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Second Balkan War

The Second Balkan War was a conflict that broke out when Bulgaria, dissatisfied with its share of the spoils of the First Balkan War, attacked its former allies, Serbia and Greece, on 16 (O.S.) / 29 (N.S.) June 1913. Serbian and Greek armies repulsed the Bulgarian offensive and counterattacked, entering Bulgaria. With Bulgaria also having previously engaged in territorial disputes with Romania and the bulk of Bulgarian forces engaged in the south, the prospect of an easy victory incited Romanian intervention against Bulgaria. The Ottoman Empire also took advantage of the situation to regain some lost territories from the previous war. When Romanian troops approached the capital Sofia, Bulgaria asked for an armistice, resulting in the Treaty of Bucharest, in which Bulgaria had to cede portions of its First Balkan War gains to Serbia, Greece and Romania. In the Treaty of Constantinople, it lost Adrianople to the Ottomans.

The political developments and military preparations for the Second Balkan War attracted an estimated 200 to 300 war correspondents from around the world.

Background
During the First Balkan War, which began in October 1912, the Balkan League (Bulgaria, Serbia, Montenegro and Greece) succeeded in driving out the Ottoman Empire from its European provinces (Albania, Macedonia, Sandžak and Thrace), leaving the Ottomans with only East Thrace. The Treaty of London, signed on 30 May 1913, which ended the war, acknowledged the Balkan states' gains west of the Enos–Midia line, drawn from Midia (Kıyıköy) on the Black Sea coast to Enos (Enez) on the Aegean Sea coast, on an uti possidetis basis, and created an independent Albania.

However, the relations between the victorious Balkan allies quickly soured over the division of the spoils, specifically in Macedonia. During the prewar negotiations that resulted in the Balkan League's establishment, a secret agreement on 13 March 1912 was signed by Serbia and Bulgaria, which determined their future boundaries, effectively sharing northern Macedonia. In case of a postwar disagreement, the area to the north of the Kriva Palanka–Ohrid line (with both cities going to the Bulgarians) had been designated as a "disputed zone" under Russian arbitration, with the area to the south of this line assigned to Bulgaria. During the war, the Serbs succeeded in capturing an area far south of the agreed border, down to the Bitola–Gevgelija line (both in Serbian hands). At the same time, the Greeks advanced north, occupying Thessaloniki shortly before the Bulgarians arrived and establishing a common Greek border with Serbia.

When Bulgarian delegates in London bluntly warned the Serbs that they must not expect Bulgarian support on their Adriatic claims, the Serbs angrily replied that that was a blatant withdrawal from the prewar agreement of mutual understanding according to the Kriva Palanka-Adriatic line of expansion. The Bulgarians insisted that the Vardar Macedonian part of the agreement remained active, and the Serbs were still obliged to surrender the area as agreed. The Serbs responded by accusing the Bulgarians of maximalism, pointing out that if they lost both northern Albania and Vardar Macedonia, their participation in the common war would have been virtually for nothing.

When Bulgaria called upon Serbia to honour the prewar agreement over northern Macedonia, the Serbs, displeased at the Great Powers' requiring them to give up their gains in north Albania, adamantly refused to evacuate any more territory. The developments ended the Serbo-Bulgarian alliance and made a future war between the two countries inevitable. Soon, minor clashes broke out along the borders of the occupation zones with the Bulgarians against the Serbs and the Greeks. Responding to the perceived Bulgarian threat, Serbia started negotiations with Greece, which also had reasons to be concerned about Bulgarian intentions.

On 19 May/1 June 1913, two days after the signing of the Treaty of London and just 28 days before the Bulgarian attack, Greece and Serbia signed a secret defensive alliance, confirming the current demarcation line between the two occupation zones as their mutual border and concluding an alliance in case of an attack from Bulgaria or from Austria-Hungary. With this agreement, Serbia succeeded in making Greece a part of its dispute over northern Macedonia, since Greece had guaranteed Serbia's current (and disputed) occupation zone in Macedonia. In an attempt to halt the Serbo-Greek rapprochement, Bulgarian Prime Minister Geshov signed a protocol with Greece on 21 May, agreeing on a permanent boundary between their respective forces, effectively accepting Greek control over southern Macedonia. However, his later dismissal ended the diplomatic targeting of Serbia.

Another point of friction arose: Bulgaria's refusal to cede the fortress of Silistra to Romania. When Romania demanded its cession after the First Balkan War, Bulgaria's foreign minister offered instead some minor border changes, which excluded Silistra, and assurances for the rights of the Kutzovlachs in Macedonia. Romania threatened to occupy Bulgarian territory by force, but a Russian proposal for arbitration prevented hostilities. In the resulting Protocol of St. Petersburg of 9 May 1913, Bulgaria agreed to give up Silistra. The resulting agreement was a compromise between the Romanian demands for the city, two triangles at the Bulgaria–Romania border and Balchik and the land between it and Romania and the Bulgarian refusal to accept any cession of its territory. However, the fact that Russia failed to protect the territorial integrity of Bulgaria made the Bulgarians uncertain of the reliability of the expected Russian arbitration of the dispute with Serbia. The Bulgarian behaviour also had a long-term impact on Russo-Bulgarian relations. The uncompromising Bulgarian position to review the prewar agreement with Serbia during a second Russian initiative for arbitration finally led Russia to cancel its alliance with Bulgaria. Both acts made conflict with Romania and Serbia inevitable.

Bulgarian war plans
In 1912, Bulgaria's national aspirations, as expressed by Tsar Ferdinand and the military leadership around him, exceeded the provisions of the 1878 Treaty of San Stefano, considered even then as maximalistic, since it included both Eastern and Western Thrace and all Macedonia with Thessaloniki, Edirne and Constantinople. Early evidence of the lack of realistic thinking in Bulgarian leadership was that although Russia had sent clear warnings expressed for the first time on 5 November 1912 (well before the First Battle of Çatalca) that if the Bulgarian army occupied Constantinople they would attack it, they continued their attempts to take the city.

Although the Bulgarian army succeeded in capturing Edirne, Tsar Ferdinand's ambition in crowning himself Emperor in Constantinople also proved unrealistic when the Bulgarian army failed to capture the city in the First Battle of Çatalca. Even worse, the concentration on capturing Thrace and Constantinople ultimately caused the loss of most of Macedonia, including Thessaloniki, and that could not be accepted, leading the Bulgarian military leadership around Tsar Ferdinand to decide upon a war against its former allies. However, with the Ottomans unwilling to accept the loss of Thrace in the east, and an enraged Romania (in the north), the decision to open war against both Greece (to the south) and Serbia (to the west) was a rather adventurous one, since in May the Ottoman Empire had urgently requested a German mission to reorganize the Ottoman army. By mid-June, Bulgaria became aware of the agreement between Serbia and Greece in case of a Bulgarian attack. On 27 June, Montenegro announced that it would side with Serbia in the event of a Serbian-Bulgarian war. On 5 February, Romania settled her differences over Transylvania with Austria-Hungary, signing a military alliance, and on 28 June, officially warned Bulgaria that it would not remain neutral in a new Balkan war.

As skirmishing continued in Macedonia, mainly between Serbian and Bulgarian troops, Tsar Nicholas II of Russia tried to stop the upcoming conflict since Russia did not wish to lose either of its Slavic allies in the Balkans. On 8 June, he sent an identical personal message to the Kings of Bulgaria and Serbia, offering to act as arbitrator according to the provisions of the 1912 Serbo-Bulgarian treaty. Serbia was asking for a revision of the original treaty since it had already lost north Albania due to the Great Powers' decision to establish the state of Albania, which had been recognized as Serbian territory under the prewar Serbo-Bulgarian treaty, in exchange for Bulgarian expansion in northern Macedonia. The Bulgarian reply to the Russian invitation contained so many conditions that it amounted to an ultimatum, leading Russian diplomats to realize that the Bulgarians had already decided to go to war with Serbia. That caused Russia to cancel the arbitration initiative and angrily repudiate its 1902 treaty of alliance with Bulgaria. Bulgaria was shattering the Balkan League, Russia's best defence against Austrian–Hungarian expansionism, a structure that had cost Russia much blood, money and diplomatic capital during the last 35 years. Russia's Foreign Minister Sazonov's exact words to Bulgaria's new Prime Minister Stoyan Danev were "Do not expect anything from us, and forget the existence of any of our agreements from 1902 until the present." Tsar Nicholas II of Russia was already angry with Bulgaria because the latter refused to honour its recently signed agreement with Romania over Silistra, which resulted from Russian arbitration. Then Serbia and Greece proposed that each of the three countries reduce its army by one-fourth as a first step to facilitate a peaceful solution, but Bulgaria rejected it.

Bulgaria was already on the track to war since a new cabinet had been formed in Bulgaria where the pacifist Geshov was replaced by the hardliner and head of a Russophile party, Danev, as premier. There is some evidence that to overcome Tsar Ferdinand's reservations over a new war against Serbia and Greece, certain personalities in Sofia threatened to overthrow him. In any case, on 16 June, the Bulgarian high command, under the direct control of Tsar Ferdinand and without notifying the government, ordered Bulgarian troops to start a surprise attack simultaneously against both the Serbian and Greek positions without declaring war and to dismiss any orders contradicting the attack order. The next day the government pressured the General Staff to order the army to cease hostilities, which caused confusion and loss of initiative and failed to remedy the state of undeclared war. In response to the government pressure, Tsar Ferdinand dismissed General Savov and replaced him with General Dimitriev as commander-in-chief.

Bulgaria intended to defeat the Serbs and Greeks and to occupy areas as large as possible before the Great Powers interfered to end the hostilities. To provide the necessary superiority in arms, the entire Bulgarian army was committed to these operations. No provisions were made in case of an (officially declared) Romanian intervention or an Ottoman counterattack, strangely assuming that Russia would assure that no attack would come from those directions, even though on 9 June Russia had angrily repudiated its Bulgarian alliance and shifted its diplomacy towards Romania (Russia already had named Romania's King Carol an honorary Russian field marshal, as a clear warning in changing its policy towards Sofia in December 1912). The plan was for a concentrated attack against the Serbian army across the Vardar plain to neutralize it and to capture northern Macedonia, together with a less concentrated one against the Greek military near Thessaloniki, which had approximately half the size of the Serbian army, to capture the city and south Macedonia. The Bulgarian high command was not sure whether their forces were enough to defeat the Greek army but believed them to be enough for defending the south front as a worst-case scenario until the arrival of reinforcements after defeating the Serbs to the north.

