characterization of some fuzzy relation equations provided with one solution on a finite set abstract in this paper some fuzzy relation equations provided with one solution on a finite set are characterized we consider fuzzy relation equations h q t with q f x x y and card x card y after recalling the definition of equivalent fuzzy relation equations we introduce the definition of equivalent ones which allows us to constrain our research without loss of generality to fuzzy relation equations where t does not have zero components

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

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

a new approach to the general minimum distance decoding problem the zero neighbors algorithm minimum distance decoding mdd for a general error correcting linear code is a hard computational problem that recently has been shown to be np hard the complexity of known decoding algorithms is determined by min 2 k 2 n k where n is the code length and k is the number of information digits two new algorithms are suggested that reduce substantially the complexity of mdd the algorithms use a new concept of zero neighbors a special set of codewords only these codewords which can be computed in advance should be stored and used in the decoding procedure the number of zero neighbors is shown to be very small compared with min 2 k 2 n k for n gg 1 and a wide range of code rates r k n for example for r approx 0 5 this number grows approximately as a square root of the number of codewords

large deviations upper bounds for the laws of matrix valued processes and non communicative entropies using ito s calculus we study the large deviations properties of the law of the spectral measure of the hermitian brownian motion we extend this strategy to the symmetric unitary and wishart processes this dynamical approach is generalized to the study of the large deviations of the non commutative laws of several independent hermitian brownian motions as a consequence we can bound from above entropies defined in the spirit of the microstates entropy introduced by voiculescu

optimal phase transitions in compressed sensing compressed sensing deals with efficient recovery of analog signals from linear encodings this paper presents a statistical study of compressed sensing by modeling the input signal as an i i d process with known distribution three classes of encoders are considered namely optimal nonlinear optimal linear and random linear encoders focusing on optimal decoders we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise respectively the optimal phase transition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms in particular we show that gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding our results also provide a rigorous justification of previous results based on replica heuristics in the weak noise regime

fuzzy modeling with tolerances abstract we consider the problem of the resolution of fuzzy equation systems a certain number of fuzzy inputs and outputs is given how can we determine a fuzzy relation which describes the behaviour of the object under consideration when attributing this problem to fuzzy equations in the classical sense it may happen that a solution does not exist therefore we introduce fuzzy tolerances as an extension of the solvability conception for this problem we get the whole solution set by means of posets moreover we consider the inverse problem how to determine tolerances to get consistency i e at least one solution for the arising equation system

numerical and applicational aspects of fuzzy relational equations we discuss numerical methods leading to a resolution of fuzzy relational equations which create a formal description of ill defined systems an applicability of the numerical approach is shown in solving some problems in fuzzy systems such as identification and control an idea of fuzzy discretization fuzzy chopping as a meaningful tool for handling various problems is also investigated

some results on the resolution of fuzzy relation equations abstract in this paper two special cases of the solution of the inverse problem are investigated the necessary and sufficient conditions for the existence of a solution such that the upper bound is the unit vector and for a unique solution of the inverse problem equal to the unit vector are presented an algorithmic approach which derives the output vector and the state matrix in each one of these cases is also proposed

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

fuzzy relational equations with generalized connectives and their applications the idea of fuzzy relational equation with generalized connectives is introduced and algorithms of resolution of this class equations are presented in details it is clearly shown that some types of well known fuzzy relational equations sup min inf max can be treated as special cases of a wise class of equations under discussion applicational aspects are carefully investigated presenting the advantages of the proposed approach

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

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

the maximum likelihood decoding threshold for cycle codes of graphs for a class mathcal c of binary linear codes we write theta mathcal c colon 0 1 to 0 1 2 for the maximum likelihood decoding threshold function of mathcal c the function whose value at r in 0 1 is the largest bit error rate p that the codes in mathcal c can tolerate with a negligible probability of maximum likelihood decoding error across a binary symmetric channel we show that if mathcal c is the class of cycle codes of graphs then theta mathcal c r le 1 sqrt r 2 2 1 r for each r and show that equality holds only when r is asymptotically achieved by the cycle codes of regular graphs

cut technique in valued relational systems mainsprings and applications abstract new technique for the research of diverse types of valued relational systems including solutions of general systems of relational equations and inequalities is based on general algebraic study of cut mapping and the reverse canonical decomposition mapping applications to relational equations reduction of relational systems decision making with fuzzy binary relations and to description of closeable and cutworthy properties of fuzzy objects are demonstrated

relational equations in totally ordered lattices and their complete resolution abstract the set of all solutions of a composite fuzzy relation equation of sanchez inform and control 30 1976 defined on finite spaces is studied by determining and characterizing all the lower solutions of such an equation

