high precision computation of riemann s zeta function by the riemann siegel formula i we present rigorous and sharp bounds for the terms and remainder in the riemann siegel formula for a general argument not necessarily on the critical line this allows for the computation of s and z t to high precision we also derive the riemann siegel formula in a new and more direct way

a note on fuzzy relational equations with min implication composition the fuzzy relational equations with min implication composition are considered where the implication operation involved is the residuum with respect to the minimum operation the solution set of such a system of equations whenever nonempty can be characterized by a minimum solution and finitely many maximal solutions it is shown that a system of fuzzy relational equations with min implication composition can be reformulated in polynomial time into a system of 0 1 mixed integer linear inequalities and consequently the structure of its solution set can be revealed in a succinct manner besides it is shown that the determination of all maximal solutions is polynomially reducible to the transversal hypergraph generation problem of which some practically well performed algorithms are available

factorization of matrices with grades we present an approach to decomposition and factor analysis of matrices with ordinal data the matrix entries are grades to which objects represented by rows satisfy attributes represented by columns e g grades to which an image is red a product has a given feature or a person performs well in a test we assume that the grades are taken from bounded scales equipped with certain aggregation operators that are involved in the decompositions particular cases of the decompositions include the well known boolean matrix decomposition and the sup t norm and inf residuum decompositions we consider the problem of decomposition of a given matrix into a product of two matrices with grades such that the number of factors i e the inner dimension be as small as possible we observe that computing such decompositions is np hard and present a greedy approximation algorithm our algorithm is based on a geometric insight provided by a theorem identifying particular rectangular shaped submatrices as optimal factors for the decompositions these factors correspond to fixpoints of certain galois connections associated with the input matrix which are called formal concepts and allow an easy interpretation of the decomposition we present illustrative examples and experimental evaluation of the algorithm

oversampled graph laplacian matrix for graph filter banks we describe a method of oversampling signals defined on a weighted graph by using an oversampled graph laplacian matrix the conventional method of using critically sampled graph filter banks has to decompose the original graph into bipartite subgraphs and a transform has to be performed on each subgraph because of the spectral folding phenomenon caused by downsampling of graph signals therefore the conventional method cannot always utilize all edges of the original graph in a single stage transformation our method is based on oversampling of the underlying graph itself and it can append nodes and edges to the graph somewhat arbitrarily we use this approach to make one oversampled bipartite graph that includes all edges of the original non bipartite graph we apply the oversampled graph with the critically sampled graph filter bank or the oversampled one for decomposing graph signals and show the performances on some experiments

sup t norm and inf residuum are one type of relational product unifying framework and consequences we present a simple framework which enables us to consider the well known sup t norm and inf residuum products of relations as two particular cases of a single more general type of product we present basic properties of the framework and consequences for the theory of fuzzy relations informally the paper implies that in many cases of fuzzy relational modeling such as in solving fuzzy relational equations there is no need to develop the methods for sup t norm and inf residuum products separately because these methods are just two particular instances of a single method

learning graphs with monotone topology properties and multiple connected components recent papers have formulated the problem of learning graphs from data as an inverse covariance estimation problem with graph laplacian constraints while such problems are convex existing methods cannot guarantee that solutions will have specific graph topology properties e g being a tree which are desirable for some applications the problem of learning a graph with topology properties is in general non convex in this paper we propose an approach to solve these problems by decomposing them into two sub problems for which efficient solutions are known specifically a graph topology inference gti step is employed to select a feasible graph topology then a graph weight estimation gwe step is performed by solving a generalized graph laplacian estimation problem where edges are constrained by the topology found in the gti step our main result is a bound on the error of the gwe step as a function of the error in the gti step this error bound indicates that the gti step should be solved using an algorithm that approximates the data similarity matrix by another matrix whose entries have been thresholded to zero to have the desired type of graph topology the gti stage can leverage existing methods which are typically based on minimizing the total weight of removed edges since the gwe stage is an inverse covariance estimation problem with linear constraints it can be solved using existing convex optimization methods we demonstrate that our approach can achieve good results for both synthetic and texture image data

compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs this paper extends previous results on wavelet filterbanks for data defined on graphs from the case of orthogonal transforms to more general and flexible biorthogonal transforms as in the recent work the construction proceeds in two steps first we design one dimensional two channel filterbanks on bipartite graphs and then extend them to multi dimensional separable two channel filterbanks for arbitrary graphs via a bipartite subgraph decomposition we specifically design wavelet filters based on the spectral decomposition of the graph and state sufficient conditions for the filterbanks to be perfect reconstruction and orthogonal while our previous designs referred to as graph qmf filterbanks are perfect reconstruction and orthogonal they are not exactly k hop localized i e the computation at each node is not localized to a small k hop neighborhood around the node in this paper we relax the condition of orthogonality to design a biorthogonal pair of graph wavelets that are k hop localized with compact spectral spread and still satisfy the perfect reconstruction conditions the design is analogous to the standard cohen daubechies feauveau s cdf construction of factorizing a maximally flat daubechies half band filter preliminary results demonstrate that the proposed filterbanks can be useful for both standard signal processing applications as well as for signals defined on arbitrary graphs

notes on solution sets of inf t fuzzy relational equations on complete brouwerian lattices and fuzzy relational equations on complete brouwerian lattices this paper clarifies several hypotheses and results given in q q xiong and x p wang solution sets of fuzzy relational equations on complete brouwerian lattices information sciences 177 21 47574767 2007 and in q q xiong and x p wang fuzzy relational equations on complete brouwerian lattices information sciences 193 0 141152 2012 which in their current form are confusing and need to be improved

associatively tied implications we say that an implication operator a on a complete lattice l is associatively tied if there is a binary operation t on l that ties a that is the identity a a gamma a t holds for all in l this property extends to multiple valued logic the following equivalence in classical logic x y z x y z we show that in this case there exists an associative binary operation ta that ties a hence the nomenclature we study properties of that ta when a is associatively tied we then seek a characterization for the validity of associative tiedness for an implication a phrased in terms of a and two adjoints of it

efficient sampling set selection for bandlimited graph signals using graph spectral proxies we study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs a frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity smoothly varying signals defined on the nodes are of particular interest in various applications and tend to be approximately bandlimited in the frequency basis sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples most approaches to this problem require a computation of the frequency basis i e the eigenvectors of the variation operator followed by a search procedure using the basis elements this can be impractical in terms of storage and time complexity for real datasets involving very large graphs we circumvent this issue in our formulation by introducing quantities called graph spectral proxies defined using the powers of the variation operator in order to approximate the spectral content of graph signals this allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements we show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited furthermore the proposed approach is valid for any choice of the variation operator thereby covering a wide range of graphs and applications we demonstrate its effectiveness through various numerical experiments

minimal solutions of general fuzzy relation equations on linear carriers an algebraic characterization abstract this paper considers a general fuzzy relation equation which has minimal solutions if it is solvable in this case an algebraic characterization is introduced which provides an interesting method to compute minimal solutions in this general setting moreover a comparison with other frameworks is also given

m channel perfect recovery of coarsened graphs and graph signals with spectral invariance and topological preservation in this paper an m channel perfect reconstruction filter bank based on a coarsening algorithm is proposed compared with most of the designs of graph filter banks that do not consider the graph reconstruction our proposed design can provide the perfect graph reconstruction as well as perfect graph signal reconstruction in the analysis part the proposed filter bank provides a coarse version of the input graph as well as coarsened graph signal spectral invariant to the input signal in the synthesis part the filter bank perfectly reconstructs the input graph as well as input signal from their coarse version the spirit of the proposed design is to partition the spectral information of the input graph and input graph signal into every channel the partition method is adjustable and can be nonuniform two intuitive schemes named sort by eigenvalue and sort by intensity for uniform partition are introduced in the proposed design the coarsening operators are obtained through an existing coarsening algorithm while the recovery operators are defined by taking the conjugate transpose of the coarsening operators besides we address the relation between the proposed design and a framework of sampling graph signals based on the discrete sampling theory it is shown that the coarsening operator in the proposed design is actually a special case of the sampling operator in the framework based on the discrete sampling theory experimental results are presented to demonstrate the effectiveness of the proposed design of filter banks

