Early years.

Born in Houston, Texas, Renfro moved with his family to Portland, Oregon, and he attended its Jefferson High School, starring as a two-way football player and track and field athlete. As a sophomore and junior, he contributed to his team achieving a 23–0 record, including consecutive Class A-1 football state championships in 1957 and 1958. The Democrats' 1958 team is regarded as one of the greatest in Oregon prep history, which had a backfield that included him at halfback, quarterback Terry Baker (1962 Heisman Trophy winner at Oregon State), halfback Mickey Hergert (one of the leading ground gainers in the nation at Lewis & Clark College) and his brother Raye Renfro at fullback (in 1958 he broke a Portland Interscholastic League record with 24 touchdowns). After Baker graduated, Renfro saw playing time at quarterback as a senior in 1959 and was also an all-state selection at running back, as Jefferson pursued a third consecutive title. They again advanced to the championship game, played before more than 21,000 at Multnomah Stadium in Portland, but lost 7–6 to Medford.  In track as a junior in 1959, he helped Jefferson win the team title by winning the 180-yard low hurdles, the 120-yard high hurdles (breaking the state record) and the broad jump state titles. In 1960, he broke two Oregon junior AAU championship records with a 120-yard high hurdles of 13.9 seconds and 19.3 seconds in the 180-yard low hurdles. Renfro also collected 32 points outscoring every team except one, recording the state record with a 24.1 broad jump and winning the low hurdles, high hurdles and the half-mile relay team. He was named outstanding athlete in the Golden West Invitational track meet, which brought annually the nation's top high school seniors. He was "Track and Field News" "High School Athlete of the Year" in 1960.  In 1983, he was inducted into the Oregon Sports Hall of Fame.

College career.

In 1960, Renfro enrolled at the University of Oregon in Eugene, where he excelled for the track team and as a two-way football player, an All-American halfback and defensive back. In football, he earned All-Pacific Coast Conference recognition three times and is one of only two-time All-American (Steve Barnett is the other player) in university history. He is also the second two-sport All-American in university history, the first one was George Shaw in football and baseball.  In 1992, Renfro was inducted into the University of Oregon Athletic Hall of Fame.

Football.

From the start of his sophomore season he received recognition as one of the best and most complete running backs in college football, being able to run inside or outside, catch, block and play special teams. In 1963, he was even named the team's backup quarterback.  One of Renfro's best collegiate games took place against Rice at Houston in 1962. Renfro shined on both offense and defense, leading the Ducks to a 31–12 victory on October 13. Rice Stadium had been a whites-only facility prior to the Ducks' visit, but university officials allowed Renfro's family to enter and sit in a special section of the stadium near the 35-yard line. According to a contemporary account in the "Register-Guard", when Renfro left the game, he was given a loud and appreciative ovation by the Rice crowd.  An Oregon sportswriter asked a Houston counterpart in the press box if an African-American had ever received such an ovation by a predominantly white crowd. The Texan enthusiastically replied, "We've never seen any player that good!" The morning after the game, a Houston paper headlined, "Renfro Runs Rice Ragged."  Renfro led Oregon in rushing three straight seasons (1961–63), finishing with a career total 1,540 yards and 23 touchdowns. He still holds the university's career record for highest average per play at 6.01 yards. He also had 41 catches for 644 yards and five touchdowns, very impressive numbers in an era when running backs usually did not catch more than 10 passes during a season.  In 1986, Renfro was enshrined in the College Football Hall of Fame and is a member of The Pigskin Club of Washington, D.C., a National Intercollegiate All-American Football Players Honor Roll. He is considered to be the best running back in the history of the University of Oregon, other than LaMichael James.

Track and field.

From 1960 through 1965, the Oregon's men's track and field team led by coach Bill Bowerman was one of the best in the United States. In 1962, Renfro contributed to the university's first NCAA Men's Division I Outdoor Track and Field Championships|NCAA team championship]] by finishing second behind teammate Jerry Tarr in the high hurdles with a 13.8 time and third in the long jump with a 25'-111⁄4" (7.905 m) leap.  That year, he was part of a world record-setting 440 yard relay team with a time of 40.0 seconds, his running mates were Tarr, Mike Gaechter and Harry Jerome. At the end of the year, he was named to the All-American track and field team in the high hurdles and the broad jump.  In 1963, he injured his knee on a foul broad jump in the NCAA Championships. In the same meet at Albuquerque, despite the injury he entered the high hurdles but failed to qualify.

Professional career.

Renfro was drafted in the second round of the 1964 NFL draft, 17th overall, by the Dallas Cowboys. The selection delayed the draft for 6 hours, while the Cowboys sent a doctor to check an injury he suffered on his wrist.  Although he had won many accolades as a running back and at the time teams put their best athletes on the offensive side of the ball, Tom Landry looking to build a dominating defense decided to start him at safety. As a rookie, he had an impressive season by leading the team with seven interceptions, the NFL in both punt return and kickoff return yardage (including a franchise record 273 total return yards against the Green Bay Packers) and culminated his great campaign with an invitation to the Pro Bowl. The 30 yards per kick-off return average he posted in his second year (1965) still remains a franchise record.  In 1966, despite Renfro coming off of an All-Pro season at safety, the Cowboys named him their starting halfback in an effort to improve its offense. He was injured in the opening game against the New York Giants and replaced by Dan Reeves, who was so impressive that Renfro was moved back to defense when he returned to action and would end up being named to the All-Pro team again.  He was switched to cornerback in his fifth season. The speedy Renfro (4.65 40-yd dash) became an exceptional threat to wide receivers, leading the NFL with 10 interceptions in 1969. Renfro was selected to the Pro Bowl in each of his first ten seasons in the league, including five All-Pro selections in 1964, 1965, 1969, 1971 and 1973. He was named the defensive MVP of the 1971 Pro Bowl.  In his 14 seasons, Renfro intercepted 52 passes, returning them for 626 yards and three touchdowns. He also returned 109 punts for 842 yards and one touchdown, 85 kickoffs for 2,246 yards and two touchdowns, along with recording 13 fumble recoveries, which he returned for 44 yards. In the 1970 NFC Championship Game, Renfro had a key interception that led to the Cowboys' game-winning touchdown over the San Francisco 49ers that helped them get to Super Bowl V, where they lost to the Baltimore Colts, 16–13. Dallas returned in Super Bowls VI, X, and XII, Renfro's last NFL game, a Cowboys' 27–10 victory over the Denver Broncos. He remains the Cowboys all-time leader in interceptions with 52 (In 1969 he led the NFL in interceptions with 10) and in career kickoff-return average (26.4 yards). His 14 seasons with the team ties him for second place in franchise history.  Renfro joined the Dallas Cowboys Ring of Honor in 1981 and was elected to the Pro Football Hall of Fame in 1996. During his time with the Cowboys, even though one of his teammates was former Olympic gold medalist Bob Hayes, many experts still considered him the best athlete on the team.

Personal life.

In 1984 Renfro coached the defensive secondary under John Hadl with the Los Angeles Express of the United States Football League (USFL). In 1986 he was part of Gene Stallings St. Louis Cardinals coaching staff as his defensive backs coach.  In the early 1980s, Renfro had financial and legal challenges. He currently works as a motivational speaker.

Melvin Lacy Renfro (born December 30, 1941) is a former American football player, a cornerback who spent his entire 14-year career in the National Football League (NFL) with the Dallas Cowboys. He is a member of the Pro Football Hall of Fame.

Melvin Lacy Renfro (* 30. Dezember 1941 in Houston, Texas, USA) ist ein ehemaliger US-amerikanischer American-Football-Spieler. Er spielte unter anderem Free Safety in der National Football League (NFL) bei den Dallas Cowboys und gilt als einer der besten Safetys aller Zeiten.


Description.

"Crataegus rhipidophylla" is a shrub or tree which can grow to tall. Its stout thorns can be up to 1.5 cm long. Leaf blades are dark green, with 2-4 pairs of acute or subacuminate lobes. A helpful characteristic trait for identification is the finely serrated lobe margin. This can help distinguish "C. rhipidophylla" from "C. monogyna" which has irregularly serrated lobe margins, with more or less coarse teeth. The basal lobes of flowering shoots leaf blades each have 6-25 teeth. Their stipules also are serrate or serrate-denticulate. Inflorescences are corymbs, 3-4.5 cm long, of 5-15 lax white flowers. The hypanthium is 3–5 mm long. Sepals are more or less narrowly triangular, 1-2.6 times as long as wide. There are 14-20 stamens with purple anthers. Apart from the serrated leaf blade lobe margins, the number of styles or pyrenes is a second useful characteristic trait for identification. "C. rhipidophylla" has flowers with 1 style (fruit with 1 pyrene) - more rarely, and at the most, 2 styles or pyrenes. This is similar to "C. monogyna" but unlike "C. laevigata" which has 2 or 3 styles or pyrenes (sometimes 1, 4 or at most 5). Fruits are either bright or dark red, 8–15 mm long and 1.3-2 times as long as wide.  Flowers are in bloom in May and June. Fruits can be seen from June to October

Taxonomy.

The type specimen for "Crataegus rhipidophylla" is a holotype named by Michel Gandoger. It was collected in 1870 at la Come, in Liergues, Rhône, France.  Despite not being one of the most common European hawthorn species, the type specimen for genus "Crataegus" L. is a "C. rhipidophylla" specimen (originally "Crataegus oxyacantha" L., "nom. rejic.")

Varieties and hybrids.

There are currently three recognized varieties:  "C. rhipidophylla" var. "ronnigeri" (K. Malý) Janjić has been suggested to be used as a valid name for "C. rhipidophylla" var. "lindmanii". Another synonym is "Crataegus lindmanii" Hrabětova  "C. rhipidophylla" var. "lindmanii" can be recognized by its erect or suberect sepals crowning the fruit.  The plant is parent to hybrids ("see table"). "C." × "macrocarpa" (with "C. laevigata") and "C." × "subsphaericea" (with "C. monogyna") are intermediates in terms of size and form between the parent species. "C." × "subsphaericea" is found outside the range of its parents.

Distribution and habitat.