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

some results on fuzzy relation equations provided with one solution in a previous paper 3 we characterized the fuzzy relation equations h q t where q f x y provided with one solution on a finite set considering card x card y or card x card y in this paper we settle the same question for card x card y and s q t o moreover we show that every fuzzy relation equation provided with one solution is equivalent to a fuzzy relation sub equation of it

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

ordered binary decision diagrams and minimal trellises ordered binary decision diagrams obdds are graph based data structures for representing boolean functions they have found widespread use in computer aided design and in formal verification of digital circuits minimal trellises are graphical representations of error correcting codes that play a prominent role in coding theory this paper establishes a close connection between these two graphical models as follows let spl cscr be a binary code of length n and let f sub c x sub 1 x sub n be the boolean function that takes the value 0 at x sub 1 x sub n if and only if x sub 1 x sub n spl isin spl cscr given this natural one to one correspondence between boolean functions and binary codes we prove that the minimal proper trellis for a code spl cscr with minimum distance d 1 is isomorphic to the single terminal obdd for its boolean indicator function f sub c x sub 1 x sub n prior to this result the extensive research during the past decade on binary decision diagrams in computer engineering and on minimal trellises in coding theory has been carried out independently as outlined in this work the realization that binary decision diagrams and minimal trellises are essentially the same data structure opens up a range of promising possibilities for transfer of ideas between these disciplines

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

on the intractability of permuting a block code to minimize trellis complexity an important problem in the theory and application of block code trellises is to find a coordinate permutation of a given code to minimize the trellis complexity we show that the problem of finding a coordinate permutation that minimizes the number of vertices at a given depth in the minimal trellis for a binary linear block code is np complete

the hardness of decoding linear codes with preprocessing the problem of maximum likelihood decoding of linear block codes is known to be hard the fact that the problem remains hard even if the code is known in advance and can be preprocessed for as long as desired in order to device a decoding algorithm is shown the hardness is based on the fact that existence of a polynomial time algorithm implies that the polynomial hierarchy collapses thus some linear block codes probably do not have an efficient decoder the proof is based on results in complexity theory that relate uniform and nonuniform complexity classes

lower solutions of systems of fuzzy equations lower solutions of a system of fuzzy equations are determined by lower solutions of each equation the result is applied to fuzzy relation equations so that they are decomposed to simple fuzzy relation equations

fixed parameter tractability and completeness i basic results for many fixed parameter problems that are trivially soluable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as k feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is soluable in time bounded by a polynomial of degree c where c is a constant independent of k we establish the main results of a completeness program which addresses the apparent fixed parameter intractability of many parameterized problems in particular we define a hierarchy of classes of parameterized problems fpt subseteq w 1 subseteq w 2 subseteq cdots subseteq w sat subseteq w p and identify natural complete problems for w t for t geq 2 in other papers we have shown many problems complete for w 1 dominating set is shown to be complete for w 2 and thus is not fixed parameter tractable unless independent set clique irredundant set and many other natural problems in w 2 are also fixed parameter tractable we also give a compendium of currently known hardness results as an appendix

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

the weighted coordinates bound and trellis complexity of block codes and periodic packings weighted entropy profiles and a new bound the weighted coordinates bound on the state complexity profile of block codes are presented these profiles and bound generalize the notion of dimension length profile dlp and entropy length profile elf to block codes whose symbols are not drawn from a common alphabet set and in particular group codes likewise the new bound may improve upon the dlp and elf bounds fur linear and nonlinear block codes over fields however it seems that the major contribution of the proposed bound is to the study of trellis complexity of block codes whose different coordinates are drawn from different alphabet sets the label code of lattice and nonlattice periodic packings usually has this property the construction of a trellis diagram for a lattice and some related bounds are generalized to periodic packings by introducing the fundamental module of the packing and using the new bound on the state complexity profile this generalization is limited to a given coordinate system we show that any bounds on the trellis structure of block codes and in particular the bound presented in this work are applicable to periodic packings

