1.
Vehicle routing problem
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It generalises the well-known travelling salesman problem. It first appeared in a paper by George Dantzig and John Ramser in 1959, often, 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 Ramsers approach using an effective greedy approach called the savings algorithm. Determining the optimal solution is an NP-hard problem in combinatorial optimization, the commercial solvers therefore tend to use heuristics due to the size & frequency of real world VRPs they need to solve. The VRP has many applications in industry. Consequently, any savings created by the VRP, even less than 5%, are significant, the VRP concerns the service of a delivery company. How things are delivered one or more depots which has a given set of home vehicles. It asks for a determination of a set of routes, S, such that all requirements and operational constraints are satisfied. 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 presence of one way streets or different costs in each direction. Each arc has an associated cost which is generally 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 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 original graph into a complete graph. For each pair of vertices i and j, there exists an arc of the 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 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 customers demands, the goal is to find optimal routes for a fleet of vehicles to visit the pickup and drop-off locations. 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
2.
Combinatorial optimization
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In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not feasible and it operates on the domain of those optimization problems, in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Some common problems involving combinatorial optimization are the travelling salesman problem, Combinatorial optimization is a subset of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in fields, including artificial intelligence, machine learning, mathematics, auction theory. It often involves determining the way to allocate resources used to find solutions to mathematical problems. Some examples of combinatorial optimization problems that fall into this framework are shortest paths and shortest path trees, flows and circulations, spanning trees, matching, however, generic search algorithms are not guaranteed to find an optimal solution, nor are they guaranteed to run quickly. Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, cook, William J. Cunningham, William H. Pulleyblank, William R. Schrijver, Alexander. Crescenzi, Pierluigi, Kann, Viggo, Halldórsson, Magnús, Karpinski, Marek, Woeginger, a Compendium of NP Optimization Problems. Das, Arnab, Chakrabarti, Bikas K, eds, Quantum Annealing and Related Optimization Methods. Das, Arnab, Chakrabarti, Bikas K. Colloquium, Quantum annealing, a First Course in Combinatorial Optimization. On the history of combinatorial optimization, in Aardal, K. Nemhauser, G. L. Weismantel, R. Handbook of Discrete Optimization. Journal of Combinatorial Optimization The Aussois Combinatorial Optimization Workshop Java Combinatorial Optimization Platform
3.
Integer programming
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An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to linear programming, in which the objective function. A special case, 0-1 integer linear programming, in which unknowns are binary, the blue lines together with the coordinate axes define the polyhedron of the LP relaxation, which is given by the inequalities without the integrality constraint. The goal of the optimization is to move the black dotted line as far upward while still touching the polyhedron, the optimal solutions of the integer problem are the points and which both have an objective value of 2. The unique optimum of the relaxation is with objective value of 2.8, note that if the solution of the relaxation is rounded to the nearest integers, it is not feasible for the ILP. The following is a reduction from minimum vertex cover to integer programming that will serve as the proof of NP-hardness, let G = be an undirected graph. The first constraint implies that at least one end point of every edge is included in this subset, therefore, the solution describes a vertex cover. Additionally given some vertex cover C, y v can be set to 1 for any v ∈ C, thus we can conclude that if we minimize the sum of y v we have also found the minimum vertex cover. Mixed integer linear programming problems in which only some of the variables, x i, are constrained to be integers. Zero-one linear programming problems in which the variables are restricted to be either 0 or 1. Note that any bounded integer variable can be expressed as a combination of binary variables, there are two main reasons for using integer variables when modeling problems as a linear program, The integer variables represent quantities that can only be integer. For example, it is not possible to build 3.7 cars, the integer variables represent decisions and so should only take on the value 0 or 1. These considerations occur frequently in practice and so integer linear programming can be used in many applications areas, mixed integer programming has many applications in industrial production, including job-shop modelling. One important example happens In agricultural production planning involves determining production yield for crops that can share resources. A possible objective is to maximize the production, without exceeding the available resources. In some cases, this can be expressed in terms of a linear program and these problems involve service and vehicle scheduling in transportation networks. For example, a problem may involve assigning buses or subways to individual routes so that a timetable can be met, here binary decision variables indicate whether a bus or subway is assigned to a route and whether a driver is assigned to a particular train or subway. The goal of these problems is to design a network of lines to install so that a set of communication requirements are met
4.
Mathematical model
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A mathematical model is a description of a system using mathematical concepts and language. The process of developing a model is termed mathematical modeling. Mathematical models are used in the sciences and engineering disciplines. Physicists, engineers, statisticians, operations research analysts, and economists use mathematical models most extensively, a model may help to explain a system and to study the effects of different components, and to make predictions about behaviour. Mathematical models can take many forms, including systems, statistical models, differential equations. These and other types of models can overlap, with a model involving a variety of abstract structures. In general, mathematical models may include logical models, in many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed, in the physical sciences, the traditional mathematical model contains four major elements. These are Governing equations Defining equations Constitutive equations Constraints Mathematical models are composed of relationships. Relationships can be described by operators, such as operators, functions, differential operators. Variables are abstractions of system parameters of interest, that can be quantified, a model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, for example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, an equation is said to be linear if it can be written with linear differential operators. In a mathematical programming model, if the functions and constraints are represented entirely by linear equations. If one or more of the functions or constraints are represented with a nonlinear equation. Nonlinearity, even in simple systems, is often associated with phenomena such as chaos. Although there are exceptions, nonlinear systems and models tend to be difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be if one is trying to study aspects such as irreversibility
5.
