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of the concept of entropy), differential entropy (a generalization of quantities of information to continuous distributions), and the conditional mutual information. Also, pragmatic information has been proposed as a measure of how much information has been used in making a decision. == Coding theory == Coding theory i... |
{1}{n}}H(X_{1},X_{2},\dots X_{n});} that is, the limit of the joint entropy per symbol. For stationary sources, these two expressions give the same result. The information rate is defined as: r = lim n → ∞ 1 n I ( X 1 , X 2 , … X n ; Y 1 , Y 2 , … Y n ) ; {\displaystyle r=\lim _{n\to \infty }{\frac {1}{n}}I(X_{1},X_{2}... |
length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε; that is, it is always possible to transmit with arbitrarily small block error. In addition, for any rate R > C, it is impossible to transmit with arbitrarily small block error. Channel coding is concerned with findi... |
of World War II in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability. Information theory leads us to believe it is much more difficult to keep secre... |
Nauta defined semiotic information theory as the study of "the internal processes of coding, filtering, and information processing.": 91 Concepts from information theory such as redundancy and code control have been used by semioticians such as Umberto Eco and Ferruccio Rossi-Landi to explain ideology as a form of mess... |
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basic form, the theorem asserts that given a field extension E/F that is finite ... |
from the base field Q {\displaystyle \mathbb {Q} } by adjoining √2, then √3, each element of K can be written as: ( a + b 2 ) + ( c + d 2 ) 3 , a , b , c , d ∈ Q . {\displaystyle (a+b{\sqrt {2}})+(c+d{\sqrt {2}}){\sqrt {3}},\qquad a,b,c,d\in \mathbb {Q} .} Its Galois group G = Gal ( K / Q ) {\displaystyle G={\text{Gal}... |
2 ) , {\displaystyle \mathbb {Q} ({\sqrt {2}}),} since g fixes √2. The subgroup {1, fg} corresponds to the subfield Q ( 6 ) , {\displaystyle \mathbb {Q} ({\sqrt {6}}),} since fg fixes √6. == Example 2 == The following is the simplest case where the Galois group is not abelian. Consider the splitting field K of the irre... |
, α 3 {\displaystyle \alpha _{1},\alpha _{2},\alpha _{3}} is (in cycle notation): f = ( 123 ) , g = ( 23 ) {\displaystyle f=(123),g=(23)} . Also, g can be considered as the complex conjugation mapping. The subgroups of G and corresponding subfields are as follows: As always, the trivial group {1} corresponds to the who... |
its value ϕ ( λ ) {\displaystyle \phi (\lambda )} , so that f ( λ ) ↦ f ( ϕ ( λ ) ) {\displaystyle f(\lambda )\mapsto f(\phi (\lambda ))} . This group is isomorphic to S 3 {\displaystyle S_{3}} (see: six cross-ratios). Let F {\displaystyle F} be the fixed field of G {\displaystyle G} , so that G a l ( E / F ) = G {\dis... |
f ∈ G = G a l ( E / F ) {\displaystyle f\in G=\mathrm {Gal} (E/F)} is completely determined by f ( 2 3 ) {\displaystyle f({\sqrt[{3}]{2}})} and that 2 = f ( 2 3 ) 3 ⟹ f = 1 {\displaystyle 2=f({\sqrt[{3}]{2}})^{3}\implies f=1} Thus, G = { 1 } {\displaystyle G=\{1\}} , is the trivial group. In particular, | G | = 1 < 3 =... |
I} the maps φ i : G → G i {\displaystyle \varphi _{i}:G\rightarrow G_{i}} are continuous, where we endow each G i {\displaystyle G_{i}} with the discrete topology. Stated differently G ≅ lim ← G i {\displaystyle G\cong \varprojlim G_{i}} as an inverse limit of topological groups (where again each G i {\displaystyle G... |
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic t... |
In addition, order theory does not restrict itself to the various classes of ordering relations, but also considers appropriate functions between them. A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found. == Basic definitions == This section in... |
relative positioning of the vertices. Orders are drawn bottom-up: if an element x is smaller than (precedes) y then there exists a path from x to y that is directed upwards. It is often necessary for the edges connecting elements to cross each other, but elements must never be located within an edge. An instructive exe... |
set of all finite subsets of a given infinite set, ordered by subset inclusion, provides one of many counterexamples. An important tool to ensure the existence of maximal elements under certain conditions is Zorn's Lemma. Subsets of partially ordered sets inherit the order. We already applied this by considering the su... |
