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300 | Only a week later, the Ciba group in Zurich, Leopold Ruzicka (1887–1976) and A. Wettstein, published their synthesis of testosterone. These independent partial syntheses of testosterone from a cholesterol base earned both Butenandt and Ruzicka the joint 1939 Nobel Prize in Chemistry. | Testosterone | 0.853807 |
301 | In 1927, the University of Chicago's Professor of Physiologic Chemistry, Fred C. Koch, established easy access to a large source of bovine testicles – the Chicago stockyards – and recruited students willing to endure the tedious work of extracting their isolates. In that year, Koch and his student, Lemuel McGee, derived 20 mg of a substance from a supply of 40 pounds of bovine testicles that, when administered to castrated roosters, pigs and rats, re-masculinized them. The group of Ernst Laqueur at the University of Amsterdam purified testosterone from bovine testicles in a similar manner in 1934, but the isolation of the hormone from animal tissues in amounts permitting serious study in humans was not feasible until three European pharmaceutical giants – Schering (Berlin, Germany), Organon (Oss, Netherlands) and Ciba – began full-scale steroid research and development programs in the 1930s. | Testosterone | 0.853807 |
302 | Quantum electrodynamics (QED), a relativistic quantum field theory of electrodynamics, is among the most stringently tested theories in physics. The most precise and specific tests of QED consist of measurements of the electromagnetic fine-structure constant, α, in various physical systems. Checking the consistency of such measurements tests the theory. Tests of a theory are normally carried out by comparing experimental results to theoretical predictions. | Tests of electromagnetism | 0.853791 |
303 | Recent developments in the field have included the generation of X-ray free electron lasers, allowing analysis of the dynamics and motion of biological molecules, and the use of structural biology in assisting synthetic biology.In the late 1930s and early 1940s, the combination of work done by Isidor Rabi, Felix Bloch, and Edward Mills Purcell led to the development of nuclear magnetic resonance (NMR). Currently, solid-state NMR is widely used in the field of structural biology to determine the structure and dynamic nature of proteins (protein NMR).In 1990, Richard Henderson produced the first three-dimensional, high resolution image of bacteriorhodopsin using cryogenic electron microscopy (cryo-EM). Since then, cryo-EM has emerged as an increasingly popular technique to determine three-dimensional, high resolution structures of biological images.More recently, computational methods have been developed to model and study biological structures. | Structural Biology | 0.853673 |
304 | For example, molecular dynamics (MD) is commonly used to analyze the dynamic movements of biological molecules. In 1975, the first simulation of a biological folding process using MD was published in Nature. Recently, protein structure prediction was significantly improved by a new machine learning method called AlphaFold. Some claim that computational approaches are starting to lead the field of structural biology research. | Structural Biology | 0.853673 |
305 | In 1912 Max Von Laue directed X-rays at crystallized copper sulfate generating a diffraction pattern. These experiments led to the development of X-ray crystallography, and its usage in exploring biological structures. In 1951, Rosalind Franklin and Maurice Wilkins used X-ray diffraction patterns to capture the first image of deoxyribonucleic acid (DNA). Francis Crick and James Watson modeled the double helical structure of DNA using this same technique in 1953 and received the Nobel Prize in Medicine along with Wilkins in 1962.Pepsin crystals were the first proteins to be crystallized for use in X-ray diffraction, by Theodore Svedberg who received the 1962 Nobel Prize in Chemistry. | Structural Biology | 0.853673 |
306 | For example, structural biology tools have enabled virologists to understand how the HIV envelope allows the virus to evade human immune responses.Structural biology is also an important component of drug discovery. Scientists can identify targets using genomics, study those targets using structural biology, and develop drugs that are suited for those targets. Specifically, ligand-NMR, mass spectrometry, and X-ray crystallography are commonly used techniques in the drug discovery process. | Structural Biology | 0.853673 |
307 | For example, researchers have used structural biology to better understand Met, a protein encoded by a protooncogene that is an important drug target in cancer. Similar research has been conducted for HIV targets to treat people with AIDS. Researchers are also developing new antimicrobials for mycobacterial infections using structure-driven drug discovery. | Structural Biology | 0.853673 |
308 | Structural biologists have made significant contributions towards understanding the molecular components and mechanisms underlying human diseases. For example, cryo-EM and ssNMR have been used to study the aggregation of amyloid fibrils, which are associated with Alzheimer's disease, Parkinson's disease, and type II diabetes. In addition to amyloid proteins, scientists have used cryo-EM to produce high resolution models of tau filaments in the brain of Alzheimer's patients which may help develop better treatments in the future. Structural biology tools can also be used to explain interactions between pathogens and hosts. | Structural Biology | 0.853673 |
309 | The Faraday paradox was a once inexplicable aspect of the reaction between nitric acid and steel. Around 1830, the English scientist Michael Faraday found that diluted nitric acid would attack steel, but concentrated nitric acid would not. The attempt to explain this discovery led to advances in electrochemistry. | Faraday paradox (electrochemistry) | 0.85354 |
