a bounded complementary sensitivity function ensures topology independent stability of homogeneous dynamical networks this paper investigates the topology independent stability of homogeneous dynamical networks composed of interconnected equal systems precisely dynamical systems with identical nominal transfer function f s are associated with the nodes of a directed graph whose arcs account for their dynamic interactions described by a common nominal transfer function g s it is shown that topology independent stability is guaranteed for all possible interconnections with interaction degree defined as the maximum number of arcs leaving a node equal at most to n if the infty norm of the complementary sensitivity function nf s g s 1 nf s g s 1 is less than 1 this bound is nonconservative in that there exist graphs with interaction degree n that are unstable for an infty norm greater than 1 when nodes and arcs transferences are affected by uncertainties with norm bound k 0 topology independent stability is robustly ensured if the infty norm is less than 1 1 2nk for symmetric systems stability is guaranteed for all topologies with interaction degree at most n if the nyquist plot of nf s g s does not intersect the real axis to the left of 1 2 the proposed results are applied to fluid networks and platoon formation

sparse sums of squares on finite abelian groups and improved semidefinite lifts let g be a finite abelian group this paper is concerned with nonnegative functions on g that are sparse with respect to the fourier basis we establish combinatorial conditions on subsets mathcal s s and mathcal t t of fourier basis elements under which nonnegative functions with fourier support mathcal s s are sums of squares of functions with fourier support mathcal t t our combinatorial condition involves constructing a chordal cover of a graph related to g and mathcal s s the cayley graph mathrm cay widehat g mathcal s cay g s with maximal cliques related to mathcal t t our result relies on two main ingredients the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern as well as a simple but key observation exploiting the structure of the fourier basis elements of g the characters of g we apply our general result to two examples first in the case where g mathbb z 2 n g z2n by constructing a particular chordal cover of the half cube graph we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most left lceil n 2 right rceil n 2 establishing a conjecture of laurent second we consider nonnegative functions of degree d on mathbb z n zn when d divides n by constructing a particular chordal cover of the dth power of the n cycle we prove that any such function is a sum of squares of functions with at most 3d log n d 3dlog n d nonzero fourier coefficients dually this shows that a certain cyclic polytope in mathbb r 2d r2d with n vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size 3d log n d 3dlog n d putting n d 2 n d2 gives a family of polytopes in mathbb r 2d r2d with linear programming extension complexity varomega d 2 d2 and semidefinite programming extension complexity o d log d o dlog d to the best of our knowledge this is the first explicit family of polytopes p d pd in increasing dimensions where mathrm xc text psd p d o mathrm xc text lp p d xcpsd pd o xclp pd where mathrm xc text psd xcpsd and mathrm xc text lp xclp are respectively the sdp and lp extension complexity

reducts and constructs in classic and dominance based rough sets approach abstract the idea of the reduct as defined in the classic rough sets approach crsa has proven to be inspiring enough to get into closely related theories including the dominance based rough sets approach drsa the procedure of reduction is generally similar to that of feature selection but narrower as it is the descriptive rather than the predictive aspect of data exploration that constitutes its principal goal crsa reducts are thus defined as minimal subsets of attributes that retain sufficiently high quality of object description developed within crsa the crsa reducts have given rise to the generalized notion of crsa constructs which have turned out to be superior to reducts in numerous practical experiments with real life data sets the generalization process is continued in this paper in which a definition of constructs in the context of drsa is introduced the definition fully analogous to that of crsa constructs differs only in that it is context based in drsa while context free in crsa consequently the presented drsa constructs are expected to have analogous properties to that of crsa constructs including superiority to drsa reducts in experiments with real life data sets

extremal positive semidefinite matrices whose sparsity pattern is given by graphs without k5 minors abstract for a graph g with p vertices the closed convex cone s 0 g consists of all real positive semidefinite p p matrices whose sparsity pattern is given by g that is those matrices with zeros in the off diagonal entries corresponding to nonedges of g the extremal rays of this cone and their associated ranks have applications to matrix completion problems maximum likelihood estimation in gaussian graphical models in statistics and gauss elimination for sparse matrices while the maximum rank of an extremal ray in s 0 g known as the sparsity order of g has been characterized for different classes of graphs we here study all possible extremal ranks of s 0 g we investigate when the geometry of the 1 cut polytope of g yields a polyhedral characterization of the set of extremal ranks of s 0 g for a graph g without k 5 minors we show that appropriately chosen normal vectors to the facets of the 1 cut polytope of g specify the off diagonal entries of extremal matrices in s 0 g we also prove that for appropriately chosen scalars the constant term of the linear equation of each facet supporting hyperplane is the rank of its corresponding extremal matrix in s 0 g furthermore we show that if g is series parallel then this gives a complete characterization of all possible extremal ranks of s 0 g consequently the sparsity order problem for series parallel graphs can be solved in terms of polyhedral geometry