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

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

on the complexity of some coding problems corresp it is shown that the problem of finding a codeword with least weight and whose weight is not a multiple of k in a binary linear code belongs to the class of nondeterministic polynomial np hard problems for any k geq 2 some other related problems are shown to belong to the same class these results were motivated by a conjecture due to berlekamp mceliece and van tilborg that the problem of finding the hamming distance of a code is np hard

bounds on the dimension of codes and subcodes with prescribed contraction index abstract let c be a linear code over gf q spanned by the rows of a matrix g of rank k a nonnegative integer is said to be the contraction index of c if a maximal set of pairwise linearly independent columns of g has k elements we derive several upper and lower bounds on the dimension of a proper subcode of c with a prescribed contraction index v q of length n minimum hamming distance d and contraction index for certain values of n and d the latter bound is shown to be tight for all q and this substantially generalizes the results obtained by delsarte and by duc for 1

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

the complexity of computing the covering radius of a code the problem of finding the covering radius of a binary linear code is shown to be nondeterministic polynomial np hard in fact a problem that is complete for the class prod 2 p in the polynomial hierarchy is shown to be reducible to the covering radius problem so that finding the covering radius is strictly harder than any np complete problem unless the polynomial hierarchy collapses with np prod 2 p

on the existence of solutions of systems of fuzzy equations abstract a necessary and sufficient condition for the existence of solutions of a fuzzy equation given by e sanchez is extended to systems of fuzzy equations

on measures of fuzziness of solutions of fuzzy relation equations with generalized connectives by using a general class of fuzzy connectives of yager fuzzy sets and systems 4 1980 235 242 pedrycz fuzzy relational equations with generalized convectives and their applications fuzzy sets and systems 10 1983 185 201 has shown that the classical fuzzy relation equations of sanchez in fuzzy automata and decision processes m m gupta g n saridis and b r gaines eds pp 221 234 north holland amsterdam 1977 can be considered as a particular case of a more extensive class of fuzzy equations for such types of equations in this paper the solutions having the greatest energy measure and the smallest possible entropy measure of fuzziness are characterized

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

on the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes it was shown earlier that for a punctured reed muller rm code or a primitive bch code which contains a punctured rm code of the same minimum distance as a large subcode the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions the problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized hamming weight hierarchy of a code and it is shown that for rm codes the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to v k wei 1991 the state complexity of the trellis diagram for the extended and permuted 64 24 bch code is discussed

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

the art of trellis decoding is fixed parameter tractable given n subspaces of a finite dimensional vector space over a fixed finite field mathbb f we wish to find a linear layout v 1 v 2 ldots v n of the subspaces such that dim v 1 v 2 cdots v i cap v i 1 cdots v n le k for all i such a linear layout is said to have width at most k when restricted to 1 dimensional subspaces this problem is equivalent to computing the trellis width or minimum trellis state complexity of a linear code in coding theory and computing the path width of an mathbb f represented matroid in matroid theory we present a fixed parameter tractable algorithm to construct a linear layout of width at most k if it exists for input subspaces of a finite dimensional vector space over mathbb f as corollaries we obtain a fixed parameter tractable algorithm to produce a path decomposition of width at most k for an input mathbb f represented matroid of path width at most k and a fixed parameter tractable algorithm to find a linear rank decomposition of width at most k for an input graph of linear rank width at most k in both corollaries no such algorithms were known previously our approach is based on dynamic programming combined with the idea developed by bodlaender and kloks 1996 for their work on path width and tree width of graphs it was previously known that a fixed parameter tractable algorithm exists for the decision version of the problem for matroid path width a theorem by geelen gerards and whittle 2002 implies that for each fixed finite field mathbb f there are finitely many forbidden mathbb f representable minors for the class of matroids of path width at most k an algorithm by hlin n 2006 can detect a minor in an input mathbb f represented matroid of bounded branch width however this indirect approach would not produce an actual path decomposition our algorithm is the first one to construct such a path decomposition and does not depend on the finiteness of forbidden minors

approximate solutions of fuzzy relational equations in the previous chs we have discussed various problems concerning solutions of fuzzy relation equations obviously an underlying assumption is that there exists a nonempty set of solutions a situation may occurr and indeed it is quite common in which no solution exists nevertheless even in this case one might be interested to obtain an approximate solution and know to which extent it can be viewed as a solution this stream of investigations is particularly interesting for applicational purposes contrary to the topics already discussed in the previous chs this is a field of research which has not been developed enough so far it concerns studies on solvability properties of fuzzy relation equations in this ch we shall try to answer by using several techniques the following basic question how difficult is it to attain a situation in which the system of equations has solutions and then how to measure this property

