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It is shown that the general theory of lifting the tensor fields from a Riemannian manifold \(M\) to its tangent bundle \(TM\) enables one to define in a natural manner the unique sympletic connection on the phase space \(T^{\ast }M\) which is induced by the Levi–Civita connection on \(M\). This is exactly the symplectic connection given also by Bordemann, Neumaier and Waldmann Commun. Math. Phys. 198, 363 (1998); J. Geom. Phys. 29, 199 (1999). Relationship between the symplectic and Riemannian geometries on \(T^{\ast }M\) and \(M\) is considered.

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Variational approach to the Bäcklund transformations is derived on the basis of strong or semi-strong necessary condition for extremum of a functional. The obtained method is applied to the sine-Gordon and Korteweg–de Vries equations.

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A method which enables one to build up explicit least-action principles in the non-projective twistor spaces is applied to the context of the theory of complexified Maxwell fields. The freedom in the choices of spinor kernels for the integrands of the universal contour integrals for interacting fields gives rise to the possibility of constructing several Lagrangian densities for the system being considered. It appears that the Lorenz-gauge condition is intrinsically tied in with the inner structure of the twistor dynamics. The configurations involving the kernels for the potential and current density turn out to suggest a natural variational prescription for deriving the equations of motion for the potential. It is shown that the equations for the fields can be derived directly from coupled statements which carry only field quantities.

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The Schwarzschild spacetime is for electromagnetic waves like a nonuniform medium with a varying refraction index. A fraction of an outgoing radiation scatters off the curvature of the geometry and can be intercepted by a gravitational center. The amount of the intercepted energy is bounded above by the backscattered energy of an initially outgoing pulse of electromagnetic radiation, which in turn depends on the initial energy, the Schwarzschild radius and the pulse location. Its magnitude depends on the frequency spectrum: it becomes negligible in the short wave limit but can be significant in the long wave regime.

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Two thin conducting, electrically neutral, parallel plates forming an isolated system in vacuum exert attracting force on each other, whose origin is the quantum electrodynamical interaction. This theoretical hypothesis, known as Casimir effect, has been also confirmed experimentally. Despite long history of the subject, no completely convincing theoretical analysis of this effect appears in the literature. Here we discuss the effect (for the scalar field) anew, on a revised physical and mathematical basis. Standard, but advanced methods of relativistic quantum theory are used. No anomalous features of the conventional approaches appear. The Casimir quantitative prediction for the force is shown to constitute the leading asymptotic term, for large separation of the plates, of the full, model-dependent expression.

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The strong coupling constant \(g_{K_0^*K\pi }\) of the scalar \(K_0^*\) meson decay to \(K\pi \) is calculated in light cone QCD sum rule. The predicted value of the coupling constant \(g_{K_0^*K\pi }\) is in a good agreement with the experimental result.

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The alternative formulated in the title has a chance to be settled, when the existence of the LSND effect is experimentally excluded or confirmed. The first option, much discussed in literature, works in the case of three active neutrinos \( \nu _e\,, \,\nu _\mu \,, \,\nu _\tau \), when among their massive states \( \nu _1\,, \,\nu _2\,, \,\nu _3\) there is no direct mixing between \( \nu _1\) and \( \nu _3 \), and the mass hierarchy \( m^2_1\lesssim m^2_2 \ll m^2_3\) holds. This option is consistent with the observed deficits of solar \( \nu _e\)’s and atmospheric \(\nu _\mu \)’s, if \( \Delta m^2_{21} \leftrightarrow \Delta m^2_{\rm sol} \) and \( \Delta m^2_{32} \leftrightarrow \Delta m^2_{\rm atm} \). On the other hand, the second option is an extension of the idea of the former to the case of four neutrinos \( \nu _s\,, \,\nu _e\,, \,\nu _\mu \,, \,\nu _\tau \) (including one sterile neutrino \( \nu _s \)), when among their massive states \(\nu _0\,, \,\nu _1\,, \,\nu _2\,, \,\nu _3\) there are no direct mixings between \( \nu _0\) and \( \nu _2 \), \( \nu _0 \) and \( \nu _3 \), \( \nu _1\) and \( \nu _3 \), and the mass hierarchy \( m^2_0\lesssim m^2_1 \ll m^2_2 \lesssim m^2_3 \) is now valid. Such an option, belonging to a class of textures widely discussed in literature, may be consistent with the observed deficits of solar \( \nu _e \)’s and atmospheric \( \nu _\mu \)’s as well as with the LSND appearance of \( \nu _e \)’s in the beam of accelerator \( \nu _\mu \)’s, if now \( \Delta m^2_{10} \leftrightarrow \Delta m^2_{\rm sol} \), \( \Delta m^2_{32} \leftrightarrow \Delta m^2_{\rm atm} \) and \( \Delta m^2_{21} \leftrightarrow \Delta m^2_{\rm LSND} \) (however, in the case of solar \(\nu _e \)’s the role of \( \nu _s \)’s in the disappearance of \( \nu _e \)’s is recently questioned). In both options, only the close neighbours in the hierarchies of massive neutrinos \(\nu _1\,, \,\nu _2\,, \,\nu _3\) and \(\nu _0\,, \,\nu _1\,, \,\nu _2\,, \,\nu _3\), respectively, mix directly. This characteristic feature of the two-mixing texture for three neutrinos or the three-mixing texture for four neutrinos may be somehow physically significant.

