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Graves' ophthalmopathy
With enlargement of the extraocular muscle at the orbital apex, the optic nerve is at risk of compression. The orbital fat or the stretching of the nerve due to increased orbital volume may also lead to optic nerve damage. The patient experiences a loss of visual acuity, visual field defect, afferent pupillary defect, ...
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Ornithine carbamoyltransferase
In milder (or partial) urea cycle enzyme deficiencies, ammonia accumulation may be triggered by illness or stress at almost any time of life, resulting in multiple mild elevations of plasma ammonia concentration . Patients with partial enzyme deficiencies may have a delay of symptoms for months or years. Indicators tha...
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Full operational capability
In military acquisition, full operating capability or full operational capability (FOC) is the completion of a development effort. This is usually preceded by an initial operating capability or initial operational capability (IOC) phase. For the United States Department of Defense military acquisition FOC is defined as...
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Tulip Period architecture
In military architecture, the Niš Fortress, located in what was a strategic city for the Ottoman Balkan provinces, dates in its current form to 1719–1723. It was built after the 1716-1718 war in which the Ottomans lost Belgrade and other territories. The Ottoman fortifications here are similar to contemporary European ...
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Area bombardment
"Carpet bombing", also known as "saturation bombing", and "obliteration bombing", refers to a type of area bombing that aims to effect complete destruction of the target area by exploding bombs in every part of it. Area bombing is contrasted with precision bombing. The latter is directed at a selected target – not nece...
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Supply train
In military contexts, a train is the logistical transport elements accompanying a military force. Often called a supply train or baggage train, it has the job of providing materiel for their associated combat forces when in the field. When focused on provision of field artillery and its ammunition, it may be termed an ...
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Supply train
These military terms predate, and do not imply a railway train, though railways are often employed for modern logistics, and can include armoured trains. For armies, this historically usually referred to forces employing wagons, horses, mules, oxen, camels, or even elephants. These can still be useful where difficult w...
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Ditch (fortification)
In military engineering, a ditch is an obstacle designed to slow down or break up an attacking force, while a trench is intended to provide cover to the defenders. In military fortifications the side of a ditch farthest from the enemy and closest to the next line of defence is known as the scarp while the side of a dit...
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Third generation warfare
In military history, the term "generations of warfare" refers to the concept of five "generations" in warfare, with each generation having different tactics, strategies, and technologies. The generations of warfare are sometimes dubbed as "4GW" or "5GW". The term originated in 1989 to describe "the changing face of war...
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Fuze
In military munitions, a fuze (sometimes fuse) is the part of the device that initiates function. In some applications, such as torpedoes, a fuze may be identified by function as the exploder. The relative complexity of even the earliest fuze designs can be seen in cutaway diagrams.
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Fuze
A fuze is a device that detonates a munition's explosive material under specified conditions. In addition, a fuze will have safety and arming mechanisms that protect users from premature or accidental detonation. For example, an artillery fuze's battery is activated by the high acceleration of cannon launch, and the fu...
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Face shield
In military or law enforcement environments, a face shield may be designed for ballistic or non-ballistic protection. The non-ballistic shield will provide no protection from projectiles shot from firearms, but is usually designed to withstand low velocity impacts, like caused by punches or thrown objects.A ballistic f...
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AN/PRC-127
In military radio communications, the AN/PRC-127 (RT-1594) Radio is a handheld transceiver operating in the 136-160 MHz range used by the US Army to aid in provisioning services, and other non-combat uses. It is a software modified version of the civilian LPI series high band Bendix-King radio. The PRC-127 radio was in...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Compound (fortification)
In military science, a compound is a type of fortification made up of walls or fences surrounding several buildings in the center of a large piece of land. The walls can either serve the purpose of being tall, thick, and impenetrable, in which case they would be made of wood, stone, or some other like substance; or dan...
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Compound (fortification)
A number of survivalists own fortified compound-like structures as a means of protection in case civilization breaks down or their government becomes abusive. The term compound is also used to refer to an unfortified enclosure, especially in Africa and Asia. See compound (enclosure).
