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In modern literature, colloquial alternative names of Alpha Centauri include Rigil Kent (also Rigel Kent and variants; ) and Toliman (the latter of which became the proper name of Alpha Centauri B on 10 August 2018 by approval of the International Astronomical Union). Rigil Kent is short for Rigil Kentaurus, which is sometimes further abbreviated to Rigil or Rigel, though that is ambiguous with Beta Orionis, which is also called Rigel. The name Toliman originates with Jacobus Golius' 1669 edition of Al-Farghani's Compendium. | https://en.wikipedia.org/wiki/Alpha_Centauri |
Tolimân is Golius' latinisation of the Arabic name الظلمان al-Ẓulmān "the ostriches", the name of an asterism of which Alpha Centauri formed the main star.During the 19th century, the northern amateur popularist Elijah H. Burritt used the now-obscure name Bungula, possibly coined from "β" and the Latin ungula ("hoof").Together, Alpha and Beta Centauri form the "Southern Pointers" or "The Pointers", as they point towards the Southern Cross, the asterism of the constellation of Crux.In Chinese astronomy, 南門 Nán Mén, meaning Southern Gate, refers to an asterism consisting of Alpha Centauri and Epsilon Centauri. Consequently, the Chinese name for Alpha Centauri itself is 南門二 Nán Mén Èr, the Second Star of the Southern Gate.To the Australian aboriginal Boorong people of northwestern Victoria, Alpha Centauri and Beta Centauri are Bermbermgle, two brothers noted for their courage and destructiveness, who speared and killed Tchingal "The Emu" (the Coalsack Nebula). The form in Wotjobaluk is Bram-bram-bult. | https://en.wikipedia.org/wiki/Alpha_Centauri |
In modern literature, some writers have mentioned this fruit: Saki uses medlars in his short stories, which often play on the decay of Edwardian society. In "The Peace of Mowsle Barton", the outwardly quiet farmstead features a medlar tree and corrosive hatred. In "The Boar Pig", the titular animal, Tarquin Superbus, is the point of contact between society ladies cheating to get into the garden party of the season and a not entirely honest young schoolgirl who lures him away by strategically throwing well-bletted medlars: "Come, Tarquin, dear old boy; you know you can't resist medlars when they're rotten and squashy." Italian novelist Giovanni Verga's naturalist narrative I Malavoglia is titled The House by the Medlar Tree in the English translation.H. C. Bailey's detective Reggie Fortune is very fond of medlars.Philip Pullman describes Sir Charles Latrom's perfume as "rotted like a medlar" in his book The Subtle Knife. | https://en.wikipedia.org/wiki/Common_medlar |
In modern literature, the trickster survives as a character archetype, not necessarily supernatural or divine, sometimes no more than a stock character. Often, the trickster is distinct in a story by their acting as a sort of catalyst; their antics are the cause of other characters' discomfiture, but they are left untouched. Shakespeare's Puck is an example of this. | https://en.wikipedia.org/wiki/Trickster_god |
Another once-famous example was the character Froggy the Gremlin on the early USA children's television show "Andy's Gang". A cigar-puffing puppet, Froggy induced the adult humans around him to engage in ridiculous and self-destructive hi-jinks.For example, many European fairy tales have a king who wants to find the best groom for his daughter by ordering several trials. No brave and valiant prince or knight manages to win them, until a poor and simple peasant comes. | https://en.wikipedia.org/wiki/Trickster_god |
With the help of his wits and cleverness, instead of fighting, they evade or fool monsters, villains and dangers in unorthodox ways. Against expectations, the most unlikely candidate passes the trials and receives the reward. More modern and obvious examples of the same type include Bugs Bunny in the USA and from Sweden the female hero in the Pippi Longstocking stories. | https://en.wikipedia.org/wiki/Trickster_god |
In modern logic theory this scenario is not a paradox. The law of implication reconciles what Uncle Joe claims are incompatible hypotheticals. This law states that "if X then Y" is logically identical to "X is false or Y is true" (¬X ∨ Y). For example, given the statement "if you press the button then the light comes on", it must be true at any given moment that either you have not pressed the button, or the light is on. In short, what obtains is not that ¬C yields a contradiction, only that it necessitates A, because ¬A is what actually yields the contradiction. In this scenario, that means Carr doesn't have to be in, but that if he isn't in, Allen has to be in. | https://en.wikipedia.org/wiki/Barbershop_paradox |
In modern logic, the term "proposition" is often used for sentences of a formal language. In this usage, propositions are formal syntactic objects which can be studied independently of the meaning they would receive from a semantics. Propositions are also called sentences, statements, statement forms, formulas, and well-formed formulas, though these terms are usually not synonymous within a single text. | https://en.wikipedia.org/wiki/Proposition_(logic) |
A formal language begins with different types of symbols. These types can include variables, operators, function symbols, predicate (or relation) symbols, quantifiers, and propositional constants. (Grouping symbols such as delimiters are often added for convenience in using the language, but do not play a logical role.) | https://en.wikipedia.org/wiki/Proposition_(logic) |
Symbols are concatenated together according to recursive rules, in order to construct strings to which truth-values will be assigned. The rules specify how the operators, function and predicate symbols, and quantifiers are to be concatenated with other strings. A proposition is then a string with a specific form. | https://en.wikipedia.org/wiki/Proposition_(logic) |
The form that a proposition takes depends on the type of logic. The type of logic called propositional, sentential, or statement logic includes only operators and propositional constants as symbols in its language. The propositions in this language are propositional constants, which are considered atomic propositions, and composite (or compound) propositions, which are composed by recursively applying operators to propositions. | https://en.wikipedia.org/wiki/Proposition_(logic) |
Application here is simply a short way of saying that the corresponding concatenation rule has been applied. The types of logics called predicate, quantificational, or n-order logic include variables, operators, predicate and function symbols, and quantifiers as symbols in their languages. The propositions in these logics are more complex. | https://en.wikipedia.org/wiki/Proposition_(logic) |
First, one typically starts by defining a term as follows: A variable, or A function symbol applied to the number of terms required by the function symbol's arity.For example, if + is a binary function symbol and x, y, and z are variables, then x+(y+z) is a term, which might be written with the symbols in various orders. Once a term is defined, a proposition can then be defined as follows: A predicate symbol applied to the number of terms required by its arity, or An operator applied to the number of propositions required by its arity, or A quantifier applied to a proposition.For example, if = is a binary predicate symbol and ∀ is a quantifier, then ∀x,y,z is a proposition. This more complex structure of propositions allows these logics to make finer distinctions between inferences, i.e., to have greater expressive power. | https://en.wikipedia.org/wiki/Proposition_(logic) |