Opposing forces
According to the Military Law of 1903, the armed forces of Bulgaria were divided into two categories: the Active Army and the National Militia. The core of the Armed forces consisted of nine infantry and one cavalry division. The Bulgarian army had a unique organization among those of Europe since each infantry division had three brigades of two regiments, composed of four battalions of six heavy companies of 250 men each, plus an independent battalion, two large artillery regiments and one cavalry regiment, giving a total of 25 very heavy infantry battalions and 16 cavalry companies per division, which was more than the equivalent of two nine-battalion divisions, the standard divisional structure in most armies, as was also the case with the Greek and Serbian militaries in 1913. Consequently, although the Bulgarian army had 599,878 men mobilized at the beginning of the First Balkan War, there were only nine organizational divisions, giving a divisional strength closer to an Army Corps than to a Division. Tactical necessities during and after the First Balkan War modified this original structure: a new 10th division was formed using two brigades from the 1st and 6th divisions, and an additional three independent brigades were formed from recruits. Nevertheless, the heavy structure generally remained. By contrast, the Greek Army of Macedonia also had nine divisions, but the total number of men under arms was only 118,000. Another decisive factor affecting the real strength of the divisions between the opposing armies was the distribution of artillery. The nine-division-strong Greek Army had 176 guns, and the ten-division-strong Serbian army had 230. The Bulgarians had 1,116, a ratio of 6:1 against the Greeks and 5:1 against the Serbian army.

There is a dispute over the strength of the Bulgarian army during the Second Balkan War. At the outbreak of the First Balkan War, Bulgaria mobilized a total of 599,878 men (366,209 in the Active Army; 53,927 in the supplementing units; 53,983 in the National Militia; 94,526 from the 1912 and 1913 levies; 14,204 volunteers; 14,424 in the border guards). The non-recoverable casualties during the First Balkan War were 33,000 men (14,000 killed and 19,000 died of disease). To replace these casualties, Bulgaria conscripted 60,000 men between the two wars, mainly from the newly occupied areas, using 21,000 of them to form the Seres, Drama and Odrin (Edirne) independent brigades. It is known that there were no demobilized men. According to the Bulgarian command, the army had 7,693 officers and 492,528 soldiers in its ranks on 16 June (including the three brigades mentioned above). This gives a difference of 99,657 men in strength between the two wars. In comparison, subtracting the actual number of casualties, including wounded and adding the newly conscripted men produces no less than 576,878 men. The army was experiencing shortages of war materials and had only 378,998 rifles at its disposal.

The 1st and 3rd armies (under generals Vasil Kutinchev and Radko Dimitriev respectively) were deployed along the old Serbian-Bulgarian borders, with the 5th Army under general Stefan Toshev around Kyustendil, and the 4th Army under General Stiliyan Kovachev in the Kočani–Radoviš area. The 2nd Army under general Nikola Ivanov was detailed against the Greek army.

The army of the Kingdom of Serbia accounted for 348,000 men (out of which 252,000 were combatants) divided into three armies with ten divisions. Its main force was deployed on the Macedonian front along the Vardar River and near Skopje. Its nominal commander-in-chief was King Peter I, with Radomir Putnik as his chief of staff and effective field commander.

By early June, the army of the Kingdom of Greece had some 142,000 armed men with nine infantry divisions and one cavalry brigade. The bulk of the army with eight divisions and a cavalry brigade (117,861 men) was gathered in Macedonia, positioned in an arc covering Thessaloniki to the north and northeast of the city, while one division and independent units (24,416 men) were left in Epirus. With the eruption of hostilities, the 8th Infantry Division (stationed in Epirus) was transferred to the front, and with the arrival of recruits, the army's strength in the Macedonian theatre increased eventually to some 145,000 men with 176 guns. King Constantine I assumed command of the Greek forces, with Lt. General Viktor Dousmanis as his chief of staff.

The Kingdom of Montenegro sent one division of 12,000 men under General Janko Vukotić to the Macedonian front.

The Kingdom of Romania had the largest army in the Balkans, although it had not seen action since the Romanian War of Independence against the Ottomans in 1878. Its peacetime strength was 6,149 officers and 94,170 men, and it was well equipped by Balkan standards, possessing 126 field batteries, fifteen howitzer and three mountain batteries made primarily by Krupp. Upon mobilizing, the Romanian army mustered 417,720 men allocated to five army corps. Some 80,000 were assembled to occupy the Southern Dobruja, while an army of 250,000 was assembled to carry the main offensive into Bulgaria.

Start of the war

The primary Bulgarian attack was planned against the Serbs with their 1st, 3rd, 4th and 5th Armies, while the 2nd army was tasked with an attack toward Greek positions around Thessaloniki. However, in the crucial opening days of the war, only the 4th Army and 2nd Army were ordered to advance. This allowed the Serbs to concentrate their forces against the attacking Bulgarians and hold their advance. The Bulgarians were outnumbered on the Greek front, and the low-level fighting soon turned into a Greek attack all along the line on 19 June. The Bulgarian forces were forced to withdraw from their positions north of Thessaloniki (except the isolated battalion stationed in the city itself, which was quickly overrun) to defensive positions between Kilkis and Struma river. The plan to rapidly destroy the Serbian army in central Macedonia by concentrated attack turned out to be unrealistic, with the Bulgarian army starting to retreat even before Romanian intervention, and the Greek advance necessitated the disengagement of forces to defend Sofia.

Bulgarian offensive against Greece
The Bulgarian 2nd army in southern Macedonia, commanded by General Ivanov, held a line from Dojran Lake southeast to Kilkis, Lachanas, Serres and then across the Pangaion Hills to the Aegean Sea. The army had been in place since May and was considered a veteran force, having fought at the siege of Edirne in the First Balkan War. Though General Ivanov, possibly to avoid any responsibility for his crushing defeat, claimed after the war that his army consisted of only 36,000 men and that many of his units were weakened, a detailed analysis concerning his units contradicted him. Ivanov's 2nd army consisted of the 3rd division minus one brigade with four regiments of four battalions (a total of 16 battalions plus the divisional artillery), the I/X brigade with the 16th and 25th regiments (total of eight battalions plus artillery), the Drama Brigade with the 69th, 75th and 7th regiments (total of 12 battalions), the Seres Brigade with 67th and 68th regiments (total of 8 battalions), the 11th division with the 55th, 56th and 57th regiments (total of 12 battalions plus the divisional artillery), the 5th Border Battalion, the 10th Independent Battalion and the 10th Cavalry Regiment of seven mounted and seven infantry companies. In total, Ivanov's force comprised 232 companies in 58 infantry battalions, a cavalry regiment (14 companies) with 175 artillery guns, numbering between 80,000 (official Bulgarian source) and 108,000 (official Greek source according to the official Bulgarian history of the war before 1932). All modern historians agree that Ivanov underestimated the number of his soldiers, but the Greek army still had a numerical superiority. The Greek Headquarters also estimated the numbers of their opponents from 80,000 to 105,000 men. A large part of Ivanov's forces, and especially the Drama Brigade and the Seres Brigade, were composed of completely untrained local recruits.

The Greek army, commanded by King Constantine I, had eight divisions and a cavalry brigade (117,861 men) with 176 artillery guns in a line extending from the Gulf of Orphanos to the Gevgelija area. Since the Greek headquarters did not know where the Bulgarian attack would occur, the Bulgarians would have temporary local superiority in the location chosen for the attack.

On 26 June, the Bulgarian army received orders to destroy the opposing Greek forces and to advance towards Thessaloniki. The Greeks stopped them, and by 29 June, an order for a general counterattack was issued. At Kilkis, the Bulgarians had constructed strong defences, including captured Ottoman guns which dominated the plain below. The Greek 4th, 2nd, and 5th divisions attacked across the plain in rushes supported by artillery. The Greek Army suffered heavy casualties but carried the trenches by the next day. On the Bulgarian left, the Greek 7th Division captured Serres and the 1st and 6th divisions captured Lachanas. The defeat of the 2nd army by the Greeks was the most serious military disaster suffered by the Bulgarians in the Second Balkan War. Bulgarian sources give 6,971 casualties, over 6,000 prisoners, and over 130 artillery pieces captured by the Greeks, who suffered 8,700 casualties. On 28 June, the retreating Bulgarian army and irregulars burned down the major city of Serres (a predominantly Greek town surrounded by both Bulgarian – to the north and west – and Greek – to the east and south – villages ), and the towns of Nigrita, Doxato and Demir Hisar, ostensibly as a retaliation for the burning of the Bulgarian town of Kilkis by the Greeks, which had taken place after the named battle, as well as the destruction of many Bulgarian villages in the region. On the Bulgarian right, Greek Evzones captured Gevgelija and the heights of Matsikovo. Consequentially, the Bulgarian line of retreat through Dojran was threatened, and Ivanov's army began a desperate retreat, threatening at times to become a rout. Reinforcements from the 14th division came too late, joining the retreat towards Strumica and the Bulgarian border. The Greeks captured Dojran on 5 July but were unable to cut off the Bulgarian retreat through Struma Pass. On 11 July, the Greeks came in contact with the Serbs and then pushed up the Struma River. Meanwhile, the Greek forces, with the support of their navy, landed in Kavala and then penetrated inland to western Thrace. On 19 July, the Greeks captured Nevrokop, and on 25 July, in another amphibious operation, entered Dedeagac (today Alexandroupoli), thus cutting off the Bulgarians completely from the Aegean sea.

Serbian front

The 4th Bulgarian army held the most important position in the attempted conquest of Serbian Macedonia. The fighting began on 29–30 June 1913, between the 4th Bulgarian army and the 1st and 3rd Serbian armies, first along the Zletovska and then after a Bulgarian retreat, along the Bregalnica. Internal confusion led to heavy Bulgarian losses in 1–3 July. The Serbs captured the whole 7th Division of the 4th Bulgarian Army, without any fight. By 8 July, the Bulgarian army had been severely defeated.

On the north, the Bulgarians started to advance towards the Serbian border town of Pirot and forced Serbian Command to send reinforcements to the 2nd army defending Pirot and Niš. This enabled Bulgarians to stop the Serbian offensive in Macedonia at Kalimanci on 18 July.

On 13 July 1913, General Mihail Savov assumed control of the 4th and 5th Bulgarian armies. The Bulgarians dug into strong positions around the village of Kalimantsi, at the Bregalnica river in the northeastern Macedonia region. On 18 July, the Serbian 3rd army attacked, closing in on Bulgarian positions. The Bulgarians held firm, and the artillery successfully broke up the Serb attacks. If the Serbs had broken through the Bulgarian defences, they might have doomed the 2nd Bulgarian Army and driven the Bulgarians entirely out of Macedonia. The defensive victory, along with the successes to the north of the 1st and 3rd armies, protected western Bulgaria from a Serbian invasion. Although this boosted the Bulgarians, the situation was critical in the south, with the Greek Army.

