On the bottom summation

We consider summation of consecutive values (φ(v), φ(v + 1), ..., φ(w) of a meromorphic function φ(z), where v, w ∈ ℤ. We assume that φ(z) satisfies a linear difference equation L(y) = 0 with polynomial coefficients, and that a summing operator for L exists (such an operator can be found—if it exists—by the Accurate Summation algorithm, or, alternatively, by Gosper’s algorithm when ordL = 1). The notion of bottom summation which covers the case where φ(z) has poles in ℤ is introduced. The text was submitted by the authors in English.     

Computing Bounds for the Star Discrepancy

We observe that the time required to compute the star discrepancy of a sequence of points in a multidimensional unit cube is prohibitive and that the best known upper bounds for the star discrepancy of (t,s)-sequences and (t,m,s)-nets are useful only for sample sizes that grow exponentially with the dimension s. Then, an algorithm to compute upper bounds for the star discrepancy of an arbitrary set of n points in the s-dimensional unit cube is proposed. For an integer k≥1, this algorithm computes in O(nslogk+2 ^s k ^s ) time and O(k ^s ) space a bound that is no better than a function depending on s and k. As an application, we give improved upper bounds for the star discrepancy of some Faure (0,m,s)-nets for s∈{7,…,20}. Received April 20, 1999; revised April 26, 2000     

On the rate of convergence of 2-term recursions in ℝ d

LetG be a compact set in ℝ ^d d≥1,M=G×G andϕ:M →G a map inC ₃(M). Suppose thatϕ has a fixed pointξ, i.e.ϕ(ξ, ξ)=ξ. We investigate the rate of convergence of the iterationx ⁿ⁺²=φ(x ⁿ⁺¹,x ⁿ) withx ⁰,x ¹∈G andx ⁿ→ξ. Iff _n=Q‖x ⁿ−ξ‖ with a suitable norm and a constantQ depending onξ, an exact representation forf _n is derived. The error terms satisfyf _2m+1≍(ƒ_2m)^γ,f _2m+2≍(ƒ_2m+1)^2γ,m≥0, with 1<gg<2, andγ=γ(x ¹,x ⁰). According to our main result we have lim_n→∞{‖x ⁿ⁺²‖/(‖x ⁿ‖)²}=Q, 0<Q<∞. This paper constitutes an extension of a part of the author’s doctoral thesis realized under the direction of Prof. E. Wirsing and Prof. A. Peyerimhoff, University of Ulm (Germany).     

The super spanning connectivity and super spanning laceability of the enhanced hypercubes

A k -container C(u,v) of a graph G is a set of k disjoint paths between u and v. A k-container C(u,v) of G is a k ^* -container if it contains all vertices of G. A graph G is k ^* -connected if there exists a k ^*-container between any two distinct vertices of G. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is Hamiltonian connected (respectively, Hamiltonian). A graph G is super spanning connected if there exists a k ^*-container between any two distinct vertices of G for every k with 1≤k≤κ(G) where κ(G) is the connectivity of G. A bipartite graph G is k ^* -laceable if there exists a k ^*-container between any two vertices from different partite set of G. A bipartite graph G is super spanning laceable if there exists a k ^*-container between any two vertices from different partite set of G for every k with 1≤k≤κ(G). In this paper, we prove that the enhanced hypercube Q _n,m is super spanning laceable if m is an odd integer and super spanning connected if otherwise.

On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫₀ ^t by(qs)ds with proportional delay qt, 0<q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p _i t and q _i t, 0<p _i ,q _i ≤1 and complex numbers a,b _i and c _i . We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the `iterated collocation solution u _it (t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M _m (t) of v(th) or M^ _m (t) of u _it (th), t∈[0,1] such that the rational approximant v(h) or u _it (h) is the (m,m)-Padé approximant to y(h) and satisfies |v(h)−y(h)|=O(h ^{2 m +1}). If they exist, then we actually give the conditions of M _m (t) and M^ _m (t), respectively. Received September 17, 1998; revised September 30, 1999     

Recursive identification for EIV ARMAX systems

The input u _k and output y _k of the multivariate ARMAX system A(z)y _k = B(z)u _k + C(z)w _k are observed with noises: u _k ^ob ≜ u _k + ε _k ^u and y _k ^ob ≜ y _k + ε _k ^y , where ε _k ^u and ε _k ^y denote the observation noises. Such kind of systems are called errors-in-variables (EIV) systems. In the paper, recursive algorithms based on observations are proposed for estimating coefficients of A(z), B(z), C(z), and the covariance matrix Rw of w _k without requiring higher than the second order statistics. The algorithms are convenient for computation and are proved to converge to the system coefficients under reasonable conditions. An illustrative example is provided, and the simulation results are shown to be consistent with the theoretical analysis.     