Science
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Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The formal sciences are often excluded as they do not depend on empirical observations, disciplines which use science, like engineering and medicine, may also be considered to be applied sciences. However, during the Islamic Golden Age foundations for the method were laid by Ibn al-Haytham in his Book of Optics. In the 17th and 18th centuries, scientists increasingly sought to formulate knowledge in terms of physical laws, over the course of the 19th century, the word science became increasingly associated with the scientific method itself as a disciplined way to study the natural world. It was during this time that scientific disciplines such as biology, chemistry, Science in a broad sense existed before the modern era and in many historical civilizations. Modern science is distinct in its approach and successful in its results, Science in its original sense was a word for a type of knowledge rather than a specialized word for the pursuit of such knowledge. In particular, it was the type of knowledge which people can communicate to each other, for example, knowledge about the working of natural things was gathered long before recorded history and led to the development of complex abstract thought. This is shown by the construction of calendars, techniques for making poisonous plants edible. For this reason, it is claimed these men were the first philosophers in the strict sense and they were mainly speculators or theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature was seen by scientists as a more appropriate interest for lower class artisans. A clear-cut distinction between formal and empirical science was made by the pre-Socratic philosopher Parmenides, although his work Peri Physeos is a poem, it may be viewed as an epistemological essay on method in natural science. Parmenides ἐὸν may refer to a system or calculus which can describe nature more precisely than natural languages. Physis may be identical to ἐὸν and he criticized the older type of study of physics as too purely speculative and lacking in self-criticism. He was particularly concerned that some of the early physicists treated nature as if it could be assumed that it had no intelligent order, explaining things merely in terms of motion and matter. The study of things had been the realm of mythology and tradition, however. Aristotle later created a less controversial systematic programme of Socratic philosophy which was teleological and he rejected many of the conclusions of earlier scientists. For example, in his physics, the sun goes around the earth, each thing has a formal cause and final cause and a role in the rational cosmic order. Motion and change is described as the actualization of potentials already in things, while the Socratics insisted that philosophy should be used to consider the practical question of the best way to live for a human being, they did not argue for any other types of applied science
6.
Engineering
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The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, the word engine itself is of even older origin, ultimately deriving from the Latin ingenium, meaning innate quality, especially mental power, hence a clever invention. The earliest civil engineer known by name is Imhotep, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. 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, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
7.
Business
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A business is an organizational entity involved in the provision of goods and services to consumers. Businesses may also be social non-profit enterprises or state-owned public enterprises operated by governments with specific social, a business owned by multiple private individuals may form as an incorporated company or jointly organise as a partnership. Countries have different laws that may ascribe different rights to the business entities. The word business can refer to an organization or to an entire market sector or to the sum of all economic activity. Compound forms such as agribusiness represent subsets of the broader meaning. Businesses aim to maximize sales to have their income exceed their expenditures, resulting in a profit, the owner operates the business alone and may hire employees. A sole proprietor has unlimited liability for all obligations incurred by the business, partnership, A partnership is a business owned by two or more people. In most forms of partnerships, each partner has unlimited liability for the debts incurred by the business, the three most prevalent types of for-profit partnerships are, general partnerships, limited partnerships, and limited liability partnerships. Corporation, The owners of a corporation have limited liability and the business has a legal personality from its owners. Corporations can be either government-owned or privately owned and they can organize either for profit or as nonprofit organizations. A privately owned, for-profit corporation is owned by its shareholders, a privately owned, for-profit corporation can be either privately held by a small group of individuals, or publicly held, with publicly traded shares listed on a stock exchange. Cooperative, Often referred to as a co-op, a cooperative is a limited-liability business that can organize as for-profit or not-for-profit, a cooperative differs from a corporation in that it has members, not shareholders, and they share decision-making authority. Cooperatives are typically classified as either consumer cooperatives or worker cooperatives, cooperatives are fundamental to the ideology of economic democracy. In contrast, unincorporated businesses or persons working on their own are not as protected. Franchises, A franchise is a system in which entrepreneurs purchase the rights to open, franchising in the United States is widespread and is a major economic powerhouse. One out of retail businesses in the United States are franchised and 8 million people are employed in a franchised business. Real estate businesses sell, invest, construct and develop properties – including land, residential homes, retailers, wholesalers, and distributors act as middlemen and get goods produced by manufacturers to the intended consumers, they make their profits by marking up their prices. Most stores and catalog companies are distributors or retailers, transportation businesses such as railways, airlines, shipping companies that deliver goods and individuals to their destinations for a fee
8.
Computer science
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Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems and its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory, are highly abstract, other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623, in 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner. He may be considered the first computer scientist and information theorist, for, among other reasons and he started developing this machine in 1834, and in less than two years, he had sketched out many of the salient features of the modern computer. A crucial step was the adoption of a card system derived from the Jacquard loom making it infinitely programmable. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information, when the machine was finished, some hailed it as Babbages dream come true. During the 1940s, as new and more powerful computing machines were developed, as it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as an academic discipline in the 1950s. The worlds first computer science program, the Cambridge Diploma in Computer Science. The first computer science program in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights and it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM704 and later the IBM709 computers, still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. During the late 1950s, the science discipline was very much in its developmental stages. Time has seen significant improvements in the usability and effectiveness of computing technology, modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base
9.
Industry
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Industry is the production of goods or related services within an economy. The major source of revenue of a group or company is the indicator of its relevant industry, when a large group has multiple sources of revenue generation, it is considered to be working in different industries. Manufacturing industry became a key sector of production and labour in European and North American countries during the Industrial Revolution, upsetting previous mercantile and this came through many successive rapid advances in technology, such as the production of steel and coal. Following the Industrial Revolution, possibly a third of the economic output are derived that is from manufacturing industries. Many developed countries and many developing/semi-developed countries depend significantly on manufacturing industry, Industries, the countries they reside in, and the economies of those countries are interlinked in a complex web of interdependence. Industries can be classified in a variety of ways, at the top level, industry is often classified according to the three-sector theory into sectors, primary, secondary, and tertiary. Some authors add quaternary or even quinary sectors, over time, the fraction of a societys industry within each sector changes. Below the economic sectors there are other more detailed industry classifications. These classification systems commonly divide industries according to functions and markets. Market-based classification systems such as the Global Industry Classification Standard and the Industry Classification Benchmark are used in finance, the International Standard Industrial Classification of all economic activities is the most complete and systematic industrial classification made by the United Nations Statistics Division. ISIC is a classification of economic activities arranged so that entities can be classified according to the activity they carry out. The Industrial Revolution led to the development of factories for large-scale production, originally the factories were steam-powered, but later transitioned to electricity once an electrical grid was developed. The mechanized assembly line was introduced to parts in a repeatable fashion. This led to significant increases in efficiency, lowering the cost of the end process, later automation was increasingly used to replace human operators. This process has accelerated with the development of the computer and the robot, historically certain manufacturing industries have gone into a decline due to various economic factors, including the development of replacement technology or the loss of competitive advantage. An example of the former is the decline in manufacturing when the automobile was mass-produced. A recent trend has been the migration of prosperous, industrialized nations towards a post-industrial society and this is manifested by an increase in the service sector at the expense of manufacturing, and the development of an information-based economy, the so-called informational revolution. In a post-industrial society, manufacturing is relocated to more favourable locations through a process of off-shoring
10.
