Related papers: On Self-Descriptive Squares
A magic square of order n is an nxn square (matrix) whose entries are distinct nonnegative integers such that the sum of the numbers of any row and column is the same number, the magic constant. In this paper we introduce the concept of…
It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for…
We define a magic square to be a square matrix whose entries are nonnegative integers and whose rows, columns, and main diagonals sum up to the same number. We prove structural results for the number of such squares as a function of the…
Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of…
It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same…
For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the…
We study a new type of sequences whose elements are defined in terms of the position, sign and magnitude of another element of the sequence. The name ultra-recursive comes from the fact that these sequences possess terms that are generated…
Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…
In recreational mathematics, a normal magic square is an $n \times n$ square matrix whose entries are distinctly the integers $1 \ldots n^2$, such that each row, column, and major and minor traces sum to one constant $\mu$. It has been…
We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
In this survey, we discuss some basic problems concerning random matrices with discrete distributions. Several new results, tools and conjectures will be presented.
We begin by introducing an extension of the traditional abundancy index to imaginary quadratic rings with unique factorization. After showing that many of the properties of the traditional abundancy index continue to hold in our extended…
In the present paper, as a generalization of the classical periodic rings, we explore those rings whose elements are additively generated by two (or more) periodic elements by calling them additively periodic. We prove that, in some major…
We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing…
The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras, which studied in a previous paper of the authors (\cite{Local}). It is also a stand-alone object admits a rich study and various…
We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees…
We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main…
The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical…
A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…