On a cosmological invariant as an observational probe in the early universe

k-essence scalar field models are usually taken to have Lagrangians of the form L = −V (φ)F(X) with F some general function of X = ▿ _μ φ▿ ^μ ϕ. Under certain conditions, this Lagrangian can take the form of that of an oscillator with time-dependent frequency. The Ermakov invariant for a time-dependent oscillator in a cosmological scenario then leads to an invariant quadratic form involving the Hubble parameter and a logarithm of the scale factor. In principle, this invariant can lead to further observational probes for the early Universe. Moreover, if such an invariant can be observationally verified, then the presence of dark energy will also be indirectly confirmed.     

Output-Sensitive Algorithm for Computing β-Skeletons

The β-skeleton is a measure of the internal shape of a planar set of points. We get an entire spectrum of shapes by varying the parameter β. For a fixed value of β, a β-skeleton is a geometric graph obtained by joining each pair of points whose β-neighborhood is empty. For β≥1, this neighborhood of a pair of points p _i ,p _j is the interior of the intersection of two circles of radius , centered at the points (1−β/2)p _i +(β/2)p _j and (β/2)p _i +(1−β/2)p _j , respectively. For β∈(0,1], it is the interior of the intersection of two circles of radius , passing through p _i and p _j . In this paper we present an output-sensitive algorithm for computing a β-skeleton in the metrics l ₁ and l _∞ for any β≥2. This algorithm is in O(nlogn+k), where k is size of the output graph. The complexity of the previous best known algorithm is in O(n ^5/2logn) [7]. Received April 26, 2000     

From liveness to promptness

Liveness temporal properties state that something “good” eventually happens, e.g., every request is eventually granted. In Linear Temporal Logic (LTL), there is no a priori bound on the “wait time” for an eventuality to be fulfilled. That is, F θ asserts that θ holds eventually, but there is no bound on the time when θ will hold. This is troubling, as designers tend to interpret an eventuality F θ as an abstraction of a bounded eventuality F ^≤k θ, for an unknown k, and satisfaction of a liveness property is often not acceptable unless we can bound its wait time. We introduce here prompt-LTL, an extension of LTL with the prompt-eventually operator F _p . A system S satisfies a prompt-LTL formula φ if there is some bound k on the wait time for all prompt-eventually subformulas of φ in all computations of S. We study various problems related to prompt-LTL, including realizability, model checking, and assume-guarantee model checking, and show that they can be solved by techniques that are quite close to the standard techniques for LTL.     

Periods in Extensions of Words

Let π(w) denote the minimum period of the word w,let w be a primitive word with period π(w) < |w|, and let z be a prefix of w. It is shown that if π(wz) = π(w), then |z| < π(w) − gcd (|w|, |z|). Detailed improvements of this result are also proven. Finally, we show that each primitive word w has a conjugate w′ = vu, where w = uv, such that π(w′) = |w′| and |u| < π(w). As a corollary we give a short proof of the fact that if u,v,w are words such that u ² is a prefix of v ², and v ² is a prefix of w ², and v is primitive, then |w| > 2|u|.     

On Chebyshev quadrature for ultraspherical weight functions

It is well-known that for the ultraspherical weight function (1-x²)^λ-1/2 there exist no Chebyshev quadrature formulae in the strict sense having n nodes, where n is sufficiently large and λ>0, whereas on the other hand for λ=0 every Gaussian quadrature formulae is a Chebyshev formula in the strict sense. In this paper we study the open question of Chebyshev quadrature for λ <0. It is shown that there exists no Chebyshev quadrature formula in the strict sense having more than two nodes for λ≤λ₀=-.30056... (for definition of λ₀ see (1.8) below). Moreover, results are obtained for Chebyshev-type formulae and Chebyshev formulae of closed type. For the remaining values of λ (λ₀<λ<0) numerical results are given.     

