interpreting perspective images a fundamental problem in computer vision is how to determine the 3 d spatial orientation of curves and surfaces appearing in an image the problem is generally underconstrained and is complicated by the fact that metric properties such as orientation and length are not invariant under projection under perspective projection the correct model for most real images the transform is nonlinear and therefore hard to invert two constructive methods are presented the first finds the orientation of parallel lines and planes by locating vanishing points and vanishing lines the second determines the orientation of planes by backprojection of two intrinsic properties of contours angle magnitude and curvature

linear objective function optimization with fuzzy relation equation constraints regarding max av composition in this paper an optimization model with a linear objective function subject to a system of the fuzzy relation equations with max av composition is presented the solution set of such a fuzzy relation equations is a non convex set in this paper firstly we discuss the feasible solution set with two schemes and secondly study relationship between maximum and minimum points and also the feasible points as well furthermore an algorithm and few concrete examples are presented in order to optimize linear objective function

carl s inequality for quasi banach spaces we prove that for any two quasi banach spaces x and y and any 0 there exists a constant 0 such that sup1 k n k ek t sup1 k n k ck t holds for all linear and bounded operators t x y here ek t is the k th entropy number of t and ck t is the k th gelfand number of t for banach spaces x and y this inequality is widely used and well known as carl s inequality for general quasi banach spaces it is a new result

resolution of a system of the max product fuzzy relation equations using lou factorization in this paper the lu factorization is extended to the fuzzy square matrix with respect to the max product composition operator called l u factorization equivalently we will find two fuzzy lower and upper triangular matrices l and u for a fuzzy square matrix a such that a l u where is the max product composition an algorithm is presented to find the matrices l and u furthermore some necessary and sufficient conditions are proposed for the existence and uniqueness of the l u factorization for a given fuzzy square matrix a an algorithm is also proposed to find the solution set of a square system of fuzzy relation equations fre using the l u factorization the algorithm finds the solution set without finding its minimal solutions and maximum solution it is shown that the two algorithms have a polynomial time complexity as o n 3 since the determination of the minimal solutions is an np hard problem the algorithm can be very important from the practical point of view

structure and motion from line segments in multiple images this paper presents a new method for recovering the three dimensional structure of a scene composed of straight line segments using the image data obtained from a moving camera the recovery algorithm is formulated in terms of an objective function which measures the total squared distance in the image plane between the observed edge segments and the projections perspective of the reconstructed lines this objective function is minimized with respect to the line parameters and the camera positions to obtain an estimate for the structure of the scene the effectiveness of this approach is demonstrated quantitatively through extensive simulations and qualitatively with actual image sequences the implementation is being made publicly available

linear optimization with an arbitrary fuzzy relational inequality in this paper optimization of a linear objective function with fuzzy relational inequality constraints is investigated whereby an arbitrary function is considered as fuzzy composition the feasible solution set is determined and compared with some common results in the literature a necessary and sufficient condition and three other necessary conditions are presented to conceptualize the feasibility of the problem it is shown that in general a lower bound is always attainable for the optimal objective value moreover we prove that the optimal solution of the problem is obtained if the problem is defined by a non decreasing or non increasing function an algorithm is presented to summarize the process and an example is described to illustrate the algorithm

numerical analysis on fuzzy relation equations with various operators abstract in this study we investigate the properties of the approximate solutions of fuzzy relation equations for composite operators defined by max min hamacher and its special case disjunctive sum conjunctive product a quasi newton method is adopted to find the solutions apart from particular issues relevant to individual operators comparisons among the various operators are made

fuzzy linear systems abstract fuzzy linear systems of equations and inequalities over a bounded chain are studied both from theoretical and computational points of view a unified approach is presented for solving such systems completed with polynomial time algorithms the main results are concerned with establishing the consistency of the system computing all kinds of solutions or marking the contradictory equations respectively inequalities if the system is inconsistent applications in fuzzy linear programming multivalued logic fuzzy matrix equations or inequalities fuzzy relation equations or inclusions fuzzy automata and medicine are presented

on the minimal solutions of max min fuzzy relational equations in this paper the minimal solutions of max min fuzzy relational equations are investigated a sufficient and necessary condition for discriminating whether a given solution is minimal or not is shown furthermore we propose a new algorithm for computing all minimal solutions less than or equal to a given one

fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max min composition in this paper we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max min composition in which fuzzy inequality replaces ordinary inequality in the constraints fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with ordinary inequality constraints denoted by x o i in this paper we have presented an algorithm to generate such optimal solutions also two examples are given to illustrate the algorithm presented and some real application of this work

solutions of fuzzy relation equations based on continuous t norms this study is concerned with fuzzy relation equations with continuous t norms in the form atr b where a and b are the fuzzy subsets of x and y respectively email protected xxy is a fuzzy relation and t is a continuous t norm the problem is how to determine a from r and b first an if and only if condition of being solvable is presented novel algorithms are then presented for determining minimal solutions when x and y are finite the proposed algorithms generate all minimal solutions for the equations making them efficient solving procedures

convexly combined fuzzy relational equations and several aspects of their application to fuzzy information processing abstract this paper presents some theoretical results on a set of the convexly combined fuzzy relational equations represented in the form x k 2 x k r y k k 1 2 k where x k and y k denote fuzzy sets r and 2 fuzzy relations the sup min compositional operator the inf max one and a fuzzy convex combinator whose complement is furthermore we propose a method to formulate the fuzzy information processing approximately by means of the convexly combined fuzzy relational equations focusing our attention on the modeling of fuzzy input output systems normality of fuzzy sets characterizing fuzzy input fuzzy logical aspects of systems and association between fuzzy information finally it is suggested that the proposed method is suitable for the modelling of several types of fuzzy systems

estimation of 3d structure and motion from image corners abstract this paper deals with motion estimation from image corner correspondences in two cases the orthogonal corner and the general corner with known space angles the contribution of the paper is in three folds first the three dimensional structure of a corner is recovered easily from its image by introducing a new coordinate system second it is shown that the one corner and two points correspondences over two views are sufficient to uniquely determine the motion i e the rotation and translation third experiments using both simulated data and real images are conducted which present good results

entropy numbers s numbers and eigenvalue problems abstract we establish inequalities between entropy numbers and approximation numbers for operators acting between banach spaces furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a banach space the results are compared with the classical inequalities of bernstein and jackson

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

multi objective optimization with a max t norm fuzzy relational equation constraint in this paper we consider minimizing multiple linear objective functions under a max t norm fuzzy relational equation constraint since the feasible domain of a max archimedean t norm relational equation constraint is generally nonconvex traditional mathematical programming techniques may have difficulty in yielding efficient solutions for such problems in this paper we apply the two phase approach utilizing the min operator and the average operator to aggregate those objectives to yield an efficient solution a numerical example is provided to illustrate the procedure

solving a linear programming problem with the convex combination of the max min and the max average fuzzy relation equations abstract in this paper we introduce a fuzzy operator constructed by the convex combination of two known operators max min and max average compositions h j zimmermann fuzzy set theory and it s application kluwer academic publishers boston dordrecht london 1999 this operator contains some properties of the two known compositions when it generates the feasible region for linear optimization problems we investigate linear optimization problems whose feasible region is the fuzzy sets defined with this operator thus firstly the structure of these fuzzy regions is considered and then a method to solve the linear optimization problems with fuzzy equation constraints regarding this operator is presented

resolution of composite interval valued fuzzy relation equations abstract in this study we investigate the properties and the solution procedure of composite interval valued fuzzy relation equations with max min operator this is a generalization of existing constant valued fuzzy relation problems the study includes both theoretical development and numerical analyses it is concluded that our proposed methodology not only solves the existing fuzzy relation equations but also reveals the necessary and sufficient conditions for more general problems therefore this study facilitates the understanding of solution structure and wider applications

entropy numbers of embeddings of schatten classes abstract let 0 p q and denote by s p n and s q n the corresponding finite dimensional schatten classes we prove optimal bounds up to constants only depending on p and q for the entropy numbers of natural embeddings between s p n and s q n this complements the known results in the classical setting of natural embeddings between finite dimensional l p spaces due to schutt edmunds triebel triebel and guedon litvak kuhn we present a rather short proof that uses all the known techniques as well as a constructive proof of the upper bound in the range n n n 2 that allows deeper structural insight and is therefore interesting in its own right our main result can also be used to provide an alternative proof of recent lower bounds in the area of low rank matrix recovery

