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Bronislav Kaminski

Bronislav Vladislavovich Kaminski (Бронисла́в Владисла́вович Ками́нский, 16 June 1899 – 28 August 1944) was a Soviet Nazi collaborator and the commander of the Kaminski Brigade, an anti-partisan and rear-security formation made up of people from the so-called Lokot Autonomy territory (1941–1943) in part of the German-occupied area of the Soviet Union. The Kaminski Brigade later became part of the Waffen-SS as the SS Sturmbrigade RONA (Russian People Liberation Army – Russkaya Osvoboditelnaya Narodnaya Armiya, RONA). (Older publications mistakenly give Kaminski's first name as Mieczyslaw. ) Under Kaminski's command, the unit committed numerous war crimes and atrocities in the German-occupied Soviet Union and in Poland. The unit is regarded as one of the most brutal units, with Kaminski himself feared by his subordinates. Kaminski's behavior made the Germans lose confidence in him: in August 1944, he was court-martialed and executed. His Brigade was later disbanded and its remaining personnel absorbed into General Andrey Vlasov's Russian Liberation Army.

Early life
Bronislav (also transliterated German-style as "Bronislaw") Kaminski was born in Vitebsk Governorate, the Russian Empire, now in Polotsk District, Belarus. His father was an ethnic Pole and his mother was an ethnic German (later termed Volksdeutsch in Nazi Germany). He studied at the Saint Petersburg Polytechnical University and then served in the Red Army during the Russian Civil War. After demobilization he returned to the institute, and after graduation worked at a chemical plant. In 1935 he was expelled from the Communist Party of the Soviet Union, and in 1937 during the Great Purge he was arrested for criticising Stalin's policy of farm collectivisation, as well as working with Germans and Poles. He was accused of "belonging to a counter-revolutionary group". In 1941 he was released from prison and settled in Bryansk, where he obtained a position as an engineer in the local distillery.

Lokot Republic leader
By October 1941, the German military advance into the Soviet Union reached the area of Lokot near the city of Bryansk, which was itself captured by German forces on 6 October 1941. In November 1941, Bronislav Kaminski, along with his close friend, a local technical school teacher, Konstantin Voskoboinik, approached the German military administration with a proposal to assist the Germans in establishing a civil administration and local police. Voskoboinik was appointed by the Germans as the Starosta (or Mayor) of the "Lokot volost" and the head of the German-controlled local militia. Kaminski became Voskoboinik's deputy. Working with Heinz Guderian the two collaborators organized a militia of 10,000 armed hands with the aim of crushing the Russian partisans. After the death in combat of Voskoboinik on 8 January 1942, Kaminski took over as Mayor and leader of the militia. In 1942 the militia was named Russian Liberation National Army (RONA) (rus. Русская Освободительная Народная Армия), which reached a force of tens of thousands. During this time Kaminski was leader of the Lokot Republic, which had half a million inhabitants in its territory.

In mid March 1942, Kaminski's representative at the German Second Panzer Army at Orel assured the commanders that Kaminski's unit was "ready to actively fight the guerrillas" as well as to carry on a propaganda campaign against "Jewish Bolshevism" and Soviet partisans. Soon after the commander of the 2nd Army Generaloberst Rudolf Schmidt, appointed Kaminski Major of the Army Rear Area 532, centered on Lokot. On 19 July 1942, after approval by the Commander of Army Group Centre, Field Marshal Günther von Kluge, Schmidt and the 532 Area commander, Kaminski received a degree of autonomy and nominal self-governing authority, under the supervision of Major von Veltheim and Colonel Rübsam. Kaminski was appointed the Chief Mayor of the republic of Lokot, and brigade commander of the local militia. He administered local government and established his own courts, jails and newspaper. Private enterprise was encouraged and collective farming abolished.

From June 1942, Kaminski's militia took part in the major action codenamed "Operation Vogelsang", as a part of General Werner von Gilsa's Kampfgruppe (taskforce) Gilsa II. In autumn 1942, Kaminski ordered a compulsory draft into the militia of able-bodied men in the area. Units were also reinforced with "volunteers" drafted from Soviet POWs at nearby Nazi concentration camps. Kaminski ordered the collection of abandoned (usually because of minor mechanical failures, or lack of fuel) Soviet tanks and armored cars. By November 1942, his unit was in possession of at least two BT-7 tanks and one 76 mm artillery piece. Owing to a lack of military uniform and boots, the Germans provided Kaminski's brigade with enough used uniforms to outfit four battalions. However, by late 1942, the militia of the Lokot Republic had expanded to the size of a 14-battalions brigade, around 8,000 men under arms. By January 1943 the brigade had 9,828 men, including an armored unit with one heavy KV-II, two medium T-34, three BT-7 and two BT-5 light tanks and three armoured cars (one BA-10, two BA-20).

The brigade's structure was reorganized in the spring of 1943. After the reorganization, the brigade consisted of 5 regiments with 3 battalions each, an anti-aircraft battalion (3 AA guns and 4 heavy machine guns) and an armored unit. A separate "guard" battalion was also created, bringing the total brigade strength up to an estimated 12 thousand men.

The brigade took part, alongside other German units, in the May–June 1943 Operation Zigeunerbaron ("Gypsy Baron"). Following this operation, the brigade was part of Operation Citadel, the massive offensive to destroy the Kursk salient. This operation was followed by similar operations, Freischütz and Tannhäuser, where the brigade, together with other units under German command, was involved in action against partisans and also took part in reprisal operations against the civilian population.

In the summer of 1943, the brigade began to suffer major desertions, due in part to the recent Soviet victories, but also due in part to the efforts of the partisans to "turn" as many of Kaminski's troops as possible. As a part of these efforts, there were several attempts on Kaminski's life. Each time, Kaminski narrowly avoided death and any captured conspirators were punished by execution. Several German officers passing through Lokot reported seeing bodies hanging from gallows outside Kaminski's headquarters. Fearing a breakdown in command, a German liaison staff was attached to Kaminski's HQ to restructure the brigade and return stability to the unit.

According to post-war Soviet estimates, up to 10,000 civilians were killed during the existence of the Kaminski formation in Lokot.

In Belarus
After the failure of the German Operation Citadel, Soviet counter offensives forced the brigade to retreat. On 29 July 1943 Kaminski issued orders for the evacuation of the property and families of the RONA brigade and the Lokot authorities. Up to 30 thousand persons (10-11 thousand of them were brigade members) were transferred by the Germans to the Lepel area of Vitebsk in Belarus by the end of August 1943. The brigade, together with "evacuated" civilians, finally settled in the Lepel area of Vitebsk. This area had active partisans, and the brigade was involved in heavy combat for the rest of the year.

During the retreat, desertions from the brigade increased greatly, and the entire formation seemed close to disintegration. When the commander of the Second Regiment, Major Tarasov, decided to join the Soviet partisans with all his regiment (he was offered amnesty if his entire regiment joined the partisans), Kaminski flew to his headquarters and according to one account, strangled him and eight others in front of his men. Despite this, up to 200 people deserted within the following two days. By the beginning of October 1943, the brigade had lost two thirds of its personnel, while still in possession of 12 tanks (8 of them T-34s), one 122 mm, three 76 mm and eight 45 mm artillery pieces.

At the end of 1943, after German withdrawal from Russia to Belarus, local Belarus police, Soviet prisoners of war and convicts freed from prison were incorporated into the brigade. The force became part of the Waffen-SS and was formed as Stormtroopers Brigade "Kaminski". On 27 January 1944, Himmler rewarded Kaminski's "achievements" by decorating him with the Iron Cross, 2nd Class, and on the same day with the Iron Cross, 1st Class.

On 15 February 1944, Kaminski issued an order for the brigade and Lokot administration to retreat further west to the Dzyatlava area in western Belarus. At this point, the brigade's ranks were replenished by the addition of police forces from Belarus. In March 1944, the brigade was renamed Volksheer-Brigade Kaminski. From 11 April 1944, it was attached to SS-Kampfgruppe von Gottberg, which also included the notorious Dirlewanger unit, and participated in a mass murder and security operations: Regenschauer (up to 7,000 "bandits" reported as killed), Frühlingsfest (7,011 "bandits" reported as killed and 1,065 weapons captured) and Kormoran (7,697 "bandits" reported as killed and 325 weapons captured). During these operations, local civilians were murdered as "suspected partisans" or deported as slave laborers, their villages burnt down.

In the SS
In June 1944, the brigade was absorbed as a part of the Waffen-SS and renamed Waffen-Sturm-Brigade RONA, with Kaminski being given the rank of Waffen-Brigadeführer der SS, the only man with such rank.

As the result of Operation Bagration, the anti-partisan activities of the brigade were halted and its personnel (6–7,000 people, though some sources state 3–4,000) were collected at the SS training camp Neuhammer. Plans were made for a non-German SS Division, and the structure was laid down for the 29.Waffen-Grenadier-Division der SS (russische Nr.1). On 1 August 1944 Kaminski received a new rank - Waffen-Brigadeführer and Major-General of the Waffen-SS.

Warsaw Uprising
In 1944 Kaminsky took part in the crushing of the Warsaw Uprising, in the Ochota area, where the Kaminski Brigade committed numerous atrocities (murder, rape and robbery). His troops committed almost 700 murders though numbering only 1% of German forces at the time of the Warsaw uprising. Kaminski thought himself to have direct authority from the SS commander Heinrich Himmler and he did not want to accept orders from SS General Erich von dem Bach-Zelewski, who was in charge of German forces in Warsaw.

The Warsaw Uprising began 1 August 1944. On 4 August 1944, a combat-ready regiment of the brigade was ordered to assist in crushing the rebellion. SS-Gruppenführer Heinz Reinefarth was placed in charge of Kampfgruppe Reinefarth, a pacification unit which consisted of the Kaminski along with the Dirlewanger and several other Ordnungspolizei and SS rear area units. Hitler personally requested Kaminski's assistance, and the latter obliged by gathering a task force of 1,700 unmarried men and sending them (some sources mentioned that they had four T-34 tanks, one SU-76 and a few artillery pieces) to Warsaw as a mixed regiment under field command of Kaminski's brigade chief-of-staff, SS-Sturmbannführer Yuri Frolov. Later in 1945 Frolov stated that the regiment had up to 1,600 men and 7 artillery pieces and 4 mortars.

Frolov noted in 1944 that Kaminski gave his men permission to loot and many did. Kaminski's brigade soon lost any combat worthiness and Kaminski himself focused entirely on collecting valuables stolen from civilian homes. Perhaps 10,000 residents of Warsaw were killed in the Ochota massacre, most murdered by Kaminski's men.

