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Medieval European Warfare: Technological, Social, and Cultural Developments
Posted on May 18, 2019 May 18, 2019 by MAMcIntosh
The Battle of Crécy (1346) between the English and French in the Hundred Years' War / Bibliothèque nationale de France, Wikimedia Commons
Developments forced a dramatic transformation in the character of warfare from antiquity, changing tactics, weaponry, and fortifications.
Edited by Matthew A. McIntosh
Journalist and Historian
Brewminate Editor-in-Chief
De re militari
Mulomedicina (1250-1375 ca.) / Biblioteca Medicea Laurenziana, Wikimedia Commons
Publius Flavius Vegetius Renatus wrote De re militari (Concerning Military Matters) possibly in the late 4th century.[2] Described by historian Walter Goffart as "the bible of warfare throughout the Middle Ages", De re militariwas widely distributed through the Latin West. While Western Europe relied on a single text for the basis of its military knowledge, the Byzantine Empire in Southeastern Europe had a succession of military writers.[3] Though Vegetius had no military experience and De re militari was derived from the works of Cato and Frontinus, his books were the standard for military discourse in Western Europe from their production until the 16th century.[4]
De re militari was divided into five books: who should be a soldier and the skills they needed to learn, the composition and structure of an army, field tactics, how to conduct and withstand sieges, and the role of the navy. According to Vegetius, infantry was the most important element of an army because it was cheap compared to cavalry and could be deployed on any terrain.[5] One of the tenets he put forward was that a general should only engage in battle when he was sure of victory or had no other choice.[6]As archaeologist Robert Liddiard explains, "Pitched battles, particularly in the eleventh and twelfth centuries, were rare."[7]
Although his work was widely reproduced, and over 200 copies, translations, and extracts survive today, the extent to which Vegetius affected the actual practice of warfare as opposed to its concept is unclear because of his habit of stating the obvious.[5] Historian Michael Clanchy noted "the medieval axiom that laymen are illiterate and its converse that clergy are literate",[8] so it may be the case that few soldiers read Vegetius' work. While their Roman predecessors were well-educated and had been experienced in warfare, the European nobility of the early Medieval period were not renowned for their education, but from the 12th century, it became more common for them to read.[9]
Some soldiers regarded the experience of warfare as more valuable than reading about it; for example, Geoffroi de Charny, a 14th century knight who wrote about warfare, recommended that his audience should learn by observing and asking advice from their superiors. While it is uncertain to what extent his work was read by the warrior class as opposed to the clergy, Vegetius remained prominent in the literature on warfare in the medieval period.[9] In 1489, King Henry VII of England commissioned the translation of De re militari into English, "so every gentleman born to arms and all manner of men of war, captains, soldiers, vituallers and all others would know how they ought to behave in the feats of wars and battles".[10]
Emergence and Evolution
The Château de Falaise in France / Photo by Ollamh, Wikimedia Commons
In Europe, breakdowns in centralized power led to the rise of a number of groups that turned to large-scale pillage as a source of income. Most notably the Vikings (but also Arabs, Mongols, Huns, and Magyars) raided significantly[11]. As these groups were generally small and needed to move quickly, building fortifications was a good way to provide refuge and protection for the people and the wealth in the region.
These fortifications evolved over the course of the Middle Ages, the most important form being the castle, a structure which has become almost synonymous with the Medieval era in the popular eye. The castle served as a protected place for the local elites. Inside a castle they were protected from bands of raiders and could send mounted warriors to drive the enemy from the area, or to disrupt the efforts of larger armies to supply themselves in the region by gaining local superiority over foraging parties that would be impossible against the whole enemy host.[12]
Celje Castle in Slovenia / Photo by xJaM, Wikimedia Commons
Fortifications were a very important part of warfare because they provided safety to the lord, his family, and his servants. They provided refuge from armies too large to face in open battle. The ability of the heavy cavalry to dominate a battle on an open field was useless against fortifications. Building siege engines was a time-consuming process, and could seldom be effectively done without preparations before the campaign. Many sieges could take months, if not years, to weaken or demoralize the defenders sufficiently. Fortifications were an excellent means of ensuring that the elite could not be easily dislodged from their lands – as Count Baldwin of Hainaut commented in 1184 on seeing enemy troops ravage his lands from the safety of his castle, "they can't take the land with them".[13][14]
Siege Warfare
In the Medieval period besieging armies used a wide variety of siege engines including: scaling ladders; battering rams; siege towers and various types of catapults such as the mangonel, onager, ballista, and trebuchet. Siege techniques also included mining in which tunnels were dug under a section of the wall and then rapidly collapsed to destabilize the wall's foundation. Another technique was to bore into the enemy walls, however this was not nearly as effective as other methods due to the thickness of castle walls.
The Walls of Dubrovnik are a series of defensive stone walls, never breached by a hostile army, that have surrounded and protected the maritime city-state of Dubrovnik (Ragusa), situated in southern Croatia. / Photo by Romanceor, Wikimedia Commons
Advances in the prosecution of sieges encouraged the development of a variety of defensive counter-measures. In particular, Medieval fortifications became progressively stronger – for example, the advent of the concentric castle from the period of the Crusades – and more dangerous to attackers – witness the increasing use of machicolations, as well the preparation of hot or incendiary substances. Arrow slits, concealed doors for sallies, and deep water wells were also integral to resisting siege at this time. Designers of castles paid particular attention to defending entrances, protecting gates with drawbridges, portcullises and barbicans. Wet animal skins were often draped over gates to repel fire. Moats and other water defences, whether natural or augmented, were also vital to defenders.
In the Middle Ages, virtually all large cities had city walls – Dubrovnik in Dalmatia is an impressive and well-preserved example – and more important cities had citadels, forts or castles. Great effort was expended to ensure a good water supply inside the city in case of siege. In some cases, long tunnels were constructed to carry water into the city. In other cases, such as the Ottoman siege of Shkodra, Venetian engineers had designed and installed cisterns that were fed by rain water channeled by a system of conduits in the walls and buildings.[15] Complex systems of underground tunnels were used for storage and communications in medieval cities like Tábor in Bohemia. Against these would be matched the mining skills of teams of trained sappers, who were sometimes employed by besieging armies.
Until the invention of gunpowder-based weapons (and the resulting higher-velocity projectiles), the balance of power and logistics definitely favored the defender. With the invention of gunpowder, the traditional methods of defence became less and less effective against a determined siege.
Manpower Structure
A Seal of the Knights Templar / Wikimedia Commons
The medieval knight was usually a mounted and armoured soldier, often connected with nobility or royalty, although (especially in north-eastern Europe) knights could also come from the lower classes, and could even be enslaved persons. The cost of their armour, horses, and weapons was great; this, among other things, helped gradually transform the knight, at least in western Europe, into a distinct social class separate from other warriors. During the crusades, holy orders of Knights fought in the Holy Land (see Knights Templar, the Hospitallers, etc.).[16]
Light cavalry consisted usually of lighter armed and armoured men, who could have lances, javelins or missile weapons, such as bows or crossbows. In much of the Middle Ages light cavalry usually consisted of wealthy commoners. Later in the Middle Ages light cavalry would also include sergeants who were men who had trained as knights but could not afford the costs associated with the title. Light cavalry were used as scouts, skirmishers or outflankers. Many countries developed their own styles of light cavalry, such as Hungarian mounted archers, Spanish jinetes, Italian and German mounted crossbowmen and English currours.
Infantry were recruited and trained in a wide variety of manners in different regions of Europe all through the Middle Ages, and probably always formed the most numerous part of a medieval field army. Many infantrymen in prolonged wars would be mercenaries. Most armies contained significant numbers of spearmen, archers and other unmounted soldiers.
Hungarian raids in the 10th century. Prior to the battle of Lechfield in 955 Medieval Europeans were vulnerable from the Nomadic style of war that came from the Hungarians. / Csanády, Wikimedia Commons
In the earliest Middle Ages it was the obligation of every noble to respond to the call to battle with his own equipment, archers, and infantry. This decentralized system was necessary due to the social order of the time, but could lead to motley forces with variable training, equipment and abilities. The more resources the noble had access to, the better his troops would typically be.
Typically the feudal armies consisted of a core of highly skilled knights and their household troops, mercenaries hired for the time of the campaign and feudal levies fulfilling their feudal obligations, who usually were little more than rabble. They could, however, be efficient in disadvantageous terrain. Towns and cities could also field militias.
As central governments grew in power, a return to the citizen and mercenary armies of the classical period also began, as central levies of the peasantry began to be the central recruiting tool. It was estimated that the best infantrymen came from the younger sons of free land-owning yeomen, such as the English archers and Swiss pikemen. England was one of the most centralized states in the Late Middle Ages, and the armies that fought the Hundred Years' War were mostly paid professionals.
In theory, every Englishman had an obligation to serve for forty days. Forty days was not long enough for a campaign, especially one on the continent. Thus the scutage was introduced, whereby most Englishmen paid to escape their service and this money was used to create a permanent army. However, almost all high medieval armies in Europe were composed of a great deal of paid core troops, and there was a large mercenary market in Europe from at least the early 12th century.[17]
As the Middle Ages progressed in Italy, Italian cities began to rely mostly on mercenaries to do their fighting rather than the militias that had dominated the early and high medieval period in this region. These would be groups of career soldiers who would be paid a set rate. Mercenaries tended to be effective soldiers, especially in combination with standing forces, but in Italy they came to dominate the armies of the city states. This made them problematic; while at war they were considerably more reliable than a standing army, at peacetime they proved a risk to the state itself like the Praetorian Guard had once been.
Mercenary-on-mercenary warfare in Italy led to relatively bloodless campaigns which relied as much on manoeuvre as on battles, since the condottieri recognized it was more efficient to attack the enemy's ability to wage war rather than his battle forces, discovering the concept of indirect warfare 500 years before Sir Basil Liddell Hart, and attempting to attack the enemy supply lines, his economy and his ability to wage war rather than risking an open battle, and manoeuvre him into a position where risking a battle would have been suicidal. Machiavelli understood this indirect approach as cowardice.[18]
The practice of carrying relics into battle is a feature that distinguishes medieval warfare from its predecessors or from early modern warfare and possibly inspired by biblical references.[19] The presence of relics was believed to be an important source of supernatural power that served both as a spiritual weapon and a form of defence; the relics of martyrs were considered by Saint John Chrysostom much more powerful than "walls, trenches, weapons and hosts of soldiers"[20]
In Italy, the carroccio or carro della guerra, the "war wagon", was an elaboration of this practice that developed during the 13th century. The carro della guerra of Milan was described in detail in 1288 by Bonvesin de la Riva in his book on the "Marvels of Milan". Wrapped in scarlet cloth and drawn by three yoke of oxen that were caparisoned in white with the red cross of Saint Ambrose, the city's patron, it carried a crucifix so massive it took four men to step it in place, like a ship's mast.[21]
