Diassociative algebras and Milnor's invariants for tangles

We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.Comment: 17 pages, many figures; v2: several typos correcte

Finite Type Invariants of Classical and Virtual Knots

We observe that any knot invariant extends to virtual knots. The isotopy classification problem for virtual knots is reduced to an algebraic problem formulated in terms of an algebra of arrow diagrams. We introduce a new notion of finite type invariant and show that the restriction of any such invariant of degree n to classical knots is an invariant of degree at most n in the classical sense. A universal invariant of degree at most n is defined via a Gauss diagram formula. This machinery is used to obtain explicit formulas for invariants of low degrees. The same technique is also used to prove that any finite type invariant of classical knots is given by a Gauss diagram formula. We introduce the notion of n-equivalence of Gauss diagrams and announce virtual counter-parts of results concerning classical n-equivalence.Comment: 22 pages, many figure

On homotopies with triple points of classical knots

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point $p$ of the cylinder is called {\em coherent} if all three branches intersect at $p$ pairwise with the same index. A {\em triple unknotting} of a classical knot $K$ is a homotopy which connects $K$ with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant $v_2(K)$ by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the homotopy.Comment: 10 pages, 13 figures, bugs in figures correcte

Quantum Mechanical/Molecular Mechanical Study on the Enantioselectivity of the Enzymatic Baeyer–Villiger Reaction of 4-Hydroxycyclohexanone

We report a combined quantum mechanical/molecular mechanical (QM/MM) study of the effect of mutations of the Phe434 residue in the active site of cyclohexanone monooxygenase (CHMO) on its enantioselectivity toward 4-hydroxycyclohexanone. In terms of our previously established model of the enzymatic Baeyer–Villiger reaction, enantioselectivity is governed by the preference toward the equatorial ((S)-selectivity) or axial ((R)-selectivity) conformation of the substituent at the C4 carbon atom of the cyclohexanone ring in the Criegee intermediate and the subsequent rate-limiting transition state for migration (TS2). We assess the enantiopreference by locating all relevant TS2 structures at the QM/MM level. In the wild-type enzyme we find that the axial conformation is energetically slightly more stable, thus leading to a small excess of (R)-product. In the Phe434Ser mutant, there is a hydrogen bond between the serine side chain and the equatorial substrate hydroxyl group that is retained during the whole reaction, and hence there is pronounced reverse (S)-enantioselectivity. Another mutant, Phe434Ile, is shown to preserve and enhance the (R)-selectivity. All these findings are in accordance with experiment. The QM/MM calculations allow us to explain the effect of point mutations on CHMO enantioselectivity for the first time at the molecular level by an analysis of the specific interactions between substrate and active-site environment in the TS2 structures that satisfy the basic stereoelectronic requirement of anti-periplanarity for the migrating σ-bond

Charge migration engineered by localisation: electron-nuclear dynamics in polyenes and glycine

We demonstrate that charge migration can be ‘engineered’ in arbitrary molecular systems if a single localised orbital – that diabatically follows nuclear displacements – is ionised. Specifically, we describe the use of natural bonding orbitals in Complete Active Space Configuration Interaction (CASCI) calculations to form cationic states with localised charge, providing consistently well-defined initial conditions across a zero point energy vibrational ensemble of molecular geometries. In Ehrenfest dynamics simulations following localised ionisation of -electrons in model polyenes (hexatriene and decapentaene) and -electrons in glycine, oscillatory charge migration can be observed for several femtoseconds before dephasing. Including nuclear motion leads to slower dephasing compared to fixed-geometry electron-only dynamics results. For future work, we discuss the possibility of designing laser pulses that would lead to charge migration that is experimentally observable, based on the proposed diabatic orbital approach

A New Type of Stereoselectivity in Baeyer–Villiger Reactions: Access to E- and Z-Olefins

A new concept for accessing configurationally defined trisubstituted olefins has been developed. Starting from a common ketone precursor of the type 4-ethylidenecyclohexanone, Baeyer–Villiger monooxygenases are employed as catalysts in diastereoselective Baeyer–Villiger reactions leading to the corresponding E- or Z-configurated lactones. Wild-type cyclohexanone monooxygenase (CHMO) as catalyst delivers the E-isomers and a directed evolution mutant the opposite Z-isomers. Subsequent transition metal-catalyzed chemical transformations of a key product containing a vinyl bromide moiety provide a variety of different trisubstituted E- or Z-olefins. A model based on QM/MM sheds light on the origin of this unusual type of diastereoselectivity. In contrast to this biocatalytic approach, traditional Baeyer–Villiger reagents such as m-CPBA fail to show any selectivity, 1:1 mixtures of E- and Z-olefins being formed

ОСОБЛИВОСТІ БІОХІМІЧНИХ ТА ІМУНОЛОГІЧНИХ МАРКЕРІВ У ДІТЕЙ З КОРОВОЮ ІНФЕКЦІЄЮ ТА В АСОЦІАЦІЇ З ГЛИСТЯНОЮ ІНВАЗІЄЮ

SUMMARY. This article presented to the extremely frequency of measles with helminthiasis in the children. We investigated the biochemical and immunological markers of Measles with helminthiasis in the children and complication of the dutarion of disease. Key words: measles, helminthiasis, biochemical investigation, children.Звертається увага на поширені патології дитячого віку – кір та глистяну інвазію. Було вивчено особливості біохімічних та імунологічних маркерів у дітей, хворих на кір та в асоціації з глистяною інвазією, яка ускладнює перебіг основного захворювання – кору. Ключові слова: кір, глистяна інвазія, біохімічне дослідження, діти

Graph complexes in deformation quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phy

Electron and nuclear dynamics following ionisation of modified bismethylene-adamantane

We have simulated the coupled electron and nuclear dynamics using the Ehrenfest method upon valence ionisation of modified bismethylene-adamantane (BMA) molecules where there is an electron transfer between the two π bonds. We have shown that the nuclear motion significantly affects the electron dynamics after a few fs when the electronic states involved are close in energy. We have also demonstrated how the non-stationary electronic wave packet determines the nuclear motion, more precisely the asymmetric stretching of the two π bonds, illustrating “charge-directed reactivity”. Taking into account the nuclear wave packet width results in the dephasing of electron dynamics with a half-life of 8 fs; this eventually leads to the equal delocalisation of the hole density over the two methylene groups and thus symmetric bond lengths