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The 4273π project or 4273pi project also offers open source educational materials for free. The course runs on low cost Raspberry Pi computers and has been used to teach adults and school pupils. 4283 is actively developed by a consortium of academics and research staff who have run research level bioinformatics using Raspberry Pi computers and the 4283π operating system.MOOC platforms also provide online certifications in bioinformatics and related disciplines, including Coursera's Bioinformatics Specialization (UC San Diego) and Genomic Data Science Specialization (Johns Hopkins) as well as EdX's Data Analysis for Life Sciences XSeries (Harvard).
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Bio informatics
| 0.837745
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1,701
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Bioinformatics is not only taught as in-person masters degree at many universities. The computational nature of bioinformatics lends it to computer-aided and online learning. Software platforms designed to teach bioinformatics concepts and methods include Rosalind and online courses offered through the Swiss Institute of Bioinformatics Training Portal. The Canadian Bioinformatics Workshops provides videos and slides from training workshops on their website under a Creative Commons license.
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Bio informatics
| 0.837745
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1,702
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Shotgun sequencing yields sequence data quickly, but the task of assembling the fragments can be quite complicated for larger genomes. For a genome as large as the human genome, it may take many days of CPU time on large-memory, multiprocessor computers to assemble the fragments, and the resulting assembly usually contains numerous gaps that must be filled in later. Shotgun sequencing is the method of choice for virtually all genomes sequenced (rather than chain-termination or chemical degradation methods), and genome assembly algorithms are a critical area of bioinformatics research.
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Bio informatics
| 0.837745
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1,703
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Major research efforts in the field include sequence alignment, gene finding, genome assembly, drug design, drug discovery, protein structure alignment, protein structure prediction, prediction of gene expression and protein–protein interactions, genome-wide association studies, the modeling of evolution and cell division/mitosis. Bioinformatics entails the creation and advancement of databases, algorithms, computational and statistical techniques, and theory to solve formal and practical problems arising from the management and analysis of biological data. Over the past few decades, rapid developments in genomic and other molecular research technologies and developments in information technologies have combined to produce a tremendous amount of information related to molecular biology. Bioinformatics is the name given to these mathematical and computing approaches used to glean understanding of biological processes. Common activities in bioinformatics include mapping and analyzing DNA and protein sequences, aligning DNA and protein sequences to compare them, and creating and viewing 3-D models of protein structures.
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Bio informatics
| 0.837745
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1,704
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Self-modifying code can alter itself in response to run time conditions in order to optimize code; this was more common in assembly language programs. Some CPU designs can perform some optimizations at run time. Some examples include out-of-order execution, speculative execution, instruction pipelines, and branch predictors. Compilers can help the program take advantage of these CPU features, for example through instruction scheduling.
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Program optimization
| 0.837667
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1,705
|
In computer science, program optimization, code optimization, or software optimization is the process of modifying a software system to make some aspect of it work more efficiently or use fewer resources. In general, a computer program may be optimized so that it executes more rapidly, or to make it capable of operating with less memory storage or other resources, or draw less power.
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Program optimization
| 0.837666
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1,706
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Between the source and compile level, directives and build flags can be used to tune performance options in the source code and compiler respectively, such as using preprocessor defines to disable unneeded software features, optimizing for specific processor models or hardware capabilities, or predicting branching, for instance. Source-based software distribution systems such as BSD's Ports and Gentoo's Portage can take advantage of this form of optimization.
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Program optimization
| 0.837666
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1,707
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Just-in-time compilers can produce customized machine code based on run-time data, at the cost of compilation overhead. This technique dates to the earliest regular expression engines, and has become widespread with Java HotSpot and V8 for JavaScript. In some cases adaptive optimization may be able to perform run time optimization exceeding the capability of static compilers by dynamically adjusting parameters according to the actual input or other factors. Profile-guided optimization is an ahead-of-time (AOT) compilation optimization technique based on run time profiles, and is similar to a static "average case" analog of the dynamic technique of adaptive optimization.
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Program optimization
| 0.837666
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1,708
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Other common trade-offs include code clarity and conciseness. There are instances where the programmer performing the optimization must decide to make the software better for some operations but at the cost of making other operations less efficient. These trade-offs may sometimes be of a non-technical nature – such as when a competitor has published a benchmark result that must be beaten in order to improve commercial success but comes perhaps with the burden of making normal usage of the software less efficient. Such changes are sometimes jokingly referred to as pessimizations.
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Program optimization
| 0.837666
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1,709
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Optimization will generally focus on improving just one or two aspects of performance: execution time, memory usage, disk space, bandwidth, power consumption or some other resource. This will usually require a trade-off – where one factor is optimized at the expense of others. For example, increasing the size of cache improves run time performance, but also increases the memory consumption.
