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17,600 | The Computer Journal is a peer-reviewed scientific journal covering computer science and information systems. Established in 1958, it is one of the oldest computer science research journals. It is published by Oxford University Press on behalf of BCS, The Chartered Institute for IT. The authors of the best paper in eac... | The Computer Journal | 0.81286 |
17,601 | In 1947, Markov and Post published independent papers showing that the word problem for semigroups cannot be effectively decided. Extending this result, Pyotr Novikov and William Boone showed independently in the 1950s that the word problem for groups is not effectively solvable: there is no effective procedure that, g... | Computability theory (computation) | 0.812857 |
17,602 | In 1936, Church and Turing were inspired by techniques used by Gödel to prove his incompleteness theorems - in 1931, Gödel independently demonstrated that the Entscheidungsproblem is not effectively decidable. This result showed that there is no algorithmic procedure that can correctly decide whether arbitrary mathemat... | Computability theory (computation) | 0.812857 |
17,603 | Rice showed that for every nontrivial class C (which contains some but not all c.e. sets) the index set E = {e: the eth c.e. set We is in C} has the property that either the halting problem or its complement is many-one reducible to E, that is, can be mapped using a many-one reduction to E (see Rice's theorem for more ... | Computability theory (computation) | 0.812857 |
17,604 | Human reproductive biology is primarily controlled through hormones, which send signals to the human reproductive structures to influence growth and maturation. These hormones are secreted by endocrine glands, and spread to different tissues in the human body. In humans, the pituitary gland synthesizes hormones used to... | Reproductive biology | 0.812852 |
17,605 | Reproductive biology includes both sexual and asexual reproduction.Reproductive biology includes a wide number of fields: Reproductive systems Endocrinology Sexual development (Puberty) Sexual maturity Reproduction Fertility | Reproductive biology | 0.812852 |
17,606 | For this discovery, he was awarded the Nobel Prize in Physics in 1913. 1919 – Albert A. Michelson makes the first interferometric measurements of stellar diameters at Mount Wilson Observatory (see history of astronomical interferometry) 1924 – Louis de Broglie postulates the wave nature of electrons and suggests that a... | Timeline of electromagnetism and classical optics | 0.812848 |
17,607 | Einstein would later receive the Nobel Prize in Physics for this discovery, which launched the quantum revolution in physics. 1911 – Superconductivity is discovered by Heike Kamerlingh Onnes, who was studying the resistivity of solid mercury at cryogenic temperatures using the recently discovered liquid helium as a ref... | Timeline of electromagnetism and classical optics | 0.812848 |
17,608 | 1806 – Alessandro Volta employs a voltaic pile to decompose potash and soda, showing that they are the oxides of the previously unknown metals potassium and sodium. These experiments were the beginning of electrochemistry. 1808 – Étienne-Louis Malus discovers polarization by reflection 1809 – Étienne-Louis Malus publis... | Timeline of electromagnetism and classical optics | 0.812848 |
17,609 | The TIMSS 2019 User Guide for the International Database describes the content and format of the data in the TIMSS 2019 International Database. The IEA has developed an application for working with data from TIMSS and other IEA large-scale assessments called the "IEA International Database (IDB) Analyzer".. This applic... | Trends in International Mathematics and Science Study | 0.812842 |
17,610 | Along with the overall students’ achievement data, TIMSS comprehensive assessments include data on student performance in various mathematics and science domains (algebra, geometry, biology, chemistry, etc.) and on performance in the problem solving challenges in each of these contexts. In addition, TIMSS provides cont... | Trends in International Mathematics and Science Study | 0.812842 |
17,611 | TIMSS Advanced, previously conducted in 1995 and 2008, was also conducted in 2015, and assessed final-year secondary students' achievement in advanced mathematics and physics. Policy-relevant data about curriculum emphasis, technology use, and teacher preparation and training accompanies the TIMSS Advanced results. The... | Trends in International Mathematics and Science Study | 0.812842 |
17,612 | The TIMSS 1999 Video Study was a study of eighth-grade mathematics and science teaching in seven countries. The study involved videotaping and analyzing teaching practices in more than one thousand classrooms. In conjunction with the IEA, the study was conducted by the US National Center for Education Statistics, and t... | Trends in International Mathematics and Science Study | 0.812842 |