two stage robust network flow and design under demand uncertainty we describe a two stage robust optimization approach for solving network flow and design problems with uncertain demand in two stage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand availability of such a recourse action allows one to come up with less conservative solutions compared to single stage optimization however this advantage often comes at a price two stage optimization is in general significantly harder than single stage optimization for network flow and design under demand uncertainty we give a characterization of the first stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is np hard even for a network flow problem on a bipartite graph we show however that if the second stage network topology is totally ordered or an arborescence then the separation problem is tractable unlike single stage robust optimization under demand uncertainty two stage robust optimization allows one to control conservatism of the solutions by means of an allowed budget for demand uncertainty using a budget of uncertainty we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector we generalize the approach to multicommodity network flow and design and give applications to lot sizing and location transportation problems by projecting out second stage flow variables we define an upper bounding problem for the two stage min max min optimization problem finally we present computational results comparing the proposed two stage robust optimization approach with single stage robust optimization as well as scenario based two stage stochastic optimization

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

a fast heuristic attribute reduction approach to ordered decision systems abstract rough set theory has shown success in being a filter based feature selection approach for analyzing information systems one of its main aims is to search for a feature subset called a reduct which preserves the classification ability of the original system in this paper we consider ordered decision systems where the preference order a fundamental concept in dominance based rough set approach plays a critical role in recent literature based on the greedy hill climbing method many heuristic attribute reduction algorithms are proposed by utilizing significance measures of attributes and they are extended to deal with ordered decision systems unfortunately they are often time consuming especially when applied to deal with large scale data sets with high dimensions to reduce the complexity a novel accelerator is introduced in heuristic algorithms from the perspectives of objects and criteria based on the new accelerator the number of objects and the dimension of criteria are lessened thus making the accelerated algorithms faster than their original counterparts while maintaining the same reducts experimental analysis shows the validity and efficiency of the proposed methods

attribute reduction in ordered decision tables via evidence theory rough set theory and dempster shafer theory of evidence are two distinct but closely related approaches to modeling and manipulating uncertain information it is quite natural to set up a hybrid model based on these two theories in this paper we investigate the problem of attribute reduction for ordered decision tables based on evidence theory belief and plausibility functions which are strongly connected with lower and upper approximation operators in dominance based rough set approach are proposed to define relative belief and plausibility reducts of ordered decision tables relationships among various types of relative reducts are thoroughly studied in consistent and inconsistent ordered decision tables a pair of numeric measures the inner and outer significance measures of a criterion is presented to search for a relative belief plausibility reduct which is meaningful for practical problems some real world tasks taken from the uci repository are employed to verify the feasibility and effectiveness of the proposed technique

decomposition in conic optimization with partially separable structure decomposition techniques for linear programming are difficult to extend to conic op timization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables however in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower dimensional cones the most important example is sparse semidefinite programming with a chordal sparsity pattern here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive semidefinite completable matrices with chordal sparsity patterns the paper describes a decomposition method that exploits partial separability in conic linear optimization the method is based on spingarn s method for equality constrained convex optimization combined with a fast interior point method for evaluating proximal operators

extremal bipartite matrices abstract let g be an undirected graph on vertices 1 n let m g be the convex cone of all positive semidefinite hermitian matrices a satisfying aij 0 if i j is not an edge of g in the case that g is a complete bipartite graph we characterize all extreme rays of m g in doing so we shall also find the maximum rank that an extreme matrix in m g can have

an efficient accelerator for attribute reduction from incomplete data in rough set framework feature selection attribute reduction from large scale incomplete data is a challenging problem in areas such as pattern recognition machine learning and data mining in rough set theory feature selection from incomplete data aims to retain the discriminatory power of original features to address this issue many feature selection algorithms have been proposed however these algorithms are often computationally time consuming to overcome this shortcoming we introduce in this paper a theoretic framework based on rough set theory which is called positive approximation and can be used to accelerate a heuristic process for feature selection from incomplete data as an application of the proposed accelerator a general feature selection algorithm is designed by integrating the accelerator into a heuristic algorithm we obtain several modified representative heuristic feature selection algorithms in rough set theory experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

dominance based rough fuzzy set approach and its application to rule induction abstract theories of fuzzy sets and rough sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness respectively marrying both leads to the hybrid notion of rough fuzzy sets in order to get a more accurate account of imperfect information in this paper our attention is paid to ordered fuzzy decision systems where condition criteria are preference ordered and decision classes are not only ordered but also fuzzy first the dominance based rough fuzzy approximations of an upward or downward cumulated fuzzy set are introduced in ordered fuzzy decision systems second lower and upper reducts relative to a certain cumulated fuzzy set are proposed to eliminate redundant criteria in the system then two approaches to attribute reduction are presented based on the discernibility matrix and the heuristic strategy respectively also decision rules are extracted directly from these approximations and some applicable and simplified decision rules are obtained according to requirements of decision makers finally a case study in bankruptcy risk analysis is used to illustrate the mechanism of the proposed methods

