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Axonal initial segment Anatomy Axonal_initial_segment > Anatomy A similar arrangement is seen in the cerebellum. Bundles of myelinated axons make up the nerve tracts in the CNS. Where these tracts cross the midline of the brain to connect opposite regions they are called commissures. The largest of these is the corpus callosum that connects the two cerebral hemispheres, and this has around 20 million axons.The structure of a neuron is seen to consist of two separate functional regions, or compartments – the cell body together with the dendrites as one region, and the axonal region as the other.
Structural recursion Examples Structural_induction > Examples the induction hypothesis). That is, p ≤ 2g − 1 and q ≤ 2h − 1 can be assumed, where g and h denotes the number of generations the father's and the mother's subtree extends over, respectively, and p and q denote the numbers of persons they show. In case g ≤ h, the whole tree extends over 1 + h generations and shows p + q + 1 persons, andi.e.
Stiff-person syndrome Signs and symptoms Stiff-person_syndrome > Signs and symptoms Chronic pain is common and worsens over time, but sometimes acute pain occurs, as well. Stress, cold weather, and infections lead to an increase in symptoms, and sleep decreases them.SPS patients experience superimposed spasms and extreme sensitivity to touch and sound.
Curtain wall (architecture) Loads Curtain_wall_(architecture) > Design concerns > Loads The loads imposed on the curtain wall are transferred to the building structure through the anchors which attach the mullions to the building. Dead loadDead load is defined as the weight of structural elements and the permanent features on the structure. In the case of curtain walls, this load is made up of the weight of the mullions, anchors and other structural components of the curtain wall, as well as the weight of the infill material. Additional dead loads imposed on the curtain wall may include sunshades or signage attached to the curtain wall.
Defect zero Ring theory interpretation Modular_representation > Ring theory interpretation Given a field K and a finite group G, the group algebra K (which is the K-vector space with K-basis consisting of the elements of G, endowed with algebra multiplication by extending the multiplication of G by linearity) is an Artinian ring. When the order of G is divisible by the characteristic of K, the group algebra is not semisimple, hence has non-zero Jacobson radical. In that case, there are finite-dimensional modules for the group algebra that are not projective modules. By contrast, in the characteristic 0 case every irreducible representation is a direct summand of the regular representation, hence is projective.
Chronic obstructive lung disease Other tests Chronic_lower_respiratory_disease > Diagnosis > Other tests A chest X-ray is not useful to establish a diagnosis of COPD but it is of use in either excluding other conditions or including comorbidities such as pulmonary fibrosis and bronchiectasis. Characteristic signs of COPD on X-ray include hyperinflation (shown by a flattened diaphragm and an increased retrosternal air space) and lung hyperlucency. A saber-sheath trachea may also be shown that is indicative of COPD.A CT scan is not routinely used except for the exclusion of bronchiectasis. An analysis of arterial blood is used to determine the need for oxygen supplementation and assess for high levels of carbon dioxide in the blood; this is recommended in those with an FEV1 less than 35% predicted, those with a peripheral oxygen saturation less than 92% and those with symptoms of congestive heart failure. WHO recommends that all those diagnosed with COPD be screened for alpha-1 antitrypsin deficiency.
Arachidonate 5-lipoxygenase Atherosclerosis Arachidonate_5-lipoxygenase > Human genetics > Atherosclerosis Bearers of two variations in the predominant five tandem repeat Sp1 binding motif (GGGCCGG) of the ALOX5 gene promoter in 470 subjects (non-Hispanic whites, 55.1%; Hispanics, 29.6%; Asian or Pacific Islander, 7.7&; African Americans, 5.3%, and others, 2.3%) were positively associated with the severity of atherosclerosis, as judged by carotid intima–media thickness measurements. Variant alleles involved deletions (one or two) or additions (one, two, or three) of Sp1 motifs to the five tandem motifs allele.
Acid Reflux Proton-pump inhibitors Gastro-esophageal_reflux_disease > Treatment > Medications > Proton-pump inhibitors When these medications are used long term, the lowest effective dose should be taken. They may also be taken only when symptoms occur in those with frequent problems. H2 receptor blockers lead to roughly a 40% improvement.
Van der Waals sphere Van der Waals equation of state Atomic_volume > Methods of determination > Van der Waals equation of state The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: where p is pressure, n is the number of moles of the gas in question and a and b depend on the particular gas, V ~ {\displaystyle {\tilde {V}}} is the volume, R is the specific gas constant on a unit mole basis and T the absolute temperature; a is a correction for intermolecular forces and b corrects for finite atomic or molecular sizes; the value of b equals the van der Waals volume per mole of the gas. Their values vary from gas to gas. The van der Waals equation also has a microscopic interpretation: molecules interact with one another.
Gravitational theory Antiquity History_of_gravitational_theory > Antiquity In On Floating Bodies, Archimedes claimed that for any object submerged in a fluid there is an equivalent upward buoyant force to the weight of the fluid displaced by the object's volume. The fluids described by Archimedes are not self-gravitating, since he assumes that "any fluid at rest is the surface of a sphere whose centre is the same as that of the Earth".Greek astronomer Hipparchus of Nicaea (c.190 – c. 120 BCE) also rejected Aristotelian physics and followed Strato in adopting some form of theory of impetus to explain motion.
Peyton Young Evolutionary game theory Peyton_Young > Contributions > Evolutionary game theory Conventional concepts of dynamic stability, including the evolutionarily stable strategy concept, identify states from which small once-off deviations are self-correcting. These stability concepts are not appropriate for analyzing social and economic systems which are constantly perturbed by idiosyncratic behavior and mistakes, and individual and aggregate shocks to payoffs. Building upon Freidlin and Wentzell's (1984) theory of large deviations for continuous time-processes, Dean Foster and Peyton Young (1990) developed the more powerful concept of stochastic stability: "The stochastically stable set is the set of states such that, in the long run, it is nearly certain that the system lies within every open set containing S as the noise tends slowly to zero" . This solution concept created a major impact in economics and game theory after Young (1993) developed a more tractable version of the theory for general finite-state Markov chains.
Classifier constructions in sign languages Summary Classifier_handshape The relative location of multiple entities can be represented iconically in two-handed constructions. Classifiers share some limited similarities with the gestures of hearing non-signers. Those who do not know the sign language can often guess the meaning of these constructions.
