id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
19,100 | This step is called discretization because it involves creating an approximate discrete model from the original continuous model. In particular, it typically translates a partial differential equation (or a system thereof) into a system of algebraic equations. The processes involved in this step are studied in the fiel... | Tangent stiffness matrix | 0.811727 |
19,101 | Scientists within the field of computational mechanics follow a list of tasks to analyze their target mechanical process: A mathematical model of the physical phenomenon is made. This usually involves expressing the natural or engineering system in terms of partial differential equations. This step uses physics to form... | Tangent stiffness matrix | 0.811727 |
19,102 | The solubility is favored by entropy of mixing (ΔS) and depends on enthalpy of dissolution (ΔH) and the hydrophobic effect. The free energy of dissolution (Gibbs energy) depends on temperature and is given by the relationship: ΔG = ΔH – TΔS. Smaller ΔG means greater solubility. Chemists often exploit differences in sol... | Rate of solution | 0.811724 |
19,103 | Others associated with or influenced by second-order cybernetics include: Henri Atlan, biophysicist influenced by Foerster. Stewart Brand, associated with Bateson and Foerster. Some of the proceeds from Brand's Whole Earth Catalogue funded the publication of Foerster's Cybernetics of Cybernetics book. Nicholas Negropon... | Second-order cybernetics | 0.811724 |
19,104 | In probability, statistics and related fields, the geometric process is a counting process, introduced by Lam in 1988. It is defined as The geometric process. Given a sequence of non-negative random variables: { X k , k = 1 , 2 , … } {\displaystyle \{X_{k},k=1,2,\dots \}} , if they are independent and the cdf of X k {\... | Geometric process | 0.811723 |
19,105 | Negation or complement ¬x0 appears as 1f1 and again as 2f5, along with 2f3 (¬x1, which did not appear at arity 1), disjunction or union x0∨x1 as 2f14, conjunction or intersection x0∧x1 as 2f8, implication x0→x1 as 2f13, exclusive-or symmetric difference x0⊕x1 as 2f6, set difference x0−x1 as 2f2, and so on. As a minor d... | Power set algebra | 0.811723 |
19,106 | Applications are far-reaching because set theory is the standard foundations of mathematics.Boolean algebra thus permits applying the methods of abstract algebra to mathematical logic and digital logic. Unlike groups of finite order, which exhibit complexity and diversity and whose first-order theory is decidable only ... | Power set algebra | 0.811723 |
19,107 | A complete homomorphism is one that preserves all sups that exist, not just the finite sups, and likewise for infs. The category CABA of all CABAs and their complete homomorphisms is dual to the category of sets and their functions, meaning that it is equivalent to the opposite of that category (the category resulting ... | Power set algebra | 0.811723 |
19,108 | A complete Boolean algebra is called completely distributive when arbitrary conjunctions distribute over arbitrary disjunctions and vice versa. A Boolean algebra is a CABA if and only if it is complete and completely distributive, giving a third definition of CABA. A fourth definition is as any Boolean algebra isomorph... | Power set algebra | 0.811723 |
19,109 | Free CABAs exist for all cardinalities of a set V of generators, namely the power set algebra 22V, this being the obvious generalization of the finite free Boolean algebras. This neatly rescues infinitary Boolean logic from the fate the Gaifman–Hales result seemed to consign it to. The nonexistence of free complete Boo... | Power set algebra | 0.811723 |
19,110 | (In place of this awkward restriction on arity we could allow any arity, leading to a different awkwardness, that the signature would then be larger than any set, that is, a proper class. One benefit of the latter approach is that it simplifies the definition of homomorphism between CABAs of different cardinality.) Suc... | Power set algebra | 0.811723 |
19,111 | This suggests that a logic with set-sized-infinitary operations may have class-many terms—just as a logic with finitary operations may have infinitely many terms. There is however another approach to introducing infinitary Boolean operations: simply drop "finitary" from the definition of Boolean algebra. A model of the... | Power set algebra | 0.811723 |
19,112 | Recall the definition of sup and inf from the section above on the underlying partial order of a Boolean algebra. A complete Boolean algebra is one every subset of which has both a sup and an inf, even the infinite subsets. Gaifman and Hales independently showed that infinite free complete Boolean algebras do not exi... | Power set algebra | 0.811723 |