Death
Heinrich Himmler used the misconduct of the Warsaw group as a pretext for having Kaminski and his leadership executed after trial by court martial in Litzmannstadt (Łódź). They were tried for stealing the property of the Reich, as the stolen property was to have been delivered to Himmler, but Kaminski and his men had attempted to keep it for themselves. Also executed with Kaminski was his brigade chief-of-staff Waffen-Obersturmbannführer Ilya Shavykin, his driver, and his brigade surgeon, F.N. Zabora, and his translator, G. Sadovsky.

The men of the brigade were given the false explanation that Kaminski had been killed by Polish partisans. When Kaminski's men rejected this explanation, the Gestapo took Kaminski's car, pushed it into a ditch, shot it up with a machine gun, and smeared goose blood all over it, as evidence. The demoralized unit was soon moved out of town and stationed to the north, far from any partisan activity.

The death of Kaminski and the unreliability of his troops as a combat unit brought the plans to expand the Kaminski Brigade to a division to an end. After Kaminski's death, his unit was placed under the command of SS-Brigadeführer and Generalmajor der Polizei Christoph Diehm.

Awards and decorations

 * Iron Cross of 1939
 * 1st Class (27 Jan 1944)
 * 2nd Class (27 Jan 1944)
 * Eastern Front Medal
 * Anti-Partisan Guerrilla Warfare Badge (31 July 1944)
 * Ostvolk Medal 1st and 2nd class (1944)
 * Wound Badge in Black

title: Bronislav Kaminski


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Johann Georg Wagler

Johann Georg Wagler (28 March 1800 – 23 August 1832) was a German herpetologist and ornithologist.

Wagler was assistant to Johann Baptist von Spix, and gave lectures in zoology at the Ludwig Maximilian University of Munich after it was moved to Munich. He worked on the extensive collections brought back from Brazil by Spix, and published partly together with him books on reptiles from Brazil. Wagler wrote Monographia Psittacorum (1832), which included the correct naming of the blue macaws.

In 1832, Wagler died of an accidental self-inflicted gunshot wound while out collecting in München-Moosach.

Life
Johann Georg Wagler was a German naturalist and scientist in the 19th century, whose works primarily focused on herpetology and ornithology (Beolens, Watkins & Grayson, 2011). Johan Georg Wagler was born on 28 March 1800, in the city of Nuremberg, where the Chancellor of the City Court was Wagler's father (Wagler, 1884). After taking up gymnastics at Nuremberg, Johann Georg Wagler began to show a unique predilection for Natural History. Johann Georg Wagler started his scientific research at the University of Erlangen in 1818. After Johann Baptist von Spix returned from his expedition to Brazil in 1819, Johann Georg Wagler, nineteen years of age at the time, was invited become Spix's assistant at the Museum of Academy of Sciences in Munich. Johann Georg Wagler later became one of Spix's colleagues (Wagler, 1884). In 1820 Johann Georg Wagler obtained the degree of Doctor of Philosophy at the University of Leipsic (Beolens, Watkins & Grayson, 2011). In 1825, after being Spix's assistant in a large number of his zoological works, Johann Georg Wagler was tasked by King Maximillian-Joseph on a special trip to visit the museums of Holland, England, and France (Wagler, 1884).

During this mission across Europe, Johann Georg Wagler was not only made the personal acquaintance of some of the most decorated naturalists and scientists of these nations at the time but also succeeded in acquiring a significant quantity of valuable specimens for the Museum and Menagerie of Munich (Beolens, Watkins & Grayson, 2011). Upon Johann Baptist von Spix's death in 1826, Johann Georg Wagler became the Director of the Zoological Museum and Ludwig Maximilian University in Munich, where Wagler continued working on Spix's extensive Brazilian collection from years prior (Beolens, B., Watkins, M., & Grayson, M, 2011). In 1827 Wagler was appointed Professor Extraordinary in the then newly instituted University of Munich (Wagler, 1884). In 1832, Johann Georg Wagler was unfortunately injured from a self-inflicted gunshot on a hunting trip, an incident that led directly to his demise (Beolens, Watkins & Grayson, 2011). During his short life span of only thirty-two years, Wagler managed to become a valuable member of the science community with many great and important contributions and discoveries.

Works
During his time, Johann Georg Wagler specialised mostly in the fields of herpetology and ornithology. Ornithology is a branch of zoology that specialises in the “methodological study and consequent knowledge of birds with all that relates to them."(Newton & Gadow, 1986). There are several aspects of ornithology that stem from the related disciplines because of the high level of visibility and the aesthetic appeal of birds (Newton, 1998). Herpetology, on the other hand, is a branch within zoology that concerns the research and study of amphibians and reptiles (Zug, Vitt & Caldwell, 2001). From his research and discoveries, Johann Georg Wagler had written and co-written multiple papers on herpetology and ornithology, of which the most famous was a book regarding ornithology named Monographia Psittacorum. Johann Georg Wagler's other books, articles and research papers are currently available for viewing at the Biodiversity Heritage Library (Beolens, Watkins & Grayson, 2011). Given his short life span, Johann Georg Wagler did not get many chances to publish, however, in 1884, a memoir of Johann Georg Wagler, Wagler's Six ornithological memoirs from the “Isis”, was edited and published (Wagler, 1884). The memoirs contained unfinished and unpublished works of Johann Georg Wagler, along with personal details of his expeditions throughout the years (Wagler, 1884).

Legacy
Wagler is commemorated in the specific names of three species of reptiles: Atractus wagleri, Podarcis waglerianus, and Tropidolaemus wagleri.

Wagler described a taxonomic arrangement of psittacine fauna, parrots and cockatoos, some of which are recognised in the systematic classification of these birds.

Awards
Despite the works and contributions in the fields of herpetology and ornithology made during his career (Beolens, Watkins & Grayson, 2011), Johann Georg Wagler did not receive any awards. His legacy and discoveries in the fields of ornithology and herpetology, however, have received appraisals from fellow scientists. Johann Georg Wagler's works have stood the test of time and continue to be used and referenced today for scientific research (Kieckbusch, Mader, Kaiser & Mecke, 2018).

Contributions
Johann Georg Wagler had made some important and valuable discoveries in the fields of herpetology and ornithology, which contributed greatly to the study of those fields in particular and to the science community as a whole (Beolens, Watkins & Grayson, 2011; Wagler, 1884). Johann Georg Wagler has been honoured and celebrated in the specific names of three species of reptiles and eight species of birds (Beolens, Watkins & Grayson, 2011). Some of the prominent names include Pteroglossus beauharnaesii Wagler 1832 (Wright, 2015; David, Wright, Elliot & Costa, 2020; Costa, Pacheco & Silveira, 2017), Cylindrophis Wagler 1828 (Kieckbusch, Mader, Kaiser & Mecke, 2018), Blanus Wagler 1830, Altractus Wagleri, Podacris Waglerianus and Tropidolaemus Wagleri (Beolens, Watkins & Grayson, 2011). Johann Georg Wagler had also made a contribution in describing the taxonomic arrangement of psittacine fauna, parrots and cockatoos, some of which are established in the systematic classification of these birds (Wagler, 1832). Johann Georg Wagler's contributions to the scientific community in general, ornithology and herpetology in particular, have withstood the test of time for more than one hundred years. Johann Georg Wagler's works are used nowadays as the foundation for new discoveries, research and experiments in the fields of ornithology and herpetology (Kieckbusch, Mader, Kaiser & Mecke, 2018).

More recently, Wagler's discoveries and works have helped scientists to categorise and research some newfound species from the Iberian Peninsula (Ceríaco & Bauer, 2018). Scientists have also recently found a new species of snake in Indonesia that shared similarities with that of Wagler's discovery in the 1800s, the fossorial snake genus Cylindrophis Wagler 1828 (Kieckbusch, Mader, Kaiser & Mecke, 2018). The fossorial snake genus Cylindrophis Wagler, 1828 currently includes 13 species and is common and widely distributed in tropical Asia. There is evidence of existence in Sri Lanka (one species), mainland Southeast Asia including south-eastern China (three species), and the Malay Archipelago (ten species). All species in the genus seem viviparous and share a myriad of morphological characteristics, such as the lack of true gastrosteges, the presence of pelvic spurs, an extremely short tail with conspicuous coloration, and contrasting light and dark ventral blotches. Since the new species was recognised as Cylindrophis melanotus, it is important to discuss the taxonomic history of C. melanotus. The name C. melanotus was first created by Johann Georg Wagler in his Observationes along with the description of C. resplendens (synonym of C. ruffus).

The name showed up in its ablative form as “Cylindrophe melanoto.” While at the time of discovery, a specimen was available to Johann Georg Wagler in the collection of the “Museo Lugdunensi Bat[avorum]” (now RMNH), Wagler also referred to a figure of “Tortricis melanoti” by Caspar Georg Carl Reinwardt (1773–1854), which was likely part of Heinrich Boie's (1794–1827) unpublished manuscript Erpétologie de Java. The nominative form “Cylindrophis melanotus” (nomen corrigendum) was instead listed by Wagler (1830) in an account for the genus Cylindrophis. Two further species, currently in synonymy with Cylindrophis melanotus, were described in the early 20th Century, C. celebensis Smith, 1927 from Sulawesi and C. heinrichi Ahl, 1933 from the Moluccan island of Halmahera. Johann Georg Wagler described his new species as possessing a light snout with dark mottling.

In 2008, a study was conducted to determine the important impacts for the nomenclature of Kinglet Calyptura Calyptura cristata as well as of its distribution (Stopiglia, Straker & Raposo, 2008). The work of Johann Georg Wagler was used to analyse the findings. A specimen (ZMB 2306) of Kinglet Calyptura Calyptura cristata retrieved in São Paulo by Friedrich Sellow (1789–1831) and Ignaz Franz J. M. von Olfers (1793–1872) was rediscovered in the ornithological collection of the Museum für Naturkunde (ZMB) in Berlin in 2007. Specimen ZMB 2306 has two labels: a label in red inscribed ‘Pipra tyrannulus Wagler and a green label from the original mount that identifies the specimen as Pipra (R.) tyrannulus. In relation to the species of Pipra tyrannulus Wagler, 1830, it appears that Wagler (1830) worked on a revision of the genus Pipra and proposed a new combination for ‘Regulus tyrannulus Lichtenstein’.

The name tyrannulus has been interpreted in two different ways throughout the course of history. Sclater considered Pipra tyrannulus Wagler, 1830, as a synonym of Calyptura cristata, whereas Hellmayr claimed Pipra tyrannulus Wagler, 1830, to be a new name for Pardalotus cristatus Vieillot, 1818. Breaking down the specimen and history, it is obvious that Wagler (1830) was first to publish the name tyrannulus. Although Johann Georg Wagler never claimed to be the author of the specific name, in accordance with the terms of the Code (ICZN 1999) he is, nevertheless, its author. Wagler never brought up tyrannulus as being a new name for cristatus. He instead mentioned Pardalotus cristatus as a synonym of Pipra tyrannulus.