"Crataegus rhipidophylla" is a subatlantic species. It has a Eurasian native range, including southern Scandinavia and the Baltic region, France, the Balkan Peninsula, Asian Turkey, Caucasia, the Crimea, and Ukraine. It grows from sea level up to 1,800 m altitude.  This species grows both on limestone and granitic or volcanic rocks. It is one of the few shade-tolerant hawthorn species, growing in shaded parts of continuous forests.

Crataegus rhipidophylla is a species of hawthorn which occurs naturally from southern Scandinavia and the Baltic region to France, the Balkan Peninsula, Turkey, Caucasia, and Ukraine. It is poorly known as a landscape and garden plant, but seems to have potential for those uses.

Der Großkelchige Weißdorn ("Crataegus rhipidophylla"), auch Langkelchiger Weißdorn genannt, ist eine Pflanzen-Art der Kernobstgewächse (Pyrinae) innerhalb der Familie der Rosengewächse (Rosaceae). Er ist eine von sechs in Mitteleuropa vorkommenden Weißdorn-Arten.


Intuition.

Intuitively, forcing consists of expanding the set theoretical universe formula_1 to a larger universe formula_2. In this bigger universe, for example, one might have many new subsets of that were not there in the old universe, and thereby violate the continuum hypothesis.  While impossible when dealing with finite sets, this is just another version of Cantor's paradox about infinity. In principle, one could consider:  identify formula_4 with formula_5, and then introduce an expanded membership relation involving "new" sets of the form formula_6. Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set, and allowing for fine control over the properties of the expanded universe.  Cohen's original technique, now called ramified forcing, is slightly different from the unramified forcing expounded here. Forcing is also equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.

Forcing posets.

A forcing poset is an ordered triple, formula_7, where formula_8 is a preorder on formula_9 that is atomless, meaning that it satisfies the following condition:  Members of formula_9 are called forcing conditions or just conditions. One reads formula_20 as "formula_21 is stronger than formula_22". Intuitively, the "smaller" condition provides "more" information, just as the smaller interval formula_23 provides more information about the number formula_24 than the interval formula_25 does.  There are various conventions in use. Some authors require formula_8 to also be antisymmetric, so that the relation is a partial order. Some use the term partial order anyway, conflicting with standard terminology, while some use the term preorder. The largest element can be dispensed with. The reverse ordering is also used, most notably by Saharon Shelah and his co-authors.

P-names.

Associated with a forcing poset formula_9 is the class formula_28 of formula_9-names. A formula_9-name is a set formula_31 of the form  This is actually a definition by transfinite recursion. More precisely, one first uses transfinite recursion to define the following hierarchy:  Then the class of formula_9-names is defined as  The formula_9-names are, in fact, an expansion of the universe. Given formula_4, one defines formula_38 to be the formula_9-name  Again, this is really a definition by transfinite recursion.

Interpretation.

Given any subset formula_41 of formula_9, one next defines the interpretation or valuation map from formula_9-names by  This is again a definition by transfinite recursion. Note that if formula_45, then formula_46. One then defines  so that formula_48.

Example.

A good example of a forcing poset is formula_49, where formula_50 and formula_51 is the collection of Borel subsets of formula_52 having non-zero Lebesgue measure. In this case, one can talk about the conditions as being probabilities, and a formula_51-name assigns membership in a probabilistic sense. Due to the ready intuition this example can provide, probabilistic language is sometimes used with other divergent forcing posets.

Countable transitive models and generic filters.

The key step in forcing is, given a formula_54 universe formula_1, to find an appropriate object formula_41 not in formula_1. The resulting class of all interpretations of formula_9-names will be a model of formula_54 that properly extends the original formula_1 (since formula_61).  Instead of working with formula_1, it is useful to consider a countable transitive model formula_63 with formula_64. "Model" refers to a model of set theory, either of all of formula_54, or a model of a large but finite subset of formula_54, or some variant thereof. "Transitivity" means that if formula_67, then formula_68. The Mostowski collapse lemma states that this can be assumed if the membership relation is well-founded. The effect of transitivity is that membership and other elementary notions can be handled intuitively. Countability of the model relies on the Löwenheim–Skolem theorem.  As formula_63 is a set, there are sets not in formula_63 – this follows from Russell's paradox. The appropriate set formula_41 to pick and adjoin to formula_63 is a generic filter on formula_9. The "filter" condition means that:  For formula_41 to be "generic" means:  The existence of a generic filter formula_41 follows from the Rasiowa–Sikorski lemma. In fact, slightly more is true: Given a condition formula_10, one can find a generic filter formula_41 such that formula_91. Due to the splitting condition on formula_92 (termed being 'atomless' above), if formula_41 is a filter, then formula_94 is dense. If formula_95, then formula_96 because formula_63 is a model of formula_54. For this reason, a generic filter is never in formula_63.

Forcing.

Given a generic filter formula_100, one proceeds as follows. The subclass of formula_9-names in formula_63 is denoted formula_103. Let  To reduce the study of the set theory of formula_105 to that of formula_63, one works with the "forcing language", which is built up like ordinary first-order logic, with membership as the binary relation and all the formula_9-names as constants.  Define formula_108 (to be read as "formula_109 forces formula_110 in the model formula_63 with poset formula_9"), where formula_21 is a condition, formula_110 is a formula in the forcing language, and the formula_115's are formula_9-names, to mean that if formula_41 is a generic filter containing formula_21, then formula_119. The special case formula_120 is often written as "formula_121" or simply "formula_122". Such statements are true in formula_105, no matter what formula_41 is.  What is important is that this external definition of the forcing relation formula_125 is equivalent to an internal definition within formula_63, defined by transfinite induction over the formula_9-names on instances of formula_128 and formula_129, and then by ordinary induction over the complexity of formulae. This has the effect that all the properties of formula_105 are really properties of formula_63, and the verification of formula_54 in formula_105 becomes straightforward. This is usually summarized as the following three key properties:  We define the forcing relation formula_141 in formula_63 by induction on the complexity of formulas, in which we first define the relation for atomic formulas by formula_143-induction and then define it for arbitrary formulas by induction on their complexity.  We first define the forcing relation on atomic formulas, doing so for both types of formulas, formula_144 and formula_145, simultaneously. This means that we define one relation formula_146 where formula_147 denotes type of formula as follows:  1. formula_148 means formula_149.  2. formula_150 means formula_151.  3. formula_152 means formula_153.  Here formula_109 is a condition and formula_155 and formula_156 are formula_92-names. Let formula_146 be a formula defined by formula_159-induction:  R1. formula_148 if and only if formula_161.  R2. formula_150 if and only if formula_163.  R3. formula_152 if and only if formula_165.  More formally, we use following binary relation formula_92-names: Let formula_167 holds for names formula_155 and formula_156 if and only if formula_170 for at least one condition formula_109. This relation is well-founded, which means that for any name formula_155 the class of all names formula_156, such that formula_167 holds, is a set and there is no function formula_175 such that formula_176.  In general a well-founded relation is not a preorder, because it might not be transitive. But, if we consider it as an "ordering", it is a relation without infinite decreasing sequences and where for any element the class of elements below it is a set.  It is easy to close any binary relation for transitivity. For names formula_155 and formula_156, formula_179 (as a map with domain formula_180) for some formula_181 such that formula_182, formula_183 and for any formula_184, formula_185 holds. Such an ordering is well-founded too.  We define the following well defined ordering on pairs of names: formula_186 if one of the following holds:  1. formula_187,  2. formula_188 and formula_189,  3. formula_188 and formula_191 and formula_192.  The relation formula_146 is defined by recursion on pairs formula_194 of names. For any pair it is defined by the same relation on "simpler" pairs. Actually, by the recursion theorem there is a formula formula_146 such that R1, R2 and R3 are theorems because its truth value at some point is defined by its truth values in "smaller" points relative to the some well-founded relation used as an "ordering". Now, we are ready to define forcing relation:  1. formula_196 means formula_197.  2. formula_198 means formula_199.  3. formula_200 means formula_201.  4. formula_202 means formula_203.  5. formula_204 means formula_205.  Actually, this is a transformation of an arbitrary formula formula_206 to the formula formula_207 where formula_109 and formula_92 are additional variables. This is the definition of the forcing relation in the universe formula_210 of all sets regardless to any countable transitive model. However, there is a relation between this "syntactic" formulation of forcing and the "semantic" formulation of forcing over some countable transitive model formula_211.  1. For any formula formula_206 there is a theorem formula_213 of the theory formula_214 (for example conjunction of finite number of axioms) such that for any countable transitive model formula_211 such that formula_216 and any atomless partial order formula_217 and any formula_92-generic filter formula_219 over formula_211  This is called the property of definability of the forcing relation.

Consistency.

The discussion above can be summarized by the fundamental consistency result that, given a forcing poset formula_9, we may assume the existence of a generic filter formula_41, not belonging to the universe formula_1, such that formula_225 is again a set-theoretic universe that models formula_54. Furthermore, all truths in formula_225 may be reduced to truths in formula_1 involving the forcing relation.  Both styles, adjoining formula_41 to either a countable transitive model formula_63 or the whole universe formula_1, are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, in which no mention of set or class models is made. This was Cohen's original method, and in one elaboration, it becomes the method of Boolean-valued analysis.  The simplest nontrivial forcing poset is formula_232, the finite partial functions from formula_233 to formula_234 under "reverse" inclusion. That is, a condition formula_21 is essentially two disjoint finite subsets formula_236 and formula_237 of formula_233, to be thought of as the "yes" and "no" parts of with no information provided on values outside the domain of formula_21. "formula_22 is stronger than formula_21" means that formula_242, in other words, the "yes" and "no" parts of formula_22 are supersets of the "yes" and "no" parts of formula_21, and in that sense, provide more information.  Let formula_41 be a generic filter for this poset. If formula_21 and formula_22 are both in formula_41, then formula_249 is a condition because formula_41 is a filter. This means that formula_251 is a well-defined partial function from formula_233 to formula_253 because any two conditions in formula_41 agree on their common domain.  In fact, formula_255 is a total function. Given formula_256, let formula_257. Then formula_258 is dense. (Given any formula_21, if formula_260 is not in formula_21's domain, adjoin a value for formula_260—the result is in formula_258.) A condition formula_264 has formula_260 in its domain, and since formula_266, we find that formula_267 is defined.  Let formula_268, the set of all "yes" members of the generic conditions. It is possible to give a name for formula_269 directly. Let  Then formula_271 Now suppose that formula_272 in formula_1. We claim that formula_274. Let  Then formula_276 is dense. (Given any formula_21, find formula_260 that is not in its domain, and adjoin a value for formula_260 contrary to the status of "formula_280".) Then any formula_281 witnesses formula_274. To summarize, formula_269 is a "new" subset of formula_233, necessarily infinite.  Replacing formula_233 with formula_286, that is, consider instead finite partial functions whose inputs are of the form formula_287, with formula_288 and formula_289, and whose outputs are formula_290 or formula_291, one gets formula_292 new subsets of formula_233. They are all distinct, by a density argument: Given formula_294, let  then each formula_296 is dense, and a generic condition in it proves that the αth new set disagrees somewhere with the formula_297th new set.  This is not yet the falsification of the continuum hypothesis. One must prove that no new maps have been introduced which map formula_233 onto formula_299, or formula_299 onto formula_292. For example, if one considers instead formula_302, finite partial functions from formula_233 to formula_299, the first uncountable ordinal, one gets in formula_225 a bijection from formula_233 to formula_299. In other words, formula_299 has "collapsed", and in the forcing extension, is a countable ordinal.  The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does not collapse cardinals. For this, a sufficient combinatorial property is that all of the antichains of the forcing poset are countable.

