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24,600 | In materials science, a porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The skeletal material is usually a solid, but structures like foams are often also... | Porous media | 0.808115 |
24,601 | There are two minimal extensions of the Virasoro algebra with N = 1 supersymmetry: the Ramond algebra and the Neveu–Schwarz algebra. They are both Lie superalgebras whose even part is the Virasoro algebra: this Lie algebra has a basis consisting of a central element C and generators Lm (for integer m) satisfying = ( m... | Ramond algebra | 0.808115 |
24,602 | In superstring theory, the fermionic fields on the closed string may be either periodic or anti-periodic on the circle around the string. States in the "Ramond sector" admit one option (periodic conditions are referred to as Ramond boundary conditions), described by the Ramond algebra, while those in the "Neveu–Schwarz... | Ramond algebra | 0.808115 |
24,603 | In statistics, isodensity lines or isodensanes are lines that join points with the same value of a probability density. Isodensanes are used to display bivariate distributions. For example, for a bivariate elliptical distribution the isodensity lines are ellipses. | Contour line | 0.808113 |
24,604 | That is, x y + y x = 0 {\displaystyle xy+yx=0\,} for odd x and y. In particular, the square of any odd element x vanishes whenever 2 is invertible: x 2 = 0. {\displaystyle x^{2}=0.} Thus a commutative superalgebra (with 2 invertible and nonzero degree one component) always contains nilpotent elements. A Z-graded antico... | Supercommutative algebra | 0.808113 |
24,605 | Grassmann algebras (also known as exterior algebras) are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that supercommute with all elements, and is a supercommutative algebra. The even subalgebra of a supercommutative algebra is always a comm... | Supercommutative algebra | 0.808113 |
24,606 | In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we have y x = ( − 1 ) | x | | y | x y , {\displaystyle yx=(-1)^{|x||y|}xy,} where |x| denotes the grade of the element and is 0 or 1 (in Z2) according to whether the grad... | Supercommutative algebra | 0.808113 |
24,607 | In 2012, the company rebranded its business, adopting a new logo dropping the word "Customer" and putting square brackets around the "24". In January 2013 it announced it would market some internally developed software products for combining chat with analytics.In May 2013 the company announced the acquisition of socia... | (24)7.ai | 0.808113 |
24,608 | 7.ai (full company name 7.ai, Inc.) is a customer service software and services company based in California that uses artificial intelligence and machine learning to provide targeted customer service. | (24)7.ai | 0.808113 |
24,609 | A key innovation was the introduction of behavior specified as a mathematical function to express what an actor does when it processes a message, including specifying a new behavior to process the next message that arrives. Behaviors provided a mechanism to mathematically model the sharing in concurrency. Behaviors als... | Unbounded nondeterminism controversy | 0.808111 |
24,610 | The actor model in computer science is a mathematical model of concurrent computation that treats an actor as the basic building block of concurrent computation. In response to a message it receives, an actor can: make local decisions, create more actors, send more messages, and determine how to respond to the next mes... | Unbounded nondeterminism controversy | 0.808111 |
24,611 | The actor model has influenced the development of the π-calculus and subsequent process calculi. In his Turing lecture, Robin Milner wrote: Now, the pure lambda-calculus is built with just two kinds of thing: terms and variables. Can we achieve the same economy for a process calculus? Carl Hewitt, with his actors model... | Unbounded nondeterminism controversy | 0.808111 |
24,612 | However, neither of these solutions provided a programming language construct that encapsulated access to shared resources. This encapsulation was later accomplished by the serializer construct ( and ). The first models of computation (e.g., Turing machines, Post productions, the lambda calculus, etc.) were based on ma... | Unbounded nondeterminism controversy | 0.808111 |
24,613 | Robin Milner's initial published work on concurrency was also notable in that it was not based on composing sequential processes. His work differed from the actor model because it was based on a fixed number of processes of fixed topology communicating numbers and strings using synchronous communication. The original c... | Unbounded nondeterminism controversy | 0.808111 |