Greek offensive
The Serbian front had become static. Seeing that the Bulgarian Army in front of him had already been defeated, King Constantine ordered the Greek Army to march further into Bulgarian territory and take the capital city of Sofia. Constantine wanted a decisive victory despite objections by his Prime Minister, Eleftherios Venizelos, who realized that the Serbs, having won their territorial objectives, now adopted a passive stance and shifted the weight of carrying the rest of the war to the Greeks. In the pass of Kresna (Battle of Kresna Gorge), the Greeks were ambushed by the Bulgarian 2nd and 4th armies, which had newly arrived from the Serbian front and had taken defensive positions there. By 21 July, the Greek army was outnumbered by the now counterattacking Bulgarians, and the Bulgarian General Staff, attempting to encircle the Greeks in a Cannae-type battle, was applying pressure on their flanks. However, after bitter fighting, the Greek side managed to break through the Kresna pass and captured Simitli on 26 July, while at the night of 27–28 July, the Bulgarian forces were pushed north to Gorna Dzhumaya (Blagoevgrad), 76 km south of Sofia. Meanwhile, the Greek forces continued their march inland into western Thrace, on 26 July, they entered Xanthi and the next day Komotini. On 28 July, the Bulgarian army, under heavy pressure, was forced to abandon Gorna Dzhumaya.

The Greek army was exhausted and faced logistical difficulties, but resisted strenuously and launched local counterattacks. By 30 July, the Bulgarian army downscaled its attacks, having to repulse Greek counterattacks on both sides. On the eastern flank, the Greek army launched a counterattack towards Mehomia through the Predela pass. The offensive was stopped by the Bulgarians on the eastern side of the pass and fighting ground to a stalemate. On the western flank, an offensive was launched against Tsarevo Selo to reach the Serbian lines. This failed, and the Bulgarian army continued advancing, especially in the south. However, after three days of fighting at the sectors of Pehchevo and Mahomia, the Greek forces retained their positions.

Romanian intervention
Romania mobilized its army on 5 July 1913, intending to seize Southern Dobruja, and declared war on Bulgaria on 10 July. In a diplomatic circular that said, "Romania does not intend either to subjugate the polity nor defeat the army of Bulgaria," the Romanian government endeavoured to allay international concerns about its motives and about increased bloodshed. According to Richard Hall, "[t]he entrance of Romania into the conflict made the Bulgarian situation untenable [and t]he Romanian thrust across the Danube was the decisive military act of the Second Balkan War."

On the day of Romania's declaration, 80,000 men of the 5th Corps under General Ioan Culcer invaded Dobruja, occupying a front from Tutrakan to Balchik. The corps cavalry occupied the port city of Varna until it was clear that there would be no Bulgarian resistance. On the night of 14–15 July, the Danube Army under Prince Ferdinand crossed into Bulgaria at Oryahovo, Gigen and Nikopol. The initial occupation completed, Romanian forces were divided into two groups: one advanced westward, towards Ferdinand (now Montana), and the other advanced southwestward, towards Sofia, the Bulgarian capital, everywhere preceded by a wide fan of cavalry troops in reconnaissance.

On 18 July, Romania took Ferdinand, and on 20 July, they occupied Vratsa, 116 km north of Sofia. On 23 July, advanced cavalry forces had entered Vrazhdebna, a suburb only 7 mi from Sofia. The Romanians and Serbs linked up at Belogradchik on 25 July, isolating the important city of Vidin. The Bulgarian rear was entirely exposed, no resistance had been offered, the capital was open to the invader, and the country's northwestern corner was cut off and surrounded. During the invasion, the fledgling Romanian Air Corps performed photoreconnaissance and propaganda leaflet drops. Sofia became the first capital city in the world to be overflown by enemy aircraft.

Romania did not count any combat casualties during its brief war. Its forces were struck by an outbreak of cholera, which cut down 1,600 men.

Ottoman intervention
The lack of resistance to the Romanian invasion convinced the Ottomans to invade the territories just ceded to Bulgaria. The main object of the invasion was the recovery of Edirne (Adrianople), which was held by Major General Vulko Velchev with a mere 4,000 troops. Most Bulgarian forces occupying East Thrace had been withdrawn to face the Serbo-Greek attack earlier in the year. On 12 July, Ottoman troops garrisoning Çatalca and Gelibolu reached the Enos–Midia line and on 20 July 1913 crossed the line and invaded Bulgaria. The entire Ottoman invasion force contained between 200,000 and 250,000 men under the command of Ahmed Izzet Pasha. The 1st Army was stationed at the eastern (Midia) end of the line. From east to west it was followed by the 2nd Army, 3rd Army and 4th Army, which was stationed at Gelibolu.

In the face of the advancing Ottomans, the outnumbered Bulgarian forces retreated to the prewar border. Edirne was abandoned on 19 July, but, since the Ottomans did not occupy it immediately, the Bulgarians re-occupied it the next day (20 July). Since it was apparent that the Ottomans were not stopping, it was abandoned a second time on 21 July and occupied by the Ottomans on 23 July. Edirne had been conquered by Sultan Murad I in the 1360s and had served as the first European capital of the Empire before the capture of Constantinople in 1453. Minister of War Enver Pasha called himself the "Second Conqueror of Edirne," although the conquering forces had met no resistance on the way to Edirne.

The Ottoman armies did not stop at the old border but crossed into Bulgarian territory. A cavalry unit advanced on Yambol and captured it on 25 July. The Ottoman invasion, more than the Romanian, incited panic among the peasantry, many of whom fled to the mountains. Among the leadership, it was recognized as a complete reversal of fortune. In the words of historian Richard Hall, "[t]he battlefields of eastern Thrace, where so many Bulgarian soldiers had died to win the First Balkan War, were again under Ottoman control." Like the Romanians, the Ottomans suffered no combat casualties but lost 4,000 soldiers to cholera. Some 8000 Armenians fighting for the Ottomans were wounded. The sacrifice of these Armenians was praised greatly in Turkish papers.

To help Bulgaria repulse the rapid Ottoman advance in Thrace, Russia threatened to attack the Ottoman Empire through the Caucasus and send its Black Sea Fleet to Constantinople; this caused Britain to intervene.

According to the 1918 book Destruction of the Thracian Bulgarians in 1913, Ottoman forces perpetrated atrocities against the Bulgarians in Eastern Thrace during the invasion and aftermath.

Armistice
As the Romanian army closed in on Sofia, Bulgaria asked Russia to mediate. On 13 July, Prime Minister Stoyan Danev resigned in the face of Russian inactivity. On 17 July, the tsar appointed Vasil Radoslavov to head a pro-German and Russophobic government. On 20 July, via Saint Petersburg, the Serbian Prime Minister Nikola Pašić invited a Bulgarian delegation to treat with the allies directly at Niš in Serbia. The Serbs and Greeks, both now on the offensive, were in no rush to conclude a peace. On 22 July, Tsar Ferdinand sent a message to King Carol via the Italian ambassador in Bucharest. The Romanian armies halted before Sofia. Romania proposed that talks be moved to Bucharest, and the delegations took a train from Niš to Bucharest on 24 July.

When the delegations met in Bucharest on 30 July, the Serbs were led by Pašić, the Montenegrins by Vukotić, the Greeks by Venizelos, the Romanians by Titu Maiorescu and the Bulgarians by Finance Minister Dimitur Tonchev. They agreed to a five-day armistice to come into effect on 31 July. Romania refused to allow the Ottomans to participate, forcing Bulgaria to negotiate with them separately.

Treaty of Bucharest

Bulgaria had agreed to cede Southern Dobruja to Romania as early as 19 July. At the peace talks in Bucharest, the Romanians, having obtained their primary objective, were a voice for moderation. The Bulgarians hoped to keep the Vardar River as the boundary between their share of Macedonia and Serbia's. The latter preferred to retain all of Macedonia as far as the Struma. Austro-Hungarian and Russian pressure forced Serbia to be satisfied with most of northern Macedonia, conceding only the town of Štip to the Bulgarians, in Pašić's words, "in honour of General Fichev," who had brought Bulgarian arms to the door of Constantinople in the first war. Ivan Fichev was chief of the Bulgarian general staff and a member of the delegation in Bucharest at the time. When Fichev explained why Bulgaria deserved Kavala, a port on the Aegean occupied by the Greeks, Venizelos is said to have responded, "General, we are not responsible. Before [29] June, we were afraid of you and offered you Serres and Drama and Kavala, but now when we see you, we assume the role of victors and will take care of our interests only." Although Austria-Hungary and Russia supported Bulgaria, the influential alliance of Germany—whose Kaiser Wilhelm II was brother-in-law to the Greek king—and France secured Kavala for Greece. Bulgaria retained the underdeveloped port of Dedeagac (Alexandroupoli).

The last day of negotiations was 8 August. On 10 August, Bulgaria, Greece, Montenegro, Romania, and Serbia signed the Treaty of Bucharest and divided Macedonia in three: Vardar Macedonia went to Serbia; the smallest part, Pirin Macedonia, to Bulgaria; and the coastal and largest part, Aegean Macedonia, to Greece. Bulgaria thus enlarged its territory by 16 percent compared to what it was before the First Balkan War, increasing its population from 4.3 to 4.7 million people. Romania enlarged its territory by 5 percent and Montenegro by 62 percent. Greece increased her population from 2.7 to 4.4 million and her territory by 68 percent. Serbia almost doubled her territory, enlarging her population from 2.9 to 4.5 million.

The Montenegrins at Bucharest were primarily interested in obtaining a favourable concession from Serbia in the former Sanjak of Novi Pazar. They did it, later confirming it in a treaty signed at Belgrade on 7 November.

Ottoman treaties
In August, Ottoman forces established a provisional government of Western Thrace at Komotini to pressure Bulgaria to make peace. Bulgaria sent a three-member delegation—General Mihail Savov and the diplomats Andrei Toshev and Grigor Nachovich—to Constantinople to negotiate peace on 6 September. The Ottoman delegation was led by Foreign Minister Mehmed Talat Bey, assisted by future Naval Minister Çürüksulu Mahmud Pasha and Halil Bey. Although Russia tried to intervene throughout August to prevent Edirne from becoming Turkish again, Toshev told the Ottomans at Constantinople that "[t]he Russians consider Constantinople their natural inheritance. Their main concern is that when Constantinople falls into their hands it shall have the largest possible hinterland. If Adrianople is in the possession of the Turks, they shall get it too."