The countable chain condition.

An (strong) antichain formula_31 of formula_9 is a subset such that if formula_311, then formula_21 and formula_22 are incompatible (written formula_314), meaning there is no formula_315 in formula_9 such that formula_317 and formula_318. In the example on Borel sets, incompatibility means that formula_319 has zero measure. In the example on finite partial functions, incompatibility means that formula_249 is not a function, in other words, formula_21 and formula_22 assign different values to some domain input.  formula_9 satisfies the countable chain condition (c.c.c.) if and only if every antichain in formula_9 is countable. (The name, which is obviously inappropriate, is a holdover from older terminology. Some mathematicians write "c.a.c." for "countable antichain condition".)  It is easy to see that formula_51 satisfies the c.c.c. because the measures add up to at most formula_291. Also, formula_327 satisfies the c.c.c., but the proof is more difficult.  Given an uncountable subfamily formula_328, shrink formula_329 to an uncountable subfamily formula_330 of sets of size formula_260, for some formula_332. If formula_333 for uncountably many formula_334, shrink this to an uncountable subfamily formula_335 and repeat, getting a finite set formula_336 and an uncountable family formula_337 of incompatible conditions of size formula_338 such that every formula_339 is in formula_340 for at most countable many formula_341. Now, pick an arbitrary formula_341, and pick from formula_337 any formula_22 that is not one of the countably many members that have a domain member in common with formula_21. Then formula_346 and formula_347 are compatible, so formula_329 is not an antichain. In other words, formula_327-antichains are countable.  The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A "maximal" antichain formula_31 is one that cannot be extended to a larger antichain. This means that every element formula_10 is compatible with some member of formula_31. The existence of a maximal antichain follows from Zorn's Lemma. Given a maximal antichain formula_31, let  Then formula_355 is dense, and formula_87 if and only if formula_357. Conversely, given a dense set formula_355, Zorn's Lemma shows that there exists a maximal antichain formula_359, and then formula_87 if and only if formula_357.  Assume that formula_9 satisfies the c.c.c. Given formula_363, with formula_364 a function in formula_225, one can approximate formula_366 inside formula_1 as follows. Let formula_368 be a name for formula_366 (by the definition of formula_225) and let formula_21 be a condition that forces formula_368 to be a function from formula_373 to formula_374. Define a function formula_375, whose domain is formula_373, by  By the definability of forcing, this definition makes sense within formula_1. By the coherence of forcing, a different formula_379 come from an incompatible formula_21. By c.c.c., formula_381 is countable.  In summary, formula_366 is unknown in formula_1 as it depends on formula_41, but it is not wildly unknown for a c.c.c.-forcing. One can identify a countable set of guesses for what the value of formula_366 is at any input, independent of formula_41.  This has the following very important consequence. If in formula_225, formula_388 is a surjection from one infinite ordinal onto another, then there is a surjection formula_389 in formula_1, and consequently, a surjection formula_391 in formula_1. In particular, cardinals cannot collapse. The conclusion is that formula_393 in formula_225.  The exact value of the continuum in the above Cohen model, and variants like formula_395 for cardinals formula_396 in general, was worked out by Robert M. Solovay, who also worked out how to violate formula_397 (the generalized continuum hypothesis), for regular cardinals only, a finite number of times. For example, in the above Cohen model, if formula_398 holds in formula_1, then formula_400 holds in formula_225.  William B. Easton worked out the proper class version of violating the formula_397 for regular cardinals, basically showing that the known restrictions, (monotonicity, Cantor's Theorem and König's Theorem), were the only formula_54-provable restrictions (see Easton's Theorem).  Easton's work was notable in that it involved forcing with a proper class of conditions. In general, the method of forcing with a proper class of conditions fails to give a model of formula_54. For example, forcing with formula_405, where formula_406 is the proper class of all ordinals, makes the continuum a proper class. On the other hand, forcing with formula_407 introduces a countable enumeration of the ordinals. In both cases, the resulting formula_225 is visibly not a model of formula_54.  At one time, it was thought that more sophisticated forcing would also allow an arbitrary variation in the powers of singular cardinals. However, this has turned out to be a difficult, subtle and even surprising problem, with several more restrictions provable in formula_54 and with the forcing models depending on the consistency of various large-cardinal properties. Many open problems remain.

Random reals.

Random forcing can be defined as forcing over the set formula_411 of all compact subsets of formula_412 of positive measure ordered by relation formula_413 (smaller set in context of inclusion is smaller set in ordering and represents condition with more information). There are two types of important dense sets:  1. For any positive integer formula_414 the set  is dense, where formula_416 is diameter of the set formula_109.  2. For any Borel subset formula_418 of measure 1, the set  is dense.  For any filter formula_219 and for any finitely many elements formula_421 there is formula_422 such that holds formula_423. In case of this ordering, this means that any filter is set of compact sets with finite intersection property. For this reason, intersection of all elements of any filter is nonempty. If formula_219 is a filter intersecting the dense set formula_425 for any positive integer formula_414, then the filter formula_219 contains conditions of arbitrarily small positive diameter. Therefore, the intersection of all conditions from formula_219 has diameter 0. But the only nonempty sets of diameter 0 are singletons. So there is exactly one real number formula_429 such that formula_430.  Let formula_431 be any Borel set of measure 1. If formula_219 intersects formula_433, then formula_434.  However, a generic filter over a countable transitive model formula_210 is not in formula_210. The real formula_429 defined by formula_219 is provably not an element of formula_210. The problem is that if formula_440, then formula_441 "formula_109 is compact", but from the viewpoint of some larger universe formula_443, formula_109 can be non-compact and the intersection of all conditions from the generic filter formula_219 is actually empty. For this reason, we consider the set formula_446 of topological closures of conditions from G. Because of formula_447 and the finite intersection property of formula_219, the set formula_449 also has the finite intersection property. Elements of the set formula_449 are bounded closed sets as closures of bounded sets. Therefore, formula_449 is a set of compact sets with the finite intersection property and thus has nonempty intersection. Since formula_452 and the ground model formula_210 inherits a metric from the universe formula_454, the set formula_449 has elements of arbitrarily small diameter. Finally, there is exactly one real that belongs to all members of the set formula_449. The generic filter formula_219 can be reconstructed from formula_429 as formula_459.  If formula_155 is name of formula_429, and for formula_462 holds formula_441"formula_464 is Borel set of measure 1", then holds  for some formula_466. There is name formula_155 such that for any generic filter formula_219 holds  Then  holds for any condition formula_109.  Every Borel set can, non-uniquely, be built up, starting from intervals with rational endpoints and applying the operations of complement and countable unions, a countable number of times. The record of such a construction is called a "Borel code". Given a Borel set formula_464 in formula_210, one recovers a Borel code, and then applies the same construction sequence in formula_474, getting a Borel set formula_475. It can be proven that one gets the same set independent of the construction of formula_476, and that basic properties are preserved. For example, if formula_477, then formula_478. If formula_464 has measure zero, then formula_475 has measure zero. This mapping formula_481 is injective.  For any set formula_431 such that formula_462 and formula_441"formula_464 is a Borel set of measure 1" holds formula_486.  This means that formula_487 is "infinite random sequence of 0s and 1s" from the viewpoint of formula_210, which means that it satisfies all statistical tests from the ground model formula_210.  So given formula_487, a random real, one can show that  Because of the mutual inter-definability between formula_487 and formula_41, one generally writes formula_494 for formula_474.  A different interpretation of reals in formula_474 was provided by Dana Scott. Rational numbers in formula_225 have names that correspond to countably-many distinct rational values assigned to a maximal antichain of Borel sets – in other words, a certain rational-valued function on formula_498. Real numbers in formula_474 then correspond to Dedekind cuts of such functions, that is, measurable functions.

Boolean-valued models.

Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In these, any statement is assigned a truth value from some complete atomless Boolean algebra, rather than just a true/false value. Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model which contains this ultrafilter, which can be understood as a new model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in an appropriate way, we can get a model that has the desired property. In it, only statements which must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property).

Meta-mathematical explanation.

In forcing, we usually seek to show that some sentence is consistent with formula_54 (or optionally some extension of formula_54). One way to interpret the argument is to assume that formula_54 is consistent and then prove that formula_54 combined with the new sentence is also consistent.  Each "condition" is a finite piece of information – the idea is that only finite pieces are relevant for consistency, since, by the compactness theorem, a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. Then we can pick an infinite set of consistent conditions to extend our model. Therefore, assuming the consistency of formula_54, we prove the consistency of formula_54 extended by this infinite set.

Logical explanation.

