1.
Computational physics
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Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of computers in science. In physics, different theories based on mathematical models provide very precise predictions on how systems behave, unfortunately, it is often the case that solving the mathematical model for a particular system in order to produce a useful prediction is not feasible. This can occur, for instance, when the solution does not have a closed-form expression, in such cases, numerical approximations are required. There is a debate about the status of computation within the scientific method, while computers can be used in experiments for the measurement and recording of data, this clearly does not constitute a computational approach. Physics problems are in very difficult to solve exactly. This is due to several reasons, lack of algebraic and/or analytic solubility, complexity, on the more advanced side, mathematical perturbation theory is also sometimes used. In addition, the computational cost and computational complexity for many-body problems tend to grow quickly, a macroscopic system typically has a size of the order of 1023 constituent particles, so it is somewhat of a problem. Solving quantum mechanical problems is generally of exponential order in the size of the system, because computational physics uses a broad class of problems, it is generally divided amongst the different mathematical problems it numerically solves, or the methods it applies. Furthermore, computational physics encompasses the tuning of the structure to solve the problems. It is possible to find a corresponding computational branch for every field in physics, for example computational mechanics. Computational mechanics consists of fluid dynamics, computational solid mechanics. One subfield at the confluence between CFD and electromagnetic modelling is computational magnetohydrodynamics, the quantum many-body problem leads naturally to the large and rapidly growing field of computational chemistry. Computational solid state physics is an important division of computational physics dealing directly with material science. A field related to computational condensed matter is computational statistical mechanics, computational statistical physics makes heavy use of Monte Carlo-like methods. More broadly, it concerns itself with in the social sciences, network theory, and mathematical models for the propagation of disease. Computational astrophysics is the application of techniques and methods to astrophysical problems. Stickler, E. Schachinger, Basic concepts in computational physics, E. Winsberg, Science in the Age of Computer Simulation
Computational physics
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Computational physics
2.
Scientific visualization
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Scientific visualization is an interdisciplinary branch of science. It is also considered a subset of computer graphics, a branch of computer science, the purpose of scientific visualization is to graphically illustrate scientific data to enable scientists to understand, illustrate, and glean insight from their data. One of the earliest examples of scientific visualisation was Maxwells thermodynamic surface. This prefigured modern scientific techniques that use computer graphics. Scientific visualization using computer graphics gained in popularity as graphics matured, primary applications were scalar fields and vector fields from computer simulations and also measured data. The primary methods for visualizing two-dimensional scalar fields are color mapping and drawing contour lines, 2D vector fields are visualized using glyphs and streamlines or line integral convolution methods. For 3D scalar fields the primary methods are volume rendering and isosurfaces, methods for visualizing vector fields include glyphs such as arrows, streamlines and streaklines, particle tracing, line integral convolution and topological methods. Later, visualization techniques such as hyperstreamlines were developed to visualize 2D, computer animation is the art, technique, and science of creating moving images via the use of computers. It is becoming common to be created by means of 3D computer graphics, though 2D computer graphics are still widely used for stylistic, low bandwidth. Sometimes the target of the animation is the computer itself, but sometimes the target is another medium and it is also referred to as CGI, especially when used in films. Computer simulation is a program, or network of computers. The simultaneous visualization and simulation of a system is called visulation, computer simulations vary from computer programs that run a few minutes, to network-based groups of computers running for hours, to ongoing simulations that run for months. Information visualization focused on the creation of approaches for conveying information in intuitive ways. The key difference between scientific visualization and information visualization is that information visualization is often applied to data that is not generated by scientific inquiry, some examples are graphical representations of data for business, government, news and social media. Interface technology and perception shows how new interfaces and an understanding of underlying perceptual issues create new opportunities for the scientific visualization community. Rendering is the process of generating an image from a model, the model is a description of three-dimensional objects in a strictly defined language or data structure. It would contain geometry, viewpoint, texture, lighting, the image is a digital image or raster graphics image. The term may be by analogy with a rendering of a scene
Scientific visualization
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A scientific visualization of a simulation of a Rayleigh–Taylor instability caused by two mixing fluids.
Scientific visualization
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Surface rendering of Arabidopsis thaliana pollen grains with confocal microscope.
Scientific visualization
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Scientific visualization of Fluid Flow: Surface waves in water
Scientific visualization
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Chemical imaging of a simultaneous release of SF 6 and NH 3.
3.
Yukawa potential
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The potential is monotone increasing in r and it is negative, implying the force is attractive. In the SI system, the unit of the Yukawa potential is, the Coulomb potential of electromagnetism is an example of a Yukawa potential with e−kmr equal to 1 everywhere. This can be interpreted as saying that the mass m is equal to 0. In interactions between a field and a fermion field, the constant g is equal to the gauge coupling constant between those fields. In the case of the force, the fermions would be a proton. Hideki Yukawa showed in the 1930s that such a potential arises from the exchange of a scalar field such as the field of a massive boson. Since the field mediator is massive the corresponding force has a certain range, if the mass is zero, then the Yukawa potential equals a Coulomb potential, and the range is said to be infinite. In fact, we have, m =0 ⇒ e − m r = e 0 =1, consequently, the equation V Yukawa = − g 2 e − m r r simplifies to the form of the Coulomb potential V Coulomb = − g 21 r. A comparison of the long range potential strength for Yukawa and Coulomb is shown in Figure 2 and it can be seen that the Coulomb potential has effect over a greater distance whereas the Yukawa potential approaches zero rather quickly. However, any Yukawa potential or Coulomb potential are non-zero for any large r, the easiest way to understand that the Yukawa potential is associated with a massive field is by examining its Fourier transform. One has V = − g 23 ∫ e i k ⋅ r 4 π k 2 + m 2 d 3 k where the integral is performed all possible values of the 3-vector momentum k. In this form, the fraction 4 π / is seen to be the propagator or Greens function of the Klein–Gordon equation, the Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The Yukawa interaction couples the fermion field ψ to the meson field ϕ with the coupling term L i n t = g ψ ¯ ϕ ψ. The scattering amplitude for two fermions, one with initial momentum p 1 and the other with momentum p 2, exchanging a meson with momentum k, is given by the Feynman diagram on the right. The Feynman rules for each associate a factor of g with the amplitude. The line in the middle, connecting the two lines, represents the exchange of a meson. The Feynman rule for an exchange is to use the propagator. Thus, we see that the Feynman amplitude for this graph is nothing more than V = − g 24 π k 2 + m 2, from the previous section, this is seen to be the Fourier transform of the Yukawa potential
Yukawa potential
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Figure 1: A comparison of Yukawa potentials where g=1 and with various values for m.
4.
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
Finite element method
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Visualization of how a car deforms in an asymmetrical crash using finite element analysis. [1]
Finite element method
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
5.
Smoothed-particle hydrodynamics
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Smoothed-particle hydrodynamics is a computational method used for simulating the dynamics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy initially for astrophysical problems and it has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a mesh-free Lagrangian method, and the resolution of the method can easily be adjusted with respect to such as the density. The smoothed-particle hydrodynamics method works by dividing the fluid into a set of discrete elements and these particles have a spatial distance, over which their properties are smoothed by a kernel function. This means that the quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within the range of the kernel. For example, using Monaghans popular cubic spline kernel the temperature at position r depends on the temperatures of all the particles within a radial distance 2 h of r. The contributions of each particle to a property are weighted according to their distance from the particle of interest, mathematically, this is governed by the kernel function. Kernel functions commonly used include the Gaussian function and the cubic spline, the latter function is exactly zero for particles further away than two smoothing lengths. This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles, similarly, the spatial derivative of a quantity can be obtained easily by virtue of the linearity of the derivative. ∇ A = ∑ j m j A j ρ j ∇ W, although the size of the smoothing length can be fixed in both space and time, this does not take advantage of the full power of SPH. By assigning each particle its own smoothing length and allowing it to vary with time, for example, in a very dense region where many particles are close together the smoothing length can be made relatively short, yielding high spatial resolution. Conversely, in low-density regions where particles are far apart and the resolution is low. Combined with an equation of state and an integrator, SPH can simulate hydrodynamic flows efficiently, however, the traditional artificial viscosity formulation used in SPH tends to smear out shocks and contact discontinuities to a much greater extent than state-of-the-art grid-based schemes. The Lagrangian-based adaptivity of SPH is analogous to the adaptivity present in grid-based adaptive mesh refinement codes, in some ways it is actually simpler because SPH particles lack any explicit topology relating them, unlike the elements in FEM. Adaptivity in SPH can be introduced in two ways, either by changing the particle smoothing lengths or by splitting SPH particles into daughter particles with smaller smoothing lengths, the first method is common in astrophysical simulations where the particles naturally evolve into states with large density differences. However, in hydrodynamics simulations where the density is constant this is not a suitable method for adaptivity. For this reason particle splitting can be employed, with conditions for splitting ranging from distance to a free surface through to material shear. Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics, the particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance tree gravity code, particle mesh, or particle-particle particle-mesh
Smoothed-particle hydrodynamics
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Fig. SPH simulation of ocean waves using FLUIDS v.1 (Hoetzlein)
6.
Monte Carlo method
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Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle and they are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are used in three distinct problem classes, optimization, numerical integration, and generating draws from a probability distribution. In principle, Monte Carlo methods can be used to any problem having a probabilistic interpretation. By the law of numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov Chain Monte Carlo sampler, the central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired distribution, by the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation, in other instances we are given a flow of probability distributions with an increasing level of sampling complexity. These models can also be seen as the evolution of the law of the states of a nonlinear Markov chain. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methodologies these mean field particle techniques rely on sequential interacting samples, the terminology mean field reflects the fact that each of the samples interacts with the empirical measures of the process. Monte Carlo methods vary, but tend to follow a particular pattern, generate inputs randomly from a probability distribution over the domain. Perform a deterministic computation on the inputs, for example, consider a circle inscribed in a unit square. Given that the circle and the square have a ratio of areas that is π/4, uniformly scatter objects of uniform size over the square. Count the number of objects inside the circle and the number of objects. The ratio of the two counts is an estimate of the ratio of the two areas, which is π/4, multiply the result by 4 to estimate π. In this procedure the domain of inputs is the square that circumscribes our circle and we generate random inputs by scattering grains over the square then perform a computation on each input. Finally, we aggregate the results to obtain our final result, there are two important points to consider here, Firstly, if the grains are not uniformly distributed, then our approximation will be poor. Secondly, there should be a number of inputs
Monte Carlo method
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Computational physics
7.
N-body simulation
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In physics and astronomy, an N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity. In physical cosmology, N-body simulations are used to study processes of non-linear structure formation such as galaxy filaments, direct N-body simulations are used to study the dynamical evolution of star clusters. The particles treated by the simulation may or may not correspond to objects which are particulate in nature. For example, an N-body simulation of a star cluster might have a particle per star and this quantity need not have any physical significance, but must be chosen as a compromise between accuracy and manageable computer requirements. These calculations are used in situations where interactions between objects, such as stars or planets, are important to the evolution of the system. The first direct N-body simulations were carried out by Sebastian von Hoerner at the Astronomisches Rechen-Institut in Heidelberg, regularization is a mathematical trick to remove the singularity in the Newtonian law of gravitation for two particles which approach each other arbitrarily close. Sverre Aarseths codes are used to study the dynamics of star clusters, planetary systems, many simulations are large enough that the effects of general relativity in establishing a Friedmann-Lemaitre-Robertson-Walker cosmology are significant. This is incorporated in the simulation as a measure of distance in a comoving coordinate system. The boundary conditions of these simulations are usually periodic, so that one edge of the simulation volume matches up with the opposite edge. N-body simulations are simple in principle, because they merely involve integrating the 6N ordinary differential equations defining the particle motions in Newtonian gravity, therefore, a number of refinements are commonly used. There are two basic approximation schemes to decrease the time for such simulations. These can reduce the complexity to O or better, at the loss of accuracy. This can dramatically reduce the number of particle pair interactions that must be computed, for simulations where particles are not evenly distributed, the well-separated pair decomposition methods of Callahan and Kosaraju yield optimal O time per iteration with fixed dimension. The gravitational field can now be found by multiplying by k →, since this method is limited by the mesh size, in practice a smaller mesh or some other technique is used to compute the small-scale forces. Sometimes an adaptive mesh is used, in which the cells are much smaller in the denser regions of the simulation. Several different gravitational perturbation algorithms are used to get accurate estimates of the path of objects in the solar system. People often decide to put a satellite in a frozen orbit and it is possible to find a frozen orbit without calculating the actual path of the satellite. Some characteristics of the paths of a system of particles can be calculated directly
N-body simulation
8.
