Aggregation/Disaggregation Method for Safety Models

The paper concerns the possibilities for mathematical modelling of safety related systems (equipment oriented on safety). Some mathematical models have been required by the present European Standards for the railway transport. We are interested in the possibility of using Markov's models to meet these Standards. In the text an example of using that method in the interlocking equipment life cycle is given. An efficient aggregation/disaggregation method for computing some characteristics of Markov chains is presented.     

Contractible subgraphs, Thomassen’s conjecture and the dominating cycle conjecture for snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is Hamiltonian), by Thomassen (every 4-connected line graph is Hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent to the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle.We use a refinement of the contractibility technique which was introduced by Ryjá?ek and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjá?ek in 1997.     

Tight β-turns in peptides. DFT-based study of infrared absorption and vibrational circular dichroism for various conformers including solvent effects

Vibrational circular dichroism (VCD) has had a large impact on configurational studies of organic molecules largely due to the theoretical advances made by Philip Stephens and co-workers. For peptides, the structural issue is not one of the configuration, but of conformation, and the flexibility of the oligomeric structure raises major computational challenges. Turns are a vital aspect of peptide and protein conformation that allow such structures to fold into a compact unit. However, unlike helices and sheets, they are not extended repeating structures, but each residue has a different local conformation. Also, when turns are part of larger peptides their termini are connected to completely different structural elements. We have done extended comparative density functional theory (DFT) computations to characterize the expected spectral contributions of selected turn structures to the amide IR and VCD spectra of peptides. The isolated vacuum results for tri-amide turns (Ac-X-Y-NH₂) of a few different sequences are compared with calculations involving correction for solvation effects. In particular, we looked at the sequence variation in spectra and structure between Ala-Ala, Aib-Gly and D-Pro-Gly for the turn-specific X–Y residues. The nature of some turn-associated, amide originating, spectral transitions are developed and tested.     

Bottlebrush‐shaped copolymers with cellulose diacetate backbone by a combination of ring opening polymerization and ATRP

Graft copolymers with cellulose diacetate (CDA) backbone and both the poly(ε‐caprolactone) and polystyrene, or poly(butyl acrylate) or PMMA grafts were prepared by two‐step process. First, ε‐caprolactone (CL) was polymerized by ring‐opening polymerization (ROP) initiated with CDA, partly funcionalized with 2‐bromo‐isobutyryl groups (degree of functionalization was 0.5). The p(CDA‐g‐CL) copolymers were used in the second step as polyfunctional macroinitiators of ATRP of the vinyl monomer, giving densely grafted copolymers with polyester and PSt, or PBuA, or PMMA grafts. The prepared copolymers were characterized by SEC, some of them also by FTIR spectroscopy and atomic force microscopy (AFM). © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 564–573, 2008     

Two extensions of asymptotic methods in controlled Markov chains

The application of the Skorokhod representation of martingales and of the local asymptotic normality to derive limit inequalities for the cost in controlled finite state Markov chains is reviewed. The inequalities are usable in self-optimizing control. The methods are taken from the references listed but, with the exception of proposition 4, the results are formulated for Markov chains for the first time.     

Quantum codes from nearly self-orthogonal quaternary linear codes

Construction X and its variants are known from the theory of classical error control codes. We present instances of these constructions that produce binary stabilizer quantum error control codes from arbitrary quaternary linear codes. Our construction does not require the classical linear code \(C\) that is used as the ingredient to satisfy the dual containment condition, or, equivalently, \(C^{\perp _h}\) is not required to satisfy the self-orthogonality condition. We prove lower bounds on the minimum distance of quantum codes obtained from our construction. We give examples of record breaking quantum codes produced from our construction. In these examples, the ingredient code \(C\) is nearly dual containing, or, equivalently, \(C^{\perp _h}\) is nearly self-orthogonal, by which we mean that \(\dim (C^{\perp _h})-\dim (C^{\perp _h}\cap C)\) is positive but small.     

On Weighted Poincaré Inequalities

We are concerned with the problem of finding sharp summability conditions on the weights which render certain weighted inequalities of Poincaré - type true. The conditions we find consist of proper integral balances between the growths of the rearrangements of the weights.     

Loops and the Lagrange Property

Let $\cal F$ be a family of finite loops closed under subloops and factor loops. Then every loop in $\cal F$ has the strong Lagrange property if and only if every simple loop in $\cal F$ has the weak Lagrange property. We exhibit several such families, and indicate how the Lagrange property enters into the problem of existence of finite simple loops.