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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. |
In mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature... |
identify elements x ∈ G {\displaystyle x\in G} with morphisms 1 → G {\displaystyle 1\rightarrow G} . It is then possible to write the axioms of a group in terms of functions (note how the existential quantifier is absent): ∀ x ∈ G , ∀ y ∈ G , ∀ z ∈ G , m ( m ( x , y ) , z ) = m ( x , m ( y , z ) ) {\displaystyle \foral... |
involving two laws +,•: R×R → R, an additive identity 0: 1 → R, a multiplicative identity 1: 1 → R, and an additive inverse for each element -: R → R. As all these functions share the same codomain R they can be glued into a single signature function 1 + 1 + R + R×R + R×R → R, with axioms to express associativity, dist... |
1 + X {\displaystyle 1+X} . Here, S e t {\displaystyle \mathrm {\bf {Set}} } denotes the category of sets, + {\displaystyle +} denotes the usual coproduct given by the disjoint union, and 1 {\displaystyle 1} is a terminal object (i.e. any singleton set). Then, the set N {\displaystyle \mathbb {N} } of natural numbers t... |
In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste ... |
equations: Because of their difficulty, these systems, with few exceptions, have been studied only since the second part of the 19th century. They have led to the development of algebraic geometry. == See also == Root-finding algorithm Properties of polynomial roots Quintic function == References == https://www.britann... |
Control theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, ... |
1877, resulting in what is now known as the Routh–Hurwitz theorem. A notable application of dynamic control was in the area of crewed flight. The Wright brothers made their first successful test flights on December 17, 1903, and were distinguished by their ability to control their flights for substantial periods (more ... |
Laplace transform, Fourier transform, Z transform, Bode plot, root locus, and Nyquist stability criterion. These lead to a description of the system using terms like bandwidth, frequency response, eigenvalues, gain, resonant frequencies, zeros and poles, which give solutions for system response and design techniques fo... |
dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information abou... |
response characteristics applied in a specification are typically percent overshoot, settling time, etc. The open-loop response characteristics applied in a specification are typically Gain and Phase margin and bandwidth. These characteristics may be evaluated through simulation including a dynamic model of the system ... |
system response neither decays nor grows over time, and has no oscillations, it is marginally stable; in this case the system transfer function has non-repeated poles at the complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real pa... |
of these variables must be controllable and observable to ensure a good behavior in the closed-loop system. That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynami... |
impossible. System identification The process of determining the equations that govern the model's dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such ... |
can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design. === Nonlinear systems control === Processes in industries like robotics and the aerospace i... |
control technique in process control. Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design. The early methods of Bode and other... |
optimal control and game theoretic problems Andrey Kolmogorov co-developed the Wiener–Kolmogorov filter in 1941. Norbert Wiener co-developed the Wiener–Kolmogorov filter and coined the term cybernetics in the 1940s. John R. Ragazzini introduced digital control and the use of Z-transform in control theory (invented by L... |
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebrai... |
similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the correspondin... |
An is a ring, which is denoted k[An]. We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k[An]. The vanishing set of S (or vanishing locus or zero set) is the set V(S) of all points in An where every polynomial in S vanishes. Symbolically, V ( S ) = ... |
be defined as the vanishing set of a prime ideal of the polynomial ring. Some authors do not make a clear distinction between algebraic sets and varieties and use irreducible variety to make the distinction when needed. === Regular functions === Just as continuous functions are the natural maps on topological spaces an... |
g ∈ k[V]. The map f → f ∘ g is a ring homomorphism from k[V′] to k[V]. Conversely, every ring homomorphism from k[V′] to k[V] defines a regular map from V to V′. This defines an equivalence of categories between the category of algebraic sets and the opposite category of the finitely generated reduced k-algebras. This ... |
algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the number i, a root of the polynomial x2 + 1, projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet. To see... |
element of k. This defines the homogeneous coordinates of a point of Pn as a sequence of n + 1 elements of the base field k, defined up to the multiplication by a nonzero element of k (the same for the whole sequence). A polynomial in n + 1 variables vanishes at all points of a line passing through the origin if and on... |
