reduction of a nilpotent fuzzy matrix abstract a problem of reducing fuzzy matrices is considered and some results are obtained with regard to nilpotent matrices first reduction of irreflexive and transitive matrices is examined and then the results are applied to nilpotent matrices nilpotent matrices are matrices which represent acyclic graphs and the graphs are used to represent consistent systems the results are useful for simplification of various systems with fuzzy transitivity

convergence of the power sequence of a nearly monotone increasing fuzzy matrix abstract by introducing a new classification the essential role of the principal diagonal elements for the convergence of the power sequence of a fuzzy matrix a is exported at the level of a itself the established theorems cover those of monotone or nearly monotone increasing fuzzy matrices the convergence index of matrices of this type is always less than or equal to n the results can be considered to be a generalization of the results obtained by some authors on the one hand and on the other it is more important that the results have laid the foundation for studying the convergence and oscillation for power sequence of an arbitrary fuzzy matrix in general furthermore as a typical case which needs to be considered the necessary and sufficient conditions for an increasing fuzzy matrix a to have the property a n 1 a n a n 1 were established and the typical form was represented

szpilrajn s theorem on fuzzy orderings fuzzy relation matrices are considered and the well known szpilrajn theorem on orderings is generalized to fuzzy orderings zadeh has already shown a generalized theorem but another generalization is given in matrix form some interesting operations on fuzzy matrices are introduced and their properties are mentioned briefly the theorem is represented in terms of the matrix operations the operations have many properties and are useful in the various applications of fuzzy matrices or fuzzy relations

on the k 2 supercongruence of van hamme abstract we prove the last remaining case of the original 13 ramanujan type supercongruence conjectures due to van hamme from 1997 the proof utilizes classical congruences and a wz pair due to guillera additionally we mention some future directions concerning this type of supercongruence

transitivity of generalized fuzzy matrices generalized fuzzy matrices are considered as matrices over a special type of semiring which is called a path algebra transitivity of generalized fuzzy matrices is defined and some properties of transitivity are shown especially properties of the composition of transitive fuzzy matrices representing transitive fuzzy relations are examined by introducing a third operation on the semiring with two operations the transitivity is closely related to metrics distances ultrametrics and so on

on the power sequence of a fuzzy matrix iii a detailed study on the power sequence of matrices of commonly used types abstract focusing on the behavior of the principal diagonal elements of ak a new classification has been introduced which is called circularly k dominating in turns out that the convergence index or the oscillating index of the power sequence of an n n fuzzy matrix of the circularly k dominating type is bounded by nk n k from above and if it is oscillating then the period index pa is a factor of k the fuzzy matrices of the 2 dominating type were discussed in detail it was shown that the 2 dominating type is a more general class than those have been discussed before and the results established for matrices of 2 dominating type is as good as the results obtained for controllable matrices therefore most commonly used types of fuzzy matrices can be examined under the framework of 2 dominating matrices and the convergence index or oscillating index can be estimated based on the results

a q analogue of the j 2 supercongruence of van hamme abstract the l 2 supercongruence of van hamme was proved by swisher recently in this paper we provide a conjectural q analogue of the l 2 supercongruence of van hamme and prove a weaker form of it by using the q wz method in the same way we prove a complete q analogue of the following congruence k 0 n 6 k 1 2 k k 3 512 n k 0 mod 4 2 n 1 2 n n which was conjectured by z w sun and confirmed by b he we also provide a conjectural q analogue of another congruence proved by swisher

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

on the sequence of consecutive powers of a fuzzy matrix with max archimedean t norms in the literature the limiting behavior of powers of a fuzzy matrix has been studied with max product composition pang and guu proposed a simple and effective characterization for the limiting behavior with the notion of asymptotic period in this paper we shall extend pang and guu s work to the realm with the max archimedean t norm composition to which the max product belongs moreover some sufficient conditions for the powers to converge in finitely many steps will be established

on the oscillating power sequence of a fuzzy matrix abstract the properties of an oscillating power sequence of an n n fuzzy matrix have been studied the necessary and sufficient conditions have been established for a power sequence to be oscillatory furthermore the oscillation index was detailed to some degree especially for some typical boolean matrices it is also shown that the period index set of the power sequences of fuzzy matrices of order n is not bounded from above by a power of n for all integers n

convergence of max arithmetic mean powers of a fuzzy matrix fuzzy matrices provide convenient representations for fuzzy relations on finite universes in the literature the powers of a fuzzy matrix with max min max product max archimedean t norm compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper the max arithmetic mean composition is considered for the fuzzy relations we show that the max arithmetic mean powers of a fuzzy matrix always are convergent

q analogues of dwork type supercongruences abstract in 1997 van hamme conjectured 13 ramanujan type supercongruences all of the 13 supercongruences have been confirmed by using a wide range of methods in 2015 swisher conjectured dwork type supercongruences related to the first 12 supercongruences of van hamme here we prove that the c 3 and j 3 supercongruences of swisher are true modulo p 3 r the original modulus is p 4 r by establishing q analogues of them our proof will use the creative microscoping method recently introduced by the author in collaboration with zudilin we also raise conjectures on q analogues of an equivalent form of the m 2 supercongruence of van hamme partially answering a question at the end of adv math 346 2019 329 358

