Elastic and structural instability of cubic Sn3N4 and C3N4 under pressure

We use in-situ high pressure angle dispersive x-ray diffraction measurements to determine the equation of state of cubic tin nitride Sn3N4 under pressure up to about 26 GPa. While we find no evidence for any structural phase transition, our estimate of the bulk modulus (B) is 145 GPa, much lower than the earlier theoretical estimates and that of other group IV-nitrides. We corroborate and understand these results with complementary first-principles analysis of structural, elastic and vibrational properties of group IV-nitrides, and predict a structural transition of Sn3N4 at a higher pressure of 88 GPa compared to earlier predictions of 40 GPa. Our comparative analysis of cubic nitrides shows that bulk modulus of cubic C3N4 is the highest (379 GPa) while it is structurally unstable and should not exist at ambient conditions.Comment: 5 pages, 4 figure

Zero-divisor graph of the rings CP(X)C_\mathscr{P}(X) and C∞P(X)C^\mathscr{P}_\infty(X)

In this article we introduce the zero-divisor graphs $\Gamma_\mathscr{P}(X)$ and $\Gamma^\mathscr{P}_\infty(X)$ of the two rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$; here $\mathscr{P}$ is an ideal of closed sets in $X$ and $C_\mathscr{P}(X)$ is the aggregate of those functions in $C(X)$, whose support lie on $\mathscr{P}$. $C^\mathscr{P}_\infty(X)$ is the $\mathscr{P}$ analogue of the ring $C_\infty (X)$. We find out conditions on the topology on $X$, under-which $\Gamma_\mathscr{P}(X)$ (respectively, $\Gamma^\mathscr{P}_\infty(X)$) becomes triangulated/ hypertriangulated. We realize that $\Gamma_\mathscr{P}(X)$ (respectively, $\Gamma^\mathscr{P}_\infty(X)$) is a complemented graph if and only if the space of minimal prime ideals in $C_\mathscr{P}(X)$ (respectively $\Gamma^\mathscr{P}_\infty(X)$) is compact. This places a special case of this result with the choice $\mathscr{P}\equiv$ the ideals of closed sets in $X$, obtained by Azarpanah and Motamedi in \cite{Azarpanah} on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals $\mathscr{P}$ and $\mathscr{Q}$ on $X$ and $Y$ respectively that the rings $C_\mathscr{P}(X)$ and $C_\mathscr{Q}(Y)$ are isomorphic if and only if $\Gamma_\mathscr{P}(X)$ and $\Gamma_\mathscr{Q}(Y)$ are isomorphic

Intrinsic characterizations of C-realcompact spaces

[EN] c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that  X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.University Grand Commission, New Delhi, research fellowship (F. No. 16-9 (June 2018)/2019 (NET/CSIR))Acharyya, SK.; Bharati, R.; Deb Ray, A. (2021). Intrinsic characterizations of C-realcompact spaces. Applied General Topology. 22(2):295-302. https://doi.org/10.4995/agt.2021.13696OJS295302222S. K. Acharyya and S. K. Ghosh, A note on functions in C(X) with support lying on an ideal of closed subsets of X, Topology Proc. 40 (2012), 297-301.S. K. Acharyya and S. K. Ghosh, Functions in C(X) with support lying on a class of subsets of X, Topology Proc. 35 (2010), 127-148.S. K. Acharyya, R. Bharati and A. Deb Ray, Rings and subrings of continuous functions with countable range, Queast. Math., to appear. https://doi.org/10.2989/16073606.2020.1752322F. Azarpanah, O. A. S. Karamzadeh, Z. Keshtkar and A. R. Olfati, On maximal ideals of $C_c(X)$ and the uniformity of its localizations, Rocky Mountain J. Math. 48, no. 2 (2018), 345-384. https://doi.org/10.1216/RMJ-2018-48-2-345P. Bhattacherjee, M. L. Knox and W. W. Mcgovern, The classical ring of quotients of $C_c(X)$, Appl. Gen. Topol. 15, no. 2 (2014), 147-154. https://doi.org/10.4995/agt.2014.3181L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand Reinhold co., New York, 1960. https://doi.org/10.1007/978-1-4615-7819-2M. Ghadermazi, O. A. S. Karamzadeh and M. Namdari, On the functionally countable subalgebras of C(X), Rend. Sem. Mat. Univ. Padova. 129 (2013), 47-69. https://doi.org/10.4171/RSMUP/129-4O. A. S. Karamzadeh and Z. Keshtkar, On c-realcompact spaces, Queast. Math. 41, no. 8 (2018), 1135-1167. https://doi.org/10.2989/16073606.2018.1441919M. Mandelkar, Supports of continuous functions, Trans. Amer. Math. Soc. 156 (1971), 73-83. https://doi.org/10.1090/S0002-9947-1971-0275367-4R. M. Stephenson Jr, Initially k-compact and related spaces, in: Handbook of Set-Theoretic Topology, ed. Kenneth Kunen and Jerry E. Vaughan. Amsterdam, North-Holland, (1984) 603-632. https://doi.org/10.1016/B978-0-444-86580-9.50016-1A. Veisi, $e_c$-filters and $e_c$-ideals in the functionally countable subalgebra of $C^*(X)$, Appl. Gen. Topol. 20, no. 2 (2019), 395-405. https://doi.org/10.4995/agt.2019.1152

