Cryptography
Cryptography or cryptology is the practice and study of techniques for secure communication in the presence of third parties called adversaries. More cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, electrical engineering, communication science, physics. Applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, military communications. Cryptography prior to the modern age was synonymous with encryption, the conversion of information from a readable state to apparent nonsense; the originator of an encrypted message shares the decoding technique only with intended recipients to preclude access from adversaries. The cryptography literature uses the names Alice for the sender, Bob for the intended recipient, Eve for the adversary. Since the development of rotor cipher machines in World War I and the advent of computers in World War II, the methods used to carry out cryptology have become complex and its application more widespread.
Modern cryptography is based on mathematical theory and computer science practice. It is theoretically possible to break such a system, but it is infeasible to do so by any known practical means; these schemes are therefore termed computationally secure. There exist information-theoretically secure schemes that provably cannot be broken with unlimited computing power—an example is the one-time pad—but these schemes are more difficult to use in practice than the best theoretically breakable but computationally secure mechanisms; the growth of cryptographic technology has raised a number of legal issues in the information age. Cryptography's potential for use as a tool for espionage and sedition has led many governments to classify it as a weapon and to limit or prohibit its use and export. In some jurisdictions where the use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation. Cryptography plays a major role in digital rights management and copyright infringement of digital media.
The first use of the term cryptograph dates back to the 19th century—originating from The Gold-Bug, a novel by Edgar Allan Poe. Until modern times, cryptography referred exclusively to encryption, the process of converting ordinary information into unintelligible form. Decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext. A cipher is a pair of algorithms that create the reversing decryption; the detailed operation of a cipher is controlled both by the algorithm and in each instance by a "key". The key is a secret a short string of characters, needed to decrypt the ciphertext. Formally, a "cryptosystem" is the ordered list of elements of finite possible plaintexts, finite possible cyphertexts, finite possible keys, the encryption and decryption algorithms which correspond to each key. Keys are important both formally and in actual practice, as ciphers without variable keys can be trivially broken with only the knowledge of the cipher used and are therefore useless for most purposes.
Ciphers were used directly for encryption or decryption without additional procedures such as authentication or integrity checks. There are two kinds of cryptosystems: asymmetric. In symmetric systems the same key is used to decrypt a message. Data manipulation in symmetric systems is faster than asymmetric systems as they use shorter key lengths. Asymmetric systems use a public key to encrypt a private key to decrypt it. Use of asymmetric systems enhances the security of communication. Examples of asymmetric systems include RSA, ECC. Symmetric models include the used AES which replaced the older DES. In colloquial use, the term "code" is used to mean any method of encryption or concealment of meaning. However, in cryptography, code has a more specific meaning, it means the replacement of a unit of plaintext with a code word. Cryptanalysis is the term used for the study of methods for obtaining the meaning of encrypted information without access to the key required to do so; some use the terms cryptography and cryptology interchangeably in English, while others use cryptography to refer to the use and practice of cryptographic techniques and cryptology to refer to the combined study of cryptography and cryptanalysis.
English is more flexible than several other languages in which crypto
Natural science
Natural science is a branch of science concerned with the description and understanding of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two main branches: physical science. Physical science is subdivided into branches, including physics, chemistry and earth science; these branches of natural science may be further divided into more specialized branches. In Western society's analytic tradition, the empirical sciences and natural sciences use tools from formal sciences, such as mathematics and logic, converting information about nature into measurements which can be explained as clear statements of the "laws of nature"; the social sciences use such methods, but rely more on qualitative research, so that they are sometimes called "soft science", whereas natural sciences, insofar as they emphasize quantifiable data produced and confirmed through the scientific method, are sometimes called "hard science".
Modern natural science succeeded more classical approaches to natural philosophy traced to ancient Greece. Galileo, Descartes and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives and presuppositions overlooked, remain necessary in natural science. Systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, minerals, so on. Today, "natural history" suggests observational descriptions aimed at popular audiences. Philosophers of science have suggested a number of criteria, including Karl Popper's controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity and quality control, such as peer review and repeatability of findings, are amongst the most respected criteria in the present-day global scientific community; this field encompasses a set of disciplines.
The scale of study can range from sub-component biophysics up to complex ecologies. Biology is concerned with the characteristics and behaviors of organisms, as well as how species were formed and their interactions with each other and the environment; the biological fields of botany and medicine date back to early periods of civilization, while microbiology was introduced in the 17th century with the invention of the microscope. However, it was not until the 19th century. Once scientists discovered commonalities between all living things, it was decided they were best studied as a whole; some key developments in biology were the discovery of genetics. Modern biology is divided into subdisciplines by the type of organism and by the scale being studied. Molecular biology is the study of the fundamental chemistry of life, while cellular biology is the examination of the cell. At a higher level and physiology look at the internal structures, their functions, of an organism, while ecology looks at how various organisms interrelate.
