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# Multidimensional matrices uniquely recovered by their lines

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## Abstract

We provide a method to determine if a q-ary multidimensional matrix is lonesum or not by using properties of line sums of lonesum multidimensional matrices. In particular, we establish a graphic method that uses edge-colored graphs to determine if a binary multidimensional matrix is lonesum or not. We also provide two methods to determine if a q-ary multidimensional matrix is lonestructure or not. The first one uses properties of line structures of lonestructure multidimensional matrices and the second one uses edge-colored directed multigraphs.

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Article
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A lonesum matrix is a matrix that can be uniquely reconstructed from its row and column sums. Kaneko defined the poly-Bernoulli numbers $B_m^{(n)}$ by a generating function, and Brewbaker computed the number of binary lonesum $m\times n$-matrices and showed that this number coincides with the poly-Bernoulli number $B_m^{(-n)}$. We compute the number of $q$-ary lonesum $m\times n$-matrices, and then provide generalized Kaneko's formulas by using the generating function for the number of $q$-ary lonesum $m\times n$-matrices. In addition, we define two types of $q$-ary lonesum matrices that are composed of strong and weak lonesum matrices, and suggest further researches on lonesum matrices. \
Chapter
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Consider the class AP(R, S) of (0, 1)-matrices with row sum vector R, column sum vector S, and zeros in all positions outside a certain set P. It is assumed that P satisfies a certain monotonicity property. We show the existence of a canonical matrix in this matrix class and give a simple algorithm for finding this matrix. Moreover, a classical interchange result of Ryser is generalized to the class AP(R, S) and the uniqueness question for the class AP(R, S) is discussed.
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Advances in Discrete Tomography and Its Applications is a unified presentation of new methods, algorithms, and select applications that are the foundations of multidimensional image reconstruction by discrete tomographic methods. The self-contained chapters, written by leading mathematicians, engineers, and computer scientists, present cutting-edge research and results in the field. Three main areas are covered: foundations, algorithms, and practical applications. Following an introduction that reports the recent literature of the field, the book explores various mathematical and computational problems of discrete tomography including new applications. Topics and Features: * introduction to discrete point X-rays * uniqueness and additivity in discrete tomography * network flow algorithms for discrete tomography * convex programming and variational methods * applications to electron microscopy, materials science, nondestructive testing, and diagnostic medicine Professionals, researchers, practitioners, and students in mathematics, computer imaging, biomedical imaging, computer science, and image processing will find the book to be a useful guide and reference to state-of-the-art research, methods, and applications.
Article
We show that the number of (0,1)-matrices with n rows and k columns uniquely reconstructable from their row and column sums is the poly-Bernoulli number B n (-k) . Combinatorial proofs for both the sieve and closed formulas are presented. In addition, we prove an analogue of Fermat’s Little Theorem: For a positive integer n and prime number p we have B n (-p) ≡2 n (modp)· Also, an analogue to Fermat’s Last Theorem is presented: For all positive integers {x,y,z} and n>1 there exist no solution to the equation B x (-n) +B y (-n) =B z (-n) ·
Chapter
This paper is concerned with a matrix A of m rows and n columns, all of whose entries are 0’s and 1’s. Let the sum of row i of A be denoted by r i (i = 1, ... , m) and let the sum of column i of A be denoted by S i (i = 1, ... ,n).
Article
For q∈C transcendental over Q, we give an algorithmic construction of an order-isomorphism between the set of H-primes of Oq(Mn(C)) and the sub-poset S of the (reverse) Bruhat order of the symmetric group S2n consisting of those permutations that move any integer by no more than n positions. Further, we describe the permutations that correspond via this bijection to rank tH-primes. More precisely, we establish the following result. Imagine that there is a barrier between positions n and n+1. Then a 2n-permutation σ∈S corresponds to a rank tH-invariant prime ideal of Oq(Mn(C)) if and only if the number of integers that are moved by σ from the right to the left of this barrier is exactly n−t. The existence of such an order-isomorphism was conjectured by Goodearl and Lenagan.
Article
This thesis shows that the number of (0,1)-matrices with n rows and k columns uniquely reconstructible from their row and column sums are the poly-Bernoulli numbers of negative index, B[subscript n superscript (-k)] . Two proofs of this main theorem are presented giving a combinatorial bijection between two poly-Bernoulli formula found in the literature. Next, some connections to Fermat are proved showing that for a positive integer n and prime number p B[subscript n superscript (-p) congruent 2 superscript n (mod p),] and that for all positive integers (x, y, z, n) greater than two there exist no solutions to the equation: B[subscript x superscript (-n)] + B[subscript y superscript (-n)] = B[subscript z superscript (-n)]. In addition directed graphs with sum reconstructible adjacency matrices are surveyed, and enumerations of similar (0,1)-matrix sets are given as an appendix. Typescript (photocopy) Thesis (M.S.)--Iowa State University, 2005. Includes bibliography.
Article
Let 𝕂 be a (commutative) field and consider a nonzero element q in 𝕂 that is not a root of unity. Goodearl and Lenagan (20028. Goodearl , K. R. , Lenagan , T. H. ( 2002 ). Prime ideals invariant under winding automorphisms in quantum matrices . Internat. J. Math 13 : 497 – 532 . [CROSSREF] View all references) have shown that the number of ℋ-primes in R = O q (ℳ n (𝕂)) that contain all (t + 1) × (t + 1) quantum minors but not all t × t quantum minors is a perfect square. The aim of this paper is to make precise their result: we prove that this number is equal to (t!) 2 S(n + 1, t + 1)2, where S(n + 1, t + 1) denotes the Stirling number of the second kind associated to n + 1 and t + 1. This result was conjectured by Goodearl, Lenagan, and McCammond. The proof involves some closed formulas for the poly-Bernoulli numbers that were established by Kaneko (199710. Kaneko , M. ( 1997 ). Poly-Bernoulli numbers . J. Théorie Nombres Bordeaux 9 : 221 – 228 . View all references) and Arakawa and Kaneko (19991. Arakawa , T. , Kaneko , M. ( 1999 ). On poly-Bernoulli numbers . Comment Math. Univ. St. Paul 48 ( 2 ): 159 – 167 . View all references).
Constructing (0, 1)-matrices with given line sums and certain fixed zeros, Advances in Discrete Tomography and Its Applications
• R A Brualdi
• G Dahl
R.A. Brualdi, G. Dahl, Constructing (0, 1)-matrices with given line sums and certain fixed zeros, Advances in Discrete Tomography and Its Applications, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2007, pp. 113–123.