Resigned to losing Edirne, the Bulgarians played for Kırk Kilise (Lozengrad in Bulgarian). Both sides made competing declarations: Savov that "Bulgaria, who defeated the Turks on all fronts, cannot end this glorious campaign with the signing of an agreement which retains none of the battlefields on which so much Bulgarian blood has been shed," and Mahmud Pasha that "[w]hat we have taken is ours." In the end, none of the battlefields were retained in the Treaty of Constantinople of 30 September. In October, Bulgarian forces finally returned south of the Rhodopes. The Radoslavov government continued negotiating with the Ottomans in the hopes of forming an alliance. These talks finally bore fruit in the Secret Bulgarian–Ottoman Treaty of August 1914.

As part of the Treaty of Constantinople, 46,764 Orthodox Bulgarians from Ottoman Thrace were exchanged for 48,570 Muslims (Turks, Pomaks, and Roma) from Bulgarian Thrace. After the exchange, according to the 1914 Ottoman census, there remained 14,908 Bulgarians belonging to the Bulgarian Exarchate in Ottoman Empire.

On 14 November 1913, Greece and the Ottomans signed a treaty in Athens, formally ending their hostilities. On 14 March 1914, Serbia signed a treaty in Constantinople, restoring relations with the Ottoman Empire and reaffirming the 1913 Treaty of London. No treaty between Montenegro and the Ottoman Empire was ever signed.

Aftermath
The Second Balkan War made Serbia the most militarily powerful state south of the Danube. Years of military investment financed by French loans borne fruit. Central Vardar and the eastern half of the Sanjak of Novi Pazar were acquired. Its territory grew in extent from 18,650 to 33,891 square miles, and its population grew by more than one and a half million. The aftermath brought harassment and oppression for many in the newly conquered lands. The freedom of association, assembly and the press guaranteed under the Serbian constitution of 1903 was not introduced into the new territories. The inhabitants were denied voting rights, ostensibly because the cultural level was considered too low, in reality, to keep the non-Serbs, who made up the majority in many areas, out of national politics. Opposition newspapers like Radicke Novine remarked that the 'new Serbs' had better political rights under the Turks. There was a destruction of Turkish buildings, schools, baths, mosques. In October and November 1913, British vice-consuls reported systematic intimidation, arbitrary detentions, beatings, rapes, village burnings and massacres by Serbs in the annexed areas. The Serbian government showed no interest in preventing further outrages or investigating those that had happened. When the Carnegie Commission, composed of an international team of experts selected for their impartiality, arrived in the Balkans, they received virtually no assistance from Belgrade.

The treaties forced the Greek Army to evacuate Western Thrace and Pirin Macedonia, which it had occupied during operations. The retreat from the areas that had to be ceded to Bulgaria, together with the loss of Northern Epirus to Albania, was not well received in Greece; from the areas occupied during the war, Greece succeeded in gaining only the territories of Serres and Kavala after diplomatic support from Germany. Serbia made additional gains in northern Macedonia and, having fulfilled its aspirations to the south, turned its attention to the north where its rivalry with Austro-Hungary over Bosnia-Herzegovina led the two countries to war a year later igniting the First World War. Italy used the excuse of the Balkan wars to keep the Dodecanese islands in the Aegean, which it had occupied during the Italo-Turkish War of 1911 over Libya, despite the agreement that ended that war in 1912.

At the strong insistence of Austria-Hungary and Italy, both hoping to control for themselves the state and thus the Otranto Straits in Adriatic, Albania acquired officially its independence according to the terms of the Treaty of London. With the delineation of the exact boundaries of the new state under the Protocol of Florence (17 December 1913), the Serbs lost their outlet to the Adriatic and the Greeks in the region of Northern Epirus (Southern Albania). This was highly unpopular with the local Greek population, who, after a revolt, managed to acquire local autonomy under the terms of the Protocol of Corfu.

After its defeat, Bulgaria became a revanchist local power looking for a second opportunity to fulfill its national aspirations. After Bucharest, the head of the Bulgarian delegation, Tonchev, remarked that "[e]ither the Powers will change [the territorial settlement], or we will destroy it." To this end, it participated in the First World War on the side of the Central Powers since its Balkan enemies (Serbia, Montenegro, Greece, and Romania) were pro-Entente (see articles on the Serbian Campaign and the Macedonian front of World War I). The resulting enormous sacrifices during World War I and renewed defeat caused Bulgaria a national trauma and new territorial losses.

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

title: Second Balkan War


text:
Talk:Dry milk

Eh? RickK 01:32, 12 Aug 2003 (UTC)

 * Apparently a reference to protein blocking, as in Western blotting. Probably needs to be deleted. Graft 01:38, 12 Aug 2003 (UTC)

title: Talk:Dry milk


text:
Talk:BattleBots

Y2K2 Mascot
The original 1999 mascot is not mentioned in this article. Here are the best quality images that exist of him LINK — Preceding unsigned comment added by 24.154.149.187 (talk) 10:03, 21 June 2015 (UTC)

Unsubstantiated Rumor Removed
I just removed an unsubstantiated rumor about the cancellation of BattleBots from the article. It read:

 * "The termination was also due to a law suit between Battlebots Inc. and Budweiser, in which Budweiser unfairly used Battlebots video footage in a commercial without credit given towards the company. During the suit, Budweiser turned to Comedy Central and gave them the ultimatum of either Battlebots is taken off the air or Budweiser removes all sponsorships and advertising from the network. Because of the amount of money given in the form of sponsorships and advertising, Comedy Central took Battlebots off the air."

Budweiser did not use BattleBots footage in their robot fighting commercial, and there has never been any corroboration of a threat to Comedy Central. BattleBots did file suit over the commercial which aired during the 2002 Superbowl. The suit was dismissed by the court in 2004 as being a fair-use parody -- two years after BattleBots went off the air. This rumor pops up from time to time, but it has no basis in fact.

Run Amok (talk) 21:31, 30 March 2009 (UTC)

--
 * Well, I think that the fair-use clause should not apply if the company filed a law suit. It happened with Youtube, so that commercial should have not been allowed to run. Either that or all Intellectual Property is covered under the fair use concerning adds and free videos. 65.13.84.34 (talk) 04:31, 23 September 2009 (UTC)

--
 * You're missing a few points:
 * The purpose of a legal challenge is to allow the courts to determine how the law is best applied to the case in question. The court determined that fair use did apply in this case.
 * The suit was filed after the commercials ran, so there was no opportunity to block showing.
 * No actual footage from BattleBots or of any actual competitors was used in the commercial. There is no similarity to showing actual video clips on You Tube.

 * Run Amok (talk) 15:24, 23 September 2009 (UTC)

Rumble Winners??
I reverted a recent edit that added info on a single winner in a single class from a single tournament. I think we either need to include all rumble winners or none. I vote for none. If we put in the sideshow 'rumble' results then we should likely add the other odd awards as well ('Best Driver', 'Best Design', 'Most Aggressive', etc.). This rapidly becomes non-encyclopedic. Discussion? —Preceding unsigned comment added by Joerger (talk • contribs) 22:21, 11 October 2010 (UTC)

 * I've been told I need to bring this up here to discuss whether to add the winners of the battlebot rumbles in with the series winners. I'm guessing it's like WWE's Royal Rumble and is only 1 a season so we only have 1 clear winner to add. Being British, the only winner I'm aware of is the winner of the 1st ever gigabot one, Razer so can any American battlebots fans add the rest of the winners? The C of E. God Save The Queen! (talk) 08:55, 12 October 2010 (UTC)

 * See my comments above. There were often multiple 'rumbles' at any given BattleBots event, and a few times there were none. They were held informally at the end of the tournament by rounding up whatever robots were still operational and had a radio channel that would not interfere with other machines in the rumble. The prestige of winning a rumble is minor compared to the main tournament and I don't believe that the addition is appropriate for an encyclopedic article. Run Amok (talk) 15:49, 12 October 2010 (UTC)

Show Genre
A recent edit changed the genre of the show from 'Reality TV' to 'Game Show'. I reverted the edit because I think it needs some discussion. The show does not fit well in any standard grouping; it could be considered coverage of a sporting event, a reality competition, an unusual game show, or a chance to laugh at the geeks (it was on Comedy Central, after all). The best, if imperfect, fir seems to be reality TV just because of the extensive biographical segments devoted to some contestants. Other opinions welcomed. Run Amok (talk) 20:08, 2 November 2010 (UTC)

starting it back up
If it was a small piece that fell off a robot and fell harmlessly it the crowd, doesn't mean anything !!!!! lots of other things fall and hits us anywhere, anytime! at school: pencils, pens, shoes, books. At home: book, shoes, bucket, boxes, and almost everything you could think of! I am part of a robotics team at a high school in Berkley, Michigan, Berkley high school team 247 da bears, and have loved all those episodes of BattleBots ever! If people go to the competitions, they did know what could happen! it is there own problem if they get hit by something! It is just like going to a WWE "fake boxing" You might get hit by something! so if people go to a BattleBots competition, they do not have the right to complain, sue or file a complaint against BattleBots!!! —Preceding unsigned comment added by 216.11.5.6 (talk) 00:11, 15 February 2011 (UTC)

 * I'm confused by your post. What small piece? What robot? What event? Starting what back up? The US legal system disagrees with your opinions on spectator safety rights, as do the organizers of the FIRST Robotics Competition in which your team participates.
 * Run Amok (talk) 17:47, 15 February 2011 (UTC)

edit should be made to show battlebots is coming back
I'm not sure how to format the edit, so maybe someone more experienced should edit the page: http://deadline.com/2015/02/battlebots-revival-reality-series-abc-summer-1201367663/ — Preceding unsigned comment added by 65.111.186.42 (talk) 21:55, 6 February 2015 (UTC)

Table issues
It looks like the article is poorly formatted. I'm just posting this in the event a bot reverts my edit because I'm a new account.DoggySoup (talk) 22:32, 24 June 2015 (UTC)

robot tech info
This article currently contains much information about the origins and history of the show but little information about the rules that govern the design and construction of the robots, no technical information about the robots, and not even a mention of the teams of people that make, repair, and operate the robots! Please improve the article by adding much more information about the robots and the people that make them.-96.233.20.34 (talk) 14:03, 28 July 2015 (UTC)

Season Article's
I think its time to integrate this article to each of their owns seasons articles. I went ahead and made each article as a redirect. I think information could be presented in a better format. For instance, There episode list is completely disorganized and unpleasing. The episode list should be a short summary of the episode "The Qualifying Rounds continued, with blank blank and blank winning their respective battles, moving on to the Round of "X". Something short and sweet. We should integrate brackets into the article as well. A section for the Round of and a section for the Qualifying, with winner on the left side and the loser on the other. Similar to this format:.