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Force multiplication
In military science, force multiplication or a force multiplier is a factor or a combination of factors that gives personnel or weapons (or other hardware) the ability to accomplish greater feats than without it. The expected size increase required to have the same effectiveness without that advantage is the multiplica...
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Logistics Management
In military science, maintaining one's supply lines while disrupting those of the enemy is a crucial—some would say the most crucial—element of military strategy, since an armed force without resources and transportation is defenseless. The historical leaders Hannibal, Alexander the Great, and the Duke of Wellington ar...
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Logistics Management
For instance, peacetime consumption of ammunition and fuel will be considerably lower than wartime consumption of these items, whereas other classes of supply such as subsistence and clothing have a relatively consistent consumption rate regardless of war or peace. Some classes of supply have a linear demand relationsh...
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Logistics Management
As equipment ages, more and more repair parts are needed over time, even when usage and quantity stay consistent. By recording and analyzing these trends over time and applying them to future scenarios, the US Armed Forces can accurately supply troops with the items necessary at the precise moment they are needed. Hist...
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Suppression fire
In military science, suppressive fire is "fire that degrades the performance of an enemy force below the level needed to fulfill its mission". When used to protect exposed friendly troops advancing on the battlefield, it is commonly called covering fire. Suppression is usually only effective for the duration of the fir...
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.30-06 Springfield
In military service, the .30-06 was used in the bolt-action M1903 Springfield rifle, the bolt-action M1917 Enfield rifle, the semi-automatic M1 Garand rifle, the semi-automatic M1941 Johnson rifle, the Famage Mauser, the Browning Automatic Rifle (BAR), and numerous machine guns, including the M1917 and M1919 series. It...
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.30-06 Springfield
All models of Gatling guns were declared obsolete by the U.S. Army in 1911, after 45 years of service.With "hot" handloads and a rifle capable of handling them, the .30-06 is capable of performance rivaling many magnum cartridges. However, when loaded more closely to the original government specs, .30-06 remains within...
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Turkey shoot
In military situations, a turkey shoot occurs when a group or team catch the enemy off-guard or outgunned to the point of the battle being extremely lopsided, as in the following famous examples: Battle of New Orleans – War of 1812 Battle of San Jacinto – Texas Revolution Charge of the Light Brigade – Crimean War Battl...
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Choke point
In military strategy, a choke point (or chokepoint) is a geographical feature on land such as a valley, defile or bridge, or maritime passage through a critical waterway such as a strait, which an armed force is forced to pass through in order to reach its objective, sometimes on a substantially narrowed front and ther...
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Swedish nuclear weapons program
In military studies concerning the deployment of nuclear weapons, Swedish plans focused on bombs dropped by ground-attack aircraft. The Swedes planned to have a relatively small number of tactical nuclear weapons; the Saab 36 bomber was supposed to be able to carry an 800 kg free falling nuclear weapon, but development...
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Swedish nuclear weapons program
Thus, even if the A36 was not designed to be an integral part of the nuclear weapons program itself, it was constructed specifically as a nuclear weapons carrier. A36 pilots were to specialize in this task.As for Robot 08, the previously classified documents provide no support for the claim that something other than ae...
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Flanking maneuver
In military tactics, a flanking maneuver is a movement of an armed force around an enemy force's side, or flank, to achieve an advantageous position over it. Flanking is useful because a force's fighting strength is typically concentrated in its front, therefore, to circumvent an opposing force's front and attack its f...
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Ground attack
In military tactics, close air support (CAS) is defined as aerial warfare actions—often air-to-ground actions such as strafes or airstrikes—by military aircraft against hostile targets in close proximity to friendly forces. A form of fire support, CAS requires detailed integration of each air mission with fire and move...
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Kill zone
In military tactics, the kill zone, also known as killing zone, is an area entirely covered by direct and effective fire, an element of ambush within which an approaching enemy force is trapped and destroyed. The objective of the ambush force is to quickly kill or capture all enemy soldiers inside the kill zone. The tr...
