Further results on controllability and the linear equation Ax = b

The linear equation Ax = b, with A an n × n matrix and b an n × l matrix over a unique factorization domain R, is related to the controllability submodule U of the pair (A, b). It is shown that the above equation has a solution lying in V if, and only if, A is unimodular as an operator on U. An example is given of a matrix which is unimodular as an operator on the controllability submodule, but not as an operator on Rⁿ and sparseness of this occurrence is discussed.     

Learning Matrix Functions over Rings

Let R be a commutative Artinian ring with identity and let X be a finite subset of R . We present an exact learning algorithm with a polynomial query complexity for the class of functions representable as f(x) = Π _i=1 ⁿ A _i (x _i ), where, for each 1 ≤ i ≤ n , A _i is a mapping A _i : X → R ^{mi× mi+1} and m ₁ = m _n+1 = 1 . We show that the above algorithm implies the following results: 1. Multivariate polynomials over a finite commutative ring with identity are learnable using equivalence and substitution queries. 2. Bounded degree multivariate polynomials over Z _n can be interpolated using substitution queries. 3. The class of constant depth circuits that consist of bounded fan-in MOD gates, where the modulus are prime powers of some fixed prime, is learnable using equivalence and substitution queries. Our approach uses a decision tree representation for the hypothesis class which takes advantage of linear dependencies. This paper generalizes the learning algorithm for automata over fields given in [BBB+]. Received January 28, 1997; revised May 29, 1997, June 18, 1997, and June 26, 1997.     

Star graph automorphisms and disjoint Hamilton cycles

The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies. We define automorphisms for star graphs St _n of degree n?1, for every positive odd integer n, which yield permutations of labels for the edges of St _n taken from the set of integers between 1 and ? n/2 ?. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known Hamilton cycle in St _n . Our method produces a better than two-fold improvement from ? ? (n)/10 ? to ? 2? (n)/9 ?, where ? is the Euler function, for the known number of edge-disjoint Hamilton cycles in St _n for all odd integers n. For prime n, the improvement is from ? n/8 ? to ? n/5 ?, and we can extend this result to the case when n is the power of a prime greater than 7.     

Construction and algebraic characterizations of a planar four-index transportation problem equivalent to a circularization network flow problem

In this study, a circularization network flow problem with m?+?n?+?2 nodes and (m?+?1)(n?+?1) arcs is described as a planar four-index transportation problem of order 1?×?m?×?n?×?1. Construction and several algebraic characterizations of the planar four-index transportation problem of order 1?×?m?×?n?×?1 are investigated using the generalized inverse and singular value decomposition of its coefficient matrix. The results are compared with some results we obtained on the transportation problem with m sources and n destinations. It is shown that these problems can be solved in terms of eigenvectors of the matrices J _m and J _n , where J _m is a m?×?m matrix whose entries are 1.     

Social choice function with subordinate relations as one variable

The paper proposes a social choice function R = g(R₁ R₂, ?,R_n, U) with a binary relation U called a subordinate relation on the group decision-makers in addition to individual preference relations R_i(i = 1, 2, ?, n). Then the individual influence to the group decision-making is connected with a relation U so that the more the members are subordinate to a member, the more its preference is reflected to the social preference. Two functions are defined as examples of such functions with subordinate relations. Four conditions, a theorem and real-world type examples to group decision-making are presented     

A more efficient algorithm for the min-plus multiplication

The (min, + ) product C of two n × n matrices A and B is defined as C _ij = min_1≦k≦n A _ik + B _kj . This paper presents an algorithm to compute the (min, +) product of two n × n matrices. The algorithm follows the approach described by Fredman, but is faster than Fredman's own algorithm: its time complexity is either O(n ³/√log₂ n) or even O(n ^2.5√log₂ n), if one adheres to the uniform-cost RAM model faithfully.

This result implies the existence of O(n ³/√log₂ n) algorithms for the problems that (min, +) matrix multiplication is equivalent to, such as the all-pairs shortest paths problem.

