2,710 research outputs found
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 of the cylinder is called {\em coherent} if all three branches
intersect at pairwise with the same index. A {\em triple unknotting} of a
classical knot is a homotopy which connects with the trivial knot and
which has as singularities only coherent triple points. We give a new formula
for the first Vassiliev invariant 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
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
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
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
Graph complexes in deformation quantization
Kontsevich's formality theorem and the consequent star-product formula rely
on the construction of an -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
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