formal concept analysis via multi adjoint concept lattices several fuzzifications of formal concept analysis have been proposed to deal with uncertain information in this paper we focus on concept lattices under a multi adjoint paradigm which enriches the language providing greater flexibility to the user in that he she can choose from a number of different connectives multi adjoint concept lattices are shown to embed different fuzzy extensions of concept lattices found in the literature the main results of the paper being the representation theorem of this paradigm and the embedding of other well known approaches

downsampling of signals on graphs via maximum spanning trees downsampling of signals living on a general weighted graph is not as trivial as of regular signals where we can simply keep every other samples in this paper we propose a simple yet effective downsampling scheme in which the underlying graph is approximated by a maximum spanning tree mst that naturally defines a graph multiresolution this mst based method significantly outperforms the two previous downsampling schemes coloring based and svd based on both random and specific graphs in terms of computations and partition efficiency quantified by the graph cuts the benefit of using mst based downsampling for recently developed critical sampling graph wavelet transforms in compression of graph signals is demonstrated

on heterogeneous formal contexts we propose a new type of fuzzification for formal concept analysis that works with heterogeneous values in a context and illustrate this with an example we formulate and prove an appropriate counterpart to the so called basic theorem of a concept lattice we show that this is a generalization of the previous approaches it covers the so called generalized concept lattice and multi adjoint t concept lattices

on the solvability of bipolar max product fuzzy relation equations with the product negation abstract this paper studies the solvability of the max product fuzzy relation equations in which a negation operator is considered specifically the residuated negation of the product t norm has been introduced in these equations in order to increase the flexibility of the standard fuzzy relation equations introduced by sanchez in 1976 the solvability and the set of solutions of these bipolar equations have been studied in different scenarios depending on the considered number of variables and equations

using concept lattice theory to obtain the set of solutions of multi adjoint relation equations an important goal in the fuzzy relation equations framework is to obtain the whole set of solutions this paper introduces algebraic properties in the multi adjoint concept lattice setting moreover using the relationship between these concept lattices and multi adjoint relation equations the complete set of solutions of these general equations is characterized and computed avoiding the necessity of considering minimal solutions

multi adjoint relation equations definition properties and solutions using concept lattices this paper generalizes fuzzy relation equations following the multi adjoint philosophy moreover the solutions of these general fuzzy relation equations and the concepts of a multi adjoint property oriented concept lattice are related and several results are obtained from the theory of concept lattices as a consequence of this relevant relation more properties about these general equations can be proven from the theory of concept lattices and the algorithms developed to compute concept lattices can be used to obtain solutions furthermore an interesting application to fuzzy logic programming has been introduced in which an important problem in this topic has been interpreted in terms of solving a multi adjoint relation equation

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

do we need minimal solutions of fuzzy relational equations in advance minimal solutions play a crucial role in description of all solutions to a fuzzy relational equation the reason is that all solutions form a convex set with respect to fuzzy set inclusion therefore having all extremal solutions we can represent the entire solution set as a union of intervals bounded from above by the greatest solution and from below by the minimal solutions however when computing the intervals we obtain many duplicate solutions the obvious question is as follows is there another way of representing the solution set for instance without the need of having all the minimal solutions in advance we provide the positive answer to this question

discrete signal processing on graphs sampling theory we propose a sampling theory for signals that are supported on either directed or undirected graphs the theory follows the same paradigm as classical sampling theory we show that perfect recovery is possible for graph signals bandlimited under the graph fourier transform the sampled signal coefficients form a new graph signal whose corresponding graph structure preserves the first order difference of the original graph signal for general graphs an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise for graphs whose graph fourier transforms are frames with maximal robustness to erasures as well as for erd s renyi graphs random sampling leads to perfect recovery with high probability we further establish the connection to the sampling theory of finite discrete time signal processing and previous work on signal recovery on graphs to handle full band graph signals we propose a graph filter bank based on sampling theory on graphs finally we apply the proposed sampling theory to semi supervised classification of online blogs and digit images where we achieve similar or better performance with fewer labeled samples compared to previous work

constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges the paper deals with the isomorphism between the constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges the essential point of the former approach encompasses the full diversification of data structures within a formal context in particular we use a different complete lattice for diverse objects a different complete lattice for diverse attributes and a different poset for diverse matrix fields the latter framework with heterogeneous hedges results in a reduction in the size of the corresponding concept lattice we present the properties of constraint heterogeneous approach that is associated with the fixpoints of hedges and we add remarks to the related studies

explicit upper bounds for the remainder term in the divisor problem we first report on computations made using the gp pari package that show that the error term x in the divisor problem is mathscr m x 4 o 0 35 x 1 4 log x when x ranges 1 081 080 10 10 where mathscr m x 4 is a smooth approximation the remaining part and in fact most of the paper is devoted to showing that delta x le 0 397 x 1 2 when x ge 5 560 and that delta x le 0 764 x 1 3 log x when x ge 9 995 several other bounds are also proposed we use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function we finally formulate a positivity conjecture concerning

tight wavelet frames on multislice graphs we present a framework for the design of wavelet transforms tailored to data defined on multislice graphs i e multiplex or dynamic graphs graphs with multiple types of interactions are ubiquitous in real life motivating the extension of wavelets to these complex domains our framework generalizes the recently proposed spectral graph wavelet transform sgwt d hammond p vandergheynst and r gribonval wavelets on graphs via spectral graph theory appl comput harmon anal vol 30 pp 129 150 mar 2011 which is designed in the spectral frequency domain of an arbitrary finite weighted graph we extend the sgwt to form a tight frame which conserves energy in the wavelet domain and define the relationship between conventional and spectral graph wavelets we then propose a design for multislice graphs that is based on the higher order singular value decomposition hosvd a powerful tool from multilinear algebra in particular the multiple adjacency matrices are stacked to form a tensor and the hosvd decomposition provides information about its third order structure analogous to that provided by matrix factorizations we obtain a set of eigennetworks and from these graph wavelets which exploit the variability across the graphs we demonstrate the feasibility of our method 1 by capturing different orientations of a gray scale image and 2 by decomposing brain signals from functional magnetic resonance imaging we show its effectiveness to identify variability across graph edges and provide meaningful decompositions

on equivalence of conceptual scaling and generalized one sided concept lattices the methods of conceptual scaling and generalized one sided concept lattices represent different possibilities on how to deal with many valued contexts we briefly describe these methods and prove that they are equivalent in particular we show that the application of these two approaches to a given many valued context yields the same closure system on the set of all objects based on this equivalence we propose a possible attribute reduction of one sided formal contexts

fuzzy inference systems preserving moser navara axioms abstract satisfiability of three naturally proposed axioms which are desirable for fuzzy inference system was discussed by moser and navara in 36 the fuzzy rule base models considered in this study were mainly the mamdani assilian ones and partly also logically motivated implicative fuzzy systems the investigation leads to the fact that often none of these fuzzy systems do satisfy the given axioms simultaneously therefore moser and navara proposed so called conditionally firing rules and proved that the simultaneous satisfaction of the three axioms is feasible under very mild and practically non restrictive conditions note that the inference mechanism tailored to the fuzzy rules in the investigation was the compositional rule of inference abb cri and similarly the concept of conditionally firing rules were stemming from the cri and the mamdani assilian model of a fuzzy rule base for the implicative interpretation of fuzzy rules the original axioms proposed by moser and navara were later on re defined by st pnicka and mandal with the aim to express the meaning of the original axioms initially proposed for the mamdani assilian rules furthermore conditionally firing implicative rules were proposed based on the so called bandler kohout subproduct abb bks inference mechanism and a study similar to the one by moser and navara was undertaken for them the study encompasses the satisfaction of the modified axioms for the use of implicative rules and the bks furthermore some additional results confirming the applicability of the conditionally firing rules satisfying the moser navara axioms are also provided including the universal approximation ability however the solvability criterions for fuzzy relational equations give a strong preference to the usage of the other combinations namely to the combination of cri with the implicative interpretation of fuzzy rules and to the combination of bks with the mamdani assilian interpretation of fuzzy rules therefore also these combinations of fuzzy inference mechanisms and interpretations of fuzzy rules should be investigated with respect to the satisfaction of moser navara axioms the main purpose of this article which extends the short conference contribution 47 by st pnicka is to provide such an investigation jointly with the review on conditionally firing rules and supplementary results

