SS-N-22 Sunburn is the NATO reporting name for two unrelated Soviet anti-ship missiles . Although the missiles were very different, distinguishing between them is difficult because their ship-mounted launching containers were identical. Confusion was exacerbated by the Soviet practice of mixing missile types within a class of ships. It was therefore not confirmed that the "SS-N-22" actually referenced two different missiles until after the fall of the Soviet Union .
55-664: One of the SS-N-22s was the P-80 Zubr , designed by a team led by Vladimir Chelomei . The Zubr was rocket-propelled and armed with a 250 kilogram warhead. It was carried by early-model Sovremenny -class destroyers and Tarantul -class corvettes . The submarine-launched version of this missile was also designated as an SS-N-22 Sunburn by NATO but was known to the Soviets as the P-100 Oniks . The other, unrelated SS-N-22
110-399: A consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure . This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ {\displaystyle \sigma } -algebra . This means that
165-460: A set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space , which assigns the conventional length , area , and volume of Euclidean geometry to suitable subsets of
220-555: A decision by the USSR State Defense Committee and People's Commissar for Aviation Industry Alexey Shakhurin , Chelomey was appointed the Director and Chief Designer of Plant N51 (its previous director Nikolay Polikarpov having died a short time before). Chelomey was to design, build, and test the first Soviet cruise missile as soon as possible. As early as December 1944, the missile, code-named 10Kh ,
275-428: A measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X {\displaystyle X} . It must assign 0 to the empty set and be ( countably ) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate
330-410: A metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance). Formally, a metric space is an ordered pair ( M , d ) {\displaystyle (M,d)} where M {\displaystyle M} is a set and d {\displaystyle d}
385-552: A professor at the School in 1952. In 1958, OKB-52 put forward a proposal for a multi-stage Intercontinental ballistic missile . Although their UR-200 rocket design was rejected in favour of Mikhail Yangel 's R-36 (NATO designation SS-9 Scarp ), their UR-100 design was accepted. Chelomey's OKB was part of the General Machine-Building Ministry headed by Sergey Afanasyev . In 1959, Chelomey
440-427: A sequence is convergence . Informally, a sequence converges if it has a limit . Continuing informally, a ( singly-infinite ) sequence has a limit if it approaches some point x , called the limit, as n becomes very large. That is, for an abstract sequence ( a n ) (with n running from 1 to infinity understood) the distance between a n and x approaches 0 as n → ∞, denoted Real analysis (traditionally,
495-575: A warhead of 320 kg. Another version of this weapon is called the ASM-MMS or Kh-41; it is intended to be launched by a Su-33 . The People's Republic of China acquired SS-N-22 launchers and missiles (specifically, the for-export 3M-80E Moskit variant) with a 1999–2000 purchase of two Sovremenny destroyers from Russia.According to Russia, the PRC funded the development of the SS-N-22 version for
550-561: Is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders . Differential equations play a prominent role in engineering , physics , economics , biology , and other disciplines. Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives)
605-537: Is a metric on M {\displaystyle M} , i.e., a function such that for any x , y , z ∈ M {\displaystyle x,y,z\in M} , the following holds: By taking the third property and letting z = x {\displaystyle z=x} , it can be shown that d ( x , y ) ≥ 0 {\displaystyle d(x,y)\geq 0} ( non-negative ). A sequence
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#1732783869397660-512: Is a branch of mathematical analysis dealing with vector-valued functions . Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have. Techniques from analysis are also found in other areas such as: The vast majority of classical mechanics , relativity , and quantum mechanics
715-402: Is an ordered list. Like a set , it contains members (also called elements , or terms ). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers . One of the most important properties of
770-558: Is based on applied analysis, and differential equations in particular. Examples of important differential equations include Newton's second law , the Schrödinger equation , and the Einstein field equations . Functional analysis is also a major factor in quantum mechanics . When processing signals, such as audio , radio waves , light waves, seismic waves , and even images, Fourier analysis can isolate individual components of
825-515: Is known or postulated. This is illustrated in classical mechanics , where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion ) may be solved explicitly. A measure on
880-499: Is now called naive set theory , and Baire proved the Baire category theorem . In the early 20th century, calculus was formalized using an axiomatic set theory . Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration , which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations . The idea of normed vector space
935-459: Is now known as Rolle's theorem . In the 14th century, Madhava of Sangamagrama developed infinite series expansions, now called Taylor series , of functions such as sine , cosine , tangent and arctangent . Alongside his development of Taylor series of trigonometric functions , he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at
990-404: Is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions ). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation , complex analysis is widely applicable to two-dimensional problems in physics . Functional analysis is a branch of mathematical analysis, the core of which is formed by
1045-542: Is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations ) for the problems of mathematical analysis (as distinguished from discrete mathematics ). Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Numerical analysis naturally finds applications in all fields of engineering and