Constituting the scientific study of matter at the atomic and molecular scale, chemistry deals with collections of atoms, such as gases, molecules and metals. The composition, statistical properties and reactions of these materials are studied. Chemistry involves understanding the properties and interactions of individual atoms and molecules for use in larger-scale applications. Most chemical processes can be studied directly in a laboratory, using a series of techniques for manipulating materials, as well as an understanding of the underlying processes. Chemistry is called "the central science" because of its role in connecting the other natural sciences. Early experiments in chemistry had their roots in the system of Alchemy, a set of beliefs combining mysticism with physical experiments; the science of chemistry began to develop with the work of Robert Boyle, the discoverer of gas, Antoine Lavoisier, who developed the theory of the Conservation of mass. The discovery of the chemical elements and atomic theory began to systematize this science, researchers developed a fundamental understanding of states of matter, chemical bonds and chemical reactions.
The success of this science led to a complementary chemical industry that now plays a significant role in the world economy. Physics embodies the study of the fundamental constituents of the universe, the forces and interactions they exert on one another, the results produced by these interactions. In general, physics is regarded as the fundamental science, because all other natural sciences use and obey the principles and laws set down by the field. Physics relies on mathematics as the logical framework for formulation and quantification of principles; the study of the principles of the universe has a long history and derives from direct observation and experimentation. The formulation of theories about the governing laws of the universe has been central to the study of physics from early on, with philosophy yielding to systematic, quantitative experimental testing and observation as the source of verification. Key historical developments in physics include Isaac Newton's theory of universal g
Engineering
Engineering is the application of knowledge in the form of science and empirical evidence, to the innovation, construction and maintenance of structures, materials, devices, systems and organizations. The discipline of engineering encompasses a broad range of more specialized fields of engineering, each with a more specific emphasis on particular areas of applied mathematics, applied science, types of application. See glossary of engineering; the term engineering is derived from the Latin ingenium, meaning "cleverness" and ingeniare, meaning "to contrive, devise". The American Engineers' Council for Professional Development has defined "engineering" as: The creative application of scientific principles to design or develop structures, apparatus, or manufacturing processes, or works utilizing them singly or in combination. Engineering has existed since ancient times, when humans devised inventions such as the wedge, lever and pulley; the term engineering is derived from the word engineer, which itself dates back to 1390 when an engine'er referred to "a constructor of military engines."
In this context, now obsolete, an "engine" referred to a military machine, i.e. a mechanical contraption used in war. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, e.g. the U. S. Army Corps of Engineers; the word "engine" itself is of older origin deriving from the Latin ingenium, meaning "innate quality mental power, hence a clever invention."Later, as the design of civilian structures, such as bridges and buildings, matured as a technical discipline, the term civil engineering entered the lexicon as a way to distinguish between those specializing in the construction of such non-military projects and those involved in the discipline of military engineering. The pyramids in Egypt, the Acropolis and the Parthenon in Greece, the Roman aqueducts, Via Appia and the Colosseum, Teotihuacán, the Brihadeeswarar Temple of Thanjavur, among many others, stand as a testament to the ingenuity and skill of ancient civil and military engineers.
Other monuments, no longer standing, such as the Hanging Gardens of Babylon, the Pharos of Alexandria were important engineering achievements of their time and were considered among the Seven Wonders of the Ancient World. The earliest civil engineer known by name is Imhotep; as one of the officials of the Pharaoh, Djosèr, he designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both military domains; the Antikythera mechanism, the first known mechanical computer, the mechanical inventions of Archimedes are examples of early mechanical engineering. Some of Archimedes' inventions as well as the Antikythera mechanism required sophisticated knowledge of differential gearing or epicyclic gearing, two key principles in machine theory that helped design the gear trains of the Industrial Revolution, are still used today in diverse fields such as robotics and automotive engineering. Ancient Chinese, Greek and Hungarian armies employed military machines and inventions such as artillery, developed by the Greeks around the 4th century BC, the trireme, the ballista and the catapult.
In the Middle Ages, the trebuchet was developed. Before the development of modern engineering, mathematics was used by artisans and craftsmen, such as millwrights, clock makers, instrument makers and surveyors. Aside from these professions, universities were not believed to have had much practical significance to technology. A standard reference for the state of mechanical arts during the Renaissance is given in the mining engineering treatise De re metallica, which contains sections on geology and chemistry. De re metallica was the standard chemistry reference for the next 180 years; the science of classical mechanics, sometimes called Newtonian mechanics, formed the scientific basis of much of modern engineering. With the rise of engineering as a profession in the 18th century, the term became more narrowly applied to fields in which mathematics and science were applied to these ends. In addition to military and civil engineering, the fields known as the mechanic arts became incorporated into engineering.
