Pebbling, Entropy and Branching Program Size Lower Bounds

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et. al. (2012). Proving a super-polynomial lower bound for the size of nondeterministic thrifty branching programs (NTBP) would separate $NL$ from $P$ for thrifty models solving the tree evaluation problem. First, we show that {\em Read-Once NTBPs} are equivalent to whole black-white pebbling algorithms thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of NTBPs called {\em Bitwise Independence}. The best known NTBPs (of size $O(k^{h/2+1})$) for the tree evaluation problem given by Cook et. al. (2012) are Bitwise Independent. As our main result, we show that any Bitwise Independent NTBP solving $TEP_{2}^{h}(k)$ must have at least $\frac{1}{2}k^{h/2}$ states. Prior to this work, lower bounds were known for NTBPs only for fixed heights $h=2,3,4$ (See Cook et. al. (2012)). We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent NTBP solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Z{\'a}k (2001) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds given by Cook et. al. (2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any $k$-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.Comment: 25 Pages, Manuscript submitted to Journal in June 2013 This version includes a proof for tight size bounds for (syntactic) read-once NTBPs. The proof is in the same spirit as the proof for size bounds for bitwise independent NTBPs present in the earlier version of the paper and is included in the journal version of the paper submitted in June 201

Transport properties of diluted magnetic semiconductors: Dynamical mean field theory and Boltzmann theory

The transport properties of diluted magnetic semiconductors (DMS) are calculated using dynamical mean field theory (DMFT) and Boltzmann transport theory. Within DMFT we study the density of states and the dc-resistivity, which are strongly parameter dependent such as temperature, doping, density of the carriers, and the strength of the carrier-local impurity spin exchange coupling. Characteristic qualitative features are found distinguishing weak, intermediate, and strong carrier-spin coupling and allowing quantitative determination of important parameters defining the underlying ferromagnetic mechanism. We find that spin-disorder scattering, formation of bound state, and the population of the minority spin band are all operational in DMFT in different parameter range. We also develop a complementary Boltzmann transport theory for scattering by screened ionized impurities. The difference in the screening properties between paramagnetic ($T>T_c$) and ferromagnetic ($T<T_c$) states gives rise to the temperature dependence (increase or decrease) of resistivity, depending on the carrier density, as the system goes from the paramagnetic phase to the ferromagnetic phase. The metallic behavior below $T_c$ for optimally doped DMS samples can be explained in the Boltzmann theory by temperature dependent screening and thermal change of carrier spin polarization.Comment: 15 pages, 15 figure

Interaction-tuned compressible-to-incompressible phase transitions in the quantum Hall systems

We analyze transitions between quantum Hall ground states at prominent filling factors $\nu$ in the spherical geometry by tuning the width parameter of the Zhang-Das Sarma interaction potential. We find that incompressible ground states evolve adiabatically under this tuning, whereas the compressible ones are driven through a first order phase transition. Overlap calculations show that the resulting phase is increasingly well described by appropriate analytic model wavefunctions (Laughlin, Moore-Read, Read-Rezayi). This scenario is shared by both odd ($\nu=1/3, 1/5, 3/5, 7/3, 11/5, 13/5$) and even denominator states ($\nu=1/2, 1/4, 5/2, 9/4$). In particular, the Fermi liquid-like state at $\nu=1/2$ gives way, at large enough value of the width parameter, to an incompressible state identified as the Moore-Read Pfaffian on the basis of its entanglement spectrum.Comment: 4 pages, 5 figures; modified version as appears in PR

Bosons in a double-well potential: Understanding the interplay between disorder and interaction in a simple model

We propose an exactly solvable model to reveal the physics of the interplay between interaction and disorder in bosonic systems. Considering interacting bosons in a double-well potential, in which disorder is mimicked by taking the energy level mismatch between the two wells to be randomly distributed, we find "two negatives make a positive" effect. While disorder or interaction by itself suppresses the phase coherence between the two wells, both together enhance the phase coherence. This model also captures several striking features of the disordered Bose-Hubbard model found in recent numerical simulations. Results at finite temperatures may help explain why a recent experiment did not find any evidence for the enhancement of phase coherence in a disordered bosonic system.Comment: Published version, 4 pages, 4 figure

Trends in Cardiac Pacemaker Batteries

Batteries used in Implantable cardiac pacemakers-present unique challenges to their developers and manufacturers in terms of high levels of safety and reliability. In addition, the batteries must have longevity to avoid frequent replacements. Technological advances in leads/electrodes have reduced energy requirements by two orders of magnitude. Micro-electronics advances sharply reduce internal current drain concurrently decreasing size and increasing functionality, reliability, and longevity. It is reported that about 600,000 pacemakers are implanted each year worldwide and the total number of people with various types of implanted pacemaker has already crossed 3 million. A cardiac pacemaker uses half of its battery power for cardiac stimulation and the other half for housekeeping tasks such as monitoring and data logging. The first implanted cardiac pacemaker used nickel-cadmium rechargeable battery, later on zinc-mercury battery was developed and used which lasted for over 2 years. Lithium iodine battery invented and used by Wilson Greatbatch and his team in 1972 made the real impact to implantable cardiac pacemakers. This battery lasts for about 10 years and even today is the power source for many manufacturers of cardiac pacemakers. This paper briefly reviews various developments of battery technologies since the inception of cardiac pacemaker and presents the alternative to lithium iodine battery for the near future