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Program optimization
| 0.837666
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1,710
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In some cases, however, optimization relies on using more elaborate algorithms, making use of "special cases" and special "tricks" and performing complex trade-offs. A "fully optimized" program might be more difficult to comprehend and hence may contain more faults than unoptimized versions. Beyond eliminating obvious antipatterns, some code level optimizations decrease maintainability.
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Program optimization
| 0.837666
|
1,711
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Pharmacogenetics is defined as the study of inherited genes causing different drug metabolisms that vary from each other, such as the rate of metabolism and metabolites. Pharmacogenomics is defined as the study of associating the drug response with one's gene. Both terms are similar in nature, so they are used interchangeably.
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Drug therapy
| 0.837665
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1,712
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The biochemistry of Polony sequencing mainly relies on the discriminatory capacities of ligases and polymerases. First, a series of anchor primers are flowed through the cells and hybridize to the synthetic oligonucleotide sequences at the immediate 3’ or 5’ end of the 17-18 bp proximal or distal genomic DNA tags. Next, an enzymatic ligation reaction of the anchor primer to a population of degenenerate nonamers that are labeled with fluorescent dyes is performed. Differentially labeled nonamers: 5' Cy5‐NNNNNNNNT 5' Cy3‐NNNNNNNNA 5' TexasRed‐NNNNNNNNC 5' 6FAM‐NNNNNNNNG The fluorophore-tagged nonamers anneal with differential success to the tag sequences according to a strategy similar to that of degenerate primers, but instead of submission to polymerases, nonamers are selectively ligated onto adjoining DNA- the anchor primer.
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Polony Sequencing
| 0.837655
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1,713
|
For instance, division of real numbers is a partial binary operation, because one can't divide by zero: a 0 {\displaystyle {\frac {a}{0}}} is undefined for every real number a {\displaystyle a} . In both model theory and classical universal algebra, binary operations are required to be defined on all elements of S × S {\displaystyle S\times S} . However, partial algebras generalize universal algebras to allow partial operations. Sometimes, especially in computer science, the term binary operation is used for any binary function.
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Binary operator
| 0.837649
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1,714
|
For example, immunohistochemistry usually uses an antibody to one or more proteins of interest that are conjugated to enzymes yielding either luminescent or chromogenic signals that can be compared between samples, allowing for localization information. Another applicable technique is cofractionation in sucrose (or other material) gradients using isopycnic centrifugation. While this technique does not prove colocalization of a compartment of known density and the protein of interest, it does increase the likelihood, and is more amenable to large-scale studies.
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Protein interactions
| 0.837618
|
1,715
|
Genome and gene sequences can be searched by a variety of tools for certain properties. Sequence profiling tools can find restriction enzyme sites, open reading frames in nucleotide sequences, and predict secondary structures. Phylogenetic trees can be constructed and evolutionary hypotheses developed using special software like ClustalW regarding the ancestry of modern organisms and the genes they express. The field of bioinformatics is now indispensable for the analysis of genes and proteins.
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Protein interactions
| 0.837618
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1,716
|
"An Introduction to Proteins" from HOPES (Huntington's Disease Outreach Project for Education at Stanford) Proteins: Biogenesis to Degradation – The Virtual Library of Biochemistry and Cell Biology
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Protein interactions
| 0.837618
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1,717
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Alternatively, E can be defined as Sym ( M ) / ⨁ n ≥ 2 Sym n ( M ) {\displaystyle \operatorname {Sym} (M)/\bigoplus _{n\geq 2}\operatorname {Sym} ^{n}(M)} where Sym ( M ) {\displaystyle \operatorname {Sym} (M)} is the symmetric algebra of M. We then have the short exact sequence 0 → M → E → p R → 0 {\displaystyle 0\to M\to E{\overset {p}{{}\to {}}}R\to 0} where p is the projection. Hence, E is an extension of R by M. It is trivial since r ↦ ( r , 0 ) {\displaystyle r\mapsto (r,0)} is a section (note this section is a ring homomorphism since ( 1 , 0 ) {\displaystyle (1,0)} is the multiplicative identity of E). Conversely, every trivial extension E of R by I is isomorphic to R ⊕ I {\displaystyle R\oplus I} if I 2 = 0 {\displaystyle I^{2}=0} . Indeed, identifying R {\displaystyle R} as a subring of E using a section, we have ( E , ϕ ) ≃ ( R ⊕ I , p ) {\displaystyle (E,\phi )\simeq (R\oplus I,p)} via e ↦ ( ϕ ( e ) , e − ϕ ( e ) ) {\displaystyle e\mapsto (\phi (e),e-\phi (e))} .One interesting feature of this construction is that the module M becomes an ideal of some new ring. In his book Local Rings, Nagata calls this process the principle of idealization.
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Algebra extension
| 0.837616
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1,718
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{\displaystyle (a,x)\cdot (b,y)=(ab,ay+bx).} Note that identifying (a, x) with a + εx where ε squares to zero and expanding out (a + εx)(b + εy) yields the above formula; in particular we see that E is a ring. It is sometimes called the algebra of dual numbers.