Profession
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The term is a truncation of the term liberal profession, which is, in turn, an Anglicization of the French term profession libérale. Medieval and early modern tradition recognized only four professions, divinity, medicine, the military officer, just as some professions rise in status and power through various stages, others may decline. Disciplines formalized more recently, such as architecture, now have equally long periods of study associated with them, originally, any regulation of the professions was self-regulation through bodies such as the College of Physicians or the Inns of Court. With the growing role of government, statutory bodies have taken on this role. It may be resisted as limiting the freedom to innovate or to practice as in their professional judgement they consider best. An example was in 2008, when the British government proposed wide statutory regulation of psychologists, besides regulating access to a profession, professional bodies may set examinations of competence and enforce adherence to an ethical code. Typically, individuals are required by law to be qualified by a professional body before they are permitted to practice in that profession. However, in countries, individuals may not be required by law to be qualified by such a professional body in order to practice. This requirement is set out by the Educational Department Bureau of Hong Kong, Professions tend to be autonomous, which means they have a high degree of control of their own affairs, professionals are autonomous insofar as they can make independent judgments about their work. This usually means the freedom to exercise their professional judgement, however, it also has other meanings. Professional autonomy is often described as a claim of professionals that has to serve primarily their own interests, one major implication of professional autonomy is the traditional ban on corporate practice of the professions, especially accounting, architecture, medicine, and law. This means that in many jurisdictions, these professionals cannot do business through regular for-profit corporations, the obvious implication of this is that all equity owners of the professional business entity must be professionals themselves. This avoids the possibility of an owner of the firm telling a professional how to do his or her job. But because professional business entities are locked out of the stock market. Professions enjoy a social status, regard and esteem conferred upon them by society. This high esteem arises primarily from the social function of their work. All professions involve technical, specialized and highly skilled work often referred to as professional expertise, training for this work involves obtaining degrees and professional qualifications without which entry to the profession is barred. Updating skills through continuing education is required through training and this power is used to control its own members, and also its area of expertise and interests
11.
Pure mathematics
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Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research, ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between arithmetic, now called number theory, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, in the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, in fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved, Pure mathematician became a recognized vocation, achievable through training. One central concept in mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central
12.
Heat equation
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The heat equation is a parabolic partial differential equation that describes the distribution of heat in a given region over time. In the physical problem of variation, u is the temperature. For the mathematical treatment it is sufficient to consider the case α =1, note that the state equation, given by the first law of thermodynamics, is written in the following form. This form is general and particularly useful to recognize which property influences which term. ρ c p ∂ T ∂ t − ∇ ⋅ = q ˙ V where q ˙ V is the heat flux. The heat equation is of importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation, in probability theory, the heat equation is connected with the study of Brownian motion via the Fokker–Planck equation. In financial mathematics it is used to solve the Black–Scholes partial differential equation, the diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes. Suppose one has a function u that describes the temperature at a given location and this function will change over time as heat spreads throughout space. The heat equation is used to determine the change in the function u over time, the rate of change of u is proportional to the curvature of u. Thus, the sharper the corner, the faster it is rounded off. Over time, the tendency is for peaks to be eroded, if u is linear in space at a given point, then u has reached steady-state and is unchanging at this point. The image to the right is animated and describes the way changes in time along a metal bar. This is essentially saying that temperature comes either from some source or from earlier in time because heat permeates but is not created from nothingness and this is a property of parabolic partial differential equations and is not difficult to prove mathematically. Another interesting property is that if u has a discontinuity at an initial time t = t0. The heat equation is used in probability and describes random walks and it is also applied in financial mathematics for this reason. The heat equation is derived from Fouriers law and conservation of energy and that is, Δ Q = c p ρ Δ u where cp is the specific heat capacity and ρ is the mass density of the material. Choosing zero energy at zero temperature, this can be rewritten as Q = c p ρ u. If no work is done and there are neither heat sources nor sinks, by conservation of energy, ∫ t − Δ t t + Δ t ∫ x − Δ x x + Δ x d ξ d τ =0
13.
Finite element method
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The finite element method is a numerical method for solving problems of engineering and mathematical physics. It is also referred to as finite element analysis, typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations, the finite element method formulation of the problem results in a system of algebraic equations. The method yields approximate values of the unknowns at discrete number of points over the domain, to solve the problem, it subdivides a large problem into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function, the global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer. To explain the approximation in this process, FEM is commonly introduced as a case of Galerkin method. The process, in language, is to construct an integral of the inner product of the residual. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE, the residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. These equation sets are the element equations and they are linear if the underlying PDE is linear, and vice versa. In step above, a system of equations is generated from the element equations through a transformation of coordinates from the subdomains local nodes to the domains global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the coordinate system. The process is carried out by FEM software using coordinate data generated from the subdomains. FEM is best understood from its application, known as finite element analysis. FEA as applied in engineering is a tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, FEA is a good choice for analyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a crash simulation it is possible to increase prediction accuracy in important areas like the front of the car. Another example would be in weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena rather than relatively calm areas
14.
Mathematical analysis
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Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. 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 or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, 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 later be called Cavalieris 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 Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. 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 definition of continuity in 1816. 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, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, 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, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. 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, also, monsters began to be investigated
15.
