Let R be a commutative ring with identity. Let Γ (R) be a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. In this paper we consider a subgraph Γ 2 (R) of Γ (R) which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph Γ 2 (R) \ J(R). In addition, it is shown that for two finite semi-local rings R and S, if R is reduced, then Γ (R) ∼ = Γ (S) if and only if R ∼ = S.

This study was aimed to evaluate the effects of near-infrared (NIR) photobiomodulation (PBM) combined with coenzyme Q10 (CoQ10) on depressive-like behavior, cerebral oxidative stress, inflammation, and apoptosis markers in mice. To induce a depressive-like model, mice were subjected to sub-chronic restraint stress for 5 consecutive days. NIR PBM (810 nm laser, 33.3 J/cm2) and/or CoQ10 (500 mg/kg/day, gavage) were administered for five days concomitantly with immobilization. Behavior was evaluated by the forced swim test (FST), tail suspension test (TST), and open field test (OFT). Mitochondrial membrane potential as well as oxidative stress, neuroinflammatory, and markers of apoptosis were evaluated in the prefrontal cortex (PFC) and hippocampus (HIP). The serum levels of pro-inflammatory cytokines, cortisol, and corticosterone were also measured. PBM or CoQ10, or the combination, ameliorated depressive-like behaviors induced by restraint stress as indicated by decreased immobility time in both the FST and TST. PBM and/or CoQ10 treatments decreased lipid peroxidation and enhanced total antioxidant capacity (TAC), GSH levels, GPx and SOD activities in both brain areas. The neuroinflammatory response in the HIP and PFC was suppressed, as indicated by decreased NF-kB, p38, and JNK levels in PBM and/or CoQ10 groups. Intrinsic apoptosis biomarkers, BAX, Bcl-2, cytochrome c release, and caspase-3 and −9, were also significantly down-regulated by both treatments. Furthermore, both treatments decreased the elevated serum levels of cortisol, corticosterone, TNF-, and IL-6 induced by restraint stress. Transcranial NIR PBM and CoQ10 therapies may be effective antidepressant strategies for the prevention of psychopathological and behavioral symptoms induced by stress.

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Objective. Transcranial photobiomodulation (tPBM) is a recently proposed non-invasive brain stimulation approach with various effects on the nervous system from the cells to the whole brain networks. Specially in the neural network level, tPBM can alter the topology and synchronizability of functional brain networks. However, the functional properties of the neural networks after tPBM are still poorly clarified. Approach. Here, we employed electroencephalography and different methods (conventional and spectral) in the graph theory analysis to track the significant effects of tPBM on the resting state brain networks. The non-parametric statistical analysis showed that just one short-term tPBM session over right medial frontal pole can significantly change both topological (i.e. clustering coefficient, global efficiency, local efficiency, eigenvector centrality) and dynamical (i.e. energy, largest eigenvalue, and entropy) features of resting state brain networks. Main results. The topological results revealed that tPBM can reduce local processing, centrality, and laterality. Furthermore, the increased centrality of central electrode was observed. Significance. These results suggested that tPBM can alter topology of resting state brain network to facilitate the neural information processing. On the other hand, the dynamical results showed that tPBM reduced stability of synchronizability and increased complexity in the resting state brain networks. These effects can be considered in association with the increased complexity of connectivity patterns among brain regions and the enhanced information propagation in the resting state brain networks. Overall, both topological and dynamical features of brain networks suggest that although tPBM decreases local processing (especially in the right hemisphere) and disrupts synchronizability of network, but it can increase the level of information transferring and processing in the brain network.

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper, we study the amalgamated duplication ring R I which is introduced by D'Anna and Fontana. It is shown that if R satisfies Serre's condition (Sn) and Ip is a maximal CohenMacaulay Rp -module for every p ∈ Spec (R), then R I satisfies Serre's condition (Sn). Moreover if R I satisfies Serre's condition (Sn), then so does R. This gives a generalization of the same result for Cohen-Macaulay rings in [D'Anna, A construction of Gorenstein rings, J. Algebra 306 (2006) 507-519]. In addition it is shown that if R is a local ring and Ann R (I) = 0, then R I is quasi-Gorenstein if and only if b R satisfies Serre's condition (S 2 ) and I is a canonical ideal of R. This result improves the result of D'Anna which is corrected by Shapiro and states that if R is a Cohen-Macaulay local ring, then R I is Gorenstein if and only if the canonical ideal of R exists and is isomorphic to I, provided Ann R (I) = 0.

Background The management of COVID-19 in organ transplant recipients is among the most imperative, yet less discussed, issues based on their immunocompromised status along with their vast post-transplant medication regimens. No conclusive study has been published to evaluate proper anti-viral and immunomodulator medications effect in treating COVID-19 patients to this date. Method This retrospective study was conducted in Shiraz Transplant Hospital, Iran from March 2020 to May 2021 and included COVID-19 diagnosed patients based on SARS-CoV-2 RT-PCR positive test who had been hospitalized for at least 48 h before enrolling in the study. Clinical and demographic information of patients, along with their treatment course and the medication used were evaluated and analyzed using multiple regression analysis. Results A total of 245 patients with a mean age of 49.59 years were included with a mortality rate of 8.16%. The administration of Remdesivir as an anti-viral drug (P value < 0.001) and Tocilizumab as an immunomodulator drug (P value < 0.001) could reduce the hospitalization period in the hospital and the intensive care unit, as well as the mortality rates significantly. Meanwhile, the patients treated with Lopinavir/Ritonavir experienced a lower chance of survival (OR < 1, P value = 0.04). No significant difference was observed between various therapeutic regimens in clinical complications such as bacterial coinfections, cardiovascular and gastrointestinal adverse reactions, and liver or kidney dysfunctions. Conclusion The administration of Remdesivir as an anti-viral and Tocilizumab as an immunomodulatory drug in solid-organ transplant recipients could be promising treatments of choice to manage COVID-19.

We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor, defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F C -pd, and we use this to give a relation between the F C -pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact, such relations force the semidualizing modules under consideration to be trivial.

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