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3,900 | the extensive use of rewriting and "symbolic evaluation". an induction heuristic based the failure of symbolic evaluation.HOL Light Written in OCaml, HOL Light is designed to have a simple and clean logical foundation and an uncluttered implementation. It is essentially another proof assistant for classical higher orde... | Automatic reasoning | 0.829354 |
3,901 | Boyer-Moore Theorem Prover (NQTHM) The design of NQTHM was influenced by John McCarthy and Woody Bledsoe. Started in 1971 at Edinburgh, Scotland, this was a fully automatic theorem prover built using Pure Lisp. The main aspects of NQTHM were: the use of Lisp as a working logic. the reliance on a principle of definition... | Automatic reasoning | 0.829354 |
3,902 | The program came up with a proof for one of the theorems in Principia Mathematica that was more efficient (requiring fewer steps) than the proof provided by Whitehead and Russell. Automated reasoning programs are being applied to solve a growing number of problems in formal logic, mathematics and computer science, logi... | Automatic reasoning | 0.829354 |
3,903 | Automated reasoning has been most commonly used to build automated theorem provers. Oftentimes, however, theorem provers require some human guidance to be effective and so more generally qualify as proof assistants. In some cases such provers have come up with new approaches to proving a theorem. Logic Theorist is a go... | Automatic reasoning | 0.829354 |
3,904 | The development of formal logic played a big role in the field of automated reasoning, which itself led to the development of artificial intelligence. A formal proof is a proof in which every logical inference has been checked back to the fundamental axioms of mathematics. All the intermediate logical steps are supplie... | Automatic reasoning | 0.829354 |
3,905 | The most developed subareas of automated reasoning are automated theorem proving (and the less automated but more pragmatic subfield of interactive theorem proving) and automated proof checking (viewed as guaranteed correct reasoning under fixed assumptions). Extensive work has also been done in reasoning by analogy us... | Automatic reasoning | 0.829354 |
3,906 | In computer science, in particular in knowledge representation and reasoning and metalogic, the area of automated reasoning is dedicated to understanding different aspects of reasoning. The study of automated reasoning helps produce computer programs that allow computers to reason completely, or nearly completely, auto... | Automatic reasoning | 0.829354 |
3,907 | In statistics, the multinomial test is the test of the null hypothesis that the parameters of a multinomial distribution equal specified values; it is used for categorical data.Beginning with a sample of N {\displaystyle ~N~} items each of which has been observed to fall into one of k {\displaystyle k} categories. It i... | Multinomial test | 0.829354 |
3,908 | There is also an emergent perspective called The Scholarship of Teaching and Learning (SoTL). Although there is debate on how to best define SoTL, one of the primary practices is for mainstream faculty members (organic, inorganic, biochemistry, etc.) to develop a more informed view of their practices, how to carry out ... | Chemistry education | 0.829353 |
3,909 | There are several different philosophical perspectives that describe how the work in chemistry education is carried out. | Chemistry education | 0.829353 |
3,910 | Researchers are asking questions such as "why do we have laboratory work in the curriculum? What is distinctive about laboratory work that cannot be met elsewhere in the curriculum?" Researchers are asking for evidence that the investment of space, time and resources in chemistry laboratories provides value to student ... | Chemistry education | 0.829353 |
3,911 | Today, the American Chemical Society on Professional Training requires students to gain 400 hours of laboratory experience, outside of introductory chemistry, to get a bachelor's degree. Similarly, the Royal Society of Chemistry requires students to gain 300 hours of laboratory experience to get a bachelor's degree.How... | Chemistry education | 0.829353 |
3,912 | After working in Liebig's laboratory, Horsford returned to the United States and helped establish the Lawrence Scientific School at Harvard University. The school was modeled after Liebig's methodology and established the first chemistry laboratory course. Two years later, Charles W. Eliot started to volunteer at the l... | Chemistry education | 0.829353 |
3,913 | The most common method of teaching chemistry is lecture with a laboratory component. Laboratory courses became a central part of the chemistry curriculum towards the end of the 19th century. The German scientist Justus von Liebig plays a major role in shifting the model of lecture with demonstrations to one that includ... | Chemistry education | 0.829353 |