In modern machines, the time to fetch a variable from the data cache is often several times longer than the time needed for basic ALU operations. A program runs faster without stalls if its memory loads can be started several cycles before the instruction that needs that variable. Complex machines can do this with a deep pipeline and "out-of-order execution" that examines and runs many instructions at once. Register machines can even do this with much simpler "in-order" hardware, a shallow pipeline, and slightly smarter compilers. | https://en.wikipedia.org/wiki/Zero_address_arithmetic |
The load step becomes a separate instruction, and that instruction is statically scheduled much earlier in the code sequence. The compiler puts independent steps in between. Scheduling memory accesses requires explicit, spare registers. | https://en.wikipedia.org/wiki/Zero_address_arithmetic |
It is not possible on stack machines without exposing some aspect of the micro-architecture to the programmer. For the expression A B -, B must be evaluated and pushed immediately prior to the Minus step. Without stack permutation or hardware multithreading, relatively little useful code can be put in between while waiting for the Load B to finish. | https://en.wikipedia.org/wiki/Zero_address_arithmetic |
Stack machines can work around the memory delay by either having a deep out-of-order execution pipeline covering many instructions at once, or more likely, they can permute the stack such that they can work on other workloads while the load completes, or they can interlace the execution of different program threads, as in the Unisys A9 system. Today's increasingly parallel computational loads suggests, however, this might not be the disadvantage it's been made out to be in the past. Stack machines can omit the operand fetching stage of a register machine. For example, in the Java Optimized Processor (JOP) microprocessor the top 2 operands of stack directly enter a data forwarding circuit that is faster than the register file. | https://en.wikipedia.org/wiki/Zero_address_arithmetic |
In modern management usage, the term data is increasingly replaced by information or even knowledge in a non-technical context. Thus data management has become information management or knowledge management. This trend obscures the raw data processing and renders interpretation implicit. | https://en.wikipedia.org/wiki/Data_lifecycle |
The distinction between data and derived value is illustrated by the information ladder. However, data has staged a comeback with the popularisation of the term big data, which refers to the collection and analyses of massive sets of data. Several organisations have established data management centers (DMC) for their operations. | https://en.wikipedia.org/wiki/Data_lifecycle |
In modern mapping, a topographic map or topographic sheet is a type of map characterized by large-scale detail and quantitative representation of relief features, usually using contour lines (connecting points of equal elevation), but historically using a variety of methods. Traditional definitions require a topographic map to show both natural and artificial features. A topographic survey is typically based upon a systematic observation and published as a map series, made up of two or more map sheets that combine to form the whole map. | https://en.wikipedia.org/wiki/Topographic_Map |
A topographic map series uses a common specification that includes the range of cartographic symbols employed, as well as a standard geodetic framework that defines the map projection, coordinate system, ellipsoid and geodetic datum. Official topographic maps also adopt a national grid referencing system. Natural Resources Canada provides this description of topographic maps:These maps depict in detail ground relief (landforms and terrain), drainage (lakes and rivers), forest cover, administrative areas, populated areas, transportation routes and facilities (including roads and railways), and other man-made features. Other authors define topographic maps by contrasting them with another type of map; they are distinguished from smaller-scale "chorographic maps" that cover large regions, "planimetric maps" that do not show elevations, and "thematic maps" that focus on specific topics.However, in the vernacular and day to day world, the representation of relief (contours) is popularly held to define the genre, such that even small-scale maps showing relief are commonly (and erroneously, in the technical sense) called "topographic".The study or discipline of topography is a much broader field of study, which takes into account all natural and man-made features of terrain. Maps were among the first artifacts to record observations about topography. | https://en.wikipedia.org/wiki/Topographic_Map |
In modern market economies, if competition is imperfect; information unevenly distributed; opportunities to acquire education and skills unequal; market failure results. Many such imperfect conditions exist in virtually every market. According to Joseph Stiglitz this means that there is an enormous potential role for government to correct such market failures.In the United States, real wages are flat over the past 40 years for occupations across income and education levels, e.g., auto mechanics, cashiers, doctors, and software engineers. However, stock ownership favors higher income and education levels, thereby resulting in disparate investment income. | https://en.wikipedia.org/wiki/Socioeconomic_inequality |
In modern mathematical language, class field theory (CFT) can be formulated as follows. Consider the maximal abelian extension A of a local or global field K. It is of infinite degree over K; the Galois group G of A over K is an infinite profinite group, so a compact topological group, and it is abelian. The central aims of class field theory are: to describe G in terms of certain appropriate topological objects associated to K, to describe finite abelian extensions of K in terms of open subgroups of finite index in the topological object associated to K. In particular, one wishes to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in this topological object for K. This topological object is the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields. The finite abelian extension corresponding to an open subgroup of finite index is called the class field for that subgroup, which gave the name to the theory. | https://en.wikipedia.org/wiki/Abelian_number_field |