eigen fuzzy sets and fuzzy relations abstract when r is a fuzzy relation between the elements of a finite set x the fuzzy sets a of x such that r a a max min composition are called eigen fuzzy sets the main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example we then state that the greatest eigen fuzzy set associated with r the transitive closure of r is exactly the one associated with r finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set

unique solvability of max min fuzzy equations and strong regularity of matrices over fuzzy algebra a matrix is said to have strongly linearly independent columns or in the case of square matrices to be strongly regular if for some vector b the system a x b has a unique solution we formulate a necessary and sufficient condition for a linear system of equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and trapezoidal property moreover an algorithm for testing these properties is reviewed

on the construction of fuzzy eigen solutions in given regions we consider the eigen set problem for fuzzy relations in a given region fuzzy interval using convenient means from the theory of posets we are able to determine the greatest and the minimal elements of the set of eigen solutions whereby the results are different for convergent and oscillating relations numerical examples for illustration are given

some properties of infinite fuzzy relational equations on complete brouwerian lattices this paper deals with fuzzy relational equations email protected x b where both a a j j j and b are known b has an irredundant continuous join decomposition x x j j j t is unknown represents sup inf composition j is an infinite index set on complete brouwerian lattices at first a necessary and sufficient condition for existence of the attainable solutions the unattainable solutions respectively and some properties of the attainable solutions the unattainable solutions respectively are formulated then the solution set of the fuzzy relational equations is described

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

increasing continuous operations in fuzzy max equations and inequalities abstract in this paper solutions of fuzzy systems of equations a x b and systems of inequalities a x b a x b are considered where a 0 1 m n b 0 1 m stands for max product and x 0 1 n is an unknown vector in this paper the families of all solutions are described during the last thirty years this problem was considered by many authors our goal is to achieve the most general assumptions for which sets of solutions systems of equations and inequalities are the union of lattice intervals especially a method of determination of all minimal solutions in such systems is indicated as a result the families of all solutions of fuzzy systems of inequalities a x b and a x b are described as the main part of these results the method of determination of the family of all minimal solutions in max system of equations with an increasing operation continuous on the second argument is presented

an alternative approach to the resolution of fuzzy relation equations abstract in this paper we propose a solution procedure by means of a quasi characteristic matrix the matrix is so defined that not only the solution set can be determined but also the type of solution set such as the non empty set full set unique minimal solution and unique solution can be identified before the solution set is determined therefore it provides an insight into the structure of fuzzy relation equations

sets of solution set invariant coefficient matrices of simple fuzzy relation equations abstract a set of coefficient matrices which are invariant to a solution set of a simple fuzzy relation equation is considered the equipollency of the cardinal numbers of solution sets of simple fuzzy relation equations for two constant vectors is also shown

solution algorithms for fuzzy relational equations with max product composition the conditions for the existence of an inverse solution to the max min composition of fuzzy relational equations have been well documented since the original work by sanchez 30 31 these same existence theorems have been extended to the t norm composition of relational equations in which the max product composition is a member 5 13 26 several studies 8 15 24 33 34 38 have shown that the max min operator may not always be the most desirable fuzzy relational composition and in fact the max product operator was superior in these instances this paper reviews the algorithms necessary to determine the complete solution of the inverse for fuzzy relational equations with max product composition

on the relation between fuzzy max archimedean t norm relational equations and the covering problem the problem of solving a system of fuzzy relational equations with max archimedean t norm composition is studied it is shown that this problem is closely related to the covering problem which belongs to the class of np hard problems it is proved that there is a one to one correspondence between the minimal solutions of the equations and the irredundant coverings as previously discovered by markovskii on the relation between equations with max product composition and the covering problem fuzzy sets and systems 153 2005 261 273 for fuzzy relational equations with max product composition since max product composition is a special case of max archimedean t norm composition this work extends markovskii s work to fuzzy relational equations with max archimedean t norm composition an extension of markovskii s algorithm is implemented yielding a processing time linearly proportional to the square of the number of minimal solutions

