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30,500 | To determine the mass of the object, a combination of reference masses was hung on one end of the beam while the object of unknown mass was hung on the other end (see balance and steelyard balance). For high precision work, such as empirical chemistry, the center beam balance is still one of the most accurate technolog... | Beam balance | 0.80494 |
30,501 | The journal covers all areas of biology and is considered the premier journal in the country within its scope. It publishes research and reviews, series articles, perspectives, clipboards, commentaries, and short communications. Special theme issues are also published. The details of the editorial board is available at... | Journal of Biosciences | 0.804938 |
30,502 | The Journal of Biosciences was established in 1934 as the Proceedings of the Indian Academy of Sciences (Section B). In 1978, the Proceedings were split into three sections: Proceedings-Animal Sciences, Proceedings-Plant Sciences, and Proceedings-Experimental Biology. The latter section was renamed Journal of Bioscienc... | Journal of Biosciences | 0.804938 |
30,503 | In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is polynomial in the input size.... | Ellipsoidal algorithm | 0.804937 |
30,504 | The new mirror was the first of its size to be cast and polished in the Steward Observatory Mirror Laboratory. The updated MMT, its name no longer an acronym, was rededicated on 20 May 2000.In late 2002, a novel deformable secondary mirror was added to the telescope. While other adaptive optics designs do their correct... | MMT Observatory | 0.804937 |
30,505 | The existence of a unique solution is very plausible by the "physical argument": any charge distribution on the boundary should, by the laws of electrostatics, determine an electrical potential as solution. However, Karl Weierstrass found a flaw in Riemann's argument, and a rigorous proof of existence was found only in... | Dirichlet Problem | 0.804937 |
30,506 | Riffle shuffles may be used to define the shuffle algebra. This is a Hopf algebra where the basis is a set of words, and the product is the shuffle product denoted by the sha symbol ш, the sum of all riffle shuffles of two words. In exterior algebra, the wedge product of a p {\displaystyle p} -form and a q {\displaysty... | Riffle shuffle permutation | 0.804936 |
30,507 | 550 — Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system. 628 — Brahmagupta writes the Brahma-sphuta-siddhanta, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed. It also gives rules for manipulating both nega... | Timeline of numerals and arithmetic | 0.804934 |
30,508 | He also wrote the Thesis on the sine and the chord and Thesis on finding the first degree sine. 15th century — Ibn al-Banna and al-Qalasadi introduced symbolic notation for algebra and for mathematics in general. 1427 — Al-Kashi completes The Key to Arithmetic containing work of great depth on decimal fractions. It app... | Timeline of numerals and arithmetic | 0.804934 |
30,509 | c. 1400 — Ghiyath al-Kashi “contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as pi. His contribution to decimal fractions is so major that for many years he was considered as their inventor. | Timeline of numerals and arithmetic | 0.804934 |
30,510 | See 0 (number). 150 BC — Jain mathematicians in India write the “Sthananga Sutra”, which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. 50 BC — Indian numerals, the first posit... | Timeline of numerals and arithmetic | 0.804934 |
30,511 | Advanced Chemistry Development (ACD/labs) is a chemoinformatics company which produces software for use in handling NMR data and predicting NMR spectra. ACD/Labs offers the Aldrich library as an add-on to their general spectrum processing software and specialized NMR software products. The NMR predictors allow improvin... | NMR spectra database | 0.804933 |
30,512 | A database that was developed and maintained by the publisher John Wiley & Sons. This database included more than 700,000 NMR, IR and MS Spectra, statistics specific to the NMR spectra are not listed. The NMR data includes 1H,13C, 11B, 15N, 17O, 19F, 29Si, and 31P. | NMR spectra database | 0.804933 |
30,513 | The Biological Magnetic Resonance Data Bank (BioMagResBank or BMRB) is sponsored by the Department of Biochemistry at the University of Wisconsin–Madison; it is dedicated to Proteins, Peptides, Nucleic Acids, and other Biomolecules. It stores a large variety of raw NMR data. | NMR spectra database | 0.804933 |
30,514 | For the purposes of the propositional calculus, propositions (utterances, sentences, assertions) are considered to be either simple or compound. Compound propositions are considered to be linked by sentential connectives, some of the most common of which are "AND", "OR", "IF ... THEN ...", "NEITHER ... NOR ...", "... I... | Propositional formula | 0.804933 |