17,613 | This assessment was first conducted in 1995, and has been administered every four years thereafter. Therefore, some of the participating educational systems have trend data across assessments from 1995 to 2019. TIMSS assesses 4th and 8th grade students, while TIMSS Advanced assesses students in the final year of second... | Trends in International Mathematics and Science Study | 0.812842 |
17,614 | TIMSS depends on the collaboration of a large number of individuals and organizations around the world including the TIMSS & PIRLS International Study Center at Boston College, IEA's offices in Amsterdam and Hamburg, Statistics Canada, and Educational Testing Service (ETS). In the United States, TIMSS is conducted by t... | Trends in International Mathematics and Science Study | 0.812842 |
17,615 | TIMSS 2019 was the seventh cycle of TIMSS and reported overall achievement as well as results according to international benchmarks, by major content domains (number, algebra, and geometry in mathematics, and earth science, biology, and chemistry in science) and by cognitive domains (knowing, applying, reasoning). TIMS... | Trends in International Mathematics and Science Study | 0.812842 |
17,616 | It acquired its current name in December 1990. The editor-in-chief is D. Wilkinson (National Institute for Medical Research). A separate section of the journal, Gene Expression Patterns, covers research on cloning and gene expression. In 2020 the journal became Cells & Development with the subtitle: 'Cell and Developme... | Mechanisms of Development | 0.812842 |
17,617 | Mechanisms of Development is a peer-reviewed scientific journal covering all aspects of developmental biology. It is the official journal of the International Society of Developmental Biologists and is published by Elsevier. The journal was established in 1972 as Cell Differentiation and was renamed Cell Differentiatio... | Mechanisms of Development | 0.812842 |
17,618 | A variety of general optimization algorithms commonly used in computer science have also been applied to the multiple sequence alignment problem. Hidden Markov models have been used to produce probability scores for a family of possible multiple sequence alignments for a given query set; although early HMM-based method... | Sequence identity | 0.812834 |
17,619 | SSAP (sequential structure alignment program) is a dynamic programming-based method of structural alignment that uses atom-to-atom vectors in structure space as comparison points. It has been extended since its original description to include multiple as well as pairwise alignments, and has been used in the constructio... | Sequence identity | 0.812834 |
17,620 | where h h = h 2 {\displaystyle hh=h^{2}} (see: Exponentiation#History of the notation). Later Whittaker & Robinson (1924) "Calculus of observations" called this quantity the modulus (of precision), but this term has dropped out of use. == References == | Precision matrix | 0.812831 |
17,621 | In statistics, the precision matrix or concentration matrix is the matrix inverse of the covariance matrix or dispersion matrix, P = Σ − 1 {\displaystyle P=\Sigma ^{-1}} . For univariate distributions, the precision matrix degenerates into a scalar precision, defined as the reciprocal of the variance, p = 1 σ 2 {\displ... | Precision matrix | 0.812831 |
17,622 | History of computer science List of pioneers in computer science | Outline of computer science | 0.812828 |
17,623 | Data structure Data type Associative array and Hash table Array List Tree String Matrix (computer science) Database | Outline of computer science | 0.812828 |
17,624 | Computer science (also called computing science) is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. One well known subject classification system for computer science is the ACM Computing Classification System devised by the Associatio... | Outline of computer science | 0.812828 |
17,625 | Computer architecture – The design, organization, optimization and verification of a computer system, mostly about CPUs and Memory subsystem (and the bus connecting them). Operating systems – Systems for managing computer programs and providing the basis of a usable system. | Outline of computer science | 0.812828 |
17,626 | Coding theory – Useful in networking, programming, system development, and other areas where computers communicate with each other. Game theory – Useful in artificial intelligence and cybernetics. Discrete Mathematics Graph theory – Foundations for data structures and searching algorithms. Mathematical logic – Boolean ... | Outline of computer science | 0.812828 |
17,627 | Automata theory – Different logical structures for solving problems. Computability theory – What is calculable with the current models of computers. Proofs developed by Alan Turing and others provide insight into the possibilities of what may be computed and what may not. List of unsolved problems in computer science C... | Outline of computer science | 0.812828 |
