For quantum systems exhibiting transition from chaotic properties to integrable ones while the control parameter is varying the order parameter is introduced. The proposed concept is based on division of the random matrix ensemble into equivalence classes of equal matrix ranks. The order parameter is proportional to the number of equivalence classes and is expressed by the jump of the cumulative distribution function at energy \(E=0\).

We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure equations and their symmetries in terms of a new product (the co-multiplication). It is shown how the Cartan–Grassmann algebra can be endowed with the structure of a Hopf algebra.

The operators \(a\) and \(a^+\) realizing the Heisenberg algebra are used to solve the spectral problems with potentials different from the harmonic oscillator. By reducing the stationary Schrödinger equation to the form of the Whittaker equation (or hypergeometric equation), and next by comparison with the \(a^+a\) product a family of the solvable potentials is obtained. The solutions of the pseudo-eigenvalue problem \((H(E_n)\psi _n=E_n\psi _n)\) with the new potentials are given. It is also shown how to construct such pseudoeigenvalue equations with eigenfunctions related to the Whittaker function. The results are used to build up Hamiltonians which are expressible by the \(a^+a\) product.

The Hilbert space of tensor functions on a homogeneous space with the compact stability group is considered. The functions are decomposed onto a sum of tensor plane waves (defined in the text), components of which are transformed by irreducible representations of the appropriate transformation group. The orthogonality relation and the completeness relation for tensor plane waves are found. The decomposition constitutes a unitary transformation, which allows to obtain the Parseval equality. The Fourier components can be calculated by means of the Fourier transformation, the form of which is given explicitly.

We show that it is possible to search for the Standard Model Higgs bosons in the \(H \to ZZ \to 4l^{\pm }\) channel at LHC using \(Z^0\) polarization. This will strengthen the conclusivity of detection by the primary method based on the reconstruction of \(4l^{\pm }\) invariant mass and provide information about Higgs boson polarization.

A model of three quarks at the corners of a Lagrange equilateral triangle is used to describe a nucleon. Dipole fits to the electromagnetic form factors are used to require an exponential radial dependence for the wave function. This guide from experiment leads to a linear confining potential using the Dirac equation to describe the dynamics of the model. This model can reproduce the proton magnetic moment, axial charge, Roper resonance energy, and the size of the dipole fits to the electromagnetic form factors, but not all simultaneously. The linear confining potential does not appear in a Schroedinger like second order differential equation involving only the large component of the Dirac equation for the proton ground state.

Effects of final-state interactions in nonleptonic decays of charmed mesons are studied in the framework of quark-diagram approach. For the case of \(u\)-\(d\)-\(s\) flavour symmetry we discuss how the inelastic coupled-channel rescattering effects (and, in particular, resonance formation in the final state) modify the input quark-diagram weak amplitudes. It is shown that such inelastic effects lead to the appearance of nonzero relative phases between various quark diagrams, thus invalidating some of the conclusions drawn in the past within the diagrammatical approach. Applicability of quark-diagram approach to the case of SU(3) symmetry-breaking in Cabibbo once- suppressed \(D^0\) decays is also studied in some detail.

General formula for inclusive gluon distribution on rapidities is obtained for the processes of multigluon production in classical instanton field with the first quantum correction. On the basis of this formula second correlation function is calculated in QCD and analyzed. The features of the correlation function behaviour can be used as a signal of instanton at HERA.

The isomorphism Hopf *-algebras between \(\kappa \)-Poincaré algebra in case \(g_{00}=0\) defined by P. Kosiński and P. Maślanka in The \(\kappa \)-Weyl Group and Its Algebra in “From Field Theory to Quantum Groups” volume on 60\(^{{\rm th}}\) anniversary of J. Lukierski, World Scientific, Singapore 1996 and “null plane” quantum Poincaré algebra by A. Ballesteros, F.J. Herranz and M.A. del Olmo “Null Plane” Quantum Poincaré Algebra, Phys. Lett.B351, 137 (1995) are defined.

A model of texture dynamics, initially constructed for charged leptons, is now described in some detail for up and down quarks of three families. It nicely correlates and reproduces all six quark masses and four mixing parameters in terms of nine constants which display some simple relations. These, if assumed, may reduce the number of free constants (and so enhance the number of predictions). Possible extensions of the model to neutrinos are briefly discussed.

C.P. Bee, D. Conti, M. Hadri, ST. Kistryn, J. Lang, O. Naviliat-Cuncic, J. Sromicki, F. Stephan, N. Bodek, J. Smyrski, A. Strzałkowski, J. Zejma, L. Grenacs, R. Abela, P. Böni, F. Foroughi, W. Zipper, A. Proykova

A polarimetry technique based on stack targets and \(\beta \)-\(\gamma \)-coincidences has been applied to the \(^{16}\)N nuclei produced in the ground state capture of negative muons on \(^{16}\)O nuclei. The performance of the polarimeter and the first measurements of \(\beta \)-asymmetry due to the longitudinal nuclear polarization are discussed.

A new, more efficient approach to include a three nucleon force into three-nucleon continuum calculations is presented. Results obtained in the new and our old approach are compared both for elastic nucleon-deuteron scattering as well as for the breakup process. The advantages of the new scheme are discussed.

A new version of the GEMINI code, including time scales and calculation of Coulomb trajectories of sequentially emitted particles, is presented. Energy spectra of charged particles and reduced velocity correlations have been calculated using the no-time-scale, and the time-scale version of the GEMINI code, to elucidate importance of the sequential decay dynamics. The reduced velocity correlations predicted by the time-scale version of the GEMINI code are in good agreement with experimental data.

Using replica approach we investigate storage capacity for “spatially” correlated patterns in diluted attractor neural networks. We investigate analog clipped-sign networks which are generalizations of the standard networks of the Hopfield type. We consider two kinds of dilution: a band, and a random (periodic) ones. “Spatial” associations of data significantly improve the storage properties of the network. The band-type dilution affects the critical capacity much weaker than the random one, especially when the stored data are strongly correlated.

Using statistical physics methods we investigate two-layered perceptrons which consist of \(N\) binary input neurons, \(K\) hidden units and a single output node. Four basic types of such networks are considered: the so-called Committee, Parity, and AND Machines which make a decision based on a majority, parity, and the logical AND rules, respectively (for these cases the weights that connect hidden units and output node are taken to be equal to one), and the General Machine where one allows all the synaptic couplings to vary. For these kinds of network we examine two types of architecture: fully connected and tree-connected ones (with overlapping and non-overlapping receptive fields, respectively). All the above mentioned machines have binary weights. Our basic interest is focused on the storage capabilities of such networks which realize \(p=\alpha N\) random, unbiased dichotomies (\(\alpha \) denotes the so-called storage ratio). The analysis is done using the annealed approximation and is valid for all values of \(K\). The critical (maximal) storage capacity of the fully connected Committee Machine reads \(\alpha _c=K\), while in the case of the tree-structure one gets \(\alpha _c=1\), independently of \(K\). The results obtained for the Parity Machine are exactly the same as those for the Committee network. The optimal storage of the AND Machine depends on the distribution of the outputs for the patterns. These associations are studied in detail. We have found also that the capacity of the General Machines remains the same as compared to systems with fixed weights between intermediate layer and the output node. Some of the findings (especially these concerning the storage capacity of the Parity Machine) are in a good agreement with known numerical results.