Some of the areas of the show should be expanded on as well, like the format and also contestant information about age, hometown, etc with the name of their bot. I will be working on this in the coming days and could use some help, as i didn't watch the first 5 seasons. Thanks. Chase (talk) 00:47, 24 June 2016 (UTC)

- I like this idea and will help out if I can. The Ultimaet Fighter is organized very well and is of similar format. https://en.wikipedia.org/wiki/The_Ultimate_Fighter#Main_seasonsMaxwell vs Euler (talk) 06:24, 25 June 2016 (UTC)

Merger proposal
I propose that BattleBots (ABC) (season 2) be merged into BattleBots. I think that the content in the "season 2" article can easily be explained in the context of BattleBots, and the BattleBots article is of a reasonable size that the merging of "season 2" will not cause any problems as far as article size or undue weight is concerned. DaltonCastle (talk) 00:36, 25 June 2016 (UTC)

I like the idea of having a separate page for each season. The information on the main battlebots page is becoming very large. As the tournament continues, more and more infor will need to be added to the tables. I propose a similar format to what is used by the UFC Ultimate Fighter seasons. https://en.wikipedia.org/wiki/The_Ultimate_Fighter#Main_seasons

They do a good job of summarizing the season on the main "The Ultimate Fighter" page and go into more details about the bouts in each episode. — Preceding unsigned comment added by Maxwell vs Euler (talk • contribs) 06:20, 25 June 2016 (UTC)

Science channel episodes
The way some of the references are written, it seems confusing whether the 16 episodes Science is airing are NEW or just the 16 episodes from ABC's 2 seasons. Is there any info that can be added that clarifies what is going on? Andyross (talk) 13:02, 9 July 2017 (UTC)

No prize(s) mentioned
There is no discussion of prizes awarded. I know from about the 2018 season the grand prize winner received the “Giant Nut.” Do any contestants win any prize money? If so, how much? Does it vary by seasons? I do not know the answers except about the grand prize, the Giant Nut. — Preceding unsigned comment added by VykrJimBob (talk • contribs) 04:18, 29 December 2020 (UTC)

 * I've added the prizes, they are very meager winnings compared to the costs the builders sustain to compete. --Dangelman (talk) 15:28, 5 March 2022 (UTC)

Split (2020): Episode list
I suggest that the episode listing and season table be split off into a separate article, as there are 10 TV seasons already. Just leave a copy of the season table here as a WP:SUMMARYSTYLE for the episode listing, and have a list of champions added for additional season summarization.

-- 67.70.26.89 (talk) 15:47, 5 January 2021 (UTC)

watch discovery make money as all contestants (including winners) lose money
I always thought there was money in this for the contestants, the show can afford kenflo as host, but there isnt. — Preceding unsigned comment added by 2600:1700:CDA0:1060:25B6:AEDB:A777:3E93 (talk) 03:24, 25 February 2022 (UTC)

Criticism towards Battlebots' malpractice
In response to the section above, I'd really like to make a criticism/controversy section about it, but no media outlet has covered it and all we have are scattered replies from teams on user-generated websites.

 * In 2020, the tournament championship winner gets about $15.000 total cash prizes
 * They are not allowed to contact big sponsors directly but have to receive sponsorship money from BB (money that is far from covering the cost of the bot and the team)
 * They do not have rights/ownership of their creations, meaning no merch revenue from BB as well as having to use different design and name for competitions outside of BB

Team Bite Force, three times champion with an unprecedented record of 26-1 W/L, said they've tried to address contract agreement improvements regarding team sustainability with the owners Trey Roski and Greg Munson but they are not open to negotiation. BB is operating the classic monopsony and union busting (aka "we don't need you, there are others who would take much less to be on TV"). The other builders agree, quietly, on their socials and podcasts, but are still chasing fame in this very expensive hobby. Jake Ewert from Team Whyachi also addressed the situation in a video in favour of Bite Force's stand. --Dangelman (talk) 16:02, 5 March 2022 (UTC)

Championship winners
I've searched everywhere and can't find anything accurate, as I just watched the 2021 Championship and know Tantrum won. There should be a link where all the winners of the Giant Nut are listed by year. Please and thanks. 2600:387:C:6B32:0:0:0:5 (talk) 05:05, 8 April 2022 (UTC)

Bite Force or Tombstone article?
Bite Force and Tombstone are like the 2 most signature bots from Battlebots so can we make articles for them HiGuys69420 (talk) 16:33, 25 December 2023 (UTC)
 * Of course you can . After all, we do have ones for Razer (robot) and Chaos 2 from Robot Wars. As long as you can find enough reliable 3rd party sources for them, go right ahead and write!  The C of E God Save the King!  ( talk ) 16:42, 25 December 2023 (UTC)
 * okie HiGuys69420 (talk) 16:53, 25 December 2023 (UTC)

title: Talk:BattleBots


text:
Lie derivative

In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted $$ \mathcal{L}_X T$$. The differential operator $$ T \mapsto \mathcal{L}_X T$$ is a derivation of the algebra of tensor fields of the underlying manifold.

The Lie derivative commutes with contraction and the exterior derivative on differential forms.

Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.

The Lie derivative of a vector field Y with respect to another vector field X is known as the "Lie bracket" of X and Y, and is often denoted [X,Y] instead of $$ \mathcal{L}_X Y$$. The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity

 * $$ \mathcal{L}_{[X,Y]} T = \mathcal{L}_X \mathcal{L}_{Y} T - \mathcal{L}_Y \mathcal{L}_X T,$$

valid for any vector fields X and Y and any tensor field T.

Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory.

Generalisations exist for spinor fields, fibre bundles with a connection and vector-valued differential forms.

Motivation
A 'naïve' attempt to define the derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directional derivative of each component with respect to the vector field. However, this definition is undesirable because it is not invariant under changes of coordinate system, e.g. the naive derivative expressed in polar or spherical coordinates differs from the naive derivative of the components in Cartesian coordinates. On an abstract manifold such a definition is meaningless and ill defined. In differential geometry, there are three main coordinate independent notions of differentiation of tensor fields: Lie derivatives, derivatives with respect to connections, and the exterior derivative of totally antisymmetric covariant tensors, i.e. differential forms. The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that tangent vector to a vector field. However a connection requires the choice of an additional geometric structure (e.g. a Riemannian metric or just an abstract connection) on the manifold. In contrast, when taking a Lie derivative, no additional structure on the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field X at a point p depends on the value of X in a neighborhood of p, not just at p itself. Finally, the exterior derivative of differential forms does not require any additional choices, but is only a well defined derivative of differential forms (including functions).

Definition
The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.

The (Lie) derivative of a function
Defining the derivative of a function $$f\colon M \to {\mathbb R} $$ on a manifold is problematic because the difference quotient $$\textstyle (f(x+h)-f(x))/h $$ cannot be determined while the displacement $$x+h$$ is undefined.

The Lie derivative of a function $$f\colon M\to {\mathbb R}$$ with respect to a vector field $$X$$ at a point $$p \in M$$ is the function
 * $$(\mathcal{L}_X f) (p) = {d \over dt} \biggr|_{t=0} \bigl(f \circ \Phi^t_X\bigr)(p) = \lim_{t\to 0} \frac{f\bigl(\Phi^t_X(p)\bigr) - f\bigl(p\bigr)}{t}$$

where $$\Phi^t_X(p)$$ is the point to which the flow defined by the vector field $$X$$ maps the point $$p$$ at time instant $$t.$$ In the vicinity of $$t=0,$$ $$\Phi^t_X(p)$$ is the unique solution of the system

\frac{d}{dt}\biggr|_t \Phi^t_X(p) = X\bigl(\Phi^t_X(p)\bigr) $$ of first-order autonomous (i.e. time-independent) differential equations, with $$\Phi^0_X(p) = p.$$

Setting $$\mathcal{L}_X f = \nabla_X f$$ identifies the Lie derivative of a function with the directional derivative, which is also denoted by $$ X(f):= \mathcal{L}_X f = \nabla_X f$$.

The Lie derivative of a vector field
If X and Y are both vector fields, then the Lie derivative of Y with respect to X is also known as the Lie bracket of X and Y, and is sometimes denoted $$[X,Y]$$. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above:

Definition in terms of flows
The Lie derivative is the speed with which the tensor field changes under the space deformation caused by the flow.

Formally, given a differentiable (time-independent) vector field $$X$$ on a smooth manifold $$M,$$ let $$\Phi^t_X : M \to M$$ be the corresponding local flow. Since $$\Phi^t_X$$ is a local diffeomorphism for each $$t$$, it gives rise to a pullback of tensor fields. For covariant tensors, this is just the multi-linear extension of the pullback map

 * $$\left(\Phi^t_X\right)^*_p : T^*_{\Phi^t_X(p)}M \to T^*_{p}M, \qquad \left(\Phi^t_X\right)^*_p \alpha (X) = \alpha\bigl(T_p \Phi^t_X(X)\bigr), \quad \alpha \in T^*_{\Phi^t_X(p)}M, X \in T_{p}M $$

For contravariant tensors, one extends the inverse
 * $$\left(T_p\Phi^t_X\right)^{-1} : T_{\Phi^t_X(p)}M \to T_{p}M$$

of the differential $$T_p\Phi^t_X $$. For every $$t,$$ there is, consequently, a tensor field $$(\Phi^t_X)^* T$$ of the same type as $$T$$'s.

If $$T$$ is an $$(r,0)$$- or $$(0,s)$$-type tensor field, then the Lie derivative $${\cal L}_XT$$ of $$T$$ along a vector field $$X$$ is defined at point $$p \in M$$ to be

 * $${\cal L}_X T(p) = \frac{d}{dt}\biggl|_{t=0} \left(\bigl(\Phi^t_X\bigr)^* T\right)_p = \frac{d}{dt}\biggl|_{t=0}\bigl(\Phi^t_X\bigr)^*_p T_{\Phi^t_X(p)}

= \lim_{t \to 0}\frac{\bigl(\Phi^t_X\bigr)^*T_{\Phi^t_X(p)} - T_p}{t}.$$

The resulting tensor field $${\cal L}_X T$$ is of the same type as $$T$$'s.