The fundamental result of general class field theory states that the group G is naturally isomorphic to the profinite completion of CK, the multiplicative group of a local field or the idele class group of the global field, with respect to the natural topology on CK related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism (the Artin reciprocity map) Gal ( L / K ) ab → C K / N L / K ( C L ) {\displaystyle \operatorname {Gal} (L/K)^{\operatorname {ab} }\to C_{K}/N_{L/K}(C_{L})} of the abelianization of the Galois group of the extension with the quotient of the idele class group of K by the image of the norm of the idele class group of L. For some small fields, such as the field of rational numbers Q {\displaystyle \mathbb {Q} } or its quadratic imaginary extensions there is a more detailed very explicit but too specific theory which provides more information. For example, the abelianized absolute Galois group G of Q {\displaystyle \mathbb {Q} } is (naturally isomorphic to) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity. This is known as the Kronecker–Weber theorem, originally conjectured by Leopold Kronecker. | https://en.wikipedia.org/wiki/Abelian_number_field |
In this case the reciprocity isomorphism of class field theory (or Artin reciprocity map) also admits an explicit description due to the Kronecker–Weber theorem. However, principal constructions of such more detailed theories for small algebraic number fields are not extendable to the general case of algebraic number fields, and different conceptual principles are in use in the general class field theory. The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field. | https://en.wikipedia.org/wiki/Abelian_number_field |
The latter property is called the global reciprocity law and is a far reaching generalization of the Gauss quadratic reciprocity law. One of the methods to construct the reciprocity homomorphism uses class formation which derives class field theory from axioms of class field theory. This derivation is purely topological group theoretical, while to establish the axioms one has to use the ring structure of the ground field.There are methods which use cohomology groups, in particular the Brauer group, and there are methods which do not use cohomology groups and are very explicit and fruitful for applications. | https://en.wikipedia.org/wiki/Abelian_number_field |
In modern mathematical language, the problem posed on the tablet is the following: a rectangle has area A = 0.75 and diagonal c = 1.25. What are the lengths a and b of the sides of the rectangle? The solution can be understood as proceeding in two stages: in stage 1, the quantity c 2 − 2 A {\displaystyle {\sqrt {c^{2}-2A}}} is computed to be 0.25. In stage 2, the well-attested Old Babylonian method of completing the square is used to solve what is effectively the system of equations b − a = 0.25, ab = 0.75. | https://en.wikipedia.org/wiki/IM_67118 |
Geometrically this is the problem of computing the lengths of the sides of a rectangle whose area A and side-length difference b−a are known, which was a recurring problem in Old Babylonian mathematics. In this case it is found that b = 1 and a = 0.75. The solution method suggests that whoever devised the solution was using the property c2 − 2A = c2 − 2ab = (b − a)2. | https://en.wikipedia.org/wiki/IM_67118 |
It must be emphasized, however, that the modern notation for equations and the practice of representing parameters and unknowns by letters were unheard of in ancient times. It is now widely accepted as a result of Jens Høyrup's extensive analysis of the vocabulary of Old Babylonian mathematics, that underlying the procedures in texts such as IM 67118 was a set of standard cut-and-paste geometric operations, not a symbolic algebra. From the vocabulary of the solution Høyrup concludes that c2, the square of the diagonal, is to be understood as a geometric square, from which an area equal to 2A is to be "cut off", that is, removed, leaving a square with side b − a. Høyrup suggests that the square on the diagonal was possibly formed by making four copies of the rectangle, each rotated by 90°, and that the area 2A was the area of the four right triangles contained in the square on the diagonal. | https://en.wikipedia.org/wiki/IM_67118 |
The remainder is the small square in the center of the figure.The geometric procedure for computing the lengths of the sides of a rectangle of given area A and side-length difference b − a was to transform the rectangle into a gnomon of area A by cutting off a rectangular piece of dimensions a×½(b − a) and pasting this piece onto the side of the rectangle. The gnomon was then completed to a square by adding a smaller square of side ½(b − a) to it. In this problem, the side of the completed square is computed to be A + 1 4 ( b − a ) 2 = 0.75 + 0.015625 = 0.875 {\displaystyle {\sqrt {A+{\tfrac {1}{4}}(b-a)^{2}}}={\sqrt {0.75+0.015625}}=0.875} . | https://en.wikipedia.org/wiki/IM_67118 |
The quantity ½(b − a)=0.125 is then added to the horizontal side of the square and subtracted from the vertical side. The resulting line segments are the sides of the desired rectangle.One difficulty in reconstructing Old Babylonian geometric diagrams is that known tablets never include diagrams in solutions—even in geometric solutions where explicit constructions are described in text—although diagrams are often included in formulations of problems. | https://en.wikipedia.org/wiki/IM_67118 |
Høyrup argues that the cut-and paste geometry would have been performed in some medium other than clay, perhaps in sand or on a "dust abacus", at least in the early stages of a scribe's training before mental facility with geometric calculation had been developed.Friberg does describe some tablets containing drawings of "figures within figures", including MS 2192 in which the band separating two concentric equilateral triangles is divided into three trapezoids. He writes "The idea of computing the area of a triangular band as the area of a chain of trapezoids is a variation on the idea of computing the area of a square band as the area of a chain of four rectangles. This is a simple idea, and it is likely that it was known by Old Babylonian mathematicians, although no cuneiform mathematical text has yet been found where this idea enters in an explicit way." | https://en.wikipedia.org/wiki/IM_67118 |
He goes on to argue that this idea is implicit in the text of IM 67118. He also invites a comparison with the diagram of YBC 7329, in which two concentric squares are shown. The band separating the squares is not subdivided into four rectangles on this tablet, but the numerical value of the area of one of the rectangles area does appear next to the figure. | https://en.wikipedia.org/wiki/IM_67118 |
In modern mathematical logic, the excluded middle has been argued to result in possible self-contradiction. It is possible in logic to make well-constructed propositions that can be neither true nor false; a common example of this is the "Liar's paradox", the statement "this statement is false", which is argued to itself be neither true nor false. Arthur Prior has argued that The Paradox is not an example of a statement that cannot be true or false. The law of excluded middle still holds here as the negation of this statement "This statement is not false", can be assigned true. | https://en.wikipedia.org/wiki/Principle_of_the_excluded_middle |
In set theory, such a self-referential paradox can be constructed by examining the set "the set of all sets that do not contain themselves". This set is unambiguously defined, but leads to a Russell's paradox: does the set contain, as one of its elements, itself? However, in the modern Zermelo–Fraenkel set theory, this type of contradiction is no longer admitted. Furthermore, paradoxes of self reference can be constructed without even invoking negation at all, as in Curry's paradox. | https://en.wikipedia.org/wiki/Principle_of_the_excluded_middle |
