On the Volume of Sections of the Cube

We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1] n . We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of ℝ n onto a k-dimensional subspace that maximizes the volume of the intersection. We find the optimal upper bound on the volume of a planar section of the cube [−1, 1] n , n ≥ 2.

Spatial Distribution of Cementite Particles in Fe-0.67% C Steel

The particles arrangement in material space is represented by point field determined by the particle reference points, i.e., particle centers which can be described by the pair-correlation function (PCF) g3(r); r - correlation distance. Information about g3(r) can be obtained by stereological method based on the PCF g2(r), which describes the point field on the planar section determined by the centers of particle planar sections. In this paper the arrangement of cementite (Fe3C) particles during coarsening in Fe - 0,67%C steel at 715ºC in a form of two materials (A, B) of different microstructure of the coarse spheroidite (with different matrix grain size and particles position) was investigated. In material A, the particles are mainly at grain (subgrain) boundaries of fine-grained matrix. In material B, particles are mainly inside grains of coarse-grained ferrite. For material A, the empirical PCF g2(r) for a long time of coarsening (600 hours) is shifted towards larger r and is more flat near the g2(r) =1 than the one of coarsening for 50 hours. For material B, the g2(r) for both annealing times are not significantly different. This is consistent with the results of the probability density function f2(d) analysis for diameter (d) of the particle sections. Obtained PCF g2(r) are similar to the PCF g2(r) for planar section of the Stienen model. This means that for both type of microstructures the PCF - g3(r) =1, i.e., particles are distributed randomly in space and the sizes of the neighboring particles are correlated with each other.

The True Size Distribution of Ain Particles Obtained from Extraction Replicas

Determination of the morphology and size distribution of particles in a multiple phase material can be accomplished using opaque planar section optical microscopy or thin foil and extraction replica TEM analysis. In all cases, measurements are obtained using two-dimensional information from a planar section or through a projection of three-dimensional space. The two-dimensional size and morphology data obtained must be converted into the true three-dimensional data to be of further use. For spherical particles, methods of two- to three-dimensional size distribution conversion have been developed for planar sections, thin foils and extraction replicas. However, A1N particles precipitated in FCC iron are rod-shaped with a square cross section. To analyze these particles, an iterative method has been developed to determine the true three-dimensional size distribution of rod-shaped particles from their projected images obtained from extraction replicas. This method assumes that the particles are randomly oriented in the matrix phase and that they have a common aspect ratio.