Efficient algorithms for three‐dimensional axial and planar random assignment problems

Beautiful formulas are known for the expected cost of random two‐dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural “Axial” and “Planar” versions, both of which are NP‐hard. For 3‐dimensional Axial random assignment instances of size n, the cost scales as Ω(1/ n), and a main result of the present paper is a linear‐time algorithm that, with high probability, finds a solution of cost O(n^–1+o(1)). For 3‐dimensional Planar assignment, the lower bound is Ω(n), and we give a new efficient matching‐based algorithm that with high probability returns a solution with cost O(n log n). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 160–196, 2015     

A Three‐Factor Product Construction for Mutually Orthogonal Latin Squares

It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, “triple products” of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish.     

Influence of Thermally Induced Chemorheological Changes on the Inflation of Spherical Elastomeric Membranes

When an elastomeric material is deformed and subjected to temperatures above some chemorheological value T _cr (near 100°C for natural rubber), its macromolecular structure undergoes time and temperature dependent chemical changes. The process continues until the temperature decreases below T _cr. Compared to the virgin material, the new material system has modified properties (often a reduced stiffness) and permanent set on removal of the applied load. A recently proposed constitutive theory is used to study the influence of chemorheological changes on the inflation of an initially isotropic spherical rubber membrane. The membrane is inflated while at a temperature below T _cr. We then look at the pressure response assuming the sphere's radius is held fixed while the temperature is increased above T _cr for a period of time and then returned to its original value. The inflation pressure during this process is expressed in terms of the temperature, representing entropic stiffening of the elastomer, and a time dependent property that represents the kinetics of the chemorheological change in the elastomer. When the membrane has been returned to its original temperature, it is shown to have a permanent set and a modified pressure-inflated radius relation. Their dependence on the initial inflated radius, material properties and kinetics of chemorheological change is studied when the underlying elastomeric networks are neo-Hookean or Mooney–Rivlin.     

Assessing and predicting changes in vegetation cover associated with military land use activities using field monitoring data at Fort Hood, Texas

We assessed short-term impacts of changes in military training load on vegetative cover at Fort Hood, TX. From 1989 to 1995, permanent field transects were monitored for vegetative cover and land use disturbance using standard army monitoring methods [Land Condition Trend Analysis (LCTA)]. Land use intensity (training load) was quantified and used to develop a model to predict future vegetation cover values. We found that standard Army monitoring methods detected changes in installation resources associated with changes in training load. Increased training loads were associated with increased measures of disturbance, decreased ground cover, and decreased aerial vegetative cover. We found that the spatial pattern of disturbance and vegetation cover remained relatively constant over the study period despite large variations in overall training load. Our model used the consistency in spatial cover patterns over time and the strong relationship between training load and vegetation cover to predict the impact of future training loads on vegetation.     

Stochastic Motion by Mean Curvature

We prove the existence of a continuously time‐varying subset K(t) of R ⁿ such that its boundary ∂K(t), which is a hypersurface, has normal velocity formally equal to the (weighted) mean curvature plus a random driving force. This is the first result in such generality combining curvature motion and stochastic perturbations. Our result holds for any C ² convex surface energy. The K(t) can have topological changes. The randomness is introduced by means of stochastic flows of diffeomorphisms generated by Brownian vector fields which are white in time but smooth in space. We work in the context of geometric measure theory, using sets of finite perimeter to represent K(t). The evolution is obtained as a limit of a time‐stepping scheme. Variational minimizations are employed to approximate the curvature motion. Stochastic calculus is used to prove global energy estimates, which in turn give a tightness statement of the approximating evolutions. (Accepted December 22, 1997)     

Finite amplitude vibrations of a Mooney-Rivlin oscillator

The solution for the finite amplitude, uniaxial motion of a Mooney-Rivlin oscillator on suspension, compression, or horizontal supports, and for arbitrary initial data is presented. The problem is unusual. Depending upon the initial data, the type of support, the amount of static stretch, and the value of a Mooney-Rivlin parameter, the solution may have one of three distinct possible periodic forms. The three cases are solved exactly, and simple bounds on the period of the finite motion are given. Some special situations are illustrated, both analytically and graphically. The effect of the amount of static stretch on the period of superimposed small amplitude oscillations also is described. The appropriate results are compared with those obtained previously for the neo-Hookean model.Dedicated to Clifford Truesdell, in admiration and gratitude, on the occasion of the twenty-fifth anniversary of the Society for Natural Philosophy.     

On the transverse vibration of a rubber string

The nearly constant pitch of the sound radiated by a sufficiently stretched rubber cord, a phenomenon observed independently by two experimenters at the turn of the century, is explained analytically for three ideal rubberlike material models. Transverse vibrational frequency data for three kinds of rubber strings, obtained with the aid of a novel laser apparatus, are compared with the analytical results. A numerical scheme is introduced to compute from the experimental data the apparent average number of links in a molecular chain of a rubberlike material from both simple tension and transverse frequency measurements. It is shown that the nearly flat frequency response is a molecular network finite extensibility effect that is controlled by the apparent number of links in a chain of the molecular network structure.     

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.