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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 first in the central cell and then mirroring the opponent's moves. | https://en.wikipedia.org/wiki/Tic-tac-toe_variants |
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 out mere correlation or even coincidence as an explanation. The high complexity of this pathology of energy metabolism and cellular function and impressively indicates how disturbances in one system can have far-reaching effects on others. | https://en.wikipedia.org/wiki/DPH_Cytopathy |
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, kinetochore-microtubule interaction, and proper timing of anaphase onset. Dynactin p27 C-terminal T186 residue is phosphorylated by cyclin-dependent kinase 1 (Cdk1) in mitosis and helps target polo-like kinase 1 (Plk1) to kinetochores during prometaphase. This activity facilitates phosphorylation of important downstream kinetochore targets (such as tension-sensing 3F3/2 phospho-epitope) of Plk1, which is important for recruitment of spindle assembly checkpoint proteins such as Mad1 and proper kinetochore-microtubule attachment. == References == | https://en.wikipedia.org/wiki/DCTN6 |
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 increasing concentrations of substrate , this type of inhibition can be reduced (due to the competitive contribution), but not entirely overcome (due to the noncompetitive component). | https://en.wikipedia.org/wiki/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 three-dimensional shape) of the enzyme so that the affinity of the substrate for the active site is reduced.These four types of inhibition can also be distinguished by the effect of increasing the substrate concentration on the degree of inhibition caused by a given amount of inhibitor. For competitive inhibition the degree of inhibition is reduced by increasing , for noncompetitive inhibition the degree of inhibition is unchanged, and for uncompetitive (also called anticompetitive) inhibition the degree of inhibition increases with . | https://en.wikipedia.org/wiki/End-product_inhibition |
In mixed martial arts and self-defense, cross-training refers to training in multiple martial arts or fighting systems to become proficient in all the phases of unarmed combat. This training is meant to overcome the shortcomings of one style by practicing another style which is strong in the appropriate area. A typical combination involves a striking-based art such as Muay Thai, combined with a grappling-based art such as wrestling and Brazilian Jiu-Jitsu. Many hybrid martial arts can be considered derivatives of such cross-training.Modern mixed martial-arts training generally involves cross-training in the different aspects and ranges of fighting. | https://en.wikipedia.org/wiki/Cross_training |
In mixed systems, such as the judiciary of Germany, a mixture of both judges and lay judges are triers of fact. | https://en.wikipedia.org/wiki/Trier_of_fact |
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 giant cells were noted to fill the alveoli of patients without destroying, infiltrating, or disturbing the normal underlying architecture, a pathologic behavior that bears some resemblance to the pneumonic variant of bronchioloalveolar carcinoma.Extensive tumor necrosis and hemorrhage is extremely common in GCCL.Although the issue has not been extensively studied in a controlled fashion, GCCLs have been noted to contain significantly elevated levels of VEGF. However, in one study where a giant-cell carcinoma tumor that had been completely excised was sectioned and examined, no qualitative or quantitative abnormalities in tissue vascularization were noted.GCCL have been noted to be encapsulated, and to be divided via septa into "pseudolobules", by a highly fibrous stroma, suggested to be produced commensurately with tumor growth. The capsule is typically infiltrated with malignant giant cells. | https://en.wikipedia.org/wiki/Giant-cell_carcinoma_of_the_lung |
In mixed winter tit flocks, seldom more than one or two marsh tits are present, and parties of this species alone are infrequent. Its performances in the bushes and branches are just as neat and agile as those of other tits; it often hangs upside down by one leg. | https://en.wikipedia.org/wiki/Marsh_tit |
In mixtures of substances, the bubble point is the saturated liquid temperature, whereas the saturated vapor temperature is called the dew point. Because the bubble and dew lines of a zeotropic mixture's temperature-composition diagram do not intersect, a zeotropic mixture in its liquid phase has a different fraction of a component than the gas phase of the mixture. On a temperature-composition diagram, after a mixture in its liquid phase is heated to the temperature at the bubble (boiling) curve, the fraction of a component in the mixture changes along an isothermal line connecting the dew curve to the boiling curve as the mixture boils. At any given temperature, the composition of the liquid is the composition at the bubble point, whereas the composition of the vapor is the composition at the dew point. | https://en.wikipedia.org/wiki/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 component reaches 1 (i.e. the zeotropic mixture is completely separated into its pure components). As shown in Figure 1, the mole fraction of component 1 decreases from 0.4 to around 0.15 as the liquid mixture boils to the gas phase. | https://en.wikipedia.org/wiki/Zeotropic_mixture |
