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They are needed to convert high voltage mains electricity into low voltage electricity which can be safely used in homes. Maxwell's formulation of the law is given in the Maxwell–Faraday equation—the fourth and final of Maxwell's equations—which states that a time-varying magnetic field produces an electric field. Together, Maxwell's equations provide a single uniform theory of the electric and magnetic fields and Maxwell's work in creating this theory has been called "the second great unification in physics" after the first great unification of Newton's law of universal gravitation.
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Introduction to electromagnetism
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In physics, fields are entities that interact with matter and can be described mathematically by assigning a value to each point in space and time. Vector fields are fields which are assigned both a numerical value and a direction at each point in space and time. Electric charges produce a vector field called the electric field. The numerical value of the electric field, also called the electric field strength, determines the strength of the electric force that a charged particle will feel in the field and the direction of the field determines which direction the force will be in.
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The discovery that certain toxic chemicals administered in combination can cure certain cancers ranks as one of the greatest in modern medicine. Childhood ALL (Acute Lymphoblastic Leukemia), testicular cancer, and Hodgkins disease, previously universally fatal, are now generally curable diseases. They have also proved effective in the adjuvant setting, in reducing the risk of recurrence after surgery for high-risk breast cancer, colon cancer, and lung cancer, among others. The overall impact of chemotherapy on cancer survival can be difficult to estimate, since improved cancer screening, prevention (e.g. anti-smoking campaigns), and detection all influence statistics on cancer incidence and mortality.
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Combination chemotherapy
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Molecular genetics has uncovered signalling networks that regulate cellular activities such as proliferation and survival. In a particular cancer, such a network may be radically altered, due to a chance somatic mutation. Targeted therapy inhibits the metabolic pathway that underlies that type of cancer's cell division.
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The nearest neighbour search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic Search Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning
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In the special case where the data is a dense 3D map of geometric points, the projection geometry of the sensing technique can be used to dramatically simplify the search problem. This approach requires that the 3D data is organized by a projection to a two-dimensional grid and assumes that the data is spatially smooth across neighboring grid cells with the exception of object boundaries. These assumptions are valid when dealing with 3D sensor data in applications such as surveying, robotics and stereo vision but may not hold for unorganized data in general. In practice this technique has an average search time of O(1) or O(K) for the k-nearest neighbor problem when applied to real world stereo vision data.
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While speech recognition is mainly based on deep learning because most of the industry players in this field like Google, Microsoft and IBM reveal that the core technology of their speech recognition is based on this approach, speech-based emotion recognition can also have a satisfactory performance with ensemble learning.It is also being successfully used in facial emotion recognition.
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As an ensemble, the Bayes optimal classifier represents a hypothesis that is not necessarily in H {\displaystyle H} . The hypothesis represented by the Bayes optimal classifier, however, is the optimal hypothesis in ensemble space (the space of all possible ensembles consisting only of hypotheses in H {\displaystyle H} ). This formula can be restated using Bayes' theorem, which says that the posterior is proportional to the likelihood times the prior: P ( h i | T ) ∝ P ( T | h i ) P ( h i ) {\displaystyle P(h_{i}|T)\propto P(T|h_{i})P(h_{i})} hence, y = a r g m a x c j ∈ C ∑ h i ∈ H P ( c j | h i ) P ( h i | T ) {\displaystyle y={\underset {c_{j}\in C}{\mathrm {argmax} }}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(h_{i}|T)}}
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The content within the book is written using a question and answer format. It contains some 250 questions, which The Science Teacher states each are answered with a "concise and well-formulated essay that is informative and readable." The Science Teacher review goes on to state that many of the answers given in the book are "little gems of science writing". The Science Teacher summarizes by stating that each question is likely to be thought of by a student, and that "the answers are informative, well constructed, and thorough".The book covers information about the planets, the Earth, the Universe, practical astronomy, history, and awkward questions such as astronomy in the Bible, UFOs, and aliens. Also covered are subjects such as the Big Bang, comprehension of large numbers, and the Moon illusion.
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A Question and Answer Guide to Astronomy
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A Question and Answer Guide to Astronomy is a book about astronomy and cosmology, and is intended for a general audience. The book was written by Pierre-Yves Bely, Carol Christian, and Jean-Rene Roy, and published in English by Cambridge University Press in 2010. It was originally written in French.
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The degree can be used to generalize Bézout's theorem in an expected way to intersections of n hypersurfaces in Pn. == Notes ==
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Degree (algebraic geometry)
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A generalization of Bézout's theorem asserts that, if an intersection of n projective hypersurfaces has codimension n, then the degree of the intersection is the product of the degrees of the hypersurfaces. The degree of a projective variety is the evaluation at 1 of the numerator of the Hilbert series of its coordinate ring. It follows that, given the equations of the variety, the degree may be computed from a Gröbner basis of the ideal of these equations.
