The focal length of an optical system is a measure of how the system converges or diverges light. A positive focal length indicates that a system converges light, while a negative focal length indicates that the system diverges light. A system with a shorter focal length bends the rays more bringing them to a focus in a shorter distance or diverging them more quickly. For the special case of a thin lens in air, a positive focal length is the distance over which collimated rays are brought to a focus, or alternatively a negative focal length indicates how far in front of the lens a point source must be located to form a collimated beam. For more general optical systems, the focal length has no intuitive meaning. In most photography and all telescopy, where the subject is infinitely far away, longer focal length leads to higher magnification and a narrower angle of view. On the other hand, in applications such as microscopy in which magnification is achieved by bringing the object close to the lens, a shorter focal length leads to higher magnification because the subject can be brought closer to the center of projection.

For a thin lens in air, the focal length is the distance from the center of the lens to the principal foci of the lens. For a converging lens, the focal length is positive, is the distance at which a beam of collimated light will be focused to a single spot. For a diverging lens, the focal length is negative, is the distance to the point from which a collimated beam appears to be diverging after passing through the lens; when a lens is used to form an image of some object, the distance from the object to the lens u, the distance from the lens to the image v, the focal length f are related by 1 f = 1 u + 1 v. The focal length of a thin convex lens can be measured by using it to form an image of a distant light source on a screen; the lens is moved. In this case 1/u is negligible, the focal length is given by f ≈ v. Determining the focal length of a concave lens is somewhat more difficult; the focal length of such a lens is considered that point at which the spreading beams of light would meet before the lens if the lens were not there.

No image is formed during such a test, the focal length must be determined by passing light through the lens, examining how much that light becomes dispersed/ bent, following the beam of light backwards to the lens's focal point. For a thick lens, or an imaging system consisting of several lenses or mirrors, the focal length is called the effective focal length, to distinguish it from other used parameters: Front focal length or front focal distance is the distance from the front focal point of the system to the vertex of the first optical surface. Back focal length or back focal distance is the distance from the vertex of the last optical surface of the system to the rear focal point. For an optical system in air, the effective focal length gives the distance from the front and rear principal planes to the corresponding focal points. If the surrounding medium is not air the distance is multiplied by the refractive index of the medium; some authors call these distances the front/rear focal lengths, distinguishing them from the front/rear focal distances, defined above.

In general, the focal length or EFL is the value that describes the ability of the optical system to focus light, is the value used to calculate the magnification of the system. The other parameters are used in determining where an image will be formed for a given object position. For the case of a lens of thickness d in air, surfaces with radii of curvature R1 and R2, the effective focal length f is given by the Lensmaker's equation: 1 f =, where n is the refractive index of the lens medium; the quantity 1/f is known as the optical power of the lens. The corresponding front focal distance is: FFD = f, {\displaystyle =f\left(1+{\frac {

The modern bilateral relationship between Russia and Ukraine formally started during the World War I as the former Russian Empire was going through its political reform. In 1920 the bilateral relationship between two countries was changed as Ukraine was conquered by the Russian Red Army. In 1990s upon the dissolution of the Soviet Union, of which both Soviet Russia and Soviet Ukraine had been formally founding constituent republics, bilateral relations were revived. Relations between the two countries collapsed since 2014 Ukrainian revolution, followed by Russia's annexation of Crimea from Ukraine, over Russia's backing for the separatists fighters of the Donetsk People's Republic and Luhansk People's Republic in a war that by early 2020 had killed more than 13,000 people and brought Western sanctions on Russia. Interactions between the two areas of Russia and Ukraine developed on a formal basis from the 17th century, but international-level relations ceased when Catherine the Great liquidated the autonomy of the Cossack Hetmanate in 1764.

For a short period of time soon after the communist 1917 October Revolution two states interacted again. In 1920, Soviet Russian forces overran Ukraine and relations between the two states transitioned from international to internal ones within the Soviet Union, founded in 1922. After the Soviet Union's dissolution in 1991, Ukraine have undergone periods of ties and outright hostility from Russia. Prior to Euromaidan, under President of Ukraine Viktor Yanukovych, relations were cooperative, with various trade agreements in place. On 1 March 2014, the Federation Council of the Russian Federal Assembly voted unanimously to allow the President of Russia enter the Russian Armed Forces on territory of Ukraine. On 3 March 2014, the Russian representative to the United Nations, Vitaly Churkin showed a letter signed by former Ukrainian President Yanukovych on 1 March 2014 and addressed to President of the Russian Federation Vladimir Putin with a request to enter the Russian Armed Forces on territory of Ukraine.