Aristotle Memory Aristotle > Natural philosophy > Psychology > Memory Recollection is thus the self-directed activity of retrieving the information stored in a memory impression. Only humans can remember impressions of intellectual activity, such as numbers and words. Animals that have perception of time can retrieve memories of their past observations.
62 (number) In mathematics 62_(number) > In mathematics the tenth member of the 7-aliquot tree (7, 8, 10, 14, 20, 22, 34, 38, 49, 62, 75, 118, 148, etc). It has an aliquot sum of 34; itself a discrete semiprime, and its aliquot sequence is: 62,34,20,22,14,10,8,7,1,0. The 20th & 21st, 72nd & 73rd, 75th & 76th digits of pi.
Particle velocity level Summary Particle_velocity_level Particle velocity (denoted v or SVL) is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string. When applied to a sound wave through a medium of a fluid like air, particle velocity would be the physical speed of a parcel of fluid as it moves back and forth in the direction the sound wave is travelling as it passes.
Murai reaction Ru(II) catalysts Murai_reaction > Mechanism > Ru(II) catalysts Following oxidative addition of H2 (6), the complex reductively eliminates the product to give the product agostically bound to the complex. Coordination of another acetophenone molecule regenerates complex 2. == References ==
Quantum uncertainty Quantum harmonic oscillator stationary states Uncertainty_Principle > Examples > Quantum harmonic oscillator stationary states Consider a one-dimensional quantum harmonic oscillator. It is possible to express the position and momentum operators in terms of the creation and annihilation operators: Using the standard rules for creation and annihilation operators on the energy eigenstates, the variances may be computed directly, The product of these standard deviations is then In particular, the above Kennard bound is saturated for the ground state n=0, for which the probability density is just the normal distribution.
Process management (computing) Process termination Process_management_(computing) > Process termination There are many reasons for process termination: Batch job issues halt instruction User logs off Process executes a service request to terminate Error and fault conditions Normal completion Time limit exceeded Memory unavailable Bounds violation; for example: attempted access of (non-existent) 11th element of a 10-element array Protection error; for example: attempted write to read-only file Arithmetic error; for example: attempted division by zero Time overrun; for example: process waited longer than a specified maximum for an event I/O failure Invalid instruction; for example: when a process tries to execute data (text) Privileged instruction Data misuse Operating system intervention; for example: to resolve a deadlock Parent terminates so child processes terminate (cascading termination) Parent request
Entropy (energy dispersal) Texts using the energy dispersal approach Entropy_(energy_dispersal) > Further reading > Texts using the energy dispersal approach Petrucci, Harwood, and Herring. General Chemistry, 9th ed. Silberberg, M.S., 2006.
History of special relativity Electromagnetic mass History_of_special_relativity > Aether and electrodynamics of moving bodies > Electromagnetic mass During his development of Maxwell's Theory, J. J. Thomson (1881) recognized that charged bodies are harder to set in motion than uncharged bodies. Electrostatic fields behave as if they add an "electromagnetic mass" to the mechanical mass of the bodies. I.e., according to Thomson, electromagnetic energy corresponds to a certain mass.
Link page Summary Link_page A link page is a type of web page that contains a list of links the website owner finds notable to mention, such as partner organizations, clients, friends, hobbies, or related projects. Links pages were popular on personal websites during the Web 1.0 era, functioning similarly to webrings as a navigation device.
Hashcash Sender's side Hashcash > Technical details > Sender's side The number of times that the sender needs to try to get a valid hash value is modeled by geometric distribution. Hence the sender will on average have to try 220 values to find a valid header. Given reasonable estimates of the time needed to compute the hash, this would take about one second to find.
Bloom filter Probability of false positives Bloom_filters > Probability of false positives Each of the k array positions computed by the hash functions is 1 with a probability as above. The probability of all of them being 1, which would cause the algorithm to erroneously claim that the element is in the set, is often given as ε = ( 1 − k n ) k ≈ ( 1 − e − k n / m ) k . {\displaystyle \varepsilon =\left(1-\left^{kn}\right)^{k}\approx \left(1-e^{-kn/m}\right)^{k}.}
Blocking antibody Cancer treatment Blocking_antibody > Uses > Cancer treatment The blocking antibody does not directly target tumor cells, but rather blocks the regulatory functions of CTLA-4, resulting in enhanced T-cell function.Some new treatments hypothesize the blocking of PD-1, a programmed cell-death protein, which will result in longer-lived T-cells. The blocking antibody BMS-936559 has been shown to bind to PD-L1 and prevent its binding to PD-1.These new treatments are not without side-effects and immune-related adverse events have been observed in a variety of patients. The tolerance that immune cells normally have to host tissues can be lost, resulting in permanent damage to host cells.
Algorithmic game theory Nisan-Ronen: a new framework for studying algorithms Algorithmic_game_theory > History > Nisan-Ronen: a new framework for studying algorithms In this model the algorithmic solution is adorned with payments to the participants and is termed a mechanism. The payments should be carefully chosen as to motivate all participants to act as the algorithm designer wishes. We apply the standard tools of mechanism design to algorithmic problems and in particular to the shortest path problem. This paper coined the term algorithmic mechanism design and was recognized by the 2012 Gödel Prize committee as one of "three papers laying foundation of growth in Algorithmic Game Theory".
Diffraction topography Basic principle of topography Diffraction_topography > Basic principle of topography When the beam hits the crystalline sample, Bragg diffraction occurs, i.e. the incident wave is reflected by the atoms on certain lattice planes of the sample, on condition that it hits those planes at the right Bragg angle. Diffraction from sample can take place either in reflection geometry (Bragg case), with the beam entering and leaving through the same surface, or in transmission geometry (Laue case). Diffraction gives rise to a diffracted beam, which will leave the sample and propagate along a direction differing from the incident direction by the scattering angle 2 θ B {\displaystyle 2\theta _{B}} .
Valence electrons Valence shell Valence_shell > Valence shell The valence shell is the set of orbitals which are energetically accessible for accepting electrons to form chemical bonds. For main-group elements, the valence shell consists of the ns and np orbitals in the outermost electron shell. For transition metals the orbitals of the incomplete (n−1)d subshell are included, and for lanthanides and actinides incomplete (n−2)f and (n−1)d subshells. The orbitals involved can be in an inner electron shell and do not all correspond to the same electron shell or principal quantum number n in a given element, but they are all at similar energies.