the study of the interval valued contexts the aim of this paper is to contribute to the interval valued fuzzy data analysis proposing a new method based on the formal concept analysis we develop a theoretical model that analyzes the interval valued information that arises from the opinion of a group of experts about the relationship between a set of objects and a set of attributes and we want to analyze this information from the point of view of the formal concept analysis to do this we are going to modelize a new l fuzzy concept theory using some results about multisets and expertons finally we include a simple example to illustrate it by showing its potential applications

spectrum adapted tight graph wavelet and vertex frequency frames we consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs while the filters used in previous spectral graph wavelet constructions are only adapted to the length of the spectrum the filters proposed in this paper are adapted to the distribution of graph laplacian eigenvalues and therefore lead to atoms with better discriminatory power our approach is to first characterize a family of systems of uniformly translated kernels in the graph spectral domain that give rise to tight frames of atoms generated via generalized translation on the graph we then warp the uniform translates with a function that approximates the cumulative spectral density function of the graph laplacian eigenvalues we use this approach to construct computationally efficient spectrum adapted tight vertex frequency and graph wavelet frames we give numerous examples of the resulting spectrum adapted graph filters and also present an illustrative example of vertex frequency analysis using the proposed construction

signal adapted tight frames on graphs the analysis of signals on complex topologies modeled by graphs is a topic of increasing importance decompositions play a crucial role in the representation and processing of such information here we propose a new tight frame design that is adapted to a class of signals on a graph the construction starts from a prototype meyer type system of kernels with uniform subbands the ensemble energy spectral density is then defined for a given set of signals defined on the graph the prototype design is then warped such that the resulting subbands capture the same amount of energy for the signal class this approach accounts at the same time for graph topology and signal features the proposed frames are constructed for three different graph signal sets and are compared with non signal adapted frames vertex localization of a set of resulting atoms is studied the frames are then used to decompose a set of real graph signals and are also used in a setting of signal denoising the results illustrate the superiority of the designed signal adapted frames over frames blind to signal characteristics in representing data and in denoising

minimal solutions of generalized fuzzy relational equations probabilistic algorithm based on greedy approach abstract the paper deals with generalized fuzzy relational equations that are defined within a recently introduced framework of sup preserving aggregation structures generalized fuzzy relational equations subsume all previously studied types of fuzzy relational equations namely those that are based on sup t norm and inf residuum compositions the paper contributes to previous studies of generalized fuzzy relational equations by presenting a method for constructing all minimal solutions and consequently for determining the whole solution set for any given generalized fuzzy relational equation that is solvable and for which every solution is bounded from below by a minimal solution moreover in the paper we present a simple probabilistic algorithm for finding all minimal solutions

a multiscale pyramid transform for graph signals multiscale transforms designed to process analog and discrete time signals and images cannot be directly applied to analyze high dimensional data residing on the vertices of a weighted graph as they do not capture the intrinsic topology of the graph data domain in this paper we adapt the laplacian pyramid transform for signals on euclidean domains so that it can be used to analyze high dimensional data residing on the vertices of a weighted graph our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling graph reduction and filtering and interpolation of signals on graphs equipped with appropriate notions of these operations we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs

bipartite approximation for graph wavelet signal decomposition to compactly represent a graph signal in the frequency domain critically sampled biorthogonal wavelet filterbanks have been proposed to decompose signals on bipartite graphs however in practice graph signals often reside on general graph structures that are not bipartite thus an original nonbipartite graph must be decomposed into a sequence of bipartite graph approximations so that the filterbanks can be applied successively for signal decomposition in this paper unlike previous proposals that are heuristic in nature we design new bipartite approximation strategies for model based and empirically derived probability distributions in the first case a signal prior assumes a gaussian markov random field gmrf model parametrized by the original graph in the second case beyond the gmrf signal prior empirical signal observations are available to compute a posterior probability distribution in both cases we optimize for energy compaction in the bipartite subgraphs with two criteria the kullback leibler divergence metric which encourages preserving the spectral characteristics of the original graph and multiplicity of eigenvalue at frequency 1 for graph laplacian which is the frequency with minimal energy discrimination the relative importance between the two criteria is determined by a proposed measure that evaluates the degree of mismatch between the signal prior and posterior given these criteria we first design a global numerical optimization and then propose a local heuristic approach for fast implementation with simplifications leading to local metric computation experimental results show that our proposed bipartite subgraph decomposition outperforms competing proposals in terms of energy compaction

