Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ \mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ \mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ \mathfrak{g} $ ) associated to $ \mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ \mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ \mathcal{G} $ . When the underlying manifold of $ \mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{\mathfrak{g}}_{\mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ \mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ \mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.     

Investigation of magnetic field effect on surface and finite-site free energy in one-dimensional Ising model of nanosystems

We investigate a one-dimensional (1-D) Ising model for finite-site systems. The finite-site free energy and the surface free energy are calculated via the transfer matrix method. We show that, at high magnetic fields, the surface free energy has an asymptotic limit. The absolute surface energy increases when the value of f (the ratio of magnetic field to nearest-neighbor interactions) increases, and for f?≥?10 approaches a constant value. For the values of f?≥?0.2, the finite-site free energy also increases, but slowly. The thermodynamic limit in which physical properties approach the bulk value is also explored.     

Spin coupling and magnetic field effects on the finite-size free energy and its non-extensivity for 1-D Ising model with nearest and next-nearest neighbor interactions in nanosystem

The effects of second-neighbor spin coupling interactions and a magnetic field are investigated on the free energies of a finite-size 1-D Ising model. For both ferromagnetic of nearest neighbor (NN) and next-nearest neighbor (NNN) spin coupling interactions, the finite-size free energy first increases and then approaches a constant value for any size of the spin chain. In contrast, when NNN and NN spin coupling interactions are antiferromagnetic and ferromagnetic, respectively, the finite-size free energy gradually decreases by increasing the competition factor and eventually vanishes for large values of it. When a magnetic field is applied, the finite-size free energy decreases with respect to the case of zero magnetic fields for both ferromagnetic and antiferromagnetic spin coupling interactions. Deviation of free energy per size for finite-size systems relative to the infinite system increases when the spin coupling interactions as well as the f parameter (the ratio of the magnetic field to NN spin coupling interaction) increase.     

Effect of calcination temperature on the photocatalytic activity of carbon‐doped titanium dioxide revealed by photoluminescence study

Carbon‐doped titania (C‐TiO₂) nanoparticles were synthesized by the sol–gel method at different calcination temperatures (300–600°C) employing titanium tetraisopropoxide (TTIP) as the titanium source and polyoxyethylene sorbitan monooleate (Tween 80) as the carbon source. The physical properties of C‐TiO₂ samples were characterized by X‐ray diffraction (XRD) and scanning electron microscopy (SEM). The photocatalytic activities were checked through the photodegradation of phenolphthalein (PHP) under ultraviolet irradiation. The UV spectrum showed that the carbon doping extends the absorption range of TiO₂ to the visible region. However, the photocatalytic activity is affected by the electron–hole recombination phenomenon, as revealed by the photoluminescence (PL) study. According to the PL spectra, carbon doping reduces the edge‐to‐edge electron–hole recombination. Nevertheless, the number of defect sites is greatly influenced by the calcination temperature of C‐TiO₂. C‐TiO₂ that was calcined at 400°C showed the highest photodegradation percentage of PHP, which was mainly attributed to the synergic effect of the low direct edge‐to‐edge electron–hole recombination, high content of defect sites, and retention of active electrons on the surface hydroxyl group.     

Facile synthesis of 1-(arylimino)naphthalen-2(1H)-ones from anilines and 2-naphthols promoted by NaBr/K2S2O8/CAN

An efficient method has been developed for the synthesis of 1-(arylimino)naphthalen-2(1H)-ones through the cascade reaction of anilines and 2-naphthols promoted by NaBr/K₂S₂O₈/Ce(NH₄)₂(NO₃)₆. Using this protocol, a series of 1-(arylimino)naphthalen-2(1H)-ones was obtained in good to excellent yields (17 examples, 70–92% yields). The reactions may proceed through the following steps: bromination of 2-naphthols by in-situ-generated bromine from NaBr and K₂S₂O₈ to afford 1-bromonaphthalen-2-ols, coupling of 1-bromonaphthalen-2-ols with anilines to afford the corresponding amines, and subsequent oxidation of the amines into the products by Ce(NH₄)₂(NO₃)_6. These newly obtained α-imine ketones have great potentials for synthesis of special optical materials bearing naphthalene moiety.     

Voltammetric Determination of Bisphenol A in Water and Juice Using a Lanthanum (III)-Doped Cobalt (II,III) Nanocube Modified Carbon Screen-Printed Electrode

A La³⁺ doped Co₃O₄ nanocube modified graphite screen-printed electrode (La³⁺-doped Co₃O₄ nanocube/SPE) was prepared and utilized for the sensitive voltammetric determination of bisphenol A. In comparison with an unmodified electrode, the presence of the La³⁺ doped Co₃O₄ nanocubes caused a significant enhancement in the peak current. Differential pulse voltammetry (DPV), cyclic voltammetry (CV), and chronoamperometry approaches were utilized as diagnostic methods. The modified SPE was used to determine bisphenol A concentrations in the range from 0.5 to 900.0?μM with a limit of detection equal to 6.1?×?10^?8 M. Real samples were effectively analyzed with the modified electrode.     

Cyclin-Dependent Kinase 4 and 6 Inhibitors in Cell Cycle Dysregulation for Breast Cancer Treatment

In cell development, the cell cycle is crucial, and the cycle progression’s main controllers are endogenous CDK inhibitors, cyclin-dependent kinases (CDKs), and cyclins. In response to the mitogenic signal, cyclin D is produced and retinoblastoma protein (Rb) is phosphorylated due to activated CDK4/CDK6. This causes various proteins required in the cell cycle progression to be generated. In addition, complexes of CDK1-cyclin A/B, CDK2-cyclin E/A, and CDK4/CDK6-cyclin D are required in each phase of this progression. Cell cycle dysregulation has the ability to lead to cancer. Based on its role in the cell cycle, CDK has become a natural target of anticancer therapy. Therefore, understanding the CDK structures and the complex formed with the drug, helps to foster the development of CDK inhibitors. This development starts from non-selective CDK inhibitors to selective CDK4/CDK6 inhibitors, and these have been applied in clinical cancer treatment. However, these inhibitors currently require further development for various hematologic malignancies and solid tumors, based on the results demonstrated. In drug development, the main strategy is primarily to prevent and asphyxiate drug resistance, thus a determination of specific biomarkers is required to increase the therapy’s effectiveness as well as patient selection suitability in order to avoid therapy failure. This review is expected to serve as a reference for early and advanced-stage researchers in designing new molecules or repurposing existing molecules as CDK4/CDK6 inhibitors to treat breast cancer.