Approximation theory
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In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by best and simpler will depend on the application, a closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. This is typically done with polynomial or rational approximations, the objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computers floating point arithmetic. This is accomplished by using a polynomial of degree, and/or narrowing the domain over which the polynomial has to approximate the function. Narrowing the domain can often be done through the use of various addition or scaling formulas for the function being approximated, modern mathematical libraries often reduce the domain into many tiny segments and use a low-degree polynomial for each segment. Once the domain and degree of the polynomial are chosen, the polynomial itself is chosen in such a way as to minimize the worst-case error. That is, the goal is to minimize the value of ∣ P − f ∣, where P is the approximating polynomial, f is the actual function. It is seen that an Nth-degree polynomial can interpolate N+1 points in a curve, such a polynomial is always optimal. It is possible to make contrived functions f for which no such polynomial exists, for example, the graphs shown to the right show the error in approximating log and exp for N =4. The red curves, for the polynomial, are level. Note that, in case, the number of extrema is N+2. Two of the extrema are at the end points of the interval, at the left, the red graph to the right shows what this error function might look like for N =4. Suppose Q is another N-degree polynomial that is an approximation to f than P. In particular, Q is closer to f than P for each value xi where an extreme of P−f occurs, but − reduces to P − Q which is a polynomial of degree N. This function changes sign at least N+1 times so, by the Intermediate value theorem, it has N+1 zeroes, one can obtain polynomials very close to the optimal one by expanding the given function in terms of Chebyshev polynomials and then cutting off the expansion at the desired degree. This is similar to the Fourier analysis of the function, using the Chebyshev polynomials instead of the trigonometric functions. If one calculates the coefficients in the Chebyshev expansion for a function, f ∼ ∑ i =0 ∞ c i T i and then cuts off the series after the T N term and that is, the first term after the cutoff dominates all later terms. The same is true if the expansion is in terms of Chebyshev polynomials, if a Chebyshev expansion is cut off after T N, the error will take a form close to a multiple of T N +1
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Numerical analysis
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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is important, for instance, in astronomy, carpentry. Numerical analysis continues this tradition of practical mathematical calculations. Much like the Babylonian approximation of the root of 2, modern numerical analysis does not seek exact answers. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors, before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead and these same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of differential equations. Car companies can improve the safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving differential equations numerically. Hedge funds use tools from all fields of analysis to attempt to calculate the value of stocks. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments, historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis, the field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago, to facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. The function values are no very useful when a computer is available. The mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of analysis, since now longer
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Probability
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Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, the higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair coin, since the coin is unbiased, the two outcomes are both equally probable, the probability of head equals the probability of tail. Since no other outcomes are possible, the probability is 1/2 and this type of probability is also called a priori probability. Probability theory is used to describe the underlying mechanics and regularities of complex systems. For example, tossing a coin twice will yield head-head, head-tail, tail-head. The probability of getting an outcome of head-head is 1 out of 4 outcomes or 1/4 or 0.25 and this interpretation considers probability to be the relative frequency in the long run of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, subjectivists assign numbers per subjective probability, i. e. as a degree of belief. The degree of belief has been interpreted as, the price at which you would buy or sell a bet that pays 1 unit of utility if E,0 if not E. The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as data to produce probabilities. The expert knowledge is represented by some prior probability distribution and these data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a probability distribution that incorporates all the information known to date. The scientific study of probability is a development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, there are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the study of probability. According to Richard Jeffrey, Before the middle of the century, the term probable meant approvable. A probable action or opinion was one such as people would undertake or hold. However, in legal contexts especially, probable could also apply to propositions for which there was good evidence, the sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes
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Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
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Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
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Fluid mechanics
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Fluid mechanics is a branch of physics concerned with the mechanics of fluids and the forces on them. Fluid mechanics has a range of applications, including for mechanical engineering, civil engineering, chemical engineering, geophysics, astrophysics. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, fluid mechanics, especially fluid dynamics, is an active field of research with many problems that are partly or wholly unsolved. Fluid mechanics can be complex, and can best be solved by numerical methods. A modern discipline, called computational fluid dynamics, is devoted to this approach to solving fluid mechanics problems, Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow. Inviscid flow was further analyzed by mathematicians and viscous flow was explored by a multitude of engineers including Jean Léonard Marie Poiseuille. Fluid statics or hydrostatics is the branch of mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium, and is contrasted with fluid dynamics, hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to some aspect of geophysics and astrophysics, to meteorology, to medicine, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the science of liquids and gases in motion. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density and it has several subdisciplines itself, including aerodynamics and hydrodynamics. Some fluid-dynamical principles are used in engineering and crowd dynamics. Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table, in a mechanical view, a fluid is a substance that does not support shear stress, that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress, the assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. This can be expressed as an equation in integral form over the control volume, the continuum assumption is an idealization of continuum mechanics under which fluids can be treated as continuous, even though, on a microscopic scale, they are composed of molecules. Fluid properties can vary continuously from one element to another and are average values of the molecular properties. The continuum hypothesis can lead to results in applications like supersonic speed flows. Those problems for which the continuum hypothesis fails, can be solved using statistical mechanics, to determine whether or not the continuum hypothesis applies, the Knudsen number, defined as the ratio of the molecular mean free path to the characteristic length scale, is evaluated. Problems with Knudsen numbers below 0.1 can be evaluated using the continuum hypothesis, the Navier–Stokes equations are differential equations that describe the force balance at a given point within a fluid
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Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
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Cryptography