Molecular dynamics
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Molecular dynamics is a computer simulation method for studying the physical movements of atoms and molecules, and is thus a type of N-body simulation. The atoms and molecules are allowed to interact for a period of time. The method was developed within the field of theoretical physics in the late 1950s but is applied today mostly in chemical physics, materials science. Following the earlier successes of Monte Carlo simulations, the method was developed by Fermi, Pasta, in 1957, Alder and Wainwright used an IBM704 computer to simulate perfectly elastic collisions between hard spheres. In 1960, Gibson et al. simulated radiation damage of solid copper by using a Born-Mayer type of repulsive interaction along with a surface force. In 1964, Rahman published landmark simulations of liquid argon that used a Lennard-Jones potential, calculations of system properties, such as the coefficient of self-diffusion, compared well with experimental data. Even before it became possible to simulate molecular dynamics with computers, the idea was to arrange them to replicate the properties of a liquid. I took a number of balls and stuck them together with rods of a selection of different lengths ranging from 2.75 to 4 inches. I tried to do this in the first place as casually as possible, working in my own office, being interrupted every five minutes or so and not remembering what I had done before the interruption. In physics, MD is used to examine the dynamics of atomic-level phenomena that cannot be observed directly, such as thin film growth and it is also used to examine the physical properties of nanotechnological devices that have not been or cannot yet be created. In principle MD can be used for ab initio prediction of protein structure by simulating folding of the chain from random coil. The results of MD simulations can be tested through comparison to experiments that measure molecular dynamics, michael Levitt, who shared the Nobel Prize awarded in part for the application of MD to proteins, wrote in 1999 that CASP participants usually did not use the method due to. A central embarrassment of molecular mechanics, namely that energy minimization or molecular dynamics generally leads to a model that is less like the experimental structure, limits of the method are related to the parameter sets used, and to the underlying molecular mechanics force fields. The neglected contributions include the conformational entropy of the polypeptide chain, Another important factor are intramolecular hydrogen bonds, which are not explicitly included in modern force fields, but described as Coulomb interactions of atomic point charges. This is an approximation because hydrogen bonds have a partially quantum mechanical and chemical nature. Furthermore, electrostatic interactions are calculated using the dielectric constant of vacuum. Using the macroscopic dielectric constant at short distances is questionable. Finally, van der Waals interactions in MD are usually described by Lennard-Jones potentials based on the Fritz London theory that is applicable in vacuum
Molecular dynamics
Molecular dynamics
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Example of a molecular dynamics simulation in a simple system: deposition of a single Cu atom on a Cu (001) surface. Each circle illustrates the position of a single atom; note that the actual atomic interactions used in current simulations are more complex than those of 2-dimensional hard spheres.
9.
Sergei K. Godunov
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Sergei Konstantinovich Godunov is professor at the Sobolev Institute of Mathematics of the Russian Academy of Sciences in Novosibirsk, Russia. Professor Godunovs most influential work is in the area of applied and it has had a major impact on science and engineering, particularly in the development of methodologies used in Computational Fluid Dynamics and other computational fields. On 1–2 May 1997 a symposium entitled, Godunov-type numerical methods, was held at the University of Michigan to honour Godunov and these methods are widely used to compute continuum processes dominated by wave propagation. On the following day,3 May, Godunov received a degree from the University of Michigan. Godunovs theorem, Linear numerical schemes for solving differential equations, having the property of not generating new extrema. Godunovs scheme is a numerical scheme for solving partial differential equations. 1946-1951 - Department of Mechanics and Mathematics, Moscow State University,1951 - Diploma, Moscow State University. 1954 - Candidate of Physical and Mathematical Sciences,1965 - Doctor of Physical and Mathematical Sciences. 1976 - Corresponding member of the USSR Academy of Sciences,1994 - Member of the Russian Academy of Sciences. 1997 - Honorary professor of the University of Michigan, krylov Prize of the USSR Academy of Sciences. 1993 - M. A. Lavrentiev Prize of the Russian Academy of Sciences, total variation diminishing Upwind scheme Godunov, Sergei K. Ph. D. Dissertation, Difference Methods for Shock Waves, Moscow State University. A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, sbornik,47, 271-306, translated US Joint Publ. Service, JPRS7225 Nov.29,1960, Godunov, Sergei K. and Romenskii, Evgenii I. Elements of Continuum Mechanics and Conservation Laws, Springer, ISBN 0-306-47735-1, Numerical Computation of Internal and External Flows, vol 2, Wiley. Sergei K. Godunov at the Mathematics Genealogy Project Godunovs Personal Web Page Sobolev Institute of Mathematics
Sergei K. Godunov
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Sergei Godunov
10.
John von Neumann
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John von Neumann was a Hungarian-American mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics, physics, economics, computing, and statistics. He published over 150 papers in his life, about 60 in pure mathematics,20 in physics, and 60 in applied mathematics and his last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain. His analysis of the structure of self-replication preceded the discovery of the structure of DNA, also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939, on the ergodic theorem, Princeton, 1931–1932. During World War II he worked on the Manhattan Project, developing the mathematical models behind the lenses used in the implosion-type nuclear weapon. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, along with theoretical physicist Edward Teller, mathematician Stanislaw Ulam, and others, he worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. Von Neumann was born Neumann János Lajos to a wealthy, acculturated, Von Neumanns place of birth was Budapest in the Kingdom of Hungary which was then part of the Austro-Hungarian Empire. He was the eldest of three children and he had two younger brothers, Michael, born in 1907, and Nicholas, who was born in 1911. His father, Neumann Miksa was a banker, who held a doctorate in law and he had moved to Budapest from Pécs at the end of the 1880s. Miksas father and grandfather were both born in Ond, Zemplén County, northern Hungary, johns mother was Kann Margit, her parents were Jakab Kann and Katalin Meisels. Three generations of the Kann family lived in apartments above the Kann-Heller offices in Budapest. In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Franz Joseph, the Neumann family thus acquired the hereditary appellation Margittai, meaning of Marghita. The family had no connection with the town, the appellation was chosen in reference to Margaret, Neumann János became Margittai Neumann János, which he later changed to the German Johann von Neumann. Von Neumann was a child prodigy, as a 6 year old, he could multiply and divide two 8-digit numbers in his head, and could converse in Ancient Greek. When he once caught his mother staring aimlessly, the 6 year old von Neumann asked her, formal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins, Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. A copy was contained in a private library Max purchased, One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor. Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911 and this was one of the best schools in Budapest, part of a brilliant education system designed for the elite
John von Neumann
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Excerpt from the university calendars for 1928 and 1928–1929 of the Friedrich-Wilhelms-Universität Berlin announcing Neumann's lectures on axiomatic set theory and logics, problems in quantum mechanics and special mathematical functions. Notable colleagues were Georg Feigl, Issai Schur, Erhard Schmidt, Leó Szilárd, Heinz Hopf, Adolf Hammerstein and Ludwig Bieberbach.
John von Neumann
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John von Neumann in the 1940s
John von Neumann
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Julian Bigelow, Herman Goldstine, J. Robert Oppenheimer and John von Neumann at the Princeton Institute for Advanced Study.
John von Neumann
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Von Neumann's gravestone
11.
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
Fluid mechanics
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Balance for some integrated fluid quantity in a control volume enclosed by a control surface.
12.
Boundary value problem
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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
Boundary value problem
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Shows a region where a differential equation is valid and the associated boundary values
13.
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
Supercomputer
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IBM 's Blue Gene/P supercomputer at Argonne National Laboratory runs over 250,000 processors using normal data center air conditioning, grouped in 72 racks/cabinets connected by a high-speed optical network
Supercomputer
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A Cray-1 preserved at the Deutsches Museum
Supercomputer
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A Blue Gene /L cabinet showing the stacked blades, each holding many processors
Supercomputer
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An IBM HS20 blade
14.
Wind tunnel
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A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. A wind tunnel consists of a passage with the object under test mounted in the middle. Air is made to move past the object by a fan system or other means. The test object, often called a wind tunnel model, is instrumented with sensors to measure aerodynamic forces, pressure distribution. The earliest wind tunnels were invented towards the end of the 19th century, in the days of aeronautic research. In that way an observer could study the flying object in action. The development of wind tunnels accompanied the development of the airplane, large wind tunnels were built during World War II. Wind tunnel testing was considered of importance during the Cold War development of supersonic aircraft. Determining such forces was required before building codes could specify the required strength of such buildings, in these studies, the interaction between the road and the vehicle plays a significant role, and this interaction must be taken into consideration when interpreting the test results. The advances in fluid dynamics modelling on high speed digital computers has reduced the demand for wind tunnel testing. However, CFD results are not completely reliable and wind tunnels are used to verify CFD predictions. Air velocity and pressures are measured in several ways in wind tunnels, air velocity through the test section is determined by Bernoullis principle. Measurement of the pressure, the static pressure, and the temperature rise in the airflow. The direction of airflow around a model can be determined by tufts of yarn attached to the aerodynamic surfaces, the direction of airflow approaching a surface can be visualized by mounting threads in the airflow ahead of and aft of the test model. Smoke or bubbles of liquid can be introduced into the upstream of the test model. Aerodynamic forces on the test model are usually measured with beam balances, connected to the test model with beams, strings, or cables. Pressure distributions can more conveniently be measured by the use of pressure-sensitive paint, the strip is attached to the aerodynamic surface with tape, and it sends signals depicting the pressure distribution along its surface. The aerodynamic properties of an object can not all remain the same for a scaled model, however, by observing certain similarity rules, a very satisfactory correspondence between the aerodynamic properties of a scaled model and a full-size object can be achieved
Wind tunnel
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NASA wind tunnel with the model of a plane.
Wind tunnel
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A model Cessna with helium-filled bubbles showing pathlines of the wingtip vortices.
Wind tunnel
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Replica of the Wright brothers' wind tunnel.
Wind tunnel
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Eiffel's wind tunnels in the Auteuil laboratory
15.
