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22,800 | In addition, by reducing prosumers' reliance on purchased products, the RepRap and the recyclebot have made it feasible for 3D printing to be used for small-scale manufacturing to aid sustainable development. It has been postulated that recycled filament production could also offer an alternative income source by the E... | Recyclebot | 0.809193 |
22,801 | Jeremy Rifkin has hypothesized that such recycling with recyclebots and distributed production with 3D printing will lead to a zero marginal cost society. The science-fiction author, Bruce Sterling wondered in Wired if recyclebots and 3D printers might be used to turn waste into guns. Recyclebots can provide a new meth... | Recyclebot | 0.809193 |
22,802 | Skolem (1922) pointed out the seeming contradiction between the Löwenheim–Skolem theorem on the one hand, which implies that there is a countable model of Zermelo's axioms, and Cantor's theorem on the other hand, which states that uncountable sets exist, and which is provable from Zermelo's axioms. "So far as I know," ... | Skolem's paradox | 0.809192 |
22,803 | Löwenheim (1915) and Skolem (1920, 1923) proved the Löwenheim–Skolem theorem. The downward form of this theorem shows that if a countable first-order axiomatisation is satisfied by any infinite structure, then the same axioms are satisfied by some countable structure. In particular, this implies that if the first-order... | Skolem's paradox | 0.809192 |
22,804 | One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets, such as the powerset of the natural numbers, the set of real numbers, and the Cantor set. An infinite set X is countable if there is a function that gives a one-to-one correspondence between X and the na... | Skolem's paradox | 0.809192 |
22,805 | Though the focus of the Morph was to demonstrate the potential of nanotechnology, it pioneered the concept of utilizing a flexible video display in a consumer electronics device. Nokia renewed their interest in flexible mobile devices again in 2011 with the Nokia Kinetic concept. Nokia unveiled the Kinetic flexible pho... | Rollable electronics | 0.80919 |
22,806 | The complete list of n for which Q(ζn) has unique factorization is 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90.Kummer found a way to deal with the failure of unique factorization. He introduced a replacement for the prime numbers in the cyclotomic integer... | Cyclotomic field | 0.80919 |
22,807 | A natural approach to proving Fermat's Last Theorem is to factor the binomial xn + yn, where n is an odd prime, appearing in one side of Fermat's equation x n + y n = z n {\displaystyle x^{n}+y^{n}=z^{n}} as follows: x n + y n = ( x + y ) ( x + ζ y ) ⋯ ( x + ζ n − 1 y ) {\displaystyle x^{n}+y^{n}=(x+y)(x+\zeta y)\cdots... | Cyclotomic field | 0.80919 |
22,808 | Although Waffles provides significant breadth, it lacks the depth of many toolkits that focus on a particular area of machine learning. The Weka (machine learning) toolkit, for example, provides many more classification algorithms than Waffles provides. Waffles only has a limited graphical interface. | Waffles (machine learning) | 0.809189 |
22,809 | The Waffles machine learning toolkit contains command-line tools for performing various operations related to machine learning, data mining, and predictive modeling. The primary focus of Waffles is to provide tools that are simple to use in scripted experiments or processes. For example, the supervised learning algorit... | Waffles (machine learning) | 0.809189 |
22,810 | Some of the advantages of Waffles in contrast with other popular open source machine learning toolkits include: Waffles automatically takes care of many issues related to data format in order to simplify its tools. Because it is implemented in C++, many of its algorithms are particularly fast. Also, the lack of depende... | Waffles (machine learning) | 0.809189 |
22,811 | Waffles is a collection of command-line tools for performing machine learning operations developed at Brigham Young University. These tools are written in C++, and are available under the GNU Lesser General Public License. | Waffles (machine learning) | 0.809189 |
22,812 | Software maintenance is a broad activity that includes error correction, enhancements of capabilities, removal of obsolete capabilities, and optimization. Because change is inevitable, mechanisms must be developed for evaluation, controlling and making modifications. Any work done on software after it is deployed is co... | Software maintenance | 0.809186 |
