1.
System
–
A system is a set of interacting or interdependent component parts forming a complex or intricate whole. Every system is delineated by its spatial and temporal boundaries, surrounded and influenced by its environment, described by its structure and purpose and expressed in its functioning. Alternatively, and usually in the context of social systems. The term system comes from the Latin word systēma, in turn from Greek σύστημα systēma, whole compounded of several parts or members, system, according to Marshall McLuhan, System means something to look at. You must have a high visual gradient to have systematization. In philosophy, prior to Descartes, there was no system, in the 19th century the French physicist Nicolas Léonard Sadi Carnot, who studied thermodynamics, pioneered the development of the concept of a system in the natural sciences. In 1824 he studied the system which he called the substance in steam engines. The working substance could be put in contact with either a boiler, in 1850, the German physicist Rudolf Clausius generalized this picture to include the concept of the surroundings and began to use the term working body when referring to the system. The biologist Ludwig von Bertalanffy became one of the pioneers of the systems theory. Norbert Wiener and Ross Ashby, who pioneered the use of mathematics to study systems, in the 1980s John H. Holland, Murray Gell-Mann and others coined the term complex adaptive system at the interdisciplinary Santa Fe Institute. Environment and boundaries Systems theory views the world as a system of interconnected parts. One scopes a system by defining its boundary, this means choosing which entities are inside the system, one can make simplified representations of the system in order to understand it and to predict or impact its future behavior. These models may define the structure and behavior of the system, Natural and human-made systems There are natural and human-made systems. Natural systems may not have an apparent objective but their behavior can be interpreted as purposefull by an observer, human-made systems are made to satisfy an identified and stated need with purposes that are achieved by the delivery of wanted outputs. Their parts must be related, they must be designed to work as a coherent entity – otherwise they would be two or more distinct systems, Theoretical framework An open system exchanges matter and energy with its surroundings. Most systems are open systems, like a car, a coffeemaker, a closed system exchanges energy, but not matter, with its environment, like Earth or the project Biosphere2 or 3. An isolated system exchanges neither matter nor energy with its environment, a theoretical example of such system is the Universe. Inputs are consumed, outputs are produced, the concept of input and output here is very broad
2.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Natural science
–
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
4.
Physics
–
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
5.
Biology
–
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
6.
Earth science
–
Earth science or geoscience is a widely embraced term for the fields of science related to the planet Earth. Earth science can be considered to be a branch of planetary science, there are both reductionist and holistic approaches to Earth sciences. The Earth sciences can include the study of geology, the lithosphere, and the structure of the Earths interior, as well as the atmosphere, hydrosphere. Typically, Earth scientists use tools from geography, chronology, physics, chemistry, biology, Geology describes the rocky parts of the Earths crust and its historic development. Major subdisciplines are mineralogy and petrology, geochemistry, geomorphology, paleontology, stratigraphy, structural geology, engineering geology, geophysics and geodesy investigate the shape of the Earth, its reaction to forces and its magnetic and gravity fields. Geophysicists explore the Earths core and mantle as well as the tectonic and seismic activity of the lithosphere, geophysics is commonly used to supplement the work of geologists in developing a comprehensive understanding of crustal geology, particularly in mineral and petroleum exploration. Soil science covers the outermost layer of the Earths crust that is subject to soil formation processes, major subdisciplines include edaphology and pedology. Ecology covers the interactions between the biota, with their natural environment and this field of study differentiates the study of the Earth, from the study of other planets in the Solar System, the Earth being the only planet teeming with life. Hydrology is a study revolved around the movement, distribution, and quality of the water and involves all the components of the cycle on the earth. Sub-disciplines of hydrology include hydrometeorology, surface hydrology, hydrogeology, watershed science, forest hydrology. Glaciology covers the icy parts of the Earth, atmospheric sciences cover the gaseous parts of the Earth between the surface and the exosphere. Major subdisciplines include meteorology, climatology, atmospheric chemistry, and atmospheric physics, plate tectonics, mountain ranges, volcanoes, and earthquakes are geological phenomena that can be explained in terms of physical and chemical processes in the Earths crust. Beneath the Earths crust lies the mantle which is heated by the decay of heavy elements. The mantle is not quite solid and consists of magma which is in a state of semi-perpetual convection and this convection process causes the lithospheric plates to move, albeit slowly. The resulting process is known as plate tectonics, plate tectonics might be thought of as the process by which the Earth is resurfaced. As the result of spreading, new crust and lithosphere is created by the flow of magma from the mantle to the near surface, through fissures. Through subduction, oceanic crust and lithosphere returns to the convecting mantle, volcanoes result primarily from the melting of subducted crust material. Crust material that is forced into the asthenosphere melts, and some portion of the material becomes light enough to rise to the surface—giving birth to volcanoes
7.
