uncertainty measures for interval set information tables based on interval δ similarity relation abstract the notion of uncertainty measure is one of the most important topics in rough set theory and has been studied in different kinds of information tables however few studies have focused on the interval set information table which is regarded as one of the generalized models of single valued information tables this paper aims at studying the uncertainty measurements for interval set information tables firstly an interval similarity relation is induced based on the similarity degree the similarity relation induces the granules which form a covering in interval set information tables secondly four types of granularity measures are defined to measure the granularity of a covering thirdly the concepts of accuracy and roughness in rough set theory are respectively extended to accuracy and roughness for interval set information tables furthermore four new combinations ofuncertainty measures by considering proposed granularity measures and accuracy and roughness are defined and analyzed theoretical analyses and experimental results illustrate that the proposed measures are effective and accurate for interval set information tables

uncertainty measurement for interval valued information systems abstract interval valued information systems are generalized models of single valued information systems accuracy and roughness are employed to depict the uncertainty of a set under an attribute subset in a pawlak rough set model based on equivalence classes information theoretic measures of uncertainty for rough sets have also been proposed however there are few studies on uncertainty measurements for interval valued information systems this paper addresses the uncertainty measurement problem in interval valued information systems the concept of the similarity degree based on the possible degree is introduced consequently the similarity relation between two interval objects are constructed by a given similarity rate based on the similarity relation similarity classes are defined under this definition accuracy and roughness are given for interval valued information systems which are generalizations of the concepts accuracy and roughness for the equivalence relation based rough set model moreover an alternative uncertainty measure called the rough degree is proposed theoretical studies and numerical experiments show that the proposed measures are effective and suitable for interval valued information systems

finite difference preconditioning cubic spline collocation method of elliptic equations we discuss a finite difference preconditioner for the c 1 interpolatory cubic spline collocation method for a uniformly elliptic operator a defined by au delta u a 0 u in omega the unit square with homogeneous dirichlet boundary conditions using the generalized field of values arguments we discuss the eigenvalues of the preconditioned matrix hat l n 2 1 hat a n 2 where hat a n 2 is the matrix of the collocation discretization operator a n 2 corresponding to a and hat l n 2 is the matrix of the finite difference operator l n 2 corresponding to the uniformly elliptic operator l given by l v delta v v in omega with homogeneous dirichlet boundary conditions finally we mention a bound of h 1 singular values of hat l 1 n 2 hat a n 2 for a general elliptic operator au delta u a 1 u x a 2 u y a 0 u in omega

relations of the nuclear norm of a tensor and its matrix flattenings abstract for a 3 tensor of dimensions i 1 i 2 i 3 we show that the nuclear norm of its every matrix flattening is a lower bound of the tensor nuclear norm and which in turn is upper bounded by min i i i j times the nuclear norm of the matrix flattening in mode j for all j 1 2 3 the results can be generalized with some modifications to n tensors with n 3 both the lower and the upper bounds for the tensor nuclear norm are sharp in the case n 3 a computable sufficient criterion for the lower bound being tight is given as well

an action learning evaluation procedure for multiple criteria decision making problems abstract in this paper we focus on an extension of the analytic hierarchy process ahp that accommodates ambiguity on the part of the decision maker dm and facilitates the exploration of the decision domain we propose a systematic action learning process that builds confidence as it converges from numeric interval estimates to numeric point estimates our multiple criteria decision making mcdm problem procedure structures the problem as a hierarchy evaluates all objects using pairwise comparisons that accommodate vagueness and ambiguity uses interval prioritization techniques and does synthesis using the linear additive value function this action learning process facilitates the understanding of key stakeholders which is imperative for the successful implementation of the subsequent decision