More generally, for every smooth 1-parameter family $$\Phi_t $$of diffeomorphisms that integrate a vector field $$X $$ in the sense that $${d \over dt}\biggr|_{t=0} \Phi_t = X \circ \Phi_0 $$, one has$$\mathcal{L}_X T = \bigl(\Phi_0^{-1}\bigr)^* {d \over dt}\biggr|_{t=0} \Phi_t^* T = - {d \over dt}\biggr|_{t=0} \bigl(\Phi_t^{-1}\bigr)^* \Phi_0^* T \,. $$

Algebraic definition
We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms:

 * Axiom 1. The Lie derivative of a function is equal to the directional derivative of the function. This fact is often expressed by the formula
 * $$\mathcal{L}_Yf=Y(f)$$

 * Axiom 2. The Lie derivative obeys the following version of Leibniz's rule: For any tensor fields S and T, we have
 * $$\mathcal{L}_Y(S\otimes T)=(\mathcal{L}_YS)\otimes T+S\otimes (\mathcal{L}_YT).$$

 * Axiom 3. The Lie derivative obeys the Leibniz rule with respect to contraction:
 * $$ \mathcal{L}_X (T(Y_1, \ldots, Y_n)) = (\mathcal{L}_X T)(Y_1,\ldots, Y_n) + T((\mathcal{L}_X Y_1), \ldots, Y_n) + \cdots + T(Y_1, \ldots, (\mathcal{L}_X Y_n)) $$

 * Axiom 4. The Lie derivative commutes with exterior derivative on functions:
 * $$ [\mathcal{L}_X, d] = 0 $$

If these axioms hold, then applying the Lie derivative $$\mathcal{L}_X$$ to the relation $$ df(Y) = Y(f) $$ shows that
 * $$\mathcal{L}_X Y (f) = X(Y(f)) - Y(X(f)),$$

which is one of the standard definitions for the Lie bracket.

The Lie derivative acting on a differential form is the anticommutator of the interior product with the exterior derivative. So if α is a differential form,
 * $$\mathcal{L}_Y\alpha=i_Yd\alpha+di_Y\alpha.$$

This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions.

Explicitly, let T be a tensor field of type (p, q). Consider T to be a differentiable multilinear map of smooth sections α1, α2, ..., αp of the cotangent bundle T∗M and of sections X1, X2, ..., Xq of the tangent bundle TM, written T(α1, α2, ..., X1, X2, ...) into R. Define the Lie derivative of T along Y by the formula

 * $$(\mathcal{L}_Y T)(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots) =Y(T(\alpha_1,\alpha_2,\ldots,X_1,X_2,\ldots))$$
 * $$- T(\mathcal{L}_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots)

- T(\alpha_1, \mathcal{L}_Y\alpha_2, \ldots, X_1, X_2, \ldots) -\ldots $$
 * $$- T(\alpha_1, \alpha_2, \ldots, \mathcal{L}_YX_1, X_2, \ldots)

- T(\alpha_1, \alpha_2, \ldots, X_1, \mathcal{L}_YX_2, \ldots) - \ldots $$

The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and the Leibniz rule for differentiation. The Lie derivative commutes with the contraction.

The Lie derivative of a differential form
A particularly important class of tensor fields is the class of differential forms. The restriction of the Lie derivative to the space of differential forms is closely related to the exterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of an interior product, after which the relationships falls out as an identity known as Cartan's formula. Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms.

Let M be a manifold and X a vector field on M. Let $$\omega \in \Lambda^{k+1}(M)$$ be a (k + 1)-form, i.e. for each $$p \in M$$, $$\omega(p)$$ is an alternating multilinear map from $$(T_p M)^{k + 1}$$ to the real numbers. The interior product of X and ω is the k-form $$i_X\omega$$ defined as

 * $$(i_X\omega) (X_1, \ldots, X_k) = \omega (X,X_1, \ldots, X_k)\,$$

The differential form $$i_X\omega$$ is also called the contraction of ω with X, and
 * $$i_X:\Lambda^{k+1}(M) \rightarrow \Lambda^k(M)$$

is a $\wedge$-antiderivation where $\wedge$ is the wedge product on differential forms. That is, $$i_X$$ is R-linear, and

 * $$i_X (\omega \wedge \eta) = (i_X \omega) \wedge \eta + (-1)^k \omega \wedge (i_X \eta)$$

for $$\omega \in \Lambda^k(M)$$ and η another differential form. Also, for a function $$f \in \Lambda^0(M)$$, that is, a real- or complex-valued function on M, one has

 * $$i_{fX} \omega = f\,i_X\omega$$

where $$f X$$ denotes the product of f and X. The relationship between exterior derivatives and Lie derivatives can then be summarized as follows. First, since the Lie derivative of a function f with respect to a vector field X is the same as the directional derivative X(f), it is also the same as the contraction of the exterior derivative of f with X:

 * $$\mathcal{L}_Xf = i_X \, df$$

For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation in X:

 * $$\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega).$$

This identity is known variously as Cartan formula, Cartan homotopy formula or Cartan's magic formula. See interior product for details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that

 * $$d\mathcal{L}_X\omega = \mathcal{L}_X(d\omega).$$

The Lie derivative also satisfies the relation

 * $$\mathcal{L}_{fX}\omega = f\mathcal{L}_X\omega + df \wedge i_X \omega .$$

Coordinate expressions
In local coordinate notation, for a type (r, s) tensor field $$T$$, the Lie derivative along $$X$$ is
 * $$\begin{align}

(\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} ={} & X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) \\ & {}-{} (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} \\ & + (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s}X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c} \end{align}$$

here, the notation $$\partial_a = \frac{\partial}{\partial x^a}$$ means taking the partial derivative with respect to the coordinate $$x^a$$. Alternatively, if we are using a torsion-free connection (e.g., the Levi Civita connection), then the partial derivative $$\partial_a$$ can be replaced with the covariant derivative which means replacing $$\partial_a X^b$$ with (by abuse of notation) $$\nabla_a X^b = X^b_{;a} := (\nabla X)_a^{\ b} = \partial_a X^b + \Gamma^b_{ac}X^c$$ where the $$\Gamma^a_{bc} = \Gamma^a_{cb}$$ are the Christoffel coefficients.

The Lie derivative of a tensor is another tensor of the same type, i.e., even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor
 * $$(\mathcal{L}_X T) ^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}\partial_{a_1}\otimes\cdots\otimes\partial_{a_r}\otimes dx^{b_1}\otimes\cdots\otimes dx^{b_s}$$

which is independent of any coordinate system and of the same type as $$T$$.

The definition can be extended further to tensor densities. If T is a tensor density of some real number valued weight w (e.g. the volume density of weight 1), then its Lie derivative is a tensor density of the same type and weight.
 * $$\begin{align}

(\mathcal {L}_X T)^{a_1 \ldots a_r}{}_{b_1 \ldots b_s} ={} &X^c(\partial_c T^{a_1 \ldots a_r}{}_{b_1 \ldots b_s}) - (\partial_c X ^{a_1}) T ^{c a_2 \ldots a_r}{}_{b_1 \ldots b_s} - \ldots - (\partial_c X^{a_r}) T ^{a_1 \ldots a_{r-1}c}{}_{b_1 \ldots b_s} + \\ &+ (\partial_{b_1} X^c) T ^{a_1 \ldots a_r}{}_{c b_2 \ldots b_s} + \ldots + (\partial_{b_s} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s-1} c} + w (\partial_{c} X^c) T ^{a_1 \ldots a_r}{}_{b_1 \ldots b_{s}} \end{align}$$

Notice the new term at the end of the expression.

For a linear connection $$\Gamma = ( \Gamma^{a}_{bc} )$$, the Lie derivative along $$X$$ is

(\mathcal{L}_X \Gamma)^{a}_{bc} = X^d\partial_d \Gamma^{a}_{bc} + \partial_b\partial_c X^a - \Gamma^{d}_{bc}\partial_d X^a + \Gamma^{a}_{dc}\partial_b X^d + \Gamma^{a}_{bd}\partial_c X^d$$

Examples
For clarity we now show the following examples in local coordinate notation.

For a scalar field $$\phi(x^c)\in\mathcal{F}(M)$$ we have:
 * $$ (\mathcal {L}_X \phi) = X(\phi) = X^a \partial_a \phi$$.

Hence for the scalar field $$\phi(x,y) = x^2 - \sin(y)$$ and the vector field $$X = \sin(x)\partial_y - y^2\partial_x$$ the corresponding Lie derivative becomes $$\begin{alignat}{3} \mathcal{L}_X\phi &= (\sin(x)\partial_y - y^2\partial_x)(x^2 - \sin(y))\\ & = \sin(x)\partial_y(x^2 - \sin(y)) - y^2\partial_x(x^2 - \sin(y))\\ & = -\sin(x)\cos(y) - 2xy^2 \\ \end{alignat}$$

For an example of higher rank differential form, consider the 2-form $$\omega = (x^2 + y^2)dx\wedge dz$$ and the vector field $$X$$ from the previous example. Then, $$\begin{align} \mathcal{L}_X\omega & = d(i_{\sin(x)\partial_y - y^2\partial_x}((x^2 + y^2)dx\wedge dz)) + i_{\sin(x)\partial_y - y^2\partial_x}(d((x^2 + y^2)dx\wedge dz)) \\ & = d(-y^2(x^2 + y^2) dz) + i_{\sin(x)\partial_y - y^2\partial_x}(2ydy\wedge dx\wedge dz) \\ & = \left(- 2xy^2 dx + (-2yx^2 - 4y^3) dy\right) \wedge dz + (2y\sin(x)dx \wedge dz + 2y^3dy \wedge dz)\\ & = \left(-2xy^2 + 2y\sin(x)\right)dx\wedge dz + (-2yx^2 - 2y^3)dy\wedge dz \end{align}$$

Some more abstract examples.
 * $$\mathcal{L}_X (dx^b) = d i_X (dx^b) = d X^b = \partial_a X^b dx^a $$.

Hence for a covector field, i.e., a differential form, $$A = A_a(x^b)dx^a$$ we have:
 * $$\mathcal{L}_X A = X (A_a) dx^a +  A_b \mathcal{L}_X (dx^b) = (X^b \partial_b A_a + A_b\partial_a (X^b))dx^a$$

The coefficient of the last expression is the local coordinate expression of the Lie derivative.