310 | In atomic, molecular, and solid-state physics, the electric field gradient (EFG) measures the rate of change of the electric field at an atomic nucleus generated by the electronic charge distribution and the other nuclei. The EFG couples with the nuclear electric quadrupole moment of quadrupolar nuclei (those with spin quantum number greater than one-half) to generate an effect which can be measured using several spectroscopic methods, such as nuclear magnetic resonance (NMR), microwave spectroscopy, electron paramagnetic resonance (EPR, ESR), nuclear quadrupole resonance (NQR), Mössbauer spectroscopy or perturbed angular correlation (PAC). The EFG is non-zero only if the charges surrounding the nucleus violate cubic symmetry and therefore generate an inhomogeneous electric field at the position of the nucleus. | Electric field gradient | 0.853502 |
311 | Discrete Mathematics: Deals with separate and distinct mathematical structures, including topics such as combinatorics, graph theory, and algorithms. 8. Decision Mathematics: Applies mathematical techniques to solve real-world problems related to optimization, networks, and decision-making. 9. Financial Mathematics: Applies mathematical concepts to analyze financial markets, investments, and risk management. | Advanced level mathematics | 0.853341 |
312 | 6. Statistics: Involves collecting, analyzing, and interpreting data, including topics like probability, hypothesis testing, regression analysis, and sampling. 7. | Advanced level mathematics | 0.853341 |
313 | Applied Mathematics: Focuses on practical applications of mathematical concepts to solve real-world problems in various fields. 5. Mechanics: Focuses on the study of motion, forces, and vectors, particularly relevant for physics or engineering interests. | Advanced level mathematics | 0.853341 |
314 | 3. Pure Mathematics: Explores advanced topics in algebra, calculus, and mathematical proofs. 4. | Advanced level mathematics | 0.853341 |
315 | ```List of subjects in A Level Mathematics``` 1. Core Mathematics: Covers foundational topics like algebra, calculus, trigonometry, and coordinate geometry. 2. Further Mathematics: Expands upon Core Mathematics with additional areas such as complex numbers, matrices, differential equations, and numerical methods. | Advanced level mathematics | 0.853341 |
316 | Prior to the 2017 reform, the basic A-Level course consisted of six modules, four pure modules (C1, C2, C3, and C4) and two applied modules in Statistics, Mechanics and/or Decision Mathematics. The C1 through C4 modules are referred to by A-level textbooks as "Core" modules, encompassing the major topics of mathematics such as logarithms, differentiation/integration and geometric/arithmetic progressions. The two chosen modules for the final two parts of the A-Level are determined either by a student's personal choices, or the course choice of their school/college, though it commonly took the form of S1 (Statistics) and M1 (Mechanics). | Advanced level mathematics | 0.853341 |
317 | Paper 1: Pure Mathematics Paper 2: Pure Mathematics and Statistics Paper 3: Pure Mathematics and Mechanics | Advanced level mathematics | 0.853341 |
318 | Paper 1: Pure Mathematics Paper 2: Content on Paper 1 plus Mechanics Paper 3: Content on Paper 1 plus Statistics | Advanced level mathematics | 0.853341 |
319 | Paper 1: Pure Mathematics 1 Paper 2: Pure Mathematics 2 Paper 3: Statistics and Mechanics | Advanced level mathematics | 0.853341 |
320 | Most students will complete three modules in one year, which will create an AS-level qualification in their own right and will complete the A-level course the following year—with three more modules. The system in which mathematics is assessed is changing for students starting courses in 2017 (as part of the A-level reforms first introduced in 2015), where the reformed specifications have reverted to a linear structure with exams taken only at the end of the course in a single sitting. In addition, while schools could choose freely between taking Statistics, Mechanics or Discrete Mathematics (also known as Decision Mathematics) modules with the ability to specialise in one branch of applied Mathematics in the older modular specification, in the new specifications, both Mechanics and Statistics were made compulsory, with Discrete Mathematics being made exclusive as an option to students pursuing a Further Mathematics course. The first assessment opportunity for the new specification is 2018 and 2019 for A-levels in Mathematics and Further Mathematics, respectively. | Advanced level mathematics | 0.853341 |
321 | Advanced Level (A-Level) Mathematics is a qualification of further education taken in the United Kingdom (and occasionally other countries as well). In the UK, A-Level exams are traditionally taken by 17-18 year-olds after a two-year course at a sixth form or college. Advanced Level Further Mathematics is often taken by students who wish to study a mathematics-based degree at university, or related degree courses such as physics or computer science. Like other A-level subjects, mathematics has been assessed in a modular system since the introduction of Curriculum 2000, whereby each candidate must take six modules, with the best achieved score in each of these modules (after any retake) contributing to the final grade. | Advanced level mathematics | 0.853341 |
322 | The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value. The problem concerns a game of chance with two players who have equal chances of winning each round. | Problem of points | 0.85334 |
323 | The problem arose again around 1654 when Chevalier de Méré posed it to Blaise Pascal. Pascal discussed the problem in his ongoing correspondence with Pierre de Fermat. Through this discussion, Pascal and Fermat not only provided a convincing, self-consistent solution to this problem, but also developed concepts that are still fundamental to probability theory. The starting insight for Pascal and Fermat was that the division should not depend so much on the history of the part of the interrupted game that actually took place, as on the possible ways the game might have continued, were it not interrupted. | Problem of points | 0.85334 |
