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26,600 | {\displaystyle U.} This topology is called the canonical LF topology and it is equal to the final topology induced by any countable set of inclusion maps L p ( K n ) → L c p ( U ) {\displaystyle L^{p}(K_{n})\to L_{c}^{p}(U)} ( n = 1 , 2 , … {\displaystyle n=1,2,\ldots } ) where K 1 ⊆ K 2 ⊆ ⋯ {\displaystyle K_{1}\subset... | Spaces of test functions and distributions | 0.807078 |
26,601 | {\displaystyle U.} The set L c p ( U ) {\displaystyle L_{c}^{p}(U)} is a vector subspace of L p ( U ) {\displaystyle L^{p}(U)} whose elements are the (equivalence classes of) compactly supported L p {\displaystyle L^{p}} functions defined on U {\displaystyle U} (or almost everywhere on U {\displaystyle U} ). Endow L c ... | Spaces of test functions and distributions | 0.807078 |
26,602 | Given 1 ≤ p ≤ ∞ , {\displaystyle 1\leq p\leq \infty ,} the vector space L c p ( U ) {\displaystyle L_{c}^{p}(U)} of compactly supported L p {\displaystyle L^{p}} functions on U {\displaystyle U} and its topology are defined as direct limits of the spaces L c p ( K ) {\displaystyle L_{c}^{p}(K)} in a manner analogous to... | Spaces of test functions and distributions | 0.807078 |
26,603 | For n = 1 , {\displaystyle n=1,} the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology: Otherwise, one can define a norm on S ( R n ) {\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})} via The Schwartz space is a Fréchet space (i.e. a complete m... | Spaces of test functions and distributions | 0.807078 |
26,604 | {\displaystyle |x|\to \infty .} These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices α {\displaystyle \alpha } and β {\displaystyle \beta } define: Then ϕ {\displaystyle \phi } is in the Schwartz space if all the values satisfy: The family of seminorms p... | Spaces of test functions and distributions | 0.807078 |
26,605 | A sequence of distributions ( T i ) i = 1 ∞ {\displaystyle (T_{i})_{i=1}^{\infty }} converges with respect to the weak-* topology on D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} to a distribution T if and only if for every test function f ∈ D ( U ) . {\displaystyle f\in {\mathcal {D}}(U).} For example, if f m:... | Spaces of test functions and distributions | 0.807078 |
26,606 | Since A {\displaystyle A} is continuous, the transpose t A: Y ′ → X ′ {\displaystyle {}^{t}A:Y'\to X'} is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology an... | Spaces of test functions and distributions | 0.807078 |
26,607 | Operations on distributions and spaces of distributions are often defined by means of the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well known in functional analysis. For inst... | Spaces of test functions and distributions | 0.807078 |
26,608 | Consequently, the image of t In , {\displaystyle {}^{t}\operatorname {In} ,} denoted by D ′ k ( U ) , {\displaystyle {\mathcal {D}}'^{k}(U),} forms a space of distributions when it is endowed with the strong dual topology of ( C c k ( U ) ) b ′ {\displaystyle \left(C_{c}^{k}(U)\right)_{b}^{\prime }} (transferred to it ... | Spaces of test functions and distributions | 0.807078 |
26,609 | {\displaystyle {\mathcal {D}}(U).} To define the space of distributions we must first define the canonical LF-topology, which in turn requires that several other locally convex topological vector spaces (TVSs) be defined first. First, a (non-normable) topology on C ∞ ( U ) {\displaystyle C^{\infty }(U)} will be defined... | Spaces of test functions and distributions | 0.807078 |
26,610 | {\displaystyle T(f_{i})\to 0.} The above characterizations can be used to determine whether or not a linear functional is a distribution, but more advanced uses of distributions and test functions (such as applications to differential equations) is limited if no topologies are placed on C c ∞ ( U ) {\displaystyle C_{c}... | Spaces of test functions and distributions | 0.807078 |
26,611 | There are regular regional and international educational medical physics activities. The oldest of these is the International College on Medical Physics at the International Centre for Theoretical Physics (ICTP), Trieste, Italy. This College has educated more than 1000 medical physicists from developing countries. | Medical physicist | 0.807076 |