dominance based rough set approach to incomplete ordered information systems dominance based rough set approach has attracted much attention in practical applications ever since its inception this theory has greatly promoted the research of multi criteria decision making problems involving preferential information this paper mainly deals with approaches to attribute reduction in incomplete ordered information systems in which some attribute values may be lost or absent by introducing a new kind of dominance relation named the characteristic based dominance relation to incomplete ordered information systems we expand the potential applications of dominance based rough set approach to eliminate information that is not essential attribute reduction in the sense of reducing attributes is needed an approach on the basis of the discernibility matrix and the discernibility function to computing all relative reducts is investigated in incomplete ordered information systems consistent incomplete ordered decision tables to reduce the computational burden a heuristic algorithm with polynomial time complexity for finding a unique relative reduct is designed by using the inner and outer significance measures of each criterion candidate moreover some numerical experiments are employed to verify the feasibility and effectiveness of the proposed algorithms

robust ordinal regression for dominance based rough set approach to multiple criteria sorting we present a new multiple criteria sorting method deriving from dominance based rough set approach drsa the preference information supplied by the decision maker dm is a set of possibly imprecise and inconsistent assignment examples on a subset of reference alternatives relatively well known to the dm to structure the data we use drsa and subsequently represent the assignment examples by all minimal sets of rules covering all alternatives from the lower approximations of class unions such a set of rules is called minimal cover set it is one of the instances of the preference model compatible with dm s preference information in this way we implement the principle of robust ordinal regression ror to decision rule preference model for each alternative we derive the necessary and possible assignments specifying the range of classes to which the alternative is assigned by all or at least one compatible set of rules respectively as well as class acceptability indices we also introduce the notion of a representative compatible minimal cover set of rules whose selection builds on the results of ror addressing the robustness concern application of the approach is demonstrated by classifying 69 land zones in 4 classes representing different risk levels

variable consistency dominance based rough set approach to preference learning in multicriteria ranking abstract we present a methodology for non statistical preference learning in multicriteria ranking based on variable consistency dominance based rough set approach vc drsa a finite set of objects to be ranked is evaluated by a set of criteria which are real valued functions with ordinal or cardinal scales given the statement of a multicriteria ranking problem the only objective information one can get is the dominance relation over the set of objects the dominance relation is however too poor because it leaves many objects incomparable to enrich this relation and make the objects more comparable a decision maker dm must supply some preference information revealing her his value system with respect to multicriteria evaluations we are considering a frequent case when the preference information has the form of pairwise comparisons of some objects relatively well known to the dm called reference objects this information is thus composed of some decision examples on the reference objects it is the input data for a method that learns the preferences of the dm since this information is prone to inconsistencies we propose to structure it using vc drsa then the pairs of objects that are sufficiently consistent serve as a basis for induction of a preference model this model has the form of a set of if then decision rules application of these rules on the whole set of objects to be ranked yields a fuzzy preference structure directed weighted graph this preference structure is then exploited using a ranking method so as to work out a final recommendation we propose a list of properties that helps to choose a proper ranking method the methodology is illustrated by an example

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

attribute reduction based on evidence theory in incomplete decision systems attribute reduction is a basic issue in knowledge representation and data mining this paper deals with attribute reduction in incomplete information systems and incomplete decision systems based on dempster shafer theory of evidence the concepts of plausibility reduct and belief reduct in incomplete information systems as well as relative plausibility reduct and relative belief reduct in incomplete decision systems are introduced it is shown that in an incomplete information system an attribute set is a belief reduct if and only if it is a classical reduct and a plausibility consistent set must be a classical consistent set in a consistent incomplete decision system the concepts of relative reduct relative plausibility reduct and relative belief reduct are all equivalent in an inconsistent incomplete decision system an attribute set is a relative plausibility reduct if and only if it is a relative reduct a plausibility consistent set must be a belief consistent set and a belief consistent set is not a plausibility consistent set in general

