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24,700 | Gerhard Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system that is sound and complete with respect to intuitionistic logic. He called this system LJ. In LK any number of formulas is allowed to appear on the conclusion side of a sequent; in contra... | Constructive logic | 0.808035 |
24,701 | In theoretical physics, one often analyzes theories with supersymmetry in which F-terms play an important role. In four dimensions, the minimal N=1 supersymmetry may be written using a superspace. This superspace involves four extra fermionic coordinates θ 1 , θ 2 , θ ¯ 1 , θ ¯ 2 {\displaystyle \theta ^{1},\theta ^{2},... | F-term | 0.808034 |
24,702 | A data dictionary or database dictionary is a file that defines the basic organization of a database. A database dictionary contains a list of all files in the database, the number of records in each file, and the names and types of each data field. Most database management systems keep the data dictionary hidden from ... | Structured Analysis | 0.808027 |
24,703 | Without a data dictionary, however, a database management system cannot access data from the database.Database users and application developers can benefit from an authoritative data dictionary document that catalogs the organization, contents, and conventions of one or more databases. This typically includes the names... | Structured Analysis | 0.808027 |
24,704 | In software engineering, structured analysis (SA) and structured design (SD) are methods for analyzing business requirements and developing specifications for converting practices into computer programs, hardware configurations, and related manual procedures. Structured analysis and design techniques are fundamental to... | Structured Analysis | 0.808027 |
24,705 | A version of Karp–Lipton theorem states that if every language in NP has polynomial size circuits then the polynomial time hierarchy collapses to SP2. This result yields a strengthening of Kannan's theorem: it is known that SP2 is not contained in SIZE(nk) for any fixed k. | S2P (complexity) | 0.808026 |
24,706 | It is immediate from the definition that SP2 is closed under unions, intersections, and complements. Comparing the definition with that of Σ 2 P {\displaystyle \Sigma _{2}^{P}} and Π 2 P {\displaystyle \Pi _{2}^{P}} , it also follows immediately that SP2 is contained in Σ 2 P ∩ Π 2 P {\displaystyle \Sigma _{2}^{P}\cap ... | S2P (complexity) | 0.808025 |
24,707 | If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants. A function u of a differential extension F of a differential field F is an elementary function over F if the function u is algebraic over F, or is an exponential, that is, ∂u = u ∂a for a ∈ F... | Elementary function (differential algebra) | 0.808025 |
24,708 | The mathematical definition of an elementary function, or a function in elementary form, is considered in the context of differential algebra. A differential algebra is an algebra with the extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written a... | Elementary function (differential algebra) | 0.808025 |
24,709 | Examples of elementary functions include: Addition, e.g. (x+1) Multiplication, e.g. (2x) Polynomial functions e tan x 1 + x 2 sin ( 1 + ( ln x ) 2 ) {\displaystyle {\frac {e^{\tan x}}{1+x^{2}}}\sin \left({\sqrt {1+(\ln x)^{2}}}\right)} − i ln ( x + i 1 − x 2 ) {\displaystyle -i\ln \left(x+i{\sqrt {1-x^{2}}}\rig... | Elementary function (differential algebra) | 0.808025 |
24,710 | It follows directly from the definition that the set of elementary functions is closed under arithmetic operations, root extraction and composition. The elementary functions are closed under differentiation. They are not closed under limits and infinite sums. Importantly, the elementary functions are not closed under i... | Elementary function (differential algebra) | 0.808025 |
24,711 | Pyrrolidonyl-β-naphthylamide (PYR) is a molecule used in microbiology to detect the presence of pyrrolidonyl peptidase. In the presence of bacteria with pyrrolidonyl peptidase, it is broken down to pyroglutamic acid and 2-naphthylamine. To detect this process, p-dimethylaminocinnamaldehyde is added and a change to a pi... | PYR test | 0.808024 |