26,612 | The presence of Medical Physicists at Expert level ('Medical Physics Experts') in healthcare in the European Union is required by the 2013/59/Euratom directive. The European Federation of Organisations for Medical Physics(EFOMP) has defined a detailed inventory of learning outcomes for Medical Physics Experts in terms ... | Medical physicist | 0.807076 |
26,613 | The majority of residencies are therapy, but diagnostic and nuclear have also been on the rise in the past several years. In the United States, professional certification is obtained from the American Board of Radiology (for all 4 areas) the American Board of Medical Physics (for MRI), the American Board of Science in ... | Medical physicist | 0.807076 |
26,614 | Assessment is provided by the completion of competencies and by a final assessment similar to the OSCE undertaken by other clinical staff. Completion of the STP leads to accreditation with the Institute of Physics and Engineering in Medicine (IPEM) and registration as a Clinical Scientist. Prior to 2011 the training ro... | Medical physicist | 0.807076 |
26,615 | A medical physicist is a health professional with specialist education and training in the concepts and techniques of applying physics in medicine and competent to practice independently in one or more of the subfields (specialties) of medical physics. A medical physicist plays a fundamental role in applying physics to... | Medical physicist | 0.807076 |
26,616 | The Australasian College of Physical Scientists and Engineers in Medicine (ACPSEM) is the professional body that oversees the education and certification of medical physicists in Australia and New Zealand and has a mission to advance services and professional standards in medical physics and biomedical engineering. | Medical physicist | 0.807076 |
26,617 | In North America, medical physics training is offered at the master's, doctorate, post-doctorate and/or residency levels. A professional doctorate has also been recently introduced as an option. Several universities offer these degrees in Canada and the United States as well. As of October 2013, over 70 universities in... | Medical physicist | 0.807076 |
26,618 | From October 2011 as part of the Modernising Scientific Careers scheme, the route to accreditation as a medical physicist in England and Wales is provided by the Scientist Training Programme (STP). This scheme is a three-year graduate program provided by the National School of Healthcare Science. Entrants are required ... | Medical physicist | 0.807076 |
26,619 | The International Conference on Theoretical, Applied, Computational and Experimental Mechanics (ICTACEM) is a professional organization in the field of Engineering. The ICTACEM was founded by (Late) Prof. P.K. Sinha of Department of Aerospace Engineering, Indian Institute of Technology Kharagpur as Conference on Theore... | ICTACEM | 0.807075 |
26,620 | In 2009 only, during July and August, students of Mathematical Grammar School have won 12 medals in International Olympiads: 5 medals in Olympiads in Mathematics, 4 medals in International Physics Olympiad, 3 medals in International Olympiad in Informatics, 3 medals in International Olympiad on Astronomy and Astrophysi... | Mathematical Grammar School | 0.807073 |
26,621 | Authors, themselves distinguished physicists from University of Cambridge, Massachusetts Institute of Technology, MIT, and Institute of Physics Belgrade and former students of Mathematical Grammar School and of professor Nataša Čaluković, claim professor Čaluković's students made 36 appearances at the International Ph... | Mathematical Grammar School | 0.807073 |
26,622 | Rastko Marinković, Vladimir Baltić, Miljan Knežević, Mladen Laudanović, Velibor Tintor, Vladimir Božin, Igor Devetak, Zlatko Filipović, Miloš Kojašević, Igor Salom, Predrag Milenović, Đorđe Krtinić, are some of the former students that held lectures, guided pupils' graduate theses, mentored special divisions, or prepar... | Mathematical Grammar School | 0.807073 |
26,623 | MG alumni holds university positions around the globe, but also managerial positions in prestigious world companies, since many of former MG graduates continue careers in business as Executive or Managing Directors of world banks and hold various business diplomas from notable institutions such as Wharton School of the... | Mathematical Grammar School | 0.807073 |
