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5,700 | Similarly, to deal with exactness, we can think of kernels and images in a function-theoretic sense. The proof will still apply to any (small) abelian category because of Mitchell's embedding theorem, which states that any small abelian category can be represented as a category of modules over some ring. | Five Lemma | 0.8256 |
5,701 | Space climate is the long-term variation in solar activity within the heliosphere, including the solar wind, the Interplanetary magnetic field (IMF), and their effects in the near-Earth environment, including the magnetosphere of Earth and the ionosphere, the upper and lower atmosphere, climate, and other related syste... | Space climate | 0.825594 |
5,702 | If G {\displaystyle G} is a semisimple linear algebraic group in G L n ( R ) {\displaystyle \mathrm {GL} _{n}(\mathbb {R} )} which is defined over the field Q {\displaystyle \mathbb {Q} } of rational numbers (i.e. the polynomial equations defining G {\displaystyle G} have their coefficients in Q {\displaystyle \mathbb ... | Lattice (discrete subgroup) | 0.825594 |
5,703 | A class of groups with similar properties (with respect to lattices) to real semisimple Lie groups are semisimple algebraic groups over local fields of characteristic 0, for example the p-adic fields Q p {\displaystyle \mathbb {Q} _{p}} . There is an arithmetic construction similar to the real case, and the dichotomy b... | Lattice (discrete subgroup) | 0.825594 |
5,704 | group under n {\displaystyle n} th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each g ∈ G ^ {\displaystyle g\in {\widehat {G}}} the exponential maps g ⋅: ( R , + ) → ( G ^ , ⋅ ): lim i q i ∈ Q ↦ lim i g q i {\displaystyle g^{\cdot }:(\mathbb {R} ,+)\to ({\wide... | Totally ordered group | 0.825588 |
5,705 | S is an axiomatic set theory set out by George Boolos in his 1989 article, "Iteration Again". S, a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. Boolos designed S to embody his understanding of the “iterative conception of set“ and the associated iterative hierarchy. S has th... | S (set theory) | 0.825587 |
5,706 | Hence S+ + Extensionality has the power of ZF. Boolos also argued that the axiom of choice does not follow from the iterative conception, but did not address whether Choice could be added to S in some way. Hence S+ + Extensionality cannot prove those theorems of the conventional set theory ZFC whose proofs require Choi... | S (set theory) | 0.825587 |
5,707 | Boolos’s name for Zermelo set theory minus extensionality was Z-. Boolos derived in S all axioms of Z- except the axiom of choice. The purpose of this exercise was to show how most of conventional set theory can be derived from the iterative conception of set, assumed embodied in S. Extensionality does not follow from ... | S (set theory) | 0.825587 |
5,708 | Hence, in 1993, Physical Review A changed its statement of scope to Atomic, Molecular and Optical Physics. In January 2007, the section of Physical Review E that published papers on classical optics was merged into Physical Review A, unifying the classical and quantum parts of optics into a single journal. In 2016, Phy... | Physical Review A | 0.825584 |
5,709 | In 1893, the Physical Review was established at Cornell University. It was taken over by the American Physical Society (formed in 1899) in 1913. In 1970, Physical Review was subdivided into Physical Review A, B, C, and D. At that time section A was subtitled Physical Review A: General Physics. In 1990 a process was sta... | Physical Review A | 0.825584 |
5,710 | As of 2020 Physical Review journals were abstracted and indexed in: Chemical Abstracts Service/CASSI Inspec Clarivate Web of Science Google Scholar INSPIRE-HEP NASA Astrophysics Data System (ADS) PubMed ScopusAccording to the Journal Citation Reports, the journal had an impact factor of 3.989 in 2020. | Physical Review A | 0.825584 |
5,711 | Physical Review A (also known as PRA) is a monthly peer-reviewed scientific journal published by the American Physical Society covering atomic, molecular, and optical physics and quantum information. As of 2021 the editor was Jan M. Rost (Max Planck Institute for the Physics of Complex Systems). | Physical Review A | 0.825584 |
5,712 | A law of Boolean algebra is an identity such as x ∨ (y ∨ z) = (x ∨ y) ∨ z between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, ... | Boolean Logic | 0.825573 |
5,713 | A tautology is a propositional formula that is assigned truth value 1 by every truth assignment of its propositional variables to an arbitrary Boolean algebra (or, equivalently, every truth assignment to the two element Boolean algebra). These semantics permit a translation between tautologies of propositional logic an... | Boolean Logic | 0.825573 |