19,113 | Example 8. The direct product of a Periodic Sequence (Example 5) with any finite but nontrivial Boolean algebra. (The trivial one-element Boolean algebra is the unique finite atomless Boolean algebra.) | Power set algebra | 0.811723 |
19,114 | This example has the same atoms and coatoms as Example 4, whence it is not atomless and therefore not isomorphic to Example 5/6. However it contains an infinite atomless subalgebra, namely Example 5, and so is not isomorphic to Example 4, every subalgebra of which must be a Boolean algebra of finite sets and their comp... | Power set algebra | 0.811723 |
19,115 | Hence Example 5 is isomorphic to a proper subalgebra of itself! Example 6, and hence Example 5, constitutes the free Boolean algebra on countably many generators, meaning the Boolean algebra of all finitary operations on a countably infinite set of generators or variables. Example 7. | Power set algebra | 0.811723 |
19,116 | For example, the truth table of x0 in the table of binary operations, namely 2f10, has period 2 (and so can be recognized as using only the first variable) even though 12 of the binary operations have period 4. When the period is 2n the operation only depends on the first n variables, the sense in which the operation i... | Power set algebra | 0.811723 |
19,117 | Periodic sequence with period a power of two. This is a proper subalgebra of Example 5 (a proper subalgebra equals the intersection of itself with its algebra). These can be understood as the finitary operations, with the first period of such a sequence giving the truth table of the operation it represents. | Power set algebra | 0.811723 |
19,118 | Example 5 resembles Example 4 in being countable, but differs in being atomless. The latter is because the conjunction of any nonzero periodic sequence x with a sequence of coprime period (greater than 1) is neither 0 nor x. It can be shown that all countably infinite atomless Boolean algebras are isomorphic, that is, ... | Power set algebra | 0.811723 |
19,119 | The period of a periodic sequence is its least witness. Negation leaves period unchanged, while the disjunction of two periodic sequences is periodic, with period at most the least common multiple of the periods of the two arguments (the period can be as small as 1, as happens with the union of any sequence and its com... | Power set algebra | 0.811723 |
19,120 | There are only countably many integers, so this infinite Boolean algebra is countable. The atoms are the powers of two, namely 1,2,4,.... Another way of describing this algebra is as the set of all finite and cofinite sets of natural numbers, with the ultimately all-ones sequences corresponding to the cofinite sets, th... | Power set algebra | 0.811723 |
19,121 | Any Boolean combination of ultimately constant sequences is ultimately constant; hence these form a Boolean algebra. We can identify these with the integers by viewing the ultimately-zero sequences as nonnegative binary numerals (bit 0 of the sequence being the low-order bit) and the ultimately-one sequences as negativ... | Power set algebra | 0.811723 |
19,122 | It is convenient when talking about a set X of natural numbers to view it as a sequence x0,x1,x2,... of bits, with xi = 1 if and only if i ∈ X. This viewpoint will make it easier to talk about subalgebras of the power set algebra 2N, which this viewpoint makes the Boolean algebra of all sequences of bits. It also fits ... | Power set algebra | 0.811723 |
19,123 | A nontrivial property is one shared by some but not all operations making up a basis. Post listed five nontrivial properties of operations, identifiable with the five Post's classes, each preserved by composition, and showed that a set of operations formed a basis if, for each property, the set contained an operation l... | Power set algebra | 0.811722 |
19,124 | Both of these bases allow Boolean algebras to be defined via a subset of the equational properties of the Boolean operations. For the lattice basis, it suffices to define a Boolean algebra as a distributive lattice satisfying x∧¬x = 0 and x∨¬x = 1, called a complemented distributive lattice. The ring basis turns a Bool... | Power set algebra | 0.811722 |
19,125 | The ring basis emerged in the 20th century with the work of Zhegalkin and Stone and became the basis of choice for algebraists coming to the subject from a background in abstract algebra. Most treatments of Boolean algebra assume the lattice basis, a notable exception being Halmos whose linear algebra background eviden... | Power set algebra | 0.811722 |
19,126 | Three bases for Boolean algebra are in common use, the lattice basis, the ring basis, and the Sheffer stroke or NAND basis. These bases impart respectively a logical, an arithmetical, and a parsimonious character to the subject. The lattice basis originated in the 19th century with the work of Boole, Peirce, and others... | Power set algebra | 0.811722 |
