Abstract:
The $\eta$ production in the $(n,n')$ bottomonium transitions $\Upsilon (n) \to \Upsilon (n') \eta, $ is studied in the method used before for dipion heavy quarkonia transitions. The widths $\Gamma_\eta(n,n')$ are calculated without fitting parameters for $n=2,3,4,5, n'=1$. Resulting $\Gamma_\eta(4,1)$ is found to be large in agreement with recent data. Multipole expansion method is shown to be inadequate for large size systems considered.

Abstract:
Dipion transitions of $\Upsilon (nS)$ with $n=5, n'=1,2,3$ are studied using the Field Correlator Method, applied previously to dipion transitions with $n=2,3,4$ The only two parameters of effective Lagrangian were fixed in that earlier study, and total widths $\Gamma_{\pi\pi} (5, n')$ as well as pionless decay widths $\Gamma_{BB} (5S), \Gamma_{BB^*} (5S), \Gamma_{B^*B^*}(5S)$ and $\Gamma_{KK} (5, n')$ were calculated and are in a reasonable agreement with experiment. The experimental $\pi\pi$ spectra for $(5,1)$ and (5,2) transitions are well reproduced taking into account FSI in the $\pi\pi$.

Abstract:
Two- and three-body decays of $\Upsilon (5S)$ into $BB, $$ BB^*, $ $ B^*B^*,$ $ B_sB_s, $ $B_sB_s^*, $ $B_s^* B_s^*$ and $BB^*\pi, B^*B^*\pi$ are evaluated using the theory, developed earlier for dipion bottomonium transitions. The theory contains only two parameters, vertex masses $M_{br}$ and $M_\omega$, known from dipion spectra and width. Predicted values of $\Gamma_{tot}(5S)$ and six partial widths $\Gamma_k(5S), k=BB, BB^*,...$ are in agreement with experiment. The decay widths $\Gamma_{5S}(\pi BB^*)$ and $\Gamma_{5S}(\pi B^*B^*)$ are also calculated and found to be of the order of 10 keV.

Abstract:
We investigate numerically lattice Weinberg - Salam model without fermions for realistic values of the fine structure constant and the Weinberg angle. We also analyze the data of the previous numerical investigations of lattice Electroweak theory. We have found that moving along the line of constant physics when the lattice spacing $a$ is decreased, one should leave the physical Higgs phase of the theory at a certain value of $a$. Our estimate of the minimal value of the lattice spacing is $a_c = [430\pm 40 {\rm Gev}]^{-1}$.

Abstract:
Dipion transitions of the subthreshold bottomonium levels $\Upsilon (nS)\to \Upsilon (n'S) \pi\pi$ with $n>n', n=2,3,4, n'=1,2$ are studied in the framework of the chiral decay Lagrangian, derived earlier. The channels $B\bar B, B\bar B^*+ c.c, B^* \bar B^*$ are considered in the intermediate state and realistic wave functions of $\Upsilon (n S),B$ and $B^*$ are used in the overlap matrix elements. Imposing the Adler zero requirement on the transition matrix element, one obtains 2d and 1d dipion spectra in reasonable agreement with experiment.

Abstract:
10 D Euclidean quantum gravity is investigated numerically using the dynamical triangulation approach. It has been found that the behavior of the model is similar to that of the lower dimensional models. However, it turns out that there are a few features that are not present in the lower dimensional models.

Abstract:
The $\eta$ production in the $(n,n')$ bottomonium transitions $\Upsilon (n) \to \Upsilon (n') \eta, $ is studied in the method used before for dipion heavy quarkonia transitions. The widths $\Gamma_\eta(n,n')$ are calculated without fitting parameters for $n=2,3,4,5, n'=1$.Resulting $\Gamma_\eta(4,1)$ is found to be large in agreement with recent data.

Abstract:
Solving numerically the equations of motion for the effective lagrangian describing supersymmetric QCD with the SU(2) gauge group, we find a menagerie of complex domain wall solutions connecting different chirally asymmetric vacua. Some of these solutions are BPS saturated walls; they exist when the mass of the matter fields does not exceed some critical value m < m* < 4.67059... There are also sphaleron branches (saddle points of the ebergy functional). In the range m* < m < m** \approx 4.83, one of these branches becomes a local minimum (which is not a BPS saturated one). At m > m*, the complex walls disappear altogether and only the walls connecting a chirally asymmetric vacuum with the chirally symmetric one survive.

Abstract:
The main objective of this presentation is to point out that the Upper bound on the cutoff in lattice Electroweak theory is still unknown. The consideration of the continuum theory is based on the perturbation expansion around trivial vacuum. The internal structure of the lattice Weinberg - Salam model may appear to be more complicated especially in the region of the phase diagram close to the phase transition between the physical Higgs phase and the unphysical symmetric phase of the lattice model, where the continuum physics is to be approached. We represent the results of our numerical investigation of the quenched model at infinite bare scalar self coupling $\lambda$. These results demonstrate that at $\lambda = \infty$ the upper bound on the cutoff is around $\frac{\pi}{a} = 1.4$ Tev. The preliminary results for finite $\lambda$ are also presented. Basing on these results we cannot yet make a definite conclusion on the maximal value of the cutoff admitted in the lattice model, although we have found that the cutoff cannot exceed the value around $1.4 \pm 0.2$ Tev for a certain particular choice of the couplings ($\lambda = 0.009$, $\beta = 12$, $\theta_W = 30^o$) for the lattices of sizes up to $12^3\times16$. We also observe that the topological defects, which are to be identified with quantum Nambu monopoles, dominate in vacuum in the vicinity of the transition. This indicates that the vacuum of the model is different from the trivial one. In addition we remind the results of the previous numerical investigations of the SU(2) gauge - Higgs model, where the maximal reported value of the cutoff was around 1.5 Tev.

Abstract:
We study the spectrum of the domain walls interpolating between different chirally asymmetric vacua in supersymmetric QCD with the SU(3) gauge group and including 2 pairs of chiral matter multiplets in fundamental and anti-fundamental representations. For small enough masses m < m* = .286... (in the units of \Lambda), there are two different domain wall solutions which are BPS-saturated and two types of ``wallsome sphalerons''. At m = m*, two BPS branches join together and, in the interval m* < m < m** = 3.704..., BPS equations have no solutions but there are solutions to the equations of motion describing a non-BPS domain wall and a sphaleron. For m > m**, there are no solutions whatsoever.