22,813 | Software maintenance in software engineering is the modification of a software product after delivery to correct faults, to improve performance or other attributes.A common perception of maintenance is that it merely involves fixing defects. However, one study indicated that over 80% of maintenance effort is used for n... | Software maintenance | 0.809186 |
22,814 | It is assumed that the system is initially in a legitimate configuration. Then the network topology is changed; the superstabilization time is the maximum time it takes for the system to reach a legitimate configuration again. Similarly, the adjustment measure is the maximum number of nodes that have to change their st... | Superstabilization | 0.809186 |
22,815 | The stabilization time of a superstabilizing algorithm is defined exactly as in the case of self-stabilizing algorithm: how long it takes to converge to a legitimate state from an arbitrary configuration. Depending on the computational model, time is measured, e.g., in synchronous communication rounds or in asynchronou... | Superstabilization | 0.809186 |
22,816 | Any self-stabilizing algorithm recovers from a change in the network topology – the system configuration after a topology change can be treated just like any other arbitrary starting configuration. However, in a self-stabilizing algorithm, the convergence after a single change in the network topology may be as slow as ... | Superstabilization | 0.809186 |
22,817 | Superstabilization is a concept of fault-tolerance in distributed computing. Superstabilizing distributed algorithms combine the features of self-stabilizing algorithms and dynamic algorithms. A superstabilizing algorithm – just like any other self-stabilizing algorithm – can be started in an arbitrary state, and it wi... | Superstabilization | 0.809186 |
22,818 | The book is generally acknowledged as a key element in awakening the genius of Ramanujan.Carr acknowledged the main sources of his book in its preface: ... In the Algebra, Theory of Equations, and Trigonometry sections, I am largely indebted to Todhunter's well-known treatises ... In the section entitled Elementary Geo... | Synopsis of Pure Mathematics | 0.809185 |
22,819 | The scientific part of the congress lasted two days, and was attended by well over 800 delegates, many of them from outside the Arab region. The conference was supported by the Dubai Health Authority, the Dubai Islamic Affairs and Charitable Activities Department, the Human Variome Project, and HUGO. Other participatin... | Centre for Arab Genomic Studies | 0.809185 |
22,820 | The Pan Arab Human Genetics series of Conference is planned to be held every alternate year by the Centre as a means of providing a platform on dialogue and education for geneticists in the region, to share their knowledge and to talk on common issues. The first edition of the conference, held in 2006, was supported by... | Centre for Arab Genomic Studies | 0.809185 |
22,821 | The Centre for Arab Genomic Studies (CAGS) is a not-for-profit study centre aimed at the characterization and prevention of genetic disorders in the Arab World. The Centre is closely associated with the Sheikh Hamdan Award for Medical Sciences. One of the major projects of CAGS is the Catalogue for Transmission Genetic... | Centre for Arab Genomic Studies | 0.809185 |
22,822 | The Catalogue for Transmission Genetics in Arabs (CTGA) database is an enormous project, which aims to create a compendium of information related to the entire range of genetic disorders reported among Arabs. This information is available in a freely accessible form at the CAGS website. As of now, the database has info... | Centre for Arab Genomic Studies | 0.809185 |
22,823 | In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. | Roulette curve | 0.809185 |
22,824 | Two years later, ScienceLogic raised an additional $15 million in funding from Intel Capital. In 2015, ScienceLogic received $43 million in Series D funding, led by Goldman Sachs In 2016, ScienceLogic announced the acquisition of AppFirst. Following the launch of the SL1 product line in 2018, which by the company and i... | ScienceLogic | 0.809185 |
22,825 | ScienceLogic ranked on Deloitte’s Technology Fast 500™ Deloitte Technology Fast 500 in 2009, 2016, and 2018. ScienceLogic was recognized by Forrester as a leader in “The Forrester Wave™: Intelligent Application and Service Monitoring, Q2 2019.” In 2019, ScienceLogic was named the #3 AIOps global vendor by Research in A... | ScienceLogic | 0.809185 |
22,826 | First, it is known that any first-order interpretation in which equality is interpreted by an equivalence relation and satisfies the substitution axioms for equality can be cut down to an elementarily equivalent interpretation on a subset of the original domain. Thus there is little additional generality in studying no... | Boolean valuation | 0.809185 |
