Etymology.

The epithets "white" for Croats and their homeland Croatia, as well "great" ("megali") for Croatia, is in relation to the symbolism used in ancient times. "White" is related to the use of colors among Eurasian peoples to indicate cardinal directions. White meant "Western Croats/Croatia", in comparison to the lands to the east where they had lived before. The epithet "great" signified "subsequently populated" land, but also "old, ancient, former" homeland for the newly arrived Croats to the Roman province of Dalmatia. Historian A. Mayorov emphasizes that the term "White Croats" and the corresponding "White Croatia" are relatively new terms that were applied historically after the Croat migrated to new territories. According to the 10th-century "De Administrando Imperio" ("DAI"), Croats who remained living in their former lands near the borders of Francia were only recently been called "White Croats."

Sources.

The 10th-century treatise "De Administrando Imperio" ("On the management of the Empire", later "DAI"), written in Greek by Constantine VII Porphyrogennetos, is the only known document that suggests "White Croatia" as the place from which Croats migrated to Dalmatia, bordering the coastline of the Adriatic Sea. In Chapter 30, under the heading "The Story of the Province of Dalmatia," it says that "the Croats at that time were dwelling beyond Bagibareia, where the Belocroats are now... The rest of the Croats stayed over near Francia, and are now called the Belocroats, that is, the White Croats, and have their own archon; they are subject to Otto, the great king of Francia, which is also Saxony, and are unbaptized, and intermarry and are friendly with the Turks".  In Chapter 31, "Of the Croats and of the Country They Now Dwell in", it says that Croats in Dalmatia "are descended from the unbaptized Croats, also called the ‘white’, who live beyond Turkey and next to Francia, and they border the Slavs, the unbaptized Serbs... ancient Croatia, also called "white", is still unbaptized to this day, as are also its neighboring Serbs... constantly plundered by the Franks and Turks and Pechenegs... live far away from sea; it takes 30 days of travel from the place where they live to the sea. The sea to which they come down to after 30 days, is that which is called dark".  In Chapter 32, "Of the Serbs and of the Country They Now Dwell in," it was said about the unbaptized ("white") Serbs, that "their neighbor is Francia, as is also Megali Croatia, the unbaptized, also called 'white.  "Croatia Alba", or White Croatia, is referred to in the Latin "Chronicle of the Priest of Duklja", compiled no earlier than the 12th century. This work refers to White Croatia as the lower part of Dalmatia ("Croatia Alba, que et inferior Dalmatia dicitur"), as opposed to "Red Croatia", which refers to upper Dalmatia ("Croatia Rubea, que et superior Dalmatia dicitur"). According to 21st-century historian A. Mayorov, the territory of this "Croatia Alba" was the most developed and densely populated and formed the core of the emerging Croatian state.  In the undated part of twelfth-century Russian "Primary Chronicle", which tells about the resettlement of the Slavs from the Danube, White Croats were mentioned once, together with Sorbs and Chorutans (Carinthians). Czechs immediately associated with the White Croats and the Sorbs. According to A. Mayorov, this account is based on Western European medieval tradition and agrees with the "Chronicle of the Priest of Duklja".

Dispute.

In 21st-century scholarship, historians do not agree on the location or even the existence of such a White Croatia. According to Majorov, in the 10th century, the ethnic Croats are believed to have been surviving in remnant communities, scattered in the West in Bohemia, with others in the East in Poland, Ukraine, and Slovakia. Given the tradition of using colours for cardinal directions, L. V. Vojtovič argued that the Great Croatia referred to in the 6th century no longer existed in the 10th century. The term White Croatia was used to refer to the Western part of its territory. Scholars such as Vatroslav Jagić have rejected the existence of an independent polity. Similarly, V. V. Sedov noted that there is no archaeological material to prove its existence. Scholars do believe that the Croats gradually moved from the East to the West and South.  Interpretations have differed over what geographic area the term "Bagibaria" refers to. Some scholars have related it to "Babia gora" near the river Vistula and Krakow in Lesser Poland, but it is more commonly considered to be a reference to Bavaria. Tibor Živković notes that this term can come from the Latin name of Bavaria ("Bagoaria" or, less probably, "Baioaria") and, therefore, the source of this information for the "DAI" could be of West European origin.  Another dispute is about the geographic reference point of the mentioned "sea to which they come down to after 30 days, is that which is called dark." Some scholars believe this is a reference to the Baltic Sea, to which people could travel in less than 15 days from Lesser Poland. Others say it is the Black Sea, to which travel would take around 30 days from Prykarpattia. The Byzantines knew of the present-day Black Sea very well, but they did not refer to it as "Black" or "Dark", but by a word meaning "Hospitable" (Εὔξεινος). They also used a different term for the word "sea" in its case ("Πόντος" ("Póntos") and not "θάλασσα" ("Thálassa")).  The "DAI" has other contradictory information. Although the Croats are described as living near the Franks in the West, they were said to be subject to repeated raids by the Pecheneg, who lived far to the East of this territory. The "DAI" says that the Pechenegs lived north of the Hungarians, and that the Croats bordered the Hungarians on the south. These chapters are known to have been based on several archival sources. Sedov and Majorov suggest that the "DAI" mistakenly referred to 7th-century locations and migrations of peoples based on the location of contemporary Croats in Bohemia when the account was compiled.

Territory versions.

White Croatia was initially thought to have been located along the river Elbe in Bohemia, and around the Vistula River and Lesser Poland. Contemporary Ukrainian and Russian scholars have tended to attribute the Croats with large and influential territories in the East. Pavel Jozef Šafárik (1795–1861) and Lubor Niederle (1865–1944) placed ancient Croatia in Eastern Galicia, and extending west to the Vistula River. N. P. Barsov (1839–1889) situated the Croats in the wide area of Carpathian Mountains, on the slopes of Tatra Mountains to the river Tisza and Prut on the South, to the Dniester to the East, and the Vistula to the North. Some scholars considered that White Croatia took in Nisa and Upper Elbe in the West, to Bug and Upper Prut and Siret in the East. In other words, much of the lands of present-day Czech Republic, Poland and Ukraine.  According to Francis Dvornik, White Croatia extended from the Southern Bug and rivers Wieprz and San along the Poland-Ukraine border, to the slopes of Carpathian Mountains, including the Northern part of Slovakia, then from the rivers Netolica and Dudleba in upper Vltava, through Cidlina to the Krkonoše Mountains to the North and North-West.  Sedov sharply criticized such assumptions: "These hypothetical constructions are now of purely historiographic interest, since they do not find any confirmation in archaeological materials". He believed that the Croats arose among the Antes. After that, they migrated west and settled in several groups in various places. He said that one of these groups were southwestern neighbours of the Dulebes living in the Northern and Southern area of Eastern Prykarpattia. He and B. O. Tymoshchuk said that the Slavic Gords in Bukovina were an offshoot band abandoned by Croats.  O. A Kupchynsʹkyĭ believed that Eastern Croats had territory from Prykarpattia (at the confluence of the rivers Laborec and Ondava at the crest of the Carpathian Mountains), valley of Beskids, western coast of the river Wisłoka, along Sandomierz valley until middle San, near Dunajec and left coast of Vistula. He said they also likely occupied the upper watershed of the Tisza river at the Ukraine-Slovakia border.  In comparison, some scholars placed White Croatia in a smaller territory, roughly from North-Eastern Bohemia to the Upper Vistula valley. This is based on the "DAI" description that they lived South-East of Bavaria, north of Hungary, and south of the White Serbs. Nada Klaić thought the Croats had migrated from Carantania, rather than from Lesser Poland. A. Mayorov distinguishes between the terms and concepts of "White Croatia" and "Great Croatia". He agrees that White Croatia appeared to have some historical presence in the Upper Elbe and Upper Vistula regions, but that Great Croatia (the pre-motherland for the Croats) was located east of the Carpathians (possibly also in Transcarpathia). Mayorov suggests that the author of the "DAI" made an attempt to reconcile the contradictions among various conflicting sources. Tibor Živković located White Croatia in Bohemia or southern Poland.

White Croatia (also Great Croatia or Chrobatia;, also ) is the region from which part of the White Croats supposedly emigrated to the Western Balkans. Some historians believe that, after the migration of the White Croats in the 7th century, their former homeland gradually lost its primacy and was influenced by other Slavic peoples, such as Czechs and Poles. Others say there was never a distinct polity known as White Croatia. According to the medieval "Chronicle of the Priest of Duklja", another area referred to as White Croatia was located along with Red Croatia in Dalmatia.

La Croatie blanche (en croate "", en polonais, ), également nommée Chrobatie, a été une région s'étendant dans les actuelles Pologne méridionale, Bohême et Slovaquie. La région à l'ouest de la Croatie blanche est connue comme étant la Serbie blanche.


Théorie des ensembles de von Neumann-Bernays-Gödel

Classes in set theory.

The uses of classes.

Classes have several uses in NBG:

Axiom schema versus class existence theorem.

Once classes are added to the language of ZFC, it is easy to transform ZFC into a set theory with classes. First, the axiom schema of class comprehension is added. This axiom schema states: For every formula formula_18 that quantifies only over sets, there exists a class formula_19 consisting of the satisfying the formula—that is, formula_20 Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified to handle classes: If two classes have the same elements, then they are identical. The other axioms of ZFC are not modified.  This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced.  To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many class existence axioms. Then these axioms are used to prove the class existence theorem which implies every instance of the axiom schema. The proof of this theorem requires only seven class existence axioms, which are used to convert the construction of a formula into the construction of a class satisfying the formula.

Axiomatization of NBG.

Classes and sets.