26,624 | From about 8,000 graduates in the 40-year history of this school, more than 400 have obtained PhD degrees and 600 have obtained Magister of Science degrees. These numbers are to be taken with precautions since this is the number of only those who reported their PhD to MG authorities. It is widely believed that much gre... | Mathematical Grammar School | 0.807073 |
26,625 | Programmers club at Faculty of Electrical Engineering was opened by the MG alumni in 1983, by MG alumnus Dejan Ristanović (graduated in 1981). At the same time was opened Programmers club of Mathematical Grammar School. Dejan Ristanović and MGB alumni founded and run "PC Press" publishing house.All of part-time ICT te... | Mathematical Grammar School | 0.807073 |
26,626 | MG graduate Diploma alone is treated as "Associate Degree" in named field, be it nuclear engineering, mathematics and IT, laser physics, or any other of 20 or so fields present in MG during its 40 years existence. However, due to old legal regulations in Serbia, graduates of MG are still obliged to start their universi... | Mathematical Grammar School | 0.807073 |
26,627 | The first professor of programming in MGB was Ljubomir Protić, professor at University of Belgrade, Faculty of Mathematics, who started teaching programming subjects from school year 1968/69. ; 33 years later, professor Protić served as Principal of Mathematical Grammar School (2001–2004). Later on, many University of ... | Mathematical Grammar School | 0.807073 |
26,628 | New curriculum was officially published in 1989, in Official Gazette of Socialist Republic of Serbia, no.2, 15 April 1989.The "experimental status" finally concluded in 1995, when the School was recognized as a "Specialized school for talented/gifted students in mathematics, informatics, physics and other natural scien... | Mathematical Grammar School | 0.807073 |
26,629 | Teaching position in Mathematical Gymnasium comprises a timetable of 18 hours per week for "ordinary professors", and up to 12 hours of teaching for "special professors" (usually called "special teachers") in special divisions (usually university employed staff, or institute researchers, former MG students from special... | Mathematical Grammar School | 0.807073 |
26,630 | Students of Mathematical Grammar School have won many other prizes in addition to the International Science Olympiads. As part of the "27 Club", which Erste Bank implemented for the third consecutive year in 2010, with the aim to promote talented young people and help them to get well-deserved public recognition for th... | Mathematical Grammar School | 0.807073 |
26,631 | Physics is taught from Newtonian mechanics, Fluid Dynamics, Waves, Optics, Electricity, Magnetism, to modern physics, usually ending with STR, Relativistic Dynamics, Quantum Theory, Quantum Mechanics, Molecular Physics, Physical Chemistry, Atomic Physics, Solid State Physics, Quantum Optics, Nuclear Physics, Nuclear En... | Mathematical Grammar School | 0.807073 |
26,632 | Mathematics is divided into fields of Algebra, Geometry, Linear Algebra, Analytical Geometry, Mathematical Analysis, Probability, Statistics, Numerical Analysis, and Selected Chapters in Mathematics. | Mathematical Grammar School | 0.807073 |
26,633 | Mathematical Gymnasium's main building has 2,144.98 m2 area, not including basement labs. It has 11 specialized cabinets (5 for informatics related subjects, 2 for physics, 1 for mathematics, 1 for chemistry, 1 for biology, 1 for foreign languages), 1 server room with equipment for administration of over 100 LAN and WL... | Mathematical Grammar School | 0.807073 |
26,634 | (Although all divisions of Mathematical Grammar School are special divisions, A- and D- divisions are "more special" than others, thus have internally reserved the term "special" for themselves.) In the current full-time engaged school staff, 23 teachers hold Doctor of Sciences degrees, 17 either PhD, Magister of Scien... | Mathematical Grammar School | 0.807073 |
26,635 | Mathematical Grammar School is unique for its success in International Science Olympiads in Mathematics, Physics, Informatics, Astronomy, Astrophysics, and Earth Sciences: its students have won more than 400 medals so far, on all 5 continents of the world.Latest examples include: 12 medals in 2009 in International Scie... | Mathematical Grammar School | 0.807073 |
