Mathematics
http://repository.iitgn.ac.in/handle/123456789/547
Sat, 17 Feb 2018 16:07:46 GMT2018-02-17T16:07:46ZGeneralized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for ζ(2m+1)
http://repository.iitgn.ac.in/handle/123456789/3460
Generalized Lambert series, Raabe's integral and a two-parameter generalization of Ramanujan's formula for ζ(2m+1)
Dixit, Atul; Gupta, Rajat; Kumar, Rahul; Maji, Bibekananda
A comprehensive study of the generalized Lambert series ∑n=1∞nN−2hexp(−anNx)1−exp(−nNx),0<a≤1, x>0, N∈N and h∈Z, is undertaken. Two of the general transformations of this series that we obtain here lead to two-parameter generalizations of Ramanujan's famous formula for ζ(2m+1), m>0 and the transformation formula for logη(z). Numerous important special cases of our transformations are derived. An identity relating ζ(2N+1),ζ(4N+1),⋯,ζ(2Nm+1) is obtained for N odd and m∈N. Certain transcendence results of Zudilin- and Rivoal-type are obtained for odd zeta values and generalized Lambert series. A criterion for transcendence of ζ(2m+1) and a Zudilin-type result on irrationality of Euler's constant γ are also given. New results analogous to those of Ramanujan and Klusch for N even, and a transcendence result involving ζ(2m+1−1N), are obtained.
Mon, 01 Jan 2018 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/34602018-01-01T00:00:00ZZero knowledge and circuit minimization
http://repository.iitgn.ac.in/handle/123456789/3443
Zero knowledge and circuit minimization
Allender, Eric; Das, Bireswar
We show that every problem in the complexity class (Statistical Zero Knowledge) is efficiently reducible to the Minimum Circuit Size Problem (). In particular Graph Isomorphism lies in .
This is the first theorem relating the computational power of Graph Isomorphism and , despite the long history these problems share, as candidate -intermediate problems.
Sun, 01 Oct 2017 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/34432017-10-01T00:00:00ZUnboundedness of Betti numbers of certain monomial curves in A4
http://repository.iitgn.ac.in/handle/123456789/3409
Unboundedness of Betti numbers of certain monomial curves in A4
Mehta, Ranjana; Saha, Joydip; Sengupta, Indranath
Bresinsky defined a class of monomial curves in A4 with the property that the minimal number of generators or the first Betti number of the defining ideal is unbounded above. We prove that the same behaviour of unboundedness is true for all the Betti numbers and construct an explicit minimal free resolution for this class.
Mon, 01 Jan 2018 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/34092018-01-01T00:00:00ZZeros of combinations of the Riemann ?-function and the confluent hypergeometric function on bounded vertical shifts
http://repository.iitgn.ac.in/handle/123456789/3359
Zeros of combinations of the Riemann ?-function and the confluent hypergeometric function on bounded vertical shifts
Dixit, Atul; Kumar, Rahul; Maji, Bibekananda; Zaharescu, Alexandru
Fri, 01 Dec 2017 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/33592017-12-01T00:00:00Z