Medieval Warfare largely predated the use of supply trains- which meant that armies had to acquire food supplies from whatever territory they were passing through, this meant that large scale looting by soldiers was unavoidable, and was actively encouraged by the 14th century with its emphasis on "Chevaunche" tactics, or use of units of light cavalry who would loot and pillage hostile territory in order to distract and demoralize the enemy while denying them their own supplies.
Through the medieval period, soldiers were responsible for supplying themselves, either through foraging, looting, or purchases. Even so, military commanders often provided their troops with food and supplies, but this would be provided in lieu of the soldiers' wages, or soldiers would be expected to pay for it from their wages, either at cost or even with a profit.[22]
In 1294, the same year John II de Balliol of Scotland refused to support Edward I of England's planned invasion of France, Edward I implemented a system in Wales and Scotland where sheriffs would acquire foodstuffs, horses and carts from merchants with compulsory sales at prices fixed below typical market prices under the Crown's rights of prise and purveyance. These goods would then be transported to Royal Magazines in southern Scotland and along the Scottish border where English conscripts under his command could purchase them. This continued during the First War of Scottish Independence which began in 1296, though the system was unpopular and was ended with Edward I's death in 1307.[22]
Starting under the rule of Edward II in 1307 and ending under the rule of Edward III in 1337, the English instead used a system where merchants would be asked to meet armies with supplies for the conscripts to purchase. This led to discontent as the merchants saw an opportunity to profiteer, forcing conscripts to pay well above normal market prices for food.[22]
As Edward III went to war with France in the Hundred Years' War (starting in 1337), the English returned to a practice of foraging and looting to meet their logistical needs. This practice lasted throughout the course of war, extending through the remainder of Edward III's reign into the reign of Henry VI.[22]
The Byzantine fleet repels the Rus' attack on Constantinople in 941. The Byzantine dromons are rolling over the Rus' vessels and smashing their oars with their spurs. / Wikimedia Commons
The waters surrounding Europe can be grouped into two types which affected the design of craft that travelled and therefore the warfare. The Mediterranean and Black Seas were free of large tides, generally calm, and weather predictable. The seas around the north and west of Europe experienced stronger and less predictable weather. The weather gauge, the advantage of having a following wind, was an important factor in naval battles, particularly to the attackers. Typically westerlies (winds blowing from west to east) dominated Europe, giving naval powers to the west an advantage.[23] Medieval sources on the conduct of medieval naval warfare are less common than those about land-based war. Most medieval chroniclers had no experience of life on the sea, and generally were not well-informed. Maritime archaeology has helped provide information.[24]
Early in the medieval period, ships in the context of warfare were used primarily for transporting troops.[25] In the Mediterranean, naval warfare in the Middle Ages was similar to that under late Roman Empire: fleets of galleys would exchange missile fire and then try to board bow first to allow marines to fight on deck. This mode of naval warfare remained basically the same into the early modern period, as, for example, at the Battle of Lepanto. Famous admirals included Roger of Lauria, Andrea Doria and Hayreddin Barbarossa.
Galleys were not suitable for the colder and more turbulent North Sea and Atlantic Ocean, although they saw occasional use. Bulkier ships were developed which were primarily sail-driven, although the long lowboard Viking-style rowed longship saw use well into the 15th century. Their main purpose in the north remained the transportation of soldiers to fight on the decks of the opposing ship (as, for example, at the Battle of Svolder or the Battle of Sluys).
A battle between the Venetian and Holy Roman fleets. detail of fresco by Spinello Aretino 1407-1408 / Wikimedia Commons
Late medieval sailing warships resembled floating fortresses, with towers in the bows and at the stern (respectively, the forecastle and aftcastle). The large superstructure made these warships quite unstable, but the decisive defeats that the more mobile but considerably lower boarded longships suffered at the hands of high-boarded cogs in the 15th century ended the issue of which ship type would dominate northern European warfare.
Introduction of Guns
The introduction of guns was the first steps towards major changes in naval warfare, but it only slowly changed the dynamics of ship-to-ship combat. The first guns on ships were introduced in the 14th century and consisted of small wrought-iron pieces placed on the open decks and in the fighting tops, often requiring only one or two men to handle them. They were designed to injure, kill or simply stun, shock and frighten the enemy prior to boarding.[26]
As guns were made more durable to withstand stronger gunpowder charges, they increased their potential to inflict critical damage to the vessel rather than just their crews. Since these guns were much heavier than the earlier anti-personnel weapons, they had to be placed lower in the ships, and fire from gunports, to avoid ships becoming unstable. In Northern Europe the technique of building ships with clinker planking made it difficult to cut ports in the hull; clinker-built (or clench-built) ships had much of their structural strength in the outer hull. The solution was the gradual adoption of carvel-built ships that relied on an internal skeleton structure to bear the weight of the ship.[27] Gunports cut in the hull of ships were not introduced until 1501, at the very start of the early modern period.[28]
Two views of a hand culverin and two small cannons from the 15th century / PHGCOM, Wikimedia Commons
The first ships to actually mount heavy cannon capable of sinking ships were galleys, with large wrought-iron pieces mounted directly on the timbers in the bow. The first example is known from a woodcut of a Venetian galley from 1486.[29]Heavy artillery on galleys was mounted in the bow which fit conveniently with the long-standing tactical tradition of attacking head-on and bow-first. The ordnance on galleys was quite heavy from its introduction in the 1480s, and capable of quickly demolishing medieval-style stone walls that still prevailed until the 16th century.[30]
This temporarily upended the strength of older seaside fortresses, which had to be rebuilt to cope with gunpowder weapons. The addition of guns also improved the amphibious abilities of galleys as they could assault supported with heavy firepower, and could be even more effectively defended when beached stern-first.[30] Galleys and similar oared vessels remained uncontested as the most effective gun-armed warships in theory until the 1560s, and in practice for a few decades more, and were actually considered a grave risk to sailing warships.[31]
Rise of Infantry
In the Medieval period, the mounted cavalry long held sway on the battlefield. Heavily armoured, mounted knights represented a formidable foe for reluctant peasant draftees and lightly armoured freemen. To defeat mounted cavalry, infantry used swarms of missiles or a tightly packed phalanx of men, techniques honed in antiquity by the Greeks.
Swiss Pikemen
The use of long pikes and densely packed foot troops was not uncommon in the Middle Ages. The Flemish footmen at the Battle of the Golden Spurs met and overcame French knights in 1302, and the Scots held their own against heavily armoured English invaders. During the St.Louis crusade, dismounted French knights formed a tight lance-and-shield phalanx to repel Egyptian cavalry. The Swiss used pike tactics in the late medieval period. While pikemen usually grouped together and awaited a mounted attack, the Swiss developed flexible formations and aggressive maneuvering, forcing their opponents to respond. The Swiss won at Morgarten, Laupen, Sempach, Grandson and Murten, and between 1450 and 1550 every leading prince in Europe (except the English and Scottish) hired Swiss pikemen, or emulated their tactics and weapons (e.g., the German Landsknechte).
Welsh and English Longbowmen
A modern replique of an English longbow / Photo by Jeanfauchelevent, Wikimedia Commons
The Welsh & English longbowman used a single-piece longbow (but some bows later developed a composite design) to deliver arrows that could penetrate contemporary mail and damage/dent plate armour. The longbow was a difficult weapon to master, requiring long years of use and constant practice. A skilled longbowman could shoot about 12 shots per minute. This rate of fire was far superior to competing weapons like the crossbow or early gunpowder weapons. The nearest competitor to the longbow was the much more expensive crossbow, used often by urban militias and mercenary forces. The crossbow had greater penetrating power, and did not require the extended years of training.
At Crécy and Agincourt bowmen unleashed clouds of arrows into the ranks of knights. At Crécy, even 5,000 Genoese crossbowmen could not dislodge them from their hill. At Agincourt, thousands of French knights were brought down by armour-piercing bodkin point arrows and horse-maiming broadheads. Longbowmen decimated an entire generation of the French nobility.
Transition to Gunpowder Warfare
Artillery in a 1490s illustration of the Siege of Orleans of 1429 / Wikimedia Commons
In 1326 the earliest known European picture of a gun appeared in a manuscript by Walter de Milemete.[32] In 1350, Petrarch wrote that the presence of cannons on the battlefield was 'as common and familiar as other kinds of arms'.[33]
Early artillery played a limited role in the Hundred Years' War, and it became indispensable in the Italian Wars of 1494–1559, marking the beginning of early modern warfare. Charles VIII, during his invasion of Italy, brought with him the first truly mobile siege train: culverins and bombards mounted on wheeled carriages, which could be deployed against an enemy stronghold immediately after arrival.
Medieval Conquerors
The initial Muslim conquests began in the 7th century after the death of the Islamic prophet Muhammad, and were marked by a century of rapid Arab expansion beyond the Arabian Peninsula under the Rashidun and Umayyad Caliphates. Under the Rashidun, the Arabs conquered the Persian Empire, along with Roman Syria and Roman Egypt during the Byzantine-Arab Wars, all within just seven years from 633 to 640. Under the Umayyads, the Arabs annexed North Africa and southern Italy from the Romans and the Arab Empire soon stretched from parts of the Indian subcontinent, across Central Asia, the Middle East, North Africa, and southern Italy, to the Iberian Peninsula and the Pyrenees.
Across the ravines lies the battlefield of Yarmouk, this picture taken about 8 miles away, from Jordan. / Wikimedia Commons
The early Arab army mainly consisted of camel-mounted infantry, alongside a few Bedouin cavalry. Constantly outnumbered by their opponent, they did however possess the advantage of strategic mobility, their camel-borne nature allowing them to constantly outmaneuver larger Byzantine and Sassanid armies to take prime defensive positions. The Rashidun cavalry, while lacking the number and mounted archery skill of their Roman and Persian counterparts was for the most part skilfully employed, and played a decisive role in many crucial battles such as Battle of Yarmouk.
In contrast, the Roman army and Persian army at the time both had large numbers of heavy infantry and heavy cavalry (cataphracts and clibanarii) that were better equipped, heavily protected, and more experienced and disciplined. The Arab invasions came at a time when both ancient powers were exhausted from the protracted Byzantine–Sassanid Wars, particularly the bitterly fought Byzantine–Sassanid War of 602–628 which had brought both empires close to collapse. Also, the typically multi-ethnic Byzantine force was always racked by dissension and lack of command unity, a similar situation also being encountered among the Sassanids who had been embroiled in a bitter civil war for a decade before the coming of the Arabs. In contrast, the Ridda Wars had forged the Caliphate's army into a united and loyal fighting force.
The Vikings were a feared force in Europe because of their savagery and speed of their attacks. Whilst seaborne raids were nothing new at the time, the Vikings refined the practice to a science through their shipbuilding, tactics and training.[34] Unlike other raiders, the Vikings made a lasting impact on the face of Europe. During the Viking age, their expeditions, frequently combining raiding and trading, penetrated most of the old Frankish empire, the British Isles, the Baltic region, Russia, and both Muslim and Christian Iberia. Many served as mercenaries, and the famed Varangian Guard, serving the Emperor of Constantinople, was drawn principally of Scandinavian warriors.