trellis complexity and minimal trellis diagrams of lattices this paper presents results on trellis complexity and low complexity trellis diagrams of lattices we establish constructive upper bounds on the trellis complexity of lattices these bounds both improve and generalize the similar results of tarokh and vardy see ibid vol 43 p 1294 1300 1997 we also construct trellis diagrams with minimum number of paths for some important lattices such trellises are called minimal the constructed trellises which are novel in many cases can be employed to efficiently decode the lattices via the viterbi algorithm in particular a general structure for minimal trellis diagrams of d sub n lattices is obtained this structure corresponds to a new code formula for d sub n moreover we develop some important duality results which are used in both deriving the upper bounds and finding the minimal trellises all the discussions are based on a universal approach to the construction and analysis of trellis diagrams of lattices using their bases

on classical analogues of free entropy dimension abstract we define a classical probability analogue of voiculescu s free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on r n we show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension first it can be viewed as a kind of fractal dimension second if one extends bochner s inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical bochner s inequalities then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in bochner s inequality for a measure regularized by convolution with the gaussian law as the regularization is removed we introduce a free analogue of the bochner inequality and study the related free entropy dimension quantity we show that it is greater or equal to the non microstates free entropy dimension

on the existence of asymptotically good linear codes in minor closed classes let c c 1 c 2 be a sequence of codes such that each c i is a linear n i k i d i code over some fixed finite field f where n i is the length of the code words k i is the dimension and d i is the minimum distance we say that c is asymptotically good if for some e 0 and for all i 0 we have n i i and min k i n i d i n i e sequences of asymptotically good codes exist we prove that if c is a class of gf p n linear codes where p is prime and n 1 closed under puncturing and shortening and if c contains an asymptotically good sequence then c must contain all gf p linear codes our proof relies on a powerful new result from matroid structure theory

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

characterizing degrees of freedom through additive combinatorics we establish a formal connection between the problem of characterizing degrees of freedom dof in constant single antenna interference channels ics with general channel matrix and the field of additive combinatorics the theory we develop is based on a recent breakthrough result by hochman 2014 in fractal geometry our first main contribution is an explicit condition on the channel matrix to admit full i e k 2 dof this condition is satisfied for almost all channel matrices we also provide a construction of corresponding full dof achieving input distributions the second main result is a new dof formula exclusively in terms of shannon entropy this formula is more amenable to both analytical statements and numerical evaluations than the dof formula by wu et al 2015 which is in terms of renyi information dimension we then use the new dof formula to shed light on the hardness of finding the exact number of dof in ics with rational channel coefficients and to improve the best known bounds on the dof of a well studied channel matrix

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

mmse dimension if n is standard gaussian the minimum mean square error mmse of estimating a random variable x based on snr x n vanishes at least as fast as 1 snr as snr we define the mmse dimension of x as the limit as snr of the product of snr and the mmse mmse dimension is also shown to be the asymptotic ratio of nonlinear mmse to linear mmse for discrete absolutely continuous or mixed distribution we show that mmse dimension equals renyi s information dimension however for a class of self similar singular x e g cantor dis tribution we show that the product of snr and mmse oscillates around information dimension periodically in snr db we also show that these results extend considerably beyond gaussian noise under various technical conditions

information dimension and the degrees of freedom of the interference channel the degrees of freedom dofs of the k user gaussian interference channel determine the asymptotic growth of the maximal sum rate as a function of the signal to noise ratio subject to a very general sufficient condition on the cross channel gains we give a formula for the dofs of the scalar interference channel as a function of the deterministic channel matrix which involves maximization of a sum of information dimensions over k scalar input distributions known special cases are recovered and even generalized in certain cases with unified proofs

discussions around voiculescu s free entropies abstract in this article we set two analogous definitions of the free entropies and introduced by voiculescu invent math 118 1994 411 132 1998 189 we discuss their relations improving the preceding results obtained in cabanal duvillard and guionnet ann probab 2001 to appear where a bound on the microstates entropy was established

