The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.     

On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

Vanishing of the Batalin–Vilkovisky Algebra Structure for TCFTs

The purpose of this short note is to show that for topological conformal field theories in dimension two (or closed string field theories) the Batalin–Vilkovisky algebra structure on the homology of the BRST-complex [G] ([Z]) is trivial after a certain element of ghost number 0 is inverted. The important point here is that we do not restrict our considerations to the tree level but incorporate the action of moduli spaces for higher genera. Received: 22 September 1998 / Accepted: 1 March 1999     

Wavelet Correlation in the One-Dimensional Projection α-Model

One-dimensional projection of two-dimensional self-affine phase space for two typical self-affine fractals is studied,and one-dimensional projection α model is obtained.About this model,starting from characterstic function,we study its correlation densities and the wavelet correption moment.The wavelet basis functions of split number Λ=2,3,4 are found out.It turns out that the various wavelet-coefficents decouple when the second order wavelet correlation moments are considered.This is a general conclusion and model independent.     

Integrability of the Bi-Yang–Baxter σ-Model

We construct a Lax pair with spectral parameter for a two-parameter doubly Poisson–Lie deformation of the principal chiral model.     

Computations of ¶in the 2-d O(N) Non-Linear σ-Model

We review the various computations of the ratio of the mass gap M to the Λ-parameter entering the perturbative computations of amplitudes at high energies in the O(n) non-linear σ-model. In particular we reproduce from original notes of H. Lehmann his computation of this ratio in the (next–to) leading order of the 1/n expansion. Received: 23 March 2000 / Accepted: 10 April 2000     

Formulation of the Snell–Descartes Laws in Terms of Geometric Algebra

Russian Physics Journal - A formulation of the laws of reflection and refraction of optical rays by the interface of two isotropic media using the mathematical apparatus of geometric algebra is...     

Uncertainty and Certainty Relations for Pauli Observables in Terms of Renyi Entropies of Order α∈ （0; 1]

We obtain uncertainty and certainty relations of state-independent form for the three Paufi observables with use of the Renyi entropies of order α∈ （0; 1]. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the condition for equality in Renyi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average Renyi a-entropy ranges for a pure measured state. A width of this band is compared with the Tsallis formulation derived previously.     

Physical Mechanism of Formation of the Bimodal Structure in the Meiyu Front System

The bimodal structure of the Meiyu front system is readdressed after Zhou et al.（2005）. The physical mechanism of the formation of the bimodal distribution is discussed. The bimodal structure of the Melyu front system considerably results from atmospheric moisture gradients, though atmospheric temperature gradients are also not negligible. According to the definition of equivalent potential temperature, and by scale analysis, we find that atmospheric equivalent potential temperature gradients, which could be regarded as an indicator of the Meiyu front system, could be mainly attributed to the variations of atmospheric potential temperature gradients with a scaling factor of 1 and moisture gradients multiplied by a scaling factor of an order of about 2.5 × 10^3, which means that small variations of atmospheric moisture gradients could lead to large variations of equivalent potential temperature gradients, and thus large variations of the Meiyu front system. Quantitative diagnostics with a mesoscale simulation data in the vicinity of the Meiyu front system show that moisture gradients contribute to equivalent potential temperature gradients more than potential temperature gradients.     

The Topological Structure of the SU（2） Chern-Simons Topological Current in the Four-Dimensional Quantum Hall Effect

In the light of the decomposition of the SU（2） gauge potential for I = 1/2, we obtain the SU（2） Chern-Simons current over S4, i.e. the vortex current in the effective field for the four-dimensional quantum Hall effect. Similar to the vortex excitations in the two-dimensional quantum Hall effect （2D FQH） which are generated from the zero points of the complex scalar field, in the 4D FQH, we show that the SU（2） Chern-Simons vortices are generated from the zero points of the two-component wave functions ψ, and their topological charges are quantized in terms of the Hopf indices and Brouwer degrees of Ф-mapping under the condition that the zero points of field ψ are regular points.     

The Topological Inner Structure of Chern-Simons Tensor Current and the World-Sheet of Strings

Using the decomposition theory of U(1) gauge potential and b-mapping topological current theory, we investigate the topological inner structure of Chern Simons tensor current. It is proven that the U(1) Chern-Simons tensor current in four-dimensional manifold is just the topological current of creating the string world-sheets.     

The Structure of S Pair in Collective States (Ⅰ)

A new S pair and seniority particle operator used to overcome the difficulties due to seniority mixing are defined in collective motion of nuclei. The structure amplitude of S pair is determined in a self-consistent manner.     

The Structure of S Pair in Collective States (Ⅱ)

Comparison between states composed of the newly defined S pairs used for dealing with seniority mixing problem and those made from the usual S pairs is given.     

“Evaluations” of Observables Versus Measurements in Quantum Theory

In Quantum Physics there are circumstances where the direct measurement of a given observable encounters difficulties; in some of these cases, however, its value can be “evaluated”, i.e. it can be inferred by measuring another observable characterized by perfect correlation with the observable of interest. Though an evaluation is often interpreted as a measurement of the evaluated observable, we prove that the two concepts cannot be identified in Quantum Physics, because the identification yields contradictions. Then, we establish the conceptual status of evaluations in Quantum Theory and how they are related to measurements.     

The Influence of the Composition on the Atomic Structure of Bi1 – xYxFeO3 Ceramics

Technical Physics - The variation of the Bi1 – xYxFeO3 ceramics atomic structure with yttrium concentration x = 0, 0.10, 0.15, and 0.20 has been analyzed. Ceramic samples have been prepared...     

Structure of the α–β Interface in a PdCu Solid Solution

Physics of the Solid State - In accordance with the orientation ratio (110), [001]β || (111), [$$1\bar {1}0$$]α established by fast electron diffraction between the ordered (β) and...     

Analysis of the Structure of the Bands in the IR Spectrum of β-D Glucose by the Regularized Method of Deconvolution

Deconvolution of the IR absorption spectrum of -D glucose in the spectral range 1500–450 cm^–1 has been carried out. The results of the deconvolution were compared with the IR and Raman spectra recorded at room and low temperatures and with the data obtained by theoretical calculations for the frequencies of the normal vibrations of the -D glucose molecule in the crystalline state. It is shown that deconvolution of the IR spectra recorded at room temperature makes it possible to separate the bands observed experimentally only at a very low temperature of the sample and a number of components that were not resolved earlier. The number of bands separated on deconvolution of the IR spectra of -D glucose in the spectral range 1500–450 cm^–1 is more than twice the number of visible absorption maxima in the usual spectrum. The results of deconvolution of the IR spectrum of -D glucose are in good agreement with the data of theoretical calculations for the frequencies of the normal vibrations of the -D glucose molecule in the crystalline state. The existence of the factor-group (Davydov) splitting of a number of frequencies of the nondegenerate fundamental vibrations of molecules in a crystal cell has been revealed in the IR spectrum of -D glucose. It was concluded that the model of an isolated molecule is insufficient for detailed theoretical interpretation of the vibrational spectra of carbohydrates.