Chemistry
–
Chemistry is a branch of physical science that studies the composition, structure, properties and change of matter. Chemistry is sometimes called the science because it bridges other natural sciences, including physics. For the differences between chemistry and physics see comparison of chemistry and physics, the history of chemistry can be traced to alchemy, which had been practiced for several millennia in various parts of the world. The word chemistry comes from alchemy, which referred to a set of practices that encompassed elements of chemistry, metallurgy, philosophy, astrology, astronomy, mysticism. An alchemist was called a chemist in popular speech, and later the suffix -ry was added to this to describe the art of the chemist as chemistry, the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία and this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, which is in turn derived from the word Chemi or Kimi, which is the ancient name of Egypt in Egyptian. Alternately, al-kīmīā may derive from χημεία, meaning cast together, in retrospect, the definition of chemistry has changed over time, as new discoveries and theories add to the functionality of the science. The term chymistry, in the view of noted scientist Robert Boyle in 1661, in 1837, Jean-Baptiste Dumas considered the word chemistry to refer to the science concerned with the laws and effects of molecular forces. More recently, in 1998, Professor Raymond Chang broadened the definition of chemistry to mean the study of matter, early civilizations, such as the Egyptians Babylonians, Indians amassed practical knowledge concerning the arts of metallurgy, pottery and dyes, but didnt develop a systematic theory. Greek atomism dates back to 440 BC, arising in works by such as Democritus and Epicurus. In 50 BC, the Roman philosopher Lucretius expanded upon the theory in his book De rerum natura, unlike modern concepts of science, Greek atomism was purely philosophical in nature, with little concern for empirical observations and no concern for chemical experiments. Work, particularly the development of distillation, continued in the early Byzantine period with the most famous practitioner being the 4th century Greek-Egyptian Zosimos of Panopolis. He formulated Boyles law, rejected the four elements and proposed a mechanistic alternative of atoms. Before his work, though, many important discoveries had been made, the Scottish chemist Joseph Black and the Dutchman J. B. English scientist John Dalton proposed the theory of atoms, that all substances are composed of indivisible atoms of matter. Davy discovered nine new elements including the alkali metals by extracting them from their oxides with electric current, british William Prout first proposed ordering all the elements by their atomic weight as all atoms had a weight that was an exact multiple of the atomic weight of hydrogen. The inert gases, later called the noble gases were discovered by William Ramsay in collaboration with Lord Rayleigh at the end of the century, thereby filling in the basic structure of the table. Organic chemistry was developed by Justus von Liebig and others, following Friedrich Wöhlers synthesis of urea which proved that organisms were, in theory
8.
Engineering
–
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
9.
Computer science
–
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
10.
Electrical engineering
–
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone. Subsequently, broadcasting and recording media made electronics part of daily life, the invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object. Electrical engineers typically hold a degree in engineering or electronic engineering. Practicing engineers may have professional certification and be members of a professional body, such bodies include the Institute of Electrical and Electronics Engineers and the Institution of Engineering and Technology. Electrical engineers work in a wide range of industries and the skills required are likewise variable. These range from basic circuit theory to the management skills required of a project manager, the tools and equipment that an individual engineer may need are similarly variable, ranging from a simple voltmeter to a top end analyzer to sophisticated design and manufacturing software. Electricity has been a subject of scientific interest since at least the early 17th century and he also designed the versorium, a device that detected the presence of statically charged objects. In the 19th century, research into the subject started to intensify, Electrical engineering became a profession in the later 19th century. Practitioners had created an electric telegraph network and the first professional electrical engineering institutions were founded in the UK. Over 50 years later, he joined the new Society of Telegraph Engineers where he was regarded by other members as the first of their cohort, Practical applications and advances in such fields created an increasing need for standardised units of measure. They led to the standardization of the units volt, ampere, coulomb, ohm, farad. This was achieved at a conference in Chicago in 1893. During these years, the study of electricity was considered to be a subfield of physics. Thats because early electrical technology was electromechanical in nature, the Technische Universität Darmstadt founded the worlds first department of electrical engineering in 1882. The first course in engineering was taught in 1883 in Cornell’s Sibley College of Mechanical Engineering. It was not until about 1885 that Cornell President Andrew Dickson White established the first Department of Electrical Engineering in the United States, in the same year, University College London founded the first chair of electrical engineering in Great Britain. Professor Mendell P. Weinbach at University of Missouri soon followed suit by establishing the engineering department in 1886
11.
Social science
–
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
12.
Economics
–
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
13.