Complexities of Homomorphism and Isomorphism for Definite Logic Programs

A homomorphism (?) of logic programs from P to P' is a function mapping Atoms(P) to Atoms(P') and it preserves complements and program clauses. For each definite program clause a←a1,...,an∈P it implies that (?)(a)←(?)(a1),...,(?)(an) is a program clause of P'. A homomorphism (?) is an isomorphism if (?) is a bijection. In this paper, the complexity of the decision problems on homomorphism and isomorphism for definite logic programs is studied. It is shown that the homomorphism problem (HOM-LP) for definite logic programs is NP-complete, and the isomorphism problem (ISO-LP) is equivalent to the graph isomorphism problem (GI).     

Generation of robust root loci for linear systems with parametric uncertainties in an ellipsoid

Given a parametric polynomial family p(s; Q) := {ⁿ _k=0 a_k (q)s^k : q ] Q}, Q R ^m , the robust root locus of p(s; Q) is defined as the two-dimensional zero set _p,Q := {s ] C:p(s; q) = 0 for some q ] Q}. In this paper we are concerned with the problem of generating robust root loci for the parametric polynomial family p(s; E) whose polynomial coefficients depend polynomially on elements of the parameter vector q ] E which lies in an m-dimensional ellipsoid E. More precisely, we present a computational technique for testing the zero inclusion/exclusion of the value set p(z; E) for a fixed point z in C, and then apply an integer-labelled pivoting procedure to generate the boundary of each subregion of the robust root locus _p,E . The proposed zero inclusion/exclusion test algorithm is based on using some simple sufficient conditions for the zero inclusion and exclusion of the value set p(z,E) and subdividing the domain E iteratively. Furthermore, an interval method is incorporated in the algorithm to speed up the process of zero inclusion/exclusion test by reducing the number of zero inclusion test operations. To illustrate the effectiveness of the proposed algorithm for the generation of robust root locus, an example is provided.     

The Inf-Sup Condition for the Mapped Q k ?P k?1 disc Element in Arbitrary Space Dimensions

 One of the most popular pairs of finite elements for solving mixed formulations of the Stokes and Navier–Stokes problem is the Q _k −P _k−1 ^disc element. Two possible versions of the discontinuous pressure space can be considered: one can either use an unmapped version of the P _k−1 ^disc space consisting of piecewise polynomial functions of degree at most k−1 on each cell or define a mapped version where the pressure space is defined as the image of a polynomial space on a reference cell. Since the reference transformation is in general not affine but multilinear, the two variants are not equal on arbitrary meshes. It is well-known, that the inf-sup condition is satisfied for the first variant. In the present paper we show that the latter approach satisfies the inf-sup condition as well for k≥2 in any space dimension. Received January 31, 2001; revised May 2, 2002 Published online: July 26, 2002     

Approximation Algorithms for Bounded Degree Phylogenetic Roots

The Degree- Δ Closest Phylogenetic k th Root Problem (ΔCPR _k ) is the problem of finding a (phylogenetic) tree T from a given graph G=(V,E) such that (1) the degree of each internal node in T is at least 3 and at most Δ, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) the number of disagreements, i.e., |E ⊕{{u,v} : u,v are leaves of T and d _T (u,v)≤k}|, is minimized, where d _T (u,v) denotes the distance between u and v in tree T. This problem arises from theoretical studies in evolutionary biology and generalizes several important combinatorial optimization problems such as the maximum matching problem. Unfortunately, it is known to be NP-hard for all fixed constants Δ,k such that either both Δ≥3 and k≥3, or Δ>3 and k=2. This paper presents a polynomial-time 8-approximation algorithm for Δ CPR ₂ for any fixed Δ>3, a quadratic-time 12-approximation algorithm for 3CPR ₃, and a polynomial-time approximation scheme for the maximization version of Δ CPR _k for any fixed Δ and k.     