the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held at Marseille, France, in June 1979. At this meeting, Dennis S. Arnon showed that George E. Collins's Cylindrical algebraic decomposition (CAD) allows the computation of the topology... |
be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst-case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequent... |
to solve specific subproblems with a better complexity. Most of these algorithms have a complexity d O ( n 2 ) {\displaystyle d^{O(n^{2})}} . Among those algorithms which solve a subproblem of the problems solved by Gröbner bases, one may cite testing whether an affine variety is empty and solving nonhomogeneous polyno... |
by a Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples have become promine... |
sides a and b. Menaechmus (c. 350 BC) considered the problem geometrically by intersecting the pair of plane conics ay = x2 and xy = ab. In the 3rd century BC, Archimedes and Apollonius systematically studied additional problems on conic sections using coordinates. Apollonius in the Conics further developed a method th... |
of Lagrange and Euler. === 19th and early 20th century === It took the simultaneous 19th-century developments of non-Euclidean geometry and Abelian integrals in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by Edmond Laguerre and Arthur Cayley, ... |
are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's Last Theorem and are also used in elliptic-curve cryptography. In parallel with the abstract tren... |
Harris, Joe (1995). Algebraic Geometry A First Course. Springer-Verlag. ISBN 978-0-387-97716-4. Zbl 0779.14001. Mumford, David (1995). Algebraic Geometry I Complex Projective Varieties (2nd ed.). Springer-Verlag. ISBN 978-3-540-58657-9. Zbl 0821.14001. Reid, Miles (1988). Undergraduate Algebraic Geometry. Cambridge Uni... |
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matrix... |
continuous operators; this identifies Mn(A) with a subalgebra of B(H⊕n). For simplicity, if we further suppose that H is separable and A ⊆ {\displaystyle \subseteq } B(H) is a unital C*-algebra, we can break up A into a matrix ring over a smaller C*-algebra. One can do so by fixing a projection p and hence its orthogon... |
Conversely, for a given left ideal I of Mn(C) the intersection of null spaces of all matrices in I gives a subspace of Cn. Under this construction, the left ideals of Mn(C) are in bijection with the subspaces of Cn. There is a bijection between the two-sided ideals of Mn(R) and the two-sided ideals of R. Namely, for ea... |
unit group of Mn(R), consisting of the invertible matrices under multiplication, is denoted GLn(R). If F is a field, then for any two matrices A and B in Mn(F), the equality AB = In implies BA = In. This is not true for every ring R though. A ring R whose matrix rings all have the mentioned property is known as a stabl... |
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental... |
space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because man... |
the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. ==== Topologies on the real and complex numbers ==== There are many ways to define a topology on R, the set of real ... |
is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics. If Γ is an ordinal number, then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ. == Continuous functions =... |
a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also... |
) ) ⊆ i n t ( f − 1 ( A ) ) {\displaystyle f^{-1}(\mathrm {int} '(A))\subseteq \mathrm {int} (f^{-1}(A))} for any subset A of X. === Properties === If f: X → Y and g: Y → Z are continuous, then so is the composition g ∘ f: X → Z. If f: X → Y is continuous and X is compact, then f(X) is compact. X is connected, then f(X... |
those sets of the form f−1(U) where U is open in X . If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S that makes f continuous. If f is in... |
sets) are X and the empty set. The only subsets of X with empty boundary are X and the empty set. X cannot be written as the union of two nonempty separated sets. The only continuous functions from X to {0,1}, the two-point space endowed with the discrete topology, are constant. Every interval in R is connected. The co... |
at most one path-component; that is, if there is a path joining any two points in X. Again, many authors exclude the empty space. Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line L* and the topologist's sine... |
least likely to be ambiguous. Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles. In all of t... |
to a continuous map defined on the whole space. == Countability axioms == An axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist. Important countability axioms for t... |
These objects arise frequently in nearly all areas of topology and analysis, and their properties are strong enough to yield many 'geometric' features. === Dynamical systems === Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with appl... |
0-387-19374-X. Kelley, John L. (1975) [1955]. General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153. Stephen Willard, General Topology, ISBN 0-486-43479-6. James Munkres, Topology, ISBN 0-13-181629-2. George F. Simmons, Introduction to Topolo... |
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links or lines). A distinction is mad... |
graph) or is incident on (for an undirected multigraph) { x , x } = { x } {\displaystyle \{x,x\}=\{x\}} which is not in { { x , y } ∣ x , y ∈ V and x ≠ y } {\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}} . To allow loops, the definitions must be expanded. For undirected simple graphs, the definition ... |
