No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Matrices of zeros and ones
... An integer matrix is called lonesum if it can be uniquely reconstructed from its row and column sums. Ryser [13,14] prove that a binary matrix (each entry is 0 or 1) is lonesum if and only if its 2 × 2 submatrices satisfy a certain condition. Brewbaker [2] exploits this result to compute the number of binary lonesum matrices for a given dimension. ...
... Brewbaker [2] exploits this result to compute the number of binary lonesum matrices for a given dimension. The results of Ryser [13,14] and Brewbaker [2] are generalised by Kim et al. [11] and Lee [12]. Kim et al. [11] present results for general integer matrices and Lee [12] extends these results to integer matrices in a multidimensional space. ...
... Using (14) and U c = U cT , we observe that U can be obtained from U T by applying another orthonormal transformation ...
... matrices with given row and column sums was addressed by Ryser [17] already in 1960; see also [3]. It is closely related to degree sequences of graphs, and-by identifying the 1-entries of such a matrix with point sets in R 2 or Z 2 -can, in retrospective, also be viewed as an anticipation of the field of discrete tomography. ...
... As Ryser [17] showed, the classical binary monoatomic case for 2 directions can be solved easily and in polynomial time. The problem becomes NP-hard, however, when more than two directions are involved [8], see also [4]. ...
... The name ColRys c (G , W ) is chosen in reverence for Ryser, as it describes a colored version of his original "monoatomic" problem studied in [17]. In fact, for c = 1, G = (Z), W = ({0, 1}), ColRys c (G , W ) asks for a 0-1 matrix with given row and column sums (computed in Z). ...
The paper deals with an inverse problem of reconstructing matrices from their marginal sums. More precisely, we are interested in the existence of r×s matrices for which only the following information is available: The entries belong to known subsets of c distinguishable abelian groups, and the row and column sums of all entries from each group are given. This generalizes Ryser’s classical problem of characterizing the set of all 0–1-matrices with given row and column sums and is a basic problem in (polyatomic) discrete tomography. We show that the problem is closely related to packings of trees in bipartite graphs, prove consistency results, give algorithms and determine its complexity. In particular, we find a somewhat unusual complexity behavior: the problem is hard for “small” but easy for “large” matrices.
... Nous pourrons aussi mentionner le travail de George G. Lorentz (1910Lorentz ( -2006 en 1949 [13] qui est connecté à celui de Alfred Rényi [11] sur les conditions qu'une fonction donnée soit reconstruite à partir de ses projections suivant certains axes (nous expliquerons en détail le terme projection). L'équivalence de ses conditions générales données dans [13] pour le cas général de l'analyse des fonctions, sera faite, mais cette fois dans un domaine restreint pour les problèmes d'analyse combinatoire sera aussi donnée par Herbert John Ryser (1923Ryser ( -1985 en 1957 [14] et en 1960 [15]. Il y a beaucoup d'autres travaux qui s'en suivront comme cela est indiqué dans le livre sur la tomographie discrète [16] dans le premier chapitre introductif. ...
... Ryser a trouvé une condition nécessaire et suffisante pour qu'une paire de vecteurs soient les deux projections orthogonales d'un ensemble discret [14]. Dans la preuve de son théorème, il a aussi décrit le premier algorithme de reconstruction pour le cas général d'un ensemble discret à partir des projections orthogonales [15,85]. Nous rappelons les définitions et les conditions de consistance quand on résout le problème de reconstruction d'une image binaire [16], ensuite nous introduisons notre approche qui utilise la notion de copule discrète. ...
This thesis studies the relationship between Computed Tomography (CT) and the notion of copula. In X-ray tomography the objective is to (re)construct an image representing the distribution of a physical quantity (density of matter) inside of an object from the radiographs obtained all around the object called projections. The link between these images and the object is described by the X-ray transform or the Radon transform. In 2D, when only two projections at two angles 0 and pi/2 (horizontal and vertical) are available, the problem can be identified as another problem in mathematics which is the determination of a joint density from its marginals, hence the notion of copula. Both problems are ill-posed in the sense of Hadamard. It requires prior information or additional criteria or constraints. The main contribution of this thesis is the use of entropy as a constraint that provides a regularized solution to this ill-posed inverse problem. Our work covers different areas. The mathematics aspects of X-ray tomography where the fundamental model to obtain projections is based mainly on the Radon transform. In general this transform does not provide all necessary projections which need to be associated with certain regularization techniques. We have two projections, which makes the problem extremely difficult, and ill-posed but noting that if a link can be done, that is, if the two projections can be equated with marginal densities and the image to reconstruct to a probability density, the problem translates into the statistical framework via Sklar's theorem. And the tool of probability theory called "copula" that characterizes all possible reconstructed images is suitable. Hence the choice of the image that will be the best and most reliable arises. Then we must find techniques or a criterion of a priori information, one of the criteria most often used, we have chosen is a criterion of entropy. Entropy is an important scientific quantity because it is used in various areas, originally in thermodynamics, but also in information theory. Different types of entropy exist (Rényi, Tsallis, Burg, Shannon), we have chosen some as criteria. Using the Rényi entropy we have discovered new copulas. This thesis provides new contributions to CT imaging, the interaction between areas that are tomography and probability theory and statistics.
