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11,600 | Materials genome is an analogy to genomes in biology, but in a conceptual sense: the many important phases, defects, and processes that make up engineered materials are the "genome" of materials science. The Materials Genome Initiative (MGI) is a federal, multi-agency effort to design, manufacture, and deploy materials... | Materials Genome | 0.817797 |
11,601 | 2D molecule editor and generator 3D geometry generation ring finding substructure search using exact structures and Smiles arbitrary target specification (SMARTS) like query language QSAR descriptor calculation fingerprint calculation, including the ECFP and FCFP fingerprints force field calculations many input-output ... | Chemistry Development Kit | 0.817797 |
11,602 | The Chemistry Development Kit (CDK) is computer software, a library in the programming language Java, for chemoinformatics and bioinformatics. It is available for Windows, Linux, Unix, and macOS. It is free and open-source software distributed under the GNU Lesser General Public License (LGPL) 2.0. | Chemistry Development Kit | 0.817797 |
11,603 | AI is a mainstay of law-related professions. Algorithms and machine learning do some tasks previously done by entry-level lawyers. While its use is common, it is not expected to replace most work done by lawyers in the near future.The electronic discovery industry uses machine learning to reduce manual searching. | AI applications | 0.817794 |
11,604 | Neural networks are used by situational awareness systems in ships and boats. There also are autonomous boats. | AI applications | 0.817794 |
11,605 | Machine learning can be used to fight against spam, scams, and phishing. It can scrutinize the contents of spam and phishing attacks to identify any malicious elements. Numerous models built on machine learning algorithms exhibit exceptional performance with accuracies over 90% in distinguishing between spam and legiti... | AI applications | 0.817794 |
11,606 | There are also prototype robot scientists, including robot-embodied ones like the two Robot Scientists, which show a form of "machine learning" not commonly associated with the term.Similarly, there is research and development of biological "wetware computers" that can learn (e.g. for use as biosensors) and/or implanta... | AI applications | 0.817794 |
11,607 | Autonomous ships that monitor the ocean, AI-driven satellite data analysis, passive acoustics or remote sensing and other applications of environmental monitoring make use of machine learning.For example, "Global Plastic Watch" is an AI-based satellite monitoring-platform for analysis/tracking of plastic waste sites to... | AI applications | 0.817794 |
11,608 | Search engines that use artificial intelligence include Google Search and Bing Chat. | AI applications | 0.817794 |
11,609 | Mark Riedl and Vadim Bulitko asserted that the essence of storytelling was experience management, or "how to balance the need for a coherent story progression with user agency, which is often at odds".While AI storytelling focuses on story generation (character and plot), story communication also received attention. In... | AI applications | 0.817794 |
11,610 | AI was incorporated into military operations in Iraq and Syria.Worldwide annual military spending on robotics rose from US$5.1 billion in 2010 to US$7.5 billion in 2015. Military drones capable of autonomous action are in wide use. Many researchers avoid military applications. | AI applications | 0.817793 |
11,611 | AI software, such as LaundroGraph which uses contemporary suboptimal datasets, could be used for anti-money laundering (AML). AI can be used to "develop the AML pipeline into a robust, scalable solution with a reduced false positive rate and high adaptability". A study about deep learning for AML identified "key challe... | AI applications | 0.817793 |
11,612 | Some of the tools also allow users to input images and various parameters e.g. to display an object or product in various environments, some can replicate artistic styles of popular artists, and some can create elaborate artistic images from rough sketches. Since their design in 2014, generative adversarial networks (G... | AI applications | 0.817793 |
11,613 | AI for Good is an ITU initiative supporting institutions employing AI to tackle some of the world's greatest economic and social challenges. For example, the University of Southern California launched the Center for Artificial Intelligence in Society, with the goal of using AI to address problems such as homelessness. ... | AI applications | 0.817793 |
11,614 | As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection r, every rigid transformation that is not a rigid motion is the product of r and a rigid motion. A glide reflection is an example of a rigid transformation that is not a rigid motion or a reflection. All groups that hav... | Euclidean spaces | 0.817782 |