∈ V 2 and x ≠ y } {\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}} , a set of edges (also called directed edges, directed links, directed lines, arrows or arcs) which are ordered pairs of vertices (that is, an edge is associated with two distinct vertices). To avoid ambiguity... |
be expanded. For directed simple graphs, the definition of E {\displaystyle E} should be modified to E ⊆ { ( x , y ) ∣ ( x , y ) ∈ V 2 } {\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}} . For directed multigraphs, the definition of ϕ {\displaystyle \phi } should be modified to ϕ : E → { ( x , y ) ∣ ( ... |
typed feature structures, which are directed acyclic graphs. Within lexical semantics, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; semantic networks are therefore important in computational linguistics. Still, other methods in phonology ... |
collaboration graphs model whether two people work together in a particular way, such as acting in a movie together. === Biology === Likewise, graph theory is useful in biology and conservation efforts where a vertex can represent regions where certain species exist (or inhabit) and the edges represent migration paths ... |
implications for theoretical chemistry. The techniques he used mainly concern the enumeration of graphs with particular properties. Enumerative graph theory then arose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937. These were generalized by De Bruijn in 1959. Cayley lin... |
complexity. A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas. The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Another important factor of common developme... |
packings, intersection graph, and other visualizations of the adjacency matrix. === Tabular: Graph data structures === The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The data structure used depends on both the graph structure an... |
polynomial time. A similar problem is finding induced subgraphs in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is als... |
and computing graph unification. The unification of two argument graphs is defined as the most general graph (or the computation thereof) that is consistent with (i.e. contains all of the information in) the inputs, if such a graph exists; efficient unification algorithms are known. For constraint frameworks which are ... |
in graph theory Publications in graph theory Graph algorithm Graph theorists === Subareas === Algebraic graph theory Geometric graph theory Extremal graph theory Probabilistic graph theory Topological graph theory Graph drawing == Notes == == References == Lowell W. Beineke; Bjarne Toft; and Robin J. Wilson: Milestones... |
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that cons... |
of R. But a univalent relation is only a partial function, while a univalent total relation is a function. The formula for totality is I ⊆ R R T . {\displaystyle I\subseteq RR^{T}.} Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation. The facility of complementary relations inspired... |
of Leibniz's known work on algebraic logic was published only in 1903 after Louis Couturat discovered it in Leibniz's Nachlass. Parkinson (1966) and Loemker (1969) translated selections from Couturat's volume into English. Modern mathematical logic began in 1847, with two pamphlets whose respective authors were George ... |
staircase logical matrix has a complement that is also a staircase, and that the theorem of N. M. Ferrers follows from interpretation of the transpose of a staircase. Riguet generated rectangular relations by taking the outer product of logical vectors; these contribute to the non-enlargeable rectangles of formal conce... |
Mathematical Society & Elsevier ISBN 0444885439 == External links == Algebraic logic at PhilPapers |
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 representatio... |
– q also produces a segment equipollent to pq. Other hypercomplex number systems also used the idea of a linear space with a basis. Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group. The mechanism of group representation became available for describi... |
specific vector space may have various natures; for example, they could be tuples, sequences, functions, polynomials, or a matrices. Linear algebra is concerned with the properties of such objects that are common to all vector spaces. === Linear maps === Linear maps are mappings between vector spaces that preserve the ... |
image T−1(0) of 0 (called kernel or null space), are linear subspaces of W and V, respectively. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums a 1 v 1 + a 2 v 2 + ⋯ + a k v k , {\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a... |
span of U1 ∪ U2. == Matrices == 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). By definiti... |
matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from W to V, there are bases such that a part of the basis of W is mapped bijectively on a part of the basis of V, and that the remaining basis elements of W, if any, a... |
include the computation of the ranks, kernels, matrix inverses. == Endomorphisms and square matrices == 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. Concerning general linear maps, linear end... |
monic polynomial of degree n, called the characteristic polynomial of the matrix (or of the endomorphism), and there are, at most, n eigenvalues. If a basis exists that consists only of eigenvectors, the matrix of f on this basis has a very simple structure: it is a diagonal matrix such that the entries on the main dia... |
