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sequent calculus | Natural deduction. Every (conditional) line has exactly one asserted proposition on the right. Sequent calculus. | wikipedia |
sequent calculus | Every (conditional) line has zero or more asserted propositions on the right.In other words, natural deduction and sequent calculus systems are particular distinct kinds of Gentzen-style systems. Hilbert-style systems typically have a very small number of inference rules, relying more on sets of axioms. Gentzen-style s... | wikipedia |
sequent calculus | Gentzen-style systems have significant practical and theoretical advantages compared to Hilbert-style systems. For example, both natural deduction and sequent calculus systems facilitate the elimination and introduction of universal and existential quantifiers so that unquantified logical expressions can be manipulated... | wikipedia |
sequent calculus | This very much parallels the way in which mathematical proofs are carried out in practice by mathematicians. Predicate calculus proofs are generally much easier to discover with this approach, and are often shorter. Natural deduction systems are more suited to practical theorem-proving. Sequent calculus systems are mor... | wikipedia |
computable real function | In mathematical logic, specifically computability theory, a function f: R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } is sequentially computable if, for every computable sequence { x i } i = 1 ∞ {\displaystyle \{x_{i}\}_{i=1}^{\infty }} of real numbers, the sequence { f ( x i ) } i = 1 ∞ {\displaystyle \{f... | wikipedia |
computable real function | A suitable generalization of the first definition is: Let D {\displaystyle D} be a subset of R n . {\displaystyle \mathbb {R} ^{n}.} | wikipedia |
computable real function | A function f: D → R {\displaystyle f\colon D\to \mathbb {R} } is sequentially computable if, for every n {\displaystyle n} -tuplet ( { x i 1 } i = 1 ∞ , … { x i n } i = 1 ∞ ) {\displaystyle \left(\{x_{i\,1}\}_{i=1}^{\infty },\ldots \{x_{i\,n}\}_{i=1}^{\infty }\right)} of computable sequences of real numbers such that (... | wikipedia |
stratified formula | In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q 1 ∧ ⋯ ∧ Q n ∧ ¬ Q n + 1 ∧ ⋯ ∧ ¬ Q n + m → P {\displaystyle Q_{1}\wedge \dots \wedge Q_{... | wikipedia |
display logic | In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof the... | wikipedia |
integer-valued function | In mathematical logic, such concepts as primitive recursive functions and μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions on Nn. Gödel numbering, defined on well-formed formulae of some formal language, is a natural-valued function. Computability theor... | wikipedia |
lindenbaum algebra | In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q exactly when p and q are provably equivalent in T). That is, two sentences are e... | wikipedia |
lindenbaum algebra | The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski. Starting in the academic year 1926-1927, Lindenbaum pioneered his method in Jan Łukasiewicz's mathematical logic seminar, and the method was popularized and generalized in subsequent decades through work by Tarski. The Lindenbaum–Tarski algebra is c... | wikipedia |
implicational propositional calculus | In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", " → {\displaystyle \rightarrow } ", etc.. | wikipedia |
primitive recursive functional | In mathematical logic, the primitive recursive functionals are a generalization of primitive recursive functions into higher type theory. They consist of a collection of functions in all pure finite types. The primitive recursive functionals are important in proof theory and constructive mathematics. They are a central... | wikipedia |
rules of passage (logic) | In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives of first-order logic. The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula in prenex normal form, and vice versa. | wikipedia |
controversy over cantor's theory | In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers. Cantor's theorem implies that there are sets having cardinality greater... | wikipedia |
controversy over cantor's theory | This argument can be improved by using a definition he gave later. The resulting argument uses only five axioms of set theory. | wikipedia |
controversy over cantor's theory | Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on mathematical infinity. For example, a line is generally presented as the infinite set of its points, and it is commonly taught that there are more real numbers than r... | wikipedia |
fuzzy logics | In mathematical logic, there are several formal systems of "fuzzy logic", most of which are in the family of t-norm fuzzy logics. | wikipedia |
decidable sublanguages of set theory | In mathematical logic, various sublanguages of set theory are decidable. These include: Sets with Monotone, Additive, and Multiplicative Functions. Sets with restricted quantifiers. == References == | wikipedia |
resilience (mathematics) | In mathematical modeling, resilience refers to the ability of a dynamical system to recover from perturbations and return to its original stable steady state. It is a measure of the stability and robustness of a system in the face of changes or disturbances. If a system is not resilient enough, it is more susceptible t... | wikipedia |