324 | While expression levels as the gene level can be more or less accurately depicted by second generation sequencing, transcript-level information still remains an important challenge. As a consequence, the role of alternative splicing in molecular biology remains largely elusive. Third generation sequencing technologies hold promising prospects in resolving this issue by enabling sequencing of mRNA molecules at their full lengths. | Third-generation sequencing | 0.853317 |
325 | Transcriptomics is the study of the transcriptome, usually by characterizing the relative abundances of messenger RNA molecules in the tissue under study. According to the central dogma of molecular biology, genetic information flows from double stranded DNA molecules to single stranded mRNA molecules where they can be readily translated into function protein molecules. By studying the transcriptome, one can gain valuable insight into the regulation of gene expressions. | Third-generation sequencing | 0.853317 |
326 | Alternative splicing (AS) is the process by which a single gene may give rise to multiple distinct mRNA transcripts and consequently different protein translations. Some evidence suggests that AS is a ubiquitous phenomenon and may play a key role in determining the phenotypes of organisms, especially in complex eukaryotes; all eukaryotes contain genes consisting of introns that may undergo AS. In particular, it has been estimated that AS occurs in 95% of all human multi-exon genes. AS has undeniable potential to influence myriad biological processes. Advancing knowledge in this area has critical implications for the study of biology in general. | Third-generation sequencing | 0.853317 |
327 | Electrokinetic phenomena refers to a variety of effects resulting from an electrical double layer. A noteworthy example is electrophoresis, where a charged particle suspended in a media will move as a result of an applied electrical field. Electrophoresis is widely used in biochemistry to distinguish molecules, such as proteins, based on size and charge. Other examples include electro-osmosis, sedimentation potential, and streaming potential. | Surface charge | 0.853223 |
328 | Associativity An operation ∗ {\displaystyle *} is associative if for every x, y and z in the algebraic structure. Left distributivity An operation ∗ {\displaystyle *} is left distributive with respect to another operation + {\displaystyle +} if for every x, y and z in the algebraic structure (the second operation is denoted here as +, because the second operation is addition in many common examples). Right distributivity An operation ∗ {\displaystyle *} is right distributive with respect to another operation + {\displaystyle +} if for every x, y and z in the algebraic structure. Distributivity An operation ∗ {\displaystyle *} is distributive with respect to another operation + {\displaystyle +} if it is both left distributive and right distributive. If the operation ∗ {\displaystyle *} is commutative, left and right distributivity are both equivalent to distributivity. | Algebraic structure | 0.853093 |
329 | An axiom of an algebraic structure often has the form of an identity, that is, an equation such that the two sides of the equals sign are expressions that involve operations of the algebraic structure and variables. If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true. Here are some common examples. Commutativity An operation ∗ {\displaystyle *} is commutative if for every x and y in the algebraic structure. | Algebraic structure | 0.853093 |
330 | Module: an abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Counting the ring operations these systems have at least three operations. Vector space: a module where the ring R is a division ring or field.Algebra over a field: a module over a field, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication. Inner product space: an F vector space V with a definite bilinear form V × V → F. | Algebraic structure | 0.853093 |
331 | {\displaystyle T'=T{\sqrt {\frac {c-v}{c+v}}}.} Here v > 0 indicates a receding source, and v < 0 indicates an approaching source. This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of c. An example is found in the cosmic microwave background radiation, which exhibits a dipole anisotropy from the Earth's motion relative to this blackbody radiation field. | Black body radiation | 0.852918 |
332 | In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values.The more inferences are made, the more likely erroneous inferences become. Several statistical techniques have been developed to address that problem, typically by requiring a stricter significance threshold for individual comparisons, so as to compensate for the number of inferences being made. | Multiple comparisons problem | 0.85289 |
333 | The electric potential at a point r in a static electric field E is given by the line integral where C is an arbitrary path from some fixed reference point to r. In electrostatics, the Maxwell-Faraday equation reveals that the curl ∇ × E {\textstyle \nabla \times \mathbf {E} } is zero, making the electric field conservative. Thus, the line integral above does not depend on the specific path C chosen but only on its endpoints, making V E {\textstyle V_{\mathbf {E} }} well-defined everywhere. The gradient theorem then allows us to write: This states that the electric field points "downhill" towards lower voltages. By Gauss's law, the potential can also be found to satisfy Poisson's equation: where ρ is the total charge density and ∇ ⋅ {\textstyle \mathbf {\nabla } \cdot } denotes the divergence. | Electrical potential difference | 0.852861 |