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Electronic Support Measures
In military telecommunications, electronic support (ES) or electronic support measures (ESM) gather intelligence through passive "listening" to electromagnetic radiations of military interest. They are an aspect of electronic warfare involving actions taken under direct control of an operational commander to detect, in...
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Electronic Support Measures
ESM collection platforms can remain electronically silent and detect and analyze RADAR transmissions beyond the RADAR detection range because of the greater power of the transmitted electromagnetic pulse with respect to a reflected echo of that pulse. United States airborne ESM receivers are designated in the AN/ALR se...
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Demolition belt
In military terminology, a demolition belt is a selected land area sown with explosive charges, mines, and other available obstacles to deny use of the land to enemy operations, and as a protection to friendly troops.There are two types of demolition belt: A primary demolition belt is a continuous series of obstacles a...
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Fingerspitzengefühl
In military terminology, it is used for the stated ability of some military commanders, such as Field-Marshal Erwin Rommel, to describe "the instinctive and immediate response to battle situations", a quality needed to maintain, with great accuracy and attention to detail, an ever-changing operational and tactical situ...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Artillery barrage
In military usage, a barrage is massed sustained artillery fire (shelling) aimed at a series of points along a line. In addition to attacking any enemy in the kill zone, a barrage intends to suppress enemy movements and deny access across that line of barrage. The impact points along the line may be 20–30 yards/meters ...
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Artillery barrage
They may target along the front line, or further into enemy back area to isolate certain enemy positions (such as a box barrage). A series of different patterns may be employed as a battle develops, with each barrage lasting only a few minutes or many hours. Barrages are usually integral with larger operations of multi...
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Artillery barrage
The barrage was developed by the British Army in the Second Boer War. It came to prominence in World War I, notably its use by the British Expeditionary Force and particularly from late 1915 onwards when the British realized that the suppressive effects of artillery to provide covering fire were the key to breaking int...
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Pratt & Whitney J75/JT4
In military use, the J75 was used on the Convair F-106 Delta Dart, Lockheed U-2, and Republic F-105 Thunderchief. It was also utilized in the prototype and experimental Avro Canada CF-105 Arrow, Lockheed A-12, Martin P6M SeaMaster, North American YF-107, and Vought XF8U-3 Crusader III. Before the arrival of the Pratt &...
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High Frequency Vibrating Equipment
In mineral processing, such as metals ore (e.g. iron, tin, tungsten, tantalum etc.) and nonferrous metals ores (e.g. lead, zinc, gold, silver and industrial sand etc.), high frequency screens have a crucial role. After the ores get comminuted, high frequency screens are used as a classifier which selects materials size...
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High Frequency Vibrating Equipment
Secondly, the fine-grained material will be unloaded in a timely manner, avoiding over-crushing caused by re-grinding. The benefits of using high frequency screens in mineral processing can meet the requirement of fineness easily for recovery and is able to achieve a smaller size separation, reducing capacity needed fo...
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Amorphous carbon
In mineralogy, amorphous carbon is the name used for coal, carbide-derived carbon, and other impure forms of carbon that are neither graphite nor diamond. In a crystallographic sense, however, the materials are not truly amorphous but rather polycrystalline materials of graphite or diamond within an amorphous carbon ma...
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Epitaxial growth
In mineralogy, epitaxy is the overgrowth of one mineral on another in an orderly way, such that certain crystal directions of the two minerals are aligned. This occurs when some planes in the lattices of the overgrowth and the substrate have similar spacings between atoms.If the crystals of both minerals are well forme...
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Silicate minerals
In mineralogy, silicate minerals are classified into seven major groups according to the structure of their silicate anion: Tectosilicates can only have additional cations if some of the silicon is replaced by an atom of lower valence such as aluminum. Al for Si substitution is common.
