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Non-ionising radiations, electromagnetic fields (EMF) such as radiofrequency (RF), or power frequency radiation have become very common in everyday life. All of these exist as low frequency radiation which can come from wireless cellular devices or through electrical appliances which induce extremely low frequency radi... | Wikipedia - Biological effects of radiation on the epigenome | null | null | null |
Non-lethal weapons are intended to minimize injury or death. While people are occasionally seriously injured or killed by these weapons, fatalities are relatively infrequent. Causes of death from non-lethal weapons are varied and occasionally uncertain. | Wikipedia - Less-lethal weapons | null | null | null |
Misplaced or ricocheting shots, pre-existing medical conditions, inadequate user training, repetitive applications and intentional misuse have been implicated in different cases where death has occurred.As different parts of the body differ in vulnerability, and because people vary in weight and fitness, any weapon pow... | Wikipedia - Less-lethal weapons | null | null | null |
Non-linear tracking algorithms use a Non-linear filter to cope with the situation where the measurements have a non-linear relationship to the final track coordinates, where the errors are non-Gaussian, or where the motion update model is non-linear. The most common non-linear filters are: the Extended Kalman filter th... | Wikipedia - Radar tracker | null | null | null |
Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, s... | Wikipedia - Logical axiom | null | null | null |
This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can eve... | Wikipedia - Logical axiom | null | null | null |
Non-logical symbols represent predicates (relations), functions and constants. It used to be standard practice to use a fixed, infinite set of non-logical symbols for all purposes: For every integer n ≥ 0, there is a collection of n-ary, or n-place, predicate symbols. Because they represent relations between n elements... | Wikipedia - Predicate Logic | null | null | null |
In this traditional approach, there is only one language of first-order logic. This approach is still common, especially in philosophically oriented books. A more recent practice is to use different non-logical symbols according to the application one has in mind. | Wikipedia - Predicate Logic | null | null | null |
Therefore, it has become necessary to name the set of all non-logical symbols used in a particular application. This choice is made via a signature.Typical signatures in mathematics are {1, ×} or just {×} for groups, or {0, 1, +, ×, <} for ordered fields. There are no restrictions on the number of non-logical symbols. | Wikipedia - Predicate Logic | null | null | null |
The signature can be empty, finite, or infinite, even uncountable. Uncountable signatures occur for example in modern proofs of the Löwenheim–Skolem theorem. Though signatures might in some cases imply how non-logical symbols are to be interpreted, interpretation of the non-logical symbols in the signature is separate ... | Wikipedia - Predicate Logic | null | null | null |
Signatures concern syntax rather than semantics. In this approach, every non-logical symbol is of one of the following types: A predicate symbol (or relation symbol) with some valence (or arity, number of arguments) greater than or equal to 0. These are often denoted by uppercase letters such as P, Q and R. Examples: I... | Wikipedia - Predicate Logic | null | null | null |
One possible interpretation is "x is a man". In Q(x,y), Q is a predicate symbol of valence 2. Possible interpretations include "x is greater than y" and "x is the father of y". | Wikipedia - Predicate Logic | null | null | null |
Relations of valence 0 can be identified with propositional variables, which can stand for any statement. One possible interpretation of R is "Socrates is a man". | Wikipedia - Predicate Logic | null | null | null |
A function symbol, with some valence greater than or equal to 0. These are often denoted by lowercase roman letters such as f, g and h. Examples: f(x) may be interpreted as "the father of x". In arithmetic, it may stand for "-x". | Wikipedia - Predicate Logic | null | null | null |
In set theory, it may stand for "the power set of x". In arithmetic, g(x,y) may stand for "x+y". In set theory, it may stand for "the union of x and y". | Wikipedia - Predicate Logic | null | null | null |
Function symbols of valence 0 are called constant symbols, and are often denoted by lowercase letters at the beginning of the alphabet such as a, b and c. The symbol a may stand for Socrates. In arithmetic, it may stand for 0. In set theory, it may stand for the empty set.The traditional approach can be recovered in th... | Wikipedia - Predicate Logic | null | null | null |
Non-mechanical work in thermodynamics is work caused by external force fields that a system is exposed to. The action of such forces can be initiated by events in the surroundings of the system, or by thermodynamic operations on the shielding walls of the system. The non-mechanical work of force fields can have either ... | Wikipedia - Pressure-volume work | null | null | null |
In thermodynamics, non-mechanical work is to be contrasted with mechanical work that is done by forces in immediate contact between the system and its surroundings. If the putative 'work' of a process cannot be defined as either long-range work or else as contact work, then sometimes it cannot be described by the therm... | Wikipedia - Pressure-volume work | null | null | null |