Log bridge Summary Log_bridge A log bridge is a timber bridge that uses logs that fall naturally or are intentionally felled or placed across streams. The first man-made bridges with significant span were probably intentionally felled trees.The use of emplaced logs is now sometimes used in temporary bridges used for logging roads, where a forest tract is to be harvested and the road then abandoned. Such log bridges have a severely limited lifetime due to soil contact and subsequent rot and wood-eating insect infestation. Longer lasting log bridges may be constructed by using treated logs and/or by providing well drained footings of stone or concrete combined with regular maintenance to prevent soil infiltration. This care in construction can be seen in the French bridge illustrated below, which has well locked dry set stone abutments and a footpath leveled with boards.
Ceramic Engineering Modern industry Ceramic-filled_epoxy > Modern industry Ceramic engines do not require a cooling system and hence allow a major weight reduction and therefore greater fuel efficiency. Fuel efficiency of the engine is also higher at high temperature, as shown by Carnot's theorem. In a conventional metallic engine, much of the energy released from the fuel must be dissipated as waste heat in order to prevent a meltdown of the metallic parts.
General-purpose operating system Memory management General-purpose_operating_system > Components > Kernel > Memory management This system of memory management is almost never seen any more, since programs often contain bugs which can cause them to exceed their allocated memory. If a program fails, it may cause memory used by one or more other programs to be affected or overwritten. Malicious programs or viruses may purposefully alter another program's memory, or may affect the operation of the operating system itself.
Radio receiver The superheterodyne design Radio_receivers > How receivers work > The superheterodyne design Without an input filter the receiver can receive incoming RF signals at two different frequencies,. The receiver can be designed to receive on either of these two frequencies; if the receiver is designed to receive on one, any other radio station or radio noise on the other frequency may pass through and interfere with the desired signal. A single tunable RF filter stage rejects the image frequency; since these are relatively far from the desired frequency, a simple filter provides adequate rejection.
Population collapse Increase the productivity of the workforce Population_collapse > National efforts to confront declining populations > Increase the productivity of the workforce Development economists would call increasing the size of the workforce "extensive growth". They would call increasing the productivity of that workforce "intensive growth". In this case, GDP growth is driven by increased output per worker, and by extension, increased GDP/capita.In the context of a stable or declining population, increasing workforce productivity is better than mostly short-term efforts to increase the size of the workforce.
Determinant (mathematics) Further methods Determinant > Calculation > Further methods For example, the Gaussian elimination (or LU decomposition) method is of order O ⁡ ( n 3 ) {\displaystyle \operatorname {O} (n^{3})} , but the bit length of intermediate values can become exponentially long. By comparison, the Bareiss Algorithm, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times n {\displaystyle n} .If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of A + uvT, where u and v are column vectors. Charles Dodgson (i.e. Lewis Carroll of Alice's Adventures in Wonderland fame) invented a method for computing determinants called Dodgson condensation. Unfortunately this interesting method does not always work in its original form.
Cognitive Model Language acquisition Cognitive_modelling > Dynamical systems > Early dynamical systems > Language acquisition This means that representations are sensitive to context, with mental representations viewed as trajectories through mental space instead of objects that are constructed and remain static. Elman networks were trained with simple sentences to represent grammar as a dynamical system. Once a basic grammar had been learned, the networks could then parse complex sentences by predicting which words would appear next according to the dynamical model.
Vaccine overload Pain mitigation for children Vaccine_hesitancy > Countermeasures > Pain mitigation for children Parents may be hesitant to have their children vaccinated due to concerns about the pain of vaccination. Several strategies can be used to reduce the child's pain. Such strategies include distraction techniques (pinwheels); deep breathing techniques; breastfeeding the child; giving the child sweet-tasting solutions; quickly administering the vaccine without aspirating; keeping the child upright; providing tactile stimulation; applying numbing agents to the skin; and saving the most painful vaccine for last. As above, the number of vaccines offered in a particular encounter is related to the likelihood of parental vaccine refusal (the more vaccines offered, the higher the likelihood of vaccine deferral). The use of combination vaccines to protect against more diseases but with fewer injections may provide reassurance to parents. Similarly, reframing the conversation with less emphasis on the number of diseases the healthcare provider is immunizing against (e.g., "we will do two injections (combined vaccinations) and an oral vaccine") may be more acceptable to parents than "we're going to vaccinate against seven diseases".
Electrolithoautotroph Summary Electrolithoautotroph An electrolithoautotroph is an organism which feeds on electricity. These organisms use electricity to convert carbon dioxide into organic matter by using electrons directly taken from solid-inorganic electron donors. Electrolithoautotrophs are microorganisms which are found in the deep crevices of the ocean. The warm, mineral-rich environment provides a rich source of nutrients.
Inferring horizontal gene transfer Evaluation Inferring_horizontal_gene_transfer > Evaluation Such HGT events can be simulated through subtree pruning and regrafting rearrangements of the species tree. However, it is important to simulate data that are realistic enough to be representative of the challenge provided by real datasets, and simulation under complex models are thus preferable.
Smart materials Types Smart_materials > Types Shape-memory alloys and shape-memory polymers are materials in which large deformation can be induced and recovered through temperature changes or stress changes (pseudoelasticity). The shape memory effect results due to respectively martensitic phase change and induced elasticity at higher temperatures. Photovoltaic materials or optoelectronics convert light to electrical current.
Desert greening Importance of trees Desert_greening > Importance of trees A main component of desert greening is the planting of trees. Trees store water, inhibit soil erosion through wind, raise water from underlying aquifers, reduce evaporation after a rain, attract animals (and thereby fertility through feces), and they can cause more rain to fall (by temperature reduction and other effects), if the planted area is large enough.All of the effects beneficial for desert-greening which trees offer can also be provided by buildings. Shading by buildings is an example for a passive effect, the pumping up of water from aquifers an example for an active effect achieved with buildings technology. An example for a building designed to offer all of the beneficial effects of natural forests in the desert is the IBTS Greenhouse.
Universal coefficient theorem Universal coefficient theorem for cohomology Universal_coefficient_theorem > Universal coefficient theorem for cohomology {\displaystyle h()()=f(x).} An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map h takes a homotopy class of maps from X to K(G, i) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.
