Nearest constrained circular words

In this paper, we study circular words arising in the development of equipment using shields in brachytherapy. This equipment has physical constraints that have to be taken into consideration. From an algorithmic point of view, the problem can be formulated as follows: Given a circular word, find a constrained circular word of the same length such that the Manhattan distance between these two words is minimal. We show that we can solve this problem in pseudo polynomial time (polynomial time in practice) using dynamic programming

Analysis of the quasi-nonlocal approximation of linear and circular chains in the plane

We give an analysis of the stability and displacement error for linear and circular atomistic chains in the plane when the atomistic energy is approximated by the Cauchy-Born continuum energy and by the quasi-nonlocal atomistic-to-continuum coupling energy. We consider atomistic energies that include Lennard-Jones type nearest neighbor and next nearest neighbor pair-potential interactions. Previous analyses for linear chains have shown that the Cauchy-Born and quasi-nonlocal approximations reproduce (up to the order of the lattice spacing) the atomistic lattice stability for perturbations that are constrained to the line of the chain. However, we show that the Cauchy-Born and quasi-nonlocal approximations give a finite increase for the lattice stability of a linear or circular chain under compression when general perturbations in the plane are allowed. We also analyze the increase of the lattice stability under compression when pair-potential energies are augmented by bond-angle energies. Our estimates of the largest strain for lattice stability (the critical strain) are sharp (exact up to the order of the lattice scale). We then use these stability estimates and modeling error estimates for the linearized Cauchy-Born and quasi-nonlocal energies to give an optimal order (in the lattice scale) {\em a priori} error analysis for the approximation of the atomistic strain in $\ell^2_\epsilon$ due to an external force.Comment: 27 pages, 0 figure

Impact of Interaction Range and Curvature on Crystal Growth of Particles Confined to Spherical Surfaces

When colloidal particles form a crystal phase on a spherical template, their packing is governed by the effective interaction between them and the elastic strain of bending the growing crystal. For example, if growth commences under appropriate conditions, and the circular crystal that forms reaches a critical size, growth continues by incorporation of defects to alleviate elastic strain. Recently it was found experimentally that, if defect formation is somehow not possible, the crystal instead continues growing in ribbons that protrude from the original crystal. Here we report on computer simulations in which we observe both the formation of ribbons at short interaction ranges and packings that incorporate defects if the interaction is longer-ranged. The ribbons only form above some critical crystal size, below which the nucleus is roughly spherically shaped. We find that the scaling of the critical crystal size differs slightly from the one proposed by the Manoharan group, and reason this is because the actual process is a two-step heterogeneous nucleation of ribbons on top of roughly circular crystals.Comment: 24 pages, 11 figure

Modified Denavit-Hartenberg parameters for better location of joint axis systems in robot arms

The Denavit-Hartenberg parameters define the relative location of successive joint axis systems in a robot arm. A recent justifiable criticism is that one of these parameters becomes extremely large when two successive joints have near-parallel rotational axes. Geometrically, this parameter then locates a joint axis system at an excessive distance from the robot arm and, computationally, leads to an ill-conditioned transformation matrix. In this paper, a simple modification (which results from constraining a transverse vector between successive joint rotational axes to be normal to one of the rotational axes, instead of both) overcomes this criticism and favorably locates the joint axis system. An example is given for near-parallel rotational axes of the elbow and shoulder joints in a robot arm. The regular and modified parameters are extracted by an algebraic method with simulated measurement data. Unlike the modified parameters, extracted values of the regular parameters are very sensitive to measurement accuracy

Circular Coloring of Random Graphs: Statistical Physics Investigation

Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study circular coloring of random graphs using the cavity method. We identify two very interesting properties of this problem. For sufficiently many color and sufficiently low temperature there is a spontaneous breaking of the circular symmetry between colors and a phase transition forwards a ferromagnet-like phase. Our second main result concerns 5-circular coloring of random 3-regular graphs. While this case is found colorable, we conclude that the description via one-step replica symmetry breaking is not sufficient. We observe that simulated annealing is very efficient to find proper colorings for this case. The 5-circular coloring of 3-regular random graphs thus provides a first known example of a problem where the ground state energy is known to be exactly zero yet the space of solutions probably requires a full-step replica symmetry breaking treatment.Comment: 19 pages, 8 figures, 3 table

Electronic structure, magnetic ordering and X-ray magnetic circular dichroism in La1−x_{1-x}Prx_xCo2_2P2_2 phosphides

The electronic structure and magnetic ordering in La$_{1-x}$Pr$_x$Co$_2$P$_2$ ($x=$0, 0.25, and 1) phosphides have been studied theoretically using the fully relativistic spin-polarized Dirac linear muffin-tin orbital (LMTO) band-structure method. The X-ray absorption and X-ray magnetic circular dichroism spectra at the Co$L_{2,3}$ and Pr$M_{4,5}$ edges have been investigated theoretically within the framework of the LSDA+$U$ method. The core-hole effect in the final state as well as the effects of the electric quadrupole $E_2$ and magnetic dipole $M_1$ transitions have been investigated. Good agreement with experimental measurements has been found.Comment: 11 pages, 8 figure