Some of Johann Georg Wagler's works in the 1800s remains an interesting topic of discussion today among scientists within the fields of ontology and herpetology. The Curl-crested Aracari Pteroglossus beauharnaesii Wagler, 1832 is one of the most scientifically significant species in the family Ramphastidae (Costa, Pacheco & Silveira, 2017). Among its distinctive characteristics, its modified curly, shiny black crown feathers are one-of-a-kind in the family and are altered to an extent not similar in any other living bird species. The crown feathers, accompanied with the distinctive white throat, were considered enough in the past to include the species in the monotypic genus Beauharnaisius Bonaparte; nevertheless, molecular research have pointed out that it is embedded within the genus Pteroglossus Illiger, sister to P. bitorquatus. It is a southern Amazonian species, showing up around the lowlands area of northern Peru, north and central Bolivia and western and central Amazonian Brazil.The correct publication date of the species’ description and the species’ epithet spelling have been matter of a recent controversy.

Some argued that Pteroglossus beauharnaesii, treated in scientific literature as originally published by Johann Georg Wagler in Isis in 1832, was published earlier as Pteroglossus beauharnaisii by Wagler in 2831 in a daily Munich newspaper, Das Ausland, on 28 April 1831. Hence, beauharnaesii might be regarded as an incorrect subsequent spelling of beauharnaisii. However, others go against that reasoning, claiming that regardless being either an incorrect subsequent spelling of Pteroglossus beauharnaisii Wagler, 1831, or a universally used junior synonym, the name Pteroglossus beauharnaesii Wagler, 1832 need to be preserved as the valid name for the species according to Articles 23.9 and 33.3.1 of the International Code of Zoological Nomenclature (1999). Pteroglossus beauharnaesii was described by Wagler in honor of Prince August Karl Eugen Napoleon Beauharnais (1810–1835), Duke of Leuchtenberg.

In the original description, Johann Georg Wagler presented the type locality only as “Brasilia, prov. Pará”; and in Wagler's earlier work in 1831, he mentioned that Pteroglossus beauharnaesii was “collected in Pará” (“bei Para erlegten”), or “in vicinity of Para”. The holotype's label shows only “Pará” as locality. The specimen that Wagler based the description of P. beauharnaesii was possibly obtained by L. Riedel in 1829 on the Rio Madeira, and thus the locality presented by Wagler was imprecise, yet also not incorrect. In fact, until 1889 most of the Brazilian Amazon region was occupied by the enormous “Province Grão-Pará", commonly referred to only as "Pará". In addition, the type locality given by Wagler possibly indicated the place from where the material collected in Amazonia during the Langsdorff expedition was shipped, the current city of Belém, which was formerly known as “Pará”, capital of the homonymous province.

During the life span of only thirty-two years, Johann Georg Wagler had managed to bring forth many important contributions and discoveries to the scientific community that stands valid and uncontested until nowadays.

Publication
Four of Wagler's books and articles are available at the Biodiversity Heritage Library:
 * Monographia Psittacorum (1832) (from: Munich). Co-writer: Wetmore, Alexander.
 * Natürliches System der Amphibien, mit vorangehender Classification der Säugethiere und Vögel. Ein Beitrag zur vergleichenden Zoologie (1830) (from: München). Co-writer: Richmond, Charles Wallace.
 * Serpentum Brasiliensium species novae ou Histoire naturelle des espèces nouvelles de serpens, recueillies et observées pendant le voyage dans l'intérieur du Brésil dans les années 1817, 1818, 1819, 1820 (1824) (from: Munich). Co-writer: Spix, Johann Baptist von.
 * Wagler's Six ornithological memoirs from the 'Isis.' (1884) (from: London). Edt: Sclater, P.L.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Johann Georg Wagler


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Greave

A greave (from the Old French greve "shin, shin armor") or jambeau is a piece of armor that protects the leg.

Description
The primary purpose of greaves is to protect the tibia from attack. The tibia, or shinbone, is very close to the skin, and is therefore extremely vulnerable to just about any kind of attack. Furthermore, a successful attack on the shin results in that leg being rendered useless, greatly hampering one's ability to maneuver in any way. Greaves were used to counteract this. They usually consisted of a metal exterior with an inner padding of felt. The felt padding was particularly important because, without it, any blow would transfer directly from the metal plating to the shin.

Bronze Age
Bronze greaves were used in Bronze Age Europe by the Mycenaean Greeks and the Central European Urnfield culture, among others.

Ancient Greece and Rome
The reference to greaves (Ancient Greek: κνημίδες) exists in various texts of classical antiquity, including The Shield of Heracles, The Iliad and The Odyssey, The Bibliotheca of Pseudo-Apollodorus, and The Aeneid. In the Illiad, the Greek forces are commonly referred to as "well-greaved Acheans" (euknēmidas Achaioi, ἐϋκνήμιδες Ἀχαιοί). The Iliad also mention the ἐπισφύρια which were either plates covering the ankle, attached to the lower edge of the greaves , or more probably a clasp fastening them round the ankle and were often silver. While these are primarily mythological texts, they still dealt with warfare and the fact that greaves were mentioned is evidence that they were indeed in use. There are also non-fictional testimonies of their use among Roman light infantry (or hastati) from Polybius up to Vegetius. These greaves are thought to have been mass-produced by the Romans using presses on sheets of metal and then attaching lining, usually leather or cloth. While it is generally assumed that greaves were always worn in pairs, there is evidence that many wore just a single greave on the left or right leg. Many skeletons have been found buried with only a single greave, including gladiators and soldiers. People may have worn a single greave as a sign of status, as opposed to any practical use.

Medieval Europe

Greaves were common until around the 9th century AD, when they largely disappeared from use. The first evidence of their reappearance is in the 1230s or 1250s, most notably the depiction of Goliath in the Trinity College Apocalypse manuscript (c. 1230). The lack of other evidence suggests that they were uncommon at the time. Almost all greaves used at this time are known as schynbalds, or greaves that only protected the shin. Illustrations showing “closed greaves”, or greaves that protected the entire leg first appear around 1290 and become popular in the 1320s. Closed greaves are made of two plates joined on the outside by hinges and fastening with buckles and straps on the inside.

Feudal Japan
Japanese greaves, known as suneate, were first introduced during the eleventh century, during the late Heian period. The earliest form consisted of three plates of metal covering the shin. By the Kamakura period (1186–1333), greaves became a standard part of Japanese armor. Around the Muromachi period (1334–1572), these took on the form of a splint mounted on a piece of fabric with mail in between the metal splint and fabric, not unlike European greaves. This is the most common form of suneate, termed shino-suneate, and saw continued use throughout the Momoyama period (1573–1602). Sometimes, cavalrymen used the older three-plate model, known as tsutsu-suneate. Like their European counterparts, most suneate contain leather padding on the interior to reduce the impact of blows and to reduce chafing.

title: Greave


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Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886.

The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion.

Since a convergent Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a non-convergent series. Despite non-convergence, the asymptotic expansion is useful when truncated to a finite number of terms. The approximation may provide benefits by being more mathematically tractable than the function being expanded, or by an increase in the speed of computation of the expanded function. Typically, the best approximation is given when the series is truncated at the smallest term. This way of optimally truncating an asymptotic expansion is known as superasymptotics. The error is then typically of the form $~ exp(−c/ε)$ where $ε$ is the expansion parameter. The error is thus beyond all orders in the expansion parameter. It is possible to improve on the superasymptotic error, e.g. by employing resummation methods such as Borel resummation to the divergent tail. Such methods are often referred to as hyperasymptotic approximations.

See asymptotic analysis and big O notation for the notation used in this article.

Formal definition
First we define an asymptotic scale, and then give the formal definition of an asymptotic expansion.

If $$\ \varphi_n\ $$ is a sequence of continuous functions on some domain, and if $$\ L\ $$ is a limit point of the domain, then the sequence constitutes an asymptotic scale if for every $n$,

 * $$\varphi_{n+1}(x) = o(\varphi_n(x)) \quad (x \to L)\ .$$

($$\ L\ $$ may be taken to be infinity.) In other words, a sequence of functions is an asymptotic scale if each function in the sequence grows strictly slower (in the limit $$\ x \to L\ $$) than the preceding function.

If $$\ f\ $$ is a continuous function on the domain of the asymptotic scale, then $f$ has an asymptotic expansion of order $$\ N\ $$ with respect to the scale as a formal series

 * $$ \sum_{n=0}^N a_n \varphi_{n}(x) $$

if

 * $$ f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = O(\varphi_{N}(x)) \quad (x \to L) $$

or the weaker condition

 * $$ f(x) - \sum_{n=0}^{N-1} a_n \varphi_{n}(x) = o(\varphi_{N-1}(x)) \quad (x \to L)\ $$

is satisfied. Here, $$o$$ is the little o notation. If one or the other holds for all $$\ N\ $$, then we write

 * $$ f(x) \sim \sum_{n=0}^\infty a_n \varphi_n(x) \quad (x \to L)\ .$$

In contrast to a convergent series for $$\ f\ $$, wherein the series converges for any fixed $$\ x\ $$ in the limit $$N \to \infty$$, one can think of the asymptotic series as converging for fixed $$\ N\ $$ in the limit $$\ x \to L\ $$ (with $$\ L\ $$ possibly infinite).

Examples

 * Gamma function (Stirling's approximation)$$ \frac{e^x}{x^x\sqrt{2\pi x}} \Gamma(x+1) \sim 1+\frac{1}{12x}+\frac{1}{288x^2}-\frac{139}{51840x^3}-\cdots\ (x \to \infty)$$
 * Exponential integral$$x e^x E_1(x) \sim \sum_{n=0}^\infty \frac{(-1)^nn!}{x^n} \ (x \to \infty) $$
 * Logarithmic integral$$\operatorname{li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^{\infty} \frac{k!}{(\ln x)^k}$$
 * Riemann zeta function$$\zeta(s) \sim \sum_{n=1}^{N}n^{-s} + \frac{N^{1-s}}{s-1} - \frac{N^{-s}}{2} + N^{-s} \sum_{m=1}^\infty \frac{B_{2m} s^{\overline{2m-1}}}{(2m)! N^{2m-1}}$$where $$B_{2m}$$ are Bernoulli numbers and $$s^{\overline{2m-1}}$$ is a rising factorial. This expansion is valid for all complex s and is often used to compute the zeta function by using a large enough value of N, for instance $$N > |s|$$.
 * Error function$$ \sqrt{\pi}x e^{x^2}{\rm erfc}(x) \sim 1+\sum_{n=1}^\infty (-1)^n \frac{(2n-1)!!}{(2x^2)^n} \ (x \to \infty)$$ where $(2n − 1)!!$ is the double factorial.