Space Shuttle
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The Space Shuttle was a partially reusable low Earth orbital spacecraft system operated by the U. S. National Aeronautics and Space Administration, as part of the Space Shuttle program. Its official program name was Space Transportation System, taken from a 1969 plan for a system of reusable spacecraft of which it was the only item funded for development, the first of four orbital test flights occurred in 1981, leading to operational flights beginning in 1982. Five complete Shuttle systems were built and used on a total of 135 missions from 1981 to 2011, the Shuttle fleets total mission time was 1322 days,19 hours,21 minutes and 23 seconds. Shuttle components included the Orbiter Vehicle, a pair of solid rocket boosters. The Shuttle was launched vertically, like a rocket, with the two SRBs operating in parallel with the OVs three main engines, which were fueled from the ET. The SRBs were jettisoned before the vehicle reached orbit, and the ET was jettisoned just before orbit insertion, at the conclusion of the mission, the orbiter fired its OMS to de-orbit and re-enter the atmosphere. The orbiter then glided as a spaceplane to a landing, usually at the Shuttle Landing Facility of KSC or Rogers Dry Lake in Edwards Air Force Base. After landing at Edwards, the orbiter was back to the KSC on the Shuttle Carrier Aircraft. The first orbiter, Enterprise, was built in 1976, used in Approach, four fully operational orbiters were initially built, Columbia, Challenger, Discovery, and Atlantis. Of these, two were lost in accidents, Challenger in 1986 and Columbia in 2003, with a total of fourteen astronauts killed. A fifth operational orbiter, Endeavour, was built in 1991 to replace Challenger, the Space Shuttle was retired from service upon the conclusion of Atlantiss final flight on July 21,2011. Nixons post-Apollo NASA budgeting withdrew support of all components except the Shuttle. The vehicle consisted of a spaceplane for orbit and re-entry, fueled by liquid hydrogen and liquid oxygen tanks. The first of four orbital test flights occurred in 1981, leading to operational flights beginning in 1982, all launched from the Kennedy Space Center, Florida. The system was retired from service in 2011 after 135 missions, the program ended after Atlantis landed at the Kennedy Space Center on July 21,2011. Major missions included launching numerous satellites and interplanetary probes, conducting space science experiments, the first orbiter vehicle, named Enterprise, was built for the initial Approach and Landing Tests phase and lacked engines, heat shielding, and other equipment necessary for orbital flight. A total of five operational orbiters were built, and of these and it was used for orbital space missions by NASA, the US Department of Defense, the European Space Agency, Japan, and Germany. The United States funded Shuttle development and operations except for the Spacelab modules used on D1, sL-J was partially funded by Japan
Space Shuttle
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Discovery lifts off at the start of STS-120.
Space Shuttle
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STS-129 ready for launch
Space Shuttle
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President Nixon (right) with NASA Administrator Fletcher in January 1972, three months before Congress approved funding for the Shuttle program
Space Shuttle
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STS-1 on the launch pad, December 1980
16.
Viscous
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The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the concept of thickness, for example. Viscosity is a property of the fluid which opposes the motion between the two surfaces of the fluid in a fluid that are moving at different velocities. For a given velocity pattern, the stress required is proportional to the fluids viscosity, a fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at low temperatures in superfluids. Otherwise, all fluids have positive viscosity, and are said to be viscous or viscid. A fluid with a high viscosity, such as pitch. The word viscosity is derived from the Latin viscum, meaning mistletoe, the dynamic viscosity of a fluid expresses its resistance to shearing flows, where adjacent layers move parallel to each other with different speeds. It can be defined through the situation known as a Couette flow. This fluid has to be homogeneous in the layer and at different shear stresses, if the speed of the top plate is small enough, the fluid particles will move parallel to it, and their speed will vary linearly from zero at the bottom to u at the top. Each layer of fluid will move faster than the one just below it, in particular, the fluid will apply on the top plate a force in the direction opposite to its motion, and an equal but opposite one to the bottom plate. An external force is required in order to keep the top plate moving at constant speed. The magnitude F of this force is found to be proportional to the u and the area A of each plate. The proportionality factor μ in this formula is the viscosity of the fluid, the ratio u/y is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates. Isaac Newton expressed the forces by the differential equation τ = μ ∂ u ∂ y, where τ = F/A. This formula assumes that the flow is moving along parallel lines and this equation can be used where the velocity does not vary linearly with y, such as in fluid flowing through a pipe. Use of the Greek letter mu for the dynamic viscosity is common among mechanical and chemical engineers. However, the Greek letter eta is used by chemists, physicists
Viscous
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Pitch has a viscosity approximately 230 billion (2.3 × 10 11) times that of water.
Viscous
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A simulation of substances with different viscosities. The substance above has lower viscosity than the substance below
Viscous
–
Example of the viscosity of milk and water. Liquids with higher viscosities make smaller splashes when poured at the same velocity.
Viscous
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Honey being drizzled.
17.
Vorticity
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Conceptually, vorticity could be determined by marking the part of continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity vector would be twice the angular velocity vector of those particles relative to their center of mass. This quantity must not be confused with the velocity of the particles relative to some other point. More precisely, the vorticity is a pseudovector field ω→, defined as the curl of the flow velocity u→ vector, the definition can be expressed by the vector analysis formula, ω → ≡ ∇ × u →, where ∇ is the del operator. The vorticity of a flow is always perpendicular to the plane of the flow. The vorticity is related to the flows circulation along a path by the Stokes theorem. Namely, for any infinitesimal surface element C with normal direction n→ and area dA, many phenomena, such as the blowing out of a candle by a puff of air, are more readily explained in terms of vorticity rather than the basic concepts of pressure and velocity. This applies, in particular, to the formation and motion of vortex rings, in a mass of continuum that is rotating like a rigid body, the vorticity is twice the angular velocity vector of that rotation. This is the case, for example, of water in a tank that has been spinning for a while around its vertical axis, the vorticity may be nonzero even when all particles are flowing along straight and parallel pathlines, if there is shear. The vorticity will be zero on the axis, and maximum near the walls, conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal irrotational vortex, where most particles rotate about some straight axis, another way to visualize vorticity is to imagine that, instantaneously, a tiny part of the continuum becomes solid and the rest of the flow disappears. If that tiny new solid particle is rotating, rather than just moving with the flow, mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by ω→, defined as the curl or rotational of the velocity field v→ describing the continuum motion. In Cartesian coordinates, ω → = ∇ × v → = × = In words, the evolution of the vorticity field in time is described by the vorticity equation, which can be derived from the Navier–Stokes equations. This is clearly true in the case of 2-D potential flow, Vorticity is a useful tool to understand how the ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a diffusion of vorticity away from the vortex cores into the flow field. This flow is accounted for by the term in the vorticity transport equation. Thus, in cases of very viscous flows, the vorticity will be diffused throughout the flow field, a vortex line or vorticity line is a line which is everywhere tangent to the local vorticity vector. Vortex lines are defined by the relation d x ω x = d y ω y = d z ω z, a vortex tube is the surface in the continuum formed by all vortex-lines passing through a given closed curve in the continuum
Vorticity
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Continuum mechanics
Vorticity
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Example flows:
18.
Conformal transformation
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In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
Conformal transformation
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A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
19.
Airfoil
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An airfoil or aerofoil is the shape of a wing, blade, or sail. An airfoil-shaped body moved through a fluid produces an aerodynamic force, the component of this force perpendicular to the direction of motion is called lift. The component parallel to the direction of motion is called drag, subsonic flight airfoils have a characteristic shape with a rounded leading edge, followed by a sharp trailing edge, often with a symmetric curvature of upper and lower surfaces. Foils of similar function designed with water as the fluid are called hydrofoils. The lift on an airfoil is primarily the result of its angle of attack, when oriented at a suitable angle, the airfoil deflects the oncoming air, resulting in a force on the airfoil in the direction opposite to the deflection. This force is known as force and can be resolved into two components, lift and drag. Most foil shapes require an angle of attack to generate lift. This turning of the air in the vicinity of the airfoil creates curved streamlines, resulting in pressure on one side. The lift force can be related directly to the average top/bottom velocity difference without computing the pressure by using the concept of circulation, a fixed-wing aircrafts wings, horizontal, and vertical stabilizers are built with airfoil-shaped cross sections, as are helicopter rotor blades. Airfoils are also found in propellers, fans, compressors and turbines, sails are also airfoils, and the underwater surfaces of sailboats, such as the centerboard and keel, are similar in cross-section and operate on the same principles as airfoils. Swimming and flying creatures and even many plants and sessile organisms employ airfoils/hydrofoils, common examples being bird wings, the bodies of fish, an airfoil-shaped wing can create downforce on an automobile or other motor vehicle, improving traction. Any object with an angle of attack in a fluid, such as a flat plate. Airfoils are more efficient lifting shapes, able to more lift. A lift and drag curve obtained in wind tunnel testing is shown on the right, the curve represents an airfoil with a positive camber so some lift is produced at zero angle of attack. With increased angle of attack, lift increases in a linear relation. At about 18 degrees this airfoil stalls, and lift falls off quickly beyond that, the drop in lift can be explained by the action of the upper-surface boundary layer, which separates and greatly thickens over the upper surface at and past the stall angle. The thicker boundary layer also causes an increase in pressure drag, so that the overall drag increases sharply near. Airfoil design is a facet of aerodynamics
Airfoil
–
Lift and Drag curves for a typical airfoil
Airfoil
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Examples of airfoils in nature and within various vehicles. Though not strictly an airfoil, the dolphin flipper obeys the same principles in a different fluid medium.
Airfoil
–
An airfoil section is displayed at the tip of this Denney Kitfox aircraft, built in 1991.
Airfoil
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Airfoil of Kamov Ka-26 helicopters
20.
Lewis Fry Richardson
–
He is also noted for his pioneering work concerning fractals and a method for solving a system of linear equations known as modified Richardson iteration. Lewis Fry Richardson was the youngest of seven born to Catherine Fry. They were a prosperous Quaker family, David Richardson operating a successful tanning, at age 12 he was sent to a Quaker boarding school, Bootham School in York, where he received an education in science, which stimulated an active interest in natural history. In 1898 he went on to Durham College of Science where he took courses in physics, chemistry, botany. He proceeded in 1900 to King’s College, Cambridge, where he was taught physics in the natural sciences tripos by J. J. Thomson, at age 47 he received a doctorate in mathematical psychology from the University of London. Richardsons working life represented his interests, National Physical Laboratory. Manager of the physical and chemical laboratory, Sunbeam Lamp Company, Meteorological Office – as superintendent of Eskdalemuir Observatory. Head of the Physics Department at Westminster Training College, principal, Paisley Technical College, now part of the University of the West of Scotland. Richardson worked from 1916 to 1919 for the Friends Ambulance Unit attached to the 16th French Infantry Division, after the war, he rejoined the Meteorological Office but was compelled to resign on grounds of conscience when it was amalgamated into the Air Ministry in 1920. He subsequently pursued a career on the fringes of the world before retiring in 1940 to research his own ideas. His pacifism had direct consequences on his research interests and he described his ideas thus, “After so much hard reasoning, may one play with a fantasy. Imagine a large hall like a theatre, except that the circles and galleries go right round through the space occupied by the stage. The walls of this chamber are painted to form a map of the globe, the ceiling represents the north polar regions, England is in the gallery, the tropics in the upper circle, Australia on the dress circle and the Antarctic in the pit. A myriad computers are at work upon the weather of the part of the map where each sits, the work of each region is coordinated by an official of higher rank. Numerous little night signs display the instantaneous values so that neighbouring computers can read them, each number is thus displayed in three adjacent zones so as to maintain communication to the North and South on the map. From the floor of the pit a tall pillar rises to half the height of the hall and it carries a large pulpit on its top. In this sits the man in charge of the theatre, he is surrounded by several assistants. One of his duties is to maintain a speed of progress in all parts of the globe
Lewis Fry Richardson
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Lewis Fry Richardson D.Sc., FRS
21.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
Three-dimensional space
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Three-dimensional Cartesian coordinate system with the x -axis pointing towards the observer. (See diagram description for correction.)
22.