resilience (mathematics) | A resilient steady state corresponds to a ball in a deep valley, so any push or perturbation will very quickly lead the ball to return to the resting point where it started. On the other hand, a less resilient steady state corresponds to a ball in a shallow valley, so the ball will take a much longer time to return to ... | wikipedia |
extraneous variables | In mathematical modeling, the dependent variable is studied to see if and how much it varies as the independent variables vary. In the simple stochastic linear model yi = a + bxi + ei the term yi is the ith value of the dependent variable and xi is the ith value of the independent variable. The term ei is known as the ... | wikipedia |
extraneous variables | This is also called a bivariate dataset, (x1, y1)(x2, y2) ...(xi, yi). The simple linear regression model takes the form of Yi = a + Bxi + Ui, for i = 1, 2, ... , n. In this case, Ui, ... ,Un are independent random variables. This occurs when the measurements do not influence each other. | wikipedia |
extraneous variables | Through propagation of independence, the independence of Ui implies independence of Yi, even though each Yi has a different expectation value. Each Ui has an expectation value of 0 and a variance of σ2. Expectation of Yi Proof: E = E = α + β x i + E = α + β x i . | wikipedia |
extraneous variables | {\displaystyle E=E=\alpha +\beta x_{i}+E=\alpha +\beta x_{i}.} The line of best fit for the bivariate dataset takes the form y = α + βx and is called the regression line. α and β correspond to the intercept and slope, respectively.In an experiment, the variable manipulated by an experimenter is something that is proven... | wikipedia |
extraneous variables | The dependent variable is the event expected to change when the independent variable is manipulated.In data mining tools (for multivariate statistics and machine learning), the dependent variable is assigned a role as target variable (or in some tools as label attribute), while an independent variable may be assigned a... | wikipedia |
morphological gradient | In mathematical morphology and digital image processing, a morphological gradient is the difference between the dilation and the erosion of a given image. It is an image where each pixel value (typically non-negative) indicates the contrast intensity in the close neighborhood of that pixel. It is useful for edge detect... | wikipedia |
place value system | In mathematical numeral systems the radix r is usually the number of unique digits, including zero, that a positional numeral system uses to represent numbers. In some cases, such as with a negative base, the radix is the absolute value r = | b | {\displaystyle r=|b|} of the base b. For example, for the decimal system ... | wikipedia |
place value system | The highest symbol of a positional numeral system usually has the value one less than the value of the radix of that numeral system. The standard positional numeral systems differ from one another only in the base they use. The radix is an integer that is greater than 1, since a radix of zero would not have any digits,... | wikipedia |
place value system | Negative bases are rarely used. In a system with more than | b | {\displaystyle |b|} unique digits, numbers may have many different possible representations. | wikipedia |
place value system | It is important that the radix is finite, from which follows that the number of digits is quite low. Otherwise, the length of a numeral would not necessarily be logarithmic in its size. (In certain non-standard positional numeral systems, including bijective numeration, the definition of the base or the allowed digits ... | wikipedia |
zero-one loss function | In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a ... | wikipedia |
zero-one loss function | The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced ... | wikipedia |
zero-one loss function | In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s... | wikipedia |
simplex algorithm | In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operat... | wikipedia |
himmelblau's function | In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms. The function is defined by: f ( x , y ) = ( x 2 + y − 11 ) 2 + ( x + y 2 − 7 ) 2 . {\displaystyle f(x,y)=(x^{2}+y-11)^{2}+(x+y^{2}-7)^{2}.\quad } It has one local maximum at x = − 0.27... | wikipedia |
fractional programming | In mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions that are in general nonlinear. The ratio to be optimized often describes some kind of efficiency of a system. | wikipedia |
linear-fractional programming | In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a speci... | wikipedia |
ackley function | In mathematical optimization, the Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation.On a 2-dimensional domain it is defined by: f ( x , y ) = − 20 exp − exp + e + 20 {\displaystyle {\begin{aligned}f... | wikipedia |
rastrigin function | In mathematical optimization, the Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function. It was first proposed in 1974 by Rastrigin as a 2-dimensional function and has been generalized by Rudolph. | wikipedia |