334 | A number of Nobel Prizes have been awarded for steroid research, including: 1927 (Chemistry) Heinrich Otto Wieland — Constitution of bile acids and sterols and their connection to vitamins 1928 (Chemistry) Adolf Otto Reinhold Windaus — Constitution of sterols and their connection to vitamins 1939 (Chemistry) Adolf Butenandt and Leopold Ružička — Isolation and structural studies of steroid sex hormones, and related studies on higher terpenes 1950 (Physiology or Medicine) Edward Calvin Kendall, Tadeus Reichstein, and Philip Hench — Structure and biological effects of adrenal hormones 1965 (Chemistry) Robert Burns Woodward — In part, for the synthesis of cholesterol, cortisone, and lanosterol 1969 (Chemistry) Derek Barton and Odd Hassel — Development of the concept of conformation in chemistry, emphasizing the steroid nucleus 1975 (Chemistry) Vladimir Prelog — In part, for developing methods to determine the stereochemical course of cholesterol biosynthesis from mevalonic acid via squalene | Steroid biosynthesis | 0.852769 |
335 | Steroids can be classified based on their chemical composition. One example of how MeSH performs this classification is available at the Wikipedia MeSH catalog. Examples of this classification include: In biology, it is common to name the above steroid classes by the number of carbon atoms present when referring to hormones: C18-steroids for the estranes (mostly estrogens), C19-steroids for the androstanes (mostly androgens), and C21-steroids for the pregnanes (mostly corticosteroids). The classification "17-ketosteroid" is also important in medicine. The gonane (steroid nucleus) is the parent 17-carbon tetracyclic hydrocarbon molecule with no alkyl sidechains. | Steroid biosynthesis | 0.852769 |
336 | In solving mathematical equations, particularly linear simultaneous equations, differential equations and integral equations, the terminology homogeneous is often used for equations with some linear operator L on the LHS and 0 on the RHS. In contrast, an equation with a non-zero RHS is called inhomogeneous or non-homogeneous, as exemplified by Lf = g,with g a fixed function, which equation is to be solved for f. Then any solution of the inhomogeneous equation may have a solution of the homogeneous equation added to it, and still remain a solution. For example in mathematical physics, the homogeneous equation may correspond to a physical theory formulated in empty space, while the inhomogeneous equation asks for more 'realistic' solutions with some matter, or charged particles. | Sides of an equation | 0.852767 |
337 | The Nullstellensatz (German for "zero-locus theorem") is a theorem, first proved by David Hilbert, which extends to the multivariate case some aspects of the fundamental theorem of algebra. It is foundational for algebraic geometry, as establishing a strong link between the algebraic properties of K {\displaystyle K} and the geometric properties of algebraic varieties, that are (roughly speaking) set of points defined by implicit polynomial equations. The Nullstellensatz, has three main versions, each being a corollary of any other. | Polynomial expression | 0.852759 |
338 | Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety. | Polynomial expression | 0.852759 |
339 | In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. | Polynomial expression | 0.852759 |
340 | The statement is: a set of polynomials S in K {\displaystyle K} has a common zero in an algebraically closed field containing K, if and only if 1 does not belong to the ideal generated by S, that is, if 1 is not a linear combination of elements of S with polynomial coefficients. The second version generalizes the fact that the irreducible univariate polynomials over the complex numbers are associate to a polynomial of the form X − α . {\displaystyle X-\alpha .} The statement is: If K is algebraically closed, then the maximal ideals of K {\displaystyle K} have the form ⟨ X 1 − α 1 , … , X n − α n ⟩ . {\displaystyle \langle X_{1}-\alpha _{1},\ldots ,X_{n}-\alpha _{n}\rangle .} | Polynomial expression | 0.852759 |
341 | Also provably unsolvable are so-called undecidable problems, such as the halting problem for Turing machines. Some well-known difficult abstract problems that have been solved relatively recently are the four-colour theorem, Fermat's Last Theorem, and the Poincaré conjecture. Computers do not need to have a sense of the motivations of mathematicians in order to do what they do. Formal definitions and computer-checkable deductions are absolutely central to mathematical science. | Mathematical problem | 0.852708 |
342 | Abstract mathematical problems arise in all fields of mathematics. While mathematicians usually study them for their own sake, by doing so, results may be obtained that find application outside the realm of mathematics. Theoretical physics has historically been a rich source of inspiration. Some abstract problems have been rigorously proved to be unsolvable, such as squaring the circle and trisecting the angle using only the compass and straightedge constructions of classical geometry, and solving the general quintic equation algebraically. | Mathematical problem | 0.852708 |
343 | The polymerase chain reaction (PCR) is a commonly used molecular biology tool for amplifying DNA, and various techniques for PCR optimization which have been developed by molecular biologists to improve PCR performance and minimize failure. | Polymerase chain reaction optimization | 0.852652 |
344 | M. vaginatus stabilizes soil using a polysaccharide sheath that binds to sand particles and absorbs water.Some of these organisms contribute significantly to global ecology and the oxygen cycle. The tiny marine cyanobacterium Prochlorococcus was discovered in 1986 and accounts for more than half of the photosynthesis of the open ocean. Circadian rhythms were once thought to only exist in eukaryotic cells but many cyanobacteria display a bacterial circadian rhythm. | Blue-green Algae | 0.852634 |
345 | One of the most critical processes determining cyanobacterial eco-physiology is cellular death. Evidence supports the existence of controlled cellular demise in cyanobacteria, and various forms of cell death have been described as a response to biotic and abiotic stresses. However, cell death research in cyanobacteria is a relatively young field and understanding of the underlying mechanisms and molecular machinery underpinning this fundamental process remains largely elusive. However, reports on cell death of marine and freshwater cyanobacteria indicate this process has major implications for the ecology of microbial communities/ Different forms of cell demise have been observed in cyanobacteria under several stressful conditions, and cell death has been suggested to play a key role in developmental processes, such as akinete and heterocyst differentiation, as well as strategy for population survival. | Blue-green Algae | 0.852634 |