... Ryser's Theorem Ryser [10] studied the structure of 0-1 matrices with given projections. We adapt his characterization of these matrices and express it in terms of tile packings. ...
... By definition, |D ∩ (I × J )| ≤ ξ I,J . Therefore we obtain the following lemma (inspired by [10]). ...
Discrete tomography deals with reconstructing finite spatial objects from their projections. The objects we study in this paper are called tilings or tile-packings, and they consist of a number of disjoint copies of a fixed tile, where a tile is defined as a connected set of grid points. A row projection specifies how many grid points are covered by tiles in a given row; column projections are defined analogously. For a fixed tile, is it possible to reconstruct its tilings from their projections in polynomial time? It is known that the answer to this question is affirmative if the tile is a bar (its width or height is 1), while for some other types of tiles ℕℙ-hardness results have been shown in the literature. In this paper we present a complete solution to this question by showing that the problem remains ℕℙ-hard for all tiles other than bars.
... It has been known for a long time that for k = 2 colors the problem can be solved in polynomial time [23]. Around a decade ago it was shown that the k-CTP is NP-hard for k 7 colors [13]. ...
... Lemma 2.1 (Gale [12] and Ryser [22]; also see [23] for the second statement). Let (r, s) be an instance of the 2-CTP such that r is nonincreasing. ...
We consider the problem of coloring a grid using k colors with the restriction that each row and each column has a specific number of cells of each color. This problem has been known as the -atom problem in the discrete tomography community. In an already classical result, Ryser obtained a necessary and sufficient condition for the existence of such a coloring when two colors are considered. This characterization yields a linear time algorithm for constructing such a coloring when it exists. Gardner et al. showed that for the problem is NP-hard. Afterward Chrobak and Dürr improved this result by proving that it remains NP-hard for . We close the gap by showing that for k=3 colors the problem is already NP-hard. In addition, we give some results on tiling tomography problems.
... , A n ]. The effective stress σ n − C p p or is defined, with σ n ∈ ℜ n as a vector of normal 2 A relation (logical, boolean or binary) matrix is a matrix with only entries of zeros or ones (Ryser, 1960). stresses (σ n = [σ n 1 , . . . ...
Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. Characteristic examples are nonautonomous and discontinuous systems due to the presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists of designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). Finally, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid’s diffusion.
... Give a (0, 1)-matrix A we say that is regular if the number of 1's is fixed in each column and has a fixed number of 1's in each row. If A is not regular we say that is irregular see [10,25,26], for more information. A sparse matrix is a (0, 1)-matrix in which most of the elements are zero. ...
The Plücker matrix BL(n,E) of the Lagrangian Grassmannian L(n,E), is determined by the linear envelope ⟨L(n,E)⟩ of the Lagrangian Grassmannian. The linear envelope ⟨L(n,E)⟩ is the intersection of linear relations of Plücker of Lagrangian Grassmannian, defined here. The Plücker matrix BL(n,E) is a direct sum of the incidence matrix of the configuration of subsets. These matrices determine the isotropy index rn and rn-atlas which are invariants associated with the symplectic vector space E.
... By mapping the individual containers against their respective payloads, we obtain a Cartesian product in which we denote the existence of a version-based relation by a unit and the lack of such a relation by a zero. This Cartesian product C e n t e r i s 2 0 2 3 is a two-dimensional binary matrix of n rows and m columns that we will call the object version grid (OVG) and which possesses several useful properties [11]. ...
Over time, persistent change may have a considerable effect on the codebases of enterprise resource planning (ERP) systems. Small changes, however, are not always expressions of intentional and incre-mental transformation. They may result from unplanned but necessary and urgent corrections for repairing undesired functionality that entered the ERP system via previously applied change. Every such repair will affect the performance and prioritization of operational processes and, by extension, influence the business that the system intends to support. We developed a change log data view to assess business impact by classifying repairs into one of two mutually exclusive classes. Repairs are either effective, meaning that the next change delivers the fix, or defective , meaning multiple successive attempts are required to satisfactorily correct the initial erroneous change. Investigating the extent to which both repair types on proprietary codebase customizations affect competence center operational performance, our view delivered two finds. First, change size is a poor indicator for repair resolution efficacy. Second, most repairs appear to be defective suggesting poor codebase change process control. Our data originated from 25 SAP landscapes, representing an average system age of around five years and an average landscape size of 57 ERP system instances.