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

interval logics and their decision procedures part i an interval logic abstract we present an interval logic called future interval logic fil for the specification and verification of concurrent systems interval logics allow reasoning to be carried out at the level of time intervals rather than instants however unlike some other interval logics the primitive objects in our semantic model for fil are not intervals but instants an intervals is formed by identifying its end points which are instants satisfying given properties the logic has an intuitive graphical representation resembling the back of the envelope timing diagrams that designers often draw to reason about concurrent interacting systems the logic is designed to be insensitive to finite stuttering a property that facilitates refinement based multi level correctness proofs and is exactly as expressive as the fragment of propositional temporal logic with until but no next as the main result of this paper we show that fil is elementarily decidable by reduction to the emptiness problem for buchi automata for most other interval logics the decision problem is at best non elementary and often undecidable we cosider an extension of fil with past operators and show that this extension leads to non elementariness in a companion paper we extend the logic to real time and investigate the decision problem for that extension

on the max contraction powers of a fuzzy matrix abstract in the literature the powers of a fuzzy matrix with max arithmetic mean convex combination of max min and max arithmetic mean convex combination of max product and max min compositions have been studied it turns out that the limiting behavior of the powers of a fuzzy matrix depends on the composition involved in this paper we consider the max contraction powers of a fuzzy matrix which is an extension of the max arithmetic mean convex combination of max min and max arithmetic mean convex combination of max product and max arithmetic mean operation we show that the powers of such a fuzzy matrix are always convergent and the limit matrix has the feature that all elements of each column are identical

residuation in fuzzy algebra and some applications abstract the system 0 1 max min has a number of well known applications many of which involve the inversion of linear relations we show the advantage of residuation theory for this process and apply it to the analysis of the problem of chebychev approximation

a note on the sequence of consecutive powers of a nonnegative matrix in max algebra abstract in this note we shall consider the sequence of consecutive powers in max algebra of a nonnegative matrix with each element less than or equal to 1 the notion of asymptotic period is defined to study the limiting behavior of this sequence a simple and effective characterization for the asymptotic period of the sequence is given

convergence of powers for a fuzzy matrix with convex combination of max min and max arithmetic mean operations fuzzy matrices have been proposed to represent fuzzy relations on finite universes since thomason s paper in 1977 showing that powers of a max min fuzzy matrix either converge or oscillate with a finite period conditions for limiting behavior of powers of a fuzzy matrix have been studied it turns out that the limiting behavior depends on the algebraic operations employed which usually in the literature includes max min max product max archimedean t norm max t norm max arithmetic mean operations respectively in this paper we consider the powers of a fuzzy matrix with convex combination of max min and max arithmetic mean operations we show that the powers of such a fuzzy matrix are always convergent

note on convergence of powers of a fuzzy matrix we show by example that a theorem in thomason j math anal appl 57 1977 476 also recently quoted in imai et al fuzzy sets and systems 109 2000 405 about the convergence of fuzzy matrices is false

some symmetric q congruences modulo the square of a cyclotomic polynomial abstract z h sun 2014 proved a symmetric congruence k 0 p 1 k 1 k f k 1 p k 0 p 1 k 1 k f k mod p 2 where p is an odd prime q is p integral p is the least non negative residue of modulo p and f k j 0 k 1 j k j f j this congruence implies several supercongruences of rodriguez villegas in this paper we give a q analogue of this congruence and prove some symmetric q congruences which also confirm two conjectures of guo and zeng 2014

interval logics and their decision procedures part ii a real time interval logic abstract in a companion paper we presented an interval logic and showed that it is elementarily decidable in this paper we extend the logic to allow reasoning about real time properties of concurrent systems we call this logic real time future interval logic rtfil we model time by the real numbers and allow our syntax to state the bounds on the duration of an interval rtfil possesses the real time interpolation property which appears to be the natural quantitative counterpart of invariance under finite stuttering as the main result of this paper we show that rtfil is decidable the decision algorithm is slightly more expensive than for the untimed logic our decidability proof is based on the reduction of the satisfiability problem for the logic to the emptiness problem for timed buchi automata the latter problem was shown decidable by alur and dill in a landmark paper in which this real time extension of automata was introduced finally we consider an extension of the logic that allows intervals to be constructed by means of real time offsets and show that even this simple extension renders the logic highly undecidable

canonical form of a transitive fuzzy matrix some properties of a transitive fuzzy matrix are examined and the canonical form of the transitive matrix is given using the properties as a special case an open problem concerning idempotent matrices of kim and roush is solved in our results a nilpotent matrix and a symmetric matrix play an important role we decompose a transitive matrix into the sum of a nilpotent matrix and a symmetric matrix then we obtain a canonical form of the transitive matrix

