Ideals in B1(X) and residue class rings of B1(X) modulo an ideal

[EN] This paper explores the duality between ideals of the ring B1(X) of all real valued Baire one functions on a topological space X and typical families of zero sets, called ZB-filters, on X. As a natural outcome of this study, it is observed that B1(X) is a Gelfand ring but non-Noetherian in general. Introducing fixed and free maximal ideals in the context of B1(X), complete descriptions of the fixed maximal ideals of both B1(X) and B1* (X) are obtained. Though free maximal ideals of B1(X) and those of B1* (X) do not show any relationship in general, their counterparts, i.e., the fixed maximal ideals obey natural relations. It is proved here that for a perfectly normal T1 space X, free maximal ideals of B1(X) are determined by a typical class of Baire one functions. In the concluding part of this paper, we study residue class ring of B1(X) modulo an ideal, with special emphasize on real and hyper real maximal ideals of B1(X).Deb Ray, A.; Mondal, A. (2019). Ideals in B1(X) and residue class rings of B1(X) modulo an ideal. Applied General Topology. 20(2):379-393. https://doi.org/10.4995/agt.2019.11417SWORD379393202A. Deb Ray and A. Mondal, On rings of Baire one functions, Applied Gen. Topol. 20, no. 1 (2019), 237-249. https://doi.org/10.4995/agt.2019.10776J. P. Fenecios and E. A. Cabral, On some properties of Baire-1 functions, Int. Journal of Math. Analysis 7, no. 8 (2013), 393-402. https://doi.org/10.12988/ijma.2013.13035L. Gillman and M. Jerison, Rings of Continuous Functions, New York: Van Nostrand Reinhold Co., 1960. https://doi.org/10.1007/978-1-4615-7819-2J. R. Munkres, Topology, Second edition, Pearson Education, Delhi, 2003.L. Vesely, Characterization of Baire-one functions between topological spaces, Acta Universitatis Carolinae. Mathematica et Physica 33, no. 2 (1992), 143-156

More on the rings B1(X)B_1(X) and B1∗(X)B_1^*(X)

This paper focuses mainly on the bounded Baire one functions. The uniform norm topology arises from the $\sup$-norm defined on the collection $B_1^*(X)$ of all bounded Baire one functions. With respect to this topology, $B_1^*(X)$ is a topological ring as well as a topological vector space. It is proved that under uniform norm topology, the set of all positive units (respectively, negative units) form an open set and as a consequence of it, every maximal ideal is closed in $B_1^*(X)$. Since the natural extension of uniform norm topology on $B_1(X)$, when $B_1^*(X) \neq B_1(X)$, does not show up these features, a topology called $m_B$-topology is defined on $B_1(X)$ suitably to achieve these results on $B_1(X)$. It is proved that the relative $m_B$ topology coincides with the uniform norm topology on $B_1^*(X)$ if and only if $B_1(X) = B_1^*(X)$. Moreover, $B_1(X)$ with $m_B$-topology is 1st countable if and only if $B_1(X) = B_1^*(X)$. The last part of the paper establishes a correspondence between the ideals of $B_1^*(X)$ and a special class of $Z_B$-filters, called $e_B$-filters on a normal topological space $X$. It is also observed that for normal spaces, the cardinality of the collection of all maximal ideals of $B_1(X)$ and those of $B_1^*(X)$ are same

Gamma Radiation Assisted Grafting of Low Surface Energy Long Chain Molecule on Various Types of Polyolefin Elastomers (POE)

Surface modification of polymers can lead to changes of properties such as surface wettability, adhesion, bioactivity and biocompatibility. Here, a long chain, low surface energy monomer, docecyl methacrylate (DMA), which is difficult to graft by chemical mean was grafted onto three different types of ‘Engages’ (a polyolefin elastomer, POE) in ethanol/water medium by gamma radiation. Total dose and DMA concentration were optimized for the desired amount of low degree grafting (less than 15wt%). The surface wetting properties towards water were decreased for all Engage surfaces after grafting. Mechanical properties of the base polymers remained almost unchanged after grafting

Transition Metal Oxide-Based Nano-materials for Energy Storage Application

With improvement of global economy, the fatigue of energy becomes inevitable in twenty-first century. It is expected that the increase of world energy requirements will be triple at the end of this century. Thus, there is an imperative need for development of renewable energy sources and storage systems. Among various energy storage systems, supercapacitors are ascertained one of the most significant storage devices. But the development of supercapacitor devices with high power and energy density are the greatest challenges for modern research. In this article, transition metal oxides such as TiO2-V2O5, NiMn2O4 etc. with porous structure are considered as high performance supercapacitors electrode. The effects of its structural, morphological and electrochemical properties have been studied extensively. A TiO2-V2O5 and NiMn2O4 based electrode delivered specific capacitance of 310 and 875 F g−1, respectively at a scan rate 2 mV s−1. This TiO2-V2O5 based asymmetric supercapacitor also exhibits excellent device performance with specific energy 20.18 W h kg−1 at specific power 5.94 kW kg−1, and retained 88.0% specific capacitance at current density of 10 A g−1 after 5000 cycles

