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28,400 | The degree of BMedSc can be awarded for students completing an intercalated degree whilst studying medicine as an intermediate award. The degree of BMedSc may also be awarded to an individual who, having followed the prescribed course of study for the degrees of MBChB, does not complete the undergraduate clinical train... | Baccalaureate degree | 0.80607 |
28,401 | The Bachelor of Engineering degree or Bachelor of Applied Science degree or Bachelor of Science in Engineering degree is a professional degree awarded to students who have completed the three or four-year course of study in engineering. Common abbreviations include BEng, BASc, AMIE and BScEng. The BAI (baccalaureus in ... | Baccalaureate degree | 0.80607 |
28,402 | Some states require elementary teachers to choose a subject major as well, and minor in education. In Canada, the bachelor of education is a two-year professional degree in which students will specialise in either elementary or secondary education, and that is taken after the completion of a three or four-year bachelor... | Baccalaureate degree | 0.80607 |
28,403 | The Bachelor of Education degree (BEd) is a four-year undergraduate professional degree offered by many American colleges and universities for those preparing to be licensed as teachers. Variants include the BEd, BAEd, BAT (Bachelor of Arts for Teaching), and BST degrees. Preparation for the MS in education, this degre... | Baccalaureate degree | 0.80607 |
28,404 | Recognized institutes of higher learning only are authorized to award degrees in Sri Lanka. Three years full-time bachelor's degree without an area of specialization is known as a general degree. A degree with a specialization (in accounting, chemistry, plant biotechnology, zoology, physics, engineering, IT, law, etc.)... | Baccalaureate degree | 0.80607 |
28,405 | Stanford University's Bachelor of Arts and Science degree is for students who are receiving one degree but who have completed two arts and sciences majors, one of which would ordinarily lead to the BA and one of which would ordinarily lead to the BSc. At many institutions one can only complete a two-degree program if t... | Baccalaureate degree | 0.80607 |
28,406 | There are various undergraduate degrees in information technology incorporating programming, database design, software engineering, networks and information systems. These programs prepare graduates for further postgraduate research degrees or for employment in a variety of roles in the information technology industry.... | Baccalaureate degree | 0.80607 |
28,407 | A managed bean – sometimes simply referred to as an MBean – is a type of JavaBean, created with dependency injection. Managed Beans are particularly used in the Java Management Extensions technology – but with Java EE 6 the specification provides for a more detailed meaning of a managed bean. The MBean represents a res... | Java Management Extensions | 0.806068 |
28,408 | When combined with the use of other isotopic labels, 15N is also a very important tracer for describing the fate of nitrogenous organic pollutants. Nitrogen-15 tracing is an important method used in biogeochemistry. The ratio of stable nitrogen isotopes, 15N/14N or δ15N, tends to increase with trophic level, such that ... | Isotope signature | 0.806065 |
28,409 | Isotope biogeochemistry has been used to investigate the timeline surrounding life and its earliest iterations on Earth. Isotopic fingerprints typical of life, preserved in sediments, have been used to suggest, but do not necessarily prove, that life was already in existence on Earth by 3.85 billion years ago.Sulfur is... | Isotope signature | 0.806065 |
28,410 | IASE: newsletters, conference proceedings, recent dissertations, and links to statistics education conferences CAUSEweb: many resources aimed at teaching undergraduate statistics classes, including activities, webinars, and a literature database SRTL: forums and publications for The International Statistical Reasoning,... | Statistical education | 0.806061 |
28,411 | The International Statistical Institute (ISI) now has one section devoted to education, the International Association for Statistical Education (IASE), which runs the International Conference on Teaching Statistics every four years as well as IASE satellite conferences around ISI and ICMI meetings. The UK established t... | Statistical education | 0.806061 |
28,412 | Statistics is often taught in departments of mathematics or in departments of mathematical sciences. At the undergraduate level, statistics is often taught as a service course. | Statistical education | 0.806061 |
28,413 | Therefore, education in statistics has strong similarities to education in empirical disciplines like psychology and chemistry, in which education is closely tied to "hands-on" experimentation. Mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small unive... | Statistical education | 0.806061 |