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

characterizing reducts in multi adjoint concept lattices abstract the construction of reducts that is minimal sets of attributes containing the main information of a database is a fundamental task in different frameworks such as in formal concept analysis fca and rough set theory rst this paper will be focused on a general fuzzy extension of fca called multi adjoint concept lattice and we present a study about the attributes generating meet irreducible elements and on the reducts in this framework from this study we introduce interesting results on the cardinality of reducts and the consequences in the classical case

completion of partial matrices to contractions an n by m partially specified complex matrix is called a partial contraction if every rectangular submatrix consisting entirely of specified entries is itself a contraction necessary and sufficient condition are given for the pattern of specified entries such that any n by m partial contraction with this pattern may be completed to a full n by m contraction

exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion a basic framework for exploiting sparsity via positive semidefinite matrix completion is presented for an optimization problem with linear and nonlinear matrix inequalities the sparsity characterized with a chordal graph structure can be detected in the variable matrix or in a linear or nonlinear matrix inequality constraint of the problem we classify the sparsity in two types the domain space sparsity d space sparsity for the symmetric matrix variable in the objective and or constraint functions of the problem which is required to be positive semidefinite and the range space sparsity r space sparsity for a linear or nonlinear matrix inequality constraint of the problem four conversion methods are proposed in this framework two for exploiting the d space sparsity and the other two for exploiting the r space sparsity when applied to a polynomial semidefinite program sdp these conversion methods enhance the structured sparsity of the problem called the correlative sparsity as a result the resulting polynomial sdp can be solved more effectively by applying the sparse sdp relaxation preliminary numerical results on the conversion methods indicate their potential for improving the efficiency of solving various problems

exact travelling wave solutions of an integrable discrete reaction diffusion equation abstract we construct exact travelling wave solutions of a discrete reaction diffusion equation describing a system of coupled bistable elements the form of the bistable potential is chosen such that in a moving frame the waveform satisfies an integrable discrete equation that can be solved explicitly we analyse the linear stability of the wave and study wave propagation failure in the weak coupling limit

inverses of banded matrices abstract we establish a correspondence between the vanishing of a certain set of minors of a matrix a and the vanishing of a related set of minors of a 1 in particular inverses of banded matrices are characterized we then use our results to find patterns for toeplitz matrices with banded inverses finally we give an interesting determinant formula for inverses of banded matrices and show that in general a banded partial matrix may be completed in a unique way to give a banded inverse of the same bandwidth

the positive definite completion problem revisited abstract in r grone c r johnson e sa h wolkowicz positive definite completions of partial hermitian matrices linear algebra appl 58 1984 109 124 the positive definite semi completion problem in which the underlying graph is chordal was solved for the positive definite case the process was constructive and the completion was obtained by completing the partial matrix an entry at a time for the positive semidefinite case they obtained completions of a particular sequence of partial positive definite matrices with the same underlying graph and noted that there is a convergent subsequence of these completions that converges to the desired completion here in the chordal case we provide a constructive solution based entirely on matrix graph theoretic methods to the positive semi definite completion problem our solution associates a specific tree called the clique tree c r johnson m lundquist matrices with chordal inverse zero patterns linear and multilinear algebra 36 1993 1 17 with the chordal graph of the given partial positive semi definite matrix this tree structure allows us to complete the matrix a block at a time as opposed to an entry at a time as in grone et al 1984 for the positive definite case in grone et al 1984 using complex analytic techniques the completion for the positive definite case was shown to be the unique determinant maximizing completion and was shown to be the unique completion that has zeros in its inverse in the positions corresponding to the unspecified entries of the partial matrix here we show the same using only matrix graph theoretic tools

dominance based rough set approach for group decisions the objective of this paper is to propose an approach to support group multicriteria classification the approach is composed of three phases the first phase exploits the knowledge provided by each decision maker to individually approximate the decision classes using rough approximation the second phase seeks to combine the outputs of individual approximation phase into a collective decision table by using an appropriate aggregation procedure the third phase uses the collective decision table in order to infer a set of collective decision rules which synthesize the judgements and perspectives of the different decision makers and to permit the classification of all decision objects the proposed approach relies on the dominance based rough set approach drsa which is used at two different levels first the drsa is used during the first phase to approximate the input data relative to each decision maker second the drsa is used during the third phase to approximate the collective decision table and generate the collective decision rules this paper presents the theoretical foundation of the proposed approach three case studies using real world data and a comparative study of recent similar proposals

a linear programming approach to large scale linear optimal control problems this paper considers the solution of large scale linear optimal control problems subject to linear control and state constraints by application of a linear programming lp based methodology the proposed algorithm is based on a particular lp method that is suitably modified and adapted to the structure of the considered discrete time dynamic problem to keep the computation time low and efficiently store the arising large but sparse matrices the efficiency of the approach is demonstrated via a practical example arising in the field of traffic control in data communication networks the algorithm is shown to solve problems involving several thousands of variables in few seconds on a workstation thus enabling real time optimal control for a number of potential practical applications

