We show that semigroups representable by triangular matrices over a fixed finite fieldform a decidable pseudovariety and provide a finite pseudoidentity basis for it.

Cover automata for finite languages have been much studied a few years ago.It turns out that a simple mathematical structure, namelysimilarity relations over a finite set of words, is underlying thesestudies. In the present work, we investigate in detail for themselvesthe properties of these relations beyond the scope of finite languages.New results with straightforward proofsare obtained in this generalized framework,and previous results concerning coverautomata are obtained as immediate consequences.

Let P be a hereditary property of words, i.e., aninfinite class of finite words such that every subword (block) ofa word belonging to P is also in P.Extending the classical Morse-Hedlund theorem, we show thateither P contains at least n+1 words of lengthn for every n or, for some N, it contains at most N words of lengthn for every n. More importantly, we prove the following quantitativeextension of this result: if Phas m ≤ n words of length n then, for every k ≥ n + m, it containsat most ⌈(m + 1)/2⌉⌈(m + 1)/2⌈ words of length k.

The problem of controlling transmission rate over a randomly varying channel is cast as a Markov decision process wherein the channel is modeled as a Markov chain. The objective is to minimize a cost that penalizes both buffer occupancy (equivalently, delay) and power. The nature of the optimal policy is characterized using techniques adapted from classical inventory control.

In this paper we investigate how it is possible to recover anautomaton from a rational expression that has been computed from thatautomaton. The notion of derived term of an expression, introduced by Antimirov,appears to be instrumental in this problem. The second important ingredient is the co-minimization of anautomaton, a dual and generalized Moore algorithm on non-deterministicautomata.
We show here that if an automaton is then sufficiently “decorated”, thecombination of these two algorithms gives the desired result. Reducing the amount of “decoration” is still the object of ongoing investigation.

We consider a storage model that can be on or off. When on, the content increases at some state-dependent rate and the system can switch to the off state at a state-dependent rate as well. When off, the content decreases at some state-dependent rate (unless it is at zero) and the system can switch to the on position at a state-dependent rate. This process is a special case of a piecewise deterministic Markov process. We identify the stationary distribution and conditions for its existence and uniqueness.

A subgraph H of a graph G is conformal if G - V(H) has aperfect matching. An orientation D of G is Pfaffian if, forevery conformal even circuit C, the number of edges of C whosedirections in D agree with any prescribed sense of orientation ofC is odd. A graph is Pfaffian if it has a Pfaffianorientation. Not every graph is Pfaffian. However, if G has aPfaffian orientation D, then the determinant of the adjacency matrixof D is the square of the number of perfect matchings of G. (Seethe book by Lovász and Plummer [Matching Theory. Annals of Discrete Mathematics, vol. 9. Elsevier Science (1986), Chap. 8.]A matching covered graph is a nontrivial connected graph inwhich every edge is in some perfect matching. The study of Pfaffianorientations of graphs can be naturally reduced to matching coveredgraphs. The properties of matching covered graphs are thus helpful inunderstanding Pfaffian orientations of graphs. For example, say thattwo orientations of a graph are similar if one can be obtainedfrom the other by reversing the orientations of all the edges in a cutof the graph. Using one of the theorems we proved in [M.H. de Carvalho, C.L. Lucchesi and U.S.R. Murty, Optimal ear decompositions of matching covered graphs. J. Combinat. Theory B85 (2002) 59–93]concerning optimal ear decompositions, we show that if a matchingcovered graph is Pfaffian then the number of dissimilar Pfaffianorientations of G is 2b(G) , where b(G) is the number of“bricks” of G. In particular, any two Pfaffian orientations of abipartite graph are similar. We deduce that the problem ofdetermining whether or not a graph is Pfaffian is as difficult as theproblem of determining whether or not a given orientation is Pfaffian,a result first proved by Vazirani and Yanakakis [Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs.Discrete Appl. Math.25 (1989) 179–180].We establish a simple property of minimal graphs without a Pfaffianorientation and use it to give an alternative proof of thecharacterization of Pfaffian bipartite graphs due to Little [ A characterization of convertible (0,1)-matrices.J. Combinat. Theory B18 (1975) 187–208] .

In this article, we give several results on (multivariate and univariate) stochastic comparisons of generalized order statistics. We give conditions on the underlying distributions and the parameters on which the generalized order statistics are based, to obtain stochastic comparisons in the stochastic, dispersive, hazard rate, and likelihood ratio orders. Our results generalize some recent results for order statistics, record values, and generalized order statistics and provide some new results for other models such as k-record values and order statistics under multivariate imperfect repair.

We study the concept of an H-partition of the vertex set of agraph G, which includes all vertex partitioning problems intofour parts which we require to be nonempty with only externalconstraints according to the structure of a model graph H, withthe exception of two cases, one that has already been classifiedas polynomial, and the other one remains unclassified. In thecontext of more general vertex-partition problems, the problemsaddressed in this paper have these properties: non-list, 4-part,external constraints only (no internal constraints), each partnon-empty. We describe tools that yield for each problemconsidered in this paper a simple and low complexitypolynomial-time algorithm.

We prove that the pseudovariety of monoids of Krohn-Rhodescomplexity at most n is not finitely based for all n>0. Morespecifically, for each pair of positive integers n,k, weconstruct a monoid of complexity n+1, all of whose k-generatedsubmonoids have complexity at most n.

