Ideal spaces

[EN] Let C∞ (X) denote the family of real-valued continuous functions which vanish at infinity in the sense that {x ∈ X : |f(x)| ≥ 1/n} is compact in X for all n ∈ N. It is not in general true that C∞ (X) is an ideal of C(X). We define those spaces X to be ideal space where C∞ (X) is an ideal of C(X). We have proved that nearly pseudocompact spaces are ideal spaces. For the converse, we introduced a property called “RCC” property and showed that an ideal space X is nearly pseudocompact if and only if X satisfies ”RCC” property. We further discussed some topological properties of ideal spaces.Mitra, B.; Chowdhury, D. (2021). Ideal spaces. Applied General Topology. 22(1):79-89. https://doi.org/10.4995/agt.2021.13608OJS7989221F. Azarpanah, M. Ghirati and A. Taherifar, Closed ideals in C(X) with different representations, Houst. J. Math. 44, no. 1 (2018), 363-383.F. Azarpanah and T. Soundarajan, When the family of functions vanishing at infinity is an ideal of C(X), Rocky Mountain J. Math. 31, no. 4 (2001), 1-8. https://doi.org/10.1216/rmjm/1021249434R. L. Blair and M. A. Swardson, Spaces with an Oz Stone-Cech compactification, Topology Appl. 36 (1990), 73-92. https://doi.org/10.1016/0166-8641(90)90037-3W. W. Comfort, On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. 131 (1968), 107-118. https://doi.org/10.1090/S0002-9947-1968-0222846-1J. M. Domínguez, J. Gómez and M. A. Mulero , Intermediate algebras between C*(X) and C(X) as rings of fractions of C*(X), Topology Appl. 77 (1997), 115-130. https://doi.org/10.1016/S0166-8641(96)00136-8R. Engelking, General Topology, Heldermann Verlag, Berlin , 1989L. Gillman and M. Jerison, Rings of Continuous Functions, University Series in Higher Math, Van Nostrand, Princeton, New Jersey,1960. https://doi.org/10.1007/978-1-4615-7819-2I. Glicksberg, Stone-Cech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369-382. https://doi.org/10.2307/1993177M. Henriksen, B. Mitra, C(X) can sometimes determine X without X being realcompact, Comment. Math. Univ. Carolina 46, no. 4 (2005), 711-720.M. Henriksen and M. Rayburn, On nearly pseudocompact spaces, Topology Appl. 11 (1980),161-172. https://doi.org/10.1016/0166-8641(80)90005-XT. Isiwata, On locally Q-complete spaces, II, Proc. Japan Acad. 35, no. 6 (1956), 263-267. https://doi.org/10.3792/pja/1195524322B. Mitra and S. K. Acharyya, Characterizations of nearly pseudocompact spaces and spaces alike, Topology Proceedings 29, no. 2 (2005), 577-594.M. C. Rayburn, On hard sets, General Topology and its Applications 6 (1976), 21-26. https://doi.org/10.1016/0016-660X(76)90004-0A. Rezaei Aliabad, F. Azarpanah and M. Namdari, Rings of continuous functions vanishing at infinity, Comm. Math. Univ. Carolinae 45, no. 3 (2004), 519-533.A. H. Stone, Hereditarily compact spaces, Amer. J. Math. 82 (1960), 900-914. https://doi.org/10.2307/2372948A. Wood Hager, On the tensor product of function rings, Doctoral dissertation, Pennsylvania State Univ., University Park, 1965

Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory

Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the ground-state electronic structure of a many-electron system. An important exact constraint for the construction of approximations is to recover the correct non-relativistic large-$Z$ expansions for the corresponding energies of neutral atoms with atomic number $Z$ and electron number $N=Z$, which are correct to leading order ($-0.221 Z^{5/3}$ and $-0.021 Z \ln Z$ respectively) even in the lowest-rung or local density approximation. We find that hydrogenic densities lead to $E_x(N,Z) \approx -0.354 N^{2/3} Z$ (as known before only for $Z \gg N \gg 1$) and $E_c \approx -0.02 N \ln N$. These asymptotic estimates are most correct for atomic ions with large $N$ and $Z \gg N$, but we find that they are qualitatively and semi-quantitatively correct even for small $N$ and for $N \approx Z$. The large-$N$ asymptotic behavior of the energy is pre-figured in small-$N$ atoms and atomic ions, supporting the argument that widely-predictive approximate density functionals should be designed to recover the correct asymptotics. It is shown that the exact Kohn-Sham correlation energy, when calculated from the pure ground-state wavefunction, should have no contribution proportional to $Z$ in the $Z\to \infty$ limit for any fixed $N$.Comment: This work has been accepted for publication at the Journal of Chemical Physics. Revisions: new Appendix A (former Appendix A is now Appendix B) discussing exact Kohn-Sham perturbation series for Ec. Added material discussing the Becke 1988 functional. More discussion of non-empirical functionals' recovery of the asymptotic series, and their accuracy in predicting atomic/molecular energie

