Interpolation by a Game

We introduce a notion of a real game (a generalisation of the Karchmer-Wigderson game (cf. [3]) and of real communication complexity, and relate this complexity to the size of monotone real formulas and circuits. We give an exponential lower bound for tree-like monotone protocols (defined in [4, Definition 2.2]) of small real communication complexity solving the monotone communication complexity problem associated with the bipartite perfect matching problem. This work is motivated by a research in interpolation theorems for prepositional logic (by a problem posed in [5, Section 8], in particular). Our main objective is to extend the communication complexity approach of [4, 5] to a wider class of proof systems. In this direction we obtain an effective interpolation in a form of a protocol of small real communication complexity. Together with the above mentioned lower bound for tree-like protocols this yields as a corollary a lower bound on the number of steps for particular semantic derivations of Hall's theorem (these include tree-like cutting planes proofs for which an exponential lower bound was demonstrated in [2]).     

On classification of conformally flat spaces

Classification of conformally flat n-dimensional pseudo-Riemannian spaces via Plebanski's method is discussed. It is based on embedding these spaces into a flat (n + 2)-dimensional space and on finding their minimal isometry groups which are subgroups of the conformal group. In particular, the case n = 4 is given in full detail and compared with incomplete results known in the literature. The found conformally flat spacetimes are identified with the associated solutions of the Einstein equations and with the spacetimes used in various cosmological considerations.     

On a certain two-sided symmetric condition in magnetic field analysis and computations

A special two-sided condition for the incremental magnetic reluctivity is introduced which guarantees the unique existence of both the weak and the approximate solutions of the nonlinear stationary magnetic field distributed on a region composed of different media, as well as a certain estimate of the error between the two solutions. The condition, being discussed from the physical as well as the mathematical point of view, can be easily verified and is fulfilled for various magnetic reluctivity models used in electrotechnical practice.     

Invariant Subspaces for Semigroups of Algebraic Operators

T. Laffey showed (Linear and Multilinear Algebra6(1978), 269–305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of Hthis result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair {A, B} of arbitrary bounded operators satisfying rank (AB−BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable.     

Homomorphic images of finite subdirectly irreducible unary algebras

We prove that a finite unary algebra with at least two operation symbols is a homomorphic image of a (finite) subdirectly irreducible algebra if and only if the intersection of all its subalgebras which have at least two elements is nonempty. While working on this paper the first and the third authors were partially supported by the Grant Agency of the Czech Republic, grant #201/02/0594, and by the institutional grant MSM0021620839; the second author was supported by the Ministry of Science, Technologies and Development of Republic of Serbia, grant no. 1227.     

Scattering and bound states of a nonrelativistic neutral spin-1/2 particle in a magnetic field

We have investigated the scattering and bound states of a nonrelativistic spin-1/2 particle in the system ofN magnetic barriers (or magnetic wells) andN magnetic delta functions. We have studied three types of problems: tunnelling with spin, band structure and change of polarization during transmission and reflection.     

EPR investigation of cell membrane fluidity upon external oxidative stimulus

The perturbation of the physical state of cell membranes triggered by an external oxidative stimulus has been studied with sperm cells which were chosen as a model system. Electron paramagnetic resonance (EPR) spectroscopy was applied and three different nitroxides bearing a paramagnetic group on the 5th, 7th and 13th carbon of the stearic acid acyl chain were used to probe different regions of the membrane. The theoretical simulations of the experimental spectra indicate distinct domains in the intact cell plasma membrane where local steric constraints impose different degrees of motional averaging experienced by the reporter group. Upon the external oxidative stimulus the spectral changes were proportional to the accumulated oxidation products and were detected only for the reporter group residing close to the lipid-water interface. The EPR spectra reveal that the motion of the reporter group has slowed down in the oxidized cells and that the oxidatively modified membrane shows up as a more rigid structure as compared with the native state. The results also indicate that the oxidation-induced spectral changes are supressed in the presence of gangliosides.