28,414 | Statistics education is the practice of teaching and learning of statistics, along with the associated scholarly research. Statistics is both a formal science and a practical theory of scientific inquiry, and both aspects are considered in statistics education. Education in statistics has similar concerns as does educa... | Statistical education | 0.806061 |
28,415 | The comparable percentages for instructors of statistics in mathematics departments were about 52% and 38%. The principle that statistics-instructors should have statistical competence has been affirmed by the guidelines of the Mathematical Association of America, which has been endorsed by the ASA. The unprofessional ... | Statistical education | 0.806061 |
28,416 | In doctoral mathematics departments, however, only about 58% of statistics course instructors had at least a master’s degree in statistics or biostatistics as their highest degree earned. In master’s-level mathematics departments, the corresponding percentage was near 44%, and in bachelor’s-level departments only 19% o... | Statistical education | 0.806061 |
28,417 | (an economist and son of Harold Hotelling). Data on the teaching of statistics in the United States has been collected on behalf of the Conference Board of the Mathematical Sciences (CBMS). Examining data from 2000, Schaeffer and Stasny reported By far the majority of instructors within statistics departments have at l... | Statistical education | 0.806061 |
28,418 | In the United States especially, statisticians have long complained that many mathematics departments have assigned mathematicians (without statistical competence) to teach statistics courses, effectively giving "double blind" courses. The principle that college-instructors should have qualifications and engagement wit... | Statistical education | 0.806061 |
28,419 | On the one hand, there is widespread agreement that the Key Stage 4 curriculum is over-crowded and that the introduction of Statistics and Data Handling may have been at the expense of time needed for practising and acquiring fluency in core mathematical manipulations. Many in higher education mathematics and engineeri... | Statistical education | 0.806061 |
28,420 | At an earlier age (typically 15–16 years) GCSE qualifications in mathematics contain "Statistics and Probability" topics on: Probability; Averages; Standard Deviation; Sampling; Cumumulative Frequency Graphs (including median and quantiles); Representing Data; Histograms. The UK's Office for National Statistics has a w... | Statistical education | 0.806061 |
28,421 | The coverage of "Further Statistics" includes: Continuous Probability Distributions; Estimation; Hypothesis Testing; One Sample Tests; Hypothesis Testing; Two Sample Tests; Goodness of Fit Tests; Experimental Design; Analysis of Variance (Anova); Statistical Process Control; Acceptance Sampling. The Centre for Innovati... | Statistical education | 0.806061 |
28,422 | In the United Kingdom, at least some statistics has been taught in schools since the 1930s. At present, A-level qualifications (typically taken by 17- to 18-year-olds) are being developed in "Statistics" and "Further Statistics". The coverage of the former includes: Probability; Data Collection; Descriptive Statistics;... | Statistical education | 0.806061 |
28,423 | In the theory of combinatorial optimization, submodular flow is a general class of optimization problems that includes as special cases the minimum-cost flow problem, matroid intersection, and the problem of computing a minimum-weight dijoin in a weighted directed graph. It was originally formulated by Jack Edmonds and... | Submodular flow | 0.806061 |
28,424 | In classical electromagnetic theory, an ideal source of light with constant intensity can be modeled by a spatially and temporally coherent electromagnetic wave of a single frequency. Such a light source can be modeled by, E ( x , t ) = E 0 sin ( k x − ω t + ϕ ) {\displaystyle E(x,t)=E_{0}\sin(kx-\omega t+\phi )} whe... | Photon Statistics | 0.806056 |
28,425 | Photon statistics is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use photodetectors to analyze the intrinsic statistical nature of photons in a light source. In these experiments, light incident on the photodetector generates photoelectrons and ... | Photon Statistics | 0.806056 |
28,426 | Light that is governed by super-Poissonian statistics exhibits a statistical distribution with variance Δ n 2 > ⟨ n ⟩ {\displaystyle {\Delta n}^{2}>\langle n\rangle } . An example of light that exhibits super-Poissonian statistics is thermal light. The intensity of thermal light fluctuates randomly and the fluctuations... | Photon Statistics | 0.806056 |