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 following: B -> VLR (uses MAP/B). Most MSCs are associated with a Visitor Location Register (VLR), making the B interface "internal". C -> HLR (uses MAP/C) Messages between MSC to HLR handled by C Interface D -> HLR (uses MAP/D) for attaching to the CS network and location update E -> MSC (uses MAP/E) for inter-MSC handover F -> EIR (uses MAP/F) for equipment identity check H -> SMS-G (uses MAP/H) for Short Message Service (SMS) over CS I -> ME (uses MAP/I) Messages between MSC to ME handled by I Interface J -> SCF (uses MAP/J) Messages between HLR to gsmSCF handled by J Interface There are also several GSM/UMTS PS interfaces in the Serving GPRS Support Node (SGSN) transported over SS7: Gr -> HLR for attaching to the PS network and location update Gd -> SMS-C for SMS over PS Gs -> MSC for combined CS+PS signaling over PS Ge -> Charging for Customised Applications for Mobile networks Enhanced Logic (CAMEL) prepaid charging Gf -> EIR for equipment identity check == References == | https://en.wikipedia.org/wiki/Mobile_application_part |
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 charger to first perform a handshake to "agree" on mutually supported charge parameters. If such a charger can't identify the connected device or determine its compatibility, it will default to normal but much slower charge parameters within the USB standard. == References == | https://en.wikipedia.org/wiki/Handshake_(computing) |
In mobile networks, the terminal adapter is used by the terminal equipment to access the mobile termination, using AT commands (see Hayes command set). In 2G (such as GSM or CDMA), the terminal adapter is a theoretically optional while in 3G (such as W-CDMA), the terminal adapter is mandatory and is part of the mobile termination. | https://en.wikipedia.org/wiki/Terminal_adapter |
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 expire the battery earlier. Lithium-ion and lithium-polymer batteries, those commonly powering portable electronics, additionally wear down more from fuller charge and deeper discharge cycles, and when unused for an extended amount of time while depleted, where self-discharging may lead to a harmful depth of discharge.Manufacturers have prevented some smartphones from operating after repairs, by associating components' unique serial numbers to the device so it will refuse to operate or disable some functionality in case of a mismatch that would occur after a replacement. Locking of the serial number was first documented in 2015 on the iPhone 6, which would become inoperable from a detected replacement of the "home" button. Later, some functionality was restricted on Apple and Samsung smartphones when a battery replacement not authorized by the vendor was detected. | https://en.wikipedia.org/wiki/PDA_Phone |
In mobile telecommunications technology, the concept of mobile signature roaming means an access point (AP) should be able to get a mobile signature from any end-user, even if the AP and the end-user have not contracted a commercial relationship with the same MSSP. Otherwise, an AP would have to build commercial terms with as many MSSPs as possible, and this might be a cost burden. This means that a mobile signature transaction issued by an application provider should be able to reach the appropriate MSSP, and this should be transparent for the AP. | https://en.wikipedia.org/wiki/Mobile_signature_roaming |
Mobile signature roaming itself requires commercial agreements between the entities that facilitate it. In this respect, we assume that various entities (including MSSPs) will join in order to define common commercial terms and rules corresponding to a mobile signature roaming Service. This is the concept of a mobile signature roaming service. | https://en.wikipedia.org/wiki/Mobile_signature_roaming |
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 coordinated resource management can be achieved through fixed, adaptive or real-time coordination with the help of additional inter-cell signaling in which the signaling rate can vary accordingly. In general, inter-cell signaling refers to the communication interface among neighboring cells and the received measurement message reports from user equipments (UEs). | https://en.wikipedia.org/wiki/Inter-cell_interference_coordination |
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 GSM handsets and network elements, but the character set is suitable only for English and a number of Western-European languages. Languages such as Chinese, Korean or Japanese must be transferred using the 16-bit UCS-2 character encoding. A limited number of languages, like Portuguese, Spanish, Turkish and a number of languages used in India written with a Brahmic scripts may use 7-bit encoding with national language shift table defined in 3GPP 23.038. For binary messages, 8-bit encoding is used. | https://en.wikipedia.org/wiki/GSM_03.38 |
