Joint 2-adic complexity is a new important index of the cryptographic security formultisequences. In this paper, we extend the usual Fourier transform to the case ofmultisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, forthe multisequences with pn-period, we discussthe relation between sequences and their Fourier coefficients. Based on the relation, wedetermine a lower bound for the number of multisequences with given joint 2-adiccomplexity.

For each poset H whose Hasse diagram is a tree of height k, we show that the largest size of a family of subsets of [n]={1,. . ., n} not containing H as an induced subposet is asymptotic to . This extends a result of Bukh [1], which in turn generalizes several known results including Sperner's theorem.

Scott conjectured in [6] that the class of graphs with no induced subdivision of a given graph is χ-bounded. We verify his conjecture for maximal triangle-free graphs.

Li, Nikiforov and Schelp [13] conjectured that any 2-edge coloured graph G with order n and minimum degree δ(G) > 3n/4 contains a monochromatic cycle of length ℓ, for all ℓ ∈ [4, ⌈n/2⌉]. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with δ(G)=3n/4 that do not contain all such cycles. Finally, we show that, for all δ>0 and n>n0(δ), if G is a 2-edge coloured graph of order n with δ(G) ≥ 3n/4, then one colour class either contains a monochromatic cycle of length at least (2/3+δ/2)n, or contains monochromatic cycles of all lengths ℓ ∈ [3, (2/3−δ)n].

We define a variant of the crossing number for an embedding of a graph G into ℝ3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the rectilinear space crossing numbers of pseudo-random graphs.

A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Št'ovíček and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colourable.

A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. We study the behaviour of measurable events in probability spaces indexed by homogeneous trees.

Precisely, we show that for every integer b ≥ 2 and every integer n ≥ 1 there exists an integer q(b,n) with the following property. If T is a homogeneous tree with branching number b and {At:t ∈ T} is a family of measurable events in a probability space (Ω,Σ,μ) satisfying μ(At)≥ϵ>0 for every t ∈ T, then for every 0<θ<ϵ there exists a strong subtree S of T of infinite height, such that for every finite subset F of S of cardinality n ≥ 1 we haveIn fact, we can take q(b,n)= ((2b−1)2n−1−1)·(2b−2)−1. A finite version of this result is also obtained.

Taking the view that infinite plays are draws, we study Conwaynon-terminating games and non-losing strategies. These admit asharp coalgebraic presentation, where non-terminating games are seen as afinal coalgebra and game contructors, such as disjunctivesum, as final morphisms. We have shown, in a previous paper,that Conway’s theory of terminating games can be rephrased naturally in terms of game(pre)congruences. Namely, various conceptually independent notions ofequivalence can be defined and shown to coincide on Conway’sterminating games. These are the equivalence induced by the ordering on surrealnumbers, the contextual equivalence determined by observingwhat player has a winning strategy, Joyal’s categoricalequivalence, and, for impartial games, the denotationalequivalence induced by Grundy semantics. In this paper, wediscuss generalizations of such equivalences to non-terminating games andnon-losing strategies. The scenario is even more rich and intriguing inthis case. In particular, we investigate efficient characterizations of the contextualequivalence, and we introduce a category of fair strategies and acategory of fair pairs of strategies, both generalizing Joyal’s categoryof Conway games and winning strategies. Interestingly, the category of fair pairs capturesthe equivalence defined by Berlekamp, Conway, Guy on loopy games.

Let H be a graph on n vertices and let the blow-up graph G[H] be defined as follows. We replace each vertex vi of H by a cluster Ai and connect some pairs of vertices of Ai and Aj if (vi,vj) is an edge of the graph H. As usual, we define the edge density between Ai and Aj asWe study the following problem. Given densities γij for each edge (i,j) ∈ E(H), one has to decide whether there exists a blow-up graph G[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic to H, i.e., no H appears as a transversal in G[H]. We call dcrit(H) the maximal value for which there exists a blow-up graph G[H] with edge densities d(Ai,Aj)=dcrit(H) ((vi,vj) ∈ E(H)) not containing H in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.

First, in the case of tree T we give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversal T in the blow-up graph. Then we give general bounds on dcrit(H) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction for H-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools.

A detachment of a hypergraph is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. For a given edge-coloured hypergraph , we prove that there exists a detachment such that the degree of each vertex and the multiplicity of each edge in (and each colour class of ) are shared fairly among the subvertices in (and each colour class of , respectively).