30,515 | Example: The map method usually is done by inspection. The following example expands the algebraic method to show the "trick" behind the combining of terms on a Karnaugh map: Minterms #3 and #7 abut, #7 and #6 abut, and #4 and #6 abut (because the table's edges wrap around). So each of these pairs can be reduced.Observ... | Propositional formula | 0.804933 |
30,516 | the result of such a calculus will be another formula (i.e. a well-formed symbol string). Eventually, however, if one wants to use the calculus to study notions of validity and truth, one must add axioms that define the behavior of the symbols called "the truth values" {T, F} ( or {1, 0}, etc.) relative to the other sy... | Propositional formula | 0.804932 |
30,517 | For more about the philosophy of the notion of IDENTITY see Leibniz's law. As noted above, Tarski considers IDENTITY to lie outside the propositional calculus, but he asserts that without the notion, "logic" is insufficient for mathematics and the deductive sciences. In fact the sign comes into the propositional calcul... | Propositional formula | 0.804932 |
30,518 | Each must have at least a subject (an immediate object of thought or observation), a verb (in the active voice and present tense preferred), and perhaps an adjective or adverb. "Dog!" probably implies "I see a dog" but should be rejected as too ambiguous. Example: "That purple dog is running", "This cow is blue", "Swit... | Propositional formula | 0.804932 |
30,519 | Example: The proposition " IF 'Winston Churchill was Chinese' THEN 'The sun rises in the east' " evaluates as a TRUTH given that 'Winston Churchill was Chinese' is a FALSEHOOD and 'The sun rises in the east' evaluates as a TRUTH.In recognition of this problem, the sign → of formal implication in the propositional calcu... | Propositional formula | 0.804932 |
30,520 | By the notion, 1 Σ {\displaystyle 1_{\Sigma }} is the testability function for the specification Σ {\displaystyle \Sigma } . The existence should not merely be asserted, should be proven rigorously. Therefore, obviously without algebraic consistency, no such function can be found, and therefore, the specification cease... | Software testability | 0.804932 |
30,521 | In this sense a pseudosphere is just a tame sphere (as opposed to wild spheres). A pseudosphere arrangement in S d {\displaystyle S^{d}} is a collection of pseudospheres that intersect along pseudospheres. Finally, the Folkman Lawrence topological representation theorem states that every oriented matroid of rank d + 1 ... | Oriented matroid | 0.804932 |
30,522 | Not all oriented matroids are representable—that is, not all have realizations as point configurations, or, equivalently, hyperplane arrangements. However, in some sense, all oriented matroids come close to having realizations are hyperplane arrangements. In particular, the Folkman–Lawrence topological representation t... | Oriented matroid | 0.804932 |
30,523 | The distinction between matroids and oriented matroids is discussed further below. Matroids are often useful in areas such as dimension theory and algorithms. Because of an oriented matroid's inclusion of additional details about the oriented nature of a structure, its usefulness extends further into several areas incl... | Oriented matroid | 0.804932 |
30,524 | The original design of ODC was to create a measurement system for software engineering using the defect stream as a source of intrinsic measurements. Thus, the attributes, either singularly, or in conjunction with one of the others provides specific measurements on certain aspects of the engineering process. These meas... | Orthogonal Defect Classification | 0.804932 |
30,525 | In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned solely with the three-dimensional case. | Plane at infinity | 0.804927 |
30,526 | R M S D = 1 N ∑ i = 1 N δ i 2 {\displaystyle \mathrm {RMSD} ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\delta _{i}^{2}}}} where δi is the distance between atom i and either a reference structure or the mean position of the N equivalent atoms. This is often calculated for the backbone heavy atoms C, N, O, and Cα or sometimes... | Root mean square deviation (bioinformatics) | 0.804927 |
30,527 | The DLVO theory of colloidal stability has been extended to take into account the effect of shear flow in fluid dynamic systems, which is relevant for many applications e.g. microfluidics, chemical reactors, atmospheric and environmental flows, in the work of Alessio Zaccone and collaborators. In this extended DLVO the... | DLVO interaction | 0.804927 |
30,528 | Nanobiotix is currently in the clinical stage of developing NBTXR3, a “radioenhancer” that uses nanotechnology to make cancer cells more receptive to radiotherapy. NBTXR3 is an aqueous suspension of crystalline metallic nanoparticles approximately 50 nanometers in diameter, designed for injection directly into a malign... | Nanobiotix | 0.804926 |
