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28,700 | Part of the seventh of Hilbert's twenty three problems posed in 1900 was to prove, or find a counterexample to, the claim that ab is always transcendental for algebraic a ≠ 0, 1 and irrational algebraic b. In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2√2. In 1919, he ga... | Gelfond–Schneider constant | 0.805915 |
28,701 | Because the term "solvable" is also used for solvable groups in group theory, there are several possible definitions of solvable Lie group. For a Lie group G {\displaystyle G} , there is termination of the usual derived series of the group G {\displaystyle G} (as an abstract group); termination of the closures of the d... | Solvable Lie group | 0.805914 |
28,702 | Let g {\displaystyle {\mathfrak {g}}} be the set of matrices on the form X = ( 0 θ x − θ 0 y 0 0 0 ) , θ , x , y ∈ R . {\displaystyle X=\left({\begin{matrix}0&\theta &x\\-\theta &0&y\\0&0&0\end{matrix}}\right),\quad \theta ,x,y\in \mathbb {R} .} Then g {\displaystyle {\mathfrak {g}}} is solvable, but not split solvable... | Solvable Lie group | 0.805914 |
28,703 | In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a delicious cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming tha... | Brouwer fixed-point | 0.805912 |
28,704 | This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it. Take an ordinary map of a country, and suppose that that map is laid out on a table insid... | Brouwer fixed-point | 0.805912 |
28,705 | The proofs of his great topological theorems are not constructive, and Brouwer's dissatisfaction with this is partly what led him to articulate the idea of constructivity. He became the originator and zealous defender of a way of formalising mathematics that is known as intuitionism, which at the time made a stand agai... | Brouwer fixed-point | 0.805912 |
28,706 | The theorem can also be found in existence proofs for the solutions of certain partial differential equations.Other areas are also touched. In game theory, John Nash used the theorem to prove that in the game of Hex there is a winning strategy for white. In economics, P. Bich explains that certain generalizations of th... | Brouwer fixed-point | 0.805912 |
28,707 | One also meets the theorem and its variants outside topology. It can be used to prove the Hartman-Grobman theorem, which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the Central Limit Theorem. | Brouwer fixed-point | 0.805912 |
28,708 | In 1930, Brouwer's fixed-point theorem was generalized to Banach spaces. This generalization is known as Schauder's fixed-point theorem, a result generalized further by S. Kakutani to set-valued functions. | Brouwer fixed-point | 0.805912 |
28,709 | A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the n-dimensional sphere to Rn has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the Lefschetz... | Brouwer fixed-point | 0.805912 |
28,710 | The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory. Brouwer's theorem is probably the most important. It is also among the foundational theorems on the topology of topological manifolds and i... | Brouwer fixed-point | 0.805912 |
28,711 | The theorem has several "real world" illustrations. Here are some examples. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does ... | Brouwer fixed-point | 0.805912 |
28,712 | The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case n = 3 first was proved by Piers Bohl in 1904 (published in Journal für die reine und angewandte Mathematik). It was later pr... | Brouwer fixed-point | 0.805912 |
28,713 | In reverse mathematics, Brouwer's theorem can be proved in the system WKL0, and conversely over the base system RCA0 Brouwer's theorem for a square implies the weak Kőnig's lemma, so this gives a precise description of the strength of Brouwer's theorem. | Brouwer fixed-point | 0.805912 |
28,714 | In mathematics, the Auslander algebra of an algebra A is the endomorphism ring of the sum of the indecomposable modules of A. It was introduced by Auslander (1974). An Artin algebra Γ is called an Auslander algebra if gl dim Γ ≤ 2 and if 0→Γ→I→J→K→0 is a minimal injective resolution of Γ then I and J are projective Γ-m... | Auslander algebra | 0.80591 |
28,715 | Many "well-behaved" spaces in mathematics are second-countable. For example, Euclidean space (Rn) with its usual topology is second-countable. Although the usual base of open balls is uncountable, one can restrict to the collection of all open balls with rational radii and whose centers have rational coordinates. This ... | Second countable | 0.805906 |