NBG has two types of objects: classes and sets. Intuitively, every set is also a class. There are two ways to axiomatize this. Bernays used many-sorted logic with two sorts: classes and sets. Gödel avoided sorts by introducing primitive predicates: formula_21 for "formula_19 is a class" and formula_23 for "formula_19 is a set" (in German, "set" is "Menge"). He also introduced axioms stating that every set is a class and that if class formula_19 is a member of a class, then formula_19 is a set. Using predicates is the standard way to eliminate sorts. Elliott Mendelson modified Gödel's approach by having everything be a class and defining the set predicate formula_27 as formula_28 This modification eliminates Gödel's class predicate and his two axioms.  Bernays' two-sorted approach may appear more natural at first, but it creates a more complex theory. In Bernays' theory, every set has two representations: one as a set and the other as a class. Also, there are two membership relations: the first, denoted by "∈", is between two sets; the second, denoted by "η", is between a set and a class. This redundancy is required by many-sorted logic because variables of different sorts range over disjoint subdomains of the domain of discourse.  The differences between these two approaches does not affect what can be proved, but it does affect how statements are written. In Gödel's approach, formula_29 where formula_19 and formula_31 are classes is a valid statement. In Bernays' approach this statement has no meaning. However, if formula_19 is a set, there is an equivalent statement: Define "set formula_33 represents class formula_19" if they have the same sets as members—that is, formula_35 The statement formula_36 where set formula_33 represents class formula_19 is equivalent to Gödel's formula_39  The approach adopted in this article is that of Gödel with Mendelson's modification. This means that NBG is an axiomatic system in first-order predicate logic with equality, and its only primitive notions are class and the membership relation.

Definitions and axioms of extensionality and pairing.

A set is a class that belongs to at least one class: formula_19 is a set if and only if formula_41. A class that is not a set is called a proper class: formula_19 is a proper class if and only if formula_43. Therefore, every class is either a set or a proper class, and no class is both (if the theory is consistent).  Gödel introduced the convention that uppercase variables range over classes, while lowercase variables range over sets. Gödel also used names that begin with an uppercase letter to denote particular classes, including functions and relations defined on the class of all sets. Gödel's convention is used in this article. It allows us to write:  The following axioms and definitions are needed for the proof of the class existence theorem.  Axiom of extensionality. If two classes have the same elements, then they are identical.  This axiom generalizes ZFC's axiom of extensionality to classes.  Axiom of pairing. If formula_49 and formula_50 are sets, then there exists a set formula_51 whose only members are formula_49 and formula_50.  As in ZFC, the axiom of extensionality implies the uniqueness of the set formula_51, which allows us to introduce the notation formula_56  Ordered pairs are defined by: Tuples are defined inductively using ordered pairs:

Class existence axioms and axiom of regularity.

Class existence axioms will be used to prove the class existence theorem: For every formula in formula_60 free set variables that quantifies only over sets, there exists a class of that satisfy it. The following example starts with two classes that are functions and builds a composite function. This example illustrates the techniques that are needed to prove the class existence theorem, which lead to the class existence axioms that are needed.  The class existence axioms are divided into two groups: axioms handling language primitives and axioms handling tuples. There are four axioms in the first group and three axioms in the second group.  Axioms for handling language primitives:  Membership. There exists a class formula_61 containing all the ordered pairs whose first component is a member of the second component.  Intersection (conjunction). For any two classes formula_19 and formula_64, there is a class formula_31 consisting precisely of the sets that belong to both formula_19 and formula_64.  Complement (negation). For any class formula_19, there is a class formula_64 consisting precisely of the sets not belonging to formula_19.  Domain (existential quantifier). For any class formula_19, there is a class formula_64 consisting precisely of the first components of the ordered pairs of formula_19.  By the axiom of extensionality, class formula_31 in the intersection axiom and class formula_64 in the complement and domain axioms are unique. They will be denoted by: formula_79 formula_80 and formula_81 respectively. On the other hand, extensionality is not applicable to formula_61 in the membership axiom since it specifies only those sets in formula_61 that are ordered pairs.  The first three axioms imply the existence of the empty class and the class of all sets: The membership axiom implies the existence of a class formula_84 The intersection and complement axioms imply the existence of formula_85, which is empty. By the axiom of extensionality, this class is unique; it is denoted by formula_86 The complement of formula_87 is the class formula_88 of all sets, which is also unique by extensionality. The set predicate formula_27, which was defined as formula_90, is now redefined as formula_91 to avoid quantifying over classes.  Axioms for handling tuples:  Product by formula_88. For any class formula_19, there is a class formula_64 consisting of the ordered pairs whose first component belongs to formula_19.  Circular permutation. For any class formula_19, there is a class formula_64 whose 3tuples are obtained by applying the circular permutation formula_99 to the 3tuples of formula_19.  Transposition. For any class formula_19, there is a class formula_64 whose 3tuples are obtained by transposing the last two components of the 3tuples of formula_19.  By extensionality, the product by formula_88 axiom implies the existence of a unique class, which is denoted by formula_107 This axiom is used to define the class formula_108 of all : formula_109 and formula_110 If formula_19 is a class, extensionality implies that formula_112 is the unique class consisting of the of formula_113 For example, the membership axiom produces a class formula_61 that may contain elements that are not ordered pairs, while the intersection formula_115 contains only the ordered pairs of formula_61.  The circular permutation and transposition axioms do not imply the existence of unique classes because they specify only the 3tuples of class formula_117 By specifying the 3tuples, these axioms also specify the for formula_118 since: formula_119 The axioms for handling tuples and the domain axiom imply the following lemma, which is used in the proof of the class existence theorem.  Tuple lemma.  Proof:Class formula_124: Apply product by formula_88 to formula_19 to produce formula_127
 Class formula_128: Apply transposition to formula_124 to produce formula_130
 Class formula_131: Apply circular permutation to formula_124 to produce formula_133
 Class formula_134: Apply circular permutation to formula_128, then apply domain to produce formula_136  One more axiom is needed to prove the class existence theorem: the axiom of regularity. Since the existence of the empty class has been proved, the usual statement of this axiom is given.  Axiom of regularity. Every nonempty set has at least one element with which it has no element in common.  This axiom implies that a set cannot belong to itself: Assume that formula_138 and let formula_139 Then formula_140 since formula_141 This contradicts the axiom of regularity because formula_49 is the only element in formula_143 Therefore, formula_144 The axiom of regularity also prohibits infinite descending membership sequences of sets: formula_145  Gödel stated regularity for classes rather than for sets in his 1940 monograph, which was based on lectures given in 1938. In 1939, he proved that regularity for sets implies regularity for classes.

Class existence theorem.

Class existence theorem. Let formula_146 be a formula that quantifies only over sets and contains no free variables other than formula_147 (not necessarily all of these). Then for all formula_148, there exists a unique class formula_19 of such that: formula_150 The class formula_19 is denoted by formula_152  The theorem's proof will be done in two steps:  Transformation rules. In rules 1 and 2 below, formula_157 and formula_158 denote set or class variables. These two rules eliminate all occurrences of class variables before an formula_11 and all occurrences of equality. Each time rule 1 or 2 is applied to a subformula, formula_160 is chosen so that formula_161 differs from the other variables in the current formula. The three rules are repeated until there are no subformulas to which they can be applied. This produces a formula that is built only with formula_154, formula_155, formula_156, formula_11, set variables, and class variables formula_166 where formula_166 does not appear before an formula_11.  Transformation rules: bound variables. Consider the composite function formula of example 1 with its free set variables replaced by formula_177 and formula_178: formula_179 The inductive proof will remove formula_180, which produces the formula formula_181 However, since the class existence theorem is stated for subscripted variables, this formula does not have the form expected by the induction hypothesis. This problem is solved by replacing the variable formula_182 with formula_183 Bound variables within nested quantifiers are handled by increasing the subscript by one for each successive quantifier. This leads to rule 4, which must be applied after the other rules since rules 1 and 2 produce quantified variables.  Proof of the class existence theorem. The proof starts by applying the transformation rules to the given formula to produce a transformed formula. Since this formula is equivalent to the given formula, the proof is completed by proving the class existence theorem for transformed formulas.  Gödel pointed out that the class existence theorem "is a metatheorem, that is, a theorem about the system [NBG], not in the system..." It is a theorem about NBG because it is proved in the metatheory by induction on NBG formulas. Also, its proof—instead of invoking finitely many NBG axioms—inductively describes how to use NBG axioms to construct a class satisfying a given formula. For every formula, this description can be turned into a constructive existence proof that is in NBG. Therefore, this metatheorem can generate the NBG proofs that replace uses of NBG's class existence theorem.  A recursive computer program succinctly captures the construction of a class from a given formula. The definition of this program does not depend on the proof of the class existence theorem. However, this proof is needed to prove that the class constructed by the program satisfies the given formula and is built using the axioms. This program is written in pseudocode that uses a Pascal-style case statement.  formula_184
 formula_185  formula_186
 formula_189  Let formula_153 be the formula of example 2. The function call formula_192 generates the class formula_193 which is compared below with formula_194 This shows that the construction of the class formula_19 mirrors the construction of its defining formula formula_194

Extending the class existence theorem.

Gödel extended the class existence theorem to formulas formula_153 containing relations over classes (such as formula_199 and the unary relation formula_200), special classes (such as formula_9), and operations (such as formula_202 and formula_203). To extend the class existence theorem, the formulas defining relations, special classes, and operations must quantify only over sets. Then formula_153 can be transformed into an equivalent formula satisfying the hypothesis of the class existence theorem.  The following definitions specify how formulas define relations, special classes, and operations:  A is defined by:  The following transformation rules eliminate relations, special classes, and operations. Each time rule 2b, 3b, or 4 is applied to a subformula, formula_160 is chosen so that formula_161 differs from the other variables in the current formula. The rules are repeated until there are no subformulas to which they can be applied. formula_217 and formula_157 denote terms.  Class existence theorem (extended version). Let formula_146 be a formula that quantifies only over sets, contains no free variables other than formula_220, and may contain relations, special classes, and operations defined by formulas that quantify only over sets. Then for all formula_221 there exists a unique class formula_19 of such that formula_150  Proof: Apply the transformation rules to formula_153 to produce an equivalent formula containing no relations, special classes, or operations. This formula satisfies the hypothesis of the class existence theorem. Therefore, for all formula_221 there is a unique class formula_19 of satisfying formula_227

Set axioms.