24,614 | Introduction to Automata Theory, Languages, and Computation is an influential computer science textbook by John Hopcroft and Jeffrey Ullman on formal languages and the theory of computation. Rajeev Motwani contributed to later editions beginning in 2000. | Introduction to Automata Theory, Languages, and Computation | 0.808109 |
24,615 | Let G {\displaystyle G} be a group, and N {\displaystyle N} a normal subgroup of G {\displaystyle G} . The canonical projection homomorphism G → G / N {\displaystyle G\rightarrow G/N} defines a bijective correspondence between the set of subgroups of G {\displaystyle G} containing N {\displaystyle N} and the set of (al... | 1st isomorphism theorem | 0.808109 |
24,616 | Automated and Computerized manufacturing allowed many new fields to emerge from Mechanical Engineering such as Industrial Engineering. Although a majority of automobiles remain to be gas powered, electric vehicles have risen as a feasible alternative.Because of the increased complexity of engineering projects, many dis... | History of mechanical engineering | 0.808109 |
24,617 | The first schools in the United States to offer an mechanical engineering education were the United States Military Academy in 1817, an institution now known as Norwich University in 1819, and Rensselaer Polytechnic Institute in 1825. Education in mechanical engineering has historically been based on a strong foundatio... | History of mechanical engineering | 0.808109 |
24,618 | These cranes were powered with human tread wheels and were based on earlier Mesopotamian water-pulley systems. The Greeks would later develop mechanical artillery independently of the Chinese. The first of these would fire darts, but advancements allowed for stone to be tossed at enemy fortifications or formations.The ... | History of mechanical engineering | 0.808109 |
24,619 | Kushite sundials applied mathematics in the form of advanced trigonometry.In Ancient Egypt, the screw pump is another example of the use of engineering to boost efficiency of water transportation. Although the Early Egyptians built colossal structures such as the pyramids, they did not develop pulleys for the lifting o... | History of mechanical engineering | 0.808109 |
24,620 | They were not the first to develop them, but did make advancements on the wheel and the chariot. They made use of pivot-able axles on their wagons, allowing easy turning. They were also one of the first armies to use the move-able siege tower and battering ram.The application of mechanical engineering can be seen in th... | History of mechanical engineering | 0.808109 |
24,621 | These allowed mechanical engineering to develop as a separate field within engineering. They brought with them manufacturing machines and the engines to power them.At the near end of the Industrial Revolution, internal combustion engine technology brought with it the piston airplane and automobile. Aerospace Engineerin... | History of mechanical engineering | 0.808109 |
24,622 | He continuously worked on improving and marketing the invention across England. At the same time, others were working on improvements to Savery's design, which did not transfer heat effectively.Thomas Newcomen would take all the advancements of the engineers and develop the Newcomen Atmospheric Engine. This new design ... | History of mechanical engineering | 0.808109 |
24,623 | Mechanical engineering is a discipline centered around the concept of using force multipliers, moving components, and machines. It utilizes knowledge of mathematics, physics, materials sciences, and engineering technologies. It is one of the oldest and broadest of the engineering disciplines. | History of mechanical engineering | 0.808109 |
24,624 | Isaac Newton formulated Newton's Laws of Motion and developed the calculus, which would become the mathematical basis of physics. Newton was reluctant to publish his works for years, but he was finally persuaded to do so by his colleagues, such as Edmond Halley. Gottfried Wilhelm Leibniz is also credited with developin... | History of mechanical engineering | 0.808109 |
24,625 | Leonardo da Vinci was a notable engineer, designing and studying many mechanical systems that were focused around transportation and warfare His designs would later be compared to early aircraft design.Although wind power provided a source of energy away from riverside estate and saw massive improvements in its harness... | History of mechanical engineering | 0.808109 |
24,626 | The Virtual Planetary Laboratory (VPL) is a virtual institute based at the University of Washington that studies how to detect exoplanetary habitability and their potential biosignatures. First formed in 2001, the VPL is part of the NASA Astrobiology Institute (NAI) and connects more than fifty researchers at twenty in... | Virtual Planetary Laboratory | 0.808107 |