By Gödel's second incompleteness theorem, one cannot prove the consistency of any sufficiently strong formal theory, such as formula_54, using only the axioms of the theory itself, unless the theory is inconsistent. Consequently, mathematicians do not attempt to prove the consistency of formula_54 using only the axioms of formula_54, or to prove that formula_509 is consistent for any hypothesis formula_510 using only formula_509. For this reason, the aim of a consistency proof is to prove the consistency of formula_509 relative to the consistency of formula_54. Such problems are known as problems of relative consistency, one of which proves  (*) formula_514  The general schema of relative consistency proofs follows. As any proof is finite, it uses only a finite number of axioms:  For any given proof, formula_54 can verify the validity of this proof. This is provable by induction on the length of the proof.  Then resolve  By proving the following  (**) formula_519  it can be concluded that  which is equivalent to  which gives (*). The core of the relative consistency proof is proving (**). A formula_54 proof of formula_523 can be constructed for any given finite subset formula_524 of the formula_54 axioms (by formula_54 instruments of course). (No universal proof of formula_523 of course.)  In formula_54, it is provable that for any condition formula_21, the set of formulas (evaluated by names) forced by formula_21 is deductively closed. Furthermore, for any formula_54 axiom, formula_54 proves that this axiom is forced by formula_16. Then it suffices to prove that there is at least one condition that forces formula_510.  In the case of Boolean-valued forcing, the procedure is similar: proving that the Boolean value of formula_510 is not formula_536.  Another approach uses the Reflection Theorem. For any given finite set of formula_54 axioms, there is a formula_54 proof that this set of axioms has a countable transitive model. For any given finite set formula_524 of formula_54 axioms, there is a finite set formula_541 of formula_54 axioms such that formula_54 proves that if a countable transitive model formula_63 satisfies formula_541, then formula_105 satisfies formula_524. By proving that there is finite set formula_548 of formula_54 axioms such that if a countable transitive model formula_63 satisfies formula_548, then formula_105 satisfies the hypothesis formula_510. Then, for any given finite set formula_524 of formula_54 axioms, formula_54 proves formula_523.  Sometimes in (**), a stronger theory formula_558 than formula_54 is used for proving formula_523. Then we have proof of the consistency of formula_509 relative to the consistency of formula_558. Note that formula_563, where formula_564 is formula_565 (the axiom of constructibility).

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory.

Forcing (deutsch auch Erzwingung oder Erzwingungsmethode) ist in der Mengenlehre eine Technik zur Konstruktion von Modellen, die hauptsächlich verwendet wird, um relative Konsistenzbeweise zu führen. Sie wurde zuerst 1963 von Paul Cohen entwickelt und verwendet, um die Unabhängigkeit des Auswahlaxioms und der Kontinuumshypothese zu beweisen. Diese Leistung ist 1966 durch die Verleihung der Fields-Medaille gewürdigt worden. Die Forcing-Methode ist von verschiedenen Mathematikern vielfach weiterentwickelt worden.


Members.

The table below lists galaxies that have been identified as associated with the IC342/Maffei 1 Group by I. D. Karachentsev. Note that Karachentsev divides this group into two subgroups centered around IC 342 and Maffei 1.  Additionally, KKH 37 is listed as possibly being a member of the IC 342 Subgroup, and KKH 6 is listed as possibly being a member of the Maffei 1 Subgroup.

Foreground dust obscuration.

As seen from Earth, the group lies near the plane of the Milky Way (a region sometimes called the Zone of Avoidance). Consequently, the light from many of the galaxies is severely affected by dust obscuration within the Milky Way. This complicates observational studies of the group, as uncertainties in the dust obscuration also affect measurements of the galaxies' luminosities and distances as well as other related quantities.  Moreover, the galaxies within the group have historically been difficult to identify. Many galaxies have only been discovered using late 20th century astronomical instrumentation. For example, while many fainter, more distant galaxies, such as the galaxies in the New General Catalogue, were already identified visually by the end of the nineteenth century, Maffei 1 and Maffei 2 were only discovered in 1968 using infrared photographic images of the region. Furthermore, it is difficult to determine whether some objects near IC 342 or Maffei 1 are galaxies associated with the IC 342/Maffei Group or diffuse foreground objects within the Milky Way that merely look like galaxies. For example, the objects MB 2 and Camelopardalis C were once thought to be dwarf galaxies in the IC 342/Maffei Group but are now known to be objects within the Milky Way.

Group formation and possible interactions with the Local Group.

Since the IC 342/Maffei Group and the Local Group are located physically close to each other, the two groups may have influenced each other's evolution during the early stages of galaxy formation. An analysis of the velocities and distances to the IC 342/Maffei Group as measured by M. J. Valtonen and collaborators suggested that IC 342 and Maffei 1 were moving faster than what could be accounted for in the expansion of the universe. They therefore suggested that IC 342 and Maffei 1 were ejected from the Local Group after a violent gravitational interaction with the Andromeda Galaxy during the early stages of the formation of the two groups.  However, this interpretation is dependent on the distances measured to the galaxies in the group, which in turn is dependent on accurately measuring the degree to which interstellar dust in the Milky Way obscures the group. More recent observations have demonstrated that the dust obscuration may have been previously overestimated, so the distances may have been underestimated. If these new distance measurements are correct, then the galaxies in the IC 342/Maffei Group appear to be moving at the rate expected from the expansion of the universe, and the scenario of a collision between the IC 342/Maffei Group and the Local Group would be implausible.

The IC 342/Maffei Group (also known as the IC 342 Group or the Maffei 1 Group) is the nearest group of galaxies to the Local Group. The group can be described as a binary group; the member galaxies are mostly concentrated around either IC 342 or Maffei 1, both of which are the brightest galaxies within the group. The group is part of the Virgo Supercluster.

Die Maffei-Gruppe, Maffei-1-Gruppe oder IC-342/Maffei-Gruppe oder auch LGG 104 ist mit einer Entfernung von rund 10 Millionen Lichtjahren die der Lokalen Gruppe nächstgelegene Galaxiengruppe.


Overview.

The portion that was found was a part of a much bigger work. What remains of the poem comes in two parts, written on two separate single leaves, usually called “fragment I” and “fragment II”, and generally dated about 1000. The date of the poem's composition is unknown. "Waldere" was first edited by George Stephens (Copenhagen, 1860), afterwards by R. Wulker in "Bibliothek der angel-sächsischen Poesie" (vol. 1, Cassel, 1881); then by Peter Holthausen in "Göteborgs högskolas årsskrift" (vol. 5, 1899), with autotype reproductions of the two leaves which have been preserved. The first main translation of "Waldere" was by Frederick Norman in 1933 and the second by Arne Zettersten in 1979. Both are accompanied by commentary. A critical edition by Jonathan B. Himes appeared in 2009.  The fragments can be situated in the epic of which they formed part because the subject, adventures surrounding the hero Walter of Aquitaine, is known in other texts: a Latin epic poem "Waltharius" by Ekkehard of Abbey of St. Gall, dating from the first half of the 10th century; fragments of a Bavarian poem dating from the first half of the 13th century; and two episodes in the Norwegian "Þiðreks saga". Incidental references to the Waldere occur in several Middle High German poems, and there is also a Polish version of the story, the earliest form of which is in "Chronicon Boguphali Episcopi", dating from the 13th or 14th century.  The poem is the only proof that is known that the Anglo-Saxon people had any knowledge of the legend of Walter of Aquitaine. This was a very popular story in the Middle Ages. It is the story of "Waldere" (Walter) and "Hildegyth" who fall in love and steal treasure from the court of Attila, where they were held hostage. Waldere and Hildegyth are sought out by two men, "Guthhere", who is the king of the Burgundians, and "Hagena". These two men are after the treasure that Waldere and Hildegyth possess. The poem is about the conflict that is about to take place between the two parties.  In one fragment someone encourages Waldere to go on fighting. In the other there is praise of a sword, followed by Waldere's praise of his own armour and his defiance of Guthhere. The speeches represented in the fragments have nothing corresponding to them in Ekkehard's text, which suggests that these are independent renderings of the same familiar source material. A passing reference— ""Win fame by valiant deeds, and may God guard thee the while""— shows that, like "Beowulf" the poem had been given a Christianized context.  The first portion is a speech given by Hildegyth trying to motivate Waldere for his upcoming fight. In this speech, Paul Cavill finds, Hildegyth tries to inspire Waldere in four main ways: "Mimming", the great sword of Waldere, that was made by the renowned smith "Weland", is praised; Waldere is reminded that the only two outcomes available to a warrior are glory or death; all the good doings of Waldere are rehearsed, as well as the loftiness of his reputation; all doubt is cleared that it is truly Guthhere at fault for engaging Waldere.  The second fragment consists mainly of Waldere challenging and taunting Guthhere, daring Guthhere to strip Waldere‘s armor from his shoulders. The end of the fragment finds Waldere putting the outcome of the fight in God’s hands (Cavill).  In the story of Walter, this fight results in harm to everyone that has participated. However, in the end the two sides come to a peaceful resolution and eventually Waldere and Hildegyth leave and get married. This ending does not appear in the remaining fragments of "Waldere".

"Waldere" or "Waldhere" is the conventional title given to two Old English fragments from a lost epic poem, discovered in 1860 by E. C. Werlauff, Librarian, in the Danish Royal Library at Copenhagen, where it is still preserved. The parchment pages had been reused as stiffening in the binding of an Elizabethan prayer book, which had presumably come to Europe following the Dissolution of the Monasteries in England in the 16th century.

Der so genannte Waldere ist die fragmentarische Überlieferung einer angelsächsischen Dichtung über den germanischen Helden Walther. Die 63 Zeilen des Waldere sind die einzige Spur der Walther-Sage in der altenglischen Literatur. Die Geschichte spielt, genauso wie die anderen, in der altenglischen Überlieferung erhaltenen germanischen Heldendichtungen (Beowulf, Deor, Widsith, Battle of Finnsburg), nicht in England, sondern schöpft aus einem Stoffrepertoire der Völkerwanderungszeit, das jahrhundertelang durch mündliche Überlieferung weitergegeben wurde. Aufgrund seiner fragmentarischen Überlieferung lässt sich nicht mit Bestimmtheit sagen, wie alt der Waldere ist, wie viele Zeilen er tatsächlich umfasste und woher das Werk ursprünglich stammte. So reichen die Angaben und Theorien zur Entstehung des Waldere denn auch vom 8. Jahrhundert bis weit ins 10. Jahrhundert. Die zwei Blätter der Waldere-Fragmente wurden erst am 12. Januar 1860 in der Königlichen Bibliothek von Kopenhagen von dem Professor und Bibliothekar Erich Christian Werlauff entdeckt. Wie die Handschriften nach Dänemark gelangten, wird wohl nicht mehr mit Sicherheit geklärt werden können. Sie könnten jedoch unerkannt Teil der Handschriften gewesen sein, die der isländische Forscher Grímur Jónsson Thorkelin 1786 im Auftrag des dänischen Königs in England gekauft hatte.