3,914 | In 2017, the Systems Thinking Into Chemistry Education (STICE) project proposed a systems thinking approach for (post)-secondary education in general chemistry education. Chemistry education has largely relied on a reductionist approach, which involves studying a complex topic as the sum of its parts. A reductionist ap... | Chemistry education | 0.829353 |
3,915 | A second perspective is defined by a self-identified group of chemical educators, faculty members and instructors who, as opposed to declaring their primary interest in a typical area of laboratory research (organic, inorganic, biochemistry, etc.), take on an interest in contributing suggestions, essays, observations, ... | Chemistry education | 0.829353 |
3,916 | The Chemical Educator: Published by Springer-Verlag from 1996 to 2002. Covers all areas of chemical education. List of scientific journals in chemistryResearch in chemistry education is also published in journals in the wider science education field. | Chemistry education | 0.829353 |
3,917 | Covers all areas of chemical education. (EiC is the RSC's educational magazine, whereas CERP is a peer-reviewed research journal). Foundations of Chemistry (FOCH): Published by Springer. | Chemistry education | 0.829353 |
3,918 | Publishes theoretical perspectives, literature reviews, and empirical papers, including systematic evaluations of innovative practice. Education in Chemistry (EiC): Published by the Royal Society of Chemistry. | Chemistry education | 0.829353 |
3,919 | Covers both school and university education. Chemical Education Journal (CEJ): Covers all areas of chemical education. Chemistry Education Research and Practice (CERP): Published by the Royal Society of Chemistry (RSC). | Chemistry education | 0.829353 |
3,920 | Several journals publish papers related to chemistry education. Some journals focus on particular education levels (schools vs. universities) while others cover all education levels. Journal articles range from reports on classroom and laboratory practices to educational research. Australian Journal of Education in Che... | Chemistry education | 0.829353 |
3,921 | The first is what one might call a practitioner's perspective, wherein the individuals who are responsible for teaching chemistry (teachers, instructors, professors) are the ones who ultimately define chemistry education by their actions. | Chemistry education | 0.829353 |
3,922 | Chemistry has applications in food, medicine, industry, the environment, and other areas. Learning chemistry allows students to learn about the scientific method and gain skills in critical thinking, deductive reasoning, problem-solving, and communication. Teaching chemistry to students at a young age can increase stud... | Chemistry education | 0.829353 |
3,923 | Chemistry education is important because the field of chemistry is fundamental to our world. The universe is subject to the laws of chemistry, while human beings depend on the orderly progress of chemical reactions within their bodies. Described as the central science, chemistry connects physical sciences with the life... | Chemistry education | 0.829353 |
3,924 | Chemistry education (or chemical education) is the study of teaching and learning chemistry. It is one subset of STEM education or discipline-based education research (DBER). Topics in chemistry education include understanding how students learn chemistry and determining the most efficient methods to teach chemistry. T... | Chemistry education | 0.829353 |
3,925 | Chemistry education research utilizes both quantitative and qualitative data collection methods. Quantitative methods typically involve collecting data that can then be analyzed using various statistical methods. Qualitative methods include interviews, observations, document analysis, journaling, and other methods comm... | Chemistry education | 0.829353 |
3,926 | Like science education researchers, CER practitioners tend to study the teaching practices of others as opposed to focusing on their own classroom practices. Chemistry education research is typically carried out in situ using human subjects from secondary and post-secondary schools. | Chemistry education | 0.829353 |
3,927 | A third perspective is chemistry education research (CER). CER is a type of discipline-based education research (DBER) focusing on the teaching and learning of chemistry. An overarching goal for chemistry education researchers is to help students develop 'expert-like' (coherent and useful) knowledge of chemistry. Thus,... | Chemistry education | 0.829353 |
3,928 | The U.S. offers chemistry education degrees at the undergraduate and graduate levels. The following degrees are offered: Bachelor of Science in Chemistry Education Master of Science in Chemistry Education PhD in Chemistry EducationAdditionally, colleges and universities offer chemistry degrees with a specialization in ... | Chemistry education | 0.829353 |