For a covariant rank 2 tensor field $$T = T_{ab}(x^c)dx^a \otimes dx^b$$ we have: $$\begin{align} (\mathcal {L}_X T) &= (\mathcal {L}_X T)_{ab} dx^a\otimes dx^b\\ &= X(T_{ab})dx^a\otimes dx^b + T_{cb} \mathcal{L}_X (dx^c) \otimes dx^b + T_{ac} dx^a \otimes \mathcal{L}_X (dx^c)\\ &= (X^c \partial_c T_{ab}+T_{cb}\partial_a X^c+T_{ac}\partial_b X^c)dx^a\otimes dx^b\\ \end{align}$$

If $$T = g$$ is the symmetric metric tensor, it is parallel with respect to the Levi-Civita connection (aka covariant derivative), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving
 * $$(\mathcal {L}_X g) = (X^c g_{ab; c} + g_{cb}X^c_{;a} + g_{ac}X^c_{; b})dx^a\otimes dx^b = (X_{b;a} + X_{a;b}) dx^a\otimes dx^b$$

Properties
The Lie derivative has a number of properties. Let $$\mathcal{F}(M)$$ be the algebra of functions defined on the manifold M. Then

 * $$\mathcal{L}_X : \mathcal{F}(M) \rightarrow \mathcal{F}(M)$$

is a derivation on the algebra $$\mathcal{F}(M)$$. That is, $$\mathcal{L}_X$$ is R-linear and

 * $$\mathcal{L}_X(fg) = (\mathcal{L}_Xf) g + f\mathcal{L}_Xg.$$

Similarly, it is a derivation on $$\mathcal{F}(M) \times \mathcal{X}(M)$$ where $$\mathcal{X}(M)$$ is the set of vector fields on M (cf. Theorem 6 from the article: Nichita, F.F. Unification Theories: New Results and Examples. Axioms 2019, 8, 60):

 * $$\mathcal{L}_X(fY) = (\mathcal{L}_Xf) Y + f\mathcal{L}_X Y$$

which may also be written in the equivalent notation

 * $$\mathcal{L}_X(f\otimes Y) = (\mathcal{L}_Xf) \otimes Y + f\otimes \mathcal{L}_X Y$$

where the tensor product symbol $$\otimes$$ is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.

Additional properties are consistent with that of the Lie bracket. Thus, for example, considered as a derivation on a vector field,

 * $$\mathcal{L}_X [Y,Z] = [\mathcal{L}_X Y,Z] + [Y,\mathcal{L}_X Z]$$

one finds the above to be just the Jacobi identity. Thus, one has the important result that the space of vector fields over M, equipped with the Lie bracket, forms a Lie algebra.

The Lie derivative also has important properties when acting on differential forms. Let α and β be two differential forms on M, and let X and Y be two vector fields. Then
 * $$\mathcal{L}_X(\alpha\wedge\beta) = (\mathcal{L}_X\alpha) \wedge\beta + \alpha\wedge (\mathcal{L}_X\beta)$$
 * $$[\mathcal{L}_X,\mathcal{L}_Y]\alpha := \mathcal{L}_X\mathcal{L}_Y\alpha-\mathcal{L}_Y\mathcal{L}_X\alpha = \mathcal{L}_{[X,Y]}\alpha$$
 * $$[\mathcal{L}_X,i_Y]\alpha = [i_X,\mathcal{L}_Y]\alpha = i_{[X,Y]}\alpha,$$ where i denotes interior product defined above and it is clear whether [·,·] denotes the commutator or the Lie bracket of vector fields.

Generalizations
Various generalizations of the Lie derivative play an important role in differential geometry.

The Lie derivative of a spinor field
A definition for Lie derivatives of spinors along generic spacetime vector fields, not necessarily Killing ones, on a general (pseudo) Riemannian manifold was already proposed in 1971 by Yvette Kosmann. Later, it was provided a geometric framework which justifies her ad hoc prescription within the general framework of Lie derivatives on fiber bundles in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.

In a given spin manifold, that is in a Riemannian manifold $$(M,g)$$ admitting a spin structure, the Lie derivative of a spinor field $$\psi$$ can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via the André Lichnerowicz's local expression given in 1963:

 * $$\mathcal{L}_X \psi := X^{a}\nabla_{a}\psi - \frac14\nabla_{a}X_{b} \gamma^{a}\gamma^{b}\psi\, ,$$

where $$\nabla_{a}X_{b} = \nabla_{[a}X_{b]}$$, as $$X = X^{a}\partial_{a}$$ is assumed to be a Killing vector field, and $$\gamma^{a}$$ are Dirac matrices.

It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for a generic vector field $$X$$, but explicitly taking the antisymmetric part of $$\nabla_{a}X_{b}$$ only. More explicitly, Kosmann's local expression given in 1972 is:

 * $$\mathcal{L}_X \psi := X^{a}\nabla_{a}\psi - \frac18\nabla_{[a}X_{b]}[\gamma^{a},\gamma^{b}]\psi\, = \nabla_X \psi - \frac14 (d X^\flat)\cdot \psi\, ,$$

where $$[\gamma^{a},\gamma^{b}]= \gamma^a\gamma^b - \gamma^b\gamma^a$$ is the commutator, $$d$$ is exterior derivative, $$X^\flat = g(X, -)$$ is the dual 1 form corresponding to $$X$$ under the metric (i.e. with lowered indices) and $$ \cdot $$ is Clifford multiplication.

It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of the connection. This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on the spinor bundle. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel.

To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article, where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called the Kosmann lift.

Covariant Lie derivative
If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle (i.e. it has horizontal and vertical components), then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle.

Now, if we're given a vector field Y over M (but not the principal bundle) but we also have a connection over the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matches Y and its vertical component agrees with the connection. This is the covariant Lie derivative.

See connection form for more details.

Nijenhuis–Lie derivative
Another generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ωk(M, TM) of differential forms with values in the tangent bundle. If K ∈ Ωk(M, TM) and α is a differential p-form, then it is possible to define the interior product iKα of K and α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative:
 * $$\mathcal{L}_K\alpha=[d,i_K]\alpha = di_K\alpha-(-1)^{k-1}i_K \, d\alpha.$$

History
In 1931, Władysław Ślebodziński introduced a new differential operator, later called by David van Dantzig that of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups of automorphisms.

The Lie derivatives of general geometric objects (i.e., sections of natural fiber bundles) were studied by A. Nijenhuis, Y. Tashiro and K. Yano.

For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940, Léon Rosenfeld —and before him (in 1921 ) Wolfgang Pauli —introduced what he called a ‘local variation’ $$\delta^{\ast}A$$ of a geometric object $$A\,$$ induced by an infinitesimal transformation of coordinates generated by a vector field $$X\,$$. One can easily prove that his $$\delta^{\ast}A$$ is $$ - \mathcal{L}_X(A)\,$$.

title: Lie derivative


text:
Dwight Presbyterian Mission

Dwight Presbyterian Mission was one of the first American missions to the Native Americans. It was established near present-day Russellville, Arkansas in 1820 to serve the Arkansas Cherokees. After the Cherokee were required to move to Indian Territory in 1828, the mission was reestablished in 1829 near present-day Marble City, Oklahoma. The mission is listed on the National Register of Historic Places.

History
A mission was requested by Tahlonteeskee, the Principal Chief of the Western Cherokees, after he had visited Brainerd Mission in 1818. The mission was founded in August 1820 on Illinois Bayou near present-day Russellville, Arkansas by Cephas Washburn. It was named for Rev. Timothy Dwight, president of Yale College and a corporate member of the American Board of Commissioners for Foreign Missions. After Tahlonteeskee's death in the spring of 1819, the mission was supported by his brother John Jolly.

By 1824, Dwight Mission was a self-contained small town on the frontier. There were at least 24 buildings, including residences for missionaries, students, staff, and visitors; support buildings that included a combination library, post office and pharmacy; a dining hall with kitchen; a storehouse; a blacksmith shop; lathe and carpenter shop; a stable and a barn.

A new treaty between the Cherokees and the United States in 1828 required the Arkansas Cherokees to move to Indian Territory. Therefore, the mission was reestablished in its present-day location, in May 1829, on Sallisaw Creek, near the city of Marble City, Oklahoma. The school opened May 1, 1830. Initially, the facility had a double log house (a building combining school and living quarters) and several log houses to house the staff. Eventually, there would be twenty-one houses, a dining hall, a barn and a number of outbuildings. The mission had more than a dozen staff members and eighty students at the peak of its activity. Its primary mission was to provide an education to Cherokee children and expose them to the Christian religion.

Samuel Worcester served at the mission in 1835, after having worked with the Cherokee as a missionary in Georgia before removal. He created the type for the Cherokee syllabary for their first newspaper, the Cherokee Phoenix. In 1837, the first Cherokee language printing press in the then Indian Territory (now Oklahoma ) was moved from Union Mission to Dwight Mission. In 1839, the majority of the Cherokee nation was removed to the area from the Southeast on the Trail of Tears.

During the American Civil War, many of the mission's buildings were burned down in warfare between pro-Confederate and pro-Union forces. The last religious service was in November 1862. Then it was abandoned. Afterwards, the property was taken over by private owners. By 1884, only two of the original buildings remained.

The Cherokee National Council decided to reopen the school in 1886. The Presbyterian Women's Board of Home Missions supplied funds to do this, and a large building was constructed to serve as a boarding school for Cherokee girls. After 1895, it became a day school, then resumed the boarding school concept in 1900, this time for boys and girls.

On January 12, 1918, a fire burned the Dwight Mission School dormitory, destroying the dormitory and killing thirteen students. The students were named as: George Wickett (Cherokee), Nighthawk Mclemore (Cherokee), Phillip Correll (Cherokee), Rufus Young (Cherokee), J. P. Chandler, Jr (Cherokee), Kenneth Crutchfield (Cherokee), Delbert Barnes (Cherokee), Wilson Beaver (Creek), George Tiger (Creek), Robert Daniels (Creek), Solomon Bruner (Creek), Newton Goins (Choctaw), and Gilbert Grants (Omaha). The boys, who were aged 9 to 17, were trapped on a screened porch. Their bodies were returned to their hometowns, and a memorial for them was erected in the Dwight Mission Cemetery.

By 1944, the facility was called Dwight Indian Training School. It was then operated by Board of National Missions of the Presbyterian Church USA, and served seventy-one students, mostly Cherokee and Choctaw.

The school was closed in 1948.

In the fall of 1950, a number of Presbyterian laymen felt that the Synod of Oklahoma should acquire the property and preserve it because of its long association with the Presbyterian Church. They formed Dwight Mission, Inc. and purchased the property on May 6, 1951, for $35,000. Dwight Mission, Inc. and the Synod of Oklahoma managed the daily programs for the next 30 years.

Dwight Mission today operates as a Presbyterian camp, retreat, and conference center between Marble City and Sallisaw, Oklahoma. It is listed on the National Register of Historic Places. Historical markers identify and explain the original location in Arkansas.

In June 2021, Dwight Mission was acquired by the Cherokee Nation.

title: Dwight Presbyterian Mission


text:
Talk:List of state leaders in 1957

What is the purpose of this page? Will there be one for every year in history? Are you planning on adding more people and countries? RickK 01:42, 12 Aug 2003 (UTC)

what does incumbents mean?
 * Office-holders.