30,529 | Nanobiotix is a biotechnology company that uses nanomedicine to develop new radiotherapy techniques for cancer patients. The company is headquartered in Paris, with additional corporate offices in New York and Massachusetts. | Nanobiotix | 0.804926 |
30,530 | The RW technique has found extensive use in manipulating antiparticles in Penning–Malmberg traps. One important application is the creation of specially tailored antiparticle beams for atomic physics experiments. Frequently one would like a beam with a large current density. In this case, one compresses the plasma with... | Rotating wall technique | 0.804926 |
30,531 | Single-component plasmas (SCP), which are a type of nonneutral plasma, have many uses, including studying a variety of plasma physics phenomena and for the accumulation, storage and delivery of antiparticles. Applications include the creation and study of antihydrogen, beams to study the interaction of positrons with o... | Rotating wall technique | 0.804926 |
30,532 | One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a on... | Slow-motion approximation | 0.804921 |
30,533 | A polymerase is an enzyme (EC 2.7.7.6/7/19/48/49) that synthesizes long chains of polymers or nucleic acids. DNA polymerase and RNA polymerase are used to assemble DNA and RNA molecules, respectively, by copying a DNA template strand using base-pairing interactions or RNA by half ladder replication. A DNA polymerase fr... | DNA-directed RNA polymerase | 0.804919 |
30,534 | Tetrahedron Letters is a weekly international journal for rapid publication of full original research papers in the field of organic chemistry. According to the Journal Citation Reports, the journal has a 2020 impact factor of 2.415. | Tetrahedron Letters | 0.804916 |
30,535 | Some mass spectrometrists use the definition that is similar to definitions used in some other fields of physics and chemistry. In this case, resolving power is defined as: R = M Δ M = r e s o l v i n g p o w e r {\displaystyle R={\cfrac {M}{\Delta M}}=\mathrm {resolving\ power} } The minimum peak separation ΔM which a... | Resolution (mass spectrometry) | 0.804916 |
30,536 | A Greek mathematician Eudoxus provided a definition for the meaning of the equality between two ratios. This definition of proportion forms the subject of Euclid's Book V, where we can read: Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whate... | Proportion (mathematics) | 0.804913 |
30,537 | A study in 2014 found that men are favored in some domains, such as tenure rates in biology, but that the majority of domains were gender-fair. The authors interpreted this to suggest that the underrepresentation of women in the professorial ranks was not solely caused by sexist hiring, promotion, and remuneration.Aude... | Women in STEM fields | 0.804912 |
30,538 | Out of STEM-intending students, 35% of women stated that their reason for leaving calculus was due to lack of understanding the material, while only 14% of men stated the same. The study reports that this difference in reason for leaving calculus is thought to develop from women's low level of confidence in their abili... | Women in STEM fields | 0.804912 |
30,539 | Although women have higher publications in Botany, men still out publish women and are often the ones cited in research papers and studies relating to the sciences. The study concluded that according to the data (shown in the table above), women in Chile that are enrolled in STEM have higher enrollment in the sciences ... | Women in STEM fields | 0.804912 |
30,540 | Nevertheless, there are many examples of women's contributions to engineering.Initially a "computer" was a person doing computations, who was often a woman. Working as a computer required conscientiousness, accuracy and speed. Some women who initially worked as human computers later advanced from doing the simpler calc... | Women in STEM fields | 0.804912 |
30,541 | There has also been an effort among historians of science to uncover under-reported contributions of women.The term STEM was first used in 2001, primarily in connection with choice of education and career. Different STEM fields have different histories, but women's participation, although limited, has been seen through... | Women in STEM fields | 0.804912 |
30,542 | UNESCO, among other agencies including the European Commission and The Association of Academies and Societies of Sciences in Asia (AASSA), have been outspoken about the underrepresentation of women in STEM fields globally.Despite their efforts to compile and interpret comparative statistics, it is necessary to exercise... | Women in STEM fields | 0.804912 |