explicit zero density estimate for the riemann zeta function near the critical line abstract in 1946 a selberg proved n t t 1 1 4 1 2 log t where n t is the number of nontrivial zeros of the riemann zeta function with and 0 t we provide an explicit version of this estimate together with an explicit approximate functional equation and an explicit upper bound for the second power moment of the zeta function on the critical line

extending classical multirate signal processing theory to graphs part ii m channel filter banks this paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper part i and studies m channel polynomial filter banks on graphs the behavior of such graph filter banks differs from that of classical filter banks in many ways the precise details depending on the eigenstructure of the adjacency matrix boldsymbol a it is shown that graph filter banks represent linear and periodically shift variant systems only when boldsymbol a satisfies the noble identity conditions developed in part i it is then shown that perfect reconstruction graph filter banks can always be developed when boldsymbol a satisfies the eigenvector structure satisfied by m block cyclic graphs and has distinct eigenvalues further restrictions on eigenvalues being unnecessary for this if boldsymbol a is actually m block cyclic then these pr filter banks indeed become practical i e arbitrary filter polynomial orders are possible and there are robustness advantages in this case the pr condition is identical to pr in classical filter banks any classical pr example can be converted to a graph pr filter bank on an m block cyclic graph it is shown that for m block cyclic graphs with all eigenvalues on the unit circle the frequency responses of filters have meaningful correspondence with classical filter banks polyphase representations are then developed for graph filter banks and utilized to develop alternate conditions for alias cancellation and perfect reconstruction again for graphs with specific eigenstructures it is then shown that the eigenvector condition on the graph can be relaxed by using similarity transforms

subgraph based filterbanks for graph signals we design a critically sampled compact support biorthogonal transform for graph signals via graph filterbanks instead of partitioning the nodes in two sets so as to remove one every two nodes in the filterbank downsampling operations the design is based on a partition of the graph in connected subgraphs coarsening is achieved by defining one supernode for each subgraph and the edges for this coarsened graph derives from the connectivity between the subgraphs unlike the one every two nodes downsampling on bipartite graphs this coarsening operation does not have an exact formulation in the graph fourier domain instead we rely on the local fourier bases of each subgraph to define filtering operations we apply successfully this method to decompose graph signals and show promising performance on compression and denoising

relational compositions in fuzzy class theory we present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes the method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations which allows transferring basic properties of sup t and inf r compositions to a family of more than 30 composition related operations including sup t and inf r images pre images cartesian products domains ranges resizes inclusion height plinth etc besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them thus trivializing a large part of the theory of fuzzy relations

triadic concept lattices in the framework of aggregation structures abstract triadic concept analysis is a method of analysis of three way tabular data describing objects by binary attributes they have under various conditions it aims to extract a set of three dimensional clusters so called triadic concepts from the data while the method was recently extended to handle the case of graded data this extension is only one of a variety of possible generalizations namely the one where the so called antitone concept forming operators are considered the generalizations not covered by the mentioned approach include isotone concept forming operators one sided concepts the use of truth stressing hedges and many others since all of the generalizations have their applications for example in factor analysis of relational data it is desirable to have a uniform mathematical foundations covering all of the cases in this paper we provide such foundations by using triadic aggregation structures as a scale of truth degrees we develop triadic concept analysis in a very general setting that covers a wide variety of its possible extensions in a uniform way

solutions and strong solutions of min product fuzzy relation inequalities with application in supply chain abstract min product fuzzy relation inequalities are introduced to describe the price requirements in the supply chain system we first investigate the resolution method and structure of the complete solution set to min product fuzzy relation inequalities motivated by the practical application we further define concept of strong solution a strong solution represents a feasible pricing scheduling enabling all the suppliers and retailers profitable in the trade activity relevant results about the strong solution are obtained based on such results a step by step algorithm is developed for finding all the strong solutions to a system of min product fuzzy relation inequalities numerical examples are given to illustrate the algorithm

