Duality in Padé-type approximation

Padé and Padé-type approximants are usually defined by replacing the function (1 − xt)⁻¹ by its Hermite (that is confluent) interpolation polynomial and then applying the functional c defined by c(xⁱ) = c_i where the c_i's are the coefficients of the series to be approximated. In this paper the functional d which, applied to (1 − xt)⁻¹, gives the same Padé or Padé-type approximant as before is studied. It can be considered as the dual of the interpolation operator applied to the functional c.     

Padé iteration method for regularization

In this study we present iterative regularization methods using rational approximations, in particular, Padé approximants, which work well for ill-posed problems. We prove that the (k, j)-Padé method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Padé method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Böckmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385–397] where this method was treated to solve ill-conditioned equation systems.     

A shortcut to asymptotics for orthogonal polynomials

We consider the asymptotic behavior of the ratios q_n+1(z)/q_n(z) of polynomials orthonormal with respect to some positive measure μ. Let the recurrence coefficients _n and β_n converge to 0 and , respectively. Then, q_n+1(z)/q_n(z) Φ(z),for n→∞ locally uniformly for , where Φ maps conformally onto the exterior of the unit disc (Nevai (1979)). We provide a new and direct proof for this and some related results due to Nevai, and apply it to convergence acceleration of diagonal Padé approximants.     

Recursive relations for block Hankel and Toeplitz systems part II: Dual recursions

The recursive relations given in Part I of this report can be interpreted as recursions for the denominators of matrix Padé approximants. In this part we shall give dual relations for the corresponding numerators and residuals.     

On the convergence of bivariate two-point Padé-type approximants

Some choices of denominators are given which ensure the geometrical convergence of certain convergence of bivariate two-point Padé-type approximants to functions being holomorphic on certain domains     

A sufficient and necessary condition for an OWA bag mapping having the self-identity

A mapping ƒ : _n=1^∞Iⁿ → I is called a bag mapping having the self-identity if for every (x₁,…,x_n) ε _i=1^∞Iⁿ we have (1) ƒ(x₁,…,x_n) = ƒ(x_i1,…,x_in) for any arrangement (i₁,…,i_n) of {1,…,n}; monotonic; (3) ƒ(x₁,…,x_n, ƒ(x₁,…,x_n)) = ƒ(x₁,…,x_n). Let {ω_i,n : I = 1,…,n;n = 1,2,…} be a family of non-negative real numbers satisfying Σ_i=1ⁿω_i,n = 1 for every n. Then one calls the mapping ƒ : _i=1^∞Iⁿ → I defined as follows an OWA bag mapping: for every (x₁,…,x_n) ε _i=1^∞Iⁿ, ƒ(x₁,…,x_n) = Σ_i=1ⁿω_i,ny_i, where y_i is the it largest element in the set {x₁,…,x_n}. In this paper, we give a sufficient and necessary condition for an OWA bag mapping having the self-identity.     

Optimization of two-point Padé-type approximants

In this paper, we first give characterization theorems for the best two-point Padé-type approximants (2PTAs) in the uniform norm. Secondly, we consider sequences of 2PTAs in a domain of the complex plane from the viewpoint of the asymptotic degree of convergence, and we also give conditions for geometric convergence.     

On a class of fractal functions with graph Hausdorff dimension 2

We prove that the graph of the continuous function

has Hausdorff dimension 2, where λ > 1, β >  > 1, (x) = 2x, 0  x  1/2, (−x) = (x) and (x + 1) = (x).     

Boundedness of semilinear Duffing equations with singularity

We prove the boundedness of all solutions for the equation x＂ ＋ V＇（x） = DxG（x,t）, where V（x） is of singular potential, i.e., limx→-1 Y（x） = ∞, and G（x, t） is bounded and periodic in t. We give sufficient conditions on V（x） and G（x, t） to ensure that all solutions are bounded.     

Odd subgraphs and matchings

Let G be a graph and f : V(G)→{1,3,5,…}. Then a subgraph H of G is called a (1,f)-odd subgraph if deg_H(x){1,3,…,f(x)} for all xV(H). If f(x)=1 for all xV(G), then a (1,f)-odd subgraph is nothing but a matching. A (1,f)-odd subgraph H of G is said to be maximum if G has no (1,f)-odd subgraph K such that |K|>|H|. We show that (1,f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1,f)-odd subgraph, which is similar to that for the order of a maximum matching.     

On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x₁+x₂−2x₃=c, which for each constant we denote by S₁(c), is the least integer, if it exists, such that every 2-coloring, Δ : [1,S₁(c)]→{0,1}, of the natural numbers admits a monochromatic solution to x₁+x₂−2x₃=c, and otherwise S₁(c)=∞. We determine the 2-color Rado number for the equation x₁+x₂−2x₃=c, when additional inequality restraints on the variables are added. In particular, the case where we require x₂<x₃<x₁, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value.     

The connectivity of a graph on uniform points on [0,1]

A random graph G_n(x) is constructed on independent random points U₁,…,U_n distributed uniformly on [0,1]^d, d1, in which two distinct such points are joined by an edge if the l_∞-distance between them is at most some prescribed value 0<x<1. The connectivity distance c_n, the smallest x for which G_n(x) is connected, is shown to satisfy

(1)

For d2, the random graph G_n(x) behaves like a d-dimensional version of the random graphs of Erdös and Rényi, despite the fact that its edges are not independent: c_n/d_n→1, a.s., as n→∞, where d_n is the largest nearest-neighbor link, the smallest x for which G_n(x) has no isolated vertices.     