Background.

In June 1998, Norwood released her second album "Never Say Never". Boosted by the success of its number-one lead single "The Boy Is Mine", a duet with singer Monica, it facilitated Norwood in becoming a viable recording artist with cross-media appeal. In total, the album sold 16 million copies worldwide and spawned seven singles, including Norwood's second number-one song, the Diane Warren-penned "Have You Ever?". Also in 1998, Norwood made her big screen debut in a supporting role in the slasher sequel, "I Still Know What You Did Last Summer", which garnered her both a Blockbuster Entertainment Award and an MTV Movie Award nomination for Best Breakthrough Female Performance. The following year, she co-starred with Diana Ross in the telefilm drama "Double Platinum" about an intense, strained relationship between a mother and daughter. Both Norwood and Ross served as executive producers of the movie, which features original songs from "Never Say Never" and Ross's "Every Day Is a New Day" (1999).  Norwood suffered a nervous breakdown in November 1999—the result of her then-hectic and unhealthy lifestyle and a failed relationship in which she had experienced emotional abuse. Frightened by the idea that a yet-to-be-made third album would not be able to live up to the success of her previous albums, Norwood went on a lengthy hiatus to reflect and take some introspective looks. "I needed to rejuvenate, get my creative juices flowing, balance my life with some privacy, to find my confidence, find my love of music again," she told "Jet" magazine in 2002. In mid-2000, she started refocusing herself on her musical career, contributing songs to albums such as "Urban Renewal" (2001) and the "Osmosis Jones" soundtrack (2001), which introduced a scratchy, evocative edge to Norwood's voice, now having a deeper and warmer tone with a textured lower register and notably stronger falsetto.

Recording.

In fall 2000, Norwood finally began conceiving ideas for a third studio album with the Atlantic label. While Rodney Jerkins, the main producer of her previous album "Never Say Never", and his Darkchild crew, including Fred Jerkins III and LaShawn Daniels, had been working on several new songs for the singer's upcoming project in hopes of recreating the winning chemistry of "Never Say Never", Norwood wanted to make sure that she was gaining more creative control over the project and thus, arranged meetings with all her writers and musicians to discuss the lyrical topics and sounds she wanted for the album. "I was involved from A-Z," she said. "Every song on the album was inspired by my life [...] I wanted to talk about how I feel on so many levels. I wanted to be in touch with all of my emotions and share them. I've taken three years off for myself and got a chance to find things I like to do, things I don't like and things I want to change about myself."  While Jerkins maintained his status as the album's executive producer, contributing most to its track listing with his team that also consisted of regular songwriters Nora Payne and Kenisha Pratt, Norwood also worked with frequent collaborators Mike City and Keith Crouch, as well as Warryn "Baby Dubb" Campbell, Stuart Brawley, Jason Derlatka, and Jerkin's cousin Robert "Big Bert" Smith, with whom she became romantically involved during the project. In addition, she also recorded with Pharrell Williams and Chad Hugo from The Neptunes. With Norwood fearing that the pair's material would not fit the concept of the album and break her type of production unit, it was left unused. Rapper Ja Rule, singer Babyface and production duo Soulshock & Karlin were reportedly also involved into the project but none of their songs eventually made the album's final track listing.  Though Norwood has acknowledged that the creative focus of the album was very much on its technical realization and its sound, she declared "Full Moon" a concept album based on the development of a male-female relationship: "It's definitely the concept for the album—me falling in love, then going through some turbulence, and then, at the end, I find the person that I really want to be with—so it's a great concept and it's a great experience that I had. I found out a lot about myself. I found a lot out about love, and I'm just happy to have that reflect in my music." Norwood decided to name the album after its title track and in reference to the previous three years of her life, stating: "I have done a complete circle and I feel whole. All of that's reflected in the music. That's why I entitled [my album] "Full Moon". It's a concept album, it's autobiographical. Everything that I've gone through in the last three years is reflected." The album was originally set to be released on November 20, 2001, but plans were scrapped.

Content.

"Full Moon" opens with "B Rocka Intro," a shortened and rearranged version of "What About Us?" that starts with a robot voice. The title of the intro references Jerkins's nickname for Brandy. The first full song on the album is title track "Full Moon," producer Mike City's only contribution to "Full Moon". A piano-dominated up-tempo song with a "lulling drumbeat and heavy bass," Norwood characterized it as urban contemporary, explaining that "Full Moon" is "pop and R&B at the same time [but] has a lot of elements to it." Lyrically, the song deals with a love at first sight during a full moon night. "I Thought", a Jerkins-crafted adamant break-up song about female empowerment, features electro bass lines and crunchy drums. Jerkins described it as an "anthem [and] a flip off" of Brandy's previous single "The Boy Is Mine". "When You Touch Me" is a ballad that revolves around the planning of a rendezvous. Initially conceived by Big Bert, it was significantly polished by Jerkins. Singer-producer Teddy Riley with whom Jerkins worked on Michael Jackson's "Invincible" (2001) during the creation of "Full Moon" appears on the talk box segment of the song.  The "hand-clapping, funky" song "Like This" sees Brandy continue to discuss her intimate desires with her lover. On "All in Me," a "futuristically funked-out" song that is built on keyboards and a sped up breakdown, the singer pleads with her lover to have faith in her, promising him that she will provide whatever he needs. Producer Rodney Jerkins decided on the inclusion of a 2-step groove section during the middle of the song, following a gig in London months before where he was inspired by artists like Craig David and Artful Dodger. "Apart," produced by Keith Crouch, blends neo soul sensibilities, airy pop, and modern R&B with Middle Eastern music. Lyrically, it has the protagnoist lamenting for her partner's attention, resulting in her decision to leave him for the better. "Can We" is a lightweight hip-hop track coated in a batter of futuristic elements on which Brandy requests her partner to clarify their problems, while committing to working through them. "What About Us?" is an offbeat, aggressive high-tech uptempo song, seemingly "set amid a steel factory's sonic churns, whirs and crunches." Brandy described it as "edgy, sexy" and a "little bit ahead of its time [...]."  On post-break song "Anybody," another "edgy, eclectic offering," Brandy reminds a lover that he is supposed to keep their hurtful relationship a secret. "It's Not Worth It" finds Brandy trying to hold her relationship together after it has deteriorated to shambles. Initially penned in 1999, Jerkins built the song around Michael Jackson's ad-libbed vocals, resulting from a joint recording session for Jackson's 2001 studio album. "He Is", the next song, is a jazzy love song with "a classy piano and sparse drum track" that was produced by Warryn Campbell. Speaking about God in third person, Brandy was unaware that the song was conceptualized as a gospel song by its writers. "Love Wouldn't Count Me Out" is a "sweeping" hymn that has the singer seeking understanding in the midst of romantic trial, while album closer "Wow" is an upbeat ballad that has Brandy singing about the joy of finally finding the perfect lover over a "sun kissed groove." International editions of "Full Moon" contain the cover versions "Another Day in Paradise" and "Die Without You", both being duets with her younger brother Ray J, as well as the beat-heavy, hand-clapping song "I Wanna Fall in Love."

Critical reception.

While "Full Moon" has earned retrospective acclaim and recognition from musicians, singers, and producers, initial media reception toawrds the album was generally mixed. It has a score of 60 out of 100 on review aggregator Metacritic, indicating "mixed or average reviews," with Album of the Year also rating the album 62 out of 100. In his review for "Entertainment Weekly", journalist Craig Seymour gave "Full Moon" an A− rating, saying that "where [Rodney] Jerkins' herky-jerky stylings come off cold on Jacko's latest, they embolden 23-year-old Brandy as she learns the difference between teen heartbreak and grown-up betrayal, [suggesting] maturity and the high price that often comes with it." Stephen Thomas Erlewine from AllMusic was critical with the album's length of over 70 minutes but considered it Norwood's most assured, risky album yet, stating: ""Full Moon" comes the closest to being a full-fledged, well-rounded album, as well as establishing a personality as a singer [...] There are plenty of moments here that are seductively smooth and even the filler goes down smoothly." He gave the album four out of five stars.  "Slant Magazine" writer Sal Cinquemani rated the album three stars out of five and compared it to Janet Jackson's 1986 album "Control", commenting: "For the most part, "Full Moon" is certainly a forward-minded album, lifting Brandy's typically schmaltzy brand of pop-R&B to a new, edgier plateau [...] The all-grown-up Miss Moesha seems to be making her final transition from sitting up in her room to sitting on top of the world." J. Victoria Sanders from "PopMatters" considered "Full Moon" "an achievement" and added: "As she proclaims her womanhood with throaty whispers and assertive wails, [...] this grown-up Brandy [...] has one thing in common with the cherubic girl she used to be: she still sings with relaxing humility and style – qualities the music world is in dire need of right about now." "Washington Post" writer Britt Robson called the album a "refreshingly sexy" record "of honest growth and modest virtues" as well as "of slight refinements and logical maturation." John Aizlewood from "The Guardian" found that "without Jerkins, Brandy stumbles more easily. At 73 minutes "Full Moon" is far too flabby, but there's nothing here to derail her."  "Billboard" praised "Full Moon" for its ballads and the leading single but was unsatisfied with the album as a whole, stating that "those expecting more from the same [as "What About Us?"] will be disappointed, it's a fairly paint-by-numbers affair." Similarly, "People" found that "the rest of "Full Moon" can’t sustain the bizarre brilliance of “What About Us?” While much of the CD brandishes a similar edge, with electronic wizardry made for headphone listening, it showcases the producing team more than its singer. Brandy has one of the more distinctive voices around, so it’s a shame that she so often gets lost in the beat-heavy mix." Devon Thomas, writer for "The Michigan Daily", was generally disappointed with the album. He said that "heavily producer-driven, the album follows the template that catapulted her sophomore album to multi-platinum status. The tradition (or condition) continues on her junior outing, [which] exhibits the same ole Jerkins production we've heard time and time before, just slightly altered (or "updated") and equipped." Critical with mainstream R&B in general, he further summed: "We know it'll be another hit, another platinum plaque for the Moe-ster, but will this album go down on any 'Best of the Decade' lists? Highly unlikely." "Rolling Stone" dismissed the album as "frantic, faceless, fake-sexy R&B," while Piers Martin from "NME" declared it a "velvet-lined bucket of slush." In his Consumer Guide, Robert Christgau gave the album a "dud" rating.