Lockheed Corporation
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The Lockheed Corporation was an American aerospace company. Lockheed was founded in 1912 and later merged with Martin Marietta to form Lockheed Martin in 1995, the Alco Hydro-Aeroplane Company was established in San Francisco in 1912 by the brothers Allan and Malcolm Loughead. Following the Model F-1, the company invested heavily in the design, however, the asking price of $2500 could not compete in a market that was saturated with post World War 1 $350 Curtiss JN-4s and De Haviland trainers. The Loughead Aircraft Manufacturing Company closed its doors in 1921, in 1926, Allan Loughead, Jack Northrop, and Kenneth Jay secured funding to form the Lockheed Aircraft Company in Hollywood. This new company utilized some of the technology originally developed for the Model S-1 to design the Vega Model. In March 1928, the relocated to Burbank, California. From 1926-28 the company produced over 80 aircraft and employed more than 300 workers who by April 1929 were building five aircraft per week, in July 1929, majority shareholder Fred Keeler sold 87% of the Lockheed Aircraft Company to Detroit Aircraft Corporation. In August 1929, Allan Lockheed resigned, the Great Depression ruined the aircraft market, and Detroit Aircraft went bankrupt. A group of headed by brothers Robert and Courtland Gross. The syndicate bought the company for a mere $40,000, ironically, Allan Lockheed himself had planned to bid for his own company, but had raised only $50,000, which he felt was too small a sum for a serious bid. In 1934, Robert E. Gross was named chairman of the new company, the Lockheed Aircraft Corporation and his brother Courtlandt S. Gross was a co-founder and executive, succeeding Robert as Chairman following his death in 1961. The company was named the Lockheed Corporation in 1977, in the 1930s, Lockheed spent $139,400 to develop the Model 10 Electra, a small twin-engined transport. The company sold 40 in the first year of production, amelia Earhart and her navigator, Fred Noonan, flew it in their failed attempt to circumnavigate the world in 1937. Subsequent designs, the Lockheed Model 12 Electra Junior and the Lockheed Model 14 Super Electra expanded their market. The Lockheed Model 14 formed the basis for the Hudson bomber and its primary role was submarine hunting. The Model 14 Super Electra were sold abroad, and more than 100 were license-built in Japan for use by the Imperial Japanese Army, the P-38 was the only American fighter aircraft in production throughout American involvement in the war, from Pearl Harbor to Victory over Japan Day. It filled ground-attack, air-to-air, and even strategic bombing roles in all theaters of the war in which the United States operated, the Lockheed Vega factory was located next to Burbanks Union Airport which it had purchased in 1940. During the war, the area was camouflaged to fool enemy aerial reconnaissance
Lockheed Corporation
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P-38J Lightning Yippee
Lockheed Corporation
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P-38 Lightning assembly line at the Lockheed plant, Burbank, California in World War II. In June 1943, this assembly line was reconfigured into a mechanized line, which more than doubled the rate of production. The transition to the new system was accomplished in only eight days. During this time production never stopped. It was continued outdoors.
Lockheed Corporation
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A Lockheed L-049 Constellation sporting the livery of Trans World Airlines at the Pima Air & Space Museum.
Lockheed Corporation
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The Lockheed U-2, which first flew in 1955, provided intelligence on Soviet bloc countries.
23.
Douglas Aircraft Company
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The Douglas Aircraft Company was an American aerospace manufacturer based in Southern California. It was founded in 1921 by Donald Wills Douglas, Sr. Douglas Aircraft Company largely operated as a division of McDonnell Douglas after the merger. MD later merged with Boeing in 1997, the Douglas Aircraft Company was founded by Donald Wills Douglas, Sr. on July 22,1921 in Santa Monica, California, following dissolution of the Davis-Douglas Company. An early claim to fame was the first circumnavigation of the world by air in Douglas airplanes in 1924. In 1923, the U. S. Army Air Service was interested in carrying out a mission to circumnavigate the Earth for the first time by aircraft, Donald Douglas proposed a modified Douglas DT to meet the Armys needs. The two-place, open cockpit DT biplane torpedo bomber had previously produced for the U. S. Navy. The DTs were taken from the lines at the companys manufacturing plants in Rock Island, Illinois and Dayton. The modified aircraft known as the Douglas World Cruiser, also was the first major project for Jack Northrop who designed the system for the series. After the prototype was delivered in November 1923, upon the completion of tests on 19 November. Due to the expedition ahead, spare parts, including 15 extra Liberty L-12 engines,14 extra sets of pontoons. These were sent to airports along the route, the last of these aircraft was delivered to the U. S. Army on 11 March 1924. After the success of the World Cruiser, the Army Air Service ordered six similar aircraft as observation aircraft. The success of the DWC established the Douglas Aircraft Company among the aircraft companies of the world. Douglas adopted a logo that showed aircraft circling a globe, replacing the original winged heart logo, the logo evolved into an aircraft, a rocket, and a globe. It was later adopted by the McDonnell Douglas Corporation, and then became the basis of the current logo of the Boeing Company after their 1997 merger, many Douglas aircraft had long service lives. Douglas Aircraft designed and built a variety of aircraft for the U. S. military, including the Navy, Army Air Forces, Marine Corps, Air Force. The company initially built torpedo bombers for the U. S. Navy, within five years, the company was building about 100 aircraft annually. Among the early employees at Douglas were Ed Heinemann, Dutch Kindelberger, and Jack Northrop, the company retained its military market and expanded into amphibian airplanes in the late 1920s, also moving its facilities to Clover Field at Santa Monica, California
Douglas Aircraft Company
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Machine tool operator at the Douglas Aircraft plant, Long Beach, California in World War II. After losing thousands of workers to military service, American manufacturers hired women for production positions, to the point where the typical aircraft plant's workforce was 40% female.
Douglas Aircraft Company
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Women at work on bomber, Douglas Aircraft Company, Long Beach, California in October 1942
Douglas Aircraft Company
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An ex-USAF C-47A Skytrain, the military version of the DC-3, on display in England in 2010. This aircraft flew from a base in Devon, England, during the Invasion of Normandy.
Douglas Aircraft Company
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Douglas DC-3
24.
McDonnell Aircraft
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The McDonnell Aircraft Corporation was an American aerospace manufacturer based in St. Louis, Missouri. McDonnell Aircraft later merged with the Douglas Aircraft Company to form McDonnell Douglas in 1967, McDonnell & Associates in Milwaukee, Wisconsin in 1928 to produce a personal aircraft for family use. The economic depression from 1929 ruined his plans and the company collapsed and he went to work for Glenn L. Martin. He left in 1938 to try again with his own firm, McDonnell Aircraft Corporation, based near St. Louis, Missouri, world War II was a major boost to the new company. It grew from 15 employees in 1939 to 5,000 at the end the war and became a significant aircraft parts producer, McDonnell also developed the LBD-1 Gargoyle guided missile. McDonnell Aircraft suffered after the war with an end of government orders and a surplus of aircraft, the advent of the Korean War helped push McDonnell into a major military fighter supply role. In 1943, McDonnell began developing jets when they were invited to bid on a US Navy contest, Dave Lewis joined the company as Chief of Aerodynamics in 1946. He led the development of the legendary F-4 Phantom II in 1954, Lewis became Executive Vice President in 1958, and finally became President and Chief Operating Officer in 1962. Lewis went on to manage Douglas Aircraft Division in 1967 after the McDonnell Douglas merger, in 1969, he returned to St. Louis as President of McDonnell Douglas. The company was now a major employer, but was having problems, with no civilian side of the company, every peacetime downturn in procurement led to lean times at McDonnell. McDonnell Aircraft and Douglas Aircraft began to sound each other out about a merger, inquiries began in 1963, Douglas offered bid invitations from December 1966 and accepted that of McDonnell. The two firms were merged on April 28,1967 as the McDonnell Douglas Corporation. In 1967, with the merger of McDonnell and Douglas Aircraft, Dave Lewis, then president of McDonnell, was named chairman of what was called the Long Beach, Lewis managed the turnaround of the division. McDonnell Douglas would later merge with Boeing in August 1997, boeings defense and space division is based in St. Louis, Missouri, U. S. and is responsible for defense and space products and services. McDonnell Douglass legacy product programs include the F-15 Eagle, AV-8B Harrier II, F/A-18 Hornet, McDonnell, nephew of founder and later President, CEO and Chair of McDonnell Douglas. Francillon, René J. McDonnell Douglas Aircraft since 1920, McDonnell Aircraft history 1939-45 McDonnell Aircraft history 1946-56 McDonnell Aircraft history 1957-67 McDonnell Gemini Space Program 1963-1966 List of all McDonnell model numbers through 1974
McDonnell Aircraft
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An FH-1 Phantom, in 1948.
McDonnell Aircraft
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McDonnell F2H Banshee, F3H Demon, and F4H Phantom II.
25.
Submarine
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A submarine is a watercraft capable of independent operation underwater. It differs from a submersible, which has more limited underwater capability, the term most commonly refers to a large, crewed vessel. It is also used historically or colloquially to refer to remotely operated vehicles and robots, as well as medium-sized or smaller vessels, such as the midget submarine. The noun submarine evolved as a form of submarine boat, by naval tradition, submarines are usually referred to as boats rather than as ships. Although experimental submarines had been built before, submarine design took off during the 19th century, Submarines were first widely used during World War I, and now figure in many navies large and small. Civilian uses for submarines include marine science, salvage, exploration and facility inspection, Submarines can also be modified to perform more specialized functions such as search-and-rescue missions or undersea cable repair. Submarines are also used in tourism, and for undersea archaeology, most large submarines consist of a cylindrical body with hemispherical ends and a vertical structure, usually located amidships, which houses communications and sensing devices as well as periscopes. In modern submarines, this structure is the sail in American usage, a conning tower was a feature of earlier designs, a separate pressure hull above the main body of the boat that allowed the use of shorter periscopes. There is a propeller at the rear, and various hydrodynamic control fins, smaller, deep-diving and specialty submarines may deviate significantly from this traditional layout. Submarines use diving planes and also change the amount of water, Submarines have one of the widest ranges of types and capabilities of any vessel. Submarines can work at greater depths than are survivable or practical for human divers, modern deep-diving submarines derive from the bathyscaphe, which in turn evolved from the diving bell. In 1578, the English mathematician William Bourne recorded in his book Inventions or Devises one of the first plans for an underwater navigation vehicle and its unclear whether he ever carried out his idea. The first submersible of whose construction there exists reliable information was designed and built in 1620 by Cornelis Drebbel and it was propelled by means of oars. By the mid-18th century, over a dozen patents for submarines/submersible boats had been granted in England, in 1747, Nathaniel Symons patented and built the first known working example of the use of a ballast tank for submersion. His design used leather bags that could fill with water to submerge the craft, a mechanism was used to twist the water out of the bags and cause the boat to resurface. In 1749, the Gentlemens Magazine reported that a design had initially been proposed by Giovanni Borelli in 1680. By this point of development, further improvement in design stagnated for over a century, until new industrial technologies for propulsion. The first military submarine was the Turtle, a hand-powered acorn-shaped device designed by the American David Bushnell to accommodate a single person and it was the first verified submarine capable of independent underwater operation and movement, and the first to use screws for propulsion
Submarine
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A Russian Navy Typhoon-class submarine underway. Also known as "Project 941".
Submarine
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Drebbel, the first navigable submarine
Submarine
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The French submarine Plongeur
Submarine
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The Nordenfelt -designed, Ottoman submarine Abdül Hamid
26.