As the presented algorithm uses operations on sets, the formal analysis of its time complexity raises a few interesting questions about the applicability of the standard RAM complexity model.     

Least-squares problem for the quaternion matrix equation AXB+CYD=E over different constrained matrices

For any A=A ₁+A ₂ j∈Q ^n×n and η∈<texlscub>i, j, k</texlscub>, denote A ^{η H} =?η A ^H η. If A ^{η H} =A, A is called an $\eta$-Hermitian matrix. If A ^{η H} =?A, A is called an η-anti-Hermitian matrix. Denote η-Hermitian matrices and η-anti-Hermitian matrices by η HQ ^n×n and η AQ ^n×n , respectively.

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least-squares solution with the least norm for the quaternion matrix equation AXB+CYD=E over X∈η HQ ^n×n and Y∈η AQ ^n×n .     

Iterative solution of a class of linear equations with application to reconstruction of three-dimensional object arrays†

An iterative algorithm baaed on probabilistic estimation is described for obtaining the minimum-norm solution of a very large, consistent, linear system of equations AX = g where A is an (m × n) matrix with non-negative elements, x and g are respectively (n × 1) and (m × 1) vectors with positive components.

This algorithm will find application in the reconstruction of three-dimensional object arrays from projections and in several other areas.     

The genuine short GRB 090227B and the disguised by excess GRB 090510

GRB 090227B and GRB 090510, traditionally classified as short gamma-ray Bursts (GRBs), indeed originate from different systems. For GRB 090227B we inferred a total energy of the e ⁺ e ^? plasma \(E_{e^ + e^ - }^{tot} \) = (2.83 ± 0.15) × 10⁵³ erg, a baryon load of B = (4.1 ± 0.05) × 10^?5, and a CircumBurstMedium (CBM) average density 〈n _CBM〉 = (1.90 ± 0.20) × 10^?5 cm^?3. From these results we have assumed the progenitor of this burst to be a symmetric neutron stars (NSs) merger with masses m = 1.34 M_⊙, radii R = 12.24 km. GRB 090510, instead, has \(E_{e^ + e^ - }^{tot} \) = (1.10 ± 0.06) × 10⁵³ erg, B = (1.45 ± 0.28) × 10^?3, implying a Lorentz factor at transparency of Γ = (6.7 ± 1.7) × 10², which are characteristic of the long GRB class, and a very high CBM density, 〈n _CBM〉 = (1.85 ± 0.14) × 10³ cm^?3. The joint effect of the high values of Γ and of 〈n _CBM〉 compresses in time and “inflates” in intensity in an extended afterglow, making appear GRB 090510 as a short burst, which we here define as “disguised short GRB by excess” occurring an overdense region with 10³ cm^?3.     

Selective sampling for approximate clustering of very large data sets

A key challenge in pattern recognition is how to scale the computational efficiency of clustering algorithms on large data sets. The extension of non‐Euclidean relational fuzzy c‐means (NERF) clustering to very large (VL = unloadable) relational data is called the extended NERF (eNERF) clustering algorithm, which comprises four phases: (i) finding distinguished features that monitor progressive sampling; (ii) progressively sampling from a N × N relational matrix R_N to obtain a n × n sample matrix R_n; (iii) clustering R_n with literal NERF; and (iv) extending the clusters in R_n to the remainder of the relational data. Previously published examples on several fairly small data sets suggest that eNERF is feasible for truly large data sets. However, it seems that phases (i) and (ii), i.e., finding R_n, are not very practical because the sample size n often turns out to be roughly 50% of n, and this over‐sampling defeats the whole purpose of eNERF. In this paper, we examine the performance of the sampling scheme of eNERF with respect to different parameters. We propose a modified sampling scheme for use with eNERF that combines simple random sampling with (parts of) the sampling procedures used by eNERF and a related algorithm sVAT (scalable visual assessment of clustering tendency). We demonstrate that our modified sampling scheme can eliminate over‐sampling of the original progressive sampling scheme, thus enabling the processing of truly VL data. Numerical experiments on a distance matrix of a set of 3,000,000 vectors drawn from a mixture of 5 bivariate normal distributions demonstrate the feasibility and effectiveness of the proposed sampling method. We also find that actually running eNERF on a data set of this size is very costly in terms of computation time. Thus, our results demonstrate that further modification of eNERF, especially the extension stage, will be needed before it is truly practical for VL data. © 2008 Wiley Periodicals, Inc.     

