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Certain identities for a continued fraction of Ramanujan
M. S. M. Naika,K. S. Bairy,S. Chandankumar 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.1
Ramanujan has recorded several continued fractions in his notebooks. In this paer, we establish several identities of a continued fraction of Ramanujan V(q). We also establish several relations between V(q) and V(qn) and several explicit evaluations of V(q).
CERTAIN MODULAR RELATIONS FOR REMARKABLE PRODUCT OF THETA-FUNCTIONS
M. S. MAHADEVA NAIKA,K. Sushan Bairy,N. P. SUMAN 장전수학회 2014 Proceedings of the Jangjeon mathematical society Vol.17 No.3
At scattered places of his second notebook, Ramanujan recorded several P Q mixed modular equations with four moduli. In this paper, we establish several new P Q mixed modular equations analogous to those recorded by Ramanujan in his notebooks. Employing these, we establish new modular relations for Ramanujan's remarkable product of theta-functions.
On some new Schläfli type mixed modular equations
M. S. M. Naika,K. S. Bairy 장전수학회 2011 Advanced Studies in Contemporary Mathematics Vol.21 No.2
On pages 86 and 88 of his rst notebook, Ramanujan recorded eleven Schla i-type modular equations for composite degrees. Out of eleven, ten have been proved by Berndt [5] using theory of modular forms. In this paper, we establish several new Schla i-type mixed modular equations.
Certain quotient of eta-function identities
M.S. Mahadeva Naika,M.C. Maheshkumar,K. Sushan Bairy 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.16 No.1
On page 212 in his lost notebok, Ramanujan defined a parameter λn by a certain quotient of Dedekind eta-functions at the argument q = exp(πpn/3). He then recorded a table of several values of λn := λn, 3. All these have been established by B. C. Berndt, H. Chan, S.-Y. Kang and L.-C. Zhang [4].λn,p at the argument q = exp(πpn/p ). We establish several interesting and new explicit evaluations for λn, p using Ramanujan-Weber class invariant,modular equations and mixed-modular equations.
General Formulas for Explicit Evaluations of Ramanujan's Cubic Continued Fraction
Naika, Megadahalli Sidda Naika Mahadeva,Maheshkumar, Mugur Chinna Swamy,Bairy, Kurady Sushan Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.3
On page 366 of his lost notebook [15], Ramanujan recorded a cubic continued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V ($q^3$) and also establish some explicit evaluations using the values of remarkable product of theta-function.
New identities for ratios of Ramanujan’s theta function
M. S. MAHADEVA NAIKA,S. Chandankumar,K. Sushan Bairy 장전수학회 2017 Advanced Studies in Contemporary Mathematics Vol.27 No.1
Ramanujan in his notebooks, has established several new modular equation which he denoted as P and Q. In this paper, we establish several new identities for ratios of Ramanujan’s theta function involving φ(q). We establish some new explicit evaluations for the ratios of Ramanujan’s theta function. We also establish some new modular relations for a continued fraction of order twelve H(q) with H(qn) for n =2, 4, 6, 8, 10, 12, 14 and 16.
Lipika Rani Bairi,Darothi Bairagi,Partha Duley,Sumantra Mandal 대한금속·재료학회 2023 METALS AND MATERIALS International Vol.29 No.7
The current study investigates the effect of long-term immersion on the corrosion response and the consequent passive filmformation process of lean duplex stainless steel 2101 in 3.5 wt% NaCl solution. The corrosion behavior of the alloy has beenmonitored through long term immersion tests at different time intervals, viz. 0, 3, 5, 10, 20, 40, and 70 days. The electrochemicaltests, such as potentiodynamic polarization and electrochemical impedance spectroscopy, have been performedafter each immersion period to understand the change in corrosion rate and passivation behaviour, respectively, with respectto immersion duration. With the increase in immersion duration, the pitting potential (Epit) gradually shifts towards a noblerdirection with a simultaneous reduction in corrosion rate (rcorr). This has been attributed to the gradual increase in the thicknessof the passive film with the increase in duration of immersion, as revealed through the EIS analysis. The passive filmis found to become nearly stable after 40 days of immersion of the specimen in 3.5 wt% NaCl solution. Interestingly, thethickness and stability of the passive film obtained after 40 days of immersion in 3.5 wt% NaCl solution are comparable tothe passive film that is formed in mildly passivating media, namely borate buffer solution. An investigation of post immersionsurface morphology has also confirmed the gradual improvement in the uniformity and compactness of the oxide-richpassive layer with increase in immersion period, which eventually led to improved corrosion resistance of the alloy in thesaline environment.