Viking longships were swift and easily manoeuvered; they could navigate deep seas or shallow rivers[34], and could carry warriors that could be rapidly deployed directly onto land due to the longships being able to land directly. The longship was the enabler of the Viking style of warfare that was fast and mobile, relying heavily on the element of surprise[35], and they tended to capture horses for mobility rather than carry them on their ships. The usual method was to approach a target stealthily, strike with surprise and then retire swiftly. The tactics used were difficult to stop, for the Vikings, like guerrilla-style raiders elsewhere, deployed at a time and place of their own choosing. The fully armoured Viking raider would wear an iron helmet and a maille hauberk, and fight with a combination of axe, sword, shield, spear or great "Danish" two-handed axe, although the typical raider would be unarmoured, carrying only a bow and arrows, a knife "seax", a shield and spear.
Full-scale replica of a Viking snekkja based in Morąg, Poland / Photo by JarmeryK, Wikimedia Commons
Almost by definition, opponents of the Vikings were ill-prepared to fight a force that struck at will, with no warning. European countries with a weak system of government would be unable to organize a suitable response and would naturally suffer the most to Viking raiders. Viking raiders always had the option to fall back in the face of a superior force or stubborn defence and then reappear to attack other locations or retreat to their bases in what is now Sweden, Denmark, Norway and their Atlantic colonies. As time went on, Viking raids became more sophisticated, with coordinated strikes involving multiple forces and large armies, as the "Great Heathen Army" that ravaged Anglo-Saxon England in the 9th century. In time, the Vikings began to hold on to the areas they raided, first wintering and then consolidating footholds for further expansion later.
With the growth of centralized authority in the Scandinavian region, Viking raids, always an expression of "private enterprise", ceased and the raids became pure voyages of conquest. In 1066, King Harald Hardråde of Norway invaded England, only to be defeated by Harold Godwinson, who in turn was defeated by William of Normandy, descendant of the Viking Rollo, who had accepted Normandy as a fief from the Frankish king. The three rulers had their claims to the English crown (Harald probably primarily on the overlord-ship of Northumbria) and it was this that motivated the battles rather than the lure of plunder.
At that point, the Scandinavians had entered their medieval period and consolidated their kingdoms of Denmark, Norway, and Sweden. This period marks the end of significant raider activity both for plunder or conquest. The resurgence of centralized authority throughout Europe limited opportunities for traditional raiding expeditions in the West, whilst the Christianisation of the Scandinavian kingdoms themselves encouraged them to direct their attacks against the still predominantly pagan regions of the eastern Baltic. The Scandinavians started adapting more continental European ways, whilst retaining an emphasis on naval power – the "Viking" clinker-built warship was used in war until the 14th century at least. However, developments in shipbuilding elsewhere removed the advantage the Scandinavian countries had previously enjoyed at sea, whilst castle building throughout frustrated and eventually ended Viking raids[36] Natural trading and diplomatic links between Scandinavia and Continental Europe ensured that the Scandinavians kept up to date with continental developments in warfare.
The Scandinavian armies of the High Middle Ages followed the usual pattern of the Northern European armies, but with a stronger emphasis on infantry. The terrain of Scandinavia favoured heavy infantry, and whilst the nobles fought mounted in the continental fashion, the Scandinavian peasants formed a well-armed and well-armoured infantry, of which approximately 30% to 50% would be archers or crossbowmen. The crossbow, the flatbow and the longbow were especially popular in Sweden and Finland.
During The Mongol invasion of Europe, Tatars, under the leadership of Kadan, experienced a major failure in March 1242 at Klis Fortress in southern Croatia.[37][38]
By 1241, having conquered large parts of Russia, the Mongols continued the invasion of Europe with a massive three-pronged advance, following the fleeing Cumans, who had established an uncertain alliance with King Bela IV of Hungary. They first invaded Poland, and finally Hungary, culminating in the crushing defeat of the Hungarians in the Battle of Mohi. The Mongol aim seems to have consistently been to defeat the Hungarian-Cuman alliance. The Mongols raided across the borders to Austria and Bohemia in the summer when the Great Khan died, and the Mongol princes returned home to elect a new Great Khan.
The Golden Horde would frequently clash with Hungarians, Lithuanians and Poles in the thirteenth century, with two large raids in the 1260s and 1280s respectively. In 1284 the Hungarians repelled the last major raid into Hungary, and in 1287 the Poles repelled a raid against them. The instability in the Golden Horde seems to have quieted the western front of the Horde. Also, the large scale invasions and raiding that had previously characterized the expansion of the Mongols was cut short probably in some part due to the death of the last great Mongol leader, Tamerlane.
The Hungarians and Poles had responded to the mobile threat by extensive fortification-building, army reform in the form of better armoured cavalry, and refusing battle unless they could control the site of the battlefield to deny the Mongols local superiority. The Lithuanians relied on their forested homelands for defence, and used their cavalry for raiding into Mongol-dominated Russia. When attacking fortresses they would launch dead or diseased animals into fortresses to help spread disease.
Battle of Kosovo, by Adam Stefanović (1870) / Wikimedia Commons
An early Turkic group, the Seljuks, were known for their cavalry archers. These fierce nomads were often raiding empires, such as the Byzantine Empire, and they scored several victories using mobility and timing to defeat the heavy cataphracts of the Byzantines.
One notable victory was at Manzikert, where a conflict among the generals of the Byzantines gave the Turks the perfect opportunity to strike. They hit the cataphracts with arrows, and outmaneuvered them, then rode down their less mobile infantry with light cavalry that used scimitars. When gunpowder was introduced, the Ottoman Turks of the Ottoman Empire hired the mercenaries that used the gunpowder weapons and obtained their instruction for the Janissaries. Out of these Ottoman soldiers rose the Janissaries (yeni ceri; "new soldier"), from which they also recruited many of their heavy infantry. Along with the use of cavalry and early grenades, the Ottomans mounted an offensive in the early Renaissance period and attacked Europe, taking Constantinople by massed infantry assaults.
Like many other nomadic peoples, the Turks featured a core of heavy cavalry from the upper classes. These evolved into the Sipahis (feudal landholders similar to western knights and Byzantine pronoiai) and Qapukulu (door slaves, taken from youth like Janissaries and trained to be royal servants and elite soldiers, mainly cataphracts).
Milner (1996), p. 63
Nicholson (2004), p. 13
Goffart (1977), p. 65
Nicholson (2004), pp. 13–14
Gillingham (1992), p. 150
Liddiard (2005), p. 79
Quoted in Nicholson (2004), p. 16
Quoted in Nicholson (2004), pp. 18–19
Parsa, Ali (2017). World History: A Concise, Selective, Interpretive History of the World. Sentia Publishing.
Bradbury, Jim (2004). The Routledge Companion to Medieval Warfare. Routledge. p. 284.
McGlynn, Sean (Jan 1994). "The Myths of Medieval Warfare". History Today. 44 (1): 32. Retrieved September 14, 2009.
Carruthers, Bob (2013). Medieval Warfare. Pen and Sword. p. 6.
Karaiskaj, Gjerak. "Furnizimi me ujë i kalasë së Shkodrës ne mesjetë." Monumentet: Materialet e sesionit III shkencor të Institutit të Monumenteve të Kulturës 11 (1985): 55-77.
Carruthers, Bob (2013). Medieval Warfare. Pen and Sword. p. 10.
http://bible.oremus.org/?passage=1+Samuel+4:3%E2%80%9311:3&version=nrsv
John Chrysostom, Laudatio martyrum Aegyptiorum, 1 PG 50 col. 694f.
Bovesin de la Riva, De Magnalibus Mediolani: Meraviglie di Milano (Milan, 1998), as reported in John Dickie, Delizia! The Epic History of Italians and Their Food (New York, 2008), p. 33.
Abels, Richard. "War in the Middle Ages: Medieval Logistics – English Experience". United States Naval Academy. Retrieved 3 October 2017.
Fernández-Armesto (1999), p. 231
Fernández-Armesto (1999), pp. 230–231
DeVries (1992), p. 283
Rodger (1997), pp. 205–6 .
Marsden (2003), pp. 137–142.
Rodger (1997), p. 207.
Lehmann (1984), p. 31
Guilmartin (1974), pp. 264–66
The British naval historian Nicholas Rodger describes this as a "crisis in naval warfare" which eventually led to the development of the galleon, which combined ahead-firing capabilities, heavy broadside guns and a considerable increase in maneuverability by introduction of more advanced sailing rigs; Rodger (2003), p. 245. For more detailed arguments concerning the development of broadside armament, see Rodger (1996).
Kelly 2004:29
Norris 2003:19
Parker, Philip (May 25, 2016). "A brief history of the Vikings". The official website for BBC History Magazine and BBC World Histories Magazine.
Walsh, David (May 2013). "Engineering the Viking Longboat". THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS.
"Medieval and Middle Ages History Timelines – Why castles were built". www.timeref.com. Retrieved 2018-05-19.
Listeš, Srećko. "Povijest Klisa". klis.hr (in Croatian). Službene stranice Općine Klis. Archived from the original on 2010-03-23. Retrieved 2010-05-16.
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Archdeacon, Thomas of Split (2006). History of the Bishops of Salona and Split – Historia Salonitanorum atque Spalatinorum pontificum (in Latin and English). Budapest: Central European University Press.
DeVries, Kelly (1992), Military Medieval Technology, Broadview Press.
Fernández-Armesto, Felipe (1999), "Naval Warfare after the Viking Age, c.1100–1500″, in Keen, Maurice (ed.), Medieval Warfare: A History, Oxford: Oxford University Press, pp. 230–252.
Gillingham, John (1992), "William the Bastard at War", in Strickland, Matthew (ed.), Anglo-Norman warfare: Studies in Late Anglo-Saxon and Anglo-Norman Military Organization and Warfare, Woodbridge: The Boydell Press, pp. 143–160.
Goffart, Walter (1977), "The date and purpose of Vegetius' De Re Militari", Traditio, xxxiii: 65–100, JSTOR 27831025
Guilmartin, John Francis, Gunpowder and Galleys: Changing Technology and Mediterranean Warfare at Sea in the Sixteenth Century. Cambridge University Press, London. 1974.
Rodger, Nicholas A. M., "The New Atlantic: Naval Warfare in the Sixteenth Century", pp. 231–47 in Hattendorf, John B. & Unger, Richard W. (editors), War at Sea in the Middle Ages and the Renaissance. Woodbridge, Suffolk. 2003.
Liddiard, Robert (2005), Castles in Context: Power, Symbolism and Landscape, 1066 to 1500, Macclesfield: Windgather Press Ltd.
Lehmann, L. Th., Galleys in the Netherlands. Meulenhoff, Amsterdam. 1984.
Marsden, Peter, Sealed by Time: The Loss and Recovery of the Mary Rose. The Archaeology of the Mary Rose, Volume 1. The Mary Rose Trust, Portsmouth. 2003.
Nicholson, Helen (2004), Medieval Warfare: Theory and Practice of War in Europe, 300–1500, Basingstoke: Palgrave Macmillan.
Publius Flavius Vegetius Renatus (1996), Milner, N. P. (ed.), Vegetius: epitome of military science, Translated Texts for Historians, xvi, Liverpool: Liverpool University Press
Rodger, Nicholas A. M., "The Development of Broadside Gunnery, 1450–1650." Mariner's Mirror 82 (1996), pp. 301–24.
Rodger, Nicholas A. M., The Safeguard of the Sea: A Naval History of Britain 660–1649. W.W. Norton & Company, New York. 1997.
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Originally published by Wikipedia, 03.21.2018, under a Creative Commons Attribution-ShareAlike 3.0 Unported license.
This entry was posted in Uncategorized and tagged History, Medieval, Middle Ages, Military, War, Weapons. Bookmark the permalink.
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I had not expected to have a Thanksgiving this year, and the day itself passed uneventfully. But I was invited to a real American Thanksgiving dinner on Saturday night, where a lively group of Finns, Brits, and Russians gathered for turkey with all the trimmings: stuffing, potatoes and gravy, sweet potatoes, corn, peas, Finnish cranberries, and four pies—two apple, two pumpkin. As the American, or maybe just as the most senior citizen, I was allowed to carve the bird.
Thanksgiving is my favorite holiday by far, not just because of the feast but also because of the requirement that we count our blessings. It is "meet and right so to do," as we Episcopalians are wont to say. I have always thought that America could be a little more aggressive in exporting this fetching holiday. Then again, maybe the idea of marketing a tribute to humility is way too oxymoronic. Anyway, I noticed that everyone at the event on Saturday night seemed to understand the spirit of the occasion, so maybe Thanksgiving is catching on internationally without our making much of an effort.
Children, most of them bilingual or trilingual, loomed large at this Thanksgiving feast. They wouldn't have sat still for a photo, so you'll have to settle for the grown-ups, pictured above. A fine time was had by all, particularly by this innocent abroad.