26,636 | During his visit to MGB Physics Lab 1, Mr. Đelić said he sees MGB students and alumni as "the best ambassadors of Serbia and of Serbian and world science". He said he appreciates top achievements in education, since he himself won French national competitions for high-school students in history and economics in 1980 an... | Mathematical Grammar School | 0.807073 |
26,637 | Preference is given to outstanding achievers at competitions and recipients of national prizes during the previous years of their education. The ranking process for candidates enrolling in the elementary school classes is similar, with mathematics entrance exam (max. 120 points), performance from the previous grades in... | Mathematical Grammar School | 0.807073 |
26,638 | Mathematical Grammar School has been visited by many distinguished guests. Among notable guests outside the world of mathematics, physics and informatics, were, during last 5 years only (i.e. after 2005), among many others: then Serbian Prime Minister Zoran Đinđić, Serbian President Boris Tadić, then Belgrade Mayor Nen... | Mathematical Grammar School | 0.807073 |
26,639 | There are approx. 160 professors employed, mostly scientists. One half of the professors come from University of Belgrade staff, Institute of Physics Belgrade, and Mathematical Institute of Serbian Academy of Sciences and Arts. | Mathematical Grammar School | 0.807073 |
26,640 | Mathematical Grammar School (Serbian: Математичка гимназија Београд, romanized: Matematička gimnazija Beograd, abbr. "MG" or "MGB"), is a special school for gifted and talented students of mathematics, physics and informatics located in Belgrade, Serbia. It is ranked number one at International Science Olympiads by the... | Mathematical Grammar School | 0.807073 |
26,641 | Nanoscale Research Letters is a peer-reviewed open access scientific journal covering research in all areas of nanotechnology and published by Springer Science+Business Media. | Nanoscale Research Letters | 0.80707 |
26,642 | Moreover, even within the term "frequentist," there are variations in interpretation, differing between philosophy and physics. The intricate details of philosophical probability interpretations are explored elsewhere. In the field of statistics, these alternative interpretations allow for the analysis of different dat... | Foundations of statistics | 0.80707 |
26,643 | While the philosophical interpretations have a long history, the specific statistical terminology is relatively recent. The terms "Bayesian" and "frequentist" became standardized in the second half of the 20th century. However, the terminology can be confusing, as the "classical" interpretation of probability aligns wi... | Foundations of statistics | 0.80707 |
26,644 | Two distinct interpretations of probability have existed for a long time, one based on objective evidence and the other on subjective degrees of belief. The debate between Gauss and Laplace could have taken place more than 200 years ago, giving rise to two competing schools of statistics. Classical inferential statisti... | Foundations of statistics | 0.80707 |
26,645 | Savage played a crucial role in popularizing de Finetti's ideas in English-speaking regions and establishing rigorous Bayesian mathematics. In 1965, Dennis Lindley's two-volume work titled "Introduction to Probability and Statistics from a Bayesian Viewpoint" played a vital role in introducing Bayesian methods to a wid... | Foundations of statistics | 0.80707 |
26,646 | During the second quarter of the 20th century, the development of classical statistics led to the emergence of two competing models for inductive statistical testing. The merits of these models were extensively debated for over 25 years until Fisher's passing. Although a hybrid approach combining elements of both metho... | Foundations of statistics | 0.80707 |
26,647 | Both the frequentist and Bayesian schools have demonstrated notable accomplishments in addressing practical challenges. Classical statistics, with its reliance on mechanical calculators and specialized printed tables, boasts a longer history of obtaining results. Bayesian methods, on the other hand, have shown remarkab... | Foundations of statistics | 0.80707 |
26,648 | A classical frequency distribution provides information about the probability of the observed data. By applying Bayes' theorem, a more abstract concept is introduced, which involves estimating the probability of a hypothesis (associated with a theory) given the data. This concept, formerly referred to as "inverse proba... | Foundations of statistics | 0.80707 |