In modern mathematics it is common to study a category by associating to every object of this category a simpler object that still retains sufficient information about the object of interest. Homotopy groups are such a way of associating groups to topological spaces. That link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be difficult to prove using only topological means. | https://en.wikipedia.org/wiki/Homotopy_group |
For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure. | https://en.wikipedia.org/wiki/Homotopy_group |
As for the example: the first homotopy group of the torus T {\displaystyle T} is because the universal cover of the torus is the Euclidean plane R 2 , {\displaystyle \mathbb {R} ^{2},} mapping to the torus T ≅ R 2 / Z 2 . {\displaystyle T\cong \mathbb {R} ^{2}/\mathbb {Z} ^{2}.} Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere S 2 {\displaystyle S^{2}} satisfies: because every loop can be contracted to a constant map (see homotopy groups of spheres for this and more complicated examples of homotopy groups). Hence the torus is not homeomorphic to the sphere. | https://en.wikipedia.org/wiki/Homotopy_group |
In modern mathematics spaces are defined as sets with some added structure. They are frequently described as different types of manifolds, which are spaces that locally approximate to Euclidean space, and where the properties are defined largely on local connectedness of points that lie on the manifold. There are however, many diverse mathematical objects that are called spaces. For example, vector spaces such as function spaces may have infinite numbers of independent dimensions and a notion of distance very different from Euclidean space, and topological spaces replace the concept of distance with a more abstract idea of nearness. | https://en.wikipedia.org/wiki/Physical_space |
In modern mathematics, a common proof involves Bézout's identity, which was unknown at Euclid's time. Bézout's identity states that if x and y are coprime integers (i.e. they share no common divisors other than 1 and −1) there exist integers r and s such that r x + s y = 1. {\displaystyle rx+sy=1.} Let a and n be coprime, and assume that n|ab. | https://en.wikipedia.org/wiki/Euclid's_lemma |
By Bézout's identity, there are r and s such that r n + s a = 1. {\displaystyle rn+sa=1.} Multiply both sides by b: r n b + s a b = b . {\displaystyle rnb+sab=b.} The first term on the left is divisible by n, and the second term is divisible by ab, which by hypothesis is divisible by n. Therefore their sum, b, is also divisible by n. | https://en.wikipedia.org/wiki/Euclid's_lemma |
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it. When a geometry is described by a set of axioms, the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. | https://en.wikipedia.org/wiki/Straight_line |
One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry, a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries, a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line. | https://en.wikipedia.org/wiki/Straight_line |
In modern mathematics, one often studies spaces whose points are themselves mathematical objects. A distance function on such a space generally aims to measure the dissimilarity between two objects. Here are some examples: Functions to a metric space. If X is any set and M is a metric space, then the set of all bounded functions f: X → M {\displaystyle f\colon X\to M} (i.e. those functions whose image is a bounded subset of M {\displaystyle M} ) can be turned into a metric space by defining the distance between two bounded functions f and g to be This metric is called the uniform metric or supremum metric. | https://en.wikipedia.org/wiki/Metric_distance |
If M is complete, then this function space is complete as well; moreover, if X is also a topological space, then the subspace consisting of all bounded continuous functions from X to M is also complete. When X is a subspace of R n {\displaystyle \mathbb {R} ^{n}} , this function space is known as a classical Wiener space. | https://en.wikipedia.org/wiki/Metric_distance |
String metrics and edit distances. There are many ways of measuring distances between strings of characters, which may represent sentences in computational linguistics or code words in coding theory. Edit distances attempt to measure the number of changes necessary to get from one string to another. | https://en.wikipedia.org/wiki/Metric_distance |
For example, the Hamming distance measures the minimal number of substitutions needed, while the Levenshtein distance measures the minimal number of deletions, insertions, and substitutions; both of these can be thought of as distances in an appropriate graph. Graph edit distance is a measure of dissimilarity between two graphs, defined as the minimal number of graph edit operations required to transform one graph into another. Wasserstein metrics measure the distance between two measures on the same metric space. | https://en.wikipedia.org/wiki/Metric_distance |
The Wasserstein distance between two measures is, roughly speaking, the cost of transporting one to the other. The set of all m by n matrices over some field is a metric space with respect to the rank distance d ( A , B ) = r a n k ( B − A ) {\displaystyle d(A,B)=\mathrm {rank} (B-A)} . The Helly metric in game theory measures the difference between strategies in a game. | https://en.wikipedia.org/wiki/Metric_distance |
In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, ..., which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent. It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is absolutely convergent. | https://en.wikipedia.org/wiki/Grandi's_series |
Otherwise these operations can alter the result of summation. Further, the terms of Grandi's series can be rearranged to have its accumulation points at any interval of two or more consecutive integer numbers, not only 0 or 1. For instance, the series 1 + 1 + 1 + 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 + 1 − ⋯ {\displaystyle 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots } (in which, after five initial +1 terms, the terms alternate in pairs of +1 and −1 terms–the infinitude of both +1’s and -1’s allows any finite number of 1’s or -1’s to be prepended, by Hilbert's paradox of the Grand Hotel) is a permutation of Grandi's series in which each value in the rearranged series corresponds to a value that is at most four positions away from it in the original series; its accumulation points are 3, 4, and 5. | https://en.wikipedia.org/wiki/Grandi's_series |
In modern mechanical watch designs, production of a highly accurate watch does not require a tourbillon. There is even debate amongst horologists as to whether tourbillons ever improved the accuracy of mechanical watches, even when first introduced, or whether the watches of the day were inherently inaccurate due to design and manufacturing techniques. A tourbillon is a valued feature of collectors' and premium-priced watches, possibly for the same reason that mechanical watches fetch a much higher price than similar quartz watches that are much more accurate.High-quality tourbillon wristwatches, usually made by the Swiss luxury watch industry, are very expensive, and typically retail for tens of thousands of dollars or euros, with much higher prices in the hundreds of thousands of dollars or euros being common. A recent renaissance of interest in tourbillons has been met by the industry with increased availability of time pieces bearing the feature, with the result that prices for basic tourbillon models have reduced somewhat in recent years. | https://en.wikipedia.org/wiki/Tourbillon |