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Radio Jackie News: Mayor of London Sadiq Khan will today launch a trial of the world's largest air pollution monitoring network.
Mayor of London Sadiq Khan will today launch a trial of the world's largest air pollution monitoring network.
It will include 100 monitors in schools, on streets lamp posts and at known hot spots.
The trial has been launched ahead of April, when the Ultra Low Emission Zone comes into force in Central London.
Mr Khan then plans to extend it to the South Circular, including parts of Wandsworth, in 2021.


text:
Peer-Reviewer
Arie Sujito
https://scholar.google.co.id/citations?user=eG2YAqYAAAAJ&hl=id
Bergerak untuk rakyat
Sugeng Bahagijo
https://scholar.google.com/citations?user=h3dJ8oEAAAAJ&hl=en
International NGO Forum on Indonesian Development, INFID.
Home > Vol 26, No 2 (2022) > Sujito
Pioneering the Road to Prosperity for the Future Workforce: Building Social Development Fund (SDF) Roadmap in Indonesia
https://doi.org/10.22146/jkap.78219
Arie Sujito(1*), Sugeng Bahagijo(2)
(1)&nbspUniversitas Gadjah Mada
(2)&nbspInternational NGO Forum on Indonesian Development, INFID.
The paper discusses possible solutions through the Social Development Fund (SDF) scheme to reduce and prevent educated unemployment problems. This paper uses critical discourse involving literature studies such as books, journals, reports, and online media. The findings of the research show that (1) there still a lot of educated workforces unemployed. This happens because of the mismatch between the capabilities and the goals of the industry. Technological advances invented new jobs. Therefore, it requires someone to adapt quickly, let alone the fact that (2) cooperation between ministries and the private sectors is not quite optimal. Other factors are, (3) SDF practices implementation by other countries, (4) resource availability in Indonesia to implement SDF. From this study concluded that SDF is a realistic alternative offer to be implemented in Indonesia, as an ecosystem that is able to ensure that existing sectors work optimally to establish the common goal of reducing the possibility of educated unemployment.
labor market; social development fund; state; unemployment
AGCAS. (2021). The Impact Of Covid-19 On Recent Graduates ' Career Decisions And Outcomes (Issue July). https://www.agcas.org.uk/write/MediaUploads/Resources/Research and knowledge/826_The_impact_of_Covid-19_on_recent_graduates_career_decisions_and_outcomes_-_July_2021.pdf
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Daniel J. Connolly, Jacquelyn Allen-Collinson & Adam B. Evans (2016) A level playing 'field'? A Bourdieusian analysis of the career aspirations of further education students on sports courses, Sport, Education and Society, 21:8, 1144-1160, DOI: 10.1080/13573322.2014.994175
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Graeber, D. (2018). Bullshit Jobs. New York: Simon & Schuster.
Green, F. (2004). Why Has Work Effort Become More Intense ? Wiley Online Library, 43(4), 709–741. https://doi.org/https://doi.org/10.1111/j.0019-8676.2004.00359.x
Green, F., Ashton, D., James, D., & Sung, J. (1999). The Role of The State in Skill Formation: Evidence from The Republic of Korea , Singapore, And Taiwan. 15(1), 82–96. https://www.jstor.org/stable/23606560.
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Howell, C. (2021). Rethinking the Role of the State in Employment Relations for a Neoliberal Era. ILR Review, 74(3), 739–772. https://doi.org/10.1177/0019793920904663
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DOI: https://doi.org/10.22146/jkap.78219
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redpajama_set_name: RedPajamaCommonCrawl