Commercial performance.

In the United States, "Full Moon" debuted on top of the US "Billboard" Top R&B/Hip-Hop Albums chart and at number two on the "Billboard" 200 in the issue dated March 13, 2002, marking Brandy's highest debut on both charts yet. It also debuted and peaked at number eleven on the "Billboard" Top Internet Albums chart. Selling approximately 155,000 copies in its first week of release, the album fell short of the "O Brother, Where Art Thou?" soundtrack (2000) by less than 4,000 copies. Spending thirty weeks on the latter chart, the album shifted about 700,000 copies within the first three months of its US release, and was eventually certified platinum by the Recording Industry Association of America (RIAA) for selling over more than 1.1 million units. With the album's promotion officially ending only six month into its release due to Brandy's pregnancy, domestic sales were not up to par with the first two albums "Brandy" and "Never Say Never" (1998), both of which had gone 4× or 5× platinum in the US, respectively.  In Canada, the album reached number eight on the Canadian Albums Chart. On July 19, 2002, it was certified gold by the Canadian Recording Industry Association (CRIA) for shipments of 50,000 copies in the country. In the United Kingdom, "Full Moon" became Brandy's first top ten album, debuting and peaking at number nine on the UK Albums Chart, with first-week sales shy of 25,000 units. Upon its release, it was immediately certified gold by the British Phonographic Industry (BPI), indicating shipments of 100,000 copies. "Full Moon" also became Brandy's second album to top the UK R&B Albums Chart. In Japan, "Full Moon" debuted at number 15 on the Oricon Albums Chart, becoming her second top 20 entry on the chart. It was eventually certified with gold by the Recording Industry Association of Japan (RIAJ) for shipments of more than 100,000 units. Elsewhere, "Full Moon" entered the top ten of the German and the Swiss Albums Chart, reaching number eight and seven respectively, becoming Brandy's first top ten entries in both countries yet.

Influence and legacy.

"Full Moon" became Norwood's second consecutive album to earn a Grammy Award nomination in the Best Contemporary R&B Album category at the 45th awards ceremony, while also garnering a Best R&B Album/Solo nod at the 2002 Soul Train Music Awards. Although initially receiving mixed reviews from music critics upon release, the album has since garnered retrospective recognition from musicians, singers, and producers, particularly within the contemporary R&B and urban contemporary gospel genres. "Full Moon" is often cited as inspiration for singers due to Norwood's vocal nuances and arrangements. Musicians Ambré, Chris Brown, Jacob Latimore, Lil Mo, Mary Mary, PJ Morton, Keke Palmer Kierra Sheard, and Tank, among many others often reference the album and its vocal work as influential. The oft-praised vocal work on the album sparked the idea of Norwood gaining the subjective nickname the "vocal bible". Norwood herself has ranked "Full Moon" among her favorites in her album discography numerous times.  Songwriter Sean Garrett credits the vocal work on the album for his approach to writing, saying "I take a lot from what [Brandy] and Rodney did on the "Full Moon" album. I was extremely impressed with it and I always try to outdo that album". B.Slade spoke of the album, commenting "Full Moon" single-handedly changed the vocal game. "It has been the template for vocal choices and background vocal arrangements [for years]." R&B singer Melanie Fiona, especially admired the singer's work on that album, dubbing Norwood the "Harmony Queen". Neo soul singer India.Arie often cites the album, particularly the song "He Is" as being the template for a wide array of singers." Canadian R&B singer Keshia Chanté credited the album for inspiring her writing for her album "Night & Day", while American singer Luke James referred to "Full Moon" as the "bible" of 2000s contemporary R&B, calling it the "blueprint of how to do vocals". British soul performer Daley included a cover version of the album cut "When You Touch Me" on his Daley, Unplugged tour; the song was also paid tribute to in gospel form by "Sunday Best" artist Y'anna Crawley. German pop singer Rüdiger Skoczowsky, who cites Brandy as one of his main vocal inspirations, included a cover of "Love Wouldn't Count Me Out" on some of his live shows.

Track listing.

Notes  Production

Full Moon is the third studio album by American R&B singer Brandy. It was released by Atlantic Records on March 5, 2002. The album was recorded primarily during the summer and fall of 2001 at The Hit Factory in Miami, amid a three-year musical hiatus following the success of her multi-platinum previous studio album "Never Say Never" (1998) and the finale of her highly successful television sitcom "Moesha" in May 2001. As with "Never Say Never", Brandy collaborated with producer Rodney Jerkins and his Darkchild production and songwriting team on the majority of the album's composition, while Mike City, Warryn Campbell, and Keith Crouch contributed additional production.

Full Moon (Engl. für Vollmond) ist das dritte Studioalbum der US-amerikanischen R&B-Sängerin Brandy. Es erschien am 25. Februar in Europa und am 5. März in Kanada und den Vereinigten Staaten und brachte mit "Another Day in Paradise", "What About Us?", "Full Moon" und "He Is" vier Singleauskopplungen hervor.


Design.

With the passage of the Second Naval Law under the direction of "Vizeadmiral" ("VAdm"—Vice Admiral) Alfred von Tirpitz in 1900, funding was allocated for a new class of battleships to succeed the ships authorized under the 1898 Naval Law. By this time, Krupp, the supplier of naval artillery to the "Kaiserliche Marine" (Imperial Navy) had developed quick-firing, guns; the largest guns that had previously incorporated the technology were the guns mounted on the "Wittelsbach"s. The Design Department of the "Reichsmarineamt" (Imperial Navy Office) adopted these guns for the new battleships, along with an increase from to for the secondary battery, owing to the increased threat from torpedo boats as torpedoes became more effective.  Though the "Braunschweig" class marked a significant improvement over earlier German battleships, its design fell victim to the rapid pace of technological development in the early 1900s. The British battleship —armed with ten 12-inch (30.5 cm) guns—was commissioned in December 1906, less than a year and a half after "Preussen" entered service. "Dreadnought"s revolutionary design rendered every capital ship of the German navy obsolete, including "Preussen".  "Preussen" was long overall and had a beam of and a draft of forward. She displaced as designed and at Full load. Her crew consisted of 35 officers and 708 enlisted men. The ship was powered by three 3-cylinder vertical triple expansion engines that drove three screws. Steam was provided by eight naval and six cylindrical boilers, all of which burned coal. "Preussen"s powerplant was rated at, which generated a top speed of. She could steam at a cruising speed of.  "Preussen"s armament consisted of a main battery of four 28 cm (11 in) SK L/40 guns in twin gun turrets, one fore and one aft of the central superstructure. Her secondary armament consisted of fourteen 17 cm (6.7 inch) SK L/40 guns and eighteen 8.8 cm (3.45 in) SK L/35 quick-firing guns. The armament suite was rounded out with six torpedo tubes, all mounted submerged in the hull. One tube was in the bow, two were on each broadside, and the final tube was in the stern. "Preussen" was protected with Krupp armor. Her armored belt was thick, the heavier armor in the central portion that protected her magazines and propulsion machinery spaces, and the thinner plating at either end of the hull. Her deck was thick. The main battery turrets had 250 mm of armor plating.

Service history.

Construction through 1907.

"Preussen" was laid down in April 1902 at the AG Vulcan shipyard in Stettin under construction number 256. The fourth unit of her class, she had been ordered under the contract name "K" as a new unit for the fleet. "Preussen" was launched on 30 October 1903, with a speech given by Chancellor Bernhard von Bülow and the christening performed by Empress Augusta Victoria. The ship was commissioned into the fleet on 12 July 1905. Sea trials lasted until September, when she formally joined II Squadron, where she replaced the battleship as the squadron flagship. "VAdm" Max von Fischel was the squadron commander at the time. The German fleet was occupied with extensive training exercises during the early 1900s. The ships were occupied with individual, division and squadron exercises throughout April 1906, the only interruption being in February, when "Preussen" carried Kaiser Wilhelm II to Copenhagen in company with the light cruiser and the torpedo boats and. Wilhelm II attended the burial of the Danish King Christian IX, who had died the previous month. Starting on 13 May, major fleet exercises took place in the North Sea and lasted until 8 June with a cruise around the Skagen into the Baltic. During Kiel Week on 21 June, "Preussen" received a gift from the provinces of West Prussia and East Prussia, in the form of a Prussian war flag. The fleet began its usual summer cruise to Norway in mid-July, and the fleet was present for the birthday of Norwegian King Haakon VII on 3 August. The German ships departed the following day for Helgoland, to join exercises being conducted there.  The fleet was back in Kiel by 15 August, where preparations for the autumn maneuvers began. On 22–24 August, the fleet took part in landing exercises in Eckernförde Bay outside Kiel. The maneuvers were paused from 31 August to 3 September when the fleet hosted vessels from Denmark and Sweden, along with a Russian squadron from 3 to 9 September in Kiel. The maneuvers resumed on 8 September and lasted five more days. The ship participated in the uneventful winter cruise into the Kattegat and Skagerrak from 8 to 16 December. The first quarter of 1907 followed the previous pattern and, on 16 February, the Active Battle Fleet was re-designated the High Seas Fleet. From the end of May to early June the fleet went on its summer cruise in the North Sea, returning to the Baltic via the Kattegat. This was followed by the regular cruise to Norway from 12 July to 10 August, after which the fleet conducted the annual autumn maneuvers, which lasted from 26 August to 6 September. The exercises included landing exercises in northern Schleswig with IX Corps. On 1 October 1907, "Konteradmiral" ("KAdm"—Rear Admiral) Ludwig von Schröder came aboard "Preussen", taking command of the squadron, as Fischel had become the chief of the "Marinestation der Nordsee" (Naval Station of the North Sea). The winter training cruise went into the Kattegat from 22 to 30 November.

1908–1914.