24,712 | Stereochemistry has important applications in the field of medicine, particularly pharmaceuticals. An often cited example of the importance of stereochemistry relates to the thalidomide disaster. Thalidomide is a pharmaceutical drug, first prepared in 1957 in Germany, prescribed for treating morning sickness in pregnan... | Stereochemistry | 0.808023 |
24,713 | Stereochemistry, a subdiscipline of chemistry, involves the study of the relative spatial arrangement of atoms that form the structure of molecules and their manipulation. The study of stereochemistry focuses on the relationships between stereoisomers, which by definition have the same molecular formula and sequence of... | Stereochemistry | 0.808023 |
24,714 | It was not until after the observations of certain molecular phenomena that stereochemical principles were developed. In 1815, Jean-Baptiste Biot's observation of optical activity marked the beginning of organic stereochemistry history. He observed that organic molecules were able to rotate the plane of polarized light... | Stereochemistry | 0.808023 |
24,715 | Cahn–Ingold–Prelog priority rules are part of a system for describing a molecule's stereochemistry. They rank the atoms around a stereocenter in a standard way, allowing the relative position of these atoms in the molecule to be described unambiguously. A Fischer projection is a simplified way to depict the stereochemi... | Stereochemistry | 0.808023 |
24,716 | In artificial intelligence and computer science, scientists study and use automated reasoning for diverse applications including automated theorem proving the formal semantics of programming languages, and formal specification in software engineering. | Insight learning | 0.808016 |
24,717 | Meta-reasoning is reasoning about reasoning. In computer science, a system performs meta-reasoning when it is reasoning about its own operation. This requires a programming language capable of reflection, the ability to observe and modify its own structure and behaviour. | Insight learning | 0.808016 |
24,718 | One way to understand this is as a consequence of the "hairy ball theorem". This property, of bald spots, is a general property of symmetric spaces in the Cartan decomposition. At each point on the manifold, the algebra of the Killing fields splits naturally into two parts, one part which is tangent to the manifold, an... | Killing vector fields | 0.808014 |
24,719 | Treating Killing's equation ∇ a X b + ∇ b X a = 0 {\displaystyle \nabla _{a}X_{b}+\nabla _{b}X_{a}=0} together with the identity ∇ a ∇ b X c = R c b a d X c . {\displaystyle \nabla _{a}\nabla _{b}X_{c}=R^{c}{}_{bad}X_{c}.} as a system of second order differential equations for X a {\displaystyle X_{a}} , we can determi... | Killing vector fields | 0.808014 |
24,720 | Maximally symmetric spaces can be considered as sub-manifolds of flat space, arising as surfaces of constant proper distance { x ∈ R p , q: η ( x , x ) = ± 1 κ 2 } {\displaystyle \{\mathbf {x} \in \mathbb {R} ^{p,q}:\eta (\mathbf {x} ,\mathbf {x} )=\pm {\frac {1}{\kappa ^{2}}}\}} which have O(p, q) symmetry. If the sub... | Killing vector fields | 0.808014 |
24,721 | {\displaystyle \partial _{\mu }\partial _{\nu }X_{\rho }=0.} Integrating and imposing the Killing equation allows us to write the general solution to X ρ {\displaystyle X_{\rho }} as X ρ = ω ρ σ x σ + c ρ {\displaystyle X^{\rho }=\omega ^{\rho \sigma }x_{\sigma }+c^{\rho }} where ω μ ν = − ω ν μ {\displaystyle \omega ^... | Killing vector fields | 0.808014 |
24,722 | A Killing field is determined uniquely by a vector at some point and its gradient (i.e. all covariant derivatives of the field at the point). The Lie bracket of two Killing fields is still a Killing field. The Killing fields on a manifold M thus form a Lie subalgebra of vector fields on M. This is the Lie algebra of th... | Killing vector fields | 0.808014 |
24,723 | This is a worth-while exercise. Alternately, one can recognize X , Y {\displaystyle X,Y} and Z {\displaystyle Z} are the generators of isometries in Euclidean space, and since the metric on the sphere is inherited from metric in Eucliden space, the isometries are inherited as well. These three Killing fields form a com... | Killing vector fields | 0.808014 |
24,724 | {\displaystyle =Z\quad =X\quad =Y.} This is the Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} . | Killing vector fields | 0.808014 |