A new formulation of the coVAT algorithm for visual assessment of clustering tendency in rectangular data

Since 1998, a graphical representation used in visual clustering called the reordered dissimilarity image or cluster heat map has appeared in more than 4000 biological or biomedical publications. These images are typically used to visually estimate the number of clusters in a data set, which is the most important input to most clustering algorithms, including the popularly chosen fuzzy c‐means and crisp k‐means. This paper presents a new formulation of a matrix reordering algorithm, coVAT, which is the only known method for providing visual clustering information on all four types of cluster structure in rectangular relational data. Finite rectangular relational data are an m× n array R of relational values between m row objects O_r and n column objects O_c. R presents four clustering problems: clusters in O_r, O_c, O_r∪c, and coclusters containing some objects from each of O_r and O_c. coVAT1 is a clustering tendency algorithm that provides visual estimates of the number of clusters to seek in each of these problems by displaying reordered dissimilarity images. We provide several examples where coVAT1 fails to do its job. These examples justify the introduction of coVAT2, a modification of coVAT1 based on a different reordering scheme. We offer several examples to illustrate that coVAT2 may detect coclusters in R when coVAT1 does not. Furthermore, coVAT2 is not limited to just relational data R. The R matrix can also take the form of feature data, such as gene microarray data where each data element is a real number: Positive values indicate upregulation, and negative values indicate downregulation. We show examples of coVAT2 on microarray data that indicate coVAT2 shows cluster tendency in these data. © 2012 Wiley Periodicals, Inc.     

BURST-ERROR-CORRECTING CODES WITH WEIGHT-CONSTRAINTS UNDER A NEW METRIC

Abstract

This paper discusses Varshamov-Gilbert bound, sphere-packing bound and some other problems of burst-error-correcting codes with weight constraints under a new metric. The new metric, introduced in an earlier paper, is defined in terms of suitable partition of the alphabet, the ring Z _q, of integers mod q. In general, different partitions of the same alphabet lead to different metrices. The partition of the alphabet {0, 1, 2, ?, q ? 1} given by {0} and {l, 2, ?, q ? 1} determines Hamming metric. Also, for the partition ρ _L = {B ₀, B ₁, ?,B?q/2?} where B ₀ = {0} and B ₀,= {i, q ?i} for i = 1, 2 ?,?q/2? the metric reduces to Lee metric.     

Approximate clustering in very large relational data

Different extensions of fuzzy c‐means (FCM) clustering have been developed to approximate FCM clustering in very large (unloadable) image (eFFCM) and object vector (geFFCM) data. Both extensions share three phases: (1) progressive sampling of the VL data, terminated when a sample passes a statistical goodness of fit test; (2) clustering with (literal or exact) FCM; and (3) noniterative extension of the literal clusters to the remainder of the data set. This article presents a comparable method for the remaining case of interest, namely, clustering in VL relational data. We will propose and discuss each of the four phases of eNERF and our algorithm for this last case: (1) finding distinguished features that monitor progressive sampling, (2) progressively sampling a square N × N relation matrix R_N until an n × n sample relation R_n passes a statistical test, (3) clustering R_n with literal non‐Euclidean relational fuzzy c‐means, and (4) extending the clusters in R_n to the remainder of the relational data. The extension phase in this third case is not as straightforward as it was in the image and object data cases, but our numerical examples suggest that eNERF has the same approximation qualities that eFFCM and geFFCM do. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 817–841, 2006.     