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Algebra extension
| 0.837616
|
1,719
|
Certain territories have 3-tiers and may also employ a State Championship.The largest single-day regional qualifying tournament is hosted by First State Robotics and First State FIRST LEGO League in Wilmington, Delaware. Taking place every January, this event holds FIRST LEGO League Explore, FIRST LEGO League Challenge, FIRST Tech Challenge, FIRST Robotics Competition, and robot sumo competitions under one roof at the University of Delaware's Bob Carpenter Center. Teams from Eastern Pennsylvania, Southern New Jersey, Delaware, and Maryland (among other regions) attend this tournament to make it the largest single-day FIRST event in the world.
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Power Puzzle
| 0.837611
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1,720
|
Teams in different parts of the world have different times allotted to complete the construction of the robot due to the varying date of qualifying tournaments but must have a minimum of 8 weeks from "Global Challenge Release" (the date, usually in August, by which the details of the missions and research project become available to the public). They go on to compete in FIRST LEGO League Challenge tournaments, similar to the FIRST Robotics Competition regionals. The initial levels of competition are managed by an Affiliate Partner Organization (commonly affiliated Universities), who are led by an Affiliate or Operational Partner Representative ("The Partner"). The Partner has complete control over all official tournaments in their region.
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Power Puzzle
| 0.837611
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1,721
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The FIRST LEGO League Challenge (formerly known as FIRST LEGO League) is an international competition organized by FIRST for elementary and middle school students (ages 9–14 in the United States and Canada, 9–16 elsewhere).Each year in August, FIRST LEGO League Challenge teams are introduced to a scientific and real-world challenge for teams to focus and research on. The robotics part of the competition involves designing and programming Lego Education robots to complete tasks. The students work out a solution to a problem related to the theme (changes every year) and then meet for regional, national and international tournaments to compete, share their knowledge, compare ideas, and display their robots. The FIRST LEGO League Challenge is a partnership between FIRST and the LEGO Group. It is the third division of FIRST LEGO League, following FIRST LEGO League Discover for ages 4-6, and FIRST Lego League Explore for ages 6-10.
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Power Puzzle
| 0.837611
|
1,722
|
Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization. Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.
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Mathematical algorithm
| 0.837601
|
1,723
|
Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and parsing techniques. Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields.
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Mathematical algorithm
| 0.837601
|
1,724
|
The analysis, and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually pseudocode is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their algorithmic efficiency is eventually put to the test using real code.
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Mathematical algorithm
| 0.837601
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1,725
|
All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–226)Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular, Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion, in particular, Gödel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936–37) in their mechanism-models of computation.
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Mathematical algorithm
| 0.837601
|
1,726
|
In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). In chemistry, it is expressed as the energy to ionize a mole of atoms or molecules, usually as kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol).Comparison of ionization energies of atoms in the periodic table reveals two periodic trends which follow the rules of Coulombic attraction: Ionization energy generally increases from left to right within a given period (that is, row). Ionization energy generally decreases from top to bottom in a given group (that is, column).The latter trend results from the outer electron shell being progressively farther from the nucleus, with the addition of one inner shell per row as one moves down the column.
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Ionization potential
| 0.837593
|
1,727
|
In physics and chemistry, ionization energy (IE) (American English spelling), ionisation energy (British English spelling) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, positive ion, or molecule. The first ionization energy is quantitatively expressed as X(g) + energy ⟶ X+(g) + e−where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. Ionization energy is positive for neutral atoms, meaning that the ionization is an endothermic process. Roughly speaking, the closer the outermost electrons are to the nucleus of the atom, the higher the atom's ionization energy.
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Ionization potential
| 0.837593
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1,728
|
There is, however, no single "correct" way of enumerating them, as it is a matter of arbitrary choice which quantities are considered "fundamental" and which as "derived". Uzan (2011) lists 22 "unknown constants" in the fundamental theories, which give rise to 19 "unknown dimensionless parameters", as follows: the gravitational constant G, the speed of light c, the Planck constant h, the 9 Yukawa couplings for the quarks and leptons (equivalent to specifying the rest mass of these elementary particles), 2 parameters of the Higgs field potential, 4 parameters for the quark mixing matrix, 3 coupling constants for the gauge groups SU(3) × SU(2) × U(1) (or equivalently, two coupling constants and the Weinberg angle), a phase for the QCD vacuum.The number of 19 independent fundamental physical constants is subject to change under possible extensions of the Standard Model, notably by the introduction of neutrino mass (equivalent to seven additional constants, i.e. 3 Yukawa couplings and 4 lepton mixing parameters).The discovery of variability in any of these constants would be equivalent to the discovery of "new physics".The question as to which constants are "fundamental" is neither straightforward nor meaningless, but a question of interpretation of the physical theory regarded as fundamental; as pointed out by Lévy-Leblond 1977, not all physical constants are of the same importance, with some having a deeper role than others. Lévy-Leblond 1977 proposed a classification schemes of three types of constants: A: physical properties of particular objects B: characteristic of a class of physical phenomena C: universal constantsThe same physical constant may move from one category to another as the understanding of its role deepens; this has notably happened to the speed of light, which was a class A constant (characteristic of light) when it was first measured, but became a class B constant (characteristic of electromagnetic phenomena) with the development of classical electromagnetism, and finally a class C constant with the discovery of special relativity.