5,714 | The semantics of propositional logic rely on truth assignments. The essential idea of a truth assignment is that the propositional variables are mapped to elements of a fixed Boolean algebra, and then the truth value of a propositional formula using these letters is the element of the Boolean algebra that is obtained b... | Boolean Logic | 0.825573 |
5,715 | Propositional logic is a logical system that is intimately connected to Boolean algebra. Many syntactic concepts of Boolean algebra carry over to propositional logic with only minor changes in notation and terminology, while the semantics of propositional logic are defined via Boolean algebras in a way that the tautolo... | Boolean Logic | 0.825573 |
5,716 | In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation. The m... | Nearly free electron model | 0.825565 |
5,717 | Medical physics is the branch of medicine and science that deals with applications of physics concepts, theories, and methods to medicine or health care. Neurology deals with the brain and the nervous system. Nephrology is the branch of medicine which deals with the kidneys. | Healthcare science | 0.825563 |
5,718 | Gynecology deals with the health of the female reproductive systems and the breasts. Hematology deals with the blood and the circulatory system. Hepatology deals with the liver, gallbladder and the biliary system. | Healthcare science | 0.825563 |
5,719 | Protein precipitation is widely used in downstream processing of biological products in order to concentrate proteins and purify them from various contaminants. For example, in the biotechnology industry protein precipitation is used to eliminate contaminants commonly contained in blood. The underlying mechanism of pre... | Protein precipitation | 0.825561 |
5,720 | In molecular biology, zinc-dependent phospholipases C is a family of bacterial phospholipases C enzymes, some of which are also known as alpha toxins. Bacillus cereus contains a monomeric phospholipase C EC 3.1.4.3 (PLC) of 245 amino-acid residues. Although PLC prefers to act on phosphatidylcholine, it also shows weak ... | Zinc-dependent phospholipase C | 0.82556 |
5,721 | In statistics, the k-nearest neighbors algorithm (k-NN) is a non-parametric supervised learning method first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover. It is used for classification and regression. In both cases, the input consists of the k closest training examples in a data... | K-nearest neighbors algorithm | 0.825536 |
5,722 | We also have to prevent data points from falling into the margin, we add the following constraint: for each i {\displaystyle i} either or These constraints state that each data point must lie on the correct side of the margin. This can be rewritten as We can put this together to get the optimization problem: minimize w... | Support Vector Machines | 0.825529 |
5,723 | Suppose now that we would like to learn a nonlinear classification rule which corresponds to a linear classification rule for the transformed data points φ ( x i ) . {\displaystyle \varphi (\mathbf {x} _{i}).} Moreover, we are given a kernel function k {\displaystyle k} which satisfies k ( x i , x j ) = φ ( x i ) ⋅ φ (... | Support Vector Machines | 0.825529 |
5,724 | For the one-versus-one approach, classification is done by a max-wins voting strategy, in which every classifier assigns the instance to one of the two classes, then the vote for the assigned class is increased by one vote, and finally the class with the most votes determines the instance classification. Directed acycl... | Support Vector Machines | 0.825529 |
5,725 | CAS provides the abstracting services of the chemical literature, and provides a numerical identifier, known as CAS registry number to each chemical substance that has been reported in the chemical literature (such as chemistry journals and patents). This information is compiled as a database and is popularly known as ... | Chemical substances | 0.825525 |
5,726 | However, there are some controversies regarding this definition mainly because the large number of chemical substances reported in chemistry literature need to be indexed. Isomerism caused much consternation to early researchers, since isomers have exactly the same composition, but differ in configuration (arrangement)... | Chemical substances | 0.825524 |
5,727 | The concept of a "chemical substance" became firmly established in the late eighteenth century after work by the chemist Joseph Proust on the composition of some pure chemical compounds such as basic copper carbonate. He deduced that, "All samples of a compound have the same composition; that is, all samples have the s... | Chemical substances | 0.825524 |