symmetry between a finite-dimensional vector space and its dual. This motivates the frequent use, in this context, of the bra–ket notation ⟨ f , x ⟩ {\displaystyle \langle f,\mathbf {x} \rangle } for denoting f(x). === Dual map === Let f : V → W {\displaystyle f:V\to W} be a linear map. For every linear form h on W, th... |
a vector v in V by ‖ v ‖ 2 = ⟨ v , v ⟩ , {\displaystyle \|\mathbf {v} \|^{2}=\langle \mathbf {v} ,\mathbf {v} \rangle ,} and we can prove the Cauchy–Schwarz inequality: | ⟨ u , v ⟩ | ≤ ‖ u ‖ ⋅ ‖ v ‖ . {\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |\leq \|\mathbf {u} \|\cdot \|\mathbf {v} \|.} In particular, ... |
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, but the constructions may be extended... |
In both cases, very large matrices are generally involved. Weather forecasting (or more specifically, parametrization for atmospheric modeling) is a typical example of a real-world application, where the whole Earth atmosphere is divided into cells of, say, 100 km of width and 100 km of height. === Fluid mechanics, flu... |
structure called a module over R, or R-module. The concepts of linear independence, span, basis, and linear maps (also called module homomorphisms) are defined for modules exactly as for vector spaces, with the essential difference that, if R is not a field, there are modules that do not have any basis. The modules tha... |
"size" of elements. The norm induces a metric, which measures the distance between elements, and induces a topology, which allows for a definition of continuous maps. The metric also allows for a definition of limits and completeness – a normed vector space that is complete is known as a Banach space. A complete metric... |
A Modern Introduction To Linear Algebra (1st ed.), 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... |
(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, McGraw–Hill, ISBN 978-0-07-038023-3 McMahon, D... |
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such rel... |
and applied it to mechanics, which led to the formulation of Lagrangian mechanics. In 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules ... |
their solutions in terms of integrals. Most ODEs that are encountered in physics are linear. Therefore, most special functions may be defined as solutions of linear differential equations (see Holonomic function). As, in general, the solutions of a differential equation cannot be expressed by a closed-form expression, ... |
depending on the context, the polynomial degree in the highest derivative of the unknown function, or its total degree in the unknown function and its derivatives. In particular, a linear differential equation has degree one for both meanings, but the non-linear differential equation y ′ + y 2 = 0 {\displaystyle y'+y^{... |
which a solution exists. Given any point ( a , b ) {\displaystyle (a,b)} in the xy-plane, define some rectangular region Z {\displaystyle Z} , such that Z = [ l , m ] × [ n , p ] {\displaystyle Z=[l,m]\times [n,p]} and ( a , b ) {\displaystyle (a,b)} is in the interior of Z {\displaystyle Z} . If we are given a differe... |
which contains p-adic numbers in an ultrametric space. Mathematical models that involve ultrametric pseudo-differential equations use pseudo-differential operators instead of differential operators. A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in ... |
CAS software can solve differential equations. These are the commands used in the leading programs: Maple: dsolve Mathematica: DSolve[] Maxima: ode2(equation, y, x) SageMath: desolve() SymPy: sympy.solvers.ode.dsolve(equation) Xcas: desolve(y'=k*y,y) == See also == == References == == Further reading == Abbott, P.; Nei... |
Information theory is the mathematical study of the quantification, storage, and communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of ele... |
be roughly subdivided into data compression (source coding) and error-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both codes and ciphers). Concepts, methods... |
information theory, opening with the assertion: "The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point." With it came the ideas of: the information entropy and redundancy of a source, and its relevance through the source coding... |
of "bits" (per symbol) because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called the shannon in his honor. Entropy is also commonly computed using the natural logarithm (base e, where e is Euler's number), which produces a measurement of entropy in nats per symbol and sometimes... |
( 1 − p ) . {\displaystyle H_{\mathrm {b} }(p)=-p\log _{2}p-(1-p)\log _{2}(1-p).} === Joint entropy === The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: (X, Y). This implies that if X and Y are independent, then their joint entropy is the sum of their individual entropi... |
A basic property of the mutual information is that I ( X ; Y ) = H ( X ) − H ( X | Y ) . {\displaystyle I(X;Y)=H(X)-H(X|Y).\,} That is, knowing Y, we can save an average of I(X; Y) bits in encoding X compared to not knowing Y. Mutual information is symmetric: I ( X ; Y ) = I ( Y ; X ) = H ( X ) + H ( Y ) − H ( X , Y ) ... |
_{x\in X}-p(x)\log {p(x)}=\sum _{x\in X}p(x)\log {\frac {p(x)}{q(x)}}.} Although it is sometimes used as a 'distance metric', KL divergence is not a true metric since it is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric). Another interpretation of the KL divergence is the "unne... |
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