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Keywords = totally acyclic complex

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22 pages, 353 KiB  
Article
Acyclic Complexes and Graded Algebras
by Chaoyuan Zhou
Mathematics 2023, 11(14), 3167; https://doi.org/10.3390/math11143167 - 19 Jul 2023
Viewed by 1267
Abstract
We already know that the noncommutative N-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and [...] Read more.
We already know that the noncommutative N-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative N-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra. Full article
16 pages, 4935 KiB  
Article
Perturbed and Unperturbed: Analyzing the Conservatively Perturbed Equilibrium (Linear Case)
by Yiming Xi, Xinquan Liu, Denis Constales and Gregory S. Yablonsky
Entropy 2020, 22(10), 1160; https://doi.org/10.3390/e22101160 - 15 Oct 2020
Cited by 8 | Viewed by 3095
Abstract
The “conservatively perturbed equilibrium” (CPE) technique for a complex chemical system is computationally analyzed in a batch reactor considering different linear mechanisms with three and four species. Contrary to traditional chemical relaxation procedures, in CPE experiments only some initial concentrations are modified; other [...] Read more.
The “conservatively perturbed equilibrium” (CPE) technique for a complex chemical system is computationally analyzed in a batch reactor considering different linear mechanisms with three and four species. Contrary to traditional chemical relaxation procedures, in CPE experiments only some initial concentrations are modified; other conditions, including the total amount of chemical elements and temperature are kept unchanged. Generally, for “unperturbed” species with initial concentrations equal to their corresponding equilibrium concentrations, unavoidable extreme values are observed during relaxation to the equilibrium. If the unperturbed species is involved in one step only, this extremum is a momentary equilibrium of the step; if the unperturbed species is involved in more reactions, the extremum is not a momentary equilibrium. The acyclic mechanism with four species may exhibit two extrema and an inflection point, which corresponds to an extremum of the rate of the species change. These facts provide essential information about the detailed mechanism of the complex reaction. Full article
(This article belongs to the Special Issue Finite-Time Thermodynamics)
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