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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 Carine may refer to: Places * Carine, Western Australia, a suburb of Perth ** Electoral district of Carine, in the Western Australian parliament * Carine, Nikšić, Montenegro * Carine (Mysia), a town of ancient Mysia, now in Turkey Owl species ... * 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
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, 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 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. The word linear comes from Latin ''linearis'', "pertaining to or resembling a line". In mathematics In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties: * Additivity: . * Homogeneity of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessarily a real ... [...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. An o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilinear Order
In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge. Wadge degrees Suppose A and B are subsets of Baire space ωω. Then A is Wadge reducible to B or A ≤W B if there is a continuous function f on ωω with A = f^ /math>. The Wadge order is the preorder or quasiorder on the subsets of Baire space. Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set A is denoted by A">math>Asub>W. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy. Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if A ≤W B and B is a countable intersection of open sets, then so is A. The same works for all levels of the Borel hierarchy and the difference hi ... [...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) = \lambda^\theta T(v), where \lambda^\theta is the image of the scalar \lambda under the automorphism. 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. ... [...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. Arithmetic Progression:-"There exit a common different that is d that we add to the previous term in output it is called arithmetic progression Finite generalized arithmetic progression A finite generalized arithmetic progression, or sometimes just generalized arithmetic pr ... [...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 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
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 relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinearity
In mathematics and science, a nonlinear 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 because 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 linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semilinear Response
Semi-linear response theory (SLRT) is an extension of linear response theory A linear response function describes the input-output relationship of a signal transducer such as a radio turning electromagnetic waves into music or a neuron turning Synapse, synaptic input into a response. Because of its many applications in infor ... (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 mesosopic conductance . The term SLRT has been coined in where it has been applied to the calculat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artificial Neuron
An artificial neuron is a mathematical function conceived as a model of biological neurons, a neural network. Artificial neurons are elementary units in an artificial neural network. The artificial neuron receives one or more inputs (representing excitatory postsynaptic potentials and inhibitory postsynaptic potentials at neural dendrites) and sums them to produce an output (or , representing a neuron's action potential which is transmitted along its axon). Usually each input is separately weighted, and the sum is passed through a non-linear function known as an activation function or transfer function. The transfer functions usually have a sigmoid shape, but they may also take the form of other non-linear functions, piecewise linear functions, or step functions. They are also often monotonically increasing, continuous, differentiable and bounded. Non-monotonic, unbounded and oscillating activation functions with multiple zeros that outperform sigmoidal and ReLU like activation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-linear Resolution
Carine may refer to: Places * Carine, Western Australia, a suburb of Perth ** Electoral district of Carine, in the Western Australian parliament * Carine, Nikšić, Montenegro * Carine (Mysia), a town of ancient Mysia, now in Turkey Owl species * Little owl (''Carine noctua'' or ''Athene noctua'') * Rodrigues scops owl (''Carine murivora'' or ''Mascarenotus murivorus'') Other uses * CARINE Carine may refer to: Places * Carine, Western Australia, a suburb of Perth ** Electoral district of Carine, in the Western Australian parliament * Carine, Nikšić, Montenegro * Carine (Mysia), a town of ancient Mysia, now in Turkey Owl species ..., a theorem prover * Carine (given name) See also * Carina (other) {{disambig, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