The fleet conducted training exercises in the Baltic in February 1908. Prince Heinrich, then the commander of the High Seas Fleet, had pressed for such a cruise the previous year, arguing that it would prepare the fleet for overseas operations and would break up the monotony of training in German waters, though tensions with Britain over the developing Anglo-German naval arms race were high. The fleet departed Kiel on 17 July, passed through the Kaiser Wilhelm Canal to the North Sea, and continued to the Atlantic. During the cruise, "Preussen" visited Las Palmas in the Canary Islands. The fleet returned to Germany on 13 August. The autumn maneuvers followed from 27 August to 12 September. Later that year, the fleet toured coastal German cities as part of an effort to increase public support for naval expenditures.  In early 1909, "Preussen" and the battleships and were sent to break paths in the sea ice off the coast of Holstein for merchant shipping. Another cruise into the Atlantic was conducted from 7 July to 1 August, during which "Preussen" stopped in El Ferrol, Spain. On the way back to Germany, the High Seas Fleet was received by the British Royal Navy in Spithead. Later that year, Admiral Henning von Holtzendorff became the fleet commander. Holtzendorff's tenure as fleet commander was marked by strategic experimentation, owing to the increased threat the latest underwater weapons posed and the fact that the new s were too wide to pass through the Kaiser Wilhelm Canal. Accordingly, the fleet was transferred from the Baltic Sea port of Kiel to the North Sea port of Wilhelmshaven on 1 April 1910.  In May 1910, the fleet conducted training maneuvers in the Kattegat. These were in accordance with Holtzendorff's strategy, which envisioned drawing the Royal Navy into the narrow waters there. The annual summer cruise was to Norway, and was followed by fleet training, during which another fleet review was held in Danzig on 29 August. After the conclusion of the maneuvers, Schröder was promoted to chief of the "Marinestation der Ostsee" (Naval Station of the Baltic Sea), his place aboard "Preussen" being taken by "VAdm" Friedrich von Ingenohl. A training cruise into the Baltic followed at the end of the year. In March 1911, the fleet held exercises in the Skagerrak and Kattegat, and the year's autumn maneuvers were confined to the Baltic and the Kattegat. Another fleet review was held during the exercises for a visiting Austro-Hungarian delegation that included Archduke Franz Ferdinand and Admiral Rudolf Montecuccoli.  In mid-1912, due to the Agadir Crisis, the summer cruise was confined to the Baltic, to avoid exposing the fleet during the period of heightened tension with Britain and France. On 30 January 1913, Holtzendorff was relieved as the fleet commander, owing in large part due to Wilhelm II's displeasure with his strategic vision; Ingenohl took Holtzendorff's place, and "KAdm" Reinhard Scheer in turn replaced Ingenohl as II Squadron commander on 4 February. In late August, the squadron steamed through the Kaiser Wilhelm Canal at the start of the autumn maneuvers to reach the island of Helgoland; the voyage through the canal was notable, because the canal had been closed for over a year while it was enlarged to allow the passage of larger dreadnought battleships. "Preussen" went into dry dock in November for periodic maintenance, and as a result, missed training exercises conducted that month.  "Preussen" participated in ceremonies at Sonderburg on 2 May to celebrate the 50th anniversary of the Battle of Dybbøl of the Second Schleswig War; she was joined by her sister ships and, the battleship, and the armored cruiser. The ship was present during the fleet cruise to Norway in July 1914, which was cut short by the July Crisis following the assassination of Archduke Franz Ferdinand the month before and subsequent rise in international tensions. On 25 July the ship's crew was made aware of Austria-Hungary's ultimatum to Serbia; "Preussen" left Norway to rendezvous with the rest of the fleet the following day. "Preussen" had been slated to be decommissioned at the end of the year, her place as the squadron flagship to be taken by the dreadnought. "Preussen" was to then replace in the Reserve Division of the Baltic Sea, but the outbreak of World War I cancelled those plans.

World War I.

After the outbreak of war in August 1914, the German command deployed II Squadron in the German Bight to defend Germany's coast from a major attack from the Royal Navy that the Germans presumed was imminent. "Preussen" and her squadron mates were stationed in the mouth of the Elbe to support the vessels on patrol duty in the Bight. Once it became clear that the British would not attack the High Seas Fleet, the Germans began a series of operations designed to lure out a portion of the numerically superior British Grand Fleet and destroy it. By achieving a rough equality of forces, the German navy could then force a decisive battle in the southern portion of the North Sea.  The first such operation in which the High Seas Fleet participated was the raid on Scarborough, Hartlepool and Whitby on 15–16 December 1914. The main fleet acted as distant support for "KAdm" Franz von Hipper's battlecruiser squadron while it raided the coastal towns. On the evening of 15 December, the fleet came to within of an isolated squadron of six British battleships. However, skirmishes between the rival destroyer screens in the darkness convinced Ingenohl, who was now the German fleet commander, that the entire Grand Fleet was deployed before him. Under orders from Wilhelm II to avoid battle if victory was not certain, Ingenohl broke off the engagement and turned the battlefleet back towards Germany. At the end of the month, Scheer was replaced by "KAdm" Felix Funke, Scheer going on to command III Battle Squadron.  On 14 March 1915, "Preussen" went to Kiel for periodic maintenance, and she was replaced as the squadron flagship by the battleship. The latter vessel held the position for the remainder of the squadron's existence, with the exceptions of 19 September to 16 October 1915, 25 February to 7 April 1916, in September that year, and from 22 January to 10 February 1917; during each of these periods, "Preussen" temporarily resumed the flagship role. Starting in April 1916, the ships of II Squadron were tasked with patrolling the Danish straits; each ship of the squadron was to rotate through the duty, the others remaining in the Elbe or serving with the main fleet. "Preussen" began her first stint on 21 April, replacing "Hessen"; she was accordingly absent during the bombardment of Yarmouth and Lowestoft three days later. On 4 May, "Preussen" was relieved and she returned to the Elbe, though she was transferred back to the straits on 21 May, remaining there until 8 June. As a result, she missed the Battle of Jutland, fought on 31 May – 1 June in the North Sea. Her sister ship "Lothringen" also missed the battle, as she had been deemed to be in too poor a condition to participate in the fleet sortie.  Jutland proved to Scheer, who was now the fleet commander, that the pre-dreadnought battleships were too vulnerable to take part in a major fleet action, and so he detached II Squadron from the High Seas Fleet. As a result, "Preussen" remained in service only as a guard ship. She saw further stints in the Danish straits from 30 June to 23 July and from 15 to 31 August. On 13 March 1917, "Preussen" left the Elbe and steamed to the Baltic, where she was temporarily used as an icebreaker to clear a path to Swinemünde. She was decommissioned on 5 August, and most of her crew were transferred to the new battlecruiser. From then to the end of the war, she served as a tender for III U-boat Flotilla based in Wilhelmshaven. "Preussen" briefly held Edouard Izac, a US Navy sailor captured after his ship was sunk by, in June 1918; Izac would go on to escape from a German prisoner of war camp and win the Medal of Honor.

Post-war career.

Following the German defeat in World War I, the German navy was reorganized as the "Reichsmarine" according to the Treaty of Versailles. The new navy was permitted to retain eight pre-dreadnought battleships under —two of which would be in reserve—for coastal defense. "Preussen" was among those ships chosen to remain on active service with the "Reichsmarine". The ship was converted into a parent ship for F-type minesweepers at the Kriegsmarinewerft in Wilhelmshaven in 1919; the ship was disarmed and platforms for holding the minesweepers were installed.  "Preussen" was commissioned with the "Reichsmarine" to support the minesweeping effort, though she proved to be too top-heavy to serve as an effective mothership, and she was soon replaced by the old light cruiser. "Preussen" was placed in reserve and was left out of service until 1929, when she was stricken from the naval register on 5 April. The "Reichsmarine" sold her to ship breakers on 25 February 1931 for 216,800 Reichsmarks. "Preussen" was subsequently broken up for scrap in Wilhelmshaven, though a length of her hull was retained as a testing target for underwater weapons, including torpedoes and mines. The section of hull was nicknamed "SMS "Vierkant" ("SMS Rectangle"). Allied bombers attacked and sank the section of "Preussen"s hull in April 1945. It was eventually raised and scrapped in late 1954.

SMS "Preussen" was the fourth of five pre-dreadnought battleships of the, built for the German "Kaiserliche Marine" (Imperial Navy). She was laid down in April 1902, was launched in October 1903, and was commissioned in July 1905. Named for the state of Prussia, the ship was armed with a battery of four guns and had a top speed of. Like all pre-dreadnoughts built at the turn of the century, "Preussen" was quickly made obsolete by the launching of the revolutionary in 1906; as a result, she saw only limited service with the German fleet.

Die SMS "Preußen" war ein Linienschiff der "Braunschweig"-Klasse der ehemaligen Kaiserlichen Marine. Benannt war es nach dem Königreich Preußen, dem damals größten Teilstaat des deutschen Kaiserreichs.


Introduction.

A particular problem in the area of liquid-state thermodynamics is the sourcing of reliable thermodynamic constants. These constants are necessary for the successful prediction of the free energy state of the system; without this information it is impossible to model the equilibrium phases of the system.  Obtaining this free energy data is not a trivial problem, and requires careful experiments, such as calorimetry, to successfully measure the energy of the system. Even when this work is performed it is infeasible to attempt to conduct this work for every single possible class of chemicals, and the binary, or higher, mixtures thereof. To alleviate this problem, free energy prediction models, such as UNIFAC, are employed to predict the system's energy based on a few previously measured constants.  It is possible to calculate some of these parameters using ab initio methods like COSMO-RS, but results should be treated with caution, because ab initio predictions can be off. Similarly, UNIFAC can be off, and for both methods it is advisable to validate the energies obtained from these calculations experimentally.

UNIFAC correlation.

The UNIFAC correlation attempts to break down the problem of predicting interactions between molecules by describing molecular interactions based upon the functional groups attached to the molecule. This is done in order to reduce the sheer number of binary interactions that would be needed to be measured to predict the state of the system.

Chemical activity.

The activity coefficient of the components in a system is a correction factor that accounts for deviations of real systems from that of an Ideal solution, which can either be measured via experiment or estimated from chemical models (such as UNIFAC). By adding a correction factor, known as the activity (formula_1, the activity of the i component) to the liquid phase fraction of a liquid mixture, some of the effects of the real solution can be accounted for. The activity of a real chemical is a function of the thermodynamic state of the system, i.e. temperature and pressure.  Equipped with the activity coefficients and a knowledge of the constituents and their relative amounts, phenomena such as phase separation and vapour-liquid equilibria can be calculated. UNIFAC attempts to be a general model for the successful prediction of activity coefficients.