Pokémon Go – how safe is this reality?
With more than 75 million downloads, the number one selling app in the Apple store, has bought with it a unique set of risks. Pokémon Go is the international craze that has increased stock prices and seen random locations become meeting places for hundreds of trainers (this is what you are called when you play Pokémon and you must 'catch them all'). Pokémon Go is an augmented reality game where players capture "pocket monsters" using a GPS map and their smartphone camera, giving the illusion they're snagging these creatures in the real world. Players are walking, driving or riding to search for the ultimate monsters, watching their screen to confirm their location, potentially stepping into traffic, driving into objects or wandering into dangerous locations, all in the pursuit of their next catch.
Law enforcement agencies across the world have issued warnings and in some areas banned the game due to the potential risk. Melbourne freeways recently displayed signage at specific locations relevant to the game warning drivers "Don't Drive and Pokémon", being caught doing so attracts four demerit points and a $466.00 fine.
The games creators have responded to concerns, with the latest game update including warnings that update and change every time the user plays the game, such as "Do not play Pokémon Go while driving," and "Do not enter dangerous areas while playing Pokémon Go."
This game, the most popular we have seen to date for the smart phone, has people engrossed in their new reality and opportunity for the venues, festivals and events to have the monsters at their locations which may attract more patrons, but with that opportunity comes the risk for users who lack situational awareness of their real world risks.