The axioms of pairing and regularity, which were needed for the proof of the class existence theorem, have been given above. NBG contains four other set axioms. Three of these axioms deal with class operations being applied to sets.  Definition. formula_228 is a function if formula_229  In set theory, the definition of a function does not require specifying the domain or codomain of the function (see Function (set theory)). NBG's definition of function generalizes ZFC's definition from a set of ordered pairs to a class of ordered pairs.  ZFC's definitions of the set operations of image, union, and power set are also generalized to class operations. The image of class formula_19 under the function formula_228 is formula_232 This definition does not require that formula_233 The union of class formula_19 is formula_235 The power class of formula_19 is formula_237 The extended version of the class existence theorem implies the existence of these classes. The axioms of replacement, union, and power set imply that when these operations are applied to sets, they produce sets.  Axiom of replacement. If formula_228 is a function and formula_33 is a set, then formula_240, the image of formula_33 under formula_228, is a set.  Not having the requirement formula_244 in the definition of formula_245 produces a stronger axiom of replacement, which is used in the following proof. Theorem (NBG's axiom of separation). If formula_19 is a set and formula_64 is a subclass of formula_193 then formula_64 is a set.
 Proof: The class existence theorem constructs the restriction of the identity function to formula_64: formula_251 Since the image of formula_19 under formula_253 is formula_64, the axiom of replacement implies that formula_64 is a set. This proof depends on the definition of image not having the requirement formula_244 since formula_257 rather than formula_258  Axiom of union. If formula_33 is a set, then there is a set containing formula_260  Axiom of power set. If formula_33 is a set, then there is a set containing formula_263  Theorem. If formula_33 is a set, then formula_266 and formula_267 are sets.
 Proof: The axiom of union states that formula_266 is a subclass of a set formula_269, so the axiom of separation implies formula_266 is a set. Likewise, the axiom of power set states that formula_267 is a subclass of a set formula_269, so the axiom of separation implies that formula_267 is a set.  Axiom of infinity. There exists a nonempty set formula_33 such that for all formula_49 in formula_33, there exists a formula_50 in formula_33 such that formula_49 is a proper subset of formula_50.  The axioms of infinity and replacement prove the existence of the empty set. In the discussion of the class existence axioms, the existence of the empty class formula_87 was proved. We now prove that formula_87 is a set. Let function formula_284 and let formula_33 be the set given by the axiom of infinity. By replacement, the image of formula_33 under formula_228, which equals formula_87, is a set.  NBG's axiom of infinity is implied by ZFC's axiom of infinity: formula_289 The first conjunct of ZFC's axiom, formula_290, implies the first conjunct of NBG's axiom. The second conjunct of ZFC's axiom, formula_291, implies the second conjunct of NBG's axiom since formula_292 To prove ZFC's axiom of infinity from NBG's axiom of infinity requires some of the other NBG axioms (see Weak axiom of infinity).

Axiom of global choice.

The class concept allows NBG to have a stronger axiom of choice than ZFC. A choice function is a function formula_5 defined on a set formula_4 of nonempty sets such that formula_7 for all formula_8 ZFC's axiom of choice states that there exists a choice function for every set of nonempty sets. A global choice function is a function formula_1 defined on the class of all nonempty sets such that formula_2 for every nonempty set formula_3 The axiom of global choice states that there exists a global choice function. This axiom implies ZFC's axiom of choice since for every set formula_4 of nonempty sets, formula_301 (the restriction of formula_1 to formula_4) is a choice function for formula_304 In 1964, William B. Easton proved that global choice is stronger than the axiom of choice by using forcing to construct a model that satisfies the axiom of choice and all the axioms of NBG except the axiom of global choice. The axiom of global choice is equivalent to every class having a well-ordering, while ZFC's axiom of choice is equivalent to every set having a well-ordering.  Axiom of global choice. There exists a function that chooses an element from every nonempty set.

History.

Von Neumann's 1925 axiom system.

Von Neumann published an introductory article on his axiom system in 1925. In 1928, he provided a detailed treatment of his system. Von Neumann based his axiom system on two domains of primitive objects: functions and arguments. These domains overlap—objects that are in both domains are called argument-functions. Functions correspond to classes in NBG, and argument-functions correspond to sets. Von Neumann's primitive operation is function application, denoted by ["a", "x"] rather than "a"("x") where "a" is a function and "x" is an argument. This operation produces an argument. Von Neumann defined classes and sets using functions and argument-functions that take only two values, "A" and "B". He defined "x" ∈ "a" if ["a", "x"] ≠ "A".  Von Neumann's work in set theory was influenced by Georg Cantor's articles, Ernst Zermelo's 1908 axioms for set theory, and the 1922 critiques of Zermelo's set theory that were given independently by Abraham Fraenkel and Thoralf Skolem. Both Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {"Z", "Z", "Z",...} where "Z" is the set of natural numbers and "Z" is the power set of "Z". They then introduced the axiom of replacement, which would guarantee the existence of such sets. However, they were reluctant to adopt this axiom: Fraenkel stated "that Replacement was too strong an axiom for 'general set theory'", while "Skolem only wrote that 'we could introduce' Replacement".  Von Neumann worked on the problems of Zermelo set theory and provided solutions for some of them:

Von Neumann's 1929 axiom system.

In 1929, von Neumann published an article containing the axioms that would lead to NBG. This article was motivated by his concern about the consistency of the axiom of limitation of size. He stated that this axiom "does a lot, actually too much." Besides implying the axioms of separation and replacement, and the well-ordering theorem, it also implies that any class whose cardinality is less than that of "V" is a set. Von Neumann thought that this last implication went beyond Cantorian set theory and concluded: "We must therefore discuss whether its [the axiom's] consistency is not even more problematic than an axiomatization of set theory that does not go beyond the necessary Cantorian framework."  Von Neumann started his consistency investigation by introducing his 1929 axiom system, which contains all the axioms of his 1925 axiom system except the axiom of limitation of size. He replaced this axiom with two of its consequences, the axiom of replacement and a choice axiom. Von Neumann's choice axiom states: "Every relation "R" has a subclass that is a function with the same domain as "R"."  Let "S" be von Neumann's 1929 axiom system. Von Neumann introduced the axiom system "S" + Regularity (which consists of "S" and the axiom of regularity) to demonstrate that his 1925 system is consistent relative to "S". He proved: These results imply: If "S" is consistent, then von Neumann's 1925 axiom system is consistent. Proof: If "S" is consistent, then "S" + Regularity is consistent (result 1). Using proof by contradiction, assume that the 1925 axiom system is inconsistent, or equivalently: the 1925 axiom system implies a contradiction. Since "S" + Regularity implies the axioms of the 1925 system (result 2), "S" + Regularity also implies a contradiction. However, this contradicts the consistency of "S" + Regularity. Therefore, if "S" is consistent, then von Neumann's 1925 axiom system is consistent.  Since "S" is his 1929 axiom system, von Neumann's 1925 axiom system is consistent relative to his 1929 axiom system, which is closer to Cantorian set theory. The major differences between Cantorian set theory and the 1929 axiom system are classes and von Neumann's choice axiom. The axiom system "S" + Regularity was modified by Bernays and Gödel to produce the equivalent NBG axiom system.

Bernays' axiom system.

In 1929, Paul Bernays started modifying von Neumann's new axiom system by taking classes and sets as primitives. He published his work in a series of articles appearing from 1937 to 1954. Bernays stated that:  Bernays handled sets and classes in a two-sorted logic and introduced two membership primitives: one for membership in sets and one for membership in classes. With these primitives, he rewrote and simplified von Neumann's 1929 axioms. Bernays also included the axiom of regularity in his axiom system.

Gödel's axiom system (NBG).

In 1931, Bernays sent a letter containing his set theory to Kurt Gödel. Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort and one membership primitive. He also weakened some of Bernays' axioms and replaced von Neumann's choice axiom with the equivalent axiom of global choice. Gödel used his axioms in his 1940 monograph on the relative consistency of global choice and the generalized continuum hypothesis.  Several reasons have been given for Gödel choosing NBG for his monograph:  Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades. In 1963, Paul Cohen proved his independence proofs for ZF with the help of some tools that Gödel had developed for his relative consistency proofs for NBG. Later, ZFC became more popular than NBG. This was caused by several factors, including the extra work required to handle forcing in NBG, Cohen's 1966 presentation of forcing, which used ZF, and the proof that NBG is a conservative extension of ZFC.

NBG, ZFC, and MK.

NBG is not logically equivalent to ZFC because its language is more expressive: it can make statements about classes, which cannot be made in ZFC. However, NBG and ZFC imply the same statements about sets. Therefore, NBG is a conservative extension of ZFC. NBG implies theorems that ZFC does not imply, but since NBG is a conservative extension, these theorems must involve proper classes. For example, NBG implies that the proper class "V" can be well-ordered and that every proper class can be put into one-to-one correspondence with "V".  One consequence of conservative extension is that ZFC and NBG are equiconsistent. Proving this uses the principle of explosion: from a contradiction, everything is provable. Assume that either ZFC or NBG is inconsistent. Then the inconsistent theory implies the contradictory statements ∅ = ∅ and ∅ ≠ ∅, which are statements about sets. By the conservative extension property, the other theory also implies these statements. Therefore, it is also inconsistent. So although NBG is more expressive, it is equiconsistent with ZFC. This result together with von Neumann's 1929 relative consistency proof implies that his 1925 axiom system with the axiom of limitation of size is equiconsistent with ZFC. This completely resolves von Neumann's concern about the relative consistency of this powerful axiom since ZFC is within the Cantorian framework.  Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa). For a survey of known results of this nature, see.  Morse–Kelley set theory has an axiom schema of class comprehension that includes formulas whose quantifiers range over classes. MK is a stronger theory than NBG because MK proves the consistency of NBG, while Gödel's second incompleteness theorem implies that NBG cannot prove the consistency of NBG.  For a discussion of some ontological and other philosophical issues posed by NBG, especially when contrasted with ZFC and MK, see Appendix C of.

Models.

ZFC, NBG, and MK have models describable in terms of the cumulative hierarchy "V" and the constructible hierarchy "L". Let "V" include an inaccessible cardinal κ, let "X" ⊆ "V", and let Def("X") denote the class of first-order definable subsets of "X" with parameters. In symbols where "formula_306" denotes the model with domain formula_307 and relation formula_11, and "formula_309" denotes the satisfaction relation:  Then:

Category theory.

The ontology of NBG provides scaffolding for speaking about "large objects" without risking paradox. For instance, in some developments of category theory, a "large category" is defined as one whose objects and morphisms make up a proper class. On the other hand, a "small category" is one whose objects and morphisms are members of a set. Thus, we can speak of the "category of all sets" or "category of all small categories" without risking paradox since NBG supports large categories.  However, NBG does not support a "category of all categories" since large categories would be members of it and NBG does not allow proper classes to be members of anything. An ontological extension that enables us to talk formally about such a "category" is the conglomerate, which is a collection of classes. Then the "category of all categories" is defined by its objects: the conglomerate of all categories; and its morphisms: the conglomerate of all morphisms from "A" to "B" where "A" and "B" are objects. On whether an ontology including classes as well as sets is adequate for category theory, see.