An example is when the wall between the system and its surrounds is not considered as idealized and vanishingly thin, so that processes can occur within the wall, such as friction affecting the transfer of matter across the wall; in this case, the forces of transfer are neither strictly long-range nor strictly due to c... | Wikipedia - Pressure-volume work | null | null | null |
Pressure–volume work is one of the two mainly considered kinds of mechanical contact work. A force acts on the interfacing wall between system and surroundings. The force is due to the pressure exerted on the interfacing wall by the material inside the system; that pressure is an internal state variable of the system, ... | Wikipedia - Pressure-volume work | null | null | null |
The work is due to change of system volume by expansion or contraction of the system. If the system expands, in the present article it is said to do positive work on the surroundings. If the system contracts, in the present article it is said to do negative work on the surroundings. | Wikipedia - Pressure-volume work | null | null | null |
Pressure–volume work is a kind of contact work, because it occurs through direct material contact with the surrounding wall or matter at the boundary of the system. It is accurately described by changes in state variables of the system, such as the time courses of changes in the pressure and volume of the system. The v... | Wikipedia - Pressure-volume work | null | null | null |
Pressure–volume work can have either positive or negative sign. Pressure–volume work, performed slowly enough, can be made to approach the fictive reversible quasi-static ideal. | Wikipedia - Pressure-volume work | null | null | null |
Non-mechanical work also contrasts with shaft work. Shaft work is the other of the two mainly considered kinds of mechanical contact work. It transfers energy by rotation, but it does not eventually change the shape or volume of the system. | Wikipedia - Pressure-volume work | null | null | null |
Because it does not change the volume of the system it is not measured as pressure–volume work, and it is called isochoric work. Considered solely in terms of the eventual difference between initial and final shapes and volumes of the system, shaft work does not make a change. During the process of shaft work, for exam... | Wikipedia - Pressure-volume work | null | null | null |
Shaft work is a kind of contact work, because it occurs through direct material contact with the surrounding matter at the boundary of the system. A system that is initially in a state of thermodynamic equilibrium cannot initiate any change in its internal energy. In particular, it cannot initiate shaft work. | Wikipedia - Pressure-volume work | null | null | null |
This explains the curious use of the phrase "inanimate material agency" by Kelvin in one of his statements of the second law of thermodynamics. Thermodynamic operations or changes in the surroundings are considered to be able to create elaborate changes such as indefinitely prolonged, varied, or ceased rotation of a dr... | Wikipedia - Pressure-volume work | null | null | null |
Thus the sign of shaft work is always negative, work being done on the system by the surroundings. Shaft work can hardly be done indefinitely slowly; consequently it always produces entropy within the system, because it relies on friction or viscosity within the system for its transfer. The foregoing comments about sha... | Wikipedia - Pressure-volume work | null | null | null |
Non-mutational stimulation of GATA2 expression and consequential aggressiveness in EVI1-positive AML appears due to the ability of EVI1, a transcription factor, to directly stimulate the expression of the GATA2 gene. The reason for the overexpression of GATA2 that begins in the early stages of prostate cancer is unclea... | Wikipedia - GATA2 | null | null | null |
Non-mutually exclusive options for data backup are: Onsite OffsiteOnsite is traditional, and one major advantage is immediate availability. | Wikipedia - Data center environmental control | null | null | null |
Non-negative matrix factorization (NMF) can take missing data while minimizing its cost function, rather than treating these missing data as zeros that could introduce biases. This makes it a mathematically proven method for data imputation. NMF can ignore missing data in the cost function, and the impact from missing ... | Wikipedia - Imputation (statistics) | null | null | null |
Non-numerical calculators include life-style and scientific calculators: Love calculator: The input is two names, and there is a button to work out the compatibility, as a percentage, of two people with these names. Formula weight calculator: The input is a chemical molecular formula, using the periodic-table symbols a... | Wikipedia - Software calculator | null | null | null |
Non-overlapping blends (also called substitution blends) have no overlap, whether phonological or orthographic: California + Mexico → Calexico beautiful + delicious → beaulicious | Wikipedia - Portmanteau word | null | null | null |
Non-parametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term non-parametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance.... | Wikipedia - Non-parametric statistic | null | null | null |