30,543 | UNESCO's A Complex Formula states that Indonesia's government has been working towards gender equality, especially through the Ministry of Education and Culture, but stereotypes about women's roles in the workplace persist. Due to traditional views and societal norms, women struggle to remain in their careers or to mov... | Women in STEM fields | 0.804912 |
30,544 | Current campaigns to increase women's participation within STEM fields include the UK's GlamSci, and Verizon's #InspireHerMind project. The US Office of Science and Technology Policy during the Obama administration collaborated with the White House Council on Women and Girls to increase the participation of women and g... | Women in STEM fields | 0.804912 |
30,545 | Organizations such as Girls Who Code, StemBox, and Stanford's Women in Data Science Initiative aim to encourage women and girls to explore male-dominated STEM fields. Many of these organizations offer summer programs and scholarships to girls interested in STEM fields. The U.S. | Women in STEM fields | 0.804912 |
30,546 | Among more experienced scientists and engineers, the gender gap in salaries is greater than for recent graduates. Salaries are highest in mathematics, computer science, and engineering, which are fields in which women are not highly represented. In Australia, a study conducted by the Australian Bureau of Statistics has... | Women in STEM fields | 0.804912 |
30,547 | As at 2004, 13.9% of students enrolled in science programs in Cambodia were female and 21% of researchers in science, technology, and innovation fields were female as of 2002. These statistics are significantly lower than those of other Asian countries such as Malaysia, Mongolia, and South Korea. According to a UNESCO ... | Women in STEM fields | 0.804912 |
30,548 | Women dominate the total number of persons with bachelor's degrees, as well as those in STEM fields defined by the National Center for Education Statistics. However, they are underrepresented in specific fields including Computer Sciences, Engineering, and Mathematics. Along with women, racial/ethnic minorities in the ... | Women in STEM fields | 0.804912 |
30,549 | According to UNESCO's data from 2012 and 2018 respectively, 40.2% of students enrolled in science programs and 49% of researchers in science, technology, and innovation in Mongolia are female. Traditionally, nomadic Mongol culture was fairly egalitarian, with both women and men raising children, tending livestock, and ... | Women in STEM fields | 0.804912 |
30,550 | According to UNESCO statistics, 30% of the Sub-Saharan tech workforce are women; this share rose to 33.5 percent in 2018. South Africa features among the top 20 countries in the world for the share of professionals with skills in artificial intelligence and machine learning, with women representing 28 percent of these ... | Women in STEM fields | 0.804912 |
30,551 | Abbiss states that "the ubiquity of computers in everyday life has seen the breaking down of gender distinctions in preferences for and the use of different applications, particularly in the use of the internet and email." Both genders have acquired skills, competencies and confidence in using a variety of technologica... | Women in STEM fields | 0.804912 |
30,552 | Additionally, 38% of female students who remained in STEM majors expressed concerns that there were other academic areas that might be a better fit for their interests. Preston's (2004) survey of 1,688 individuals who had left sciences also showed that 30 percent of the women endorsed "other fields more interesting" as... | Women in STEM fields | 0.804912 |
30,553 | A fact sheet published by UNESCO in March 2015 presented worldwide statistics of women in the STEM fields, with a focus on Asia and the Pacific region. It reports that, worldwide, 30 percent of researchers are women; as of 2018, this share had increased to 33 percent. In these areas, East Asia, the Pacific, South Asia ... | Women in STEM fields | 0.804912 |
30,554 | A Statistics Canada study from 2019 found that first-year women make up 44% of STEM students, compared with 64% of non-STEM students. Those women who transfer out of STEM courses usually go to a related field, such as health care or finance. A study conducted by the University of British Columbia discovered that only 2... | Women in STEM fields | 0.804912 |
30,555 | The differences in the intended majors between male and female first-time freshmen directly relate to the differences in the fields in which men and women earn their degree. At the post-secondary level, women are less likely than men to earn a degree in mathematics, physical sciences, or computer sciences and engineeri... | Women in STEM fields | 0.804912 |
30,556 | Feminist scholars postulate that boys are more likely to gain spatial skills outside the classroom because they are culturally and socially encouraged to build and work with their hands. Research shows that girls can develop these same skills with the same form of training.A 1996 USA study of college freshmen by the Hi... | Women in STEM fields | 0.804912 |