factorization properties of lattices over the integers abstract let a be a nonsingular n n matrix over the integers l l a denotes the lattice whose elements are combinations with integer coefficients of the rows of a l is cyclic if it can be defined in the modular form l x x i a i x i 0 mod d where the a i s and d are integers and 0 a i d let l l 1 l 2 b be lattices over the integers l l 1 l 2 is a factorization of l if every element of l is a combination of the rows of b such that the vector of combination coefficients is in l 1 and b is a nonsingular n n matrix the following results are proved every lattice can be expressed as a product of cyclic factors in polynomial time every cyclic lattice can be factored into simple term explained in the text factors in polynomial time every simple lattice can be factored into prime factors in polynomial time if a prime factorization of the determinant of its basis is given in addition we provide polynomial algorithms for the following problems transform a cyclic lattice given by a basis into a modular form and vice versa find a basis of a finite modular lattice given in modular form

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

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

some results in the setting of fuzzy relation equations theory abstract in this paper we point out certain theoretical aspects of fuzzy relation equations connecting some known results between composite and o fuzzy equations several theorems are obtained and some examples are provided too

on the pless construction and ml decoding of the 48 24 12 quadratic residue code we present a method for maximum likelihood decoding of the 48 24 12 quadratic residue code this method is based on projecting the code onto a subcode with an acyclic tanner graph and representing the set of coset leaders by a trellis diagram this results in a two level coset decoding which can be considered a systematic generalization of the wagner rule we show that unlike the 24 12 8 golay code the 48 24 12 code does not have a pless construction which has been an open question in the literature it is determined that the highest minimum distance of a 48 24 binary code having a pless 1986 construction is 10 and up to equivalence there are three such codes

a fixed charge model with fuzzy inequality constraints composed by max product operator abstract a fixed charge model subject to a system of fuzzy relation inequalities with max product composite operator is presented the feasible domain of the model is first investiged of which the necessary and sufficient condition is proposed if it is infeasible a minimum calibration procedure is provided to have at least one solution due to the nonconvexity of its feasible domain a solution procedure is designed to have a minimum objective value of which the initial cost is considered an illustrative example is presented

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

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

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

a decomposition theory for binary linear codes the decomposition theory of matroids initiated by paul seymour in the 1980s has had an enormous impact on research in matroid theory this theory when applied to matrices over the binary field yields a powerful decomposition theory for binary linear codes in this paper we give an overview of this code decomposition theory and discuss some of its implications in the context of the recently discovered formulation of maximum likelihood ml decoding of a binary linear code over a binary input discrete memoryless channel as a linear programming problem we translate matroid theoretic results of grotschel and truemper from the combinatorial optimization literature to give examples of nontrivial families of codes for which the ml decoding problem can be solved in time polynomial in the length of the code one such family is that consisting of codes for which the codeword polytope is identical to the koetter vontobel fundamental polytope derived from the entire dual code cperp however we also show that such families of codes are not good in a coding theoretic sense either their dimension or their minimum distance must grow sublinearly with code length as a consequence we have that decoding by linear programming when applied to good codes cannot avoid failing occasionally due to the presence of pseudocode words

graphical codes revisited the set of all even subgraphs of a connected graph g on p vertices with q edges forms a binary linear code c c sub e g with parameters q q p 1 g where g is the girth of g such codes were studied systematically by bredeson and hakimi 1967 and hakimi and bredeson 1968 who were concerned with the problems of augmenting c to a larger q k g code and of efficiently decoding such augmented graphical codes we give a new approach to these problems by requiring the augmented codes to be graphical on one hand we present two construction methods which turn out to contain the methods proposed by hakimi and bredeson as special cases as we show this not only gives a better understanding of their construction it also results in augmenting codes of larger dimension we look at the case of 1 error correcting graphical codes in some detail in particular we show how to obtain the extended hamming codes as purely graphical codes by our approach on the other hand we follow a suggestion of ntafos and hakimi 1981 and use techniques from combinatorial optimization to give decoding procedures for graphical codes which turn out to be considerably more efficient than the approach via majority logic decoding proposed by bredeson and hakimi we also consider the decoding problem for the even graphical code based on the complete graph k sub 2n in more detail we discuss an efficient hardware implementation of an encoding decoding scheme for these codes and show that things may be arranged in such a way that one can also correct all adjacent double errors finally we discuss nonlinear graphical codes