28,716 | In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T {\displaystyle T} is second-countable if there exists some countable collection U = { U i } i = 1 ∞ {\displaystyle {\mathcal {U}}=\{U_{i}\}_... | Second countable | 0.805906 |
28,717 | A number of stable low valent magnesium compounds containing a metal-metal, Mg-Mg, bond, where magnesium exhibits the formal oxidation state of +1 are known. These compounds generally have the formula L2Mg2, where L represents a bulky ligand. The first examples of these stable magnesium(I) compounds were reported in 20... | Low valent magnesium compound | 0.805906 |
28,718 | The SAB, an international collection of experts from the fields of medicinal chemistry, pharmacology, and chemical biology, was created to provide the Portal with expert advice. The SAB considers submitted chemical probes and makes recommendations for their use. In addition to considering chemical probes, SAB members a... | Chemical Probes Portal | 0.805904 |
28,719 | The Chemical Probes Portal is an open, online resource whose purpose is to identify and make available high quality chemical probes for use in biological research and drug discovery. While chemical probes can be valuable tools to elucidate signal transduction pathways and to validate new drug targets, many of the probe... | Chemical Probes Portal | 0.805904 |
28,720 | In 1925, Frank Plumpton Ramsey argued that it is not needed. However in the second edition of Principia Mathematica (1927, page xiv) and in Ramsey's 1926 paper it is stated that certain theorems about real numbers could not be proved using Ramsey's approach. Most later mathematical formalisms (Hilbert's Formalism or Br... | Axiom of reducibility | 0.805902 |
28,721 | He then notes the work of the set theorists Zermelo, Fraenkel and Schoenflies, in which "one understands by "set" nothing but an object of which one knows no more and wants to know no more than what follows about it from the postulates. The postulates are to be formulated in such a way that all the desired theorems of... | Axiom of reducibility | 0.805902 |
28,722 | The term wetware is used to describe the protocols and molecular devices used in molecular biology and synthetic biology. Where biological components and systems are treated in a similar manner to software, and similar development models and methodologies are applied, the term 'wetware' can be used to imply an approach... | Wetware (biology) | 0.8059 |
28,723 | The National Science Foundation (NSF) funded Wiki project Open Wetware (OWW) provides a resource for reagent, project and laboratory notebook sharing.A somewhat related NSF consortium Synthetic Biology Engineering Research Center (SynBERC) constructs and distributes wetware. == References == | Wetware (biology) | 0.8059 |
28,724 | Ruthenium complexes typically adopt oxidation states II and III The geometry assumed by most ruthenium complexes is octahedral, which differs from the square planar molecular geometry typical for platinum(II). The presence of six ligands allows for tuning of the complexes' electronic and steric properties. Its partiall... | Ruthenium anti-cancer drugs | 0.805893 |
28,725 | RAPTA compounds are ruthenium–arene complexes bearing the 1,3,5-triaza-7-phosphatricyclo-decane ligand. The complex has a piano stool geometry. The PTA ligand confers water solubility, and the two chloride ligands are labile. RAPTA compounds have low general toxicity that apparently reduces the side-effects associated ... | Ruthenium anti-cancer drugs | 0.805893 |
28,726 | This phenomenon remains a theory, and has been primarily demonstrated in vitro.Although this theory is attractive, convenient, and grounded in fundamental ruthenium chemistry, this theory falls apart when investigated under in vivo. A direct contradiction of this theory was proven using XANES and BOLD-100. This study e... | Ruthenium anti-cancer drugs | 0.805893 |
28,727 | The higher order thinking skills that students have the opportunity to develop during inquiry activities will assist in the critical thinking skills that they will be able to transfer to other subjects. As shown in the section above on the neuroscience of inquiry learning, it is significant to scaffold students to teac... | Inquiry-based learning | 0.805891 |
28,728 | Theorem: Given a cyclic group G of size |G|. If a deterministic polynomial time algorithm A computes the discrete logarithm for a 1/poly(n) fraction of all inputs (where n = log |G| is the input size), then there is a randomized polynomial time algorithm for discrete logarithm for all inputs. Given a generator g of a c... | Random self-reducibility | 0.805891 |
