A Generalization of m m -topology and U U -topology on rings of measurable functions

For a measurable space ($X,\mathcal{A}$), let $\mathcal{M}(X,\mathcal{A})$ be the corresponding ring of all real valued measurable functions and let $\mu$ be a measure on ($X,\mathcal{A}$). In this paper, we generalize the so-called $m_{\mu}$ and $U_{\mu}$ topologies on $\mathcal{M}(X,\mathcal{A})$ via an ideal $I$ in the ring $\mathcal{M}(X,\mathcal{A})$. The generalized versions will be referred to as the $m_{\mu_{I}}$ and $U_{\mu_{I}}$ topology, respectively, throughout the paper. $L_{I}^{\infty} \left(\mu\right)$ stands for the subring of $\mathcal{M}(X,\mathcal{A})$ consisting of all functions that are essentially $I$-bounded (over the measure space ($X,\mathcal{A}, \mu$)). Also let $I_{\mu} (X,\mathcal{A}) = \big \{ f \in \mathcal{M}(X,\mathcal{A}) : \, \text{for every} \, g \in \mathcal{M}(X,\mathcal{A}), fg \, \, \text{is essentially} \, I$-$\text{bounded} \big \}$. Then $I_{\mu} (X,\mathcal{A})$ is an ideal in $\mathcal{M}(X,\mathcal{A})$ containing $I$ and contained in $L_{I}^{\infty} \left(\mu\right)$. It is also shown that $I_{\mu} (X,\mathcal{A})$ and $L_{I}^{\infty} \left(\mu\right)$ are the components of $0$ in the spaces $m_{\mu_{I}}$ and $U_{\mu_{I}}$, respectively. Additionally, we obtain a chain of necessary and sufficient conditions as to when these two topologies coincide

Electronic substitution effect on the ground and excited state properties of indole chromophore: A computational study

Indole, being the main chromophore of amino acid tryptophan and several other biologically relevant molecules like serotonin, melatonin, has prompted considerable theoretical and experimental interest. The current work focuses on the investigation of photophysical and photochemical properties of indole and indole derivatives e.g. tryptophan, serotonin and melatonin using theoretical and computational methodologies. Having three close-lying excited electronic states, the vibronic coupling effect becomes extremely important yet challenging for the photophysics and photochemistry of indole. Here, we have used density functional theory (DFT) extensively and evaluated the performance of DFT in compared to available experimental and ab initio results from literature. The benchmarking of the method is followed by investigation of the chemical and geometrical effects of ring substitution in indole. A bathochromic shift has been observed in the HOMO-LUMO gap as well as vertical excitation energy from indole to melatonin. While the contribution of the in-plane small adjacent groups increases the electron density of the indole ring, the out-of-plane long substituent groups have minor effect. The comparison of singlet-triplet gaps suggests highest probability of inter-system crossing for tryptophan which is in line with previous experiment. The absorption spectra calculated including the vibronic coupling are in good agreement with experiment. These results can be used to estimate the error in photophysical observables of indole derivatives calculated considering indole as prototypical system. This study also demonstrates the merits and demerits of using DFT functionals to compute the photophysical properties of indole derivatives

A Computational Study on Light-Induced Spin Crossover in [Fe(Tp)(CN)3]-2 in Search of Potential Building Block for Single-Molecule Magnet

Light-induced spin crossover (LISCO) in transition metal complexes has drawn attention to the researcher due to its various application in science and technologies. The interplay of LISCO with single molecular magnetism (SMM) is interesting in view of its application towards photoregulated storage devices, magnetic photoswitches. Herein, we have studied the interplay between LISCO and SMM of a Fe(II) complex ([Fe(Tp)(CN)3]-2) which is an ultimate miniature of potential building block of SMM using density functional theory (DFT) and time-dependent DFT method. The molecular structure and energy in low-spin singlet, high-spin quintet as well as intermediate spin triplet is calculated. It is found that the molecule is stable in its LS state but can undergo spin crossover upon irradiation of UV-vis light via triplet excited states. The singlet excited states are close-lying, forming a band structure. The detailed mechanism of LISCO is proposed based on the calculated potential energy cuts and spin-orbit coupling values. While the LS state of the complex has Ms=0 and diamagnetic, the HS state has Ms=±2 and paramagnetic. The calculations suggest a positive zero field splitting parameter and a reasonably small E/D value with a high magnetic relaxation barrier of 96 cm-1. Therefore, for a good SMM, the complex has to be trapped in its HS state after the SCO and reverse spin-crossover (rSCO) has to be stopped. On the other hand, the complex can be used as photoregulated magnetic switch if both the SCO and rSCO happens at the similar time scale

Impact of Cooperation and Intra-Specific Competition of Prey on the Stability of Prey–Predator Models with Refuge

The main objective of this study is to find out the influences of cooperation and intra-specific competition in the prey population on escaping predation through refuge and the effect of the two intra-specific interactions on the dynamics of prey–predator systems. For this purpose, two mathematical models with Holling type II functional response functions were proposed and analyzed. The first model includes cooperation among prey populations, whereas the second one incorporates intra-specific competition. The existence conditions and stability of different equilibrium points for both models were analyzed to determine the qualitative behaviors of the systems. Refuge through intra-specific competition has a stabilizing role, whereas cooperation has a destabilizing role on the system dynamics. Periodic oscillations were observed in both systems through Hopf bifurcation. From the analytical and numerical findings, we conclude that intra-specific competition affects the prey population and continuously controls the refuge class under a critical value, and thus, it never becomes too large to cause predator extinction due to food scarcity. Conversely, cooperation leads the maximal number of individuals to escape predation through the refuge so that predators suffer from low predation success