We offer for consideration and study the concept of a pre-minimal pair. Also we examine extensions of valuations of a field
F
to valuations of the rational function field
F
(
x
) in one variable.
K-Completions of T0-Spaces Ershov, Yu. L.
Algebra and logic,
2022/7, Letnik:
61, Številka:
3
Journal Article
Recenzirano
For a wide category
K
, we introduce the notions of a
K
-precomplete map and of a
K-
subspace. Based on these, we create a uniform method for constructing
K
-completions of
T
0
-spaces.
On Brown Polynomials. II Ershov, Yu. L.
Siberian mathematical journal,
09/2019, Letnik:
60, Številka:
5
Journal Article
Recenzirano
The results of the author’s previous article are improved. We also establish that each irreducible polynomial over a Henselian valued field is a Brown polynomial.
An analysis is performed for data on the composition of cationic fragments formed during the pulsed photolysis of protonated form of dimethylaminoazobenzene dye in a rarefied helium atmosphere at
T
... ≈ 40 K. It is shown that the interpretation of the mechanism of photolysis, which is based on the traditional representation of this dye as a monomeric azonium cation, is incorrect. An adequate mechanism of the process is obtained using a new concept, according to which the ground state of protonated dimethylaminoazobenzene and its substituted derivatives are rydimers with two protonated amino groups. It was established that the photoexcitation of these rydimers in an atmosphere of rarefied helium results their deprotonation, accompanied by the formation of electrically neutral monomers and monomers with positive charges on
sp
2
orbitals. Both types of monomers react in a helium atmosphere with released protons, splitting into different cationic fragments. Formulas for the structure of fragments and schemes of monomer fragmentation are presented.
A comprehensive analysis of the causes, extent, and consequences of interregional differences in production and consumption indicators cannot be carried out based on a limited range of the most ...readily accessible statistical indicators. The most common of them—per capita GRP, average wages, and per capita cash incomes—are insufficient for obtaining objective conclusions and identifying all the factors that determine interregional differentiation; moreover, they lead to an overestimated scale of this differentiation. In addition, to estimate the extent of donor and receiver, it is also clearly insufficient to compare data on the volume of revenues in the budget system and expenditures of regional consolidated budgets. The article shows how drastic differences in interregional differentiation indicators can be when they are determined by diverse approaches.
Data on the spectroscopy of photoexcitation of
trans
-to-
cis
isomerization and fragmentation of protonated
trans
-azobenzene (ABH
+
) in an inert gas atmosphere are analyzed. Results from gas-phase ...isomerization are compared to ones from a liquid-phase reaction in aqueous sulfuric acid that results in the formation of cyclic benzoccinnoline. There is no cyclization upon gas-phase
trans
-to-
cis
isomerization in a rarefied nitrogen medium at
T
= 300 K, due to the high rate of inverse
cis
-to-
trans
isomerization. It is shown that transformations in both phases proceed with the participation of phenylaminyl type cations
trans
- and
cis
-ABH
+
, which are chromogens. Photofragmentation of
trans
-ABH
+
in an atmosphere of cryogenically cooled helium (
T
= 40 K) proceeds with the formation of fragment phenyl and phenyl diazonium cations and the participation of phenyl amine cations. Mechanisms of photofragmentation are proposed, including intermediate acts with the participation of H atoms formed by dissociation of the NH group. A fast attack of H atoms along the phenyl ring bonded to the NH group results in the breaking of the C–N bond, eliminating the possibility of
trans
-to-
cis
isomerization in a cryogenic helium atmosphere.
A critical analysis of the UV–Vis absorption spectra of azobenzene (AB) and dimethylaminoazobenzene (DAB) used as an optical probe was performed based on the key empirical regularities of the
n
→ π* ...transitions of nitrogen- and carbonyl-containing compounds. The spectrum of DAB has an
n
→ π* band located in the same range ν = 28 000–33 000 cm
–1
(λ = 350–300 nm) as the
n
→ π* band of protonated azobenzene AB
+
H, but differs in the presence of a vibronic structure. It was also shown that the low-frequency
n
→ π* band belonging to AB lies in the range where the π → π* band of AB is located with a maximum at a wavelength λ
max
≈ 320–340 nm (ν
max
= 30 300–29 400 cm
–1
), in contrast to the conventional assignment of this band to the Vis band at 440 nm. This
n
→ π * band is not found in the spectra of AB solutions in polar media, but appears in the nonpolar medium due to the presence of a vibronic structure.