28,729 | Problems that require some privacy in the data (typically cryptographic problems) can use randomization to ensure that privacy. In fact, the only provably secure cryptographic system (the one-time pad) has its security relying totally on the randomness of the key data supplied to the system. The field of cryptography u... | Random self-reducibility | 0.805891 |
28,730 | Lubell's proof of this result on set families is a double counting argument on permutations, used to prove an inequality rather than an equality. Erdős–Ko–Rado theorem, an upper bound on intersecting families of sets, proven by Gyula O. H. Katona using a double counting inequality. Proofs of Fermat's little theorem. | Double counting (proof technique) | 0.805889 |
28,731 | The number of vertex-edge incidences in the graph may be counted in two different ways: by summing the degrees of the vertices, or by counting two incidences for every edge. Therefore where e {\displaystyle e} is the number of edges. The sum of the degrees of the vertices is therefore an even number, which could not ha... | Double counting (proof technique) | 0.805889 |
28,732 | Another theorem that is commonly proven with a double counting argument states that every undirected graph contains an even number of vertices of odd degree. That is, the number of vertices that have an odd number of incident edges must be even. In more colloquial terms, in a party of people some of whom shake hands, a... | Double counting (proof technique) | 0.805889 |
28,733 | For each possible size k {\displaystyle k} , the number of ways in which a committee of k {\displaystyle k} people can be formed from n {\displaystyle n} people is the binomial coefficient Therefore the total number of possible committees is the sum of binomial coefficients over k = 0 , 1 , 2 , … , n {\displaystyle k=0... | Double counting (proof technique) | 0.805889 |
28,734 | Nontechnical. Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). | Principle of conservation of energy | 0.805888 |
28,735 | Physics for Scientists and Engineers (6th ed.). Brooks/Cole. ISBN 978-0-534-40842-8. | Principle of conservation of energy | 0.805888 |
28,736 | Oxtoby & Nachtrieb (1996). Principles of Modern Chemistry, 3rd ed. Saunders College Publishing. | Principle of conservation of energy | 0.805888 |
28,737 | (1996). Fundamentals of College Physics, 2nd ed. William C. Brown Publishers. | Principle of conservation of energy | 0.805888 |
28,738 | A gentle introduction. Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). | Principle of conservation of energy | 0.805888 |
28,739 | Smith, C. (1998). The Science of Energy: Cultural History of Energy Physics in Victorian Britain. | Principle of conservation of energy | 0.805888 |
28,740 | Brown, T.M. (1965). "Resource letter EEC-1 on the evolution of energy concepts from Galileo to Helmholtz". American Journal of Physics. | Principle of conservation of energy | 0.805888 |
28,741 | (For macroscopic systems, this effect is usually too small to measure.) The disposition of this energy gap is not well-understood; some physicists believe that the energy is transferred to or from the macroscopic environment in the course of the measurement process, while others believe that the observable energy is on... | Principle of conservation of energy | 0.805888 |
28,742 | In quantum mechanics, the energy of a quantum system is described by a self-adjoint (or Hermitian) operator called the Hamiltonian, which acts on the Hilbert space (or a space of wave functions) of the system. If the Hamiltonian is a time-independent operator, emergence probability of the measurement result does not ch... | Principle of conservation of energy | 0.805888 |
28,743 | It mentions the work of Brahmagupta and Bhāskara II on the subject of zero. He mentions Madhava of Sangamagrama and Aryabhata and illustrates the - historically first exact - formula for calculating the π (pi).Du Sautoy then considers the Middle East: the invention of the new language of algebra and the evolution of a ... | The Story of Maths | 0.805887 |
28,744 | He mentions Qin Jiushao. He describes Indian mathematicians’ invention of trigonometry; their introduction of a symbol for the number zero and their contribution to the new concepts of infinity and negative numbers. It shows Gwalior Fort where zero is inscribed on its walls. | The Story of Maths | 0.805887 |