346 | Helicopter dynamics is a field within aerospace engineering concerned with theoretical and practical aspects of helicopter flight. Its comprises helicopter aerodynamics, stability, control, structural dynamics, vibration, and aeroelastic and aeromechanical stability.By studying the forces in helicopter flight, improved helicopter designs can be made. In 2013, stereophotogrammetry was used to measure the dynamics of a Robinson R44 helicopter during the hover. | Helicopter dynamics | 0.85258 |
347 | For detailed discussions of some solution methods see: Tschirnhaus transformation (general method, not guaranteed to succeed); Bezout method (general method, not guaranteed to succeed); Ferrari method (solutions for degree 4); Euler method (solutions for degree 4); Lagrange method (solutions for degree 4); Descartes method (solutions for degree 2 or 4);A quartic equation a x 4 + b x 3 + c x 2 + d x + e = 0 {\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0} with a ≠ 0 {\displaystyle a\neq 0} may be reduced to a quadratic equation by a change of variable provided it is either biquadratic (b = d = 0) or quasi-palindromic (e = a, d = b). Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions. | Polynomial equation | 0.852539 |
348 | For example, x 5 − 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coefficients and y 4 + x y 2 − x 3 3 + x y 2 + y 2 + 1 7 = 0 {\displaystyle y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0} is a multivariate polynomial equation over the rationals. Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations). | Polynomial equation | 0.852539 |
349 | In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0 {\displaystyle P=0} where P is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term algebraic equation refers only to univariate equations, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables. In the case of several variables (the multivariate case), the term polynomial equation is usually preferred to algebraic equation. | Polynomial equation | 0.852539 |
350 | In computer science (specifically computational complexity theory), the worst-case complexity measures the resources (e.g. running time, memory) that an algorithm requires given an input of arbitrary size (commonly denoted as n in asymptotic notation). It gives an upper bound on the resources required by the algorithm. In the case of running time, the worst-case time complexity indicates the longest running time performed by an algorithm given any input of size n, and thus guarantees that the algorithm will finish in the indicated period of time. The order of growth (e.g. linear, logarithmic) of the worst-case complexity is commonly used to compare the efficiency of two algorithms. The worst-case complexity of an algorithm should be contrasted with its average-case complexity, which is an average measure of the amount of resources the algorithm uses on a random input. | Worst case complexity | 0.852491 |
351 | "Some researchers suggest that AI designers specify their desired goals by listing forbidden actions or by formalizing ethical rules (as with Asimov's Three Laws of Robotics). However, Russell and Norvig argued that this approach overlooks the complexity of human values: "It is certainly very hard, and perhaps impossible, for mere humans to anticipate and rule out in advance all the disastrous ways the machine could choose to achieve a specified objective. "Additionally, even if an AI system fully understands human intentions, it may still disregard them, because following human intentions may not be its objective (unless it is already fully aligned). | Alignment problem | 0.852467 |
352 | In the field of artificial intelligence (AI), AI alignment research aims to steer AI systems towards humans' intended goals, preferences, or ethical principles. An AI system is considered aligned if it advances the intended objectives. A misaligned AI system pursues some objectives, but not the intended ones.It can be challenging for AI designers to align an AI system because it can be difficult for them to specify the full range of desired and undesired behaviors. To avoid this difficulty, they typically use simpler proxy goals, such as gaining human approval. | Alignment problem | 0.852467 |
353 | As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. | Group axioms | 0.85238 |
354 | Certain abelian group structures had been used implicitly in Carl Friedrich Gauss's number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers.The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. | Group axioms | 0.85238 |
355 | After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.The third field contributing to group theory was number theory. | Group axioms | 0.85238 |
356 | At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θ n = 1 {\displaystyle \theta ^{n}=1} (1854) gives the first abstract definition of a finite group.Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. | Group axioms | 0.85238 |
357 | (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4, MR 1269324, OCLC 36131259 Weinberg, Steven (1972), Gravitation and Cosmology, New York: John Wiley & Sons, ISBN 0-471-92567-5. Welsh, Dominic (1989), Codes and Cryptography, Oxford: Clarendon Press, ISBN 978-0-19-853287-3. | Group axioms | 0.85238 |
358 | Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Mathematical Association of America, ISBN 978-0-88385-511-9. Shatz, Stephen S. (1972), Profinite Groups, Arithmetic, and Geometry, Princeton University Press, ISBN 978-0-691-08017-8, MR 0347778 Simons, Jack (2003), An Introduction to Theoretical Chemistry, Cambridge University Press, ISBN 978-0-521-53047-7 Solomon, Ronald (2018), "The classification of finite simple groups: A progress report", Notices of the AMS, 65 (6): 1, doi:10.1090/noti1689 Stewart, Ian (2015), Galois Theory (4th ed. | Group axioms | 0.85238 |
