Lifted Multiplicity Codes and the Disjoint Repair Group Property

Lifted Reed Solomon Codes (Guo, Kopparty, Sudan 2013) were introduced in the context of locally correctable and testable codes. They are multivariate polynomials whose restriction to any line is a codeword of a Reed-Solomon code. We consider a generalization of their construction, which we call lifted multiplicity codes. These are multivariate polynomial codes whose restriction to any line is a codeword of a multiplicity code (Kopparty, Saraf, Yekhanin 2014). We show that lifted multiplicity codes have a better trade-off between redundancy and a notion of locality called the t-disjoint-repair-group property than previously known constructions. More precisely, we show that, for t <=sqrt{N}, lifted multiplicity codes with length N and redundancy O(t^{0.585} sqrt{N}) have the property that any symbol of a codeword can be reconstructed in t different ways, each using a disjoint subset of the other coordinates. This gives the best known trade-off for this problem for any super-constant t < sqrt{N}. We also give an alternative analysis of lifted Reed Solomon codes using dual codes, which may be of independent interest

Exploring A Multi-Scale Method for Molecular Simulations in Continuum Solvent Model: Explicit Simulation of Continuum Solvent As An Incompressible Fluid

A multi-scale framework was recently proposed for more realistic molecular dynamics simulations in continuum solvent models by coupling a molecular mechanics treatment of solute with a fluid mechanics treatment of solvent, where we formulated the physical model and developed a numerical fluid dynamics integrator. In this study, we incorporated the fluid dynamics integrator with the Amber simulation engine to conduct atomistic simulations of biomolecules. At this stage of the development, only nonelectrostatic interactions, i.e., van del Waals and hydrophobic interactions are included in the multi-scale model. Nevertheless numerical challenges exist in accurately interpolating the highly nonlinear van del Waals term when solving the finite-difference fluid dynamics equations. We were able to bypass the challenge rigorously by merging the van del Waals potential and pressure together when solving the fluid dynamics equations and by considering its contribution in the free-boundary condition analytically. The multi-scale simulation engine was first validated by reproducing the solute-solvent interface of a single atom with analytical solution. Next, we performed the relaxation simulation of a restrained symmetrical monomer and observed a symmetrical solvent interface at equilibrium with detailed surface features resembling those found on the solvent excluded surface. Four typical small molecular complexes were then tested, both volume and force balancing analysis showing that these simple complexes can reach equilibrium within the simulation time window. Finally, we studied the quality of the multi-scale solute-solvent interfaces for the four tested dimer complexes and found they agree well with the boundaries as sampled in the explicit water simulations.Comment: 34 pages, 6 figures, 1 tabl

Modeling of fibrous biological tissues with a general invariant that excludes compressed fibers

Dispersed collagen fibers in fibrous soft biological tissues have a significant effect on the overall mechanical behavior of the tissues. Constitutive modeling of the detailed structure obtained by using advanced imaging modalities has been investigated extensively in the last decade. In particular, our group has previously proposed a fiber dispersion model based on a generalized structure tensor. However, the fiber tension–compression switch described in that study is unable to exclude compressed fibers within a dispersion and the model requires modification so as to avoid some unphysical effects. In a recent paper we have proposed a method which avoids such problems, but in this present study we introduce an alternative approach by using a new general invariant that only depends on the fibers under tension so that compressed fibers within a dispersion do not contribute to the strain-energy function. We then provide expressions for the associated Cauchy stress and elasticity tensors in a decoupled form. We have also implemented the proposed model in a finite element analysis program and illustrated the implementation with three representative examples: simple tension and compression, simple shear, and unconfined compression on articular cartilage. We have obtained very good agreement with the analytical solutions that are available for the first two examples. The third example shows the efficacy of the fibrous tissue model in a larger scale simulation. For comparison we also provide results for the three examples with the compressed fibers included, and the results are completely different. If the distribution of collagen fibers is such that it is appropriate to exclude compressed fibers then such a model should be adopted