28,745 | He describes Isaac Newton’s development of math and physics crucial to understanding the behaviour of moving objects in engineering. He covers the Leibniz and Newton calculus controversy and the Bernoulli family. He further covers Leonhard Euler, the father of topology, and Gauss' invention of a new way of handling equ... | The Story of Maths | 0.805887 |
28,746 | From the seventeenth century, Europe replaced the Middle East as the engine house of mathematical ideas. Du Sautoy visits Urbino to introduce perspective using mathematician and artist, Piero della Francesca's The Flagellation of Christ.Du Sautoy proceeds to describes René Descartes realisation that it was possible to ... | The Story of Maths | 0.805887 |
28,747 | The symmetry of certain geometric objects was the key. Galois' work was picked up by André Weil who built algebraic geometry, a whole new language. Weil's work connected number theory, algebra, topology and geometry. Finally du Sautoy mentions Weil's part in the creation of the fictional mathematician Nicolas Bourbaki ... | The Story of Maths | 0.805887 |
28,748 | The final section briefly covers algebraic geometry. Évariste Galois had refined a new language for mathematics. Galois believed mathematics should be the study of structure as opposed to number and shape. Galois had discovered new techniques to tell whether certain equations could have solutions or not. | The Story of Maths | 0.805887 |
28,749 | For 30 years Hilbert believed that mathematics was a universal language powerful enough to unlock all the truths and solve each of his 23 Problems. Yet, even as Hilbert was stating We must know, we will know, Kurt Gödel had shattered this belief; he had formulated the Incompleteness Theorem based on his study of Hilber... | The Story of Maths | 0.805887 |
28,750 | Next Marcus discusses Henri Poincaré's work on the discipline of 'Bendy geometry'. If two shapes can be moulded or morphed to each other's shape then they have the same topology. Poincaré was able to identify all possible two-dimensional topological surfaces; however in 1904 he came up with a topological problem, the P... | The Story of Maths | 0.805887 |
28,751 | When studying probability theory the blocks method consists of splitting a sample into blocks (groups) separated by smaller subblocks so that the blocks can be considered almost independent. The blocks method helps proving limit theorems in the case of dependent random variables. The blocks method was introduced by S. ... | Randomized block design | 0.805884 |
28,752 | Handbook of Discrete Optimization. Elsevier: 1–68. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. Introduction to Algorithms. | Bellman–Ford algorithm | 0.805884 |
28,753 | ISBN 978-1-84800-997-4. Schrijver, Alexander (2005). "On the history of combinatorial optimization (till 1960)" (PDF). | Bellman–Ford algorithm | 0.805884 |
28,754 | Each node sends its table to all neighboring nodes. When a node receives distance tables from its neighbors, it calculates the shortest routes to all other nodes and updates its own table to reflect any changes.The main disadvantages of the Bellman–Ford algorithm in this setting are as follows: It does not scale well. ... | Bellman–Ford algorithm | 0.805884 |
28,755 | This algebra has a central extension called the Virasoro algebra that is important in two-dimensional conformal field theory and string theory. Note that by restricting n to 1,0,-1, one gets a subalgebra. Taken over the field of complex numbers, this is just the Lie algebra s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}... | Witt algebra | 0.805881 |
28,756 | A basis for the Witt algebra is given by the vector fields L n = − z n + 1 ∂ ∂ z {\displaystyle L_{n}=-z^{n+1}{\frac {\partial }{\partial z}}} , for n in Z {\displaystyle \mathbb {Z} } . The Lie bracket of two basis vector fields is given by = ( m − n ) L m + n . {\displaystyle =(m-n)L_{m+n}.} | Witt algebra | 0.805881 |
28,757 | Over a field k of characteristic p>0, the Witt algebra is defined to be the Lie algebra of derivations of the ring k/zpThe Witt algebra is spanned by Lm for −1≤ m ≤ p−2. | Witt algebra | 0.805881 |
28,758 | In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations ... | Witt algebra | 0.805881 |
28,759 | In order to study classical physics rigorously and methodically, we need to classify systems. Based on previous discussion, we can classify physical systems into holonomic systems and non-holonomic systems. One of the conditions for the applicability of many theorems and equations is that the system must be a holonomic... | Holonomic constraint | 0.80588 |
