Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono’s resul...

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In a recent paper Folsom and Ono constructed a grid of Poincaré series of weights 3/2 and 1/2. They conjectured that the coefficients of the holomorphic parts of these series are rational integers. We prove that these coefficients are indeed rational numbers with bounded denominators.

The concept of right (left) quotient (or residual) of an ideal η by anideal ν of an L-subring μ of a ring R is introduced. The right (left) quotients areshown to be ideals of μ . It is proved that the right quotient [η :r ν ] of an idealη by an ideal ν of an L-subring μ is the largest ideal of μ such that[η :r ν ]ν ⊆ η . Most of the results pertaining to the notion of quotients(or residual) of ...

We study the impact of induced pressure fields on water flow and salt rejection in nanopores produced MoS2 membranes. observe that permeability are not impacted by distance between pores. This result contradicts continuous fluid mechanics calculations microfilters, which indicates existence hydrodynamic interactions adjacent pores increase mobility. Our results suggest at this nanoscale, do aff...

The concepts of free modules, projective modules, injective modules and the likeform an important area in module theory. The notion of free fuzzy modules was introducedby Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameriintroduced the concept of projective and injective L-modules. In this paper we give analternate definition for projective L-modules. We prove that e...