COVERS OF GENERALIZED QUADRANGLES

We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order.Proc. Indian Acad. Sci. Math. Sci.126(2016), 591–612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from θ-covers, and consider related problems.

RETRACTED ARTICLE: Growth property at infinity of harmonic functions

This paper gives the growth property of certain harmonic functions at infinity in an n-dimensional cone, which generalize the results obtained by Huang and Qiao (Abstr. Appl. Anal. 2012:203096, 2012), Xu et al. (Bound. Value Probl. 2013:262, 2013), Yang and Ren (Proc. Indian Acad. Sci. Math. Sci. 124(2): 175-178, 2014) and Zhao and Yamada (J. Inequal. Appl. 2014:497, 2014) to the conical case.

FACTORIALS AND THE RAMANUJAN FUNCTION

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.

RETRACTED ARTICLE: A note on the boundary behavior for a modified Green function in the upper-half space

Motivated by (Xu et al. in Bound. Value Probl. 2013:262, 2013) and (Yang and Ren in Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014), in this paper we aim to construct a modified Green function in the upper-half space of the n-dimensional Euclidean space, which generalizes the boundary property of general Green potential.

A non-commutative Sobolev estimate and its application to spectral synthesis

In [M. K. Vemuri, Realizations of the canonical representation, Proc. Indian Acad. Sci. Math. Sci., 118 (2008), 115 –131], it was shown that the spectral synthesis problem for the Alpha transform is closely related to the problem of classifying realizations of the canonical representation (of the Heisenberg group). In this paper, it is shown that discrete sets are sets of spectral synthesis for the Alpha transform.

On the 𝐿1-Convergence of Modified Cosine Sums

Concerning the 𝐿1-convergence of modified cosine sums introduced by Kumari and Ram [Indian J. Pure Appl. Math. 19: 1101–1104, 1988], we generalise the results of Kumari, Ram [Indian J. Pure Appl. Math. 19: 1101–1104, 1988] and Bor [Proc. Indian Acad. Sci. Math. Sci. 102: 235–238, 1992]. Also a necessary and sufficient condition for the 𝐿1-convergence of cosine series has been deduced as a corollary for the said class.

Limitation theorems for some methods of summability

The object of this paper is to establish limitation theorems for the ordinary and also absolute generalized Nörlund methods which include some known results as special cases. We shall give a different proof of the recent result of S. Narang (Proc. Indian Acad. Sci. Sect. A 88 (1979), 115–123), and we get a generalization of the result of G. Das (J. London Math. Soc. 41 (1966), 685–692) which states the summability factors of the absolute Nörlund methods.