Worked example
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. Thus, for example, one may start with the ordinary series

 * $$\frac{1}{1-w}=\sum_{n=0}^\infty w^n.$$

The expression on the left is valid on the entire complex plane $$w\ne 1$$, while the right hand side converges only for $$|w|< 1$$. Multiplying by $$e^{-w/t}$$ and integrating both sides yields

 * $$\int_0^\infty \frac{e^{-\frac{w}{t}}}{1-w}\, dw = \sum_{n=0}^\infty t^{n+1} \int_0^\infty e^{-u} u^n\, du,$$

after the substitution $$u=w/t$$ on the right hand side. The integral on the left hand side, understood as a Cauchy principal value, can be expressed in terms of the exponential integral. The integral on the right hand side may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion

 * $$e^{-\frac{1}{t}} \operatorname{Ei}\left(\frac{1}{t}\right) = \sum_{n=0}^\infty n! t^{n+1}. $$

Here, the right hand side is clearly not convergent for any non-zero value of t. However, by truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of $$\operatorname{Ei} \left (\tfrac{1}{t} \right )$$ for sufficiently small t. Substituting $$x=-\tfrac{1}{t}$$ and noting that $$\operatorname{Ei}(x)=-E_1(-x)$$ results in the asymptotic expansion given earlier in this article.

Integration by parts
Using integration by parts, we can obtain an explicit formula $$\operatorname{Ei}(z) = \frac{e^{z}} {z} \left (\sum _{k=0}^{n} \frac{k!} {z^{k}} + e_{n}(z)\right), \quad e_{n}(z) \equiv (n + 1)!\ ze^{-z}\int _{ -\infty }^{z} \frac{e^{t}} {t^{n+2}}\,dt$$For any fixed $$z$$, the absolute value of the error term $$|e_n(z)|$$ decreases, then increases. The minimum occurs at $$n\sim |z|$$, at which point $$\vert e_{n}(z)\vert \leq \sqrt{\frac{2\pi } {\vert z\vert }}e^{-\vert z\vert }$$. This bound is said to be "asymptotics beyond all orders".

Uniqueness for a given asymptotic scale
For a given asymptotic scale $$\{\varphi_n(x)\}$$ the asymptotic expansion of function $$f(x)$$ is unique. That is the coefficients $$\{a_n\}$$ are uniquely determined in the following way: $$\begin{align} a_0 &= \lim_{x \to L} \frac{f(x)}{\varphi_0(x)} \\ a_1 &= \lim_{x \to L} \frac{f(x) - a_0 \varphi_0(x)} {\varphi_1(x)} \\ & \;\;\vdots \\ a_N &= \lim_{x \to L} \frac {f(x) - \sum_{n=0}^{N-1} a_n \varphi_n(x)} {\varphi_N(x)} \end{align}$$ where $$L$$ is the limit point of this asymptotic expansion (may be $$\pm \infty$$).

Non-uniqueness for a given function
A given function $$f(x)$$ may have many asymptotic expansions (each with a different asymptotic scale).

Subdominance
An asymptotic expansion may be an asymptotic expansion to more than one function.

Related fields

 * Asymptotic analysis
 * Singular perturbation

Asymptotic methods

 * Watson's lemma
 * Mellin transform
 * Laplace's method
 * Stationary phase approximation
 * Method of dominant balance
 * Method of steepest descent

title: Asymptotic expansion


text: User talk:68.85.125.50

title: User talk:68.85.125.50


text:
Les Champs magnétiques

Les Champs magnétiques  (The Magnetic Fields) is a 1920 book by André Breton and Philippe Soupault. It is famous as the first work of literary Surrealism. The authors used a surrealist automatic writing technique.

The book is considered Surrealist, rather than Dadaist, because it attempts to create something new rather than react to an existing work.

Les Champs magnetiques is characterised by rich textured language that often seems to border on the nonsensical. This is considered a "normal" result of automatic writing and is considerably more logical than the output from other Surrealist techniques, such as "exquisite corpse" (a method whereby each of a group of collaborators, in sequence, adds words or images to a composition).

A typical paragraph in (an English-language version of) Les Champs magnetiques is:

 * It was the end of sorrow lies. The rail stations were dead, flowing like bees stung from honeysuckle. The people hung back and watched the ocean, animals flew in and out of focus. The time had come. Yet king dogs never grow old – they stay young and fit, and someday they might come to the beach and have a few drinks, a few laughs, and get on with it. But not now. The time had come; we all knew it. But who would go first?

The division between chapters was the point where the writers stopped writing at the end of the day. The next chapter was started the following morning.

Breton gave many interviews about the creation of the book.

title: Les Champs magnétiques


text:
Sambar

Sambar or Sambhar may refer to:

 * Sambar deer, a species of deer
 * Sambar (dish), a lentil-based dish common in India and Sri Lanka
 * Sambar (film), an Indian Malayalam-language film by Navas Kallara
 * Sambhar, Rajasthan, a city and a municipality in Rajasthan, India
 * Sambhar Salt Lake, a lake in Rajasthan
 * Subaru Sambar, a kei-class van
 * David Sambar, a British American Lebanese international investment banker

title: Sambar


text:
Talk:Carstensz Pyramid

Talk moved to Puncak Jaya

title: Talk:Carstensz Pyramid


text:
DSCH

DSCH may stand for:

 * DSCH motif, a musical motif used by the Soviet composer Dmitri Shostakovich to represent himself. DSCH is also the name of a journal and a publishing house devoted to the composer.
 * Concerto DSCH, a ballet made by Alexei Ratmansky on New York City Ballet to Shostakovich's Concerto No. 2 in F Major, Op. 102.
 * Downlink Shared Channel, a channel used by the telecommunications industry to manage downloads to UE.
 * dsch, a tetragraph used in the German language to transcribe the /d͡ʒ/ phoneme.

title: DSCH


text:
Talk:Goseck Circle

'''What about the four excavated graves from nearby Eulau, dated to ~2600 BC ? Perhaps they merit some sort of link ?'''

Who are these 'British loyalists' who claim to dispute the dating of the Gosek Circle? Googling "Goseck circle" produces no results at all. A reference would be appreciated. Fondest wishes adamsan 08:01, 8 May 2004 (UTC)

 * As no response from the author is forthcoming, I shall remove the unsubstantiated (and rather eccentric) assertions in the article. adamsan 19:18, 20 Jun 2004 (UTC)

--- Someone with an account should upload the picture http://de.wikipedia.org/wiki/Bild:Goseck-2.jpg from the german wikipedia (public domain). The yellow lines show the the direction of sun rise and set at the winter solstice. The vertical line shows the astronomical meridian.
 * Done as requested. Nagelfar 08:56, 5 December 2005 (UTC)

--- Removing this paragraph: The press dubbed Goseck "the German Stonehenge," stating that it "precedes the final stage of Stonehenge by three millennia": "Discovery of the German Stonehenge proves that the Germans were at least as advanced as the British in the Stone Age" [1]. Irrelevant comparison, and ridiculous comparison. And a useless bit of info about press and nationalism. --Dumbo1 01:30, 6 June 2006 (UTC)
 * "although many British enthusiasts refuse the use of the term henge for anything outside Britain. (See henge.)" : The term of 'henge' should be explained in archaeological terms (archaeologically, is Goseck a 'henge'? Or not?) and POV removed.--89.49.33.72 17:02, 30 March 2007 (UTC)

 * I've removed the bit about British enthusiasts, as it is obsolete and the current definition doesn't say only in Britain, but does still call for a bank. I've also removed the bit about the Nebra disk as that is what, 3500 years younger than the Gosseck Circle? I'm still worried about the calendar stuff, looks like speculation,Doug Weller (talk) 10:39, 5 April 2008 (UTC)

Sunrise and sunset of winter solstice Vs Summer and winter solstices
The diagram of the Goseck circle describes the two southern entrances as indicating the rise and set of sun at the winter soltice whereas the linked wikipedia article about the Goseck circle says that these entrances mark the sun's position at the summer and winter solstices. This discrepancy needs to be resolved. 92.21.59.90 (talk) Moved from article by Thesevenseas (talk) 19:34, 23 December 2009 (UTC)
 * the image file says in german: "Sonnenobservatorium von Goseck. Aufsicht mit Darstellung von Sonnenauf- und untergang zur Wintersonnende". loosely translated: "Solar Observatory of Goseck. Supervision with representation of sunrise and sunset for Winter solstice." this does not jive with the Discovery section in the goseck article. i am moving this discussion from the winter solstice talk page to the goseck circle talk page. please update Talk:winter solstice if a conclusion is reached99.140.187.120 (talk) 17:50, 28 December 2009 (UTC)

The problem is with our (home-made, unreferenced) drawing. This plaque indicates that (at least in the interpretation of the people who put together the visitor information), the circle did include markers for both solstices, as well as some "spring festival", apparently of arbitrary (not astronomically significant) date. --dab (𒁳) 10:11, 31 January 2013 (UTC)

"observatory"
The site is often described as a "solar observatory". But this happens mostly in shoddy and journalistic sources. Technically, anyone watching a sunrise is engaged in "solar observation". But the term is altogether too grand, and suggests some sort of pseudoarchaeological fringe theory. What was "observed" was not the Sun as a heavenly body (which the modern term "solar observatory" would imply), e.g. the sun spots, spectrum, etc. The "observation" is limited to marking the point of sunrise/set on solstice days. Thousands of churches are aligned towards these points without being called "solar observatory". I appreciate that this is significant in the history of astronomy because it is the first such alignment known, ever. It is still jarring to throw around misleading descriptions like "solar observatory". The archaeologists mostly avoid doing this. The Germans have the very fitting term of Kalenderbauten, "a structure built with the purpose of serving as a calendar". --dab (𒁳) 15:03, 31 January 2013 (UTC)
 * The English translation of "Kalenderbauten"; is that perhaps "solar observatory"? The churches that you mention are already called "churches". To change their name might offend some. For all we know, this circle could have been a temple, but "solar observatory" is all that we can assume.

Kortoso (talk) 18:41, 14 November 2013 (UTC)

Related to your topic, and may need UPDATE checks occasionally
Hi, in addition to your comments, the article uses a ref that agrees with you that Goseck is NOT DEFINITELY an observatory (or Solar calendar), its discoverers presented that ONLY as a preliminary theory not as a solidly known fact, so I changed the wording to fit the citation given.