In mobile telephony a bearer service is a link between two points, which is defined by a certain set of characteristics. Whenever user equipment (UE) is being provided with any service (CS/PS service), the service has to be associated with a Radio Bearer specifying the configuration for layer 2 and physical layer in order to have its QoS clearly defined. Radio bearers are channels offered by Layer 2 to higher layers for the transfer of either user or control data. In other words, Layer 2 offers to the upper layers the service of information transmission between the UE and the UTRAN by means of the Radio Bearers (RBs) and Signaling Radio Bearers (SRBs). Therefore, the service access points between Layer 2 and upper layers are RBs. | https://en.wikipedia.org/wiki/Radio_Bearer_in_UMTS |
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 treatment. CSC and follow-up care for combat stress casualties are held as close as possible to and maintain close association with the member's unit, and are an integral part of the entire healing process. A visit from a member of the individual's unit during restoration is effective in keeping a bond with the organization. | https://en.wikipedia.org/wiki/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 does not want to let their comrades down. | https://en.wikipedia.org/wiki/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. | https://en.wikipedia.org/wiki/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 likely they will recover. They are more likely to develop a chronic psychiatric illness and get evacuated from the war. | https://en.wikipedia.org/wiki/Combat_stress_reaction |
In mobile warfare, such as the German Blitzkrieg, salients were more likely to be made into pockets which became the focus of annihilation battles. A pocket carries connotations that the encircled forces have not allowed themselves to be encircled intentionally, as they may when defending a fortified position, which is usually called a siege. This is a similar distinction to that made between a skirmish and pitched battle. | https://en.wikipedia.org/wiki/Salients,_re-entrants_and_pockets |
In mobile, web, and general application design, a similar symbol is sometimes used as an interface element, where it is called a hamburger icon. The element typically indicates that a navigation menu can be accessed when the element is activated; the bars of the symbol may be seen as stylized menu items, and some variations of this symbols add more bars, or bullet points to each bar, to enhance this visual similarity. Usage of this symbol dates back to the early computer interfaces developed at Xerox PARC in the 1980s. | https://en.wikipedia.org/wiki/Triple_bar |
It is also similar to the icon frequently used to indicate justified text alignment. It is an oft-used component of Google's Material Design guidelines and many Android apps and web apps that follow these guidelines make use of the hamburger menu. == References == | https://en.wikipedia.org/wiki/Triple_bar |
In mobile-telephone technology, the UniPro protocol stack follows the architecture of the classical OSI Reference Model. In UniPro, the OSI Physical Layer is split into two sublayers: Layer 1 (the actual physical layer) and Layer 1.5 (the PHY Adapter layer) which abstracts from differences between alternative Layer 1 technologies. The actual physical layer is a separate specification as the various PHY options are reused in other MIPI Alliance specifications. The UniPro specification itself covers Layers 1.5, 2, 3, 4 and the DME (Device Management Entity). | https://en.wikipedia.org/wiki/UniPro_protocol_stack |
The Application Layer (LA) is out of scope because different uses of UniPro will require different LA protocols. The Physical Layer (L1) is covered in separate MIPI specifications in order to allow the PHY to be reused by other (less generic) protocols if needed. OSI Layers 5 (Session) and 6 (Presentation) are, where applicable, counted as part of the Application Layer. | https://en.wikipedia.org/wiki/UniPro_protocol_stack |
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 Maltz. It is one of the most popular mobility models to evaluate mobile ad hoc network (MANET) routing protocols, because of its simplicity and wide availability. | https://en.wikipedia.org/wiki/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 and the random direction model are variants of the random waypoint model. | https://en.wikipedia.org/wiki/Random_waypoint_model |
In mobility management, the restricted random waypoint model is a random model for the movement of mobile users, similar to the random waypoint model, but where the waypoints are restricted to fall within one of a finite set of sub-domains. It was originally introduced by Blaževic et al. in order to model intercity examples and later defined in a more general setting by Le Boudec et al. | https://en.wikipedia.org/wiki/Restricted_random_waypoint_model |