5,728 | A common application is public-key cryptography, whose algorithms commonly employ arithmetic with integers having hundreds of digits. Another is in situations where artificial limits and overflows would be inappropriate. It is also useful for checking the results of fixed-precision calculations, and for determining opt... | Arbitrary precision arithmetic | 0.825521 |
5,729 | Nature-inspired metaheuristics in general have attracted criticism in the research community for hiding their lack of novelty behind metaphors. The firefly algorithm has been criticized as differing from the well-established particle swarm optimization only in a negligible way. | Firefly algorithm | 0.825504 |
5,730 | Thus the total number of objective function evaluations is (number of generations) × (number of fireflies). The main update formula for any pair of two fireflies x i {\displaystyle \mathbf {x} _{i}} and x j {\displaystyle \mathbf {x} _{j}} is x i t + 1 = x i t + β exp ( x j t − x i t ) + α t ϵ t {\displaystyle \math... | Firefly algorithm | 0.825504 |
5,731 | In mathematical optimization, the firefly algorithm is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behavior of fireflies. | Firefly algorithm | 0.825504 |
5,732 | Therefore, there exists a fixed point in r {\displaystyle r} and a Nash equilibrium.When Nash made this point to John von Neumann in 1949, von Neumann famously dismissed it with the words, "That's trivial, you know. That's just a fixed-point theorem." (See Nasar, 1998, p. 94.) | Nash Equilibrium | 0.825498 |
5,733 | The existence of a Nash equilibrium is equivalent to r {\displaystyle r} having a fixed point. Kakutani's fixed point theorem guarantees the existence of a fixed point if the following four conditions are satisfied. Σ {\displaystyle \Sigma } is compact, convex, and nonempty. | Nash Equilibrium | 0.825498 |
5,734 | Nash's original proof (in his thesis) used Brouwer's fixed-point theorem (e.g., see below for a variant). This section presents a simpler proof via the Kakutani fixed-point theorem, following Nash's 1950 paper (he credits David Gale with the observation that such a simplification is possible). To prove the existence of... | Nash Equilibrium | 0.825498 |
5,735 | As the cross product of a finite number of compact convex sets, Δ {\displaystyle \Delta } is also compact and convex. Applying the Brouwer fixed point theorem to f {\displaystyle f} and Δ {\displaystyle \Delta } we conclude that f {\displaystyle f} has a fixed point in Δ {\displaystyle \Delta } , call it σ ∗ {\displays... | Nash Equilibrium | 0.825498 |
5,736 | Σ {\displaystyle \Sigma } is nonempty as long as players have strategies. Condition 2. and 3. are satisfied by way of Berge's maximum theorem. Because u i {\displaystyle u_{i}} is continuous and compact, r ( σ i ) {\displaystyle r(\sigma _{i})} is non-empty and upper hemicontinuous. | Nash Equilibrium | 0.825498 |
5,737 | However, as a theoretical concept in economics and evolutionary biology, the NE has explanatory power. The payoff in economics is utility (or sometimes money), and in evolutionary biology is gene transmission; both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim tha... | Nash Equilibrium | 0.825498 |
5,738 | In molecular biology, protein aggregation is a phenomenon in which intrinsically-disordered or mis-folded proteins aggregate (i.e., accumulate and clump together) either intra- or extracellularly. Protein aggregates have been implicated in a wide variety of diseases known as amyloidoses, including ALS, Alzheimer's, Par... | Protein aggregates | 0.825498 |
5,739 | Beginning with 2007, Probability and Mathematical Statistics became an affiliated journal of the Institute of Mathematical Statistics. PMS (ISSN 0208-4147) is indexed by Scopus, MathSciNet, Index Copernicus and Journal Citation Reports (IF=0.617). PMS is an open-access journal. | Probability and Mathematical Statistics | 0.825491 |
5,740 | Probability and Mathematical Statistics is a peer-reviewed scientific journal covering mathematical aspects of the probability theory. It was founded in 1980 as the initiative of the Wrocław probability community led by Kazimierz Urbanik and Czesław Ryll-Nardzewski, and statistics community represented by Witold Klonec... | Probability and Mathematical Statistics | 0.825491 |
5,741 | The definitions of the thermodynamic potentials may be differentiated and, along with the first and second laws of thermodynamics, a set of differential equations known as the fundamental equations follow. (Actually they are all expressions of the same fundamental thermodynamic relation, but are expressed in different ... | Thermodynamic potentials | 0.825479 |
