Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks

We develop a number of statistical aspects of symmetric groups (mostly dealing with the distribution of cycles in various subsets of Sn), asymptotic properties of (ordinary) characters of symmetric groups, and estimates for the multiplicities of root number functions of these groups. As main applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, a finiteness result for the number of Fuchsian presentations of such a group (resolving a long-standing problem of Roger Lyndon), as well as a proof of a well-known conjecture of Roichman concerning the mixing time of random walks on symmetric groups.     

The number of order-preserving maps between fences and crowns

We compute the number of order-preserving and -reversing maps between posets in the class of fences (zig-zags) and crowns (cycles).     

Supersequences, rearrangements of sequences, and the spectrum of bases in additive number theory

The set of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be represented as the sum of h elements of A. If an∼αnh for some real number α>0, then α is called an additive eigenvalue of order h. The additive spectrum of order h is the set N(h) consisting of all additive eigenvalues of order h. It is proved that there is a positive number ηh?1/h! such that N(h)=(0,ηh) or N(h)=(0,ηh]. The proof uses results about the construction of supersequences of sequences with prescribed asymptotic growth, and also about the asymptotics of rearrangements of infinite sequences. For example, it is proved that there does not exist a strictly increasing sequence of integers such that bn∼n² and B contains a subsequence such that bnk∼k³.     

Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1?r,s?g, the minimal size μ_G(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.     

Uniquely factorizable elements and solvability of finite groups

In a finite group G every element can be factorized in such a way that there is one factor for each prime divisor p of | G |, and the order of this factor is p^α for some integer α ≧ 0. We define g ∈G to be uniquely factorizable if it has just one such factorization (whose factors must be pairwise commuting). We consider the existence of uniquely factorizable elements and its relation to the solvability of the group. We prove that G is solvable if and only if the set of all uniquely factorizable elements of G is the Fitting subgroup of G. We also prove various sufficient conditions for the non-existence of uniquely factorizable elements in non-solvable groups. Received: 9 June 2005     

Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture   for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N$\mathbb{N}$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–?ech compactification βN$\beta \mathbb{N}$. This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture  . The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–?ech compactification of N$\mathbb{N}$.     

The generalized order-k Fibonacci–Pell sequence by matrix methods

In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order-k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F–P sequence by matrix methods. Further, we give the generating function and combinatorial representations of these numbers. Also we present an algorithm for computing the sums of the generalized order-k Pell numbers, as well as the Pell numbers themselves.     

On chaotic extensions of dynamical systems

In this paper, inspired by some results in linear dynamics, we will show that every dynamical system (X,f), where f is a continuous self-map on a separable metric space X, can be extended to a chaotic (in the sense of Devaney) dynamical system in an isometric way.     

A cohomological index of Fuller type for set-valued dynamical systems

In the paper we develop the theory of a cohomological index of the Fuller type detecting periodic orbits of a set-valued dynamical system generated by a differential inclusion or a differential equation without the uniqueness of solutions. The theory presented is applied to establish a general result on the existence of bifurcation of periodic orbits from an equilibrium point of a differential inclusion.     

Sell’s conjecture for non-autonomous dynamical systems

This paper is dedicated to the study of the G. Sell’s conjecture for general non-autonomous dynamical systems. We give a positive answer for this conjecture and we apply this result to different classes of non-autonomous evolution equations: Ordinary Differential Equations, Functional Differential Equations and Semi-linear Parabolic Equations.     

An application of Fourier transforms on finite Abelian groups to an enumeration arising from the Josephus problem

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We analyze an enumeration associated with the Josephus problem by applying a Fourier transform to a multivariate generating function. This yields a formula for the enumeration that reduces to a simple expression under a condition we call local prime abundance. Under this widely held condition, we prove (Corollary 3.4) that the proportion of Josephus permutations in the symmetric group Sn that map t to k (independent of the choice of t and k) is 1/n. Local prime abundance is intimately connected with a well-known result of S.S. Pillai, which we exploit for the purpose of determining when it holds and when it fails to hold. We pursue the first case where it fails, reducing an intractable DFT computation of the enumeration to a tractable one. A resulting computation shows that the enumeration is nontrivial for this case.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=DnZi-Znuk-A.     

Studying discrete dynamical systems through differential equations

In this paper we consider dynamical systems generated by a diffeomorphism F defined on U an open subset of Rn, and give conditions over F which imply that their dynamics can be understood by studying the flow of an associated differential equation, , also defined on U. In particular the case where F has n−1 functionally independent first integrals is considered. In this case X is constructed by imposing that it shares with F the same set of first integrals and that the functional equation μ(F(x))=det(DF(x))μ(x), x∈U, has some non-zero solution, μ. Several examples for n=2,3 are presented, most of them coming from several well-known difference equations.     

Positively expansive dynamical systems

We introduce the notions of weakly and strongly positively expansive (wPE and sPE, respectively) discrete dynamical systems. Both are topological generalizations of the well-known metric notion of positive expansiveness (PE). We prove that the three notions are identical on compact metrizable spaces, but not on noncompact spaces. We investigate properties of PE, wPE, and sPE dynamical systems and show how they are related. Finally, we show that the possible dynamics for wPE and sPE homeomorphisms are severely limited, and we classify sPE homeomorphisms.     

Homotopy Conley index for discrete multivalued dynamical systems

We define an index of Conley type for a certain class of upper semicontinuous multivalued dynamical systems. We use the Szymczak functor and apply techniques introduced by Reineck, Mrozek and Srzednicki for the index over the base. Moreover we introduce the notion of the homotopy partial functor for the usc maps. We show that the index possesses Wa?ewski and homotopy properties. We also give four examples that exhibit the benefits of our index over the cohomological index defined by Mrozek and Kaczyński.     

Bowen-Franks groups of reducible bimodal subshifts of finite type

We study the Bowen-Franks groups of subshifts of finite type associated with reducible bimodal periodic kneading sequences pairs.     

Jacobi continued fraction and Hankel determinants of the Thue-Morse sequence

We study the Jacobi continued fraction and the Hankel determinants of the Thue-Morse sequence and obtain several interesting properties. In particular, a formal power series φ(x) is being discovered, having the property that the Hankel transforms of φ(x) and of φ(x²) are identical.     

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

We define the multiple zeta function of the free Abelian group Z^d as

ζ_Z^d(s₁,…,s_d)=∑_|Z^d:H|<∞α₁(H)^−s₁?α_d(H)^−s_d,