perfect reconstruction two channel wavelet filter banks for graph structured data in this work we propose the construction of two channel wavelet filter banks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph these graph based functions are referred to as graph signals as we build a framework in which many concepts from the classical signal processing domain such as fourier decomposition signal filtering and downsampling can be extended to graph domain especially we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals this property of bipartite graphs allows us to design critically sampled two channel filter banks and we propose quadrature mirror filters referred to as graph qmf for bipartite graph which cancel aliasing and lead to perfect reconstruction for arbitrary graphs we present a bipartite subgraph decomposition which produces an edge disjoint collection of bipartite subgraphs graph qmfs are then constructed on each bipartite subgraph leading to multi dimensional separable wavelet filter banks on graphs our proposed filter banks are critically sampled and we state necessary and sufficient conditions for orthogonality aliasing cancellation and perfect reconstruction the filter banks are realized by chebychev polynomial approximations

extending classical multirate signal processing theory to graphs part i fundamentals signal processing on graphs finds applications in many areas in recent years renewed interest on this topic was kindled by two groups of researchers narang and ortega constructed two channel filter banks on bipartitie graphs described by laplacians sandryhaila and moura developed the theory of linear systems filtering and frequency responses for the case of graphs with arbitrary adjacency matrices and showed applications in signal compression prediction etc inspired by these contributions this paper extends classical multirate signal processing ideas to graphs the graphs are assumed to be general with a possibly nonsymmetric and complex adjacency matrix the paper revisits ideas such as noble identities aliasing and polyphase decompositions in graph multirate systems drawing such a parallel to classical systems allows one to design filter banks with polynomial filters with lower complexity than arbitrary graph filters it is shown that the extension of classical multirate theory to graphs is nontrivial and requires certain mathematical restrictions on the graph thus classical noble identities cannot be taken for granted similarly one cannot claim that the so called delay chain system is a perfect reconstruction system as in classical filter banks it will also be shown that m partite extensions of the bipartite filter bank results will not work for m channel filter banks but a more restrictive condition called m block cyclic property should be imposed such graphs are studied in detail a detailed theory for m channel filter banks is developed in a companion paper

special properties closures and interiors of crisp and fuzzy relations this paper offers optimal definitions of the special properties which relations on a set may possess and which combine to define the classically important types of preorders both kinds of orders tolerances and equivalences the useful concept of local properties is introduced the definitions are shown to unify in a highly convenient way fuzzy and crisp considerations the concepts of closure and interior of a relation with respect to a property are defined necessary and sufficient conditions for the full existence of specific closures and interiors are given and formulae for all of them are presented and validated a widely important area of application of the closure and interior operators is discussed finally fast algorithms are given for the computer calculation of the more troublesome closures and a standardized procedure for the analysis of local orders is presented

extensions of fuzzy relational compositions based on generalized quantifiers abstract the aim of this paper is to extend the fuzzy relational compositions motivated by their applications to medical diagnosis problem by bandler and kohout these compositions employ two quantifiers the universal quantifier and the existential quantifier as there exists a big gap between these two quantifiers that may be appropriately filled in by intermediate generalized quantifiers we take this as a motivation for our study in particular we introduce the concept of fuzzy relational compositions based on generalized quantifiers furthermore we show that the properties that are typically valid for standard fuzzy relational compositions are also valid for the compositions based on generalized quantifiers yet sometimes in a weaken form apart from fuzzy relational compositions the use of generalized quantifiers is also applied to images and preimages of fuzzy sets under fuzzy relations

l concept analysis with positive and negative attributes we describe an extension of formal fuzzy concept analysis allowing a user to choose which attributes are viewed as positive and which are viewed as negative the two sets are then handled using a combination of previously studied antitone concept forming operators and isotone concept forming operators respectively the two main outputs of formal concept analysis namely concept lattices and attribute implications in the setting of positive and negative attributes are presented an analogy of the main theorem of concept lattices and a relationship between the new concept lattice and the previously studied concept lattices is showed we introduce basic syntactic and semantic notions for attribute implications called fuzzy containment implications we consider two settings one where the sets of positive and negative attributes are crisp sets and a generalization where the two sets are fuzzy sets