5,742 | Since 1990, there is an International Conference on Algorithmic Learning Theory (ALT), called Workshop in its first years (1990–1997). Between 1992 and 2016, proceedings were published in the LNCS series. Starting from 2017, they are published by the Proceedings of Machine Learning Research. The 34th conference will be... | Algorithmic learning theory | 0.825478 |
5,743 | Algorithmic learning theory is a mathematical framework for analyzing machine learning problems and algorithms. Synonyms include formal learning theory and algorithmic inductive inference. Algorithmic learning theory is different from statistical learning theory in that it does not make use of statistical assumptions a... | Algorithmic learning theory | 0.825478 |
5,744 | x 1 , … , x n ≥ 0 , {\displaystyle \mathrm {HSIC_{Lasso}} :\min _{\mathbf {x} }{\frac {1}{2}}\left\|{\bar {\mathbf {L} }}-\sum _{k=1}^{n}x_{k}{\bar {\mathbf {K} }}^{(k)}\right\|_{F}^{2}+\lambda \|\mathbf {x} \|_{1},\quad {\mbox{s.t. }}\ x_{1},\ldots ,x_{n}\geq 0,} where ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} is the Fro... | Feature selection | 0.82547 |
5,745 | A metaheuristic is a general description of an algorithm dedicated to solve difficult (typically NP-hard problem) optimization problems for which there is no classical solving methods. Generally, a metaheuristic is a stochastic algorithm tending to reach a global optimum. There are many metaheuristics, from a simple lo... | Feature selection | 0.82547 |
5,746 | The above may then be written as an optimization problem: m R M R = max x ∈ { 0 , 1 } n . {\displaystyle \mathrm {mRMR} =\max _{x\in \{0,1\}^{n}}\left.} The mRMR algorithm is an approximation of the theoretically optimal maximum-dependency feature selection algorithm that maximizes the mutual information between the j... | Feature selection | 0.82547 |
5,747 | mRMR is a typical example of an incremental greedy strategy for feature selection: once a feature has been selected, it cannot be deselected at a later stage. While mRMR could be optimized using floating search to reduce some features, it might also be reformulated as a global quadratic programming optimization problem... | Feature selection | 0.82547 |
5,748 | Search approaches include: Exhaustive Best first Simulated annealing Genetic algorithm Greedy forward selection Greedy backward elimination Particle swarm optimization Targeted projection pursuit Scatter search Variable neighborhood searchTwo popular filter metrics for classification problems are correlation and mutual... | Feature selection | 0.82547 |
5,749 | According to Klein's definition, "a geometry is the study of the invariant properties of a spacetime, under transformations within itself." Therefore, the geometry of the 5th dimension studies the invariant properties of such space-time, as we move within it, expressed in formal equations. | Five-dimensional space | 0.82547 |
5,750 | The first approach is space-time-matter, which utilizes an unrestricted group of 5D coordinate transforms to derive new solutions of the Einstein's field equations that agree with the corresponding classical solutions in 4D spacetime. Another 5D representation describes quantum physics from a thermal-space-time ensembl... | Five-dimensional space | 0.82547 |
5,751 | A five-dimensional space is a space with five dimensions. In mathematics, a sequence of N numbers can represent a location in an N-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics. Whether or not the univ... | Five-dimensional space | 0.82547 |
5,752 | Examples of reform curricula introduced in response to the 1989 NCTM standards and the reasons for initial criticism: Mathland (no longer offered) Investigations in Numbers, Data, and Space, criticized for not containing explicit instruction of the standard algorithms Core-Plus Mathematics Project, criticized for faili... | Math wars | 0.825466 |
5,753 | High-quality research does not support the exclusive use of either approach." The Panel effectively called for an end to the Math Wars, concluding that research showed "conceptual understanding, computational and procedural fluency, and problem-solving skills are equally important and mutually reinforce each other. Deb... | Math wars | 0.825466 |
5,754 | Most of the modern Rubella vaccines (including the combination vaccine MMR) contain the RA 27/3 strain, which was developed by Stanley Plotkin and Leonard Hayflick at the Wistar Institute in Philadelphia. The vaccine was attenuated and prepared in the WI-38 normal human diploid cell strain which was developed by Hayfli... | Rubella vaccine | 0.82546 |