well conditioned fractional collocation methods using fractional birkhoff interpolation basis the purpose of this paper is twofold firstly we provide explicit and compact formulas for computing both caputo and modified riemann liouville rl fractional pseudospectral differentiation matrices f psdms of any order at general jacobi gauss lobatto jgl points we show that in the caputo case it suffices to compute f psdm of order 0 1 to compute that of any order k with integer k 0 while in the modified rl case it is only necessary to evaluate a fractional integral matrix of order 0 1 secondly we introduce suitable fractional jgl birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties i the matrix generated from the new basis yields the exact inverse of f psdm at interior jgl points ii the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal and iii the resulted linear system is well conditioned in the caputo case while in the modified rl case the eigenvalues of the coefficient matrix are highly concentrated in both cases the linear systems of the collocation schemes using the new basis can be solved by an iterative solver within a few iterations notably the inverse can be computed in a very stable manner so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations fdes it is also noteworthy that the choice of certain special jgl points with parameters related to the order of the equations can ease the implementation we highlight that the use of the bateman s fractional integral formulas and fast transforms between jacobi polynomials with different parameters is essential for our algorithm development

the piecewise linear finite volume scheme the best known lowest order preconditioner for the d2dx2 chebyshev spectral operator the low order 3 node finite volume fv preconditioning technique is considered for its efficiency to spectrally solve the l u d 2udx 2 f problem the fourier spectrum of the associated preconditioning operator is analytically determined and compared with the known fourier spectra of the corresponding finite difference fd and finite element fe preconditioning operators moreover following what was done in p haldenwang g labrosse s abboudi m o deville chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation j comput phys 55 1984 115 128 for the fd case the fe and fv preconditioning operator chebyshev spectra are analytically determined first without reference to boundary conditions and then confirmed when dirichlet boundary conditions are imposed the convergence of the chebyshev towards the fourier spectra is established all this analysis leads to conclude that the best of the known lowest order preconditioners is provided by the piecewise linear finite volume scheme

knowledge acquisition in incomplete information systems a rough set approach abstract this paper deals with knowledge acquisition in incomplete information systems using rough set theory the concept of similarity classes in incomplete information systems is first proposed two kinds of partitions lower and upper approximations are then formed for the mining of certain and association rules in incomplete decision tables one type of optimal certain and two types of optimal association decision rules are generated two new quantitative measures random certainty factor and random coverage factor associated with each decision rule are further proposed to explain relationships between the condition and decision parts of a rule in incomplete decision tables the reduction of descriptors and induction of optimal rules in such tables are also examined

systematic studies on three way decisions with interval valued decision theoretic rough sets decision theoretic rough sets dtrs are a representative rough set model the loss function is a pivotal ingredient of dtrs which is associated with the decision maker s evaluation considering the value of loss function with the imprecise evaluation interval valued dtrs ivdtrs and its mechanism in this paper are explored first we construct a basic model of ivdtrs the comparison between dtrs and ivdtrs is discussed in the frame of ivdtrs we then focus on deriving three way decisions with the aid of two conventional methods i e a certain ranking method and a degree of possibility ranking method respectively the certain ranking method converts an interval value into single and derives decision rules under a certain risk attitude of decision maker the degree of possibility ranking method assumes the flexibility of interval and utilizes the preference between interval values all the combinations and their prerequisites are summarized in which we obtain two types of decision rules based on the above analysis we further propose an optimization method for three way decisions with ivdtrs which is designed to minimize the overall uncertainty based on the shannon entropy we also compare these methods based on standard data sets finally the criteria for choosing a suitable method to three way decisions with ivdtrs are generated these results can support decision making in the uncertain environment

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

interval ordered information systems interval information systems are generalized models of single valued information systems by introducing a dominance relation to interval information systems we propose a ranking approach for all objects based on dominance classes and establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation furthermore we discuss interval ordered decision tables and dominance rules to simplify knowledge representation and extract much simpler dominance rules we propose attribute reductions of interval ordered information systems and decision tables that eliminate only the information that are not essential from the viewpoint of the ordering of objects or dominance rules the approaches show how to simplify an interval ordered information system and find dominance rules directly from an interval ordered decision table these results will be helpful for decision making analysis in interval information systems