3,929 | The Plant Cell is a monthly peer-reviewed scientific journal of plant sciences, especially the areas of cell and molecular biology, genetics, development, and evolution. It is published by the American Society of Plant Biologists. The editor-in-chief is Blake Meyers (Donald Danforth Plant Science Center). The journal w... | Plant Cell | 0.829338 |
3,930 | This work was translated into German by W. Ostwald (who styled its author the "founder of chemical energetics") in 1891 and into French by H. le Chatelier in 1899.Gibbs's "Equilibrium" paper is considered one of the greatest achievements in physical science in the 19th century and one of the foundations of the science ... | On the Equilibrium of Heterogeneous Substances | 0.829318 |
3,931 | Gibbs first contributed to mathematical physics with two papers published in 1873 in the Transactions of the Connecticut Academy of Arts and Sciences on "Graphical Methods in the Thermodynamics of Fluids," and "Method of Geometrical Representation of the Thermodynamic Properties of Substances by means of Surfaces." His... | On the Equilibrium of Heterogeneous Substances | 0.829318 |
3,932 | It introduced concepts such as chemical potential, phase rule, and others, which form the basis for modern physical chemistry. American writer Bill Bryson describes Gibbs's Equilibrium paper as "the Principia of thermodynamics".On the Equilibrium of Heterogeneous Substances, was originally published in a relatively obs... | On the Equilibrium of Heterogeneous Substances | 0.829318 |
3,933 | In the history of thermodynamics, On the Equilibrium of Heterogeneous Substances is a 300-page paper written by American chemical physicist Willard Gibbs. It is one of the founding papers in thermodynamics, along with German physicist Hermann von Helmholtz's 1882 paper "Thermodynamik chemischer Vorgänge." Together they... | On the Equilibrium of Heterogeneous Substances | 0.829318 |
3,934 | A Cal-Tech physicist invited to observe Brown's disk device in the early '50s noted during the demonstration that its motivation force was the well-known phenomenon of "electric wind", and not anti-gravity, saying, “I’m afraid these gentlemen played hooky from their high school physics classes…”. Research into the phen... | Electrogravitics | 0.829301 |
3,935 | In 1929, Brown published "How I Control Gravitation" in Science and Invention where he claimed the capacitors were producing a mysterious force that interacted with the pull of gravity. He envisioned a future where, if his device could be scaled up, "Multi-impulse gravitators, weighing hundreds of tons, may propel the ... | Electrogravitics | 0.829301 |
3,936 | Campbell pointed out to a Wired magazine reporter that creating a true vacuum similar to space for the test requires tens of thousands of dollars in equipment. Byron Preiss, in his 1985 book on the current science and future of the Solar System titled The Planets, commented that electrogravitics development seemed to b... | Electrogravitics | 0.829301 |
3,937 | Typically, a workflow for applying machine learning to biological data goes through four steps: Recording, including capture and storage. In this step, different information sources may be merged into a single set. Preprocessing, including cleaning and restructuring into a ready-to-analyze form. | Machine learning in bioinformatics | 0.8293 |
3,938 | This domain, particularly phylogenetic tree reconstruction, uses the features of machine learning techniques. Phylogenetic trees are schematic representations of the evolution of organisms. Initially, they were constructed using features such as morphological and metabolic features. Later, due to the availability of ge... | Machine learning in bioinformatics | 0.8293 |
3,939 | The increase of experimentally characterized ribosomally synthesized and post-translationally modified peptides (RiPPs), together with the availability of information on their sequence and chemical structure, selected from databases such as BAGEL, BACTIBASE, MIBIG, and THIOBASE, provide the opportunity to develop machi... | Machine learning in bioinformatics | 0.8293 |
3,940 | Text Nailing, an alternative approach to machine learning, capable of extracting features from clinical narrative notes was introduced in 2017. This technique has been applied to the search for novel drug targets, as this task requires the examination of information stored in biological databases and journals. Annotati... | Machine learning in bioinformatics | 0.8293 |
3,941 | The increase in biological publications increased the difficulty in searching and compiling relevant available information on a given topic. This task is known as knowledge extraction. It is necessary for biological data collection which can then in turn be fed into machine learning algorithms to generate new biologica... | Machine learning in bioinformatics | 0.8293 |