19,127 | The operations need not be all explicitly stated. A basis is any set from which the remaining operations can be obtained by composition. A "Boolean algebra" may be defined from any of several different bases. | Power set algebra | 0.811722 |
19,128 | This works equally well for bit vectors of finite and infinite length, the only rule being that the bit positions all be indexed by the same set in order that "corresponding position" be well defined. The atoms of such an algebra are the bit vectors containing exactly one 1. In general the atoms of a Boolean algebra ar... | Power set algebra | 0.811722 |
19,129 | Example 2. All bit vectors of a given length form a Boolean algebra "pointwise", meaning that any n-ary Boolean operation can be applied to n bit vectors one bit position at a time. For example, the ternary OR of three bit vectors each of length 4 is the bit vector of length 4 formed by oring the three bits in each of ... | Power set algebra | 0.811722 |
19,130 | Another infinitary class intermediate between Boolean algebras and complete Boolean algebras is the notion of a sigma-algebra. This is defined analogously to complete Boolean algebras, but with sups and infs limited to countable arity. That is, a sigma-algebra is a Boolean algebra with all countable sups and infs. Beca... | Power set algebra | 0.811722 |
19,131 | This resembles Example 7 in having both atoms and an atomless subalgebra, but differs in having only finitely many atoms. Example 8 is in fact an infinite family of examples, one for each possible finite number of atoms. These examples by no means exhaust the possible Boolean algebras, even the countable ones. Indeed, ... | Power set algebra | 0.811722 |
19,132 | Ultimately periodic sequences, sequences that become periodic after an initial finite bout of lawlessness. They constitute a proper extension of Example 5 (meaning that Example 5 is a proper subalgebra of Example 7) and also of Example 4, since constant sequences are periodic with period one. Sequences may vary as to w... | Power set algebra | 0.811722 |
19,133 | The algebra of binary digits or bits 0 and 1 under the logical operations including disjunction, conjunction, and negation. Applications include the propositional calculus and the theory of digital circuits. The algebra of sets under the set operations including union, intersection, and complement. | Power set algebra | 0.811722 |
19,134 | Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions.Boolean algebra is a mathematically rich branch of abstract algebra. Stanford Encyclopaedia of Philosophy defines Boolean algebra as 'the algebra of two-valued logic with only sentential... | Power set algebra | 0.811722 |
19,135 | More recent applications include using forensic genetic identification to identify species in cases of poaching. Wildlife DNA registers are used to regulate trade of protected species, species laundering, and poaching. Conservation genetics techniques can be used alongside a variety of scientific disciplines. For examp... | Conservation genetics | 0.811719 |
19,136 | If the allele at A promotes fitness "one point", while the allele at B reduces fitness "one point", but the two genes are inherited together, then selection cannot favour the allele at A while penalising the allele at B – the fitness balance is "zero points". Recombination can swap out alternative alleles at A and B, a... | Conservation genetics | 0.811719 |
19,137 | Genetic diversity is the total number of genetic characteristics in a species. It can be measured in several ways: observed heterozygosity, expected heterozygosity, the mean number of alleles per locus, or the percentage of polymorphic loci. Genetic diversity on the population level is a crucial focus for conservation ... | Conservation genetics | 0.811719 |
19,138 | Conservation genetics is an interdisciplinary subfield of population genetics that aims to understand the dynamics of genes in a population for the purpose of natural resource management and extinction prevention. Researchers involved in conservation genetics come from a variety of fields including population genetics,... | Conservation genetics | 0.811719 |
19,139 | Museums are used because there is a wide range of species that are available to scientists all over the world. The problem with museums is that, historical perspectives are important because understanding how species reacted to changes in conditions in the past is a key to understanding reactions of similar species in ... | Conservation genetics | 0.811719 |
19,140 | Conservation issues related to the introduction of genetically modified organisms. The interaction between environmental contaminants and the biology and health of an organism, including changes in mutation rates and adaptation to local changes in the environment (e.g. industrial melanism). New techniques for noninvasi... | Conservation genetics | 0.811719 |