24,725 | The flow generated by ∂ θ {\displaystyle \partial _{\theta }} goes from north to south; points at the north pole spread apart, those at the south come together. Any transformation that moves points closer or farther apart cannot be an isometry; therefore, the generator of such motion cannot be a Killing field. The gene... | Killing vector fields | 0.808014 |
24,726 | Killing tensor fields are symmetric tensor fields T such that the trace-free part of the symmetrization of ∇ T {\displaystyle \nabla T\,} vanishes. Examples of manifolds with Killing tensors include the rotating black hole and the FRW cosmology. Killing vector fields can also be defined on any manifold M (possibly with... | Killing vector fields | 0.808014 |
24,727 | In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by (Conway 1968, 1969). The largest of the Conway groups, Co0, is the group of automorphisms of the Leech lattice Λ with respect to addition and... | Conway groups | 0.808013 |
24,728 | Otherwise, they argue, it is extremely difficult to explain how children, within the first five years of life, routinely master the complex, largely tacit grammatical rules of their native language. Additionally, the evidence of such rules in their native language is all indirect— adult speech to children cannot encomp... | Language learning | 0.808011 |
24,729 | The reduction in conditional distribution allows for the unconditional distribution to remain the same, which is the only thing that matters to exclude superluminal communication. Also note that the conditional distribution would be different from the unconditional distribution in classical physics as well. But measuri... | Popper's experiment | 0.808009 |
24,730 | The expected additional momentum scatter which Popper wrongly attributed to the Copenhagen interpretation would allow faster-than-light communication, which is excluded by the no-communication theorem in quantum mechanics. Note however that both Collet and Loudon and Qureshi compute that scatter decreases with decreasi... | Popper's experiment | 0.808009 |
24,731 | This is what quantum mechanics predicts. Popper had said "...if the Copenhagen interpretation is correct, then any increase in the precision in the measurement of our mere knowledge of the particles going through slit B should increase their scatter." This particular aspect can be experimentally tested. | Popper's experiment | 0.808009 |
24,732 | This is what was actually seen in the experiment of Kim and Shih. Popper's proposed experiment, if carried out in this way, is incapable of testing the Copenhagen interpretation of quantum mechanics. On the other hand, if slit A is gradually narrowed, the momentum spread of particle 2 (conditioned on the detection of p... | Popper's experiment | 0.808009 |
24,733 | Thus, particle 2 does not acquire any extra momentum spread than it already had. This is the prediction of standard quantum mechanics. So, the momentum spread of particle 2 will always be smaller than what was contained in the original beam. | Popper's experiment | 0.808009 |
24,734 | Popper's experiment is an experiment proposed by the philosopher Karl Popper to test aspects of the uncertainty principle in quantum mechanics. | Popper's experiment | 0.808009 |
24,735 | It maintains that observations lead to a wavefunction collapse, thereby suggesting the counter-intuitive result that two well separated, non-interacting systems require action-at-a-distance. Popper argued that such non-locality conflicts with common sense, and would lead to a subjectivist interpretation of phenomena, d... | Popper's experiment | 0.808009 |
24,736 | In fact, as early as 1934, Popper started criticising the increasingly more accepted Copenhagen interpretation, a popular subjectivist interpretation of quantum mechanics. Therefore, in his most famous book Logik der Forschung he proposed a first experiment alleged to empirically discriminate between the Copenhagen Int... | Popper's experiment | 0.808009 |
24,737 | However, Short's argument implies that if the source were improved, we should see a spread in the momentum of particle 2.Sancho carried out a theoretical analysis of Popper's experiment, using the path-integral approach, and found a similar kind of narrowing in the momentum spread of particle 2, as was observed by Kim ... | Popper's experiment | 0.808009 |