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I elaborate on the argument that the violation of Hara’s theorem for conserved current requires that the current is not sufficiently well localized. It is also stressed that whatever sign of asymmetry is measured in the \({\mit \Xi } ^0 \to {\mit \Lambda } \gamma \) decay, one of the following three statements must be incorrect: (1) Hara’s theorem is satisfied, (2) vector meson dominance is applicable to weak radiative hyperon decays, and (3) basic structure of our quark-model description of nuclear parity violation is correct.

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This is somewhat extended version of the talk given at the Gran Sasso Summer Institute: Massive Neutrinos in Physics and Astrophysics. It describes general ideas about mirror world, extra spatial dimensions and dinosaur extinction. Some suggestions are made how these seemingly different things can be related to each other.

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The problem of the quantum vacuum in a one-dimensional cavity is discussed. We put forward a new method to solve Moore’s equation and search for fundamental classical basis. This new method is applicable to problems with general wall trajectories and enables us to calculate both phase functions and their derivatives by iterations. We calculate energy densities and total energies for oscillating cavities. In particular, off resonant motions are studied.

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We show that the Hamiltonian dynamics of the self-interacting, Abelian \(p\)-form theory in \(D=2p+2\) dimensional space-time gives rise to the quasi-local structure. Roughly speaking, it means that the field energy is localized but on closed \(2p\)-dimensional surfaces (quasi-localised). From the mathematical point of view this approach is implied by the boundary value problem for the corresponding field equations. Various boundary problems, e.g. Dirichlet or Neumann, lead to different Hamiltonian dynamics. Physics seems to prefer gauge-invariant, positively defined Hamiltonians which turn out to be quasi-local. Our approach is closely related with the standard two-potential formulation and enables one to generate e.g. duality transformations in a perfectly local way (but with respect to a new set of nonlocal variables). Moreover, the form of the quantization condition displays very similar structure to that of the symplectic form of the underlying \(p\)-form theory expressed in the quasi-local language.

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The projected shell model is applied to the nucleus \(^{129}\)La. The results of theoretical calculations about the one-quasiproton bands are compared with experimental data, the agreement with the yrast \(\pi h_{11\,/\,2}\) band and \(\pi g_{\,7\,/\,2}\) band is satisfactory. We also assign the \(\pi g_{\,7\,/\,2}\otimes [\nu h_{\,11\,/\,2}]^2\) configuration with an oblate shape for one of bands in \(^{129}\)La.

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Differential cross-sections for the \(^{56}\)Fe\((p,d)^{55}\)Fe reaction have been measured at 28 MeV proton energy using 15 UD Pelletron accelerator at Nuclear Science Centre, New Delhi. The data has been analysed using the zero range Distorted Wave Born Approximation (DWBA) using the standard Woods–Saxon optical model potential and the non local range corrections. The spectroscopic factors for the 5/2\(^-\) (2.144 MeV), 1/2\(^+\) (4.450 MeV), 3/2\(^+\) (4.825 MeV) and 5/2\(^-\) (7.610 MeV) excited states have been newly measured.

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The rapidity (pseudorapidity) distribution of produced particles in high-energy nucleus–nucleus collisions are studied by the thermalized cylinder model. In the calculation, two rapidity (pseudorapidity) distribution formulas for an isotropic thermal source are used. The calculated results are compared with the experimental data of Au–Au collisions at 11.5\(A\) and 10.8\(A\) GeV/\(c\) and Pb–Pb collisions at 158\(A\) GeV/\(c\). The formula for rapidity (pseudorapidity) agrees well with data on rapidity (pseudorapidity) distribution, whereas approximating rapidity (pseudorapidity) by pseudorapidity (rapidity) leads to discrepancies in the fragmentation regions. The fit parameter \(\Delta y\) representing the rapidity interval over which isotropic sources are distributed seems to be independent on the kinds of concerned produced particles and the centrality cut in the fitted data.

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Using a microscopic phase-space model of the membrane system, the boundary condition at a membrane is derived. According to the condition, the substance flow across the membrane is proportional to the difference of the substance concentrations at the opposite membrane surfaces. The Green’s function of the diffusion equation is found for the derived boundary condition and the exact solution of the equation is given.

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We study the formation of first bound objects in the Universe after recombination. We trace the evolution of a spherically symmetric density perturbation in the \({\mit \Omega }=1\) Cold Dark Matter (CDM) model with baryon and dark matter contributions, respectively, \({\mit \Omega }_b=0.1\) and \({\mit \Omega }_{\rm dm}=0.9\). Physical processes in the collapsing gas relevant to various stages of the nonlinear collapse of low mass objects are considered. We find that the first density perturbations which collapse to form luminous objects have baryon mass in the range \(10^3\)–\(10^4 M_{\odot }\). The final collapse of these objects is triggered by the cooling due to H\(_2\) molecules and it starts early at redshifts \(z \sim 20\). The role of the initial baryon overdensity in the collapse of density perturbations in CDM model is studied.