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Cryptography or cryptology is the practice and study of techniques for secure communication in the presence of third parties called adversaries. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, Applications of cryptography include ATM cards, computer passwords, and electronic commerce. Cryptography prior to the age was effectively synonymous with encryption. The originator of an encrypted message shared the decoding technique needed to recover the information only with intended recipients. The cryptography literature often uses Alice for the sender, Bob for the intended recipient and it is theoretically possible to break such a system, but it is infeasible to do so by any known practical means. The growth of technology has raised a number of legal issues in the information age. Cryptographys potential for use as a tool for espionage and sedition has led governments to classify it as a weapon and to limit or even prohibit its use. 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 also plays a major role in digital rights management and copyright infringement of digital media. Until modern times, cryptography referred almost exclusively to encryption, which is the process of converting ordinary information into unintelligible text, 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 encryption, the detailed operation of a cipher is controlled both by the algorithm and in each instance by a key. The key is a secret, usually a short string of characters, historically, ciphers were often used directly for encryption or decryption without additional procedures such as authentication or integrity checks. There are two kinds of cryptosystems, symmetric and asymmetric, in symmetric systems the same key is used to encrypt and decrypt a message. Data manipulation in symmetric systems is faster than asymmetric systems as they generally use shorter key lengths, asymmetric systems use a public key to encrypt a message and a private key to decrypt it. Use of asymmetric systems enhances the security of communication, examples of asymmetric systems include RSA, and ECC. Symmetric models include the commonly used AES which replaced the older DES, in colloquial use, the term code is often used to mean any method of encryption or concealment of meaning. However, in cryptography, code has a specific meaning. It means the replacement of a unit of plaintext with a code word, English is more flexible than several other languages in which cryptology is always used in the second sense above. RFC2828 advises that steganography is sometimes included in cryptology, the study of characteristics of languages that have some application in cryptography or cryptology is called cryptolinguistics
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Applicable mathematics
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Applicable mathematics is a subdiscipline of applied mathematics. It can be difficult to find a definition of what the subject consists of. It often includes areas other than the applied areas of mathematics that have become increasingly important in applications. Even fields such as number theory that are part of mathematics are now important in applications. Individuals studying the topic can focus on economics, social sciences, engineering, coding, the subject of applicable mathematics was studied greatly by Hua Luogeng. There is no consensus as to what applicable mathematics is, as there are problems about what the various branches of applied mathematics are. Categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. For example, from this viewpoint, an ecologist or geographer using population models and applying known mathematics would not be doing applied, other authors prefer describing applicable mathematics as a union of new mathematical applications with the traditional fields of applied mathematics. With this outlook, the applied mathematics and applicable mathematics are thus interchangeable
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Biology
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Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, evolution, distribution, identification and taxonomy. Modern biology is a vast and eclectic field, composed of branches and subdisciplines. However, despite the broad scope of biology, there are certain unifying concepts within it that consolidate it into single, coherent field. In general, biology recognizes the cell as the unit of life, genes as the basic unit of heredity. It is also understood today that all organisms survive by consuming and transforming energy and by regulating their internal environment to maintain a stable, the term biology is derived from the Greek word βίος, bios, life and the suffix -λογία, -logia, study of. The Latin-language form of the term first appeared in 1736 when Swedish scientist Carl Linnaeus used biologi in his Bibliotheca botanica, the first German use, Biologie, was in a 1771 translation of Linnaeus work. In 1797, Theodor Georg August Roose used the term in the preface of a book, karl Friedrich Burdach used the term in 1800 in a more restricted sense of the study of human beings from a morphological, physiological and psychological perspective. The science that concerns itself with these objects we will indicate by the biology or the doctrine of life. Although modern biology is a recent development, sciences related to. Natural philosophy was studied as early as the ancient civilizations of Mesopotamia, Egypt, the Indian subcontinent, however, the origins of modern biology and its approach to the study of nature are most often traced back to ancient Greece. While the formal study of medicine back to Hippocrates, it was Aristotle who contributed most extensively to the development of biology. Especially important are his History of Animals and other works where he showed naturalist leanings, and later more empirical works that focused on biological causation and the diversity of life. Aristotles successor at the Lyceum, Theophrastus, wrote a series of books on botany that survived as the most important contribution of antiquity to the plant sciences, even into the Middle Ages. Scholars of the medieval Islamic world who wrote on biology included al-Jahiz, Al-Dīnawarī, who wrote on botany, biology began to quickly develop and grow with Anton van Leeuwenhoeks dramatic improvement of the microscope. It was then that scholars discovered spermatozoa, bacteria, infusoria, investigations by Jan Swammerdam led to new interest in entomology and helped to develop the basic techniques of microscopic dissection and staining. Advances in microscopy also had a impact on biological thinking. In the early 19th century, a number of biologists pointed to the importance of the cell. Thanks to the work of Robert Remak and Rudolf Virchow, however, meanwhile, taxonomy and classification became the focus of natural historians
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Vladimir Arnold
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Vladimir Igorevich Arnold was a Soviet and Russian mathematician. Arnold was also known as a popularizer of mathematics, through his lectures, seminars, and as the author of several textbooks and popular mathematics books, he influenced many mathematicians and physicists. Many of his books were translated into English, Vladimir Igorevich Arnold was born on 12 June 1937 in Odessa, Soviet Union. His father was Igor Vladimirovich Arnold, a mathematician and his mother was Nina Alexandrovna Arnold, an art historian. This is the Kolmogorov–Arnold representation theorem, after graduating from Moscow State University in 1959, he worked there until 1986, and then at Steklov Mathematical Institute. He became an academician of the Academy of Sciences of the Soviet Union in 1990, Arnold can be said to have initiated the theory of symplectic topology as a distinct discipline. The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms, Arnold worked at the Steklov Mathematical Institute in Moscow and at Paris Dauphine University up until his death. As of 2006 he was reported to have the highest citation index among Russian scientists, to his students and colleagues Arnold was known also for his sense of humour. In accordance with this principle I shall formulate some problems. ”Arnold died of pancreatitis on 3 June 2010 in Paris. He was buried on June 15 in Moscow, at the Novodevichy Monastery, in a telegram to Arnolds family, Russian President Dmitry Medvedev stated, “The death of Vladimir Arnold, one of the greatest mathematicians of our time, is an irretrievable loss for world science. It is difficult to overestimate the contribution made by academician Arnold to modern mathematics, teaching had a special place in Vladimir Arnolds life and he had great influence as an enlightened mentor who taught several generations of talented scientists. The memory of Vladimir Arnold will forever remain in the hearts of his colleagues, friends and students, as well as everyone who knew and admired this brilliant man. ”Arnold is well known for his writing style, combining mathematical rigour with physical intuition. His defense is that his books are meant to teach the subject to those who wish to understand it. Arnold was a critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. Arnold was very interested in the history of mathematics and he liked to study the classics, most notably the works of Huygens, Newton and Poincaré, and many times he reported to have found in their works ideas that had not been explored yet. Arnold worked on systems theory, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics. Moser and Arnold expanded the ideas of Kolmogorov and gave rise to what is now known as KAM Theory, KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period of time, and specifies what the conditions for this are. In 1965, Arnold attended René Thoms seminar on catastrophe theory, after this event, singularity theory became one of the major interests of Arnold and his students