attribute reduction based on generalized fuzzy evidence theory in fuzzy decision systems attribute reduction is viewed as an important issue in data mining and knowledge representation this paper studies attribute reduction in fuzzy decision systems based on generalized fuzzy evidence theory the definitions of several kinds of attribute reducts are introduced the relationships among these reducts are then investigated in a fuzzy decision system it is proved that the concepts of fuzzy positive region reduct lower approximation reduct and generalized fuzzy belief reduct are all equivalent the concepts of fuzzy upper approximation reduct and generalized fuzzy plausibility reduct are equivalent and a generalized fuzzy plausibility consistent set must be a generalized fuzzy belief consistent set in a consistent fuzzy decision system an attribute set is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy plausibility reduct but in an inconsistent fuzzy decision system a generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct in general

on generalized fuzzy belief functions in infinite spaces determined by a fuzzy implication operator a general type of fuzzy belief structure and its induced dual pair of fuzzy belief and plausibility functions in infinite universes of discourse are first defined relationship between the belief structure based and the belief space based fuzzy dempster shafer models is then established it is shown that the lower and upper fuzzy probabilities induced by the fuzzy belief space yield a dual pair of fuzzy belief and plausibility functions for any fuzzy belief structure there must exist a fuzzy belief space such that the fuzzy belief and plausibility functions defined by the given fuzzy belief structure are just the lower and upper fuzzy probabilities induced by the fuzzy belief space respectively essential properties of the fuzzy belief and plausibility functions are also examined the fuzzy belief and plausibility functions are respectively a fuzzy monotone choquet capacity and a fuzzy alternating choquet capacity of infinite order

local attribute reductions for decision tables abstract attribute reduction is among the most important areas of research in rough sets this paper investigates the types of local attribute reduction for decision tables we propose the concepts of lth decision class lower approximation reduction lth decision class reduction and lth decision class reduction for decision tables and provide their corresponding reduction algorithms via discernibility matrices we also establish the relationship between positive region reduction and the lth decision class reduction and report a case study using the university of california irvine dataset to verify the theoretical results

the rough membership functions on four types of covering based rough sets and their applications abstract pawlak s rough membership functions not only give numerical characterizations of pawlak s rough set approximations but also establishes the relationships between pawlak s rough sets and fuzzy sets or probabilistic rough sets however it is noteworthy that pawlak s rough membership functions have limitations when handling incomplete data that exist widely in the real world as will be shown in this paper one way to overcome this is to construct rough membership functions for covering based rough sets in this paper we first use an example in evidence based medicine to illustrate how to use pawlak s rough membership function on numerically characterizing decisions under the circumstances where data are complete then we construct covering based rough membership functions for four types of covering based rough sets which were examined by zhu and wang in ieee transactions on knowledge and data engineering 19 8 2007 1131 1144 and information sciences 201 2012 80 92 and use them to characterize these covering based rough sets numerically finally we present theoretical backgrounds for these covering based rough membership functions and illustrate how to apply them on numerically characterizing decisions under the circumstances where data are incomplete

mihaly matrix completions moments and sums of hermitian squares by mihaly bakonyi and hugo j woerdeman princeton series in applied mathematics princeton university press 2011 xii 518 pp hardback isbn 978 0 691 12889 4

extremal positive semidefinite doubly stochastic matrices abstract let k n denote the closed convex set of all n by n positive semidefinite doubly stochastic matrices the extreme points of k n have not been determined in this paper we find some extreme points our results are based primarily on rank and sparsity pattern we have a complete classification in the case n 4

positive semidefinite matrices with a given sparsity pattern abstract this paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries one of the main purposes is to obtain information and if possible describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros in particular we study those patterns for which the ranks of extremal matrices are less than a given number

dominance based rough set approach and knowledge reductions in incomplete ordered information system many methods based on the rough set to deal with incomplete information systems have been proposed in recent years however they are only suitable for the incomplete systems with regular attributes whose domains are not preference ordered this paper thus attempts to present research focusing on a complex incomplete information system the incomplete ordered information system in such incomplete information systems all attributes are considered as criterions a criterion indicates an attribute with preference ordered domain to conduct classification analysis in the incomplete ordered information system the concept of similarity dominance relation is first proposed two types of knowledge reductions are then formed for preserving two different notions of similarity dominance relations with introduction of the approximate distribution reduct into the incomplete ordered decision system the judgment theorems and discernibility matrixes associated with four novel approximate distribution reducts are obtained a numerical example is employed to substantiate the conceptual arguments

