Narrow Results

We investigate the descriptional complexity of the ${\nu }_i$- and ${\alpha }_i$-products with $0\le i\le 2$ of two automata, for reset, permutation, permutation-reset, and finite automata in general. This is a continuation of the recent studies on the state complexity of the well-known cascade product undertaken in [7, 8]. Here we show that in almost all cases, except for the direct product (${\nu }_0$) and the cascade product (${\alpha }_0$) for certain types of automata operands, the whole range of state complexities, namely the interval $[1,nm]$, where $n$ is the state complexity of the left operand and $m$ that of the right one, is attainable. To this end we prove a simulation result on products of automata that allows us to reduce the products of automata in question to the ${\nu }_0$, ${\alpha }_0$, and a double sided ${\alpha }_0$-product.

We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has α states, for all n and α satisfying n ≤ α ≤ 2ⁿ. A number α not satisfying this condition is called a magic number (for n). It was shown that no magic numbers exist for general regular languages, whereas trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, star-free languages, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages, showing that there are only trivial magic numbers, when they exist. For finite languages we obtain some partial results showing that certain numbers are non-magic.

We investigate the state complexity of languages resulting from the cascade product of two minimal deterministic finite automata with $n$ and $m$ states, respectively. More precisely we study the magic number problem of the cascade product operation and show what range of complexities can be produced in case the left automaton is unary, that is, has only a singleton letter alphabet. Here we distinguish the cases when the involved automata are reset automata, permutation automata, permutation-reset automata, or do not have any restriction on their structure. It turns out that the picture on the obtained state complexities of the cascade product is diverse, and for all cases, except where the left automaton is a unary permutation(-reset) or a deterministic finite automaton without structural restrictions, and the right one is a reset automaton or a deterministic finite automaton without structural restrictions, we are able to identify state sizes that cannot be reached — these numbers are called “magic.”

The significant features of exchange symmetry are displayed by simple systems such as two identical, spinless fermions in a one-dimensional well with infinite walls. The conclusion is that the maxima of probability of the antisymmetrized wave function of these two fermions lie at the same positions as if a repulsive force (of unknown nature) was applied between these two fermions. This conclusion is combined with the solution of a mathematical problem dealing with the equilibrium of identical repulsive particles (of one or two kinds) on one or more spheres like neutrons and protons on nuclear shells. Such particles are at equilibrium only for specific numbers of particles and, in addition, if these particles lie on the vertices of regular polyhedra or their derivative polyhedra. Finally, this result leads to a pictorial representation of the structure of all closed shell nuclei. This representation could be used as a laboratory for determining nuclear properties and corresponding wave functions.

The system of Kohn–Sham equations is solved self-consistently for the two-dimensional, spatially separated electrons and holes. We find the series of magic numbers for the total angular momentum of the electrons and holes in a strong magnetic field. The change of the angular momentum of the charge carriers is shown to lead to the persistent current oscillations.

We present a new theoretical framework for modeling the fusion process of Lennard–Jones (LJ) clusters. Starting from the initial tetrahedral cluster configuration, adding new atoms to the system and absorbing its energy at each step, we find cluster growing paths up to the cluster size of 150 atoms. We demonstrate that in this way all known global minima structures of the LJ-clusters can be found. Our method provides an efficient tool for the calculation and analysis of atomic cluster structure. With its use we justify the magic number sequence for the clusters of noble gas atoms and compare it with experimental observations. We report the striking correspondence of the peaks in the dependence of the second derivative of the binding energy per atom on cluster size calculated for the chain of the LJ-clusters based on the icosahedral symmetry with the peaks in the abundance mass spectra experimentally measured for the clusters of noble gas atoms. Our method serves as an efficient alternative to the global optimization techniques based on the Monte-Carlo simulations and it can be applied for the solutions of a broad variety of problems in which atomic cluster structure is important.

A new systematics from the separation energy of deuteron is used to examine the magicity of stable as well as nuclei towards dripline.

The shell and pairing correction energies are calculated for heavy and superheavy nuclei (SHN) by means of the Strutinsky's method. The single-particle (s.p.) energy levels are obtained from the diagonalization of the Woods–Saxon s.p. Hamiltonian in the deformed harmonic oscillator basis for both neutrons and protons. The residual pairing interaction is calculated by means of the usual Bardeen–Cooper–Schrieffer (BCS) approximation. A two-dimensional deformation space describing axially and reflection-symmetric shapes of nuclei has been used. Based on the shell and pairing correction energies, the signatures of the magic numbers appear at the spherical shell closures Z = 82, 114, 164 and N = 126, 184, 228 and 308. There are also signatures for some other shell closures at, e.g., Z = 108 and N = 162 which appear only when the deformation degrees of freedom is taken into account.