Psychology
–
Psychology is the science of behavior and mind, embracing all aspects of conscious and unconscious experience as well as thought. It is a discipline and a social science which seeks to understand individuals and groups by establishing general principles. In this field, a professional practitioner or researcher is called a psychologist and can be classified as a social, behavioral, Psychologists explore behavior and mental processes, including perception, cognition, attention, emotion, intelligence, phenomenology, motivation, brain functioning, and personality. This extends to interaction between people, such as relationships, including psychological resilience, family resilience, and other areas. Psychologists of diverse orientations also consider the unconscious mind, Psychologists employ empirical methods to infer causal and correlational relationships between psychosocial variables. Psychology has been described as a hub science, with psychological findings linking to research and perspectives from the sciences, natural sciences, medicine, humanities. By many accounts psychology ultimately aims to benefit society, the majority of psychologists are involved in some kind of therapeutic role, practicing in clinical, counseling, or school settings. Many do scientific research on a range of topics related to mental processes and behavior. The word psychology derives from Greek roots meaning study of the psyche, the Latin word psychologia was first used by the Croatian humanist and Latinist Marko Marulić in his book, Psichiologia de ratione animae humanae in the late 15th century or early 16th century. In 1890, William James defined psychology as the science of mental life and this definition enjoyed widespread currency for decades. Also since James defined it, the more strongly connotes techniques of scientific experimentation. Folk psychology refers to the understanding of people, as contrasted with that of psychology professionals. The ancient civilizations of Egypt, Greece, China, India, historians note that Greek philosophers, including Thales, Plato, and Aristotle, addressed the workings of the mind. As early as the 4th century BC, Greek physician Hippocrates theorized that mental disorders had physical rather than supernatural causes, in China, psychological understanding grew from the philosophical works of Laozi and Confucius, and later from the doctrines of Buddhism. This body of knowledge involves insights drawn from introspection and observation and it frames the universe as a division of, and interaction between, physical reality and mental reality, with an emphasis on purifying the mind in order to increase virtue and power. Chinese scholarship focused on the advanced in the Qing Dynasty with the work of Western-educated Fang Yizhi, Liu Zhi. Distinctions in types of awareness appear in the ancient thought of India, a central idea of the Upanishads is the distinction between a persons transient mundane self and their eternal unchanging soul. Divergent Hindu doctrines, and Buddhism, have challenged this hierarchy of selves, yoga is a range of techniques used in pursuit of this goal
14.
Sociology
–
Sociology is the study of social behaviour or society, including its origins, development, organisation, networks, and institutions. It is a science that uses various methods of empirical investigation and critical analysis to develop a body of knowledge about social order, disorder. Many sociologists aim to research that may be applied directly to social policy and welfare. Subject matter ranges from the level of individual agency and interaction to the macro level of systems. The traditional focuses of sociology include social stratification, social class, social mobility, religion, secularization, law, sexuality, the range of social scientific methods has also expanded. Social researchers draw upon a variety of qualitative and quantitative techniques, the linguistic and cultural turns of the mid-twentieth century led to increasingly interpretative, hermeneutic, and philosophic approaches towards the analysis of society. There is often a great deal of crossover between social research, market research, and other statistical fields, Sociology is distinguished from various general social studies courses, which bear little relation to sociological theory or to social-science research-methodology. The US National Science Foundation classifies sociology as a STEM field, Sociological reasoning pre-dates the foundation of the discipline. Social analysis has origins in the stock of Western knowledge and philosophy. The origin of the survey, i. e, there is evidence of early sociology in medieval Arab writings. The word sociology is derived from both Latin and Greek origins, the Latin word, socius, companion, the suffix -logy, the study of from Greek -λογία from λόγος, lógos, word, knowledge. It was first coined in 1780 by the French essayist Emmanuel-Joseph Sieyès in an unpublished manuscript, Sociology was later defined independently by the French philosopher of science, Auguste Comte, in 1838. Comte used this term to describe a new way of looking at society, Comte had earlier used the term social physics, but that had subsequently been appropriated by others, most notably the Belgian statistician Adolphe Quetelet. Comte endeavoured to unify history, psychology and economics through the understanding of the social realm. Comte believed a positivist stage would mark the final era, after conjectural theological and metaphysical phases, Comte gave a powerful impetus to the development of sociology, an impetus which bore fruit in the later decades of the nineteenth century. To say this is not to claim that French sociologists such as Durkheim were devoted disciples of the high priest of positivism. To be sure, beginnings can be traced back well beyond Montesquieu, for example, Marx rejected Comtean positivism but in attempting to develop a science of society nevertheless came to be recognized as a founder of sociology as the word gained wider meaning. For Isaiah Berlin, Marx may be regarded as the father of modern sociology
15.
Dynamical system
–
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the models that describe the swinging of a clock pendulum, the flow of water in a pipe. At any given time, a system has a state given by a tuple of real numbers that can be represented by a point in an appropriate state space. The evolution rule of the system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a time interval only one future state follows from the current state. However, some systems are stochastic, in random events also affect the evolution of the state variables. In physics, a system is described as a particle or ensemble of particles whose state varies over time. In order to make a prediction about the future behavior. Dynamical systems are a part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly process. The concept of a system has its origins in Newtonian mechanics. To determine the state for all future times requires iterating the relation many times—each advancing time a small step, the iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, given a point it is possible to determine all its future positions. Before the advent of computers, finding an orbit required sophisticated mathematical techniques, numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, the difficulties arise because, The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions, to address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent, the operation for comparing orbits to establish their equivalence changes with the different notions of stability. The type of trajectory may be more important than one particular trajectory, some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class, classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes
16.
Game theory
–
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
17.