24,738 | The two-particle entangled state is not the state of two individual particles. Our experimental result is emphatically NOT a violation of the uncertainty principle which governs the behavior of an individual quantum." This led to a renewed heated debate, with some even going to the extent of claiming that Kim and Shih'... | Popper's experiment | 0.808009 |
24,739 | Rather, the momentum spread of particle 2 (observed in coincidence with particle 1 passing through slit A) was narrower than its momentum spread in the initial state. They concluded that:"Popper and EPR were correct in the prediction of the physical outcomes of their experiments. However, Popper and EPR made the same e... | Popper's experiment | 0.808009 |
24,740 | Kripps theoretically analyzed Popper's experiment and predicted that narrowing slit A would lead to momentum spread increasing at slit B. Kripps also argued that his result was based just on the formalism of quantum mechanics, without any interpretational problem. Thus, if Popper was challenging anything, he was challe... | Popper's experiment | 0.808009 |
24,741 | Many viewed Popper's experiment as a crucial test of quantum mechanics, and there was a debate on what result an actual realization of the experiment would yield. In 1985, Sudbery pointed out that the EPR state, which could be written as ψ ( y 1 , y 2 ) = ∫ − ∞ ∞ e i k y 1 e − i k y 2 d k {\displaystyle \psi (y_{1},y_{... | Popper's experiment | 0.808009 |
24,742 | In soil science, hydrology and agricultural sciences, water content has an important role for groundwater recharge, agriculture, and soil chemistry. Many recent scientific research efforts have aimed toward a predictive-understanding of water content over space and time. Observations have revealed generally that spatia... | Soil moisture measurement | 0.808006 |
24,743 | The Journal of Cheminformatics is a peer-reviewed open access scientific journal that covers cheminformatics and molecular modelling. It was established in 2009 with David Wild (Indiana University) and Christoph Steinbeck (then at EMBL-EBI) as founding editors-in-chief, and was originally published by Chemistry Central... | Journal of Cheminformatics | 0.808005 |
24,744 | This is a prototypical example of a function space. One may similarly define 1 / f: ( x 1 , … , x n ) ↦ 1 / f ( x 1 , … , x n ) , {\displaystyle 1/f:(x_{1},\ldots ,x_{n})\mapsto 1/f(x_{1},\ldots ,x_{n}),} which is a function only if the set of the points (x1, …,xn) in the domain of f such that f(x1, …, xn) ≠ 0 contains... | Multivariate function | 0.808003 |
24,745 | The usual operations of arithmetic on the reals may be extended to real-valued functions of several real variables in the following way: For every real number r, the constant function is everywhere defined. For every real number r and every function f, the function: has the same domain as f (or is everywhere defined if... | Multivariate function | 0.808003 |
24,746 | Assuming an n-dimensional analogue of a rectangular Cartesian coordinate system, these partial derivatives can be used to form a vectorial linear differential operator, called the gradient (also known as "nabla" or "del") in this coordinate system: ∇ f ( x ) = ( ∂ ∂ x 1 , ∂ ∂ x 2 , … , ∂ ∂ x n ) f ( x ) {\displaystyle ... | Multivariate function | 0.808003 |
24,747 | In economics, a utility representation theorem asserts that, under certain conditions, a preference ordering can be represented by a real-valued utility function, such that option A is preferred to option B if and only if the utility of A is larger than that of B. | Preference representation theorem | 0.808001 |
24,748 | A utility representation theorem gives conditions on a preference relation, that are sufficient for the existence of a utility representation. Often, one would like the representing function u to satisfy additional conditions, such as continuity. This requires additional conditions on the preference relation. | Preference representation theorem | 0.808001 |
24,749 | He gives conditions for existence of a utility function representation on a probability space, that is upper semicontinuous or lower semicontinuous in the order topology. Herden defines a utility function representation of a weak preorder ⪰ {\displaystyle \succeq } as an isotone function u: ( X , ⪰ ) → ( R , ≥ ) {\disp... | Preference representation theorem | 0.808001 |