I also edited today to fit the fact that that source claiming Goseck as "oldest" may be {outdated} because the following stone circles (see bold text) were dicovered circa 2010 which are thus far are dated to 20,000-5,000BCE, and also are said by their modern-age discoverers to have "likely" alignments to the sun's solstice (used as calendars to time the planting/harvesting cycles ACCORDING TO the theories of each henge's discoverers/archeologists), whilst some of the following ALSO have the cup-and-ring marks & ceremonial burials found at at UK's henges (so it'd be one shit of a coincidence if 2 cultures who never met BOTH built circles of uprighted stones, used for solar astronomy, cupmarks, AND as ceremonial burial sites, but until further analysis is done in a GLOBAL/international context, we can't put that part into WP articles yet): Atlit Yam, Nabta Playa, & an as-yet unnamed henge for which one can google "stone circle Galilee" -- this last one being a MUCH older circle of uprighted stones in Israel (originally dated near the age of Atlit Yam's henge, but now re-dated to 20k-10k BCE, but as submarine salvage work is epensive & slow-going, and "oldest" (up to 20,000 BCE!) tends to be deepest, it and Atlit Yam are so old that they're SUBMERGED...so these may take the longest to excavate. ALL of these, and Goseck, are only POSSIBLE observatories until a Scientific Consensus is formed. However, noteworthy is that younger types of megaliths are also shared between UK & Kanaan (Israel/Lebanon region), e.g. dolmens circa 3,000-1,500 BCE in Israel (older in UK, and they eist all across Europe such as in Italy & Russia), and the cairn/tumulus of Rogem Hiri (exposed layer dated to circa 3,000 BCE, with a gate that, again, "LIKELY" was aligned to sunrise on summer solstice), but an unexcavated layer estimated to be 4,000 BCE is under its 40,000,000 kg of stones that are EXPOSED and 3,000 BCE (so, just as Salisbury stonehenge has 2 simple wood posts carbon-dated to 8,000 BCE in its parking lot & it took awhile to find those 2 wood post-holes at Salisbury, in contrast to the very obvious & easily-seen stone circle at Salisbury, we may have a continuous culture at Rogem Hiri much older than the thus-far EXPOSED layer), so these younger types of non-henge megaliths may indicate an ONGOING searoute tween the UK and the Levant, but we can't exxactly put that into articles until further research confirms it...but it's exciting as we know Phoenicians such as Himilco & Hanno (Iron Age) are known to have reached the UK, along that searoute is Karnak France's megaliths (same name as a city in Egypt, Karnak), Cadiz is a Phoenician-founded city in Western Spain with MEGALITH present in Cadiz as well, Cadiz is near the native-Spanish COPPER mining city of Tartessos (likely exported copper to Kanaan in Bronze or even Chalcolithic Age), so this searoute Himilco finally RECORDED is "possibly" much older than Himilco himself & was long-known to his Kanaanite forefathers...unless we COINCIDENTALLY have 2 cultures [UK & Mideast] who just HAPPENED to both use, circa 8,000 BCE or earlier, cupmarks, ceremonial burial & "likely" solar tracking at (in & near) uprighted stone circles, right near the same era, and then they ALSO both definitely used dolmens & cairns in later eras ;-) ).

Phrasing
This sentence seems a bit unclear. "Excavators also found the remains of what may have been ritual fires, animal and human bones, and a headless skeleton near the southeastern gate...." The words "what may have been" could refer to all the things listed, so are the human bones not actually proven to be human? If they are then that phrase should be moved to after the items that are confirmed and then,ritual fires' included at the end of the sentence. 86.128.240.4 (talk) 01:21, 18 May 2014 (UTC)

title: Talk:Goseck Circle


text:
Cardioid

In geometry, a cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle.

The name was coined by Giovanni Salvemini in 1741 but the cardioid had been the subject of study decades beforehand. Although named for its heart-like form, it is shaped more like the outline of the cross-section of a round apple without the stalk.

A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid (any 2d plane containing the 3d straight line of the microphone body). In three dimensions, the cardioid is shaped like an apple centred around the microphone which is the "stalk" of the apple.

Equations
Let $$a$$ be the common radius of the two generating circles with midpoints $$(-a,0), (a,0)$$, $$\varphi$$ the rolling angle and the origin the starting point (see picture). One gets the x(\varphi) &= 2a (1 - \cos\varphi)\cdot\cos\varphi \, \\ y(\varphi) &= 2a (1 - \cos\varphi)\cdot\sin\varphi \, \qquad 0\le \varphi < 2\pi \end{align}$$ and herefrom the representation in
 * parametric representation: $$\begin{align}
 * polar coordinates: $$r(\varphi) = 2a (1 - \cos\varphi).$$
 * Introducing the substitutions $$\cos\varphi = x/r$$ and $r = \sqrt{x^2 + y^2}$ one gets after removing the square root the implicit representation in Cartesian coordinates: $$\left(x^2 + y^2\right)^2 + 4 a x \left(x^2 + y^2\right) - 4a^2 y^2 = 0.$$

Proof for the parametric representation
A proof can be established using complex numbers and their common description as the complex plane. The rolling movement of the black circle on the blue one can be split into two rotations. In the complex plane a rotation around point $$0$$ (the origin) by an angle $$\varphi$$ can be performed by multiplying a point $$z$$ (complex number) by $$ e^{i\varphi}$$. Hence
 * the rotation $$\Phi_+$$ around point $$a$$ is$$:z \mapsto a + (z - a)e^{i\varphi}$$,
 * the rotation $$\Phi_-$$ around point $$-a$$ is: $$z \mapsto -a + (z + a)e^{i\varphi}$$.

A point $$p(\varphi)$$ of the cardioid is generated by rotating the origin around point $$a$$ and subsequently rotating around $$-a$$ by the same angle $$\varphi$$: $$p(\varphi) = \Phi_ - (\Phi_+(0)) = \Phi_-\left(a - ae^{i\varphi}\right) = -a + \left( a - ae^{i\varphi} + a\right)e^{i\varphi} = a\;\left(-e^{i2\varphi} + 2e^{i\varphi} - 1\right).$$ From here one gets the parametric representation above: $$\begin{array}{cclcccc} x(\varphi) &=& a\;(-\cos(2\varphi) + 2\cos\varphi - 1) &=& 2a(1 - \cos\varphi)\cdot\cos\varphi & & \\ y(\varphi) &=& a\;(-\sin(2\varphi) + 2\sin\varphi)    &=& 2a(1 - \cos\varphi)\cdot\sin\varphi &.& \end{array}$$ (The trigonometric identities $$e^{i\varphi} = \cos\varphi + i\sin\varphi, \ (\cos\varphi)^2 + (\sin\varphi)^2 = 1,$$ $$\cos(2\varphi) = (\cos\varphi)^2 - (\sin\varphi)^2, $$ and $$\sin (2\varphi) = 2\sin\varphi\cos\varphi$$ were used.)

Metric properties
For the cardioid as defined above the following formulas hold: The proofs of these statements use in both cases the polar representation of the cardioid. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area)
 * area $$A = 6\pi a^2$$,
 * arc length $$L = 16 a$$ and
 * radius of curvature $$\rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \, . $$

$$

$$

$$

Chords through the cusp

 * C1: Chords through the cusp of the cardioid have the same length $$4a$$.
 * C2: The midpoints of the chords through the cusp lie on the perimeter of the fixed generator circle (see picture).

Proof of C1
The points $$P: p(\varphi),\; Q: p(\varphi + \pi)$$ are on a chord through the cusp (=origin). Hence $$\begin{align} |PQ| &= r(\varphi) + r(\varphi + \pi) \\ &= 2a (1 - \cos\varphi) + 2a (1 - \cos(\varphi + \pi)) = \cdots = 4a \end{align}.$$

Proof for C2
For the proof the representation in the complex plane (see above) is used. For the points $$P:\ p(\varphi) = a\,\left(-e^{i2\varphi} + 2e^{i\varphi} - 1\right)$$ and $$Q:\ p(\varphi + \pi) = a\,\left(-e^{i2(\varphi + \pi)} + 2e^{i(\varphi + \pi)} - 1\right) = a\,\left(-e^{i2\varphi} - 2e^{i\varphi} - 1\right),$$

the midpoint of the chord $$PQ$$ is $$M:\ \tfrac{1}{2}(p(\varphi) + p(\varphi + \pi)) = \cdots = -a - ae^{i2\varphi}$$ which lies on the perimeter of the circle with midpoint $$-a$$ and radius $$a$$ (see picture).

Cardioid as inverse curve of a parabola

 * A cardioid is the inverse curve of a parabola with its focus at the center of inversion (see graph)

For the example shown in the graph the generator circles have radius $a = \frac{1}{2}$. Hence the cardioid has the polar representation $$r(\varphi) = 1 - \cos\varphi$$ and its inverse curve $$r(\varphi) = \frac{1}{1 - \cos\varphi},$$ which is a parabola (s. parabola in polar coordinates) with the equation $x = \tfrac{1}{2}\left(y^2 - 1\right)$ in Cartesian coordinates.

Remark: Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the vertex of the parabola, then the result is a cissoid of Diocles.

Cardioid as envelope of a pencil of circles
In the previous section if one inverts additionally the tangents of the parabola one gets a pencil of circles through the center of inversion (origin). A detailed consideration shows: The midpoints of the circles lie on the perimeter of the fixed generator circle. (The generator circle is the inverse curve of the parabola's directrix.)

This property gives rise to the following simple method to draw a cardioid:
 * 1) Choose a circle $$c$$ and a point $$O$$ on its perimeter,
 * 2) draw circles containing $$O$$ with centers on $$c$$, and
 * 3) draw the envelope of these circles.

$$

Cardioid as envelope of a pencil of lines
A similar and simple method to draw a cardioid uses a pencil of lines. It is due to L. Cremona:

 * 1) Draw a circle, divide its perimeter into equal spaced parts with $$2N$$ points (s. picture) and number them consecutively.
 * 2) Draw the chords: $$(1,2), (2,4), \dots, (n,2n), \dots, (N,2N), (N+1,2), (N+2,4), \dots $$. (That is, the second point is moved by double velocity.)
 * 3) The envelope of these chords is a cardioid.

Proof
The following consideration uses trigonometric formulae for $$\cos\alpha + \cos\beta$$, $$\sin\alpha + \sin\beta$$, $$1 + \cos 2\alpha $$, $$\cos 2\alpha$$, and $$\sin 2\alpha$$. In order to keep the calculations simple, the proof is given for the cardioid with polar representation $$r = 2(1 \mathbin{\color{red}+} \cos\varphi)$$ (§ Cardioids in different positions).