11,615 | On the other hand, David Hilbert proposed a set of axioms, inspired by Euclid's postulates. They belong to synthetic geometry, as they do not involve any definition of real numbers. Later G. D. Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use real numbers (see Birkhoff's axioms and Tarski's axioms)... | Euclidean spaces | 0.817782 |
11,616 | Since the introduction, at the end of 19th century, of non-Euclidean geometries, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as... | Euclidean spaces | 0.817782 |
11,617 | Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension.Despite the wide use of Descartes' approach, which was ca... | Euclidean spaces | 0.817782 |
11,618 | In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances. Euclidean g... | Euclidean spaces | 0.817782 |
11,619 | Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of ... | Euclidean spaces | 0.817782 |
11,620 | Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry and number theory. For example, the Fermat's Last Theorem can be stated "a Fermat curve of degree higher than two has no point in the affine plane over the rationals." Geometry in affine s... | Euclidean spaces | 0.817782 |
11,621 | For example, a circle and a line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces a... | Euclidean spaces | 0.817782 |
11,622 | A Euclidean space is an affine space equipped with a metric. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field, they allow doing geometry in other contexts. As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the comple... | Euclidean spaces | 0.817782 |
11,623 | In Geometric Algebra, Emil Artin has proved that all these definitions of a Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Art... | Euclidean spaces | 0.817782 |
11,624 | Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated ... | Non-well-founded set theory | 0.817772 |
11,625 | The Radiometer Assessment using Vertically Aligned Nanotubes (RAVAN) was an experiment launched into low Earth orbit on November 11, 2016. RAVAN's goal is to test new technologies that help measure Earth's radiation imbalance, which is the difference between the amount of energy received from the Sun that reaches Earth... | Radiometer Assessment using Vertically Aligned Nanotubes | 0.817767 |
11,626 | Rules about selection operators play the most important role in query optimization. Selection is an operator that very effectively decreases the number of rows in its operand, so if the selections in an expression tree are moved towards the leaves, the internal relations (yielded by subexpressions) will likely shrink. | Relational logic | 0.817764 |
11,627 | s j n ) | j ∈ 1... m } {\displaystyle S=\{(s_{j1},s_{j2},...s_{jn})|j\in 1...m\}} where we commonly interpret m to be the number of rows in a table and n to be the number of columns. All entries in each column have the same type. Five primitive operators of Codd's algebra are the selection, the projection, the Cartesia... | Relational logic | 0.817764 |
11,628 | Relational algebra operates on homogeneous sets of tuples S = { ( s j 1 , s j 2 , . . . | Relational logic | 0.817764 |
11,629 | Relational algebra received little attention outside of pure mathematics until the publication of E.F. Codd's relational model of data in 1970. Codd proposed such an algebra as a basis for database query languages. (See section Implementations.) | Relational logic | 0.817764 |
11,630 | In practice the classical relational algebra described above is extended with various operations such as outer joins, aggregate functions and even transitive closure. | Relational logic | 0.817764 |
11,631 | While the simulations appear to agree broadly with observations, their interpretation is complicated by the understanding of how dense accumulations of dark matter spur galaxy formation. In particular, many more small haloes form than we see in astronomical observations as dwarf galaxies and globular clusters. This is ... | Structure formation | 0.817754 |
11,632 | These perturbations are important, as they are responsible for the subtle physics that result in the cosmic microwave background anisotropy. In this epoch, the amplitude of perturbations that enter the horizon oscillate sinusoidally, with dense regions becoming more rarefied and then becoming dense again, with a freque... | Structure formation | 0.817754 |
11,633 | The physics of structure formation in this epoch is particularly simple, as dark matter perturbations with different wavelengths evolve independently. As the Hubble radius grows in the expanding universe, it encompasses larger and larger disturbances. During matter domination, all causal dark matter perturbations grow ... | Structure formation | 0.817754 |
11,634 | On even larger scales, galaxy clusters and sheet-like structures of galaxies are separated by enormous voids containing few galaxies. Structure formation attempts to model how these structures were formed by gravitational instability of small early ripples in spacetime density or another emergence.The modern Lambda-CDM... | Structure formation | 0.817754 |