24,750 | He proved that, for every partially-ordered space ( X , ≻ ) {\displaystyle (X,\succ )} that is perfectly-separable, there exists a utility function that is upper-semicontinuous in any topology stronger than the upper order topology. : Sec.4 An analogous statement states the existence of a utility function that is lower... | Preference representation theorem | 0.808001 |
24,751 | The width of ⪰ {\displaystyle \succeq } (the largest size of a set in which all elements are incomparable) is finite. The number of utility functions in the representation is at most the width of ⪰ {\displaystyle \succeq } .Note that the guaranteed functions are semicontinuous, but not necessarily continuous, even if a... | Preference representation theorem | 0.808001 |
24,752 | : Thm 2 All the representations guaranteed by the above theorems might contain infinitely many utilities, and even uncountably many utilities. In practice, it is often important to have a finite MUR - a MUR with finitely many utilities. Evren and Ok prove there exists a finite MUR where all utilities are upper semicont... | Preference representation theorem | 0.808001 |
24,753 | This process required careful calibration and exact TEM optics. Other examples include the use of EELS to measure local temperature using change of gas density, and resistivity changes.Optimal resolution in a TEM is achieved when spherical aberrations are corrected with objective lens. However, due to the geometry of m... | Electron lens | 0.808 |
24,754 | Zassenhaus (1948) showed that conversely any group that is the extension of Zn by a finite group acting faithfully is an affine space group. Combining these results shows that classifying space groups in n dimensions up to conjugation by affine transformations is essentially the same as classifying isomorphism classes ... | Crystallographic group | 0.807995 |
24,755 | Some methods analyze microbes in situ rather than extract them. In biogeophysics, the effects of microbes on properties of geological materials are remotely probed using electrical signals. Microbes can be tagged using a stable isotope such as carbon-13 and then re-injected in the ground to see where they go. A "push-p... | Deep biosphere | 0.807995 |
24,756 | Methods from modern molecular biology allow the extraction of nucleic acids, lipids and proteins from cells, DNA sequencing, and the physical and chemical analysis of molecules using mass spectrometry and flow cytometry. A lot can be learned about the microbial communities using these methods even when the individuals ... | Deep biosphere | 0.807995 |
24,757 | ≅ {\displaystyle \cong } 1. May denote an isomorphism between two mathematical structures, and is read as "is isomorphic to". 2. In geometry, may denote the congruence of two geometric shapes (that is the equality up to a displacement), and is read "is congruent to". | Mathematical symbol | 0.807991 |
24,758 | Denotes an identity, that is, an equality that is true whichever values are given to the variables occurring in it. 2. In number theory, and more specifically in modular arithmetic, denotes the congruence modulo an integer. | Mathematical symbol | 0.807991 |
24,759 | In probability and statistics, may specify the probability distribution of a random variable. For example, X ∼ N ( 0 , 1 ) {\displaystyle X\sim N(0,1)} means that the distribution of the random variable X is standard normal. 6. | Mathematical symbol | 0.807991 |
24,760 | On a blackboard and in some mathematical texts, this may be abbreviated as x = d e f E {\displaystyle x\,{\stackrel {\mathrm {def} }{=}}\,E} or x ≜ E . {\displaystyle x\triangleq E.} This is related to the concept of assignment in computer science, which is variously denoted (depending on the programming language used)... | Mathematical symbol | 0.807991 |
24,761 | 5. In Euclidean geometry and more generally in affine geometry, P Q → {\displaystyle {\overrightarrow {PQ}}} denotes the vector defined by the two points P and Q, which can be identified with the translation that maps P to Q. The same vector can be denoted also Q − P {\displaystyle Q-P} ; see Affine space. ↦ Used for d... | Mathematical symbol | 0.807991 |