Ship
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Historically, a ship was a sailing vessel with at least three square-rigged masts and a full bowsprit. Ships are generally distinguished from boats, based on size, shape, Ships have been important contributors to human migration and commerce. They have supported the spread of colonization and the trade, but have also served scientific, cultural. After the 16th century, new crops that had come from, Ship transport is responsible for the largest portion of world commerce. As of 2016, there were more than 49,000 merchant ships, of these 28% were oil tankers, 43% were bulk carriers, and 13% were container ships. Military forces operate vessels for naval warfare and to transport and support forces ashore, the top 50 navies had a median fleet of 88 surface vessels each, according to various sources. There is no definition of what distinguishes a ship from a boat. Ships can usually be distinguished from boats based on size and the ability to operate independently for extended periods. A legal definition of ship from Indian case law is a vessel that carries goods by sea, a common notion is that a ship can carry a boat, but not vice versa. American and British 19th Century maritime law distinguished vessels from other craft, ships and boats fall in one legal category, a number of large vessels are usually referred to as boats. Other types of vessel which are traditionally called boats are Great Lakes freighters, riverboats. Though large enough to carry their own boats and heavy cargoes, in most maritime traditions ships have individual names, and modern ships may belong to a ship class often named after its first ship. The first known vessels date back about 10,000 years ago, the first navigators began to use animal skins or woven fabrics as sails. Affixed to the top of a pole set upright in a boat and this allowed men to explore widely, allowing for the settlement of Oceania for example. By around 3000 BC, Ancient Egyptians knew how to assemble wooden planks into a hull and they used woven straps to lash the planks together, and reeds or grass stuffed between the planks helped to seal the seams. Sneferus ancient cedar wood ship Praise of the Two Lands is the first reference recorded to a ship being referred to by name, the ancient Egyptians were perfectly at ease building sailboats. A remarkable example of their skills was the Khufu ship. Aksum was known by the Greeks for having seaports for ships from Greece, a panel found at Mohenjodaro depicted a sailing craft
Ship
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Italian full-rigged ship Amerigo Vespucci in New York Harbor, 1976
Ship
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A raft is among the simplest boat designs.
Ship
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Roman trireme mosaic from Carthage, Bardo Museum, Tunis.
Ship
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A Japanese atakebune from the 16th century
27.
Aircraft
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An aircraft is a machine that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the lift of an airfoil. The human activity that surrounds aircraft is called aviation, crewed aircraft are flown by an onboard pilot, but unmanned aerial vehicles may be remotely controlled or self-controlled by onboard computers. Aircraft may be classified by different criteria, such as type, aircraft propulsion, usage. Each of the two World Wars led to technical advances. Consequently, the history of aircraft can be divided into five eras, Pioneers of flight, first World War,1914 to 1918. Aviation between the World Wars,1918 to 1939, Second World War,1939 to 1945. Postwar era, also called the jet age,1945 to the present day, aerostats use buoyancy to float in the air in much the same way that ships float on the water. They are characterized by one or more large gasbags or canopies, filled with a relatively low-density gas such as helium, hydrogen, or hot air, which is less dense than the surrounding air. When the weight of this is added to the weight of the aircraft structure, a balloon was originally any aerostat, while the term airship was used for large, powered aircraft designs – usually fixed-wing. In 1919 Frederick Handley Page was reported as referring to ships of the air, in the 1930s, large intercontinental flying boats were also sometimes referred to as ships of the air or flying-ships. – though none had yet been built, the advent of powered balloons, called dirigible balloons, and later of rigid hulls allowing a great increase in size, began to change the way these words were used. Huge powered aerostats, characterized by an outer framework and separate aerodynamic skin surrounding the gas bags, were produced. There were still no fixed-wing aircraft or non-rigid balloons large enough to be called airships, then several accidents, such as the Hindenburg disaster in 1937, led to the demise of these airships. Nowadays a balloon is an aerostat and an airship is a powered one. A powered, steerable aerostat is called a dirigible, sometimes this term is applied only to non-rigid balloons, and sometimes dirigible balloon is regarded as the definition of an airship. Non-rigid dirigibles are characterized by a moderately aerodynamic gasbag with stabilizing fins at the back and these soon became known as blimps. During the Second World War, this shape was adopted for tethered balloons, in windy weather
Aircraft
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NASA test aircraft
Aircraft
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The Mil Mi-8 is the most-produced helicopter in history
Aircraft
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"Voodoo" a modified P 51 Mustang is the 2014 Reno Air Race Champion
Aircraft
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A hot air balloon in flight
28.
Yacht
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A yacht /ˈjɒt/ is a recreational boat or ship. In modern use of the term, yachts differ from working ships mainly by their leisure purpose, there are two different classes of yachts, sailing and power boats. With the rise of the steamboat and other types of powerboat, sailing vessels in general came to be perceived as luxury, later the term came to encompass large motor boats for primarily private pleasure purposes as well. Yacht lengths normally range from 10 metres up to dozens of meters, a luxury craft smaller than 12 metres is more commonly called a cabin cruiser or simply a cruiser. A superyacht generally refers to any yacht above 24 m and a megayacht generally refers to any yacht over 50 metres and this size is small in relation to typical cruise liners and oil tankers. A few countries have a special flag worn by recreational boats or ships, although inspired by the national flag, the yacht ensign does not always correspond with the civil or merchant ensign of the state in question. Yacht ensigns differ from merchant ensigns in order to signal that the yacht is not carrying cargo that requires a customs declaration, carrying commercial cargo on a boat with a yacht ensign is deemed to be smuggling in many jurisdictions. Until the 1950s, almost all yachts were made of wood or steel, although wood hulls are still in production, the most common construction material is fibreglass, followed by aluminium, steel, carbon fibre, and ferrocement. The use of wood has changed and is no longer limited to traditional board-based methods, wood is mostly used by hobbyists or wooden boat purists when building an individual boat. Apart from materials like carbon fibre and aramid fibre, spruce veneers laminated with epoxy resins have the best weight-to-strength ratios of all boatbuilding materials. Sailing yachts can range in length from about 6 metres to well over 30 metres. Most privately owned yachts fall in the range of about 7 metres -14 metres, in the United States, sailors tend to refer to smaller yachts as sailboats, while referring to the general sport of sailing as yachting. Within the limited context of racing, a yacht is any sailing vessel taking part in a race. Many modern racing yachts have efficient sail-plans, most notably the Bermuda rig. This capability is the result of a sail-plan and hull design oriented towards this capability, day sailing yachts are usually small, at under 6 metres in length. Sometimes called sailing dinghies, they often have a keel, centreboard. Most day sailing yachts do not have a cabin, as they are designed for hourly or daily use and not for overnight journeys. They may have a cabin, where the front part of the hull has a raised solid roof to provide a place to store equipment or to offer shelter from wind or spray
Yacht
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Sailing Yacht "Zapata II"
Yacht
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The "Lazzara" 80' "Alchemist" runs at full speed up the California Coast
Yacht
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A yacht in Lorient, Brittany, France
Yacht
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Aerial view of a yacht club and marina - Yacht Harbour Residence "Hohe Düne" in Rostock, Germany.
29.
Boundary layer
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In the Earths atmosphere, the atmospheric boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the close to the wing. Laminar boundary layers can be classified according to their structure. When a fluid rotates and viscous forces are balanced by the Coriolis effect, in the theory of heat transfer, a thermal boundary layer occurs. A surface can have multiple types of boundary layer simultaneously, the viscous nature of airflow reduces the local velocities on a surface and is responsible for skin friction. The layer of air over the surface that is slowed down or stopped by viscosity, is the boundary layer. There are two different types of boundary layer flow, laminar and turbulent, laminar Boundary Layer Flow The laminar boundary is a very smooth flow, while the turbulent boundary layer contains swirls or eddies. The laminar flow creates less skin friction drag than the turbulent flow, Boundary layer flow over a wing surface begins as a smooth laminar flow. As the flow continues back from the edge, the laminar boundary layer increases in thickness. Turbulent Boundary Layer Flow At some distance back from the leading edge, the low energy laminar flow, however, tends to break down more suddenly than the turbulent layer. The aerodynamic boundary layer was first defined by Ludwig Prandtl in a paper presented on August 12,1904 at the third International Congress of Mathematicians in Heidelberg and this allows a closed-form solution for the flow in both areas, a significant simplification of the full Navier–Stokes equations. The majority of the transfer to and from a body also takes place within the boundary layer. The pressure distribution throughout the layer in the direction normal to the surface remains constant throughout the boundary layer. The thickness of the velocity boundary layer is defined as the distance from the solid body at which the viscous flow velocity is 99% of the freestream velocity. Displacement Thickness is an alternative definition stating that the boundary layer represents a deficit in mass compared to inviscid flow with slip at the wall. It is the distance by which the wall would have to be displaced in the case to give the same total mass flow as the viscous case. The no-slip condition requires the flow velocity at the surface of an object be zero. The flow velocity will then increase rapidly within the layer, governed by the boundary layer equations
Boundary layer
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Ludwig Prandtl
30.
University of Stuttgart
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The University of Stuttgart is a university located in Stuttgart, Germany. It was founded in 1829 and is organized into 10 faculties and it is one of the top nine leading technical universities in Germany with highly ranked programs in civil, mechanical, industrial and electrical engineering. The academic tradition of the University of Stuttgart goes back to its probably most famous student, Gottlieb Daimler. These four universities, in combination with RWTH Aachen are the top five universities of the aforementioned TU9, from 1770 to 1794, the Karlsschule was the first university in Stuttgart. Located in Stuttgart-Hohenheim, it has since 1818 been the University of Hohenheim and is not related to the University of Stuttgart, what is now the University of Stuttgart was founded in 1829, and celebrated its 175th anniversary in 2004. Because of the importance of the technical sciences and instruction in these fields. In 1900 it was awarded the right to grant doctoral degrees in the technical disciplines, the development of the courses of study at the Technical College of Stuttgart led to its renaming in 1967 to the present-day Universität Stuttgart. With this change of name came along a built-up of new fields, such as history of science and technology and the sciences. Since the end of the 1950s, a part of the university has located in the suburb of Stuttgart-Vaihingen. Most technical subjects are located in Vaihingen, while the humanities, the sciences, architecture. As of 2014, University of Stuttgart is ranked 85th in the world in the field of Engineering & Technology according to QS World University Rankings
University of Stuttgart
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Mensa building at the main campus
University of Stuttgart
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Campus at Vaihingen
University of Stuttgart
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International Centrum at the University of Stuttgart
University of Stuttgart
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Keplerstraße 11 ("K1", right) and 17 ("K2", left) in the city center
31.
MIT
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The Massachusetts Institute of Technology is a private research university in Cambridge, Massachusetts, often cited as one of the worlds most prestigious universities. Researchers worked on computers, radar, and inertial guidance during World War II, post-war defense research contributed to the rapid expansion of the faculty and campus under James Killian. The current 168-acre campus opened in 1916 and extends over 1 mile along the bank of the Charles River basin. The Institute is traditionally known for its research and education in the sciences and engineering, and more recently in biology, economics, linguistics. Air Force and 6 Fields Medalists have been affiliated with MIT, the school has a strong entrepreneurial culture, and the aggregated revenues of companies founded by MIT alumni would rank as the eleventh-largest economy in the world. In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a Conservatory of Art and Science, but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, Rogers, a professor from the University of Virginia, wanted to establish an institution to address rapid scientific and technological advances. The Rogers Plan reflected the German research university model, emphasizing an independent faculty engaged in research, as well as instruction oriented around seminars, two days after the charter was issued, the first battle of the Civil War broke out. After a long delay through the war years, MITs first classes were held in the Mercantile Building in Boston in 1865, in 1863 under the same act, the Commonwealth of Massachusetts founded the Massachusetts Agricultural College, which developed as the University of Massachusetts Amherst. In 1866, the proceeds from sales went toward new buildings in the Back Bay. MIT was informally called Boston Tech, the institute adopted the European polytechnic university model and emphasized laboratory instruction from an early date. Despite chronic financial problems, the institute saw growth in the last two decades of the 19th century under President Francis Amasa Walker. Programs in electrical, chemical, marine, and sanitary engineering were introduced, new buildings were built, the curriculum drifted to a vocational emphasis, with less focus on theoretical science. The fledgling school still suffered from chronic financial shortages which diverted the attention of the MIT leadership, during these Boston Tech years, MIT faculty and alumni rebuffed Harvard University president Charles W. Eliots repeated attempts to merge MIT with Harvard Colleges Lawrence Scientific School. There would be at least six attempts to absorb MIT into Harvard, in its cramped Back Bay location, MIT could not afford to expand its overcrowded facilities, driving a desperate search for a new campus and funding. Eventually the MIT Corporation approved an agreement to merge with Harvard, over the vehement objections of MIT faculty, students. However, a 1917 decision by the Massachusetts Supreme Judicial Court effectively put an end to the merger scheme, the neoclassical New Technology campus was designed by William W. Bosworth and had been funded largely by anonymous donations from a mysterious Mr. Smith, starting in 1912. In January 1920, the donor was revealed to be the industrialist George Eastman of Rochester, New York, who had invented methods of production and processing
MIT
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Stereographic card showing an MIT mechanical drafting studio, 19th century (photo by E.L. Allen, left/right inverted)
MIT
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Massachusetts Institute of Technology
MIT
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A 1905 map of MIT's Boston campus
MIT
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Plaque in Building 6 honoring George Eastman, founder of Eastman Kodak, who was revealed as the anonymous "Mr. Smith" who helped maintain MIT's independence
32.