19,141 | Population genetic structure of natural and managed populations, including identification of Evolutionary Significant Units (ESUs) and management units for conservation. Assessing genetic variation within a species or population, including small or endangered populations, and estimates such as effective population size... | Conservation genetics | 0.811719 |
19,142 | New technology in conservation genetics has many implications for the future of conservation biology. At the molecular level, new technologies are advancing. Some of these techniques include the analysis of minisatellites and MHC. These molecular techniques have wider effects from clarifying taxonomic relationships, as... | Conservation genetics | 0.811719 |
19,143 | By rearranging terms we can see that D ¯ = 1 n ∑ i ( Y i 2 − Y i 1 ) = 1 n ∑ i Y i 2 − 1 n ∑ i Y i 1 = Y ¯ 2 − Y ¯ 1 , {\displaystyle {\bar {D}}={\frac {1}{n}}\sum _{i}(Y_{i2}-Y_{i1})={\frac {1}{n}}\sum _{i}Y_{i2}-{\frac {1}{n}}\sum _{i}Y_{i1}={\bar {Y}}_{2}-{\bar {Y}}_{1},} where n is the number of pairs. Thus the mea... | Paired difference test | 0.811718 |
19,144 | In statistics, a paired difference test is a type of location test that is used when comparing two sets of paired measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to i... | Paired difference test | 0.811718 |
19,145 | In E C T S {\displaystyle {\mathsf {ECTS}}} plus finite exponentiation, the recursion principle is a theorem. Moreover, enumerable forms of the pigeon hole principle can now also be proven, e.g. that on a finitely indexed set, every auto-injection is also a surjection. At last, the finitely indexed discrete sets are ju... | Constructive set theory | 0.811717 |
19,146 | W K L {\displaystyle {\mathrm {WKL} }} is also equivalent to the Brouwer fixed point theorem and other theorems regarding values of continuous functions on the reals. The fixed point theorem in turn implies the intermediate value theorem, but always be aware that these claims may depend on the formulation, as the class... | Constructive set theory | 0.811717 |
19,147 | Going beyond E C S T {\displaystyle {\mathsf {ECST}}} with regards to arithmetic, the axiom granting definition of set functions via iteration-step set functions must be added: For any set y {\displaystyle y} , set z ∈ y {\displaystyle z\in y} and f: y → y {\displaystyle f\colon y\to y} , there must also exist a functi... | Constructive set theory | 0.811717 |
19,148 | In turn the theorem formulations regarding finite objects tends to not differ from their classical counterparts. Given a model of all natural numbers, the equivalent for predicates, namely Markov's principle, does not automatically hold, but may be considered as an additional principle. In an inhabited domain and using... | Constructive set theory | 0.811717 |
19,149 | Here the double-negation captures that the disjunction statement now provenly can never be ruled out or rejected, even in cases where the disjunction may not be provable (for example, by demonstrating one of the disjuncts, thus deciding P {\displaystyle P} ) from the assumed axioms. More generally, constructive mathema... | Constructive set theory | 0.811717 |
19,150 | Adopting the standard definition of set equality via extensionality, the full Axiom of Choice is such a non-constructive principle that implies P E M {\displaystyle {\mathrm {PEM} }} for the formulas permitted in one's adopted Separation schema, by Diaconescu's theorem. Similar results hold for the Axiom of Regularity ... | Constructive set theory | 0.811717 |
19,151 | As in the classical theory, Dedekind cuts are characterized using subsets of algebraic structures such as Q {\displaystyle {\mathbb {Q} }}: The properties of being inhabited, numerically bounded above, "closed downwards" and "open upwards" are all bounded formulas with respect to the given set underlying the algebraic ... | Constructive set theory | 0.811717 |
19,152 | For example, it is independent whether all such x {\displaystyle x} have 2 | ω | ≤ | x | {\displaystyle 2^{|\omega |}\leq |x|} , nor does it prove that 2 | ω | < 2 | x | {\displaystyle 2^{|\omega |}<2^{|x|}} , see continuum hypothesis and the related Easton's theorem. Full Separation is equivalent to just assuming that... | Constructive set theory | 0.811717 |
19,153 | Consider three different undecidable proposition, none of which provenly imply another. They can ben used to define three subclasses of the singleton 1 {\displaystyle 1} , none of which are provenly the same. In this view, the powerclass P 1 {\displaystyle {\mathcal {P}}_{1}} of the singleton, usually denoted by Ω {\di... | Constructive set theory | 0.811717 |
19,154 | All statements validated in the subcountable model of the set theory can be proven exactly via C Z F {\displaystyle {\mathsf {CZF}}} plus the choice principle Π Σ {\displaystyle \Pi \Sigma } - A C {\displaystyle \mathrm {AC} } , stated further above. As noted, theories like C Z F {\displaystyle {\mathsf {CZF}}} , and a... | Constructive set theory | 0.811717 |