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Minimal model (physics)
In minimal models, the central charge of the Virasoro algebra takes values of the type c p , q = 1 − 6 ( p − q ) 2 p q . {\displaystyle c_{p,q}=1-6{(p-q)^{2} \over pq}\ .} where p , q {\displaystyle p,q} are coprime integers such that p , q ≥ 2 {\displaystyle p,q\geq 2} . Then the conformal dimensions of degenerate rep...
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Minimal model (physics)
{\displaystyle h_{r,s}=h_{q-r,p-s}=h_{r+q,s+p}\ .} The spectrums of minimal models are made of irreducible, degenerate lowest-weight representations of the Virasoro algebra, whose conformal dimensions are of the type h r , s {\displaystyle h_{r,s}} with 1 ≤ r ≤ q − 1 , 1 ≤ s ≤ p − 1 .
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Minimum deviation
In minimum deviation, the refracted ray in the prism is parallel to its base. In other words, the light ray is symmetrical about the axis of symmetry of the prism. Also, the angles of refractions are equal i.e. r1 = r2.
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Minimum deviation
And, the angle of incidence and angle of emergence equal each other (i = e). This is clearly visible in the graph below. The formula for minimum deviation can be derived by exploiting the geometry in the prism.
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Potassium dicyanoaurate
In mining of gold from dilute sources, gold is selectively extracted by dissolution in aqueous solutions of cyanide, provided by dissolving sodium cyanide, potassium cyanide and/or calcium cyanide. The reaction for the dissolution of gold, the "Elsner Equation", is: 4 Au + 8 KCN + O2 + 2 H2O → 4 K + 4 KOHIn this proces...
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Tailings
In mining, tailings or tails are the materials left over after the process of separating the valuable fraction from the uneconomic fraction (gangue) of an ore. Tailings are different from overburden, which is the waste rock or other material that overlies an ore or mineral body and is displaced during mining without be...
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Investment analysis
In mining, valuation is the process of determining the value or worth of a mining property - i.e. as distinct from a listed mining corporate. Mining valuations are sometimes required for IPOs, fairness opinions, litigation, mergers and acquisitions, and shareholder-related matters. In valuing a mining project or mining...
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Investment analysis
"CIMVal" generally applied by the Toronto Stock Exchange, is widely recognised as a "standard" for the valuation of mining projects. (CIMVal: Canadian Institute of Mining, Metallurgy and Petroleum on Valuation of Mineral Properties ) The Australasian equivalent is VALMIN; the Southern African is SAMVAL. These standards...
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Investment analysis
The price of a "Junior" mining stock, typically having one asset, will at its early stages be linked to the result of its feasibility study; later, the price will be a function of that mine's viability and value, largely applying the above techniques. A "Major", on the other hand, has numerous properties, and the conte...
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Low-energy electron microscopy
In mirror electron microscopy, electrons are slowed in the retarding field of the condenser lens to the limit of the instrument and thus, only allowed to interact with the “near-surface” region of the sample. It is very complicated to understand the exact contrast variations come from, but the important things to point...
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Quantum Cryptography
Examples of tasks in mistrustful cryptography are commitment schemes and secure computations, the latter including the further examples of coin flipping and oblivious transfer. Key distribution does not belong to the area of mistrustful cryptography. Mistrustful quantum cryptography studies the area of mistrustful cryp...
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Quantum Cryptography
In contrast to quantum key distribution where unconditional security can be achieved based only on the laws of quantum physics, in the case of various tasks in mistrustful cryptography there are no-go theorems showing that it is impossible to achieve unconditionally secure protocols based only on the laws of quantum ph...
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Tic-tac-toe variants
In misère tic-tac-toe, the player wins if the opponent gets n in a row. This game is also known as avoidance tic tac toe, toe-tac-tic, inverse tic tac toe, or reverse tic tac toe. A 3×3 game is a draw. More generally, the first player can draw or win on any board (of any dimension) whose side length is odd, by playing ...