28,427 | Light that is governed by sub-Poisson statistics cannot be described by classical electromagnetic theory and is defined by Δ n 2 < ⟨ n ⟩ {\displaystyle {\Delta n}^{2}<\langle n\rangle } . The advent of ultrafast photodetectors has made it possible to measure the sub-Poissonian nature of light. An example of light exhib... | Photon Statistics | 0.806056 |
28,428 | Three regimes of statistical distributions can be obtained depending on the properties of the light source: Poissonian, super-Poissonian, and sub-Poissonian. The regimes are defined by the relationship between the variance and average number of photon counts for the corresponding distribution. Both Poissonian and super... | Photon Statistics | 0.806056 |
28,429 | The Journal of Mathematical Psychology is a peer-reviewed scientific journal established in 1964. It covers all areas of mathematical and theoretical psychology, including sensation and perception, psychophysics, learning and memory, problem solving, judgment and decision-making, and motivation. It is the official jour... | Journal of Mathematical Psychology | 0.806055 |
28,430 | The problem belongs to the complexity class P. This can be proven using the Gale–Ryser theorem, i.e., one has to validate the correctness of n {\displaystyle n} inequalities. | Bipartite realization problem | 0.806052 |
28,431 | The bipartite realization problem is a classical decision problem in graph theory, a branch of combinatorics. Given two finite sequences ( a 1 , … , a n ) {\displaystyle (a_{1},\dots ,a_{n})} and ( b 1 , … , b n ) {\displaystyle (b_{1},\dots ,b_{n})} of natural numbers, the problem asks whether there is labeled simple ... | Bipartite realization problem | 0.806052 |
28,432 | Closeness is a basic concept in topology and related areas in mathematics. Intuitively we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a metric space where a notion of distance between elements of the space is defined, but it can be generalized to topologica... | Closeness (mathematics) | 0.806047 |
28,433 | if p {\displaystyle p} is close to A {\displaystyle A} then A ≠ ∅ {\displaystyle A\neq \emptyset } if p {\displaystyle p} is close to A {\displaystyle A} and B ⊃ A {\displaystyle B\supset A} then p {\displaystyle p} is close to B {\displaystyle B} if p {\displaystyle p} is close to A ∪ B {\displaystyle A\cup B} then p ... | Closeness (mathematics) | 0.806047 |
28,434 | If the initial state of O {\displaystyle O} is | init ⟩ {\displaystyle |{\text{init}}\rangle } , some degrees of freedom in O {\displaystyle O} become correlated with the state of S {\displaystyle S} after the measurement, and this correlation can take one of two values: | O ↑ ⟩ {\displaystyle |O_{\uparrow }\rangle } o... | Relational quantum mechanics | 0.806043 |
28,435 | The question is whether RQM denies any objective reality, or otherwise stated: there is only a subjectively knowable reality. Rovelli limits the scope of this claim by stating that RQM relates to the variables of a physical system and not to constant, intrinsic properties, such as the mass and charge of an electron. In... | Relational quantum mechanics | 0.806043 |
28,436 | The only group of interpretations of quantum mechanics with which RQM is almost completely incompatible is that of hidden variables theories. RQM shares some deep similarities with other views, but differs from them all to the extent to which the other interpretations do not accord with the "relational world" put forwa... | Relational quantum mechanics | 0.806043 |
28,437 | {\displaystyle |Q_{b}^{(j)}\rangle =\sum _{i}U_{bc}^{ij}|Q_{c}^{(i)}\rangle .} And the conventional probability rule of quantum mechanics states that if two sets of basis vectors are in the relation above, then the probability p i j {\displaystyle p^{ij}} is p i j = | ⟨ Q c ( i ) | Q b ( j ) ⟩ | 2 = | U b c i j | 2 . {... | Relational quantum mechanics | 0.806043 |
28,438 | If we further assume that the relations { ∧ , ∨ } {\displaystyle \left\{\land ,\lor \right\}} are defined for all Q i {\displaystyle Q_{i}} , then W ( S ) {\displaystyle W\left(S\right)} is an orthomodular lattice, while all the possible unions of sets of complete questions form a Boolean algebra with the Q c ( i ) {\d... | Relational quantum mechanics | 0.806043 |
28,439 | Bohm's interpretation of QM does not sit well with RQM. One of the explicit hypotheses in the construction of RQM is that quantum mechanics is a complete theory, that is it provides a full account of the world. Moreover, the Bohmian view seems to imply an underlying, "absolute" set of states of all systems, which is al... | Relational quantum mechanics | 0.806043 |
28,440 | Hypothesis 2: the completeness of quantum mechanics. There are no hidden variables or other factors which may be appropriately added to quantum mechanics, in light of current experimental evidence.Thus, if a state is to be observer-dependent, then a description of a system would follow the form "system S is in state x ... | Relational quantum mechanics | 0.806043 |