Nonparametric regression and semiparametric regression methods have been developed based on kernels, splines, and wavelets. Data envelopment analysis provides efficiency coefficients similar to those obtained by multivariate analysis without any distributional assumption. KNNs classify the unseen instance based on the ... | Wikipedia - Non-parametric statistic | null | null | null |
Non-pharmacological interventions provide key tools for the management of breathlessness. Potentially beneficial approaches include active management of psychosocial issues (anxiety, depression, etc.), and implementation of self-management strategies, such as physical and mental relaxation techniques, pacing techniques... | Wikipedia - Respiratory distress | null | null | null |
Non-pharmacological techniques include Lamaze breathing, acupuncture, acupressure, LeBoyer technique, transcutaneous nerve stimulation, massage, hydrotherapy, vertical positioning, presence of a support person, intradermal water injections, and biofeedback amongst many more.Water immersion in the first stage of labor m... | Wikipedia - Obstetric anesthesiology | null | null | null |
Non-pharmacological treatment of PAM consists of utilizing the Artemisia annua plant as an herbal remedy. The basis for this reasoning is because A. annua acts as the plant source for Artemisinin-based combination therapy (ACT), a commonly used pharmacological treatment of PAM. However, the WHO currently does not suppo... | Wikipedia - Pregnancy-associated malaria | null | null | null |
Non-pharmacological treatments are effective in treating autoimmune disease and contribute to a sense of well-being. Women can: Eat healthy, well-balanced meals. A healthy diet limits saturated fat, trans fat, cholesterol, salt, and added sugars. People may alleviate symptoms of inflammation by following the Autoimmune... | Wikipedia - Autoimmune disease in women | null | null | null |
Those with autoimmune diseases should focus on consuming foods that are very fresh and nutritious. Engage in regular physical activity without overdoing it. Patients should speak with a clinician about what types of physical activity is appropriate. | Wikipedia - Autoimmune disease in women | null | null | null |
A gradual and gentle exercise program often works well for people with long-lasting muscle and joint pain. For example, yoga or tai chi may be helpful. Get enough rest. | Wikipedia - Autoimmune disease in women | null | null | null |
Rest allows body tissues and joints the time they need to repair. Sleeping is a great way to maintain health of the mind and body. Lack of sleep, along with elevated stress levels may cause symptoms to worsen. | Wikipedia - Autoimmune disease in women | null | null | null |
Without proper rest, the body's immune defense remains inadequate. Many people need at least 7 to 9 hours of sleep each day to feel well-rested. Reduce stress. | Wikipedia - Autoimmune disease in women | null | null | null |
Stress and anxiety can trigger symptoms to flare up with some autoimmune diseases. Simplifying daily stressors will help alleviate symptoms and contribute to a sense of well-being. Meditation, self-hypnosis, and guided imagery, may be effective in reducing stress, pain, and boost people's ability to cope with other eff... | Wikipedia - Autoimmune disease in women | null | null | null |
Instructional materials can guide people in learning these activities. Some include self-help books, audio sources, tapes, or consulting with an instructor. Joining a support group or talking with a counselor might also help manage stress and cope with the disease. | Wikipedia - Autoimmune disease in women | null | null | null |
Non-physiological Trochlear nerve malfunction Vestibular Pathology (Ménière's disease, SCDS (superior canal dehiscence syndrome), BPPV, vestibular neuritis) Exposure to strong magnetic fields (as in MRI machines) Long-term exposure to low light conditions or darkness, called miner's nystagmus after 19th-century coal mi... | Wikipedia - Physiologic nystagmus | null | null | null |
Non-precision gear couplings use backlash to allow for slight angular misalignment. However, backlash is undesirable in precision positioning applications such as machine tool tables. It can be minimized by tighter design features such as ball screws instead of leadscrews, and by using preloaded bearings. A preloaded b... | Wikipedia - Backlash (engineering) | null | null | null |
There can be significant backlash in unsynchronized transmissions because of the intentional gap between gears in dog clutches. The gap is necessary so that the driver or electronics can engage the gears easily while synchronizing the engine speed with the driveshaft speed. If there was a smaller clearance, it would be... | Wikipedia - Backlash (engineering) | null | null | null |
Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space, and separate from space. Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion, or anything external. Furthermore, it assumes that spac... | Wikipedia - Space Time Continuum | null | null | null |
In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are called x, y, and z. A position in spacetime is called an event, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). An event... | Wikipedia - Space Time Continuum | null | null | null |
Unlike the analogies used in popular writings to explain events, such as firecrackers or sparks, mathematical events have zero duration and represent a single point in spacetime. Although it is possible to be in motion relative to the popping of a firecracker or a spark, it is not possible for an observer to be in moti... | Wikipedia - Space Time Continuum | null | null | null |
