In the framework of many-body quantum mechanics, models solvable by the Bethe ansatz can be contrasted with free fermion models. One can say that the dynamics of a free model is one-body reducible: the many-body wave function for fermions (bosons) is the anti- symmetrized (symmetrized) product of one-body wave functions. Models solvable by the Bethe ansatz are not free: the two-body sector has a non-trivial scattering matrix, which in general depends on the momenta. On the other hand, the dynamics of the models solvable by the Bethe ansatz is two-body reducible: the many-body scattering matrix is a product of two-body scattering matrices.
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"Pickover Publications". Retrieved August 14, 2008. Additional visualization work includes topics that involve breathing motions of proteins, snow-flake like patterns for speech sounds,"On the use of symmetrized dot patterns for the visual characterization of speech waveforms and other sampled data". Retrieved August 19, 2008.
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For the representation to be unique, the kernels must be symmetrical in the n variables \tau. If it is not symmetrical, it can be replaced by a symmetrized kernel, which is the average over the n! permutations of these n variables τ. If N is finite, the series is said to be truncated.
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For univariate distributions that are symmetric about one median, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population median; for non-symmetric distributions, the Hodges–Lehmann estimator is a robust and highly efficient estimator of the population pseudo-median, which is the median of a symmetrized distribution and which is close to the population median. The Hodges–Lehmann estimator has been generalized to multivariate distributions.
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Rayner/Tuckett, p. 23-4 The next deeper, third stratum is one where different classes are identified (thus containing a fair amount of asymmetrical thinking) but in which...parts of a class are always taken as the whole class — symmetrization (plus a degree of timelessness).Rayner/Tuckett, p. 24 The fourth stratum is defined by the fact that there is formation of wider classes which are also symmetrized, while asymmetry becomes less and less.
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Atomic orbitals may be defined more precisely in formal quantum mechanical language. They are an approximate solution to the Schrodinger equation for the electrons bound to the atom by the electric field of the atom's nucleus. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, is approximated by an expansion (see configuration interaction expansion and basis set) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions.
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The Supreme Court rejected the theory on a rule of reason analysis, noting that there are any number of legitimate business strategies that involve buying large quantities of raw materials. A plaintiff alleging predatory buying must therefore prove—and Ross-Simmons had not—that the defendant caused the price to rise, and that the defendant is likely to recoup the costs incurred in such a scheme. The Court's decision symmetrized its case law, with Weyerhaeuser and Brooke Group Ltd. v. Brown & Williamson Tobacco Corp.
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The formula of the third Theil index has some similarity with the Hoover index (as explained in the related articles). As in case of the Hoover index, the symmetrized Theil index does not change when swapping the incomes with the income earners. How to generate that third Theil index by means of a spreadsheet computation directly from distribution data is shown below. An important property of the Theil index which makes its application popular is its decomposability into the between-group and within-group component.
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Neither does using the Theil index necessarily imply that a very low inequality (low redundancy, high entropy) is "good", because high entropy is associated with slow, weak and inefficient resource allocation processes. There are three variants of the Theil index. When applied to income distributions, the first Theil index relates to systems within which incomes are stochastically distributed to income earners, whereas the second Theil index relates to systems within which income earners are stochastically distributed to incomes. A third "symmetrized" Theil index is the arithmetic average of the two previous indices.
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The basis light-front quantization (BLFQ) approach uses expansions in products of single-particle basis functions to represent the Fock-state wave functions. Typically, the longitudinal (x^-) dependence is represented in the DLCQ basis of plane waves, and the transverse dependence is represented by two- dimensional harmonic oscillator functions. The latter are ideal for applications to confining cavities and are consistent with light-front holographic QCD. The use of products of single particle basis functions is also convenient for incorporation of boson and fermion statistics, because the products are readily (anti)symmetrized.
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The creation and annihilation operators are introduced to add or remove a particle from the many-body system. These operators lie at the core of the second quantization formalism, bridging the gap between the first- and the second-quantized states. Applying the creation (annihilation) operator to a first-quantized many-body wave function will insert (delete) a single-particle state from the wave function in a symmetrized way depending on the particle statistics. On the other hand, all the second-quantized Fock states can be constructed by applying the creation operators to the vacuum state repeatedly.
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In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture. Often the mapping from functions on phase space to operators is called the Weyl transform or Weyl quantization, whereas the inverse mapping, from operators to functions on phase space, is called the Wigner transform. This mapping was originally devised by Hermann Weyl in 1927 in an attempt to map symmetrized classical phase space functions to operators, a procedure known as Weyl quantization. It is now understood that Weyl quantization does not satisfy all the properties one would require for quantization and therefore sometimes yields unphysical answers.
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