17,628 | Outline of software engineering Formal methods – Mathematical approaches for describing and reasoning about software design. Software engineering – The principles and practice of designing, developing, and testing programs, as well as proper engineering practices. Algorithm design – Using ideas from algorithm theory to... | Outline of computer science | 0.812828 |
17,629 | Evolutionary computing - Biologically inspired algorithms. Natural language processing - Building systems and algorithms that analyze, understand, and generate natural (human) languages. Robotics – Algorithms for controlling the behaviour of robots. | Outline of computer science | 0.812828 |
17,630 | Outline of artificial intelligence Artificial intelligence – The implementation and study of systems that exhibit an autonomous intelligence or behavior of their own. Automated reasoning – Solving engines, such as used in Prolog, which produce steps to a result given a query on a fact and rule database, and automated t... | Outline of computer science | 0.812828 |
17,631 | Networking – Algorithms and protocols for reliably communicating data across different shared or dedicated media, often including error correction. Computer security – Practical aspects of securing computer systems and computer networks. Cryptography – Applies results from complexity, probability, algebra and number th... | Outline of computer science | 0.812828 |
17,632 | Computational science – constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems. Numerical analysis – Approximate numerical solution of mathematical problems such as root-finding, integration, the solution of ordinary differential equations; the... | Outline of computer science | 0.812828 |
17,633 | Parallel computing - The theory and practice of simultaneous computation; data safety in any multitasking or multithreaded environment. Concurrency (computer science) – Computing using multiple concurrent threads of execution, devising algorithms for solving problems on multiple processors to achieve maximal speed-up c... | Outline of computer science | 0.812828 |
17,634 | In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles (covariant classical field theory). The variational bicomplex is a coc... | Variational bicomplex | 0.812822 |
17,635 | Biochemical and Organic Simulation System (BOSS) is a general-purpose molecular modeling program that performs molecular mechanics calculations, Metropolis Monte Carlo statistical mechanics simulations, and semiempirical Austin Model 1 (AM1), PM3, and PDDG/PM3 quantum mechanics calculations. The molecular mechanics cal... | BOSS (molecular mechanics) | 0.812821 |
17,636 | OPLS force field inventor Geometry optimization Semiempirical quantum chemistry MC simulations for pure liquids, solutions, clusters or gas-phase systems Free energies are computed from statistical perturbation (free energy perturbation (FEP)) theory TIP3P, TIP4P, and TIP5P water models | BOSS (molecular mechanics) | 0.812821 |
17,637 | In mid 2005, VINNOVA, the Knut and Alice Wallenberg Foundation and the Foundation for Strategic Research (SSF) established the Swedish research node of the SGC. Experimental activities started at the Karolinska Institutet in Stockholm, led by Pär Nordlund and Johan Weigelt. Together, the three SGC laboratories contribu... | Structural Genomics Consortium | 0.812821 |
17,638 | The new goal for structures was 650. The SGC focused considerable activities in the areas of ubiquitination, protein phosphorylation, small G-proteins and epigenetics, and also initiated an effort in the structural biology of integral membrane proteins. In this phase, the SGC determined the structures of 665 human prot... | Structural Genomics Consortium | 0.81282 |
17,639 | The for-profit spin-off companies M4K Pharma (Medicines for Kids), M4ND Pharma (Medicines for Neurological Diseases) and M4ID Pharma (Medicines for Infectious Diseases) do not file patents and practise open science. The M4 companies are wholly owned by a Canadian charity Agora Open Science Trust whose mandate is to sha... | Structural Genomics Consortium | 0.81282 |
17,640 | Target 2035 is an open science movement with the goal of creating chemical and/or biological tools for the entire proteome by 2035. The launch in November 2020 and monthly webinars have and continue to be free to attend. Supporting projects currently underway include the SGC’s epigenetics chemical probe program, the NI... | Structural Genomics Consortium | 0.81282 |
17,641 | The Unrestricted Leveraging of Targets for Research Advancement and Drug Discovery (ULTRA-DD) program, funded by the European Commission’s Innovative Medicines Initiative (IMI), aims to identify and validate under-explored targets in auto-immune and inflammatory disease models. Patient-derived cell lines are screened a... | Structural Genomics Consortium | 0.81282 |
