For $d \in \mathbb{N}$ the well-known Schur-Cohn region $\mathcal{E}_d$ consists of all $d$-dimensional vectors $(a_1,\ldots,a_d)\in\mathbb{R}^d$ corresponding to monic polynomials $X^d+a_1X^{d-1}+\cdots+a_{d-1}X+a_d$ whose roots all lie in the open unit disk. This region has been extensively studied over decades. Recently, Akiyama and Peth\H{o} considered the subsets $\mathcal{E}_d^{(s)}$ of the Schur-Cohn region that correspond to polynomials of degree $d$ with exactly $s$ pairs of nonreal roots. They were especially interested in the $d$-dimensional Lebesgue measures $v_d^{(s)}:=\lambda_d(\mathcal{E}_d^{(s)})$ of these sets and their arithmetic properties, and gave some fundamental results. Moreover, they posed two conjectures that we prove in the present paper. Namely, we show that in the totally complex case $d=2s$ the formula \[ \frac{v_{2s}^{(s)}}{v_{2s}^{(0)}} = 2^{2s(s-1)}\binom {2s}s \] holds for all $s\in\mathbb{N}$ and in the general case the quotient $v_d^{(s)}/v_d^{(0)}$ is an integer for all choices $d\in \mathbb{N}$ and $s\le d/2$. We even go beyond that and prove explicit formul\ae{} for $v_d^{(s)} / v_d^{(0)}$ for arbitrary $d\in \mathbb{N}$, $s\le d/2$. The ingredients of our proofs comprise Selberg type integrals, determinants like the Cauchy double alternant, and partial Hilbert matrices.

The asymptotic form of the sum ∑n i=0 i p ( n+i i ) is established in two quite different ways—by means of the longstanding EulerMaclaurin summation formula, and then via a direct (and somewhat more contemporary) proof. ∗The first author is supported by the Austrian Science Fund Grants FWF-S9610 and FWF-W1230.

Let $\mathcal{E}_d^{(s)}$ denote the set of coefficient vectors $(a_1,\dots,a_d)\in \mathbb{R}^d$ of contractive polynomials $x^d+a_1x^{d-1}+\dots+a_d\in \mathbb{R}[x]$ that have exactly $s$ pairs of complex conjugate roots and let $v_d^{(s)}=\lambda_d(\mathcal{E}_d^{(s)})$ be its ($d$-dimensional) Lebesgue measure. We settle the instance $s=1$ of a conjecture by Akiyama and Pethő, stating that the ratio $v_d^{(s)}/v_d^{(0)}$ is an integer for all $d\ge 2s.$ Moreover we establish the surprisingly simple formula $v_d^{(1)}/v_d^{(0)} = (P_d(3)-2d-1)/4,$ where $P_d(x)$ are the Legendre polynomials.

The asymptotic form of the sum) Sigma(n)(i=0)i(P)((n+i)(i))is established in two quite different ways-by means of the longstanding Euler-Maclaurin summation formula, and then via a direct (and some-what more contemporary) proof.

Numeration and Substitution June 4～8, 2012. edited by Shigeki Akiyama, Valerie Berthe, Hui Rao and Takao Komatsu. The papers presented in this volume of RIMS Kokyuroku Bessatsu are in final form and refereed.

Let d≥ 1 be an integer and r=(r_0,…,r_d-1) ∈𝐑^d. The shift radix system τ_𝐫: ℤ^d →ℤ^d is defined by τ_ r( z)=(z_1,…,z_d-1,-⌊ r z⌋)^t ( z=(z_0,…,z_d-1)^t). τ_𝐫 has the finiteness property if each z∈ℤ^d is eventually mapped to 0 under iterations of τ_𝐫. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t. Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of τ_𝐫 with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems.

Let d >= 1 be an integer and r = (r(1), ... , r(d)) is an element of R-d. We define the shift radix system tau(r) : Z(d) -> Z(d) bytau(r)(a) = (a(2), ... , a(d), -left perpendicularraright perpendicular) (a = (a(1), ... , a(d))).The shift radix system tau(r) has the finiteness property if each a is an element of Z(d) is eventually mapped to 0 under iterations of tau(r).The mapping tau(r) can be written as tau(r)(a) = R(r)a + v(a), where R(r) is a d x d matrix and v is a correction term. It has been conjectured that the fact that tau(r) has the finiteness property implies that all eigenvalues of R(r) are strictly smaller than one in modulus.The aim of the present paper is to prove this conjecture for the case d = 3.

Recently, Akiyama et al. introduced so-called shift radix systems. These simple dynamical systems form a common generalization of several well-known notions of Humber systems like beta numeration and canonical number systems. To the present paper we generalize shift radix systems Os follows: for (r(1),...,r(d)) is an element of C-d we study mappings Z[i](d) -> Z[i](d) given by(x(1),...,x(d)) -> (x(2),...,x(d), - [r(1)x(1) +...+r(d)x(d)]).where for x is an element of C we set [x] = [Rx] + i[(sic)x]. We study basic dynamical properties of this class of mappings and relate them to known notions of number systems.

where rz is the scalar product of the vectors r and z. If each orbit of τ r ends up at 0, we call τ r a shift radix system. It is a well-known fact that each orbit of τ r ends up periodically if the polynomial t d +r d-1 t d-1 +⋯+r 0 associated to r is contractive. On the other hand, whenever this polynomial has at least one root outside the unit disc, there exist starting vectors that give rise to unbounded orbits. The present paper deals with the remaining situations of periodicity properties of the mappings τ r for vectors r associated to polynomials whose roots have modulus less than or equal to one with equality in at least one case. We show that for a large class of vectors r belonging to the above class the ultimate periodicity of the orbits of τ r is equivalent to the fact that τ s is a shift radix system or has another prescribed orbit structure for a certain parameter s related to r. These results are combined with new algorithmic results in order to characterize vectors r of the above class that give rise to ultimately periodic orbits of τ r for each starting value. In particular, we work out the description of these vectors r for the case d=3. This leads to sets which seem to have a very intricate structure.