28,760 | In fact, power consumption is regarded as the limiting factor in the continuing scaling of CMOS technology. To respond to this challenge, in the last decade or so, intensive research has been put into developing computed-aided design (CAD) tools that address the problem of power optimization. Initial efforts were direc... | Power optimization (EDA) | 0.805879 |
28,761 | Power optimization is the use of electronic design automation tools to optimize (reduce) the power consumption of a digital design, such as that of an integrated circuit, while preserving the functionality. | Power optimization (EDA) | 0.805879 |
28,762 | The main levels include: Circuit Level Power Estimation, using a circuit simulator such as SPICE Static Power Estimation does not use the input vectors, but may use the input statistics. Analogous to static timing analysis. Logic-Level Power Estimation, often linked to logic simulation. Analysis at the Register-Transfe... | Power optimization (EDA) | 0.805878 |
28,763 | The increasing speed and complexity of today’s designs implies a significant increase in the power consumption of very-large-scale integration (VLSI) chips. To meet this challenge, researchers have developed many different design techniques to reduce power. The complexity of today’s ICs, with over 100 million transisto... | Power optimization (EDA) | 0.805878 |
28,764 | Logic synthesis can also be optimized in many ways to keep power consumption under control. Details of the following steps can have a significant impact on power optimization: Clock gating Logic Factorization Path Balancing Technology Mapping State Encoding Finite-State Machine Decomposition Retiming | Power optimization (EDA) | 0.805878 |
28,765 | Quantum information processing. Attosecond physics.Another important issue is the spectral selectivity of two photon coherent control. These concepts can be applied to single pulse Raman spectroscopy and microscopy.As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and ... | Coherent control | 0.805876 |
28,766 | Optimal control as applied in coherent control seeks the optimal control field for steering a quantum system to its objective. For state-to-state control the objective is defined as the maximum overlap at the final time T with the state | ϕ f ⟩ {\displaystyle |\phi _{f}\rangle }: J = | ⟨ ψ ( T ) | ϕ f ⟩ | 2 {\displayst... | Coherent control | 0.805876 |
28,767 | Active inference is closely related to the good regulator theorem and related accounts of self-organisation, such as self-assembly, pattern formation, autopoiesis and practopoiesis. It addresses the themes considered in cybernetics, synergetics and embodied cognition. Because free energy can be expressed as the expecte... | Free energy principle | 0.805872 |
28,768 | Active inference has been used to address a range of issues in cognitive neuroscience, brain function and neuropsychiatry, including action observation, mirror neurons, saccades and visual search, eye movements, sleep, illusions, attention, action selection, consciousness, hysteria and psychosis. Explanations of action... | Free energy principle | 0.805872 |
28,769 | This upper bound is known as the free energy, and we can accordingly characterize perception as the minimization of the free energy with respect to inbound sensory information, and action as the minimization of the same free energy with respect to outbound action information. This holistic dual optimization is characte... | Free energy principle | 0.805872 |
28,770 | Comparing the two models reveals a notable similarity between their respective results while also highlighting a remarkable discrepancy, whereby – in the standard version of the SAIM – the model's focus is mainly upon the excitatory connections, whereas in the PE-SAIM, the inhibitory connections are leveraged to make a... | Free energy principle | 0.805872 |
28,771 | The model takes into account the transmission of prediction errors to the same level or a level above, in order to minimise the energy function that indicates the difference between the data and its cause, or, in other words, between the generative model and the posterior. To increase validity, they also incorporated n... | Free energy principle | 0.805872 |
28,772 | Rates of groundwater recharge are difficult to quantify. This is because other related processes, such as evaporation, transpiration (or evapotranspiration) and infiltration processes must first be measured or estimated to determine the balance. There are no widely applicable method available that can directly and accu... | Deep drainage | 0.805869 |