Let be a hypergraph with vertex partition {V1,. . .,Vn}, |Vi| = pi for 1 ≤ i ≤ n such that there are λi edges of size hi incident with every hi vertices, at most one vertex from each part for 1 ≤ i ≤ m (so no edge is incident with more than one vertex of a part). We use our detachment theorem to show that the obvious necessary conditions for to be expressed as the union 1 ∪ ··· ∪ k of k edge-disjoint factors, where for 1 ≤ i ≤ k, i is ri-regular, are also sufficient. Baranyai solved the case of h1 = ··· = hm, λ1 = ··· = λm = 1, p1 = ··· = pm, r1 = ··· = rk. Berge and Johnson (and later Brouwer and Tijdeman, respectively) considered (and solved, respectively) the case of hi = i, 1 ≤ i ≤ m, p1 = ··· = pm = λ1 = ··· = λm = r1 = ··· = rk = 1. We also extend our result to the case where each i is almost regular.

We study translations of dyadic first-order sentences into equalities between relationalexpressions. The proposed translation techniques (which work also in the conversedirection) exploit a graphical representation of formulae in a hybrid of the twoformalisms. A major enhancement relative to previous work is that we can cope with therelational complement construct and with the negation connective. Complementation ishandled by adopting a Smullyan-like uniform notation to classify and decompose relationalexpressions; negation is treated by means of a generalized graph-representation offormulae in ℒ+, and through a series of graph-transformation rules whichreflect the meaning of connectives and quantifiers.

Frequency hopping sequences sets are required in frequency hopping code division multiple access systems. For the anti-jamming purpose, frequency hopping sequences are required to have a large linear span. In this paper, by using a permutation polynomial δ(x) over a finite field, we transform several optimal sets of frequency hopping sequences with small linear span into ones with large linear span. The exact values of the linear span are presented by using the methods of counting the terms of the sequences representations. The results show that the transformed frequency hopping sequences are optimal with respect to the Peng-Fan bound, and can resist the analysis of Berlekamp-Massey algorithm.

In a previous paper [L. Giambruno and S. Mantaci, Theoret. Comput. Sci.411 (2010) 1785–1792] a bideterministic transducer is defined forthe bidirectional deciphering of words by the method introduced by Girod [IEEECommun. Lett. 3 (1999) 245–247]. Such a method is defined usingprefix codes. Moreover a coding method, inspired by the Girod’s one, is introduced, and atransducer that allows both right-to-left and left-to-right decoding by this method isdefined. It is proved also that this transducer is minimal. Here we consider the number ofstates of such a transducer, related to some features of the considered prefix codeX. We find some bounds of such a number of states in relation withdifferent notions of “size” of X. In particular, we give an exact formulafor the number of states of transducers associated to maximal prefix codes. We moreoverconsider two special cases of codes: maximal uniform codes and a class of codes, that wename string-codes. We show that they represent, for maximal codes, the extreme cases withregard to the number of states in terms of different sizes. Moreover we prove that prefixcodes corresponding to isomorphic trees have transducers that are isomorphic as unlabeledgraphs.

A closed λ-term M is easy if, for anyother closed term N, the lambda theory generated byM = N is consistent. Recently, it has been introduceda general technique to prove the easiness of λ-terms through thesemantical notion of simple easiness. Simple easiness implies easiness and allows to proveconsistency results via construction of suitable filter models ofλ-calculus living in the category of complete partial orderings: givena simple easy term M and an arbitrary closed term N, itis possible to build (in a canonical way) a non-trivial filter model which equates theinterpretation of M and N. The question whether easinessimplies simple easiness constitutes Problem 19 in the TLCA list of open problems. In thispaper we negatively answer the question providing a non-empty co-r.e. (complement of arecursively enumerable) set of easy, but not simple easy, λ-terms.

Starting from late 90’s the public administration has started to employ a quite relevantamount of its budget in developing ICT solutions to better deliver services to citizens.In spite of this effort many statistics show that the mere availability of ICT basedservices does not guarantee per se their usage. Citizens have continued to largely accessservices through “traditional” means. In our study we suggest that the highlightedsituation is partly due to the fact that relevant domain dependent requirements, mainlyrelated to the delivery process of e-government digital services, are often ignored in thedevelopment of e-government solutions. We provide here a domain related quality frameworkand encoded it in a set of formal statements, so that we can apply automatic verificationtechniques to assess and improve ICT solutions adopted by public administrations. Thepaper discusses both the defined quality framework and the tool chain we developed toenable automatic assessment of ICT solutions. The tool chain is based on a denotationalmapping of business process modeling notation elements into process algebraic descriptionsand to the encoding of quality requirements in linear temporal logic formulas. Theresulting approach has been applied to real case studies with encouraging results.