17,642 | The protein family approach is supported by publicly available bioinformatics tools (ChromoHub, UbiHub), family-based protein production and biochemistry, crystallography and structure determination, biophysics, and cell biology (for example target engagement assays). The SGC has (so far) contributed ~120 chemical prob... | Structural Genomics Consortium | 0.81282 |
17,643 | Structural biology of human proteins – The SGC has so far contributed over 2000 protein structures of human proteins of potential relevance for drug discovery into the public domain since 2003. Structures that constitute complexes with synthetic small molecules is aided by a partnership with the Diamond synchrotron in ... | Structural Genomics Consortium | 0.81282 |
17,644 | In optimization, a descent direction is a vector p ∈ R n {\displaystyle \mathbf {p} \in \mathbb {R} ^{n}} that points towards a local minimum x ∗ {\displaystyle \mathbf {x} ^{*}} of an objective function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } . Computing x ∗ {\displaystyle \mathbf {x} ^{*}} by an... | Descent direction | 0.812819 |
17,645 | Impedance control: This type of control is suitable for environment interaction and object manipulation, such as in robotics. Electronic gearing (or cam profiling): The position of a slave axis is mathematically linked to the position of a master axis. A good example of this would be in a system where two rotating drum... | Motion control system | 0.812817 |
17,646 | In 1898, the council of the British Association passed a resolution that "Mr E. T. Whittaker be requested to draw up a report on the planetary theory". A year later, Whittaker delivered his report, titled “Report on the progress of the solution of the problem of three bodies”, in a lecture to the Association, who publi... | Analytical Dynamics of Particles and Rigid Bodies | 0.812811 |
17,647 | Whittaker was 31 years old and working as a lecturer at Trinity College, Cambridge when the book was first published, less than ten years after he graduated from Cambridge University in 1895. Whittaker was branded Second Wrangler in his Cambridge Tripos examination upon graduation in 1895 and elected as a Fellow of Tri... | Analytical Dynamics of Particles and Rigid Bodies | 0.812811 |
17,648 | Among the most heavily altered, chapter fifteen went from fourteen sections to twenty-two while chapter sixteen doubled its section count from nine to eighteen.Most of the differences between the second and third editions were adding outlines of and references to works published after the book's second edition. The edi... | Analytical Dynamics of Particles and Rigid Bodies | 0.812811 |
17,649 | In September 1925, Dirac received proofs of a seminal paper by Werner Heisenberg on the new physics. Soon he realised that the key idea in Heisenberg's paper was the anti-commutativity of dynamical variables and remembered that the analogous mathematical construction in classical mechanics was Poisson brackets.In a 198... | Analytical Dynamics of Particles and Rigid Bodies | 0.812811 |
17,650 | He wanted to see a satisfactory derivation of the former and the origins of the latter. Carathéodory explained some fundamental details of the canonical transformations and referred Einstein to E. T. Whittaker's Analytical Dynamics. Einstein was trying to solve the problem of "closed time-lines" or the geodesics corres... | Analytical Dynamics of Particles and Rigid Bodies | 0.812811 |
17,651 | Chapter nine discusses action principles, such as the principle of least action and the principle of least curvature. Chapters ten through twelve, the final three chapters of part one, discuss Hamiltonian dynamics at length.Chapter thirteen begins part two and focuses on the applications of the material in part one to ... | Analytical Dynamics of Particles and Rigid Bodies | 0.812811 |
17,652 | A Treatise on the Analytical Dynamics of Particles and Rigid Bodies is a treatise and textbook on analytical dynamics by British mathematician Sir Edmund Taylor Whittaker. Initially published in 1904 by the Cambridge University Press, the book focuses heavily on the three-body problem and has since gone through four ed... | Analytical Dynamics of Particles and Rigid Bodies | 0.812811 |
17,653 | Polynomial regression models are usually fit using the method of least squares. The least-squares method minimizes the variance of the unbiased estimators of the coefficients, under the conditions of the Gauss–Markov theorem. The least-squares method was published in 1805 by Legendre and in 1809 by Gauss. | Polynomial fitting | 0.81281 |
17,654 | Having enough such pairs, using Gaussian elimination, one can get products of certain r and of the corresponding s to be squares at the same time. A slightly stronger condition is needed—that they are norms of squares in our number fields, but that condition can be achieved by this method too. Each r is a norm of a − r... | Number Field Sieve | 0.812803 |
17,655 | In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log2 n⌋ + 1 bits) is of the form exp ( ( ( 64 / 9 ) 1 / 3 + o ( 1 ) ) ( log n ) 1 / 3 ( log log... | Number Field Sieve | 0.812803 |
