Coreduction Homology Algorithm

This paper presents a new reduction algorithm for the efficient computation of the homology of cubical sets and polotypes. The algorithm—particularly strong for low-dimensional sets embedded in high dimensions—runs in linear time. The paper presents the theoretical background of the algorithm, the algorithm itself, experimental results based on an implementation for cubical sets as well as some theoretical complexity estimates. Both authors are partially supported by Polish MNSzW, Grant N201 037 31/3151.     

Rank-two vector bundles on Hirzebruch surfaces

We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.     

On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.     

Stereoselective synthesis of and atropisomerism in 4-pyridyl-3-(1-pyridinio)-3,4-trans-1,2,3,4-tetrahydropyridines and their transformation products

A series of 2-oxo-4-pyridyl-3-(1-pyridinio)-5-cyano-3,4-trans-1,2,3,4-tetrahydropyridine-6-6-olates were prepared by condensation of pyridinium ylides with α,β-unsaturated carbonyl compounds or more conveniently by a three-component condensation of pyridinium ylides, pyridine aldehydes, and ethyl cyanoacetate 3 and/or 6 . This analogues of the above tetrahydropyridines were prepared in a similar way starting from suitable substrates. Spectroscopic data revealed that the reaction leads to trans-isomers around the C³-C⁴ bond and is atroposelective. The conformation of and tautomerism in the tetrahydropyridines are discussed in the light of ¹H NMR data. The reaction of 5-cyano-3-(3-methyl-1-pyridinio)-2-oxo-4-(3-pyridyl)-1,2,3,4-tetrahydropyridine-6-thiolate 10c with phenacyl bromide was found to give 3-hydroxy-5-oxo-7-(3-pyridyl)-6-(3-methyl-1-pyridinio)-3-phenyl-8-cyano-6,7-trans-2,3,6,7-tetrahydrothiazol[3,2a]pyridine bromide, the crystal and molecular structure of which has been determined by X-ray crystallographic analysis.     

Enantiomorphism in sodium chlorate crystals

Creation of enantiomorphous forms in sodium chlorate and enantiomorphous nature of surface micromorphology were studied by optical microscopy. It was shown that when the small number of crystals nucleates spontaneously in unstirred, unseeded solutions then exclusively left- (L) or right-handed (D) crystals can be formed. At the large number of nucleated crystals the probability of the creation of L and D form is nearly the same. L and D crystals can be distinguished by the shape of growth or dissolution centers.     

On the nature of growth hillocks on sodium chlorate crystals

Optical examination of as-grown {100} surfaces of sodium chlorate crystals grown from aqueous solution revealed the presence of elliptical growth hillocks. The hillocks were present on both enantiomorphous forms and originated from dislocations, inclusions, and microcrystals attached to the growing surface. The value of the surface entropy factor equal to 4.55 at 313 K suggests that crystals grow via/or with the participation of dislocation mechanism, and the hillocks are dislocation growth centers. Compound mechanism controlled growth of some crystals because edge nucleation and dislocation centers operated simultaneously on the same surfaces.     

Arnold Diffusion,Quantitative Estimates,and Stochastic Behavior in the Three-Body Problem

We consider a class of autonomous Hamiltonian systems subject to small, time-periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non-zero. We describe a topological method to establish orbits which diffuse in energy for every suitably small perturbation parameter . The method yields quantitative estimates:

• (i) the existence of orbits along which the energy drifts by an amount independent of ε; the time required by such orbits to drift is ;
• (ii) the existence of orbits along which the energy makes chaotic excursions;
• (iii) explicit estimates for the Hausdorff dimension of the set of such chaotic orbits;
• (iv) the existence of orbits along which the time evolution of energy approaches a stopped diffusion process (Brownian motion with drift), as ε tends to 0. For each ε fixed, the set of initial conditions of the orbits that yield the diffusion process has positive Lebesgue measure, and in the limit the measure of these sets approaches 0. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion for appropriate sets of initial conditions.

A key feature of our topological method is that it can be implemented in computer-assisted proofs. We give an application to a concrete model of the planar elliptic restricted three-body problem, on the motion of an infinitesimal body relative to the Neptune-Triton system. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Analysis of the hydrogen bonding in (2-hydroxypropyl)cellulose-poly(acrylic acid) composites by Raman spectroscopy

Novel anisotropic polymer composites obtained by the photopolymerisation of acrylic acid (AA) in lyotropic solutions of (2-hydroxypropyl)cellulose (HPC) were investigated. Raman spectroscopy was used in the study of hydrogen bonding formed by carboxylic groups of poly(acrylic acid) (pAA) between themselves and with the hydroxyl groups of HPC macromolecules. These investigations show that the supramolecular structure of pAA obtained by photopolymerisation contains mainly inner oligomeric forms and cyclic dimer forms, as a consequence of the “matrix polymerisation” of AA. The presence of HPC affects the process of the photopolymerisation of AA, by a redistribution of hydrogen bonding within the system, leading to a different supramolecular structure of pAA. With increasing amounts of HPC the inner oligomeric forms characteristic of bulk pAA disappear. In the composites, the pAA chains probably form the terminal oligomeric structure, with some free COOH groups, as well as hydrogen bonds with HPC.     

Synthesis of silica in elastomer's matrix

We tried to synthesise silica's fillers in polar and nonpolar elastomers. Di-, tri- and tetraalkoxysilanes were used as a precursor of silica. Such functionality allowed us to obtain a filler with different ratio of crosslinking. We could expect some interactions between elastomer and silanes, which contained additional functional groups. We investigated vulcanizates by using the following methods: crosslink density in toluene and 10% solution of ethylenediamine in toluene, mechanical properties, IR microscopy, DMA. It appeared that alkoxysilanes influenced advantageously on elastomers and properties.