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Superposition (other)
A superposition is a linear combination. It can also refer to: Math and science Math * The superposition principle of Linear differential equation, linear differential equations (LDEs), that a linear combination of solutions to an LDE is itself another solution ** Superposition theorem for electric circuits ** Quantum superposition, in quantum physics * The Kolmogorov–Arnold representation theorem, Kolmogorov–Arnold superposition theorem, representing a multivariate function as a superposition of univariate functions * Superposition calculus, used in logic for equational first-order reasoning Chemistry * Dalton's law Geology * Law of superposition in geology and archaeology Political science * Parallel voting, Superposition or parallel voting, a mixture of two electoral systems that do not interact Music and art *"Superposition (song), Superposition", a song by Young the Giant from the 2018 album ''Mirror Master'' *"Superposition", a song by Daniel Caesar fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ''ax'' + ''by'', where ''a'' and ''b'' are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field (mathematics), field, with some generalizations given at the end of the article. Definition Let ''V'' be a vector space over the field ''K''. As usual, we call elements of ''V'' ''vector space, vectors'' and call elements of ''K'' ''scalar (mathematics), scalars''. If v1,...,v''n'' are vectors and ''a''1,...,''a''''n'' are scalars, then the ''linear combination of those vectors with those scalars as coefficients'' is :a_1 \mathbf v_1 + a_2 \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superposition Principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input ''A'' produces response ''X'', and input ''B'' produces response ''Y'', then input (''A'' + ''B'') produces response (''X'' + ''Y''). A function F(x) that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties: additivity F(x_1 + x_2) = F(x_1) + F(x_2) and homogeneity F(ax) = a F(x) for scalar . This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Differential Equation
In mathematics, a linear differential equation is a differential equation that is linear equation, linear in the unknown function and its derivatives, so it can be written in the form a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b(x) where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable . Such an equation is an ordinary differential equation (ODE). A ''linear differential equation'' may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives. Types of solution A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of antiderivative, integrals. This is also true for a linear equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superposition Theorem
In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ''ax'' + ''by'', where ''a'' and ''b'' are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article. Definition Let ''V'' be a vector space over the field ''K''. As usual, we call elements of ''V'' ''vectors'' and call elements of ''K'' '' scalars''. If v1,...,v''n'' are vectors and ''a''1,...,''a''''n'' are scalars, then the ''linear combination of those vectors with those scalars as coefficients'' is :a_1 \mathbf v_1 + a_2 \mathbf v_2 + a_3 \mathbf v_3 + \cdots + a_n \mathbf v_n. There is some ambiguity in the use of the term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Superposition
Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system. An example is a qubit used in quantum information processing. A qubit state is most generally a superposition of the basis states , 0 \rangle and , 1 \rangle: : , \Psi \rangle = c_0, 0\rangle + c_1, 1\rangle, where , \Psi \rangle is the quantum state of the qubit, and , 0 \rangle, , 1 \rangle denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes c_0 and c_1 that both are complex numbers. Here , 0 \rangle corresponds to the classical 0 bit, and , 1 \r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov–Arnold Representation Theorem
In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function f\colon ,1n\to \R can be represented as a superposition of continuous single-variable functions. The works of Vladimir Arnold and Andrey Kolmogorov established that if ''f'' is a multivariate continuous function, then ''f'' can be written as a finite composition of continuous functions of a single variable and the binary operation of addition. More specifically, : where \phi_\colon ,1to \R and \Phi_\colon \R \to \R. There are proofs with specific constructions. It solved a more constrained form of Hilbert's thirteenth problem, so the original Hilbert's thirteenth problem is a corollary. In a sense, they showed that the only true continuous multivariate function is the sum, since every other continuous function can be written using univariate continuous functions and summing. History The Kolmogorov–Arnold repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superposition Calculus
The superposition calculus is a calculus for reasoning in equational logic. It was developed in the early 1990s and combines concepts from first-order resolution with ordering-based equality handling as developed in the context of (unfailing) Knuth–Bendix completion. It can be seen as a generalization of either resolution (to equational logic) or unfailing completion (to full clausal logic). Like most first-order calculi, superposition tries to show the ''unsatisfiability'' of a set of first-order clauses, i.e. it performs proofs by refutation. Superposition is refutation complete—given unlimited resources and a ''fair'' derivation strategy, from any unsatisfiable clause set a contradiction will eventually be derived. Many (state-of-the-art) theorem provers for first-order logic are based on superposition (e.g. the E equational theorem prover), although only a few implement the pure calculus. Implementations * E * SPASS * Vampire * Waldmeisterbr>(official web page ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dalton's Law
Dalton's law (also called Dalton's law of partial pressures) states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. This empirical law was observed by John Dalton in 1801 and published in 1802.J. Dalton (1802)"Essay IV. On the expansion of elastic fluids by heat,"''Memoirs of the Literary and Philosophical Society of Manchester'', vol. 5, pt. 2, pages 595–602; see page 600. Dalton's law is related to the ideal gas laws. Formula Mathematically, the pressure of a mixture of non- reactive gases can be defined as the summation: p_\text = \sum_^n p_i = p_1+p_2+p_3+\cdots+p_n where , , ..., represent the partial pressures of each component. p_ = p_\text x_i where is the mole fraction of the ''i''th component in the total mixture of ''n'' components. Volume-based concentration The relationship below provides a way to determine the volume-based concentration of any individual gaseous compone ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Superposition
The law of superposition is an axiom that forms one of the bases of the sciences of geology, archaeology, and other fields pertaining to geological stratigraphy. In its plainest form, it states that in undeformed stratigraphic sequences, the oldest strata will lie at the bottom of the sequence, while newer material stacks upon the surface to form new deposits over time. This is paramount to stratigraphic dating, which requires a set of assumptions, including that the law of superposition holds true and that an object cannot be older than the materials of which it is composed. To illustrate the practical applications of superposition in scientific inquiry, sedimentary rock that has not been deformed by more than 90° will exhibit the oldest layers on the bottom, thus enabling paleontologists and paleobotanists to identify the relative ages of any fossils found within the strata, with the remains of the most archaic lifeforms confined to the lowest. These findings can inform t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Voting
In political science, parallel voting or superposition refers to the use of two or more Electoral system, electoral systems to elect different members of a legislature. More precisely, an electoral system is a superposition if it is a mixture of at least two tiers, which do not interact with each other in any way; one part of a legislature is elected using one method, while another part is elected using a different method, with all voters participating in both. Thus, the final results can be found by calculating the results for each system separately based on the votes alone, then adding them together. A system is called fusion (not to be confused with Electoral fusion in the United States, electoral fusion) or Majority bonus system, majority bonus, another independent mixture of two system but without two tiers. Superposition (parallel voting) is also not the same as "Coexistence (electoral systems), coexistence", which when different districts in the same election use different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superposition (song)
"Superposition" is a song by American alternative rock band Young the Giant, promoted as the second single off of the band's fourth album, ''Mirror Master''. It was released on August 23, 2018, through Elektra Records. Background and recording Beyond the original version, two remakes were released: the first one was dubbed "Superposition (Reprise)", recorded in a single day in the Layman Drug Company Studio in Nashville, and uploaded to YouTube on January 16, 2019 while the second was named "Superposition (Reflection)" and posted online on April 12, 2019. On social media posts, the band has described "Superposition" as "a song about quantum physics that has defied odds". On the ''Reprise'' version, they stated it is "a deconstructed negative of the original: a flipper zip on the double helix". Furthermore, on the ''Reflection'' version, they added it "is about recognizing the constant that hides behind infinite variations". Unusual in modern popular music, "Superposition" promi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