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

Given a finite set of matrices with integer entries,consider the question of determining whether the semigroup they generated 1) is free; 2) contains the identity matrix; 3) contains the null matrix or 4) is a group.Even for matrices of dimension 3,questions 1) and 3) are undecidable. For dimension2, they are still open as far as we know.Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a givennon singular matrixbelongs to a given finitely generated semigroup.

We propose an analytically tractable model of loading-dependent cascading failure that captures some of the salient features of large blackouts of electric power transmission systems. This leads to a new application and derivation of the quasibinomial distribution and its generalization to a saturating form with an extended parameter range. The saturating quasibinomial distribution of the number of failed components has a power-law region at a critical loading and a significant probability of total failure at higher loadings.


We study simulation of gate circuits in the infinite algebra oftransients recently introduced by Brzozowski and Ésik. A transientis a word consisting of alternating 0s and 1s; it represents achanging signal. In the algebra of transients, gates processtransients instead of 0s and 1s. Simulation in this algebra iscapable of counting signal changes and detecting hazards. We studytwo simulation algorithms: a general one that works with any initialstate, and a special one that applies only if the initial state isstable. We show that the two algorithms agree in the stable case. Wealso show that the general algorithm is insensitive to the removal ofstate variables that are not feedback variables. We prove thesufficiency of simulation: all signal changes occurring in binaryanalysis are predicted by the general algorithm. Finally, we showthat simulation can be more pessimistic than binary analysis, if wiredelays are not taken into account. We propose a circuit model that weconjecture to be sufficient for proving the equivalence of simulationand binary analysis for feedback-free circuits.


A Σ-labeled n-poset is an (at most) countable set,labeled in the set Σ, equipped with n partial orders.The collection of all Σ-labeled n-posets is naturallyequipped with n binary product operations andnω-ary product operations.Moreover, the ω-ary product operationsgive rise to nω-power operations.We show that those Σ-labeled n-posets that can be generated fromthe singletons by the binary and ω-aryproduct operations form the free algebra on Σin a variety axiomatizable by an infinite collection of simpleequations. When n = 1, this variety coincides with the class ofω-semigroups of Perrin and Pin.Moreover, we show that those Σ-labeledn-posets that can be generated fromthe singletons by the binary product operations andthe ω-power operations form the free algebra on Σin a related variety that generalizes Wilke's algebras.We also give graph-theoretic characterizationsof those n-posets contained in the above free algebras. Our resultsserve as a preliminary study to a development of a theory ofhigher dimensional automata and languages on infinitaryassociative structures.

Let T sH be the graph obtained from a given graph H by subdividing each edge s times. Motivated by a problem raised by Igor Pak [Mixing time and long paths in graphs, in Proc. of the 13th annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002) 321–328], we prove that, for any graph H, there exist graphs G with O(s) edges that are Ramsey with respect to T sH .

In this article, we obtain, in a unified way, a closed-form analytic expression, in terms of roots of the so-called characteristic equation of the stationary waiting-time distribution for the GIX/R/1 queue, where R denotes the class of distributions whose Laplace–Stieltjes transforms are rational functions (ratios of a polynomial of degree at most n to a polynomial of degree n). The analysis is not restricted to generalized distributions with phases such as Coxian-n (Cn) but also covers nonphase-type distributions such as deterministic (D). In the latter case, we get approximate results. Numerical results are presented only for (1) the first two moments of waiting time and (2) the probability that waiting time is zero. It is expected that the results obtained from the present study should prove to be useful not only for practitioners but also for queuing theorists who would like to test the accuracies of inequalities, bounds, or approximations.

Motivated by a problem posed by Hamming in 1980, we define even codes.They are Huffman type prefix codes with the additional property of beingable to detect the occurrence of an odd number of 1-bit errors in the message.We characterize optimal even codes and describe a simple method forconstructing the optimal codes. Further, we compare optimal even codeswith Huffman codes for equal frequencies. We show that the maximum encodingin an optimal even code is at most two bits larger than the maximum encodingin a Huffman tree. Moreover, it is always possible to choose an optimal evencode such that this difference drops to 1 bit. We compare averagesizes and show that the average size of an encoding in a optimal even treeis at least 1/3 and at most 1/2 of a bit larger thanthat of a Huffman tree. These values represent the overhead in the encodingsizes for having the ability to detect an odd number of errors in themessage. Finally, we discuss the case of arbitrary frequencies and describe some results for this situation.

Alignment of sequences is widely used for biological sequence comparisons, and only biological events like mutations, insertions and deletions are considered. Other biological events like inversions are not automatically detected by the usual alignment algorithms, thus some alternative approaches have been tried in order to include inversions or other kinds of rearrangements. Despite many important results in the last decade, the complexity of the problem of alignment with inversions is still unknown. In 1992, Schöniger and Waterman proposed the simplification hypothesis that the inversions do not overlap. They also presented an O(n6) exact solution for the alignment with non-overlapping inversions problem and introduced a heuristic for it that brings the average case complexity down. (In this work, n is the maximal length of both sequences that are aligned.) The present paper gives two exact algorithms for the simplified problem. We give a quite simple dynamic program with O(n4)-time and O(n2)-space complexity for alignments with non-overlapping inversions and exhibit a sparse and exact implementation version of this procedure that uses much less resources for some applications with real data.