3,942 | Other applications include medical image classification, genomic sequence analysis, as well as protein structure classification and prediction. Deep learning has been applied to regulatory genomics, variant calling and pathogenicity scores. Natural language processing and text mining have helped to understand phenomena... | Machine learning in bioinformatics | 0.8293 |
3,943 | In general, a machine learning system can usually be trained to recognize elements of a certain class given sufficient samples. For example, machine learning methods can be trained to identify specific visual features such as splice sites.Support vector machines have been extensively used in cancer genomic studies. In ... | Machine learning in bioinformatics | 0.8293 |
3,944 | Duplicate data is a significant issue in bioinformatics. Publicly available data may be of uncertain quality. Errors during experimentation. | Machine learning in bioinformatics | 0.8293 |
3,945 | Divisive algorithms begin with the whole set and proceed to divide it into successively smaller clusters. Hierarchical clustering is calculated using metrics on Euclidean spaces, the most commonly used is the Euclidean distance computed by finding the square of the difference between each variable, adding all the squar... | Machine learning in bioinformatics | 0.8293 |
3,946 | Convolutional neural networks (CNN) are a class of deep neural network whose architecture is based on shared weights of convolution kernels or filters that slide along input features, providing translation-equivariant responses known as feature maps. CNNs take advantage of the hierarchical pattern in data and assemble ... | Machine learning in bioinformatics | 0.8293 |
3,947 | Clustering - the partitioning of a data set into disjoint subsets, so that the data in each subset are as close as possible to each other and as distant as possible from data in any other subset, according to some defined distance or similarity function - is a common technique for statistical data analysis. Clustering ... | Machine learning in bioinformatics | 0.8293 |
3,948 | Satria et al. demonstrated the utility of such analyses by reconstructing a global map of secondary metabolic diversity across taxonomy to identify the uncharted biosynthetic potential of 1.2 million biosynthetic gene clusters. This opens up new possibilities to accelerate natural product discovery and offers a first s... | Machine learning in bioinformatics | 0.8293 |
3,949 | Artificial neural networks in bioinformatics have been used for: Comparing and aligning RNA, protein, and DNA sequences. Identification of promoters and finding genes from sequences related to DNA. Interpreting the expression-gene and micro-array data. Identifying the network (regulatory) of genes. | Machine learning in bioinformatics | 0.8293 |
3,950 | An important part of bioinformatics is the management of big datasets, known as databases of reference. Databases exist for each type of biological data, for example for biosynthetic gene clusters and metagenomes. | Machine learning in bioinformatics | 0.8293 |
3,951 | A phylogenetic convolutional neural network (Ph-CNN) is a novel convolutional neural network architecture proposed by Fioranti et al. to classify metagenomics data. In this approach, phylogenetic data is endowed with patristic distance (the sum of the lengths of all branches connecting two operational taxonomic units )... | Machine learning in bioinformatics | 0.8293 |
3,952 | The algorithm for the prediction of cross-links and deciphering the complete chemical structure of RiPP has been implemented for lanthipeptides, lasso peptides, cyanobactins, and thiopeptides. The prediction of lanthionine linkages in lanthipeptides was carried out using machine learning. A dataset of 93 lanthipeptides... | Machine learning in bioinformatics | 0.8293 |
3,953 | The National Center for Biotechnology Information (NCBI) provides a large suite of online resources for biological information and data, including the GenBank nucleic acid sequence database and the PubMed database of citations and abstracts for published life science journals. Augmenting many of the Web applications ar... | Machine learning in bioinformatics | 0.8293 |
3,954 | RiPPs analysis tools such as antiSMASH and RiPP-PRISM use HMM of modifying enzymes present in biosynthetic gene clusters in RiPP to predict the RiPP subclass. Unlike these tools, RiPPMiner uses a machine learning model, trained with 513 RiPPs, that uses the amino acid sequence of the RiPP gene uniquely to identify and ... | Machine learning in bioinformatics | 0.8293 |
3,955 | Prior to machine learning, researchers needed to conduct this prediction manually. This trend began in 1951 when Pauling and Corey released their work on predicting the hydrogen bond configurations of a protein from a polypeptide chain. Automatic feature learning reaches an accuracy of 82-84%. The current state-of-the-... | Machine learning in bioinformatics | 0.8293 |