5,755 | In spite of these limitations, most generally used practical thermometers are of the empirically based kind. Especially, it was used for calorimetry, which contributed greatly to the discovery of thermodynamics. Nevertheless, empirical thermometry has serious drawbacks when judged as a basis for theoretical physics. Em... | Kinetic temperature | 0.825452 |
5,756 | In the United States, the Fahrenheit scale is the most widely used. On this scale the freezing point of water corresponds to 32 °F and the boiling point to 212 °F. The Rankine scale, still used in fields of chemical engineering in the US, is an absolute scale based on the Fahrenheit increment. | Kinetic temperature | 0.825452 |
5,757 | In physics, the internationally agreed conventional temperature scale is called the Kelvin scale. It is calibrated through the internationally agreed and prescribed value of the Boltzmann constant, referring to motions of microscopic particles, such as atoms, molecules, and electrons, constituent in the body whose temp... | Kinetic temperature | 0.825452 |
5,758 | For experimental physics, hotness means that, when comparing any two given bodies in their respective separate thermodynamic equilibria, any two suitably given empirical thermometers with numerical scale readings will agree as to which is the hotter of the two given bodies, or that they have the same temperature. This ... | Kinetic temperature | 0.825452 |
5,759 | Elements of Classical Thermodynamics for Advanced Students of Physics, original publication 1957, reprint 1966, Cambridge University Press, Cambridge UK. Quinn, T.J. (1983). | Kinetic temperature | 0.825452 |
5,760 | (1949). An Advanced Treatise on Physical Chemistry, volume 1, Fundamental Principles. The Properties of Gases, Longmans, Green & Co., London, pp. | Kinetic temperature | 0.825452 |
5,761 | "Cooling molecules the optoelectric way". Physics Today. 66 (1): 12–14. | Kinetic temperature | 0.825452 |
5,762 | (1965). Gibbs vs Boltzmann entropies, American Journal of Physics, 33(5), 391–398. Middleton, W.E.K. | Kinetic temperature | 0.825452 |
5,763 | Experimentally, it can be approached very closely but not actually reached, as recognized in the third law of thermodynamics. It would be impossible to extract energy as heat from a body at that temperature. Temperature is important in all fields of natural science, including physics, chemistry, Earth science, astronom... | Kinetic temperature | 0.825452 |
5,764 | From that probability distribution function, the average kinetic energy (per particle) of a monatomic ideal gas is E k = 1 2 m v rms 2 = 3 2 k B T , {\displaystyle E_{\text{k}}={\frac {1}{2}}mv_{\text{rms}}^{2}={\frac {3}{2}}k_{\text{B}}T,} where the Boltzmann constant kB is the ideal gas constant divided by the Avogad... | Kinetic temperature | 0.825452 |
5,765 | Compatibility is a term used by geochemists to describe how elements partition themselves in the solid and melt within Earth's mantle. In geochemistry, compatibility is a measure of how readily a particular trace element substitutes for a major element within a mineral. Compatibility of an ion is controlled by two thin... | Compatibility (geochemistry) | 0.825451 |
5,766 | These phenomena can be quantified by looking at radioactive decay records of isotopes in these basalts, which is a valuable tool for mantle geochemists. More specifically, the geochemistry of serpentinites along the ocean floor, specifically subduction zones, can be examined using compatibility of specific trace elemen... | Compatibility (geochemistry) | 0.825451 |
5,767 | An algebraic lattice is a complete lattice that is algebraic as a poset.Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic la... | Lattice Homomorphism | 0.825449 |
5,768 | In domain theory, it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of continuous posets, consisting of posets where every element can be obtained as the supremum of a directed set of elements that are way-below the element. If one can additionally ... | Lattice Homomorphism | 0.825449 |
5,769 | Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join. Pointless topology Lattice of subgroups Spectral space Invariant subspace Closure operator Abstract interpretation Subsumption lattice Fuzzy set theory Algebraizations of first-order logic Semantics of prog... | Lattice Homomorphism | 0.825449 |
5,770 | In both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics. In the context of electromagnetism, this remark applies to not only the dynamics of free charges... | Constitutive equations | 0.825443 |
5,771 | In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applie... | Constitutive equations | 0.825443 |