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Computational engineering
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Not to be confused with computer engineering. CSE has been described as the mode of discovery. In many fields, computer simulation is integral and therefore essential to business and it is typically offered as a masters or doctorate program at several institutions. The most widely used programming language in the community is FORTRAN. Recently, C++ and C have increased in popularity over FORTRAN, due to the wealth of legacy code in FORTRAN and its simpler syntax, the scientific computing community has been slow in completely adopting C++ as the lingua franca. Python along with external libraries has gain some popularity as a free and Copycenter alternative to MATLAB
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Supercomputer
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A supercomputer is a computer with a high level of computing performance compared to a general-purpose computer. Performance of a supercomputer is measured in floating-point operations per second instead of instructions per second. As of 2015, there are supercomputers which can perform up to quadrillions of FLOPS and it tops the rankings in the TOP500 supercomputer list. Sunway TaihuLights emergence is also notable for its use of indigenous chips, as of June 2016, China, for the first time, had more computers on the TOP500 list than the United States. However, U. S. built computers held ten of the top 20 positions, in November 2016 the U. S. has five of the top 10, throughout their history, they have been essential in the field of cryptanalysis. The use of multi-core processors combined with centralization is an emerging trend, the history of supercomputing goes back to the 1960s, with the Atlas at the University of Manchester and a series of computers at Control Data Corporation, designed by Seymour Cray. These used innovative designs and parallelism to achieve superior computational peak performance, Cray left CDC in 1972 to form his own company, Cray Research. Four years after leaving CDC, Cray delivered the 80 MHz Cray 1 in 1976, the Cray-2 released in 1985 was an 8 processor liquid cooled computer and Fluorinert was pumped through it as it operated. It performed at 1.9 gigaflops and was the second fastest after M-13 supercomputer in Moscow. Fujitsus Numerical Wind Tunnel supercomputer used 166 vector processors to gain the top spot in 1994 with a speed of 1.7 gigaFLOPS per processor. The Hitachi SR2201 obtained a performance of 600 GFLOPS in 1996 by using 2048 processors connected via a fast three-dimensional crossbar network. The Intel Paragon could have 1000 to 4000 Intel i860 processors in various configurations, the Paragon was a MIMD machine which connected processors via a high speed two dimensional mesh, allowing processes to execute on separate nodes, communicating via the Message Passing Interface. Approaches to supercomputer architecture have taken dramatic turns since the earliest systems were introduced in the 1960s, early supercomputer architectures pioneered by Seymour Cray relied on compact innovative designs and local parallelism to achieve superior computational peak performance. However, in time the demand for increased computational power ushered in the age of massively parallel systems, supercomputers of the 21st century can use over 100,000 processors connected by fast connections. The Connection Machine CM-5 supercomputer is a parallel processing computer capable of many billions of arithmetic operations per second. Throughout the decades, the management of heat density has remained a key issue for most centralized supercomputers, the large amount of heat generated by a system may also have other effects, e. g. reducing the lifetime of other system components. There have been diverse approaches to management, from pumping Fluorinert through the system. Systems with a number of processors generally take one of two paths
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Simulation
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Simulation is the imitation of the operation of a real-world process or system over time. The model represents the system itself, whereas the simulation represents the operation of the system over time, Simulation is used in many contexts, such as simulation of technology for performance optimization, safety engineering, testing, training, education, and video games. Often, computer experiments are used to study simulation models, Simulation is also used with scientific modelling of natural systems or human systems to gain insight into their functioning, as in economics. Simulation can be used to show the real effects of alternative conditions. Physical simulation refers to simulation in which objects are substituted for the real thing. These physical objects are chosen because they are smaller or cheaper than the actual object or system. Simulation Fidelity is used to describe the accuracy of a simulation, Fidelity is broadly classified as 1 of 3 categories, low, medium, and high. Simulation in failure analysis refers to simulation in which we create environment/conditions to identify the cause of equipment failure and this was the best and fastest method to identify the failure cause. A computer simulation is an attempt to model a real-life or hypothetical situation on a computer so that it can be studied to see how the system works, by changing variables in the simulation, predictions may be made about the behaviour of the system. It is a tool to investigate the behaviour of the system under study. A good example of the usefulness of using computers to simulate can be found in the field of network traffic simulation, in such simulations, the model behaviour will change each simulation according to the set of initial parameters assumed for the environment. Computer simulation is used as an adjunct to, or substitution for. Several software packages exist for running computer-based simulation modeling that makes all the modeling almost effortless, modern usage of the term computer simulation may encompass virtually any computer-based representation. The computer simulates the subject machine, accordingly, in theoretical computer science the term simulation is a relation between state transition systems, useful in the study of operational semantics. Less theoretically, an application of computer simulation is to simulate computers using computers. For example, simulators have been used to debug a microprogram or sometimes commercial application programs, simulators may also be used to interpret fault trees, or test VLSI logic designs before they are constructed. Symbolic simulation uses variables to stand for unknown values, in the field of optimization, simulations of physical processes are often used in conjunction with evolutionary computation to optimize control strategies. Simulation is extensively used for educational purposes and it is frequently used by way of adaptive hypermedia
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Natural science
