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1,700 | 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 ... | Bio informatics | 0.837745 |
1,701 | 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 ... | Bio informatics | 0.837745 |
1,702 | 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 cont... | Bio informatics | 0.837745 |
1,703 | 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 divis... | Bio informatics | 0.837745 |
1,704 | 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 predi... | Program optimization | 0.837667 |
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 operatin... | Program optimization | 0.837666 |
1,706 | 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... | Program optimization | 0.837666 |
1,707 | 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 ... | Program optimization | 0.837666 |
1,708 | 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 ... | Program optimization | 0.837666 |
1,709 | 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 ... | Program optimization | 0.837666 |
1,710 | 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 ... | Program optimization | 0.837666 |
1,711 | 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 intercha... | Drug therapy | 0.837665 |
1,712 | 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... | Polony Sequencing | 0.837655 |
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 ... | Binary operator | 0.837649 |
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 oth... | 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 sof... | Protein interactions | 0.837618 |
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 | Protein interactions | 0.837618 |
1,717 | 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... | Algebra extension | 0.837616 |
1,718 | {\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. | 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... | Power Puzzle | 0.837611 |
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 becom... | Power Puzzle | 0.837611 |
1,721 | 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-wo... | 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. | 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, combinato... | 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 a... | Mathematical algorithm | 0.837601 |
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 p... | 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... | 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) + en... | Ionization potential | 0.837593 |
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 grav... | 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 spa... | 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. | Protein targeting | 0.837538 |
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 ... | 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 aris... | 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... | Quadratic eigenvalue problem | 0.837523 |
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 ... | Charge parity | 0.837493 |
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, i... | Lamport's Bakery algorithm | 0.837489 |
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 s... | 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. | 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. | 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-c... | 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... | 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 ... | 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 form... | 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 archi... | 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 computa... | 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 prob... | 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 ... | 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 ... | Logic learning machine | 0.837463 |
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... | 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 in... | 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 extensivel... | 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 sys... | 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 valua... | 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. ... | 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 condi... | 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 algeb... | 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. ... | 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 wer... | 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 ... | 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 cen... | 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 sol... | 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 ca... | 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 techn... | 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: co... | 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. Th... | 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 f... | 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... | 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 de... | 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... | 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.... | 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... | 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 represen... | 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 ... | 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 s... | 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 wh... | 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, ... | 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... | 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 equation... | 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... | 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 equati... | 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 value... | 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 ... | 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 h... | Free subgroup | 0.837294 |
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