scholarly journals Note on Relative Homological Dimension

1958 ◽  
Vol 13 ◽  
pp. 89-94 ◽  
Author(s):  
G. Hochschild

Let R be a ring with identity element 1, and let S be a subring of R containing 1. We consider R-modules on which 1 acts as the identity map, and we shall simultaneously regard such R-modules as S-modules in the natural way. In [4], we have defined the relative analogues of the functors of Cartan-Eilenberg [1], and we have briefly treated the corresponding relative analogues of module dimension and global ring dimension.

2020 ◽  
Vol 27 (03) ◽  
pp. 621-642
Author(s):  
Driss Bennis ◽  
J.R. García Rozas ◽  
Lixin Mao ◽  
Luis Oyonarte

In Enochs’ relative homological dimension theory occur the (co)resolvent and (co)proper dimensions, which are defined by proper and coproper resolutions constructed by precovers and preenvelopes, respectively. Recently, some authors have been interested in relative homological dimensions defined by just exact sequences. In this paper, we contribute to the investigation of these relative homological dimensions. First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs. Then relative global dimensions are studied, which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories. At the end of this paper, relative derived functors are studied and generalizations of some known results of balance for relative homology are established.


2017 ◽  
Vol 10 (03) ◽  
pp. 1750058 ◽  
Author(s):  
Yeni Susanti ◽  
Joerg Koppitz

An involuted semilattice [Formula: see text] is a semilattice [Formula: see text] with an identity element [Formula: see text] and with an involution [Formula: see text] satisfying [Formula: see text] and [Formula: see text]. We consider involuted semilattices [Formula: see text] with an identity [Formula: see text] such that there is a subsemilattice [Formula: see text] without [Formula: see text] with the property that any [Formula: see text] belongs to exactly one of the following four sets : [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. In this paper, we introduce an associative binary operation [Formula: see text] on [Formula: see text] in the following quite natural way: [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] and characterize all endomorphisms of the orthodox semigroup [Formula: see text].


2001 ◽  
Vol 33 (5) ◽  
pp. 520-526 ◽  
Author(s):  
JOHN S. KAUTA

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.


Author(s):  
Nikolay S. Savkin

Introduction. Radical pessimism and militant anti-natalism of Arthur Schopenhauer and David Benathar create an optimistic philosophy of life, according to which life is not meaningless. It is given by nature in a natural way, and a person lives, studies, works, makes a career, achieves results, grows, develops. Being an active subject of his own social relations, a person does not refuse to continue the race, no matter what difficulties, misfortunes and sufferings would be experienced. Benathar convinces that all life is continuous suffering, and existence is constant dying. Therefore, it is better not to be born. Materials and Methods. As the main theoretical and methodological direction of research, the dialectical materialist and integrative approaches are used, the realization of which, in conjunction with the synergetic technique, provides a certain result: is convinced that the idea of anti-natalism is inadequate, the idea of giving up life. A systematic approach and a comprehensive assessment of the studied processes provide for the disclosure of the contradictory nature of anti-natalism. Results of the study are presented in the form of conclusions that human life is naturally given by nature itself. Instincts, needs, interests embodied in a person, stimulate to active actions, and he lives. But even if we finish off with all of humanity by agreement, then over time, according to the laws of nature and according to evolutionary theory, man will inevitably, objectively, and naturally reappear. Discussion and Conclusion. The expected effect of the idea of inevitability of rebirth can be the formation of an optimistic orientation of a significant part of the youth, the idea of continuing life and building happiness, development. As a social being, man is universal, and the awareness of this universality allows one to understand one’s purpose – continuous versatile development.


Author(s):  
Elizabeth Schechter

This chapter addresses the intuitive fascination of the split-brain phenomenon. According to what I call the standard explanation, it is because we ordinarily assume that people are psychologically unified, while split-brain subjects are not psychologically unified, which suggests that we might not be unified either. I offer a different interpretation. One natural way of grappling with people’s failures to conform to various assumptions we make about them is to conceptualize them as having multiple minds. Such multiple-minds models take their most dramatic form in narrative art as duality myths. The split-brain cases grip people in part because the subjects strike them as living embodiments of such myths.


Author(s):  
Stephen Piddock ◽  
Ashley Montanaro

AbstractA family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $$k \geqslant 2$$ k ⩾ 2 and all local dimensions $$d \geqslant 2$$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $$d \geqslant 2$$ d ⩾ 2 (spin $$\geqslant 1/2$$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $$d=3$$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.


Chemistry ◽  
2021 ◽  
Vol 3 (3) ◽  
pp. 821-830
Author(s):  
Davide De Simeis ◽  
Stefano Serra ◽  
Alessandro Di Fonzo ◽  
Francesco Secundo

Natural flavor and fragrance market size is expected to grow steadily due to the rising consumer demand of natural ingredients. This market request is guided by the general opinion that the production of natural compounds leads to a reduction of pollution, with inherent advantages for the environment and people’s health. The biotransformation reactions have gained high relevance in the production of natural products. In this context, few pieces of research have described the role of microalgae in the oxidation of terpenoids. In this present study, we questioned the role of microalgal based oxidation in the synthesis of high-value flavors and fragrances. This study investigated the role of three different microalgae strains, Chlorella sp. (211.8b and 211.8p) and Chlorococcum sp. (JB3), in the oxidation of different terpenoid substrates: α-ionone, β-ionone, theaspirane and valencene. Unfortunately, the experimental data showed that the microalgal strains used are not responsible for the substrate oxidation. In fact, our experiments demonstrate that the transformation of the four starting compounds is a photochemical reaction that involves the oxygen as oxidant. Even though these findings cast a shadow on the use of these microorganisms for an industrial purpose, they open a new possible strategy to easily obtain nootkatone in a natural way by just using an aqueous medium, oxygen and light.


2021 ◽  
Vol 11 (4) ◽  
pp. 1616
Author(s):  
Antonina Rita Limongi ◽  
Emanuele Viviano ◽  
Maria De Luca ◽  
Rosa Paola Radice ◽  
Giuliana Bianco ◽  
...  

The need to safeguard our planet by reducing carbon dioxide emissions has led to a significant development of research in the field of alternative energy sources. Hydrogen has proved to be the most promising molecule, as a fuel, due to its low environmental impact. Even if various methods already exist for producing hydrogen, most of them are not sustainable. Thus, research focuses on the biological sector, studying microalgae, and other microorganisms’ ability to produce this precious molecule in a natural way. In this review, we provide a description of the biochemical and molecular processes for the production of biohydrogen and give a general overview of one of the most interesting technologies in which hydrogen finds application for electricity production: fuel cells.


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