Poverty in Tribal Dominated Economy: Dimensions and Proximate Determinants

Inequality in individual income, assets or occupational status among society work as a hindrance to realize objective of balanced economic development. Also with the modernisation of society and integration of economies the question of social fluidity is becoming essential aspect of development. Certainly the issue of disparity and its impact on individual present and future has a bearing on intergenerational mobility. Historically some groups belong to lower strata of society due to economic and or social discrimination leading to lower income and asset possession as well as capability formation. It is often found that backward social classes are excluded from the process of capability formation and income-earning opportunities due to various forms of discrimination. This exclusion and backwardness transcends the boundary of the current generation and spills over to successive generations as well. As a result Intergenerational Mobility in terms of both Education and Occupation is very low among backward classes, trapping them in multidimensional poverty that continues across generations. This paper focuses on one of the most backward region of West Bengal, the district of Purulia which is a resource poor district dominated by Scheduled Tribes. Similar in socio-economic and geographic profile to its contagious areas of Jharkhand rather than that of West Bengal, this district has witnessed unprecedented severity of poverty and occupational stagnancy. We have tried to link these two issues by examining the deprivation aspect with question of intergenerational mobility. We measure poverty in Purulia district at the block level composing indices like Modified Capability Poverty Measure and Human Poverty Index (HPI) to get the intra district scenario. Then the intergenerational mobility in education and occupation status have been taken into account to explain the present scenario of the district

Sarcoma botryoides in a 14 year old girl: a rare case

Embryonal rhabdomyosarcoma (Sarcoma botryoides) of the uterine cervix is an uncommon entity. Because of extreme rarity its discussion has mainly been in the light of individual case reports. We report a case of a 14-year old female who presented with irregular vaginal bleeding and cervical polyp. Her biopsy specimen confirmed sarcoma botryoides and she underwent abdominal hysterectomy with wide excision of vaginal cuff after a multidisciplinary consultation

On rings of Baire one functions

[EN] This paper introduces the ring of all real valued Baire one functions, denoted by B1(X) and also the ring of all real valued bounded Baire one functions, denoted by B∗1(X). Though the resemblance between C(X) and B1(X) is the focal theme of this paper, it is observed that unlike C(X) and C∗(X) (real valued bounded continuous functions), B∗1 (X) is a proper subclass of B1(X) in almost every non-trivial situation. Introducing B1-embedding and B∗1-embedding, several analogous results, especially, an analogue of Urysohn’s extension theorem is established.The authors are thankful to Professor S. K. Acharyya, Retired Professor, Department of Pure Mathematics, University of Calcutta for his continuous support and encouragement. We also acknowledge the learned referee for his valuable comments. The second author is supported by Council of Scientific and Industrial Research, HRDG, India. Sanction No.- 09/028(0998)/2017-EMR-1.Deb Ray, A.; Mondal, A. (2019). On rings of Baire one functions. Applied General Topology. 20(1):237-249. https://doi.org/10.4995/agt.2019.10776SWORD237249201J. P. Fenecios and E. A. Cabral, On some properties of Baire-1 functions, Int. Journal of Math. Analysis 7, no. 8 (2013), 393-402. https://doi.org/10.12988/ijma.2013.13035I. Gelfand and A. Kolmogoroff, On rings of continuous functions on topological spaces, Dokl. Akad. Nauk SSSR 22 (1939), 11-15.L. Gillman and M. Jerison, Rings of Continuous Functions, New York: Van Nostrand Reinhold Co., 1960. https://doi.org/10.1007/978-1-4615-7819-2E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948), 54-99. https://doi.org/10.1090/S0002-9947-1948-0026239-9P. Y. Lee, W. K. Tang and D. Zhao, An equivalent definition of functions of the first Baire class, Proc. Amer. Math. Soc. 129, no. 8 (2000), 2273-2275. https://doi.org/10.1090/S0002-9939-00-05826-3M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-481. https://doi.org/10.1090/S0002-9947-1937-1501905-7L. Vesely, Characterization of Baire-one functions between topological spaces, Acta Universitatis Carolinae Mathematica et Physica 33, no. 2 (1992), 143-156.D. Zhao, Functions whose composition with Baire class one functions are Baire class one, Soochow Journal of Mathematics 33, no. 4 (2007), 543-551

BRUCELLAR UVEITIS IN EASTERN INDIA

Objective: Brucellosis- one of the major zoonotic diseases, still remains an uncontrolled problem, in regions of high endemicity. Ophthalmic brucellosis is not studied and overlooked in most developing countries.  Considering the severe outcome of undiagnosed ophthalmic brucellosis, in this paper we made attempts to find out whether this disease still remains a health problem in a South East Asian developing country, where study of this disease is largely neglected.  Methods This study was carried out over a one year period from January 2015 to December 2015. Blood samples were collected from clinically confirmed cases of uveitis and they were subjected to five serological and one genus specific molecular investigations, for the detection of Brucella infection.Results Out of 20 uveitis cases, 4 (20%) cases confirmed as brucellar uveitis, by serological tests followed by PCR confirmation. After treatment of brucellosis all the four patients were recovered uneventfully.Conclusion: Brucella infection involving the eye is still a significant problem in South East Asian countries; hence in all uveitis cases in this reason brucellosis should be excluded by available laboratory tests.Government of India (Dept. of Biotechnology the major zoonotic diseases, still remains an uncontrolled problem, in regions of high endemicity. Ophthalmic brucellosis is not studied and overlooked in most developing countries.  Considering the severe outcome of undiagnosed ophthalmic brucellosis, in this paper we made attempts to find out whether this disease still remains a health problem in a South East Asian developing country, where study of this disease is largely neglected.   Methods This study was carried out over a one year period from January 2015 to December 2015. Blood samples were collected from clinically confirmed cases of uveitis and they were subjected to five serological and one genus specific molecular investigations, for the detection of Brucella infection.Results Out of 20 uveitis cases, 4 (20%) cases confirmed as brucellar uveitis, by serological tests followed by PCR confirmation. After treatment of brucellosis all the four patients were recovered uneventfully.Conclusion: Brucella infection involving the eye is still a significant problem in South East Asian countries; hence in all uveitis cases in this reason brucellosis should be excluded by available laboratory tests