Effect of Geometry and Infill on Strength of 3D Printed Planar Matrices for Matting Applications

3D (three-dimensional) printing was used as a rapid prototyping tool to determine the influence of cell geometry and infill materials on the physical properties of geometrically patterned matrices while subjected to compressive stress. Matrices of comparable patterns but varied scales and densities were fabricated from acrylonitrile butadiene styrene (ABS) plastic using fused deposition modeling (FDM) 3D printing. The test results confirm that some matrices reinforced by infill with sand, gravel, and mixtures of the two show better compressive strength than conventional concrete, and may find application in matting for airfield damage repair. The cell matrix geometry that demonstrated maximum strength (comparable with conventional concrete) was a hexagonal geometry with a relative density to solid plastic of 0.32 infilled with a mixture of sand and gravel. Additional data suggests that at larger scales, maximum strength comparable with conventional concrete could be achieved with even lower relative density.

Some patterned matrices with independent entries

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the [Formula: see text]th moment of the limit equals a weighted sum over different types of pair-partitions of the set [Formula: see text] and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.

Bulk behavior of Schur–Hadamard products of symmetric random matrices

We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur–Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSD of Schur–Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur–Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.

Algorithms for Finding Inverse of Two Patterned Matrices overZp

Circulant matrix families have become an important tool in network engineering. In this paper, two new patterned matrices overZpwhich include row skew first-plus-last right circulant matrix and row first-plus-last left circulant matrix are presented. Their basic properties are discussed. Based on Newton-Hensel lifting and Chinese remaindering, two different algorithms are obtained. Moreover, the cost in terms of bit operations for each algorithm is given.

SPECTRAL PROPERTIES OF RANDOM TRIANGULAR MATRICES

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.