Grumman Aircraft
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The Grumman Aircraft Engineering Corporation, later Grumman Aerospace Corporation, was a leading 20th century U. S. producer of military and civilian aircraft. Founded on December 6,1929, by Leroy Grumman and partners, factory in Baldwin on Long Island, New York. All of the early Grumman employees were former Loening employees, the company was named for Grumman because he was its largest investor. The company filed as a business on December 5,1929, keeping busy by welding aluminum tubing for truck frames, the company eagerly pursued contracts with the US Navy. Grumman designed the first practical floats with a landing gear for the Navy. The first Grumman aircraft was also for the Navy, the Grumman FF-1 and this was followed by a number of other successful designs. Grumman ranked 22nd among United States corporations in the value of wartime production contracts, Grummans first jet aircraft was the F9F Panther, it was followed by the upgraded F9F/F-9 Cougar, and the less well known F-11 Tiger in the 1950s. The companys big postwar successes came in the 1960s with the A-6 Intruder and E-2 Hawkeye and in the 1970s with the Grumman EA-6B Prowler, Grumman products were prominent in the film Top Gun and numerous World War II naval and Marine Corps aviation films. The U. S. Navy still employs the Hawkeye as part of Carrier Air Wings on board aircraft carriers, Grumman was the chief contractor on the Apollo Lunar Module that landed men on the moon. The firm received the contract on November 7,1962, as the Apollo program neared its end, Grumman was one of the main competitors for the contract to design and build the Space Shuttle, but lost to Rockwell International. The company ended up involved in the program nonetheless, as a subcontractor to Rockwell, providing the wings. In 1969 the company changed its name to Grumman Aerospace Corporation, the company built the Grumman Long Life Vehicle, a light transport mail truck designed for and used by the United States Postal Service. The LLV entered service in 1986, Gulfstream business jets continue to be currently manufactured by Gulfstream Aerospace which is a wholly owned subsidiary of General Dynamics. For much of the Cold War period Grumman was the largest corporate employer on Long Island, Grummans products were considered so reliable and ruggedly built that the company was often referred to as the Grumman Iron Works. At its peak in 1986 it employed 23,000 people on Long Island, a portion of the airport property has been used for the Grumman Memorial Park. Northrop Grummans remaining business at the Bethpage campus is the Battle Management and Engagement Systems Division, under the Grumman Olson brand it made the P-600 and P-6800 step vans for UPS. Grumman manufactured fire engines under the name Firecat and aerial tower trucks under the Aerialcat name, the company entered the fire apparatus business in 1976 with its purchase of Howe Fire Apparatus and ended operations in 1992. Grumman canoes were developed in 1944 as World War II was winding down, Company executive William Hoffman used the companys aircraft aluminum to replace the traditional wood design
Grumman Aircraft
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Grumman Historical Marker
Grumman Aircraft
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Grumman Corporation
Grumman Aircraft
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Apollo Spacecraft: Apollo Lunar Module Diagram
Grumman Aircraft
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F-14 Tomcat at Grumman Memorial Park, Calverton, New York
33.
New York University
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New York University is a private nonprofit research university based in New York City. Founded in 1831, NYU is considered one of the worlds most influential research universities, University rankings compiled by Times Higher Education, U. S. News & World Report, and the Academic Ranking of World Universities all rank NYU amongst the top 32 universities in the world. NYU is a part of the creativity, energy and vibrancy that is Manhattan, located with its core in Greenwich Village. Among its faculty and alumni are 37 Nobel Laureates, over 30 Pulitzer Prize winners, over 30 Academy Award winners, alumni include heads of state, royalty, eminent mathematicians, inventors, media figures, Olympic medalists, CEOs of Fortune 500 companies, and astronauts. NYU alumni are among the wealthiest in the world, according to The Princeton Review, NYU is consistently considered by students and parents as a Top Dream College. Albert Gallatin, Secretary of Treasury under Thomas Jefferson and James Madison, declared his intention to establish in this immense, a system of rational and practical education fitting and graciously opened to all. A three-day-long literary and scientific convention held in City Hall in 1830 and these New Yorkers believed the city needed a university designed for young men who would be admitted based upon merit rather than birthright or social class. On April 18,1831, an institution was established, with the support of a group of prominent New York City residents from the merchants, bankers. Albert Gallatin was elected as the institutions first president, the university has been popularly known as New York University since its inception and was officially renamed New York University in 1896. In 1832, NYU held its first classes in rented rooms of four-story Clinton Hall, in 1835, the School of Law, NYUs first professional school, was established. American Chemical Society was founded in 1876 at NYU and it became one of the nations largest universities, with an enrollment of 9,300 in 1917. NYU had its Washington Square campus since its founding, the university purchased a campus at University Heights in the Bronx because of overcrowding on the old campus. NYU also had a desire to follow New York Citys development further uptown, NYUs move to the Bronx occurred in 1894, spearheaded by the efforts of Chancellor Henry Mitchell MacCracken. The University Heights campus was far more spacious than its predecessor was, as a result, most of the universitys operations along with the undergraduate College of Arts and Science and School of Engineering were housed there. NYUs administrative operations were moved to the new campus, but the schools of the university remained at Washington Square. In 1914, Washington Square College was founded as the undergraduate college of NYU. In 1935, NYU opened the Nassau College-Hofstra Memorial of New York University at Hempstead and this extension would later become a fully independent Hofstra University. In 1950, NYU was elected to the Association of American Universities, in the late 1960s and early 1970s, financial crisis gripped the New York City government and the troubles spread to the citys institutions, including NYU
New York University
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Albert Gallatin
New York University
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New York University
New York University
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The University Heights campus, now home to Bronx Community College
New York University
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The Silver Center c. 1900
34.
Cartesian coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
Cartesian coordinate system
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The right hand rule.
Cartesian coordinate system
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Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinate system
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3D Cartesian Coordinate Handedness
35.
Overflow (software)
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OVERFLOW - the OVERset grid FLOW solver - is a software package for simulating fluid flow around solid bodies using computational fluid dynamics. It is a compressible 3-D flow solver that solves the time-dependent, Reynolds-averaged, OVERFLOW was developed as part of a collaborative effort between NASAs Johnson Space Center in Houston, Texas and NASA Ames Research Center in Moffett Field, California. The driving force behind this work was the need for evaluating the flow about the Space Shuttle launch vehicle, scientists use OVERFLOW to better understand the aerodynamic forces on a vehicle by evaluating the flowfield surrounding the vehicle. OVERFLOW has also used to simulate the effect of debris on the space shuttle launch vehicle. Computational fluid dynamics Official NASA OVERFLOW CFD Code web site Article on OVERFLOW from NASA Insights
Overflow (software)
–
This image depicts the flowfield around the Space Shuttle Launch Vehicle traveling at Mach 2.46 and at an altitude of 66,000 feet (20,000 m). The surface of the vehicle is colored by the pressure coefficient, and the gray contours represent the density of the surrounding air, as calculated using the OVERFLOW codes.
36.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.
37.
Enthalpy
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Enthalpy /ˈɛnθəlpi/ is a measurement of energy in a thermodynamic system. It is the thermodynamic quantity equivalent to the heat content of a system. It is equal to the energy of the system plus the product of pressure. Enthalpy is defined as a function that depends only on the prevailing equilibrium state identified by the systems internal energy, pressure. The unit of measurement for enthalpy in the International System of Units is the joule, but other historical, conventional units are still in use, such as the British thermal unit and the calorie. At constant pressure, the enthalpy change equals the energy transferred from the environment through heating or work other than expansion work, the total enthalpy, H, of a system cannot be measured directly. The same situation exists in classical mechanics, only a change or difference in energy carries physical meaning. Enthalpy itself is a potential, so in order to measure the enthalpy of a system, we must refer to a defined reference point, therefore what we measure is the change in enthalpy. The ΔH is a change in endothermic reactions, and negative in heat-releasing exothermic processes. For processes under constant pressure, ΔH is equal to the change in the energy of the system. This means that the change in enthalpy under such conditions is the heat absorbed by the material through a reaction or by external heat transfer. Enthalpies for chemical substances at constant pressure assume standard state, most commonly 1 bar pressure, standard state does not, strictly speaking, specify a temperature, but expressions for enthalpy generally reference the standard heat of formation at 25 °C. Enthalpy of ideal gases and incompressible solids and liquids does not depend on pressure, unlike entropy, real materials at common temperatures and pressures usually closely approximate this behavior, which greatly simplifies enthalpy calculation and use in practical designs and analyses. The word enthalpy stems from the Ancient Greek verb enthalpein, which means to warm in and it combines the Classical Greek prefix ἐν- en-, meaning to put into, and the verb θάλπειν thalpein, meaning to heat. The word enthalpy is often attributed to Benoît Paul Émile Clapeyron. This misconception was popularized by the 1927 publication of The Mollier Steam Tables, however, neither the concept, the word, nor the symbol for enthalpy existed until well after Clapeyrons death. The earliest writings to contain the concept of enthalpy did not appear until 1875, however, Gibbs did not use the word enthalpy in his writings. The actual word first appears in the literature in a 1909 publication by J. P. Dalton
Enthalpy
Enthalpy
–
Fig.1 During steady, continuous operation, an energy balance applied to an open system equates shaft work performed by the system to heat added plus net enthalpy added
Enthalpy
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Fig.3 Two open systems in the steady state. Fluid enters the system (dotted rectangle) at point 1 and leaves it at point 2. The mass flow is. a: schematic diagram of the throttling process. b: schematic diagram of a compressor. A power P is applied and a heat flow is released to the surroundings at ambient temperature T a.
38.