359 | Rosen, Kenneth H. (2000), Elementary Number Theory and its Applications (4th ed. ), Addison-Wesley, ISBN 978-0-201-87073-2, MR 1739433. | Group axioms | 0.85238 |
360 | Naber, Gregory L. (2003), The Geometry of Minkowski Spacetime, New York: Dover Publications, ISBN 978-0-486-43235-9, MR 2044239. Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. | Group axioms | 0.85238 |
361 | Kuga, Michio (1993), Galois' Dream: Group Theory and Differential Equations, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3688-3, MR 1199112. Kurzweil, Hans; Stellmacher, Bernd (2004), The Theory of Finite Groups, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-40510-0, MR 2014408. Lay, David (2003), Linear Algebra and Its Applications, Addison-Wesley, ISBN 978-0-201-70970-4. | Group axioms | 0.85238 |
362 | 588–596, ISBN 0-201-02918-9. Gollmann, Dieter (2011), Computer Security (2nd ed. ), West Sussex, England: John Wiley & Sons, Ltd., ISBN 978-0-470-74115-3 Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1. | Group axioms | 0.85238 |
363 | Dudek, Wiesław A. (2001), "On some old and new problems in n-ary groups" (PDF), Quasigroups and Related Systems, 8: 15–36, MR 1876783. Eliel, Ernest; Wilen, Samuel; Mander, Lewis (1994), Stereochemistry of Organic Compounds, Wiley, ISBN 978-0-471-01670-0 Ellis, Graham (2019), "6.4 Triangle groups", An Invitation to Computational Homotopy, Oxford University Press, pp. | Group axioms | 0.85238 |
364 | (2002), Universal Algebra and Applications in Theoretical Computer Science, London: CRC Press, ISBN 978-1-58488-254-1. Dove, Martin T (2003), Structure and Dynamics: An Atomic View of Materials, Oxford University Press, p. 265, ISBN 0-19-850678-3. | Group axioms | 0.85238 |
365 | (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie, 42 (2): 475–507, arXiv:math.MG/9911185, MR 1865535. Coornaert, M.; Delzant, T.; Papadopoulos, A. | Group axioms | 0.85238 |
366 | (1993), Group Theory and Chemistry, New York: Dover Publications, ISBN 978-0-486-67355-4. Borel, Armand (1991), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (2nd ed. | Group axioms | 0.85238 |
367 | Any finite abelian group is isomorphic to a product of finite cyclic groups; this statement is part of the fundamental theorem of finitely generated abelian groups. Any group of prime order p {\displaystyle p} is isomorphic to the cyclic group Z p {\displaystyle \mathrm {Z} _{p}} (a consequence of Lagrange's theorem). Any group of order p 2 {\displaystyle p^{2}} is abelian, isomorphic to Z p 2 {\displaystyle \mathrm {Z} _{p^{2}}} or Z p × Z p {\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}} . But there exist nonabelian groups of order p 3 {\displaystyle p^{3}} ; the dihedral group D 4 {\displaystyle \mathrm {D} _{4}} of order 2 3 {\displaystyle 2^{3}} above is an example. | Group axioms | 0.85238 |
368 | A Lie group is a group that also has the structure of a differentiable manifold; informally, this means that it looks locally like a Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth. A standard example is the general linear group introduced above: it is an open subset of the space of all n {\displaystyle n} -by- n {\displaystyle n} matrices, because it is given by the inequality where A {\displaystyle A} denotes an n {\displaystyle n} -by- n {\displaystyle n} matrix.Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time, are basic symmetries of the laws of mechanics. | Group axioms | 0.85238 |
369 | Finally, the inverse of a / b {\displaystyle a/b} is b / a {\displaystyle b/a} , therefore the axiom of the inverse element is satisfied. The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if division by other than zero is possible, such as in Q {\displaystyle \mathbb {Q} } – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities. | Group axioms | 0.85238 |
370 | This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.A group action gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups. | Group axioms | 0.85238 |
371 | In particular, it includes the study of equations that involve nth roots and, more generally, algebraic expressions. This makes the term algebraic equation ambiguous outside the context of the old problem. So the term polynomial equation is generally preferred when this ambiguity may occur, specially when considering multivariate equations. | Polynomial equations | 0.852345 |
372 | The term "algebraic equation" dates from the time when the main problem of algebra was to solve univariate polynomial equations. This problem was completely solved during the 19th century; see Fundamental theorem of algebra, Abel–Ruffini theorem and Galois theory. Since then, the scope of algebra has been dramatically enlarged. | Polynomial equations | 0.852345 |
373 | The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets). Univariate algebraic equations over the rationals (i.e., with rational coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form of radical expressions, like x = 1 + 5 2 {\displaystyle x={\frac {1+{\sqrt {5}}}{2}}} for the positive solution of x 2 − x − 1 = 0 {\displaystyle x^{2}-x-1=0} . The ancient Egyptians knew how to solve equations of degree 2 in this manner. | Polynomial equations | 0.852345 |
374 | In particular the equation P = Q {\displaystyle P=Q} is equivalent to P − Q = 0 {\displaystyle P-Q=0} . It follows that the study of algebraic equations is equivalent to the study of polynomials. A polynomial equation over the rationals can always be converted to an equivalent one in which the coefficients are integers. | Polynomial equations | 0.852345 |
375 | A Diophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is the study of the solutions in an algebraically closed field of multivariate polynomial equations. Two equations are equivalent if they have the same set of solutions. | Polynomial equations | 0.852345 |
