Sum of Products of Read-Once Formulas

We study limitations of polynomials computed by depth two circuits built over read-once formulas (ROFs). In particular, 1. We prove an exponential lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [CC,2009]. 2. We obtain an exponential lower bound on the size of arithmetic circuits computing sum of products of restricted ROFs of unbounded depth computing the permanent of an n by n matrix. The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by sqrt(n). Additionally, we restrict the product fan in to be bounded by a sub linear function. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial. 3. We also show an exponential lower bound for the above model against a polynomial in VP. 4. Finally we observe that the techniques developed yield an exponential lower bound on the size of sums of products of syntactically multilinear arithmetic circuits computing a product of variable disjoint linear forms where the bottom sum gate and product gates at the second level have fan in bounded by a sub linear function. Our proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [STOC 2004]

Lower Bounds for Multilinear Order-Restricted ABPs

Proving super-polynomial lower bounds on the size of syntactic multilinear Algebraic Branching Programs (smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in {x_1,...,x_n} appear along any source to sink path in an smABP can be viewed as a permutation in S_n. In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted: 1) Strict circular-interval ABPs: For every sub-program the index set of variables occurring in it is contained in some circular interval of {1,..., n}. 2) L-ordered ABPs: There is a set of L permutations (orders) of variables such that every source to sink path in the smABP reads variables in one of these L orders, where L 0. We prove exponential (i.e., 2^{Omega(n^delta)}, delta>0) lower bounds on the size of above models computing an explicit multilinear 2n-variate polynomial in VP. As a main ingredient in our lower bounds, we show that any polynomial that can be computed by an smABP of size S, can be written as a sum of O(S) many multilinear polynomials where each summand is a product of two polynomials in at most 2n/3 variables, computable by smABPs. As a corollary, we show that any size S syntactic multilinear ABP can be transformed into a size S^{O(sqrt{n})} depth four syntactic multilinear Sigma Pi Sigma Pi circuit where the bottom Sigma gates compute polynomials on at most O(sqrt{n}) variables. Finally, we compare the above models with other standard models for computing multilinear polynomials

Note on the advisability of recording higher derivatives of NMR absorption lines

When the amplitude of modulation, used to obtain the 1st deriv. of the NMR absorption line-​shape function after the phase-​sensitive detector, is not small, the exptl. recorded curve is actually a composite of the line-​shape function and the 1st deriv., and higher derivs. Errors introduced by assuming that the curve is the 1st deriv. curve alone and a method of eliminating this error by anal. of higher deriv. curves are discussed

Structure and stability of glucoamylase II from Aspergillus niger: a circular dichroism study

Glucoamylase II (EC 3.2.1.3) from Aspergillus niger has 31 % α-helix, 36 % β-structure and rest aperiodic structure at pH 4.8 as analysed by the method of Provencher and Glockner (1981,Biochemistry, 20,33). In the near ultra-violet circular dichroism spectrum the enzyme exhibits peaks at 304, 289, 282 and 257 nm and troughs at 285, 277 and 265 nm respectively. The enzyme activity and structure showed greater stability at pH 4.8 than at pH 7.0, were highly sensitive to alkaline pH but less sensitive to acid pH values. The enzyme retained most of its catalytic activity and structure even on partial removal of carbohydrate moieties by periodate treatment but was less stable at higher temperatures and storage at 30‡C. Reduction of the periodate treated enzyme did not reverse the loss of stability. Binding of the synthetic substrate, p-nitrophenyl-α-D-glucoside, perturbed the environment around aromatic amino acids and caused a decrease in the ordered structure

Pressure dependence of the chlorine NQR in three solid chloro anisoles

The 35Cl Nuclear Quadrupole Resonance (NQR) frequency (νQ) and spin lattice relaxation time (T1) in the three anisoles 2,3,4-trichloroanisole, 2,3,6-trichloroanisole and 3,5-dichloroanisole have been measured as a function of pressure upto 5.1 kbar at 300 K, and the data have been analysed to estimate the temperature coefficients of the NQR frequency at constant volume. All the three compounds show a non linear variation of the NQR frequency with pressure, the rate of which is positive and decreases with increasing pressure. In case of 3,5-dichloroanisole the value becomes negative in the higher range of pressure studied. The spin lattice relaxation time T1 in all the three compounds shows a weak dependence on pressure, indicating that the relaxation is mainly due to the torsional motions