5,772 | The Chow variety Chow(d,P3) is a projective algebraic variety which parametrizes degree d curves in P3. It is constructed as follows. Let C be a curve of degree d in P3, then consider all the lines in P3 that intersect the curve C. This is a degree d divisor DC in G(2, 4), the Grassmannian of lines in P3. When C varies... | Moduli problem | 0.825438 |
5,773 | Moduli of algebraic curves Moduli stack of elliptic curves Moduli spaces of K-stable Fano varieties Modular curve Picard functor Moduli of semistable sheaves on a curve Kontsevich moduli space Moduli of semistable sheaves | Moduli problem | 0.825438 |
5,774 | Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells. The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smalle... | Stress resultants | 0.825431 |
5,775 | This example yields a Boolean algebra with 16 elements; in general, for finite n, the free Boolean algebra with n generators has 2n atoms, and therefore 2 2 n {\displaystyle 2^{2^{n}}} elements. If there are infinitely many generators, a similar situation prevails except that now there are no atoms. Each element of the... | Free Boolean algebra | 0.825426 |
5,776 | This collection is countable. In fact, while the free Boolean algebra with n generators, n finite, has cardinality 2 2 n {\displaystyle 2^{2^{n}}} , the free Boolean algebra with ℵ 0 {\displaystyle \aleph _{0}} generators, as for any free algebra with ℵ 0 {\displaystyle \aleph _{0}} generators and countably many finita... | Free Boolean algebra | 0.825426 |
5,777 | The free Boolean algebra with κ generators, where κ is a finite or infinite cardinal number, may be realized as the collection of all clopen subsets of {0,1}κ, given the product topology assuming that {0,1} has the discrete topology. For each α<κ, the αth generator is the set of all elements of {0,1}κ whose αth coordin... | Free Boolean algebra | 0.825426 |
5,778 | To relate the two, we introduce a functor U: BA → Set that "forgets" the algebraic structure, mapping algebras and homomorphisms to their underlying sets and functions. If we interpret the top arrow as a diagram in BA and the bottom triangle as a diagram in Set, then this diagram properly expresses that every function ... | Free Boolean algebra | 0.825426 |
5,779 | It is also easily shown that a free Boolean algebra with generating set X, as defined originally, is isomorphic to FX, so the two definitions agree. One shortcoming of the above definition is that the diagram doesn't capture that f′ is a homomorphism; since it is a diagram in Set each arrow denotes a mere function. We ... | Free Boolean algebra | 0.825426 |
5,780 | The idea is that once one chooses where to send the elements of X, the laws for Boolean algebra homomorphisms determine where to send everything else in the free algebra FX. If FX contained elements inexpressible as combinations of elements of X, then f′ wouldn't be unique, and if the elements of X weren't sufficiently... | Free Boolean algebra | 0.825426 |
5,781 | In the language of category theory, free Boolean algebras can be defined simply in terms of an adjunction between the category of sets and functions, Set, and the category of Boolean algebras and Boolean algebra homomorphisms, BA. In fact, this approach generalizes to any algebraic structure definable in the framework ... | Free Boolean algebra | 0.825426 |
5,782 | The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John is tall" and "Mary is rich". These generate a Boolean algebra with four atoms, namely: John is tall, and Mary is rich; John is tall, and Mary is not rich; John is not tall, and Mary is rich; Jo... | Free Boolean algebra | 0.825426 |
5,783 | In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that: Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and The generators are as independent as possible, in the sense that t... | Free Boolean algebra | 0.825426 |
5,784 | Epigenetic changes in eukaryotic biology serve to regulate the process of cellular differentiation. During morphogenesis, totipotent stem cells become the various pluripotent cell lines of the embryo, which in turn become fully differentiated cells. A single fertilised egg cell, the zygote, gives rise to the many diffe... | Botany | 0.825424 |
5,785 | Epigenetics is the study of heritable changes in gene function that cannot be explained by changes in the underlying DNA sequence but cause the organism's genes to behave (or "express themselves") differently. One example of epigenetic change is the marking of the genes by DNA methylation which determines whether they ... | Botany | 0.825424 |