The series of events can be linked together to form a line which represents a particle's progress through spacetime. That line is called the particle's world line. : 105 Mathematically, spacetime is a manifold, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, a gl... | Wikipedia - Space Time Continuum | null | null | null |
A scale factor, c {\displaystyle c} (conventionally called the speed-of-light) relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly 300,000 kilometres or 190,000 miles in space being equivalent to one second in time), along with the fact that spacetime is a man... | Wikipedia - Space Time Continuum | null | null | null |
In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events is being measured. This usage differs significantly from the ordinary English meaning of the term. Reference frames are inherently nonlocal constructs, and according to this usage of the term, it does... | Wikipedia - Space Time Continuum | null | null | null |
In Fig. 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks is used to determine the ... | Wikipedia - Space Time Continuum | null | null | null |
The term observer refers to the entire ensemble of clocks associated with one inertial frame of reference. : 17–22 In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, howev... | Wikipedia - Space Time Continuum | null | null | null |
To synchronize the clocks, in the data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks. In many books on special relativity, especially older ones, the word "observer" is used in the more o... | Wikipedia - Space Time Continuum | null | null | null |
It is usually clear from context which meaning has been adopted. Physicists distinguish between what one measures or observes (after one has factored out signal propagation delays), versus what one visually sees without such corrections. Failure to understand the difference between what one measures/observes versus wha... | Wikipedia - Space Time Continuum | null | null | null |
Non-restoring division uses the digit set {−1, 1} for the quotient digits instead of {0, 1}. The algorithm is more complex, but has the advantage when implemented in hardware that there is only one decision and addition/subtraction per quotient bit; there is no restoring step after the subtraction, which potentially cu... | Wikipedia - Non-restoring division | null | null | null |
Example: If the −1 digits of Q {\displaystyle Q} are stored as zeros (0) as is common, then P {\displaystyle P} is Q {\displaystyle Q} and computing M {\displaystyle M} is trivial: perform a one's complement (bit by bit complement) on the original Q {\displaystyle Q} . Finally, quotients computed by this algorithm are ... | Wikipedia - Non-restoring division | null | null | null |
Non-softening infill material is material that does not change in shear characteristics under the influence of water nor under the influence of shear displacement. The material may break but no greasing effect will occur. The material particles can roll but this is considered to be of minor influence because, after sma... | Wikipedia - Sliding criterion (Geotechnical engineering) | null | null | null |
Non-spiking cells, spiking cells, and their measurement Not all the cells of the nervous system produce the type of spike that define the scope of the spiking neuron models. For example, cochlear hair cells, retinal receptor cells, and retinal bipolar cells do not spike. Furthermore, many cells in the nervous system ar... | Wikipedia - Integrate-and-fire model | null | null | null |
Neuronal activity can be measured with different experimental techniques, such as the "Whole cell" measurement technique, which captures the spiking activity of a single neuron and produces full amplitude action potentials. With extracellular measurement techniques an electrode (or array of several electrodes) is locat... | Wikipedia - Integrate-and-fire model | null | null | null |
Extracellular measurement has several advantages: Is easier to obtain experimentally Is robust and lasts for a longer time Can reflect the dominant effect, especially when conducted in an anatomical region with many similar cells.Overview of neuron models Neuron models can be divided into two categories according to th... | Wikipedia - Integrate-and-fire model | null | null | null |
Some models in this category predict only the moment of occurrence of output spike (also known as "action potential"); other models are more detailed and account for sub-cellular processes. The models in this category can be either deterministic or probabilistic. Natural stimulus or pharmacological input neuron models ... | Wikipedia - Integrate-and-fire model | null | null | null |
The input stage of these models is not electrical, but rather has either pharmacological (chemical) concentration units, or physical units that characterize an external stimulus such as light, sound or other forms of physical pressure. Furthermore, the output stage represents the probability of a spike event and not an... | Wikipedia - Integrate-and-fire model | null | null | null |
To accelerate the convergence to a unified theory, we list several models in each category, and where applicable, also references to supporting experiments. Aims of neuron models Ultimately, biological neuron models aim to explain the mechanisms underlying the operation of the nervous system. | Wikipedia - Integrate-and-fire model | null | null | null |