A new quantum claw-finding algorithm for three functions

Fork functionsf ₁, ...f _k, ak-tuple (x ₁, ...x _k) such thatf ₁(x ₁)=...=f _k(x _k) is called a claw off ₁, ...,f _k. In this paper, we construct a new quantum claw-finding algorithm for three functions that is efficient when the numberM of intermediate solutions is small. The known quantum claw-finding algorithm for three functions requiresO(N ^7/8 logN) queries to find a claw, but our algorithm requiresO(N ^3/4 logN) queries ifM ≤ √N andO(N ^7/12 M ^1/3 logN) queries otherwise. Thus, our algorithm is more efficient ifM≤N ^7/8. Kazuo Iwama, Ph.D.: Professor of Informatics, Kyoto University, Kyoto 606-8501, Japan. Received BE, ME, and Ph.D. degrees in Electrical Engineering from Kyoto University in 1978, 1980 and 1985, respectively. His research interests include algorithms, complexity theory and quantum computation. Editorial board of Information Processing Letters and Parallel Computing. Council Member of European Association for Theoretical Computer Science (EATCS). Akinori Kawachi: Received B.Eng. and M.Info. from Kyoto University in 2000 and 2002, respectively. His research interests are quantum computation and distributed computation.     

Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues

The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA ₁, A₂, …, A_p, is the least integert for wich there existt diadsu ^(r) v ^(r)τ, τ=1,2,...,t, such that . IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log² k) steps with max(n ²+p (n+k), k² n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log² k) parallel steps and by a sequential algorithm inn(1+k ²)+p (n+k)+0 (k ³) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown. Dedicated to Professor S. Faedo on his 70th birthday Part of the results of this paper has been presented at the Oberwolfach Conference on Komplexitatstheorie, November 1983     

The Computational Complexity of Some Julia Sets

Numerous computer programs have been written to compute sets of points which approximate Julia sets [4]. Usually, no error estimations are added so that it remains unclear, how good such approximations are. Furthermore, high precision pictures are unreliable because of rounding errors, since the realizing computer programs use fixed length floating point numbers. Computable error estimation w.r.t. the Hausdorff metric d_H means that the set is recursive [10]. Many Julia sets J are recursive [11]. Recursive compact subsets of the Euclidean plane have a computable Turing machine time complexity [10]. In this paper we prove that the Julia set of a complex function f(z) = z² + c for c < 1/4 can be computed locally in time O(k²M(k)) (where M(k) is a time bound for multiplication of k-bit integers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide for a single point whether it belongs to a grid K_k (2^−k · )² such that d_H(K_k,J) ≤ = 2^−k.     

Minimum Cost Source Location Problems with Flow Requirements

In this paper, we consider source location problems and their generalizations with three connectivity requirements (arc-connectivity requirements λ and two kinds of vertex-connectivity requirements κ and ), where the source location problems are to find a minimum-cost set S⊆V in a given graph G=(V,A) with a capacity function u:A→ℝ₊ such that for each vertex v∈V, the connectivity from S to v (resp., from v to S) is at least a given demand d ⁻(v) (resp., d ⁺(v)). We show that the source location problem with edge-connectivity requirements in undirected networks is strongly NP-hard, which solves an open problem posed by Arata et al. (J. Algorithms 42: 54–68, 2002). Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of cln D for some constant c, unless every problem in NP has an O(N ^{log log N} )-time deterministic algorithm. Here D denotes the sum of given demands. We also devise (1+ln D)-approximation algorithms for all the extended source location problems if we have the integral capacity and demand functions. By the inapproximable results above, this implies that all the source location problems are Θ(ln ∑ _v∈V (d ⁺(v)+d ⁻(v)))-approximable. An extended abstract of this paper appeared in Sakashita et al. (Proceedings of LATIN 2006, Chile, LNCS, vol. 3887, pp. 769–780, March 2006).     

Time slot scheduling of compatible jobs

A version of weighted coloring of a graph is introduced which is motivated by some types of scheduling problems: each node v of a graph G corresponds to some operation to be processed (with a processing time w(v)), edges represent nonsimultaneity requirements (incompatibilities). We have to assign each operation to one time slot in such a way that in each time slot, all operations assigned to this slot are compatible; the length of a time slot will be the maximum of the processing times of its operations. The number k of time slots to be used has to be determined as well. So, we have to find a k-coloring = of G such that w(S ₁) + ⋅s +w(S _k ) is minimized where w(S _i ) = max {w(v) :v∊V}. Properties of optimal solutions are discussed, and complexity and approximability results are presented. Heuristic methods are given for establishing some of these results. The associated decision problems are shown to be NP-complete for bipartite graphs, for line-graphs of bipartite graphs, and for split graphs.