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

fuzzy preference based rough sets preference analysis is an important task in multi criteria decision making the rough set theory has been successfully extended to deal with preference analysis by replacing equivalence relations with dominance relations the existing studies involving preference relations cannot capture the uncertainty presented in numerical and fuzzy criteria in this paper we introduce a method to extract fuzzy preference relations from samples characterized by numerical criteria fuzzy preference relations are incorporated into a fuzzy rough set model which leads to a fuzzy preference based rough set model the measure of attribute dependency of the pawlak s rough set model is generalized to compute the relevance between criteria and decisions the definitions of upward dependency downward dependency and global dependency are introduced algorithms for computing attribute dependency and reducts are proposed and experimentally evaluated by using two publicly available data sets

on representing the positive semidefinite cone using the second order cone the second order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity second order cone formulations can also be solved more efficiently than semidefinite programming problems in general we consider the following question posed by lewis and glineur parrilo saunderson is it possible to express the general positive semidefinite cone using second order cones we provide a negative answer to this question and show that the 3 times 3 positive semidefinite cone does not admit any second order cone representation in fact we show that the slice consisting of 3 times 3 positive semidefinite hankel matrices does not admit a second order cone representation our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3 times 3 positive semidefinite cone whose second order cone rank grows to infinity

three layer granular structures and three way informational measures of a decision table abstract attribute reduction in rough set theory serves as a fundamental topic for information processing and its basis is usually a decision table d table d table attribute reduction concerns three hierarchical types and only classification based reduction is related to information theoretic representation aiming at inducing comprehensive d table attribute reduction with hierarchies and information this paper concretely constructs a d table s three layer granular structures and three way informational measures via granular computing and bayes theorem with regard to the d table the micro bottom meso middle and macro top are hierarchically organized according to the formal structure and systematic granularity then different layers produce different three way informational measures by developing bayes theorem thus three way weighted entropies originate from three way probabilities at the micro bottom and further evolve from the meso middle to the macro top and their granulation monotonicity and evolution systematicness are acquired furthermore three way informational measures are analyzed by three layer granular structures to achieve their hierarchical evolution superiority and algorithms finally structural and informational results are effectively illustrated by a d table example this study establishes d table s hierarchical structures to reveal constructional mechanisms and systematic relationships of informational measures the obtained results underlie the d table s hierarchical systematic and informational attribute reduction and they also enrich the three way decisions theory

on the analytic treatment of non integrable difference equations guided by general features of the solution of non integrable difference equations that emerge from kam theory a local perturbation theory is developed this captures both qualitatively and quantitatively all the essential features of the solution the method is applied to the standard map of chirikov 1979

stability analysis of the discrete landau ginsburg equation the stability of time independent solutions of a class of discrete nonlinear equations is investigated by extending a method developed earlier to study the stability of the static solutions of the continuous landau ginsburg equation a simple necessary condition for stability is found and it is shown that all nonlinear wave solutions are unstable while soliton and kink solutions may be stable a further method is introduced which shows that the soliton solution is in fact unstable whilst the kink is marginally stable

sequential covering rule induction algorithm for variable consistency rough set approaches we present a general rule induction algorithm based on sequential covering suitable for variable consistency rough set approaches this algorithm called vc domlem can be used for both ordered and non ordered data in the case of ordered data the rough set model employs dominance relation and in the case of non ordered data it employs indiscernibility relation vc domlem generates a minimal set of decision rules these rules are characterized by a satisfactory value of the chosen consistency measure we analyze properties of induced decision rules and discuss conditions of correct rule induction moreover we show how to improve rule induction efficiency due to application of consistency measures with desirable monotonicity properties

an optimal control approach to dynamic routing in networks this paper explores the application of optimal control theory to the problem of dynamic routing in networks the approach derives from a continuous state space model for dynamic routing and an associated linear optimal control problem with linear state and control variable inequality constraints the conceptual form of an algorithm is presented for finding a feedback solution to the optimal control problem when the inputs are assumed to be constant in time the algorithm employs a combination of necessary conditions dynamic programming and linear programming to construct a set of convex polyhedral cones which cover the admissible state space with optimal controls an implementable form of the algorithm along with a simple example is presented for a special class of single destination networks

minimal separators of 2 chordal graphs abstract as a generalization of chordal graph the notion of 2 chordal graph arises naturally from the consideration of matrix valued matrix completion problems the structure of minimal vertex separators of 2 chordal graphs is determined moreover except in trivial cases it is possible to choose a clique complete subgraph as a minimal vertex separator as an application of this result and using the techniques in 5 the characterizations of 2 chordal graphs in terms of an elimination order and a list of forbidden subgraphs begun in 6 is given

incremental update of approximations in dominance based rough sets approach under the variation of attribute values dominance based rough sets approach drsa has received much attention since it is able to acquire knowledge from information with preference ordered attribute domains and decision classes in many real life applications the information systems may evolve over time dynamically in a dynamic information system the obtained knowledge e g approximations in drsa need to be updated for decision making and other related tasks as a useful technique the incremental update can be applied to process dynamic information with revising the obtained knowledge in this paper we propose an incremental approach for maintaining approximations of drsa when attribute values vary over time some numerical examples illustrate that the incremental approach can renew approximations of drsa without beginning from scratch experimental evaluations show that the incremental algorithm can effectively reduce the computational time in comparison with the non incremental one when the ratio of the attribute values varied is less than a threshold