... In the rest of the paper we assume that the integral vectors R and S are compatible in Problem 1. This problem was first solved independently by Ryser [16,17] and Gale [18]. Ryser's approach is based on the construction of a maximal matrix 3 and shifting certain ones to the right within rows to attain the correct column sums. ...
A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points the average distance is typically given by integration over the focal set. The paper contains a survey on the applications of taxicab distance mean functions and generalized conics' theory in geometric tomography: bisection of the focal set and reconstruction problems by coordinate X-rays. The theoretical results are illustrated by implementations in Maple, methods and examples as well.
... A (0, 1)-matrix is a matrix in which each element is either 0 or is 1. A (k, ℓ)-matrix is a (0, 1)-matrix with k ones in each row and ℓ ones in each column, for uses and applications of this type of matrices see [1], [6], [14] and [16]. A sparse matrix is a (0, 1)-matrix if many of its elements are zero. ...
In this article a family of recursive and self-similar matrices is constructed. It is shown that the Pl\"ucker matrix of the Isotropic Grassmannian variety is a direct sum of this class of matrices.
... The original problem in binary tomography is then to construct an m × n binary matrix with specific row and column sums. Ryser, and independently Gale, determined necessary and sufficient conditions for the existence of a binary matrix with specified row or column sums, along with a polynomial time algorithm for constructing a solution [10,24,25]. Furthermore, different solutions are related by a series of transformations; we refer to as Ryser interchanges. ...
The classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a 10 × 10 grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data, represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem: the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.
... Give a (0, 1)-matrix A we say that is regular if the number of 1's is fixed in each column and has a fixed number of 1's in each row. If A is not regular we say that is irregular see [25] and [26], for more information. A sparse matrix is a (0, 1)-matrix in which most of the elements are zero. ...
In this paper it is shown that Family of Linear Relations of Contraction (FRLC) are the only ones, up to linear combination, that vanish the Lagrangian-Grassmannian. It is shown that the Pl\"ucker matrix of the Lagrangian-Grassmannian is a direct sum of incidence matrix, regular and sparce with entries in the set { 0, 1}.
... Chan [7] , Chan et al. [8] , 9 ], Corless [11] , Corless and Thornton [13] , 14 ], Fasi and Porzio [18] , Feng and Fan [19] , Thornton [42] ) as well as to the closely related problem of studying the zeros of polynomials with coefficients belonging to a fixed population (see e.g. Borwein and Jorgenson [3] , Borwein and Pinner [4] , Christensen [ 12,23,29 ], Odlyzko and Ponnen [27] ). ...
In this paper, for certain type of structured {0,1,−1}–matrices, we give a complete description of the inner Bohemian inverses over any population containing the set {0,1,−1}. In addition, when the population is exactly {0,1,−1}, we provide explicit formulas for the number of inner Bohemian inverses of these type of matrices.
... Ryser [20] has also introduced an interesting entity called the structure matrix of a directed graph. A structure matrix of a directed graph over [n] is an integer-valued (n + 1)× (n + 1) matrix that is constructed directly from the vectors r and c and gives important insights to the structure of the graphs within the edge-type class. ...
The method of types presented by Csiszar and Korner is a central tool used to develop and analyze the basic properties and constraints on sequences of data over finite alphabets. A central problem considered using these tools is that of data compression, and specifically lossy data compression. In this work we consider this very problem, however, instead of sequences of data we consider directed graphs. We show that given a more natural distortion measure, fitting the data structure of a directed graph, the method of types cannot be applied. The suggested distortion measure aims to preserves the local structure of a directed graph. We build on the recent work of Barvinok and extend the method of types to the two dimensional setting of directed graphs. We see that the extension is quite natural in many ways. Given this extension we provide a lower and upper bound on the rate-distortion problem of lossy compression given the suggested distortion measure.
... We say that the sequences d and k are configurable if H d,k = ∅. A criterion for configurability in terms of the majorization of vectors appears to have been obtained independently in early work by Gale and Ryser [30,31]. ...
Many empirical networks are intrinsically polyadic, with interactions occurring within groups of agents of arbitrary size. There are, however, few flexible null models that can support statistical inference in polyadic networks. We define a class of null random hypergraphs that hold constant both the node degree and edge dimension sequences, thereby generalizing the classical dyadic configuration model. We provide a Markov Chain Monte Carlo scheme for sampling from these models and discuss connections and distinctions between our proposed models and previous approaches. We then illustrate the application of these models through a triplet of data-analytic vignettes. We start with two classical topics in network science—triadic clustering and degree-assortativity. In each, we emphasize the importance of randomizing over hypergraph space rather than projected graph space, showing that this choice can dramatically alter both the quantitative and qualitative outcomes of statistical inference. We then define and study the edge intersection profile of a hypergraph as a measure of higher-order correlation between edges, and derive asymptotic approximations for this profile under the stub-labeled null. We close with suggestions for multiple avenues of future work. Taken as a whole, our experiments emphasize the ability of explicit, statistically grounded polyadic modelling to significantly enhance the toolbox of network data science.