Do we really want an article containing every officholder in every country for every position for every year in history, or are we planning on just having this one page? And why 1957 in particular? RickK 01:44, 12 Aug 2003 (UTC)

 * This article was just an idea I had and I picked a random date for it. I think they might eventually be quite useful. I think this sort of information is quite relevant to how we link to year articles.  For instance someone reading about the Great Smog of 1952 would quite likely be interested to see who was British PM in that year, more so than what was published that year, for instance. This is just the sort of obscure, but useful, information that wikipedia will want to gather as it heads for the million article mark, information that would be hard to find anywhere else. It can usefully illustrate trends, for instance the 1980's cabal of conservatives leaders such as Reagan, Thatcher, Kohl, Mulroney etc., who were all in office at the same time. Lists like US Senators in 1957 can certainly be regarded to be as important, or even more important, than such things as hit songs or the winner of that year's horse races. - SimonP 02:48, Aug 12, 2003 (UTC)

 * Given the experimental nature, perhaps this should be off in your sandbox until it's a bit more fully developed... Daniel Quinlan 07:01, Aug 12, 2003 (UTC)

 * I think this is a great idea in general, but I would just merge the content with the year pages, just like births and deaths are. After all it's not like the yearpages are unmanageable monsters yet, plenty of room there. If they balloon into huge mammoths, then split them. US Senators of 57 should go to their own page, but heads of state of the world into the year-page proper. -- Cimon Avaro

 * I also think it's a great idea but is simple plain-text a good support? I mean, there were a debate wheter battles should be listed by War, year, location... Looks like a database was be better. --Ann O'nyme 14:13, 17 Aug 2003 (UTC)

 * I think that putting the data in the year articles is a great idea and would make the year articles even better. --Dante Alighieri 08:25, 19 Aug 2003 (UTC)

Above from VfD. Simon - what do you think of the idea of merging this with 1957? Martin 18:53, 19 Aug 2003 (UTC)

Umm, hello? Is there anybody there? I already did it.
I already merged the info in this page to the 1957 yearpage. (I did remove the speculative links, which can be restored if you think them necessary). -- Cimon Avaro on a pogostick

 * Ahh right - I looked at this page, but not at 1957... cheers. Martin

Formatting?
I like this page; its a good prototype. However, we should try to coordinate the formats, as we have different ones for this page, the 2003 page, and the 1816 page (those are the 3 complete (or mostly complete) ones to date). The 2003 page has the entires organized in different geographical groups, which isnt absolutely necessary, but can be helpful especially for earler years with more unfamiliar entries (the German states spread across the 1816 article, for example, should be grouped together). However, that page also has a separate group for non-heads of state, which i think should be incorporated in to the others (the whole reason, I thought, for changing "heads of state" to "incumbents"). This page does that well. The other thing I would suggest is turn the titles into links, where possible. It could be helpful if in the UK entry "Prime Minister" linked to the article for British Prime Ministrers. This would be especially helpful for more confusing entries, like the USSR; many people would want to know what the difference is between the First Secretary and the two chairmen. Links to those titles would help. Theres a separate discussion page at Talk:Incumbents by year. Thats my 2 cents. -R. fiend

I agree with most of your thoughts. I was basically trying to follow the 2003 model, but without the regionalizing or the international organizations. I do agree about the German states - perhaps they should go as subsets of Germanic Confederation - Franz I of Austria, President of the Germanic Confederation (1815-1835). I agree that links are good - I just wasn't sure which links had articles, or where the articles would be, so I was just trying to finish the whole business. john 04:12, 25 Apr 2004 (UTC)

title: Talk:List of state leaders in 1957


text:
Is-a

In knowledge representation and ontology components, including for object-oriented programming and design, is-a (also written as is_a or is a) is a subsumptive relationship between abstractions (e.g., types, classes), wherein one class A is a subclass of another class B (and so B is a superclass  of A). In other words, type A is a subtype of type B when A's specification implies B's specification. That is, any object (or class) that satisfies A's specification also satisfies B's specification, because B's specification is weaker.

For example, a cat 'is a' animal, but not vice versa. All cats are animals, but not all animals are cats. Behaviour that is relevant to all animals is defined on an animal class, whereas behaviour that is relevant only for cats is defined in a cat class. By defining the cat class as 'extending' the animal class, all cats 'inherit' the behaviour defined for animals, without the need to explicitly code that behaviour for cats.

Related concepts
The is-a relationship is to be contrasted with the has-a (has_a or has a) relationship between types (classes); confusing the relations has-a and is-a is a common error when designing a model (e.g., a computer program) of the real-world relationship between an object and its subordinate. The is-a relationship may also be contrasted with the instance-of relationship between objects (instances) and types (classes): see Type–token distinction.

To summarize the relations, there are:

 * hyperonym–hyponym (supertype/superclass–subtype/subclass) relations between types (classes) defining a taxonomic hierarchy, where
 * for a subsumption relation: a hyponym (subtype, subclass) has a type-of (is-a) relationship with its hyperonym (supertype, superclass);
 * holonym–meronym (whole/entity/container–part/constituent/member) relations between types (classes) defining a possessive hierarchy, where
 * for an aggregation (i.e. without ownership) relation:
 * a holonym (whole) has a has-a relationship with its meronym (part),
 * for a composition (i.e. with ownership) relation:
 * a meronym (constituent) has a part-of relationship with its holonym (entity),
 * for a containment relation:
 * a meronym (member) has a member-of relationship with its holonym (container);
 * concept–object (type–token) relations between types (classes) and objects (instances), where
 * a token (object) has an instance-of relationship with its type (class).

Examples of subtyping
Subtyping enables a given type to be substituted for another type or abstraction. Subtyping is said to establish an is-a relationship between the subtype and some existing abstraction, either implicitly or explicitly, depending on language support. The relationship can be expressed explicitly via inheritance in languages that support inheritance as a subtyping mechanism.

C++
The following C++ code establishes an explicit inheritance relationship between classes B and A, where B is both a subclass and a subtype of A, and can be used as an A wherever a B is specified (via a reference, a pointer or the object itself).

Python
The following python code establishes an explicit inheritance relationship between classes B and A, where B is both a subclass and a subtype of A, and can be used as an A wherever a B is required.

The following example, type(a) is a "regular" type, and type(type(a)) is a metatype. While as distributed all types have the same metatype (PyType_Type, which is also its own metatype), this is not a requirement. The type of classic classes, known as types.ClassType, can also be considered a distinct metatype.

Java
In Java, is-a relation between the type parameters of one class or interface and the type parameters of another are determined by the extends and implements clauses.

Using the Collections classes, ArrayList implements List, and List extends Collection. So ArrayList is a subtype of List, which is a subtype of Collection. The subtyping relationship is preserved between the types automatically. When defining an interface, PayloadList, that associates an optional value of generic type P with each element, its declaration might look like:

The following parameterizations of PayloadList are subtypes of List:

Liskov substitution principle
Liskov substitution principle explains a property, "If for each object o1 of type S there is an object o2 of type T such that for all programs P defined in terms of T, the behavior of P is unchanged when o1 is substituted for o2 then S is a subtype of T,". Following example shows a violation of LSP.

Here is perhaps an example of violation of LSP:

From a programing point of view, the Square class may be implemented by inheriting from the Rectangle class.

However, this violates LSP even though the is-a relationship holds between Rectangle and Square

Consider the following example, where function g does not work if a Square is passed in, and so the open-closed principle might be considered to have been violated.

Conversely, if one considers that the type of a shape should only be a constraint on the relationship of its dimensions, then it is the assumption in g that SetHeight will change height, and area, but not width that is invalid, not just for true squares, but even potentially for other rectangles that might be coded so as to preserve area or aspect ratio when height changes.

title: Is-a


text:
Moses Blah

Moses Zeh Blah (18 April 1947 – 1 April 2013) was a Liberian politician. He served as the 28th vice president of Liberia under President Charles Taylor and became the 23rd president of Liberia on 11 August 2003, following Taylor's resignation. He served as president for two months, until 14 October 2003, when a United Nations-backed transitional government, headed by Gyude Bryant, was established and Bryant was sworn in as Chairman of the Transitional Government of Liberia.

Career
Blah was born in Toweh Town, Liberia, a Gio-speaking hamlet in north-eastern Nimba County, close to the border with the Ivory Coast. He joined with Taylor because of a shared hatred of President Samuel Doe, who had killed Blah's wife along with hundreds of others in an ethnic-related massacre. He trained with Taylor in a Libyan guerrilla camp and served with him as a general during Liberia's civil war in the 1990s. He held the post of ambassador to Libya and Tunisia after Taylor was elected in 1997. In July 2000 Blah was appointed as Vice President after the death of Enoch Dogolea who was rumoured to have been poisoned.

Blah was known as a quiet and unassuming man, driving his own jeep around town rather than using a motorcade and driver, and wearing flowing African robes instead of the normal olive green military uniform. He was constantly annoyed by the presence of bodyguards following him around.

In June 2003, Taylor had left the country for peace talks in Ghana, and while there he was indicted by the war crimes tribunal in Sierra Leone. Blah was urged by the United States to take power from Taylor during his absence, but Blah made no such attempt. After Taylor's return, Blah was held under house arrest for ten days, but was subsequently absolved and reinstated as vice president.

When Taylor resigned in August of that year, Blah briefly succeeded him as president. He was condemned by Liberian rebel groups for his close ties to Taylor; they charged that he would simply continue Taylor's practices. Blah responded by calling the rebels "brothers" and saying "Let bygones be bygones. If there is power, we can share it." He invited the rebels to negotiate in his own house.

On 7 April 2008, Blah said that he had been sent a subpoena to testify at Taylor's trial before the Special Court for Sierra Leone in The Hague. He said that he would testify and "speak the truth", and he testified on 14 May 2008, describing child soldiers and the relationship between Taylor and Foday Sankoh. On 1 February 2009, Blah was accused of taking part in the murder of RUF commander Sam Bockarie, by a witness narrative to the commissioners of Liberia’s Truth and Reconciliation Commission (TRC). The witness, a senior commander of the defunct Anti-Terrorist Unit (ATU) of exiled president Charles Taylor, claimed Vice President Blah was part of the conspiracy and participated in the killing of Bockarie in the town of Tiaplay in Nimba County.

Blah died early on 1 April 2013, at age 65, at the John F. Kennedy Hospital in Monrovia.

title: Moses Blah


text:
Derivation

Derivation may refer to:

Language

 * Morphological derivation, a word-formation process
 * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars

Law

 * Derivative work, in copyright law
 * Derivation proceeding, a proceeding in United States patent law

Music

 * The creation of a derived row, in the twelve-tone musical technique

Science and mathematics

 * Derivation (differential algebra), a unary function satisfying the Leibniz product law
 * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the preceding sentences in the sequence by a rule of inference
 * An after-the-fact justification for an action, in the work of sociologist Vilfredo Pareto

title: Derivation


text:
Leibniz' law

Leibniz' law may refer to:

 * The product rule
 * General Leibniz rule, a generalization of the product rule
 * Identity of indiscernibles

title: Leibniz' law