Information entropy,rough entropy and knowledge granulation in incomplete information systems

Rough set theory is a relatively new mathematical tool for use in computer applications in circumstances that are characterized by vagueness and uncertainty. Rough set theory uses a table called an information system, and knowledge is defined as classifications of an information system. In this paper, we introduce the concepts of information entropy, rough entropy, knowledge granulation and granularity measure in incomplete information systems, their important properties are given, and the relationships among these concepts are established. The relationship between the information entropy E(A) and the knowledge granulation GK(A) of knowledge A can be expressed as E(A)+GK(A) = 1, the relationship between the granularity measure G(A) and the rough entropy E _r(A) of knowledge A can be expressed as G(A)+E _r(A) = log₂|U|. The conclusions in Liang and Shi (2004 Liang, J.Y. and Shi, Z.Z. 2004. The information entropy, rough entropy and knowledge granulation in rough set theory. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12(1): 37–46. [Crossref], [Web of Science ®] , [Google Scholar]) are special instances in this paper. Furthermore, two inequalities ? log₂ GK(A) ≤ G(A) and E _r(A) ≤ log₂(|U|(1 ? E(A))) about the measures GK, G, E and E _r are obtained. These results will be very helpful for understanding the essence of uncertainty measurement, the significance of an attribute, constructing the heuristic function in a heuristic reduct algorithm and measuring the quality of a decision rule in incomplete information systems.     

On Parallel Integer Merging

The problem of merging two sorted arrays A = (a₁, a₂, ..., a_n1) and B = (b₁, b₂, ..., b_n2) is considered. For input elements that are drawn from a domain of integers [1...s] we present an algorithm that runs in O(log log log s) time using n/log log log s CREW PRAM processors (optimal speed-up) and O(ns^ε) space, where n = n₁ + n₂. For input elements that are drawn from a domain of integers [1...n] we present a second algorithm that runs in O(α(n)) time (where α(n) is the inverse of Ackermann′s function) using n/α(n) CREW PRAM processors and linear space. This second algorithm is non-uniform; however, it can be made uniform at a price of a certain loss of speed, or by using a CRCW PRAM.     

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×???×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box?(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}An axis-parallel k-dimensional box is a Cartesian product R ₁×R ₂×⋅⋅⋅×R _k where R _i (for 1≤i≤k) is a closed interval of the form [a _i ,b _i ] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a ?1+\frac1clogn?^d-1\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1} approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard.     

Ring models for delay-differential systems

This paper studies the algebraic structure of linear systems defined over R[λ], the ring of polynomials in λ with real coefficients. Natural definitions of controllability and observability are introduced and properties of R[λ]-transfer matrix realizations are discussed. It is shown that (A_n×n,D_n×m) is a controllable R[λ]-matrix pair if and only if for each set of polynomialsβ₁,β₂,…,β_n, in R[λ] there exists an R[λ] feedback matrixF such that detsI?A?BF]=∏i=1n(s+β_i). By regarding λ as a suitably defined delay operator, it is explained how this result might be applied to delay-differential systems in order to control dynamic response.     

A Time-Optimal Multiple Search Algorithm on Enhanced Meshes,with Applications

Given a sorted sequence A = a₁, a₂, ..., a_n of items from a totally ordered universe, along with an arbitrary sequence Q = q₁, q₂, ..., q_m (1 ≤ m ≤ n) of queries, the multiple search problem involves computing for every q_j (1 ≤ j ≤ m) the unique a_i for which a_i−1 ≤ q_j < a_i. In this paper we propose a time-optimal algorithm to solve the multiple search problem on meshes with multiple broadcasting. More specifically, on a [formula] × [formula] mesh with multiple broadcasting, our algorithm runs in [formula] time which is shown to be time-optimal. We also present some surprising applications of the multiple search algorithm to computer graphics, image processing, robotics, and computational geometry.