26,649 | Fisher's approach had certain philosophical advantages, while Neyman and Pearson emphasized rigorous mathematics. Hypothesis testing remains a subject of controversy for some users, but the most widely accepted alternative method, confidence intervals, is based on the same mathematical principles. Due to the historical... | Foundations of statistics | 0.80707 |
26,650 | These arguments occurred 15 years after textbooks began teaching a hybrid theory of statistical testing. Fisher and Neyman held different perspectives on the foundations of statistics (though they both opposed the Bayesian viewpoint): The interpretation of probability The disagreement between Fisher's inductive reasoni... | Foundations of statistics | 0.80707 |
26,651 | The duration of the dispute allowed for a comprehensive discussion of various fundamental issues in the field of statistics. During this exchange, Fisher also discussed the requirements for inductive inference, specifically criticizing cost functions that penalize erroneous judgments. Neyman countered by mentioning the... | Foundations of statistics | 0.80707 |
26,652 | The likelihood principle has become problematic for both major philosophical schools of statistics, weakening rather than favoring either. Its staunchest proponents argue that it provides a more solid foundation for statistics compared to the alternatives presented by Bayesian and frequentist approaches. These supporte... | Foundations of statistics | 0.80707 |
26,653 | The likelihood principle, according to Bayesian statistics, implies that information about the experimental design used to collect evidence does not factor into the statistical analysis of the data. Some Bayesians, including Savage, acknowledge this implication as a vulnerability. | Foundations of statistics | 0.80707 |
26,654 | Bayesians, on the other hand, embrace the principle as it aligns with their philosophical standpoint (perhaps in response to frequentists' discomfort). The likelihood approach is compatible with Bayesian statistical inference, where the posterior Bayes distribution for a parameter is derived by multiplying the prior di... | Foundations of statistics | 0.80707 |
26,655 | In common usage, likelihood is often considered synonymous with probability. However, according to statistics, this is not the case. In statistics, probability refers to variable data given a fixed hypothesis, whereas likelihood refers to variable hypotheses given a fixed set of data. For instance, when making repeated... | Foundations of statistics | 0.80707 |
26,656 | However, none of the main contributors were directly involved in the further development of the hybrid approach currently taught in introductory statistics.Statistics subsequently branched out into various directions, including decision theory, Bayesian statistics, exploratory data analysis, robust statistics, and non-... | Foundations of statistics | 0.80707 |
26,657 | ), published by Edward Arnold. In his book Statistics as Principled Argument, Robert P. Abelson presents the perspective that statistics serve as a standardized method for resolving disagreements among scientists, who could otherwise engage in endless debates about the merits of their respective positions. From this st... | Foundations of statistics | 0.80707 |
26,658 | For a concise introduction to the fundamentals of statistics, refer to Stuart, A.; Ord, J.K. (1994). "Ch. 8 – Probability and statistical inference" in Kendall's Advanced Theory of Statistics, Volume I: Distribution Theory (6th ed. | Foundations of statistics | 0.80707 |
26,659 | Bayesians share a common stance against the limitations of frequentism, but they are divided into various philosophical camps (empirical, hierarchical, objective, personal, and subjective), each emphasizing different aspects. A philosopher of statistics from the frequentist perspective has observed a shift from the sta... | Foundations of statistics | 0.80707 |
26,660 | It implies that sufficiently good data will bring previously disparate observers to an agreement. It makes full use of available information, and it produces decisions having the least possible error rate." "Bayesian statistics is about making probability statements, frequentist statistics is about evaluating probabili... | Foundations of statistics | 0.80707 |
