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7,100 | The behavior of the system can be determined by knowing the seed state and the algorithm used. These methods are often quicker than getting "true" randomness from the environment.The many applications of randomness have led to many different methods for generating random data. These methods may vary as to how unpredict... | Random chance | 0.823139 |
7,101 | The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of calculus had a positive impact on the formal study of randomness. | Random chance | 0.823138 |
7,102 | A symposium series followed in 1949, and a sister journal, the Journal of General Virology, in 1967. The society purchased its own headquarters in Reading in 1971, after initially sharing accommodation with the Biochemical Society in London. In 2014 the Society moved to Charles Darwin House, London, sharing the premise... | Microbial Genomics | 0.823134 |
7,103 | The society was founded on 16 February 1945 as the Society for General Microbiology. Its first president was Alexander Fleming. The Society's first academic meeting was in July 1945 and its first journal, the Journal of General Microbiology (later renamed Microbiology), was published in 1947. | Microbial Genomics | 0.823134 |
7,104 | The Society currently organises a large Annual Conference and a number of smaller Focused Meetings, which cover a specific microbiology discipline. It publishes a magazine, Microbiology Today (formerly SGM Quarterly), and academic journals in virology and microbiology: Microbiology International Journal of Systematic a... | Microbial Genomics | 0.823134 |
7,105 | The Sir Howard Dalton Young Microbiologist of the Year Competition is presented annually to a Society member who is a PhD student or early-career postdoctoral researcher. The competition is judged on the participants' oral or poster presentations at Society conferences. The prize was renamed in 2009 in honour of the la... | Microbial Genomics | 0.823134 |
7,106 | The lecture is usually published in a society journal. Marjory Stephenson was the second president of the Society (1947–1949) and a distinguished pioneer of chemical microbiology.The Fleming Prize Lecture is awarded annually to recognise outstanding research in any branch of microbiology by a microbiologist in the earl... | Microbial Genomics | 0.823134 |
7,107 | In 2015, the prize was renamed the Microbiology Society Prize Medal. The Marjory Stephenson Prize is awarded annually for an outstanding contribution of current importance in microbiology. The winner receives £1000 and gives a lecture on his/her work at a Society meeting. | Microbial Genomics | 0.823134 |
7,108 | The Microbiology Society awards a range of prizes in recognition of significant contributions to microbiology. In 2009, the Society announced the Society for General Microbiology Prize Medal, awarded annually to a microbiologist of international standing whose work has had a far-reaching impact beyond microbiology. The... | Microbial Genomics | 0.823134 |
7,109 | The Microbiology Society (previously the Society for General Microbiology) is a learned society based in the United Kingdom with a worldwide membership based in universities, industry, hospitals, research institutes and schools. It is the largest learned microbiological society in Europe. Interests of its members inclu... | Microbial Genomics | 0.823134 |
7,110 | Solar radius is a unit of distance used to express the size of stars in astronomy relative to the Sun. The solar radius is usually defined as the radius to the layer in the Sun's photosphere where the optical depth equals 2/3: 695,700 kilometres (432,300 miles) is approximately 10 times the average radius of Jupiter, 1... | Sun's radius | 0.823128 |
7,111 | Small science refers (in contrast to big science) to science performed in a smaller scale, such as by individuals, small teams or within community projects. Bodies which fund research, such as the National Science Foundation, DARPA, and the EU with its Framework programs, have a tendency to fund larger-scale research p... | Small science | 0.823125 |
7,112 | Many historical examples show that results of Small Science can have enormous impacts: Galois theory, one of the foundational theories of abstract algebra was developed by Évariste Galois within just weeks. Albert Einstein developed his theory of special relativity as a hobby while working full-time in a patent office.... | Small science | 0.823125 |
7,113 | Women played many roles in this movement within American society. Some were able to find supervisory jobs, or jobs as professor in natural history at school districts, or institutions of higher learning. Some women helped to create the movement itself, like Anna Botsford Comstock, and also teachers were able to " natur... | Nature study | 0.823125 |