In mock trial, students take responsibility for the prosecution/plaintiff or defense case in a trial presented using fabricated evidence, and role-players as witnesses and faculty or volunteers as judge or jury. It evaluates the participants’ skills in argument, evidence handling, and examination of witnesses, but omits jury selection and strategic matters. Mock trial differs from moot court in that moot court practices appellate argument, and so involves no handling of witnesses or evidence, but rather is an exercise in legal research and oral advocacy. | https://en.wikipedia.org/wiki/Trial_advocacy |
In modal logic and the philosophy of language, a term is said to be a rigid designator or absolute substantial term when it designates (picks out, denotes, refers to) the same thing in all possible worlds in which that thing exists. A designator is persistently rigid if it also designates nothing in all other possible worlds. A designator is obstinately rigid if it designates the same thing in every possible world, period, whether or not that thing exists in that world. Rigid designators are contrasted with connotative terms, non-rigid or flaccid designators, which may designate different things in different possible worlds. | https://en.wikipedia.org/wiki/Rigid_designators |
In modal logic and the philosophy of language, a vivid designator is a term which is believed to designate the same thing in all possible worlds and nothing else where such an object does not exist in a possible world. It is the analogue, in the sense of believing, of a rigid designator, which is (refers to) the same in all possible worlds, rather than is just believed to be so. | https://en.wikipedia.org/wiki/Vivid_designator |
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 formulas with first-order correspondents. Since it is undecidable, by Chagrova's theorem, whether an arbitrary modal formula has a first-order correspondent, there are formulas with first-order frame conditions that are not Sahlqvist (see the examples below). Hence Sahlqvist formulas define only a (decidable) subset of modal formulas with first-order correspondents. | https://en.wikipedia.org/wiki/Sahlqvist_formula |
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\leftrightarrow \Box B}}.} Alternatively, one can give a dual definition of L by which L is classical if and only if it contains (as axiom or theorem) ◻ A ↔ ¬ ◊ ¬ A {\displaystyle \Box A\leftrightarrow \lnot \Diamond \lnot A} and is closed under the rule A ↔ B ◊ A ↔ ◊ B . | https://en.wikipedia.org/wiki/Classical_modal_logic |
{\displaystyle {\frac {A\leftrightarrow B}{\Diamond A\leftrightarrow \Diamond B}}.} The weakest classical system is sometimes referred to as E and is non-normal. Both algebraic and neighborhood semantics characterize familiar classical modal systems that are weaker than the weakest normal modal logic K. Every regular modal logic is classical, and every normal modal logic is regular and hence classical. | https://en.wikipedia.org/wiki/Classical_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)\to \Box C}}.} Every normal modal logic is regular, and every regular modal logic is classical. | https://en.wikipedia.org/wiki/Regular_modal_logic |
In modal logic, modal collapse is the condition in which every true statement is necessarily true, and vice versa; that is to say, there are no contingent truths, or to put it another way, that "everything exists necessarily". In the notation of modal logic, this can be written as ϕ ↔ ◻ ϕ {\displaystyle \phi \leftrightarrow \Box \phi } . In the context of philosophy, the term is commonly used in critiques of ontological arguments for the existence of God and the principle of divine simplicity. | https://en.wikipedia.org/wiki/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 leads to the breakdown of Gödel's argument. == References == | https://en.wikipedia.org/wiki/Modal_collapse |
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 and the objects in the first-order language are the accessible worlds. The logical connectives from propositional logic remain untouched and the modal operators are transformed into first-order formulas according to their semantics. | https://en.wikipedia.org/wiki/Standard_translation |
In modal logic, the lozenge expresses that there is "possibility." For example, the expression ◊ P {\displaystyle \lozenge P} expresses that it is possible that P {\displaystyle P} is true. | https://en.wikipedia.org/wiki/Lozenge_(shape) |
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://en.wikipedia.org/wiki/Modal_depth |
In modal logic, the necessity of identity is the thesis that for every object x and object y, if x and y are the same object, it is necessary that x and y are the same object. The thesis is best known for its association with Saul Kripke, who published it in 1971, although it was first derived by the logician Ruth Barcan Marcus in 1947, and later, in simplified form, by W. V. O. Quine in 1953. | https://en.wikipedia.org/wiki/Necessity_of_identity |