Equation of the tangent of the cardioid with polar representation $r = 2(1 + )$
From the parametric representation $$\begin{align} x(\varphi) &= 2(1 + \cos\varphi) \cos \varphi, \\ y(\varphi) &= 2(1 + \cos\varphi) \sin \varphi \end{align}$$

one gets the normal vector $$\vec n = \left(\dot y, -\dot x\right)^\mathsf{T}$$. The equation of the tangent $$\dot y(\varphi) \cdot (x - x(\varphi)) - \dot x(\varphi) \cdot (y - y(\varphi)) = 0$$ is: $$(\cos2\varphi + \cos \varphi)\cdot x + (\sin 2\varphi + \sin \varphi)\cdot y = 2(1 + \cos \varphi)^2 \, .$$

With help of trigonometric formulae and subsequent division by $\cos\frac{1}{2}\varphi$, the equation of the tangent can be rewritten as: $$\cos(\tfrac{3}{2}\varphi) \cdot x + \sin\left(\tfrac{3}{2}\varphi\right) \cdot y = 4 \left(\cos\tfrac{1}{2}\varphi\right)^3 \quad 0 < \varphi < 2\pi,\ \varphi \ne \pi .$$

Equation of the chord of the circle with midpoint $$ and radius $3$
For the equation of the secant line passing the two points $$(1 + 3\cos\theta, 3\sin\theta),\ (1 + 3\cos{\color{red}2}\theta, 3\sin{\color{red}2}\theta))$$ one gets: $$(\sin\theta - \sin 2\theta) x + (\cos 2\theta - \sin \theta) y = -2\cos \theta - \sin(2\theta) \, .$$

With help of trigonometric formulae and the subsequent division by $\sin\frac{1}{2}\theta$ the equation of the secant line can be rewritten by: $$\cos\left(\tfrac{3}{2}\theta\right) \cdot x + \sin\left(\tfrac{3}{2}\theta\right) \cdot y = 4 \left(\cos\tfrac{1}{2}\theta\right)^3 \quad 0 < \theta < 2\pi .$$

Conclusion
Despite the two angles $$\varphi, \theta$$ have different meanings (s. picture) one gets for $$\varphi = \theta $$ the same line. Hence any secant line of the circle, defined above, is a tangent of the cardioid, too:
 * The cardioid is the envelope of the chords of a circle.

Remark:

The proof can be performed with help of the envelope conditions (see previous section) of an implicit pencil of curves: $$F(x, y, t) = \cos\left(\tfrac{3}{2}t\right) x + \sin\left(\tfrac{3}{2}t\right) y - 4 \left(\cos\tfrac{1}{2}t\right)^3 = 0 $$

is the pencil of secant lines of a circle (s. above) and $$F_t(x, y, t) = - \tfrac{3}{2}\sin\left(\tfrac{3}{2}t\right) x + \tfrac{3}{2}\cos \left(\tfrac{3}{2}t\right) y + 3\cos\left(\tfrac{1}{2}t\right) \sin t = 0\, .$$

For fixed parameter t both the equations represent lines. Their intersection point is $$x(t) = 2(1 + \cos t)\cos t,\quad y(t) = 2(1 + \cos t)\sin t,$$

which is a point of the cardioid with polar equation $$r = 2(1 + \cos t).$$

Cardioid as caustic of a circle
The considerations made in the previous section give a proof that the caustic of a circle with light source on the perimeter of the circle is a cardioid.
 * If in the plane there is a light source at a point $$Z$$ on the perimeter of a circle which is reflecting any ray, then the reflected rays within the circle are tangents of a cardioid.

$$

Remark: For such considerations usually multiple reflections at the circle are neglected.

Cardioid as pedal curve of a circle
The Cremona generation of a cardioid should not be confused with the following generation:

Let be $$k$$ a circle and $$O$$ a point on the perimeter of this circle. The following is true:
 * The foots of perpendiculars from point $$O$$ on the tangents of circle $$k$$ are points of a cardioid.

Hence a cardioid is a special pedal curve of a circle.

Proof
In a Cartesian coordinate system circle $$k$$ may have midpoint $$(2a,0)$$ and radius $$2a$$. The tangent at circle point $$(2a + 2a\cos\varphi, 2a\sin \varphi)$$ has the equation $$(x - 2a) \cdot \cos\varphi + y\cdot\sin\varphi = 2a\, .$$ The foot of the perpendicular from point $$O$$ on the tangent is point $$(r\cos \varphi, r\sin \varphi)$$ with the still unknown distance $$r$$ to the origin $$O$$. Inserting the point into the equation of the tangent yields $$(r\cos\varphi - 2a)\cos\varphi + r\sin^2\varphi = 2a \quad \rightarrow \quad r = 2a(1 + \cos \varphi) $$ which is the polar equation of a cardioid.

Remark: If point $$O$$ is not on the perimeter of the circle $$k$$, one gets a limaçon of Pascal.

The evolute of a cardioid
The evolute of a curve is the locus of centers of curvature. In detail: For a curve $$\vec x(s) = \vec c(s)$$ with radius of curvature $$\rho(s)$$ the evolute has the representation $$\vec X(s) = \vec c(s) + \rho(s)\vec n(s).$$ with $$\vec n(s)$$ the suitably oriented unit normal.

For a cardioid one gets:
 * The evolute of a cardioid is another cardioid, one third as large, and facing the opposite direction (s. picture).

Proof
For the cardioid with parametric representation $$x(\varphi) = 2a (1 - \cos\varphi)\cos\varphi = 4a \sin^2\tfrac{\varphi}{2}\cos\varphi\, ,$$ $$y(\varphi) = 2a (1 - \cos\varphi)\sin\varphi = 4a \sin^2\tfrac{\varphi}{2}\sin\varphi$$ the unit normal is $$\vec n(\varphi) = (-\sin\tfrac{3}{2}\varphi, \cos\tfrac{3}{2}\varphi)$$ and the radius of curvature $$\rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \,. $$ Hence the parametric equations of the evolute are $$X(\varphi) = 4a \sin^2\tfrac{\varphi}{2}\cos\varphi-\tfrac{8}{3}a\sin\tfrac{\varphi}{2}\cdot \sin\tfrac{3}{2} \varphi = \cdots = \tfrac{4}{3}a\cos^2\tfrac{\varphi}{2}\cos\varphi - \tfrac{4}{3}a \,, $$ $$Y(\varphi) = 4a \sin^2\tfrac{\varphi}{2} \sin\varphi + \tfrac{8}{3}a \sin\tfrac{\varphi}{2} \cdot\cos\tfrac{3}{2} \varphi = \cdots = \tfrac{4}{3}a \cos^2\tfrac{\varphi}{2} \sin\varphi \,. $$ These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by $$-\tfrac{4}{3} a$$.

(Trigonometric formulae were used: $$ \sin\tfrac{3}{2}\varphi = \sin\tfrac{\varphi}{2}\cos\varphi + \cos\tfrac{\varphi}{2}\sin\varphi\ ,\ \cos\tfrac{3}{2}\varphi = \cdots, \   \sin\varphi = 2\sin\tfrac{\varphi}{2}\cos\tfrac{\varphi}{2}, \   \cos\varphi= \cdots \ . $$)

Orthogonal trajectories
An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. For cardioids the following is true:

(The second pencil can be considered as reflections at the y-axis of the first one. See diagram.)

Proof
For a curve given in polar coordinates by a function $$r(\varphi)$$ the following connection to Cartesian coordinates hold: $$\begin{align} x(\varphi) &= r(\varphi)\cos\varphi\, ,\\ y(\varphi) &= r(\varphi)\sin\varphi \end{align} $$

and for the derivatives $$\begin{align} \frac{dx}{d\varphi} &= r'(\varphi)\cos\varphi - r(\varphi)\sin\varphi\, ,\\ \frac{dy}{d\varphi} &= r'(\varphi)\sin\varphi + r(\varphi)\cos\varphi\,. \end{align}$$

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point $$(r(\varphi), \varphi)$$: $$\frac{dy}{dx} = \frac{r'(\varphi)\sin\varphi + r(\varphi)\cos\varphi}{r'(\varphi)\cos\varphi - r(\varphi)\sin\varphi}.$$

For the cardioids with the equations $$r=2a(1-\cos\varphi) \; $$ and $$r = 2b(1 + \cos\varphi)\ $$ respectively one gets: $$\frac{dy_a}{dx} = \frac{\cos(\varphi) - \cos(2\varphi)}{\sin(2\varphi) - \sin(\varphi)} $$ and $$ \frac{dy_b}{dx} = -\frac{\cos(\varphi) + \cos(2\varphi)}{\sin(2\varphi) + \sin(\varphi)}\ .$$

(The slope of any curve depends on $$\varphi$$ only, and not on the parameters $$a$$ or $$b$$!)

Hence $$\frac{dy_a}{dx}\cdot \frac{dy_b}{dx} = \cdots = -\frac{\cos^2\varphi-\cos^2 (2\varphi)}{\sin^2 (2\varphi)-\sin^2\varphi} = -\frac{-1 + \cos^2\varphi + 1 - \cos^2 2\varphi}{\sin^2 (2\varphi) - \sin^2(\varphi)} = -1\, .$$ That means: Any curve of the first pencil intersects any curve of the second pencil orthogonally.

In different positions
Choosing other positions of the cardioid within the coordinate system results in different equations. The picture shows the 4 most common positions of a cardioid and their polar equations.

In complex analysis
In complex analysis, the image of any circle through the origin under the map $$z \to z^2$$ is a cardioid. One application of this result is that the boundary of the central period-1 component of the Mandelbrot set is a cardioid given by the equation $$ c \,=\, \frac{1 - \left(e^{it} - 1\right)^2}{4}.$$

The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies is an approximate cardioid.

Caustics
Certain caustics can take the shape of cardioids. The catacaustic of a circle with respect to a point on the circumference is a cardioid. Also, the catacaustic of a cone with respect to rays parallel to a generating line is a surface whose cross section is a cardioid. This can be seen, as in the photograph to the right, in a conical cup partially filled with liquid when a light is shining from a distance and at an angle equal to the angle of the cone. The shape of the curve at the bottom of a cylindrical cup is half of a nephroid, which looks quite similar.

title: Cardioid


text:
Timeshare

A timeshare (sometimes called a vacation ownership or vacation club) is a property with a divided form of ownership or use rights. These properties are typically resort condominium units, in which multiple parties hold rights to use the property, and each owner of the same accommodation is allotted their period of time. Units may be sold as a partial ownership, lease, or "right to use", in which case the latter holds no claim to ownership of the property. The ownership of timeshare programs is varied, and has been changing over the decades.

History
The term "timeshare" was coined in the United Kingdom in the early 1960s, expanding on a vacation system that became popular after World War II. Vacation home sharing, also known as holiday home sharing, involved four European families that would purchase a vacation cottage jointly, each having exclusive use of the property for one of the four seasons. They rotated seasons each year, so each family enjoyed the prime seasons equally. This concept was mostly used by related families because joint ownership requires trust and no property manager was involved. However, not many families vacation for an entire season at a time; so the vacation home sharing properties were often vacant for long periods.

British businesses decided to go one step further and divide a resort room into 1/50th ownership, have two weeks each year for repairs and upgrades, and charge a maintenance fee to each owner. It took almost a decade for timeshares in Europe to evolve into a smoothly run, successful business venture.