(Gives gender – specific) The sceptre has the same symbolism as most rods and staves: that of a phallic object.
In a woman's dream it suggests the power for action.
In a man's dream it signifies the power to take authority.


2fluid had final concepts approved this week for the rebranding of health food company Best of Health.
The rebranding, worth a £250 000 design fee to the Glasgow-based consultancy, includes a revamped visual identity, brand guidelines, packaging, brochures, advertising and website work, for use across European, British and US markets. Products carrying the new branding will be on-shelf from late June.
2fluid managing director Jacqueline Doherty says that the revamp was necessary because Best of Health did not focus on branding when it launched.
'[The client] took its product to market quickly and is only a year old, but is [now] building up a real momentum. We've put clear guidelines in place to ensure everything co-ordinates visually and have thought about branding in the context of its business needs over the next five years,' she says.
The consultancy was briefed to develop a visual identity that could be applied to products ranging from food supplements to beauty products, says 2fluid creative director Dino Squillino.
'The challenge was taking what they had and moving it on in a way that became useable across all formats, from packaging to the website,' he says.
'Initially, the logo was grass green but we've taken it away from its association with green, because [Best of Health] isn't just about eating vegetables.
'We chose yellow because it speaks about inner aspirations about health and communicates a warm glow,' he adds.
2fluid, which was appointed in March, won the business following a five-way, unpaid, creative pitch against both British and US consultancies.
Best of Health operates as Radiant Health in the US.


text:
Jardin des Abbesses je veřejný park, který se nachází v Paříži na Montmartru v 18. obvodu.

Charakteristika
Původně se jednalo o zahradu léčivých rostlin. Vstup je přes Passage des Abbesses a navazuje na Square Jehan-Rictus. Název zahrady je odvozen od abatyší kláštera Montmartre.

Odkazy

Reference

Externí odkazy

  Park na stránkách města Paříže

Abbesses
Montmartre

redpajama_set_name: RedPajamaWikipedia


text:
Megha Fiber Glass Industries Limited (MFGIL)
MFGIL is a recognized manufacturer of Glass Fibre Reinforced Polyester (GRP) pipes on continuous mandrel process, which represents state of art in GRP pipe production technology. The unit also manufacturers Glass Reinforced Epoxy (GRE) pipes.
In the role of an EPC contractor we are involved in the laying, jointing, testing, and commissioning processes.
GRP Pipes
MFGIL-GRP is the first plant in South India to manufacture Glass Reinforced Polyester (GRP) pipes, with a considerably large diameter (up to 2600 mm) on CFW machine. This process allows the use of continuous glass fibre reinforcements in the circumferential direction. The production activities are carried out under controlled conditions with continuously advancing Mandrel process.
MFGIL can manufacture GRP pipes from a nominal diameter of 300 mm to 2600 mm, and fittings with different stiffness and pressure classes by complying with national and international standards.
MFGIL has an in-house pipe design, R&D department, and a complete test facility. The unit has a fully integrated facility of manufacturing and testing of GRP pipes, as per quality standards. MFGIL has a production capacity of about 300 km. per year.
GRE Pipes
MFGIL manufactures GRE pipes on helical filament winding process. It is the first plant in South India to manufacture API monogram GRE pipes with diameters ranging from 50 mm to 300 mm diameter. They are extensively used in oil & gas sector. In this process, filament is wound in a helix angle which gives more mechanical strength. The complete process is controlled through SCADA system.
MFGIL manufactures GRE pipes of different pressure classes while complying with national and international standards. The GRE pipes are manufactured on a fixed mandrel in lengths ranging from a minimum of 3 meters to a maximum of 9 meters. MFGIL has an in-house pipe design, R&D department, and a complete test facility.
MFGIL also houses a fully integrated facility for manufacturing and testing of GRE pipes as per quality standards. The GRE plant has a production capacity of 150 km. per year.


text:
Q: ConcurrentLinkedQueue won't remove I'd like to know in which case a ConcurrentLinkedQueue can't remove an element from its Collections. The item exists, it is inside the collection but calling
SomeConcurrentLinkedQueue.remove(item)

won't remove this item from the collection. Is the collection locked or something ?

A: If ConcurrentLinkedQueue.remove(item) returns false, this means that item didn't compare equal to any of the queue's entries. If that's not what you're expecting, check the item's equals() method for errors.
Another possibility (pointed out by @Louis Wasserman) is that the queue contains multiple occurrences of item. ConcurrentLinkedQueue.remove(item) would only remove one of them.

redpajama_set_name: RedPajamaStackExchange


Written by Peter Grant. Published on May 31st 2013.
ONCE upon a time there was a BBC Three series called Conviction which was a forerunner of many of today's gritty crime dramas.
But another seed was sown there in 2004. Now Conviction is a feature film with a stark title - Blood.
There is a significant Merseyside tie: nearly all exterior - and some interior - scenes were shot on the Wirral and the Vines pub in Lime Street, although this is not a Liverpool film.
Blood is a tale of two brothers, both cops, played by Paul Bettany as weary Joe and our very own multi-versatile Stephen Graham as upbeat Chrissie.
They don't look remotely like siblings, but we suspend that belief, for both are on the same wavelength ... one instigated by their retired police chief dad, Lenny, played by multi award-winning actor Brian Cox.
Cox's character still thinks he runs the department - with his iron fist, no-holds-barred, hit-first-ask-questions-later mentality. Sadly, he has Alzheimers and, in one cruel scene, he turns up at his old office, uninvited. Bewildered colleagues look on and he berates the state of his desk and it is left to his sons put their deluded dad right.
The film opens with a grisly shot of a brutally murdered schoolgirl. In reality, her body would have been covered up, the crime scene cordened off, and not left open to a crowd of angry onlookers. That is just one of a flew dubious scenarios, but the film is packed with enough tension for us all to move on.
The girl has a tattoo that says "4 Real" It's a clue and one of many that unfold.
Prime suspect is Jason Buleigh, a sinister looking born again Christian (played by Ben Crompton) who looks set to walk. Unfortunately, for him, the two brothers are passionate about justice, no matter what methods, (reminiscent of The Sweeney with John Thaw at the helm). And both have family issues: extreme violent behaviour being one.
A confrontational scene on Hilbre Island is chilling and you can almost feel the wind in your hair as the bleak encounter goes beyond the point of no return. It's 3D with a difference.
But is the blood of the title thicker than water?
Stephen Graham's Chrissie is a highlight of this fast paced thriller with plenty of psychological turns.
The acting is first class and when you see that superb producer Nicola Schindler is part of the creative team it bodes well.
There are some graphic close-ups, but you can look away at those moments and soak in the well-written screenplay by Brian Gallagher or observe Nick Murphy's direction of this BBC co-production, which shows Wallasey Town Hall and Leasowe Lighthouse in a new light.
Look out for Liverpool's Natasha Little - another Merseyside star of the future - and let's hope the newly set up Wirral Film Office takes up where this film leaves off. Liverpool has a lot of the attention in the past. Maybe the tide is turning.
Opens Friday at selected cinemas including The Light, Kings Parade, New Brighton.
You're being a tad kind Granty. Stephen Graham usually sounds like a plate of steaming Scouse but here is more like Brian Sewell on benzodiazopines. Very strange. Scenes are disjointed and script lacks insight. Nice views though!


text:
Q: Parametrization of the solutions of $a^2+b^2=1$ in the $p$-adic integers I was trying to solve the equality $a^2+b^2=1$ in $\mathbb{Z}_p$, the $p$-adic integers.
If $p =1$ mod $4$ then $\mathbb{Z}_p$ admits an element $i$ such that $i^2=-1$, using this we can define two functions
\begin{equation} cos(x)=\frac{e^{ix}+e^{-ix}}{2} \quad sin(x)=\frac{e^{ix}-e^{-ix}}{2i} \end{equation}
exactly as in the case of complex numbers and show they include all the solutions to the above equation.
If $p=2$ then by considering the equality modulo $4$ it is easy to see that one between $a,b$ must be a unit and the other must be divisible by $4$, say $b=4x$. Then we have $a=\pm\sqrt{1-16x^2}$ and we can express this using the power series
\begin{equation} (1+z)^{\alpha}=\sum_{k=0}^{\infty}\binom{\alpha}{k}\frac{z^k}{k!} \end{equation}
by taking $\alpha=1/2$, since $z=-16x^2$ we have convergence.
I have a problem with the case $p=3$ mod $4$. In this situation we do not have a square root of $-1$ so we cannot define $cos$ and $sin$ as for the first case. Moreover we do not have necessarily that one between $a$ and $b$ must be divisible by $p$, therefore the argument I proposed for $p=2$ cannot be repeated.
Do you have an idea of how to present all the possible solutions of this equation in the remaining case?