In modal logic, the window operator △ {\displaystyle \triangle } is a modal operator with the following semantic definition: M , w ⊨ △ ϕ ⟺ ∀ u , M , u ⊨ ϕ ⇒ R w u {\displaystyle M,w\models \triangle \phi \iff \forall u,M,u\models \phi \Rightarrow Rwu} for M = ( W , R , f ) {\displaystyle M=(W,R,f)} a Kripke model and w , u ∈ W {\displaystyle w,u\in W} . Informally, it says that w "sees" every φ-world (or every φ-world is seen by w). This operator is not definable in the basic modal logic (i.e. some propositional non-modal language together with a single primitive "necessity" (universal) operator, often denoted by ' ◻ {\displaystyle \square } ', or its existential dual, often denoted by ' ◊ {\displaystyle \Diamond } '). Notice that its truth condition is the converse of the truth condition for the standard "necessity" operator. For references to some of its applications, see the References section. | https://en.wikipedia.org/wiki/Window_operator |
In modal logic, there is an important distinction between what is logically necessary to be true and what is true but not logically necessary to be so. One common form is replacing p → q {\displaystyle p\rightarrow q} with p → ◻ q {\displaystyle p\rightarrow \Box q} . In the first statement, q {\displaystyle q} is true given p {\displaystyle p} but is not logically necessary to be so. A common example in everyday life might be the following: Mickey Mouse is the President of the United States. | https://en.wikipedia.org/wiki/Modal_fallacy |
The President is at least 35 years old. Thus, Mickey Mouse is necessarily 35 years or older.Why is this false? The conclusion is false, since, even though Mickey Mouse is over 35 years old, there is no logical necessity for him to be. | https://en.wikipedia.org/wiki/Modal_fallacy |
Even though it is certainly true in this world, a possible world can exist in which Mickey Mouse is not yet 35 years old. If instead of adding a stipulation of necessity, the argument just concluded that Mickey Mouse is 35 or older, it would be valid. Norman Swartz gave the following example of how the modal fallacy can lead one to conclude that the future is already set, regardless of one's decisions; this is based on the "sea battle" example used by Aristotle to discuss the problem of future contingents in his On Interpretation:Two admirals, A and B, are preparing their navies for a sea battle tomorrow. | https://en.wikipedia.org/wiki/Modal_fallacy |
The battle will be fought until one side is victorious. But the 'laws' of the excluded middle (no third truth-value) and of non-contradiction (not both truth-values), mandate that one of the propositions, 'A wins' and 'B wins', is true (always has been and ever will be) and the other is false (always has been and ever will be). Suppose 'A wins' is today true. | https://en.wikipedia.org/wiki/Modal_fallacy |
Then whatever A does (or fails to do) today will make no difference; similarly, whatever B does (or fails to do) today will make no difference: the outcome is already settled. Or again, suppose 'A wins' is today false. Then no matter what A does today (or fails to do), it will make no difference; similarly, no matter what B does (or fails to do), it will make no difference: the outcome is already settled. | https://en.wikipedia.org/wiki/Modal_fallacy |
Thus, if propositions bear their truth-values timelessly (or unchangingly and eternally), then planning, or as Aristotle put it 'taking care', is illusory in its efficacy. The future will be what it will be, irrespective of our planning, intentions, etc.Suppose that the statement "A wins" is given by A {\displaystyle A} and "B wins" is given by B {\displaystyle B} . It is true here that only one of the statements "A wins" or "B wins" must be true. | https://en.wikipedia.org/wiki/Modal_fallacy |
In other words, only one of ⋄ A {\displaystyle \diamond A} or ⋄ B {\displaystyle \diamond B} is true. In logic syntax, this is equivalent to A ∨ B {\displaystyle A\lor B} (either A {\displaystyle A} or B {\displaystyle B} is true) ¬ ⋄ ( A ∧ B ) {\displaystyle \lnot \diamond (A\land B)} (it is not possible that A {\displaystyle A} and B {\displaystyle B} are both true at the same time) The fallacy here occurs because one assumes that ⋄ A {\displaystyle \diamond A} and ⋄ B {\displaystyle \diamond B} implies ◻ A {\displaystyle \Box A} and ◻ B {\displaystyle \Box B} . | https://en.wikipedia.org/wiki/Modal_fallacy |
Thus, one believes that, since one of both events is logically necessarily true, no action by either can change the outcome. Swartz also argued that the argument from free will suffers from the modal fallacy. == References == | https://en.wikipedia.org/wiki/Modal_fallacy |
In modal tunings, the strings are tuned to form a chord which is not definitively minor or major. These tunings may facilitate very easy chords and unique sounds when the open strings are used as drones. Often these tunings form a suspended chord on the open strings. A well known user of modal tunings is Sonic Youth. | https://en.wikipedia.org/wiki/Drop_B_tuning |
In model checking, a branch of computer science, linear time properties are used to describe requirements of a model of a computer system. Example properties include "the vending machine does not dispense a drink until money has been entered" (a safety property) or "the computer program eventually terminates" (a liveness property). Fairness properties can be used to rule out unrealistic paths of a model. For instance, in a model of two traffic lights, the liveness property "both traffic lights are green infinitely often" may only be true under the unconditional fairness constraint "each traffic light changes colour infinitely often" (to exclude the case where one traffic light is "infinitely faster" than the other).Formally, a linear time property is an ω-language over the power set of "atomic propositions". | https://en.wikipedia.org/wiki/Linear_time_property |