A potent betulinic acid analogue ascertains an antagonistic mechanism between autophagy and proteasomal degradation pathway in HT-29 cells

Betulinic acid (BA), a member of pentacyclic triterpenes has shown important biological activities like anti-bacterial, anti-malarial, anti-inflammatory and most interestingly anticancer property. To overcome its poor aqueous solubility and low bioavailability, structural modifications of its functional groups are made to generate novel lead(s) having better efficacy and less toxicity than the parent compound. BA analogue, 2c was found most potent inhibitor of colon cancer cell line, HT-29 cells with IC50 value 14.9 μM which is significantly lower than standard drug 5-fluorouracil as well as parent compound, Betulinic acid. We have studied another mode of PCD, autophagy which is one of the important constituent of cellular catabolic system as well as we also studied proteasomal degradation pathway to investigate whole catabolic pathway after exploration of 2c on HT-29 cells. Mechanism of autophagic cell death was studied using fluorescent dye like acridine orange (AO) and monodansylcadaverin (MDC) staining by using fluorescence microscopy. Various autophagic protein expression levels were determined by Western Blotting, qRT-PCR and Immunostaining. Confocal Laser Scanning Microscopy (CLSM) was used to study the colocalization of various autophagic proteins. These were accompanied by formation of autophagic vacuoles as revealed by FACS and transmission electron microscopy (TEM). Proteasomal degradation pathway was studied by proteasome-Glo™ assay systems using luminometer.The formation of autophagic vacuoles in HT-29 cells after 2c treatment was determined by fluorescence staining – confirming the occurrence of autophagy. In addition, 2c was found to alter expression levels of different autophagic proteins like Beclin-1, Atg 5, Atg 7, Atg 5-Atg 12, LC3B and autophagic adapter protein, p62. Furthermore we found the formation of autophagolysosome by colocalization of LAMP-1 with LC3B, LC3B with Lysosome, p62 with lysosome. Finally, as proteasomal degradation pathway downregulated after 2c treatment colocalization of ubiquitin with lysosome and LC3B with p62 was studied to confirm that protein degradation in autophagy induced HT-29 cells follows autolysosomal pathway. In summary, betulinic acid analogue, 2c was able to induce autophagy in HT-29 cells and as proteasomal degradation pathway downregulated after 2c treatment so protein degradation in autophagy induced HT-29 cell

Mutations in HYAL2, Encoding Hyaluronidase 2, Cause a Syndrome of Orofacial Clefting and Cor Triatriatum Sinister in Humans and Mice.

Orofacial clefting is amongst the most common of birth defects, with both genetic and environmental components. Although numerous studies have been undertaken to investigate the complexities of the genetic etiology of this heterogeneous condition, this factor remains incompletely understood. Here, we describe mutations in the HYAL2 gene as a cause of syndromic orofacial clefting. HYAL2, encoding hyaluronidase 2, degrades extracellular hyaluronan, a critical component of the developing heart and palatal shelf matrix. Transfection assays demonstrated that the gene mutations destabilize the molecule, dramatically reducing HYAL2 protein levels. Consistent with the clinical presentation in affected individuals, investigations of Hyal2-/- mice revealed craniofacial abnormalities, including submucosal cleft palate. In addition, cor triatriatum sinister and hearing loss, identified in a proportion of Hyal2-/- mice, were also found as incompletely penetrant features in affected humans. Taken together our findings identify a new genetic cause of orofacial clefting in humans and mice, and define the first molecular cause of human cor triatriatum sinister, illustrating the fundamental importance of HYAL2 and hyaluronan turnover for normal human and mouse development

Bi doped CeO 2 oxide supported gold nanoparticle catalysts for the aerobic oxidation of alcohols

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