24,627 | In the prehistoric period, communication was done visually and aurally and involved touching, either delicately or forcefully, as well as movements and gestures. The earliest graphics known to anthropologists studying prehistoric periods are cave paintings and markings on boulders, bone, ivory, and antlers, which were ... | Graphical representation | 0.808102 |
24,628 | Laaksonen, A. (2017). Guide to Competitive Programming (Undergraduate Topics in Computer Science). | Competitive programming | 0.808101 |
24,629 | Participation in programming contests may increase student enthusiasm for computer science studies. The skills acquired in ICPC-like programming contests also improve career prospects, as they help to pass the "technical interviews", which often require candidates to solve complex programming and algorithmic problems o... | Competitive programming | 0.808101 |
24,630 | PPA has applications in a wide range of areas, including astronomy, archaeology, geography, ecology, biology, and epidemiology. A few topics in the last area are discussed here. A case control study compares the point patterns of organisms both with and without some condition to determine if there were significant diff... | Point pattern | 0.808097 |
24,631 | A database engine (or storage engine) is the underlying software component that a database management system (DBMS) uses to create, read, update and delete (CRUD) data from a database. Most database management systems include their own application programming interface (API) that allows the user to interact with their ... | Storage engine | 0.808096 |
24,632 | CAS References consists of bibliographic information and abstracts for all articles in chemical journals worldwide, and chemistry-related articles from all scientific journals, patents, and other scientific publications. | Chemical Abstracts Service | 0.808096 |
24,633 | SciFinder is a database of chemical and bibliographic information. Originally it was available only as a client application (for both Windows and MacOS operating systems), a web version was released in 2008. By that time it had a graphical interface, and was able to do graphical searches for chemical structures and rea... | Chemical Abstracts Service | 0.808096 |
24,634 | (b) By definition, each iteration removes a minimum cut with <= n/2 edges. Theorems 1 and 2a provide a strong indication of a final cluster's homogeneity. Doing better approaches the case where all vertices of a cluster are connected, which is both too stringent and also NP-hard. Theorem 2b indicates separation since a... | HCS clustering algorithm | 0.808092 |
24,635 | (b) The number of edges removed by each iteration of the HCS algorithm is at most linear. Proof: (a) From Theorem 1 we know that every vertex has degree >= n/2. Therefore, the number of edges in a highly connected graph must be at least (n × n/2)/2, where we sum the degrees of each vertex and divide by 2. | HCS clustering algorithm | 0.808092 |
24,636 | So if G is highly connected, every vertex has degree >= n/2. There is a famous theorem in graph theory that says that if every vertex has degree >= n/2, then the diameter of G (the longest path between any two nodes) <= 2. Theorem 2 (a) The number of edges in a highly connected graph is quadratic. | HCS clustering algorithm | 0.808092 |
24,637 | The clusters produced by the HCS clustering algorithm possess several properties, which can demonstrate the homogeneity and separation of the solution. Theorem 1 The diameter of every highly connected graph is at most two. Proof: Let n=|G|. If G has a vertex x with degree <= n/2, then G has a minimum cut (that isolates... | HCS clustering algorithm | 0.808092 |
24,638 | "Survey of clustering algorithms." Neural Networks, IEEE Transactions The CLICK clustering algorithm is an adaptation of HCS algorithm on weighted similarity graphs, where the weight is assigned with a probability flavor. https://www.researchgate.net/publication/259350461_Partitioning_Biological_Networks_into_Highly_Co... | HCS clustering algorithm | 0.808092 |
24,639 | Linear Algebra and its Applications is a biweekly peer-reviewed mathematics journal published by Elsevier and covering matrix theory and finite-dimensional linear algebra. | Linear Algebra and Its Applications | 0.808091 |
24,640 | Self-assembling peptides are a category of peptides which undergo spontaneous assembling into ordered nanostructures. Originally described in 1993, these designer peptides have attracted interest in the field of nanotechnology for their potential for application in areas such as biomedical nanotechnology, tissue cell c... | Self-assembling peptide | 0.80809 |
24,641 | Self-propelled particles interact with each other, which can lead to the emergence of collective behaviours. These collective behaviours mimic the self-organization observed with the flocking of birds, the swarming of bugs, the formation of sheep herds, etc. To understand the ubiquity of such phenomena, physicists have... | Self-propelled particles | 0.808087 |
