Quantum Isometry Group for Spectral Triples with Real Structure

Given a spectral triple of compact type with a real structure in the sense of [Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of Connes' original definition to accommodate examples coming from quantum group theory) and references therein, we prove that there is always a universal object in the category of compact quantum group acting by orientation preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]) and also preserving the real structure of the spectral triple. This gives a natural definition of quantum isometry group in the context of real spectral triples without fixing a choice of 'volume form' as in [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]

Molecular motors: design, mechanism and control

Biological functions in each animal cell depend on coordinated operations of a wide variety of molecular motors. Some of the these motors transport cargo to their respective destinations whereas some others are mobile workshops which synthesize macromolecules while moving on their tracks. Some other motors are designed to function as packers and movers. All these motors require input energy for performing their mechanical works and operate under conditions far from thermodynamic equilibrium. The typical size of these motors and the forces they generate are of the order of nano-meters and pico-Newtons, respectively. They are subjected to random bombardments by the molecules of the surrounding aqueous medium and, therefore, follow noisy trajectories. Because of their small inertia, their movements in the viscous intracellular space exhibits features that are characteristics of hydrodynamics at low Reynold's number. In this article we discuss how theoretical modeling and computer simulations of these machines by physicists are providing insight into their mechanisms which engineers can exploit to design and control artificial nano-motors.Comment: 11 pages, including 8 embedded EPS figures; Invited article, accepted for Publication in "Computing in Science and Engineering" (AIP & IEEE

100 years of Einstein's theory of Brownian motion: from pollen grains to protein trains

Experimental verification of the theoretical predictions made by Albert Einstein in his paper, published in 1905, on the molecular mechanisms of Brownian motion established the existence of atoms. In the last 100 years discoveries of many facets of the ubiquitous Brownian motion has revolutionized our fundamental understanding of the role of {\it thermal fluctuations} in the exotic structures and complex dynamics exhibited by soft matter like, for example, colloids, gels, etc. The domain of Brownian motion transcends the traditional disciplinary boundaries of physics and has become an area of multi-disciplinary research. Brownian motion finds applications also in earth and environmental sciences as well as life sciences. Nature exploits Brownian motion for running many dynamical processes that are crucial for sustaining life. In the first one-third of this article I present a brief historical survey of the initial period, including works of Brown and Einstein. In the next one-third I introduce the main concepts and the essential theoretical techniques used for studying translational as well as rotational Brownian motions and the effects of time-independent potentials. In the last one-third of this article I discuss some contemporary problems on Brownian motion in time-dependent potentials, namely, {\it stochastic resonance} and {\it Brownian ratchet}, two of the hottest topics in this area of interdisciplinary research.Comment: 15 pages, LATEX, Based on the inaugural lecture in the Horizon Lecture Series organized by the Physics Society of I.I.T. Kanpur, in the "World Year of Physics 2005