graph wavelets for multiscale community mining we develop a signal processing approach to the multiscale detection of communities in networks that is of groups of nodes well connected together the method relies on carefully engineered wavelets on graphs to introduce the notion of scale and to obtain a local view of the graph from each node computing the correlations between wavelets centered at different nodes one has access to a notion of similarity between nodes thereby enabling a clustering procedure that groups nodes according to their community at the scale of analysis by using a collection of random vectors to estimate the correlation between the nodes we show that the method is suitable for the analysis of large graphs furthermore we introduce a notion of partition stability and a statistical test allowing us to assess which scales of analysis of the network are relevant the effectiveness of the method is discussed first on multiscale graph benchmarks then on real data of social networks and on models for signal processing on graphs

parameterizing the semantics of fuzzy attribute implications by systems of isotone galois connections we study the semantics of fuzzy if then rules called fuzzy attribute implications parameterized by systems of isotone galois connections the rules express dependences between fuzzy attributes in object attribute incidence data the proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches we formalize the general parameterizations propose bivalent and graded notions of semantic entailment of fuzzy attribute implications show their characterization in terms of least models and complete axiomatization and provide characterization of bases of fuzzy attribute implications derived from data

local set based graph signal reconstruction signal processing on graph is attracting more and more attentions for a graph signal in the low frequency subspace the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal in this paper the concept of local set is introduced and two local set based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data in each iteration one of the proposed methods reweights the sampled residuals for different vertices while the other propagates the sampled residuals in their respective local sets these algorithms are built on frame theory and the concept of local sets based on which several frames and contraction operators are proposed we then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method furthermore the correspondence between graph signal sampling and time domain irregular sampling is analyzed comprehensively which may be helpful to future works on graph signals computer simulations are conducted the experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries imprecise priori knowledge of cutoff frequency and noisy scenarios

explicit zero density for the riemann zeta function abstract let n t denote the number of nontrivial zeros of the riemann zeta function with real part greater than and imaginary part between 0 and t we provide explicit upper bounds for n t commonly referred to as a zero density result in 1937 ingham showed the following asymptotic result n t o t 8 3 1 log t 5 ramare recently proved an explicit version of this estimate we discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants

on multi adjoint concept lattices based on heterogeneous conjunctors sets of attributes and objects in fuzzy formal concept analysis are usually different and hence it might not make sense to evaluate them on the same carrier in this context the operators used to obtain the concept lattice could be defined by associating different lattices to attributes and objects several reasons exist for which we need to evaluate the sets of attributes and objects in the same carrier following this direction we introduce a new definition of a concept lattice where objects and attributes are evaluated on the same lattice l although operators evaluating objects and attributes in different carriers are used moreover we study the relationship between this new concept lattice and the alternative one which can be obtained directly by using different carriers for the sets of attributes and objects

concept lattices of isotone vs antitone galois connections in graded setting mutual reducibility revisited it is well known that concept lattices of isotone and antitone galois connections induced by an ordinary binary relation and its complement are isomorphic via a natural isomorphism mapping extents to themselves and intents to their complements it is also known that in a fuzzy setting this and similar kinds of reduction fail to hold in this note we show that when the usual notion of a complement based on a residuum w r t 0 is replaced by a new one based on residua w r t arbitrary truth degrees the above mentioned reduction remains valid for ordinary relations the new and the usual complement coincide the result we present reveals a new deeper root of the reduction it is not the availability of the law of double negation but rather the fact that negations are implicitly present in the construction of concept lattices of isotone galois connections

concept lattices defined from implication operators abstract we introduce the work with two extensions of the concept definition of wille to the fuzzy theory showing the relationship between them we continue with the study of fuzzy material implications and their associated operators of kohout and bandler that lump together the definitions given until this moment to conclude we show an example where we compare the results obtained from the l fuzzy concepts using the fuzzy implications of lukasiewicz kleene dienes and godel brower