3,956 | Natural language processing algorithms personalized medicine for patients who suffer genetic diseases, by combining the extraction of clinical information and genomic data available from the patients. Institutes such as Health-funded Pharmacogenomics Research Network focus on finding breast cancer treatments.Precision ... | Machine learning in bioinformatics | 0.8293 |
3,957 | In addition, due to the huge number of genes on which data is collected by the microarray, winnowing the large amount of irrelevant data to the task of expressed gene identification is challenging. Machine learning presents a potential solution as various classification methods can be used to perform this identificatio... | Machine learning in bioinformatics | 0.8293 |
3,958 | Microarrays, a type of lab-on-a-chip, are used for automatically collecting data about large amounts of biological material. Machine learning can aid in analysis, and has been applied to expression pattern identification, classification, and genetic network induction. This technology is especially useful for monitoring... | Machine learning in bioinformatics | 0.8293 |
3,959 | The framework combines: identifying a few significant features by a massively parallel forward variable selection procedure mapping the selected species on a phylogenetic tree, and predicting functional profiles by functional gene enrichment analysis from metagenomic 16S rRNA data.They demonstrated performance by analy... | Machine learning in bioinformatics | 0.8293 |
3,960 | Despite their importance, machine learning tools related to metagenomics have focused on the study of gut microbiota and the relationship with digestive diseases, such as inflammatory bowel disease (IBD), Clostridioides difficile infection (CDI), colorectal cancer and diabetes, seeking better diagnosis and treatments. ... | Machine learning in bioinformatics | 0.8293 |
3,961 | Metagenomics is the study of microbial communities from environmental DNA samples. Currently, limitations and challenges predominate in the implementation of machine learning tools due to the amount of data in environmental samples. Supercomputers and web servers have made access to these tools easier. The high dimensi... | Machine learning in bioinformatics | 0.8293 |
3,962 | This multi-layered approach allows such systems to make sophisticated predictions when appropriately trained. These methods contrast with other computational biology approaches which, while exploiting existing datasets, do not allow the data to be interpreted and analyzed in unanticipated ways. In recent years, the siz... | Machine learning in bioinformatics | 0.8293 |
3,963 | Machine learning in bioinformatics is the application of machine learning algorithms to bioinformatics, including genomics, proteomics, microarrays, systems biology, evolution, and text mining.Prior to the emergence of machine learning, bioinformatics algorithms had to be programmed by hand; for problems such as protei... | Machine learning in bioinformatics | 0.8293 |
3,964 | The differences between them are the following: Classification/recognition outputs a categorical class, while prediction outputs a numerical valued feature. The type of algorithm, or process used to build the predictive models from data using analogies, rules, neural networks, probabilities, and/or statistics.Due to th... | Machine learning in bioinformatics | 0.8293 |
3,965 | MIBiG, the minimum information about a biosynthetic gene cluster specification, provides a standard for annotations and metadata on biosynthetic gene clusters and their molecular products. MIBiG is a Genomic Standards Consortium project that builds on the minimum information about any sequence (MIxS) framework.MIBiG fa... | Machine learning in bioinformatics | 0.8293 |
3,966 | In this type of machine learning task, the output is a discrete variable. One example of this type of task in bioinformatics is labeling new genomic data (such as genomes of unculturable bacteria) based on a model of already labeled data. | Machine learning in bioinformatics | 0.8293 |
3,967 | However, not all the genes in a given input sequence can be identified through homology alone, due to limits in the size of the database of known and annotated gene sequences. Therefore, an intrinsic search is needed where a gene prediction program attempts to identify the remaining genes from the DNA sequence alone.Ma... | Machine learning in bioinformatics | 0.8293 |
3,968 | While genomic sequence data has historically been sparse due to the technical difficulty of sequencing a piece of DNA, the number of available sequences is growing exponentially. However, while raw data is becoming increasingly available and accessible, biological interpretation of this data is occurring at a much slow... | Machine learning in bioinformatics | 0.8293 |