Outline of physical science
–
Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a physical science, in natural science, hypotheses must be verified scientifically to be regarded as scientific theory. Validity, accuracy, and social mechanisms ensuring quality control, such as review and repeatability of findings, are amongst the criteria. Natural science can be broken into two branches, life science, for example biology and physical science. Each of these branches, and all of their sub-branches, are referred to as natural sciences, physics – natural and physical science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the analysis of nature, conducted in order to understand how the universe behaves. Branches of astronomy Chemistry – studies the composition, structure, properties, branches of chemistry Earth science – all-embracing term referring to the fields of science dealing with planet Earth. Earth science is the study of how the natural environment works and it includes the study of the atmosphere, hydrosphere, lithosphere, and biosphere. Branches of Earth science History of physical science – history of the branch of science that studies non-living systems. It in turn has many branches, each referred to as a physical science, however, the term physical creates an unintended, somewhat arbitrary distinction, since many branches of physical science also study biological phenomena. History of astrodynamics – history of the application of ballistics and celestial mechanics to the problems concerning the motion of rockets. History of astrometry – history of the branch of astronomy that involves precise measurements of the positions and movements of stars, History of cosmology – history of the discipline that deals with the nature of the Universe as a whole. History of physical cosmology – history of the study of the largest-scale structures, History of planetary science – history of the scientific study of planets, moons, and planetary systems, in particular those of the Solar System and the processes that form them. History of neurophysics – history of the branch of biophysics dealing with the nervous system, History of chemical physics – history of the branch of physics that studies chemical processes from the point of view of physics. History of computational physics – history of the study and implementation of algorithms to solve problems in physics for which a quantitative theory already exists. History of condensed matter physics – history of the study of the properties of condensed phases of matter. History of cryogenics – history of the cryogenics is the study of the production of low temperature. History of biomechanics – history of the study of the structure and function of biological systems such as humans, animals, plants, organs, History of fluid mechanics – history of the study of fluids and the forces on them
18.
Defining equation (physics)
–
In physics, defining equations are equations that define new quantities in terms of base quantities. This article uses the current SI system of units, not natural or characteristic units, physical quantities and units follow the same hierarchy, chosen base quantities have defined base units, from these any other quantities may be derived and have corresponding derived units. Defining quantities is analogous to mixing colours, and could be classified a similar way, primary colours are to base quantities, as secondary colours are to derived quantities. Mixing colours is analogous to combining quantities using mathematical operations, the choice of a base system of quantities and units is arbitrary, but once chosen it must be adhered to throughout all analysis which follows for consistency. It makes no sense to mix up different systems of units, choosing a system of units, one system out of the SI, CGS etc. is like choosing whether use paint or light colours. Much of physics requires definitions to be made for the equations to make sense, theoretical implications, Definitions are important since they can lead into new insights of a branch of physics. Two such examples occurred in classical physics, ease of comparison, They allow comparisons of measurements to be made when they might appear ambiguous and unclear otherwise. Example A basic example is mass density and it is not clear how compare how much matter constitutes a variety of substances given only their masses or only their volumes. Making such comparisons without using mathematics logically in this way would not be as systematic, functions may be incorporated into a definition, in for calculus this is necessary. Quantities may also be complex-valued for theoretical advantage, but for a measurement the real part is relevant. For more advanced treatments the equation may have to be written in an equivalent, often definitions can start from elementary algebra, then modify to vectors, then in the limiting cases calculus may be used. The various levels of maths used typically follows this pattern, for vector equations, sometimes the defining quantity is in a cross or dot product and cannot be solved for explicitly as a vector, but the components can. Examples Electric current density is an example spanning all of these methods, see the classical mechanics section below for nomenclature and diagrams to the right. Elementary algebra Operations are simply multiplication and division, equations may be written in a product or quotient form, both of course equivalent. Vector algebra There is no way to divide a vector by a vector, elementary calculus The arithmetic operations are modified to the limiting cases of differentiation and integration. Equations can be expressed in these equivalent and alternative ways, vector calculus Tensor analysis Vectors are rank-1 tensors. The formulae below are no more than the equations in the language of tensors. Sometimes there is still freedom within the chosen units system, to one or more quantities in more than one way
19.
Boundary value problem
–
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them, problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems, the analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed and this means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of differential equations is devoted to proving that boundary value problems arising from scientific. Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions, boundary value problems are similar to initial value problems. Finding the temperature at all points of a bar with one end kept at absolute zero. If the problem is dependent on both space and time, one could specify the value of the problem at a point for all time or at a given time for all space. Concretely, an example of a value is the problem y ″ + y =0 to be solved for the unknown function y with the boundary conditions y =0, y =2. Without the boundary conditions, the solution to this equation is y = A sin + B cos . From the boundary condition y =0 one obtains 0 = A ⋅0 + B ⋅1 which implies that B =0, from the boundary condition y =2 one finds 2 = A ⋅1 and so A =2. One sees that imposing boundary conditions allowed one to determine a unique solution, a boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of a rod is held at absolute zero. A boundary condition which specifies the value of the derivative of the function is a Neumann boundary condition. For example, if there is a heater at one end of a rod, then energy would be added at a constant rate. If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition. Summary of boundary conditions for the function, y, constants c 0 and c 1 specified by the boundary conditions
20.