Previously such models were very rare, either antique or new. Any watch with a tourbillon will cost a great deal more than an equivalent piece without the feature. The prices of Swiss models typically start at $40,000 and the prices of more expensive tourbillon watches can reach six figures. | https://en.wikipedia.org/wiki/Tourbillon |
The prices of some Chinese models can range from hundreds of dollars to nearly $5000.Modern implementations typically allow the tourbillon to be seen through a window in the watch face. In addition to the decorative effect, a tourbillon can act as a second hand for some watches, if the tourbillon rotates exactly once per minute. | https://en.wikipedia.org/wiki/Tourbillon |
Some tourbillons rotate faster than this (Greubel Forsey's 24 second tourbillon for example). Also, many quotidian watches feature their oscillating balance wheel. Sometimes termed, appropriately enough, the "open heart", these are sometimes misrepresented by unscrupulous dealers as a tourbillon (and "tourbillon-style" by ethical ones). | https://en.wikipedia.org/wiki/Tourbillon |
In modern mechanized farming, generally a farmer will use two harrows, one after the other. The disk harrow is used first to slice up the large clods left by the mould-board plough, followed by the spring-tooth harrow. To save time and fuel they may be pulled by one tractor; the disk hitched to the tractor, and the spring-tooth hitched to, and directly behind, the disk. The result is a smooth field with powdery dirt at the surface. | https://en.wikipedia.org/wiki/Spike_harrow |
In modern media, images of partial and full nudity are used in advertising to draw attention. In the case of attractive models this attention is due to the visual pleasure the images provide; in other cases it is due to the relative rarity of such images. The use of nudity in advertising tends to be carefully controlled to avoid the impression that a company whose product is being advertised is indecent or unrefined. There are also (self-imposed) limits on what advertising media such as magazines will allow. | https://en.wikipedia.org/wiki/Depictions_of_nudity |
The success of sexually provocative advertising is claimed in the truism "sex sells." However, responses to nudity in American advertisements have been more mixed; nudity in the advertisements of Calvin Klein, Benetton, and Abercrombie & Fitch, for example, has provoked negative as well as positive responses. An example of an advertisement featuring male full frontal nudity is one for M7 fragrance. Many magazines refused to place the ad, so there was also a version with a more modest photograph of the same model. | https://en.wikipedia.org/wiki/Depictions_of_nudity |
In modern medicine there are multiple options to limit the effect of cancer treatment on fertility. One of the preventative measures in females is transposition of gonadal organs further from local therapeutic agents with a success rate over 90%. Another less invasive method used for many years is lead shielding of gonadal region in both males and females as a protective measure against radiotherapy.In prepubescent males novel techniques such as testicular tissue extraction and cryopreservation as well as in vitro maturation of spermatogonia which can then be transferred to native tissue after the treatment are being heavily researched. | https://en.wikipedia.org/wiki/Side_effects_of_radiotherapy_on_fertility |
The most common solution is cryopreservation of sperms in post pubertal males and cryopreservation of oocytes or embryos for females with smaller age constrain compared to males who can then utilise multiple assisted reproductive techniques (ART) methods such as intrauterine insemination, IVF or ICSI as an alternative resource for preservation of fertility.Future approach to this problem focuses on cytoprotective strategies using hormonal treatment to alter HPG-A to guard reproductive organs from radiotherapy. By disrupting the gametogenesis or decreasing the sensitivity of germ cells scientists could acquire quiescent state less susceptible to side effects of cancer treatment. == References == | https://en.wikipedia.org/wiki/Side_effects_of_radiotherapy_on_fertility |
In modern merchandising, distribution responsibilities are absent, and focus is placed on planning and analysis. A separate team is tasked with distribution. Large organizations separate merchandisers by type. There are retail merchandisers and product merchandisers. | https://en.wikipedia.org/wiki/Fashion_merchandising |
Retail merchandisers manage store allocation and must maximize sales. Product merchandisers manage the flow of materials to suppliers and then the flow of product to stores. Product merchandisers then pass control of product to the retail merchandisers. | https://en.wikipedia.org/wiki/Fashion_merchandising |
Modern Structure Many large organizations have concluded that distribution requires highly detailed work and that it is necessary to have a team specifically for that purpose. This is due to the fine details of allocation, which require focus on aspects such as colour and sizes for a specific store. This approach not only minimizes costs, but also extends to areas like better control of the overall process. | https://en.wikipedia.org/wiki/Fashion_merchandising |
Organizations that do not conduct distribution this way risk losing control of their stock at both the highest and lowest level. This is a result of the lack of uniformity and oversight. | https://en.wikipedia.org/wiki/Fashion_merchandising |
The distribution team specializes not only in managing distribution, but they are also focused on sales and profit. They employ detailed, accurate information about distribution points sourced from product planners. They possess the ability to manage dynamic stock demands. | https://en.wikipedia.org/wiki/Fashion_merchandising |
They partner with buyers and merchandisers for any necessary repeat buying. Though they are positioned to manage stock, they still operate within the limits of the buying plan, and merchandisers ensure they remain within this realm. Buyers provide guidelines for distribution, such as the type of stores where product should be distributed; for example, a product may have only been acquired for the top 3 stores. | https://en.wikipedia.org/wiki/Fashion_merchandising |