The first timeshare in the United States was started in 1974 by Caribbean International Corporation (CIC), based in Fort Lauderdale, Florida. It offered what it called a 25-year vacation license rather than ownership. The company owned two other resorts the vacation license holder could alternate their vacation weeks with: one in St. Croix and one in St. Thomas; both in the U.S. Virgin Islands. The Virgin Islands properties began their timeshare sales in 1973.

The contract was simple and straightforward: The company, CIC, promised to maintain and provide the specified accommodation type (a studio, one bedroom, or two bedroom unit) for use by the "license owner" for a period of 25 years (from 1974 to 1999, for example) in the specified season and number of weeks agreed upon, with only two extra charges: a $15.00 per diem (per night) rate, frozen at that cost for the life of the contract, and a $25.00 switching fee, should the licensee decide to use their time at one of the other resorts. The contract was based on the fact that the cost of the license, and the small per diem, compared with the projected increase in the cost of hotel rates over 25 years to over $100.00 per night, would save the license owner many vacation dollars over the span of the license agreement. Between 1974 and 1999, in the United States, inflation boosted the current cost of the per diem to $52.00, validating the cost savings assumption. The license owner was allowed to rent, or give their week away as a gift in any particular year. The only stipulation was that the $15.00 per diem must be paid every year whether the unit was occupied or not. This "must be paid yearly fee" would become the roots of what is known today as "maintenance fees", once the Florida Department of Real Estate became involved in regulating timeshares.

The timeshare concept in the United States caught the eye of many entrepreneurs due to the enormous profits to be made by selling the same room 52 times to 52 different owners at an average price in 1974–1976 of $3,500.00 per week. Shortly thereafter, the Florida Real Estate Commission stepped in, enacting legislation to regulate Florida timeshares, and make them fee simple ownership transactions. This meant that in addition to the price of the owner's vacation week, a maintenance fee and a homeowners association had to be initiated. This fee simple ownership also spawned timeshare location exchange companies, such as Interval International and RCI, so owners in any given area could exchange their week with owners in other areas.

Cancellations, or rescission, of the timeshare contract, remain the industry's biggest problems to date; the difficulty has been the subject of comedy in popular entertainment.

Legislation
The industry is regulated in all countries where resorts are located. In Europe, it is regulated by European and by national legislation. In 1994, the European Communities adopted "The European Directive 94/47/EC of the European Parliament and Council on the protection of purchasers in respect of certain aspects of contracts relating to the purchase of the right to use immovable properties on a timeshare basis", which was subject to recent review, and resulted in the adoption on January 14, 2009, on European Directive 2008/122/EC.

Established regulations in Mexico
On May 17, 2010, Mexico’s Ministry of Economy through the General Directorate of Standards established new regulations and requirements for developers of timeshare services. The new regulations are outlined in the Official Mexican Norm (NOM), which consists of a series of official standards and regulations applicable to diverse activities in Mexico. The following institutions were involved during the new standardization:
 * Mexican Resort Development Association (AMDETUR)
 * National Fund for Tourism Development (FONATUR)
 * Federal Consumer Protection Office (PROFECO)
 * Ministry of Economy (SE)
 * Secretary of Tourism (SECTUR)

NOM is officially called: "NOM-029-SCFI-2010, Commercial Practices and Information Requirements for the Rendering of Timeshare Service". It established the following standards:
 * Marketing companies are not allowed to offer gifts and solicit for prospective timeshare owners without clearly specifying the real purpose of the offer.
 * The requirements to cancel a timeshare contract must be more practical and less burdensome.
 * NOM recognizes the privacy rights of timeshare consumers. It is strictly prohibited for the timeshare provider to dispose of the consumer's personal information without written consent.
 * Verbal promises must be written and established in the original timeshare contract.
 * The timeshare provider must comply with all obligations written in the timeshare contract, as well as the internal rules of the timeshare resort.
 * The charges that are intended to be made to the consumer must be plainly and clearly defined on the timeshare application forms, including the membership cost, and all extra fees (maintenance fees/exchange club fees).

To make the new regulations applicable to any person or entity that provides timeshares, the definition of a timeshare service provider was substantially extended and clarified. If the timeshare provider does not follow the rules decreed in NOM, the consequences may be substantial, and may include financial penalties that can range from $50 to $200,000.

Methods of use
Owners can:

 * Use their usage time
 * Rent out their owned usage
 * Give it as a gift
 * Donate it to a charity (should the charity choose to accept the burden of the associated maintenance payments)
 * Exchange internally within the same resort or resort group
 * Exchange externally into thousands of other resorts
 * Sell it either through traditional or online advertising, or by using a licensed broker. Timeshare contracts allow transfer through sale, but it is rarely accomplished. According to the EU Directive, Long-term holiday ownership products in the EU, which includes timeshare fixed weeks, points and fractional ownerships cannot be sold as resellable assets.

Recently, with most point systems, owners may elect to:

 * Assign their usage time to the point system to be exchanged for airline tickets, hotels, travel packages, cruises, amusement park tickets
 * Instead of renting all their actual usage time, rent part of their points without actually getting any usage time and use the rest of the points
 * Rent more points from either the internal exchange entity or another owner to get a larger unit, more vacation time, or to a better location
 * Save or move points from one year to another

Some developers, however, may limit which of these options are available at their respective properties.

Owners can elect to stay at their resort during the prescribed period, which varies depending on the nature of their ownership. In many resorts, they can rent out their week or give it as a gift to friends and family.

Exchanging timeshares
Used as the basis for attracting mass appeal to purchasing a timeshare, is the idea of owners exchanging their week, either independently or through exchange agencies. The two largest—often mentioned in media—are RCI and Interval International (II), which combined, have over 7,000 resorts. They have resort affiliate programs, and members can only exchange with affiliated resorts. It is most common for a resort to be affiliated with only one of the larger exchange agencies, although resorts with dual affiliations are not uncommon. The timeshare resort one purchases determines which of the exchange companies can be used to make exchanges. RCI and II charge a yearly membership fee, and additional fees for when they find an exchange for a requesting member, and bar members from renting weeks for which they already have exchanged.

Owners can also exchange their weeks or points through independent exchange companies. Owners can exchange without needing the resort to have a formal affiliation agreement with the companies, if the resort of ownership agrees to such arrangements in the original contract.

Due to the promise of exchange, timeshares often sell regardless of the location of their deeded resort. What is not often disclosed is the difference in trading power depending on the location, and season of the ownership. If a resort is in a prime vacation region, it will exchange extremely well depending on the season and week that is assigned to the particular unit trying to make an exchange. However, timeshares in highly desirable locations and high season time slots are the most expensive in the world, subject to demand typical of any heavily trafficked vacation area. An individual who owns a timeshare in the American desert community of Palm Springs, California, in the middle of July or August will possess a much reduced ability to exchange time, because fewer come to a resort at a time when the temperatures are in excess of 110 F.

Deeded versus right-to-use contracts
A major difference in types of vacation ownership is between deeded and right-to-use contracts.

With deeded contracts the use of the resort is usually divided into week-long increments and are sold as real property via fractional ownership. As with any other piece of real estate, the owner may do whatever is desired: use the week, rent it, give it away, leave it to heirs, or sell the week to another prospective buyer. The owner is also liable for an equal portion of the real estate taxes, which usually are collected with condominium maintenance fees. The owner can potentially deduct some property-related expenses, such as real estate taxes from taxable income.

Deeded ownership can be as complex as outright property ownership in that the structure of deeds vary according to local property laws. Leasehold deeds are common and offer ownership for a fixed period of time after which the ownership reverts to the freeholder. Occasionally, leasehold deeds are offered in perpetuity, however many deeds do not convey ownership of the land, but merely the apartment or unit (housing) of the accommodation.

With right-to-use contracts, a purchaser has the right to use the property in accordance with the contract, but at some point the contract ends and all rights revert to the property owner. Thus, a right-to-use contract grants the right to use the resort for a specific number of years. In many countries there are severe limits on foreign property ownership; thus, this is a common method for developing resorts in countries such as Mexico. Care should be taken with this form of ownership as the right to use often takes the form of a club membership or the right to use the reservation system, where the reservation system is owned by a company not in the control of the owners. The right to use may be lost with the demise of the controlling company, because a right to use purchaser's contract is usually only good with the current owner, and if that owner sells the property, the lease holder could be out of luck depending on the structure of the contract, and/or current laws in foreign venues.

Fixed-week ownership
The formerly common unit of sale is a fixed week; the resort will have a calendar enumerating the weeks roughly starting with the first calendar week of the year. An owner may own a deed to use a unit for a single specified week; for example, week 51 typically includes Christmas. An individual who owns Week 26 at a resort can use only that week in each year.

Floating-week ownership
Sometimes units are sold as floating weeks, in which a contract specifies the number of weeks held by each owner and from which weeks the owner may select for his stay. An example of this may be a floating summer week, in which the owner may choose any single week during the summer. In such a situation, there is likely to be greater competition during weeks featuring holidays, while lesser competition is likely when schools are still in session. Some floating contracts exclude major holidays so they may be sold as fixed weeks.

Rotating or flex-week ownership
Some are sold as rotating weeks, commonly referred to as flex weeks. In an attempt to give all owners a chance for the best weeks, the weeks are rotated forward or backward through the calendar, so in year 1 the owner may have use of week 25, then week 26 in year 2, and then week 27 in year 3. This method gives each owner a fair opportunity for prime weeks, but unlike its name, it is not flexible.

Points programs
A variant form of real estate-based timeshare that combines features of deeded timeshare with right-to-use offerings was developed by Disney Vacation Club (DVC) in 1991. Purchasers of DVC timeshare interests, whom DVC calls members receive a deed conveying an undivided real property interest in a timeshare unit. Each DVC member's property interest is accompanied by an annual allotment of vacation points in proportion to the size of the property interest. DVC's vacation points system is marketed as highly flexible and may be used in different increments for vacation stays at DVC resorts in a variety of accommodations from studios to three-bedroom villas. DVC's vacation points can be exchanged for vacations worldwide in non-Disney resorts, or may be banked into or borrowed from future years.

DVC's deeded/vacation point structure, which has been used at all of its timeshare resorts, has been adopted by other large timeshare developers including the Hilton Grand Vacations Company, the Marriott Vacation Club, the Holiday Inn Club Vacations, the Hyatt Residence Club and Accor in France.

Resort-based points programs are also sold as deeded and as right to use. Points programs annually give the owner a number of points equal to the level of ownership. The owner in a points program can then use these points to make travel arrangements within the resort group. Many points programs are affiliated with large resort groups offering a large selection of options for destination. Many resort point programs provide flexibility from the traditional week stay. Resort point program members, such as WorldMark by Wyndham and Diamond Resorts International, may request from the entire available inventory of the resort group.