17,656 | Suppose f is a k-degree polynomial over Q (the rational numbers), and r is a complex root of f. Then, f(r) = 0, which can be rearranged to express rk as a linear combination of powers of r less than k. This equation can be used to reduce away any powers of r with exponent e ≥ k. For example, if f(x) = x2 + 1 and r is t... | Number Field Sieve | 0.812803 |
17,657 | Several groups are beyond proof-of-principle, and are working to scale up cost-effectively. Electrofuels have the potential to be disruptive if carbon-neutral electrofuels are cheaper than petroleum fuels, and if chemical feedstocks produced by electrosynthesis are cheaper than those refined from crude oil. Electrofuel... | Electrofuel | 0.812803 |
17,658 | Israeli Professor Yoram Palti, professor of physiology and biophysics at the Israel Institute of Technology, is the company's founder and chief technology officer. Novocure Ltd. owns 145 patents. | Alternating electric field therapy | 0.8128 |
17,659 | One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. Because of their non-empirical nature, formal sciences ar... | Formal sciences | 0.812794 |
17,660 | Principal branches of formal sciences are: logic (also a branch of philosophy), mathematics, computer science. | Formal sciences | 0.812794 |
17,661 | Non-B DNA can have significant implications for DNA biology and human health. For example, Z-DNA has been implicated in immunity and autoimmune diseases, such as lupus and arthritis. H-DNA has been implicated in genomic instability and cancer, and G-quadruplexes have been linked to telomere maintenance, oncogene activa... | Non B-DNA | 0.812791 |
17,662 | To put this in context of the next proof, the individual elements of the Gauss sum are in the cyclotomic field L = Q ( ζ p ) {\displaystyle L=\mathbb {Q} (\zeta _{p})} but the above formula shows that the sum itself is a generator of the unique quadratic field contained in L. Again, since the quadratic Gauss sum is an ... | Proofs of quadratic reciprocity | 0.812787 |
17,663 | The idea for the general proof follows the above supplemental case: Find an algebraic integer that somehow encodes the Legendre symbols for p, then find a relationship between Legendre symbols by computing the qth power of this algebraic integer modulo q in two different ways, one using Euler's criterion the other usin... | Proofs of quadratic reciprocity | 0.812787 |
17,664 | In the ring of integers O L = Z {\displaystyle {\mathcal {O}}_{L}=\mathbb {Z} } , choose any unramified prime ideal β of lying over q, and let ϕ ∈ Gal ( L / Q ) {\displaystyle \phi \in \operatorname {Gal} (L/\mathbb {Q} )} be the Frobenius automorphism associated to β; the characteristic property of ϕ {\displaystyle... | Proofs of quadratic reciprocity | 0.812787 |
17,665 | The proof of Quadratic Reciprocity using Gauss sums is one of the more common and classic proofs. These proofs work by comparing computations of single values in two different ways, one using Euler's Criterion and the other using the Binomial theorem. As an example of how Euler's criterion is used, we can use it to giv... | Proofs of quadratic reciprocity | 0.812787 |
17,666 | Of the elementary combinatorial proofs, there are two which apply types of double counting. One by Gotthold Eisenstein counts lattice points. Another applies Zolotarev's lemma to ( Z / p q Z ) × {\displaystyle (\mathbb {Z} /pq\mathbb {Z} )^{\times }} , expressed by the Chinese remainder theorem as ( Z / p Z ) × × ( Z /... | Proofs of quadratic reciprocity | 0.812787 |
17,667 | Using Euler's criterion, it follows that We can then say that But we can also compute τ p ( mod p ) {\textstyle \tau ^{p}{\pmod {p}}} using the binomial theorem. Because the cross terms in the binomial expansion all contain factors of p, we find that τ p ≡ ζ 8 p + ζ 8 − p ( mod p ) {\textstyle \tau ^{p}\equiv \zeta _{8... | Proofs of quadratic reciprocity | 0.812787 |
17,668 | Let ζ 8 = e 2 π i / 8 {\textstyle \zeta _{8}=e^{2\pi i/8}} , a primitive 8th root of unity and set τ = ζ 8 + ζ 8 − 1 {\textstyle \tau =\zeta _{8}+\zeta _{8}^{-1}} . Since ζ 8 2 = i {\textstyle \zeta _{8}^{2}=i} and ζ 8 − 2 = − i {\textstyle \zeta _{8}^{-2}=-i} we see that τ 2 = 2 {\textstyle \tau ^{2}=2} . Because τ {\... | Proofs of quadratic reciprocity | 0.812787 |
17,669 | In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published. | Proofs of quadratic reciprocity | 0.812787 |
17,670 | In this way, when a positive sample is added to the sensor, a characteristic, "signature-like" change in electrical potential occurs. BERA is the core technology behind the recently launched pan-European FOODSCAN project, about pesticide and food risk assessment in Europe. BERA has been used for the detection of human ... | Intracellular recording | 0.812784 |
