Abstract:
The asymmetric unit of the title compound, C17H15NO4, contains two independent molecules with similar geometric parameters. In both molecules, the conformation of the cyclohexene ring is half-chair, while the pyrrolidinone ring adopts an envelope conformation with the γ-carbon atom of the α-pyrrolidinone ring as the flap. In the crystal, O—H...O hydrogen bonds between the carboxylic and carbonyl groups link alternate independent molecules into chains propagating in the b-axis direction. The crystal packing also features weak C—H...π interactions.

Abstract:
The title compound, C26H28N2O9·1.5H2O, the product of an acid-catalysed Wagner–Meerwein skeletal rearrangement, crystallizes as a sesquihydrate with the O atom of one of the two independent water molecules occupying a special position on a twofold axis. The organic molecule comprises a fused pentacyclic system containing two five-membered rings (cyclopentane and tetrahydrofuran) and three six-membered rings (piperidinone, tetrahydropyridine and benzene). The five-membered rings have the usual envelope conformations, and the central six-membered piperidinone and tetrahydropyridine rings adopt boat and sofa conformations, respectively. In the crystal, there are three independent O—H...O hydrogen bonds, which link the organic molecules and water molecules into complex two-tier layers parallel to (001). The layers are further linked into a three-dimensional framework by attractive intermolecular carbonyl–carbonyl interactions.

Abstract:
The title compound, C27H29NO11, is a product of the tandem `domino' Diels–Alder reaction. The molecule comprises a fused hexacyclic system containing four five-membered rings (two dihydrofuran and two tetrahydrofuran) in the usual envelope conformations and two six-membered rings (tetrahydropyridinone and piperidine) adopting slightly flattened boat and chair conformations, respectively. The dispositions of the carboxylate substituents relative to each other are determined by both steric reasons and intermolecular C—H...O hydrogen bonding and attractive antiparallel C=O...C=O interactions [C...O = 2.995 (2) ].

Abstract:
The title compound, C17H18NO+·I , is an adduct resulting from an intramolecular Diels–Alder reaction of methallyl chloride with 3,4-dihydro-1-furylisoquinoline. The cation comprises a fused pentacyclic system containing three five-membered rings (dihydropyrrole, dihydrofuran and tetrahydrofuran) and two six-membered rings (tetrahydropyridine and benzene). The five-membered rings have the usual envelope conformations, and the central six-membered tetrahydropyridine ring adopts the unsymmetrical half-boat conformation. In the crystal, cations and iodide anions are bound by weak intermolecular hydrogen-bonding interactions into a three-dimensional framework.

Abstract:
We investigate numerically lattice Weinberg - Salam model without fermions for realistic values of the fine structure constant and the Weinberg angle. We also analyze the data of the previous numerical investigations of lattice Electroweak theory. We have found that moving along the line of constant physics when the lattice spacing $a$ is decreased, one should leave the physical Higgs phase of the theory at a certain value of $a$. Our estimate of the minimal value of the lattice spacing is $a_c = [430\pm 40 {\rm Gev}]^{-1}$.

Abstract:
10 D Euclidean quantum gravity is investigated numerically using the dynamical triangulation approach. It has been found that the behavior of the model is similar to that of the lower dimensional models. However, it turns out that there are a few features that are not present in the lower dimensional models.

Abstract:
The main objective of this presentation is to point out that the Upper bound on the cutoff in lattice Electroweak theory is still unknown. The consideration of the continuum theory is based on the perturbation expansion around trivial vacuum. The internal structure of the lattice Weinberg - Salam model may appear to be more complicated especially in the region of the phase diagram close to the phase transition between the physical Higgs phase and the unphysical symmetric phase of the lattice model, where the continuum physics is to be approached. We represent the results of our numerical investigation of the quenched model at infinite bare scalar self coupling $\lambda$. These results demonstrate that at $\lambda = \infty$ the upper bound on the cutoff is around $\frac{\pi}{a} = 1.4$ Tev. The preliminary results for finite $\lambda$ are also presented. Basing on these results we cannot yet make a definite conclusion on the maximal value of the cutoff admitted in the lattice model, although we have found that the cutoff cannot exceed the value around $1.4 \pm 0.2$ Tev for a certain particular choice of the couplings ($\lambda = 0.009$, $\beta = 12$, $\theta_W = 30^o$) for the lattices of sizes up to $12^3\times16$. We also observe that the topological defects, which are to be identified with quantum Nambu monopoles, dominate in vacuum in the vicinity of the transition. This indicates that the vacuum of the model is different from the trivial one. In addition we remind the results of the previous numerical investigations of the SU(2) gauge - Higgs model, where the maximal reported value of the cutoff was around 1.5 Tev.

Abstract:
We investigate lattice Weinberg - Salam model without fermions for the value of the Weinberg angle $\theta_W \sim 30^o$, and bare fine structure constant around $\alpha \sim 1/150$. We consider the value of the scalar self coupling corresponding to bare Higgs mass around 150 GeV. The effective constraint potential for the zero momentum scalar field is used in order to investigate phenomena existing in the vicinity of the phase transition between the physical Higgs phase and the unphysical symmetric phase of the lattice model. This is the region of the phase diagram, where the continuum physics is to be approached. We compare the above mentioned effective potential (calculated in selected gauges) with the effective potential for the value of the scalar field at a fixed space - time point. We also calculate the renormalized fine structure constant using the correlator of Polyakov lines and compare it with the one - loop perturbative estimate.

Abstract:
We investigate numerically 10 - dimensional Euclidean quantum gravity (with discretized Einstein - Hilbert action) in the framework of the dynamical triangulation approach. For the considered values of the gravitational coupling we observed two phases, the behavior of which is found to be similar to that of the crumpled and elongated phases of 3, 4 and 5 dimensional models. Surprisingly, (for the observed lattice sizes) the natural state of the 10 D system (when the Einstein - Hilbert action is turned off) is found to resemble branched polymer while in the low dimensional systems the natural state belongs to the crumpled phase.

Abstract:
Invariants of $G_2$ and $Spin(7)$, both acting on several copies of octonions, have been decribed in \cite{schw2} over a ground field of characteristic zero. In the current manuscript, we extend this result to an arbitrary infinite field of odd characteristic. More precisely, we prove that the corresponding algebras of invariants are generated by the same invariants of degree at most $4$ as in the case of a field of characteristic zero.