30,557 | There were three different situations, designed to test the impact of stereotype on performance in math. One group of women were informed that men had previously out-performed women on the same calculus test they were about to take. The next group was told men and women had performed at the same level. | Women in STEM fields | 0.804912 |
30,558 | Some studies propose the explanation that STEM fields (and especially fields like physics, math and philosophy) are considered by both teachers and students to require more innate talent than skills that can be learned. Combined with a tendency to view women as having less of the required innate abilities, researchers ... | Women in STEM fields | 0.804912 |
30,559 | Researchers have investigated the usefulness of self-affirmation in alleviating stereotype threat. One study found that women who affirmed a personal value prior to experiencing stereotype threat performed as well on a math test as men and as women who did not experience stereotype threat. A subsequent study found that... | Women in STEM fields | 0.804912 |
30,560 | Female participants who read about successful women, even though these successes were not directly related to performance in math, performed better on a subsequent math test than participants who read about successful corporations rather than successful women. A study investigating the role of textbook images on scienc... | Women in STEM fields | 0.804912 |
30,561 | In 2013 in the EU as an average men scientists and engineers made up 4.1% of total labour force, while women made up only 2.8%. In more than half of the countries women make up less than 45% of scientists and engineers. The situation has improved, as between 2008 and 2011 the number of women amongst employed scientists... | Women in STEM fields | 0.804912 |
30,562 | In terms of the most prestigious awards in STEM fields, fewer have been awarded to women than to men. Between 1901 and 2017 the female:total ratio of Nobel Prizes were 2:207 for physics, 4:178 for chemistry, 12:214 for physiology/medicine, and 1:79 for economic sciences. The ratios for other fields were 14:114 in liter... | Women in STEM fields | 0.804912 |
30,563 | A sphere that has the Cartesian equation x2 + y2 + z2 = c2 has the simple equation r = c in spherical coordinates. Two important partial differential equations that arise in many physical problems, Laplace's equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. The angular porti... | Angular coordinate | 0.804912 |
30,564 | Therefore, the test correctly identifies 80% of non-use for non-users, but also generates 20% false positives, or false positive rate (FPR) = 0.20, for non-users. Assuming 0.05 prevalence, meaning 5% of people use cannabis, what is the probability that a random person who tests positive is really a cannabis user? The P... | Bayes' Rule | 0.80491 |
30,565 | In part VI of "A Dynamical Theory of the Electromagnetic Field", subtitled "Electromagnetic theory of light", Maxwell uses the correction to Ampère's Circuital Law made in part III of his 1862 paper, "On Physical Lines of Force", which is defined as displacement current, to derive the electromagnetic wave equation. He ... | A Dynamical Theory of the Electromagnetic Field | 0.804909 |
30,566 | The Gaussian or normal probability distribution plays an important role in the relationship between variance and entropy: it is a problem of the calculus of variations to show that this distribution maximizes entropy for a given variance, and at the same time minimizes the variance for a given entropy. In fact, for any... | Hirschman uncertainty | 0.804908 |
30,567 | In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. This is stronger t... | Hirschman uncertainty | 0.804908 |
30,568 | Let R be a ring (associative, with 1, not necessarily commutative). A (right) pp-n-formula is a formula in the language of (right) R-modules of the form where ℓ , n , m {\displaystyle \ell ,n,m} are natural numbers, A {\displaystyle A} is an ( ℓ + n ) × m {\displaystyle (\ell +n)\times m} matrix with entries from R, an... | Ziegler spectrum | 0.804907 |
30,569 | {\displaystyle L=K/p(x)\cong \prod _{i=1}^{m}K/p_{i}(x).} We will find this decomposition without knowing the factorization. First, we write L explicitly as an algebra over Q {\displaystyle \mathbb {Q} }: we pick a random element α ∈ L {\displaystyle \alpha \in L} , which generates L {\displaystyle L} over Q {\displays... | Polynomial factorization | 0.804907 |
30,570 | Kronecker's classical method is interesting only from a historical point of view; modern algorithms proceed by a succession of: Square-free factorization Factorization over finite fieldsand reductions: From the multivariate case to the univariate case. From coefficients in a purely transcendental extension to the multi... | Polynomial factorization | 0.804907 |