Kinematics
–
Kinematics as a field of study is often referred to as the geometry of motion and as such may be seen as a branch of mathematics. The study of the influence of forces acting on masses falls within the purview of kinetics, for further details, see analytical dynamics. Kinematics is used in astrophysics to describe the motion of celestial bodies, in mechanical engineering, robotics, and biomechanics kinematics is used to describe the motion of systems composed of joined parts such as an engine, a robotic arm or the human skeleton. Kinematic analysis is the process of measuring the quantities used to describe motion. In addition, kinematics applies geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A. M, ampères cinématique, which he constructed from the Greek κίνημα kinema, itself derived from κινεῖν kinein. Kinematic and cinématique are related to the French word cinéma, particle kinematics is the study of the trajectory of a particle. The position of a particle is defined to be the vector from the origin of a coordinate frame to the particle. If the tower is 50 m high, then the vector to the top of the tower is r=. In the most general case, a coordinate system is used to define the position of a particle. However, if the particle is constrained to move in a surface, all observations in physics are incomplete without those observations being described with respect to a reference frame. The position vector of a particle is a vector drawn from the origin of the frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin, the magnitude of the position vector |P| gives the distance between the point P and the origin. | P | = x P2 + y P2 + z P2, the direction cosines of the position vector provide a quantitative measure of direction. It is important to note that the vector of a particle isnt unique. The position vector of a particle is different relative to different frames of reference. The velocity of a particle is a quantity that describes the direction of motion. More mathematically, the rate of change of the vector of a point
21.
Differential operator
–
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an operation that accepts a function. This article considers mainly linear operators, which are the most common type, however, non-linear differential operators, such as the Schwarzian derivative also exist. Assume that there is a map A from a function space F1 to another function space F2, the most common differential operator is the action of taking the derivative itself. Common notations for taking the first derivative with respect to a variable x include, d d x, D, D x, and ∂ x. When taking higher, nth order derivatives, the operator may also be written, d n d x n, D n, the derivative of a function f of an argument x is sometimes given as either of the following, ′ f ′. The D notations use and creation is credited to Oliver Heaviside, one of the most frequently seen differential operators is the Laplacian operator, defined by Δ = ∇2 = ∑ k =1 n ∂2 ∂ x k 2. Another differential operator is the Θ operator, or theta operator, as in one variable, the eigenspaces of Θ are the spaces of homogeneous polynomials. In writing, following common mathematical convention, the argument of an operator is usually placed on the right side of the operator itself. Such a bidirectional-arrow notation is used for describing the probability current of quantum mechanics. The differential operator del, also called nabla operator, is an important vector differential operator and it appears frequently in physics in places like the differential form of Maxwells equations. In three-dimensional Cartesian coordinates, del is defined, ∇ = x ^ ∂ ∂ x + y ^ ∂ ∂ y + z ^ ∂ ∂ z. Del is used to calculate the gradient, curl, divergence, and Laplacian of various objects. This definition therefore depends on the definition of the scalar product. In the functional space of functions, the scalar product is defined by ⟨ f, g ⟩ = ∫ a b f g ¯ d x. If one moreover adds the condition that f or g vanishes for x → a and x → b and this formula does not explicitly depend on the definition of the scalar product. It is therefore chosen as a definition of the adjoint operator. When T ∗ is defined according to formula, it is called the formal adjoint of T. A self-adjoint operator is an equal to its own adjoint
22.
Mathematical optimization
–
In mathematics, computer science and operations research, mathematical optimization, also spelled mathematical optimisation, is the selection of a best element from some set of available alternatives. The generalization of optimization theory and techniques to other formulations comprises an area of applied mathematics. Such a formulation is called a problem or a mathematical programming problem. Many real-world and theoretical problems may be modeled in this general framework, typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the space or the choice set. The function f is called, variously, a function, a loss function or cost function, a utility function or fitness function, or, in certain fields. A feasible solution that minimizes the objective function is called an optimal solution, in mathematics, conventional optimization problems are usually stated in terms of minimization. Generally, unless both the function and the feasible region are convex in a minimization problem, there may be several local minima. While a local minimum is at least as good as any nearby points, a global minimum is at least as good as every feasible point. In a convex problem, if there is a minimum that is interior, it is also the global minimum. Optimization problems are often expressed with special notation, consider the following notation, min x ∈ R This denotes the minimum value of the objective function x 2 +1, when choosing x from the set of real numbers R. The minimum value in case is 1, occurring at x =0. Similarly, the notation max x ∈ R2 x asks for the value of the objective function 2x. In this case, there is no such maximum as the function is unbounded. This represents the value of the argument x in the interval, John Wiley & Sons, Ltd. pp. xxviii+489. (2008 Second ed. in French, Programmation mathématique, Théorie et algorithmes, Editions Tec & Doc, Paris,2008. Nemhauser, G. L. Rinnooy Kan, A. H. G. Todd, handbooks in Operations Research and Management Science. Amsterdam, North-Holland Publishing Co. pp. xiv+709, J. E. Dennis, Jr. and Robert B
23.
