The structure of gauge-invariant ideals of labelled graph C∗C^*-algebras

In this paper, we consider the gauge-invariant ideal structure of a $C^*$-algebra $C^*(E,\mathcal{L},\mathcal{B})$ associated to a set-finite, receiver set-finite and weakly left-resolving labelled space $(E,\mathcal{L},\mathcal{B})$, where $\mathcal{L}$ is a labelling map assigning an alphabet to each edge of the directed graph $E$ with no sinks. Under the assumption that an accommodating set $\mathcal{B}$ is closed under taking relative complement, it is obtained that there is a one to one correspondence between the set of all hereditary saturated subsets of $\mathcal{B}$ and the gauge-invariant ideals of $C^*(E,\mathcal{L},\mathcal{B})$. For this, we introduce a quotient labelled space $(E,\mathcal{L},[\mathcal{B}]_R)$ arising from an equivalence relation $\sim_R$ on $\mathcal{B}$ and show the existence of the $C^*$-algebra $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ generated by a universal representation of $(E,\mathcal{L},[\mathcal{B}]_R)$. Also the gauge-invariant uniqueness theorem for $C^*(E,\mathcal{L},[\mathcal{B}]_R)$ is obtained. For simple labelled graph $C^*$-algebras $C^*(E,\mathcal{L},\bar{\mathcal{E}})$, where $\bar{\mathcal{E}}$ is the smallest accommodating set containing all the generalized vertices, it is observed that if for each vertex $v$ of $E$, a generalized vertex $[v]_l$ is finite for some $l$, then $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ is simple if and only if $(E,\mathcal{L},\bar{\mathcal{E}})$ is strongly cofinal and disagreeable. This is done by examining the merged labelled graph $(F,\mathcal{L}_F)$ of $(E,\mathcal{L})$ and the common properties that $C^*(E,\mathcal{L},\bar{\mathcal{E}})$ and $C^*(F,\mathcal{L},\bar{\mathcal{F}})$ share

Thermodynamic Volume and the Extended Smarr Relation

We continue to explore the scaling transformation in the reduced action formalism of gravity models. As an extension of our construction, we consider the extended forms of the Smarr relation for various black holes, adopting the cosmological constant as the bulk pressure as in some literatures on black holes. Firstly, by using the quasi-local formalism for charges, we show that, in a general theory of gravity, the volume in the black hole thermodynamics could be defined as the thermodynamic conjugate variable to the bulk pressure in such a way that the first law can be extended consistently. This, so called, thermodynamic volume can be expressed explicitly in terms of the metric and field variables. Then, by using the scaling transformation allowed in the reduced action formulation, we obtain the extended Smarr relation involving the bulk pressure and the thermodynamic volume. In our approach, we do not resort to Euler's homogeneous scaling of charges while incorporating the would-be hairy contribution without any difficulty.Comment: 1+21 pages, plain LaTeX; v2 typo fixed and references adde

Molecular Mechanisms of Skin Aging and Rejuvenation

The aging process in the skin is complex and influenced by more intrinsic and extrinsic factors than any other body organ. The effects of these two types of factors overlap for the most part. The combined effects of these two aging processes also affect dermal matrix alterations. The main clinical signs of skin aging include wrinkling and irregular pigmentation, which are influenced by a combination of intrinsic and extrinsic (e.g., UV radiation, heat, smoking, and pollutants) factors. Histologically, collagen decreases, and the dermis is replaced by abnormal elastic fibers as a cause of wrinkle formation through the loss of skin elasticity. There have been numerous studies of skin aging performed to elucidate the underlying molecular mechanisms and to develop various antiaging therapeutics and preventive strategies. We summarized the molecular mechanisms and treatments of skin aging. Mainly UV radiation induces ROS formation and DNA damage, leading to increased production of MMPs and decreased production of collagen in keratinocytes and fibroblasts, which reflect the central aspects of skin aging. Besides UV radiation exposure, extrinsic factors including tobacco smoking, exposure to environmental pollutants, infrared radiation, and heat contribute to premature skin aging. Like UV radiation, these factors cause ROS formation and increase expression of MMPs, thus accelerating skin aging by inducing extracellular matrix (ECM) degradation. Accumulated collagen fibrils inhibit the new collagen synthesis and account for the further degradation of the ECM through this positive feedback loop. Accumulating evidence for molecular mechanisms of skin aging should provide clinicians with an expanding spectrum of therapeutic targets in the treatment of skin aging