30,571 | For example, the fundamental theorem of algebra, which states that every polynomial with complex coefficients has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms) into linear factors over the complex field C. Similarly, over the field of reals, the irredu... | Polynomial factorization | 0.804907 |
30,572 | But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficient... | Polynomial factorization | 0.804907 |
30,573 | In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer... | Polynomial factorization | 0.804907 |
30,574 | If f: X 0 → Y {\displaystyle f:X_{0}\to Y} is a smooth surjective submersion of smooth manifolds, then X 0 × Y X 0 ⊂ X 0 × X 0 {\displaystyle X_{0}\times _{Y}X_{0}\subset X_{0}\times X_{0}} is an equivalence relation since Y {\displaystyle Y} has a topology isomorphic to the quotient topology of X 0 {\displaystyle X_{0... | Groupoid | 0.804907 |
30,575 | Moreover, in this case, the fundamental groupoid and the fundamental groups are equivalent as categories (see the section below for the general theory). An important extension of this idea is to consider the fundamental groupoid π 1 ( X , A ) {\displaystyle \pi _{1}(X,A)} where A ⊂ X {\displaystyle A\subset X} is a cho... | Groupoid | 0.804907 |
30,576 | A groupoid is an algebraic structure ( G , ∗ ) {\displaystyle (G,\ast )} consisting of a non-empty set G {\displaystyle G} and a binary partial function ' ∗ {\displaystyle \ast } ' defined on G {\displaystyle G} . | Groupoid | 0.804907 |
30,577 | This generalized representation theorem can be expressed as a category-theoretic duality between distributive lattices and spectral spaces (sometimes called coherent spaces, but not the same as the coherent spaces in linear logic), topological spaces in which the compact open sets are closed under intersection and form... | Birkhoff's representation theorem | 0.804903 |
30,578 | This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations co... | Birkhoff's representation theorem | 0.804903 |
30,579 | Theorem. Any finite distributive lattice L is isomorphic to the lattice of lower sets of the partial order of the join-irreducible elements of L.That is, there is a one-to-one order-preserving correspondence between elements of L and lower sets of the partial order. The lower set corresponding to an element x of L is s... | Birkhoff's representation theorem | 0.804903 |
30,580 | In any partial order, the lower sets form a lattice in which the lattice's partial ordering is given by set inclusion, the join operation corresponds to set union, and the meet operation corresponds to set intersection, because unions and intersections preserve the property of being a lower set. Because set unions and ... | Birkhoff's representation theorem | 0.804903 |
30,581 | Such spaces are known as Priestley spaces. Further, certain bitopological spaces, namely pairwise Stone spaces, generalize Stone's original approach by utilizing two topologies on a set to represent an abstract distributive lattice. Thus, Birkhoff's representation theorem extends to the case of infinite (bounded) distr... | Birkhoff's representation theorem | 0.804903 |
30,582 | In an infinite distributive lattice, it may not be the case that the lower sets of the join-irreducible elements are in one-to-one correspondence with lattice elements. Indeed, there may be no join-irreducibles at all. This happens, for instance, in the lattice of all natural numbers, ordered with the reverse of the us... | Birkhoff's representation theorem | 0.804903 |
30,583 | There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but in a distributive lattice the two types of elements coincide. For, suppose that x is join-irreducible, and that x ≤ y ∨ z. This inequality is equivalent to the statement that x = x ∧ (y ∨ z), and by the dist... | Birkhoff's representation theorem | 0.804903 |
30,584 | The 16 sets that are associated with divisors of 120 are the lower sets of this smaller partial order, subsets of elements such that if x ≤ y and y belongs to the subset, then x must also belong to the subset. From any lower set L, one can recover the associated divisor by computing the least common multiple of the pri... | Birkhoff's representation theorem | 0.804903 |
30,585 | Birkhoff's theorem, as stated above, is a correspondence between individual partial orders and distributive lattices. However, it can also be extended to a correspondence between order-preserving functions of partial orders and bounded homomorphisms of the corresponding distributive lattices. The direction of these map... | Birkhoff's representation theorem | 0.804903 |
30,586 | Birkhoff's representation theorem may also be generalized to finite structures other than distributive lattices. In a distributive lattice, the self-dual median operation m ( x , y , z ) = ( x ∨ y ) ∧ ( x ∨ z ) ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ) {\displaystyle m(x,y,z)=(x\vee y)\wedge (x\vee z)\wedge (y\ve... | Birkhoff's representation theorem | 0.804903 |