26,661 | Some of these issues have been subject to unresolved debate for up to two centuries.Bandyopadhyay & Forster describe four statistical paradigms: classical statistics (or error statistics), Bayesian statistics, likelihood-based statistics, and the use of the Akaike Information Criterion as a statistical basis. Leonard J... | Foundations of statistics | 0.80707 |
26,662 | Therefore, statistics also encompasses the study of identifying statistical laws. The foundations of statistics involve the epistemological debate, describing how inductive inference from data should be conducted. Statistical inference addresses various issues, including Bayesian inference versus frequentist inference;... | Foundations of statistics | 0.80707 |
26,663 | For instance, it is commonly believed that taller parents are more likely to have taller children. However, it's important to note that individual parent-child pairings can deviate from these expectations, with the child potentially exceeding or falling short of the anticipated height based on their parent's stature. R... | Foundations of statistics | 0.80707 |
26,664 | It is an integral part of machine learning, bioinformatics, genomics, economics, and more. Statistics focuses on the quantitative characteristics of numerous repeatable phenomena. This is because certain conclusions in some fields are difficult to express with certainty, unlike mathematical formulas or theorems. | Foundations of statistics | 0.80707 |
26,665 | For instance, a weather forecast indicating a 90% probability of rain means it will likely rain, while a 5% probability means it is unlikely to rain. The actual outcome, whether it rains or not, can only be determined after the event. Statistics is also fundamental to other disciplines of science that involve predictin... | Foundations of statistics | 0.80707 |
26,666 | Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data, and is used to solve practical problems and draw conclusions. When analyzing data, the approaches used can lead to different conclusions on the same data. For example, weather forecasts often var... | Foundations of statistics | 0.80707 |
26,667 | Model complexity is a trade-off and less subjective approaches such as the Akaike information criterion and Bayesian information criterion aim to strike a balance.Concerns have been raised even about simple regression models used in the social sciences, as a multitude of assumptions underlying model validity are often ... | Foundations of statistics | 0.80707 |
26,668 | Statisticians are aware of the challenges in establishing causation, often stating that "correlation does not imply causation," which is more of a limitation in modeling than a mathematical constraint. As statistics and data sets have become more complex, questions have arisen regarding the validity of models and the i... | Foundations of statistics | 0.80707 |
26,669 | Inferential statistics relies on statistical models, and classical hypothesis testing, for instance, has often relied on the assumption of data normality. To reduce reliance on this assumption, robust and nonparametric statistics have been developed. Bayesian statistics, on the other hand, interpret new observations ba... | Foundations of statistics | 0.80707 |
26,670 | The report's recommendations included instructional content and skills: Five New Basics: English, Mathematics, Science, Social Studies, Computer ScienceOther Curriculum Matters: Develop proficiency, rigor, and skills in Foreign Languages, Performing Arts, Fine Arts, Vocational Studies, and the pursuit of higher-level e... | 21st century skills | 0.807069 |
26,671 | The crystal structures of the THF complexes of the Reformatsky reagents tert-butyl bromozincacetate and ethyl bromozincacetate have been determined. Both form cyclic eight-membered dimers in the solid state, but differ in stereochemistry: the eight-membered ring in the ethyl derivative adopts a tub-shaped conformation ... | Reformatsky reaction | 0.807067 |
26,672 | In most logics, weakening is either an inference rule or a metatheorem if the logic doesn't have an explicit rule. Notable exceptions are: Relevance logic, where every premise is necessary for the conclusion. Linear logic, which lacks monotonicity and idempotency of entailment. | Monotonic logic | 0.807067 |