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Universal constant
| 0.837592
|
1,729
|
Some physicists have explored the notion that if the dimensionless physical constants had sufficiently different values, our Universe would be so radically different that intelligent life would probably not have emerged, and that our Universe therefore seems to be fine-tuned for intelligent life. However, the phase space of the possible constants and their values is unknowable, so any conclusions drawn from such arguments are unsupported. The anthropic principle states a logical truism: the fact of our existence as intelligent beings who can measure physical constants requires those constants to be such that beings like us can exist. There are a variety of interpretations of the constants' values, including that of a divine creator (the apparent fine-tuning is actual and intentional), or that the universe is one universe of many in a multiverse (e.g. the many-worlds interpretation of quantum mechanics), or even that, if information is an innate property of the universe and logically inseparable from consciousness, a universe without the capacity for conscious beings cannot exist.
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Universal constant
| 0.837592
|
1,730
|
Minimotif Miner is a bioinformatics tool that searches protein sequence queries for a known protein targeting sequence motifs. Phobius predicts signal peptides based on a supplied primary sequence. SignalP predicts signal peptide cleavage sites. LOCtree predicts the subcellular localization of proteins.
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Protein targeting
| 0.837538
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1,731
|
Optimization has been widely used in civil engineering. Construction management and transportation engineering are among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures, resource leveling, water resource allocation, traffic management and schedule optimization.
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Optimization algorithm
| 0.837528
|
1,732
|
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.
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Optimization algorithm
| 0.837528
|
1,733
|
Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing: M q ″ ( t ) + C q ′ ( t ) + K q ( t ) = 0 {\displaystyle Mq''(t)+Cq'(t)+Kq(t)=0} Where q ( t ) ∈ R n {\displaystyle q(t)\in \mathbb {R} ^{n}} , and M , C , K ∈ R n × n {\displaystyle M,C,K\in \mathbb {R} ^{n\times n}} . If all quadratic eigenvalues of Q ( λ ) = λ 2 M + λ C + K {\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K} are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as q ( t ) = ∑ j = 1 2 n α j x j e λ j t = X e Λ t α {\displaystyle q(t)=\sum _{j=1}^{2n}\alpha _{j}x_{j}e^{\lambda _{j}t}=Xe^{\Lambda t}\alpha } Where Λ = Diag ( ) ∈ R 2 n × 2 n {\displaystyle \Lambda ={\text{Diag}}()\in \mathbb {R} ^{2n\times 2n}} are the quadratic eigenvalues, X = ∈ R n × 2 n {\displaystyle X=\in \mathbb {R} ^{n\times 2n}} are the 2 n {\displaystyle 2n} right quadratic eigenvectors, and α = ⊤ ∈ R 2 n {\displaystyle \alpha =^{\top }\in \mathbb {R} ^{2n}} is a parameter vector determined from the initial conditions on q {\displaystyle q} and q ′ {\displaystyle q'} . Stability theory for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.
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Quadratic eigenvalue problem
| 0.837523
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1,734
|
In physics, the C parity or charge parity is a multiplicative quantum number of some particles that describes their behavior under the symmetry operation of charge conjugation. Charge conjugation changes the sign of all quantum charges (that is, additive quantum numbers), including the electrical charge, baryon number and lepton number, and the flavor charges strangeness, charm, bottomness, topness and Isospin (I3). In contrast, it doesn't affect the mass, linear momentum or spin of a particle.
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Charge parity
| 0.837493
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1,735
|
Lamport's bakery algorithm is a computer algorithm devised by computer scientist Leslie Lamport, as part of his long study of the formal correctness of concurrent systems, which is intended to improve the safety in the usage of shared resources among multiple threads by means of mutual exclusion. In computer science, it is common for multiple threads to simultaneously access the same resources. Data corruption can occur if two or more threads try to write into the same memory location, or if one thread reads a memory location before another has finished writing into it. Lamport's bakery algorithm is one of many mutual exclusion algorithms designed to prevent concurrent threads entering critical sections of code concurrently to eliminate the risk of data corruption.
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Lamport's Bakery algorithm
| 0.837489
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1,736
|
These other types of substances, such as ionic compounds and network solids, are organized in such a way as to lack the existence of identifiable molecules per se. Instead, these substances are discussed in terms of formula units or unit cells as the smallest repeating structure within the substance. Examples of such substances are mineral salts (such as table salt), solids like carbon and diamond, metals, and familiar silica and silicate minerals such as quartz and granite. One of the main characteristics of a molecule is its geometry often called its structure. While the structure of diatomic, triatomic or tetra-atomic molecules may be trivial, (linear, angular pyramidal etc.) the structure of polyatomic molecules, that are constituted of more than six atoms (of several elements) can be crucial for its chemical nature.