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DPH Cytopathy
In mitochondrial ‘NTBI glycolysis cytopathy’ (working diagnosis, abbreviation 'NG cytopathy'), markedly impaired glucose metabolism (glycolysis) is synergistically associated with H63D syndrome in the form of mitochondrial dysfunction that is not yet fully understood. The clinically observable patterns and values rule ...
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DCTN6
In mitosis, unlike dynactin or dynein perturbation that causes mitotic spindle disarrangement and mitotic arrest, dynactin p27/p25 depletion does not affect mitotic spindle formation, pole focusing or dynein/dynactin targeting to kinetochores. However, dynactin p27/p25 are required for normal chromosome alignment, kine...
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End-product inhibition
In mixed inhibition the inhibitor may bind to the enzyme whether or not the substrate has already bound. Hence mixed inhibition is a combination of competitive and noncompetitive inhibition. Furthermore, the affinity of the inhibitor for the free enzyme and the enzyme-substrate complex may differ. : 136–139 By increasi...
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End-product inhibition
: 381–382 Although it is possible for mixed-type inhibitors to bind in the active site, this type of inhibition generally results from an allosteric effect where the inhibitor binds to a different site on an enzyme. Inhibitor binding to this allosteric site changes the conformation (that is, the tertiary structure or t...
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Giant-cell carcinoma of the lung
In mixed tumors, giant cells are more likely to be found in higher proportions at the edge of a tumor. When extensive necrosis is present, it is possible for a giant-cell tumor to have only a thin rim of viable cells remaining at the perimeter of the mass.In one early case series, abundant production of loose malignant...
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Zeotropic mixture
Unlike azeotropic mixtures, there is no azeotropic point at any temperature on the diagram where the bubble line and dew lines would intersect. Thus, the composition of the mixture will always change between the bubble and dew point component fractions upon boiling from a liquid to a gas until the mass fraction of a co...
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Mobile application part
In mobile cellular telephony networks like GSM and UMTS the SS7 application MAP is used. Voice connections are Circuit Switched (CS) and data connections are Packet Switched (PS) applications. Some of the GSM/UMTS Circuit Switched interfaces in the Mobile Switching Center (MSC) transported over SS7 include the followin...
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Handshake (computing)
In mobile device chargers offering special quick-charge abilities to supported devices, the charging process will switch up to a higher output voltage for increased power transfer. But this could cause serious damage to an unsupported device or even result in a fire. It is therefore very important for the device and ch...
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PDA Phone
In mobile phones released since the second half of the 2010s, operational life span commonly is limited by built-in batteries which are not designed to be interchangeable. The life expectancy of batteries depends on usage intensity of the powered device, where activity (longer usage) and tasks demanding more energy exp...
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Inter-cell interference coordination
In mobile telecommunications, inter-cell interference coordination (ICIC) techniques apply restrictions to the radio resource management (RRM) block, improving favorable channel conditions across subsets of users that are severely impacted by the interference, and thus attaining high spectral efficiency. This coordinat...
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GSM 03.38
In mobile telephony GSM 03.38 or 3GPP 23.038 is a character encoding used in GSM networks for SMS (Short Message Service), CB (Cell Broadcast) and USSD (Unstructured Supplementary Service Data). The 3GPP TS 23.038 standard (originally GSM recommendation 03.38) defines GSM 7-bit default alphabet which is mandatory for G...
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Combat stress reaction
In mobile war requiring rapid and frequent movement, treatment of many combat stress cases takes place at various battalion or regimental headquarters or logistical units, on light duty, rather than in medical units, whenever possible. This is a key factor and another area where the small-unit leader helps in the treat...
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Combat stress reaction
A service member experiencing combat stress reaction is having a crisis, and there are two basic elements to that crisis working in opposite directions. On the one hand, the service member is driven by a strong desire to seek safety and to get out of an intolerable environment. On the other hand, the service member doe...
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Combat stress reaction
They want to return to their unit. If a service member starts to lose contact with their unit when he enters treatment, the impulse to get out of the war and return to safety takes over. They feel that they've failed their comrades who have already rejected them as unworthy.