26,673 | The base case does not necessarily begin with n = 0 {\displaystyle n=0} , but often with n = 1 {\displaystyle n=1} , and possibly with any fixed natural number n = N {\displaystyle n=N} , establishing the truth of the statement for all natural numbers n ≥ N {\displaystyle n\geq N} . The method can be extended to prove ... | Induction hypothesis | 0.807066 |
26,674 | are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said perpendicular. Two segments AB and AC that share a common endpoint A are perpendicular or form a right angle if the vectors A B → {\displaystyle {\overrightarrow {AB}... | Euclidean manifold | 0.807066 |
26,675 | They occur also in configuration spaces of physical systems. Beside Euclidean geometry, Euclidean spaces are also widely used in other areas of mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. | Euclidean manifold | 0.807066 |
26,676 | Since ancient Greeks, Euclidean space is used for modeling shapes in the physical world. It is thus used in many sciences such as physics, mechanics, and astronomy. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as architecture, geodesy, topography, nav... | Euclidean manifold | 0.807066 |
26,677 | Again, this yields the Euler method.A similar computation leads to the midpoint method and the backward Euler method. (4) Finally, one can integrate the differential equation from t 0 {\displaystyle t_{0}} to t 0 + h {\displaystyle t_{0}+h} and apply the fundamental theorem of calculus to get: y ( t 0 + h ) − y ( t 0 )... | Euler approximations | 0.807065 |
26,678 | A coalgebra C with a linear map from C×A to B is said to measure A to B if it preserves the algebra product and identity (in the coalgebra sense). If we think of the elements of C as linear maps from A to B, this means that c(a1a2) = Σc1(a1)c2(a2) where Σc1⊗c2 is the coproduct of c, and c multiplies identities by the c... | Measuring coalgebra | 0.807062 |
26,679 | The group-like elements of a measuring coalgebra from A to B are the homomorphisms from A to B. The primitive elements of a measuring coalgebra from A to B are the derivations from A to B. If A is the algebra of continuous real functions on a compact Hausdorff space X, and B is the real numbers, then the measuring coal... | Measuring coalgebra | 0.807062 |
26,680 | In algebra, a measuring coalgebra of two algebras A and B is a coalgebra enrichment of the set of homomorphisms from A to B. In other words, if coalgebras are thought of as a sort of linear analogue of sets, then the measuring coalgebra is a sort of linear analogue of the set of homomorphisms from A to B. In particular... | Measuring coalgebra | 0.807062 |
26,681 | In particle physics, there is a debate as to whether some elementary particles are not bodies, but are points without extension in physical space within spacetime, or are always extended in at least one dimension of space as in string theory or M theory. | Concrete object | 0.807062 |
26,682 | Since the linear relationship between two Nye tensor components and densities of geometrically necessary dislocations is usually under-determined, the total density of geometrically necessary dislocations is minimized subject to this relationship. This lower bound solution represents the minimum geometrically necessary... | Geometrically necessary dislocations | 0.807056 |
26,683 | Geometrically necessary dislocations are like-signed dislocations needed to accommodate for plastic bending in a crystalline material. They are present when a material's plastic deformation is accompanied by internal plastic strain gradients. They are in contrast to statistically stored dislocations, with statistics of... | Geometrically necessary dislocations | 0.807056 |
26,684 | Due to the zero solution for equations with j = k {\displaystyle j=k} are zero and the symmetry of j {\displaystyle j} and k {\displaystyle k} , only nine independent equations remain of all twenty-seven possible permutations of i , j , k {\displaystyle i,j,k} . The Nye's tensor α i j {\displaystyle \alpha _{ij}} can b... | Geometrically necessary dislocations | 0.807056 |
26,685 | In particle physics, the Goldberger–Wise mechanism is a popular mechanism that determines the size of the fifth dimension in Randall–Sundrum models. The mechanism uses a scalar field that propagates throughout the five-dimensional bulk. On each of the branes that end the fifth dimension (frequently referred to as the P... | Goldberger–Wise mechanism | 0.807054 |