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Molecular chemistry
| 0.837485
|
1,737
|
Similarly, theories from classical physics can be used to predict many ionic structures. With more complicated compounds, such as metal complexes, valence bond theory is less applicable and alternative approaches, such as the molecular orbital theory, are generally used. See diagram on electronic orbitals.
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Molecular chemistry
| 0.837485
|
1,738
|
Charged polyatomic collections residing in solids (for example, common sulfate or nitrate ions) are generally not considered "molecules" in chemistry. Some molecules contain one or more unpaired electrons, creating radicals. Most radicals are comparatively reactive, but some, such as nitric oxide (NO) can be stable.
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Molecular chemistry
| 0.837485
|
1,739
|
If it is longer, the constant used in GenerateComplexString can always be changed appropriately. )The above proof uses a contradiction similar to that of the Berry paradox: "1The 2smallest 3positive 4integer 5that 6cannot 7be 8defined 9in 10fewer 11than 12twenty 13English 14words". It is also possible to show the non-computability of K by reduction from the non-computability of the halting problem H, since K and H are Turing-equivalent.There is a corollary, humorously called the "full employment theorem" in the programming language community, stating that there is no perfect size-optimizing compiler.
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Algorithmic complexity theory
| 0.837478
|
1,740
|
For dynamical systems, entropy rate and algorithmic complexity of the trajectories are related by a theorem of Brudno, that the equality K ( x ; T ) = h ( T ) {\displaystyle K(x;T)=h(T)} holds for almost all x {\displaystyle x} .It can be shown that for the output of Markov information sources, Kolmogorov complexity is related to the entropy of the information source. More precisely, the Kolmogorov complexity of the output of a Markov information source, normalized by the length of the output, converges almost surely (as the length of the output goes to infinity) to the entropy of the source.
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Algorithmic complexity theory
| 0.837478
|
1,741
|
Theorem: There exists a constant L (which only depends on S and on the choice of description language) such that there does not exist a string s for which the statement K(s) ≥ L (as formalized in S)can be proven within S.Proof Idea: The proof of this result is modeled on a self-referential construction used in Berry's paradox. We firstly obtain a program which enumerates the proofs within S and we specify a procedure P which takes as an input an integer L and prints the strings x which are within proofs within S of the statement K(x) ≥ L. By then setting L to greater than the length of this procedure P, we have that the required length of a program to print x as stated in K(x) ≥ L as being at least L is then less than the amount L since the string x was printed by the procedure P. This is a contradiction. So it is not possible for the proof system S to prove K(x) ≥ L for L arbitrarily large, in particular, for L larger than the length of the procedure P, (which is finite).
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Algorithmic complexity theory
| 0.837478
|
1,742
|
By the above theorem (§ Compression), most strings are complex in the sense that they cannot be described in any significantly "compressed" way. However, it turns out that the fact that a specific string is complex cannot be formally proven, if the complexity of the string is above a certain threshold. The precise formalization is as follows.
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Algorithmic complexity theory
| 0.837478
|
1,743
|
It is straightforward to compute upper bounds for K(s) – simply compress the string s with some method, implement the corresponding decompressor in the chosen language, concatenate the decompressor to the compressed string, and measure the length of the resulting string – concretely, the size of a self-extracting archive in the given language. A string s is compressible by a number c if it has a description whose length does not exceed |s| − c bits. This is equivalent to saying that K(s) ≤ |s| − c. Otherwise, s is incompressible by c. A string incompressible by 1 is said to be simply incompressible – by the pigeonhole principle, which applies because every compressed string maps to only one uncompressed string, incompressible strings must exist, since there are 2n bit strings of length n, but only 2n − 1 shorter strings, that is, strings of length less than n, (i.e. with length 0, 1, ..., n − 1). For the same reason, most strings are complex in the sense that they cannot be significantly compressed – their K(s) is not much smaller than |s|, the length of s in bits. To make this precise, fix a value of n. There are 2n bitstrings of length n. The uniform probability distribution on the space of these bitstrings assigns exactly equal weight 2−n to each string of length n. Theorem: With the uniform probability distribution on the space of bitstrings of length n, the probability that a string is incompressible by c is at least 1 − 2−c+1 + 2−n. To prove the theorem, note that the number of descriptions of length not exceeding n − c is given by the geometric series: 1 + 2 + 22 + ... + 2n − c = 2n−c+1 − 1.There remain at least 2n − 2n−c+1 + 1bitstrings of length n that are incompressible by c. To determine the probability, divide by 2n.
|
Algorithmic complexity theory
| 0.837478
|
1,744
|
Theorem: There exist strings of arbitrarily large Kolmogorov complexity. Formally: for each natural number n, there is a string s with K(s) ≥ n.Proof: Otherwise all of the infinitely many possible finite strings could be generated by the finitely many programs with a complexity below n bits. Theorem: K is not a computable function. In other words, there is no program which takes any string s as input and produces the integer K(s) as output.
|
Algorithmic complexity theory
| 0.837478
|
1,745
|
Switching Neural Networks made use of Boolean algebra to build sets of intelligible rules able to obtain very good performance. In 2014, an efficient version of Switching Neural Network was developed and implemented in the Rulex suite with the name Logic Learning Machine. Also, an LLM version devoted to regression problems was developed.