Linear equation
–
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A simple example of an equation with only one variable, x, may be written in the form, ax + b =0, where a and b are constants. The constants may be numbers, parameters, or even functions of parameters. Linear equations can have one or more variables. An example of an equation with three variables, x, y, and z, is given by, ax + by + cz + d =0, where a, b, c, and d are constants and a, b. Linear equations occur frequently in most subareas of mathematics and especially in applied mathematics, an equation is linear if the sum of the exponents of the variables of each term is one. Equations with exponents greater than one are non-linear, an example of a non-linear equation of two variables is axy + b =0, where a and b are constants and a ≠0. It has two variables, x and y, and is non-linear because the sum of the exponents of the variables in the first term and this article considers the case of a single equation for which one searches the real solutions. All its content applies for complex solutions and, more generally for linear equations with coefficients, a linear equation in one unknown x may always be rewritten a x = b. If a ≠0, there is a solution x = b a. The origin of the name comes from the fact that the set of solutions of such an equation forms a straight line in the plane. Linear equations can be using the laws of elementary algebra into several different forms. These equations are referred to as the equations of the straight line. In what follows, x, y, t, and θ are variables, in the general form the linear equation is written as, A x + B y = C, where A and B are not both equal to zero. The equation is written so that A ≥0, by convention. The graph of the equation is a line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, if B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis, is C/B, and the slope of the line is −A/B. The general form is written as, a x + b y + c =0
24.
Nonlinear system
–
In mathematics and physical sciences, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, physicists and mathematicians, nonlinear systems may appear chaotic, unpredictable or counterintuitive, contrasting with the much simpler linear systems. In other words, in a system of equations, the equation to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as non-linear, regardless of whether or not known linear functions appear in the equations. In particular, an equation is linear if it is linear in terms of the unknown function and its derivatives. As nonlinear equations are difficult to solve, nonlinear systems are approximated by linear equations. This works well up to some accuracy and some range for the input values and it follows that some aspects of the behavior of a nonlinear system appear commonly to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is not random. For example, some aspects of the weather are seen to be chaotic and this nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term nonlinear science for the study of nonlinear systems and this is disputed by others, Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. In mathematics, a function f is one which satisfies both of the following properties, Additivity or superposition, f = f + f, Homogeneity. Additivity implies homogeneity for any rational α, and, for continuous functions, for a complex α, homogeneity does not follow from additivity. For example, a map is additive but not homogeneous. The equation is called homogeneous if C =0, if f contains differentiation with respect to x, the result will be a differential equation. Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials to zero, for example, x 2 + x −1 =0. For a single equation, root-finding algorithms can be used to find solutions to the equation. However, systems of equations are more complicated, their study is one motivation for the field of algebraic geometry. It is even difficult to decide whether a given system has complex solutions
25.
Chaos theory
–
Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. This happens even though these systems are deterministic, meaning that their behavior is fully determined by their initial conditions. In other words, the nature of these systems does not make them predictable. This behavior is known as chaos, or simply chaos. The theory was summarized by Edward Lorenz as, Chaos, When the present determines the future, Chaotic behavior exists in many natural systems, such as weather and climate. It also occurs spontaneously in some systems with components, such as road traffic. This behavior can be studied through analysis of a mathematical model, or through analytical techniques such as recurrence plots. Chaos theory has applications in several disciplines, including meteorology, sociology, physics, environmental science, computer science, engineering, economics, biology, ecology, the theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, self-assembly process. Chaos theory concerns deterministic systems whose behavior can in principle be predicted, Chaotic systems are predictable for a while and then appear to become random. Some examples of Lyapunov times are, chaotic electrical circuits, about 1 millisecond, weather systems, a few days, in chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast and this means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random, in common usage, chaos means a state of disorder. However, in theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition originally formulated by Robert L, in these cases, while it is often the most practically significant property, sensitivity to initial conditions need not be stated in the definition. If attention is restricted to intervals, the second property implies the other two, an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list. Sensitivity to initial conditions means that each point in a system is arbitrarily closely approximated by other points with significantly different future paths. Thus, a small change, or perturbation, of the current trajectory may lead to significantly different future behavior. C. Entitled Predictability, Does the Flap of a Butterflys Wings in Brazil set off a Tornado in Texas, the flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena
26.