... We say that H is a matrix approximate lower diagonal form if in the lower left corner of H is has an identity matrix as its submatrix. Following [9], let A be a (0, 1)-matrix of order p × q, the sum of row i of A be denoted by r i where i = 1, . . . , p and the sum of column j of A be denoted by s j where j = 1, . . . ...
In this paper we give a recursive algorithm to construct two families of (0,1)-matrices, one sparse regular and the other dense. We study various properties of the two families of (0,1)-matrices built with our algorithm. We present a new construction of two clases of isodual linear codes, one is the low density generator matrix codes and other is the dense linear codes, for both codes we obtain the polynomial of the distribution of weights, a bound for the minimum distance and we apply to these codes the efficient encoders based on approximate lower triangulations developed by Richardson-Urbanke. We identify the unique (0,1)-matrices, up basis change, associated with the geometry of the Lagrangian-Grassmannian variety.
... This problem is addressed in the literature 9 , and several known algorithms tackle it. Rycer [Ryc60] was the first to study such matrices. Some combinatorial properties of Rycer matrices were studied in [Bru80] by connecting them to bipartite matrices and hyper-graphs. ...
Inspired by the decomposition in the hybrid quantum-classical optimization algorithm we introduced in arXiv:1902.04215, we propose here a new (fully classical) approach to solving certain non-convex integer programs using Graver bases. This method is well suited when (a) the constraint matrix A has a special structure so that its Graver basis can be computed systematically, (b) several feasible solutions can also be constructed easily and (c) the objective function can be viewed as many convex functions quilted together. Classes of problems that satisfy these conditions include Cardinality Boolean Quadratic Problems (CBQP), Quadratic Semi-Assignment Problems (QSAP) and Quadratic Assignment Problems (QAP). Our Graver Augmented Multi-seed Algorithm (GAMA) utilizes augmentation along Graver basis elements (the improvement direction is obtained by comparing objective function values) from these multiple initial feasible solutions. We compare our approach with a best-in-class commercially available solver (Gurobi). Sensitivity analysis indicates that the rate at which GAMA slows down as the problem size increases is much lower than that of Gurobi. We find that for several instances of practical relevance, GAMA not only vastly outperforms in terms of time to find the optimal solution (by two or three orders of magnitude), but also finds optimal solutions within minutes when the commercial solver is not able to do so in 4 or 10 hours (depending on the problem class) in several cases.
... Swaps operate on row and column pairs with a specific organization of values, shown in Ryser, among others, has shown that all matrices with identical margin sums can be produced with swap operations [19]. When the sequential swap procedure is implemented correctly, the consecutive matrices form a Markov Chain that has uniform stationary distribution [11,20]. ...
Assessing the significance of patterns in presence-absence data is an important question in ecological data analysis, e.g., when studying nestedness. Significance testing can be performed with the commonly used fixed-fixed models, which preserve the row and column sums while permuting the data. The manuscript considers the properties of fixed-fixed models and points out how their strict constraints can lead to limited randomizability. The manuscript considers the question of relaxing row and column sun constraints of the fixed-fixed models. The Rasch models are presented as an alternative with relaxed constraints and sound statistical properties. Models are compared on presence-absence data and surprisingly the fixed-fixed models are observed to produce unreasonably optimistic measures of statistical significance, giving interesting insight into practical effects of limited randomizability.
... > r m > 0; si > s 2 > ... > s w > 0. When this is the case we say the class %(R } S) is normalized. We find a formula for the minimal term rank of %(R, S) analogous to formulas for maximal term rank, minimal and maximal trace, and minimal column width already developed by Ryser and Fulkerson (1,3,4,5,6). (For definitions and a more complete bibliography see (6).) ...
Let be the class of m × n matrices all of whose entries are either 0 or 1 where every matrix A in the class satisfies the conditions that row i of A has r i ones, i = 1, 2, . . . , m ; and column j of A has s j ones, j = 1, 2, . . . , n . We let R = ( r 1 . . . , r m ), S = ( s 1 , . . . , s n ), and assume that r 1 ≥ r 2 ≥ . . . ≥ r m ≥ 0; s 1 ≥ s 2 ≥ . . . ≥ s n > 0. When this is the case we say the class is normalized.
... An interesting connected problem is the characterization of pairs of different directed graphs having a pair of prescribed indegree and outdegree sequences [8,9,10,11,12,14,15,20,40,72,76,81]. ...