22,827 | One example of many-sorted logic is for planar Euclidean geometry. There are two sorts; points and lines. There is an equality relation symbol for points, an equality relation symbol for lines, and a binary incidence relation E which takes one point variable and one line variable. The intended interpretation of this la... | Boolean valuation | 0.809184 |
22,828 | Theorems (and derived rules of inference) are written with $p statements; for example: $( Prove a theorem $) th1 $p |- t = t $= $( Here is its proof: $) tt tze tpl tt weq tt tt weq tt a2 tt tze tpl tt weq tt tze tpl tt weq tt tt weq wim tt a2 tt tze tpl tt tt a1 mp mp $. Note the inclusion of the proof in the $p statem... | Metamath | 0.809183 |
22,829 | The Metamath website hosts several databases that store theorems derived from various axiomatic systems. Most databases (.mm files) have an associated interface, called an "Explorer", which allows one to navigate the statements and proofs interactively on the website, in a user-friendly way. Most databases use a Hilber... | Metamath | 0.809183 |
22,830 | The Metamath website also hosts a few older databases which are not maintained anymore, such as the "Hilbert Space Explorer", which presents theorems pertaining to Hilbert space theory which have now been merged into the Metamath Proof Explorer, and the "Quantum Logic Explorer", which develops quantum logic starting wi... | Metamath | 0.809183 |
22,831 | The largest database of proved theorems follows conventional ZFC set theory and classic logic.The Metamath language design (employed to state the definitions, axioms, inference rules and theorems) is focused on simplicity; it contains only a handful of keywords. Proofs are checked using an algorithm based on variable s... | Metamath | 0.809183 |
22,832 | The database contains developments, among other fields, of set theory (ordinals and cardinals, recursion, equivalents of the axiom of choice, the continuum hypothesis...), the construction of the real and complex number systems, order theory, graph theory, abstract algebra, linear algebra, general topology, real and co... | Metamath | 0.809183 |
22,833 | User can enter the demonstration(proof) in two modes: forward and backward relative to the statement to prove. Milpgame checks if a statement is well formed (has a syntactic verifier). It can save unfinished proofs without the use of dummylink theorem. The demonstration is shown as tree, the statements are shown using ... | Metamath | 0.809183 |
22,834 | Because Metamath has a very generic concept of what a proof is (namely a tree of formulas connected by inference rules) and no specific logic is embedded in the software, Metamath can be used with species of logic as different as Hilbert-style logics or sequents-based logics or even with lambda calculus. However, Metam... | Metamath | 0.809183 |
22,835 | It is the conclusion of the theorem opreq2i. The theorem opreq2i states that if A = B, then (C F A) = (C F B). This theorem would never appear under this cryptic form in a textbook but its literate formulation is banal: when two quantities are equal, one can replace one by the other in an operation. | Metamath | 0.809183 |
22,836 | Steps 1 and 2 of the theorem 2p2e4 in the Metamath Proof Explorer (set.mm) are depicted left. Let's explain how Metamath uses its substitution algorithm to check that step 2 is the logical consequence of step 1 when you use the theorem opreq2i. Step 2 states that ( 2 + 2 ) = ( 2 + ( 1 + 1 ) ). | Metamath | 0.809183 |
22,837 | All Metamath proof steps use a single substitution rule, which is just the simple replacement of a variable with an expression and not the proper substitution described in works on predicate calculus. Proper substitution, in Metamath databases that support it, is a derived construct instead of one built into the Metama... | Metamath | 0.809183 |
22,838 | Metamath is a formal language and an associated computer program (a proof assistant) for archiving and verifying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others.As of ... | Metamath | 0.809183 |
22,839 | There is also a project by William Hale to add a graphical user interface to Metamath called Mmide. Paul Chapman in its turn is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Milpgame is a proof assistant and a checker (it... | Metamath | 0.809183 |
22,840 | Mel O'Cat designed a system called Mmj2, which provides a graphic user interface for proof entry. The initial aim of Mel O'Cat was to allow the user to enter the proofs by simply typing the formulas and letting Mmj2 find the appropriate inference rules to connect them. In Metamath on the contrary you may only enter the... | Metamath | 0.809183 |