24,642 | Another shape which is crucial in the history of direct proof is the circle, which was crucial for the design of arenas and water tanks. This meant that ancient geometry (and Euclidean Geometry) discussed circles. | Direct proof | 0.808086 |
24,643 | In days gone by, mathematics and proof was often intertwined with practical questions – with populations like the Egyptians and the Greeks showing an interest in surveying land. This led to a natural curiosity with regards to geometry and trigonometry – particularly triangles and rectangles. These were the shapes which... | Direct proof | 0.808086 |
24,644 | After some simplifying, a 2 + 2 a b + b 2 = 2 a b + c 2 . {\displaystyle a^{2}+2ab+b^{2}=2ab+c^{2}.} Removing the ab that appears on both sides gives a 2 + b 2 = c 2 , {\displaystyle a^{2}+b^{2}=c^{2},} which proves Pythagoras' theorem. ∎ | Direct proof | 0.808086 |
24,645 | In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually axioms, existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then... | Direct proof | 0.808086 |
24,646 | A critical set of phenomena that ties together various aspects of health interventions, such as drug sensitivity screening, cancer or autoimmune susceptibility screening, infectious disease prevalence and application of pharmacologic or nutrition therapies, is the systems biology of the immune response. For example, th... | Genetic susceptibility | 0.808084 |
24,647 | Examples of ways in which genetics may affect an individual's response to drugs include: drug transporters, metabolism and drug interactions. Pharmacogenetics may be used in the near future by public health practitioners to determine the best candidates for certain drugs, thereby reducing much of the guesswork in presc... | Genetic susceptibility | 0.808084 |
24,648 | One of the many facets involved in public health genomics is that of bioethics. This has been highlighted in a study in 2005 by Cogent Research, that found when American citizens were asked what they thought the strongest drawback was in using genetic information, they listed "misuse of information/invasion of privacy"... | Genetic susceptibility | 0.808084 |
24,649 | As we continue to have advances in the study of human genetics, we hope to one day incorporate it into the day-to-day practice of healthcare. Understanding one's own genetic blueprint can empower oneself to take an active role in promoting their own health.Genomics and understanding of disease susceptibility can help v... | Genetic susceptibility | 0.808084 |
24,650 | The term genomics, referring to the organism's whole genome, is also used to refer to gene informatics, or the collection and storage of genetic data, including the functional information associated with the genes, and the analysis of the data as combinations, patterns and networks by computer algorithms. Systems biolo... | Genetic susceptibility | 0.808084 |
24,651 | Soil is made up of a multitude of physical, chemical, and biological entities, with many interactions occurring among them. Soil is a variable mixture of broken and weathered minerals and decaying organic matter. Together with the proper amounts of air and water, it supplies, in part, sustenance for plants as well as m... | Soil ecosystem | 0.808082 |
24,652 | Soil ecology is the study of the interactions among soil organisms, and between biotic and abiotic aspects of the soil environment. It is particularly concerned with the cycling of nutrients, formation and stabilization of the pore structure, the spread and vitality of pathogens, and the biodiversity of this rich biolo... | Soil ecosystem | 0.808082 |
24,653 | This method of recording and reproducing image with coherent light allows three-dimensional images. It was not until 1971 when the procedure gained practical importance, he received the Nobel Prize for Physics. 1949 In Germany, FM broadcasting starts regular program operation. | Timeline of electrical and electronic engineering | 0.808081 |
24,654 | 1924 – Directive shortwave antenna (Yagi–Uda antenna) 1924 – Enrico Fermi's major contribution to semiconductor statistics 1924–1941 – Development of electronic television 1925 – Bell Telephone Laboratories 1928 – One-way police radio communication 1929 – Shannon Scheme for the electrification of the Irish Free State 1... | Timeline of electrical and electronic engineering | 0.808081 |
24,655 | Pyruvate is an important chemical compound in biochemistry. It is the output of the metabolism of glucose known as glycolysis. One molecule of glucose breaks down into two molecules of pyruvate, which are then used to provide further energy, in one of two ways. Pyruvate is converted into acetyl-coenzyme A, which is the... | Pyruvic acid | 0.808079 |
