Abstract:
It is shown that the periodic DNLS, with cubic nonlinearity, possesses gap solutions, i. e. standing waves, with the frequency in a spectral gap, that are exponentially localized in spatial variable. The proof is based on the linking theorem in combination with periodic approximations.

Abstract:
We present basic results, known and new, on nontrivial solutions of periodic stationary nonlinear Schr\"odinger equations. We also sketch an application to nonlinear optics and discuss some open problems.

Abstract:
The sensitivity to the $Z^\prime$ couplings of the processes $e^+e^-\to l^+l^-,\bar{b}b$ and $\bar{c}c$ at the linear collider with $\sqrt{s}=500 GeV$ with initial beam polarization, for typical extended model examples are studied. To this aim, the suitable integrated, polarized, observables directly related to the helicity cross sections that carry information on the individual $Z^\prime$ chiral couplings to fermions are used. We discuss the derivation of separate, model-independent limits on the couplings in the case of no observed indirect $Z^\prime$ signal within the expected experimental accuracy. In the hypothesis that such signals were, indeed, observed we assess the expected accuracy on the numerical determination of such couplings and the consequent range of $Z^\prime$ masses where the individual models can be distinguished from each other as the source of the effect.

Abstract:
Fermion compositness, and other new physics which can be described by an exchange of very massive particles ($Z^\prime$ boson, leptoquarks, sparticles with R-parity violating couplings), can be manifest itself as the presence of a strong four-fermion contact interaction. For the processes $e^+e^-\to \mu^+\mu^-$, $\tau^+\tau^-$, $\bar{b}b$ and $\bar{c}c$ at a future $e^+e^-$ linear collider (LC) with $\sqrt{s}=0.5$ TeV, we examine the sensitivity of the helicity cross sections to four-fermion contact interactions. If longitudinal polarization of the electron beam were available, two polarized integrated cross sections would offer the opportunity to separate the helicity cross sections and, in this way, to derive model-independent bounds on the relevant parameters. The measurement of these polarized cross sections with optimal kinematical cuts could significantly increase the sensitivity of helicity cross sections to contact interaction parameters and could give crucial information on the chiral structure of such new interactions. In addition, we consider the application of the proposed approach to the search for manifestations of a $Z^\prime$ for typical extended model examples.

Abstract:
The four-fermion contact-interaction searches in the process $e^+e^-\to\mu^+\mu^-$ at a future $e^+e^-$ Linear Collider with c.m. energy $\sqrt{s}=0.5$ TeV and with both beams longitudinally polarized are studied. We evaluate the corresponding model-independent constraints on the coupling constants, emphasizing the role of beam polarization, and make a comparison with the case of Bhabha scattering.

Abstract:
We discuss electron-electron contact-interaction searches in the process e^+e^- -->e^+e^- at a future e^+e^- Linear Collider with c.m. energy \sqrt{s}=0.5 TeV and with both beams longitudinally polarized. Our analysis is based on the measurement of the polarized differential cross sections, and allows to simultaneously take into account the general set of contact interaction couplings as independent, non-zero, parameters thus avoiding simplifying, model-dependent, assumptions. We evaluate the corresponding model-independent constraints on the coupling constants, emphasizing the role of beam polarization, and make a comparison with the case of e^+e^- -->\mu^+\mu^-.

Abstract:
The book is devoted to study so-called irregular subsets of the Grassmannian manifold $G^{n}_{k}(V)$ (this class of sets was introduced by author). In the previous variant of the book we restrict ourself only to the case when $V$ is an $n$-dimensional vector space under the field $R$. Now we consider irregular subsets of $G^{n}_{k}(V)$, where $V$ is an $n$-dimensional vector space under arbitrary field.

Abstract:
Let $\Omega$ be a non-singular syplectic form on some vector space $V$. Denote by $S^{n}_{k}(\Omega)$ the set of all $k$-dimensional planes $s$ in $V$ such that the restriction of $\Omega$ onto $s$ is singular. For the cases when $k=2,n-2$ a simple geometric characteristic of $S^{n}_{k}(\Omega)$ will be described.

Abstract:
We discuss the potential of NLC500 and NLC1000 to probe Z-Z' mixing and mass by the reaction e^+e^- -> W^+W^- with longitudinally polarized electrons. We perform a model-independent analysis of the deviations from the Standard Model, and apply it to a specific class of extended weak gauge models. Results indicate that the corresponding bounds at the NLC500 complement the present ones obtained from LEP1, and rapidly become quite stringent at the higher energy of the NLC1000. Also, we emphasize the importance of initial beam polarization in improving the sensitivity to mixing.