on the asymptotic period of powers of a fuzzy matrix in our prior study we have examined in depth the notion of an asymptotic period of the power sequence of an nxn fuzzy matrix with max archimedean t norms and established a characterization for the power sequence of an nxn fuzzy matrix with an asymptotic period using analytical decomposition methods in this paper by using graph theoretical tools we further give an alternative proof for this characterization with the notion of an asymptotic period using graph theoretical tools we additionally show a new characterization for the limit behaviour and then derive some results for the power sequence of an nxn fuzzy matrix with an asymptotic period

a q microscope for supercongruences abstract by examining asymptotic behavior of certain infinite basic q hypergeometric sums at roots of unity that is at a q microscopic level we prove polynomial congruences for their truncations the latter reduce to non trivial super congruences for truncated ordinary hypergeometric sums which have been observed numerically and proven rarely a typical example includes derivation from a q analogue of ramanujan s formula n 0 4 n 2 n 2 n n 2 2 8 n 3 2 n 8 n 1 2 3 of the two supercongruences s p 1 p 3 p mod p 3 and s p 1 2 p 3 p mod p 3 valid for all primes p 3 where s n denotes the truncation of the infinite sum at the n th place and 3 stands for the quadratic character modulo 3

powers of matrices over distributive lattices a review in this paper we review the results on powers of matrices over a distributive lattice we stress the significance of the graph theoretical approach and show how several previous results for matrices of special types can be obtained in a unified way

some congruences involving fourth powers of central q binomial coefficients we prove some congruences on sums involving fourth powers of central q binomial coefficients as a conclusion we confirm the following supercongruence observed by long pacific j math 249 2011 405 418 sum limits k 0 p r 1 2 displaystyle 4k 1 over 256 k left matrix 2k cr k right 4 equiv p r quad left bmod p r 3 right where p 5 is a prime and r is a positive integer our method is similar to but a little different from the wz method used by zudilin to prove ramanujan type supercongruences

dioids and semirings links to fuzzy sets and other applications besides the classical algebraic structures of groups rings and fields which have long been the almost exclusive reference concepts used in mathematical modelling other algebraic structures such as dioids and semirings have emerged in the last two or three decades in connection with modelling and solving a rich variety of non classical problems e g in decision analysis fuzzy set theory operations research automatic control and mathematical physics the present paper aims at providing an overview of applications of dioid and semiring structures stressing links with fuzzy sets and emphasizing linear algebraic problems solving linear systems computing eigenvalues and eigenvectors non classical path finding problems using algebras of endomorphisms and connections between dioid structure and nonlinear analysis in view of solving problems in mathematical physics

convergence of products of fuzzy matrices convergence of powers of a fuzzy matrix has been studied in the literature in this paper we shall extend the scope to the products of a finite number of fuzzy matrices we shall demonstrate that the outcomes of products of fuzzy matrices will be more complicated than the cases of powers of a fuzzy matrix two notions of convergence are established weak convergence and strong convergence the main results to guarantee the weak convergence of products of fuzzy matrices are due to the concepts of transitive and compact properties of fuzzy matrices which can be viewed as a generalization of the transitive and compact properties of a fuzzy matrix sufficient conditions similar to the pinching theorem in calculus are given as well for strong convergence

the complexity of propositional linear temporal logics the complexity of satisfiability and determination of truth in a particular finite structure are considered for different propositional linear temporal logics it is shown that these problems are np complete for the logic with f and are pspace complete for the logics with f x with u with u s x operators and for the extended logic with regular operators given by wolper

on the max nilpotent t norm powers of a fuzzy matrix fuzzy matrices have been proposed to represent fuzzy relations in finite universes various studies have evaluated the powers of a fuzzy matrix with max min max product max archimedean t norm max t norm max arithmetic mean compositions indicating that the limiting behavior of the powers of a fuzzy matrix depends on its composition in this paper max nilpotent composition is considered for the fuzzy relations we demonstrate that the max nilpotent powers of a fuzzy matrix either converge or oscillate in a finite period moreover the max nilpotent t norm powers of a fuzzy matrix a are p periodic if and only if the powers of an associated boolean matrix a c are p periodic finally necessary and sufficient conditions are proposed for nilpotent fuzzy matrices that exhibit max nilpotent composition

convergence of powers of a fuzzy transitive matrix convergence of powers of a fuzzy transitive matrix is considered and some conditions for convergence are given under the max min operation on fuzzy matrices a fuzzy transitive matrix is a matrix which represents a fuzzy transitive relation and has many interesting properties the conditions for convergence of fuzzy matrices are examined under a special operation which is essential to reduction of fuzzy matrices or fuzzy systems given results show graph theoretic properties of fuzzy matrices and are useful when we consider convergence of systems represented by fuzzy matrices

on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

convergence eigen fuzzy sets and stability analysis of relational matrices abstract if stable control applications are formulated with a max product composition then instability in the relational matrix under any conditions is undesirable this paper reviews the stability analysis of relational matrices combined with the max min composition 19 and then presents an analysis of the stability of relational matrices combined with the max product composition this analysis includes results defining the convergence properties of the relational matrix determination of the eigen fuzzy sets of the stable matrices and algorithmic solutions for several important cases a method is also presented to stabilize unstable relational matrices for some control applications