3,969 | Systems biology focuses on the study of emergent behaviors from complex interactions of simple biological components in a system. Such components can include DNA, RNA, proteins, and metabolites.Machine learning has been used to aid in modeling these interactions in domains such as genetic networks, signal transduction ... | Machine learning in bioinformatics | 0.8293 |
3,970 | In 2021 Dhungel et al., designed an R package called MegaR. This package allows working with 16S rRNA and whole metagenomic sequences to make taxonomic profiles and classification models by machine learning models. MegaR includes a comfortable visualization environment to improve the user experience. Machine learning i... | Machine learning in bioinformatics | 0.8293 |
3,971 | For example, in 2018, Fioravanti et al. developed an algorithm called Ph-CNN to classify data samples from healthy patients and patients with IBD symptoms (to distinguish healthy and sick patients) by using phylogenetic trees and convolutional neural networks.In addition, random forest (RF) methods and implemented impo... | Machine learning in bioinformatics | 0.8293 |
3,972 | Machine learning methods for the analysis of neuroimaging data are used to help diagnose stroke. Historically multiple approaches to this problem involved neural networks.Multiple approaches to detect strokes used machine learning. As proposed by Mirtskhulava, feed-forward networks were tested to detect strokes using n... | Machine learning in bioinformatics | 0.8293 |
3,973 | Machine learning algorithms in bioinformatics can be used for prediction, classification, and feature selection. Methods to achieve this task are varied and span many disciplines; most well known among them are machine learning and statistics. Classification and prediction tasks aim at building models that describe and... | Machine learning in bioinformatics | 0.8293 |
3,974 | Consequently, as can be shown, the zeros of the following polynomials are not expressible by sums, products, and radicals. For the latter polynomial, this fact is known as the Abel–Ruffini theorem: f(X) = X5 − 4X + 2 (and E = Q), f(X) = X n + an−1X n−1 + ⋯ + a0 (where f is regarded as a polynomial in E(a0, ..., an−1), ... | Field (mathematics) | 0.829294 |
3,975 | The latter condition is always satisfied if E has characteristic 0. For a finite Galois extension, the Galois group Gal(F/E) is the group of field automorphisms of F that are trivial on E (i.e., the bijections σ: F → F that preserve addition and multiplication and that send elements of E to themselves). The importance ... | Field (mathematics) | 0.829294 |
3,976 | Galois theory studies algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication. An important notion in this area is that of finite Galois extensions F / E, which are, by definition, those that are separable and normal. The primitive element theorem shows that ... | Field (mathematics) | 0.829294 |
3,977 | For fields that are not algebraically closed (or not separably closed), the absolute Galois group Gal(F) is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs all finite separable extensions of F. By elementary means, the group Gal(Fq) can be shown to be the Prüfe... | Field (mathematics) | 0.829294 |
3,978 | This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor. The non-existence of an odd-dimensional division algebra is more classical. It can be deduced from the hairy ball theorem illustrated at the right. | Field (mathematics) | 0.829294 |
3,979 | This result is known as the Frobenius theorem. The octonions O, for which multiplication is neither commutative nor associative, is a normed alternative division algebra, but is not a division ring. | Field (mathematics) | 0.829294 |
3,980 | Dropping one or several axioms in the definition of a field leads to other algebraic structures. As was mentioned above, commutative rings satisfy all field axioms except for the existence of multiplicative inverses. Dropping instead commutativity of multiplication leads to the concept of a division ring or skew field;... | Field (mathematics) | 0.829294 |
3,981 | Applied to the above sentence φ, this shows that there is an isomorphism ulim p → ∞ F ¯ p ≅ C . {\displaystyle \operatorname {ulim} _{p\to \infty }{\overline {\mathbf {F} }}_{p}\cong \mathbf {C} .} The Ax–Kochen theorem mentioned above also follows from this and an isomorphism of the ultraproducts (in both cases over... | Field (mathematics) | 0.829294 |
3,982 | Moreover, any fixed statement φ holds in C if and only if it holds in any algebraically closed field of sufficiently high characteristic.If U is an ultrafilter on a set I, and Fi is a field for every i in I, the ultraproduct of the Fi with respect to U is a field. It is denoted by ulimi→∞ Fi,since it behaves in several... | Field (mathematics) | 0.829294 |
3,983 | In model theory, a branch of mathematical logic, two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely. The mathematical statements in question are required to be first-order sentences (involving 0, 1, the addition and multiplication).... | Field (mathematics) | 0.829294 |