Directed evolution Screening Directed_evolution > Principles > Detecting fitness differences > Screening An alternative to selection is a screening system. Each variant gene is individually expressed and assayed to quantitatively measure the activity (most often by a colourgenic or fluorogenic product). The variants are then ranked and the experimenter decides which variants to use as templates for the next round of DE.
Radiative heat transfer Summary Radiation_properties : 73–86 Particle motion results in charge-acceleration or dipole oscillation which produces electromagnetic radiation. Infrared radiation emitted by animals (detectable with an infrared camera) and cosmic microwave background radiation are examples of thermal radiation. If a radiation object meets the physical characteristics of a black body in thermodynamic equilibrium, the radiation is called blackbody radiation. Planck's law describes the spectrum of blackbody radiation, which depends solely on the object's temperature. Wien's displacement law determines the most likely frequency of the emitted radiation, and the Stefan–Boltzmann law gives the radiant intensity.Thermal radiation is also one of the fundamental mechanisms of heat transfer.
Biological applications of bifurcation theory Biological networks and dynamical systems Biological_applications_of_bifurcation_theory > Biological networks and dynamical systems Biological networks originate from evolution and therefore have less standardized components and potentially more complex interactions than networks designed by humans, such as electrical networks. At the cellular level, components of a network can include a large variety of proteins, many of which differ between organisms. Network interactions occur when one or more proteins affect the function of another through transcription, translation, translocation, phosphorylation, or other mechanisms.
Dynamic IP address Address translation Static_IP_address > Address translation In residential networks, NAT functions are usually implemented in a residential gateway. In this scenario, the computers connected to the router have private IP addresses and the router has a public address on its external interface to communicate on the Internet. The internal computers appear to share one public IP address.
Intrinsic reaction coordinate Degrees of freedom Energy_profile_(chemistry) > Potential energy surfaces > Degrees of freedom These internal coordinates may be represented by simple stretch, bend, torsion coordinates, or symmetry-adapted linear combinations, or redundant coordinates, or normal modes coordinates, etc. For a system described by n-internal coordinates a separate potential energy function can be written with respect to each of these coordinates by holding the other n – 1 parameters at a constant value allowing the potential energy contribution from a particular molecular motion (or interaction) to be monitored while the other n – 1 parameters are defined. Consider a diatomic molecule AB which can macroscopically visualized as two balls (which depict the two atoms A and B) connected through a spring which depicts the bond. As this spring (or bond) is stretched or compressed, the potential energy of the ball-spring system (AB molecule) changes and this can be mapped on a 2-dimensional plot as a function of distance between A and B, i.e. bond length.
The Loop (CTA) Station listing The_Loop_(CTA) > Station listing This lists each station beginning at the northwest corner and moving counterclockwise around the loop: south along Wells Street, east along Van Buren Street, north along Wabash Avenue, and west along Lake Street.
Common Variable Immunodeficiency Causes Common_Variable_Immunodeficiency > Causes Genetic mutations can be identified as the cause of disease in about 10% of people, while familial inheritance accounts for 10–25% of cases. Rather than arising from a single genetic mutation, CVID seems to result from a variety of mutations that all contribute to a failure in antibody production. Mutations in the genes encoding ICOS, TACI, CD19, CD20, CD21, CD80, and BAFFR have been identified as causative of CVID.
Macro Assembler Terminology Cross_assembler > Terminology A macro assembler is an assembler that includes a macroinstruction facility so that (parameterized) assembly language text can be represented by a name, and that name can be used to insert the expanded text into other code. Open code refers to any assembler input outside of a macro definition. A cross assembler (see also cross compiler) is an assembler that is run on a computer or operating system (the host system) of a different type from the system on which the resulting code is to run (the target system). Cross-assembling facilitates the development of programs for systems that do not have the resources to support software development, such as an embedded system or a microcontroller.
Circuit design Design Circuit_design > Design Finally, the individual circuit components are chosen to carry out each function in the overall design; at this stage, the physical layout and electrical connections of each component are also decided, this layout commonly taking the form of artwork for the production of a printed circuit board or Integrated circuit. This stage is typically highly time-consuming because of the vast array of choices available. A practical constraint on the design at this stage is standardization;.
Taylor polynomials Geometric series Taylor_polynomials > List of Maclaurin series of some common functions > Geometric series The geometric series and its derivatives have Maclaurin series 1 1 − x = ∑ n = 0 ∞ x n 1 ( 1 − x ) 2 = ∑ n = 1 ∞ n x n − 1 1 ( 1 − x ) 3 = ∑ n = 2 ∞ ( n − 1 ) n 2 x n − 2 . {\displaystyle {\begin{aligned}{\frac {1}{1-x}}&=\sum _{n=0}^{\infty }x^{n}\\{\frac {1}{(1-x)^{2}}}&=\sum _{n=1}^{\infty }nx^{n-1}\\{\frac {1}{(1-x)^{3}}}&=\sum _{n=2}^{\infty }{\frac {(n-1)n}{2}}x^{n-2}.\end{aligned}}} All are convergent for \| x \| < 1 {\displaystyle \|x\|<1} . These are special cases of the binomial series given in the next section.
Wood screw Historical use Wood_screw > Differentiation between bolt and screw > Terminology > Historical use The 'carriage' bolt was derived from this and was employed more to speed up manufacturing than achieve a different function. The carriage bolt passes through both pieces of materials and employs a nut to provide the clamping force. Both are still, however, bolts.
Methodology of science Scientific inquiry Scientific_process > Scientific inquiry Scientific inquiry generally aims to obtain knowledge in the form of testable explanations that scientists can use to predict the results of future experiments. This allows scientists to gain a better understanding of the topic under study, and later to use that understanding to intervene in its causal mechanisms (such as to cure disease). The better an explanation is at making predictions, the more useful it frequently can be, and the more likely it will continue to explain a body of evidence better than its alternatives. The most successful explanations – those which explain and make accurate predictions in a wide range of circumstances – are often called scientific theories.Most experimental results do not produce large changes in human understanding; improvements in theoretical scientific understanding typically result from a gradual process of development over time, sometimes across different domains of science.
Adaptive immune response Evolution Active_immunity > Evolution Like TCR and Ig, the MHC is found only in jawed vertebrates. Genes involved in antigen processing and presentation, as well as the class I and class II genes, are closely linked within the MHC of almost all studied species. Lymphoid cells can be identified in some pre-vertebrate deuterostomes (i.e., sea urchins).