28,441 | RQM, however, takes the point illustrated by this problem at face value. Instead of trying to modify quantum mechanics to make it fit with prior assumptions that we might have about the world, Rovelli says that we should modify our view of the world to conform to what amounts to our best physical theory of motion. Just... | Relational quantum mechanics | 0.806043 |
28,442 | Alternatively, we could claim that quantum mechanics is not a complete theory, and that by adding more structure we could arrive at a universal description (the troubled hidden variables approach). Yet another option is to give a preferred status to a particular observer or type of observer, and assign the epithet of c... | Relational quantum mechanics | 0.806043 |
28,443 | Thus the standard mathematical formulation of quantum mechanics allows different observers to give different accounts of the same sequence of events. There are many ways to overcome this perceived difficulty. | Relational quantum mechanics | 0.806043 |
28,444 | According to O {\displaystyle O} , at t 2 {\displaystyle t_{2}} , the system S {\displaystyle S} is in a determinate state, namely spin up. And, if quantum mechanics is complete, then so is this description. But, for O ′ {\displaystyle O'} , S {\displaystyle S} is not uniquely determinate, but is rather entangled with ... | Relational quantum mechanics | 0.806043 |
28,445 | The first two postulates are motivated entirely by experimental results, while the third postulate, although it accords perfectly with what we have discovered experimentally, is introduced as a means of recovering the full Hilbert space formalism of quantum mechanics from the other two postulates. The two empirical pos... | Relational quantum mechanics | 0.806043 |
28,446 | Relational quantum mechanics arose from a comparison of the quandaries posed by the interpretations of quantum mechanics with those resulting from Lorentz transformations prior to the development of special relativity. Rovelli suggested that just as pre-relativistic interpretations of Lorentz's equations were complicat... | Relational quantum mechanics | 0.806043 |
28,447 | A "measurement event" is thus described as an ordinary physical interaction where two systems become correlated to some degree with respect to each other. The proponents of the relational interpretation argue that this approach resolves some of the traditional interpretational difficulties with quantum mechanics. By gi... | Relational quantum mechanics | 0.806043 |
28,448 | RQM argues that this is a complete picture of the world because the notion of "state" is always relative to some observer. There is no privileged, "real" account. The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, with respect to the o... | Relational quantum mechanics | 0.806043 |
28,449 | Relational quantum mechanics (RQM) is an interpretation of quantum mechanics which treats the state of a quantum system as being observer-dependent, that is, the state is the relation between the observer and the system. This interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and has since been ex... | Relational quantum mechanics | 0.806043 |
28,450 | The consistency inherent in the quantum formalism, exemplified by the "M-operator" defined above, guarantees that there will be no contradictions between records. The interaction between O ′ {\displaystyle O'} and whatever he chooses to measure, be it the S + O {\displaystyle S+O} compound system or O {\displaystyle O}... | Relational quantum mechanics | 0.806043 |
28,451 | Thus the relational interpretation, by shedding the notion of an "absolute state" of the system, allows for an analysis of the EPR paradox which neither violates traditional locality constraints, nor implies superluminal information transfer, since we can assume that all observers are moving at comfortable sub-light ve... | Relational quantum mechanics | 0.806043 |
28,452 | So, at time t 2 {\displaystyle t_{2}} , Alice knows the value of M A ( α ) {\displaystyle M_{A}(\alpha )}: the spin of her particle, relative to herself. But, since the particles are in a singlet state, she knows that M A ( α ) + M A ( β ) = 0 , {\displaystyle M_{A}(\alpha )+M_{A}(\beta )=0,} and so if she measures her... | Relational quantum mechanics | 0.806043 |
28,453 | As far as Alice is concerned, the specific results obtained on Bob's wing of the experiment are indeterminate for her, although she will know that Bob has a definite result. In order to find out what result Bob has, she has to interact with him at some time t 3 {\displaystyle t_{3}} in their future light cones, through... | Relational quantum mechanics | 0.806043 |
28,454 | In the experiment, students that were presented with worked examples needed less learning time to obtain the procedural skill and conceptual understanding of Geometry. In the second experiment, the authors aimed at avoiding the negative effects that occurred due to the lack of understanding of the purpose of worked exa... | Worked-example effect | 0.80604 |