this operation preserves the theorems of partial orders. For a given mathematical result, one can just invert the order and replace all definitions by their duals and one obtains another valid theorem. This is important and useful, since one obtains two theorems for the price of one. Some more details and examples can ... |
i.e. when they are the same up to renaming of elements. Order isomorphisms are functions that define such a renaming. An order-isomorphism is a monotone bijective function that has a monotone inverse. This is equivalent to being a surjective order-embedding. Hence, the image f(P) of an order-embedding is always isomorp... |
a strict weak ordering. Requiring two scores to be separated by a fixed threshold before they may be compared leads to the concept of a semiorder, while allowing the threshold to vary on a per-item basis produces an interval order. An additional simple but useful property leads to so-called well-founded, for which all ... |
A simple example are upper sets; i.e. sets that contain all elements that are above them in the order. Formally, the upper closure of a set S in a poset P is given by the set {x in P | there is some y in S with y ≤ x}. A set that is equal to its upper closure is called an upper set. Lower sets are defined dually. More ... |
in turn induce ≤ as their specialization order, the finest such topology is the Alexandrov topology, given by taking all upper sets as opens. Conversely, the coarsest topology that induces the specialization order is the upper topology, having the complements of principal ideals (i.e. sets of the form {y in X | y ≤ x} ... |
In this context the works of George Boole are of great importance. Moreover, works of Charles Sanders Peirce, Richard Dedekind, and Ernst Schröder also consider concepts of order theory. Contributors to ordered geometry were listed in a 1961 textbook: It was Pasch in 1882, who first pointed out that a geometry of order... |
A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure A is a non-associative algebra over a field K if it is a vector space over K and is equipped with a K-bilinear binary multiplication ... |
authors. Jordan identity: (x2y)x = x2(yx) or (xy)x2 = x(yx2) depending on authors. Alternative: (xx)y = x(xy) (left alternative) and (yx)x = y(xx) (right alternative). Flexible: (xy)x = x(yx). nth power associative with n ≥ 2: xn−kxk = xn for all integers k so that 0 < k < n. Third power associative: x2x = xx2. Fourth ... |
equivalent. === Associator === The associator on A is the K-multilinear map [ ⋅ , ⋅ , ⋅ ] : A × A × A → A {\displaystyle [\cdot ,\cdot ,\cdot ]:A\times A\times A\to A} given by [x,y,z] = (xy)z − x(yz). It measures the degree of nonassociativity of A {\displaystyle A} , and can be used to conveniently express some possi... |
Jordan algebra can be constructed this way. Those that can are called special. Alternative algebras are algebras satisfying the alternative property. The most important examples of alternative algebras are the octonions (an algebra over the reals), and generalizations of the octonions over other fields. All associative... |
all non-associative monomials, finite formal products of elements of X retaining parentheses. The product of monomials u, v is just (u)(v). The algebra is unital if one takes the empty product as a monomial. Kurosh proved that every subalgebra of a free non-associative algebra is free. == Associated algebras == An alge... |
not hold, in general, for non-associative algebras. The best-known example is, perhaps the Albert algebra, an exceptional Jordan algebra that is not enveloped by the canonical construction of the enveloping algebra for Jordan algebras. == See also == List of algebras Commutative non-associative magmas, which give rise ... |
nearly associative. Translated by Smith, Harry F. ISBN 0-12-779850-1. |
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical ... |
of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven. === Banach spaces === General Banach spaces are more complicated than Hilbert spaces, and cannot ... |
of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointw... |
a characterization of Banach spaces in which various forms of the law of large numbers hold. Noncommutative geometry. Developed by Alain Connes, partly building on earlier notions, such as George Mackey's approach to ergodic theory. Connection with quantum mechanics. Either narrowly defined as in mathematical physics, ... |