Name decoration Summary Name_mangling Any object code produced by compilers is usually linked with other pieces of object code (produced by the same or another compiler) by a type of program called a linker. The linker needs a great deal of information on each program entity. For example, to correctly link a function it needs its name, the number of arguments and their types, and so on.
Cancer biomarkers Risk assessment Cancer_biomarker > Role of biomarkers in cancer research and medicine > Uses of biomarkers in cancer medicine > Risk assessment Cancer biomarkers, particular those associated with genetic mutations or epigenetic alterations, often offer a quantitative way to determine when individuals are predisposed to particular types of cancers. Notable examples of potentially predictive cancer biomarkers include mutations on genes KRAS, p53, EGFR, erbB2 for colorectal, esophageal, liver, and pancreatic cancer; mutations of genes BRCA1 and BRCA2 for breast and ovarian cancer; abnormal methylation of tumor suppressor genes p16, CDKN2B, and p14ARF for brain cancer; hypermethylation of MYOD1, CDH1, and CDH13 for cervical cancer; and hypermethylation of p16, p14, and RB1, for oral cancer.
Trilaciclib Summary Trilaciclib Trilaciclib, sold under the brand name Cosela, is a medication used to reduce the frequency of chemotherapy-induced bone marrow suppression.The most common side effects include fatigue; low levels of calcium, potassium and phosphate; increased levels of an enzyme called aspartate aminotransferase; headache; and infection in the lungs (pneumonia).Trilaciclib may help protect bone marrow cells from damage caused by chemotherapy by inhibiting cyclin-dependent kinase 4/6, a type of enzyme. Trilaciclib is the first therapy in its class and was approved for medical use in the United States in February 2021. The U.S. Food and Drug Administration considers it to be a first-in-class medication.Chemotherapy drugs are designed to kill cancer cells but can damage normal tissues as well.
Helek Summary Helek A helek is 31/3 seconds or 1/18 minute. The helek derives from a small Babylonian time period called a she, meaning '"barleycorn", itself equal to 1/72 of a Babylonian time degree (1° of celestial rotation). 360 degrees × 72 shes per degree / 24 hours = 1080 shes per hour.The Hebrew calendar defines its mean month to be exactly equal to 29 days 12 hours and 793 halakim, which is 29 days 12 hours 44 minutes and 31/3 seconds. It defines its mean year as exactly 235/19 times this amount, or 365 days, 5 hours, 55 minutes, and 25 and 25/57 seconds (approximately 365.2468222 days).
Fracture in polymers Fatigue Fracture_in_polymers > Fracture Mechanics in Polymers > Fatigue Four regimes were proposed: the steady-state regime, the linear regime, the power-law regime and the fourth regime where crack grows catastrophically. In the power-law regime, the relation is found to be similar to the Paris law, which was found by Paris et al. in metals. Hysteresis heating and Chain scission Fatigue in polymers, controlled by cyclical loading, is caused by two general mechanisms: hysteresis heating and chain scission.
Metalog distribution Applications Metalog_distribution > Applications Metalogs have also been used to fit output data from simulations in order to represent those outputs as closed-form continuous distributions (both CDFs and PDFs). Used in this way, they are typically more stable and smoother than histograms. Sums of lognormals.
Quinolinic acid Parkinson's disease Quinolinic_acid > Clinical implications > Conditions related to neuronal death > Parkinson's disease Quinolinic acid neurotoxicity is thought to play a role in Parkinson's disease. Studies show that quinolinic acid is involved in the degeneration of the dopaminergic neurons in the substantia nigra (SN) of Parkinson's disease patients. SN degeneration is one of the key characteristics of Parkinson's disease. Microglia associated with dopaminergic cells in the SN produce quinolinic acid at this location when scientists induce Parkinson's disease symptoms in macaques. Quinolinic acid levels are too high at these sites to be controlled by KYNA, causing neurotoxicity to occur.
Email privacy Encryption Email_privacy > Technological workarounds > Encryption According to Hilarie Orman, mail encryption was first developed in the mid-1980s. She states that mail encryption is a powerful tool that protects one's email privacy. Although it is widely available, it is rarely used, with the majority of email sent at risk of being read by third parties. In general, encryption provides protection against malicious entities. However, a court order might force the responsible parties to hand over decryption keys, with a notable example being Lavabit. Encryption can be performed at different levels of the email protocol.
Ovarian epithelial cancer Pathophysiology Ovary_cancer > Pathophysiology This structure needs to be repaired by dividing cells in the ovary. Continuous ovulation for a long time means more repair of the ovary by dividing cells, which can acquire mutations in each division.Overall, the most common gene mutations in ovarian cancer occur in NF1, BRCA1, BRCA2, and CDK12. Type I ovarian cancers, which tend to be less aggressive, tend to have microsatellite instability in several genes, including both oncogenes (most notably BRAF and KRAS) and tumor suppressors (most notably PTEN).
General-purpose operating system Memory management General-purpose_operating_system > Components > Kernel > Memory management This is called a segmentation violation or Seg-V for short, and since it is both difficult to assign a meaningful result to such an operation, and because it is usually a sign of a misbehaving program, the kernel generally resorts to terminating the offending program, and reports the error. Windows versions 3.1 through ME had some level of memory protection, but programs could easily circumvent the need to use it. A general protection fault would be produced, indicating a segmentation violation had occurred; however, the system would often crash anyway.
Wall-associated kinase Structure Wall-associated_kinase > Structure All the five WAK proteins have highly conserved serine/threonine protein kinase domain (86% similarity) on the cytoplasmic side and an extracellular domain (only 40% to 64% similarity in amino acid sequences). Moreover, All of the isomers of WAK proteins have epidermal growth factor (EGF) like repeats located at the amino-terminal side. Six cysteins (located in the EGF repeats) positions are well-kept in all the five WAKs, however, protein-protein interactions of WAks are still unknown.All WAKs (WAKs 1-5) have Asp/Asn hydroxylation site (Cxx(4)xCxC; Prosite PS00010) overlapping with calcium binding EGF domains where both hydroxylated and nonhydroxylated forms of coagulation proteases have equal affinities for calcium at physiological concentrations. Hydroxyl group may be involved in hydrogen bonding in protein-protein interactions mediated by the EGF-like domain.WAKs' association with cell wall is very strong (having covalent link to pectin), such that its release from the cell wall requires enzymatic digestion.