28,455 | Pillay (1994) found that worked examples showing 3 intermediate problem stages were more effective than showing only one but suggested that the distance between stages needs to be small enough to allow students to connect them without having to create their own linkages. Schwonke et al. (2009), in two experiments with ... | Worked-example effect | 0.80604 |
28,456 | The worked-example effect suggests that learning by studying worked examples is more effective than problem solving, and a number of research studies have demonstrated this effect. Sweller and Cooper were not the first to use this form of instruction, but certainly they were the first to describe it from a cognitive lo... | Worked-example effect | 0.80604 |
28,457 | It is suggested that worked examples be faded over time to be replaced with problems for practice (Renkl, Atkinson & Maier, 2000). Thus it is important to consider the learner as well as the media while developing worked examples, else learners may not perform as expected. Since worked-out examples include the steps to... | Worked-example effect | 0.80604 |
28,458 | Although a number of research studies have shown that worked examples have positive effects on learners, Kalyuga et al. (2000, 2001a, b) showed that the worked-example effect is related to the expertise level of a learner, referred to as expertise reversal effect. In three studies, (Kalyuga et al. 2000, 2001a, b) used ... | Worked-example effect | 0.80604 |
28,459 | Studying worked examples is an effective instructional strategy to teach complex problem-solving skills (van Merriënboer, 1997). This is because example-based instruction provides expert mental models, to explain the steps of a solution for novices. Worked examples, like the example above, are commonly found in mathema... | Worked-example effect | 0.80604 |
28,460 | In statistics, a location parameter of a probability distribution is a scalar- or vector-valued parameter x 0 {\displaystyle x_{0}} , which determines the "location" or shift of the distribution. In the literature of location parameter estimation, the probability distributions with such parameter are found to be formal... | Location parameters | 0.80604 |
28,461 | The Saint Petersburg State University of Aerospace Instrumentation (Russian: Санкт-Петербургский государственный университет аэрокосмического приборостроения) is a university with 13 faculties in Saint Petersburg, Russia. Faculties are devoted to innovation management, aerospace engineering, electronic engineering, ene... | Saint Petersburg State University of Aerospace Instrumentation | 0.806038 |
28,462 | mzXML is a XML (eXtensible Markup Language) based common file format for proteomics mass spectrometric data. This format was developed at the Seattle Proteome Center/Institute for Systems Biology while the HUPO-PSI was trying to specify the standardized mzData format, and is still in use in the proteomics community. | Mass spectrometry data format | 0.806037 |
28,463 | Over the years, different manufacturers of mass spectrometers have developed various proprietary data formats for handling such data which makes it difficult for academic scientists to directly manipulate their data. To address this limitation, several open, XML-based data formats have recently been developed by the Tr... | Mass spectrometry data format | 0.806037 |
28,464 | Mass spectrometry is a scientific technique for measuring the mass-to-charge ratio of ions. It is often coupled to chromatographic techniques such as gas- or liquid chromatography and has found widespread adoption in the fields of analytical chemistry and biochemistry where it can be used to identify and characterize s... | Mass spectrometry data format | 0.806037 |
28,465 | Natural products chemistry is a distinct area of chemical research which was important in the development and history of chemistry. Isolating and identifying natural products has been important to source substances for early preclinical drug discovery research, to understand traditional medicine and ethnopharmacology, ... | Natural product | 0.806033 |
28,466 | However, to be available for further study, all natural products must yield to isolation and purification. This may suffice if isolation provides appropriate quantities of the natural product for the intended purpose (e.g. as a drug to alleviate disease). Drugs such as penicillin, morphine, and paclitaxel proved to be ... | Natural product | 0.806033 |
28,467 | In the field of software engineering, the interface segregation principle (ISP) states that no code should be forced to depend on methods it does not use. ISP splits interfaces that are very large into smaller and more specific ones so that clients will only have to know about the methods that are of interest to them. ... | Interface segregation principle | 0.80603 |