17,671 | Minimum Information (MI) standards or reporting guidelines specify the minimum amount of meta data (information) and data required to meet a specific aim or aims in a clinical study. The "Minimum Information about a Neuroscience investigation" (MINI) family of reporting guideline documents aims to provide a consistent ... | Intracellular recording | 0.812784 |
17,672 | In 1901 he was awarded by the first Nobel Prize in Chemistry "in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions". After van 't Hoff, chemical kinetics deals with the experimental determination of reaction rates from which rat... | Chemical reaction kinetics | 0.812784 |
17,673 | The mathematical models that describe chemical reaction kinetics provide chemists and chemical engineers with tools to better understand and describe chemical processes such as food decomposition, microorganism growth, stratospheric ozone decomposition, and the chemistry of biological systems. These models can also be ... | Chemical reaction kinetics | 0.812784 |
17,674 | It is possible to make predictions about reaction rate constants for a reaction from free-energy relationships. The kinetic isotope effect is the difference in the rate of a chemical reaction when an atom in one of the reactants is replaced by one of its isotopes. Chemical kinetics provides information on residence tim... | Chemical reaction kinetics | 0.812784 |
17,675 | The activation energy for a chemical reaction can be provided when one reactant molecule absorbs light of suitable wavelength and is promoted to an excited state. The study of reactions initiated by light is photochemistry, one prominent example being photosynthesis. | Chemical reaction kinetics | 0.812784 |
17,676 | Textbooks and scientific publications related to computer science and numerical computation often use pseudocode in description of algorithms, so that all programmers can understand them, even if they do not all know the same programming languages. In textbooks, there is usually an accompanying introduction explaining ... | Pseudo code | 0.812783 |
17,677 | The stars and bars method is often introduced specifically to prove the following two theorems of elementary combinatorics concerning the number of solutions to an equation. | Stars and bars (combinatorics) | 0.812782 |
17,678 | If one wishes to count the number of ways to distribute seven indistinguishable one dollar coins among Amber, Ben, and Curtis so that each of them receives at least one dollar, one may observe that distributions are essentially equivalent to tuples of three positive integers whose sum is 7. (Here the first entry in the... | Stars and bars (combinatorics) | 0.812782 |
17,679 | For any pair of positive integers n and k, the number of k-tuples of positive integers whose sum is n is equal to the number of (k − 1)-element subsets of a set with n − 1 elements. For example, if n = 10 and k = 4, the theorem gives the number of solutions to x1 + x2 + x3 + x4 = 10 (with x1, x2, x3, x4 > 0) as the bin... | Stars and bars (combinatorics) | 0.812782 |
17,680 | For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of cardinality n taken from a set of size k, or equivalently, the number of multisets of cardinality k − 1 taken from a set of size n + 1. For example, if n = 10 and k = 4, the t... | Stars and bars (combinatorics) | 0.812782 |
17,681 | In this case, the weakened restriction of non-negativity instead of positivity means that we can place multiple bars between stars, before the first star and after the last star. For example, when n = 7 and k = 5, the tuple (4, 0, 1, 2, 0) may be represented by the following diagram: To see that there are ( n + k − 1 k... | Stars and bars (combinatorics) | 0.812782 |
17,682 | In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", "balls and bars", and "dots and dividers") is a graphical aid for deriving certain combinatorial theorems. It was popularized by William Feller in his classic book on probability. It can be used to solve many simple counting p... | Stars and bars (combinatorics) | 0.812782 |
17,683 | Both SLA and LMA are frequently used in plant ecology and biology. SLA is one of the components in plant growth analysis, and mathematically scales positively and linearly with the relative growth rate of a plant. LMA mathematically scales positively with the investments plants make per unit leaf area (amount of protei... | Specific leaf area | 0.812782 |
17,684 | Suppose there are n objects (represented here by stars) to be placed into k bins, such that all bins contain at least one object. The bins are distinguishable (say they are numbered 1 to k) but the n stars are not (so configurations are only distinguished by the number of stars present in each bin). A configuration is ... | Stars and bars (combinatorics) | 0.812782 |
17,685 | Many elementary word problems in combinatorics are resolved by the theorems above. | Stars and bars (combinatorics) | 0.812782 |
17,686 | He said that it was the responsibility of mathematical physics to reconstitute those principles, or to find a replacement for them (the greater goal being to return the field to unity), given that it had played the main role in questioning them only after consolidating them to begin with. Moreover, it was the value of ... | The Value of Science | 0.812782 |