1100-458: The n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . For instance, the Lebesgue measure of the interval [ 0 , 1 ] {\displaystyle \left[0,1\right]} in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically,
1155-465: The Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. This began when Fermat and Descartes developed analytic geometry , which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and
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#17327838693971210-630: The Kyiv University . He also attended lectures by Tullio Levi-Civita in the Ukrainian SSR Academy of Sciences. Namely in this time Chelomey became interested in mechanics and in the theory of oscillations and remained interested the rest of his life. In 1937, Chelomey graduated from the institute with honours. After that he worked there as a lecturer, defending a dissertation for the Candidate of Science (in 1939). From
1265-700: The N1 should be fueled with RP-1 / LOX or Hypergolic propellant resulted in Glushko and Korolev refusing to work with each other, causing Glushko to instead offer his RD-253 rocket engine to Chelomey, who adopted it for his UR-500 . On 3 August 1964 the Central Committee of the Communist Party and the USSR Council of Ministers adopted and signed Decree #655-268 On Work on Research on
1320-480: The People's Liberation Army Navy (PLAN). It has the designation 3M-80MBE, and this version differs from earlier versions mainly in having increased range (now beyond 220 km; 240 km has been quoted). These new missiles will be first installed on board the second pair of Sovremenny class destroyers . A total of 500 SS-N-22 AShM were ordered by the PRC for the four Russian-built Sovremenny class destroyers . With
1375-750: The UR-500 (also known as Proton ) took place in early 1965. Although it was never used to send cosmonauts to the Moon as Chelomey had hoped, Proton became the staple heavy lift launch vehicle of the Soviet/Russian fleet and would be used over the years for planetary probes, space stations, geosynchronous satellites, and more. Chelomey's OKB also designed anti-satellite weapons such as Polyot . Unlike earlier satellites, Chelomey's Polyot-1 (1963) and Polyot-2 (1964) were equipped with an propulsion bus which enabled them to change their orbits. He also headed
1430-408: The calculus of variations , ordinary and partial differential equations , Fourier analysis , and generating functions . During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of a mathematical function . Real analysis began to emerge as an independent subject when Bernard Bolzano introduced
1485-471: The "Moon race". Chelomey proposed that the powerful UR-500 be used to launch a small two-man craft on a lunar flyby, and managed to gain support for his proposal by employing Nikita Khrushchev 's son, Sergei Khrushchev. He also claimed the UR-500 could be used to launch a military space station. An argument between Sergey Korolev and rocket engine designer Valentin Glushko over personal issues and whether
1540-441: The "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers . It is useful in many branches of mathematics, including algebraic geometry , number theory , applied mathematics ; as well as in physics , including hydrodynamics , thermodynamics , mechanical engineering , electrical engineering , and particularly, quantum field theory . Complex analysis
1595-480: The "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences , including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity , smoothness and related properties of real-valued functions. Complex analysis (traditionally known as
1650-631: The 17th century during the Scientific Revolution , but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics . For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy . (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of
1705-488: The 1980s Chelomey's OKB proposed non-realised 15-ton Uragan spaceplane based on Zenit-2 launcher. SS-N-3 Shaddock He is reported to have led the construction of the P-80 Zubr anti-ship missile, and it may be presumed that he was involved in the related P-100 Oniks missile. Chelomey died in Moscow on 8 December 1984 at the age of 70, suffering a fatal stroke while on the phone with his wife. He had been admitted to
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1760-410: The Moon and Outer Space , which redefined Chelomey's and Korolev's roles in the space program: Korolev was now responsible for development of the N1 , which was chosen to accomplish a crewed lunar landing , while Chelomei was assigned to the development of the UR-500 which was chosen to perform a crewed circumlunar flight . The projects continued to work separately side-by-side. The first launch of
1815-463: The Soviet ekranoplan Lun . Vladimir Chelomei Vladimir Nikolayevich Chelomey or Chelomei ( Russian : Влади́мир Никола́евич Челоме́й , Ukrainian : Володи́мир Микола́йович Челоме́й ; 30 June 1914 – 8 December 1984) was a Soviet engineer and designer in the missile program of the former Soviet Union . He invented the first Soviet pulse jet engine and was responsible for developing
1870-446: The attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. Also, various pathological objects , (such as nowhere continuous functions , continuous but nowhere differentiable functions , and space-filling curves ), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure , Cantor developed what
1925-697: The beginning of the Great Patriotic War in 1941, Chelomey worked at the Baranov Central Institute of Aviation Motor Building ( TsIAM ) in Moscow , where he created the first Soviet pulsating air jet engine in 1942, independently of similar contemporary developments in Nazi Germany . In the summer of 1944, it became known that Nazi Germany used V-1 cruise missiles against Southern England . On 9 October 1944, following
1980-572: The concept of the Cauchy sequence , and started the formal theory of complex analysis . Poisson , Liouville , Fourier and others studied partial differential equations and harmonic analysis . The contributions of these mathematicians and others, such as Weierstrass , developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration , and made significant advances in complex analysis. Towards