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

α dominance relation and rough sets in interval valued information systems though rough set has been widely used to study systems characterized by insufficient and incomplete information its performance in dealing with initial interval valued data needs to be seriously considered for improving the suitability and scalability the aim of this paper is to present a parameterized dominance based rough set approach to interval valued information systems first by considering the degree that an interval valued data is dominating another one we propose the concept of dominance relation second we present the dominance based rough set model in interval valued decision systems finally we introduce lower and upper approximate reducts into dominance based rough set for simplifying decision rules we also present the judgement theorems and discernibility functions which describe how lower and upper approximate reducts can be calculated this study suggests potential application areas and new research trends concerning rough set approach to interval valued information systems

matrices attaining the minimum semidefinite rank of a chordal graph every positive semidefinite matrix whose graph g is chordal may be represented as a sum of rank 1 positive semidefinite matrices whose graphs are subgraphs of g we show that if the matrix is of minimum rank then this representation is unique this result is used to give a full characterization of the completely positive matrices of minimum rank with a given chordal graph every such matrix is shown to have a unique easily computable minimal rank 1 representation and cp rank equal to its rank we also characterize all chordal graphs with the property that every minimum rank doubly nonnegative matrix realization of the graph is completely positive

the linear saturated decentralized strategy for constrained flow control is asymptotically optimal we present an algorithm for constrained network flow control in the presence of an unknown demand our algorithm is decentralized in the sense that it is implemented by a team of agents each controlling just the flow on a single arc of the network based only on the buffer levels at the nodes at the extremes of the arc while ignoring the actions of other agents and the network topology we prove that our algorithm is also stabilizing and steady state optimal specifically we show that it asymptotically produces the minimum norm flow we finally generalize our algorithm to networks with a linear dynamics and we prove that certain least square optimality properties still hold

knowledge reduction in random information systems via dempster shafer theory of evidence knowledge reduction is one of the main problems in the study of rough set theory this paper deals with knowledge reduction in random information systems based on dempster shafer theory of evidence the concepts of belief and plausibility reducts in random information systems are first introduced it is proved that both of belief reduct and plausibility reduct are equivalent to classical reduct in random information systems the relative belief and plausibility reducts in consistent and inconsistent random decision systems are then proposed and compared to the relative reduct and relationships between the new reducts and some existing ones are examined

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

chordal decomposition in operator splitting methods for sparse semidefinite programs we employ chordal decomposition to reformulate a large and sparse semidefinite program sdp either in primal or dual standard form into an equivalent sdp with smaller positive semidefinite psd constraints in contrast to previous approaches the decomposed sdp is suitable for the application of first order operator splitting methods enabling the development of efficient and scalable algorithms in particular we apply the alternating direction method of multipliers admm to solve decomposed primal and dual standard form sdps each iteration of such admm algorithms requires a projection onto an affine subspace and a set of projections onto small psd cones that can be computed in parallel we also formulate the homogeneous self dual embedding hsde of a primal dual pair of decomposed sdps and extend a recent admm based algorithm to exploit the structure of our hsde the resulting hsde algorithm has the same leading order computational cost as those for the primal or dual problems only with the advantage of being able to identify infeasible problems and produce an infeasibility certificate all algorithms are implemented in the open source matlab solver cdcs numerical experiments on a range of large scale sdps demonstrate the computational advantages of the proposed methods compared to common state of the art solvers

approximate distribution reducts in inconsistent interval valued ordered decision tables abstract many methods based on the rough set theory to deal with information systems have been proposed in recent decades in practice some information systems are based on dominance relations and may be inconsistent because of various factors moreover taking the imprecise evaluations and assignments in the description of objects into account single valued information systems have been generalized to interval valued information systems in this paper by introducing a dominance relation to interval valued ordered information systems we establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation to extract the minimal decision rules approximate distribution reducts are proposed in inconsistent interval valued ordered decision tables this paper presents a theoretical method based on the discernibility matrix to enumerate all reducts and a practical approach on the basis of significance to find one reduct and two equivalent definitions of approximate distribution reducts are also introduced in addition numerical examples are employed to examine the validity of the approaches proposed in this paper

a direct proof for the matrix decomposition of chordal structured positive semidefinite matrices agler helton mccullough and rodman proved that a graph is chordal if and only if any positive semidefinite psd symmetric matrix whose nonzero entries are specified by a given graph can be decomposed as a sum of psd matrices corresponding to the maximal cliques this decomposition is recently exploited to solve positive semidefinite programming efficiently their proof is based on a characterization for psd matrix completion of a chordal structured matrix due to grone johnson sa and wolkowicz this note gives a direct and simpler proof for the result of agler et al which leads to an alternative proof of grone et al

