Faster identification of optimal contraction sequences for tensor networks

The efficient evaluation of tensor expressions involving sums over multiple indices is of significant importance to many fields of research, including quantum many-body physics, loop quantum gravity, and quantum chemistry. The computational cost of evaluating an expression may depend strongly upon the order in which the index sums are evaluated, and determination of the operation-minimising contraction sequence for a single tensor network (single term, in quantum chemistry) is known to be NP-hard. The current preferred solution is an exhaustive search, using either an iterative depth-first approach with pruning or dynamic programming and memoisation, but these approaches are impractical for many of the larger tensor network Ansaetze encountered in quantum many-body physics. We present a modified search algorithm with enhanced pruning which exhibits a performance increase of several orders of magnitude while still guaranteeing identification of an optimal operation-minimising contraction sequence for a single tensor network. A reference implementation for MATLAB, compatible with the ncon() and multienv() network contractors of arXiv:1402.0939 and arXiv:1310.8023 respectively, is supplied.Comment: 25 pages, 12 figs, 2 tables, includes reference implementation of algorithm, v2.01. Update corrects the display of contraction sequences involving single-tensor traces (i.e. where an index in the input appears twice on the same tensor

Bends in the plane with variable curvature

Explicit formulae for planar variable curvature bends are constructed using Euler’s method of natural equations. The bend paths are expressed in terms of special functions. It is shown that the length of the different bend types varies linearly with increasing radius and that the curvature of variable curvature bends can be expressed as a multiple of the curvature of a circle

Immunoregulatory soluble CTLA-4 modifies effector T cell responses in systemic lupus erythematosus

Acknowledgments This work was supported by Arthritis Research UK (Grant no. 19282). We are grateful to Dr. Nick Fluck for his invaluable support in recruiting patients for the study, and Mrs. Vivien Vaughan for her invaluable expertise in recruiting study participants and maintaining ethical documentation.Peer reviewedPublisher PD

Inherent Structures for Soft Long-Range Interactions in Two-Dimensional Many-Particle Systems

We generate inherent structures, local potential-energy minima, of the "$k$-space overlap potential" in two-dimensional many-particle systems using a cooling and quenching simulation technique. The ground states associated with the $k$-space overlap potential are stealthy ({\it i.e.,} completely suppress single scattering of radiation for a range of wavelengths) and hyperuniform ({\it i.e.,} infinite wavelength density fluctuations vanish). However, we show via quantitative metrics that the inherent structures exhibit a range of stealthiness and hyperuniformity depending on the fraction of degrees of freedom that are constrained. Inherent structures in two dimensions typically contain five-particle rings, wavy grain boundaries, and vacancy-interstitial defects. The structural and thermodynamic properties of inherent structures are relatively insensitive to the temperature from which they are sampled, signifying that the energy landscape is relatively flat and devoid of deep wells. Using the nudged-elastic-band algorithm, we construct paths from ground-state configurations to inherent structures and identify the transition points between them. In addition, we use point patterns generated from a random sequential addition (RSA) of hard disks, which are nearly stealthy, and examine the particle rearrangements necessary to make the configurations absolutely stealthy. We introduce a configurational proximity metric to show that only small local, but collective, particle rearrangements are needed to drive initial RSA configurations to stealthy disordered ground states. These results lead to a more complete understanding of the unusual behaviors exhibited by the family of "collective-coordinate" potentials to which the $k$-space overlap potential belongs.Comment: 36 pages, 16 figure

The effects of different additives on the dielectric relaxation and the dynamic mechanical properties of urethane dimethacrylate

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73548/1/j.1365-2842.2000.00491.x.pd

The effects of moisture on the dielectric relaxation of urethane dimethacrylate polymer and composites

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/75059/1/j.1365-2842.2001.00669.x.pd