An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

We show here a 2(Omega(root d.log N)) size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N = d(3) in our case) with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Sigma(i) Pi(j) Q(ij), where the Q(ij)'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that Sigma(i,j) (Number of monomials of Q(ij)) >= 2(Omega(root d.log N)). The above mentioned family, which we refer to as the Nisan-Wigderson design-based family of polynomials, is in the complexity class VNP. Our work builds on the recent lower bound results 1], 2], 3], 4], 5] and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of 6] and the N-Omega(log log (N)) lower bound in the independent work of 7].

Rayleigh-Taylor instability in a thermocline based thermal storage tank

Numerical simulations are performed to study the stability characteristics of a molten salt thermocline storage unit. Perturbations are introduced into a stable flow field in such a way as to make the top-fluid heavier than the fluid at the bottom, thereby causing a possible instability in the system. The evolution pattern of the various disturbances are examined in detail. Disturbances applied for short duration get decayed before they could reach the thermocline, whereas medium and long duration disturbances evolve into a ``falling spike'' or ``stalactite-like'' structure and destabilize the thermocline. Rayleigh Taylor instability is observed inside the storage tank. The effect of the duration, velocity and temperature of the disturbance on thermocline thickness and penetration length are studied. A quadratic time dependence of penetration length was observed. New perspectives on thermocline breakdown phenomena are obtained from the numerical flow field. (C) 2015 Elsevier Masson SAS. All rights reserved.