3,984 | The subset consisting of O and I (highlighted in red in the tables at the right) is also a field, known as the binary field F2 or GF(2). In the context of computer science and Boolean algebra, O and I are often denoted respectively by false and true, and the addition is then denoted XOR (exclusive or). In other words, ... | Field (mathematics) | 0.829294 |
3,985 | If a ≠ 0, then the equation ax = bhas a unique solution x in a field F, namely x = a − 1 b . {\displaystyle x=a^{-1}b.} This immediate consequence of the definition of a field is fundamental in linear algebra. For example, it is an essential ingredient of Gaussian elimination and of the proof that any vector space has ... | Field (mathematics) | 0.829294 |
3,986 | The field F is usually rather implicit since its construction requires the ultrafilter lemma, a set-theoretic axiom that is weaker than the axiom of choice. In this regard, the algebraic closure of Fq, is exceptionally simple. It is the union of the finite fields containing Fq (the ones of order qn). For any algebraica... | Field (mathematics) | 0.829294 |
3,987 | Any field F has an algebraic closure, which is moreover unique up to (non-unique) isomorphism. It is commonly referred to as the algebraic closure and denoted F. For example, the algebraic closure Q of Q is called the field of algebraic numbers. | Field (mathematics) | 0.829294 |
3,988 | A field is algebraically closed if it does not have any strictly bigger algebraic extensions or, equivalently, if any polynomial equation fn xn + fn−1xn−1 + ⋯ + f1x + f0 = 0, with coefficients fn, ..., f0 ∈ F, n > 0,has a solution x ∊ F. By the fundamental theorem of algebra, C is algebraically closed, i.e., any polyno... | Field (mathematics) | 0.829294 |
3,989 | This section lists some results that have been proved using the classification of finite simple groups. The Schreier conjecture The Signalizer functor theorem The B conjecture The Schur–Zassenhaus theorem for all groups (though this only uses the Feit–Thompson theorem). A transitive permutation group on a finite set wi... | Classification of finite simple groups | 0.829293 |
3,990 | Given a larger, orchestrated proof, dealing with many of these special cases can be postponed until the most powerful assumptions can be applied. The price paid under this revised strategy is that these first generation theorems no longer have comparatively short proofs, but instead rely on the complete classification.... | Classification of finite simple groups | 0.829293 |
3,991 | As a result, many of the pieces of the theorem were proved using techniques that were overly general. Because the conclusion was unknown, the first generation proof consists of many stand-alone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of nume... | Classification of finite simple groups | 0.829293 |
3,992 | The most important thing is that the correct, final statement of the theorem is now known. Simpler techniques can be applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those who worked on the first generation proof did not know how many sporadic groups there were, an... | Classification of finite simple groups | 0.829293 |
3,993 | The proof of the theorem, as it stood around 1985 or so, can be called first generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called a second-generation classification proof. This effort, called "revisionism", was originally led by Daniel ... | Classification of finite simple groups | 0.829293 |
3,994 | This was essentially done by the Trichotomy theorem. Classification of simple groups of characteristic 2 type. This was handled by the Gilman–Griess theorem, with 3-elements replaced by p-elements for odd primes. | Classification of finite simple groups | 0.829293 |
3,995 | This was essentially solved by Aschbacher's trichotomy theorem for groups with e(G)=3. The main change is that 2-local 3-rank is replaced by 2-local p-rank for odd primes. Centralizers of 3-elements in standard form. | Classification of finite simple groups | 0.829293 |
3,996 | The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified by Aschbacher in 1978 Groups with a strongly p-embedded subgroup for p odd The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors. Thi... | Classification of finite simple groups | 0.829293 |
3,997 | Classification of groups of odd type. The problem is to show that if a group has a centralizer of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution theorem. | Classification of finite simple groups | 0.829293 |
3,998 | A group is of characteristic 2 type if the generalized Fitting subgroup F*(Y) of every 2-local subgroup Y is a 2-group. As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification ... | Classification of finite simple groups | 0.829293 |
3,999 | The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered... | Classification of finite simple groups | 0.829293 |
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