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We study a natural geometric variant of the classic Knapsack problem called 2D-Knapsack: we are given a set of axis-parallel rectangles and a rectangular bounding box, and the goal is to pack as many of these rectangles inside the box…
Given a set of squares and a strip of bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are…
In the Colored Bin Packing problem a set of items with varying weights and colors must be packed into bins of uniform weight limit such that no two items of the same color may be packed adjacently within a bin. We solve this problem for the…
This paper addresses the equal circle packing problem, and proposes an efficient Quasi-physical Quasi-human (QPQH) algorithm. QPQH is based on a modified Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm which we call the local BFGS and a…
We consider a recently proposed generalisation of the abelian hidden subgroup problem: the shifted subset problem. The problem is to determine a subset S of some abelian group, given access to quantum states of the form |S+x>, for some…
We study, via the replica method of disordered systems, the packing problem of hard-spheres with a square-well attractive potential when the space dimensionality, d, becomes infinitely large. The phase diagram of the system exhibits…
We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of…
We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called SSC and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show…
Accurate photometric and kinematic modelling of disc galaxies requires the inclusion of radiative transfer models. Due to the complexity of the radiative transfer equation (RTE), sophisticated techniques are required. Various techniques…
Defect-free atom arrays have emerged as a powerful and versatile platform for quantum sciences and technologies, offering high programmability and promising scalability. The arrays can be prepared by rearranging atoms from a partially…
We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have…
A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.
We present a unified treatment of the Fourier spectra of spherically symmetric nonlocal diffusion operators. We develop numerical and analytical results for the class of kernels with weak algebraic singularity as the distance between source…
The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by…
First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general…
Inspired by the linear programming method developed by Cohn and Elkies (Ann. Math. 157(2): 689-714, 2003), we introduce a new linear programming method to solve the sphere packing problem. More concretely, we consider sequences of auxiliary…
Order picking is the problem of collecting a set of products in a warehouse in a minimum amount of time. It is currently a major bottleneck in supply-chain because of its cost in time and labor force. This article presents two exact and…
The Container Relocation Problem (CRP) is concerned with finding a sequence of moves of containers that minimizes the number of relocations needed to retrieve all containers respecting a given order of retrieval. While the problem is known…
Optimal packing of spheres in $\mathbb R^d$ is studied by optimization of the energy $E$ (effective conductivity) of composites with ideally conducting spherical inclusions. It is demonstrated that the minimum of $E$ over locations of…
We have adapted Coupled Escape Probability, a new exact method of solving radiative transfer problems, for use in asymmetrical spherical situations. Our model is intended specifically for use in modeling optically thick cometary comae,…