A: For any ring $R$, let $$V(R):= \{(a,b) \in R \times R: a^2+b^2=1\}.$$
Short answer: For $p \equiv 1 \; \mathrm{mod} \; 4$, there is a one-to-one parametrisation of $V(\mathbb Z_p)$ by $\mathbb Z_p^\times$; otherwise, there is a one-to-one parametrisation of $V(\mathbb Z_p)$ by $P_1(\mathbb Q_p)$ (projective line), which is basically given by Lubin in the other answer.
Long answer: A wonderful thing about the variety you're interested in is that it has the structure of an algebraic group. Namely, you can parametrise  $V(K)$ for any field $K$ of characteristic $\neq 2$ -- e.g. $K=\mathbb Q_p$ -- as the group $SO_2(K)$, which is a non-split torus (the unit circle in the quadratic extension $K(\sqrt{-1})$) in case your ground field contains no square root of $-1$, and which is a split torus, i.e. isomorphic to the multiplicative group $K^\times$ otherwise. Cf. https://math.stackexchange.com/a/3049844/96384 and from there let's go to the integral solutions.
First, in the case that we have a primitive fourth root of unity $i \in K$, it's shown in the above link that the map
$$\phi: x \mapsto \left(\frac12(x +x^{-1}), \frac{i}{2}(x-x^{-1})\right)$$
is a bijective map from $K^\ast$ to $V(K)$ (the inverse is $(a,b) \mapsto a+ib$)). Now for $K$ a finite extension of some $\mathbb Q_p$, with ring of integers $\mathcal{O}_K$, then if $p \neq 2$ it's easy to see (from the ultrametric maximum principle) that $\phi(x)$ is in $V(\mathcal{O}_K)$ if and only if $x \in \mathcal{O}_K^\ast$, meaning that the restriction of $\phi$ to $\mathcal{O}_K^\ast$ is a

one-to-one parametrisation $\phi: \mathcal{O}_K^\ast \rightarrow V(\mathcal{O}_K)$

(and further transports the group structure to the one of $SO_2$).
Your attempt with cosine and sine is close to that, it is basically $\phi \circ exp (i\cdot \;)$. However, as alluded to in the comments, the exponential map has smaller domain and range; for $K=\mathbb Q_p$, this map only converges on $p\mathbb Z_p$, and only parametrises the part of $V(\mathbb Z_p)$ which via $\phi$ corresponds to the principal units $(1+p\mathbb Z_p, \cdot) \subsetneq \mathbb Z_p^\ast$.

In the case that $K$ does not contain a square root of $-1$, on first sight the above link is not very helpful because it says that now the group is parametrised by the norm-$1$-group of the quadratic extension $L:=K(i)$,
$$\psi: N:= \{z\in L: N_{L\vert K}(z)=1\} \stackrel{\simeq}\rightarrow V(K)$$
$$z =a+ib \;(a,b \in K) \mapsto (a,b)$$
(And again, this is even a group isomorphism if we give the right hand side the group structure of $SO_2$). But if one unravels that, one basically falls back to the underlying set being all $(a,b) \in K\times K$ with $a^2+b^2=1$.
However, from here one can give a purely algebraic explanation for Lubin's parametrisation :
It is well-known (basic case of Hilbert 90, although certainly there's some elementary argument in this case) that $N = \{\dfrac{z}{\sigma(z)}: z \in L^\ast \}$, where $\sigma$ is the conjugation map $i \mapsto -i$. That means we have a surjective (!) map:
$$p: L^\ast \twoheadrightarrow N$$
$$ z \mapsto \dfrac{z}{\sigma(z)}$$
Writing $z=x+iy$ with $x,y \in K$ we have
$$\dfrac{z}{\sigma(z)} =\dfrac{x+iy}{x-iy} = \dfrac{x^2+2ixy-y^2}{x^2+y^2} = \dfrac{x^2-y^2}{x^2+y^2}+\dfrac{2xy}{x^2+y^2} \cdot i$$
and in case $x \neq 0$ this is
$$x\cdot \dfrac{1-t^2}{1+t^2}+x \cdot\dfrac{2t}{1+t^2} \cdot i$$
with $t:=y/x \in K$, i.e. $x+iy = x \cdot(1+it)$. But since $p$ is a group homomorphism with obvious kernel $K^\ast$, the composed map induces a bijection
$$\psi \circ p : \{1+it: t\in K\} \rightarrow V(K) \setminus \{(-1,0)\}$$
(not a group homomorphism anymore, because the first set is just a non-multiplicatively closed set of representatives of $(L^\ast/K^\ast) \setminus \{iK^\ast\}$) which of course we can further precompose with $t \mapsto 1+it$, which gives Lubin's parametrisation (just with a sign switch and $a$ and $b$ switched maybe). More conceptually, after identifying $(L^\ast/K^\ast)$ with the projective space $P_1(K)$ via $x+iy \mapsto [x:y]$ (homogeneous coordinates), this is a bijective map
$$P_1(K) \stackrel{\simeq}\rightarrow V(K)$$
$$[1:t] \mapsto (\dfrac{1-t^2}{1+t^2}, \dfrac{2t}{1+t^2}),$$
$$[0:y] \mapsto (-1,0).$$
Now what about $V(\mathcal{O}_K)$ in this case? Well something nice happens: Check for yourself (I did it with $p$-adic valuations) that whatever $t \in K$ is, both $\dfrac{1-t^2}{1+t^2}$ and $\dfrac{2t}{1+t^2}$ are actually in $\mathcal{O}_K$. That means that in this case, somewhat surprisingly, $V(K) = V(\mathcal{O}_K)$ and we have that

one-to-one parametrisation $K \rightarrow V(\mathcal{O}_K) \setminus \{\text{point}\}$,

$$t \mapsto (\dfrac{1-t^2}{1+t^2} ,\dfrac{2t}{1+t^2}).$$

A: Here's an answer that is probably in perfect equivalence to that of Torsten, but rather more elementary.
Why not use the well-known formulas,
\begin{align}
a&=\frac{2t}{t^2+1}\\
b&=\frac{t^2-1}{t^2+1}\,
\end{align}
You can derive them for yourself by taking the line through $(0,-1)$ with slope $t$ and finding its other intersection with the circle. You catch all points on the circle but $(0,1)$.
I think it's shameful that this parametrization is not routinely shown to students in first-year Calculus.
EDIT, Addendum:
You've asked in a comment how to know that the parametrization catches all points on the curve $X^2+Y^2=1$. Take one such, say $P=$ your point $(a,b)$. Draw the line ($Y=\mu X+\beta$) between $(0,-1)$ and $P$. It's immediate that $\beta=-1$ and $\mu=\frac{b+1}a$.
But now you're done, subject to the verification that when you substitute $\mu$ for $t $ in the formulas, you do in fact get $a$ and $b$. I won't insult you by doing the algebra for you—just don't forget that $a^2+b^2=1$ at the crucial point in the computation.