5,786 | Following the evolution of the seed habit, seed plants diversified, giving rise to a number of now-extinct groups, including seed ferns, as well as the modern gymnosperms and angiosperms. Gymnosperms produce "naked seeds" not fully enclosed in an ovary; modern representatives include conifers, cycads, Ginkgo, and Gneta... | Botany | 0.825424 |
5,787 | The chloroplasts of plants have a number of biochemical, structural and genetic similarities to cyanobacteria, (commonly but incorrectly known as "blue-green algae") and are thought to be derived from an ancient endosymbiotic relationship between an ancestral eukaryotic cell and a cyanobacterial resident.The algae are ... | Botany | 0.825424 |
5,788 | Phytochemistry is a branch of plant biochemistry primarily concerned with the chemical substances produced by plants during secondary metabolism. Some of these compounds are toxins such as the alkaloid coniine from hemlock. Others, such as the essential oils peppermint oil and lemon oil are useful for their aroma, as f... | Botany | 0.825424 |
5,789 | Inheritance in plants follows the same fundamental principles of genetics as in other multicellular organisms. Gregor Mendel discovered the genetic laws of inheritance by studying inherited traits such as shape in Pisum sativum (peas). What Mendel learned from studying plants has had far-reaching benefits outside of bo... | Botany | 0.825424 |
5,790 | Fluid dynamic gauging can be conducted as an in-line measuring technique, but is more commonly used as a research tool. The technique was originally developed to measure the buildup or removal of the fouling layers commonly encountered in the process industry (such as in the heat treatment of dairy products). More rece... | Fluid dynamic gauge | 0.825422 |
5,791 | The volume and surface area of a cylinder, cone sphere and torus are calculated using pi. Pi is also used in calculating planetary orbit times, gaussian curves and alternating current. In calculus, there are infinite series that involve pi and pi is used in trigonometry. | Project Mathematics! | 0.825422 |
5,792 | A clay tablet shows that the Babylonians knew of Pythagorean triples 1200 years before Pythagoras, but nobody knows if they knew the more-general Pythagorean theorem. The Chinese proof uses four similar triangles to prove the theorem. Today, we know of the Pythagorean theorem because of Euclid's Elements, a set of 13 b... | Project Mathematics! | 0.825422 |
5,793 | In 1988, The Theorem of Pythagoras was the first video produced by the series and reviews the Pythagorean theorem. For all right triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a2 + b2 = c2). The theorem is named after Pythagoras of ancient Greece. Pythagorean triples ... | Project Mathematics! | 0.825422 |
5,794 | How did the ancients dig the Tunnel of Samos from two opposite sides of a mountain in 500 BCE? And how were they able to meet under the mountain? Maybe they used geometry and trigonometry. | Project Mathematics! | 0.825422 |
5,795 | Describes the addition formulas of sines and cosines and discusses the history of Ptolemy's Almagest. It also goes into details of Ptolemy's Theorem. Animation shows how sines and cosines relate to harmonic motion. | Project Mathematics! | 0.825422 |
5,796 | The Project Mathematics! series of videos is a teaching aid for teachers to help students understand the basics of geometry and trigonometry. The series was developed by Tom M. Apostol and James F. Blinn, both from the California Institute of Technology. Apostol led the production of the series, while Blinn provided th... | Project Mathematics! | 0.825422 |
5,797 | The journal is abstracted and indexed in Chemical Abstracts Service, Chemistry Citation Index, Compendex, Current Contents/Engineering, Computing & Technology, Current Contents/Physical, Chemical & Earth Sciences, Inspec, Journal Citation Reports/Science Edition, Materials Science Citation Index, and the Science Citati... | Macromolecular Reaction Engineering | 0.825419 |
5,798 | Macromolecular Reaction Engineering is intended for polymer scientists, chemists, physicists, materials scientists, theoreticians, and chemical engineers. The journal covers recent and significant results of academic and industrial research in the field of interest, encompassing all related topics - this includes polym... | Macromolecular Reaction Engineering | 0.825419 |
5,799 | Macromolecular Reaction Engineering is a peer-reviewed scientific journal published monthly by Wiley-VCH. The journal covers academic and industrial research in the field of polymer reaction engineering, which includes polymer science. It emerged from a section that was part of Macromolecular Materials and Engineering.... | Macromolecular Reaction Engineering | 0.825419 |
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