York University. Lecture videos on functional analysis by Greg Morrow Archived 2017-04-01 at the Wayback Machine from University of Colorado Colorado Springs |
In mathematics, a loop in a topological space X is a continuous function f from the unit interval I = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point. A loop may also be seen as a continuous map f from the pointed unit circle S1 into X, because S1 may be... |
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane. The characteristic property of linear functions is that when the input variable is changed, the change in the output is prop... |
point ( x , y ) = ( − b a , 0 ) . {\displaystyle (x,y)=(-{\tfrac {b}{a}},0).} The x-intercept value x = − b a , {\displaystyle x=-{\tfrac {b}{a}},} the solution of the equation f ( x ) = 0 , {\displaystyle f(x)=0,} is also called the root or zero of f ( x ) . {\displaystyle f(x).} == Slope == The slope of a nonvertical... |
+ b {\displaystyle f(x)=ax+b} for b = f ( 0 ) {\displaystyle b=f(0)} . == Slope-intercept, point-slope, and two-point forms == A given linear function f ( x ) {\displaystyle f(x)} can be written in several standard formulas displaying its various properties. The simplest is the slope-intercept form: f ( x ) = a x + b {... |
+ b {\displaystyle y=f(x)=ax+b} . In the xy-coordinate plane, the possible values of ( x , y ) {\displaystyle (x,y)} form a line, the graph of the function f ( x ) {\displaystyle f(x)} . If B = 0 {\displaystyle B=0} in the original equation, the resulting line x = C A {\displaystyle x={\tfrac {C}{A}}} is vertical, and ... |
meat to the butcher). Thus we should restrict our function f ( x ) {\displaystyle f(x)} to the domain 0 ≤ x ≤ 2 {\displaystyle 0\leq x\leq 2} . Also, we could choose y as the independent variable, and compute x by the inverse linear function: x = g ( y ) = − 1 2 y + 2 {\displaystyle x=g(y)=-{\tfrac {1}{2}}y+2} over the... |
Mathematical physics is the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of ph... |
ergodic theory and some parts of probability theory. There are increasing interactions between combinatorics and physics, in particular statistical physics. == Usage == The usage of the term "mathematical physics" is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of phys... |
and these ultimately were reintroduced or became available to the West in the 12th century and during the Renaissance. In the first decade of the 16th century, amateur astronomer Nicolaus Copernicus proposed heliocentrism, and published a treatise on it in 1543. He retained the Ptolemaic idea of epicycles, and merely s... |
time and space could now be thought as axes belonging to the same plane. This essential mathematical framework is at the base of all modern physics and used in all further mathematical frameworks developed in next centuries. By the middle of the 17th century, important concepts such as the fundamental theorem of calcul... |
and Magnetism in 1828, which in addition to its significant contributions to mathematics made early progress towards laying down the mathematical foundations of electricity and magnetism. A couple of decades ahead of Newton's publication of a particle theory of light, the Dutch Christiaan Huygens (1629–1695) developed ... |
modeled by the Dutch Hendrik Lorentz [1853–1928]. In 1887, experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motion into the aether prompted aether's shortening, too, as modeled in the Lorentz contraction. It was hypothesized that the aether thus kept Maxwell's elect... |
relativity—even massless energy exerts gravitational effect by its mass equivalence locally "curving" the geometry of the four, unified dimensions of space and time.) === Quantum === Another revolutionary development of the 20th century was quantum theory, which emerged from the seminal contributions of Max Planck (185... |
Aharonov (1932–) Sheldon Glashow (1932–) Steven Weinberg (1933–2021) Ludvig Dmitrievich Faddeev (1934–2017) David Ruelle (1935–) Yakov Grigorevich Sinai (1935–) Vladimir Igorevich Arnold (1937–2010) Arthur Michael Jaffe (1937–) Roman Wladimir Jackiw (1939–) Leonard Susskind (1940–) Rodney James Baxter (1940–) Michael V... |
ISBN 0-8053-7002-1 Menzel, Donald H. (1961), Mathematical Physics, Dover Publications, ISBN 0-486-60056-4 {{citation}}: ISBN / Date incompatibility (help) Riley, Ken F.; Hobson, Michael P.; Bence, Stephen J. (2006), Mathematical Methods for Physics and Engineering (3rd ed.), Cambridge University Press, ISBN 978-0-521-8... |