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Logic learning machine
| 0.837463
|
1,746
|
The Switching Neural Network approach was developed in the 1990s to overcome the drawbacks of the most commonly used machine learning methods. In particular, black box methods, such as multilayer perceptron and support vector machine, had good accuracy but could not provide deep insight into the studied phenomenon. On the other hand, decision trees were able to describe the phenomenon but often lacked accuracy.
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Logic learning machine
| 0.837463
|
1,747
|
Logic learning machine (LLM) is a machine learning method based on the generation of intelligible rules. LLM is an efficient implementation of the Switching Neural Network (SNN) paradigm, developed by Marco Muselli, Senior Researcher at the Italian National Research Council CNR-IEIIT in Genoa. LLM has been employed in many different sectors, including the field of medicine (orthopedic patient classification, DNA micro-array analysis and Clinical Decision Support Systems ), financial services and supply chain management.
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Logic learning machine
| 0.837463
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1,748
|
Like other machine learning methods, LLM uses data to build a model able to perform a good forecast about future behaviors. LLM starts from a table including a target variable (output) and some inputs and generates a set of rules that return the output value y {\displaystyle y} corresponding to a given configuration of inputs. A rule is written in the form: if p r e m i s e then c o n s e q u e n c e {\displaystyle {\textbf {if}}{\text{ }}premise{\text{ }}{\textbf {then}}{\text{ }}consequence} where consequence contains the output value whereas premise includes one or more conditions on the inputs. According to the input type, conditions can have different forms: for categorical variables the input value must be in a given subset: x 1 ∈ { A , B , C , .
|
Logic learning machine
| 0.837463
|
1,749
|
Expectation (or mean), variance and covariance Jensen's inequality General moments about the mean Correlated and uncorrelated random variables Conditional expectation: law of total expectation, law of total variance Fatou's lemma and the monotone and dominated convergence theorems Markov's inequality and Chebyshev's inequality
|
Outline of probability
| 0.837443
|
1,750
|
Events in probability theory Elementary events, sample spaces, Venn diagrams Mutual exclusivity
|
Outline of probability
| 0.837443
|
1,751
|
Central limit theorem and Laws of large numbers Illustration of the central limit theorem and a 'concrete' illustration Berry–Esséen theorem Law of the iterated logarithm
|
Outline of probability
| 0.837443
|
1,752
|
A proof of the central limit theorem
|
Outline of probability
| 0.837443
|
1,753
|
(Related topics: set theory, simple theorems in the algebra of sets)
|
Outline of probability
| 0.837443
|
1,754
|
Stochastic calculus Diffusions Brownian motion Wiener equation Wiener process
|
Outline of probability
| 0.837443
|
1,755
|
Sample spaces, σ-algebras and probability measures Probability space Sample space Standard probability space Random element Random compact set Dynkin system Probability axioms Event (probability theory) Complementary event Elementary event "Almost surely"
|
Outline of probability
| 0.837443
|
1,756
|
The attitude of mind is of the form "How certain is it that the event will occur?" The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
|
Outline of probability
| 0.837443
|
1,757
|
Martingale central limit theorem Azuma's inequality
|
Outline of probability
| 0.837443
|
1,758
|
Independence (probability theory) The Borel–Cantelli lemmas and Kolmogorov's zero–one law
|
Outline of probability
| 0.837443
|
1,759
|
Convergence in distribution and convergence in probability, Convergence in mean, mean square and rth mean Almost sure convergence Skorokhod's representation theorem
|
Outline of probability
| 0.837443
|
1,760
|
Conditional probability The law of total probability Bayes' theorem
|
Outline of probability
| 0.837443
|
1,761
|
Conditional probability Conditioning (probability) Conditional expectation Conditional probability distribution Regular conditional probability Disintegration theorem Bayes' theorem Rule of succession Conditional independence Conditional event algebra Goodman–Nguyen–van Fraassen algebra
|
Outline of probability
| 0.837443
|
1,762
|
According to ABI, the SOLiD 3plus platform yields 60 gigabases of usable DNA data per run. Due to the two base encoding system, an inherent accuracy check is built into the technology and offers 99.94% accuracy. The chemistry of the systems also means that it is not hindered by homopolymers unlike the Roche 454 FLX system and so large and difficult homopolymer repeat regions are no longer a problem to sequence.
|
ABI Solid Sequencing
| 0.837428
|
1,763
|
It has in the past been combined with array technology (ChIP-chip) with some success. Next gen sequencing can also be applied in this area. Methylation immunoprecipitation (MeDIP) can also be performed and also on arrays. The ability to learn more about methylation and TF binding sites on a genome wide scale is a valuable resource and could teach us much about disease and molecular biology in general.
|
ABI Solid Sequencing
| 0.837428
|
1,764
|
The DNA Learning Center (DNALC) is a genetics learning center affiliated with the Cold Spring Harbor Laboratory, in Cold Spring Harbor, New York. It is the world's first science center devoted entirely to genetics education and offers online education, class field trips, student summer day camps, and teacher training. The DNALC's family of internet sites cover broad topics including basic heredity, genetic disorders, eugenics, the discovery of the structure of DNA, DNA sequencing, cancer, neuroscience, and plant genetics. The center developed a website called DNA Subway for the iPlant Collaborative.
|
Dolan DNA Learning Center
| 0.837402
|
1,765
|
In model theory, the term "submodel" is often used as a synonym for substructure, especially when the context suggests a theory of which both structures are models. In the presence of relations (i.e. for structures such as ordered groups or graphs, whose signature is not functional) it may make sense to relax the conditions on a subalgebra so that the relations on a weak substructure (or weak subalgebra) are at most those induced from the bigger structure. Subgraphs are an example where the distinction matters, and the term "subgraph" does indeed refer to weak substructures. Ordered groups, on the other hand, have the special property that every substructure of an ordered group which is itself an ordered group, is an induced substructure.
|
Substructure (mathematics)
| 0.837371
|
1,766
|
In mathematical logic, an (induced) substructure or (induced) subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are restricted to the substructure's domain. Some examples of subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of algebras over a field, or induced subgraphs. Shifting the point of view, the larger structure is called an extension or a superstructure of its substructure.
|
Substructure (mathematics)
| 0.837371
|
1,767
|
Much of the past research into the composition and toxicity of UCM hydrocarbons has been conducted by the Petroleum and Environmental Geochemistry Group (PEGG) at the University of Plymouth, UK. As well as the hydrocarbon UCM, oils also contain more polar compounds such as those containing oxygen, sulphur or nitrogen. These compounds can be very soluble in water and hence bioavailable to marine and aquatic organisms. Polar UCMs are present within produced waters from oil rigs and from oil sands processing.
|
Unresolved complex mixture
| 0.837364
|
1,768
|
In physics, a force field is a vector field corresponding with a non-contact force acting on a particle at various positions in space. Specifically, a force field is a vector field F → {\displaystyle {\vec {F}}} , where F → ( x → ) {\displaystyle {\vec {F}}({\vec {x}})} is the force that a particle would feel if it were at the point x → {\displaystyle {\vec {x}}} .
|
Force field (physics)
| 0.837354
|
1,769
|
Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations. Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues. George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).
|
Algebra
| 0.837314
|
1,770
|
The idea of a determinant was developed by Japanese mathematician Seki Kōwa in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph-Louis Lagrange in his 1770 paper "Réflexions sur la résolution algébrique des équations" devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents.
|
Algebra
| 0.837314
|
1,771
|
François Viète's work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, René Descartes published La Géométrie, inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century.
|
Algebra
| 0.837314
|
1,772
|
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z).
|
Algebra
| 0.837314
|
1,773
|
Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors of a plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra. Binary operations: The notion of addition (+) is generalized to the notion of binary operation (denoted here by ∗). The notion of binary operation is meaningless without the set on which the operation is defined.
|
Algebra
| 0.837314
|
1,774
|
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are the listed fundamental concepts in abstract algebra. Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: collections of objects called elements. All collections of the familiar types of numbers are sets.
|
Algebra
| 0.837314
|
1,775
|
Herstein, I. N. (1964). Topics in Algebra. Ginn and Company. ISBN 0-471-02371-X.
|
Algebra
| 0.837314
|
1,776
|
"The Sources of Al-Khowārizmī's Algebra". Osiris. 1: 263–277.
|
Algebra
| 0.837314
|
1,777
|
The two preceding examples define the same polynomial function. Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that cannot be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.
|
Algebra
| 0.837314
|
1,778
|
It has been suggested that elementary algebra should be taught to students as young as eleven years old, though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States. However, in some US schools, algebra instruction starts in ninth grade.
|
Algebra
| 0.837314
|
1,779
|
With a qualifier, there is the same distinction: Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves). With an article, it means an instance of some algebraic structure, like a Lie algebra, an associative algebra, or a vertex operator algebra. Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.
|
Algebra
| 0.837314
|
1,780
|
Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations.