376 | The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, with rational coefficients). Galois theory was introduced by Évariste Galois to specify criteria for deciding if an algebraic equation may be solved in terms of radicals. In field theory, an algebraic extension is an extension such that every element is a root of an algebraic equation over the base field. Transcendental number theory is the study of the real numbers which are not solutions to an algebraic equation over the rationals. | Polynomial equations | 0.852345 |
377 | There is a vast body of methods for solving various kinds of differential equations, both numerically and analytically. A particular class of problem that can be considered to belong here is integration, and the analytic methods for solving this kind of problems are now called symbolic integration. Solutions of differential equations can be implicit or explicit. | Solving equations | 0.852289 |
378 | Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra. | Solving equations | 0.852289 |
379 | Smaller systems of linear equations can be solved likewise by methods of elementary algebra. For solving larger systems, algorithms are used that are based on linear algebra. See Gaussian elimination | Solving equations | 0.852289 |
380 | Cystic fibrosis, for example, is caused by mutations in a single gene called CFTR and is inherited as a recessive trait.Other diseases are influenced by genetics, but the genes a person gets from their parents only change their risk of getting a disease. Most of these diseases are inherited in a complex way, with either multiple genes involved, or coming from both genes and the environment. As an example, the risk of breast cancer is 50 times higher in the families most at risk, compared to the families least at risk. | Introduction to genetics | 0.852251 |
381 | Genetics is the study of genes and tries to explain what they are and how they work. Genes are how living organisms inherit features or traits from their ancestors; for example, children usually look like their parents because they have inherited their parents' genes. Genetics tries to identify which traits are inherited and to explain how these traits are passed from generation to generation. Some traits are part of an organism's physical appearance, such as eye color, height or weight. | Introduction to genetics | 0.852251 |
382 | The combination of mutations creating new alleles at random, and natural selection picking out those that are useful, causes an adaptation. This is when organisms change in ways that help them to survive and reproduce. Many such changes, studied in evolutionary developmental biology, affect the way the embryo develops into an adult body. | Introduction to genetics | 0.852251 |
383 | A population of organisms evolves when an inherited trait becomes more common or less common over time. For instance, all the mice living on an island would be a single population of mice: some with white fur, some gray. If over generations, white mice became more frequent and gray mice less frequent, then the color of the fur in this population of mice would be evolving. In terms of genetics, this is called an increase in allele frequency. | Introduction to genetics | 0.852251 |
384 | This is because of the large number of genes involved; this makes the trait very variable and people are of many different heights. Despite a common misconception, the green/blue eye traits are also inherited in this complex inheritance model. Inheritance can also be complicated when the trait depends on the interaction between genetics and environment. For example, malnutrition does not change traits like eye color, but can stunt growth. | Introduction to genetics | 0.852251 |
385 | Another, less common, not-in-place, version of quicksort uses O(n) space for working storage and can implement a stable sort. The working storage allows the input array to be easily partitioned in a stable manner and then copied back to the input array for successive recursive calls. Sedgewick's optimization is still appropriate. | Quicksort | 0.852218 |
386 | Later, Hoare learned about ALGOL and its ability to do recursion that enabled him to publish an improved version of the algorithm in ALGOL in Communications of the Association for Computing Machinery, the premier computer science journal of the time. The ALGOL code is published in Communications of the ACM (CACM), Volume 4, Issue 7 July 1961, pp 321 Algorithm 63: partition and Algorithm 64: Quicksort. Quicksort gained widespread adoption, appearing, for example, in Unix as the default library sort subroutine. | Quicksort | 0.852218 |
387 | {\displaystyle T(n)=O(n)+2T\left({\frac {n}{2}}\right).} The master theorem for divide-and-conquer recurrences tells us that T(n) = O(n log n). | Quicksort | 0.852218 |
388 | In graph theory, the degree diameter problem is the problem of finding the largest possible graph G (in terms of the size of its vertex set V) of diameter k such that the largest degree of any of the vertices in G is at most d. The size of G is bounded above by the Moore bound; for 1 < k and 2 < d only the Petersen graph, the Hoffman-Singleton graph, and possibly one more graph (not yet proven to exist) of diameter k = 2 and degree d = 57 attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound. | Degree diameter problem | 0.852116 |
389 | Topological vector space: a vector space whose M has a compatible topology. Normed vector space: a vector space with a compatible norm. If such a space is complete (as a metric space) then it is called a Banach space. Hilbert space: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure. Vertex operator algebra Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology. | Algebraic structures | 0.852059 |
390 | Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure. Topological group: a group with a topology compatible with the group operation. Lie group: a topological group with a compatible smooth manifold structure. | Algebraic structures | 0.852059 |