That is, the property contains sequences of sets of propositions, each sequence known as a "word". Every property can be rewritten as "P and Q both occur" for some safety property P and liveness property Q. An invariant for a system is something that is true or false for a particular state. Invariant properties describe an invariant that every reachable state of a model must satisfy, while persistence properties are of the form "eventually forever some invariant holds". | https://en.wikipedia.org/wiki/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 between 0 and 1 before termination. The more general formalism used in Safety and liveness properties can handle this. | https://en.wikipedia.org/wiki/Linear_time_property |
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 intersection and the sum of two zones. It is, for example, used in the Uppaal model checker; where it is also distributed as an independent library.More precisely, there is a notion of canonical DBM; there is a one-to-one relation between canonical DBMs and zones and from each DBM a canonical equivalent DBM can be efficiently computed. Thus, equality of zone can be tested by checking for equality of canonical DBMs. | https://en.wikipedia.org/wiki/Zone_(convex_polytope) |
In model checking, a field of computer science, a region is a convex polytope in R d {\displaystyle \mathbb {R} ^{d}} for some dimension d {\displaystyle d} , and more precisely a zone, satisfying some minimality property. The regions partition R d {\displaystyle \mathbb {R} ^{d}} . The set of zones depends on a set K {\displaystyle K} of constraints of the form x ≤ c {\displaystyle x\leq c} , x ≥ c {\displaystyle x\geq c} , x 1 ≤ x 2 + c {\displaystyle x_{1}\leq x_{2}+c} and x 1 ≥ x 2 + c {\displaystyle x_{1}\geq x_{2}+c} , with x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} some variables, and c {\displaystyle c} a constant. | https://en.wikipedia.org/wiki/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 futures. Intuitively, it means that any timed propositional temporal logic-formula, or timed automaton or signal automaton using only the constraints of K {\displaystyle K} can not distinguish both vectors. | https://en.wikipedia.org/wiki/Region_(model_checking) |
The set of region allows to create the region automaton, which is a directed graph in which each node is a region, and each edge r → r ′ {\displaystyle r\to r'} ensure that r ′ {\displaystyle r'} is a possible future of r {\displaystyle r} . Taking a product of this region automaton and of a timed automaton A {\displaystyle {\mathcal {A}}} which accepts a language L {\displaystyle L} creates a finite automaton or a Büchi automaton which accepts untimed L {\displaystyle L} . In particular, it allows to reduce the emptiness problem for A {\displaystyle {\mathcal {A}}} to the emptiness problem for a finite or Büchi automaton. This technique is used for example by the software UPPAAL. | https://en.wikipedia.org/wiki/Region_(model_checking) |
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, TPTL furthermore allows to give a time limit for q to occur. | https://en.wikipedia.org/wiki/Timed_propositional_temporal_logic |
In model checking, a subfield of computer science, a clock is a mathematical object used to model time. More precisely, a clock measures how much time passed since a particular event occurs, in this sense, a clock is more precisely an abstraction of a stopwatch. In a model of some particular program, the value of the clock may either be the time since the program was started, or the time since a particular event occurred in the program. | https://en.wikipedia.org/wiki/Clock_(model_checking) |
Those clocks are used in the definition of timed automaton, signal automaton, timed propositional temporal logic and clock temporal logic. They are also used in programs such as UPPAAL which implement timed automata.Generally, the model of a system uses many clocks. Those multiple clocks are required in order to track a bounded number of events. | https://en.wikipedia.org/wiki/Clock_(model_checking) |
All of those clocks are synchronized. That means that the difference in value between two fixed clocks is constant until one of them is restarted. In the language of electronics, it means that clock's jitter is null. | https://en.wikipedia.org/wiki/Clock_(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 set of real numbers to letters. This allow the use of formalisms similar to the ones of automata theory to deal with continuous signals. | https://en.wikipedia.org/wiki/Signal_(model_checking) |