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel-Choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

La théorie des ensembles de von Neumann–Bernays–Gödel, abrégée en NBG ou théorie des classes, est une théorie axiomatique essentiellement équivalente à la théorie ZFC de Zermelo-Fraenkel avec axiome du choix (et avec les mêmes variantes possibles), mais dont le pouvoir expressif est plus riche. Elle peut s’énoncer en un "nombre fini d’axiomes", et donc sans schéma, au contraire de ZFC (voir schéma d'axiomes de compréhension et schéma d'axiomes de remplacement). Ceci n’est possible que grâce à une modification du langage de la théorie, qui permet de parler directement de classe, une notion par ailleurs utile en théorie des ensembles et qui apparaissait déjà, de façon assez informelle, dans les écrits de Georg Cantor dès avant 1900.


History.

Bad Vilbel was founded in 774 (first written document) but much older artefacts were found in the area. In 1848 during railway works, a Roman villa was unearthed with a Thermae and a Mosaic. A replica of this mosaic is presented in a modern exhibition in the spa gardens.

20th Century.

The town Vilbel got the label "Bad" (spa) in 1948 for its numerous mineral springs. The health spa operations stopped in the 1960s but the mineral water industry connected more springs of the Wetterau by pipelines to the bottling plant of "Hassia" in Bad Vilbel. The hessian government reform formed 1971/72 Bad Vilbel (with Heilsberg), Dortelweil, Gronau and Massenheim to the new city Bad Vilbel. Since 1997 great areas have been developed for living and business, like the residential area of Dortelweil-West or the commercial park "Quellenpark" between Bad Vilbel, Massenheim and Dortelweil.

Mayor.

Thomas Stöhr (born 1966) (CDU) was elected in March 2003. He was re-elected in February 2010 and in March 2016.

Transport.

"Bad Vilbel" has four railway stations (Bad Vilbel, Bad Vilbel Süd and Dortelweil on the Main–Weser Railway and Gronau on the Vilbel–Glauburg-Stockheim railway) served by Frankfurt's local transport network (S-Bahn line S 6 and RE Line 34). It has access to the A661 autobahn and the highway B3.  Furthermore, Bad Vilbel has several local bus lines called "Vilbus" connecting the city center to the neighborhoods. Additional connections to Frankfurt am Main with the bus line "30" and with Offenbach am Main with the "fast bus" line "X97".

International relations.

Bad Vilbel is twinned with:  The Glossop/Bad Vilbel Twinning Association was established in 1985 and the formal twinning ceremonies were held in Glossop and Bad Vilbel in 1987.

Bad Vilbel is a spa town in Hesse (Hessen), Germany, famous for its many mineral water springs. Bad Vilbel is the largest town in the Wetteraukreis district and part of the Frankfurt Rhein-Main urban area with its city center being located 8 km northeast of downtown Frankfurt am Main at the banks of the river Nidda.

Bad Vilbel est une ville du land de Hesse en Allemagne, connue pour ses sources d'eau minérale. Elle est située à au nord-est de Francfort-sur-le-Main et est arrosée par la rivière Nidda.


Performance history.

It was commissioned to celebrate the marriage of the Infanta Maria Teresa Rafaela of Spain to Louis, Dauphin of France. "La princesse de Navarre" opened the wedding festivities, while another new Rameau opera, "Platée", closed them. The piece takes the form of a "comédie-ballet", effectively a play with a large amount of incidental music, recalling the collaborations of Molière and Lully in the 17th century.  Voltaire was a great admirer of Rameau, even considering him too good a composer for such a task. Nevertheless, Rameau wrote around an hour of music for the play, including an overture and three "divertissements" (musical interludes which ended each act). Little of it has much bearing on the main action of the drama which concerns the complicated love life of the eponymous mediaeval princess. Voltaire found Rameau to be a demanding and critical collaborator, leading the dramatist to declare: "Poor Rameau is mad...Rameau is as great an eccentric as he is a musician". The production was a spectacular one, involving no less than 180 "extras". Voltaire complained about the acoustics of the hall in which it was staged claiming "the ceiling was so high that the actors appeared pygmies and they couldn't be heard". Nevertheless, the music was a critical success. Much of the material was reworked to produce another opera, "Les fêtes de Ramire", later the same year.

Roles.

Source: "Oeuvres complètes de Voltaire", Théatre, ninth tome, Société Littéraire-typographique, 1784 (digitalized by Google) (accessed 21 October 2010)

Singing characters.

Fifteen women and twenty-five men

Synopsis.

Act 1.

Don Pedro, King of Castile, has taken captive Constance, Princess of Navarre. She manages to escape from prison in disguise and takes refuge with Don Morillo, but she is recognised by the young Alamir. Alamir falls in love with Constance and rejects Don Morillo’s daughter Sanchette. The act ends with a divertissement in which warriors, gypsies and astrologers entertain the lords and ladies.

Act 2.

The king demands the return of Constance but Alamir vows to protect her. Alamir’s army, in alliance with the French, defeats the Castilians. Constance feels herself falling more and more in love with Alamir but she is reluctant to hurt Sanchette and sees Alamir’s lack of noble birth as a barrier to their union. The Graces distract her from her cares with a divertissement.

Act 3.

Alamir returns from battle in triumph and reveals his true identity as the Duke of Foix. He and Constance can now marry and the play ends with a divertissement led by Cupid against the backdrop of the Pyrenees.

La princesse de Navarre ("The Princess of Navarre") is a "comédie-ballet" with music by Jean-Philippe Rameau and words by Voltaire, first performed on 23 February 1745 at La Grande Ecurie, Versailles.

La Princesse de Navarre est une comédie-ballet de Voltaire, en un prologue et trois actes, avec musique de Jean-Philippe Rameau.


History.

Establishment.

The club was established in 1967 by Mahmoud Khayami with financial help from the Iran Khodro factory. The main objective was to promote their factory's products, and improve their reputation. They were able to lure great players such as Ali Parvin, Majid Halvaei and Amir Abedini, and several of the Persepolis players of the time. In 1969 the team became the Tehran City League's champions and the team appeared to have a bright future. The team was dissolved in 1970 after a series of difficulties between the players and management. Most of the players went back to Persepolis.

Rebirth of Paykan.

Paykan still operated without a football team specifically in the 1980s but carried on with basketball, handball, and volleyball teams. Paykan did not have a football team until the year 2000, when they were able to buy Bahman's shares and participate in the top level of Iranian football after 30 years. The team participated in the 2000–01 Azadegan League with Bahman's squad but with a different coach.  Among Iranian football clubs, the team is known to have one of the better organizations in terms of facilities and management. The team had little success and were relegated from the Iran Pro League in the 2004–05 season. They were back in the IPL for the 06–07 season where they finished a respectable 7th and showed their ability to compete at the highest standard.

Relocation to Qazvin.

In 2008 Paykan was moved to Qazvin because the city of Tehran had many football teams with low attendance figures. They started the season well but toward the end their results got worse. The team changed their coach in the last weeks of the 2008–09 season and finished in mid table. Although they finished in the top half of the table in 2009–10 season the club chairman decided to replace Hamid Derakhshan with Mohammad Ahmadzadeh as the head coach. Paykan were relegated in 2010–2011, but the following year they earned their promotion back to the Iran Pro League for the 2012–2013 season.

Tehran and Qods.

Paykan moved back to Tehran and started playing in the newly renovated Takhti Stadium. Paykan was relegated to Azadegan League again in April 2013. Before the start of the 2013–14 season Paykan once again relocate and this time to the city of Qods. In the 2013–2014 season Paykan finished 2nd in the Azadegan League, thus giving them a shot at Iran Pro League promotion. Paykan defeated Siah Jamegan 3–1 on aggregate and advanced to play Iran Pro League club Fajr Sepasi in a promotion play-off. The first game in Shiraz ended 0–0 with neither team getting the upper hand. But, in the return leg in Qods, Paykan achieved promotion with a lone goal in the 65th minute. Following the good support from the people of Qods, Paykan announced they will be playing in that city for the 2014–15 Persian Gulf Pro League season. At the end of the 2014–15 season, Paykan finished 15th and was relegated.  In the 2015–16 Azadegan League season, Paykan finished first and were directly promoted to the Persian Gulf Pro League. In the winter of 2017, after a good first half of the season, Paykan became the first ever Persian Gulf Pro League side to sign an Afghan player, when they signed Afghan national team captain Faysal Shayesteh.

Sponsorship.

Official sponsors.

For the 2007/08 season it has been announced that Paykan's official kit sponsor will be ISACO.

Players.

First-team squad.

Updated as December 10, 2018<BR>

Loan list.

"For recent transfers, see List of Iranian football transfers winter 2018–19".

Former players.

"For notable former players, see."

Paykan Football Club () is a football team based in Tehran, Iran. The team is sponsored by Iran's main automobile manufacturer Iran Khodro and is named after one of its older products; the Paykan car. Paykan F.C. is the football club of the multisport "Paykan Sport Club" which also includes Basketball and volleyball teams.

Le Paykan Qazvin Football Club est un club iranien de football fondé en 1967 et dont le stade est situé à Qazvin. Hossein Faraki est l'entraineur depuis début. Le sponsor de l'équipe est Iran Khodro, principal constructeur automobile du pays, ce qui explique qu'elle porte le nom de la voiture phare de l'entreprise : la Paykan, dont la production s'est arrêtée en 2005.


Etymology.

The name of the town derives from the curved riverbed of the river "Kriva". The old name of Kriva Palanka was "Egri Dere", in Turkish - "Curved River". The Turkish word "dere" was later substituted with "Palanka".

Geography and climate.