system of fuzzy relation equations with inf composition complete set of solutions the problem of solvability of a system of equations with inf composition is considered on finite universes equations are expressed using operations of a bl algebra we study complete set of solutions of the respective system in the particular case one equation and in the general case in both cases various conditions of solvability are found and proved we characterize all maximal solutions and prove that under certain conditions each solution of the system an equation is less than or equal to a respective maximal one as a result we are able to characterize the complete set of solutions examples of single equations and their systems are considered and complete sets of solutions are found for each

some properties of sup min fuzzy relational equations on infinite domains abstract this paper deals with a sup min fuzzy relational equation on infinite domains some properties of its solution set will be obtained in particular a necessary and sufficient condition describing the attainable solution resp the unattainable solution is formulated in the end the solution set will be given under some conditions

min max programming problem subject to addition min fuzzy relation inequalities a bittorrent like peer to peer file sharing system can be reduced into a system of addition min fuzzy relation inequalities addition min is a new composition and the solution set of addition min fuzzy relation inequalities differs much from that of general max t norm fuzzy relation inequalities or equations in order to avoid network congestion and improve the stability of data transmission a min max programming problem is proposed subject to addition min fuzzy relation inequalities in this paper based on some relevant theorems on resolution a novel algorithm is developed step by step to find an optimal solution of the proposed problem moreover an application example is given to illustrate the feasibility and efficiency of the algorithm finally some further discussions are made concerning the optimal solution of the proposed problem

solution sets of finite fuzzy relation equations with sup inf composition over bounded brouwerian lattices this paper considers the resolution of finite fuzzy relation equations with sup inf composition over a bounded brouwerian lattice the solution sets of finite fuzzy relation equations on a bounded brouwerian lattice are described in a similar way as those of linear spaces of n dimensional vectors in linear algebra

finite fuzzy relation equations with unique solution in complete brouwerian lattices abstract we characterize finite fuzzy relation equations having a unique solution for sake of completeness we first recall some results already established in foregoing papers in the context of the linear lattices new results are established in the general setting of the complete brouwerian lattices suitable examples illustrate the theorems obtained

fuzzy correspondence inequations and equations the paper deals with fuzzy correspondences i e with mappings from a direct product of sets into a complete lattice fuzzy control problems connected with fuzzy correspondence inequations and equations are considered a fuzzy correspondence is fixed while the solutions are investigated for the input and output fuzzy sets which are unknown first we prove that these solutions can be analyzed and formulated in crisp framework solving the corresponding cut problems further the space of solutions of an inequation is proved to be a complete lattice the same holds for the space of solutions of the corresponding equation in case the membership values belong to an infinitely distributive lattice frame the solutions can be found in specified intervals moreover we were able to locate in such intervals minimal and maximal solutions of these problems

t eigen fuzzy sets abstract this paper concerns the study of the solutions of sup t equations such that atr a where t is a triangular norm and r is a fuzzy relation by generalizing a result from sanchez 12 we determine the greatest t eigen fuzzy set of a fuzzy relation a t eigen fuzzy set of r being a fuzzy set a which is a solution of atr a we also study some algebraic properties for the class of the t eigen sets of a fuzzy relation and for the set of fuzzy relations leaving invariant a given fuzzy set finally properties related with the transitive closure are established

optimization of fuzzy relation equations with max product composition abstract an optimization problem with a linear objective function subject to a system of fuzzy relation equations using max product composition is considered since the feasible domain is non convex traditional linear programming methods cannot be applied we study this problem and capture some special characteristics of its feasible domain and the optimal solutions some procedures for reducing the original problem are presented the problem is transformed into a 0 1 integer program which is then solved by the branch and bound method for illustration purpose an example of the procedures is provided

unbiased estimation and statistical analysis of 3 d rigid motion from two views the problem of estimating 3d rigid motion from point correspondences over two views is formulated as nonlinear least squares ls optimization and the statistical behaviors of the errors in the solution are analyzed by introducing a realistic model of noise described in terms of the covariance matrices of n vectors it is shown that the ls solution based on the epipolar constraint is statistically biased the geometry of this bias is described in both quantitative and qualitative terms finally an unbiased estimation scheme is presented and random number simulations are conducted to observe its effectiveness