22,841 | "Distributed neural network underlying musical sight-reading and keyboard performance". Science. 257 (5066): 106–109. | Sight reading | 0.809183 |
22,842 | A central problem in algorithmic graph theory is the shortest path problem. One of the generalizations of the shortest path problem is known as the single-source-shortest-paths (SSSP) problem, which consists of finding the shortest paths from a source vertex s {\displaystyle s} to all other vertices in the graph. There... | Parallel single-source shortest path algorithm | 0.809182 |
22,843 | Embedding latent space and multimodal embedding models have found numerous applications across various domains: Information Retrieval: Embedding techniques enable efficient similarity search and recommendation systems by representing data points in a compact space. Natural Language Processing: Word embeddings have revo... | Latent manifold | 0.80918 |
22,844 | To embed multimodal data, specialized architectures such as deep multimodal networks or multimodal transformers are employed. These architectures combine different types of neural network modules to process and integrate information from various modalities. The resulting embeddings capture the complex relationships bet... | Latent manifold | 0.80918 |
22,845 | The CERN Neutrinos to Gran Sasso (CNGS) project was a physics project of the European Organization for Nuclear Research (CERN). The aim of the project was to analyse the hypothesis of neutrino oscillation by directing a beam of neutrinos from CERN's facilities to the detector of the OPERA experiment at the Gran Sasso N... | CNGS experiment | 0.809179 |
22,846 | In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands). Band theory derives these bands and band gaps by examining the... | Theory of electrons in solids | 0.809179 |
22,847 | However, once subsets of a given set X (in Cantor's case, X = R) are considered, the universe may need to be a set of subsets of X. (For example, a topology on X is a set of subsets of X.) The various sets of subsets of X will not themselves be subsets of X but will instead be subsets of PX, the power set of X. This ma... | Universe (mathematics) | 0.809178 |
22,848 | In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that contain (as elements) all sets for which one hopes to p... | Universe (mathematics) | 0.809178 |
22,849 | LambdaNative is an open-source cross-platform development environment written on top of the Gambit-C Scheme implementation of the Scheme programming language, supporting the creation of native applications on a range of mobile, desktop and embedded platforms. The underlying Scheme programming language has a long tradit... | LambdaNative | 0.809176 |
22,850 | Logic gates evaluate Boolean algebra, typically using electronic switches controlled by logical inputs connected in parallel or series. Diode logic can only implement OR and AND, because inverters (NOT gates) require an active device. | Diode logic | 0.809172 |
22,851 | The Curtin–Hammett principle has been invoked to explain regioselectivity in the acylation of 1,2-diols. Ordinarily, the less-hindered site of an asymmetric 1,2-diol would experience more rapid esterification due to reduced steric hindrance between the diol and the acylating reagent. Developing a selective esterificati... | Curtin–Hammett principle | 0.809172 |
22,852 | In particle physics, isospin multiplets are families of hadrons with approximately equal masses. All particles within a multiplet, have the same spin, parity, and baryon numbers, but differ in electric charges. Isospin formally behaves as an angular momentum operator and thus satisfies the appropriate canonical commuta... | Isospin multiplet | 0.809171 |
22,853 | The journal publishes "communications" (articles for rapid publication), "reviews" (state-of-the-art accounts of a research field), "mini-reviews" (research highlights in an emerging area of materials science, usually from the past 2–3 years) and "focus articles" (educational articles providing an overview of a concept... | Materials Horizons | 0.80917 |
22,854 | The journal is indexed in the Science Citation Index. Selective content is also indexed in Polymer Library, Inspec, Biotechnology and Bioengineering Abstracts, METADEX, Mechanical Engineering Abstracts, Solid State and Superconductivity Abstracts, Metal Abstracts and CSA Technology Research Database, and CABI. | Materials Horizons | 0.80917 |
22,855 | Materials Horizons is a bimonthly peer-reviewed scientific journal that covers research across the breadth of materials science at the interface between chemistry, physics, biology and engineering. The current editor-in-chief is Martina Stenzel. The journal was established in 2014. A sister journal Nanoscale Horizons w... | Materials Horizons | 0.80917 |