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Combat stress reaction
The potential is for the service member to become more and more emotionally invested in keeping their symptoms so they can stay in a safe environment. Much of this is done outside the service member's conscious awareness, but the result is the same. The more out of touch the service member is with their unit, the less ...
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Random waypoint model
In mobility management, the random waypoint model is a random model for the movement of mobile users, and how their location, velocity and acceleration change over time. Mobility models are used for simulation purposes when new network protocols are evaluated. The random waypoint model was first proposed by Johnson and...
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Random waypoint model
In random-based mobility simulation models, the mobile nodes move randomly and freely without restrictions. To be more specific, the destination, speed and direction are all chosen randomly and independently of other nodes. This kind of model has been used in many simulation studies. Two variants, the random walk model...
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Sahlqvist formula
In modal logic, Sahlqvist formulas are a certain kind of modal formula with remarkable properties. The Sahlqvist correspondence theorem states that every Sahlqvist formula is canonical, and corresponds to a first-order definable class of Kripke frames. Sahlqvist's definition characterizes a decidable set of modal formu...
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Classical modal logic
In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality of the modal operators ◊ A ↔ ¬ ◻ ¬ A {\displaystyle \Diamond A\leftrightarrow \lnot \Box \lnot A} that is also closed under the rule A ↔ B ◻ A ↔ ◻ B . {\displaystyle {\frac {A\leftrightarrow B}{\Box A\leftrightarro...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Regular modal logic
In modal logic, a regular modal logic is a modal logic containing (as axiom or theorem) the duality of the modal operators: ◊ A ↔ ¬ ◻ ¬ A {\displaystyle \Diamond A\leftrightarrow \lnot \Box \lnot A} and closed under the rule ( A ∧ B ) → C ( ◻ A ∧ ◻ B ) → ◻ C . {\displaystyle {\frac {(A\land B)\to C}{(\Box A\land \Box B...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Modal collapse
For example, Gödel's ontological proof contains ϕ → ◻ ϕ {\displaystyle \phi \rightarrow \Box \phi } as a theorem, which combined with the axioms of system S5 leads to modal collapse. Since some regard divine freedom as essential to the nature of God, and modal collapse as negating the concept of free will, this then le...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Standard translation
In modal logic, standard translation is a logic translation that transforms formulas of modal logic into formulas of first-order logic which capture the meaning of the modal formulas. Standard translation is defined inductively on the structure of the formula. In short, atomic formulas are mapped onto unary predicates ...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Modal depth
In modal logic, the modal depth of a formula is the deepest nesting of modal operators (commonly ◻ {\displaystyle \Box } and ◊ {\displaystyle \Diamond } ). Modal formulas without modal operators have a modal depth of zero.
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Linear time property
Temporal logics such as linear temporal logic describe types of linear time properties using formulae. This article is about propositional linear-time properties and cannot handle predicates about program states, so it cannot define a property like: the current value of y determines the number of times that x toggles b...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Zone (convex polytope)
In model checking, a field of computer science, a difference bound matrix (DBM) is a data structure used to represent some convex polytopes called zones. This structure can be used to efficiently implement some geometrical operations over zones, such as testing emptyness, inclusion, equality, and computing the intersec...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Region (model checking)
The regions are defined such that if two vectors x → {\displaystyle {\vec {x}}} and x → ′ {\displaystyle {\vec {x}}'} belong to the same region, then they satisfy the same constraints of K {\displaystyle K} . Furthermore, when those vectors are considered as a tuple of clocks, both vectors have the same set of possible...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Timed propositional temporal logic
In model checking, a field of computer science, timed propositional temporal logic (TPTL) is an extension of propositional linear temporal logic (LTL) in which variables are introduced to measure times between two events. For example, while LTL allows to state that each event p is eventually followed by an event q, TPT...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Signal (model checking)
In model checking, a subfield of computer science, a signal or timed state sequence is an extension of the notion of words in a formal language, in which letters are continuously emitted. While a word is traditionally defined as a function from a set of non-negative integers to letters, a signal is a function from a se...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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State transition system
In model checking, a transition system is sometimes defined to include an additional labeling function for the states as well, resulting in a notion that encompasses that of Kripke structure.Action languages are extensions of transition systems, adding a set of fluents F, a set of values V, and a function that maps F ×...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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The fundamental theorem of ultraproducts
In model theory and set theory, the direct limit of a sequence of ultrapowers is often considered. In model theory, this construction can be referred to as an ultralimit or limiting ultrapower. Beginning with a structure, A 0 {\displaystyle A_{0}} and an ultrafilter, D 0 , {\displaystyle {\mathcal {D}}_{0},} form an ul...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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The fundamental theorem of ultraproducts
Then repeat the process to form A 2 , {\displaystyle A_{2},} and so forth. For each n {\displaystyle n} there is a canonical diagonal embedding A n → A n + 1 .