In the paper we report on the parallel enumeration of the degree sequences (their number is denoted by G(n)) and zero-free degree sequences (their number is denoted by (G z (n)) of simple graphs on n=30 and n=31 vertices. Among others we obtained that the number of zero-free degree sequences of graphs on G z (30)=5876236938019300 and G z (31)=22974847474172374. Due to Corollary 21 in [A. Iványi et al., ibid. 3, No. 2, 230–268 (2011; Zbl 06315785)] these results give the number of degree sequences of simple graphs on 30 and 31 vertices. For Part I see [A. Iványi et al., ibid. 4, No. 2, 260–288 (2012; Zbl 06315452)].
... , n, with equality holding when k = n. The necessity of these conditions is fairly obvious, and proofs of their sufficiency have been given using a variety of different methods [1,2,4,10,22,23]. The purpose of this note is to exhibit Landau's theorem as an instance of the "duality principle" of linear programming, and to point out that this approach suggests an extension of Landau's result going beyond the well-known generalizations due to J. W. Moon [20,19]. ...
H. G. Landau has characterized those integer-sequences S=(s 1 ,s 2 ,⋯,s n ) which can arise as score-vectors in an ordinary round-robin tournament among n contestants [17]. If s 1 ≤s 2 ≤⋯≤s n , the relevant conditions are expressed simply by the inequalities: ∑ i=1 k s i ≥k 2,(1) for k=1,2,⋯,n, with equality holding when k=n. The necessity of these conditions is fairly obvious, and proofs of their sufficiency have been given using a variety of different methods [1, 2, 4, 10, 22, 23]. The purpose of this note is to exhibit Landau’s theorem as an instance of the “duality principle” of linear programming, and to point out that this approach suggests an extension of Landau’s result going beyond the well-known generalizations due to J. W. Moon [20, 19].
... The determination of the maximal width sequence (1.3) for such a class 23 involves deep issues. For instance, in the first example mentioned, i(l) = 2 or 3 according as a finite projective plane of order n does not or does exist (6). Specifically, the complement of a finite plane has 1-width 3, whereas other matrices in the class have 1-width 2. This state of affairs is of course decidedly in contrast with the situation for the minimal width sequence (1.1) for 23. ...
: The study of alpha-widths of (0, 1)-matrices (AD-274 181) continued, the emphasis being on those special classes of b by v (0, 1)-matrices having k 1's per row and 4 1's per column. It is assumed throughout that the class parameters b, v, k, r satisfy the inequality (b-r)(v-k-1) less than or equal to v - 1. Such a class has special combinatorial interest. For example, complements of finite projective planes and of Steiner triple systems have parameters satisfying this inequality. Several theorems are proved concerning the width sequence for a matrix in such a class. Insofar as possible, these results are used to obtain information concerning the maximal width sequence for the class. Perhaps the major general result established is that jumps in the width sequence for a matrix in the class, or in the maximal width sequence for the class, are either 1 or 2.
... Proof. See [6,9,10,11,12]. ...
A lonesum matrix is a matrix that can be uniquely reconstructed from its
row and column sums. Kaneko defined the poly-Bernoulli numbers
by a generating function, and Brewbaker computed the number
of binary lonesum -matrices and showed that this number
coincides with the poly-Bernoulli number . We compute the
number of q-ary lonesum -matrices, and then provide
generalized Kaneko's formulas by using the generating function for the
number of q-ary lonesum -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. \
... , X m of subsets of [n], i ∈ X k ⇐⇒ A ij = 1, where A ∈ M n×m (Z 2 ). An interesting paper examining incidence matrices is [Rys60]. ...
We prove assorted properties of matrices over
, and outline the complexity of the concepts required to prove these properties. The goal of this line of research is to establish the proof complexity of matrix algebra. It also presents a different approach to linear algebra: one that is formal, consisting in algebraic manipulations according to the axioms of a ring, rather than the traditional semantic approach via linear transformations.
... Theorem, which states that 19.I(R,S)I/> 1 if and only if S is weakly majorized by the conjugate of R [1,3]. While the determination of the precise number of matrices in the class, as Ryser put twice in [4,5], is much more difficult, and "this number is undoubtedly an extremely intricate function of R and S". A lower bound when R=S=(k,k, .... k) was witnessed in 1982 [8] (n!) k I~(R, S)l 1> (k~)-~ ...
This paper gives a reduced formula for the precise number of matrices in AA(R,S), the class of matrices of zeros and ones with row and column sum vectors R and S, respectively. With the new formula, the computing time is greatly shortened.