A points program member may often request fractional weeks as well as full or multiple week stays. The number of points required to stay at the resort in question will vary based on a points chart. The points chart will allow for factors such as:
 * Popularity of the resort
 * Size of the accommodations
 * Number of nights
 * Desirability of the season

Types and sizes of accommodations
Timeshare properties tend to be apartment style accommodations ranging in size from studio units (with room for two), to three and four bedroom units. These larger units can usually accommodate large families comfortably. Units normally include fully equipped kitchens with a dining area, dishwasher, televisions, DVD players, etc. It is not uncommon to have washers and dryers in the unit or accessible on the resort property. The kitchen area and amenities will reflect the size of the particular unit in question.

Units are usually listed by how many the unit will sleep and how many the unit will sleep privately. Traditionally, but not exclusively: Timeshares are sold worldwide, and every venue has its own unique descriptions.
 * Sleeps 2/2 would normally be a one bedroom or studio
 * Sleeps 6/4 would normally be a two bedroom with a sleeper sofa

Sleep privately usually refers to the number of guests who will not have to walk through another guest's sleeping area to use a restroom. Timeshare resorts tend to be strict on the number of guests permitted per unit.

Unit size affects the cost and demand at any given resort. The same does not hold true comparing resorts in different locations. A one-bedroom unit in a desirable location may still be more expensive and in higher demand than a two-bedroom accommodation in a resort with less demand. An example of this may be a one-bedroom at a desirable beach resort compared to a two-bedroom unit at a resort located inland from the same beach.

Sales incentives
Timeshare sales staffs often provide incentives, including free or deeply discounted hotel rooms, tickets, and gifts, for the prospective buyer to take a tour of the property.

The tour
The prospective buyers (or "prospects") are seated in a hospitality room with many tables and chairs to accommodate families. The prospects are assigned a tour guide, who may be a licensed real estate agent. Procedures vary from venue to venue. After a warm-up period and some coffee or snack, a podium speaker welcomes the prospects to the resort, and shows a promotional video.

The prospects are then invited to take a tour of the property. Depending on the resort's available inventory, the tour will include an accommodation that the tour guide or agent feels will best fit the prospect's family's needs. After the tour, the group returns to the hospitality room for a verbal sales presentation. During the presentation, they are handed the resort exchange book from RCI, Interval International, or whatever exchange company is associated with that particular resort property. The prospects are asked about the places they would like to visit if they were timeshare owners. The rest of the presentation is designed around the responses the prospective buyers give to that question.

If the guide is licensed, the prospect is quoted the retail price of the particular unit that best seemed to fit the prospective buyer's needs. If the tour guide is not a licensed agent, a licensed agent steps in to present the price. If the prospect replies with "no", or "I would like to think about it", the prospect will then be given a new incentive to buy. This incentive will usually be a discounted price that is only "good today" (which is untrue). If again, the reply is "no", or "I would like to think about it", the sales agent asks the prospect to please talk to one of the managers before the prospect leaves.

A sales manager, assistant manager or project director may be called to the table for "T.O.", for "takeover", to find an incentive, usually in the form of a smaller, less expensive unit, or a traded in unit from another owner. A common T.O. tactic is to claim that a lower price is exclusive to a specific buyer. If one incentive doesn't move a prospect to purchase, another will follow shortly, until the prospect has either purchased, convinced the sales crew that they are uninterested, or has left on their own.

Cancellation of a timeshare contract
Timeshare sales are often high pressure and fast moving. Some people sign a contract, then want to cancel it shortly thereafter.

U.S. Federal Trade Commission mandates a "cooling off period" that allows people to cancel some types of purchases without penalty within three days. Additionally, almost all U.S. states have laws that specifically govern cancellation of timeshare contracts. In Florida, a new timeshare owner can cancel the purchase within ten days. The law differs by jurisdiction as to whether out-of-state purchasers are subject to the rescission period of their state of residence, or the rescission period of the state where the timeshare purchase was made.

Another common practice is to have the prospective buyer sign a "cancellation waiver", using it as an excuse to lower the price of the timeshare in exchange for the buyer waiving cancellation rights (or paying a penalty, such as losing 10% of the purchase price, if the sale is cancelled). However, such a waiver is not legally enforceable anywhere in Mexico or the United States.

There are companies that specialize in timeshare cancellation, although some cancellation companies are suspected of fraud.

Market value
A resale value of a timeshare unit varies between the timeshare systems and properties. While it's possible for certain units in certain timeshare systems to appreciate in value over time, this is a rather rare event. Most properties in most timeshare systems are worth a fraction of the original price after the purchase, and often carry no value at all.

Deeds with severe restrictions on resale are typically given out for free. For example, upon resale of most Holiday Inn Club Vacations properties, the new owner can no longer use the allotted point value to make reservations at any HICV resort. They are limited to their home resort only, and booking window is shorter.

Deeds with no resale restrictions may carry some resale value, depending on the location, season, unit size, or the allotted points value. For example, large units in mountain resorts during ski season and large penthouse units in popular destinations will have some resale value. Likewise, small units in destinations saturated by timeshares, or with expensive maintenance fees compared to the value of offered points in their respective system, will have no resale value at all.

List of timeshare systems with severe resale restrictions

 * Holiday Inn Club Vacations (excluding Gatlinburg, Lake Geneva and Myrtle Beach)
 * Westgate Resorts (including some HGV-managed deeds at Elara purchased from Westgate)

Right of first refusal
Most timeshare companies retain the right of first refusal for most of their properties. If that is a case, each owner is required to submit their purchase agreement contract to the developer and ask for a waiver. The developer may decide to exercise ROFR to purchase the subject property, thus refusing the sale to the original buyer. A transaction is typically exempted from ROFR if the purchaser is a direct family member.

Even though resales come at a fraction of original costs, developers are often reluctant to exercise the ROFR. Many sought-after deeds change hands uninterrupted. When a developer exercises the ROFR, they become the owner and must pay the same maintenance fees and property taxes to the resort's HOA, so they must maintain a healthy balance between inventory ready for sale and maintenance costs.

List of timeshare systems with right of first refusal on resales

 * Disney Vacation Club
 * Hilton Grand Vacations Club (excluding HGV at the Flamingo, and all affiliate resorts)
 * Marriott Vacation Club

Criticism
The United States Federal Trade Commission provides consumers with information regarding timeshare pricing and other related information. Also known as Universal Lease Programs (ULPs), timeshares are considered to be securities under the law.

Many timeshare owners complain about the annual maintenance fee (which includes property taxes) being too high.

Timeshare developers contend that pricing compared to staying at hotels in the long term is projected to be lower to the timeshare owner. However, a hotel guest does not have a monthly vacation mortgage payment, upfront cost, fixed schedule, maintenance fees, and preset vacation locations. Many owners also complain that the increasing cost of timeshares and accompanying maintenance and exchange fees are rising faster than hotel rates in the same areas.

The industry's reputation has been severely injured by the comparison of the timeshare salesman to the used car salesman, because of the sales pressure put on the prospective buyer to "buy today". "The discounted price I quoted you is only good if you buy today", is the industry standard's pitch to close the sale on the first visit to the resort. Many have left a timeshare tour complaining of being exhausted by the barrage of salespeople they had to deal with before they finally exited the tour. The term "TO", or "turn over" man, was coined in the land industry, and quickly evolved to the timeshare industry. Once the original tour guide or salesman gives the prospective buyer the pitch and price, the "TO" is sent in to drop the price and secure the down payment.

Exiting a timeshare without the cooperation of the developer is not easy, and often requires the assistance of a specialist. This specialist may be a broker who works both sides of the market, or an attorney who can litigate the contract. While some timeshare exit specialists are legitimate, the business is plagued by scammers who cold-call owners and solicit payment in advance for nonexistent services. The original buyer of a timeshare generally takes a substantial loss when a timeshare is successfully resold. In 2017, Spanish police arrested 35 people in connection with a fraudulent timeshare resale operation on the Costa del Sol that had taken an estimated £11m profit from 500 victims.

title: Timeshare


text:
Talk:Mount Kenya

Link
The link at the end of the main article to: Mount Kenya Trust page Appears to have nothing to do with the mountain. It is all about credit cards and credit history. Maybe it should be removed.

 * This seems to have been removed at some point. --Mehmet Karatay 11:53, 12 May 2007 (UTC)

More pictures?
There is another picture of Mount Kenya available on Commons, Image:Mount Kenya.jpg. Would it fit in this article?--ZorroIII 21:28, 24 October 2005 (UTC)

 * I think so. One would only have to remove the picture of Fujiyama ("Mt. Kenya as it might have looked at one time") and put a relevant picture in its place. 173.174.85.88 (talk) 15:55, 13 September 2017 (UTC) Eric

External links modified (February 2018)
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I have just modified 8 external links on Mount Kenya. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20100821014732/http://www.ewpnet.com/maps/mkinside.htm to http://www.ewpnet.com/maps/mkinside.htm
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Name
I removed " (see the various local names below, some of which are similar). Wangari Maathai tells the following story about the naming: Krapf and Johannes Rebmann asked their guide, a member of the Kamba community, who was carrying a gourd, what they called the mountain, and the guide, believing that the Germans were referring to the gourd, replied kĩĩ-nyaa, which became the name of the mountain and then the country. ."

This is an excellent example of the kind of reference Wikipedia warns us not to use: it is from a primary source, and in that source the author is telling a story he was told for which he does not have direct evidence. Further there are hundreds of identical stories, e.g. The Wikipedia article kangaroo says, "A common myth about the kangaroo's English name is that it was a Guugu Yimithirr phrase for 'I don't know' or 'I don't understand' "; "In researching [country name] derivations, we discovered countless examples of folk etymology — popular but unscientific, and almost certainly incorrect, attempts to explain the origins of words. Aruba, for example, has been said to take its name from a Spanish phrase, “oro huba,” supposedly meaning “there is gold” — but “huba” isn’t a Spanish word, and the actual translation would be “hay oro.” The Moldova River, for which the country of Moldova is named, is sometimes said to be called that in honor of a hunting dog named Molda that drowned in its waters. For serious etymologists, that dog won’t hunt."

The removed text also does not match the name-origin myth given at Kenya which says it means "ostirch" or "ostirch fethers." Nick Beeson (talk) 17:47, 15 October 2020 (UTC)

There is something Wikipedia generalised
You guys generalised that all people of kenya are kikuyu and believe in Ngai,please change that article. 197.179.178.206 (talk) 05:04, 9 January 2022 (UTC)

 * @197.179.178.206 "you guys generalised"
 * how ironic 102.222.7.241 (talk) 17:18, 11 January 2023 (UTC)

title: Talk:Mount Kenya