26,686 | The biopharmaceutical industry uses predictive informatics (a superset of chemoinformatics) to integrate information resources to transform data into knowledge in order to make better decisions faster in the area of drug lead identification and optimization. | Predictive informatics | 0.807054 |
26,687 | Scientists involved in systems biology employ predictive informatics to integrate complex data about the interactions in biological systems from diverse experimental sources. | Predictive informatics | 0.807054 |
26,688 | In mathematics, vector algebra may mean: Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space. The algebraic operations in vector calculus, namely the specific additional structure of vectors in 3-dimensional Euclidean space R 3 {\displaystyle \mathb... | Vector algebra | 0.807053 |
26,689 | We briefly review the HSW coding theorem (the statement of the achievability of the Holevo information rate I ( X ; B ) {\displaystyle I(X;B)} for communicating classical data over a quantum channel). We first review the minimal amount of quantum mechanics needed for the theorem. We then cover quantum typicality, and f... | Classical capacity | 0.807053 |
26,690 | We now prove the HSW theorem with Sen's non-commutative union bound. We divide up the proof into a few parts: codebook generation, POVM construction, and error analysis. Codebook Generation. We first describe how Alice and Bob agree on a random choice of code. | Classical capacity | 0.807053 |
26,691 | We usually denote the elements of a POVM as { Λ m } m {\displaystyle \left\{\Lambda _{m}\right\}_{m}} . These operators should satisfy positivity and completeness in order to form a valid POVM: Λ m ≥ 0 ∀ m {\displaystyle \Lambda _{m}\geq 0\ \ \ \ \forall m} ∑ m Λ m = I . {\displaystyle \sum _{m}\Lambda _{m}=I.} The pro... | Classical capacity | 0.807053 |
26,692 | In order to prove the HSW coding theorem, we really just need a few basic things from quantum mechanics. First, a quantum state is a unit trace, positive operator known as a density operator. Usually, we denote it by ρ {\displaystyle \rho } , σ {\displaystyle \sigma } , ω {\displaystyle \omega } , etc. The simplest mod... | Classical capacity | 0.807053 |
26,693 | A lot of the computer software explicitly developed for school lessons has not yet been implemented due to its complexity. Thereby, the subject is either not at all integrated into the curriculum or does not pass the step of an interpretation of analogue images. In fact, the subject of remote sensing requires a consoli... | Infrared remote sensing | 0.807048 |
26,694 | quantum number Pauli exclusion principle quantum indeterminacy uncertainty principle wavefunction collapse zero-point energy bound state coherent state squeezed coherent state density state Fock state, Fock space vacuum state quasinormal mode no-cloning theorem quantum entanglement | List of mathematical topics in quantum theory | 0.807048 |
26,695 | quantum computing qubit qutrit pure qubit state quantum dot Kane quantum computer quantum cryptography quantum decoherence quantum circuit universal quantum computer measurement based Quantum Computing timeline of quantum computing | List of mathematical topics in quantum theory | 0.807048 |
26,696 | bra–ket notation canonical commutation relation complete set of commuting observables Heisenberg picture Hilbert space Interaction picture Measurement in quantum mechanics quantum field theory quantum logic quantum operation Schrödinger picture semiclassical statistical ensemble wavefunction wave–particle duality Wight... | List of mathematical topics in quantum theory | 0.807048 |
26,697 | Feynman diagram One-loop Feynman diagram Schwinger's quantum action principle Propagator Annihilation operator S-matrix Standard Model Local quantum physics Nonlocal Effective field theory Correlation function (quantum field theory) Renormalizable Cutoff Infrared divergence, infrared fixed point Ultraviolet divergence ... | List of mathematical topics in quantum theory | 0.807048 |
26,698 | Copenhagen interpretation locality principle Bell's theorem Bell test loopholes CHSH inequality hidden variable theory path integral formulation, quantum action Bohm interpretation many-worlds interpretation Tsirelson's bound | List of mathematical topics in quantum theory | 0.807048 |
26,699 | Quantum group Hopf algebra Noncommutative quantum field theory | List of mathematical topics in quantum theory | 0.807048 |
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