During the February–March 2014 Crimean crisis Ukraine lost control of its government buildings and military bases in Crimea to unmarked soldiers and local pro-Russian militias. This started on 27 February; the same day the Crimean parliament replaced the local government with one who wanted Crimean unification with Russia. This government organized the 2014 Crimean status referendum on 14 March 2014 in which the voters voted overwhelmingly to join Russia. On 17 March 2014, Crimea declared its independence. On 18 March 2014, a treaty on incorporating Crimea and Sevastopol into Russia was signed in Moscow and in five days the "Constitutional Law on admitting to the Russian Federation the Republic of Crimea and establishing within the Russian Federation the New Constituent Entities the Republic of Crimea and the City of Federal Importance Sevastopol" was pushed through the Russian parliament, signed by the Russian President and entered into force. On 19 March 2014 all Armed Forces of Ukraine are withdrawn from Crimea.

On 17 April 2014, President Putin stated that the Russian military had backed Crimean separatist militias, stating that Russia's intervention was necessary "to ensure proper conditions for the people of Crimea to be able to express their will". Throughout March and April 2014, pro-Russian unrest spread in Ukraine, with pro-Russian groups proclaiming "People's Republics" in the oblasts of Donetsk and Luhansk, as of 2017 both outside the control of the Ukrainian government. In response, Ukraine initiated multiple international-court litigations against Russia, as well as suspending all types of military cooperation and military exports. Many countries and international organizations applied sanctions against the Russian Federation and against Ukrainian citizens involved in and responsible for the escalation. Military clashes between pro-Russian rebels and the Ukrainian Armed Forces began in the east of Ukraine in April 2014. On 5 September 2014 the Ukrainian government and representatives of the self-proclaimed Donetsk People's Republic and the Luhansk People's Republic signed a tentative truce.

The ceasefire imploded amidst intense new fighting in January 2015. A new ceasefire agreement has operated since mid-February 2015, but this agreement failed to stop the fighting. In January 2018 the Verkhovna Rada passed a law defining areas seized by the Donetsk People's Republic and the Luhansk People's Republic as "temporarily occupied by Russia", the law called Russia an "aggressor" state. Russia has been accused by NATO and Ukraine of engaging in direct military operations to support the Donetsk People's Republic and the Luhansk People's Republic. Russia denies this, but in December 2015, Russian Federation President Vladimir Putin admitted that Russian military intelligence officers were operating in Ukraine, insisting though that they were not the same as regular troops. Russia has admitted. On 10 February 2015, in response to Russian military intervention, the Ukrainian parliament registered a draft decree on suspending diplomatic relations with Russian Federation. Although this suspension did not materialize, Ukrainian official Dmytro Kuleba acknowledged early April 2016 that diplomatic relations had been reduced "almost to zero".

Late 2017 Ukrainian Foreign Minister Pavlo Klimkin stated that "there are no diplomatic relations with Russia in

The American Committee for the Defense of Leon Trotsky was a pseudo-judicial process set up by American Trotskyists as a front organization following the first of the Moscow Trials. It had official imprimatur from any government, it was composed of historians, journalists and other notable figures, including Edmund Wilson, Suzanne La Follette, Louis Hacker, Norman Thomas, John Dos Passos, Reinhold Niebuhr, George Novack, Franz Boas, John Chamberlain and Sidney Hook. John Dewey seventy-eight years old, agreed to head its Commission of Inquiry. In March 1937, The American Committee for the Defense of Leon Trotsky initiated the so-called Dewey Commission; the inquiry was named after John Dewey. Its other members were an authority on Latin-American affairs. A sub-commission, comprising the first five commission members listed above, conducted thirteen hearings at Trotsky's home in Coyoacan, Mexico, D. F. from April 10 to April 17, 1937. Leon Trotsky was defended by the lawyer Albert Goldman. John Finerty acted as the commission's legal counsel.

During the course of the "trial", committee member Mauritz A. Hallgren an editor of The Nation magazine, made headlines with a public resignation from the committee published in the pages of the New York Times. Hallgren charged that the American Committee for the Defense of Leon Trotsky had "become an instrument of the Trotskyists for political intervention against the Soviet Union." Hallgren's January 27, 1937, letter of resignation was published as a 1-cent pamphlet by the Communist Party's International Publishers. Albert Einstein, although noting that Trotsky deserved the opportunity to prove his innocence, was critical of the Dewey inquiry: "The question is raised because Trotsky is an active and adroit politician, who might well search for an effective platform for the presentation and promulgation of his political goals in the public sphere.... I'm afraid that the only result would be Trotsky's own self-promotion without the possibility of a well-grounded judgment." Following months of investigation, the Dewey Commission made its findings public in New York on September 21, 1937.

The commission purported to clear Trotsky of all charges made during the Moscow Trials and, exposed the scale of the alleged frame-up of all other defendants during these trials. Among its conclusions, it stated that "the conduct of the Moscow trials was such as to convince any unprejudiced person that no effort was made to ascertain the truth. Leon Trotsky, I Stake My Life! Trotsky's Address to the NY Hippodrome Meeting. New York: Pioneer Publishers, 1937. Mass Meeting Called by the American Committee for the Defense of Leon Trotsky, to Answer His Accusers - At the Hippodrome, New York City, February 5th, 1937: Stenographic Report. New York: American Committee for the Defense of Leon Trotsky, 1937