11,635 | When the perturbations have grown sufficiently, a small region might become substantially denser than the mean density of the universe. At this point, the physics involved becomes substantially more complicated. When the deviations from homogeneity are small, the dark matter may be treated as a pressureless fluid and e... | Structure formation | 0.817754 |
11,636 | The very early universe is still a poorly understood epoch, from the viewpoint of fundamental physics. The prevailing theory, cosmic inflation, does a good job explaining the observed flatness, homogeneity and isotropy of the universe, as well as the absence of exotic relic particles (such as magnetic monopoles). Anoth... | Structure formation | 0.817754 |
11,637 | The final stage in evolution comes when baryons condense in the centres of galaxy haloes to form galaxies, stars and quasars. Dark matter greatly accelerates the formation of dense haloes. As dark matter does not have radiation pressure, the formation of smaller structures from dark matter is impossible. This is becaus... | Structure formation | 0.817754 |
11,638 | The chemistry of alkali metal germanides, involving the germanide ion Ge4− and other cluster (Zintl) ions such as Ge2−4, Ge4−9, Ge2−9, and 6−, is largely analogous to that of the corresponding silicides. : 393 Alkali metal stannides are mostly ionic, sometimes with the stannide ion (Sn4−), and sometimes with more compl... | Group 1 elements | 0.817751 |
11,639 | Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques, but in many cases it can be necessary to devise from scratch theories permitting control of nonlinear sy... | Control theorist | 0.81775 |
11,640 | Many active and historical figures made significant contribution to control theory including Pierre-Simon Laplace invented the Z-transform in his work on probability theory, now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the Laplace transform which is named aft... | Control theorist | 0.81775 |
11,641 | Research shows that when birth weights of infants are greater than the 90th percentile of the growth chart for babies of the same gestational age, they are considered large for gestational age or LGA. This indicates that these babies are weighing more than 90% of babies of the same gestational age.Many factors account ... | Prenatal nutrition | 0.817748 |
11,642 | With the rapid evolution of deep learning algorithms and DLPs, many architectures have been explored. Roughly, DLPs can be classified into three categories based on their implementation: digital circuits, analog circuits, and hybrid circuits. As the pure analog DLPs are rarely seen, we introduce the digital DLPs and hy... | Deep learning accelerator | 0.817745 |
11,643 | Atomically thin semiconductors are considered promising for energy-efficient deep learning hardware where the same basic device structure is used for both logic operations and data storage. In 2020, Marega et al. published experiments with a large-area active channel material for developing logic-in-memory devices and ... | Deep learning accelerator | 0.817745 |
11,644 | A deep learning processor (DLP), or a deep learning accelerator, is an electronic circuit designed for deep learning algorithms, usually with separate data memory and dedicated instruction set architecture. Deep learning processors range from mobile devices, such as neural processing units (NPUs) in Apple iPhones or Hu... | Deep learning accelerator | 0.817745 |
11,645 | Inspired from the pioneer work of DianNao Family, many DLPs are proposed in both academia and industry with design optimized to leverage the features of deep neural networks for high efficiency. Only at ISCA 2016, three sessions, 15% (!) of the accepted papers, are all architecture designs about deep learning. Such eff... | Deep learning accelerator | 0.817745 |
11,646 | The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: the sphere, the connected sum of g tori for g ≥ 1, the connected sum of k real projective planes for k ≥ 1.The surfaces in the first two families are orientable. It is co... | Classification theorem for surfaces | 0.817737 |
11,647 | Mace stands for "Models And Counter-Examples", and is a model finder. Most automated theorem provers try to perform a proof by refutation on the clause normal form of the proof problem, by showing that the combination of axioms and negated conjecture can never be simultaneously true, i.e. does not have a model. A model... | Models And Counter-Examples | 0.817734 |
11,648 | A set of c-squares (contiguous or non contiguous) can be represented as a concatenated list of individual square codes, separated by the "pipe" (|) character, thus: 7500:110:3|7500:110:1|1500:110:3|1500:110:1 (etc.). This set of squares can then serve as an indication of a dataset extent, similar in function (but simpl... | C-squares | 0.81773 |