24,762 | 3. Algebraic closure: If F is a field, then F ¯ {\displaystyle {\overline {F}}} is its algebraic closure, that is, the smallest algebraically closed field that contains F. For example, Q ¯ {\displaystyle {\overline {\mathbb {Q} }}} is the field of all algebraic numbers. | Mathematical symbol | 0.807991 |
24,763 | 4. In Cryptography often denotes an error in place of a regular value. | Mathematical symbol | 0.807991 |
24,764 | In multilinear algebra, geometry, and multivariable calculus, denotes the wedge product or the exterior product. ⊻ Exclusive or: if E and F are two Boolean variables or predicates, E ⊻ F {\displaystyle E\veebar F} denotes the exclusive or. Notations E XOR F and E ⊕ F {\displaystyle E\oplus F} are also commonly used; se... | Mathematical symbol | 0.807991 |
24,765 | 3. In topology, denotes the wedge sum of two pointed spaces. ∧ 1. | Mathematical symbol | 0.807991 |
24,766 | ∮ Often used, typically in physics, instead of ∫ {\displaystyle \textstyle \int } for line integrals over a closed curve. ∬, ∯ Similar to ∫ {\displaystyle \textstyle \int } and ∮ {\displaystyle \textstyle \oint } for surface integrals. ∇ {\displaystyle {\boldsymbol {\nabla }}} or ∇ → {\displaystyle {\vec {\nabla }}} Na... | Mathematical symbol | 0.80799 |
24,767 | In general, the total synthesis of natural products is a non-commercial research activity, aimed at deeper understanding of the synthesis of particular natural product frameworks, and the development of fundamental new synthetic methods. Even so, it is of tremendous commercial and societal importance. By providing chal... | Natural products | 0.807989 |
24,768 | The concept of natural products dates back to the early 19th century, when the foundations of organic chemistry were laid. Organic chemistry was regarded at that time as the chemistry of substances that plants and animals are composed of. It was a relatively complex form of chemistry and stood in stark contrast to inor... | Natural products | 0.807989 |
24,769 | Additionally, when children do understand that they are being corrected, they don't always reproduce accurate restatements. Yet, barring situations of medical abnormality or extreme privation, all children in a given speech-community converge on very much the same grammar by the age of about five years. An especially d... | Language acquisition | 0.807988 |
24,770 | In addition to word-order violations, other more ubiquitous results of a first-merge stage would show that children's initial utterances lack the recursive properties of inflectional morphology, yielding a strict Non-inflectional stage-1, consistent with an incremental Structure-building model of child language. Genera... | Language acquisition | 0.807988 |
24,771 | In a study conducted by Newman et al., the relationship between cognitive neuroscience and language acquisition was compared through a standardized procedure involving native speakers of English and native Spanish speakers who all had a similar length of exposure to the English language (averaging about 26 years). It w... | Language acquisition | 0.807988 |
24,772 | There have been debates in linguistics, philosophy, psychology, and genetics, with some scholars arguing that language is fully or mostly innate, but the research evidence points to genetic factors only working in interaction with environmental ones.Although it is difficult to determine without invasive measures which ... | Language acquisition | 0.807988 |
24,773 | According to several linguists, neurocognitive research has confirmed many standards of language learning, such as: "learning engages the entire person (cognitive, affective, and psychomotor domains), the human brain seeks patterns in its searching for meaning, emotions affect all aspects of learning, retention and rec... | Language acquisition | 0.807988 |
24,774 | Fetus auditory learning through environmental habituation has been seen in a variety of different modes, such as fetus learning of familiar melodies (Hepper, 1988), story fragments (DeCasper & Spence, 1986), recognition of mother's voice (Kisilevsky, 2003), and other studies showing evidence of fetal adaptation to nati... | Language acquisition | 0.807988 |