Nuclear level densities of ${}^{207}$Pb and ${}^{89}$Y are calculated using the Lipkin–Nogami (LN) method and Bradeen–Cooper–Schrieffer (BCS) model. It is revealed that the calculated nuclear level densities are highly matched with the experimental data of Oslo group. The excitation energy and entropy are calculated for mentioned nuclei. In the case of two studied nuclei the characteristic of being magic for the number of neutrons or protons causes the decrease of the excitation energy and entropy contribution of magic system at low temperatures.

We formalized the nuclear mass problem in the inverse problem framework. This approach allows us to infer the underlying model parameters from experimental observation, rather than to predict the observations from the model parameters. The inverse problem was formulated for the numerically generalized semi-empirical mass formula of Bethe and von Weizsäcker. It was solved in a step-by-step way based on the AME2012 nuclear database. The established parametrization describes the measured nuclear masses of 2564 isotopes with a maximum deviation less than 2.6MeV, starting from the number of protons and number of neutrons equal to 1.

The explicit form of unknown functions in the generalized mass formula was discovered in a step-by-step way using the modified least ${\chi }^2$ procedure, that realized in the algorithms which were developed by Lubomir Aleksandrov to solve the nonlinear systems of equations via the Gauss–Newton method, lets us to choose the better one between two functions with same ${\chi }^2$. In the obtained generalized model, the corrections to the binding energy depend on nine proton (2, 8, 14, 20, 28, 50, 82, 108, 124) and ten neutron (2, 8, 14, 20, 28, 50, 82, 124, 152, 202) magic numbers as well on the asymptotic boundaries of their influence. The obtained results were compared with the predictions of other models.

In nuclear theory, there is always a quest for possible extensions of the nuclear landscape and extending our knowledge to the limits of nuclear existence. In this study, we examine the stability and structural properties of a wide range of nuclei in super- and ultra-heavy region in a phenomenological semi-microscopic approach. we calculated the shell correlation energy, residual pairing correction energy, two-nucleon separation energy and two-nucleon energy gap for 3670 even–even nuclei along $\beta$-stability line and two-neutron driplines in the ranges $70\le Z\le 274$ with $80\le N\le 548$ and $70\le Z\le 212$ with $126\le N\le 548$, respectively. To assure reliability and confidence of the new results in the ultra-heavy region, we extended the search space to include heavy and super-heavy nuclei. We report 83 double magic nuclei and address the predominance of proton and neutron magic numbers. Our calculations reproduced known results on nuclear magicity and present strong evidences on islands of stability and magic numbers in super- and ultra-heavy regions. We also address shifts in nuclear magicity along the nuclear landscape close to the $\beta$-stability line and close to the neutron rich regions.

The significant features of exchange symmetry are displayed by simple systems such as two identical, spinless fermions in a one-dimensional well with infinite walls. The conclusion is that the maxima of probability of the antisymmetrized wave function of these two fermions lie at the same positions as if a repulsive force (of unknown nature) was applied between these two fermions. This conclusion is combined with the solution of a mathematical problem dealing with the equilibrium of identical repulsive particles (of one or two kinds) on one or more spheres like neutrons and protons on nuclear shells. Such particles are at equilibrium only for specific numbers of particles and, in addition, if these particles lie on the vertices of regular polyhedra or their derivative polyhedra. Finally, this result leads to a pictorial representation of the structure of all closed shell nuclei. This representation could be used as a laboratory for determining nuclear properties and corresponding wave functions.

We formalized the nuclear mass problem in the inverse problem framework. This approach allows us to infer the underlying model parameters from experimental observation, rather than to predict the observations from the model parameters. The inverse problem was formulated for the numerically generalized the semi-empirical mass formula of Bethe and von Weizsäcker. It was solved in step by step way based on the AME2012 nuclear database.

The solution of the over determined system of nonlinear equations has been obtained with the help of the Aleksandrov's auto-regularization method of Gauss-Newton type for ill-posed problems. In the obtained generalized model the corrections to the binding energy depend on nine proton (2, 8, 14, 20, 28, 50, 82, 108, 124) and ten neutron (2, 8, 14, 20, 28, 50, 82, 124, 152, 202) magic numbers as well on the asymptotic boundaries of their influence. These results help us to evaluate the borders of the nuclear landscape and show their limit. The efficiency of the applied approach was checked by comparing relevant results with the results obtained independently.

It is emerging from data that excited 0⁺ states in nuclei are critical signatures of structure. Some examples of excited 0⁺ states that are reasonably well understood are followed by details, from recent experimental work, of excited 0⁺ states that are causing major reassessment of some models.