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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The fundamental theorem of ultraproducts
{\displaystyle A_{n}\to A_{n+1}.} At limit stages, such as A ω , {\displaystyle A_{\omega },} form the direct limit of earlier stages. One may continue into the transfinite.
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Hereditary property
In model theory and universal algebra, a class K of structures of a given signature is said to have the hereditary property if every substructure of a structure in K is again in K. A variant of this definition is used in connection with Fraïssé's theorem: A class K of finitely generated structures has the hereditary pr...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Tarski's exponential function problem
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable.
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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U-rank
In model theory, a branch of mathematical logic, U-rank is one measure of the complexity of a (complete) type, in the context of stable theories. As usual, higher U-rank indicates less restriction, and the existence of a U-rank for all types over all sets is equivalent to an important model-theoretic condition: in this...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Reduced product
In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let {Si | i ∈ I} be a family of structures of the same signature σ indexed by a set I, and let U be a filter on I. The domain of the reduced product is the quotie...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Reduced product
Relations are interpreted by R ( ( a i 1 ) / ∼ , … , ( a i n ) / ∼ ) ⟺ { i ∈ I ∣ R S i ( a i 1 , … , a i n ) } ∈ U . {\displaystyle R((a_{i}^{1})/{\sim },\dots ,(a_{i}^{n})/{\sim })\iff \{i\in I\mid R^{S_{i}}(a_{i}^{1},\dots ,a_{i}^{n})\}\in U.} For example, if each structure is a vector space, then the reduced product...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Hrushovski construction
In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure ≤ {\displaystyle \leq } rather than ⊆ {\displaystyle \subseteq } . It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Spectrum of a theory
In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities. More precisely, for any complete theory T in a language we write I(T, κ) for the number of models of T (up to isomorphism) of cardinality κ. The spectrum problem is...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Field (mathematics)
Moreover, any fixed statement φ holds in C if and only if it holds in any algebraically closed field of sufficiently high characteristic.If U is an ultrafilter on a set I, and Fi is a field for every i in I, the ultraproduct of the Fi with respect to U is a field. It is denoted by ulimi→∞ Fi,since it behaves in several...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Field (mathematics)
Applied to the above sentence φ, this shows that there is an isomorphism ulim p → ∞ ⁡ F ¯ p ≅ C . {\displaystyle \operatorname {ulim} _{p\to \infty }{\overline {\mathbf {F} }}_{p}\cong \mathbf {C} .} The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Elementary substructure
In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences. If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-...
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Ehrenfeucht–Mostowski theorem
In model theory, a field within mathematical logic, the Ehrenfeucht–Mostowski theorem (Ehrenfeucht & Mostowski 1956) gives conditions for the existence of a model with indiscernibles.
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Model-complete theory
In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson.
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus
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Atomic model (mathematical logic)
In model theory, a subfield of mathematical logic, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula. Such types are called principal types, and the formulas that axiomatize them are called complete formulas.
https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus