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7,300 | Within Z, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses in Z. If the ring or algebra is finite, however, then all elements a which are not zero divisors do have a (left and right) inverse. For, first observe that the map f(x) = ax must be injective: f(x) = f(y) impl... | Arithmetic inverse | 0.822911 |
7,301 | The extended Euclidean algorithm may be used to compute it. The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements x, y such that xy = 0. A square matrix has an inverse if and only if its determinant has an inverse ... | Arithmetic inverse | 0.822911 |
7,302 | In the mathematical subfield of graph theory a level structure of an undirected graph is a partition of the vertices into subsets that have the same distance from a given root vertex. | Level structure | 0.822906 |
7,303 | In the field of molecular physics, microwave spectroscopy is commonly used to probe the rotation of molecules. | Microwave spectroscopy | 0.8229 |
7,304 | In the field of condensed matter physics, microwave spectroscopy is used to detect dynamic phenomena of either charges or spins at GHz frequencies (corresponding to nanosecond time scales) and energy scales in the µeV regime. Matching to these energy scales, microwave spectroscopy on solids is often performed as a func... | Microwave spectroscopy | 0.8229 |
7,305 | Nitrogen is the chemical element with atomic number 7, the symbol N and atomic mass 14.00674 u. Elemental nitrogen is a colorless, odorless, tasteless and mostly inert diatomic gas at standard conditions, constituting 78.08% by volume of Earth's atmosphere. The element nitrogen was discovered as a separable component o... | Period 2 element | 0.822898 |
7,306 | In plane geometry, the square root of 6 can be constructed via a sequence of dynamic rectangles, as illustrated here.In solid geometry, the square root of 6 appears as the longest distances between corners (vertices) of the double cube, as illustrated above. The square roots of all lower natural numbers appear as the d... | Square root of 6 | 0.822895 |
7,307 | It takes two more digits (2.4495) to reduce the error by about half. The approximation 218/89 (≈ 2.449438...) is nearly ten times better: despite having a denominator of only 89, it differs from the correct value by less than 1/20,000, or less than one part in 47,000. Since 6 is the product of 2 and 3, the square root ... | Square root of 6 | 0.822895 |
7,308 | {\displaystyle 6^{\frac {1}{2}}.} It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are: 2.44948974278317809819728407470589139196594748065667012843269....which can be rounded up to 2.45 to within about 99.98% accuracy (about 1 part in 4800); that is, it differs from the c... | Square root of 6 | 0.822895 |
7,309 | In the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation: v RMS = 3 R T M {\displaystyle v_{\text{RMS}}={\sqrt {3RT \over M}}} where R represents the gas constant, 8.314 J/(mol·K), T i... | Quadratic mean | 0.822894 |
7,310 | Omics based methods use omics technologies, such as chemoproteomics, reverse genetics and genomics, transcriptomics, and proteomics, to identify the potential targets of the compound of interest. Reverse genetics and genomics approaches, for instance, uses genetic perturbation (e.g. CRISPR-Cas9 or siRNA) in combination... | Mechanism of action | 0.822888 |
7,311 | In some literature articles, the terms "mechanism of action" and "mode of action" are used interchangeably, typically referring to the way in which the drug interacts and produces a medical effect. However, in actuality, a mode of action describes functional or anatomical changes, at the cellular level, resulting from ... | Mechanism of action | 0.822888 |
7,312 | The Margolus–Levitin theorem sets a bound on the maximum computational speed per unit of energy: 6 × 1033 operations per second per joule. This bound, however, can be avoided if there is access to quantum memory. Computational algorithms can then be designed that require arbitrarily small amounts of energy/time per one... | Limits to computation | 0.822882 |
7,313 | This is a special case of Weil's conjecture on Tamagawa numbers, which asserts the equality of similar such infinite products of arithmetic quantities, localized at each prime p, and a geometrical quantity: the reciprocal of the volume of a certain locally symmetric space. In the case of the Basel problem, it is the hy... | Archimedes' constant | 0.822881 |
7,314 | The Riemann zeta function ζ(s) is used in many areas of mathematics. When evaluated at s = 2 it can be written as Finding a simple solution for this infinite series was a famous problem in mathematics called the Basel problem. Leonhard Euler solved it in 1735 when he showed it was equal to π2/6. Euler's result leads to... | Archimedes' constant | 0.822881 |
7,315 | Many of the appearances of π in the formulae of mathematics and the sciences have to do with its close relationship with geometry. However, π also appears in many natural situations having apparently nothing to do with geometry. In many applications, it plays a distinguished role as an eigenvalue. | Archimedes' constant | 0.822881 |
7,316 | In modern mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as number theory and statistics. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of science. Sever... | Archimedes' constant | 0.822881 |
7,317 | The extensive computations involved have also been used to test supercomputers. Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology,... | Archimedes' constant | 0.822881 |
7,318 | The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706.The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20... | Archimedes' constant | 0.822881 |
7,319 | The number π (; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two i... | Archimedes' constant | 0.822881 |
7,320 | This theorem is ultimately connected with the spectral characterization of π as the eigenvalue associated with the Heisenberg uncertainty principle, and the fact that equality holds in the uncertainty principle only for the Gaussian function. Equivalently, π is the unique constant making the Gaussian normal distributio... | Archimedes' constant | 0.822881 |
7,321 | The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution. The Gaussian function, which is the probability density function of the normal... | Archimedes' constant | 0.822881 |
7,322 | Although not a physical constant, π appears routinely in equations describing fundamental principles of the universe, often because of π's relationship to the circle and to spherical coordinate systems. A simple formula from the field of classical mechanics gives the approximate period T of a simple pendulum of length ... | Archimedes' constant | 0.822881 |
7,323 | The constant π also appears naturally in Fourier series of periodic functions. Periodic functions are functions on the group T =R/Z of fractional parts of real numbers. The Fourier decomposition shows that a complex-valued function f on T can be written as an infinite linear superposition of unitary characters of T. Th... | Archimedes' constant | 0.822881 |
7,324 | Several infinite series are described, including series for sine (which Nilakantha attributes to Madhava of Sangamagrama), cosine, and arctangent which are now sometimes referred to as Madhava series. The series for arctangent is sometimes called Gregory's series or the Gregory–Leibniz series. Madhava used infinite ser... | Archimedes' constant | 0.822881 |
7,325 | In machine learning, pattern recognition is the assignment of a label to a given input value. In statistics, discriminant analysis was introduced for this same purpose in 1936. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes (for example, d... | Pattern Recognition | 0.822874 |
7,326 | KDD and data mining have a larger focus on unsupervised methods and stronger connection to business use. Pattern recognition focuses more on the signal and also takes acquisition and signal processing into consideration. It originated in engineering, and the term is popular in the context of computer vision: a leading ... | Pattern Recognition | 0.822874 |
7,327 | Pattern recognition is the automated recognition of patterns and regularities in data. While similar, pattern recognition (PR) is not to be confused with pattern machines (PM) which may possess (PR) capabilities but their primary function is to distinguish and create emergent pattern. PR has applications in statistical... | Pattern Recognition | 0.822874 |
7,328 | It follows that they can be defined, specified and studied in terms of linear maps. This is also the case of homographies and Möbius transformations, when considered as transformations of a projective space. Until the end of the 19th century, geometric spaces were defined by axioms relating points, lines and planes (sy... | Linear Algebra | 0.82287 |
7,329 | Most of the theory of abelian groups may be extended to modules over a principal ideal domain. In particular, over a principal ideal domain, every submodule of a free module is free, and the fundamental theorem of finitely generated abelian groups may be extended straightforwardly to finitely generated modules over a p... | Linear Algebra | 0.82287 |
7,330 | (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed. ), Springer, ISBN 978-1-4020-5494-5 Golan, Johnathan S. (August 1995), Foundations of Linear Algebra, Kluwer, ISBN 0-7923-3614-3 Greub, Werner H. | Linear Algebra | 0.82287 |
7,331 | (1989), Lectures on Linear Algebra, Dover Publications, ISBN 978-0-486-66082-0 Glazman, I. M.; Ljubic, Ju. I. (2006), Finite-Dimensional Linear Analysis, Dover Publications, ISBN 978-0-486-45332-3 Golan, Johnathan S. | Linear Algebra | 0.82287 |
7,332 | Mikaelian, Vahagn H. (2020) . Linear Algebra: Theory and Algorithms. Yerevan, Armenia: Self-published – via ResearchGate. Sharipov, Ruslan, Course of linear algebra and multidimensional geometry Treil, Sergei, Linear Algebra Done Wrong | Linear Algebra | 0.82287 |
7,333 | ), CRC Press, ISBN 978-1-4398-0040-9 Sadun, Lorenzo (2008), Applied Linear Algebra: the decoupling principle (2nd ed. ), AMS, ISBN 978-0-8218-4441-0 Strang, Gilbert (2016), Introduction to Linear Algebra (5th ed. ), Wellesley-Cambridge Press, ISBN 978-09802327-7-6 The Manga Guide to Linear Algebra (2012), by Shin Takah... | Linear Algebra | 0.82287 |
7,334 | 1977) , Pearson Higher Education, ISBN 978-0130413437. Poole, David (2010), Linear Algebra: A Modern Introduction (3rd ed. ), Cengage – Brooks/Cole, ISBN 978-0-538-73545-2 Ricardo, Henry (2010), A Modern Introduction To Linear Algebra (1st ed. | Linear Algebra | 0.82287 |
7,335 | (2006), Linear Algebra With Applications (7th ed. ), Pearson Prentice Hall, ISBN 978-0-13-185785-8 Murty, Katta G. (2014) Computational and Algorithmic Linear Algebra and n-Dimensional Geometry, World Scientific Publishing, ISBN 978-981-4366-62-5. | Linear Algebra | 0.82287 |
7,336 | ), Prentice Hall, ISBN 978-0-13-229654-0 Lay, David C. (2005), Linear Algebra and Its Applications (3rd ed. ), Addison Wesley, ISBN 978-0-321-28713-7 Leon, Steven J. | Linear Algebra | 0.82287 |
7,337 | OL 30872051M. Kolman, Bernard; Hill, David R. (2007), Elementary Linear Algebra with Applications (9th ed. | Linear Algebra | 0.82287 |
7,338 | Linear Algebra (4th ed.). Ann Arbor, Michigan: Orthogonal Publishing. | Linear Algebra | 0.82287 |
7,339 | Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed. ), Wiley International Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed. ), Chapman and Hall/CRC, ISBN 978-1420095388 Bretscher, Otto (2004), Linear Algebra with... | Linear Algebra | 0.82287 |
7,340 | Most physical phenomena are modeled by partial differential equations. To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting cells. For linear systems this interaction involves linear functions. For nonlinear systems, this interaction is often approximated ... | Linear Algebra | 0.82287 |
7,341 | A complete metric space along with the additional structure of an inner product (a conjugate symmetric sesquilinear form) is known as a Hilbert space, which is in some sense a particularly well-behaved Banach space. Functional analysis applies the methods of linear algebra alongside those of mathematical analysis to st... | Linear Algebra | 0.82287 |
7,342 | A finite set of linear equations in a finite set of variables, for example, x1, x2, ..., xn, or x, y, ..., z is called a system of linear equations or a linear system.Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such... | Linear Algebra | 0.82287 |
7,343 | Nearly all scientific computations involve linear algebra. Consequently, linear algebra algorithms have been highly optimized. BLAS and LAPACK are the best known implementations. For improving efficiency, some of them configure the algorithms automatically, at run time, for adapting them to the specificities of the com... | Linear Algebra | 0.82287 |
7,344 | Vector spaces that are not finite dimensional often require additional structure to be tractable. A normed vector space is a vector space along with a function called a norm, which measures the "size" of elements. The norm induces a metric, which measures the distance between elements, and induces a topology, which all... | Linear Algebra | 0.82287 |
7,345 | The first four axioms mean that V is an abelian group under addition. An element of a specific vector space may have various nature; for example, it could be a sequence, a function, a polynomial or a matrix. Linear algebra is concerned with those properties of such objects that are common to all vector spaces. | Linear Algebra | 0.82287 |
7,346 | Until the 19th century, linear algebra was introduced through systems of linear equations and matrices. In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general (not limited to the finite-dimensional case), and conceptually simpler, although more abs... | Linear Algebra | 0.82287 |
7,347 | This section presents several related topics that do not appear generally in elementary textbooks on linear algebra, but are commonly considered, in advanced mathematics, as parts of linear algebra. | Linear Algebra | 0.82287 |
7,348 | Around this date, it appeared that one may also define geometric spaces by constructions involving vector spaces (see, for example, Projective space and Affine space). It has been shown that the two approaches are essentially equivalent. In classical geometry, the involved vector spaces are vector spaces over the reals... | Linear Algebra | 0.82287 |
7,349 | This was one of the main motivations for developing linear algebra. Most geometric transformation, such as translations, rotations, reflections, rigid motions, isometries, and projections transform lines into lines. | Linear Algebra | 0.82287 |
7,350 | There is a strong relationship between linear algebra and geometry, which started with the introduction by René Descartes, in 1637, of Cartesian coordinates. In this new (at that time) geometry, now called Cartesian geometry, points are represented by Cartesian coordinates, which are sequences of three real numbers (in... | Linear Algebra | 0.82287 |
7,351 | More precisely, if S is a linearly independent set, and T is a spanning set such that S ⊆ T, then there is a basis B such that S ⊆ B ⊆ T. Any two bases of a vector space V have the same cardinality, which is called the dimension of V; this is the dimension theorem for vector spaces. Moreover, two vector spaces over the... | Linear Algebra | 0.82287 |
7,352 | The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries w... | Linear Algebra | 0.82287 |
7,353 | He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the... | Linear Algebra | 0.82287 |
7,354 | Linear algebra grew with ideas noted in the complex plane. For instance, two numbers w and z in C {\displaystyle \mathbb {C} } have a difference w – z, and the line segments wz and 0(w − z) are of the same length and direction. The segments are equipollent. | Linear Algebra | 0.82287 |
7,355 | In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule. Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy.In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new top... | Linear Algebra | 0.82287 |
7,356 | The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations.Systems o... | Linear Algebra | 0.82287 |
7,357 | The modeling of ambient space is based on geometry. Sciences concerned with this space use geometry widely. This is the case with mechanics and robotics, for describing rigid body dynamics; geodesy for describing Earth shape; perspectivity, computer vision, and computer graphics, for describing the relationship between... | Linear Algebra | 0.82287 |
7,358 | Functional analysis studies function spaces. These are vector spaces with additional structure, such as Hilbert spaces. Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, quantum mechanics (wave functions). | Linear Algebra | 0.82287 |
7,359 | Fearnley-Sander, Desmond, "Hermann Grassmann and the Creation of Linear Algebra", American Mathematical Monthly 86 (1979), pp. 809–817. Grassmann, Hermann (1844), Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik... | Linear Algebra | 0.82287 |
7,360 | ), Springer, ISBN 978-0-387-96412-6 Marcus, Marvin; Minc, Henryk (2010), A Survey of Matrix Theory and Matrix Inequalities, Dover Publications, ISBN 978-0-486-67102-4 Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-4... | Linear Algebra | 0.82287 |
7,361 | ), Pearson, ISBN 978-0-13-486024-4 Horn, Roger A.; Johnson, Charles R. (February 23, 1990), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 Horn, Roger A.; Johnson, Charles R. (June 24, 1994), Topics in Matrix Analysis, Cambridge University Press, ISBN 978-0-521-46713-1 Lang, Serge (March 9, 2004), ... | Linear Algebra | 0.82287 |
7,362 | : Prentice-Hall, Inc., MR 0276251 Halmos, Paul R. (August 20, 1993), Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-90093-3 Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (September 7, 2018), Linear Algebra (5th ed. | Linear Algebra | 0.82287 |
7,363 | (October 16, 1981), Linear Algebra, Graduate Texts in Mathematics (4th ed. ), Springer, ISBN 978-0-8018-5414-9 Hoffman, Kenneth; Kunze, Ray (1971), Linear algebra (2nd ed. ), Englewood Cliffs, N.J. | Linear Algebra | 0.82287 |
7,364 | Bhatia, Rajendra (November 15, 1996), Matrix Analysis, Graduate Texts in Mathematics, Springer, ISBN 978-0-387-94846-1 Demmel, James W. (August 1, 1997), Applied Numerical Linear Algebra, SIAM, ISBN 978-0-89871-389-3 Dym, Harry (2007), Linear Algebra in Action, AMS, ISBN 978-0-8218-3813-6 Gantmacher, Felix R. (2005), A... | Linear Algebra | 0.82287 |
7,365 | Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an inner product. The inner product is an example of a bilinear form, and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an inner product is a map ⟨ ... | Linear Algebra | 0.82287 |
7,366 | (2013) . Elementary Linear Algebra. University of Queensland, Brisbane, Australia: Self-published. | Linear Algebra | 0.82287 |
7,367 | Interactive Linear Algebra. Georgia Institute of Technology, Atlanta, Georgia: Self-published. Matthews, Keith R. | Linear Algebra | 0.82287 |
7,368 | Linear Algebra (4th ed.). Ann Arbor, Michigan: Orthogonal Publishing. ISBN 978-1-944325-11-4. | Linear Algebra | 0.82287 |
7,369 | Elements of Abstract and Linear Algebra. University of Miami, Coral Gables, Florida: Self-published. Hefferon, Jim (2020). | Linear Algebra | 0.82287 |
7,370 | Beezer, Robert A. (2009) . A First Course in Linear Algebra. Gainesville, Florida: University Press of Florida. | Linear Algebra | 0.82287 |
7,371 | A linear endomorphism is a linear map that maps a vector space V to itself. If V has a basis of n elements, such an endomorphism is represented by a square matrix of size n. With respect to general linear maps, linear endomorphisms and square matrices have some specific properties that make their study an important par... | Linear Algebra | 0.82287 |
7,372 | Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra. Let V be a finite-dimensional vector space over a field F, and (v1, v2, ..., vm) be a basis of V (thus m is the dimension of V). | Linear Algebra | 0.82287 |
7,373 | MIT Linear Algebra Video Lectures, a series of 34 recorded lectures by Professor Gilbert Strang (Spring 2010) International Linear Algebra Society "Linear algebra", Encyclopedia of Mathematics, EMS Press, 2001 Linear Algebra on MathWorld Matrix and Linear Algebra Terms on Earliest Known Uses of Some of the Words of Ma... | Linear Algebra | 0.82287 |
7,374 | Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F, a linear map (also called, in some contexts, linear transformation or linear mapping) is a map T: V → W {\displaystyle T:V\to W} that is compatible with addition and scalar multiplica... | Linear Algebra | 0.82287 |
7,375 | Linear algebra is used in almost all areas of mathematics, thus making it relevant in almost all scientific domains that use mathematics. These applications may be divided into several wide categories. | Linear Algebra | 0.82287 |
7,376 | Linear algebra is the branch of mathematics concerning linear equations such as: a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,} linear maps such as: ( x 1 , … , x n ) ↦ a 1 x 1 + ⋯ + a n x n , {\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},} and their representat... | Linear Algebra | 0.82287 |
7,377 | Leduc, Steven A. (May 1, 1996), Linear Algebra (Cliffs Quick Review), Cliffs Notes, ISBN 978-0-8220-5331-6 Lipschutz, Seymour; Lipson, Marc (December 6, 2000), Schaum's Outline of Linear Algebra (3rd ed. ), McGraw-Hill, ISBN 978-0-07-136200-9 Lipschutz, Seymour (January 1, 1989), 3,000 Solved Problems in Linear Algebra... | Linear Algebra | 0.82287 |
7,378 | In multilinear algebra, one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the dual space, the vector space V* consisting of linear maps f: V → F where F is the field of scalars. Multilinea... | Linear Algebra | 0.82287 |
7,379 | An evolution algebra over a field is an algebra with a basis on which multiplication is defined by the product of distinct basis terms being zero and the square of each basis element being a linear form in basis elements. A real evolution algebra is one defined over the reals: it is non-negative if the structure consta... | Genetic algebra | 0.82286 |
7,380 | Train algebras were introduced by Etherington (1939, section 4) as special cases of baric algebras. Let c 1 , … , c n {\displaystyle c_{1},\ldots ,c_{n}} be elements of the field K with 1 + c 1 + ⋯ + c n = 0 {\displaystyle 1+c_{1}+\cdots +c_{n}=0} . The formal polynomial x n + c 1 w ( x ) x n − 1 + ⋯ + c n w ( x ) n {\... | Genetic algebra | 0.82286 |
7,381 | Genetic algebras were introduced by Schafer (1949) who showed that special train algebras are genetic algebras and genetic algebras are train algebras. | Genetic algebra | 0.82286 |