Model parameters.

The UNIFAC model splits up the activity coefficient for each species in the system into two components; a combinatorial formula_2 and a residual component formula_3. For the formula_4-th molecule, the activity coefficients are broken down as per the following equation:  In the UNIFAC model, there are three main parameters required to determine the activity for each molecule in the system. Firstly there are the group surface area formula_6 and volume contributions formula_7 obtained from the Van der Waals surface area and volumes. These parameters depend purely upon the individual functional groups on the host molecules. Finally there is the binary interaction parameter formula_8, which is related to the interaction energy formula_9 of molecular pairs (equation in "residual" section). These parameters must be obtained either through experiments, via data fitting or molecular simulation.

Combinatorial.

The combinatorial component of the activity is contributed to by several terms in its equation (below), and is the same as for the UNIQUAC model.  where formula_11 and formula_12 are the molar weighted segment and area "fractional" components for the formula_4-th molecule in the total system and are defined by the following equation; formula_14 is a compound parameter of formula_15, formula_16 and formula_17. formula_16 is the coordination number of the system, but the model is found to be relatively insensitive to its value and is frequently quoted as a constant having the value of 10.  formula_20 and formula_21 are calculated from the group surface area and volume contributions formula_6 and formula_7 (Usually obtained via tabulated values) as well as the number of occurrences of the functional group on each molecule formula_24 such that:

Residual.

The residual component of the activity formula_26 is due to interactions between groups present in the system, with the original paper referring to the concept of a "solution-of-groups". The residual component of the activity for the formula_4-th molecule containing formula_28 unique functional groups can be written as follows:  where formula_30 is the activity of an isolated group in a solution consisting only of molecules of type formula_4. The formulation of the residual activity ensures that the condition for the limiting case of a single molecule in a pure component solution, the activity is equal to 1; as by the definition of formula_30, one finds that formula_33 will be zero. The following formula is used for both formula_34 and formula_30  In this formula formula_37 is the summation of the area fraction of group formula_38, over all the different groups and is somewhat similar in form, but not the same as formula_11. formula_40 is the group interaction parameter and is a measure of the interaction energy between groups. This is calculated using an Arrhenius equation (albeit with a pseudo-constant of value 1). formula_41 is the group mole fraction, which is the number of groups formula_28 in the solution divided by the total number of groups.  formula_45 is the energy of interaction between groups "m" and "n", with SI units of joules per mole and "R" is the ideal gas constant. Note that it is not the case that formula_46, giving rise to a non-reflexive parameter. The equation for the group interaction parameter can be simplified to the following:  Thus formula_48 still represents the net energy of interaction between groups formula_38 and formula_28, but has the somewhat unusual units of absolute temperature (SI kelvins). These interaction energy values are obtained from experimental data, and are usually tabulated.

The UNIFAC method (UNIQUAC Functional-group Activity Coefficients) is a semi-empirical system for the prediction of non-electrolyte activity in non-ideal mixtures. UNIFAC uses the functional groups present on the molecules that make up the liquid mixture to calculate activity coefficients. By using interactions for each of the functional groups present on the molecules, as well as some binary interaction coefficients, the activity of each of the solutions can be calculated. This information can be used to obtain information on liquid equilibria, which is useful in many thermodynamic calculations, such as chemical reactor design, and distillation calculations.

UNIFAC (Abkürzung für Universal Quasichemical Functional Group Activity Coefficients) ist ein Verfahren zur Abschätzung von Aktivitätskoeffizienten, das in der Verfahrenstechnik und der Technischen Chemie häufig eingesetzt wird.


History.

Early years.

Jamiat "emerged" in 1972 from among "the informal Islamist groupings that had existed since the 1960s". Led by Burhanuddin Rabbani, a professor of Islamic theology at Kabul University, it was inspired by Abul A'la Maududi and his Jamaat-e-Islami Pakistan. When Rabbani's arrest was ordered by Mohammad Daoud Khan in 1973, it was to Pakistan that Rabbani fled, and Jamaat-e-Islami who initially hosted him there. (Later Jamait lost the backing of Jamaat-e-Islami to the more purist "Hezb-i Islami".)  In Pakistan, Professor Rabbani gathered important people and continued to build the party. Sayed Noorullah Emad, who was then a young Muslim at Kabul University became its general secretary and, later, its deputy chief. Some of its prominent commanders included Ustad Zabihullah, Ahmad Shah Massoud, Ismail Khan, Atta Muhammad Nur, Mullah Naqib and Dr. Fazlullah. Ahmad Shah Massoud directed the military wing of the party.

Massoud–Hekmatyar split.

Ahmad Shah Massoud and Gulbuddin Hekmatyar were both early followers of Rabbani, being Kabul University students at the time. Hekmatyar broke away from Jamiat in 1976 to found his own party: Hezb-e Islami.  The two groups formed the two main tendencies of the Islamist movement in Afghanistan, and after the April 1978 coup and the brutality of the invading Soviet Army, the two strongest Afghan mujahideen groups in the 1980s.

Jamiat.

Rabbani and the Jamiat advocated "building of a widely based movement that would create popular support", a gradualist strategy of infiltration of society and the state apparatus to gain power. Jamiat was dominated by Tajiks but had a greater `tribal and regional cross section` than other groups, and was willing to seek "common ground" with non-Islamists. It gained prominence because of the battlefield success of Ahmad Shah Massoud.

Hezb-i Islami.

Hezb-i Islami was overwhelmingly Ghilzai Pashtun, and backed by Pakistan president Zia ul-Haq. Its leader, Hekmatyar, was "implacably hostile to any form of compromise" favoring violent armed conflict. As a result, Hezb-i Islami, and not Jamiat, gained the support of the Muslim Brotherhood, Jamaat-e-Islami Pakistan, and Saudi Arabian networks.

Soviet invasion.

After the Soviet invasion in Afghanistan in 1979, Massoud organised a mujahideen group in Parwan Province to fight against the Communist government and their Soviet allies. This group grew to control multiple provinces and include thousands of fighters. The Soviet Army launched a series of major offensives to attempt to destroy their forces, but they were unable to engage most of Massoud's men.

Battles between Jamiat and Hezb-i Islami.

After the withdrawal of the Soviet troops in 1989, the mujahideen groups continued to wear down government forces. However they also fought among each other: in June 1990, battles between Jamiat and Gulbuddin's Hezb in Logar and Parwan caused hundreds of casualties on each side.

1992–2001.

In 1992 the communist government collapsed entirely. Jamiat's forces were among the first to enter Kabul. Meanwhile, a peace and power-sharing agreement among the leadership of the Afghan political party leaders led to a tentative agreement to appoint Burhanuddin Rabbani, who had spent the civil war in exile, as interim president. The peace agreement was called the Peshawar Accords.  Gulbuddin Hekmatyar however, did not support the peace agreement despite the fact that he was repeatedly offered the position of prime minister. Subsequently his Hezb-i Islami attacked the new interim government and the capital of Kabul with tens of thousands of rockets. As Hezb-i Wahdat and Ittihad-i Islami started a second war in 1992 and Dostum's Junbish-i Milli joined Hekmatyar in 1994, Kabul witnessed a gruesome war with massive civilian casualties and destruction of much of the city. In 1995 the Islamic State of Afghanistan government also with Jamiat forces retained control of Kabul, pushing back a coalition of Hekmatyar's Hezb-i Islami, the Hizb-i-Wahdat and Abdul Rashid Dostum's Jumbish-i-Milli Islami.  By 1995, the Taliban, which had seized control of much of southern Afghanistan with comparative ease the previous year, were advancing on Kabul. Jamiat rejected Taliban demands that they surrender, and the Taliban rejected Jamiat's offer to join a peaceful political process leading towards a general election. In March 1995, Massoud handed the Taliban their first major loss, however, with the aid of Saudi and Pakistani backing, they regrouped and launched and offensive in mid-1996. Massoud ordered the retreat of his troops among them Jamiat to avoid another bloodbath.  Following the capture of Kabul by the Taliban, the major mujahideen factions put aside their feuds and formed the United Islamic Front for the Salvation of Afghanistan (the United Front), commonly known in the west as the Northern Alliance, with Rabbani, officially becoming its political leader. Other Jamiat members took up senior positions within the United Front government: Yunus Qanuni served as Interior Minister and Dr. Abdullah became Foreign Minister, for example.  On 9 September 2001, just two days before the September 11 attacks in the United States, Massoud was assassinated by two suicide bombers, probably at the instigation of al-Qaeda. Immediately afterwards Taliban forces launched a major offensive against United Front positions. Mohammed Qasim Fahim was chosen to succeed Massoud as leader of Jamiat's military wing and repulsed the Taliban offensive. With extensive assistance from an American-led coalition in October and November 2001 (see War in Afghanistan (2001–present)), United Front forces recaptured most of Afghanistan.  Jamiat's founder and leader, Burhanuddin Rabbani, was assassinated in 2011. His son, Salahuddin Rabbani, has since led the party.

Jamayat-E-Islami (also rendered as Jamiat-e-Islami and Jamiati Islami; ; "Islamic Society"), sometimes shortened to Jamiat, is a Muslim political party in Afghanistan. The oldest Muslim political party in Afghanistan, it was originally formed as a student political society at Kabul University. The majority of the party are ethnic Tajiks of northern and western Afghanistan. It has a communitarian ideology based on Islamic law. During the Soviet–Afghan War and the following Afghan Civil War against the communist government, Jamiat-e Islami was one of the most powerful of the mujahideen groups. Burhanuddin Rabbani led the party (including its predecessors) from 1968 to 2011, and served as President of the Islamic State of Afghanistan from 1992 to 2001 (in exile from 1996).

Die Dschamiat-i Islāmī-yi Afghanistān () ist eine überwiegend von ethnischen Tadschiken geprägte afghanische Partei. Sie gehört neben der Ittahād-i Islāmī, der Hizb-i Islāmī (Hekmatyār) und der Hizb-i Islāmī (Chalis) zu den vier großen islamistisch ausgerichteten Parteien Afghanistans.