24,656 | Multiple inheritance can solve this, by allowing subclasses to have multiple parents, and thus a few parent classes can be considered rather than every child, without disrupting any existing hierarchy. An example is a taxonomy where different vegetables inherit from their ancestor species. Attempting to compare the tas... | Combinatorial explosion (communication) | 0.808076 |
24,657 | A Latin square of order n is an n × n array with entries from a set of n elements with the property that each element of the set occurs exactly once in each row and each column of the array. An example of a Latin square of order three is given by, A common example of a Latin square would be a completed Sudoku puzzle. A... | Combinatorial explosion (communication) | 0.808075 |
24,658 | This is a generalization of the Hobby–Rice theorem, and it is used to get an exact division of a cake.Each problem can be solved by the next problem: Discrete splitting can be solved by continuous splitting, since a discrete necklace can be converted to a coloring of the real interval {\displaystyle } in which each in... | Necklace splitting problem | 0.80807 |
24,659 | The case k = 2 {\displaystyle k=2} can be proved by the Borsuk-Ulam theorem.When k {\displaystyle k} is an odd prime number, the proof involves a generalization of the Borsuk-Ulam theorem.When k {\displaystyle k} is a composite number, the proof is as follows (demonstrated for the measure-splitting variant). Suppose k ... | Necklace splitting problem | 0.80807 |
24,660 | Simonyi credits Gábor Tardos with noticing that the result above is a direct generalization of Alon's original necklace theorem in the case k = 2. Either the necklace has a (t − 1)-split, or it does not. If it does, there is nothing to prove. If it does not, we may add beads of a fictitious colour to the necklace, and ... | Necklace splitting problem | 0.80807 |
24,661 | In psychology, the term medical model refers to the assumption that psychopathology is the result of one's biology, that is to say, a physical/organic problem in brain structures, neurotransmitters, genetics, the endocrine system, etc., as with traumatic brain injury, Alzheimer's disease, or Down's syndrome. The medica... | Disease model | 0.80807 |
24,662 | According to the medical model, medical treatment, wherever possible, should be directed at the underlying pathology in an attempt to correct the abnormality and cure the disease. In regard to many mental illnesses, for example, the assumption is that the cause of the disorder lies in abnormalities within the affected ... | Disease model | 0.80807 |
24,663 | In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradie... | Gradient method | 0.808067 |
24,664 | Several basic algorithms related to the MMS are: Computing the 1-of-n MMS of a given agent. This is an NP-hard optimization problem, but it has several approximation algorithms; see Multiway number partitioning and Identical-machines scheduling. Deciding whether a given allocation is MMS-fair is co-NP complete for agen... | Pairwise maximin share | 0.808066 |
24,665 | Lattice model may refer to: Lattice model (physics), a physical model that is defined on a periodic structure with a repeating elemental unit pattern, as opposed to the continuum of space or spacetime Lattice model (finance), a "discrete-time" model of the varying price over time of the underlying financial instrument,... | Lattice model | 0.808063 |
24,666 | The Journal of Biological Education is a quarterly peer-reviewed academic journal covering biology education. It was established in 1967, is owned by the Royal Society of Biology, and is published by Routledge. The editor-in-chief is Ian Kinchin (University of Surrey). According to the Journal Citation Reports, the jou... | Journal of Biological Education | 0.808062 |
24,667 | Since 1957 many papers were published on the k shortest path routing problem. Most of the fundamental works were done between 1960s and 2001. Since then, most of the research has been on the problem's applications and its variants. In 2010, Michael Günther et al. published a book on Symbolic calculation of k-shortest p... | K shortest path routing | 0.808059 |
24,668 | Another example is the use of k shortest paths algorithm to track multiple objects. The technique implements a multiple object tracker based on the k shortest paths routing algorithm. A set of probabilistic occupancy maps is used as input. An object detector provides the input. The complete details can be found at "Com... | K shortest path routing | 0.808059 |
24,669 | The k shortest path routing is a good alternative for: Geographic path planning Network routing, especially in optical mesh network where there are additional constraints that cannot be solved by using ordinary shortest path algorithms. Hypothesis generation in computational linguistics Sequence alignment and metabolic... | K shortest path routing | 0.808059 |
