Assisted Common Information: Further Results

We presented assisted common information as a generalization of G\'acs-K\"orner (GK) common information at ISIT 2010. The motivation for our formulation was to improve upperbounds on the efficiency of protocols for secure two-party sampling (which is a form of secure multi-party computation). Our upperbound was based on a monotonicity property of a rate-region (called the assisted residual information region) associated with the assisted common information formulation. In this note we present further results. We explore the connection of assisted common information with the Gray-Wyner system. We show that the assisted residual information region and the Gray-Wyner region are connected by a simple relationship: the assisted residual information region is the increasing hull of the Gray-Wyner region under an affine map. Several known relationships between GK common information and Gray-Wyner system fall out as consequences of this. Quantities which arise in other source coding contexts acquire new interpretations. In previous work we showed that assisted common information can be used to derive upperbounds on the rate at which a pair of parties can {\em securely sample} correlated random variables, given correlated random variables from another distribution. Here we present an example where the bound derived using assisted common information is much better than previously known bounds, and in fact is tight. This example considers correlated random variables defined in terms of standard variants of oblivious transfer, and is interesting on its own as it answers a natural question about these cryptographic primitives.Comment: 8 pages, 3 figures, 1 appendix; to be presented at the IEEE International Symposium on Information Theory, 201

Detection Limit for Optically Sensing Specific Protein Interactions in Free-solution

Optical molecular sensing techniques are often limited by the refractive index change associated with the probed interactions. In this work, we present a closed form analytical model to estimate the magnitude of optical refractive index change arising from protein-protein interactions. The model, based on the Maxwell Garnett effective medium theory and first order chemical kinetics serves as a general framework for estimating the detection limits of optical sensing of molecular interactions. The model is applicable to situations where one interacting species is immobilized to a surface, as commonly done, or to emerging techniques such as Back-Scattering Interferometry (BSI) where both interacting species are un-tethered. Our findings from this model point to the strong role of as yet unidentified factors in the origin of the BSI signal resulting in significant deviation from linear optical response.Comment: 7 Page Manuscript + 14 Page Supplementary Informatio

Cosmology and thermodynamics of FRW universe with bulk viscous stiff fluid

We consider a cosmological model dominated by stiff fluid with a constant bulk viscosity. We classify all the possible cases of the universe predicted by the model and analyzing the scale factor, density as well as the curvature scalar. We find that when the dimensionless constant bulk viscous parameter is in the range $0 < \bar\zeta <6$ the model began with a Big Bang, and make a transition form the decelerating expansion epoch to an accelerating epoch, then tends to the de Sitter phase as $t\to \infty$. The transition into the accelerating epoch would be in the recent past, when $4<\bar\zeta<6.$ For $\bar\zeta>6$ the model doesn't have a Big Bang and suffered an increase in the fluid density and scalar curvature as the universe expands, which are eventually saturates as the scale factor $a \to \infty$ in the future. We have analyzed the model with statefinder diagnostics and find that the model is different from $\Lambda$CDM model but approaches $\Lambda$CDM point as $a \to \infty.$ We have also analyzed the status of the generalized second law of thermodynamics with apparent horizon as the boundary of the universe and found that the law is generally satisfied when $0 \leq \bar\zeta <6$ and for $\bar\zeta >6$ the law is satisfied when the scale factor is larger than a minimum value

Projective modules over overrings of polynomial rings

Let A be a commutative Noetherian ring of dimension d and let P be a projective R=A[X_1,\ldots,X_l,Y_1,\ldots,Y_m,\frac {1}{f_1\ldots f_m}]-module of rank r\geq max {2,dim A+1, where f_i\in A[Y_i]. Then (i) \EL^1(R\op P) acts transitively on Um(R\oplus P). In particular, P is cancellative. (ii) If A is an affine algebra over a field, then P has a unimodular element. (iii) The natural map \Phi_r : GL_r(R)/EL^1_r(R) \ra K_1(R) is surjective. (iv) Assume f_i is a monic polynomial. Then \Phi_{r+1} is an isomorphism. In the case of Laurent polynomial ring (i.e. f_i=Y_i), (i) is due to Lindel, (ii) is due to Bhatwadekar, Lindel and Rao and (iii, iv) is due to Suslin