7,114 | Nature Study could be found in both urbanized, highly populated cities and in rural school systems because of the involvement of scientists in designing and implementing curriculum. For example, Wilbur Jackman created an outline of nature study with "life and its phenomena" which examined how the study of plants and an... | Nature study | 0.823125 |
7,115 | Nature-study is not the study of a science, as of botany, entomology, geology, and the like. That is, it takes the things at hand and endeavors to understand them, without reference primarily to the systematic order or relationships of objects. | Nature study | 0.823125 |
7,116 | Hence, small science can be seen as an important factor for bringing together science and society. According to the Chronicle for Higher Education, James M. Caruthers, a professor of chemical engineering at Purdue University, data from Big Science is highly organized on the front end where researchers define it before ... | Small science | 0.823125 |
7,117 | Medicine focuses on curing diseases rather than improving one's health.In social sciences biomedicine is described somewhat differently. Through an anthropological lens biomedicine extends beyond the realm of biology and scientific facts; it is a socio-cultural system which collectively represents reality. While biomed... | Biomedicine | 0.823123 |
7,118 | Biomedicine is the cornerstone of modern health care and laboratory diagnostics. It concerns a wide range of scientific and technological approaches: from in vitro diagnostics to in vitro fertilisation, from the molecular mechanisms of cystic fibrosis to the population dynamics of the HIV virus, from the understanding ... | Biomedicine | 0.823123 |
7,119 | Biomedicine (also referred to as Western medicine, mainstream medicine or conventional medicine) is a branch of medical science that applies biological and physiological principles to clinical practice. Biomedicine stresses standardized, evidence-based treatment validated through biological research, with treatment adm... | Biomedicine | 0.823123 |
7,120 | Biochemistry is the science of the chemical processes which takes place within living organisms. Living organisms need essential elements to survive, among which are carbon, hydrogen, nitrogen, oxygen, calcium, and phosphorus. These elements make up the four macromolecules that living organisms need to survive: carbohy... | Biomedicine | 0.823123 |
7,121 | Molecular biology is the process of synthesis and regulation of a cell's DNA, RNA, and protein. Molecular biology consists of different techniques including Polymerase chain reaction, Gel electrophoresis, and macromolecule blotting to manipulate DNA.Polymerase chain reaction is done by placing a mixture of the desired ... | Biomedicine | 0.823123 |
7,122 | However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory, ... | Geometric object | 0.823111 |
7,123 | In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry... | Geometric object | 0.823111 |
7,124 | The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were arti... | Geometric object | 0.823111 |
7,125 | Projective geometry studies properties of shapes which are unchanged under projections and sections, especially as they relate to artistic perspective.Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovic... | Geometric object | 0.823111 |
7,126 | The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was th... | Geometric object | 0.823111 |
7,127 | The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on... | Geometric object | 0.823111 |
7,128 | 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyam (1048–11... | Geometric object | 0.823111 |
7,129 | Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. In Eucl... | Geometric object | 0.823111 |
7,130 | Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.Tilings, or tessellations, have been used in art throughout history. Islamic... | Geometric object | 0.823111 |
7,131 | Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.Artists have long used concepts of proportion in design. Vitruvius developed a complicated theo... | Geometric object | 0.823111 |
7,132 | In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas)... | Geometric object | 0.823111 |
7,133 | The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to the Sulba Sutras. According to (Hayashi 2005, p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Bab... | Geometric object | 0.823111 |
7,134 | 287–212 BC) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of pi. He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution. Indian ... | Geometric object | 0.823111 |
7,135 | Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes (c. | Geometric object | 0.823111 |
7,136 | Eudoxus (408–c. 355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolu... | Geometric object | 0.823111 |
7,137 | South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first u... | Geometric object | 0.823111 |
7,138 | For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures antici... | Geometric object | 0.823111 |
7,139 | The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronom... | Geometric object | 0.823111 |