A matrix family is called Bohemian if its entries come from a fixed finite discrete (and hence bounded) set, usually integers, called the “population” P. We look at Bohemian matrices, specifically those with entries from {-1,0,+1}. The name is a mnemonic for Bounded Height Matrix of Integers. Such families arise in many applications (e.g. compressed sensing) and the properties of matrices selected “at random” from such families are of practical and mathematical interest. An overview of some of our original interest in Bohemian matrices can be found in [6, 7]. In this paper we present a Bohemian Matrices tour, exposing their appearance in the past, their promising present and their hopeful future.
Abstract Hypergraphs offer a natural modeling language for studying polyadic interactions between sets of entities. Many polyadic interactions are asymmetric, with nodes playing distinctive roles. In an academic collaboration network, for example, the order of authors on a paper often reflects the nature of their contributions to the completed work. To model these networks, we introduce annotated hypergraphs as natural polyadic generalizations of directed graphs. Annotated hypergraphs form a highly general framework for incorporating metadata into polyadic graph models. To facilitate data analysis with annotated hypergraphs, we construct a role-aware configuration null model for these structures and prove an efficient Markov Chain Monte Carlo scheme for sampling from it. We proceed to formulate several metrics and algorithms for the analysis of annotated hypergraphs. Several of these, such as assortativity and modularity, naturally generalize dyadic counterparts. Other metrics, such as local role densities, are unique to the setting of annotated hypergraphs. We illustrate our techniques on six digital social networks, and present a detailed case-study of the Enron email data set.
Let a finite set be given. The taxicab distance sum function is defined as the sum of the taxicab distances from the elements (focuses) of the so-called focal set F. The sublevel sets of the taxicab distance sum function are called generalized conics because the boundary points have the same average taxicab distance from the focuses. In case of a classical conic (ellipse) the focal set contains exactly two different points and the distance taken to be averaged is the Euclidean one. The sublevel sets of the taxicab distance sum function can be considered as its generalizations. We prove some geometric (convexity), algebraic (semidefinite representation) and extremal (the problem of the minimizer) properties of the generalized conics and the taxicab distance sum function. We characterize its minimizer and we give an upper and lower bound for the extremal value. A continuity property of the mapping sending a finite subset F to the taxicab distance sum function is also formulated. Finally we present some applications in discrete tomography. If the rectangular grid determined by the coordinates of the elements in is given then the number of points in F along the directions parallel to the sides of the grid is a kind of tomographic information. We prove that it is uniquely determined by the function measuring the average taxicab distance from the focal set F and vice versa. Using the method of the least average values we present an algorithm to reconstruct F with a given number of points along the directions parallel to the sides of the grid.
We develop a Bayesian methodology for systemic risk assessment in financial networks such as the interbank market. Nodes represent participants in the network, and weighted directed edges represent liabilities. Often, for every participant, only the total liabilities and total assets within this network are observable. However, systemic risk assessment needs the individual liabilities. We propose a model for the individual liabilities, which, following a Bayesian approach, we then condition on the observed total liabilities and assets and, potentially, on certain observed individual liabilities. We construct a Gibbs sampler to generate samples from this conditional distribution. These samples can be used in stress testing, giving probabilities for the outcomes of interest. As one application we derive default probabilities of individual banks and discuss their sensitivity with respect to prior information included to model the network. An R package implementing the methodology is provided.
This paper was accepted by Noah Gans, stochastic models and simulation.
When an image is given with only some measurable data, e.g., projections, the most important task is to reconstruct it, i.e., to find an image that provides the measured data. These tomographic problems are frequently used in the theory and applications of image processing. In this paper, memetic algorithms are investigated on triangular grids for the reconstruction of binary images using their three and six direction projections. The algorithm generates an initial population using the network flow algorithm for two of the input projections. The reconstructed images evolve towards an optimal solution or close to the optimal solution, by using crossover operators and guided mutation operators. The quality of the images is improved by using switching components and compactness operator.
An accurate knowledge of the complex microstructure of a heterogeneous material is crucial for its performance prediction, prognosis and optimization. X-ray tomography has provided a nondestructive means for microstructure characterization in 3D and 4D (i.e. structural evolution over time), in which a material is typically reconstructed from a large number of tomographic projections using filtered-back-projection (FBP) method or algebraic reconstruction techniques (ART). Here, we present in detail a stochastic optimization procedure that enables one to accurately reconstruct material microstructure from a small number of absorption contrast x-ray tomographic projections. This discrete tomography reconstruction procedure is in contrast to the commonly used FBP and ART, which usually requires thousands of projections for accurate microstructure rendition. The utility of our stochastic procedure is first demonstrated by reconstructing a wide class of two-phase heterogeneous materials including sandstone and hard-particle packing from simulated limited-angle projections in both cone-beam and parallel beam projection geometry. It is then applied to reconstruct tailored Sn-sphere-clay-matrix systems from limited-angle cone-beam data obtained via a lab-scale tomography facility at Arizona State University and parallel-beam synchrotron data obtained at Advanced Photon Source, Argonne National Laboratory. In addition, we examine the information content of tomography data by successively incorporating larger number of projections and quantifying the accuracy of the reconstructions. We show that only a small number of projections (e.g. 20-40, depending on the complexity of the microstructure of interest and desired resolution) are necessary for accurate material reconstructions via our stochastic procedure, which indicates its high efficiency in using limited structural information. The ramifications of the stochastic reconstruction procedure in 4D materials science are also discussed.