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text:
Ru-El Sailor Exonerated!
Finally home, finally free.
Author: Nikita Srivastava ('19)
Right after Ru-El Sailor's release. Andrew Radin ('18), Ru-El Sailor, and Jennifer Bergeron.
Today Ru-El Sailor is a free man, after spending 15 years in prison for a murder he did not commit. Over the years, Sailor continuously maintained and fought to prove his innocence. Then, finally, on March 28, 2018, the Cuyahoga County Court vacated his sentence.
How Sailor got Wrongly Convicted
In November 2002, Sailor was hanging out with his friends at a bar on the East Side of Cleveland. Across town, Nicole and Cordell Hubbard got into a dispute with Omar Clark . The matter got out of hands – threats, guns, and then shots rang out, leaving Clark dead. Cuyahoga County prosecutors roped in Sailor who Cordell Hubbard's best friend at the time, wrongly believing that Sailor was the second man in this fatal shooting. Sailor testified that he was not the shooter nor was involved in this violent outburst. However, after a trial that included shady eye witness testimony that could not place Sailor at the scene, a jury still convicted Sailor. The court sentenced him to 28 years to life with the possibility of parole.
Making the Case for Sailor's Innocence
Sailor maintained his innocence throughout his conviction and time in prison. Eventually, his pleas made their way to the Conviction Integrity Unit in Cleveland and the Ohio Innocence Project in Cincinnati.
OIP Fellows: Caitlyn Idoine ('19), Jesse Knowlden ('19), Andrew Radin ('18), and Ruby Wilz ('18).
Andrew Radin ('18) and Ruby Wilz ('18) were the first OIP student-fellows to work on Sailor's case. Radin knew three pages into Sailor's case file that Sailor was innocent. "The actual shooter admitted to the crime, and Ru-El was not even at the scene of the crime. Also, a friend of Ru-El's came forward, stating Ru-El was not there," Radin told me. Furthermore, his co-defendant testified at sentencing and had written several affidavits saying he wasn't there. After Sailor met with OIP Staff Attorney Jennifer Bergeron, OIP agreed to represent him in the fall of 2016 with attorney Kimberly Corral who worked on his case for 5 years.
Ru-El and Amy Spence right after his release.
Sailor's wonderfully fierce fiancé, Amy Spence, brought a lot of attention to Sailor's case. She talked about Sailor's case at town hall meetings, bringing light to Sailor's wrongful conviction.. Before his exoneration, fellows Radin and Wilz had all the information they needed to prove his innocence. All they had to do was wait for the court to make a decision. Throughout the year, Sailor's belief in their abilities kept Radin and Wilz going. After their fellowship ended, the case then became OIP fellows Caitlyn Idoine ('19) and Jesse Knowlden ('19). Eventually the fellows hard-work paid off – Sailor was granted his freedom!
On the day of Sailor's exoneration, the 17th floor of the Justice Center in Cleveland was packed with Sailor's loved ones, supporters, and OIP fellows. Because the tiny courtroom could barely fit Sailor's legal team and family, a large group stood in the adjacent hallway and watched the live stream footage of the hearing from Cleveland Channel 8's website. The night before Sailor's exoneration, Bergeron, Radin, and Wilz spent time with the Sailor family. They knew how important it was for the Sailor family to see Sailor's sentence vacated. Although it was frustrating for them not to be in the courtroom, Radin and Wilz both knew the importance of the family presence in the courtroom. Both Radin and Wilz told me Sailor's release was the most important thing that day! They were happy to see him come out! Sailor's family arrived in a party bus all excited to take Sailor home.
Exonerees: King, Wheatt, Mcclendon, Glover, and Gillipse.
"The judge's order means that Ru-El is a free man," Radin told me. Moreover, Sailor is walking out to a different world.
Radin, Sailor, and Wilz.
"Adjusting to life will be Ru-El's biggest challenge. A lot has changed in 15 years. But, as evidenced by his support on March 28, he has a family that has been hoping and praying for this day on his side. In addition, he now joins the wonderful group of OIP exonerees who will take them in as their brother. He is so lucky!" Wilz told me. Sailor may face some financial difficulties or find it difficult to use his new iPhone, but he's home.
He is free.
Click here to read more about Ru-El Sailor's Exoneration.
Visit the Ohio Innocence Project's Facebook page to see Sailor's exoneration.
Click here to support Ru-El Sailor's homecoming.
Nikita Srivastava is a 2L at the University of Cincinnati College of Law. Currently, she is a fellow for the Ohio Innocence Project and Vice President of UC Law's Trial Team. Also, she is Secretary of UCLW and Vice President of Criminal Law Society.
Author Nikita_SrivastavaPosted on 2 Apr 2018 5 Jan 2021 Categories Civil Rights, criminal justice, lawyers, legal profession, Ohio Innocence Project, Social Justice, student leaders, Wrongful Convictions
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The Washington Military Department is Connecting to FirstNet
WASHINGTON, September 21, 2021
Washington Military Department Joins FirstNet® – America's Public Safety Network
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The Washington Military Department is connecting its disaster response and emergency management teams to FirstNet – the only nationwide, high-speed broadband communications platform dedicated to and purpose-built for America's first responders and the extended public safety community. With this tech advancement, the state's first responders and emergency management teams will be equipped with new capabilities and reliable access to critical information while in the field, allowing them to better serve those who live, work and visit Washington.
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text:
The political tide turns
Brazilians want change. That could deprive Dilma Rousseff of a second term
The AmericasMay 24th 2014 edition
BRAZIL likes to think of itself as o país do futebol—the football country. So it is extraordinary that just three weeks before the World Cup kicks off in São Paulo, a recent poll found less than half of Brazilians saying they were happy to host it. True, this may change once the tournament gets going, especially if fears of transport chaos prove misplaced. Yet that poll result betrays not just public anger at the inflated cost of the tournament, but also wider grumpiness.
Although the cup will doubtless see a few protests—some have already begun—the public mood will be tested more clearly in the presidential election on October 5th. Many pundits, especially outsiders, take it for granted that Dilma Rousseff, the president, will win a second term. That is what the polls have long suggested.
Certainly, Ms Rousseff has some formidable electoral assets. She will be able to point to full employment, a continuing (albeit now small) rise in real wages and the exit from poverty of some 40m Brazilians in the close-to-12 years of rule by her centre-left Workers' Party (PT). Because she presides over a broad coalition, she will have more free television time than her opponents when campaigning starts in earnest in August, as well as more money. In Luiz Inácio Lula da Silva, her predecessor and ally, she has the country's most potent political campaigner. And incumbency is a big advantage in Latin America: only two sitting presidents have lost re-election bids in the region since 1985.
Even so, Ms Rousseff and the PT are on the defensive. The economy is lacklustre; inflation is stubbornly around 6%; consumer and business confidence are falling. Ms Rousseff is an exemplary democrat and a tough, determined fighter, but she lacks Lula's political touch. Her term in the Planalto Palace has produced neither great achievements nor a compelling narrative. She has had to see off a push by a minority within the PT to ditch her and draft Lula in as its candidate (something he has rightly rejected). This month the party aired a backward-looking television commercial warning voters not to allow "the ghosts of the past" to threaten progress—a reference to the centre-right Brazilian Social Democratic Party (PSDB) of Aécio Neves, her closest rival. The PSDB governed in tougher economic times in 1995-2002 (and laid the foundations of later prosperity).
The PT won the past two elections by championing those striving to leave poverty—helped by Bolsa Família, a trademark cash-transfer programme received by 14.1m families—while tagging the PSDB as heartless "neo-liberals". This electoral success led André Singer, a political scientist who worked in Lula's government, to claim that "Lulismo" had created a lasting political realignment in the country.
But the terms of the political debate in Brazil are changing—partly as a result of Lula's achievements. Health care has replaced employment as the main public concern. Brazilians want faster growth and, above all, better public services and less corruption. Such issues brought more than a million people on to the streets in spontaneous protests last June. "Lulismo" has no clear answer to their problems.
This year's race will also be harder to portray as a straight fight between PT and PSDB, thanks to the presence of Eduardo Campos. He is a centrist whose running mate, Marina Silva, an environmentalist, is a former PT leader who can make a plausible claim to be Lula's true heir.
Crucially, around 70% of those asked by pollsters say they want change, not continuity. The government's approval rating plunged from 65% to 30% after the June protests; after recovering somewhat, it is now back down to 35%, according to Datafolha, a polling firm. Mr Singer conceded in a recent column in Folha de São Paulo, a newspaper, that an "amber light is flashing" for Ms Rousseff.
If she is on course to lose the election, neither of her opponents is yet close to winning it. But both Mr Neves and Mr Campos are starting to rise in the polls (see chart). Both are former state governors who are not yet national political figures. They are counting on the campaign to make them so—perhaps helped by additional television time from last-minute defections from the governing coalition.
Ms Rousseff looks certain to be forced into a run-off. To win a second round, her opponent, whoever he is, must convince the mass of poorer Brazilians that he would not threaten Bolsa Família or the social gains Lula brought about. Having banished the "ghosts of the past" he must also persuade the voters—and especially the record number of potential abstainers—that on the issues that now matter most he is a more plausible standard-bearer of change than she is. That is not easy, but it is far from impossible.
This article appeared in the The Americas section of the print edition under the headline "The political tide turns"
More from The Americas
Driven away
Why Ford is leaving Brazil
Waiting for whiplash
Joe Biden will shift gears in Latin America
Green wave, blue breakwater
Argentina's legalisation of abortion will provoke a backlash


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Desambiguação


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