Ontological catalogue Selected bibliography Theory_of_Categories > Selected bibliography Aristotle, 1953. Metaphysics. Ross, W. D., trans. Oxford University Press.
Pythagorean hammers Antiquity Pythagorean_hammers > Reception > Antiquity The crucial point is that Pythagoras was initially prompted by sensory perception to formulate his question and hypotheses, and through empirical testing of hypotheses, he arrived at irrefutable certainty. The path to knowledge went from sensory perception to the initial hypothesis, which turned out to be erroneous, then to the formation of a correct opinion, and finally to its verification. Boethius acknowledges the necessity and value of sensory perception and opinion formation on the path to insight, although as a Platonist, he is inherently skeptical of sensory perception due to its proneness to error.
ParABS system Summary ParABS_system The parABS system is a broadly conserved molecular mechanism for plasmid partitioning and chromosome segregation in bacteria. Originally identified as a genetic element required for faithful partitioning of low-copy-number plasmids, it consists of three components: the ParA ATPase, the ParB DNA-binding protein, and the cis-acting parS sequence. The parA and parB genes are typically found in the same operon, with parS elements located within or adjacent to this operon. Collectively, these components function to ensure accurate partitioning of plasmids or whole chromosomes between bacterial daughter cells prior to cell division.
Banach Spaces Banach's theorems Banach_norm > General theory > Banach's theorems The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where X {\displaystyle X} is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood U {\displaystyle U} of 0 {\displaystyle \mathbf {0} } in X {\displaystyle X} such that all T {\displaystyle T} in F {\displaystyle F} are uniformly bounded on U , {\displaystyle U,} This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps. This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from M 1 ⊕ ⋯ ⊕ M n {\displaystyle M_{1}\oplus \cdots \oplus M_{n}} onto X {\displaystyle X} sending m 1 , ⋯ , m n {\displaystyle m_{1},\cdots ,m_{n}} to the sum m 1 + ⋯ + m n . {\displaystyle m_{1}+\cdots +m_{n}.}
Surface filtration Concentration Surface_filtration > Techniques to improve performance > Concentration The volume of the fluid is reduced by allowing permeate flow to occur. Solvent, solutes, and particles smaller than the membrane pore size pass through the membrane, while particles larger than the pore size are retained, and thereby concentrated. In bioprocessing applications, concentration may be followed by diafiltration.
Azomethine ylide 1,3-dipolar cycloaddition reactions Azomethine_ylide > 1,3-dipolar cycloaddition reactions While dipolarophiles are typically α,β-unsaturated carbonyl compounds, there have been many recent advances in developing new types of dipolarophiles.When the dipole and dipolarophile are part of the same molecule, an intramolecular cyclization reaction can lead to a polycyclic product of considerable complexity. If the dipolarophile is tethered to a carbon of the dipole, a fused bicycle is formed. If it is tethered to the nitrogen, a bridged structure results. The intramolecular nature of the reaction can also be useful in that regioselectivity is often constrained. Another advantage to intramolecular reactions is that the dipolarophile need not be electron-deficient—many examples of cyclization reactions with electron-rich, alkyl-substituted dipolarophiles have been reported, including the synthesis of martinellic acid shown below.
Special Theory of Relativity How far can you travel from the Earth? Special_Theory_of_Relativity > Dynamics > How far can you travel from the Earth? This is described by: where v(t) is the velocity at a time t, a is the acceleration of the spaceship and t is the coordinate time as measured by people on Earth. Therefore, after one year of accelerating at 9.81 m/s2, the spaceship will be travelling at v = 0.712c and 0.946c after three years, relative to Earth. After three years of this acceleration, with the spaceship achieving a velocity of 94.6% of the speed of light relative to Earth, time dilation will result in each second experienced on the spaceship corresponding to 3.1 seconds back on Earth.
Coordinate acceleration Summary Proper_acceleration This weight, in turn, is produced by fictitious forces or "inertial forces" which appear in all such accelerated coordinate systems, in a manner somewhat like the weight produced by the "force of gravity" in systems where objects are fixed in space with regard to the gravitating body (as on the surface of the Earth). The total (mechanical) force that is calculated to induce the proper acceleration on a mass at rest in a coordinate system that has a proper acceleration, via Newton's law F = ma, is called the proper force. As seen above, the proper force is equal to the opposing reaction force that is measured as an object's "operational weight" (i.e. its weight as measured by a device like a spring scale, in vacuum, in the object's coordinate system). Thus, the proper force on an object is always equal and opposite to its measured weight.
Thermodynamic Equilibrium Relation of exchange equilibrium between systems Thermodynamical_equilibrium > Relation of exchange equilibrium between systems When two systems are in contact equilibrium with respect to a particular kind of permeability, they have common values of the intensive variable that belongs to that particular kind of permeability. Examples of such intensive variables are temperature, pressure, chemical potential.
Electricity Meter Electronic Electrical_meter > Operation > Electronic These inputs are then processed using a digital signal processor to calculate the various metering parameters. The largest source of long-term errors in the meter is drift in the preamp, followed by the precision of the voltage reference.
Boundary layer control Summary Natural_laminar_flow Turbulent boundary layers are more resistant to separation. The energy in a boundary layer may need to be increased to keep it attached to its surface. Fresh air can be introduced through slots or mixed in from above.
Electrocatalyst Background and theory Electrocatalyst > Background and theory In these systems, each of the two electrodes and its associated half-cell would require its own specialized electrocatalyst.Half-reactions involving multiple steps, multiple electron transfers, and the evolution or consumption of gases in their overall chemical transformations, will often have considerable kinetic barriers. Furthermore, there is often more than one possible reaction at the surface of an electrode. For example, during the electrolysis of water, the anode can oxidize water through a two electron process to hydrogen peroxide or a four electron process to oxygen. The presence of an electrocatalyst could facilitate either of the reaction pathways.
Bunch–Davies vacuum Summary Bunch–Davies_vacuum_state The Bunch–Davies vacuum can also be described as being generated by an infinite time trace from the condition that the scale of quantum fluctuations is much smaller than the Hubble scale. The state possesses no quanta at the asymptotic past infinity.The Bunch–Davies state is the zero-particle state as seen by a geodesic observer, that is, an observer who is in free fall in the expanding state. The state explains the origin of cosmological perturbation fluctuations in inflationary models.
