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Hurwitz type results for sums of squares and sums of triangular numbers
D. D. Somashekara,K. N. VIDYA 장전수학회 2019 Advanced Studies in Contemporary Mathematics Vol.29 No.4
Let rk(n) denote the number of representations of n as a sum of k squares and tk(n) denote the number of representations of n as a sum of k triangular numbers. Let r,(n) denote the number of representations of n as a sum of times a square and times another square and t,(n) denote the number of representations of n as a sum of times a triangular number and times another triangular number. We prove a number of results in which the generating function of r2(2k(an + b)), r4(2k(an + b)), r1,3(2k(an + b)), t2(2k(an + b)), t1,2(2k(an + b)), for various integer values of k, a and b, is a simple infinite product. We also obtain a relation between t1,3(3k+1n + 3k + 3k−1 · · · +3 + 1) and t1,3(n).
On some identities of Ramanujan found in his lost notebook
D.D. Somashekara,D. Mamta 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.16 No.2
In this paper we obtain an interesting identity using the well-knownq-binomial the-orem and Heine's transformation. Using this identity, we derive some identities ofRamanujan found in his lost notebook and also obtain some identities which areanalogous to the identities of Ramanujan along with some special cases.
HURWITZ TYPE RESULTS FOR CERTAIN REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES
D. D. Somashekara,THULASI.M.B. 장전수학회 2022 Proceedings of the Jangjeon mathematical society Vol.25 No.3
Let rα,β(n) denote the number of representations of n as a sum of α times a square and β times another square. In the recent past, a number of authors have obtained Hurwitz type results for representations of integers as sums of squares. Motivated by their works, in this paper we prove many results in which the generating function of r2,3(λk(an + b)), r2,4(an + b) and r1,5(λk(an + b)) for various non-negative integer values of λ, k, a and b are infinite products. To obtain our main results we use the theta function identities of Ramanujan found in chapter 16 of his second notebook.
On square sum labelings of graphs
D. D. Somashekara,C. R. Veena 장전수학회 2012 Proceedings of the Jangjeon mathematical society Vol.15 No.1
Abstract. Let G = (V,E) be an (n,m)-graph. A graph G is said to admit a square sum labeling, if its vertices can be labeled with distinct non-negative integers so that the values on the edges, obtained as the sum of the labels of their end vertices, are the first m squares. In this paper we show that certain classes of graphs admit square sum labeling and also that certain classes of graphs do not admit square sum labeling.
On some applications of the new forms of the reciprocity theorems
D. D. Somashekara,S. L. Shalini 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.3
In his lost notebook, Ramanujan has stated a beautiful two variable reciprocity theorem. In the recent past, three and four variable generalizations of the reciprocity theorem of Ramanujan were established. In this paper, we present a number of interesting applications of the new symmetric forms of the reciprocity theorems.
ON SOME PARTIAL THETA FUNCTION IDENTITIES OF RAMANUJAN FOUND IN HIS LOST NOTEBOOK
D. D. Somashekara,T. Kim,H. I. Kwon,S. L. Shalini 장전수학회 2015 Advanced Studies in Contemporary Mathematics Vol.25 No.1
Ramanujan has recorded several identities involving partial theta functions in his lost notebook. In 1981, Andrews has given proofs for sixteen of those identities by employing his general identity. In this paper, we obtain ten of those identities by an alternate method. We also deduce several interesting special cases.
On the reciprocity theorem of Ramanujan and its Generalizations
D. D. Somashekara,K. N. Murthy,S. L. Shalini 장전수학회 2012 Proceedings of the Jangjeon mathematical society Vol.15 No.3
In his lost notebook, Ramanujan has stated a beautiful two variable reciprocity theorem. In the recent past, three and four variable generalizations of the reciprocity theorem of Ramanujan were established. In this paper, we give an unified approach to the proofs of all the three reciprocity theorems. Some new q-gamma, q-beta and eta-function identities were also be deduced from the four variable reciprocity theorem.
On skew-generalised energy of digraphs
D. D. Somashekara,H. E. Ravi 장전수학회 2020 Proceedings of the Jangjeon mathematical society Vol.23 No.4
In this paper we introduce the concept of skew-generalised energy of directed graphs. We then obtain upper and lower bounds for skew-generalised energy of digraphs. Then we compute the skew-generalised energy of some graphs such as star digraph, complete bipartite digraph, the (Sm ^ P2) digraph, (n, 2n − 3) strong vertex graceful digraph and a crown digraph.
On pentagonal sum labelings of graphs
D. D. Somashekara,C. R. Veena 장전수학회 2012 Advanced Studies in Contemporary Mathematics Vol.22 No.3
Let G = (V,E) be an (n,m)-graph. A graph G is said to admit a pentagonal sum labeling, if its vertices can be labeled with distinct non-negative integers so that the values on the edges, obtained as the sum of the labels of their end vertices, are the first m pentagonal numbers. In this paper we show that certain classes of graphs admit pentagonal sum labeling and also that certain classes of graphs do not admit pentagonal sum labeling.
Sanjib Kumar Panda,Somashekara Nirvanashetty,Vivek A. Parachur,Nilima Mohanty 한국식품영양과학회 2020 Journal of medicinal food Vol.23 No.9
Premature ejaculation (PE) is one of the foremost sex-related health problems among men. The global occurrence ranges from 20% to 30%, according to various studies. PE has a great impact on the men's quality of life, with deleterious effects such as embarrassment, frustration, and feeling of incompetence. Considering the necessity of treatment of PE, this study was planned to compare the efficacy and safety of OLNP-05 versus placebo for treating subjects suffering from PE. In this randomized clinical study, 60 men with PE were enrolled and randomly assigned to receive either OLNP-05 or placebo one capsule twice daily for a period of 8 weeks. Subjects were evaluated during visits on day 1, day 28, and day 56. Mean change from baseline in intravaginal ejaculatory latency time (IELT), improvement in premature ejaculation profile (PEP), and Clinical Global Impression-Improvement scale (CGI-I) were used to assess the efficacy of treatment. P-value <.05 was considered significant. At the end of the treatment, the improvement in IELT score in the OLNP-05 group was remarkably higher than the placebo. Subjects in the OLNP-05 treatment group also reported significantly greater improvement in PEP subscale score. Majority of OLNP-05-treated subjects were found to be in the “much improved” category as per CGI-I assessment. The result confirms the safety and efficacy of OLNP-05, therefore suggesting that OLNP-05 may be a safe and effective intervention for the management of PE.