Kriva Palanka has a temperate continental climate with moderately cold winter, warm summer, and relatively warm autumn. The higher parts of the municipality on Osogovo mountain are affected by the alpine climate. The average annual temperature is 10.2 C. The warmest month is July with an average temperature of 20.0 C and the coldest month is January with an average temperature of -0.3 C. Compared to the surrounding region, the Kriva Palanka area gets significant rainfall. This is due to the high altitude setting, which stimulates orographic precipitation of moisture carried by the western and southern winds. The average date of first snow cover in the area is 30 November. The areas over 1700 meters above sea level have quite low mean annual temperatures, therefore the peaks of Ruen and Carev Vrv are under snow cover from October until early June. Ruen remains under snow even in July.

History.

The town is one of the youngest in North Macedonia. It was settled by Ottoman vizier Bayram Pasha and originally served as a fort to strengthen that region. The formation of the town is noted in Turkish documents from 1633. This includes a stone plate with Arabic encryption on the entrance of the fortress. The plate was later moved to the mosque which was situated in the center of the town and now is kept in the Museum of Macedonia in Skopje.  Its favorable position enabled this small settlement to become a larger town. There was a significant Christian influence in the town which led to the development and revival of the town in a Christian and orthodox manner. A class of rich and influential traders and artisans was formed who traveled outside the Ottoman Empire and brought new ideas of the 19th century to the city.  During this period, one of the first Macedonian revivalists Joachim Krchovski, who by the help of the local traders in 1814-1819, in Budim printed five famous books in lively folk speech. Joachim not only addressed the need for literacy, but had a crucial role in the establishment of the church - cell school in 1817, which was placed in Enger's house, in which vicinity later with great efforts by the city aristocrats and the donor David Jerej, in 1833 the church of St. Dimitrija would be built.  As a result of church-educational activity in Kriva Palanka, as early as 1833 functions the epitropic - church community, which in 1861 was transformed into a church - school community, and already the same year will be self-governed without renouncement of the Skopje Patriarchal mitropolitan.  From 1877 to 1912, Kriva Palanka was part of the Kosovo Vilayet of the Ottoman Empire, while after the Balkan Wars it became part of the Kingdom of Serbia.  After the First World War, in 1919-1920 when Kriva Palanka for the first time obtained the status of municipality as part of the Vardar Banovina of the Kingdom of Serbs, Croats and Slovenes, the town received its first native president (mayor) and its physical appearance, which has mostly been maintained as it is today. In that time Kriva Palanka became richer with a few significant facilities: The Municipal Building, The Officer's House, primary school Partizan, the old Hospital, and the first regulation of the riverbed of Kriva Reka from Osichka Maala to Numulija. In addition, the construction of the first power plant in North-East Macedonia began on the river Durachka, but the construction was stopped later on.  From 1941 to 1944, during the Axis occupation of Yugoslavia, Kriva Palanka, as most of Vardar Macedonia, was annexed by the Kingdom of Bulgaria.

Population.

The city has a favorable demographic picture with population growth. Notable is the high birthrate among the Macedonians, given that in the city and several surrounding villages, many families of Macedonian nationality live with 4 or more children.

Religion.

The population of Kriva Palanka is mostly of the Orthodox faith. Orthodoxy is deeply entered into the life and traditions of the citizens of Kriva Palanka for which testify the large number of churches and monasteries that are located in the city and in the immediate vicinity. Kriva Palanka is well known in North Macedonia for one of the most beautiful monastery complexes, the monastery of Saint Joachim Osogovski.

Economy.

Economically, trade is most developed in Kriva Palanka due to the proximity to the Bulgarian border, and the rest are engaged in agriculture (mostly cultivated potato, to a lesser extent corn, wheat, barley, rye, oats, etc..), then mining is also important with the nearby mines Toranica (lead and zinc) and Bentomak. In the center of Kriva Palanka there are several traditional craft stores that are still actively working. There are also grocery stores, and on the hills near the town there are many sawmills and cutmills for firewood and furniture. In the city operate many kindergartens, three primary schools and a high school. Due to its location on the European Corridor G-8, since a long time ago is planned an extension of the railroad to Kriva Palanka and the Bulgarian border, which is currently built to the village of Beljakovce. In near future will begin the project for gasification of the town.

Twin cities.

List of Kriva Palanka's sister and twin cities:

Kriva Palanka ( ) is a town located in the northeastern part of North Macedonia. It has 14,558 inhabitants. The town of Kriva Palanka is the seat of Kriva Palanka Municipality which has almost 21,000 inhabitants.

Kriva Palanka (en macédonien : Крива Паланка, prononcé ) est une municipalité et une ville du nord-est de la Macédoine du Nord. La municipalité comptait habitants en 2002 et couvre. La ville en elle-même comptait alors habitants, le reste de la population étant réparti dans les villages alentour.)


Definitions.

Let "S" be a partially ordered set. Two elements "a" and "b" of a partially ordered set are called comparable if "a" ≤ "b" or "b" ≤ "a". If two elements are not comparable, they are called incomparable; that is, "x" and "y" are incomparable if neither "x" ≤ "y" nor "y" ≤ "x".  A chain in "S" is a subset "C" of "S" in which each pair of elements is comparable; that is, "C" is totally ordered. An antichain in "S" is a subset "A" of "S" in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in "A".

Height and width.

A maximal antichain is an antichain that is not a proper subset of any other antichain. A maximum antichain is an antichain that has cardinality at least as large as every other antichain. The "width" of a partially ordered set is the cardinality of a maximum antichain. Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into "k" chains then the width of the order must be at most "k" (if the antichain has more than "k" elements, by the pigeonhole principle, there would be 2 of its elements belonging to the same chain, contradiction). Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width. Similarly, one can define the "height" of a partial order to be the maximum cardinality of a chain. Mirsky's theorem states that in any partial order of finite height, the height equals the smallest number of antichains into which the order may be partitioned.

Sperner families.

An antichain in the inclusion ordering of subsets of an "n"-element set is known as a Sperner family. The number of different Sperner families is counted by the Dedekind numbers, the first few of which numbers are Even the empty set has two antichains in its power set: one containing a single set (the empty set itself) and one containing no sets.

Join and meet operations.

Any antichain "A" corresponds to a lower set In a finite partial order (or more generally a partial order satisfying the ascending chain condition) all lower sets have this form. The union of any two lower sets is another lower set, and the union operation corresponds in this way to a join operation on antichains: Similarly, we can define a meet operation on antichains, corresponding to the intersection of lower sets: The join and meet operations on all finite antichains of finite subsets of a set "X" define a distributive lattice, the free distributive lattice generated by "X". Birkhoff's representation theorem for distributive lattices states that every finite distributive lattice can be represented via join and meet operations on antichains of a finite partial order, or equivalently as union and intersection operations on the lower sets of the partial order.

Computational complexity.

A maximum antichain (and its size, the width of a given partially ordered set) can be found in polynomial time. Counting the number of antichains in a given partially ordered set is #P-complete.

In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.

En mathématiques, plus précisément en théorie des ordres, une antichaîne est une partie d'un ensemble partiellement ordonné dont les éléments sont deux à deux incomparables.


Dedekind numbers.

The number of different Sperner families on a set of "n" elements is counted by the Dedekind numbers, the first few of which are Although accurate asymptotic estimates are known for larger values of "n", it is unknown whether there exists an exact formula that can be used to compute these numbers efficiently.  The collection of all Sperner families on a set of "n" elements can be organized as a free distributive lattice, in which the join of two Sperner families is obtained from the union of the two families by removing sets that are a superset of another set in the union.

Bounds on the size of a Sperner family.

Sperner's theorem.

The "k"-element subsets of an "n"-element set form a Sperner family, the size of which is maximized when "k" = "n"/2 (or the nearest integer to it). Sperner's theorem states that these families are the largest possible Sperner families over an "n"-element set. Formally, the theorem states that, for every Sperner family "S" over an "n"-element set,

LYM inequality.

The Lubell–Yamamoto–Meshalkin inequality provides another bound on the size of a Sperner family, and can be used to prove Sperner's theorem. It states that, if "a" denotes the number of sets of size "k" in a Sperner family over a set of "n" elements, then

Clutters.

A clutter is a family of subsets of a finite set such that none contains any other; that is, it is a Sperner family. The difference is in the questions typically asked. Clutters are an important structure in the study of combinatorial optimization.  If formula_7 is a clutter, then the blocker of "H", denoted by formula_8, is the clutter with vertex set "V" and edge set consisting of all minimal sets formula_9 so that formula_10 for every formula_11. It can be shown that formula_12, so blockers give us a type of duality. We define formula_13 to be the size of the largest collection of disjoint edges in "H" and formula_14 to be the size of the smallest edge in formula_8. It is easy to see that formula_16.

Minors.

There is a minor relation on clutters which is similar to the minor relation on graphs. If formula_7 is a clutter and formula_30, then we may delete "v" to get the clutter formula_31 with vertex set formula_32 and edge set consisting of all formula_11 which do not contain "v". We contract "v" to get the clutter formula_34. These two operations commute, and if "J" is another clutter, we say that "J" is a minor of "H" if a clutter isomorphic to "J" may be obtained from "H" by a sequence of deletions and contractions.

In combinatorics, a Sperner family (or Sperner system; named in honor of Emanuel Sperner), or clutter, is a family F of subsets of a finite set "E" in which none of the sets contains another. Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of "E". A Sperner family is also sometimes called an independent system or irredundant set.

En combinatoire, une famille de Sperner (ou système de Sperner), appelé en l'honneur d'Emanuel Sperner, est un hypergraphe ("E", F) (c'est-à-dire un ensemble "E" et un ensemble F de parties de "E") dans lequel aucun élément de F ne contient un autre. Formellement,


Geography and climate.

The district spreads from 27'40" E to 28'17" E and 80'17"N to 84'35"N. The mountain range Ganesh Himal is the predominate mountain range located within Dhading. Some of the peaks are over. The plus mountain Manaslu is clearly visible from much of Dhading, although it is located within the bounds of Gorkha. The transnational Prithivi Highway connecting Kathmandu and Pokhara runs through the southern portion of the district making for easy access to the Kathmandu valley. The road parallels the Trishuli River. The western border with Gorkha is bisected by the Budhigandaki River.  The district is bounded by  Dhading is the only district of Nepal which ranges from the mountain Ganesh Himal to the Churevawar pradesh of Terai (Chitwan). The district, with Dhading Besi as its district headquarters, covers an area of and has a population (2001) of 338,658. The mountain range "Ganesh" is the predominated mountain range located within Dhading. All of the peaks are over with some approaching. The plus mountain "Manaslu" is clearly visible from much of Dhading, although it is located within the bounds of Gorkha. The transnational "King Prithivi Highway" connecting Kathmandu and Pokhara runs through the southern portion of the district making for easy access too the Kathmandu valley. The road parallels the Trishuli River. Dhading is 80% farmland and 20% forest. The western border with Gorkha is bisected by the Budhigandaki River, one of the prettiest rivers of Nepal.  The people of the district are primarily Bhramin and Chetri in the south and Tamang and Gurung in the north, with much of the center Newari. Gurkha route, the birthplace of founder of Nepal King Prithivi Naryan Shah crosses through Dhading.