19,155 | The latter is a formulation of choice. Barr's theorem states that any topos admits a surjection from a Boolean topos onto it, relating to classical statements being provable intuitionistically. In type theory, the expression " x → y {\displaystyle x\to y} " exists on its own and denotes function spaces, a primitive not... | Constructive set theory | 0.811717 |
19,156 | Every such topos that interprets Z F {\displaystyle {\mathsf {ZF}}} is of course a model of these weaker theories, but locally Cartesian closed pretoposes have been defined that e.g. interpret theories with Exponentiation but reject full Separation and Powerset. A form of P E M {\displaystyle {\mathrm {PEM} }} correspo... | Constructive set theory | 0.811717 |
19,157 | So on another end, on the Brouwerian intuitionist side ( I N T {\displaystyle {\mathsf {INT}}} ), there are bar induction, the decidable fan theorem F A N Δ {\displaystyle {\mathrm {FAN} }_{\Delta }} saying decidable bars are uniform, which are amongst the weakest often discussed principles, Kripke's schema (with count... | Constructive set theory | 0.811717 |
19,158 | Relatedly, Aczel's theory constructive Zermelo-Fraenkel C Z F {\displaystyle {\mathsf {CZF}}} has been interpreted in a Martin-Löf type theories, as sketched in the section on C Z F {\displaystyle {\mathsf {CZF}}} . In this way, theorems provable in this and weaker set theories are candidates for a computer realization... | Constructive set theory | 0.811717 |
19,159 | L∞(R) is the ||·||∞-closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the spectral theorem for self-adjoint operators. The projections of a finite factor form a continuous geometry. | Von Neumann group algebra | 0.811716 |
19,160 | There is also a natural partial order on the set of equivalence classes of projections, induced by the partial order ≤ of projections. If M is a factor, ≤ is a total order on equivalence classes of projections, described in the section on traces below. A projection (or subspace belonging to M) E is said to be a finite ... | Von Neumann group algebra | 0.811716 |
19,161 | If E1 ⊥ E2 and F1 ⊥ F2, then E1 + E2 ~ F1 + F2. Additivity would not generally hold if one were to require unitary equivalence in the definition of ~, i.e. if we say E is equivalent to F if u*Eu = F for some unitary u. The Schröder–Bernstein theorems for operator algebras gives a sufficient condition for Murray-von Neu... | Von Neumann group algebra | 0.811716 |
19,162 | The basic theory of projections was worked out by Murray & von Neumann (1936). Two subspaces belonging to M are called (Murray–von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if M "knows" that the subspaces... | Von Neumann group algebra | 0.811716 |
19,163 | The tensor product of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert spaces.By forgetting about the topology on a von Neumann algebra, we can consider i... | Von Neumann group algebra | 0.811716 |
19,164 | A von Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology. The von Neumann algebra generated by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators. | Von Neumann group algebra | 0.811716 |
19,165 | Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject. A factor is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators. A finite von Neumann algebra is one which is the direct integral of finite fa... | Von Neumann group algebra | 0.811716 |
19,166 | Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other. Nevertheless, Voiculescu has shown that the class of non-amenable factors coming from the group-measure sp... | Von Neumann group algebra | 0.811716 |
19,167 | For example, the Nicholson–Bailey model for a host-parasite interaction is given by N t + 1 = λ N t e − a P t {\displaystyle N_{t+1}=\lambda N_{t}e^{-aP_{t}}} P t + 1 = N t ( 1 − e − a P t ) , {\displaystyle P_{t+1}=N_{t}(1-e^{-aP_{t}}),} with N t {\displaystyle N_{t}} representing the hosts, and P t {\displaystyle P_{... | Difference equation | 0.811716 |
19,168 | Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a s... | Barycentric-sum problem | 0.811714 |
19,169 | It has been introduced by Ordaz, inspired in a theorem of Hamidoune: every sequence of length n + k − 1 {\displaystyle n+k-1} in Z n {\displaystyle Z_{n}} contains a k-barycentric sequence. Notice that a k-barycentric sequence in Z n {\displaystyle Z_{n}} , with k a multiple of n, is a sequence with zero-sum. The zero-... | Barycentric-sum problem | 0.811714 |
19,170 | In 2007, the company acquired a controlling interest in India-based Matrix Laboratories Limited, a top producer of active pharmaceutical ingredients (APIs) for generic drugs, and the generics business of Germany-based Merck KGaA. Through these acquisitions, the company grew from the third-largest generic and pharmaceut... | Mylan | 0.811704 |