Irreversible process
–
In science, a process that is not reversible is called irreversible. This concept arises frequently in thermodynamics, a system that undergoes an irreversible process may still be capable of returning to its initial state, however, the impossibility occurs in restoring the environment to its own initial conditions. An irreversible process increases the entropy of the universe, however, because entropy is a state function, the change in entropy of the system is the same whether the process is reversible or irreversible. The second law of thermodynamics can be used to determine whether a process is reversible or not, all complex natural processes are irreversible. A certain amount of energy will be used as the molecules of the working body do work on each other when they change from one state to another. Many biological processes that were thought to be reversible have been found to actually be a pairing of two irreversible processes. Thermodynamics defines the behaviour of large numbers of entities, whose exact behavior is given by more specific laws. The irreversibility of thermodynamics must be statistical in nature, that is, that it must be highly unlikely, but not impossible. The German physicist Rudolf Clausius, in the 1850s, was the first to quantify the discovery of irreversibility in nature through his introduction of the concept of entropy. For example, a cup of hot coffee placed in an area of temperature will transfer heat to its surroundings. However, that same initial cup of coffee will never absorb heat from its surroundings causing it to grow even hotter with the temperature of the room decreasing, therefore, the process of the coffee cooling down is irreversible unless extra energy is added to the system. However, a paradox arose when attempting to reconcile microanalysis of a system with observations of its macrostate, many processes are mathematically reversible in their microstate when analyzed using classical Newtonian mechanics. His formulas quantified the work done by William Thomson, 1st Baron Kelvin who had argued that, in 1890, he published his first explanation of nonlinear dynamics, also called chaos theory. Sensitivity to initial conditions relating to the system and its environment at the compounds into an exhibition of irreversible characteristics within the observable. In the physical realm, many processes are present to which the inability to achieve 100% efficiency in energy transfer can be attributed. The following is a list of events which contribute to the irreversibility of processes. The internal energy of the gas remains the same, while the volume increases, the original state cannot be recovered by simply compressing the gas to its original volume, since the internal energy will be increased by this compression. The original state can only be recovered by then cooling the re-compressed system, the diagram to the right applies only if the first expansion is free
27.
Linearization
–
In mathematics linearization refers to finding the linear approximation to a function at a given point. In the study of systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in such as engineering, physics, economics. Linearizations of a function are lines—usually lines that can be used for purposes of calculation, in short, linearization approximates the output of a function near x = a. However, what would be an approximation of 4.001 =4 +.001. For any given function y = f, f can be approximated if it is near a known differentiable point, the most basic requisite is that L a = f, where L a is the linearization of f at x = a. The point-slope form of an equation forms an equation of a line, given a point, the general form of this equation is, y − K = M. Using the point, L a becomes y = f + M, because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to f at x = a. While the concept of local linearity applies the most to points arbitrarily close to x = a, the slope M should be, most accurately, the slope of the tangent line at x = a. Visually, the diagram shows the tangent line of f at x. At f, where h is any positive or negative value. The final equation for the linearization of a function at x = a is, the derivative of f is f ′, and the slope of f at a is f ′. To find 4.001, we can use the fact that 4 =2. The linearization of f = x at x = a is y = a +12 a, substituting in a =4, the linearization at 4 is y =2 + x −44. In this case x =4.001, so 4.001 is approximately 2 +4.001 −44 =2.00025. The true value is close to 2.00024998, so the linearization approximation has an error of less than 1 millionth of a percent. Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest, in stability analysis of autonomous systems, one can use the eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium
28.
Differential equation
–
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
29.
Newton's method
–
In numerical analysis, Newtons method, named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. If the function satisfies the assumptions made in the derivation of the formula, geometrically, is the intersection of the x-axis and the tangent of the graph of f at. The process is repeated as x n +1 = x n − f f ′ until an accurate value is reached. This algorithm is first in the class of Householders methods, succeeded by Halleys method, the method can also be extended to complex functions and to systems of equations. This x-intercept will typically be an approximation to the functions root than the original guess. Suppose f, → ℝ is a function defined on the interval with values in the real numbers ℝ. The formula for converging on the root can be easily derived, suppose we have some current approximation xn. Then we can derive the formula for an approximation, xn +1 by referring to the diagram on the right. The equation of the tangent line to the curve y = f at the point x = xn is y = f ′ + f, the x-intercept of this line is then used as the next approximation to the root, xn +1. In other words, setting y to zero and x to xn +1 gives 0 = f ′ + f, Solving for xn +1 gives x n +1 = x n − f f ′. We start the process off with some arbitrary initial value x0, the method will usually converge, provided this initial guess is close enough to the unknown zero, and that f ′ ≠0. More details can be found in the section below. The Householders methods are similar but have higher order for even faster convergence, however, his method differs substantially from the modern method given above, Newton applies the method only to polynomials. He does not compute the successive approximations xn, but computes a sequence of polynomials, finally, Newton views the method as purely algebraic and makes no mention of the connection with calculus. Newton may have derived his method from a similar but less precise method by Vieta, a special case of Newtons method for calculating square roots was known much earlier and is often called the Babylonian method. Newtons method was used by 17th-century Japanese mathematician Seki Kōwa to solve single-variable equations, Newtons method was first published in 1685 in A Treatise of Algebra both Historical and Practical by John Wallis. In 1690, Joseph Raphson published a description in Analysis aequationum universalis. Finally, in 1740, Thomas Simpson described Newtons method as a method for solving general nonlinear equations using calculus
30.