network decentralized control strategies for stabilization we consider the problem of stabilizing a class of systems formed by a set of decoupled subsystems nodes interconnected through a set of controllers arcs controllers are network decentralized i e they use information exclusively from the nodes they interconnect this condition requires a block structured feedback matrix having the same structure as the transpose of the overall input matrix of the system if the subsystems do not have common unstable eigenvalues we demonstrate that the problem is solvable in the general case we provide sufficient conditions for solvability when subsystems are identical and each input agent controls a pair of subsystems with input matrices having opposite sign flow networks we prove that stabilization is possible if and only if the system is connected with the external environment our proofs are constructive and lead to structured linear matrix inequalities lmis

decentralized quantized control for ncss under periodic protocol this paper is concerned with the problem of designing decentralized quantized controller for networked control systems ncss that are subject to time varying transmission intervals packet dropout communication constraints and limited bandwidth the convex overapproximation method proposed in donkers and heemels 2011 is used to handle with time varying transmission intervals packet dropout process in this paper is modeled as both independent identically distributed process and homogeneous markov jumping process the uniform quantizer is adopted here to deal with limited bandwidth in quantization strategy which switches between zooming out and zooming in modes a variable called proportionality coefficient of saturation value is used to handle with the complex coupling relationship between system states and quantized variables periodic protocol is used here to deal with communication constraints under which only one node can access the network and send its information per transmission within this setup we derive stability results in terms of linear matrix inequalities lmis an illustrative example is given to show the usefulness of the proposed framework for the stability analysis of some classes of ncss

positive approximation an accelerator for attribute reduction in rough set theory feature selection is a challenging problem in areas such as pattern recognition machine learning and data mining considering a consistency measure introduced in rough set theory the problem of feature selection also called attribute reduction aims to retain the discriminatory power of original features many heuristic attribute reduction algorithms have been proposed however quite often these methods are computationally time consuming to overcome this shortcoming we introduce a theoretic framework based on rough set theory called positive approximation which can be used to accelerate a heuristic process of attribute reduction based on the proposed accelerator a general attribute reduction algorithm is designed through the use of the accelerator several representative heuristic attribute reduction algorithms in rough set theory have been enhanced note that each of the modified algorithms can choose the same attribute reduct as its original version and hence possesses the same classification accuracy experiments show that these modified algorithms outperform their original counterparts it is worth noting that the performance of the modified algorithms becomes more visible when dealing with larger data sets

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

instability of spatially quasi periodic states of the ginzburg landau equation the ginzburg landau gl equation with real coefficients is a model equation appearing in superconductor physics and near critical hydrodynamic stability problems the stationary gl equation has a two parameter i 1 i 2 family of spatially quasi periodic qp states with frequencies omega 1 omega 2 and frequency map with determinant delta k partial omega 1 omega 2 partial i 1 i 2 in this paper the linear stability of these qp states is studied and an expression for the stability exponent is obtained which has a novel geometric interpretation in terms of delta k when delta k 0 is a necessary but not sufficient condition for linear stability there is an interesting relation between delta k and the kam persistence theorem for invariant toroids

sparse matrix decompositions and graph characterizations abstract zeros in positive definite correlation matrices arise frequently in probability and statistics and are intimately related to the notion of stochastic independence the question of when zeros i e sparsity in a positive definite matrix a are preserved in its cholesky decomposition and vice versa was addressed by paulsen et al v i paulsen s c power r r smith schur products and matrix completions j funct anal 85 1989 151 178 in particular they prove that for the pattern of zeros in a to be retained in the cholesky decomposition of a the pattern of zeros in a has to necessarily correspond to a chordal or decomposable graph associated with a specific type of vertex ordering this result therefore also yields a characterization of chordal graphs in terms of sparse positive definite matrices and has proved to be extremely useful in probabilistic and statistical analysis of markov random fields now consider a positive definite matrix a and its cholesky decomposition given by a ldl t where l is lower triangular with unit diagonal entries and d a diagonal matrix with positive entries in this paper we prove that a necessary and sufficient condition for zeros i e sparsity in a positive definite matrix a to be preserved in its associated cholesky matrix l and in addition also preserved in the inverse of the cholesky matrix l 1 is that the pattern of zeros corresponds to a co chordal or homogeneous graph associated with a specific type of vertex ordering we proceed to provide a second characterization of this class of graphs in terms of determinants of submatrices that correspond to cliques in the graph these results add to the growing body of literature in the field of sparse matrix decompositions and also prove to be critical ingredients in the probabilistic analysis of an important class of markov random fields