Blood gas Summary Blood_gas_analysis A blood gas test or blood gas analysis tests blood to measure blood gas tension values, it also measures blood pH, and the level and base excess of bicarbonate. The source of the blood is reflected in the name of each test; arterial blood gases come from arteries, venous blood gases come from veins and capillary blood gases come from capillaries. The blood gas tension levels of partial pressures can be used as indicators of ventilation, respiration and oxygenation. Analysis of paired arterial and venous specimens can give insights into the aetiology of acidosis in the newborn.
Liver fibrosis Summary Hepatic_cirrhosis Early symptoms may include tiredness, weakness, loss of appetite, unexplained weight loss, nausea and vomiting, and discomfort in the right upper quadrant of the abdomen. As the disease worsens, symptoms may include itchiness, swelling in the lower legs, fluid build-up in the abdomen, jaundice, bruising easily, and the development of spider-like blood vessels in the skin. The fluid build-up in the abdomen develop spontaneous infections.
Antigenic variation HIV-1 Antigenic_variation > In viruses > HIV-1 Although the CTL response in the acute phase is directed against limited number of epitopes, the epitopic repertoire increases with time due to viral escape. Additionally amino acid co-evolution is a challenging issue that needs to be addressed. For example, a substitution in a particular site results in a secondary or compensatory mutation in another site.
B+ tree Intervals in internal nodes B+_tree > Structure > Intervals in internal nodes Internal nodes higher in the tree can then construct their own intervals, which recursively aggregate the intervals contained in their own child internal nodes. Eventually, the root of a B+ Tree represents the whole range of values in the tree, where every internal node represents a subinterval. For this recursive interval information to be retained, internal nodes must additionally contain m − 1 {\displaystyle m-1} copies of keys l i {\displaystyle l_{i}} for i ∈ {\displaystyle i\in } representing the least element within the interval covered by the child with index i (which may itself be an internal node, or a leaf).
Glossary of electrical and electronics engineering O Glossary_of_electrical_and_electronics_engineering > O one-line diagram A simplified schematic diagram of a power system. on-premises wiring Telecommunications wiring owned by the customer. open-circuit test A test, of a transformer or other device, with no load connected.
Procedural justice The participation model Procedural_justice > Evaluating the fairness of different procedural systems > The participation model The idea of the participation model is that a fair procedure is one that affords those who are affected by an opportunity to participate in the making of the decision. In the context of a trial, for example, the participation model would require that the defendant be afforded an opportunity to be present at the trial, to put on evidence, cross examination witnesses, and so forth.
Alternative periodic tables Spatial Types_of_periodic_tables > Typology > Spatial This can aid in uncovering hidden connections and correlations among the elements. Flexibility and customization: Periodic tables of three or more dimensions offer flexibility in terms of their design and customization. Researchers, educators, or scientists can adapt the dimensions and properties represented based on their specific needs and objectives.
Hydrodynamic stability Laboratory and computational experiments Hydrodynamic_instability > Analysing flow stability > Laboratory and computational experiments Laboratory experiments are a very useful way of gaining information about a given flow without having to use more complex mathematical techniques. Sometimes physically seeing the change in the flow over time is just as useful as a numerical approach and any findings from these experiments can be related back to the underlying theory. Experimental analysis is also useful because it allows one to vary the governing parameters very easily and their effects will be visible. When dealing with more complicated mathematical theories such as Bifurcation theory and Weakly nonlinear theory, numerically solving such problems becomes very difficult and time-consuming but with the help of computers this process becomes much easier and quicker. Since the 1980s computational analysis has become more and more useful, the improvement of algorithms which can solve the governing equations, such as the Navier–Stokes equation, means that they can be integrated more accurately for various types of flow.
Inorganic polymer Transition-metal-containing polymers Inorganic_polymer > Transition-metal-containing polymers Inorganic polymers also include materials with transition metals in the backbone. Examples are Polyferrocenes, Krogmann's salt and Magnus's green salt.
Glucose oxidation reaction Anoxic regeneration of NAD+ Glycolysis > Post-glycolysis processes > Anoxic regeneration of NAD+ One method of doing this is to simply have the pyruvate do the oxidation; in this process, pyruvate is converted to lactate (the conjugate base of lactic acid) in a process called lactic acid fermentation: Pyruvate + NADH + H+ → lactate + NAD+This process occurs in the bacteria involved in making yogurt (the lactic acid causes the milk to curdle). This process also occurs in animals under hypoxic (or partially anaerobic) conditions, found, for example, in overworked muscles that are starved of oxygen. In many tissues, this is a cellular last resort for energy; most animal tissue cannot tolerate anaerobic conditions for an extended period of time. Some organisms, such as yeast, convert NADH back to NAD+ in a process called ethanol fermentation.
Quantum radar Concept behind a microwave-range model Quantum_radar > Concept behind a microwave-range model A microwave-range model of a quantum radar was proposed in 2015 by an international team and is based on the protocol of Gaussian quantum illumination. The basic concept is to create a stream of entangled visible-frequency photons and split it in half. One half, the "signal beam", goes through a conversion to microwave frequencies in a way that preserves the original quantum state. The microwave signal is then sent and received as in a normal radar system.
Vibrational transition Summary Vibrational_transition Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state. Simultaneous excitation of a vibration and rotations gives rise to vibration–rotation spectra.
Long non coding RNA In aging and disease Long_noncoding_RNA > In aging and disease Likewise, genome-wide association studies identified a region associated with coronary artery disease that encompassed a long ncRNA, ANRIL. ANRIL is expressed in tissues and cell types affected by atherosclerosis and its altered expression is associated with a high-risk haplotype for coronary artery disease.The complexity of the transcriptome, and our evolving understanding of its structure may inform a reinterpretation of the functional basis for many natural polymorphisms associated with disease states. Many SNPs associated with certain disease conditions are found within non-coding regions and the complex networks of non-coding transcription within these regions make it particularly difficult to elucidate the functional effects of polymorphisms.
Ion Optics Cylinder lens Ion_Optics > Cylinder lens Three cylinder lenses achieve the change of the magnification while holding the object and image positions because there are two gaps that work as lenses. Although the voltages have to change depending on the electron kinetic energy, the voltage ratio is kept constant when the optical parameters are not changed. While a charged particle is in an electric field force acts upon it.