28,468 | π-systems are more commonly used in the study of probability theory than in the general field of measure theory. This is primarily due to probabilistic notions such as independence, though it may also be a consequence of the fact that the π-𝜆 theorem was proven by the probabilist Eugene Dynkin. Standard measure theory... | Pi-system | 0.806029 |
28,469 | A 𝜆-system on Ω {\displaystyle \Omega } is a set D {\displaystyle D} of subsets of Ω , {\displaystyle \Omega ,} satisfying Ω ∈ D , {\displaystyle \Omega \in D,} if A ∈ D {\displaystyle A\in D} then Ω ∖ A ∈ D , {\displaystyle \Omega \setminus A\in D,} if A 1 , A 2 , A 3 , … {\displaystyle A_{1},A_{2},A_{3},\ldots } is ... | Pi-system | 0.806029 |
28,470 | {\displaystyle \sigma (E_{1},E_{2},\ldots ).} So instead we may examine the union of all 𝜎-algebras generated by finitely many sets ⋃ n σ ( E 1 , … , E n ) . {\textstyle \bigcup _{n}\sigma (E_{1},\ldots ,E_{n}).} This forms a π-system that generates the desired 𝜎-algebra. Another example is the collection of all inte... | Pi-system | 0.806029 |
28,471 | π-systems are also useful for checking independence of random variables. This is desirable because in practice, π-systems are often simpler to work with than 𝜎-algebras. For example, it may be awkward to work with 𝜎-algebras generated by infinitely many sets σ ( E 1 , E 2 , … ) . | Pi-system | 0.806029 |
28,472 | The importance of π-systems arises from the fact that if two probability measures agree on a π-system, then they agree on the 𝜎-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated 𝜎-algebra as well. This is the case ... | Pi-system | 0.806029 |
28,473 | The missing part of the calorimeter was installed in the fall of 2010. T2K far detector is the large Super-Kamiokande detector, which has been running since 1996 and studying proton lifetime and oscillations of atmospheric, solar and accelerator neutrinos.T2K experiment started to take neutrino data for a physics analy... | T2K experiment | 0.806028 |
28,474 | Parexel International is an American provider of biopharmaceutical services. It conducts clinical trials on behalf of its pharmaceutical clients to expedite the drug approval process. It is the second largest clinical research organization in the world and has helped develop approximately 95% of the 200 top-selling bio... | Calyx AI | 0.806027 |
28,475 | September 2020: Parexel completes purchase of Roam Analytics, integrating its natural language processing and software development capabilities into the AI Labs. February 2020: Parexel acquires Model Answers, a consultancy firm based in Brisbane, Queensland, Australia. July 2021: Parexel is acquired by EQT IX fund and ... | Calyx AI | 0.806027 |
28,476 | January 2003: Parexel acquires FWPS Group Limited, a provider of software for clinical trial management systems in Birmingham, UK, for approximately $11.9 million in cash and shares. March 2004: Parexel buys the remaining majority of outstanding shares of 3Clinical Research AG, a clinical research organization with exp... | Calyx AI | 0.806027 |
28,477 | from British pharmaceutical company GlaxoWellcome Inc. July 2001: Parexel acquires Edyabe, a clinical research organization in Latin America with offices in Argentina and Brazil, for approximately $1.6 million in cash. October 2002: Parexel acquires Invantage Inc, a privately held company based in Cambridge, MA providi... | Calyx AI | 0.806027 |
28,478 | November 1997: Parexel acquires substantially all of the assets of Hayden Image Processing Group, a Colorado corporation developing software for analyzing and measuring high resolution medical images, and announces its strategic alliance with The IRIS Group S.A. based in Belgium, specializing in intelligent optical cha... | Calyx AI | 0.806027 |
28,479 | In March 2006, a Parexel-run trial on behalf of TeGenero, the now bankrupt German biotechnology firm, on its anti-inflammatory drug TGN1412 to treat rheumatoid arthritis, multiple sclerosis or leukaemia, caused severe inflammation and multiple organ failure in six healthy volunteers at a facility based at Northwick Par... | Calyx AI | 0.806027 |
28,480 | Since the ground field of X {\displaystyle X} ( R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ) is complete, X ∗ {\displaystyle X^{*}} is a Banach space. The topology on X ∗ {\displaystyle X^{*}} induced by ‖ ⋅ ‖ {\displaystyle \|\cdot \|} turns out to be stronger than the weak-* topology on X ∗ . {\... | Dual norm | 0.806025 |
28,481 | {\displaystyle X^{*}.} (See Theorems 1 and 2 below.) The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces. | Dual norm | 0.806025 |
28,482 | It follows, in particular, that every non-reflexive Banach space has some bounded linear functional that does not achieve its norm on the closed unit ball. However, the Bishop–Phelps theorem guarantees that the set of bounded linear functionals that achieve their norm on the unit sphere of a Banach space is a norm-dens... | Dual norm | 0.806025 |