In this paper binary matrices are reconstructed from two orthogonal projections. In the reconstruction process particle swarm optimization is used. Since most of the reconstruction problems are NP-hard, the use of a global optimization process gave promising results. The task is to adjust the particle swarm algorithm to the problem of the reconstruction of binary matrices. The main focus is on the reconstruction of hv-convex matrices which consist of either one or more connected components. At the end of the paper the results are shown and the appropriate conclusions are drawn.
Ant Colony Optimization (ACO) algorithms have been applied to get the solution of many hard discrete optimization problems. But ACO algorithms have not been applied to Discrete Tomography (DT) problems yet. In this paper, we propose a framework of ACO meta-heuristic for DT problems. Some variations in the framework have also been discussed.
In this initial chapter, we shall present many of the fundamental results and techniques of the theory of inequalities. Some of the results are important in themselves, and some are required for use in subsequent chapters; others are included, as are multiple proofs, on the basis of their elegance and unusual flavor [1].
Binary tomography is a hard and challenging task in image processing. The usage of non-traditional grids may have several benefits in this task also, due to the fact that their symmetric properties differ from the properties of the square grid. There are various approaches to do binary tomography, e.g., network flow algorithms, genetic and meme tic algorithms, simulated annealing. On the triangular grid tomography algorithms are based on 3 and 6 projection directions. The first algorithms have used projections by lanes (orthogonal to coordinate axes). In this paper, we present an algorithm that uses 3 projections parallel to the coordinate axes. By our experimental results it can be seen that this newly proposed algorithm is effective to get good results in a short time.
: Let A be an m by n (0, 1)-matrix, and suppose that E* is an m by epsilon submatrix of A having the property that each row of E* contains at least alpha 1's. The epsilon columns of E* are said to form an alpha-set of representatives for A. Let epsilon(alpha) be the minimal number of columns of A that form an alpha-set of representatives. The integer epsilon(alpha) is called the alpha-width of A. If A has alpha-width epsilon(alpha), select an m by epsilon(alpha) submatrix E* of A having the property that the number delta(alpha) of rows of E* containing exactly alpha 1'S is as small as possible. The integer delta(alpha) is called the alpha-height of A.
If is an n X n matrix, the permanent of A, Per A, is defined by
1
where the sum is over all permutations. If A is doubly stochastic (i.e., nonnegative with row and column sums all equal to 1), then it has been conjectured that Per A ⩾ n !/ n ⁿ . When confronted with a vaguely similar problem about determinants, M. Kac (1) observed that the computation of minima can often be aided by knowledge of various averages. In this spirit we calculate here the average permanent of a class of doubly stochastic matrices and thereby obtain upper bounds for the minima. These turn out to be surprisingly sharp.
In a previous paper (1) the notion of the α-width ∈ A (α) of a (0, 1)-matrix A was introduced, and a formula for the minimal α-width taken over the class of all (0, 1)-matrices having the same row and column sums as A , was obtained. The main tool in arriving at this formula was a block decomposition theorem (1, Theorem 2.1; repeated below as Theorem 2.1) that established the existence, in the class generated by A , of certain matrices having a simple block structure. The block decomposition theorem does not itself directly involve the notion of minimal α-width, but rather centres around a related class concept, that of multiplicity. We review both of these notions in § 2, together with some other pertinent definitions and results.
In this paper we use a generalized form of Polya's theorem (1) to obtain generating functions for the number of ordinary graphs with given partition and for the number of bicoloured graphs with given bipartition. Both the points and lines of the graphs are taken as unlabelled. These graph enumeration problems were proposed by Harary in his review article (4). Read (7, 8) solved the problem for unlabelled general graphs and labelled ordinary graphs.
X-ray tomography has provided a non-destructive means for microstructure characterization in three and four dimensions. A stochastic procedure to accurately reconstruct material microstructure from limited-angle X-ray tomographic projections is presented and its utility is demonstrated by reconstructing a variety of distinct heterogeneous materials and elucidating the information content of different projection data sets. A small number of projections (e.g. 20–40) are necessary for accurate reconstructions via the stochastic procedure, indicating its high efficiency in using limited structural information.
New mixed-integer linear programming formulations are presented for the quadratic assignment problem, based on splittings of the coefficient matrices. Computational results are reported for medium-sized problem instances in the QAPLIB collection.
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.