Reconstruction of infinite locally finite connected graphs

Let G be an infinite locally finite connected graph. We study the reconstructibility of G in relation to the structure of its end set . We prove that an infinite locally finite connected graph G is reconstructible if there exists a finite family (Ω_i)_0i (n2) of pairwise finitely separable subsets of such that, for all x,y,x′,y′V(G) and every isomorphism f of G−{x,y} onto G−{x′,y′} there is a permutation π of {0,…,n−1} such that for 0i<n. From this theorem we deduce, as particular consequences, that G is reconstructible if it satisfies one of the following properties: (i) G contains no end-respecting subdivision of the dyadic tree and has at least two ends of maximal order; (ii) the set of thick ends or the one of thin ends of G is finite and of cardinality greater than one. We also prove that if almost all vertices of G are cutvertices, then G is reconstructible if it contains a free end or if it has at least a vertex which is not a cutvertex.     

Pancyclic orderings of in-tournaments

An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. In this paper, pancyclic orderings of a strong in-tournament D are investigated. This is a labeling, say x₁,x₂,…,x_n, of the vertex set of D such that D[{x₁,x₂,…,x_t}] is Hamiltonian for t=3,4,…,n. Moreover, we consider the related problem on weak pancyclic orderings, where the same holds for t4 and x₁ belongs to an arbitrary oriented 3-cycle. We present sharp lower bounds for the minimum indegree ensuring the existence of a pancyclic or a weak pancyclic ordering in strong in-tournaments.     

Existence results for doubly near resolvable (υ, 3, 2)-BIBDs

Let V be a set of υ elements. A (1, 2; 3, υ, 1)-frame F is a square array of side v which satisfies the following properties. We index the rows and columns of F with the elements of V, V={x₁,x₂,…,x_υ}. (1) Each cell is either empty or contains a 3-subset of V. (2) Cell (x_i, x_i) is empty for i=1, 2,…, υ. (3) Row x_i of F contains each element of V−{x_i} once and column x_i of F contains each element of V−{x_i} once. (4) The collection of blocks obtained from the nonempty cells of F is a (υ, 3, 2)-BIBD. A (1, 2; 3, υ, 1)-frame is a doubly near resolvable (υ, 3, 2)-BIBD. In this paper, we first present a survey of existence results on doubly near resolvable (υ, 3, 2)-BIBDs and (1, 2; 3, υ, 1)-frames. We then use frame constructions to provide a new infinite class of doubly near resolvable (υ, 3, 2)-BIBDs by constructing (1, 2; 3, υ, 1)-frames.     

Language-theoretic complexity of disjunctive sequences

A sequence over an alphabet ∑ is called disjunctive if it contains all possible finite strings over ∑ as its substrings. Disjunctive sequences have been recently studied in various contexts. They abound in both category and measure senses. In this paper we measure the complexity of a sequence x by the complexity of the language P(x) consisting of all prefixes of x. The languages P(x) associated to disjunctive sequences can be arbitrarily complex. We show that for some disjunctive numbers x the language P(x) is context-sensitive, but no language P(x) associated to a disjunctive number can be context-free. We also show that computing a disjunctive number x by rationals corresponding to an infinite subset of P(x) does not decrease the complexity of the procedure, i.e. if x is disjunctive, then P(x) does not have an infinite context-free subset. This result reinforces, in a way, Chaitin's thesis (1969) according to which perfect sets, i.e. sets for which there is no way to compute infinitely many of its members essentially better (simpler/quicker) than computing the whole set, do exist. Finally we prove the existence of the following language-theoretic complexity gap: There is no x ε ∑^w such that P(x) is context-free but not regular. If S(x), the set of all finite substrings of a sequence x ε ∑^w, is slender, then the set of all prefixes of x is regular, that is P(x) is regular if and only if S(x) is slender.     

All solutions of a class of difference equations are truncated periodic

We propose the difference equation x_n+1 = x_n − f(x_n−k) as a model for a single neuron with no internal decay, where f satisfies the McCulloch-Pitts nonlinearity. It is shown that every solution is truncated periodic with the minimal period 2(2l + 1) for some l ≥ 0 such that (k - l)/(2l + 1) is a nonnegative even integer. The potential application of our results to neural networks is obvious.     

路的字典积的邻和可区别边染色

图G的正常[k]-边染色σ是指颜色集合为[k]={1,2,…,k}的G的一个正常边染色.用w_σ（x）表示顶点x关联边的颜色之和,即w_σ（x）=∑_e∋x σ（e）,并称w_σ（x）关于σ的权.图G的k-邻和可区别边染色是指相邻顶点具有不同权的正常[k]-边染色,最小的k值称为G的邻和可区别边色数,记为χ'_∑（G）.现得到了路P_n与简单连通图H的字典积P_n[H]的邻和可区别边色数的精确值,其中H分别为正则第一类图、路、完全图的补图.     

A note on the determinant formulas computation of generalized inverse matrix Padé approximation

In this paper, the computation of two special determinants which appear in the construction of a generalized inverse matrix Padé approximation of type [n/2k] (described in [Linear Algebra Appl. 322 (2001) 141]) for a given power series is investigated. Here a common computational approach of determinant can not be used. The main tool to be used to do the two special determinants is the well-known Schur complement theorem.     

A note on G-intersecting families

Consider a graph G and a k-uniform hypergraph on common vertex set [n]. We say that is G-intersecting if for every pair of edges in there are vertices xX and yY such that x=y or x and y are joined by an edge in G. This notion was introduced by Bohman, Frieze, Ruszinkó and Thoma who proved a natural generalization of the Erd s–Ko–Rado Theorem for G-intersecting k-uniform hypergraphs for G sparse and k=O(n^1/4). In this note, we extend this result to .