Random-access memory
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Random-access memory is a form of computer data storage which stores frequently used program instructions to increase the general speed of a system. A random-access memory device allows data items to be read or written in almost the same amount of time irrespective of the location of data inside the memory. RAM contains multiplexing and demultiplexing circuitry, to connect the lines to the addressed storage for reading or writing the entry. Usually more than one bit of storage is accessed by the same address, in todays technology, random-access memory takes the form of integrated circuits. RAM is normally associated with types of memory, where stored information is lost if power is removed. Other types of non-volatile memories exist that allow access for read operations. These include most types of ROM and a type of memory called NOR-Flash. Integrated-circuit RAM chips came into the market in the early 1970s, with the first commercially available DRAM chip, early computers used relays, mechanical counters or delay lines for main memory functions. Ultrasonic delay lines could only reproduce data in the order it was written, drum memory could be expanded at relatively low cost but efficient retrieval of memory items required knowledge of the physical layout of the drum to optimize speed. Latches built out of vacuum tube triodes, and later, out of transistors, were used for smaller and faster memories such as registers. Such registers were relatively large and too costly to use for large amounts of data, the first practical form of random-access memory was the Williams tube starting in 1947. It stored data as electrically charged spots on the face of a cathode ray tube, since the electron beam of the CRT could read and write the spots on the tube in any order, memory was random access. The capacity of the Williams tube was a few hundred to around a thousand bits, but it was smaller, faster. In fact, rather than the Williams tube memory being designed for the SSEM, magnetic-core memory was invented in 1947 and developed up until the mid-1970s. It became a form of random-access memory, relying on an array of magnetized rings. By changing the sense of each rings magnetization, data could be stored with one bit stored per ring, since every ring had a combination of address wires to select and read or write it, access to any memory location in any sequence was possible. Magnetic core memory was the form of memory system until displaced by solid-state memory in integrated circuits. Data was stored in the capacitance of each transistor, and had to be periodically refreshed every few milliseconds before the charge could leak away
Random-access memory
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Example of writable volatile random-access memory: Synchronous Dynamic RAM modules, primarily used as main memory in personal computers, workstations, and servers.
Random-access memory
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These IBM tabulating machines from the 1930s used mechanical counters to store information
Random-access memory
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A portion of a core memory with a modern flash RAM SD card on top
Random-access memory
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1 Megabit chip – one of the last models developed by VEB Carl Zeiss Jena in 1989
39.
Reynolds number
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The Reynolds number is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. It has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds, who popularized its use in 1883. A similar effect is created by the introduction of a stream of higher velocity fluid and this relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, the Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation. Such scaling is not linear and the application of Reynolds numbers to both situations allows scaling factors to be developed, the Reynolds number can be defined for several different situations where a fluid is in relative motion to a surface. These definitions generally include the properties of density and viscosity, plus a velocity. This dimension is a matter of convention – for example radius and diameter are equally valid to describe spheres or circles, for aircraft or ships, the length or width can be used. For flow in a pipe or a sphere moving in a fluid the internal diameter is used today. Other shapes such as pipes or non-spherical objects have an equivalent diameter defined. For fluids of variable density such as gases or fluids of variable viscosity such as non-Newtonian fluids. The velocity may also be a matter of convention in some circumstances, in practice, matching the Reynolds number is not on its own sufficient to guarantee similitude. Fluid flow is chaotic, and very small changes to shape. Nevertheless, Reynolds numbers are an important guide and are widely used. Osborne Reynolds famously studied the conditions in which the flow of fluid in pipes transitioned from laminar flow to turbulent flow, when the velocity was low, the dyed layer remained distinct through the entire length of the large tube. When the velocity was increased, the broke up at a given point. The point at which this happened was the point from laminar to turbulent flow. From these experiments came the dimensionless Reynolds number for dynamic similarity—the ratio of forces to viscous forces
Reynolds number
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Sir George Stokes, introduced Reynolds numbers
Reynolds number
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Osborne Reynolds popularised the concept
Reynolds number
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The Moody diagram, which describes the Darcy–Weisbach friction factor f as a function of the Reynolds number and relative pipe roughness.
40.
Discretization
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In mathematics, discretization concerns the process of transferring continuous functions, models, and equations into discrete counterparts. This process is carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the case of discretization in which the number of discrete classes is 2. Discretization is also related to mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand, discretization is not always distinguished from quantization in any clearly defined way. The two terms share a semantic field, the same is true of discretization error and quantization error. Mathematical methods relating to discretization include the Euler–Maruyama method and the zero-order hold, discretization is also concerned with the transformation of continuous differential equations into discrete difference equations, suitable for numerical computing. It can, however, be computed by first constructing a matrix, the discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of G with the upper-right partition of G, Q d = T. Now we want to discretise the above expression and we assume that u is constant during each timestep. Exact discretization may sometimes be due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate model, based on that for small timesteps e A T ≈ I + A T. The approximate solution then becomes, x ≈ x + T B u Other possible approximations are e A T ≈ −1 and e A T ≈ −1, each of them have different stability properties. The last one is known as the transform, or Tustin transform. In statistics and machine learning, discretization refers to the process of converting continuous features or variables to discretized or nominal features and this can be useful when creating probability mass functions. Discrete space Time-scale calculus Discrete event simulation Stochastic simulation Finite volume method for unsteady flow Discrete time, introduction to random signals and applied Kalman filtering. Philadelphia, PA, USA, Saunders College Publishing, computing integrals involving the matrix exponential. Digital control and estimation, a unified approach
Discretization
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A solution to a discretized partial differential equation, obtained with the finite element method.
41.
Flux limiters
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Flux limiters are used in high resolution schemes – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations. Use of flux limiters, together with a high resolution scheme. In general, the term flux limiter is used when the acts on system fluxes. They are used in high resolution schemes for solving problems described by PDEs, for smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher order approximations without introducing spurious oscillations. Consider the 1D semi-discrete scheme below, d u i d t +1 Δ x i =0, R i = u i − u i −1 u i +1 − u i. The limiter function is constrained to be greater than or equal to zero, therefore, when the limiter is equal to zero, the flux is represented by a low resolution scheme. Similarly, when the limiter is equal to 1, it is represented by a high resolution scheme, the various limiters have differing switching characteristics and are selected according to the particular problem and solution scheme. No particular limiter has been found to work well for all problems, HQUICK ϕ h q =2, lim r → ∞ ϕ h q =4. Koren – third-order accurate for sufficiently smooth data ϕ k n = max, minmod – symmetric ϕ m m = max, lim r → ∞ ϕ m m =1. Monotonized central – symmetric ϕ m c = max, lim r → ∞ ϕ m c =2. Osher ϕ o s = max, lim r → ∞ ϕ o s = β. ospre – symmetric ϕ o p =1.5, smart ϕ s m = max, lim r → ∞ ϕ s m =4. Superbee – symmetric ϕ s b = max, lim r → ∞ ϕ s b =2, Sweby – symmetric ϕ s w = max, lim r → ∞ ϕ s w = β. UMIST ϕ u m = max, lim r → ∞ ϕ u m =2, Van Albada 1 – symmetric ϕ v a 1 = r 2 + r r 2 +1, lim r → ∞ ϕ v a 1 =1. Van Albada 2 – alternative form used on high spatial order schemes ϕ v a 2 =2 r r 2 +1, lim r → ∞ ϕ v a 2 =0. Van Leer – symmetric ϕ v l = r + | r |1 + | r |, all the above limiters indicated as being symmetric, exhibit the following symmetry property, ϕ r = ϕ. This is a property as it ensures that the limiting actions for forward and backward gradients operate in the same way. Unless indicated to the contrary, the above functions are second order TVD. This means that they are designed such that they pass through a region of the solution, known as the TVD region
Flux limiters
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Admissible limiter region for second-order TVD schemes.
42.
Total variation diminishing
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In numerical methods, total variation diminishing is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten, a numerical method is said to be total variation diminishing if, T V ≤ T V. A numerical scheme is said to be monotonicity preserving if the properties are maintained, If u n is monotonically increasing in space. Harten 1983 proved the following properties for a scheme, A monotone scheme is TVD. In Computational Fluid Dynamics, TVD scheme is employed to capture sharper shock predictions without any misleading oscillations when variation of field variable “Ø” is discontinuous, to capture the variation fine grids are needed and the computation becomes heavy and therefore uneconomic. The use of coarse grids with central difference scheme, upwind scheme, hybrid difference scheme, TVD scheme enables sharper shock predictions on coarse grids saving computation time and as the scheme preserves monotonicity there are no spurious oscillations in the solution. Note that f + is the function when the flow is in positive direction i. e. from left to right. So, f r + is a function of ϕ P − ϕ L ϕ R − ϕ L. Likewise when the flow is in direction, P is negative. Monotone schemes are attractive for solving engineering and scientific problems because they do not produce non-physical solutions, Godunovs theorem proves that linear schemes which preserve monotonicity are, at most, only first order accurate. Higher order linear schemes, although more accurate for smooth solutions, are not TVD, to overcome these drawbacks, various high-resolution, non-linear techniques have been developed, often using flux/slope limiters. Flux limiters Godunovs theorem High-resolution scheme MUSCL scheme Sergei K. Godunov Total variation Hirsch, Numerical Computation of Internal and External Flows, Vol 2, Wiley. Computational Gas Dynamics, Cambridge University Press, toro, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag. Anderson, D. A. and Pletcher, R. H. Computational Fluid Mechanics and Heat Transfer, principles of Computational Fluid Dynamics, Springer-Verlag. Anil W. Date Introduction to Computational Fluid Dynamics, Cambridge University Press
Total variation diminishing
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A picture showing the control volume with velocities at the faces,nodes and the distance between them, where 'P' is the node at the center.
43.
Wavelet
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A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a brief oscillation like one recorded by a seismograph or heart monitor, generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing. Wavelets can be combined, using a reverse, shift, multiply and integrate technique called convolution, for example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly a 32nd note. If this wavelet were to be convolved with a created from the recording of a song. Mathematically, the wavelet will correlate with the if the unknown signal contains information of similar frequency. This concept of correlation is at the core of many applications of wavelet theory. As a mathematical tool, wavelets can be used to extract information from different kinds of data, including – but certainly not limited to – audio signals. Sets of wavelets are generally needed to analyze data fully, a set of complementary wavelets will decompose data without gaps or overlap so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful in wavelet based compression/decompression algorithms where it is desirable to recover the information with minimal loss. This is accomplished through coherent states, the word wavelet has been used for decades in digital signal processing and exploration geophysics. The equivalent French word ondelette meaning small wave was used by Morlet, Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of representation for continuous-time signals. Almost all practically useful discrete wavelet transforms use discrete-time filterbanks and these filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response or infinite impulse response filters, the product of the uncertainties of time and frequency response scale has a lower bound. Thus, in the scaleogram of a wavelet transform of this signal, such an event marks an entire region in the time-scale plane. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle, Wavelet transforms are broadly divided into three classes, continuous, discrete and multiresolution-based. In continuous wavelet transforms, a signal of finite energy is projected on a continuous family of frequency bands. For instance the signal may be represented on every frequency band of the form for all frequencies f >0
Wavelet
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Seismic wavelet
Wavelet
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Meyer
44.
Probability density function
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In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. The probability density function is everywhere, and its integral over the entire space is equal to one. The terms probability distribution function and probability function have also sometimes used to denote the probability density function. However, this use is not standard among probabilists and statisticians, further confusion of terminology exists because density function has also been used for what is here called the probability mass function. In general though, the PMF is used in the context of random variables. Suppose a species of bacteria typically lives 4 to 6 hours, what is the probability that a bacterium lives exactly 5 hours. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000, instead we might ask, What is the probability that the bacterium dies between 5 hours and 5.01 hours. Lets say the answer is 0.02, next, What is the probability that the bacterium dies between 5 hours and 5.001 hours. The answer is probably around 0.002, since this is 1/10th of the previous interval, the probability that the bacterium dies between 5 hours and 5.0001 hours is probably about 0.0002, and so on. In these three examples, the ratio / is approximately constant, and equal to 2 per hour, for example, there is 0.02 probability of dying in the 0. 01-hour interval between 5 and 5.01 hours, and =2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours, therefore, in response to the question What is the probability that the bacterium dies at 5 hours. A literally correct but unhelpful answer is 0, but an answer can be written as dt. This is the probability that the bacterium dies within a window of time around 5 hours. For example, the probability that it lives longer than 5 hours, there is a probability density function f with f =2 hour−1. The integral of f over any window of time is the probability that the dies in that window. A probability density function is most commonly associated with absolutely continuous univariate distributions, a random variable X has density fX, where fX is a non-negative Lebesgue-integrable function, if, Pr = ∫ a b f X d x. That is, f is any function with the property that. In the continuous univariate case above, the measure is the Lebesgue measure
Probability density function
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Boxplot and probability density function of a normal distribution N (0, σ 2).