391 | This method gives more flexibility to the artificial intelligences, often resulting in stronger and more generalizable models. In 2017, the first-ever MPNN model, a deep tensor neural network, was used to calculate the properties of small organic molecules. Such technology was commercialized, leading to the development of Matlantis in 2022, which extracts properties through both the forward and backward passes. Matlantis, which can simulate 72 elements, handle up to 20,000 atoms at a time, and execute calculations up to 20,000,000 times faster than density functional theory with almost indistinguishable accuracy, showcases the power of machine learning potentials in the age of artificial intelligence. == References == | Machine learning potential | 0.85204 |
392 | These models, called message-passing neural networks (MPNNs), are graph neural networks. Treating molecules as three-dimensional graphs (where atoms are nodes and bonds are edges), the model intakes feature vectors describing the atoms, and iteratively updates these feature vectors as information about neighboring atoms is processed through message functions and convolutions. These feature vectors are then used to predict the final potentials. | Machine learning potential | 0.85204 |
393 | From this data, a separate atomic neural network is trained for each element; each atomic neural network is evaluated whenever that element occurs in the given structure, and then the results are pooled together at the end. This process - in particular, the atom-centered symmetry functions, which convey translational, rotational, and permutational invariances - has greatly improved machine learning potentials by significantly constraining the neural networks' search space. Other models use a similar process but emphasize bonds over atoms, using pair symmetry functions and training one neural network per atom pair.Still other models, rather than using predetermined symmetry-dictating functions, prefer to learn their own descriptors instead. | Machine learning potential | 0.85204 |
394 | There exist some nonlocal models, but these have been experimental for almost a decade. For most systems, reasonable cutoff radii enable highly accurate results.Almost all neural networks intake atomic coordinates and output potential energies. For some, these atomic coordinates are converted into atom-centered symmetry functions. | Machine learning potential | 0.85204 |
395 | These models thus remained confined to academia. Modern neural networks construct highly-accurate, computationally-light potentials because theoretical understanding of materials science was increasingly built into their architectures and preprocessing. Almost all are local, accounting for all interactions between an atom and its neighbor up to some cutoff radius. | Machine learning potential | 0.85204 |
396 | Beginning in the 1990s, researchers have employed machine learning programs to construct interatomic potentials, mapping atomic structures to their potential energies. Such machine learning potentials promised to fill the gap between density functional theory, a highly-accurate but computationally-intensive simulation program, and empirically derived or intuitively-approximated potentials, which were far computationally lighter but substantially less accurate. Improvements in artificial intelligence technology have served to heighten the accuracy of MLPs while lowering their computational cost, increasing machine learning's role in fitting potentials.Machine learning potentials began by using neural networks to tackle low dimensional systems. While promising, these models could not systematically account for interatomic energy interactions; they could be applied to small molecules in a vacuum and molecules interacting with frozen surfaces, but not much else, and even in these applications often relied on force fields or potentials derived empirically or with simulations. | Machine learning potential | 0.85204 |
397 | For instance, the Higman embedding theorem can be used to construct a group containing an isomorphic copy of every finitely presented group with solvable word problem. It seems natural to ask whether this group can have solvable word problem. But it is a consequence of the Boone-Rogers result that: Corollary: There is no universal solvable word problem group. | Word problem for groups | 0.85202 |
398 | The criterion given above, for the solvability of the word problem in a single group, can be extended by a straightforward argument. This gives the following criterion for the uniform solvability of the word problem for a class of finitely presented groups: To solve the uniform word problem for a class K of groups, it is sufficient to find a recursive function f ( P , w ) {\displaystyle f(P,w)} that takes a finite presentation P for a group G and a word w {\displaystyle w} in the generators of G, such that whenever G ∈ K: f ( P , w ) = { 0 if w ≠ 1 in G undefined/does not halt if w = 1 in G {\displaystyle f(P,w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ G\end{cases}}} Boone-Rogers Theorem: There is no uniform partial algorithm that solves the word problem in all finitely presented groups with solvable word problem.In other words, the uniform word problem for the class of all finitely presented groups with solvable word problem is unsolvable. This has some interesting consequences. | Word problem for groups | 0.85202 |
399 | The oldest result relating algebraic structure to solvability of the word problem is Kuznetsov's theorem: A recursively presented simple group S has solvable word problem.To prove this let ⟨X|R⟩ be a recursive presentation for S. Choose a ∈ S such that a ≠ 1 in S. If w is a word on the generators X of S, then let: S w = ⟨ X | R ∪ { w } ⟩ . {\displaystyle S_{w}=\langle X|R\cup \{w\}\rangle .} There is a recursive function f ⟨ X | R ∪ { w } ⟩ {\displaystyle f_{\langle X|R\cup \{w\}\rangle }} such that: f ⟨ X | R ∪ { w } ⟩ ( x ) = { 0 if x = 1 in S w undefined/does not halt if x ≠ 1 in S w . | Word problem for groups | 0.85202 |
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