On Computing the Size of a Lattice and Related Decision Problems

The problem of determining the size of a finite concept lattice is shown to be #P-complete. Since any finite lattice can be represented as a concept lattice, the problem of determining the size of a lattice given by the ordered sets of its irreducibles is also #P-complete. Some results about NP-completeness or polynomial tractability of decision problems related to concepts with bounded extent, intent, and the sum of both are given. These problems can be reformulated as decision problems about lattice elements generated by a certain amount of irreducibles.     

On optimization of metal forming with adaptable characteristics

In the present paper we consider a problem of choosing the rational way to carry on the metal processing (the problem of stochastic optimization) and the problem of determing the unknown characteristics of parameters described with random variables.     

Current tensor with heavy photon for double hard photon emission by a longitudinally polarized electron

The electron current tensor for the scattering of a heavy photon on a longitudinally polarized electron with the emission of two hard real photons is considered. The contributions of collinear and semicollinear kinematics are computed. The result allows one to calculate the corresponding contribution to the second-order radiative correction to the deep inelastic scattering or electron-positron annihilation cross sections with next-to-leading-order accuracy. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 12, 845–850 (25 June 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in L²(-p, p)\mathcal{L}^2(-\pi, \pi). Our method can be applied to a wide range of evolution problems given by PT-symmetric operators.     

Graded quasi-Lie algebras

This paper is concerned with a new class of graded algebras naturally uniting within a single framework various deformations of the Witt, Virasoro and other Lie algebras based on twisted and deformed derivations, as well as color Lie algebras and Lie superalgebras. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

On the balancing principle for some problems of Numerical Analysis

We discuss a choice of weight in penalization methods. The motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution. One example of such improvement is a regularization, where a penalization substitutes an ill-posed problem for a well-posed one. In modern numerical methods for PDEs a penalization is used, for example, to enforce a continuity of an approximate solution on non-matching grids. A choice of penalty weight should provide a balance between error components related with convergence and stability, which are usually unknown. In this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components. It is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates. Finally, we successfully apply our strategy for self-regularization of Volterra-type severely ill-posed problems, such as the sideways heat equation, and for the choice of a weight in interior penalty discontinuous approximation on non-matching grids. Numerical experiments on a series of model problems support theoretical results.     

Dendrimers with tetrabenzoporphyrin cores: near infrared phosphors for in vivo oxygen imaging

Pd complexes of tetrabenzoporphyrins (PdTBPs) are strong near infra-red phosphors, suitable for tissue oxygen imaging by phosphorescence quenching. For in vivo use, PdTBP phosphors have to be water-soluble and protected from interactions with biomacromolecules in the blood. An approach to the construction of biologically compatible PdTBP-phosphors based on dendritic encapsulation is proposed. A series of polyglutamic PdTBP-dendrimers was synthesized and their optical and quenching properties were studied. While retaining the excellent optical characteristics (λ_max Q=630 nm, φ_phos=12%) of basic PdTBPs and in addition being highly water soluble, PdTBP-dendrimers still exhibit excessively high oxygen quenching constants (2000-3000 mmHg⁻¹ s⁻¹) and form complexes with lipophilic proteins, such as serum albumin. Modification of the dendrimer periphery with polyethyleneglycol chains somewhat reduces the quenching constants but does not prevent binding of the phosphors to albumin.