Colton, David; Kress, Rainer (2013), Integral Equation Methods in Scattering Theory, Society for Industrial and Applied Mathematics, ISBN 978-1-611973-15-0 Ciarlet, Philippe G. (1988–2000), Mathematical Elasticity, Vol 1–3, Elsevier Galdi, Giovanni P. (2011), An Introduction to the Mathematical Theory of the Navier-Sto... |
John (2018), Mathematical Foundations of Quantum Mechanics, Princeton University Press, ISBN 978-0-691-17856-1 Weyl, Hermann (2014), The Theory of Groups and Quantum Mechanics, Martino Fine Books, ISBN 978-1614275800 Ynduráin, Francisco J. (2006), The Theory of Quark and Gluon Interactions (4th ed.), Springer, Bibcode:... |
Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entails use of algebraic nota... |
is always 1 (e.g. x 0 {\displaystyle x^{0}} is always rewritten to 1). However 0 0 {\displaystyle 0^{0}} , being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents. === Alternative notation === Other types of notation are used in... |
{\displaystyle 3x} (where 3 is a numerical coefficient). Multiplied terms are simplified using exponents. For example, x × x × x {\displaystyle x\times x\times x} is represented as x 3 {\displaystyle x^{3}} Like terms are added together, for example, 2 x 2 + 3 a b − x 2 + a b {\displaystyle 2x^{2}+3ab-x^{2}+ab} is writ... |
that when multiplying or dividing by a negative number, the inequality symbol must be flipped. ==== Properties of equality ==== By definition, equality is an equivalence relation, meaning it is reflexive (i.e. b = b {\displaystyle b=b} ), symmetric (i.e. if a = b {\displaystyle a=b} then b = a {\displaystyle b=a} ), an... |
known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement x + 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false... |
is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable. This problem and its solution are as follows: In words: the child is 4 years old. ... |
on the associated plot of the equations. For other ways to solve this kind of equations, see below, System of linear equations. === Quadratic equations === A quadratic equation is one which includes a term with an exponent of 2, for example, x 2 {\displaystyle x^{2}} , and no term with higher exponent. The name derives... |
] = 0. {\displaystyle [x-(-1)][x-(-1)]=0.} ==== Complex numbers ==== All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching ... |
{\displaystyle x^{\frac {3}{2}}} . So a common form of a radical equation is x m n = a {\displaystyle {\sqrt[{n}]{x^{m}}}=a} (equivalent to x m n = a {\displaystyle x^{\frac {m}{n}}=a} ) where m and n are integers. It has real solution(s): For example, if: ( x + 5 ) 2 / 3 = 4 {\displaystyle (x+5)^{2/3}=4} then x + 5 = ... |
same solution as in the previous method is obtained. { x = 2 y = 3. {\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}} This is not the only way to solve this specific system; in this case as well, y could have been solved before x. === Other types of systems of linear equations === ==== Inconsistent systems ==== In th... |
an overdetermined system has any solutions, necessarily some equations are linear combinations of the others. == See also == History of algebra Binary operation Gaussian elimination Mathematics education Number line Polynomial Cancelling out Tarski's high school algebra problem == References == Leonhard Euler, Elements... |
In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies N ( x y ) = N ( x ) N ( y ) {\displaystyle N(xy)=N(x)N(y)} for all x and y in A. A composition algebra includes an involution called a conjugation: x ↦ x ∗... |
Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra. The squaring function N(x) = x2 on the real number field forms the primordial composition algebra. When the field K is taken to be real numbers R, then there are just six other real composition algebras.: 166 In two, four, and eight ... |
when he showed that Dickson doubling could be applied to any field with the squaring function to construct binarion, quaternion, and octonion algebras with their quadratic forms. Nathan Jacobson described the automorphisms of composition algebras in 1958. The classical composition algebras over R and C are unital algeb... |
Physics is the scientific study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. It is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist. Physics is one of th... |
