Preserving some numerical correspondencesbetween Hilbert space effects

In this paper we introduce some natural numerical quantities which measure differentkinds of correspondences between Hilbert space effects. We show that every objective map of the space ?(H) of all effects on the Hilbert space H which preserves any of these quantities is implemented by a unitary or antiunitary operator on H.     

Methods to determine activation energies for the two kinetic states of the oscillatory Bray-Liebhafsky reaction

Six different methods are suggested to calculate the apparent activation energies of the overall decomposition of hydrogen peroxide, both by oxidative and reductive pathways, in the Bray-Liebhafsky oscillatory chemical system. The activation energies of the reduction pathway lie between 31 and 35 kJ/mol and those of the oxidation one between 75 and 78 kJ/mol.     

Noncommutative version of Nikodym boundedness theorem for uniform space-valued functions

A Nikodym boundedness-type theorem with necessary and sufficient conditions for a family of functions defined on a ()-difference-poset and with values in a uniform space is proved. For a special important case — orthomodular lattice-the conditions are relaxed.     

The 1,4-cyclohexanedione-bromate-acid oscillatory system I. Its organic chemistry

The organic chemistry of the 1,4-cyclohexanedione (CHD)-bromate-sulfuric acid oscillatory system has been revealed by following the reaction of 1,4-CHD with bromate using a GC/MS technique. We could identify 1,4-dihydroxybenzene as an intermediate, 1,4-benzoquinone as the main oxidation, and mono- and dibromocyclohexanedione as the main bromination products. Acid bromate does not cleave the alicyclic ring.     

Protein Alpha Shape Similarity Analysis (PASSA): a new method for mapping protein binding sites. Application in the design of a selective inhibitor of tyrosine kinase 2

We have developed a method that we have called Protein Alpha Shape Similarity Analysis (PASSA), that identifies interaction sites that can be utilised to achieve selectivity towards a protein. We have shown that this method is able to identify residues of tyrosine kinases that interact with known selective inhibitors using the following test cases: Abelson (Abl) kinase in complex with STI-571 and Janus kinase 2 (Jak2) in complex with AG-490. The 3D structures of the tyrosine kinase domains of Tyrosine kinase 2 (Tyk2) and Jak2 have been predicted by homology modelling. Computational docking of AG-490 and a set of tyrphostins known not to inhibit Jak2 indicated that our homology models are able to separate inhibitors from non-inhibitors. PASSA has also been used to identify unique properties of Tyk2. According to our results, interactions with hydrogen acceptors and donors on the following residues can be utilised to achieve selectivity towards Tyk2: Y955, E1053, D1062 and S1063. These residues are placed close to non-conserved hydrophobic pockets. The PASSA results, together with results from Multiple Copy Simultaneous Search (MCSS) were used to suggest functional groups of a selective Tyk2 inhibitor.     

Synthesis of Photoaffinity Labelling Analogues of the Peptide Hormone Bradykinin

Fourteen peptides, analogues of bradykinin and (des-Arg⁹)bradykinin, have been synthesized by the solid-phase method or have been obtained through chemical modification. In all these peptides an aromatic residue has been substituted by (4′-NO₂)Phe, (4′-NH₂)Phe, (4′-N₃)Phe or (4′-NH₂-3′, 5′-I₂)Phe. These peptides will be used as photoaffinity labelling or affinity labelling probes for peptide hormone receptors.     

Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations

A recurring theme in attempts to break the curse of dimensionality in the numerical approximation of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class \(\Sigma _n\) of functions, which can be written as a sum of rank-one tensors using a total of at most \(n\) parameters, and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side \(f\) of the elliptic PDE can be approximated with a certain rate \(\mathcal {O}(n^{-r})\) in the norm of \({\mathrm H}^{-1}\) by elements of \(\Sigma _n\), then the solution \(u\) can be approximated in \({\mathrm H}^1\) from \(\Sigma _n\) to accuracy \(\mathcal {O}(n^{-r'})\) for any \(r'\in (0,r)\). Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second “basis-free” model of tensor-sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent to which such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.     

Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows

Endre Süli ^{We develop a posteriori upper and lower error bounds for mixedfinite-element approximations of a general family of steady,viscous, incompressible quasi-Newtonian flows in a bounded Lipschitzdomain ; thefamily includes degenerate models such as the power law model,as well as non-degenerate ones such as the Carreau model. Theunified theoretical framework developed herein yields residual-baseda posteriori bounds which measure the error in the approximationof the velocity in the W1, r() norm and that of the pressurein the Lr'() norm, 1/r + 1/r' = 1, r (1, ).}     

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

^** Email: Paul.Houston{at}mcs.le.ac.uk^*** Email: Janice.Robson{at}comlab.ox.ac.uk^**** Email: Endre.Suli{at}comlab.ox.ac.uk We develop a one-parameter family of hp-version discontinuousGalerkin finite element methods, parameterised by [–1,1], for the numerical solution of quasilinear elliptic equationsin divergence form on a bounded open set ^d, d 2. In particular,we consider the analysis of the family for the equation –·{µ(x, |u|)u} = f(x) subject to mixed Dirichlet–Neumannboundary conditions on . It is assumed that µ is a real-valuedfunction, µ C( x [0, )), and thereexist positive constants m_µ and M_µ such that m_µ(t– s) µ(x, t)t – µ(x, s)s M_µ(t– s) for t s 0 and all x . Using a result from the theory of monotone operators for any valueof [–1, 1], the corresponding method is shown to havea unique solution u_DG in the finite element space. If u C¹() H^k(), k 2, then with discontinuous piecewise polynomials ofdegree p 1, the error between u and u_DG, measured in the brokenH¹()-norm, is (h^s–1/p^k–3/2), where 1 s min {p+ 1, k}.