22,856 | The perplexity PP of a discrete probability distribution p is a concept widely used in information theory, machine learning, and statistical modeling. It is defined as P P ( p ) := 2 H ( p ) = 2 − ∑ x p ( x ) log 2 p ( x ) = ∏ x p ( x ) − p ( x ) {\displaystyle {\mathit {PP}}(p):=2^{H(p)}=2^{-\sum _{x}p(x)\log _{2}p(... | Perplexity | 0.80917 |
22,857 | The lowest perplexity that has been published on the Brown Corpus (1 million words of American English of varying topics and genres) as of 1992 is indeed about 247 per word, corresponding to a cross-entropy of log2247 = 7.95 bits per word or 1.75 bits per letter using a trigram model. While this figure represented the ... | Perplexity | 0.80917 |
22,858 | Since 2007, significant advancements in language modeling have emerged, particularly with the advent of deep learning techniques. Perplexity per word, a measure that quantifies the predictive power of a language model, has remained central to evaluating models like transformers, BERT, GPT-2, and others. This measure is... | Perplexity | 0.80917 |
22,859 | In natural language processing, a corpus is a set of sentences or texts, and a language model is a probability distribution over entire sentences or texts. Consequently, in NLP, the more commonly used measure is perplexity per word, defined as: where s 1 , . . . | Perplexity | 0.80917 |
22,860 | In elementary network theory, a network is represented by a graph G = ( V , E ) {\displaystyle G=(V,E)} in which V {\displaystyle V} is the set of nodes and E {\displaystyle E} the links between nodes, typically represented as a tuple of nodes u , v ∈ V {\displaystyle u,v\in V} . While this basic formalization is usefu... | Multidimensional network | 0.809168 |
22,861 | In a multidimensional network in which different dimensions of connection have different real-world values, statistics characterizing the distribution of links to the various classes are of interest. Thus it is useful to consider two metrics that assess this: dimension connectivity and edge-exclusive dimension connecti... | Multidimensional network | 0.809168 |
22,862 | Diffusion processes are widely used in physics to explore physical systems, as well as in other disciplines as social sciences, neuroscience, urban and international transportation or finance. Recently, simple and more complex diffusive processes have been generalized to multilayer networks. One result common to many s... | Multidimensional network | 0.809168 |
22,863 | Among other network statistics, many centrality measures rely on the ability to assess shortest paths from node to node. Extending these analyses to a multidimensional network requires incorporating additional connections between nodes into the algorithms currently used (e.g., Dijkstra's). Current approaches include co... | Multidimensional network | 0.809168 |
22,864 | Like many other network statistics, the meaning of a clustering coefficient becomes ambiguous in multidimensional networks, due to the fact that triples may be closed in different dimensions than they originated. Several attempts have been made to define local clustering coefficients, but these attempts have highlighte... | Multidimensional network | 0.809168 |
22,865 | When we seek to probe connectivity between nodes, are we comparing the same nodes across dimensions, or different nodes within dimensions, or a combination? What are the consequences of variations in each of these statistics on other network properties? In one study, assortativity was found to decrease robustness in a ... | Multidimensional network | 0.809168 |
22,866 | In network theory, multidimensional networks, a special type of multilayer network, are networks with multiple kinds of relations. Increasingly sophisticated attempts to model real-world systems as multidimensional networks have yielded valuable insight in the fields of social network analysis, economics, urban and int... | Multidimensional network | 0.809168 |
22,867 | As a result, even though the material has a relatively high Young's modulus, devices manufactured on ultrathin substrates exhibit a decrease in bending stiffness and allow bending to a small radius of curvature without fracturing. Thin devices have been developed as a result of significant advancements in the field of ... | Stretchable electronics | 0.809165 |
22,868 | Rigid electronics doesn't typically conform well to soft, biological organisms and tissue. Since stretchable electronics is not limited by this, some researchers try to implement it as sensors for touch, or tactile sensing. One way of achieving this is to make an array of conductive OFET (Organic Field Effect Transisto... | Stretchable electronics | 0.809165 |