28,773 | Physical methods use the principles of soil physics to estimate recharge. The direct physical methods are those that attempt to actually measure the volume of water passing below the root zone. Indirect physical methods rely on the measurement or estimation of soil physical parameters, which along with soil physical pr... | Deep drainage | 0.805869 |
28,774 | The cheap and simple water aspirator is commonly used in chemistry and biology laboratories and consists of a tee fitting attached to a tap and has a hose barb at one side. The flow of water passes through the straight portion of the tee, which has a restriction at the intersection, where the hose barb is attached. The... | Eductor-jet pump | 0.805868 |
28,775 | Given two sequences S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} , the idea is to measure the cost of obtaining S 2 {\displaystyle S_{2}} from S 1 {\displaystyle S_{1}} using operators from the algebra. Let A = a 1 , a 2 , … a n {\displaystyle A={a_{1},a_{2},\ldots a_{n}}} be a sequence of operators such tha... | Optimal matching | 0.805865 |
28,776 | Optimal matching is a sequence analysis method used in social science, to assess the dissimilarity of ordered arrays of tokens that usually represent a time-ordered sequence of socio-economic states two individuals have experienced. Once such distances have been calculated for a set of observations (e.g. individuals in... | Optimal matching | 0.805865 |
28,777 | Mathematics of Operations Research is a quarterly peer-reviewed scientific journal established in February 1976. It focuses on areas of mathematics relevant to the field of operations research such as continuous optimization, discrete optimization, game theory, machine learning, simulation methodology, and stochastic m... | Mathematics of Operations Research | 0.805863 |
28,778 | Erhan Cinlar served from 1987 to 1992, and was followed by Jan Karel Lenstra (1993-1998). Next was Gérard Cornuéjols (1999-2003) and Nimrod Megiddo (2004-2009). Finally came Uri Rothblum (2009-2012), Jim Dai (2012-2018), and the current editor-in-chief Katya Scheinberg (2019–present). The journal's three initial sectio... | Mathematics of Operations Research | 0.805863 |
28,779 | The following papers have been cited most frequently: Roger B. Myerson, "Optimal Auction Design", vol 6:1, 58-73 A. Ben-Tal and Arkadi Nemirovski, "Robust Convex Optimization", vol 23:4, 769-805 M. R. Garey, D. S. Johnson, and Ravi Sethi, "The Complexity of Flowshop and Jobshop Scheduling", vol 1:2, 117-129 | Mathematics of Operations Research | 0.805863 |
28,780 | Semantic analysis (computational) within applied linguistics and computer science, is a composite of semantic analysis and computational components. Semantic analysis refers to a formal analysis of meaning, and computational refers to approaches that in principle support effective implementation in digital computers. | Semantic analysis (computational) | 0.805862 |
28,781 | It can be written as 4{{{{4}}}}4, a more clear representation of the denotetration used to generate the number. Kruskal's tree theorem is a sequence relating to graphs. TREE(3) is larger than Graham's number. Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest named number.... | Astronomically large | 0.805861 |
28,782 | The question then becomes: on which level of iterated logarithms do to compare two numbers? There is a sense in which one may want to consider 10 10 10 {\displaystyle 10^{10^{10}}} and 10 10 9 {\displaystyle 10^{10^{9}}} to be "close in magnitude". The relative error between these two numbers is large, and the relative... | Astronomically large | 0.805861 |
28,783 | In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance ( 1 / r {\displaystyle 1/r} ), such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanical calculations on... | Laplace expansion (potential) | 0.80586 |
28,784 | The derivation of this expansion is simple. By the law of cosines, 1 | r − r ′ | = 1 r 2 + ( r ′ ) 2 − 2 r r ′ cos γ = 1 r 1 + h 2 − 2 h cos γ with h := r ′ r . {\displaystyle {\frac {1}{|\mathbf {r} -\mathbf {r} '|}}={\frac {1}{\sqrt {r^{2}+(r')^{2}-2rr'\cos \gamma }}}={\frac {1}{r{\sqrt {1+h^{2}-2h\cos \gamma }}}... | Laplace expansion (potential) | 0.80586 |
28,785 | The fermentation process of wine impacts the productivity of wine-related industries as well as the quality of the wine. Data science techniques, such as the k-means algorithm, and classification techniques based on the concept of biclustering have been used to study the process of fermentation in order to predict prob... | Data mining in agriculture | 0.805858 |
