In textile industry, marker planning is one of the main operations in the cutting fabric stage. Marker packing is usually used to maximize cloth exploitation and minimize its waste. In this research, a method is used based on new found meta-heuristic imperialist competitive algorithm (ICA) and Bottom-Left-Fill Algorithm (BLF) to achieve optimal marker packing. Function of the proposed method wa...

HIGHLIGHTS AxonPacking: Open-source software for simulating white matter microstructure.Validation on a theoretical disk packing problem.Reproducible and stable for various densities and diameter distributions.Can be used to study interplay between myelin/fiber density and restricted fraction. Quantitative Magnetic Resonance Imaging (MRI) can provide parameters that describe white matter micros...

A sequence π = (d1, . . . , dn) is graphic if there is a simple graph G with vertex set {v1, . . . , vn} such that the degree of vi is di. We say that graphic sequences π1 = (d (1) 1 , . . . , d (1) n ) and π2 = (d (2) 1 , . . . , d (2) n ) pack if there exist edge-disjoint n-vertex graphs G1 and G2 such that for j ∈ {1, 2}, dGj (vi) = d (j) i for all i ∈ {1, . . . , n}. Here, we prove several ...

The set of disk packing exponents is an interval equal to (£, 2] or [E, 2]. The set of triangle packing exponents is [log23,2]. The analogy strongly suggests that £ is attained and that E=S, the osculatory exponent whose value is known to lie in the interval 1.0300197<5<1.314534. 1. Literature survey and statement of theorems. Let U denote a disk of diameter 1 in the Euclidean plane. A countabl...

A sequence π = (d1, . . . , dn) is graphic if there is a simple graph G with vertex set {v1, . . . , vn} such that the degree of vi is di. We say that graphic sequences π1 = (d (1) 1 , . . . , d (1) n ) and π2 = (d (2) 1 , . . . , d (2) n ) pack if there exist edge-disjoint n-vertex graphs G1 and G2 such that for j ∈ {1, 2}, dGj (vi) = d (j) i for all i ∈ {1, . . . , n}. Here, we prove several ...

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19] who generalized the notion of s-dimensional measure to include non-intege...