The team also supports the goals of an organization through being instrumental in responding to trends. The nature of modern analysis has allowed many merchandisers to plan as much as four seasons ahead, and they are expected to apply the data. This further increases the demands placed on their roles and emphasizes the need to task out minor details that do not require their input or much of their supervision.Fashion merchandisers follow the five rights of merchandising, or 5Rs, to ensure that they properly meet the needs of consumers; thus, turning a profit.The five rights of merchandising include: the right merchandise at the right price at the right time in the right place in the right quantities.By researching and answering the five rights of merchandising, fashion merchandisers can gain an understanding of what products consumers want, when and where they wish to make purchases, and what prices will have the highest demand. Both fashion retailers and manufacturers utilize the 5Rs. | https://en.wikipedia.org/wiki/Fashion_merchandising |
In modern military engineering, a berm is the earthen or sod wall or parapet, especially a low earthen wall adjacent to a ditch. The digging of the ditch (often by a bulldozer or military engineering vehicle) can provide the soil from which the berm is constructed. Walls constructed in this manner are an obstacle to vehicles, including most armoured fighting vehicles but are easily crossed by infantry. Because of the ease of construction, such walls can be made hundreds or thousands of kilometres long. A prominent example of such a berm is the 2,700 km (1,700 mi) Moroccan Western Sahara Wall. | https://en.wikipedia.org/wiki/Berm |
In modern mixed economies such as those of the OECD countries, it is currently the most common form of work arrangement. Although most labour is organised as per this structure, the wage work arrangements of CEOs, professional employees, and professional contract workers are sometimes conflated with class assignments, so that "wage labour" is considered to apply only to unskilled, semi-skilled or manual labour. | https://en.wikipedia.org/wiki/Wage_labor |
In modern moral psychology, morality is sometimes considered to change through personal development. Several psychologists have produced theories on the development of morals, usually going through stages of different morals. Lawrence Kohlberg, Jean Piaget, and Elliot Turiel have cognitive-developmental approaches to moral development; to these theorists morality forms in a series of constructive stages or domains. | https://en.wikipedia.org/wiki/Moral_consciousness |
In the Ethics of care approach established by Carol Gilligan, moral development occurs in the context of caring, mutually responsive relationships which are based on interdependence, particularly in parenting but also in social relationships generally. Social psychologists such as Martin Hoffman and Jonathan Haidt emphasize social and emotional development based on biology, such as empathy. Moral identity theorists, such as William Damon and Mordechai Nisan, see moral commitment as arising from the development of a self-identity that is defined by moral purposes: this moral self-identity leads to a sense of responsibility to pursue such purposes. | https://en.wikipedia.org/wiki/Moral_consciousness |
Of historical interest in psychology are the theories of psychoanalysts such as Sigmund Freud, who believe that moral development is the product of aspects of the super-ego as guilt-shame avoidance. Theories of moral development therefore tend to regard it as positive moral development: the higher stages are morally higher, though this, naturally, involves a circular argument. The higher stages are better because they are higher, but the better higher because they are better. | https://en.wikipedia.org/wiki/Moral_consciousness |
As an alternative to viewing morality as an individual trait, some sociologists as well as social- and discursive psychologists have taken upon themselves to study the in-vivo aspects of morality by examining how persons conduct themselves in social interaction.A new study analyses the common perception of a decline in morality in societies worldwide and throughout history. Adam M. Mastroianni and Daniel T. Gilbert present a series of studies indicating that the perception of moral decline is an illusion and easily produced, with implications for misallocation of resources, underuse of social support, and social influence. To begin with, the authors demonstrate that people in no less than 60 nations hold the belief that morality is deteriorating continuously, and this conviction has been present for the last 70 years. | https://en.wikipedia.org/wiki/Moral_consciousness |
Subsequently, they indicate that people ascribe this decay to the declining morality of individuals as they age and the succeeding generations. Thirdly, the authors demonstrate that people's evaluations of the morality of their peers have not decreased over time, indicating that the belief in moral decline is an illusion. Lastly, the authors explain a basic psychological mechanism that uses two well-established phenomena (distorted exposure to information and distorted memory of information) to cause the illusion of moral decline. The authors present studies that validate some of the predictions about the circumstances in which the perception of moral decline is attenuated, eliminated, or reversed (e.g., when participants are asked about the morality of people closest to them or people who lived before they were born). | https://en.wikipedia.org/wiki/Moral_consciousness |
In modern multitasking environment, an application process usually has in its address space (or spaces) chunks of memory of following types: Machine code, including: program's own code (historically known as code segment or text segment); shared libraries. Data, including: initialized data (data segment); uninitialized (but allocated) variables; run-time stack; heap; shared memory and memory mapped files.Some parts of address space may be not mapped at all. Some systems have a "split" memory architecture where machine code, constants, and data are in different locations, and may have different address sizes. For example, PIC18 microcontrollers have a 21-bit program counter to address machine code and constants in Flash memory, and 12-bit address registers to address data in SRAM. | https://en.wikipedia.org/wiki/Absolute_address |
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en) and ἁρμονία (harmonía). | https://en.wikipedia.org/wiki/Enharmonic |
In modern mysticism, the infinity symbol has become identified with a variation of the ouroboros, an ancient image of a snake eating its own tail that has also come to symbolize the infinite, and the ouroboros is sometimes drawn in figure-eight form to reflect this identification—rather than in its more traditional circular form.In the works of Vladimir Nabokov, including The Gift and Pale Fire, the figure-eight shape is used symbolically to refer to the Möbius strip and the infinite, as is the case in these books' descriptions of the shapes of bicycle tire tracks and of the outlines of half-remembered people. Nabokov's poem after which he entitled Pale Fire explicitly refers to "the miracle of the lemniscate". Other authors whose works use this shape with its symbolic meaning of the infinite include James Joyce, in Ulysses, and David Foster Wallace, in Infinite Jest. | https://en.wikipedia.org/wiki/Infinity_symbol |
