The (ir)regularity of Tor and Ext

We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules in high homological degrees. We show a similar result for Tor, under the additional hypothesis that high enough Tor modules are supported in dimension at most one; we then provide examples showing that the behaviour could be pretty hectic when the latter condition is not satisfied.Comment: 24 pages, Comments and suggestions are welcom

Generalized analytic model for rotational and anisotropic metasolids

An analytical approach is presented to model a metasolid accounting for anisotropic effects and rotational mode. The metasolid is made of either cylindrical or spherical hard inclusions embedded in a stiff matrix via soft claddings, and the analytical approach to study the composite material is a generalization of the method introduced by Liu \textit{et al.} [Phys. Rev. B, 71, 014103 (2005)]. It is shown that such a metasolid exhibits negative mass densities near the translational-mode resonances, and negative density of moment of inertia near the rotational resonances. The results obtained by this analytical and continuum approach are compared with those from discrete mass-spring model, and the validity of the later is discussed. Based on derived analytical expressions, we study how different resonance frequencies associated with different modes vary and are placed with respect to each other, in function of the mechanical properties of the coating layer. We demonstrate that the resonances associated with additional modes taken into account, that is, axial translation for cylinders, and rotations for both cylindrical and spherical systems, can occur at lower frequencies compared to the previously studied plane-translational modes.Comment: 30 pages, 10 figure

Nonlocal description of sound propagation through an array of Helmholtz resonators

A generalized macroscopic nonlocal theory of sound propagation in rigid-framed porous media saturated with a viscothermal fluid has been recently proposed, which takes into account both temporal and spatial dispersion. Here, we consider applying this theory capable to describe resonance effects, to the case of sound propagation through an array of Helmholtz resonators whose unusual metamaterial properties such as negative bulk moduli, have been experimentally demonstrated. Three different calculations are performed, validating the results of the nonlocal theory, relating to the frequency-dependent Bloch wavenumber and bulk modulus of the first normal mode, for 1D propagation in 2D or 3D periodic structures.Comment: 19 page

Nonlocal dynamics of dissipative phononic fluids

We describe the nonlocal effective properties of a two-dimensional dissipative phononic crystal made by periodic arrays of rigid and motionless cylinders embedded in a viscothermal fluid such as air. The description is based on a nonlocal theory of sound propagation in stationary random fluid/rigid media that was proposed by Lafarge and Nemati [Wave Motion 50, 1016 (2013)WAMOD90165-212510.1016/j.wavemoti.2013.04.007]. This scheme arises from a deep analogy with electromagnetism and a set of physics-based postulates including, particularly, the action-response procedures, whereby the effective density and bulk modulus are determined. Here, we revisit this approach, and clarify further its founding physical principles through presenting it in a unified formulation together with the two-scale asymptotic homogenization theory that is interpreted as the local limit. Strong evidence is provided to show that the validity of the principles and postulates within the nonlocal theory extends to high-frequency bands, well beyond the long-wavelength regime. In particular, we demonstrate that up to the third Brillouin zone including the Bragg scattering, the complex and dispersive phase velocity of the least-attenuated wave in the phononic crystal which is generated by our nonlocal scheme agrees exactly with that reproduced by a direct approach based on the Bloch theorem and multiple scattering method. In high frequencies, the effective wave and its associated parameters are analyzed by treating the phononic crystal as a random medium.United States. Office of Naval Research (N00014-13-1-0631

Linear Truncations Package for Macaulay2

We introduce the Macaulay2 package $\mathtt{LinearTruncations}$ for finding and studying the truncations of a multigraded module over a standard multigraded ring that have linear resolutions

Black Holes Algorithm: A Swarm Algorithm inspired of Black Holes for Optimization Problems

In this paper a swarms algorithms, for optimization problem is proposed. This algorithm is inspired of black holes. A black hole is a region of space-time whose gravitational field is so strong that nothing which enters it, not even light, can escape. Every black hole has mass, and charge.  In this Algorithm we suppose each solution of problem as a black hole and use of gravity force for global search and electrical force for local search. The proposed method is verified using several benchmark problems commonly used in the area of optimization. The experimental results on different benchmarks indicate that the performance of the proposed algorithm is better than    PSO (Particle Swarms Optimization), AFS (Artifitial Fish Swarm Algorithm) and RBH-PSO (random black hole particle swarm optimization Algorithm).DOI: http://dx.doi.org/10.11591/ij-ai.v2i3.322