rastrigin function | The generalized version was popularized by Hoffmeister & Bäck and Mühlenbein et al. Finding the minimum of this function is a fairly difficult problem due to its large search space and its large number of local minima. On an n {\displaystyle n} -dimensional domain it is defined by: f ( x ) = A n + ∑ i = 1 n {\displays... | wikipedia |
rosenbrock function | In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It is also known as Rosenbrock's valley or Rosenbrock's banana function. The global minimum is inside a long, narrow, parab... | wikipedia |
rosenbrock function | To converge to the global minimum, however, is difficult. The function is defined by f ( x , y ) = ( a − x ) 2 + b ( y − x 2 ) 2 {\displaystyle f(x,y)=(a-x)^{2}+b(y-x^{2})^{2}} It has a global minimum at ( x , y ) = ( a , a 2 ) {\displaystyle (x,y)=(a,a^{2})} , where f ( x , y ) = 0 {\displaystyle f(x,y)=0} . Usually, ... | wikipedia |
criss-cross algorithm | In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and nonlinear objective functions; there are criss-cross algorithms for linear-fractional programming... | wikipedia |
ellipsoidal algorithm | In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is polynomial in the input size.... | wikipedia |
firefly algorithm | In mathematical optimization, the firefly algorithm is a metaheuristic proposed by Xin-She Yang and inspired by the flashing behavior of fireflies. | wikipedia |
network simplex algorithm | In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied... | wikipedia |
perturbation function | In mathematical optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form of shifting the constraints.In some texts the value function is called t... | wikipedia |
push–relabel maximum flow algorithm | In mathematical optimization, the push–relabel algorithm (alternatively, preflow–push algorithm) is an algorithm for computing maximum flows in a flow network. The name "push–relabel" comes from the two basic operations used in the algorithm. Throughout its execution, the algorithm maintains a "preflow" and gradually c... | wikipedia |
push–relabel maximum flow algorithm | The generic algorithm has a strongly polynomial O(V 2E) time complexity, which is asymptotically more efficient than the O(VE 2) Edmonds–Karp algorithm. Specific variants of the algorithms achieve even lower time complexities. The variant based on the highest label node selection rule has O(V 2√E) time complexity and i... | wikipedia |
revised simplex algorithm | In mathematical optimization, the revised simplex method is a variant of George Dantzig's simplex method for linear programming. The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constrai... | wikipedia |
geometry of special relativity | In mathematical physics, Minkowski space (or Minkowski spacetime) () combines inertial space and time manifolds (x,y) with a non-inertial reference frame of space and time (x',t') into a four-dimensional model relating a position (inertial frame of reference) to the field. A four-vector (x,y,z,t) consists of a coordina... | wikipedia |
geometry of special relativity | The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz, Henri Poincaré, and others, and said it "was grown on experimental physical grounds." Minkowsk... | wikipedia |
geometry of special relativity | While the individual components in Euclidean space and time might differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events. Minkowski space differs from four-dimensional Euclidean space insofar as it treats time diff... | wikipedia |
geometry of special relativity | It is generated by rotations, reflections and translations. When time is appended as a fourth dimension, the further transformations of translations in time and Lorentz boosts are added, and the group of all these transformations is called the Poincaré group. Minkowski's model follows special relativity where motion ca... | wikipedia |
geometry of special relativity | Spacetime is equipped with an indefinite non-degenerate bilinear form, variously called the Minkowski metric, the Minkowski norm squared or Minkowski inner product depending on the context. The Minkowski inner product is defined so as to yield the spacetime interval between two events when given their coordinate differ... | wikipedia |
lattice model (physics) | In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lat... | wikipedia |
lattice model (physics) | Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inver... | wikipedia |
lattice model (physics) | The solution of these models has given insights into the nature of phase transitions, magnetization and scaling behaviour, as well as insights into the nature of quantum field theory. Physical lattice models frequently occur as an approximation to a continuum theory, either to give an ultraviolet cutoff to the theory t... | wikipedia |
lattice model (physics) | However, digital physics considers nature fundamentally discrete at the Planck scale, which imposes upper limit to the density of information, aka Holographic principle. More generally, lattice gauge theory and lattice field theory are areas of study. Lattice models are also used to simulate the structure and dynamics ... | wikipedia |
constructive quantum field theory | In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requires new mathematics, in a sense analogous to classical real analysis, putting calculus on a mathematically rigorous fou... | wikipedia |
