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Two types of operators (I, II) are constructed in terms of bosonic quantum (\(q\)-)oscillator operators. These operators satisfy a relation analogous to the \(q\)-commutation relation for fermionic quantum (\(q\)-)oscillators. Both types of operators (I, II) can be identified with the fermionic harmonic oscillator in the limit \(q \to 1\) by exploiting the presence of certain arbitrary functions of the deformation parameter \(q\). For I they may be identified as parameters of U(1) transformations while for II they correspond to parameters of GL(2,R) transformations. Appropriate fermionic number operators in the limit \(q \to 1\) are also constructed.

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The group theoretical foundation of some conservation laws in the Maxwell electrodynamics in vacuum is considered. In particular the “mysterious” conservation laws of Lipkin [4] are found to be a consequence of the Noether theorem.

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Reduction of \(sl(\infty ; \mathcal {C})\) self-dual Yang–Mills equations to the heavenly equations is proposed.

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We study the critical properties of the two-dimensional simplicial gravity with the \(R^2\) term in the action. Changing the value of the coupling constant of this term we observe a sharp transition between the surfaces with low and high Haussdorff dimension. This transition seems not to be leading to a phase transition. For the whole range of the coupling constant a value of \(\gamma _{\rm str}\) is consistent with the pure gravity value \(-\)1/2.

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Bayesian probability theory is applied to approximate solving of the inverse problems. In order to solve the moment problem with the noisy data, the entropic prior is used. The expressions for the solution and its error bounds are presented. When the noise level tends to zero, the Bayesian solution tends to the classic maximum entropy solution in the \(L_1\) norm. The way of using spline prior is also shown.

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In this note we attempt to explain the L3 \(l^+l^-\gamma \gamma \) events in technicolor models. We find that the four L3 events are in reasonable agreement with the signature characterized by the process of \(Z \to \rho ^0P^0 \to l^+l^-P^0(\gamma \gamma )\).

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Following the idea that very short range correlations in momentum space characteristic for intermittency reflect the power law distribution of space-time region of hadron emission we investigate the origin of the power law dependence. We use the current ensemble model of pion production to describe the boson sources and show the relation between the space-time distribution of the boson sources and intermittency exponents obtained from multiplicities in momentum space.

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The set of 16 first-order relativistic equations derived from the two-body Dirac equation is discussed for a system of two spin-1/2 particles moving in an external potential. Scalar and vector wave-function components are used in place of double-bispinor components. In general, the case of different masses is considered. In the case of equal masses, this set is explicitly reduced to four second-order ecjuations involving “large–large” components, if terms quartic in momenta are neglected.

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It is shown that the ad hoc assumptions of Yukawa’s bilocal theory can be derived from first principles if one takes into account the full symmetry group of internal and external symmetries represented in a complex space.

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Microscopic static calculations of the mean-square charge radii of even nuclei are presented. It is shown that the quadrupole pairing forces influence the magnitude of the isotopic shift of \(\langle r^2\rangle \). The experimental data are better reproduced with quadrupole pairing interaction then with the monopole pairing only.