However, several approaches can be distinguished, from more realistic models (e.g., mechanistic models) to more pragmatic models (e.g., phenomenological models). Modeling helps to analyze experimental data and address questions. Models are also important in the context of restoring lost brain functionality through neur... | Wikipedia - Integrate-and-fire model | null | null | null |
Non-static environments, such as those containing other vehicles or pedestrians, continue to present research challenges. SLAM with DATMO is a model which tracks moving objects in a similar way to the agent itself. | Wikipedia - Simultaneous localization and mapping | null | null | null |
Non-vegetated or constructed surfaces absorb incoming solar radiation striking that energy and re-radiating it as infrared heat with long waveforms. This is sensible heat in that it can be sensed. Temperature is changed without a change of state. By contrast latent heat (hidden heat) results from a change of state with... | Wikipedia - Transpirational cooling (biological) | null | null | null |
For example as radiant energy warms a body of water it raises the temperature generating sensible heat. Water evaporated from the body of water changes state as latent heat. To change one gram of liquid water to vapour requires 540 calories of heat, and if that water vapour condenses back to liquid water 540 calories a... | Wikipedia - Transpirational cooling (biological) | null | null | null |
One climate mitigation pathway is for water vapour to carry energy back into the atmosphere where some of that energy will dissipate into space. Earth’s energy budget reveals the pathways of solar energy to earth, its cycling in earth systems and atmosphere, and release back into space. There is an average of 340.4 wat... | Wikipedia - Transpirational cooling (biological) | null | null | null |
To maintain a stable climate the same amount of energy must return to space. While increased levels of greenhouse gasses retain more heat, there are other pathways that can influence this energy balance. | Wikipedia - Transpirational cooling (biological) | null | null | null |
Understanding these dynamics provides more pathways to moderate the climate than simply relying on emissions reductions and sequestration alone. Referencing the NASA earth’s energy budget, an example is reducing the 398.2 watts/m2 emitted by the surface, by extending terrestrial and marine vegetative cover as a percent... | Wikipedia - Transpirational cooling (biological) | null | null | null |
Non-viral methods of nucleic acid transfection involved injecting a naked DNA plasmid into cell for incorporation into the genome. This method used to be relatively ineffective with low frequency of integration, however, efficiency has since greatly improved, using methods to enhance the delivery of the gene of interes... | Wikipedia - Designer baby | null | null | null |
The method is believed to function due to the formation of pores across the membrane, but although these are temporary, electroporation results in a high rate of cell death which has limited its use. An improved version of this technology, electron-avalanche transfection, has since been developed, which involves shorte... | Wikipedia - Designer baby | null | null | null |
The device generates a force to penetrate the cell membrane, allowing the DNA to enter whilst retaining the metal particle. Oligonucleotides are used as chemical vectors for gene therapy, often used to disrupt mutated DNA sequences to prevent their expression. Disruption in this way can be achieved by introduction of s... | Wikipedia - Designer baby | null | null | null |
Non-volatile spin-logic devices to enable scaling are being extensively studied. Spin-transfer, torque-based logic devices that use spins and magnets for information processing have been proposed. These devices are part of the ITRS exploratory road map. Logic-in memory applications are already in the development stage.... | Wikipedia - Spintronics | null | null | null |
Noncoding RNAs belong to several groups and are involved in many cellular processes. These range from ncRNAs of central importance that are conserved across all or most cellular life through to more transient ncRNAs specific to one or a few closely related species. The more conserved ncRNAs are thought to be molecular ... | Wikipedia - Noncoding RNA | null | null | null |
Noncommutative ring theory began with extensions of the complex numbers to hypercomplex numbers, specifically William Rowan Hamilton's quaternions in 1843. Many other number systems followed shortly. In 1844, Hamilton presented biquaternions, Cayley introduced octonions, and Grassman introduced exterior algebras. James... | Wikipedia - Abstract Algebra | null | null | null |
William Kingdon Clifford introduced split-biquaternions in 1873. In addition Cayley introduced group algebras over the real and complex numbers in 1854 and square matrices in two papers of 1855 and 1858.Once there were sufficient examples, it remained to classify them. | Wikipedia - Abstract Algebra | null | null | null |
In an 1870 monograph, Benjamin Peirce classified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of an associative algebra. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined the Peirce decomposition. | Wikipedia - Abstract Algebra | null | null | null |
Frobenius in 1878 and Charles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras over R {\displaystyle \mathbb {R} } were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimple Lie algebras could be decomposed into s... | Wikipedia - Abstract Algebra | null | null | null |
Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called the Wedderburn principal theorem and Artin–Wedderburn theorem.For commutative rings, several areas togeth... | Wikipedia - Abstract Algebra | null | null | null |
Jacobi and Eisenstein at around the same time proved a cubic reciprocity law for the Eisenstein integers. The study of Fermat's last theorem led to the algebraic integers. In 1847, Gabriel Lamé thought he had proven FLT, but his proof was faulty as he assumed all the cyclotomic fields were UFDs, yet as Kummer pointed o... | Wikipedia - Abstract Algebra | null | null | null |
In 1846 and 1847 Kummer introduced ideal numbers and proved unique factorization into ideal primes for cyclotomic fields. Dedekind extended this in 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product of prime ideals, a precursor of the theory of Dedekind doma... | Wikipedia - Abstract Algebra | null | null | null |
Riemann's methods relied on an assumption he called Dirichlet's principle, which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing the direct method in the calculus of variations. In the 1860s and 1870s, Clebsch, Gordan, Brill, and especially M. Noether studi... | Wikipedia - Abstract Algebra | null | null | null |
In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ring R {\displaystyle \mathbb {R} } , although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on... | Wikipedia - Abstract Algebra | null | null | null |
Lasker proved a special case of the Lasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection of primary ideals. Macauley proved the uniqueness of this decomposition. Overall, this work led to the development of algebraic geometry.In 1801 Gauss introduced binary quadratic forms over ... | Wikipedia - Abstract Algebra | null | null | null |
He further defined the discriminant of these forms, which is an invariant of a binary form. Between the 1860s and 1890s invariant theory developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the Jacobian and the Hessian for binary quartic forms and cubic forms. | Wikipedia - Abstract Algebra | null | null | null |
In 1868 Gordan proved that the graded algebra of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis. Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 to Hilbert's b... | Wikipedia - Abstract Algebra | null | null | null |
The first axiomatic definition was given by Abraham Fraenkel in 1914. His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element. In addition, he ha... | Wikipedia - Abstract Algebra | null | null | null |
These allowed Fraenkel to prove that addition was commutative. Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one.In 1920, Emmy Noether, in co... | Wikipedia - Abstract Algebra | null | null | null |
The following year she published a landmark paper called Idealtheorie in Ringbereichen (Ideal theory in rings'), analyzing ascending chain conditions with regard to (mathematical) ideals. The publication gave rise to the term "Noetherian ring", and several other mathematical objects being called Noetherian. | Wikipedia - Abstract Algebra | null | null | null |
Noted algebraist Irving Kaplansky called this work "revolutionary"; results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom. Artin, inspired by Noether’s work, came up with the descending chain condition. These definitions marked the birth of abstract ring ... | Wikipedia - Abstract Algebra | null | null | null |
Noncommutativity presents a major challenge for probabilistic interpretation of quantum stochastic differential equations due to non-existence of conditional expectations for general pairs of quantum observables. Belavkin resolved this issue by discovering the error-perturbation uncertainty relation and formulating the... | Wikipedia - Belavkin equation | null | null | null |
Nonconstructive proof Existence theorem Intuitionistic logic Intuitionistic type theory Type theory Lambda calculus Church–Rosser theorem Simply typed lambda calculus Typed lambda calculus Curry–Howard isomorphism Calculus of constructions Constructivist analysis Lambda cube System F Introduction to topos theory LF (lo... | Wikipedia - List of mathematical logic topics | null | null | null |
None of the assumptions above are likely to hold for natural populations. Nevertheless, the Lotka–Volterra model shows two important properties of predator and prey populations and these properties often extend to variants of the model in which these assumptions are relaxed: Firstly, the dynamics of predator and prey p... | Wikipedia - Lotka–Volterra equation | null | null | null |
Making the environment better for the prey benefits the predator, not the prey (this is related to the paradox of the pesticides and to the paradox of enrichment). A demonstration of this phenomenon is provided by the increased percentage of predatory fish caught had increased during the years of World War I (1914–18),... | Wikipedia - Lotka–Volterra equation | null | null | null |
A further example is provided by the experimental iron fertilization of the ocean. In several experiments large amounts of iron salts were dissolved in the ocean. The expectation was that iron, which is a limiting nutrient for phytoplankton, would boost growth of phytoplankton and that is would sequester carbon dioxide... | Wikipedia - Lotka–Volterra equation | null | null | null |
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