Canal building was an important engineering work during the early phases of the Industrial Revolution. John Smeaton was the first self-proclaimed civil engineer and is regarded as the "father" of civil engineering, he was an English civil engineer responsible for the design of bridges, canals and lighthouses. He was a capable mechanical engineer and an eminent physicist. Using a model water wheel, Smeaton conducted experiments for seven years, determining ways to increase efficiency. Smeaton introduced iron gears to water wheels. Smeaton made mechanical improvements to the Newcomen steam engine. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of'hydraulic lime' and developed a technique involving dovetailed blocks of granite in the building of the lighthouse, he is important in the history, rediscovery of, development of modern cement, because he identified the compositional requirements needed to obtain "hydraulicity" in lime.
Mathematics
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
World War II
World War II known as the Second World War, was a global war that lasted from 1939 to 1945. The vast majority of the world's countries—including all the great powers—eventually formed two opposing military alliances: the Allies and the Axis. A state of total war emerged, directly involving more than 100 million people from over 30 countries; the major participants threw their entire economic and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. World War II was the deadliest conflict in human history, marked by 50 to 85 million fatalities, most of whom were civilians in the Soviet Union and China, it included massacres, the genocide of the Holocaust, strategic bombing, premeditated death from starvation and disease, the only use of nuclear weapons in war. Japan, which aimed to dominate Asia and the Pacific, was at war with China by 1937, though neither side had declared war on the other. World War II is said to have begun on 1 September 1939, with the invasion of Poland by Germany and subsequent declarations of war on Germany by France and the United Kingdom.
From late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Finland and the Baltic states. Following the onset of campaigns in North Africa and East Africa, the fall of France in mid 1940, the war continued between the European Axis powers and the British Empire. War in the Balkans, the aerial Battle of Britain, the Blitz, the long Battle of the Atlantic followed. On 22 June 1941, the European Axis powers launched an invasion of the Soviet Union, opening the largest land theatre of war in history; this Eastern Front trapped most crucially the German Wehrmacht, into a war of attrition. In December 1941, Japan launched a surprise attack on the United States as well as European colonies in the Pacific. Following an immediate U. S. declaration of war against Japan, supported by one from Great Britain, the European Axis powers declared war on the U.
S. in solidarity with their Japanese ally. Rapid Japanese conquests over much of the Western Pacific ensued, perceived by many in Asia as liberation from Western dominance and resulting in the support of several armies from defeated territories; the Axis advance in the Pacific halted in 1942. Key setbacks in 1943, which included a series of German defeats on the Eastern Front, the Allied invasions of Sicily and Italy, Allied victories in the Pacific, cost the Axis its initiative and forced it into strategic retreat on all fronts. In 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained its territorial losses and turned toward Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in Central China, South China and Burma, while the Allies crippled the Japanese Navy and captured key Western Pacific islands; the war in Europe concluded with an invasion of Germany by the Western Allies and the Soviet Union, culminating in the capture of Berlin by Soviet troops, the suicide of Adolf Hitler and the German unconditional surrender on 8 May 1945.
Following the Potsdam Declaration by the Allies on 26 July 1945 and the refusal of Japan to surrender under its terms, the United States dropped atomic bombs on the Japanese cities of Hiroshima and Nagasaki on 6 and 9 August respectively. With an invasion of the Japanese archipelago imminent, the possibility of additional atomic bombings, the Soviet entry into the war against Japan and its invasion of Manchuria, Japan announced its intention to surrender on 15 August 1945, cementing total victory in Asia for the Allies. Tribunals were set up by fiat by the Allies and war crimes trials were conducted in the wake of the war both against the Germans and the Japanese. World War II changed the political social structure of the globe; the United Nations was established to foster international co-operation and prevent future conflicts. The Soviet Union and United States emerged as rival superpowers, setting the stage for the nearly half-century long Cold War. In the wake of European devastation, the influence of its great powers waned, triggering the decolonisation of Africa and Asia.
Most countries whose industries had been damaged moved towards economic expansion. Political integration in Europe, emerged as an effort to end pre-war enmities and create a common identity; the start of the war in Europe is held to be 1 September 1939, beginning with the German invasion of Poland. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred and the two wars merged in 1941; this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935; the British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the fo
Vehicle routing problem
The vehicle routing problem is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?". It generalises the well-known travelling salesman problem, it first appeared in a paper by George Dantzig and John Ramser in 1959, in which first algorithmic approach was written and was applied to petrol deliveries. The context is that of delivering goods located at a central depot to customers who have placed orders for such goods; the objective of the VRP is to minimize the total route cost. In 1964, Clarke and Wright improved on Dantzig and Ramser's approach using an effective greedy approach called the savings algorithm. Determining the optimal solution to VRP is NP-hard, so the size of problems that can be solved, using mathematical programming or combinatorial optimization may be limited. Therefore, commercial solvers tend to use heuristics due to the size and frequency of real world VRPs they need to solve.
