Balancing Bounded Treewidth Circuits

Algorithmic tools for graphs of small treewidth are used to address questions in complexity theory. For both arithmetic and Boolean circuits, it is shown that any circuit of size $n^{O(1)}$ and treewidth $O(\log^i n)$ can be simulated by a circuit of width $O(\log^{i+1} n)$ and size $n^c$, where $c = O(1)$, if $i=0$, and $c=O(\log \log n)$ otherwise. For our main construction, we prove that multiplicatively disjoint arithmetic circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in arithmetic formulas of depth $O(k^2\log n)$. From this we derive the analogous statement for syntactically multilinear arithmetic circuits, which strengthens a theorem of Mahajan and Rao. As another application, we derive that constant width arithmetic circuits of size $n^{O(1)}$ can be balanced to depth $O(\log n)$, provided certain restrictions are made on the use of iterated multiplication. Also from our main construction, we derive that Boolean bounded fan-in circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in formulas of depth $O(k^2\log n)$. This strengthens in the non-uniform setting the known inclusion that $SC^0 \subseteq NC^1$. Finally, we apply our construction to show that {\sc reachability} for directed graphs of bounded treewidth is in $LogDCFL$

A case of multiple fibroid uterus, complete placenta praevia, antepartum haemorrhage, myomectomy and obstetric hysterectomy: a near miss

An interesting case of young unbooked, unregistered, primigravida with multiple fibroid uterus, placenta praevia type IV, presented with APH in haemorrhagic Shock, impacted large fibroid in Lower Uterine Segment and Multiple fibroids on Anterior wall, fundus for which myomectomy was performed. Preterm Caesarean Section with extremely low birth weight infant delivered followed by life-saving Emergency Obstetric hysterectomy. She had uneventful recovery without any complications of massive transfusion or surgery. Case is being reported as an obstetric near miss

Dark Energy and the Statistical Study of the Observed Image Separations of the Multiply Imaged Systems in the CLASS Statistical Sample

The present day observations favour a universe which is flat, accelerated and composed of $\sim 1/3$ matter (baryonic + dark) and $\sim 2/3$ of a negative pressure component, usually referred to as dark energy or quintessence. The Cosmic Lens All Sky Survey (CLASS), the largest radio-selected galactic mass scale gravitational lens search project to date, has resulted in the largest sample suitable for statistical analyses. In the work presented here, we exploit observed image separations of the multiply imaged lensed radio sources in the sample. We use two different tests: (1) image separation distribution function $n(\Delta\theta)$ of the lensed radio sources and (2) {\dtheta}_{\mathrm{pred}} vs {\dtheta}_{\mathrm{obs}} as observational tools to constrain the cosmological parameters $w$ and \Om. The results are in concordance with the bounds imposed by other cosmological tests.Comment: 20 pages latex; Modified " Results and Discussion " section, new references adde

The Bouncing Jet: A Newtonian Liquid Rebounding off a Free Surface

We find that a liquid jet can bounce off a bath of the same liquid if the bath is moving horizontally with respect to the jet. Previous observations of jets rebounding off a bath (e.g. Kaye effect) have been reported only for non-Newtonian fluids, while we observe bouncing jets in a variety of Newtonian fluids, including mineral oil poured by hand. A thin layer of air separates the bouncing jet from the bath, and the relative motion replenishes the film of air. Jets with one or two bounces are stable for a range of viscosity, jet flow rate and velocity, and bath velocity. The bouncing phenomenon exhibits hysteresis and multiple steady states.Comment: 9 pages, 7 figures. submitted to Physical Review

A New Lower Bound on the Maximum Number of Satisfied Clauses in Max-SAT and its Algorithmic Applications

A pair of unit clauses is called conflicting if it is of the form $(x)$, $(\bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for each UCF CNF formula with $m$ clauses we can simultaneously satisfy at least \pp m clauses, where \pp =(\sqrt{5}-1)/2. We improve the Lieberherr-Specker bound by showing that for each UCF CNF formula $F$ with $m$ clauses we can find, in polynomial time, a subformula $F'$ with $m'$ clauses such that we can simultaneously satisfy at least \pp m+(1-\pp)m'+(2-3\pp)n"/2 clauses (in $F$), where $n"$ is the number of variables in $F$ which are not in $F'$. We consider two parameterized versions of MAX-SAT, where the parameter is the number of satisfied clauses above the bounds $m/2$ and $m(\sqrt{5}-1)/2$. The former bound is tight for general formulas, and the later is tight for UCF formulas. Mahajan and Raman (J. Algorithms 31, 1999) showed that every instance of the first parameterized problem can be transformed, in polynomial time, into an equivalent one with at most $6k+3$ variables and $10k$ clauses. We improve this to $4k$ variables and $(2\sqrt{5}+4)k$ clauses. Mahajan and Raman conjectured that the second parameterized problem is fixed-parameter tractable (FPT). We show that the problem is indeed FPT by describing a polynomial-time algorithm that transforms any problem instance into an equivalent one with at most $(7+3\sqrt{5})k$ variables. Our results are obtained using our improvement of the Lieberherr-Specker bound above

Doping effects in the coupled, two-leg spin ladder BiCu2PO6

We report preparation, x-ray diffraction, magnetic susceptibility chi(T) and heat capacity Cp(T) measurements on the undoped samples as also samples with Zn-doped (S = 0) at Cu site, Ni doped (S = 1) at Cu site, and Ca-doped (holes) at Bi site in the coupled two-leg spin ladder system BiCu2PO6. While, Zn shows complete solid solubility, Ni could be doped to about 20% and Ca to about 15%. Magnetization and heat capacity data in the undoped compound point towards the existence of frustration effects. In all the samples, the chi(T) at low temperature increases with doping content. The Zn-induced susceptibility is smaller than that due to effective S=1/2 moments possibly due to frustrating next-nearest-neighbor interactions along the leg. For Zn content x > 0.01, chi(T) deviates from the Curie-law at low temperatures. The magnetic specific heat data Cm(T) for the Zn-doped samples show weak anomalies at low temperature in agreement with chi(T) behavior. The anomalies are suggestive of spin freezing at low-T. In contrast, prominent effects are observed in chi(T) and Cm(T) on Ni-doped samples. The zero-field-cooled (ZFC) and field-cooled (FC) chi(T) data are different from each other at low temperature unlike that for Zn doped samples, clearly indicating a transition to a spin-glass like phase. No anomalies were found in Ca- or Pb-doped samples.Comment: 16 pages, 9 figures, Submitted to J. Phy. Cond. Matte

CP violation in the decay mode B→πγγB\to \pi \gamma \gamma

Within the framework of Standard Model, the exclusive decay mode $B\to \pi \gamma \gamma$ is studied. Although the usual short distance contribution is small compared to the similar $B\to K\gamma\gamma$ mode, the process offers the possibility of studying the CP violation, a feature absent in the $B \to K$ counterpart.Comment: 11 page latex file including 2 ps figures. Typos corrected, minor changes. To appear in PR