Integrable and Chaotic Systems Associated with Fractal Groups

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.     

Generating subtour elimination constraints for the TSP from pure integer solutions

The traveling salesman problem (TSP) is one of the most prominent combinatorial optimization problems. Given a complete graph \(G = (V, E)\) and non-negative distances d for every edge, the TSP asks for a shortest tour through all vertices with respect to the distances d. The method of choice for solving the TSP to optimality is a branch and cut approach. Usually the integrality constraints are relaxed first and all separation processes to identify violated inequalities are done on fractional solutions. In our approach we try to exploit the impressive performance of current ILP-solvers and work only with integer solutions without ever interfering with fractional solutions. We stick to a very simple ILP-model and relax the subtour elimination constraints only. The resulting problem is solved to integer optimality, violated constraints (which are trivial to find) are added and the process is repeated until a feasible solution is found. In order to speed up the algorithm we pursue several attempts to find as many relevant subtours as possible. These attempts are based on the clustering of vertices with additional insights gained from empirical observations and random graph theory. Computational results are performed on test instances taken from the TSPLIB95 and on random Euclidean graphs.     

Synthesis and antiviral activity of nonannulated tetrazolylpyrimidines

Chemistry of Heterocyclic Compounds - Nonannulated tetrazolylpyrimidines in the structure of which the heterocyclic fragments are separated by hydrazinocarbonylmethyl, methylpyrazolyl groups or a...     

Polarization-insensitive ultralow-power second-harmonic generation frequency-resolved optical gating

We demonstrate polarization-insensitive ultralow-power second-harmonic generation frequency-resolved optical gating (FROG) measurements with a fiber-pigtailed, aperiodically poled lithium niobate waveguide. By scrambling the polarization much faster than the measurement integration time, we eliminate the impairment that frequency-independent random polarization fluctuations induce in FROG measurements. As a result we are able to retrieve intensity and phase profiles of few hundred femtosecond optical pulses with 50 MHz repetition rates at 5.2 nW coupled average power without control of the input polarization.     

Automated structure verification based on a combination of 1D (1)H NMR and 2D (1)H - (13)C HSQC spectra

A method for structure validation based on the simultaneous analysis of a 1D (1)H NMR and 2D (1)H - (13)C single-bond correlation spectrum such as HSQC or HMQC is presented here. When compared with the validation of a structure by a 1D (1)H NMR spectrum alone, the advantage of including a 2D HSQC spectrum in structure validation is that it adds not only the information of (13)C shifts, but also which proton shifts they are directly coupled to, and an indication of which methylene protons are diastereotopic. The lack of corresponding peaks in the 2D spectrum that appear in the 1D (1)H spectrum, also gives a clear picture of which protons are attached to heteroatoms. For all these benefits, combined NMR verification was expected and found by all metrics to be superior to validation by 1D (1)H NMR alone. Using multiple real-life data sets of chemical structures and the corresponding 1D and 2D data, it was possible to unambiguously identify at least 90% of the correct structures. As part of this test, challenging incorrect structures, mostly regioisomers, were also matched with each spectrum set. For these incorrect structures, the false positive rate was observed as low as 6%.     

Conformational transformations in chiral polythiophene derivatives aggregated in polymethacrylate matrices: Experiments and molecular dynamics simulations

Chiral derivatives of poly(3-(S-3',7'-dimethyloctyl)thiophene) embedded in either poly(methyl methacrylate) (PMMA) or poly(butyl methacrylate) (PBMA) were studied by polarization optical microscopy, UV absorption and circular dichroism (CD) spectroscopy as well as simulated using the Molecular Dynamics procedure. CD spectra change on heating to mirror spectra; for PMMA-containing blends the changes in heating/cooling cycles are reversible. The signs of CD spectra depend on free volume, the packing of side groups in particular. Since in the melt there are no restrictions on the positions of the chiral groups, those groups appears with equal probabilities at any locations with respect to the thiophene ring. By contrast, in clusters restrictions imposed by other alkyl chains result in positioning of chiral methyl groups preferentially on the left or right sides of the thiophene rings. The consequence is the formation of right or left twists in molecular backbones.     

A simple way to novel 2-dichloromethyl-5-aryl-2H-tetrazoles

A series of 2-dichloromethyl-5-aryl-2H-tetrazoles ( 3a–i ) were synthesized by the reaction of 5-aryl-NH-tetrazoles with trichloromethane in strong organic basic condition. The reaction represents a single-stage access to a new class of promising compounds, which not only can have biological activity, but can also serve as intermediates for further alkylation of halogen and the introduction of various fragments into the structure. The compounds obtained were fully characterized by means of ESI–MS, ¹H and ¹³C{¹H} NMR spectroscopies, as well as by single-crystal X-ray diffraction (for 3b ) the structural parameters as well as the vibrational corrections to enthalpies and total energies of the investigated systems were calculated using DFT B3LYP/6-31+G** approach.     

Bis(methanol‐κO)dioxido[3,3′‐(1H‐1,2,4‐triazole‐3,5‐diyl)diphenolato‐κ3O,N4,O′]uranium(VI) methanol monosolvate

The structure of the title compound, [U(C₁₄H₉N₃O₂)O₂(CH₃OH)₂]·CH₃OH, is the first to be reported for an actinide complex including triazole ligands. The U^VI atom exhibits a pentagonal–bipyramidal NO₆ coordination environment, involving two axial oxide ligands [U=O = 1.766 (3) and 1.789 (3) Å], four equatorial O atoms [U—O = 2.269 (3)–2.448 (3) Å] from the ligand and the two coordinated methanol molecules, and one equatorial N atom [U—N = 2.513 (4) Å] from the ligand. In the crystal structure, the complex molecules are linked via intermolecular N—H...O and O—H...O hydrogen bonds to form a two‐dimensional structure.     

On Dehn Functions of Infinite Presentations of Groups

We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, for two well-known periodic groups. In particular, we prove that the (standard) Dehn function of a 2-group Γ of intermediate growth, defined by a system of defining relators due to Lysenok, is bounded from above by C₁x² log₂ x, where C₁ > 1 is a constant. We also show that the (standard) Dehn function of a free m-generator Burnside group B(m, n) of exponent n ≥ 2⁴⁸, where n is either odd or divisible by 2⁹, defined by a minimal system of defining relators, is bounded from above by the subquadratic function x^19/12. Received: September 2007, Revision: March 2008, Accepted: March 2008     

An efficient synthesis of a novel analog of octreotide with an unnatural l-lysine-like tetrazolyl amino acid

A new cyclic octreotide-like octapeptide was prepared by incorporation of an unnatural tetrazolyl amino acid, an analog of Fmoc-l-lysine, into the peptide chain. The new tetrazolyl amino acid, (S)-2-{[2-(9H-fluoren-9-yl)acetoxy]amino}-6-(1H-tetrazol-1-yl)hexanoic acid, was obtained by azidation of Fmoc-l-lysine trifluoroacetate with sodium azide in the presence of triethyl orthoformate in glacial acetic acid. The linear peptide sequence was prepared using efficient Fmoc/t-Bu solid-phase peptide synthesis (SPPS). Cyclization of the octapeptide was carried out via oxidation with iodine. The structure and purity of the cyclic octapeptide were confirmed by LC–MS, MALDI-TOF MS/MS analysis as well as 1D/2D ¹H and ¹³C NMR spectroscopy.