7,140 | Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics, econometrics, and bioinformatics, among others. In particular, differential geometry is of importance to mathematical physics due to Albert Einstein's general relativity postulation that t... | Geometric object | 0.823111 |
7,141 | Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hidden-line removal, and linear programming.Although being a young area of geometry, it has many application... | Geometric object | 0.823111 |
7,142 | Wheels incur mechanical and other disadvantages in certain environments and situations that would represent a decreased fitness when compared with limbed locomotion. These disadvantages suggest that, even barring the biological constraints discussed above, the absence of wheels in multicellular life may not be the "mis... | Rotating locomotion in living systems | 0.823106 |
7,143 | The absence of wheels in nature is frequently attributed to constraints imposed by biology: natural selection constrains the evolutionary paths available to species, and the processes by which multicellular organisms grow and develop may not permit the construction of a functioning wheel. | Rotating locomotion in living systems | 0.823106 |
7,144 | The exception to this is the flagellum, the only known example of a freely rotating propulsive system in biology; in the evolution of flagella, individual components were recruited from older structures, where they performed tasks unrelated to propulsion. The basal body that is now the rotary motor, for instance, might... | Rotating locomotion in living systems | 0.823106 |
7,145 | Techniques and methods from one area can be applied to prove results in other related areas. Patterns from one mathematical object can be generalized to other similar objects in the same class.On the other hand, abstraction can also be disadvantageous in that highly abstract concepts can be difficult to learn. A degree... | Mathematical abstraction | 0.823103 |
7,146 | Finally Felix Klein's "Erlangen program" identified the underlying theme of all of these geometries, defining each of them as the study of properties invariant under a given group of symmetries. This level of abstraction revealed connections between geometry and abstract algebra.In mathematics, abstraction can be advan... | Mathematical abstraction | 0.823103 |
7,147 | In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann and Gauss, who generalised the concepts of geometry to develop non-Euclidean geometries. Later in the 19th century, mathematicians ... | Mathematical abstraction | 0.823103 |
7,148 | Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in a... | Mathematical abstraction | 0.823103 |
7,149 | In mathematics, the dyadic cubes are a collection of cubes in Rn of different sizes or scales such that the set of cubes of each scale partition Rn and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularly harmonic analysis) as a way of disc... | Dyadic cubes | 0.823093 |
7,150 | In theoretical computer science, in particular in formal language theory, Kleene's algorithm transforms a given nondeterministic finite automaton (NFA) into a regular expression. Together with other conversion algorithms, it establishes the equivalence of several description formats for regular languages. Alternative p... | Kleene's algorithm | 0.823089 |
7,151 | The automaton shown in the picture can be described as M = (Q, Σ, δ, q0, F) with the set of states Q = { q0, q1, q2 }, the input alphabet Σ = { a, b }, the transition function δ with δ(q0,a)=q0, δ(q0,b)=q1, δ(q1,a)=q2, δ(q1,b)=q1, δ(q2,a)=q1, and δ(q2,b)=q1, the start state q0, and set of accept states F = { q1 }.Kleen... | Kleene's algorithm | 0.823088 |
7,152 | Minmax (sometimes Minimax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for minimizing the possible loss for a worst case (maximum loss) scenario. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. O... | Minmax algorithm | 0.823076 |
7,153 | Supervised analysis for class prediction involves use of techniques such as linear regression, k-nearest neighbor, learning vector quantization, decision tree analysis, random forests, naive Bayes, logistic regression, kernel regression, artificial neural networks, support vector machines, mixture of experts, and super... | DNA Microarray | 0.823075 |
7,154 | Based on goals of the physics program the ND consist of Electromagnetic calorimeter 168 rectangular NaI(Tl) scintillation counters total mass of NaI(Tl) is 2.6 t solid angle coverage is 65% of 4π sr minimum thickness is 32 cm or 12 radiation length energy resolution for photons is σ/E = 4% / √ECharged particle coordina... | ND Experiment | 0.823065 |
7,155 | Neutral Detector (ND) is a detector for particle physics experiments created by the team of physicists in the Budker Institute of Nuclear Physics, Novosibirsk, Russia. Experiments with the ND were conducted from 1982 to 1987 at the e+e− storage ring VEPP-2M in the energy range 2E=0.5-1.4 GeV. | ND Experiment | 0.823065 |