constructive quantum field theory | Attempts to put quantum field theory on a basis of completely defined concepts have involved most branches of mathematics, including functional analysis, differential equations, probability theory, representation theory, geometry, and topology. It is known that a quantum field is inherently hard to handle using convent... | wikipedia |
constructive quantum field theory | The existence theorems for quantum fields can be expected to be very difficult to find, if indeed they are possible at all. One discovery of the theory that can be related in non-technical terms, is that the dimension d of the spacetime involved is crucial. Notable work in the field by James Glimm and Arthur Jaffe show... | wikipedia |
constructive quantum field theory | Along with work of their students, coworkers, and others, constructive field theory resulted in a mathematical foundation and exact interpretation to what previously was only a set of recipes, also in the case d < 4. Theoretical physicists had given these rules the name "renormalization," but most physicists had been s... | wikipedia |
constructive quantum field theory | The traditional basis of constructive quantum field theory is the set of Wightman axioms. Osterwalder and Schrader showed that there is an equivalent problem in mathematical probability theory. The examples with d < 4 satisfy the Wightman axioms as well as the Osterwalder–Schrader axioms. They also fall in the related ... | wikipedia |
covariant classical field theory | In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the variational bicomplex are the correct domain for such a descriptio... | wikipedia |
four-dimensional chern-simons theory | In mathematical physics, four-dimensional Chern–Simons theory, also known as semi-holomorphic or semi-topological Chern–Simons theory, is a quantum field theory initially defined by Nikita Nekrasov, rediscovered and studied by Kevin Costello, and later by Edward Witten and Masahito Yamazaki. It is named after mathemati... | wikipedia |
geometric quantisation | In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theor... | wikipedia |
energy quantization | In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theor... | wikipedia |
energy quantization | The method proceeds in two stages. First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation relations corresponding exactly to the classical Poisson-bracket r... | wikipedia |
noncommutative field theory | In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutativ... | wikipedia |
noncommutative field theory | Various lower limits have been claimed for the noncommutative scale, (i.e. how accurately positions can be measured) but there is currently no experimental evidence in favour of such a theory or grounds for ruling them out. One of the novel features of noncommutative field theories is the UV/IR mixing phenomenon in whi... | wikipedia |
light scattering in liquids and solids | In mathematical physics, scattering theory is a framework for studying and understanding the interaction or scattering of solutions to partial differential equations. In acoustics, the differential equation is the wave equation, and scattering studies how its solutions, the sound waves, scatter from solid objects or pr... | wikipedia |
light scattering in liquids and solids | In regular quantum mechanics, which includes quantum chemistry, the relevant equation is the Schrödinger equation, although equivalent formulations, such as the Lippmann-Schwinger equation and the Faddeev equations, are also largely used. The solutions of interest describe the long-term motion of free atoms, molecules,... | wikipedia |
light scattering in liquids and solids | The scenario is that several particles come together from an infinite distance away. These reagents then collide, optionally reacting, getting destroyed or creating new particles. The products and unused reagents then fly away to infinity again. | wikipedia |
light scattering in liquids and solids | (The atoms and molecules are effectively particles for our purposes. Also, under everyday circumstances, only photons are being created and destroyed.) The solutions reveal which directions the products are most likely to fly off to and how quickly. They also reveal the probability of various reactions, creations, and ... | wikipedia |
six-dimensional holomorphic chern–simons theory | In mathematical physics, six-dimensional holomorphic Chern–Simons theory or sometimes holomorphic Chern–Simons theory is a gauge theory on a three-dimensional complex manifold. It is a complex analogue of Chern–Simons theory, named after Shiing-Shen Chern and James Simons who first studied Chern–Simons forms which appe... | wikipedia |
schrödinger functional | In mathematical physics, some approaches to quantum field theory are more popular than others. For historical reasons, the Schrödinger representation is less favored than Fock space methods. In the early days of quantum field theory, maintaining symmetries such as Lorentz invariance, displaying them manifestly, and pro... | wikipedia |
spacetime algebra | In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4). According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space that a... | wikipedia |