They proposed ideas verified by reason and observation, and many of their hypotheses proved successful in experiment; for example, atomism was found to be correct approximately 2000 years after it was proposed by Leucippus and his pupil Democritus. === Aristotle and Hellenistic physics === During the classical period i... |
dominant Aristotelian approach to science although much of his work was focused on Christian theology. In the sixth century, Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest. Islamic scholarship inherited Aristotelian physics from the Greeks and durin... |
in classical mechanics in certain situations. Classical mechanics predicted that the speed of light depends on the motion of the observer, which could not be resolved with the constant speed predicted by Maxwell's equations of electromagnetism. This discrepancy was corrected by Einstein's theory of special relativity, ... |
to an acceleration), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics), the latter include such branches as hydrostatics, hydrodynamics an... |
quantum theory is concerned with the discrete nature of many phenomena at the atomic and subatomic level and with the complementary aspects of particles and waves in the description of such phenomena. The theory of relativity is concerned with the description of phenomena that take place in a frame of reference that is... |
imaging (MRI) and transistors. Feynman has noted that experimentalists may seek areas that have not been explored well by theorists. === Scope and aims === Physics covers a wide range of phenomena, from elementary particles (such as quarks, neutrinos, and electrons) to the largest superclusters of galaxies. Included in... |
mechanics remain unsolved; examples include the formation of sandpiles, nodes in trickling water, the shape of water droplets, mechanisms of surface tension catastrophes, and self-sorting in shaken heterogeneous collections. These complex phenomena have received growing attention since the 1970s for several reasons, in... |
in nuclear medicine and magnetic resonance imaging, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology. ==== Atomic, molecular, and optical ==== Atomic, molecular, and optical physics (AMO) is the study of matter—matter and light—matter interactions on the scale of single atoms... |
statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics. The discovery by Karl Jansky in 1931 that radio signals were emitted by celestial bodies initiated the science of radio astronomy. Most recently, the frontiers of astronomy have been exp... |
physics experiments are numerical data, with their units of measure and estimates of the errors in the measurements. Technologies based on mathematics, like computation have made computational physics an active area of research. Ontology is a prerequisite for physics, but not for mathematics. It means physics is ultima... |
other static structures. The understanding and use of acoustics results in sound control and better concert halls; similarly, the use of optics creates better optical devices. An understanding of physics makes for more realistic flight simulators, video games, and movies, and is often critical in forensic investigation... |
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic c... |
Write f λ − ∏ i = 1 d ( x − u λ , i ) = ∑ j = 0 d − 1 r λ , j ⋅ x j ∈ R [ x ] {\displaystyle f_{\lambda }-\prod _{i=1}^{d}(x-u_{\lambda ,i})=\sum _{j=0}^{d-1}r_{\lambda ,j}\cdot x^{j}\in R[x]} with r λ , j ∈ R {\displaystyle r_{\lambda ,j}\in R} . Let I be the ideal in R generated by the r λ , j {\displaystyle r_{\lamb... |
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word crystallography is derived from the Ancient Greek word κρύσταλλος (krústallos; "clear ice, rock-crystal"), and γράφειν (gráphein; "to write"). In July 2012, the United Nations recognised the imp... |
include electrons or neutrons. Crystallographers often explicitly state the type of beam used, as in the terms X-ray diffraction, neutron diffraction and electron diffraction. These three types of radiation interact with the specimen in different ways. X-rays interact with the spatial distribution of electrons in the s... |
from a body-centered cubic (bcc) structure called ferrite to a face-centered cubic (fcc) structure called austenite when it is heated. The fcc structure is a close-packed structure unlike the bcc structure; thus the volume of the iron decreases when this transformation occurs. Crystallography is useful in phase identif... |