7,382 | A gametic algebra is a finite-dimensional real algebra for which all structure constants lie between 0 and 1. | Genetic algebra | 0.82286 |
7,383 | Not every profinite group occurs as an absolute Galois group of a field. The étale fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety. The fundamental groups of algebraic topology, however, are in ge... | Profinite groups | 0.822853 |
7,384 | The topology obtained on Gal ( L / K ) {\displaystyle \operatorname {Gal} (L/K)} is known as the Krull topology after Wolfgang Krull. Waterhouse (1974) showed that every profinite group is isomorphic to one arising from the Galois theory of some field K , {\displaystyle K,} but one cannot (yet) control which field K ... | Profinite groups | 0.822853 |
7,385 | The topology on this profinite group is the same as the topology arising from the p {\displaystyle p} -adic valuation on Z p . {\displaystyle \mathbb {Z} _{p}.} The group of profinite integers Z ^ {\displaystyle {\widehat {\mathbb {Z} }}} is the profinite completion of Z . | Profinite groups | 0.822853 |
7,386 | Finite groups are profinite, if given the discrete topology. The group of p {\displaystyle p} -adic integers Z p {\displaystyle \mathbb {Z} _{p}} under addition is profinite (in fact procyclic). It is the inverse limit of the finite groups Z / p n Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} } where n {\displaystyle ... | Profinite groups | 0.822853 |
7,387 | Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems.To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjectiv... | Profinite groups | 0.822853 |
7,388 | In addition to the essentials of mechanics and geometry, a suitable scaffold construct will possess surface properties that are optimized for the attachment and migration of the cell types of particular interest. Generally, it has been found that scaffolds that closely resemble the natural environments of the tissue be... | Protein adsorption | 0.82285 |
7,389 | Biomolecular Adsorption Database (BAD) is a freely available online database with experimental protein adsorption data collected from the literature. The database can be used for the selection of materials for microfluidic device fabrication and for the selection of optimum operating conditions of lab-on-a-chip devices... | Protein adsorption | 0.82285 |
7,390 | In terms of biomedical engineering applications, micromachining techniques are often used to increase protein adhesion to implants in the hopes of shortening recovery time. The technique of laserpatterning introduces grooves and surface roughness that will influence adhesion, migration and alignment. Grit-blasting, a m... | Protein adsorption | 0.82285 |
7,391 | It is the unique combination of amino acids that gives a protein its properties. In terms of surface chemistry, protein adsorption is a critical phenomenon that describes the aggregation of these molecules on the exterior of a material. The tendency for proteins to remain attached to a surface depends largely on the ma... | Protein adsorption | 0.82285 |
7,392 | In protein structures determined by X-ray crystallography, poor peptide-plane geometry has been described as a common problem; many structures need correction by peptide-plane flips or peptide bond flips. == References == | Peptide plane flipping | 0.822845 |
7,393 | Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many case... | Mathematical model | 0.82284 |
7,394 | For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering, physics models are often made by mathematical methods such as finite element analysis. Different mathematical models use different geometries that are not necessarily accurate de... | Mathematical model | 0.82284 |
7,395 | The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on a computer, a model that is computationally... | Mathematical model | 0.82284 |
7,396 | Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used i... | Mathematical model | 0.82284 |
7,397 | Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. | Mathematical model | 0.82284 |
7,398 | Tangential speed v {\displaystyle v} (Latin letter v), rotational frequency ν {\displaystyle \nu } , and radial distance r {\displaystyle r} , are related by the following equation: An algebraic rearrangement of this equation allows us to solve for rotational frequency: Thus, the tangential speed will be directly propo... | Rotational frequency | 0.822838 |
7,399 | This branch of computer science aims to manage networks between computers worldwide. | Computer sciences | 0.822836 |
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