7,156 | Historically, mathematics was most important in the natural sciences and engineering. However, since World War II, fields outside the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory, which grew out of economic considerations. Further, the utilization... | Techno-mathematics | 0.823061 |
7,157 | Since about 1990, FDTD techniques have emerged as the primary means to model many scientific and engineering problems addressing electromagnetic wave interactions with material structures. An effective technique based on a time-domain finite-volume discretization procedure was introduced by Mohammadian et al. in 1991. ... | Electromagnetic simulation | 0.823061 |
7,158 | FDTD is the only technique where one person can realistically implement oneself in a reasonable time frame, but even then, this will be for a quite specific problem. Since it is a time-domain method, solutions can cover a wide frequency range with a single simulation run, provided the time step is small enough to satis... | Electromagnetic simulation | 0.823061 |
7,159 | In 2D and no polarization terms present, Maxwell's equations can then be formulated as: where u, A, B, and C are defined as In this representation, g ¯ {\displaystyle {\bar {g}}} is the forcing function, and is in the same space as u ¯ {\displaystyle {\bar {u}}} . It can be used to express an externally applied field o... | Electromagnetic simulation | 0.823061 |
7,160 | Maxwell's equations can be formulated as a hyperbolic system of partial differential equations. This gives access to powerful techniques for numerical solutions. It is assumed that the waves propagate in the (x,y)-plane and restrict the direction of the magnetic field to be parallel to the z-axis and thus the electric ... | Electromagnetic simulation | 0.823061 |
7,161 | Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describ... | Techno-mathematics | 0.823061 |
7,162 | Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability and decision theory, and makes extensive use of scientifi... | Techno-mathematics | 0.823061 |
7,163 | Such descriptions can lead to applicable mathematics being seen as a collection of mathematical methods such as real analysis, linear algebra, mathematical modelling, optimisation, combinatorics, probability and statistics, which are useful in areas outside traditional mathematics and not specific to mathematical physi... | Techno-mathematics | 0.823061 |
7,164 | Sometimes, the term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today, although there is no consensus as to a precise definition.Mathematicians often distinguish be... | Techno-mathematics | 0.823061 |
7,165 | In March 2011, it was decided to list Osram on the stock market in the autumn, but CEO Peter Löscher said Siemens intended to retain a long-term interest in the company, which was already independent from the technological and managerial viewpoints. In September 2011, Siemens, which had been responsible for constructin... | Siemens | 0.823057 |
7,166 | In the realm of electrical products, the division offers low-voltage switching, measuring and control equipment, distribution systems, and medium voltage switchgear.In the renewable energy industry, the company provides a portfolio of products and services to help build and operate microgrids of any size. It provides g... | Siemens | 0.823057 |
7,167 | On 18 March 2003, the company unveiled the latest in its series of Xelibri fashion phones.In 2004, the wind energy company Bonus Energy in Brande, Denmark was acquired, forming Siemens Wind Power division. Also in 2004, Siemens invested in Dasan Networks (South Korea, broadband network equipment) acquiring ~40% of the ... | Siemens | 0.823056 |
7,168 | Common descent Evidence of common descent Speciation Co-operation (evolution) Adaptive radiation Coevolution Divergent evolution Convergent evolution Parallel evolution Evolutionary developmental biology Evolutionary biology Evolutionary history of life Human evolution Evolutionary taxonomy | Outline of genetics | 0.823056 |
7,169 | Classical genetics Developmental genetics Conservation genetics Ecological genetics Evolutionary genetics Genetic engineering Metagenics Genetic epidemiology Archaeogenetics Archaeogenetics of the Near East Genetics of intelligence Genetic testing Genomics Human genetics Human evolutionary genetics Human mitochondrial ... | Outline of genetics | 0.823056 |
7,170 | The following outline is provided as an overview of and topical guide to genetics: Genetics – science of genes, heredity, and variation in living organisms. Genetics deals with the molecular structure and function of genes, and gene behavior in context of a cell or organism (e.g. dominance and epigenetics), patterns of... | Outline of genetics | 0.823056 |
7,171 | History of genetics Ancient Concepts of Heredity Experiments on Plant Hybridization History of evolutionary thought History of genetic engineering History of genomics History of paleontology History of plant systematics Neanderthal genome project Timeline of paleontology | Outline of genetics | 0.823056 |