2035-586: The concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems , a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. From Jain literature, it appears that Hindus were in possession of
2090-439: The context of real and complex numbers and functions . Analysis evolved from calculus , which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry ; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space ) or specific distances between objects (a metric space ). Mathematical analysis formally developed in
2145-720: The development of the Proton satellite . In the 1970s Chelomey's OKB proposed non-realised Proton-based 20-ton LKS (Kosmolyot) spaceplane and worked on the Almaz military orbital stations (flown as Salyut 2 , Salyut 3 and Salyut 5 ) which also became the basis for the Salyut, Mir and Zvezda civil space stations. To support his Almaz stations, Chelomey designed the TKS , as a large alternative to Soyuz . The TKS never flew crewed as planned but derivatives flew as modules on Salyut 7 and Mir. In
2200-410: The empty set, countable unions , countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice . Numerical analysis
2255-489: The end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts , in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions . Around that time,
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2310-629: The exception of the first 20, these are the 3M-80MBE variant. The PRC has stockpiled roughly 15 SS-N-22 missiles per launch tube (each destroyer having a total of 8 launchers in two quadruple configurations). It is speculated that the PLAN intends to use the missiles against carrier battle groups deployed by the United States Navy in the event of a confrontation with the Republic of China ( Taiwan ). Six SS-N-22 launchers were also present on
2365-483: The formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in 433 BCE . Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what
2420-408: The hospital three days prior after breaking his leg in his dacha. He was buried at Novodevichy Cemetery . Mathematical analysis Analysis is the branch of mathematics dealing with continuous functions , limits , and related theories, such as differentiation , integration , measure , infinite sequences , series , and analytic functions . These theories are usually studied in
2475-520: The modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals . Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y . He also introduced
2530-469: The physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. Vector analysis , also called vector calculus ,
2585-570: The study of differential and integral equations . Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves . This includes the study of the notions of Fourier series and Fourier transforms ( Fourier analysis ), and of their generalizations. Harmonic analysis has applications in areas as diverse as music theory , number theory , representation theory , signal processing , quantum mechanics , tidal analysis , and neuroscience . A differential equation
2640-625: The study of vector spaces endowed with some kind of limit-related structure (e.g. inner product , norm , topology , etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous , unitary etc. operators between function spaces. This point of view turned out to be particularly useful for
2695-480: The tangents of curves. Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system , is considered to be the establishment of mathematical analysis. It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus , which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as
2750-743: The world's first anti-ship cruise missiles and the ICBM program of the Soviet Union such as the UR-100 , UR-200 , UR-500 and UR-700 . Chelomey was born to a Ukrainian family in Siedlce , Lublin Governorate , Russian Empire (now Poland ). At the age of three months, his family fled to Poltava , Ukraine , when the Eastern Front of World War I came close to Siedlce. When Chelomey
2805-553: Was 12 years old, the family moved again to Kyiv . In 1932, Chelomey was admitted to the Kyiv Polytechnic Institute (later the basis of Kyiv Aviation Institute), where he showed himself as a student with outstanding talent. In 1936, his first book Vector Analysis was published. Studying at the institute, Chelomey also attended lectures on mathematical analysis , theory of differential equations , mathematical physics , theory of elasticity and mechanics in
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#17327838693972860-563: Was appointed the Chief Designer of Aviation Equipment. OKB-52, along with designing ICBMs, started to work on spacecraft, and in 1961 began work on a design for a much more powerful ICBM, the UR-500 , although it was rather quickly rejected as impractical to use as a missile. In 1962, Chelomey became an Academician of the USSR Academy of Sciences , Mechanics Department. Chelomey became Sergey Korolev 's internal competitor in
2915-412: Was in the air, and in the 1920s Banach created functional analysis . In mathematics , a metric space is a set where a notion of distance (called a metric ) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line , the complex plane , Euclidean space , other vector spaces , and the integers . Examples of analysis without
2970-632: Was test fired from Petlyakov Pe-8 and Tupolev Tu-2 aircraft. Following his success with the 10Kh, the USSR Special Design Bureau on designing pilot-less aircraft (OKB-52) was established under Chelomey's. In 1955, Chelomey was appointed the Chief Designer of the OKB-52 , where he continued to work on cruise missiles. Chelomey continued his scientific research, earning a doctorate in science from Bauman Moscow Higher Technical School . After his dissertation defense in 1951, he became
3025-499: Was the Raduga P-270 Moskit . It was ramjet -propelled (though launched by a small solid-fuel rocket), and was carried by later-model Sovremenny class destroyers , Tarantul class corvettes , and several smaller warships. This weapon has a top speed of Mach 3. The high speed of the missile means a typical response time for the target of only 25 to 30 seconds, giving a target little time to react. Moskit can be armed with
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