Rivers.

The main river of the district is Budi Gandaki which comes from Ganesh Himal and passes through Arughat Bazaar and Salyantar and meets Ankhu khola at Narsingha Dhamm. Budhi Gandaki separates the district from Gorkha district. There are 25 small rivers, the main being Charoudi, Malekhu, Galtukhola, Belkhukhola, Chirandikhola, Maheshkhola, Thopal, Manukhola, Kastekhola, Mastekhola. Besides these, there are over 1743 smaller rivers, springs and seasonal streams.

Demography.

At the time of the 2011 Nepal census, Dhading District had a population of 336,067. Of these, 70.6% spoke Nepali, 19.9% Tamang, 2.5% Chepang, 2.1% Gurung, 1.9% Magar, 0.9% Newari, 0.7% Ghale and 0.5% Kumal as their first language.  25.5% of the population in the district spoke Nepali, 0.5% Tamang and 0.5% English as their second language.  The people of the district are primarily Brahmin and Chhetri in the south and Ghale, Gurung and Tamang in the north, with much of the center Newari. The famous King Prithivi Naryan Shah/Gurkha route crosses through Dhading.

Religious temple.

Dhading District has many religious temples. Among them is Tripurasundari Mai which lies in the northern part of the district. Siddha than in Siddhalekh Rural Municipality is a Hindu religious site. Others are Bhairabi Temple in Sunaula Bazar. Sri Nrsimha Dham Kshetra in Salyantar is a historical religious site for vedic sanatan Hindu people, where Jagannath Foundation - Sri Rupanuga Para Vidyapeeth, Bimala Devi temple, Shesa Temple, Sada Shiva and ancient Nrsimha Deva temple, and Ganga Jamuna temple are attraction. Kot Devi is a religious place in Jwalamukhi Rural Municipality, Maidi.

Administration.

Dhanding district consists of 2 municipalities and 11 rural municipals.

Dhading District ( ), a part of Bagmati Pradesh, is one of the seventy-seven districts of Nepal. The district, with Dhading Besi as its district headquarters, covers an area of, had a population of 338,658 in 2001 and 336,067 in 2011.

Le district de Dhading – en népalais : – est l'un des 75 districts du Népal. Il est rattaché à la zone de la Bagmati et à la région de développement Centre. La population du district s'élevait à en 2011.


History.

Early history.

The area now occupied by Barry has seen human activity in many periods of history. Mesolithic or Middle Stone Age microlith flint tools have been found at Friars Point on Barry Island and near Wenvoe and Neolithic or New Stone Age polished stone axe-heads were discovered in St. Andrews Major. A cinerary urn (pottery urn buried with cremation ashes) was found on Barry Island during excavations of Bronze Age barrows and two more were found in a barrow at Cold Knap Point. A large defended enclosure or Iron Age promontory hillfort was located at the Bulwarks at Porthkerry and there was evidence of the existence of an early Iron Age farmstead during construction of Barry College off Colcot Road.  In Roman times farmsteads existed on the site of Barry Castle and Biglis and there were verbal reports of discovery of a cemetery including lead coffins with scallop-shell decoration. Both St. Baruc's Chapel and St. Nicholas Church have re-used Roman bricks and tiles incorporated in their building fabric and a Roman villa was discovered in Llandough. In 1980 a Roman building consisting of 22 rooms and cellars in four ranges around a central courtyard was excavated at Glan-y-môr and is believed to be a third-century building associated with naval activity, maybe a supply depot.  The Vikings launched raids in the area and Barry Island was known to be a raider base in 1087. Flat Holm and Steep Holm islands in the Bristol Channel have their name Holm name derived from a Scandinavian word for an island in an estuary. The excavation of the Glan-y-môr site revealed the site had been reused in the 6th and 7th century and also between AD 830 and 950 as a dry stone sub-rectangular building with a turf or thatched roof.

Medieval Barry.

The main feature of the area at this time was the island in the Bristol Channel, separated from the mainland by a tidal estuary. It is described in Giraldus Cambrensis or Gerald of Wales' ("Journey through Wales", 1191). He states that Barry derives its name from St. Baruc whose remains are deposited in a chapel on the island. The local noble family who owned the island and the adjoining estates took the name of de Barri from the island.  Following the Norman conquest of England the area was divided into manors with the Barry area split into two large lordships, Penmark and Dinas Powys. Penmark was split into the sub-manors of Fonmon, West Penmark and Barry. Dinas Powys was split into the sub-manors of Cadoxton and ('Highlight'). The sub-manor of Barry was granted by the de Umfraville family to the de Barri family and the seat of the manor was Barry Castle, located on high ground overlooking the Bristol Channel, a site occupied in Roman times by a native homestead. The castle was a small fortified manor house, built to replace an earlier earthwork. By the late 13th century the castle had two stone buildings on the east and west sides of a courtyard. Early in the 14th century the castle was strengthened by the addition of a large hall and gatehouse on its south side, the ruins of which are all that survive today. By now Barry had grown into a village and port with its own church and watermill but in the 14th century its population was drastically reduced by the Black Death and the consequences of the rebellion of Owain Glyndŵr. It took the population some 300 years to recover and once more hold the title of village, essentially a sparsely populated area with a few scattered farms and much of the land a marsh that a small river flowed through. By 1622 the pattern of fields, where enclosure was almost complete, around Barry village was pretty much as it was to remain until the growth of the modern town. According to the 1673 hearth tax list the parish contained thirteen houses.  Whitehouse Cottage, the oldest existing inhabited house in modern Barry, dates from the late 1500s with the east end of the building added in around 1600. It overlooks the sea at Cold Knap.

Industrial history.

By 1871 the population of Barry was over 100, with 21 buildings, the new estate-owning Romilly family being involved in the buildup of the village but it remained a largely agricultural community. It grew when it was developed as a coal port in the 1880s. The coal trade was growing faster than the facilities at Tiger Bay in Cardiff ever could and so a group of colliery owners formed the Barry Railway Company and chose to build the docks at Barry. Work commenced in 1884 and the first dock basin was opened in 1889 to be followed by two other docks and extensive port installations. The Barry Railway brought coal down from the South Wales Valleys to the new docks whose trade grew from one million tons in the first year, to over nine million tons by 1903. The port was crowded with ships and had flourishing ship repair yards, cold stores, flour mills and an ice factory. By 1913, Barry was the largest coal exporting port in the world. Behind the docks rose the terraced houses of Barry which, with Cadoxton, soon formed a sizeable town. The railways which had played a major part in the development of the dock helped make Barry Island a popular resort. Barry Memorial Hall on Gladstone Road was inaugurated in November 1932, and obtained its name to honour those locals who lost their lives in World War I.  During its industrial peak a number of ships sank off the Barry coast.

Barry Scrapyard.

Following the rise of diesel and electric power on the UK railways, the marshalling yards at Barry Docks became the largest repository of steam engines awaiting scrapping in the UK. Dai Woodham owned the Woodham Brothers Scrap yard and he allowed rail preservation organisations to buy back the locomotives at the scrap value, allowing around 200 of the 300 locomotives to be saved for future generations, although during the years of storage many were vandalised or looted by souvenir hunters. When interviewed just before his death, Woodham was reluctant to take full credit for this and pointed out that the town of Barry with its redundant sidings was the major factor in allowing these locomotives to be saved.

Modern times.

Barry is the administrative centre of the Vale of Glamorgan, and home to Barry Town United F.C..  The road from Bonvilston was originally the B4266, as only Pontypridd Road within the town still is, and the road from Highlight Park right through the Vale to Bridgend was the B4265, as beyond Cardiff International Airport it still is. Since the 1970s, parts of these roads are numbered A4226, with the result that the A4226 radiates from Weycock Cross roundabout in "three" directions.  Although still a port, Barry is more of a manufacturing town and as a service centre for the Vale of Glamorgan. Barry Docks and the adjoining industrial area form the largest employment centre in the town. The docks, whose road links were dramatically improved with the opening of the Docks Link Road in 1981, now have direct road access with the M4 motorway. The docks can handle vessels up to 23,000 tonnes and the first-class tidal position close to the deep-water channel of the Severn Estuary, allows for scheduled sailings. With its transit sheds, warehouses and open storage, the docks are equipped to handle bulk cargoes but with the scrapping of their former electric cranes, ships’ own derricks have to be used or cranes hired in by ABP as required. Two roll on/roll off berths are available and have been used by routes to Ireland and West Africa. As at January 2016, Intermodal rail freight traffic is being operated from No. 2 Dock. With a new presence on the Mole in No. 1 Dock and the provision of a concrete slipway from it, leisure rowing and dinghy sailing is available (2016).  The town has a town council which is controlled by the Labour Party. The majority of industrial firms are located in the dock area. The largest are the chemical producing concerns such as Cabot Carbon and Dow Corning who not long ago completed the development of the largest silicones plant in Europe. Other main employers in Barry Docks are Jewson Builders' Merchants, Western Welding and Engineering, Bumnelly, and Associated British Ports Holdings who, since 1982 have run the docks as successors to the British Transport Docks Board.  To the west of Barry is Porthkerry Park. This is a large area of open space, with woodlands, streams, and access to a pebbly beach. In the park is the former Barry Railway Company viaduct with 13 arched spans standing 110 ft high. Following the closure of the Vale of Glamorgan line to passengers between Barry and Bridgend in 1964, it was reopened on 10 June 2005 and for most of its 19 miles, provides a scenic view and link to Llantwit Major and beyond to Bridgend.

Barry Island.