In model checking, a subfield of computer science, a timed word is an extension of the notion of words, in a formal language, in which each letter is associated with a positive time tag. The sequence of time tags must be non-decreasing, which intuitively means that letters are received. For example, a system receiving a word over a network may associate to each letter the time at which the letter is received. The non-decreasing condition here means that the letters are received in the correct order. A timed language is a set of timed words. | https://en.wikipedia.org/wiki/Timed_language |
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 × S to V. | https://en.wikipedia.org/wiki/State_transition_system |
In model checking, the Metric Interval Temporal Logic (MITL) is a fragment of Metric Temporal Logic (MTL). This fragment is often preferred to MTL because some problems that are undecidable for MTL become decidable for MITL. | https://en.wikipedia.org/wiki/Metric_interval_temporal_logic |
In model rocketry, a parachute, streamer or other recovery device or method deploys at apogee, but high-power rockets may employ more complex recovery systems since altitudes can be much higher than their counterparts. In a high-power rocket, an altimeter or electronic timer may deploy a drogue parachute (which stabilizes the rocket in descent) or a controlled freefall (where the fore and aft sections are merely separated by a tether or umbilical cord, often made of tubular nylon). These recovery events can be brought about by small explosive charges (black powder or Pyrodex) or pressurized gasses (e.g., CO2). At an altitude predetermined by the hobbyist, an altimeter deploys a main parachute that slows the rocket to a safe recovery speed. The most common varieties of altimeter use accelerometers, barometric sensors or a combination of both. | https://en.wikipedia.org/wiki/High_Power_Rocketry |
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x1, x2,…, xn that are true of a set of n-tuples of an L-structure M {\displaystyle {\mathcal {M}}} . Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure M {\displaystyle {\mathcal {M}}} . The question of which types represent actual elements of M {\displaystyle {\mathcal {M}}} leads to the ideas of saturated models and omitting types. | https://en.wikipedia.org/wiki/Type_(model_theory) |
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 ultrapower, A 1 . {\displaystyle A_{1}.} | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/The_fundamental_theorem_of_ultraproducts |
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 property if every finitely generated substructure is again in K. See age. | https://en.wikipedia.org/wiki/Hereditary_property |
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://en.wikipedia.org/wiki/Tarski's_exponential_function_problem |
In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by Vaught (1963, p. 309), states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is ( ω 2 , ω 1 ) ↠ ( ω 1 , ω ) {\displaystyle (\omega _{2},\omega _{1})\twoheadrightarrow (\omega _{1},\omega )} . The axiom of constructibility implies that Chang's conjecture fails. | https://en.wikipedia.org/wiki/Chang's_conjecture |
Silver proved the consistency of Chang's conjecture from the consistency of an ω1-Erdős cardinal. Hans-Dieter Donder showed a weak version of the reverse implication: if CC is not only consistent but actually holds, then ω2 is ω1-Erdős in K. More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim that every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ). The consistency of ( ω 3 , ω 2 ) ↠ ( ω 2 , ω 1 ) {\displaystyle (\omega _{3},\omega _{2})\twoheadrightarrow (\omega _{2},\omega _{1})} was shown by Laver from the consistency of a huge cardinal. | https://en.wikipedia.org/wiki/Chang's_conjecture |
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 case, superstability. | https://en.wikipedia.org/wiki/U-rank |
In model theory, a branch of mathematical logic, a C-minimal theory is a theory that is "minimal" with respect to a ternary relation C with certain properties. Algebraically closed fields with a (Krull) valuation are perhaps the most important example. This notion was defined in analogy to the o-minimal theories, which are "minimal" (in the same sense) with respect to a linear order. | https://en.wikipedia.org/wiki/C-minimal_theory |
In model theory, a branch of mathematical logic, a complete first-order theory T is called stable in λ (an infinite cardinal number), if the Stone space of every model of T of size ≤ λ has itself size ≤ λ. T is called a stable theory if there is no upper bound for the cardinals κ such that T is stable in κ. The stability spectrum of T is the class of all cardinals κ such that T is stable in κ. For countable theories there are only four possible stability spectra. The corresponding dividing lines are those for total transcendentality, superstability and stability. This result is due to Saharon Shelah, who also defined stability and superstability. | https://en.wikipedia.org/wiki/Stability_spectrum |
In model theory, a branch of mathematical logic, a complete theory T is said to satisfy NIP ("not the independence property") if none of its formulae satisfy the independence property—that is, if none of its formulae can pick out any given subset of an arbitrarily large finite set. | https://en.wikipedia.org/wiki/NIP_(model_theory) |