convergence analysis of spectral collocation methods for a singular differential equation solutions of partial differential equations with coordinate singularities often have special behavior near the singularities which forces them to be smooth special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency it has been observed numerically in the past that for a scheme to attain high accuracy it is unnecessary to impose all the pole conditions the constraints representing the special solution behavior near singularities in this paper we provide a theoretical justification for this observation specifically we consider an existing approach which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used gauss lobatto collocation scheme spectral convergence of the legendre and chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries

uncertainly measures of rough set prediction abstract the main statistics used in rough set data analysis the approximation quality is of limited value when there is a choice of competing models for predicting a decision variable in keeping within the rough set philosophy of non invasive data analysis we present three model selection criteria using information theoretic entropy in the spirit of the minimum description length principle our main procedure is based on the principle of indifference combined with the maximum entropy principle thus keeping external model assumptions to a minimum the applicability of the proposed method is demonstrated by a comparison of its error rates with results of c4 5 using 14 published data sets

on the subdifferential of symmetric convex functions of the spectrum for symmetric and orthogonally decomposable tensors abstract the subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis optimization and automatic control theory convex optimization over spaces of tensors is now gaining much interest due to its potential applications in signal processing statistics and engineering the goal of this paper is to present an extension of the approach by lewis 16 for the analysis of the subdifferential of certain convex functions of the spectrum of symmetric tensors we give a complete characterization of the subdifferential of schatten type tensor norms for symmetric tensors some partial results in this direction are also given for orthogonally decomposable tensors

incremental updating of rough approximations in interval valued information systems under attribute generalization interval valued information system ivis is a generalized model of single valued information system in which the attribute values of objects are all interval values instead of single values the attribute set in ivis is not static but rather dynamically changing over time with the collection of new information which results in the continuous updating of rough approximations for rough set based data analysis in this paper on the basis of the similarity based rough set model in ivis we develop incremental approaches for updating rough approximations in ivis under attribute generalization which refers to the dynamic changing of attributes firstly increment relationships between the original rough approximations and the updated ones when adding or deleting an attribute set are analyzed respectively and the incremental mechanisms for updating rough approximations in ivis are introduced which carry out the computation using the previous results from the original data set along with new results then the corresponding incremental algorithms are designed based on the proposed mechanisms finally comparative experiments on data sets from uci as well as artificial data sets are conducted respectively experimental results show that the proposed incremental algorithms can effectively reduce the running time for the computation of rough approximations in comparison with the static algorithm

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

set valued ordered information systems set valued ordered information systems can be classified into two categories disjunctive and conjunctive systems through introducing two new dominance relations to set valued information systems we first introduce the conjunctive disjunctive set valued ordered information systems and develop an approach to queuing problems for objects in presence of multiple attributes and criteria then we present a dominance based rough set approach for these two types of set valued ordered information systems which is mainly based on substitution of the indiscernibility relation by a dominance relation through the lower upper approximation of a decision some certain possible decision rules from a so called set valued ordered decision table can be extracted finally we present attribute reduction also called criteria reduction in ordered information systems approaches to these two types of ordered information systems and ordered decision tables which can be used to simplify a set valued ordered information system and find decision rules directly from a set valued ordered decision table these criteria reduction approaches can eliminate those criteria that are not essential from the viewpoint of the ordering of objects or decision rules

dynamic maintenance of approximations in set valued ordered decision systems under the attribute generalization set valued information systems are important type of data tables and generalized models of single valued information systems approximations are the focal point of approaches to knowledge discovery based on rough set theory which can be used to extract and represent the hidden knowledge in the form of decision rules attribute generalization refers to dynamic change of the attribute set in an information system with respect to the requirements of real life applications in this paper we focus on maintaining approximations dynamically in set valued ordered decision systems under the attribute generalization firstly a matrix based approach for computing approximations of upward and downward unions of decision classes is constructed by introducing the dominant and dominated matrices with respect to the dominance relation then incremental approaches for updating approximations are proposed which involves several modifications to relevant matrices without having to retrain from the start on all accumulated training data finally comparative experiments on data sets from uci as well as synthetic data sets show the proposed incremental updating methods are efficient and effective for dynamic attribute generalization