11,649 | Additional areas where a grid-based approach to spatial indexing can be beneficial can be for the representation of data "footprints" in support of spatial search, data binning to reduce complex and potentially voluminous data into "blocks" which then can be more easily compared and summarised, and the potential for a ... | C-squares | 0.81773 |
11,650 | In October 1998, the PDB was transferred to the Research Collaboratory for Structural Bioinformatics (RCSB); the transfer was completed in June 1999. The new director was Helen M. Berman of Rutgers University (one of the managing institutions of the RCSB, the other being the San Diego Supercomputer Center at UC San Die... | Protein Data Bank | 0.81773 |
11,651 | The Protein Data Bank was announced in October 1971 in Nature New Biology as a joint venture between Cambridge Crystallographic Data Centre, UK and Brookhaven National Laboratory, US. Upon Hamilton's death in 1973, Tom Koeztle took over direction of the PDB for the subsequent 20 years. In January 1994, Joel Sussman of ... | Protein Data Bank | 0.81773 |
11,652 | The Protein Data Bank (PDB) is a database for the three-dimensional structural data of large biological molecules, such as proteins and nucleic acids. The data, typically obtained by X-ray crystallography, NMR spectroscopy, or, increasingly, cryo-electron microscopy, and submitted by biologists and biochemists from aro... | Protein Data Bank | 0.81773 |
11,653 | The weaker form of the theorem can be proved from the undecidability of the halting problem as follows. Assume that we have a sound (and hence consistent) and complete axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. | Halting problem | 0.817722 |
11,654 | Since soundness implies consistency, this weaker form can be seen as a corollary of the strong form. It is important to observe that the statement of the standard form of Gödel's First Incompleteness Theorem is completely unconcerned with the truth value of a statement, but only concerns the issue of whether it is poss... | Halting problem | 0.817722 |
11,655 | The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard sta... | Halting problem | 0.817722 |
11,656 | The Virus Pathogen Database and Analysis Resource (ViPR) is an integrative and comprehensive publicly available database and analysis resource to search, analyze, visualize, save and share data for viral pathogens in the U.S. National Institute of Allergy and Infectious Diseases (NIAID) Category A-C Priority Pathogen l... | Virus Pathogen Database and Analysis Resource | 0.817715 |
11,657 | EIV (or E6(-26)), which has maximal compact subgroup F4, trivial fundamental group cyclic and outer automorphism group of order 2.The EIV form of E6 is the group of collineations (line-preserving transformations) of the octonionic projective plane OP2. It is also the group of determinant-preserving linear transformatio... | E6 (mathematics) | 0.817708 |
11,658 | There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form (see bel... | E6 (mathematics) | 0.817708 |
11,659 | Precisely: 2E6,sc(q) is a covering of 2E6(q), and 2E6,ad(q) lies in its automorphism group; when q+1 is not divisible by 3, all three coincide, and otherwise (when q is congruent to 2 mod 3), the degree of 2E6,sc(q) over 2E6(q) is 3 and 2E6(q) is of index 3 in 2E6,ad(q), which explains why 2E6,sc(q) and 2E6,ad(q) are o... | E6 (mathematics) | 0.817708 |
11,660 | Beyond this "split" (or "untwisted") form of E6, there is also one other form of E6 over the finite field Fq, known as 2E6, which is obtained by twisting by the non-trivial automorphism of the Dynkin diagram of E6. Concretely, 2E6(q), which is known as a Steinberg group, can be seen as the subgroup of E6(q2) fixed by t... | E6 (mathematics) | 0.817708 |
11,661 | By means of a Chevalley basis for the Lie algebra, one can define E6 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as "untwisted") adjoint form of E6. Over an algebraically closed field,... | E6 (mathematics) | 0.817708 |
11,662 | Similar to how the algebraic group G2 is the automorphism group of the octonions and the algebraic group F4 is the automorphism group of an Albert algebra, an exceptional Jordan algebra, the algebraic group E6 is the group of linear automorphisms of an Albert algebra that preserve a certain cubic form, called the "dete... | E6 (mathematics) | 0.817708 |
11,663 | One way of achieving this is through breaking to SO(10) × U(1). The adjoint 78 representation breaks, as explained above, into an adjoint 45, spinor 16 and 16 as well as a singlet of the SO(10) subalgebra. Including the U(1) charge we have 78 → 45 0 ⊕ 16 − 3 ⊕ 16 ¯ 3 + 1 0 . | E6 (mathematics) | 0.817708 |
11,664 | Another is that whereas 2E6,sc(q) and 2E6,ad(q) are the Fq-points of an algebraic group, the group in question also depends on q (e.g., the points over Fq2 of the same group are the untwisted E6,sc(q2) and E6,ad(q2)). The groups E6(q) and 2E6(q) are simple for any q, and constitute two of the infinite families in the c... | E6 (mathematics) | 0.817708 |
