Productivity of Coreflective Subcategories of Semitopological Groups

The present paper generalizes to semitopological and quasitopological groups some results achieved by Horst Herrlich and the second author for topological groups. The results concern preserving products in coreflective subcategories. Unlike in paratopological or topological groups, there are non-finitely productive bicoreflective subcategories of quasitopological groups. We desctribe bicoreflective subcategories of semitopological groups that are either finitely productive or productive or their productivity number is submeasurable. To achieve the results, several general factorization theorems for maps on products will be proved with the help of modification of NSS property of groups.     

Higher Laplace–Beltrami Operators on Bounded Symmetric Domains

It was conjectured by the first author and Peetre that the higher Laplace–Beltrami operators generate the whole ring of invariant operators on bounded symmetric domains. We give a proof of the conjecture for domains of rank ≤ 6 by using a graph manipulation of K¨ahler curvature tensor. We also compute higher order terms in the asymptotic expansions of the Bergman kernels and the Berezin transform on bounded symmetric domain.     

Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata A and B is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on A and B and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata A and B are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of A and B with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.     

Thermal noise computation in gravitational wave interferometers from first principles

We propose a universal method of computation of thermal noise in mirrors of gravitational wave interferometers based on first principles. We imagine a situation where a mirror is part of a Fabry–Perot cavity. The movement of the mirror's surface produces variation of the eigen frequency of the cavity, which is computed by evaluating the variation of the energy stored in cavity. We consider two particular examples: first, the thermal noise from a dielectric slab inside the Fabry–Perot cavity, and second, the polarization-dependent thermal noise in the folded cavity.     

The importance of structure in incomplete factorization preconditioners

In this paper, we consider level-based preconditioning, which is one of the basic approaches to incomplete factorization preconditioning of iterative methods. It is well-known that while structure-based preconditioners can be very useful, excessive memory demands can limit their usefulness. Here we present an improved strategy that considers the individual entries of the system matrix and restricts small entries to contributing to fewer levels of fill than the largest entries. Using symmetric positive-definite problems arising from a wide range of practical applications, we show that the use of variable levels of fill can yield incomplete Cholesky factorization preconditioners that are more efficient than those resulting from the standard level-based approach. The concept of level-based preconditioning, which is based on the structural properties of the system matrix, is then transferred to the numerical incomplete decomposition. In particular, the structure of the incomplete factorization determined in the symbolic factorization phase is explicitly used in the numerical factorization phase. Further numerical results demonstrate that our level-based approach can lead to much sparser but efficient incomplete factorization preconditioners.     

Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes

In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results for the unknown quadratic payoffs to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

     

Bound Motion of Bodies and Paticles in the Rotating Systems

The Lagrange theory of particle motion in the noninertial systems is applied to the Foucault pendulum, isosceles triangle pendulum and the general triangle pendulum swinging on the rotating Earth. As an analogue, planet orbiting in the rotating galaxy is considered as the giant galactic gyroscope. The Lorentz equation and the Bargmann-Michel-Telegdi equations are generalized for the rotation system. The knowledge of these equations is inevitable for the construction of LHC where each orbital proton “feels” the Coriolis force caused by the rotation of the Earth.     

Theoretical study of molecular properties of low-lying electronic excited states of H2O and H2S

Geometries, excitation energies, dipole moments and dipole polarisability tensor components of the ground and four lowest excited states ³ B ₁, ¹ B ₁, ³ A ₂, ¹ A ₂ of the H₂O and H₂S molecules were calculated at the CASSCF, CASPT2, CCSD and CCSD(T) level of approximation. Vertical excitation and equilibrium transition energies of these states, having the Rydberg character, are reported too. Properties of both molecules in the ground and in low lying excited states are compared and discussed from the point of view of their molecular electronic structure. Upon excitation we observe dramatic changes of dipole moments and polarisabilities with respect to the ground state. We stress the change of the polarity of H₂O in all excited states accompanied by the enhancement of the dipole polarisability by an order of magnitude. Large, even if less pronounced, are changes of electric properties of H₂S in its excited states. Dipole moments and dipole polarisabilities of ³ B ₁, ¹ B ₁ states of H₂S and H₂O behave quite analogously in comparison to their respective ground state. The general pattern of properties for both molecules in their ³ A ₂ and ¹ A ₂ excited states is more different due to a pronounced participation of the sulphur d-orbitals in these states of the H₂S molecule.     

DLC coating of textile blood vessels using PLD

Textile blood vessels with a length of 30 cm were coated with amorphous diamond-like carbon (DLC) layers with thicknesses up to 200 nm. The layers were created by pulsed laser deposition in vacuum or argon ambient. The percentage of sp³ carbon was evaluated using X-ray photoelectron spectroscopy, X-ray excited Auger electron spectroscopy and Raman spectroscopy. Depending on the deposition conditions the sp³ content varied from ∼40% to 60%. The adhesion of the DLC layers to the textile vessels was checked. The preliminary biocompatibility results from in vivo tests with sheep are also given.