24,775 | Language acquisition has been studied from the perspective of developmental psychology and neuroscience, which looks at learning to use and understand language parallel to a child's brain development. It has been determined, through empirical research on developmentally normal children, as well as through some extreme ... | Language acquisition | 0.807988 |
24,776 | In statistical hypothesis testing, the error exponent of a hypothesis testing procedure is the rate at which the probabilities of Type I and Type II decay exponentially with the size of the sample used in the test. For example, if the probability of error P e r r o r {\displaystyle P_{\mathrm {error} }} of a test decay... | Error exponents in hypothesis testing | 0.807988 |
24,777 | Let T be a semigroup. A semigroup S that is a homomorphic image of a subsemigroup of T is said to be a divisor of T. The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup S is a divisor of a finite alternating wreath product of finite simple groups, each a divisor of S, and finite aperiodic ... | Krohn–Rhodes complexity | 0.807988 |
24,778 | The advantages of the CCM declarative formulation and the availability of off-the-shelf solvers have led to a large variety of natural language processing tasks being formulated within the framework, including semantic role labeling, syntactic parsing, coreference resolution, summarization, transliteration, natural lan... | Constrained conditional model | 0.807987 |
24,779 | Given a set of feature functions { ϕ i ( x , y ) } {\displaystyle \{\phi _{i}(x,y)\}} and a set of constraints { C i ( x , y ) } {\displaystyle \{C_{i}(x,y)\}} , defined over an input structure x ∈ X {\displaystyle x\in X} and an output structure y ∈ Y {\displaystyle y\in Y} , a constraint conditional model is characte... | Constrained conditional model | 0.807987 |
24,780 | Formulating problems as constrained optimization problems over the output of learned models has several advantages. It allows one to focus on the modeling of problems by providing the opportunity to incorporate domain-specific knowledge as global constraints using a first order language. Using this declarative framewor... | Constrained conditional model | 0.807987 |
24,781 | A constrained conditional model (CCM) is a machine learning and inference framework that augments the learning of conditional (probabilistic or discriminative) models with declarative constraints. The constraint can be used as a way to incorporate expressive prior knowledge into the model and bias the assignments made ... | Constrained conditional model | 0.807987 |
24,782 | Making decisions in many domains (such as natural language processing and computer vision problems) often involves assigning values to sets of interdependent variables where the expressive dependency structure can influence, or even dictate, what assignments are possible. These settings are applicable not only to Struc... | Constrained conditional model | 0.807987 |
24,783 | Suppose that ( S 1 , μ 1 ) {\displaystyle (S_{1},\mu _{1})} and ( S 2 , μ 2 ) {\displaystyle (S_{2},\mu _{2})} are two 𝜎-finite measure spaces and F: S 1 × S 2 → R {\displaystyle F:S_{1}\times S_{2}\to \mathbb {R} } is measurable. Then Minkowski's integral inequality is (Stein 1970, §A.1), (Hardy, Littlewood & Pólya 1... | Minkowski inequality | 0.807987 |
24,784 | A twice continuously differentiable function f {\displaystyle f} on a compact domain X {\displaystyle X} that satisfies f ″ ( x ) > 0 {\displaystyle f''(x)>0} for all x ∈ X {\displaystyle x\in X} is strongly convex. The proof of this statement follows from the extreme value theorem, which states that a continuous funct... | Convex surface | 0.807987 |
24,785 | Assuming still that the function is twice continuously differentiable, one can show that the lower bound of ∇ 2 f ( x ) {\displaystyle \nabla ^{2}f(x)} implies that it is strongly convex. Using Taylor's Theorem there exists such that Then by the assumption about the eigenvalues, and hence we recover the second strong c... | Convex surface | 0.807987 |
24,786 | Composition algebras of dimension 8 are called octonion algebras. They are neither associative nor commutative.For consistent terminology, algebras of dimension 1 have been called unarion, and those of dimension 2 binarion.Every composition algebra is an alternative algebra.Using the doubled form ( _: _ ): A × A → K by... | Multiplicative quadratic form | 0.807987 |