Locality-Sensitive Hashing Summary Locality-preserving_hashing It differs from conventional hashing techniques in that hash collisions are maximized, not minimized. Alternatively, the technique can be seen as a way to reduce the dimensionality of high-dimensional data; high-dimensional input items can be reduced to low-dimensional versions while preserving relative distances between items. Hashing-based approximate nearest-neighbor search algorithms generally use one of two main categories of hashing methods: either data-independent methods, such as locality-sensitive hashing (LSH); or data-dependent methods, such as locality-preserving hashing (LPH).Locality-preserving hashing was initially devised as a way to facilitate data pipelining in implementations of massively parallel algorithms that use randomized routing and universal hashing to reduce memory contention and network congestion.
Specific conductance Complex resistivity and conductivity Electrically_conductive > Complex resistivity and conductivity When analyzing the response of materials to alternating electric fields (dielectric spectroscopy), in applications such as electrical impedance tomography, it is convenient to replace resistivity with a complex quantity called impedivity (in analogy to electrical impedance). Impedivity is the sum of a real component, the resistivity, and an imaginary component, the reactivity (in analogy to reactance). The magnitude of impedivity is the square root of sum of squares of magnitudes of resistivity and reactivity. Conversely, in such cases the conductivity must be expressed as a complex number (or even as a matrix of complex numbers, in the case of anisotropic materials) called the admittivity.
Mathematical research Areas of mathematics Fields_of_mathematics > Areas of mathematics Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes. Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields differential calculus and integral calculus, is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables.
Straight-pull rifles Bolt knob Bolt_action > Operating the bolt > Bolt knob The bolt knob is the part of the bolt handle that the user grips when loading and reloading the firearm and thereby acts as a cocking handle. On many older firearms, the bolt knob is welded to the bolt handle, and as such becoming an integral part of the bolt handle itself. On many newer firearms, the bolt knob is instead threaded onto the handle, allowing the user to change the original bolt knob for an aftermarket one, either for aesthetical reasons, achieving better grip or similar. The type of threads used vary between firearms.
Rainer Blatt Research Rainer_Blatt > Research Experimental physicist Rainer Blatt has carried out trail-blazing experiments in the fields of precision spectroscopy, quantum metrology and quantum information processing. He works with atoms caught in ion traps which he manipulates using laser beams. This work is based on suggestions made in the mid-1990s by theorists Ignacio Cirac and Peter Zoller. In 2004, using their suggested set-up, Blatt’s working group, at the same time as a team at the National Institute of Standards and Technology in Boulder, Colorado, USA, succeeded for the first time in transferring the quantum information of one atom in a totally controlled manner onto another atom (teleportation).
Biological data Applications of Deep Learning to Biological Data Biological_data > Applications of Deep Learning to Biological Data From that point onwards, different analyses may be performed, such as GE profiling splicing junction prediction, and protein-protein interaction evaluation may all be performed.Reinforcement learning, a term stemming from behavioral psychology, is a method of problem solving by learning things through trial and error. Reinforcement learning can be applied to biological data, in the field of omics, by using RL to predict bacterial genomes.Other studies have shown that reinforcement learning can be used to accurately predict biological sequence annotation.Deep Learning (DL) architectures are also useful in training biological data. For instance, DL architectures that target pixel levels of biological images have been used to identify the process of mitosis in histological images of the breast. DL architectures have also been used to identify nuclei in images of breast cancer cells.
Markov Decision Process Reinforcement learning Policy_iteration > Extensions and generalizations > Reinforcement learning Reinforcement learning uses MDPs where the probabilities or rewards are unknown.For this purpose it is useful to define a further function, which corresponds to taking the action a {\displaystyle a} and then continuing optimally (or according to whatever policy one currently has): Q ( s , a ) = ∑ s ′ P a ( s , s ′ ) ( R a ( s , s ′ ) + γ V ( s ′ ) ) . {\displaystyle \ Q(s,a)=\sum _{s'}P_{a}(s,s')(R_{a}(s,s')+\gamma V(s')).\ } While this function is also unknown, experience during learning is based on ( s , a ) {\displaystyle (s,a)} pairs (together with the outcome s ′ {\displaystyle s'} ; that is, "I was in state s {\displaystyle s} and I tried doing a {\displaystyle a} and s ′ {\displaystyle s'} happened"). Thus, one has an array Q {\displaystyle Q} and uses experience to update it directly. This is known as Q-learning.
Chemoprophylaxis Summary Chemoprophylaxis Chemoprevention or chemoprophylaxis refers to the administration of a medication for the purpose of preventing disease or infection. Antibiotics, for example, may be administered to patients with disorders of immune system function to prevent bacterial infections (particularly opportunistic infection). Antibiotics may also be administered to healthy individuals to limit the spread of an epidemic, or to patients who have repeated infections (such as urinary tract infections) to prevent recurrence. It may also refer to the administration of heparin to prevent deep venous thrombosis in hospitalized patients.
Dynamic Programming Control theory Dynamic_Programming > Overview > Control theory In practice, this generally requires numerical techniques for some discrete approximation to the exact optimization relationship. Alternatively, the continuous process can be approximated by a discrete system, which leads to a following recurrence relation analog to the Hamilton–Jacobi–Bellman equation: J k ∗ ( x n − k ) = min u n − k { f ^ ( x n − k , u n − k ) + J k − 1 ∗ ( g ^ ( x n − k , u n − k ) ) } {\displaystyle J_{k}^{\ast }\left(\mathbf {x} _{n-k}\right)=\min _{\mathbf {u} _{n-k}}\left\{{\hat {f}}\left(\mathbf {x} _{n-k},\mathbf {u} _{n-k}\right)+J_{k-1}^{\ast }\left({\hat {\mathbf {g} }}\left(\mathbf {x} _{n-k},\mathbf {u} _{n-k}\right)\right)\right\}} at the k {\displaystyle k} -th stage of n {\displaystyle n} equally spaced discrete time intervals, and where f ^ {\displaystyle {\hat {f}}} and g ^ {\displaystyle {\hat {\mathbf {g} }}} denote discrete approximations to f {\displaystyle f} and g {\displaystyle \mathbf {g} } . This functional equation is known as the Bellman equation, which can be solved for an exact solution of the discrete approximation of the optimization equation.

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