|
Algebra
| 0.837314
|
1,781
|
The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, "algebra" names a broad part of mathematics. As a single word with an article or in the plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the context.
|
Algebra
| 0.837314
|
1,782
|
This includes but is not limited to the theory of equations. At the beginning of the 20th century, algebra evolved further by considering operations that act not only on numbers but also on elements of so-called mathematical structures such as groups, fields and vector spaces. This new algebra was called modern algebra by van der Waerden in his eponymous treatise, whose name has been changed to Algebra in later editions.
|
Algebra
| 0.837314
|
1,783
|
This is still what historians of mathematics generally mean by algebra.In mathematics, the meaning of algebra has evolved after the introduction by François Viète of symbols (variables) for denoting unknown or incompletely specified numbers, and the resulting use of the mathematical notation for equations and formulas. So, algebra became essentially the study of the action of operations on expressions involving variables.
|
Algebra
| 0.837314
|
1,784
|
The use of the word "algebra" for denoting a part of mathematics dates probably from the 16th century. The word is derived from the Arabic word al-jabr that appears in the title of the treatise Al-Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala (The Compendious Book on Calculation by Completion and Balancing), written circa 820 by Al-Kwarizmi. Al-jabr referred to a method for transforming equations by subtracting like terms from both sides, or passing one term from one side to the other, after changing its sign. Therefore, algebra referred originally to the manipulation of equations, and, by extension, to the theory of equations.
|
Algebra
| 0.837314
|
1,785
|
The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed Σn2, Σn3 and used the method of successive approximation to determine square roots.
|
Algebra
| 0.837314
|
1,786
|
His book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe. Yet another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function.
|
Algebra
| 0.837314
|
1,787
|
His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".According to Jeffrey Oaks and Jean Christianidis neither Diophantus no Al-Khwarizmi should be called "father of algebra" .Pre-modern algebra was developed and used by merchants and surveyors as part of what Jens Høyrup called "subscientific" tradition. Diophantus used this method of algebra in his book, in particular for indeterminate problems, while Al-Khwarizmi wrote one of the first books in arabic about this method.Another Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation.
|
Algebra
| 0.837314
|
1,788
|
It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs while treating algebra as an independent discipline in its own right.
|
Algebra
| 0.837314
|
1,789
|
Later, Persian and Arab mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".
|
Algebra
| 0.837314
|
1,790
|
780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta, continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brāhmasphuṭasiddhānta are on a higher level. For example, the first complete arithmetic solution written in words instead of symbols, including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta, published in 628 AD.
|
Algebra
| 0.837314
|
1,791
|
These texts deal with solving algebraic equations, and have led, in number theory, to the modern notion of Diophantine equation. Earlier traditions discussed above had a direct influence on the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c.
|
Algebra
| 0.837314
|
1,792
|
The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica.
|
Algebra
| 0.837314
|
1,793
|
The roots of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam.By the time of Plato, Greek mathematics had undergone a drastic change.
|
Algebra
| 0.837314
|
1,794
|
Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions. Relational algebra: a set of finitary relations that is closed under certain operators.Many mathematical structures are called algebras: Algebra over a field or more generally algebra over a ring.Many classes of algebras over a field or over a ring have a specific name: Associative algebra Non-associative algebra Lie algebra Composition algebra Hopf algebra C*-algebra Symmetric algebra Exterior algebra Tensor algebra In measure theory, Sigma-algebra Algebra over a set In category theory F-algebra and F-coalgebra T-algebra In logic, Relation algebra, a residuated Boolean algebra expanded with an involution called converse. Boolean algebra, a complemented distributive lattice. Heyting algebra
|
Algebra
| 0.837314
|
1,795
|
Universal algebra, in which properties common to all algebraic structures are studied. Algebraic number theory, in which the properties of numbers are studied from an algebraic point of view. Algebraic geometry, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.
|
Algebra
| 0.837314
|
1,796
|
Commutative algebra, the study of commutative rings. Computer algebra, the implementation of algebraic methods as algorithms and computer programs. Homological algebra, the study of algebraic structures that are fundamental to study topological spaces.
|
Algebra
| 0.837314
|
1,797
|
Abstract algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated. Linear algebra, in which the specific properties of linear equations, vector spaces and matrices are studied. Boolean algebra, a branch of algebra abstracting the computation with the truth values false and true.
|
Algebra
| 0.837314
|
1,798
|
Some subareas of algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name. Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics.
|
Algebra
| 0.837314
|
1,799
|
The group (Z,+) of integers is free of rank 1; a generating set is S = {1}. The integers are also a free abelian group, although all free groups of rank ≥ 2 {\displaystyle \geq 2} are non-abelian. A free group on a two-element set S occurs in the proof of the Banach–Tarski paradox and is described there. On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order. In algebraic topology, the fundamental group of a bouquet of k circles (a set of k loops having only one point in common) is the free group on a set of k elements.
|
Free subgroup
| 0.837294
|
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