19,171 | In non-parametric statistics, the Theil–Sen estimator is a method for robustly fitting a line to sample points in the plane (simple linear regression) by choosing the median of the slopes of all lines through pairs of points. It has also been called Sen's slope estimator, slope selection, the single median method, the ... | Median slope | 0.811704 |
19,172 | For many RNA molecules, the secondary structure is highly important to the correct function of the RNA — often more so than the actual sequence. This fact aids in the analysis of non-coding RNA sometimes termed "RNA genes". One application of bioinformatics uses predicted RNA secondary structures in searching a genome ... | RNA secondary structure | 0.811702 |
19,173 | In the thermoelectric literature, the term band structure engineering refers to the manipulation of material structure and chemistry to achieve a high density of states near the Fermi level. More specifically, thermoelectric materials are intentionally doped to exhibit only partially filled bands at the Fermi level, re... | Electronic entropy | 0.8117 |
19,174 | The regulator, a calculation of volume in 'logarithmic space' as divided by the logarithms of the units of the cyclotomic field, can be set against the quantities from the L(1) recognisable as logarithms of cyclotomic units. There result formulae stating that the class number is determined by the index of the cyclotomi... | Class number formula | 0.811699 |
19,175 | This is the case of the above, with Gal(K/Q) an abelian group, in which all the ρ can be replaced by Dirichlet characters (via class field theory) for some modulus f called the conductor. Therefore all the L(1) values occur for Dirichlet L-functions, for which there is a classical formula, involving logarithms. By the ... | Class number formula | 0.811699 |
19,176 | In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. | Class number formula | 0.811699 |
19,177 | Thermodynamics describes the bulk behavior of the body, not the microscopic behaviors of the very large numbers of its microscopic constituents, such as molecules. The behavior of these microscopic particles is described by, and the laws of thermodynamics are derived from, statistical mechanics. The study of thermodyna... | Material Engineering | 0.811699 |
19,178 | Materials science is a highly active area of research. Together with materials science departments, physics, chemistry, and many engineering departments are involved in materials research. Materials research covers a broad range of topics; the following non-exhaustive list highlights a few important research areas. | Material Engineering | 0.811699 |
19,179 | In mathematics and optimization, a pseudo-Boolean function is a function of the form f: B n → R , {\displaystyle f:\mathbf {B} ^{n}\to \mathbb {R} ,} where B = {0, 1} is a Boolean domain and n is a nonnegative integer called the arity of the function. A Boolean function is then a special case, where the values are also... | Pseudo-Boolean function | 0.811698 |
19,180 | Genomics is the field of genetics concerned with structural and functional studies of the genome. A genome is all the DNA contained within an organism or a cell including nuclear and mitochondrial DNA. The human genome is the total collection of genes in a human being contained in the human chromosome, composed of over... | Human geneticist | 0.811697 |
19,181 | Population genetics is the branch of evolutionary biology responsible for investigating processes that cause changes in allele and genotype frequencies in populations based upon Mendelian inheritance. Four different forces can influence the frequencies: natural selection, mutation, gene flow (migration), and genetic dr... | Human geneticist | 0.811697 |
19,182 | Medical genetics is the branch of medicine that involves the diagnosis and management of hereditary disorders. Medical genetics is the application of genetics to medical care. It overlaps human genetics, for example, research on the causes and inheritance of genetic disorders would be considered within both human genet... | Human geneticist | 0.811697 |
19,183 | Quantum materials is an umbrella term in condensed matter physics that encompasses all materials whose essential properties cannot be described in terms of semiclassical particles and low-level quantum mechanics. These are materials that present strong electronic correlations or some type of electronic order, such as s... | Quantum materials | 0.811693 |
19,184 | Eagle was a 184-node IBM RS/6000 SP operated by the Computer Science and Mathematics Division of ORNL. It had 176 Winterhawk-II “thin” nodes, each with four 375 MHz Power3-II processors and 2GB of memory. Eagle also had eight Winterhawk-II “wide” nodes - each with two 375 MHz Power3-II processors and 2 GB of memory—for... | National Center for Computational Sciences | 0.811693 |