28,786 | Fruit defects are often recorded (for a multitude of reasons, sometimes for insurance reasons when exporting fruit overseas). It may be done manually or through computer vision (detecting surface defects when grading fruit). Spray diaries are a legal requirement in many countries and at the very least record the date o... | Data mining in agriculture | 0.805858 |
28,787 | A platform of artificial neural network-based models with sensitivity analysis and optimization algorithms was used successfully to integrate published data on the responses of broiler chickens to threonine. Analyses of the artificial neural network models for weight gain and feed efficiency from a compiled data set su... | Data mining in agriculture | 0.805858 |
28,788 | A group method of data handling-type neural network (GMDH-type network) with an evolutionary method of genetic algorithm was used to predict the metabolizable energy of feather meal and poultry offal meal based on their protein, fat, and ash content. Published data samples were collected from literature and used to tra... | Data mining in agriculture | 0.805858 |
28,789 | Polymerase chain reaction-single strand conformation polymorphism (PCR-SSCP) method was used to determine the growth hormone (GH), leptin, calpain, and calpastatin polymorphism in Iranian Baluchi male sheep. An artificial neural network (ANN) model was developed to describe average daily gain (ADG) in lambs from input ... | Data mining in agriculture | 0.805858 |
28,790 | Data mining in agriculture is a recent research topic consisting of the application of data mining and data science techniques to agriculture. Recent technologies are able to provide extensive data on agricultural-related activities, which can then be analyzed in order to find relevant information. A related, but not e... | Data mining in agriculture | 0.805858 |
28,791 | In atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of atoms with light quanta. The Bohr school required that only those quantities that were in principle measurable by spectroscopy should appear in the theory. | Matrix Mechanics | 0.805857 |
28,792 | In quantum mechanics in the Heisenberg picture the state vector, |ψ⟩ does not change with time, while an observable A satisfies the Heisenberg equation of motion, The extra term is for operators such as which have an explicit time dependence, in addition to the time dependence from the unitary evolution discussed. The ... | Matrix Mechanics | 0.805857 |
28,793 | To make the transition to standard quantum mechanics, the most important further addition was the quantum state vector, now written |ψ⟩, which is the vector that the matrices act on. Without the state vector, it is not clear which particular motion the Heisenberg matrices are describing, since they include all the moti... | Matrix Mechanics | 0.805857 |
28,794 | In quantum mechanics, any unitary symmetry transformation yields a conservation law, since if the matrix U has the property that so it follows that and that the time derivative of U is zero—it is conserved. The eigenvalues of unitary matrices are pure phases, so that the value of a unitary conserved quantity is a compl... | Matrix Mechanics | 0.805857 |
28,795 | In quantum mechanics, the quantum analog G is now a Hermitian matrix, and the equations of motion are given by commutators, The infinitesimal canonical motions can be formally integrated, just as the Heisenberg equation of motion were integrated, where U = eiGs and s is an arbitrary parameter. The definition of a quant... | Matrix Mechanics | 0.805857 |
28,796 | For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert's book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of q... | Matrix Mechanics | 0.805857 |
28,797 | Up until this time, matrices were seldom used by physicists; they were considered to belong to the realm of pure mathematics. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in his work on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra... | Matrix Mechanics | 0.805857 |
28,798 | Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg's paper.A follow-on paper was submitted for publication before the end... | Matrix Mechanics | 0.805857 |
28,799 | Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in the bright-line spectrum. The quantities in Heisenberg's formulation were the classical position and momentum, but n... | Matrix Mechanics | 0.805857 |
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