28,483 | R.C. James proved James's theorem in 1964, which states that a Banach space X {\displaystyle X} is reflexive if and only if every bounded linear function f ∈ X ∗ {\displaystyle f\in X^{*}} achieves its norm on the closed unit ball. | Dual norm | 0.806025 |
28,484 | The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.The time evolution of a quantum state is desc... | Quantum systems | 0.806022 |
28,485 | After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\displaystyle P_{\lambda }\psi /{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in the general ca... | Quantum systems | 0.806022 |
28,486 | Quantum mechanics has had enormous success in explaining many of the features of our universe, with regard to small-scale and discrete quantities and interactions which cannot be explained by classical methods. Quantum mechanics is often the only theory that can reveal the individual behaviors of the subatomic particle... | Quantum systems | 0.806022 |
28,487 | Another method is called "semi-classical equation of motion", which applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. This approach is particularly important in the field of quantum chaos. | Quantum systems | 0.806022 |
28,488 | Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a par... | Quantum systems | 0.806022 |
28,489 | It was the central topic in the famous Bohr–Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Newer inter... | Quantum systems | 0.806022 |
28,490 | In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a (separable) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector is postulated to be normalized under the Hilbert space inner product, that is, it ... | Quantum systems | 0.806022 |
28,491 | Measures of dependence based on quantiles are always defined. Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiased, or asymptotically consistent, based on the spatial structure of the population from which the data wer... | Positively correlated | 0.80602 |
28,492 | The degree of dependence between variables X and Y does not depend on the scale on which the variables are expressed. That is, if we are analyzing the relationship between X and Y, most correlation measures are unaffected by transforming X to a + bX and Y to c + dY, where a, b, c, and d are constants (b and d being pos... | Positively correlated | 0.80602 |
28,493 | In mechanical engineering, a kinematic diagram or kinematic scheme (also called a joint map or skeleton diagram) illustrates the connectivity of links and joints of a mechanism or machine rather than the dimensions or shape of the parts. Often links are presented as geometric objects, such as lines, triangles or square... | Kinematic diagram | 0.80602 |
28,494 | Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on more mathematical topics of computing, and includes the theory of computation. It is difficult to circumscribe the theoretical areas precisely. The ACM's Special Interest Group on Algorithms and Computation Theory... | Scientific discipline | 0.806018 |
28,495 | Ecology (from Greek: οἶκος, "house", or "environment"; -λογία, "study of") is a branch of biology concerning interactions among organisms and their biophysical environment, which includes both biotic and abiotic components. Topics of interest include the biodiversity, distribution, biomass, and populations of organisms... | Scientific discipline | 0.806018 |
28,496 | Zoology () is the branch of biology that studies the animal kingdom, including the structure, embryology, evolution, classification, habits, and distribution of all animals, both living and extinct, and how they interact with their ecosystems. The term is derived from Ancient Greek ζῷον, zōion, i.e. "animal" and λόγος,... | Scientific discipline | 0.806017 |
28,497 | The formal sciences are the branches of science that are concerned with formal systems, such as logic, mathematics, theoretical computer science, information theory, systems theory, decision theory, statistics. Unlike other branches, the formal sciences are not concerned with the validity of theories based on observati... | Scientific discipline | 0.806017 |
28,498 | Geology (from the Ancient Greek γῆ, gē ("earth") and -λoγία, -logia, ("study of", "discourse")) is an Earth science concerned with the solid Earth, the rocks of which it is composed, and the processes by which they change over time. Geology can also include the study of the solid features of any terrestrial planet or n... | Scientific discipline | 0.806017 |
28,499 | Chemistry (the etymology of the word has been much disputed) is the science of matter and the changes it undergoes. The science of matter is also addressed by physics, but while physics takes a more general and fundamental approach, chemistry is more specialized, being concerned by the composition, behavior (or reactio... | Scientific discipline | 0.806017 |
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