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This is a generalization of Bézout's theorem (For a proof, see Hilbert series and Hilbert polynomial § Degree of a projective variety and Bézout's theorem). The degree is not an intrinsic property of the variety, as it depends on a specific embedding of the variety in an affine or projective space. The degree of a hypersurface is equal to the total degree of its defining equation.
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In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in general position. For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components. For (irreducible) varieties, if one takes into account the multiplicities and, in the affine case, the points at infinity, the hypothesis of general position may be replaced by the much weaker condition that the intersection of the variety has the dimension zero (that is, consists of a finite number of points).
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Actin was first observed experimentally in 1887 by W.D. Halliburton, who extracted a protein from muscle that 'coagulated' preparations of myosin that he called "myosin-ferment". However, Halliburton was unable to further refine his findings, and the discovery of actin is credited instead to Brunó Ferenc Straub, a young biochemist working in Albert Szent-Györgyi's laboratory at the Institute of Medical Chemistry at the University of Szeged, Hungary. Following up on the discovery of Ilona Banga & Szent-Györgyi in 1941 that the coagulation only occurs in some myosin extractions and was reversed upon the addition of ATP, Straub identified and purified actin from those myosin preparations that did coagulate.
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It is possible that actin could be applied to nanotechnology as its dynamic ability has been harnessed in a number of experiments including those carried out in acellular systems. The underlying idea is to use the microfilaments as tracks to guide molecular motors that can transport a given load. That is actin could be used to define a circuit along which a load can be transported in a more or less controlled and directed manner.
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Actin is used in scientific and technological laboratories as a track for molecular motors such as myosin (either in muscle tissue or outside it) and as a necessary component for cellular functioning. It can also be used as a diagnostic tool, as several of its anomalous variants are related to the appearance of specific pathologies. Nanotechnology. Actin-myosin systems act as molecular motors that permit the transport of vesicles and organelles throughout the cytoplasm.
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Actin can spontaneously acquire a large part of its tertiary structure. However, the way it acquires its fully functional form from its newly synthesized native form is special and almost unique in protein chemistry. The reason for this special route could be the need to avoid the presence of incorrectly folded actin monomers, which could be toxic as they can act as inefficient polymerization terminators. Nevertheless, it is key to establishing the stability of the cytoskeleton, and additionally, it is an essential process for coordinating the cell cycle.CCT is required in order to ensure that folding takes place correctly.
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A number of natural toxins that interfere with actin's dynamics are widely used in research to study actin's role in biology. Latrunculin – a toxin produced by sponges – binds to G-actin preventing it from joining microfilaments. Cytochalasin D – produced by certain fungi – serves as a capping factor, binding to the (+) end of a filament and preventing further addition of actin molecules. In contrast, the sponge toxin jasplakinolide promotes the nucleation of new actin filaments by binding and stabilzing pairs of actin molecules. Phalloidin – from the "death cap" mushroom Amanita phalloides – binds to adjacent actin molecules within the F-actin filament, stabilizing the filament and preventing its depolymerization.Phalloidin is often labelled with fluorescent dyes to visualize actin filaments by fluorescence microscopy.
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This is the topic of the scientific field of structural biology, which employs techniques such as X-ray crystallography, NMR spectroscopy, cryo-electron microscopy (cryo-EM) and dual polarisation interferometry, to determine the structure of proteins. Protein structures range in size from tens to several thousand amino acids.
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Protein Structure
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Faraday paradox (electrochemistry)
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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.
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6. Statistics: Involves collecting, analyzing, and interpreting data, including topics like probability, hypothesis testing, regression analysis, and sampling. 7.
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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.
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3. Pure Mathematics: Explores advanced topics in algebra, calculus, and mathematical proofs. 4.
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```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.
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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).
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Paper 1: Pure Mathematics Paper 2: Pure Mathematics and Statistics Paper 3: Pure Mathematics and Mechanics
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Paper 1: Pure Mathematics Paper 2: Content on Paper 1 plus Mechanics Paper 3: Content on Paper 1 plus Statistics
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Paper 1: Pure Mathematics 1 Paper 2: Pure Mathematics 2 Paper 3: Statistics and Mechanics
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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.
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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.
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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.
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Algebraic structure
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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.
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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.
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Algebraic structure
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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.
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Multiple comparisons problem
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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
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Steroid biosynthesis
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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.
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Steroid biosynthesis
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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.
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Sides of an equation
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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 .}
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Polynomial expression
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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.
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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.
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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.
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Polynomial expression
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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.
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Blue-green Algae
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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.
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Blue-green Algae
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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.
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Polynomial equation
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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).
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Polynomial equation
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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.
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Polynomial equation
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"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).
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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.
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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.
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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.
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(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.
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(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.
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(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.
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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.
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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.
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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.
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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.
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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.
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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.
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(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.
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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.
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Rosen, Kenneth H. (2000), Elementary Number Theory and its Applications (4th ed. ), Addison-Wesley, ISBN 978-0-201-87073-2, MR 1739433.
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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.
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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.
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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.
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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.
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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.
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Polynomial equations
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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.