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

an accelerated approach for solving fuzzy relation equations with a linear objective function in literature the optimization model with a linear objective function subject to fuzzy relation equations has been converted into a 0 1 integer programming problem by fang and li 1999 they proposed a jump tracking branch and bound method to solve this 0 1 integer programming problem in this paper we propose an upper bound for the optimal objective value based on this upper bound and rearranging the structure of the problem we present a backward jump tracking branch and bound scheme for solving this optimization problem a numerical example is provided to illustrate our scheme furthermore testing examples show that the performance of our scheme is superior to the procedure in the paper by fang and li several testing examples show that our initial upper bound is sharp

s norm aggregation of infinite collections abstract the aggregation of membership values by means of the iterate application of an s norm is of cardinal relevance for the development of any general theory of approximate reasoning which relies on a sup t model for relation composition here we present a definition of s norm aggregation for possibly infinite collections which can be demonstrated to generalize previous approaches consistently it provides a common framework in which classical sup t and finite s t compositions can be analyzed as well as other significant cases general properties of the s norm aggregation concept its specialization for the class of continuous s norms and its relationship with fuzzy measure theory are discussed

an algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max prod composition abstract fuzzy sets as the feasible region for optimization problems is an interesting and on going research topic s c fang g li solving fuzzy relations equations with a linear objective function fuzzy sets syst 103 1999 107 113 7 j lu s c fang solving nonlinear optimization problems with fuzzy relation constraints fuzzy sets syst 119 2001 1 20 16 e khorram a ghodpusian linear objective function optimization with fuzzy relation constraints regarding max av composition appl math comput in press doi 10 1016 j amc 2005 04 021 in this paper we focus on these kind problems in which the solutions region is the fuzzy relation equation with max prod composition and the objective function is linear whereas one of the major difficulties in such problems is non convexity of the feasible region it is preferable to study these regions in the first step hence we have primarily investigated two methods and their relationship and then we have determined the feasible region via them after determining the feasible set we have given an algorithm to optimize the linear objective function on such these regions finally we have presented two examples to illustrate the methods and algorithms

determination of camera location from 2 d to 3 d line and point correspondences a method for the determination of camera location from two dimensional 2 d to three dimensional 3 d straight line or point correspondences is presented with this method the computations of the rotation matrix and the translation vector of the camera are separable first the rotation matrix is found by a linear algorithm using eight or more line correspondences or by a nonlinear algorithm using three or more line correspondences where the line correspondences are either given or derived from point correspondences then the translation vector is obtained by solving a set of linear equations based on three or more line correspondences or two or more point correspondences eight 2 d to 3 d line correspondences or six 2 d to 3 d point correspondences are needed for the linear approach three 2 d to 3 d line or point correspondences for the nonlinear approach good results can be obtained in the presence of noise if more than the minimum required number of correspondences are used

linear independence in bottleneck algebras abstract let r be a linearly ordered set a bottleneck algebra is the set r together with two operations a b max a b and a b min a b in this paper we suggest various definitions of linear independence of sets of n tuples over r study their characteristics and interrelations

some properties of minimal solutions for a fuzzy relation equation abstract some properties of the solution set and minimal solutions of a fuzzy relation equation are considered in this paper we show the necessary and sufficient condition for existence of a minimal solution of a finite fuzzy relation equation defined on infinite index sets

on fuzzy relational equations and the covering problem previous studies have shown that fuzzy relational equations fres based on either the max continuous archimedean t norm or the max arithmetic mean composition can be transformed into the covering problem which is an np hard problem exploiting the properties common to the continuous archimedean t norm and the arithmetic mean this study proposes a generalization of them as the u norm enabling fres that are based on the max continuous u norm composition also to be transformed into the covering problem this study also proposes a procedure for transforming the covering problem into max product fres consequently max continuous u norm fres can be solved by extending any procedure for solving either the covering problem or max product fres

an efficient procedure for solving a fuzzy relational equation with max archimedean t norm composition in the literature a necessary condition for minimal solutions of a fuzzy relational equation with max product composition shows that each of its components is either zero or the corresponding component s value of the greatest solution in this paper we first extend this necessary condition to the situation with max archimedean triangular norm t norm composition based on this necessary condition we then propose rules to reduce the problem size so that the complete set of minimal solutions can be computed efficiently furthermore rather than work with the actual equations we employ a simple matrix whose elements capture all of the properties of the equations in finding the minimal solutions numerical examples with specific cases of the max archimedean t norm composition are provided to illustrate the procedure