measures for evaluating the decision performance of a decision table in rough set theory as two classical measures approximation accuracy and consistency degree can be employed to evaluate the decision performance of a decision table however these two measures cannot give elaborate depictions of the certainty and consistency of a decision table when their values are equal to zero to overcome this shortcoming we first classify decision tables in rough set theory into three types according to their consistency and introduce three new measures for evaluating the decision performance of a decision rule set extracted from a decision table we then analyze how each of these three measures depends on the condition granulation and decision granulation of each of the three types of decision tables experimental analyses on three practical data sets show that the three new measures appear to be well suited for evaluating the decision performance of a decision rule set and are much better than the two classical measures

positive approximation and converse approximation in interval valued fuzzy rough sets methods of fuzzy rule extraction based on rough set theory are rarely reported in incomplete interval valued fuzzy information systems thus this paper deals with such systems instead of obtaining rules by attribute reduction which may have a negative effect on inducting good rules the objective of this paper is to extract rules without computing attribute reducts the data completeness of missing attribute values is first presented positive and converse approximations in interval valued fuzzy rough sets are then defined and their important properties are discussed two algorithms based on positive and converse approximations namely mine rules based on the positive approximation mrbpa and mine rules based on the converse approximation mrbca are proposed for rule extraction the two algorithms are evaluated by several data sets from the uc irvine machine learning repository the experimental results show that mrbpa and mrbca achieve better classification performances than the method based on attribute reduction

rough approximation quality revisited in rough set theory the approximation quality is the traditional measure to evaluate the classification success of attributes in terms of a numerical evaluation of the dependency properties generated by these attributes in this paper we re interpret the classical in terms of a classic measure based on sets the marczewski steinhaus metric and also in terms of proportional reduction of errors pre measures we also exhibit infinitely many possibilities to define like statistics which are meaningful in situations different from the classical one and provide tools to ascertain the statistical significance of the proposed measures which are valid for any kind of sample copyright 2001 elsevier science b v

converse approximation and rule extraction from decision tables in rough set theory in this paper the concept of a granulation order is proposed in an information system the converse approximation of a target concept under a granulation order is defined and some of its important properties are obtained which can be used to characterize the structure of a set approximation for a subset of the universe in an information system its converge degree is monotonously increasing under a granulation order this means that a proper family of granulations can be chosen for a target concept approximation according to user requirements as an application of the converse approximation an algorithm based on the converse approximation called rebca is designed for decision rule extraction from a decision table which has a time complexity of o m2 c 2 u log 2 u and its practical applications are illustrated by two examples

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

rough sets methodology for sorting problems in presence of multiple attributes and criteria abstract we consider a sorting classification problem in the presence of multiple attributes and criteria called the mac however the domains of attributes are not preference ordered while the domains of criteria scales are totally ordered by preference relations among the attributes we distinguish between qualitative attributes and quantitative attributes in order to construct a comprehensive preference model that could be used to support the sorting task we consider preferential information of the decision maker dm in the form of assignment examples i e exemplary assignments of some reference actions to the decision classes the preference model inferred from these examples is a set of if then decision rules the rules are derived from rough approximations of decision classes made up of reference actions they satisfy conditions of completeness and dominance and manage with possible ambiguity inconsistencies in the set of examples our idea of rough approximations involves three relations together indiscernibility similarity and dominance defined on qualitative and quantitative attributes and on criteria respectively the usefulness of this approach is illustrated by an example

rough set approach to incomplete information systems abstract in the paper we present rough set approach to reasoning in incomplete information systems we propose reduction of knowledge that eliminates only that information which is not essential from the point of view of classification or decision making in our approach we make only one assumption about unknown values the real value of a missing attribute is one from the attribute domain however we do not assume which one we show how to find decision rules directly from such an incomplete decision table which are as little non deterministic as possible and have minimal number of conditions