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Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on observational and empirical evidence. Mechanisms such as review and repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two branches, life science and physical science. Physical science is subdivided into branches, including physics, space science, chemistry and these branches of natural science may be further divided into more specialized branches. Modern natural science succeeded more classical approaches to natural philosophy, usually traced to ancient Greece, galileo, Descartes, Francis Bacon, and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and so on. Today, natural history suggests observational descriptions aimed at popular audiences, philosophers of science have suggested a number of criteria, including Karl Poppers controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity, accuracy, and quality control, such as peer review and this field encompasses a set of disciplines that examines phenomena related to living organisms. The scale of study can range from sub-component biophysics up to complex ecologies, biology is concerned with the characteristics, classification 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, zoology, and medicine date back to periods of civilization. However, it was not until the 19th century that became a unified science. Once scientists discovered commonalities between all living things, it was decided they were best studied as a whole, modern biology is divided into subdisciplines by the type of organism and by the scale being studied. Molecular biology is the study of the chemistry of life, while cellular biology is the examination of the cell. At a higher level, anatomy and physiology looks at the internal structures, constituting the scientific study of matter at the atomic and molecular scale, chemistry deals primarily with collections of atoms, such as gases, molecules, crystals, and metals. The composition, statistical properties, transformations and reactions of these materials are studied, chemistry also 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, chemistry is often 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, the science of chemistry began to develop with the work of Robert Boyle, the discoverer of gas, and Antoine Lavoisier, who developed the theory of the Conservation of mass. The success of science led to a complementary chemical industry that now plays a significant role in the world economy
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World War II
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World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945, although related conflicts began earlier. It involved the vast majority of the worlds countries—including all of the great powers—eventually forming two opposing alliances, the Allies and the Axis. It was the most widespread war in history, and directly involved more than 100 million people from over 30 countries. Marked by mass deaths of civilians, including the Holocaust and the bombing of industrial and population centres. These made World War II the deadliest conflict in human history, from late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, and 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, Poland, Finland, Romania and the Baltic states. In December 1941, Japan attacked the United States and European colonies in the Pacific Ocean, and quickly conquered much of the Western Pacific. The Axis advance halted in 1942 when Japan lost the critical Battle of Midway, near Hawaii, in 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained all of its territorial losses and invaded Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in South Central China and Burma, while the Allies crippled the Japanese Navy, thus ended the war in Asia, cementing the total victory of the Allies. World War II altered the political alignment and social structure of the world, the United Nations was established to foster international co-operation and prevent future conflicts. The victorious great powers—the United States, the Soviet Union, China, the United Kingdom, the Soviet Union and the United States emerged as rival superpowers, setting the stage for the Cold War, which lasted for the next 46 years. Meanwhile, the influence of European great powers waned, while the decolonisation of Asia, most countries whose industries had been damaged moved towards economic recovery. Political integration, especially in Europe, emerged as an effort to end pre-war enmities, the start of the war in Europe is generally held to be 1 September 1939, beginning with the German invasion of Poland, Britain and France declared war on Germany two days later. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or even 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 simultaneously and 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 forces of Mongolia and the Soviet Union from May to September 1939, the exact date of the wars end is also not universally agreed upon. It was generally accepted at the time that the war ended with the armistice of 14 August 1945, rather than the formal surrender of Japan
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Game theory
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Game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers. Game theory is used in economics, political science, and psychology, as well as logic, computer science. Originally, it addressed zero-sum games, in one persons gains result in losses for the other participants. Today, game theory applies to a range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals. Modern game theory began with the idea regarding the existence of equilibria in two-person zero-sum games. Von Neumanns original proof used Brouwer fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this provided an axiomatic theory of expected utility. This theory was developed extensively in the 1950s by many scholars, Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole in 2014, John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of modern, the first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1713. In this letter, Waldegrave provides a mixed strategy solution to a two-person version of the card game le Her. James Madison made what we now recognize as an analysis of the ways states can be expected to behave under different systems of taxation. In 1913 Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels and it proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems, the Danish mathematician Zeuthen proved that the mathematical model had a winning strategy by using Brouwers fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false. Game theory did not really exist as a field until John von Neumann published a paper in 1928. Von Neumanns original proof used Brouwers fixed-point theorem on continuous mappings into compact convex sets and his paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern
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Theoretical computer science
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It is not easy to circumscribe the theoretical areas precisely. Work in this field is often distinguished by its emphasis on mathematical technique, despite this broad scope, the theory people in computer science self-identify as different from the applied people. Some characterize themselves as doing the science underlying the field of computing, other theory-applied people suggest that it is impossible to separate theory and application. This means that the theory people regularly use experimental science done in less-theoretical areas such as software system research. It also means there is more cooperation than mutually exclusive competition between theory and application. These developments have led to the study of logic and computability. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon, in the same decade, Donald Hebb introduced a mathematical model of learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks, in 1971, Stephen Cook and, working independently, Leonid Levin, proved that there exist practically relevant problems that are NP-complete – a landmark result in computational complexity theory. With the development of mechanics in the beginning of the 20th century came the concept that mathematical operations could be performed on an entire particle wavefunction. In other words, one could compute functions on multiple states simultaneously, modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed. An algorithm is a procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, a data structure is a particular way of organizing data in a computer so that it can be used efficiently. Different kinds of structures are suited to different kinds of applications. For example, databases use B-tree indexes for small percentages of data retrieval and compilers, data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms, some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Storing and retrieving can be carried out on data stored in main memory and in secondary memory. A problem is regarded as inherently difficult if its solution requires significant resources, the theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage
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Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
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Social science