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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.
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Polynomial equations
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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.
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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.
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Polynomial equations
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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.
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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.
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Solving equations
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Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra.
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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
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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.
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Introduction to genetics
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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.
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Introduction to genetics
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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.
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Introduction to genetics
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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.
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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.
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Introduction to genetics
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Subsequent authors have greatly extended Dehn's algorithm and applied it to a wide range of group theoretic decision problems.It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable. It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by William Boone in 1958.The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra.
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Word problem for groups
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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 .
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There are a number of results that relate solvability of the word problem and algebraic structure. The most significant of these is the Boone-Higman theorem: A finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group.It is widely believed that it should be possible to do the construction so that the simple group itself is finitely presented. If so one would expect it to be difficult to prove as the mapping from presentations to simple groups would have to be non-recursive. The following has been proved by Bernhard Neumann and Angus Macintyre: A finitely presented group has solvable word problem if and only if it can be embedded in every algebraically closed groupWhat is remarkable about this is that the algebraically closed groups are so wild that none of them has a recursive presentation.
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The following will be proved as an example of the use of this technique: Theorem: A finitely presented residually finite group has solvable word problem.Proof: Suppose G = ⟨X|R⟩ is a finitely presented, residually finite group. Let S be the group of all permutations of N, the natural numbers, that fixes all but finitely many numbers then: S is locally finite and contains a copy of every finite group. The word problem in S is solvable by calculating products of permutations. There is a recursive enumeration of all mappings of the finite set X into S. Since G is residually finite, if w is a word in the generators X of G then w ≠ 1 in G if and only of some mapping of X into S induces a homomorphism such that w ≠ 1 in S.Given these facts, algorithm defined by the following pseudocode: For every mapping of X into S If every relator in R is satisfied in S If w ≠ 1 in S return 0 End if End if End for defines a recursive function h such that: h ( x ) = { 0 if x ≠ 1 in G undefined/does not halt if x = 1 in G {\displaystyle h(x)={\begin{cases}0&{\text{if}}\ x\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ x=1\ {\text{in}}\ G\end{cases}}} This shows that G has solvable word problem.
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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.
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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.
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Classification of malware codes such as computer viruses, computer worms, trojans, ransomware and spywares with the usage of machine learning techniques, is inspired by the document categorization problem. Ensemble learning systems have shown a proper efficacy in this area.
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Ensemble learning
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Bayesian model averaging (BMA) makes predictions by averaging the predictions of models weighted by their posterior probabilities given the data. BMA is known to generally give better answers than a single model, obtained, e.g., via stepwise regression, especially where very different models have nearly identical performance in the training set but may otherwise perform quite differently. The question with any use of Bayes' theorem is the prior, i.e., the probability (perhaps subjective) that each model is the best to use for a given purpose. Conceptually, BMA can be used with any prior.
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R: at least three packages offer Bayesian model averaging tools, including the BMS (an acronym for Bayesian Model Selection) package, the BAS (an acronym for Bayesian Adaptive Sampling) package, and the BMA package. Python: scikit-learn, a package for machine learning in Python offers packages for ensemble learning including packages for bagging, voting and averaging methods. MATLAB: classification ensembles are implemented in Statistics and Machine Learning Toolbox.
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According to Snopes.com, more recent (1999 and 1988) versions identify the problem as a question in "a physics degree exam at the University of Copenhagen" and the student was Niels Bohr, and includes the following answers: Tying a piece of string to the barometer, lowering the barometer from the roof to the ground, and measuring the length of the string and barometer. Dropping the barometer off the roof, measuring the time it takes to hit the ground, and calculating the building's height assuming constant acceleration under gravity. When the sun is shining, standing the barometer up, measuring the height of the barometer and the lengths of the shadows of both barometer and building, and finding the building's height using similar triangles. Tying a piece of string to the barometer, and swinging it like a pendulum both on the ground and on the roof, and from the known pendulum length and swing period, calculate the gravitational field for the two cases.
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Neither of two available options (pass or fail) was morally acceptable.By mutual agreement with the student and the examiner, Calandra gave the student another opportunity to answer, warning the student the answer would require demonstrating some knowledge of physics. The student came up with several possible answers, but settled on dropping the barometer from the top of the building, timing its fall, and using the equation of motion d = 1 2 a t 2 {\displaystyle d={\tfrac {1}{2}}{a}t^{2}} to derive the height. The examiner agreed that this satisfied the requirement and gave the student “almost full credit”.When Calandra asked about the other answers, the student gave the examples: using the proportion between the lengths of the building's shadow and that of the barometer to calculate the building's height from the height of the barometer using the barometer as a measuring rod to mark off its height on the wall while climbing the stairs, then counting the number of marks suspending the barometer from a string to create a pendulum, then using the pendulum to measure the strength of Earth's gravity at the top and bottom of the building, and calculating the height of the building from the difference in the two measurements (see Newton's law of universal gravitation)There were, the student said, many other possible solutions.
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