Related papers: Efficient Generation of Binary Magic Squares
We describe a dynamic programming algorithm for exact counting and exact uniform sampling of matrices with specified row and column sums. The algorithm runs in polynomial time when the column sums are bounded. Binary or non-negative integer…
An associative magic square is a magic square such that the sum of any 2 cells at symmetric positions with respect to the center is constant. The total number of associative magic squares of order 7 is enormous, and thus, it is not…
Holonomic equations are recursive equations which allow computing efficiently numbers of combinatoric objects. R{\'e}my showed that the holonomic equation associated with binary trees yields an efficient linear random generator of binary…
We consider the problem of finding 4 rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background and exhibit one such quadruple.
There has been a rise in the popularity of algebraic methods for graph algorithms given the development of the GraphBLAS library and other sparse matrix methods. An exemplar for these approaches is Breadth-First Search (BFS). The algebraic…
Despite the astonishing progress in generative AI, 4D dynamic object generation remains an open challenge. With limited high-quality training data and heavy computing requirements, the combination of hallucinating unseen geometry together…
We use the $H$-matrix technology to compute the approximate square root of a covariance matrix in linear cost. This allows us to generate normal and log-normal random fields on general point sets with optimal cost. We derive rigorous error…
Monte Carlo simulations are one of the major tools in statistical physics, complex system science, and other fields, and an increasing number of these simulations is run on distributed systems like clusters or grids. This raises the issue…
We develop an algorithm for i) computing generalized regular $k$-point grids, ii) reducing the grids to their symmetrically distinct points, and iii) mapping the reduced grid points into the Brillouin zone. The algorithm exploits the…
In this paper, we study the concept of "binary color-coded magic squares" by assigning two distinct colors to the even and odd numbers within a magic square. We investigate the uniqueness of patterns within these squares using three…
Matrix multiplication is a fundamental computation in many scientific disciplines. In this paper, we show that novel fast matrix multiplication algorithms can significantly outperform vendor implementations of the classical algorithm and…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
High dimensional correlated binary data arise in many areas, such as observed genetic variations in biomedical research. Data simulation can help researchers evaluate efficiency and explore properties of different computational and…
In this paper we examine the descriptive potential of a combinatorial data structure known as "Generating Set" in constructing the boundary maps of a simplicial complex. By refining the approach of \cite{Dumas} in generating these maps, we…
From the standard procedure for constructing Feynman vacuum graphs in $\phi^4$ theory from the generating functional $Z$, we find a relation with sets of certain combinatorial matrices, which allows us to generate the set of all Feynman…
Binary embedding of high-dimensional data aims to produce low-dimensional binary codes while preserving discriminative power. State-of-the-art methods often suffer from high computation and storage costs. We present a simple and fast…
We show a simple method for constructing larger matrices but preserving the spectral radius. This yields a sufficient criteria for two square matrices of arbitrary dimension have the same spectral radius, a way to compare spectral radii of…
The Boolean product $R = P \cdot Q$ of two $\{ 0, 1\} \; m \times m \; $ matrices is $$R(j,k) = 1 \; \mathrm{\ IF\ for\ some\ } \; t \; \,P(j, t) = Q(t, k) = 1\; \; \mathrm{ELSE\ } \, R(j, k) = 0. $$ The near-optimal design reduces the…
The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we…
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.