22,869 | Self-adjointness is substantially more restricting than these three properties. The famous spectral theorem holds for self-adjoint operators. In combination with Stone's theorem on one-parameter unitary groups it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-par... | Closed operator | 0.809162 |
22,870 | The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. | Closed operator | 0.809162 |
22,871 | The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for quantum mechanics. The theory's development is due to John von Neumann and Marshall Stone. Von Neumann introduced using graphs to analyze unbounded operators in 1932. | Closed operator | 0.809162 |
22,872 | By definition, an operator T is an extension of an operator S if Γ(S) ⊆ Γ(T). An equivalent direct definition: for every x in the domain of S, x belongs to the domain of T and Sx = Tx.Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at Discontinuous linear ... | Closed operator | 0.809162 |
22,873 | A symmetric operator is often studied via its Cayley transform. An operator T on a complex Hilbert space is symmetric if and only if its quadratic form is real, that is, the number ⟨ T x ∣ x ⟩ {\displaystyle \langle Tx\mid x\rangle } is real for all x in the domain of T.A densely defined closed symmetric operator T is ... | Closed operator | 0.809162 |
22,874 | Magnesium compounds are compounds formed by the element magnesium (Mg). These compounds are important to industry and biology, including magnesium carbonate, magnesium chloride, magnesium citrate, magnesium hydroxide (milk of magnesia), magnesium oxide, magnesium sulfate, and magnesium sulfate heptahydrate (Epsom salts... | Magnesium compounds | 0.809158 |
22,875 | Two insects, the silkworm and the western honey bee, have been domesticated for over 5,000 years, often for commercial use. The silkworm is raised for the silk threads wound around its pupal cocoon; the western honey bee, for honey, and, lately, for pollination of crops.Several other invertebrates have been domesticate... | Domestication of plants | 0.809151 |
22,876 | Next, transferring these advantages to otherwise vulnerable crop plants. Rice, which is one of the most vulnerable crops in terms of drought, has been successfully improved by the addition of the Barley hva1 gene into the genome using transgenetics. Drought resistance can also be improved through changes in a plant's r... | Domestication of plants | 0.809151 |
22,877 | Both the chicken in Southeast Asia and the cat in Egypt were domesticated 4,000 years ago.The sudden appearance of the domestic dog (Canis lupus familiaris) in the archaeological record then led to a rapid shift in the evolution, ecology, and demography of both humans and numerous species of animals and plants. It was ... | Domestication of plants | 0.809151 |
22,878 | The archaeological and genetic data suggest that long-term bidirectional gene flow between wild and domestic stocks – including donkeys, horses, New and Old World camelids, goats, sheep, and pigs – was common. Given its importance to humans and its value as a model of evolutionary and demographic change, domestication ... | Domestication of plants | 0.809151 |
22,879 | In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, writing the sequence of prime numbers as (p1, p2, p3, ...) = (2, 3, 5, ...), pn is a st... | Strong prime | 0.809147 |
22,880 | In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory. | Strong prime | 0.809147 |
22,881 | Without the aid of a computer, this number would be a strong prime in the cryptographic sense because 439351292910452432574786963588089477522344330 has the large prime factor 1747822896920092227343 (and in turn the number one less than that has the large prime factor 1683837087591611009), 439351292910452432574786963588... | Strong prime | 0.809146 |
22,882 | In cryptography, a prime number p is said to be "strong" if the following conditions are satisfied. p is sufficiently large to be useful in cryptography; typically this requires p to be too large for plausible computational resources to enable a cryptanalyst to factorise products of p with other strong primes. p − 1 ha... | Strong prime | 0.809146 |
22,883 | Ramsey sentences are formal logical reconstructions of theoretical propositions attempting to draw a line between science and metaphysics. A Ramsey sentence aims at rendering propositions containing non-observable theoretical terms (terms employed by a theoretical language) clear by substituting them with observational... | Theoretical term | 0.809143 |
22,884 | The BEST theorem shows that the number of Eulerian circuits in directed graphs can be computed in polynomial time, a problem which is #P-complete for undirected graphs. It is also used in the asymptotic enumeration of Eulerian circuits of complete and complete bipartite graphs. | BEST theorem | 0.809142 |