In modern neoclassical economics, exchange value itself is no longer explicitly theorised. The reason is that the concept of money-price is deemed sufficient in order to understand trading processes and markets. Exchange value thus becomes simply the price for which a good will trade in a given market which is identical to what Marx refers to as price. These trading processes are no longer understood in economics as social processes involving human giving and taking, getting and receiving, but as technical processes in which rational, self-interested economic actors negotiate prices based on subjective perceptions of utility. | https://en.wikipedia.org/wiki/Exchange_value |
Market realities are therefore understood in terms of supply and demand curves which sets price at a level where supply equals demand. Professor John Eatwell criticizes this approach as follows: Since the markets are driven by average opinion about what average opinion will be, an enormous premium is placed on any information or signals that might provide a guide to the swings in average opinion and as to how average opinion will react to changing events. | https://en.wikipedia.org/wiki/Exchange_value |
These signals have to be simple and clear-cut. Sophisticated interpretations of the economic data would not provide a clear lead. So the money markets and foreign exchange markets become dominated by simple slogans—larger fiscal deficits lead to higher interest rates, an increased money supply results in higher inflation, public expenditure bad, private expenditure good—even when those slogans are persistently refuted by events. | https://en.wikipedia.org/wiki/Exchange_value |
To these simplistic rules of the game there is added a demand for governments to publish their own financial targets, to show that their policy is couched within a firm financial framework. The main purpose of insisting on this government commitment to financial targeting is to aid average opinion in guessing how average opinion will expect the government to respond to changing economic circumstances and how average opinion will react when the government fails to meet its goals. So "the markets" are basically a collection of overexcited young men and women, desperate to make money by guessing what everyone else in the market will do. Many have no more claim to economic rationality than tipsters at the local racetrack and probably rather less specialist knowledge. | https://en.wikipedia.org/wiki/Exchange_value |
In modern news, social media has become a major source of amateur reporting. The Arab Spring is believed to have been aided by citizen journalism that was reported using and disseminated by Facebook and Twitter. The Top 15 Most Popular Social Media Sites range from 15 million users to 900 million users, and as their user base grows stronger, amateur reporting's base expands as well. | https://en.wikipedia.org/wiki/Mass_amateurization |
In modern notation, a breve is commonly represented in either of two ways: by a hollow oval note head, like a whole note, with one or two vertical lines on either side, as on the left and right of the image, or as the rectangular shape also found in older notation, shown in the middle of the image.Because it lasts longer than a bar in most modern time signatures in common use, the breve is rarely encountered except in English music, where the half-note is often used as the beat unit. | https://en.wikipedia.org/wiki/Double_whole_note |
In modern notation, the key signature for music in a minor key is typically based on the accidentals of the natural minor scale, not on those of the harmonic or melodic minor scales. For example, a piece in E minor will have one sharp in its key signature because the E natural minor scale has one sharp (F♯). Major and minor keys that share the same key signature are relative to each other. For instance, F major is the relative major of D minor since both have key signatures with one flat. Since the natural minor scale is built on the 6th degree of the major scale, the tonic of the relative minor is a major sixth above the tonic of the major scale. For instance, B minor is the relative minor of D major because the note B is a major sixth above D. As a result, the key signatures of B minor and D major both have two sharps (F♯ and C♯). | https://en.wikipedia.org/wiki/Melodic_minor_scale |
In modern operating systems, non-system (i.e. user-mode) applications are prevented from accessing any memory locations not explicitly authorized by the virtual memory controller (called memory management unit (MMU)). In addition to containing damage that may be caused by software flaws and allowing more efficient use of physical memory, this architecture forms an integral part of the security of the operating system. However, kernel-mode drivers, many hardware devices, and user-mode vulnerabilities allow direct, unimpeded access of the physical memory address space. The physical address space includes all of the main system memory, as well as memory-mapped buses and hardware devices (which are controlled by the operating system through reads and writes as if they were ordinary RAM). | https://en.wikipedia.org/wiki/DMA_attack |
The OHCI 1394 specification allows devices, for performance reasons, to bypass the operating system and access physical memory directly without any security restrictions. But SBP2 devices can easily be spoofed, making it possible to trick an operating system into allowing an attacker to both read and write physical memory, and thereby to gain unauthorised access to sensitive cryptographic material in memory.Systems may still be vulnerable to a DMA attack by an external device if they have a FireWire, ExpressCard, Thunderbolt or other expansion port that, like PCI and PCI Express in general, connects attached devices directly to the physical rather than virtual memory address space. Therefore, systems that do not have a FireWire port may still be vulnerable if they have a PCMCIA/CardBus/PC Card or ExpressCard port that would allow an expansion card with a FireWire to be installed. | https://en.wikipedia.org/wiki/DMA_attack |
In modern operating systems, rasterization is normally provided by a shared library common to many applications. Such a shared library may be built into the operating system or the desktop environment, or may be added later. In principle, each application may use a different font rasterization library, but in practice most systems attempt to standardize on a single library. Microsoft Windows has supported subpixel rendering since Windows XP. | https://en.wikipedia.org/wiki/Font_rendering |
On the other hand, the standard Microsoft rasterizer without ClearType is an example of one that prioritizes type designer's intent of clarity; by forcing text into integral coordinate positions, following the type designer's intent of hinting, and even not antialiasing certain fonts at certain sizes, following the type designer's intent of the gasp table, it becomes easier to read on the screen, but may appear somewhat different when printed. This has changed with Direct2D/DirectWrite shipping on Windows 7 and Windows Vista platform update, allowing subpixel text positioning to 1/16 pixel sizes.Mac OS X's Quartz is distinguished by the use of subpixel positioning; it does not force glyphs into exact pixel locations, instead using various antialiasing techniques, including subpixel rendering, to position characters and lines to appear further from the type designer's intent of hinting and closer to the original outline. The result is that the on-screen display looks extremely similar to printed output, but can occasionally be difficult to read at smaller point sizes. | https://en.wikipedia.org/wiki/Font_rendering |