resolution of finite fuzzy relation equations abstract general schemes for solving fuzzy relation equations with finite sets are derived in this paper the relationship between the results derived here and those of previous publications regarding this subject is also discussed

finite fuzzy relation equations with a unique solution in linear lattices abstract we characterize fuzzy relation equations possessing a unique solution the equations presented here involve fuzzy sets and fuzzy relations assuming values in a linear lattice and defined on finite sets a suitable example also shows that our main result generally does not hold if the fuzzy equations are assigned on complete brouwerian lattices not necessarily linear

on direct and inverse problems for fuzzy equation systems with tolerances abstract we investigate the resolution of fuzzy relational equation systems with tolerances which are a certain extension of fuzzy equations considered f i in 3 5 the extension of the concept of higashi and klir 3 enables us to describe the set of solutions to our problem for given tolerances by means of posets in a second part we investigate an inverse problem given upper lower tolerances how to determine lower upper tolerances such that the arising problem becomes consistent numerical examples are given

conditions for the existence of the least solution and minimal solutions to fuzzy relation equations over complete brouwerian lattices this paper discusses properties and existence conditions of minimal solutions to sup inf fuzzy relation equations over complete brouwerian lattices the relationship between minimal solutions to a system of fuzzy relation equations and minimal solutions to a fuzzy relation equation in the system is illustrated by two examples necessary and sufficient conditions for the existence of the least solution and minimal solutions to a system of fuzzy relation equations are given

sets of solution set equivalent coefficient matrices of fuzzy relation equations abstract a set of coefficient matrices which are equivalent to the solution set of a fuzzy relation equation is considered it is also shown that the cardinal numbers of solution sets of fuzzy relation equations for two constant vectors with the same partition on the index set are equal to each other

solving nonlinear optimization problems with fuzzy relation equation constraints abstract an optimization model with a nonlinear objective function subject to a system of fuzzy relation equations is presented since the solution set of the fuzzy relation equations is in general a non convex set when it is not empty conventional nonlinear programming methods are not ideal for solving such a problem in this paper a genetic algorithm ga is proposed this ga is designed to be domain specific by taking advantage of the structure of the solution set of fuzzy relation equations the individuals from the initial population are chosen from the feasible solution set and are kept within the feasible region during the mutation and crossover operations the construction of test problems is also developed to evaluate the performance of the proposed algorithm

further contributions to the study of finite fuzzy relations equations abstract an interesting property of the schein rank of a fuzzy matrix stressed by k h kim and f w roush is formalized as definition of the basis of a set of fuzzy vectors defined over a commutative semiring with respect to another set of fuzzy vectors we apply this definition and another related concept in the setting of finite fuzzy relation equations in particular we give a sufficient condition for the existence of the smallest solution of a max min fuzzy equation

an efficient solution procedure for fuzzy relation equations with max product composition we study a system of fuzzy relation equations with max product composition and present an efficient solution procedure to characterize the whole solution set by finding the maximum solution as well as the complete set of minimal solutions instead of solving the problem combinatorially the procedure identifies the nonminimal solutions and eliminates them from the set of minimal solutions

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

fuzzy relation equations on a finite set abstract the family of all the solutions of a fuzzy relation equation on a finite set is considered it is characterized by the set of all the lower solutions which can be obtained by a combinatorial algorithm

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions

posynomial geometric programming problem subject to max min fuzzy relation equations we discuss a class of posynomial geometric programming problem pgpf aimed at minimizing a posynomial subject to fuzzy relational equations with max min composition by introducing auxiliary variables we convert the pgpf into an equivalent programming problem whose objective function is a non decreasing function with an auxiliary variable we show that an optimal solution consists of a maximum feasible solution and one of the minimal feasible solutions by an equivalent programming problem in addition we introduce some rules for simplifying the problem then by using a branch and bound method and fuzzy relational equations fre path we present an algorithm to obtain an optimal solution to the pgpf finally numerical examples are provided to illustrate the steps of the procedure