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Semilinear Map, Semilinear
Semilinear or semi-linear (literally, "half linear") may refer to: Mathematics * Antilinear map, also called a "semilinear map" * Semilinear order * Semilinear map * Semilinear set * Semilinearity (operator theory) * Semilinear equation, a type of differential equation which is linear in the highest order derivative(s) of the unknown function * Various forms of "mild" nonlinearity are referred to as "semilinear" Other * Semilinear response, physics * Artificial neuron, also called a "semi-linear unit" * Semi-linear resolution * A mixture of linear and nonlinear gameplay in video games may be referred to as "semi-linear gameplay" {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear
In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d. Linearity of a mapping is closely related to '' proportionality''. Examples in physics include the linear relationship of voltage and current in an electrical conductor ( Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are '' nonlinear''. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. Linearity of a polynomial means that its de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antilinear Map
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y \in V and every complex number s, where \overline denotes the complex conjugate of s. Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity. Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces. Definitions and characterizations A function is called or if it is additive and conjugate homogeneous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilinear Order in video games may be referred to as "semi-linear gameplay"
{{disambig ...
Semilinear or semi-linear (literally, "half linear") may refer to: Mathematics * Antilinear map, also called a "semilinear map" * Semilinear order * Semilinear map * Semilinear set * Semilinearity (operator theory) * Semilinear equation, a type of differential equation which is linear in the highest order derivative(s) of the unknown function * Various forms of "mild" nonlinearity are referred to as "semilinear" Other * Semilinear response, physics * Artificial neuron, also called a "semi-linear unit" * Semi-linear resolution * A mixture of linear and nonlinear gameplay A video game with nonlinear gameplay presents players with challenges that can be completed in a number of different sequences. Each may take on (or even encounter) only some of the challenges possible, and the same challenges may be played in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilinear Map
In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means " field automorphism of ''K''". Explicitly, it is a function that is: * additive with respect to vector addition: T(v+v') = T(v)+T(v') * there exists a field automorphism ''θ'' of ''K'' such that T(\lambda v) = \theta(\lambda) T(v). If such an automorphism exists and ''T'' is nonzero, it is unique, and ''T'' is called ''θ''-semilinear. Where the domain and codomain are the same space (i.e. ), it may be termed a semilinear transformation. The invertible semilinear transforms of a given vector space ''V'' (for all choices of field automorphism) form a group, called the general semilinear group and denoted \operatorname(V), by analogy with and extending the general linear group. The special case where the field is the complex numbers \mathbb and the autom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilinear Set
In mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple common differences – whereas an arithmetic progression is generated by a single common difference, a generalized arithmetic progression can be generated by multiple common differences. For example, the sequence 17, 20, 22, 23, 25, 26, 27, 28, 29, \dots is not an arithmetic progression, but is instead generated by starting with 17 and adding either 3 ''or'' 5, thus allowing multiple common differences to generate it. A semilinear set generalizes this idea to multiple dimensions – it is a set of vectors of integers, rather than a set of integers. Finite generalized arithmetic progression A finite generalized arithmetic progression, or sometimes just generalized arithmetic progression (GAP), of dimension ''d'' is defined to be a set of the form :\ where x_0,x_1,\dots,x_d,L_1,\dots,L_d\in\mathbb. The product L_1L_2\cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilinearity (operator Theory)
In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s by Jacq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinearity
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilinear Response
Semi-linear response theory (SLRT) is an extension of linear response theory (LRT) for mesoscopic circumstances: LRT applies if the driven transitions are much weaker/slower than the environmental relaxation/dephasing effect, while SLRT assumes the opposite conditions. SLRT uses a resistor network analogy (see illustration) in order to calculate the rate of energy absorption: The driving induces transitions between energy levels, and connected sequences of transitions are essential in order to have a non-vanishing result, as in the theory of percolation. Applications The original motivation for introducing SLRT was the study of mesoscopic conductance . The term SLRT has been coined in where it has been applied to the calculation of energy absorption by metallic grains. Later the theory has been applied for analysing the rate of heating of atoms in vibrating traps . Definition of semilinear response Consider a system that is driven by a source f(t) that has a power spectrum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artificial Neuron
An artificial neuron is a mathematical function conceived as a model of a biological neuron in a neural network. The artificial neuron is the elementary unit of an ''artificial neural network''. The design of the artificial neuron was inspired by biological neural circuitry. Its inputs are analogous to excitatory postsynaptic potentials and inhibitory postsynaptic potentials at neural dendrites, or . Its weights are analogous to synaptic weights, and its output is analogous to a neuron's action potential which is transmitted along its axon. Usually, each input is separately weighted, and the sum is often added to a term known as a ''bias'' (loosely corresponding to the threshold potential), before being passed through a nonlinear function known as an activation function. Depending on the task, these functions could have a sigmoid shape (e.g. for binary classification), but they may also take the form of other nonlinear functions, piecewise linear functions, or step fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-linear Resolution
CARINE (Computer Aided Reasoning Engine) is a first-order classical logic automated theorem prover. It was initially built for the study of the enhancement effects of the strategies delayed clause-construction (DCC) and attribute sequences (ATS) in a depth-first search based algorithm. CARINE's main search algorithm is semi-linear resolution (SLR) which is based on an iteratively-deepening depth-first search (also known as depth-first iterative-deepening (DFID)) and used in theorem provers like THEO. SLR employs DCC to achieve a high inference rate, and ATS to reduce the search space. Delayed Clause Construction (DCC) Delayed Clause Construction is a stalling strategy that enhances a theorem prover's performance by reducing the work to construct clauses to a minimum. Instead of constructing every conclusion (clause) of an applied inference rule, the information to construct such clause is temporarily stored until the theorem prover decides to either discard the clause or construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