This paper continues a study appearing in (5) of the combinatorial properties of 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 and let the sum of column i of A be noted by s t . We call R = ( r 1 , … , r m ) the row sum vector and S = ( s 1 , … , s n ) the column sum vector of A . The vectors R and S determine a class consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. Simple arithmetic properties of R and S are necessary and sufficient for the existence of a class (1 ; 5).
The theory developed for the study of flows in networks (2; 3; 4; 5; 6; 7) sometimes provides a useful tool for dealing with certain kinds of combinatorial problems, as has been previously indicated in (3; 4; 6; 7). In particular, Hall-type theorems for the existence of systems of distinct representatives which contain a prescribed set of marginal elements (10; 11), or, more generally, whose intersection with each member of a given partition of the fundamental set has a cardinality between prescribed lower and upper bounds (9), can be obtained in this way (7).
The problem of arranging v elements into v sets in such a way that every set contains exactly k distinct elements and that every pair of sets has exactly λ = k(k — l)/(v — 1) elements in common, where 0 < » < k < v , is equivalent to finding a normal integral v by v matrix A such that A T A = B , where B is the v by v matrix having k in every position on the main diagonal and λ in all other positions (10). Utilizing the fact that for the existence of a λ, k, v design it is necessary that I (the v by v identity matrix) represent B rationally, (2) and (3) have proved the non-existence of certain λ, k, v designs. Neither of the proofs utilize the fact that it is necessary that A be normal. However, Albert (1) for the projective plane case and Hall and Ryser (5) for the general design proved that if there exists a rational A such that A T A = B then there exists a normal rational matrix satisfying the same equation. Thus the requirement of normality does not exclude any λ, k, v which were not previously excluded.
There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical of these are the supply-demand theorem due to Gale (4), which states a condition for the existence of a flow satisfying demands at certain nodes from supplies at other nodes, and the Hoffman circulation theorem (received by the present author in private communication), which states a condition for the existence of a circulatory flow in a network in which each arc has associated with it not only an upper bound for the arc flow, but a lower bound as well. If the constraints on flows are integral (for example, if the bounds on arc flows for the circulation theorem are integers), it is also true that integral flows meeting the requirements exist provided any flow does so.
The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M . A generalization of this conjecture is proved in § 2.
The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T .
1. Introduction. Let Q be a matrix of order v, all of whose entries are 0's and l's. Let the total number of l's in Q be t , and let the absolute value of the determinant of Q be denoted by |det Q |. In this paper we study the problem of determining the maximum of |det Q | for fixed t and v. It turns out that this problem is closely related to the v , k , λ problem, which has been extensively studied of late.
Let it be required to arrange v elements into v sets such that each set contains exactly k distinct elements and such that each pair of sets has exactly λ elements in common (0 < λ < k < v). This problem we refer to as the v, k,λ combinatorial problem.
Combinatorial configurations may generally be phrased in terms of arrangements of objects into sets subject to certain conditions. In view of this, the question arises as to whether given a set S and its power-set U s (the class of all subsets of S ), it might be possible to structure U s in a combinatorially significant manner. This paper proposes and investigates one such structuring achieved by defining a distance function over U S .
Given A, B in U s , define their distance by
where N(E) denotes the number of elements in E, + ∞ being an admissible value.
Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k . An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].
This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by r i and let the sum of column j of A be denoted by S j . We call R = (r 1 , … , r m ) the row sum vector and S = (s 1 . . , s n ) the column sum vector of A . The vectors R and S determine a class
1.1
consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).
If S 1 , S 2 , S 3 , … , S n are subsets of a set M then it is known that a necessary and sufficient condition that it is possible to choose representatives at such that a i is in S i for ( i = 1, 2, 3, … , n ) and such that a i ≠ a j for i ≠ j , is that for k = 1, 2, 3, … , n , the union of any k of the sets S 1 , S 2 , … , S n , contains at least k elements. The theorem has a number of consequences amongst which we list the following.
Let A 1 , A 2 ,… , A n be a finite collection of subsets (not necessarily distinct) of a set A . By a transversal of A 1 , A 2 ,… , A n we shall mean a set of n distinct elements a 1 , a 2 ,… , a n of A such that, for some permutation i ¹ i 2 , … , i n of the integers 1, 2, … , n ,
More generally, we shall say that the set {a ¹ , a 2 , … , a r }, (r ≤ n) is a partial transversal oi A ¹ , A 2 , … A n of length r if (i) a 1 , a 2 , … , a r are distinct elements of A and (ii) there exists a set of distinct integers i 1 , i 2 … , i r such that
A well-known theorem of P. Hall (2) states that the sets A 1 , A 2 … , A n have a transversal (of length n) if, and only if, every k of them contain collectively at least k distinct elements (k = 1, 2, … , n) .