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

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

optimal error estimates of the chebyshev legendre spectral method for solving the generalized burgers equation in this paper the chebyshev legendre collocation method is applied to the generalized burgers equation optimal error estimate of the method is proved for the problem with the dirichlet boundary conditions also a legendre galerkin chebyshev collocation method is given for the generalized burgers equation the scheme is basically formulated in the legendre spectral form but with the nonlinear term being treated by the chebyshev collocation method so that the scheme can be implemented at chebyshev gauss lobatto points efficiently optimal order convergence is also obtained through coupling estimates in the l2 norm and the h1 norm

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

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

fuzzy rough sets assisted attribute selection attribute selection as refers to the problem of selecting those input attributes or features that are most predictive of a given outcome a problem encountered in many areas such as machine learning pattern recognition and signal processing unlike other dimensionality reduction methods attribute selectors preserve the original meaning of the attributes after reduction this has found application in tasks that involve datasets containing huge numbers of attributes in the order of tens of thousands which for some learning algorithms might be impossible to process further recent examples include text processing and web content classification as techniques have also been applied to small and medium sized datasets in order to locate the most informative attributes for later use one of the many successful applications of rough set theory has been to this area the rough set ideology of using only the supplied data and no other information has many benefits in as where most other methods require supplementary knowledge however the main limitation of rough set based attribute selection in the literature is the restrictive requirement that all data is discrete in classical rough set theory it is not possible to consider real valued or noisy data this paper investigates a novel approach based on fuzzy rough sets fuzzy rough feature selection frfs that addresses these problems and retains dataset semantics frfs is applied to two challenging domains where a feature reducing step is important namely web content classification and complex systems monitoring the utility of this approach is demonstrated and is compared empirically with several dimensionality reducers in the experimental studies frfs is shown to equal or improve classification accuracy when compared to the results from unreduced data classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method in addition it is shown that frfs is more powerful than the other as techniques in the comparative study

operator norm inequalities between tensor unfoldings on the partition lattice interest in higher order tensors has recently surged in data intensive fields with a wide range of applications including image processing blind source separation community detection and feature extraction a common paradigm in tensor related algorithms advocates unfolding or flattening the tensor into a matrix and applying classical methods developed for matrices despite the popularity of such techniques how the functional properties of a tensor changes upon unfolding is currently not well understood in contrast to the body of existing work which has focused almost exclusively on matricizations we here consider all possible unfoldings of an order k tensor which are in one to one correspondence with the set of partitions of 1 k we derive general inequalities between the lp norms of arbitrary unfoldings defined on the partition lattice in particular we demonstrate how the spectral norm p 2 of a tensor is bounded by that of its unfoldings and obtain an improved upper bound on the ratio of the frobenius norm to the spectral norm of an arbitrary tensor for specially structured tensors satisfying a generalized definition of orthogonal decomposability we prove that the spectral norm remains invariant under specific subsets of unfolding operations

preconditioning chebyshev spectral collocation by finite difference operators in 1979 orszag proposed a finite difference preconditioning of the chebyshev collocation discretization of the poisson equation in 1984 haldenwang labrosse abboudi and deville gave analytic formulae for the eigenvalues of this preconditioned operator in the one dimensional case experimental results over many years have shown the effectiveness of this procedure and appropriate bounds on the eigenvalues in two dimensions however there have been no mathematical proofs describing the behavior of the eigenvalues in two or more dimensions in this work we consider the generalized field of values u hat a nu u l nu where hat a n is the matrix of the chebyshev collocation scheme and l n is the matrix of the finite difference operator for the case of the chebyshev collocation of the helmoltz operator au delta u au a ge 0 preconditioned by the finite difference operator associated with the helmholtz operator bu delta u bu b ge 0 we prove that there are two constants 0 0 and u hat a nu u l nu le lambda 1 these results extend to higher dimensions and to bounds on l n 1 hat a n 1 w and hat a n 1 l n 1 w in the general case where au delta u a 1u x a 2u y au