17,687 | The failure of the laws entails the failure of the principles, because they must account for the results of experiment. To abolish these principles, products of the scientific thought of several centuries, without finding a new explanation that encompasses them (in the same manner that the "Physics of principles" encom... | The Value of Science | 0.812782 |
17,688 | In fact, it was difficult to do otherwise: they had discovered experimental facts which the principles could not account for, and which they evidently could not ignore. Poincaré himself remained relatively optimistic regarding the evolution of physics with respect to these severe experimental difficulties. He had littl... | The Value of Science | 0.812782 |
17,689 | His demonstration rests on the theory of The Mathematical Continuum (1893), one of Poincaré's earlier publications. Finally, Poincaré advances the idea of a fundamental relationship between the sciences of geometry and analysis. According to him, intuition has two major roles: to permit one to choose which route to fol... | The Value of Science | 0.812782 |
17,690 | This latter had in 1900 published the spectral laws governing blackbody radiation, which were the foundation of quantum mechanics. In 1905, the same year as the publication of The Value of Science, Albert Einstein published a decisive article on the photoelectric effect, which he based on the work of Planck. Despite th... | The Value of Science | 0.812782 |
17,691 | Poincaré argued that the advancement of the physical sciences would have to consider a new kind of determinism, giving a new place to chance. And in effect, the history of twentieth century physics is marked by a paradigm where probability reigns. In The Value of Science, Poincaré writes and repeats his enthusiasm for ... | The Value of Science | 0.812782 |
17,692 | However, in conjunction with the development of thermodynamics (at the time disputed), physicists began developing an energy-based physics. In both its mathematics and its fundamental ideas, this new physics seemed to contradict the Newtonian concept of particle interactions. Poincaré terms this the first crisis of mat... | The Value of Science | 0.812782 |
17,693 | In addition, physics offers solutions and reasoning - thus the development of infinitesimal calculus by Isaac Newton within the framework of Newtonian mechanics. Mathematical physics finds its scientific origins in the study of celestial mechanics. Initially, it was a consolidation of several fields of physics that dom... | The Value of Science | 0.812782 |
17,694 | The language of mathematics not only permits one to express scientific advancements, but also to take a step back to comprehend the broader world of nature. Mathematics demonstrates the extent of the specific and limited discoveries made by physicists. On the other hand, physics has a key role for the mathematician - a... | The Value of Science | 0.812782 |
17,695 | His ideas of a three dimensional space are given significance in this context. Poincaré states that mathematics (analysis) and physics are in the same spirit, that the two disciplines share a common aesthetic goal and that both can liberate humanity from its simple state. In a more pragmatic way, the interdependence of... | The Value of Science | 0.812782 |
17,696 | The primary objective of mathematical physics is not invention or discovery, but reformulation. It is an activity of synthesis, which permits one to assure the coherence of theories current at a given time. Poincaré recognized that it is impossible to systematize all of physics of a specific time period into one axioma... | The Value of Science | 0.812782 |
17,697 | According to him, it is not necessary that mathematics incorporate physics into itself, but must develop as an asset unto itself. This asset would be above all a tool: in the words of Poincaré, mathematics is "the only language in which could speak" to understand each other and to make themselves heard. This language ... | The Value of Science | 0.812782 |
17,698 | In the second part of his book, Poincaré studies the links between physics and mathematics. His approach, at once historical and technical, illustrates the preceding general ideas. Even though he was rarely an experimenter, Poincaré recognizes and defends the importance of experimentation, which must remain a pillar of... | The Value of Science | 0.812782 |
17,699 | In mathematics, the max–min inequality is as follows: For any function f: Z × W → R , {\displaystyle \ f:Z\times W\to \mathbb {R} \ ,} sup z ∈ Z inf w ∈ W f ( z , w ) ≤ inf w ∈ W sup z ∈ Z f ( z , w ) . {\displaystyle \sup _{z\in Z}\inf _{w\in W}f(z,w)\leq \inf _{w\in W}\sup _{z\in Z}f(z,w)\ .} When equality holds one ... | Max–min inequality | 0.812777 |
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