n=2 superconformal algebra | In mathematical physics, the 2D N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional conformal field theory. It has important applications in mirror symmetry. It was introduced by M. Ademollo, L. Brink, and A. D'Adda et al. ... | wikipedia |
dirac algebra | In mathematical physics, the Dirac algebra is the Clifford algebra Cl 1 , 3 ( C ) {\displaystyle {\text{Cl}}_{1,3}(\mathbb {C} )} . This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of the gamma matrices, which rep... | wikipedia |
dirac algebra | For this article we fix the signature to be mostly minus, that is, ( + , − , − , − ) {\displaystyle (+,-,-,-)} . The Dirac algebra is then the linear span of the identity, the gamma matrices γ μ {\displaystyle \gamma ^{\mu }} as well as any linearly independent products of the gamma matrices. This forms a finite-dimens... | wikipedia |
dirac delta functions | In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.The current understanding of the unit impul... | wikipedia |
duffin–kemmer–petiau algebra | In mathematical physics, the Duffin–Kemmer–Petiau algebra (DKP algebra), introduced by R.J. Duffin, Nicholas Kemmer and G. Petiau, is the algebra which is generated by the Duffin–Kemmer–Petiau matrices. These matrices form part of the Duffin–Kemmer–Petiau equation that provides a relativistic description of spin-0 and ... | wikipedia |
operator-valued distribution | In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 after Haag–Ruelle scatter... | wikipedia |
almost mathieu operator | In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by ( n ) = u ( n + 1 ) + u ( n − 1 ) + 2 λ cos ( 2 π ( ω + n α ) ) u ( n ) , {\displaystyle (n)=u(n+1)+u(n-1)+2\lambda \cos(2\pi (\omega +n\alpha ))u(n),\,} acting as a self-adjoint operator on the Hilber... | wikipedia |
almost mathieu operator | In pure mathematics, its importance comes from the fact of being one of the best-understood examples of an ergodic Schrödinger operator. For example, three problems (now all solved) of Barry Simon's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator. In phys... | wikipedia |
quantum spacetime | In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some vari... | wikipedia |
quantum spacetime | The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko as a way to eliminate infinities from quantum field theory. The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field can be regarded as moving in a quantum spaceti... | wikipedia |
conformal algebra | In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group, known as the conformal group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poin... | wikipedia |
conformal algebra | Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations. They called a generic expression of conformal symmetry a spherical wave transformation. General relativity in two spacetime dimensions also enjoys conformal symmetry. | wikipedia |
quantum kz equations | In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the N-point functions, the ... | wikipedia |
two-dimensional yang–mills theory | In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined Yang–Mills measure, meaning that the (Euclidean) path integral can be interpreted as a measure on the set of conne... | wikipedia |
unit interval graph | In mathematical psychology, indifference graphs arise from utility functions, by scaling the function so that one unit represents a difference in utilities small enough that individuals can be assumed to be indifferent to it. In this application, pairs of items whose utilities have a large difference may be partially o... | wikipedia |
hecke algebra of a pair | In mathematical representation theory, the Hecke algebra of a pair (g,K) is an algebra with an approximate identity, whose approximately unital modules are the same as K-finite representations of the pairs (g,K). Here K is a compact subgroup of a Lie group with Lie algebra g. | wikipedia |
cohen algebra | In mathematical set theory, a Cohen algebra, named after Paul Cohen, is a type of Boolean algebra used in the theory of forcing. A Cohen algebra is a Boolean algebra whose completion is isomorphic to the completion of a free Boolean algebra (Koppelberg 1993). | wikipedia |
jónsson function | In mathematical set theory, an ω-Jónsson function for a set x of ordinals is a function f: ω → x {\displaystyle f:^{\omega }\to x} with the property that, for any subset y of x with the same cardinality as x, the restriction of f {\displaystyle f} to ω {\displaystyle ^{\omega }} is surjective on x {\displaystyle x} .... | wikipedia |
jónsson function | Erdős and Hajnal (1966) showed that for every ordinal λ there is an ω-Jónsson function for λ. Kunen's proof of Kunen's inconsistency theorem uses a Jónsson function for cardinals λ such that 2λ = λℵ0, and Kunen observed that for this special case there is a simpler proof of the existence of Jónsson functions. Galvin an... | wikipedia |
illness narrative | In mathematical sociology, the theory of comparative narratives was devised in order to describe and compare the structures (expressed as "and" in a directed graph where multiple causal links incident into a node are conjoined) of action-driven sequential events.Narratives so conceived comprise the following ingredient... | wikipedia |
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