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Decomposed
Biology and ecology Decomposition is the process through which organic matter is broken down into simpler molecules. Decomposition, decompose may also refer to: Chemistry *Chemical decomposition or analysis, in chemistry, is the fragmentation of a chemical compound into elements or smaller compounds **Thermal decomposition, a chemical decomposition caused by heat Mathematics * Doob decomposition of an integrable, discrete-time stochastic process * Doob–Meyer decomposition of a continuous-time sub- or supermartingale *Hahn decomposition of a measure space *Hahn–Jordan decomposition of a signed measure *Helmholtz decomposition, decomposition of a vector field *Indecomposability (other) *Indecomposable continuum *Lebesgue's decomposition theorem, decomposition of a measure *Lie group decomposition, used to analyse the structure of Lie groups and associated objects *Manifold decomposition, decomposition of manifolds **JSJ decomposition, or toral decomposition, a decompo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold Decomposition
In topology, a branch of mathematics, a manifold ''M'' may be decomposed or split by writing ''M'' as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form ''M''. Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold. In particular, it has been useful in attempts to classify 3-manifolds and also in proving the higher-dimensional Poincaré conjecture. The table below is a summary of the various manifold-decomposition techniques. The column labeled "''M''" indicates what kind of manifold can be decomposed; the column labeled "How it is decomposed" indicates how, starting with a manifold, one can decompose it into smaller pieces; the column labeled "The pieces" i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Decomposition
In 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. Example In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A \mathbf = \mathbf, the matrix ''A'' can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix ''L'' and an upper triangular matrix ''U''. The systems L(U \mathbf) = \mathbf and U \mathbf = L^ \mathbf require fewer additions and multiplications to solve, compared with the original system A \mathbf = \mathbf, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses ''A'' as ''QR'' with ''Q'' an orthogonal matrix and ''R'' an u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomposition
Decomposition or rot is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. 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 or detritivores. 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. Decomposition can also be a gradual process for organisms that have extended periods of dormancy. One can differentiate abiotic decomposition from biotic decomposition ( biodegradation). The former means "the degra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Decomposition
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many '' primary ideals'' (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational ( curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz. As an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the form -\nabla \phi + \nabla \times \mathbf, where \phi is a scalar field called "scalar potential", and is a vector field, called a vector potential. Statement of the theorem Let \mathbf be a vector field on a bounded domain V\subseteq\mathbb^3, which is twice continuou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Organic Matter
Organic matter, organic material, or natural organic matter refers to the large source of carbon-based compounds found within natural and engineered, terrestrial, and aquatic environments. It is matter composed of organic compounds that have come from the feces and remains of organisms such as plants and animals. Organic molecules can also be made by chemical reactions that do not involve life. Basic structures are created from cellulose, tannin, cutin, and lignin, along with other various proteins, lipids, and carbohydrates. Organic matter is very important in the movement of nutrients in the environment and plays a role in water retention on the surface of the planet. Formation Living organisms are composed of organic compounds. In life, they secrete or excrete organic material into their environment, shed body parts such as leaves and roots and after organisms die, their bodies are broken down by bacterial and fungal action. Larger molecules of organic matter can be forme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Decomposition
Chemical decomposition, or chemical breakdown, is the process or effect of simplifying a single chemical entity (normal molecule, reaction intermediate, etc.) into two or more fragments. 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. 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 su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermal Decomposition
Thermal 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 or other chemical reaction. Decomposition temperature definition A simple substance (like water) may exist in equilibrium with its thermal decomposition products, effectively halting the decomposition. The equilibrium fraction of decomposed molecules increases with the temperature. Examples * Calcium carbonate (limestone or chalk) decomposes into calcium oxide and carbon dioxide when heated. The chemical reaction is as follows: ::CaCO3 → CaO + CO2 :The reaction is used to make quick lime, which is an industrially important product. :A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doob–Meyer Decomposition Theorem
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. It is named for Joseph L. Doob and Paul-André Meyer. History In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales. He conjectured a continuous time version of the theorem and in two publications in 1962 and 1963 Paul-André Meyer proved such a theorem, which became known as the Doob-Meyer decomposition. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.Protter 2005 Class D supermartingales A càdlàg supermartingale Z is of Class D if Z_0=0 and the collection : \ is uniformly integrable.Protter (2005) The theorem Let Z be a cadlag supermartingale of class D. Then there exists a unique, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue's Decomposition Theorem
In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures \mu and \nu on a measurable space (\Omega,\Sigma), there exist two σ-finite signed measures \nu_0 and \nu_1 such that: * \nu=\nu_0+\nu_1\, * \nu_0\ll\mu (that is, \nu_0 is absolutely continuous with respect to \mu) * \nu_1\perp\mu (that is, \nu_1 and \mu are singular). These two measures are uniquely determined by \mu and \nu. Refinement Lebesgue's decomposition theorem can be refined in a number of ways. First, the decomposition of the singular part of a regular Borel measure on the real line can be refined: :\, \nu = \nu_ + \nu_ + \nu_ where * ''ν''cont is the absolutely continuous part * ''ν''sing is the singular continuous part * ''ν''pp is the pure point part (a discrete measure). Second, absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decomposition (computer Science)
Decomposition in computer science, also known as factoring, is breaking a complex problem or system into parts that are easier to conceive, understand, program, and maintain. Overview There are different types of decomposition defined in computer sciences: * In structured programming, ''algorithmic decomposition'' breaks a process down into well-defined steps. * Structured analysis breaks down a software system from the system context level to system functions and data entities as described by Tom DeMarco. * ''Object-oriented decomposition'', on the other hand, breaks a large system down into progressively smaller classes or objects that are responsible for some part of the problem domain. * According to Booch, algorithmic decomposition is a necessary part of object-oriented analysis and design, but object-oriented systems start with and emphasize decomposition into objects.Grady Booch (1994). ''Object-oriented Analysis and Design'' (2nd ed.). Redwood Cita, CA: Benjamin/Cummi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