24,670 | The configuration linear program (configuration-LP) is a linear programming technique used for solving combinatorial optimization problems. It was introduced in the context of the cutting stock problem. Later, it has been applied to the bin packing and job scheduling problems. In the configuration-LP, there is a variab... | Configuration linear program | 0.808058 |
24,671 | NPB 3 retained the MPI implementation from NPB 2 and came in more flavors, namely OpenMP, Java and High Performance Fortran. These new parallel implementations were derived from the serial codes in NPB 2.3 with additional optimizations. NPB 3.1 and NPB 3.2 added three more benchmarks, which, however, were not available... | NAS Parallel Benchmarks | 0.808053 |
24,672 | Soil and Water Bioengineering is a discipline of civil engineering. It pursues technological, ecological, economic as well as design goals and seeks to achieve these primarily by making use of living materials, i.e. seeds, plants, part of plants and plant communities, and employing them in near–natural constructions wh... | Soil bioengineering | 0.808052 |
24,673 | The Genomic Standards Consortium (GSC) is an initiative working towards richer descriptions of our collection of genomes, metagenomes and marker genes. Established in September 2005, this international community includes representatives from a range of major sequencing and bioinformatics centres (including NCBI, EMBL, ... | Genomic Standards Consortium | 0.808044 |
24,674 | In December 2018 the BSC added to its website an online glossary of hundreds of words and short phrases used in the field of biological staining. This was revised and enlarged in 2021 and 2023.The annual meetings of the BSC are usually held on the first or second weekend in June, in cities easily accessible by air from... | Biological Stain Commission | 0.80804 |
24,675 | In recent years, together with Colour Science Analytical at the University of Leeds, UK, the BSC is trying to identify several dyes sold with the name "alcian blue", none of which are the textile dye that had that name in the 1950s and 1960s (Ali et al 2022). Biological Stains, a book about the chemistry, testing and a... | Biological Stain Commission | 0.80804 |
24,676 | The assays and other tests used in the commission's laboratory are all published, making the required standards known to both vendors and users of biological stains (Penney et al. 2002a,b). The commission's web site includes instructions for submitting samples of stains for testing and certification.Since 1925 the BSC ... | Biological Stain Commission | 0.80804 |
24,677 | These tests are published (Penney et al. 2002a). Changes to tests and additions to the list of stains eligible for certification are published from time to time in Biotechnic & Histochemistry and are summarized on the commission's web site. A BSC-certified stain rarely costs more than a non-certified one with the same ... | Biological Stain Commission | 0.80804 |
24,678 | Educators have shown some interest and described projects and experiences with transformation geometry for children from kindergarten to high school. In the case of very young age children, in order to avoid introducing new terminology and to make links with students' everyday experience with concrete objects, it was s... | Transformation geometry | 0.808038 |
24,679 | Such transformation geometry lessons present an alternate view that contrasts with classical synthetic geometry. When students then encounter analytic geometry, the ideas of coordinate rotations and reflections follow easily. All these concepts prepare for linear algebra where the reflection concept is expanded. | Transformation geometry | 0.808038 |
24,680 | Thus inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students. Experiments with concrete symmetry groups make way for abstract group theory. Other concrete activities use computations with complex numbers, hypercomplex numbers, or matrices to express transfo... | Transformation geometry | 0.808038 |
24,681 | For instance, consider reflection in a vertical line and a line inclined at 45° to the horizontal. One can observe that one composition yields a counter-clockwise quarter-turn (90°) while the reverse composition yields a clockwise quarter-turn. Such results show that transformation geometry includes non-commutative pro... | Transformation geometry | 0.808038 |
24,682 | An exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parall... | Transformation geometry | 0.808038 |
24,683 | For nearly a century this approach remained confined to mathematics research circles. In the 20th century efforts were made to exploit it for mathematical education. Andrei Kolmogorov included this approach (together with set theory) as part of a proposal for geometry teaching reform in Russia. These efforts culminated... | Transformation geometry | 0.808038 |
24,684 | In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geome... | Transformation geometry | 0.808038 |