Broyden's method
–
In numerical analysis, Broydens method is a quasi-Newton method for finding roots in k variables. It was originally described by C. G. Broyden in 1965, newtons method for solving f =0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian is a difficult and expensive operation, the idea behind Broydens method is to compute the whole Jacobian only at the first iteration, and to do a rank-one update at the other iterations. The Jacobian matrix is determined based on the secant equation in the finite difference approximation, J n ≃ f − f. The above equation is underdetermined when k is greater than one and we may then proceed in the Newton direction, x n +1 = x n − J n −1 f. A similar technique can be derived by using a different modification to Jn −1. Many other quasi-Newton schemes have been suggested in optimization, where one seeks a maximum or minimum by finding the root of the first derivative, the Jacobian of the gradient is called Hessian and is symmetric, adding further constraints to its update. Broyden has defined not only two methods, but a class of methods. Other members of this class have been added by other authors, the Davidon–Fletcher–Powell update is the only member of this class being published before the two members defined by Broyden. Schuberts or sparse Broyden algorithm – a modification for sparse Jacobian matrices, klement – uses fewer iterations to solve many equation systems
31.
Jet engine
–
A jet engine is a reaction engine discharging a fast-moving jet that generates thrust by jet propulsion. This broad definition includes airbreathing jet engines and non-airbreathing jet engines, in general, jet engines are combustion engines. In common parlance, the jet engine loosely refers to an internal combustion airbreathing jet engine. These typically feature an air compressor powered by a turbine. Jet aircraft use such engines for long-distance travel, early jet aircraft used turbojet engines which were relatively inefficient for subsonic flight. Modern subsonic jet aircraft usually use more complex high-bypass turbofan engines and these engines offer high speed and greater fuel efficiency than piston and propeller aeroengines over long distances. Jet engines date back to the invention of the aeolipile before the first century AD and this device directed steam power through two nozzles to cause a sphere to spin rapidly on its axis. So far as is known, it did not supply mechanical power, instead, it was seen as a curiosity. However, although powerful, at reasonable flight speeds rockets are very inefficient. The earliest attempts at airbreathing jet engines were hybrid designs in which a power source first compressed air. In one such system, called a thermojet by Secondo Campini but more commonly, motorjet, examples of this type of design were the Caproni Campini N.1, and the Japanese Tsu-11 engine intended to power Ohka kamikaze planes towards the end of World War II. None were entirely successful and the N.1 ended up being slower than the design with a traditional engine. If aircraft performance were ever to increase beyond such a barrier and this was the motivation behind the development of the gas turbine engine, commonly called a jet engine. The key to a jet engine was the gas turbine. The gas turbine was not an idea developed in the 1930s, the first gas turbine to successfully run self-sustaining was built in 1903 by Norwegian engineer Ægidius Elling. Limitations in design and practical engineering and metallurgy prevented such engines reaching manufacture, the main problems were safety, reliability, weight and, especially, sustained operation. The first patent for using a gas turbine to power an aircraft was filed in 1921 by Frenchman Maxime Guillaume and his engine was an axial-flow turbojet. Alan Arnold Griffith published An Aerodynamic Theory of Turbine Design in 1926 leading to work at the RAE
32.
Thermodynamic cycle
–
In the process of passing through a cycle, the working fluid may convert heat from a warm source into useful work, and dispose of the remaining heat to a cold sink, thereby acting as a heat engine. Conversely, the cycle may be reversed and use work to move heat from a cold source, during a closed cycle, the system returns to its original thermodynamic state of temperature and pressure. Process quantities, such as heat and work are process dependent, ein might be the work and heat input during the cycle and Eout would be the work and heat output during the cycle. The first law of thermodynamics also dictates that the net heat input is equal to the net work output over a cycle, the repeating nature of the process path allows for continuous operation, making the cycle an important concept in thermodynamics. Thermodynamic cycles are often represented mathematically as quasistatic processes in the modeling of the workings of an actual device, two primary classes of thermodynamic cycles are power cycles and heat pump cycles. Power cycles are cycles which convert some heat input into a mechanical work output, cycles composed entirely of quasistatic processes can operate as power or heat pump cycles by controlling the process direction. On a pressure-volume diagram or temperature-entropy diagram, the clockwise and counterclockwise directions indicate power and heat pump cycles, because the net variation in state properties during a thermodynamic cycle is zero, it forms a closed loop on a PV diagram. A PV diagrams Y axis shows pressure and X axis shows volume, if the cyclic process moves clockwise around the loop, then W will be positive, and it represents a heat engine. If it moves counterclockwise, then W will be negative, and this does not exclude energy transfer as work. Isothermal, The process is at a constant temperature during that part of the cycle and this does not exclude energy transfer as heat or work. Isobaric, Pressure in that part of the cycle will remain constant and this does not exclude energy transfer as heat or work. Isochoric, The process is constant volume and this does not exclude energy transfer as heat or work. Isentropic, The process is one of constant entropy and this excludes the transfer of heat but not work. Thermodynamic power cycles are the basis for the operation of heat engines, power cycles can be organized into two categories, real cycles and ideal cycles. Cycles encountered in real world devices are difficult to analyze because of the presence of complicating effects, power cycles can also be divided according to the type of heat engine they seek to model. The most common used to model internal combustion engines are the Otto cycle, which models gasoline engines, and the Diesel cycle. There is no difference between the two except the purpose of the refrigerator is to cool a very small space while the heat pump is intended to warm a house. Both work by moving heat from a space to a warm space