The VRP has many obvious applications in industry. In fact, the use of computer optimization programs can give savings of 5% to a company as transportation is a significant component of the cost of a product - indeed, the transportation sector makes up 10% of the EU's GDP. Any savings created by the VRP less than 5%, are significant; the VRP concerns the service of a delivery company. How things are delivered from one or more depots which has a given set of home vehicles and operated by a set of drivers who can move on a given road network to a set of customers, it asks for a determination of a set of routes, S, such that all customers' requirements and operational constraints are satisfied and the global transportation cost is minimized. This cost may be monetary, distance or otherwise; the road network can be described using a graph where the arcs are roads and vertices are junctions between them. The arcs may be directed or undirected due to the possible presence of one way streets or different costs in each direction.
Each arc has an associated cost, its length or travel time which may be dependent on vehicle type. To know the global cost of each route, the travel cost and the travel time between each customer and the depot must be known. To do this our original graph is transformed into one where the vertices are the customers and depot, the arcs are the roads between them; the cost on each arc is the lowest cost between the two points on the original road network. This is easy to do as shortest path problems are easy to solve; this transforms the sparse original graph into a complete graph. For each pair of vertices i and j, there exists an arc of the complete graph whose cost is written as C i j and is defined to be the cost of shortest path from i to j; the travel time t i j is the sum of the travel times of the arcs on the shortest path from i to j on the original road graph. Sometimes it is impossible to satisfy all of a customer's demands and in such cases solvers may reduce some customers' demands or leave some customers unserved.
To deal with these situations a priority variable for each customer can be introduced or associated penalties for the partial or lack of service for each customer given The objective function of a VRP can be different depending on the particular application of the result but a few of the more common objectives are: Minimize the global transportation cost based on the global distance travelled as well as the fixed costs associated with the used vehicles and drivers Minimize the number of vehicles needed to serve all customers Least variation in travel time and vehicle load Minimize penalties for low quality service Several variations and specializations of the vehicle routing problem exist: Vehicle Routing Problem with Pickup and Delivery: A number of goods need to be moved from certain pickup locations to other delivery locations. The goal is to find optimal routes for a fleet of vehicles to visit the pickup and drop-off locations. Vehicle Routing Problem with LIFO: Similar to the VRPPD, except an additional restriction is placed on the loading of the vehicles: at any delivery location, the item being delivered must be the item most picked up.
This scheme reduces the loading and unloading times at delivery locations because there is no need to temporarily unload items other than the ones that should be dropped off. Vehicle Routing Problem with Time Windows: The delivery locations have time windows within which the deliveries must be made. Capacitated Vehicle Routing Problem: CVRP or CVRPTW; the vehicles have limited carrying capacity of the goods. Vehicle Routing Problem with Multiple Trips: The vehicles can do more than one route. Open Vehicle Routing Problem: Vehicles are not required to return to the depot. Several software vendors have built software products to solve the various VRP problems. Numerous articles are available for more detail on their research and results. Although VRP is related to the Job Shop Scheduling Problem, the two problems are solved using different techniques. There are three main different approaches to modelling the VRP Vehicle flow formulations—this uses integer variables associated with each arc that count the number of times that the edge is traversed by a vehicle.
It is used for basic VRPs. This is good for cases where the solution cost can be expressed as the sum of any costs associat
Mathematical analysis
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, analytic functions. These theories are studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Analysis may be distinguished from geometry. Mathematical analysis formally developed in 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. Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of 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 AD to find the area of a circle. Zu Chongzhi established a method that would be called Cavalieri's principle to find the volume of a sphere in the 5th century; the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series, his followers at 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. Descartes and Fermat independently developed analytic geometry, a few decades 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 the calculus of variations and partial differential equations, Fourier analysis, 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 mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become 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 used in earlier work by Euler. Instead, Cauchy formulated calculus in terms of geometric infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y, he introduced the concept of the Cauchy sequence, started the formal theory of complex analysis. Poisson, Liouville and others studied partial differential equations and harmonic analysis; the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis.
In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, introduced the "epsilon-delta" definition of limit. Mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind 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, developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions. "monsters" began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, Baire proved the Baire category theorem.
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, Hilbert introduced Hilbert spaces to solve integral equations; the idea of normed vector space was in the air, in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. Much of analysis happens in some metric space. Examples of analysis without a metric include functional analysis. Formally, a metric space is an ordered pair where M is a set
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