22,885 | The BEST theorem is due to van Aardenne-Ehrenfest and de Bruijn (1951), §6, Theorem 6. Their proof is bijective and generalizes the de Bruijn sequences. In a "note added in proof", they refer to an earlier result by Smith and Tutte (1941) which proves the formula for graphs with deg(v)=2 at every vertex. | BEST theorem | 0.809142 |
22,886 | {\displaystyle \operatorname {ec} (G)=t_{w}(G)\prod _{v\in V}{\bigl (}\deg(v)-1{\bigr )}!.} Here tw(G) is the number of arborescences, which are trees directed towards the root at a fixed vertex w in G. The number tw(G) can be computed as a determinant, by the version of the matrix tree theorem for directed graphs. It ... | BEST theorem | 0.809142 |
22,887 | We denote the indegree of a vertex v by deg(v). The BEST theorem states that the number ec(G) of Eulerian circuits in a connected Eulerian graph G is given by the formula ec ( G ) = t w ( G ) ∏ v ∈ V ( deg ( v ) − 1 ) ! . | BEST theorem | 0.809142 |
22,888 | In graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in directed (oriented) graphs. The name is an acronym of the names of people who discovered it: de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte. | BEST theorem | 0.809142 |
22,889 | In mathematics, the minimum k-cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI d... | Minimum k-cut | 0.809142 |
22,890 | All the above cases may be unified and generalized in universal algebra. | Kernel (universal algebra) | 0.809141 |
22,891 | Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider topological groups or topological vector spaces, which are equipped with a topology. In this case, we would expect the homomorphism f to preserve this additional structure; in the topol... | Kernel (universal algebra) | 0.809141 |
22,892 | In population genetics, the concept of heterozygosity is commonly extended to refer to the population as a whole, i.e., the fraction of individuals in a population that are heterozygous for a particular locus. It can also refer to the fraction of loci within an individual that are heterozygous. Typically, the observed ... | Heterozygous DNA | 0.809141 |
22,893 | The ideas behind the MFS were developed primarily by V. D. Kupradze and M. A. Alexidze in the late 1950s and early 1960s. However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s, followed by a number of papers by Mathon, Johnston and Graeme Fairwe... | Method of fundamental solution | 0.80914 |
22,894 | In scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis function. The MFS was developed to overcome the major drawbacks in the boundary element method (BEM) which also uses the fundam... | Method of fundamental solution | 0.80914 |
22,895 | Since the publication of the hammerhead and P4-6 structures, numerous major contributions to the field have been made. Some of the most noteworthy examples include the structures of the Group I and Group II introns, and the Ribosome. The first three structures were produced using in vitro transcription, and that NMR ha... | RNA Tertiary Structure | 0.809138 |
22,896 | In addition to the advances being made in global structure determination via crystallography, the early 1990s also saw the implementation of NMR as a powerful technique in RNA structural biology. Investigations such as this enabled a more precise characterization of the base pairing and base stacking interactions which... | RNA Tertiary Structure | 0.809138 |
22,897 | In the mid-1960s, the role of tRNA in protein synthesis was being intensively studied. In 1965, Holley et al. purified and sequenced the first tRNA molecule, initially proposing that it adopted a cloverleaf structure, based largely on the ability of certain regions of the molecule to form stem loop structures. The isol... | RNA Tertiary Structure | 0.809138 |
22,898 | The earliest work in RNA structural biology coincided, more or less, with the work being done on DNA in the early 1950s. In their seminal 1953 paper, Watson and Crick suggested that van der Waals crowding by the 2`OH group of ribose would preclude RNA from adopting a double helical structure identical to the model they... | RNA Tertiary Structure | 0.809138 |
22,899 | They most often stabilize RNA duplex interactions in loops and helices, such as in the core of group II introns.An interesting example of A-minor is its role in anticodon recognition. The ribosome must discriminate between correct and incorrect codon-anticodon pairs. It does so, in part, through the insertion of adenin... | RNA Tertiary Structure | 0.809138 |
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