In model theory, a branch of mathematical logic, an elementary class (or axiomatizable class) is a class consisting of all structures satisfying a fixed first-order theory. | https://en.wikipedia.org/wiki/Axiomatizable_class |
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 quotient of the Cartesian product ∏ i ∈ I S i {\displaystyle \prod _{i\in I}S_{i}} by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if { i ∈ I: a i = b i } ∈ U {\displaystyle \left\{i\in I:a_{i}=b_{i}\right\}\in U} If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the original Cartesian product. If U is an ultrafilter, the reduced product is an ultraproduct. Operations from σ are interpreted on the reduced product by applying the operation pointwise. | https://en.wikipedia.org/wiki/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 is a vector space with addition defined as (a + b)i = ai + bi and multiplication by a scalar c as (ca)i = c ai. | https://en.wikipedia.org/wiki/Reduced_product |
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 created, called the generic or rich model. The specifics of ≤ {\displaystyle \leq } determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture. | https://en.wikipedia.org/wiki/Hrushovski_construction |
In model theory, a branch of mathematical logic, the diagram of a structure is a simple but powerful concept for proving useful properties of a theory, for example the amalgamation property and the joint embedding property, among others. | https://en.wikipedia.org/wiki/Diagram_(mathematical_logic) |
In model theory, a branch of mathematical logic, the notion of an existentially closed model (or existentially complete model) of a theory generalizes the notions of algebraically closed fields (for the theory of fields), real closed fields (for the theory of ordered fields), existentially closed groups (for the theory of groups), and dense linear orders without endpoints (for the theory of linear orders). | https://en.wikipedia.org/wiki/Existentially_closed_model |
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 to describe the possible behaviors of I(T, κ) as a function of κ. It has been almost completely solved for the case of a countable theory T. | https://en.wikipedia.org/wiki/Spectrum_of_a_theory |
In model theory, a branch of mathematical logic, the Łoś–Vaught test is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic, this means that for every sentence, the theory contains either the sentence or its negation but not both. | https://en.wikipedia.org/wiki/Łoś–Vaught_test |
In model theory, a branch of mathematical logic, two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely. The mathematical statements in question are required to be first-order sentences (involving 0, 1, the addition and multiplication). A typical example, for n > 0, n an integer, is φ(E) = "any polynomial of degree n in E has a zero in E"The set of such formulas for all n expresses that E is algebraically closed. The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero. | https://en.wikipedia.org/wiki/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 ways as a limit of the fields Fi: Łoś's theorem states that any first order statement that holds for all but finitely many Fi, also holds for the ultraproduct. | https://en.wikipedia.org/wiki/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 all primes p) ulimp Qp ≅ ulimp Fp((t)).In addition, model theory also studies the logical properties of various other types of fields, such as real closed fields or exponential fields (which are equipped with an exponential function exp: F → F×). | https://en.wikipedia.org/wiki/Field_(mathematics) |
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-order σ-formula φ(a1, …, an) with parameters a1, …, an from N is true in N if and only if it is true in M. If N is an elementary substructure of M, then M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M. A substructure N of M is elementary if and only if it passes the Tarski–Vaught test: every first-order formula φ(x, b1, …, bn) with parameters in N that has a solution in M also has a solution in N when evaluated in M. One can prove that two structures are elementarily equivalent with the Ehrenfeucht–Fraïssé games. Elementary embeddings are used in the study of large cardinals, including rank-into-rank. | https://en.wikipedia.org/wiki/Elementary_substructure |
In model theory, a branch of mathematics, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by Shelah (1990), and elimination of imaginaries was introduced by Poizat (1983). | https://en.wikipedia.org/wiki/Imaginary_element |
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model). | https://en.wikipedia.org/wiki/Nonstandard_model |
In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model theory. They were introduced by Saharon Shelah. | https://en.wikipedia.org/wiki/Abstract_elementary_class |
In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren, who observed that tame AECs were much easier to handle than general AECs. | https://en.wikipedia.org/wiki/Tame_abstract_elementary_class |
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://en.wikipedia.org/wiki/Ehrenfeucht–Mostowski_theorem |
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