The Barry Island peninsula was an island until the 1880s when it was linked to the mainland as the town of Barry expanded. This was partly due to the opening of Barry Dock by the Barry Railway Company. Established by David Davies, the docks now link up the gap which used to form Barry Island.  There is a railway station to access the island at Barry Docks, there is also a heritage railway station which houses refurbished steam passenger trains. The railway is open to the public and holds events involving a large steam engine replica of Thomas the Tank Engine.  Barry Island is now known for its beach and Barry Island Pleasure Park. From 1966, the island was home to a Butlins Holiday camp, which was closed in 1987 and taken over by Majestic Holidays who renamed it Barry Island Resort. Between Butlins' closure and Majestic's reopening the camp was used as for filming scenes in the "Shangri-La" holiday camp from the "Doctor Who" serial "Delta and the Bannermen". The camp closed in 1996 after Majestic had a disagreement with the local council, who refused an entertainments licence unless work was carried out to improve the now 30-year-old site. It was redeveloped for housing between 1997–2003 with the remaining two camp buildings and outdoor pool demolished in early 2005.  The preserved Vale of Glamorgan Railway runs on Barry Island.  The BBC sitcom "Gavin & Stacey" was filmed in many areas of Barry and Barry Island. Long-running medical drama "Casualty" which is filmed in Cardiff, regularly films scenes in and around Barry Island.

Politics and administration.

UK parliamentary constituency.

From the 1536 Act of Union, Glamorgan was represented in parliament by one member, elected by the freeholders in the county. In 1885, the constituency was split into three with the creation of East Glamorganshire, Mid Glamorganshire and South Glamorganshire. The Representation of the People Act 1918 created the Llandaff and Barry constituency. Sir William Cope (Conservative) won the 1918 general election. Labour regained the seat at the 1929 general election when Charles Ellis Lloyd was returned but two years later lost the seat to the Conservatives' Patrick Munro.  After Munro's death in 1942 Cyril Lakin won the by-election for the Conservatives. Arwyn Lynn Ungoed-Thomas (Labour) won the seat at the 1945 general election. The Llandaff and Barry constituency was abolished by the Representation of the People Act 1948. and replaced by the Barry parliamentary constituency. This seat was first contested in the 1950 United Kingdom general election when Dorothy Rees (Labour) was elected. She lost the seat to Sir Herbert Raymond Gower (Conservative) at the 1951 general election. He held the seat until its abolition in 1983.  It was replaced by the Vale of Glamorgan constituency which Sir Herbert Raymond Gower (Conservative) won at the 1983 general election. He remained as MP until his death in 1989. At the subsequent by-election the seat was won by John Smith (Labour). At the 1992 general election Walter Sweeney (Conservative) won it by only 19 votes. That made it the most marginal seat in Britain. John Smith won it back at the 1997 general election. In the 2010 General election there was a 6.1% swing from labour to conservative. This has now handed the seat to Alun Cairns a (Conservative) MP.

Welsh Assembly.

Barry is part of the Vale of Glamorgan Assembly constituency and part of the South Wales Central Assembly region. Jane Hutt has been the Vale of Glamorgan Assembly member since the inception of the Welsh Assembly.

Local councils.

Barry was incorporated as a municipal borough by Royal Charter in September 1939. The Borough was the successor to Barry and Cadoxton Local Board (1888–1894) and Barry Urban District Council (1894–1939). The area covered by the borough comprised Barry, Cadoxton-juxta-Barry, Merthyr Dyfan and parts of Penmark, Porthkerry and Sully. In 1974, it was abolished and its functions taken over by the Vale of Glamorgan District Council and South Glamorgan County Council.  The local council, Barry Town Council, is the largest town council in Wales. It has given Olympic silver medalist David Davies freedom of the town, the first freedom granted since 1958. The mayor since May 2017 is Nic Hodges who represents Baruc Ward and is a Plaid Cymru councillor, Barry's first Plaid Cymru mayor. The town council has no overall control with Labour, Plaid Cymru and Conservative councillors.  The local unitary authority, created in 1995, is the Vale of Glamorgan Council which has its administrative headquarters in Barry. There are 23 wards electing 47 councillors, including eight wards in Barry (electing 18 county councillors) which comprise Baruc (2 councillors), Buttrills (2), Cadoc (3), Castleland (2), Court (2), Gibbonsdown (2), Dyfan (2) and Illtyd (3).

Climate.

As with the rest of the British Isles and Wales, Barry experiences a maritime climate with cool summers and mild winters, and often high winds. It is amongst the sunnier of Welsh locations, due to its southerly and coastal position. The nearest official weather observation station is at Cardiff Airport near Rhoose, about west of the town centre.

Education.

Secondary schools.

Barry has four secondary schools. Since 1993, Bryn Hafren and Barry Comprehensive School have worked together to provide the co-educational Barry Sixth Form. In July 2018, as part of the 'Transforming Secondary Education in Barry' scheme, both Barry Comprehensive School and Bryn Hafren closed their doors to make way for two new mixed-sex secondary schools which opened in September 2018. The old Barry Comprehensive School site has now become Whitmore High School and Bryn Hafren has now become Pencoedtre High School.

Primary schools.

Primary education (5–11) in Barry is provided by a number of community, Welsh-language, and faith-based schools spread throughout the town.

Transport.

The main forms of public transport in the town are bus and rail. Barry is served by Cardiff Bus which operates services to Llantwit Major, Penarth, Cardiff International Airport and Cardiff City Centre as well as operating town circular services. Barry's King's Square bus station is located on King's Square in the town centre. The A4050 road connects Barry to Culverhouse Cross Interchange and the rest of west Cardiff.  There are four railway stations in the town: Barry, Barry Docks, Barry Island and Cadoxton. These are operated and served by Transport for Wales and are on the Valley Lines network, a commuter rail network focused on Cardiff. Services operate westbound to Bridgend via Llantwit Major and Rhoose Cardiff International Airport, and eastbound to Cardiff Queen Street via Dinas Powys, Cardiff Grangetown and Cardiff Central. The latter service can continue to either Merthyr Tydfil, Pontypridd and/or Aberdare.  Barry is located less than 3 miles (4 kilometres) east of Cardiff International Airport.

Freedom of the Town.

The Following People and Military Units have received the Freedom of the Town of Barry.

Barry ( ) is a town in the Vale of Glamorgan, Wales, on the north coast of the Bristol Channel approximately south-southwest of Cardiff. Barry is a seaside resort, with attractions including several beaches and the resurrected Barry Island Pleasure Park. According to Office for National Statistics 2016 estimate data, the population of Barry was 54,673.

Barry ("Y Barri" en gallois) est une ville du Vale of Glamorgan ("Bro Morgannwg" en gallois), "county borough" du pays de Galles. Située à quelques kilomètres de Cardiff, la capitale du pays de Galles, Barry est située sur la côte sud. Il s'agit d'une station balnéaire très fréquentée, avec des atouts tels qu'une plage et le parc d'attractions Barry Island Pleasure Park.


Early life.

Stadler was born in San Diego His father started him in golf at age four, and he displayed a talent for golf early in life. Stadler attended La Jolla High School He won the 1973 U.S. Amateur, while attending the University of Southern California, where he was a teammate of future PGA Tour winners Mark Pfeil and Scott Simpson. Stadler was an All-American all four years – first-team his sophomore and junior years; second-team his freshman and senior years. Stadler finished college in 1975 and turned professional in 1976.

Professional career.

Stadler won his first two PGA Tour events in 1980, at the Bob Hope Desert Classic and the Greater Greensboro Open. His career year was 1982 when he won four PGA Tour events including the Masters Tournament after a playoff with Dan Pohl and the World Series of Golf at the end of the year. Stadler won the money list for the only time. His next win was at the 1984 Byron Nelson Classic.  Despite playing relatively well, Stadler did not win a PGA Tour event for over 7 years (May 1984 – November 1991) during the heart of his career. During this period he recorded six runner-up performances and dozens of top-10s on the PGA Tour. He had more success at winning international tournaments. He won the 1985 Swiss Open on the European Tour and the 1987 Dunlop Phoenix Tournament on the Japan Golf Tour. He had great success at the Scandinavian Enterprise Open too, an official event on the European Tour, finishing runner-up at the 1983 and 1986 events until finally winning in 1990. His winless steak in America was broken at the final event of the 1991 season, defeating Russ Cochran in a playoff at the Tour Championship.  Stadler won the B.C. Open in 2003, becoming the first player over age 50 to win a PGA Tour event in 28 years and the first player ever to win on the PGA Tour after he had won on the Champions Tour. He won 13 PGA Tour events in all, and played on the 1983 and 1985 Ryder Cup teams.  He appeared as himself, with a speaking role in the 1996 film "Tin Cup".  Stadler began playing on the Champions Tour upon becoming eligible in June 2003. His greatest successes came during his first two years of eligibility; he was the leading money winner in his first full year on that tour in 2004. Stadler underwent total left-hip-replacement surgery in Los Angeles on September 15, 2010, which limits his playing time.  Very popular with the galleries, Stadler is affectionately called "The Walrus" for his portly build and ample mustache. He currently lives in Denver, Colorado. His son Kevin is also a PGA Tour champion. His brother Gary Stadler is a Billboard-charting recording artist.  Stadler announced that the 2014 Masters Tournament, his 38th and in which he played with Kevin, was his last.

Professional wins (31).

PGA Tour wins (13).

PGA Tour playoff record (3–3)

European Tour wins (3).

European Tour playoff record (1–1)

Champions Tour wins (9).

Champions Tour playoff record (1–2)

Major championships.

Wins (1).

Defeated Pohl with par on first extra hole.

Results timeline.

CUT = missed the halfway cut (3rd round cut in 1975 and 1985 Open Championships)
 WD = withdrew
 "T" indicates a tie for a place.

Results in The Players Championship.

CUT = missed the halfway cut "T" indicates a tie for a place

Results in World Golf Championships.

Cancelled due to 9/11
 QF, R16, R32, R64 = Round in which player lost in match play
 NT = No tournament

U.S. national team appearances.

Amateur  Professional

Craig Robert Stadler (born June 2, 1953) is an American professional golfer who has won numerous tournaments at both the PGA Tour and Champions Tour level, including one major championship, the 1982 Masters Tournament.