The International Tables for Crystallography is an eight-book series that outlines the standard notations for formatting, describing and testing crystals. The series contains books that covers analysis methods and the mathematical procedures for determining organic structure through x-ray crystallography, electron diff... |
The word 'algebra' is used for various branches and structures of mathematics. For their overview, see Algebra. The name comes from the famous 10th century book Al-Jabr by Al-Khwarizmi. == The bare word "algebra" == The bare word "algebra" may refer to: Elementary algebra Abstract algebra Algebra over a field In univer... |
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which ar... |
n-by-n matrices, and are numerically related via index juggling, the difference in their transformation laws indicates it would be improper to add them together. The total number of indices (m) required to identify each component uniquely is equal to the dimension or the number of ways of an array, which is why a tenso... |
(subscript). As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array T {\displaystyle T} that transforms under a change of basis matrix R = ( R i j ) {\displaystyle R=\left(R_{i}^{j}\right)} by T ^ = R − 1 T R {\displaystyle {\hat {T}}=R^{-1}TR} . For the individual matrix en... |
(Tv)^{i}=T_{j}^{i}v^{j}} . These components transform contravariantly, since ( T v ^ ) i ′ = T ^ j ′ i ′ v ^ j ′ = [ ( R − 1 ) i i ′ T j i R j ′ j ] [ ( R − 1 ) k j ′ v k ] = ( R − 1 ) i i ′ ( T v ) i . {\displaystyle \left({\widehat {Tv}}\right)^{i'}={\hat {T}}_{j'}^{i'}{\hat {v}}^{j'}=\left[\left(R^{-1}\right)_{i}^{i... |
f ] {\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]} R j 1 ′ j 1 ⋯ R j q ′ j q . {\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.} The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci. An equivalent definition of a ... |
_{q{\text{ copies}}}\rightarrow \mathbf {R} ,} where V∗ is the corresponding dual space of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the real numbers, R {\displaystyle \mathbb {R} } . More generally, V can be taken over any field F (e.g. the complex numbers), wit... |
obtained from a basis {ei} for V and its dual basis {εj}, i.e. T = T j 1 … j q i 1 … i p e i 1 ⊗ ⋯ ⊗ e i p ⊗ ε j 1 ⊗ ⋯ ⊗ ε j q . {\displaystyle T=T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\;\mathbf {e} _{i_{1}}\otimes \cdots \otimes \mathbf {e} _{i_{p}}\otimes {\boldsymbol {\varepsilon }}^{j_{1}}\otimes \cdots \otimes {\b... |
specific models of those categories. === Tensor fields === In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space. This was the setting of Ricci's original work. In modern mathematical terminology such an objec... |
mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect: I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously... |
order of the tensor. For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, M)-entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension o... |
is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stres... |
a tensor of different type. === Tensor product === The tensor product takes two tensors, S and T, and produces a new tensor, S ⊗ T, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e., ( S ⊗ T ) ( v 1 , … , v n , v ... |
as a linear combination T = v 1 ⊗ w 1 ⊗ α 1 + v 2 ⊗ w 2 ⊗ α 2 + ⋯ + v N ⊗ w N ⊗ α N . {\displaystyle T=v_{1}\otimes w_{1}\otimes \alpha _{1}+v_{2}\otimes w_{2}\otimes \alpha _{2}+\cdots +v_{N}\otimes w_{N}\otimes \alpha _{N}.} The contraction of T on the first and last slots is then the vector α 1 ( v 1 ) w 1 + α 2 ( v... |
mass of varying stress quantities, each requiring 9 quantities to describe. Thus, a second-order tensor is needed. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surfac... |
=== The properties of tensors, especially tensor decomposition, have enabled their use in machine learning to embed higher dimensional data in artificial neural networks. This notion of tensor differs significantly from that in other areas of mathematics and physics, in the sense that a tensor is usually regarded as a ... |
More generally, if the Cartesian coordinates x, y, z undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the determinant of the coordinate transformation, so that the integral remains invariant, by the change of variables formula ... |
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