In the present paper we study sequences defined by the recurrence relation [Formula: see text] for n ≥ 0, where [Formula: see text] the golden ratio. These sequences are related to shift radix systems as well as to β-expansions with respect to Salem numbers.

A particular class of binomial coefficient identities, which involve Harmonic numbers, can be generated in two different ways, recursively. In this paper we present several approaches to the recurrence equations which underpin such formulations. A vaxiety of arguments from different branches of combinatorial theory are used.

Let k := Q( √−D ) be an imaginary quadratic number field and Zk be the corresponding ring of integers. We consider the family of relative Thue equations Ft(x, y) = x 3 − (t− 1)xy − (t + 2)xy − y = ` with t, ` ∈ Zk, t / ∈ Z and |`| ≤ |2t + 1|. Let k(α) be the cubic extension of k generated by a root α of the polynomial ft(x) = Ft(x, 1), and let Zk(α) be its ring of integers. A pair (x, y) with x, y ∈ Zk is a solution of the Thue equation if and only if the element γ = x − αy ∈ Zk(α) has a norm satisfying |Nk(α)/k(γ)| ≤ |2t + 1|. We determine all elements of Zk(α) having norms less than or equal to |2t+1|. Further we solve the above Thue equation for all t ∈ Zk, t / ∈ Z with <t = − 12 and all |`| ≤ |2t + 1|.

In this paper, we give exact and asymptotic approximations for the variance of the external path length in a symmetric Patricia trie. The problem was open up to now. We prove that for the binary Patricia trie, the variance is asymptotically equal to 0.37 ... n+n P (log2 n) where n is the number of stored records and P(x) is a periodic function with a very small amplitude. This result is next used to show that from the practical (average) viewpoint, the Patricia trie does not need to be restructured in order to keep it balanced. In general, we ask to what extent simpler and more direct algorithms (for digital search tries) can be expected in practice to match the performance of more complicated, worst-case asymptotically better ones.

We consider a class of probabilistic counting algorithms parameter-ized by an integer d≥ 0 that estimate the number of elements N in a large set. Our algorithms generalize an idea of Flajolet and Martin who limited themselves to the case d=0. As noted by Brassard and Bratley "it is far from obvious how to carry out a more precise analysis of the unbiased estimate of N ...". We present a novel and complete analysis of these new counting algorithms that — to the best of our knowledge — cannot be obtained by an extension of the analysis by Flajolet and Martin. We present results concerning the average value, the variance and the limiting generating function of an estimate of N. Moreover, our novel approach is not limited to probabilistic counting algorithms, and it can be applied in the investigation of several other "splitting algorithms" such as selecting the loser within a group of people, estimating the number of ques-tions necessary to identify the number of distinct objects, searching algorithms based on digital tries, approximate counting, electing d finalists in a contest (cf. polling system), and so forth.

Given m, n ≥ 2, we prove that, for sufficiently large y, the sum 1 n +···+ y n is not a product of m consecutive integers. We also prove that for m ≠ n we have 1 m +···+ x m ≠ 1 n +···+ y n , provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are ‘almost’ indecomposable, a result of independent interest.

In this paper distribution results are proved on the cost of insertion in digital search trees, (binary) tries and Patricia tries. A method from the calculus of finite differences is used to achieve asymptotic results.

Let M3 := {−4 + 9i √ 2,−3 + 9i √ 2,−2 + 9i √ 2,−1 + 9i √ 2,−5 + 8i √ 2,−4 + 8i √ 2,−3 + 8i √ 2, − 2 + 8i √ 2,−1 + 8i √ 2,−6 + 7i √ 2,−5 + 7i √ 2,−4 + 7i √ 2,−3 + 7i √ 2,−2 + 7i √ 2,−1 + 7i √ 2, − 6 + 6i √ 2,−5 + 6i √ 2,−4 + 6i √ 2,−3 + 6i √ 2,−2 + 6i √ 2,−1 + 6i √ 2,−6 + 5i √ 2,−5 + 5i √ 2, − 4 + 5i √ 2,−3 + 5i √ 2,−2 + 5i √ 2,−1 + 5i √ 2,−6 + 4i √ 2,−5 + 4i √ 2,−4 + 4i √ 2,−3 + 4i √ 2, − 2 + 4i √ 2,−1 + 4i √ 2,−5 + 3i √ 2,−4 + 3i √ 2,−3 + 3i √ 2,−2 + 3i √ 2,−1 + 3i √ 2,−4 + 2i √ 2, − 3 + 2i √ 2,−2 + 2i √ 2,−1 + 2i √ 2}