19,185 | Wombat is a single-rack cluster from HPE based on the 64-bit ARM architecture instead of traditional x86-based architecture. This system is available to support computer science research projects aimed at exploring the ARM architecture. The Wombat cluster has 16 compute nodes, four of which have two AMD GPU accelerator... | National Center for Computational Sciences | 0.811693 |
19,186 | Each blade contained 4 nodes. Every node had 2 sockets with 8 physical cores each. Intel’s HyperThreading (HT) Technology allowed each physical core to work as 2 logical cores so each node could function as if it had 32 cores. In total, the Eos compute partition contained 11,776 traditional processor cores (23,552 logi... | National Center for Computational Sciences | 0.811693 |
19,187 | Nuclear Energy – The development of new nuclear reactors that employ advanced fuel cycles and adhere to modern safety and nonproliferation constraints requires complex modelling and simulations. Often, the complexity of these simulations necessitates the use of supercomputers that can ensure accuracy of models. Physics... | National Center for Computational Sciences | 0.811693 |
19,188 | Engineering – OLCF resources like Summit are being used for engineering applications such as simulations of gas turbines and combustion engines. Fusion – Understanding the behavior of fusion plasmas and simulating various device aspects gives researchers insight into the construction of ITER, a prototype fusion power p... | National Center for Computational Sciences | 0.811693 |
19,189 | Biology – With OLCF supercomputing resources, researchers can use knowledge of the molecular scale to develop new drugs and medical therapies, study complex biological systems, and model gene regulation. Chemistry – Supercomputers like Summit can explore the intricacies of matter at the atomic level, allowing for first... | National Center for Computational Sciences | 0.811693 |
19,190 | The National Center for Computational Sciences (NCCS) is a United States Department of Energy (DOE) Leadership Computing Facility that houses the Oak Ridge Leadership Computing Facility (OLCF), a DOE Office of Science User Facility charged with helping researchers solve challenging scientific problems of global interes... | National Center for Computational Sciences | 0.811693 |
19,191 | Phoenix was a Cray X1E provided as a primary system in NCCS. The original X1 was installed in 2003 and went through several upgrades, arriving at its final configuration in 2005. From October 2005 until 2008, it provided almost 17 million processor-hours. The system supported more than 40 large projects in research are... | National Center for Computational Sciences | 0.811693 |
19,192 | Oak Ridge National Laboratory (ORNL) joined with three other national laboratories and seven universities to submit the Partnership in Computational Science (PICS) proposal to the US Department of Energy as part of the High-Performance Computing and Communications Initiative.With the High-End Computing Revitalization A... | National Center for Computational Sciences | 0.811693 |
19,193 | For example, let τ = 2i, then λ(2i) = (−1 + √2)4 which implies g′2, g′3, and therefore g′23 − 27g′32 of the formula above are all algebraic numbers if τ involves an imaginary quadratic field. In fact, it yields the integer j(2i) = 663 = 287496. In contrast, the modular discriminant Δ ( τ ) = g 2 ( τ ) 3 − 27 g 3 ( τ ) ... | Elliptic Curve | 0.811692 |
19,194 | Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q. In particular, the group E(K) of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due... | Elliptic Curve | 0.811692 |
19,195 | In characteristic 2, even this much is not possible, and the most general equation is y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 {\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}} provided that the variety it defines is non-singular. If characteristic were not an obstruction, each equation woul... | Elliptic Curve | 0.811692 |
19,196 | However, all real polynomials factorize completely into linear factors over the complex numbers, since the field of complex numbers is the algebraic closure of the reals. So, the elliptic curve may be written as y 2 = x ( x − 1 ) ( x − λ ) {\displaystyle y^{2}=x(x-1)(x-\lambda )} One finds that g 2 ′ = 4 3 3 ( λ 2 − λ ... | Elliptic Curve | 0.811692 |
19,197 | Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K. If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form y... | Elliptic Curve | 0.811692 |
19,198 | The definition of elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps, self-intersections, or isolated points. Algebraically, this holds if and only if the discriminant, Δ {\displaystyle \Delta } , is not equal to zero. | Elliptic Curve | 0.811692 |
19,199 | Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry. In this context, an elliptic curve is a plane curve defined by an equation of the form y 2 =... | Elliptic Curve | 0.811692 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.