24,685 | The Pythagorean theorem that a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} can be proven without words.One method of doing so is to visualise a larger square of sides a + b {\displaystyle a+b} , with four right-angled triangles of sides a {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} in its corner... | Visual proof | 0.808038 |
24,686 | Mixture of experts (MoE) is a machine learning technique where multiple expert networks (learners) are used to divide a problem space into homogeneous regions. It differs from ensemble techniques in that typically only one or a few expert models will be run, rather than combining results from all models. An example fro... | Hierarchical mixture of experts | 0.808036 |
24,687 | Masoudnia, Saeed; Ebrahimpour, Reza (12 May 2012). "Mixture of experts: a literature survey". Artificial Intelligence Review. 42 (2): 275–293. doi:10.1007/s10462-012-9338-y. S2CID 3185688. | Hierarchical mixture of experts | 0.808036 |
24,688 | For instance, the intersection graph of line segments in one dimension is an interval graph; the intersection graph of unit disks in the plane is a unit disk graph. The Circle packing theorem states that the intersection graphs of non-crossing circles are exactly the planar graphs. Scheinerman's conjecture (proven in 2... | Euclidean graph | 0.808036 |
24,689 | The skeleton of any convex polyhedron is a planar graph, and the skeleton of any k-dimensional convex polytope is a k-connected graph. Conversely, Steinitz's theorem states that any 3-connected planar graph is the skeleton of a convex polyhedron; for this reason, this class of graphs is also known as the polyhedral gra... | Euclidean graph | 0.808036 |
24,690 | A planar straight-line graph is a graph in which the vertices are embedded as points in the Euclidean plane, and the edges are embedded as non-crossing line segments. Fáry's theorem states that any planar graph may be represented as a planar straight line graph. A triangulation is a planar straight line graph to which ... | Euclidean graph | 0.808036 |
24,691 | Negated propositions are comparably weak, in that the classically valid De Morgan's law, granting a disjunction from a single negative hypothetical, does not automatically hold constructively. See also the L L P O {\displaystyle {\mathrm {LLPO} }} for a variant concerning two existentially closed decidable predicates. ... | Constructive logic | 0.808035 |
24,692 | Incorporating the predicate ψ {\displaystyle \psi } and currying, the following generalization also entails the relation between implication and conjunction in the predicate calculus, discussed below. ( ∀ x ϕ ( x ) → ( ψ ( x ) → φ ) ) ↔ ( ( ∃ x ϕ ( x ) ∧ ψ ( x ) ) → φ ) {\displaystyle {\big (}\forall x\ \phi (x)\to (\p... | Constructive logic | 0.808035 |
24,693 | If the domain of discourse is not empty and ϕ {\displaystyle \phi } is moreover independent of x {\displaystyle x} , such principles are equivalent to formulas in the propositional calculus. Here, the formula then just expresses the identity ( ϕ → φ ) → ( ϕ → φ ) {\displaystyle (\phi \to \varphi )\to (\phi \to \varphi ... | Constructive logic | 0.808035 |
24,694 | Any finite Heyting algebra that is not equivalent to a Boolean algebra defines (semantically) an intermediate logic. On the other hand, validity of formulae in pure intuitionistic logic is not tied to any individual Heyting algebra but relates to any and all Heyting algebras at the same time. | Constructive logic | 0.808035 |
24,695 | Semantical systems claiming to capture such intuitions, due to offering meaningful concepts of “constructive truth” (rather than merely validity or provability), are Kurt Gödel’s dialectica interpretation, Stephen Cole Kleene’s realizability, Yurii Medvedev’s logic of finite problems, or Giorgi Japaridze’s computabilit... | Constructive logic | 0.808035 |
24,696 | Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation.Several systems of semantics for intuitionistic logic have b... | Constructive logic | 0.808035 |
24,697 | Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and ... | Constructive logic | 0.808035 |
24,698 | The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics. Recently, a Tarski-like model theory was proved complete by Bob Constable, but with a different notion of completeness than classically. Unproved statements in in... | Constructive logic | 0.808035 |
24,699 | These are fundamentally consequences of the law of bivalence, which makes all such connectives merely Boolean functions. The law of bivalence is not required to hold in intuitionistic logic. As a result, none of the basic connectives can be dispensed with, and the above axioms are all necessary. So most of the classica... | Constructive logic | 0.808035 |
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