11,665 | The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27-dimensional (complex), and a basis is given by the 27 lines on a cubic surface. The dual representation, ... | E6 (mathematics) | 0.817708 |
11,666 | In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e 6 {\displaystyle {\mathfrak {e}}_{6}} , all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing c... | E6 (mathematics) | 0.817708 |
11,667 | Two 16-dimensional subalgebras that transform as a Weyl spinor of spin ( 10 ) {\displaystyle \operatorname {spin} (10)} and its complex conjugate. These have a non-zero last entry. 1 generator which is their chirality generator, and is the sixth Cartan generator.One choice of simple roots for E6 is given by the rows ... | E6 (mathematics) | 0.817708 |
11,668 | An alternative (6-dimensional) description of the root system, which is useful in considering E6 × SU(3) as a subgroup of E8, is the following: All 4 × ( 5 2 ) {\displaystyle 4\times {\begin{pmatrix}5\\2\end{pmatrix}}} permutations of ( ± 1 , ± 1 , 0 , 0 , 0 , 0 ) {\displaystyle (\pm 1,\pm 1,0,0,0,0)} preserving the ze... | E6 (mathematics) | 0.817708 |
11,669 | The Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU(3) × SU(3) × SU(3) subalgebra. Other maximal subalgebras which have an importance in physics (see below) and can be read off the Dynkin diagram, are the algebras of SO(10) × U(1) and SU(6) × SU(2). In addition to ... | E6 (mathematics) | 0.817708 |
11,670 | Tegmark, whose background and scientific research have been in the fields of theoretical astrophysics and cosmology, mixes autobiography and humor into his analysis of the universe. The book begins with an account of a bicycle accident in Stockholm in which Tegmark was killed—in some theoretical parallel universes, tho... | Our Mathematical Universe | 0.817706 |
11,671 | Reviews of the book are generally enthusiastic about Tegmark's ability to make scientific topics enjoyable and intelligible. For example, according to a reviewer for The Guardian:Tegmark, a professor of physics at MIT, writes at the cutting edge of cosmology and quantum theory in friendly and relaxed prose, full of ent... | Our Mathematical Universe | 0.817706 |
11,672 | Our Mathematical Universe: My Quest for the Ultimate Nature of Reality is a 2014 nonfiction book by the Swedish-American cosmologist Max Tegmark. Written in popular science format, the book interweaves what a New York Times reviewer called "an informative survey of exciting recent developments in astrophysics and quant... | Our Mathematical Universe | 0.817706 |
11,673 | In the field of theoretical computer science, lazy initialization (also called a lazy array) is a technique to design data structures that can work with memory that does not need to be initialized. Specifically, assume that we have access to a table T of n uninitialized memory cells (numbered from 1 to n), and want to ... | Lazy initialization | 0.817705 |
11,674 | The smoothness (also called clumpiness) is a measure of the fraction of light in an object which is in small scale structures. It is an analog of the standard deviation for a 2-dimensional image. In astronomy galaxies which are elliptical have a low smoothness as there is little in the way of small scale structure with... | CAS parameters | 0.817703 |
11,675 | The asymmetry index is a measure of how symmetric an object is. It is defined by rotating and subtracting an image by 180 deg from its center. This parameter has been used in astronomy to determine galaxy mergers and its history. The asymmetry is a measure of skewness in terms of the 2-dimensional distribution of light... | CAS parameters | 0.817703 |
11,676 | The CAS (concentration, asymmetry, smoothness) parameters are a tool originally developed in astronomy to characterize the shapes and images of objects with some central concentration. Each parameter is a single number which represents some aspect of the structure of the object under study. These parameters were origin... | CAS parameters | 0.817703 |
11,677 | The idea is that ϕ {\displaystyle \phi } allows one to consider V {\displaystyle V} and W {\displaystyle W} as "the same" vector space, and by following this analogy, then one can transport an inner product from one space to the other. A more elaborated example comes from differential topology, in which the notion of s... | Transport of structure | 0.8177 |
11,678 | In mathematics, particularly in universal algebra and category theory, transport of structure refers to the process whereby a mathematical object acquires a new structure and its canonical definitions, as a result of being isomorphic to (or otherwise identified with) another object with a pre-existing structure. Defini... | Transport of structure | 0.8177 |
11,679 | Some of the important factors influencing solubility are: Concentration Temperature Solvent mixture composition Polarity Ionic strengthSo one may identify two main families of crystallization processes: Cooling crystallization Evaporative crystallizationThis division is not really clear-cut, since hybrid systems exist,... | Crystallization | 0.817698 |