30,587 | Another result analogous to Birkhoff's representation theorem, but applying to a broader class of lattices, is the theorem of Edelman (1980) that any finite join-distributive lattice may be represented as an antimatroid, a family of sets closed under unions but in which closure under intersections has been replaced by ... | Birkhoff's representation theorem | 0.804903 |
30,588 | Many lattices can be defined in such a way that the elements of the lattice are represented by sets, the join operation of the lattice is represented by set union, and the meet operation of the lattice is represented by set intersection. For instance, the Boolean lattice defined from the family of all subsets of a fini... | Birkhoff's representation theorem | 0.804903 |
30,589 | The School of Biological Sciences is one of the academic units of the University of California, Irvine (UCI). The school is divided into four departments: developmental and cell biology, ecology and evolutionary biology, molecular biology and biochemistry, and neurobiology and behavior. With over 3,700 students it is i... | UCI School of Biological Sciences | 0.804903 |
30,590 | In physics, rotational–vibrational coupling occurs when the rotation frequency of a system is close to or identical to a natural internal vibration frequency. The animation on the right shows ideal motion, with the force exerted by the spring and the distance from the center of rotation increasing together linearly wit... | Rovibrational coupling | 0.8049 |
30,591 | The discrepancy of N {\displaystyle N} arbitrary points t 1 , … , t N − 1 ∈ [ 0 , 1 ) k {\displaystyle \mathbf {t} _{1},\dots ,\mathbf {t} _{N-1}\in [0,1)^{k}} satisfies D N ( t 0 , t 1 , … , t N − 1 ) ≥ π 2 N ( ( π + 1 ) l − 1 ) ∏ j = 1 k m a x ( 1 , h j ) | ∑ n = 0 N − 1 e ( h ⋅ t n ) | {\displaystyle D_{N}(\mathbf {... | Inversive congruential generator | 0.8049 |
30,592 | The Statistics Online Computational Resource (SOCR) is an online multi-institutional research and education organization. SOCR designs, validates and broadly shares a suite of online tools for statistical computing, and interactive materials for hands-on learning and teaching concepts in data science, statistical analy... | Statistics Online Computational Resource | 0.804897 |
30,593 | Pop!_OS is a Linux distribution developed by System76 based on Ubuntu, using the GNOME desktop environment. It is intended for use by "developers, makers, and computer science professionals". Pop!_OS provides full disk encryption by default as well as streamlined window management, workspaces, and keyboard shortcuts fo... | System76 | 0.804896 |
30,594 | The complex is prepared by reducing chloroauric acid with triphenylphosphine in 95% ethanol: HAuCl4 + H2O + 2 PPh3 → (Ph3P)AuCl + Ph3PO + 3 HClPh3PAuCl can also be prepared by treating a thioether complex of gold like (dimethyl sulfide)gold(I) chloride, , with triphenylphosphine.The complex adopts a linear coordination... | C18H15AuClP | 0.804896 |
30,595 | Monge–Ampère equations arise naturally in several problems in Riemannian geometry, conformal geometry, and CR geometry. One of the simplest of these applications is to the problem of prescribed Gauss curvature. Suppose that a real-valued function K is specified on a domain Ω in Rn, the problem of prescribed Gauss curva... | Monge-Ampère equation | 0.804895 |
30,596 | The most complete results so far have been obtained when the equation is elliptic. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 and later by André-Marie Ampère in 1... | Monge-Ampere equation | 0.804895 |
30,597 | Moreover, factorials naturally appear in formulae from quantum and statistical physics, where one often considers all the possible permutations of a set of particles. In statistical mechanics, calculations of entropy such as Boltzmann's entropy formula or the Sackur–Tetrode equation must correct the count of microstate... | Factorial function | 0.804893 |
30,598 | An algorithm for this by Arnold Schönhage begins by finding the list of the primes up to n {\displaystyle n} , for instance using the sieve of Eratosthenes, and uses Legendre's formula to compute the exponent for each prime. Then it computes the product of the prime powers with these exponents, using a recursive algori... | Factorial function | 0.804893 |
30,599 | One property of the gamma function, distinguishing it from other continuous interpolations of the factorials, is given by the Bohr–Mollerup theorem, which states that the gamma function (offset by one) is the only log-convex function on the positive real numbers that interpolates the factorials and obeys the same funct... | Factorial function | 0.804893 |
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