7,172 | DNA sequencing Medical genetics Genomics Evolutionary ideas of the Renaissance and Enlightenment On the Origin of Species Charles Darwin The eclipse of Darwinism | Outline of genetics | 0.823056 |
7,173 | chromosome sex-linked shotgun sequencing single-nucleotide polymorphisms (SNPs) somatic cells Southern blot spectral karyotype (SKY) substitution suicide gene syndrome technology transfer transgenic trisomy tumor suppressor gene vector Western blot yeast artificial chromosome (YAC) | Outline of genetics | 0.823056 |
7,174 | Molecules amino acids Nucleobase Adenine Cytosine Guanine Thymine Uracil Adenovirus Antibody Bacteria Codon Deoxyribonucleic acid (DNA) Messenger RNA mRNA Enzyme Exon Intron nucleotide allele animal model antisense apoptosis autosomal dominant autosome bacterial artificial chromosome (BAC) base pair birth defect bone m... | Outline of genetics | 0.823056 |
7,175 | Introduction to genetics Genetics Chromosome DNA Genetic diversity Genetic drift Genetic variation Genome Heredity Mutation Nucleotide RNA Introduction to evolution Evolution Modern evolutionary synthesis Transmutation of species Natural selection Extinction Adaptation Polymorphism (biology) Gene flow Biodiversity Biog... | Outline of genetics | 0.823056 |
7,176 | The standards for Monte Carlo experiments in statistics were set by Sawilowsky. In applied statistics, Monte Carlo methods may be used for at least four purposes: To compare competing statistics for small samples under realistic data conditions. Although type I error and power properties of statistics can be calculated... | Monte Carlo Simulation | 0.823052 |
7,177 | As long as the function in question is reasonably well-behaved, it can be estimated by randomly selecting points in 100-dimensional space, and taking some kind of average of the function values at these points. By the central limit theorem, this method displays 1 / N {\displaystyle \scriptstyle 1/{\sqrt {N}}} convergen... | Monte Carlo Simulation | 0.823052 |
7,178 | The network performance is then evaluated and, if results are not satisfactory, the network design goes through an optimization process. In reliability engineering, Monte Carlo simulation is used to compute system-level response given the component-level response. In signal processing and Bayesian inference, particle f... | Monte Carlo Simulation | 0.823052 |
7,179 | In geostatistics and geometallurgy, Monte Carlo methods underpin the design of mineral processing flowsheets and contribute to quantitative risk analysis. In fluid dynamics, in particular rarefied gas dynamics, where the Boltzmann equation is solved for finite Knudsen number fluid flows using the direct simulation Mont... | Monte Carlo Simulation | 0.823052 |
7,180 | Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations that produce photo-realistic images of virtual 3D models, with applications in video games, architecture, design, ... | Monte Carlo Simulation | 0.823052 |
7,181 | In electrical engineering and computer science, Lloyd's algorithm, also known as Voronoi iteration or relaxation, is an algorithm named after Stuart P. Lloyd for finding evenly spaced sets of points in subsets of Euclidean spaces and partitions of these subsets into well-shaped and uniformly sized convex cells. Like th... | Lloyd's algorithm | 0.82305 |
7,182 | It is important for the convergence of the finite element methods that these elements be well shaped; in the case of triangles, often elements that are nearly equilateral triangles are preferred. Lloyd's algorithm can be used to smooth a mesh generated by some other algorithm, moving its vertices and changing the conne... | Lloyd's algorithm | 0.82305 |
7,183 | It is particularly well-suited to picking sample positions for dithering. Lloyd's algorithm is also used to generate dot drawings in the style of stippling. In this application, the centroids can be weighted based on a reference image to produce stipple illustrations matching an input image.In the finite element method... | Lloyd's algorithm | 0.82305 |
7,184 | Additionally, Cartesian coordinates are highly correlated, that is, the Hessian matrix has many non-diagonal terms that are not close to zero. This can lead to numerical problems in the optimization, because, for example, it is difficult to obtain a good approximation to the Hessian matrix and calculating it precisely ... | Geometry optimization | 0.823049 |
7,185 | The simplest chain-of-state method is the linear synchronous transit (LST) method. It operates by taking interpolated points between the reactant and product geometries and choosing the one with the highest energy for subsequent refinement via a local search. The quadratic synchronous transit (QST) method extends LST b... | Geometry optimization | 0.823049 |