11,680 | Crystallization is also a chemical solid–liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid crystalline phase occurs. In chemical engineering, crystallization occurs in a crystallizer. Crystallization is therefore related to precipitation, although the result is not... | Crystallization | 0.817698 |
11,681 | Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal, then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If... | Euclid's Elements | 0.817697 |
11,682 | Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles, the number 1 was sometimes treated separately from other... | Euclid's Elements | 0.817697 |
11,683 | It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier.As was common... | Euclid's Elements | 0.817697 |
11,684 | Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass and straightedge. His constructive approach appears even in his geometry's post... | Euclid's Elements | 0.817697 |
11,685 | Book 1 contains 5 postulates (including the infamous parallel postulate) and 5 common notions, and covers important topics of plane geometry such as the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures. Book 2 contains... | Euclid's Elements | 0.817697 |
11,686 | Book 5, on proportions of magnitudes, gives the highly sophisticated theory of proportion probably developed by Eudoxus, and proves properties such as "alternation" (if a: b :: c: d, then a: c :: b: d). Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures. Book 7 ... | Euclid's Elements | 0.817697 |
11,687 | However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements, encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Furthermore, its logical, axiomatic approach and ... | Euclid's Elements | 0.817697 |
11,688 | Vincent Millay wrote in her sonnet "Euclid alone has looked on Beauty bare", "O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized!". Albert Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring... | Euclid's Elements | 0.817697 |
11,689 | The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you... | Euclid's Elements | 0.817697 |
11,690 | The Elements (Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions.... | Euclid's Elements | 0.817697 |
11,691 | Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians.Proclus (412–485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements: "Euclid, who put together the Elements, collecting many of Eudoxu... | Euclid's Elements | 0.817697 |
11,692 | Eventually in 1829, mathematician Nikolai Lobachevsky published a description of acute geometry (or hyperbolic geometry), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth p... | Euclid's Elements | 0.817697 |
11,693 | Out of the group 5 elements, only vanadium has been identified as playing a role in the biological chemistry of living systems, but even it plays a very limited role in biology, and is more important in ocean environments than on land. Vanadium, essential to ascidians and tunicates as vanabins, has been known in the bl... | Group V elements | 0.817696 |
11,694 | Like other groups, the members of this family show patterns in its electron configuration, especially the outermost shells. (The expected 4d3 5s2 configuration for niobium is a very low-lying excited state at about 0.14 eV.) Most of the chemistry has been observed only for the first three members of the group (the chem... | Group V elements | 0.817696 |
11,695 | Stress tensor may refer to: Cauchy stress tensor, in classical physics Stress deviator tensor, in classical physics Piola–Kirchhoff stress tensor, in continuum mechanics Viscous stress tensor, in continuum mechanics Stress–energy tensor, in relativistic theories Maxwell stress tensor, in electromagnetism Electromagneti... | Stress tensor | 0.817696 |
11,696 | The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. Starting in the academic year 1926-1927, Lindenbaum pioneered his method in Jan Łukasiewicz's mathematical logic seminar, and the method was popularized and generalized in subsequent decades through work by Tarski. The Lindenbaum–Tarski algebra is c... | Lindenbaum algebra | 0.817696 |
11,697 | In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q exactly when p and q are provably equivalent in T). That is, two sentences are e... | Lindenbaum algebra | 0.817696 |
11,698 | Heyting algebras and interior algebras are the Lindenbaum–Tarski algebras for intuitionistic logic and the modal logic S4, respectively. A logic for which Tarski's method is applicable, is called algebraizable. There are however a number of logics where this is not the case, for instance the modal logics S1, S2, or S3,... | Lindenbaum algebra | 0.817696 |
11,699 | The areas below date from no earlier than the mid-twentieth century, even if they are based on older material. For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of computability dates only from the... | Combinatorial number theory | 0.817696 |
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