24,787 | Composition algebras of dimension 2 are either quadratic field extensions of K or isomorphic to K ⊕ K. Composition algebras of dimension 4 are called quaternion algebras. They are associative but not commutative. | Multiplicative quadratic form | 0.807987 |
24,788 | Every unital composition algebra over a field K can be obtained by repeated application of the Cayley–Dickson construction starting from K (if the characteristic of K is different from 2) or a 2-dimensional composition subalgebra (if char(K) = 2). The possible dimensions of a composition algebra are 1, 2, 4, and 8. 1-d... | Multiplicative quadratic form | 0.807987 |
24,789 | Nathan Jacobson described the automorphisms of composition algebras in 1958.The classical composition algebras over R and C are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others. : 463–81 | Multiplicative quadratic form | 0.807987 |
24,790 | In 1923 the case of real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras). In 1931 Max Zorn introduced a gamma (γ) into the multiplication rule in the Dickson construction to generate split-octonions. Adrian Albert also used the gamma in 1942 when he showed that Dickso... | Multiplicative quadratic form | 0.807987 |
24,791 | About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra: Historically, the first non-associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting comp... | Multiplicative quadratic form | 0.807987 |
24,792 | The composition of sums of squares was noted by several early authors. Diophantus was aware of the identity involving the sum of two squares, now called the Brahmagupta–Fibonacci identity, which is also articulated as a property of Euclidean norms of complex numbers when multiplied. Leonhard Euler discussed the four-sq... | Multiplicative quadratic form | 0.807987 |
24,793 | In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N(y)} for all x and y in A. A composition algebra includes an involution called a conjugation: x ↦ x ∗... | Multiplicative quadratic form | 0.807987 |
24,794 | CDER is divided into 8 sections with different responsibilities: Office of New DrugsThis office is responsible for oversight of clinical trials and other studies during drug development, and for the evaluation of new drug applications The Office of New Drugs is divided into several departments based on the indication o... | Center for Drug Evaluation and Research | 0.807981 |
24,795 | The growing crisis around HIV testing and treatment and an inter-agency dispute between officials from the former Bureau of Drugs and officials from the former Bureau of Biologics over whether to approve Genentech's Activase (tissue plasminogen activator) led to the split.In its original form, CDER was composed of six ... | Center for Drug Evaluation and Research | 0.807981 |
24,796 | In 1966, the division was reorganized to create the Office of New Drugs, which was responsible for reviewing new drug applications and clinical testing of drugs.In 1982, when the beginning of the biotechnology revolution blurred the line between a drug and a biologic, the Bureau of Drugs was merged with the FDA's Burea... | Center for Drug Evaluation and Research | 0.807981 |
24,797 | 1905.1 provides a tool to get a global view of the network topology regardless of the technologies running in the home/office network. The abstraction layer generates different topology messages to build this protocol's topology: Discovery (message type 0x0000) to detect direct 1905.1 neighbors Notification (message ty... | IEEE 1905 | 0.807979 |
24,798 | The architecture designed for the abstraction layer is based on two 1905.1 service access points accessible to upper layers: a 1905.1 MAC SAP and a 1905.1 ALME SAP. The ALME is a unique management entity supporting different media dependent management entities and a flow-based forwarding table. A 1905.1 protocol is use... | IEEE 1905 | 0.807979 |
24,799 | Mathematical models need to integrate the increasing volume of data being generated on host-pathogen interactions. Many theoretical studies of the population dynamics, structure and evolution of infectious diseases of plants and animals, including humans, are concerned with this problem.Research topics include: antigen... | Mathematical modelling of infectious diseases | 0.807977 |
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