7,186 | Geometry optimization is then a mathematical optimization problem, in which it is desired to find the value of r for which E(r) is at a local minimum, that is, the derivative of the energy with respect to the position of the atoms, ∂E/∂r, is the zero vector and the second derivative matrix of the system, ( ∂ 2 E ∂ r i ... | Geometry optimization | 0.823049 |
7,187 | The geometry of a set of atoms can be described by a vector of the atoms' positions. This could be the set of the Cartesian coordinates of the atoms or, when considering molecules, might be so called internal coordinates formed from a set of bond lengths, bond angles and dihedral angles. Given a set of atoms and a vect... | Geometry optimization | 0.823049 |
7,188 | However, in case that energy is expressed with standard force fields, computationally efficient methods have been developed able to derive analytically the Hessian matrix in Cartesian coordinates while preserving a computational complexity of the same order to that of gradient computations. Internal coordinates tend to... | Geometry optimization | 0.823049 |
7,189 | As described above, some method such as quantum mechanics can be used to calculate the energy, E(r) , the gradient of the PES, that is, the derivative of the energy with respect to the position of the atoms, ∂E/∂r and the second derivative matrix of the system, ∂∂E/∂ri∂rj, also known as the Hessian matrix, which descri... | Geometry optimization | 0.823049 |
7,190 | A so-called local optimization requires an initial guess of the transition state that is very close to the true transition state. Very close typically means that the initial guess must have a corresponding Hessian matrix with one negative eigenvalue, or, the negative eigenvalue corresponding to the reaction coordinate ... | Geometry optimization | 0.823049 |
7,191 | Some degrees of freedom can be eliminated from an optimization, for example, positions of atoms or bond lengths and angles can be given fixed values. Sometimes these are referred to as being frozen degrees of freedom. Figure 1 depicts a geometry optimization of the atoms in a carbon nanotube in the presence of an exter... | Geometry optimization | 0.823049 |
7,192 | As an example, when optimizing the geometry of a water molecule, one aims to obtain the hydrogen-oxygen bond lengths and the hydrogen-oxygen-hydrogen bond angle which minimize the forces that would otherwise be pulling atoms together or pushing them apart. The motivation for performing a geometry optimization is the ph... | Geometry optimization | 0.823049 |
7,193 | In the field of computational chemistry, energy minimization (also called energy optimization, geometry minimization, or geometry optimization) is the process of finding an arrangement in space of a collection of atoms where, according to some computational model of chemical bonding, the net inter-atomic force on each ... | Geometry optimization | 0.823049 |
7,194 | Geometry optimization is fundamentally different from a molecular dynamics simulation. The latter simulates the motion of molecules with respect to time, subject to temperature, chemical forces, initial velocities, Brownian motion of a solvent, and so on, via the application of Newton's laws of Motion. This means that ... | Geometry optimization | 0.823049 |
7,195 | Other names for a double layer are electrostatic double layer, electric double layer, plasma double layers. The term ‘electrostatic shock’ in the magnetosphere has been applied to electric fields oriented at an oblique angle to the magnetic field in such a way that the perpendicular electric field is much stronger than... | Double layer (plasma physics) | 0.823047 |
7,196 | The Erdős–Anning theorem states that a set of points with integer distances must either be finite or lie on a single line. However, there are other infinite sets of points with rational distances. For instance, on the unit circle, let S be the set of points ( cos θ , sin θ ) {\displaystyle (\cos \theta ,\sin \theta... | Erdős–Ulam problem | 0.823047 |
7,197 | In 1946, Stanislaw Ulam asked whether there exists a set of points at rational distances from each other that forms a dense subset of the Euclidean plane. While the answer to this question is still open, József Solymosi and Frank de Zeeuw showed that the only irreducible algebraic curves that contain infinitely many po... | Erdős–Ulam problem | 0.823047 |
7,198 | By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization. (By convention, 1 is the empty product.) Testing whether the integer is prime can be done in polynomial time, for example, by the AKS primality test. If composite, however, the polynomial time tests give no insight into h... | Factoring problem | 0.823046 |
7,199 | For current computers, GNFS is the best published algorithm for large n (more than about 400 bits). For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time. This will have significant implications for cryptography if quantum computation becomes scalable. | Factoring problem | 0.823046 |
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