ANECESSARY AND REQUIRED CONDITION OF HAMBURGER MOMENT PROBLEM

Objective: This present paper deals with necessary condition of Hamburger moment problem and polynomial which is not identically and non-negative sequence and semi-definite nature of a moment sequence. Materials and Methods: If we suppose that (sn)n>=0 is a sequence of real numbers, the moment problem on I consists of solving the following three problems: There exists a positive measure on I with moment(sn)n>=0. This positive measure uniquely determined by the moments(sn)n>=0. The moment problem on [0,1) is referred to as Hausdroff moment problem and the moment problem on R is called Hamburger moment problem and the [0, ∞) is called Stieltjes moment problem.   Results: For n be an arbitrary non-negative integer, and sub-interval tn in every sub-interval is not greater than such that . The function V(t) in terms of operator M is ifα(t) had infinitely many points of non-decrease, then for every positive polynomial P(t) not identically zero, 21 TT n n μ t d t 2 t d t t t t 1p 0 i TT n i 1 i n 1 i 21 . . 2 μ μ μ t V M m 20 n n0 k k k 0t d t P μ t P M Conclusion: For increasing function α(t) has a finite number of points of non-increase. Every non-negative sequence is either definite or semi-definite

Invited review: Physics potential of the ICAL detector at the India-based Neutrino Observatory (INO)

The upcoming 50 kt magnetized iron calorimeter (ICAL) detector at the India-based Neutrino Observatory (INO) is designed to study the atmospheric neutrinos and antineutrinos separately over a wide range of energies and path lengths. The primary focus of this experiment is to explore the Earth matter effects by observing the energy and zenith angle dependence of the atmospheric neutrinos in the multi-GeV range. This study will be crucial to address some of the outstanding issues in neutrino oscillation physics, including the fundamental issue of neutrino mass hierarchy. In this document, we present the physics potential of the detector as obtained from realistic detector simulations. We describe the simulation framework, the neutrino interactions in the detector, and the expected response of the detector to particles traversing it. The ICAL detector can determine the energy and direction of the muons to a high precision, and in addition, its sensitivity to multi-GeV hadrons increases its physics reach substantially. Its charge identification capability, and hence its ability to distinguish neutrinos from antineutrinos, makes it an efficient detector for determining the neutrino mass hierarchy. In this report, we outline the analyses carried out for the determination of neutrino mass hierarchy and precision measurements of atmospheric neutrino mixing parameters at ICAL, and give the expected physics reach of the detector with 10 years of runtime. We also explore the potential of ICAL for probing new physics scenarios like CPT violation and the presence of magnetic monopoles