45.
N-body problem
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In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, in the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is more difficult to solve. Having done so, he and others soon discovered over the course of a few years, Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits. Thus came the awareness and rise of the problem in the early 17th century. Ironically, this conformity led to the wrong approach, after Newtons time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of gravitational interactive forces. Newton said in his Principia, paragraph 21, And hence it is that the force is found in both bodies. The Sun attracts Jupiter and the planets, Jupiter attracts its satellites. Two bodies can be drawn to other by the contraction of rope between them. Newton concluded via his third law of motion according to this Law all bodies must attract each other. This last statement, which implies the existence of gravitational forces, is key. The problem of finding the solution of the n-body problem was considered very important. Indeed, in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, in case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to Poincaré, even though he did not solve the original problem, the version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for n =3. The n-body problem considers n point masses mi, i =1,2, …, n in a reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction. Each mass mi has a position vector qi, Newtons second law says that mass times acceleration mi d2qi/dt2 is equal to the sum of the forces on the mass
N-body problem
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The Real Motion v.s. Kepler's Apparent Motion
N-body problem
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Restricted 3-Body Problem
46.
Two-phase flow
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In fluid mechanics, two-phase flow is a flow of gas and liquid usually in a pipe. Two-phase flow is an example of multiphase flow. Two-phase flow can occur in various forms, the widely-accepted method to categorize two-phase flows is to consider the velocity of each phase as if there is not other phases available. The parameter is a concept called Superficial velocity. Historically, probably the most commonly studied cases of two-phase flow are in power systems. Coal and gas-fired power stations used very large boilers to produce steam for use in turbines, in such cases, pressurised water is passed through heated pipes and it changes to steam as it moves through the pipe. The design of boilers requires an understanding of two-phase flow heat-transfer and pressure drop behaviour. Even more critically, nuclear reactors use water to heat from the reactor core using two-phase flow. A great deal of study has been performed on the nature of flow in such cases, so that engineers can design against possible failures in pipework, loss of pressure. Another case where two-phase flow can occur is in pump cavitation, here a pump is operating close to the vapor pressure of the fluid being pumped. If pressure drops further, which can happen locally near the vanes for the pump, for example, then a change can occur. Similar effects can occur on marine propellors, wherever it occurs. When the vapor bubble collapses, it can produce very large pressure spikes, the above two-phase flow cases are for a single fluid occurring by itself as two different phases, such as steam and water. The term two-phase flow is applied to mixtures of different fluids having different phases, such as air and water, or oil. Sometimes even three-phase flow is considered, such as in oil, other interesting areas where two-phase flow is studied includes in climate systems such as clouds, and in groundwater flow, in which the movement of water and air through the soil is studied. Other examples of two-phase flow include bubbles, rain, waves on the sea, foam, fountains, mousse, cryogenics, several features make two-phase flow an interesting and challenging branch of fluid mechanics, Surface tension makes all dynamical problems nonlinear. In the case of air and water at temperature and pressure. Similar differences are typical of water liquid/water vapor densities, the sound speed changes dramatically for materials undergoing phase change, and can be orders of magnitude different
Two-phase flow
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Different modes of two-phase flows.
47.
Ordinary differential equations
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In mathematics, an ordinary differential equation is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the partial differential equation which may be with respect to more than one independent variable. ODEs that are linear equations have exact closed-form solutions that can be added and multiplied by coefficients. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, Ordinary differential equations arise in many contexts of mathematics and science. Mathematical descriptions of change use differentials and derivatives, often, quantities are defined as the rate of change of other quantities, or gradients of quantities, which is how they enter differential equations. Specific mathematical fields include geometry and analytical mechanics, scientific fields include much of physics and astronomy, meteorology, chemistry, biology, ecology and population modelling, economics. Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, dAlembert, in general, F is a function of the position x of the particle at time t. The unknown function x appears on both sides of the equation, and is indicated in the notation F. In what follows, let y be a dependent variable and x an independent variable, the notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. Given F, a function of x, y, and derivatives of y, then an equation of the form F = y is called an explicit ordinary differential equation of order n. The function r is called the term, leading to two further important classifications, Homogeneous If r =0, and consequently one automatic solution is the trivial solution. The solution of a homogeneous equation is a complementary function. The additional solution to the function is the particular integral. The general solution to an equation can be written as y = yc + yp. Non-linear A differential equation that cannot be written in the form of a linear combination, a number of coupled differential equations form a system of equations. In column vector form, = These are not necessarily linear, the implicit analogue is, F =0 where 0 = is the zero vector. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations and this distinction is not merely one of terminology, DAEs have fundamentally different characteristics and are generally more involved to solve than ODE systems. Given a differential equation F =0 a function u, I ⊂ R → R is called the solution or integral curve for F, if u is n-times differentiable on I, and F =0 x ∈ I
Ordinary differential equations
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
48.
Newton's method
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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
Newton's method
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The tangent lines of x 3 - 2 x + 2 at 0 and 1 intersect the x -axis at 1 and 0 respectively, illustrating why Newton's method oscillates between these values for some starting points.
Newton's method
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The function ƒ is shown in blue and the tangent line is in red. We see that x n +1 is a better approximation than x n for the root x of the function f.
49.
Aorta
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The aorta is the main artery in the human body, originating from the left ventricle of the heart and extending down to the abdomen, where it splits into two smaller arteries. The aorta distributes oxygenated blood to all parts of the body through the systemic circulation, in anatomical sources, the aorta is usually divided into sections. One way of classifying a part of the aorta is by anatomical compartment, the aorta then continues downward as the abdominal aorta diaphragm to the aortic bifurcation. Another system divides the aorta with respect to its course and the direction of blood flow, in this system, the aorta starts as the ascending aorta then travels superiorly from the heart and then makes a hairpin turn known as the aortic arch. Following the aortic arch, the aorta then travels inferiorly as the descending aorta, the descending aorta has two parts. The aorta begins to descend in the cavity, and consequently is known as the thoracic aorta. After the aorta passes through the diaphragm, it is known as the abdominal aorta, the aorta ends by dividing into two major blood vessels, the common iliac arteries and a smaller midline vessel, the median sacral artery. The ascending aorta begins at the opening of the valve in the left ventricle of the heart. It runs through a common pericardial sheath with the pulmonary trunk and these two blood vessels twist around each other, causing the aorta to start out posterior to the pulmonary trunk, but end by twisting to its right and anterior side. The transition from ascending aorta to aortic arch is at the reflection on the aorta. At the root of the aorta, the lumen has three small pockets between the cusps of the aortic valve and the wall of the aorta, which are called the aortic sinuses or the sinuses of Valsalva. The left aortic sinus contains the origin of the coronary artery. Together, these two arteries supply the heart, the posterior aortic sinus does not give rise to a coronary artery. For this reason the left, right and posterior aortic sinuses are also called left-coronary, right-coronary and non-coronary sinuses, in addition to these blood vessels, the aortic arch crosses the left main bronchus. Between the aortic arch and the trunk is a network of autonomic nerve fibers. The left vagus nerve, which passes anterior to the arch, gives off a major branch, the recurrent laryngeal nerve. It then runs back to the neck, the aortic arch has three major branches, from proximal to distal, they are the brachiocephalic trunk, the left common carotid artery, and the left subclavian artery. The brachiocephalic trunk supplies the right side of the head and neck as well as the arm and chest wall
Aorta
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A pig's aorta cut open showing also some leaving arteries.
Aorta
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Course of the aorta in the thorax (anterior view), starting posterior to the main pulmonary artery, then anterior to the right pulmonary arteries, the trachea and the esophagus, then turning posteriorly to course dorsally to these structures.
Aorta
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Major Aorta anatomy displaying Ascending Aorta, Brachiocephalic trunk, Left Common Carotid Artery, Left Subclavian Artery, Aortic Isthmus, Aortic Arch and Descending Thoracic Aorta
50.
Shape optimization
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Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints, in many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain. Topology optimization is, in addition, concerned with the number of connected components/boundaries belonging to the domain, such methods are needed since typically shape optimization methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them. Topological optimization techniques can then help work around the limitations of pure shape optimization, mathematically, shape optimization can be posed as the problem of finding a bounded set Ω, minimizing a functional F, possibly subject to a constraint of the form G =0. Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem, Shape optimization is an infinite-dimensional optimization problem. Furthermore, the space of allowable shapes over which the optimization is performed does not admit a vector space structure, Shape optimization problems are usually solved numerically, by using iterative methods. That is, one starts with a guess for a shape. To solve a shape optimization problem, one needs to find a way to represent a shape in the computer memory, one approach is to follow the boundary of the shape. For that, one can sample the shape boundary in a dense and uniform manner. Then, one can evolve the shape by gradually moving the boundary points and this is called the Lagrangian approach. Another approach is to consider a function defined on a box around the shape, which is positive inside of the shape, zero on the boundary of the shape. One can then evolve this function instead of the shape itself, one can consider a rectangular grid on the box and sample the function at the grid points. As the shape evolves, the points do not change. This approach, of using a grid, is called the Eulerian approach. The idea of using a function to represent the shape is at the basis of the level set method, a third approach is to think of the shape evolution as of a flow problem. Mathematically, if Ω0 is the shape, and Ω t is the shape at time t, one considers the diffeomorphisms f t, Ω0 → Ω t. The idea is again that shapes are difficult entities to be dealt with directly, then the Gâteaux or shape derivative of F at Ω0 with respect to the shape is the limit of d F = lim s →0 F − F s if this limit exists. This gives an idea of gradient descent, where the boundary ∂ Ω is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional
Shape optimization
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Example: Shape optimization as applied to building geometry. Example provided courtesy of Formsolver.com
Shape optimization
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Example: Optimization shape families resulting from differing goal parameters. Example provided courtesy of Formsolver.com
51.
Dortmund University of Technology
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TU Dortmund University is a university in Dortmund, North Rhine-Westphalia, Germany with over 30,000 students, and over 3,000 staff. It is situated in the Ruhr area, the fourth largest urban area in Europe, the university is highly ranked in terms of its research performance in the areas of physics, electrical engineering, chemistry and economics. The University of Dortmund was founded in 1968, during the decline of the coal and its establishment was seen as an important move in the economic change from heavy industry to technology. The universitys main areas of research are the sciences, engineering, pedagogy/teacher training in a wide spectrum of subjects, special education. The University of Dortmund was originally designed to be a technical university, in 2006, The University of Dortmund hosted the 11th Federation of International Robot-soccer Association RoboWorld Cup. The universitys robot soccer team, the Dortmund Droids, became world champion in the RoboWorld Cup 2002. Following the Zeitgeist of the late 1960s in Germany, the university was built auf der grünen Wiese about 2 miles outside of downtown Dortmund, one of the most prominent buildings in the university is the Mathetower, which houses the faculty of Mathematics. The first point of registration for. de-domains was at the Dortmund University Department of Computer Science, former president of Germany, Johannes Rau was awarded an honorary degree from the university in 2004. Carl Djerassi was awarded an honorary degree for his science-in-fiction in 2009. ESDP-Network ConRuhr Official homepage of the TU Dortmund University
Dortmund University of Technology
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Dortmund University's Mathetower
Dortmund University of Technology
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Official logo of the TU Dortmund University
Dortmund University of Technology
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Student hostels
Dortmund University of Technology
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Campus Food Court
52.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. Hani envisioned the new body as both a citadel of popular sovereignty, and the means of realizing a peaceful revolution, the National Diet Library opened in June 1948 in the present-day State Guest-House with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori, the philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo