* – subtracting each data value from its variable's measured mean so that its empirical mean (average) is zero – and, possibly, normalizing each variable's variance to make it equal to 1; see Z-scores. The results of a PCA are usually discussed in terms of *component scores*, sometimes called *factor scores* (the transformed variable values corresponding to a particular data point), and *loadings* (the weight by which each standardized original variable should be multiplied to get the component score). If component scores are standardized to unit variance, loadings must contain the data variance in them (and that is the magnitude of eigenvalues). If component scores are not standardized (therefore they contain the data variance) then loadings must be unit-scaled, ("normalized") and these weights are called eigenvectors; they are the cosines of orthogonal rotation of variables into principal components or back.

PCA is the simplest of the true eigenvector-based multivariate analyses. Often, its operation can be thought of as revealing the internal structure of the data in a way that best explains the variance in the data. If a multivariate dataset is visualised as a set of coordinates in a high-dimensional data space (1 axis per variable), PCA can supply the user with a lower-dimensional picture, a projection of this object when viewed from its most informative viewpoint^{[citation needed]}. This is done by using only the first few principal components so that the dimensionality of the transformed data is reduced.

PCA is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.

PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset.

PutrefactionPutrefaction is the fifth stage of death, following pallor mortis, algor mortis, rigor mortis, and livor mortis. This process references the breaking down of a body of a human or animal post-mortem (meaning after death). In broad terms, it can be viewed as the decomposition of proteins, and the eventual breakdown of the cohesiveness between tissues, and the liquefaction of most organs. This is caused by the decomposition of organic matter by bacterial or fungal digestion, which causes the release of gases that infiltrate the body's tissues, and leads to the deterioration of the tissues and organs.

The approximate time it takes putrefaction to occur is dependent on various factors. Internal factors that affect the rate of putrefaction include the age at which death has occurred, the overall structure and condition of the body, the cause of death, and external injuries arising before or after death. External factors include environmental temperature, moisture and air exposure, clothing, burial factors, and light exposure.

The first signs of putrefaction are signified by a greenish discoloration on the outside of the skin on the abdominal wall corresponding to where the large intestine begins, as well as under the surface of the liver.

Certain substances, such as carbolic acid, arsenic, strychnine, and zinc chloride, can be used to delay the process of putrefaction in various ways based on their chemical make up.

Body farms are facilities which study the process of human decomposition as well as how environmental factors affect the rate of putrefaction.

PyrolysisPyrolysis is the thermal decomposition of materials at elevated temperatures in an inert atmosphere. It involves the change of chemical composition and is irreversible. The word is coined from the Greek-derived elements pyro "fire" and lysis "separating".

Pyrolysis is most commonly used to the treatment of organic materials. It is one of the processes involved in charring wood. In general, pyrolysis of organic substances produces volatile products and leaves a solid residue enriched in carbon, char. Extreme pyrolysis, which leaves mostly carbon as the residue, is called carbonization.

The process is used heavily in the chemical industry, for example, to produce ethylene, many forms of carbon, and other chemicals from petroleum, coal, and even wood, to produce coke from coal. Aspirational applications of pyrolysis would convert biomass into syngas and biochar, waste plastics back into usable oil, or waste into safely disposable substances.

QR decompositionIn linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm.

Singular value decomposition

In linear algebra, the **singular-value decomposition** (**SVD**) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics.

Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix. The diagonal entries of are known as the **singular values** of . The columns of and the columns of are called the **left-singular vectors** and **right-singular vectors** of , respectively.

The singular-value decomposition can be computed using the following observations:

Applications that employ the SVD include computing the pseudoinverse, least squares fitting of data, multivariable control, matrix approximation, and determining the rank, range and null space of a matrix.

Spectral theoremIn mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.

The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.

Augustin-Louis Cauchy proved the spectral theorem for self-adjoint matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about determinants. The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory.

This article mainly focuses on the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.

Thermal decompositionThermal decomposition, or thermolysis, is a chemical decomposition caused by heat. The decomposition temperature of a substance is the temperature at which the substance chemically decomposes.

The reaction is usually endothermic as heat is required to break chemical bonds in the compound undergoing decomposition. If decomposition is sufficiently exothermic, a positive feedback loop is created producing thermal runaway and possibly an explosion.

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