Craig Robert Stadler, né le à San Diego en Californie, est un golfeur professionnel américain. Il a joué principalement sur les circuits du PGA Tour sur lequel il a remporté treize victoires entre 1980 et 2003, dont le Masters en 1982, et du Champions Tour sur lequel il a gagné deux majeurs sénior, dont le Sénior Players Championship un mois après avoir obtenu le droit de jouer sur le Champions Tour. Il a au total gagné entre 2003 et 2013 neuf tournois sur ce circuit, dont cinq pour la seule année 2004. Il met fin à sa carrière après le Masters 2014, après trente-cinq participations consécutive dans le majeur de l'année. Son fils, Kevin Stadler, est lui aussi un golfeur professionnel qui évolue sur le PGA Tour.


History.

The county was formed in 1840 as Meegisee County. Meegisee (meaning "eagle"), was the name of a Chippewa chief who signed the 1821 Treaty of Chicago and the 1826 Treaty of Mississinewas.  It was renamed Antrim County in 1843, one of the Irish or Scots Irish names given to five renamed Michigan counties at that time, supposedly in deference to the increasing number of settlers of Irish and Scots Irish heritage in Michigan at that time. In the text of the 1843 legislative act, the name was misspelled as "Antim".  Separate county government was organized in 1863. The county seat was originally located in Elk Rapids, but was moved to Bellaire in 1904 after 25 years of litigation. In 1950 its population was 10,721.

Geography.

According to the U.S. Census Bureau, the county has a total area of, of which is land and (21%) is water.  The county is considered to be part of Northern Michigan. Glaciers shaped the area, creating a unique regional ecosystem. A large portion of the area is Grayling outwash plain, which consists of broad outwash plain including sandy ice-disintegration ridges; jack pine barrens, some white pine-red pine forest, and northern hardwood forest. Large lakes were created by glacial action.

Demographics.

As of the 2010 United States Census, there were 23,580 people, 9,890 households, and 6,925 families in the county. The population density was 49 people per square mile (19/km2). There were 17,824 housing units at an average density of 37 per square mile (45/km2). 96.8% of the population were White, 1.0% Native American, 0.2% Asian, 0.2% Black or African American, 0.4% of some other race and 1.4% of two or more races. 1.7% were Hispanic or Latino (of any race). 20.2% were of German, 13.4% English, 8.9% Irish, 6.9% French, French Canadian or Cajun, 6.9% Polish and 6.4% American ancestry.  There were 9,222 households out of which 26% had children under the age of 18 living with them, 57.30% were married couples living together, 8.30% had a female householder with no husband present, and 30% were non-families. 25.3% of all households were made up of individuals and 11.50% had someone living alone who was 65 years of age or older. The average household size was 2.36 and the average family size was 2.78.  The county population contained 21.10% under the age of 18, 6.30% from 19 to 24, 3.9% from 25 to 44, 31.1% from 45 to 64, and 22.2% who were 65 years of age or older. The median age was 47 years. For every 100 females there were 99.80 males.

Government.

Antrim County has been reliably Republican since its organization. Since 1884 its voters have selected the Republican Party nominee in 94% (32 of 34) of the national elections through 2016.  Antrim County operates the County jail, maintains rural roads, operates the major local courts, records deeds, mortgages, and vital records, administers public health regulations, and participates with the state in the provision of social services. The county board of commissioners controls the budget and has limited authority to make laws or ordinances. In Michigan, most local government functions – police and fire, building and zoning, tax assessment, street maintenance etc. – are the responsibility of individual cities and townships.

Antrim County is a county located in the U.S. state of Michigan. As of the 2010 census, the population was 23,580. The county seat is Bellaire. The name is taken from County Antrim in Northern Ireland.

Le comté de Benzie ("Benzie County" en anglais) est situé dans le nord-ouest de la péninsule inférieure de l'État du Michigan. Son siège est à Beulah. Selon le recensement de 2000, sa population est de.


Meaning.

The lyrics, written by Roger Waters, deal with war (particularly the Falklands War) and criticism of British Prime Minister Margaret Thatcher, as well as general criticisms of the greed and corruption that Waters saw as dangers to society. It also shows the corruptible and fruitless labour of post-war America, Europe and Japan. The wording is such that it mainly tells of the changing of global trade and that a new leader is emerging in the consumer goods industry such as Japan.  Despite the political content of the album and the specific references in other songs to public figures of the time, the "John" of the title is not intended to refer to any particular person named John. It is being used in the British colloquial sense as a placeholder name, where "John" can be employed in the same way as "mate", "pal", "buddy", or "Guv" to refer to anyone to whom one is speaking, particularly if the speaker does not know their name. At the time, this usage of "John" as a general means of address to others would have been particularly associated with blue-collar workers, who were the people being most strongly affected by the changes to manufacturing and trade referred to in the song.

Music video.

In The Final Cut Video EP for the song depicts a Japanese boy walking through a factory searching for a soldier. The child is confronted by factory workers playing cards and geisha girls before he falls to his death from a scaffold and is discovered by a World War II veteran (played by Alex McAvoy, who also played the schoolteacher in "Pink Floyd — The Wall"). The video was directed by Waters' then brother-in-law, Willie Christie.

Composition.

It is the only track on the album not to feature exclusively Waters on lead vocals. Unlike the majority of other tracks on "The Final Cut", "Not Now John" takes an upbeat, driving, tempo — and hard rock style — for much of its duration. Gilmour and Waters split vocals duties, similar to the song "Comfortably Numb" from "The Wall", and they represent different "characters" or points of view — Gilmour is the self-serving ignorant layperson while Waters is the intellectual, responsible observer of the world's woes.  "Not Now John" is notable for being one of the few Pink Floyd tracks to feature strong profanity, as the word "fuck" occurs in the album version of the song seven times: six times as part of the phrase "fuck all that", and near the end of the song.

Single.

"Not Now John" was released as a single on 3 May 1983. The words "fuck all that" were overdubbed as "stuff all that" by Gilmour, Waters, and the female backing singers, while the "Where's-the-bar?" lyric is sung in Italian, Greek and French, as the single fades out before the English iteration.  "The Hero's Return" was released as the B-side, featuring an additional verse not included on the album. A 12" single was released in the UK, featuring the two 7" tracks on side 1 and the album version of "Not Now John" on side 2. The single hit number 30 in the UK and number seven on the US Mainstream Rock Tracks chart.

Reception.

In a review for "The Final Cut" on release, Kurt Loder of "Rolling Stone" described "Not Now John" as "one of the most ferocious performances Pink Floyd has ever put on record." In a retrospective review of "The Final Cut", Rachel Mann of "The Quietus" described "Not Now John" as "fun, but musically crass and obvious," further saying "this is Surrey Blues rock as vapid as the views it seeks to satirize."

"Not Now John" is a song by the progressive rock band Pink Floyd, written by Roger Waters. It appears on the album "The Final Cut". The track is the only one on the album featuring the lead vocals of David Gilmour, found in the verses, with Roger Waters singing the refrains and interludes, and was the only single released from the album. It reached No. 30 in the UK Singles Chart.

Not Now John est une chanson du groupe rock progressif britannique Pink Floyd. C'est le douzième titre de l'album "The Final Cut" paru en 1983. Elle parut également en single. La chanson parle en général de la guerre ainsi que Margaret Thatcher, Premier ministre du Royaume-Uni à l'époque. Roger Waters la décrit comme étant dangereuse pour la société.


Discussion.

Since antiquity, the problem of predicting the motions of the heavenly bodies has been simplified by reducing it to one of a single body in orbit about another. In calculating the position of the body around its orbit, it is often convenient to begin by assuming circular motion. This first approximation is then simply a constant angular rate multiplied by an amount of time. There are various methods of proceeding to correct the approximate circular position to that produced by elliptical motion, many of them complex, and many involving solution of Kepler's equation. In contrast, the equation of the center is one of the easiest methods to apply.  In cases of small eccentricity, the position given by the equation of the center can be nearly as accurate as any other method of solving the problem. Many orbits of interest, such as those of bodies in the Solar System or of artificial Earth satellites, have these nearly-circular orbits. As eccentricity becomes greater, and orbits more elliptical, the equation's accuracy declines, failing completely at the highest values, hence it is not used for such orbits.  The equation in its modern form can be truncated at any arbitrary level of accuracy, and when limited to just the most important terms, it can produce an easily calculated approximation of the true position when full accuracy is not important. Such approximations can be used, for instance, as starting values for iterative solutions of Kepler's equation, or in calculating rise or set times, which due to atmospheric effects cannot be predicted with much precision.  The ancient Greeks, in particular Hipparchus, knew the equation of the center as "prostaphaeresis", although their understanding of the geometry of the planets' motion was not the same. The word "equation" (Latin, "aequatio, -onis") in the present sense comes from astronomy. It was specified and used by Kepler, as "that variable quantity determined by calculation which must be added or subtracted from the mean motion to obtain the true motion." In astronomy, the term equation of time has a similar meaning. The equation of the center in modern form was developed as part of perturbation analysis, that is, the study of the effects of a third body on two-body motion.

Series expansion.

In Keplerian motion, the coordinates of the body retrace the same values with each orbit, which is the definition of a periodic function. Such functions can be expressed as periodic series of any continuously increasing angular variable, and the variable of most interest is the mean anomaly,. Because it increases uniformly with time, expressing any other variable as a series in mean anomaly is essentially the same as expressing it in terms of time. Because the eccentricity,, of the orbit is small in value, the coefficients of the series can be developed in terms of powers of. Note that while these series can be presented in truncated form, they represent a sum of an "infinite" number of terms.  The series for, the true anomaly can be expressed most conveniently in terms of, and Bessel functions of the first kind,  where  The result is in radians.  The Bessel functions can be expanded in powers of by,  and by,  Substituting and reducing, the equation for becomes (truncated at order ),  and by the definition, moving to the left-hand side,  gives the equation of the center.  This equation is sometimes derived in an alternate way and presented in terms of powers of with coefficients in functions of (truncated at order ), which is identical to the above form.  For small, the series converges rapidly. If exceeds 0.6627..., it diverges for some values of, first discovered by Pierre-Simon Laplace.

In two-body, Keplerian orbital mechanics, the equation of the center is the angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the same period. It is defined as the difference true anomaly,, minus mean anomaly,, and is typically expressed a function of mean anomaly,, and orbital eccentricity,.

En astronomie, l’équation du centre traduit, dans le cadre du mouvement elliptique, la différence entre l'anomalie vraie "v" et l'anomalie moyenne "M".