UU-topology and mm-topology on the ring of Measurable Functions, generalized and revisited

Let $\mathcal{M}(X,\mathcal{A})$ be the ring of all real valued measurable functions defined over the measurable space $(X,\mathcal{A})$. Given an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ and a measure $\mu:\mathcal{A}\to[0,\infty]$, we introduce the $U_\mu^I$-topology and the $m_\mu^I$-topology on $\mathcal{M}(X,\mathcal{A})$ as generalized versions of the topology of uniform convergence or the $U$-topology and the $m$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively. With $I=\mathcal{M}(X,\mathcal{A})$, these two topologies reduce to the $U_\mu$-topology and the $m_\mu$-topology on $\mathcal{M}(X,\mathcal{A})$ respectively, already considered before. If $I$ is a countably generated ideal in $\mathcal{M}(X,\mathcal{A})$, then the $U_\mu^I$-topology and the $m_\mu^I$-topology coincide if and only if $X\setminus \bigcap Z[I]$ is a $\mu$-bounded subset of $X$. The components of $0$ in $\mathcal{M}(X,\mathcal{A})$ in the $U_\mu^I$-topology and the $m_\mu^I$-topology are realized as $I\cap L^\infty(X,\mathcal{A},\mu)$ and $I\cap L_\psi(X,\mathcal{A},\mu)$ respectively. Here $L^\infty(X,\mathcal{A},\mu)$ is the set of all functions in $\mathcal{M}(X,\mathcal{A})$ which are essentially $\mu$-bounded over $X$ and $L_\psi(X,\mathcal{A},\mu)=\{f\in \mathcal{M}(X,\mathcal{A}): ~\forall g\in\mathcal{M}(X,\mathcal{A}), f.g\in L^\infty(X,\mathcal{A},\mu)\}$. It is established that an ideal $I$ in $\mathcal{M}(X,\mathcal{A})$ is dense in the $U_\mu$-topology if and only if it is dense in the $m_\mu$-topology and this happens when and only when there exists $Z\in Z[I]$ such that $\mu(Z)=0$. Furthermore, it is proved that $I$ is closed in $\mathcal{M}(X,\mathcal{A})$ in the $m_\mu$-topology if and only if it is a $Z_\mu$-ideal in the sense that if $f\equiv g$ almost everywhere on $X$ with $f\in I$ and $g\in\mathcal{M}(X,\mathcal{A})$, then $g\in I$

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

Rings and subrings of continuous functions with countable range

Intermediate rings of the functionally countable subalgebra of C(X) (i.e., the rings Ac(X) lying between Cc (X) and Cc(X)), where X is a Hausdorff zero-dimensional space, are studied in this article. It is shown that the structure space of each Ac(X) is homeomorphic to 0X, the Banaschewski compactication of X. From this a main result of [A. Veisi, ec-lters and ec-ideals in the functionally countable subalgebra C (X), Appl. Gen. Topol. 20(2) (2019), 395{405] easily follows. The countable counterpart of the m-topology and U-topology on C(X), namely mc-topology and Uc-topology, respectively, are introduced and using these, new characterizations of P-spaces and pseudocompact spaces are found out. More- over, X is realized to be an almost P-space when and only when each maximal ideal/z-ideal in Cc(X) become a z0-ideal. This leads to a characterization of Cc(X) among its intermediate rings for the case that X is an almost P-space.  Noetherianness/Artinianness of Cc(X) and a few chosen subrings of Cc(X) are examined and nally, a complete description of z0-ideals in a typical ring Ac(X) via z0-ideals in Cc(X) is established

Photocontrolled nuclear-targeted drug delivery by single component photoresponsive fluorescent organic nanoparticles of acridin-9-methanol

We report for the first time an organic nanoparticle based nuclear-targeted photoresponsive drug delivery system (DDS) for regulated anticancer drug release. Acridin-9-methanol fluorescent organic nanoparticles used in this DDS performed three important roles: (i) ″nuclear-targeted nanocarrier″ for drug delivery, (ii) ″phototrigger″ for regulated drug release, and (iii) fluorescent chromophore for cell imaging. In vitro biological studies reveal acridin-9-methanol nanoparticles of 60 nm size to be very efficient in delivering the anticancer drug chlorambucil into the target nucleus, killing the cancer cells upon irradiation. Such targeted organic nanoparticles with good biocompatibility, cellular uptake property, and efficient photoregulated drug release ability will be of great benefit in the field of targeted intracellular controlled drug release