Computable Linear Orders and Limitwise Monotonic Functions

In this paper, we describe the technique of extremely monotonic functions in the theory of computable linear orders. The basic definitions of extremely monotonic functions and their generalizations are given, and a number of their basic properties and applications are presented

Spectral Universality of Linear Orders with One Binary Relation

© 2020, Pleiades Publishing, Ltd. We show the spectral universality of the class of structures that are linear orders with an additional binary relation and hence with an n-ary relation for each n ≥ 2. To this end, we study the category of the structures. We then obtain some other algorithmic properties of the category by using the notion of a computable functor which was studied in recent papers by other authors

Rings of matrix invariants in positive characteristic

AbstractDenote by Rn,m the ring of invariants of m-tuples of n×n matrices (m,n⩾2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char(K)=0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata–Higman theorem and the degree bound for generators of Rn,m. We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0<char(K))⩽n, then Rn,m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R2,m is determined for the case char(K)=2: it consists of 2m+m−1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char(K)≠2. We prove that the characterization of the Rn,m that are complete intersections, known before when char(K)=0, is valid for any infinite K. We give a Cohen–Macaulay presentation of R2,4, and analyze the difference between the cases char(K)=2 and char(K)≠2

Composition and reactivity of aminolysis products of phenyl glycidyl ether with benzylamine

Composition of aminolysis products of phenyl glycidyl ether with benzylamine in various conditions was studied. The ratio of 1-(benzylamino)-3-phenoxypropan-2-ol and 1,1'-(benzylazanediyl)bis(3-phenoxypropan-2-ol) does not considerably depend on the nature of the solvent and is basically determined by ratio of initial reagents. 2,6-Bis(phenoxymethyl)morpholine was obtained by dehydration of aminodiol in conditions of Mitsunobu reaction with subsequent reductive debenzylation. © 2017, Pleiades Publishing, Ltd

Composition and reactivity of aminolysis products of phenyl glycidyl ether with benzylamine

Composition of aminolysis products of phenyl glycidyl ether with benzylamine in various conditions was studied. The ratio of 1-(benzylamino)-3-phenoxypropan-2-ol and 1,1'-(benzylazanediyl)bis(3-phenoxypropan-2-ol) does not considerably depend on the nature of the solvent and is basically determined by ratio of initial reagents. 2,6-Bis(phenoxymethyl)morpholine was obtained by dehydration of aminodiol in conditions of Mitsunobu reaction with subsequent reductive debenzylation. © 2017, Pleiades Publishing, Ltd

Regioselective [3+2]-cycloaddition of trimethylvinylsilane to 4,5-dihydro-5-methyl-3H-[2-benzazepine-3′-spiro-cyclohexane] n-oxide [2]

[No abstract available

[3+2] Cycloaddition of dimethyl acetylenedicarboxylate, methyl acrylate, and ethyl acrylate to 4,5-dihydro-5-methyl-3h-spiro[benz-2-azepine-3,1′- cyclohexane] N-oxide

The cycloaddition of methyl acrylate and ethyl acrylate to 4,5-dihydro-5-methyl-3H-spiro[benz-2-azepine-3,1′-cyclohexane] N-oxide proceeds without either regiospecificity or stereospecificity. Eight geometrical isomers of spiro[isoxazolidino[3,2-a]benz-2-azepine-5,1′-cyclohexane] were formed, of which several were isolated as pure samples. The cycloaddition of dimethyl acetylenedicarboxylate proceeds stereoselectively, leading to spiro[isoxazolino[3,2-a]benz-2-azepine-5,1′-cyclohexane] with cis arrangement of the protons at C(7) and C(11b)