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Social science is a major category of academic disciplines, concerned with society and the relationships among individuals within a society. It in turn has many branches, each of which is considered a social science, the social sciences include economics, political science, human geography, demography, psychology, sociology, anthropology, archaeology, jurisprudence, history, and linguistics. The term is sometimes used to refer specifically to the field of sociology. A more detailed list of sub-disciplines within the sciences can be found at Outline of social science. Positivist social scientists use methods resembling those of the sciences as tools for understanding society. In modern academic practice, researchers are often eclectic, using multiple methodologies, the term social research has also acquired a degree of autonomy as practitioners from various disciplines share in its aims and methods. Social sciences came forth from the philosophy of the time and were influenced by the Age of Revolutions, such as the Industrial Revolution. The social sciences developed from the sciences, or the systematic knowledge-bases or prescriptive practices, the beginnings of the social sciences in the 18th century are reflected in the grand encyclopedia of Diderot, with articles from Jean-Jacques Rousseau and other pioneers. The growth of the sciences is also reflected in other specialized encyclopedias. The modern period saw social science first used as a distinct conceptual field, Social science was influenced by positivism, focusing on knowledge based on actual positive sense experience and avoiding the negative, metaphysical speculation was avoided. Auguste Comte used the term science sociale to describe the field, taken from the ideas of Charles Fourier, following this period, there were five paths of development that sprang forth in the social sciences, influenced by Comte on other fields. One route that was taken was the rise of social research, large statistical surveys were undertaken in various parts of the United States and Europe. Another route undertaken was initiated by Émile Durkheim, studying social facts, a third means developed, arising from the methodological dichotomy present, in which social phenomena were identified with and understood, this was championed by figures such as Max Weber. The fourth route taken, based in economics, was developed and furthered economic knowledge as a hard science, the last path was the correlation of knowledge and social values, the antipositivism and verstehen sociology of Max Weber firmly demanded this distinction. In this route, theory and prescription were non-overlapping formal discussions of a subject, around the start of the 20th century, Enlightenment philosophy was challenged in various quarters. The development of social science subfields became very quantitative in methodology, examples of boundary blurring include emerging disciplines like social research of medicine, sociobiology, neuropsychology, bioeconomics and the history and sociology of science. Increasingly, quantitative research and qualitative methods are being integrated in the study of action and its implications. In the first half of the 20th century, statistics became a discipline of applied mathematics
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Economics
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Economics is a social science concerned chiefly with description and analysis of the production, distribution, and consumption of goods and services according to the Merriam-Webster Dictionary. Economics focuses on the behaviour and interactions of economic agents and how economies work, consistent with this focus, textbooks often distinguish between microeconomics and macroeconomics. Microeconomics examines the behaviour of elements in the economy, including individual agents and markets, their interactions. Individual agents may include, for example, households, firms, buyers, macroeconomics analyzes the entire economy and issues affecting it, including unemployment of resources, inflation, economic growth, and the public policies that address these issues. Economic analysis can be applied throughout society, as in business, finance, health care, Economic analyses may also be applied to such diverse subjects as crime, education, the family, law, politics, religion, social institutions, war, science, and the environment. At the turn of the 21st century, the domain of economics in the social sciences has been described as economic imperialism. The ultimate goal of economics is to improve the conditions of people in their everyday life. There are a variety of definitions of economics. Some of the differences may reflect evolving views of the subject or different views among economists, to supply the state or commonwealth with a revenue for the publick services. Say, distinguishing the subject from its uses, defines it as the science of production, distribution. On the satirical side, Thomas Carlyle coined the dismal science as an epithet for classical economics, in this context and it enquires how he gets his income and how he uses it. Thus, it is on the one side, the study of wealth and on the other and more important side, a part of the study of man. He affirmed that previous economists have usually centred their studies on the analysis of wealth, how wealth is created, distributed, and consumed, but he said that economics can be used to study other things, such as war, that are outside its usual focus. This is because war has as the goal winning it, generates both cost and benefits, and, resources are used to attain the goal. If the war is not winnable or if the costs outweigh the benefits. Some subsequent comments criticized the definition as overly broad in failing to limit its subject matter to analysis of markets, there are other criticisms as well, such as in scarcity not accounting for the macroeconomics of high unemployment. The same source reviews a range of included in principles of economics textbooks. Among economists more generally, it argues that a particular definition presented may reflect the direction toward which the author believes economics is evolving, microeconomics examines how entities, forming a market structure, interact within a market to create a market system
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United Kingdom
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The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom or Britain, is a sovereign country in western Europe. Lying off the north-western coast of the European mainland, the United Kingdom includes the island of Great Britain, Northern Ireland is the only part of the United Kingdom that shares a land border with another sovereign state—the Republic of Ireland. The Irish Sea lies between Great Britain and Ireland, with an area of 242,500 square kilometres, the United Kingdom is the 78th-largest sovereign state in the world and the 11th-largest in Europe. It is also the 21st-most populous country, with an estimated 65.1 million inhabitants, together, this makes it the fourth-most densely populated country in the European Union. The United Kingdom is a monarchy with a parliamentary system of governance. The monarch is Queen Elizabeth II, who has reigned since 6 February 1952, other major urban areas in the United Kingdom include the regions of Birmingham, Leeds, Glasgow, Liverpool and Manchester. The United Kingdom consists of four countries—England, Scotland, Wales, the last three have devolved administrations, each with varying powers, based in their capitals, Edinburgh, Cardiff and Belfast, respectively. The relationships among the countries of the UK have changed over time, Wales was annexed by the Kingdom of England under the Laws in Wales Acts 1535 and 1542. A treaty between England and Scotland resulted in 1707 in a unified Kingdom of Great Britain, which merged in 1801 with the Kingdom of Ireland to form the United Kingdom of Great Britain and Ireland. Five-sixths of Ireland seceded from the UK in 1922, leaving the present formulation of the United Kingdom of Great Britain, there are fourteen British Overseas Territories. These are the remnants of the British Empire which, at its height in the 1920s, British influence can be observed in the language, culture and legal systems of many of its former colonies. The United Kingdom is a country and has the worlds fifth-largest economy by nominal GDP. The UK is considered to have an economy and is categorised as very high in the Human Development Index. It was the worlds first industrialised country and the worlds foremost power during the 19th, the UK remains a great power with considerable economic, cultural, military, scientific and political influence internationally. It is a nuclear weapons state and its military expenditure ranks fourth or fifth in the world. The UK has been a permanent member of the United Nations Security Council since its first session in 1946 and it has been a leading member state of the EU and its predecessor, the European Economic Community, since 1973. However, on 23 June 2016, a referendum on the UKs membership of the EU resulted in a decision to leave. The Acts of Union 1800 united the Kingdom of Great Britain, Scotland, Wales and Northern Ireland have devolved self-government
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