The Quartz renderer has, since macOS Mojave, removed subpixel rendering, relying purely on greyscale anti-aliasing instead. This change is acceptable to HiDPI "retina" screens, but make text on external monitors harder to read.Most other systems use the FreeType library, which depending on the settings, can fall anywhere between Microsoft's and Apple's implementations; it supports hinting and anti-aliasing, and optionally performs subpixel rendering and positioning. FreeType also offers some features not present in either implementation such as color-balanced subpixel rendering and gamma correction.Applications may also bring their own font rendering solutions. Graphics frameworks like Skia Graphics Engine (used by Chrome) occasionally use their own font renderer. Video games and other 3D applications may also need faster, GPU-based renderers such as various SDF-based renderers and "Slug". | https://en.wikipedia.org/wiki/Font_rendering |
In modern parallel computing systems, memory consistency must be maintained to avoid undesirable outcomes. Strict consistency models like sequential consistency are intuitively composed but can be quite restrictive in terms of performance as they would disable instruction level parallelism which is widely applied in sequential programming. To achieve better performance, some relaxed models are explored and release consistency is an aggressive relaxing attempt. | https://en.wikipedia.org/wiki/Release_Consistency |
In modern particle accelerators at high energies, the predictions of special relativity are routinely confirmed, and are necessary for the design and theoretical evaluation of collision experiments, especially in the ultrarelativistic limit. For instance, time dilation must be taken into account to understand the dynamics of particle decay, and the relativistic velocity addition theorem explains the distribution of synchrotron radiation. Regarding the relativistic energy-momentum relations, a series of high precision velocity and energy-momentum experiments have been conducted, in which the energies employed were necessarily much higher than the experiments mentioned above. | https://en.wikipedia.org/wiki/Tests_of_relativistic_energy_and_momentum |
In modern particle physics, forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying gauge bosons. With the development of quantum field theory and general relativity, it was realized that force is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). The conservation of momentum can be directly derived from the homogeneity or symmetry of space and so is usually considered more fundamental than the concept of a force. Thus the currently known fundamental forces are considered more accurately to be "fundamental interactions". | https://en.wikipedia.org/wiki/Physical_force |
: 199–128 While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time, which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. | https://en.wikipedia.org/wiki/Physical_force |
World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines and, in the case of virtual particle exchange, are absorbed at an adjacent vertex. The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and antineutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force. | https://en.wikipedia.org/wiki/Physical_force |
In modern peacekeeping operations, which involve both civilian and military operations, more comprehensive (not just military) doctrines are now emerging such as the 2008 United Nations peacekeeping operations' "Capstone Doctrine" which speaks to integrated civilian and military operations. | https://en.wikipedia.org/wiki/Religious_doctrine |
In modern personal computers, almost every peripheral device uses an adapter to communicate with a system bus, for example: Display adapters used to transmit signals to a display device Universal Serial Bus (USB) adapters for printers, keyboards and mice, among others Network adapters used to connect a computer to a network Host bus adapters used to connect hard disks or other storage | https://en.wikipedia.org/wiki/Adapter_(computing) |
In modern philosophical use, the term phenomena means things as they are experienced through the senses and processed by the mind as distinct from things in and of themselves (noumena). In his inaugural dissertation, titled On the Form and Principles of the Sensible and Intelligible World, Immanuel Kant (1770) theorizes that the human mind is restricted to the logical world and thus can only interpret and understand occurrences according to their physical appearances. He wrote that humans could infer only as much as their senses allowed, but not experience the actual object itself. This may make sense in terms of a communications-channel (epistemology) feeding from an ensemble of inputs (ontology) yet not in the sense of applying wise imagination (à la Albert Einstein, to partial success). Thus, the term phenomenon refers to any incident deserving of inquiry and investigation, especially processes and events which are particularly unusual or of distinctive importance. | https://en.wikipedia.org/wiki/Scientific_phenomenon |
In modern philosophy, nominalism was revived by Thomas Hobbes and Pierre Gassendi.In contemporary analytic philosophy, it has been defended by Rudolf Carnap, Nelson Goodman, H. H. Price, and D. C. Williams.Lately, some scholars have been questioning what kind of influences nominalism might have had in the conception of modernity and contemporaneity. According to Michael Allen Gillespie, nominalism profoundly influences these two periods. Even though modernity and contemporaneity are secular eras, their roots are firmly established in the sacred. | https://en.wikipedia.org/wiki/Medieval_nominalism |
Furthermore, "Nominalism turned this world on its head," he argues. "For the nominalists, all real being was individual or particular and universals were thus mere fictions. "Another scholar, Victor Bruno, follows the same line. | https://en.wikipedia.org/wiki/Medieval_nominalism |
According to Bruno, nominalism is one of the first signs of rupture in the medieval system. "The dismembering of the particulars, the dangerous attribution to individuals to a status of totalization of possibilities in themselves, all this will unfold in an existential fissure that is both objective and material. The result of this fissure will be the essays to establish the nation state." | https://en.wikipedia.org/wiki/Medieval_nominalism |
In modern philosophy, the coherence theory of truth was defended by Baruch Spinoza, Immanuel Kant, Johann Gottlieb Fichte, Karl Wilhelm Friedrich Schlegel, Georg Wilhelm Friedrich Hegel and Harold Henry Joachim (who is credited with the definitive formulation of the theory). However, Spinoza and Kant have also been interpreted as defenders of the correspondence theory of truth. In contemporary philosophy, several epistemologists have significantly contributed to and defended the theory, primarily Brand Blanshard (who gave the earliest characterization of the theory in contemporary times) and Nicholas Rescher. | https://en.wikipedia.org/wiki/Coherence_theory_of_truth |
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