The conjugating map for commutative groups of circle diffeomorphisms

For a single aperiodic, orientation preserving diffeomorphism on the circle, all known local results on the differentiability of the conjugating map are also known to be global results. We show that this does not hold for commutative groups of diffeomorphisms. Given a set of rotation numbers, we construct commuting diffeomorphisms inC ^2-ε for all ε>0 with these rotation numbers that are not conjugate to rotations. On the other hand, we prove that for a commutative subgroupF ⊂C ^1+β, 0<β<1, containing diffeomorphisms that are perturbations of rotations, a conjugating maph exists as long as the rotation numbers of this subset jointly satisfy a Diophantine condition.     

Periodic points and rotation numbers for area preserving diffeomorphisms of the plane

Letf be an orientation preserving diffeomorphism ofR ² which preserves area. We prove the existence of infinitely many periodic points with distinct rotation numbers around a fixed point off, provided only thatf has two fixed points whose infinitesimal rotation numbers are not both 0. We also show that if a fixed pointz off is enclosed in a “simple heteroclinic cycle” and has a non-zero infinitesimal rotation numberr, then for every non-zero rational numberp/q in an interval with endpoints 0 andr, there is a periodic orbit inside the heteroclinic cycle with rotation numberp/q aroundz.     

Degree-Continuous Graphs

A graph G is degree-continuous if the degrees of every two adjacent vertices of G differ by at most 1. A finite nonempty set S of integers is convex if k S for every integer k with min(S)kmax(S). It is shown that for all integers r > 0 and s 0 and a convex set S with min(S) = r and max(S) = r+s, there exists a connected degree-continuous graph G with the degree set S and diameter 2s+2. The minimum order of a degree-continuous graph with a prescribed degree set is studied. Furthermore, it is shown that for every graph G and convex set S of positive integers containing the integer 2, there exists a connected degree-continuous graph H with the degree set S and containing G as an induced subgraph if and only if max(S)(G) and G contains no r-regular component where r = max(S).     

Crossing numbers of sequences of graphs II: Planar tiles

We describe a method of creating an infinite family of crossing‐critical graphs from a single small planar map, the tile, by gluing together many copies of the tile together in a circular fashion. This method yields all known infinite families of k‐crossing‐critical graphs. Furthermore, the method yields new infinite families, which extend from (4,6) to (3.5,6) the interval of rationals r for which there is, for some k, an infinite sequence of k‐crossing‐critical graphs all having average degree r. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 332–341, 2003     

Characterizations of trees with equal domination parameters

Let G = (V, E) be a graph. A set S ⊆ V is a restrained dominating set, if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γ_r(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a weak dominating set of G if, for every u in V − S, there exists a v ∈ S such that uv ∈ E and deg u ≥ deg v. The weak domination number of G, denoted by γ_w(G), is the minimum cardinality of a weak dominating set of G. In this article, we provide a constructive characterization of those trees with equal independent domination and restrained domination numbers. A constructive characterization of those trees with equal independent domination and weak domination numbers is also obtained. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 142–153, 2000     

Genus 2 Belyĭ surfaces with a unicellular uniform dessin

A bicoloured graph embedded in a compact oriented surface and dividing it into a union of simply connected components (faces) is known as a dessin d’enfant. It is well known that such a graph determines a complex structure on the underlying topological surface, but a given compact Riemann surface may correspond to different dessins. In this paper we deal with all unicellular (one-faced) uniform dessins of genus 2 and their underlying Riemann surfaces. A dessin is called uniform if white vertices, black vertices and faces have constant degree, say p, q and r respectively. A uniform dessin d’enfant of type (p, q, r) on a given surface S corresponds to the inclusion of the torsion-free Fuchsian group K uniformizing S inside a triangle group Δ(p, q, r). Hence the existence of different uniform dessins on S is related to the possible inclusion of K in different triangle groups. The main result of the paper states that two unicellular uniform dessins belonging to the same genus 2 surface must necessarily be isomorphic or obtained by renormalisation. The problem is approached through the study of the face-centers of the dessins. The displacement of such a point by the elements of K must belong to a prescribed discrete set of (hyperbolic) distances determined by the signature (p, q, r). Therefore looking for face-centers amounts to finding points correctly displaced by every element of K.     

Uniform Distributions on the Natural Numbers

We compare the following three notions of uniformity for a finitely additive probability measure on the set of natural numbers: that it extend limiting relative frequency, that it be shift-invariant, and that it map every residue class mod m to 1/m. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set L of extensions of limiting relative frequency is a proper subset of the set S of shift-invariant measures and that S is a proper subset of the set R of measures which map residue classes uniformly. Moreover, we show that there are subsets G of ℕ for which the range of possible values μ(G) for μ∈L is properly contained in the set of values obtained when μ ranges over S, and that there are subsets G which distinguish S and R analogously.     

Recognition by the Set of Element Orders of Symmetric Groups of Degree r and r+1 for Prime r

We prove that the symmetric group S _r of prime degree r17 is recognizable by its element order set. A test for recognizability is obtained for the symmetric group S _r+1 with r17 prime.     

Set estimation under convexity type assumptions

The problem of estimating a set S from a random sample of points taken within S is considered. It is assumed that S is r-convex, which means that a ball of radius r can go around from outside the set boundary. Under this assumption, the r-convex hull of the sample is a natural estimator of S. We obtain convergence rates for this estimator under both the distance in measure and the Hausdorff metric between sets. It is also proved that the boundary of the estimator consistently estimates the boundary of S, in Hausdorff's sense.     

Involutions on surfaces with pg?=?q?=?0 and K2?=?3

We study surfaces of general type S with p _g  = 0 and K ² = 3 having an involution i such that the bicanonical map of S is not composed with i. It is shown that, if S/i is not rational, then S/i is birational to an Enriques surface or it has Kodaira dimension 1 and the possibilities for the ramification divisor of the covering map S → S/i are described. We also show that these two cases do occur, providing an example. In this example S has a hyperelliptic fibration of genus 3 and the bicanonical map of S is of degree 2 onto a rational surface.     

Exact Pairs for Abstract Bounded Reducibilities

In an attempt to give a unified account of common properties of various resource bounded reducibilities, we introduce conditions on a binary relation ≤_r between subsets of the natural numbers, where ≤_r is meant as a resource bounded reducibility. The conditions are a formalization of basic features shared by most resource bounded reducibilities which can be found in the literature. As our main technical result, we show that these conditions imply a result about exact pairs which has been previously shown by Ambos-Spies [2] in a setting of polynomial time bounds: given some recursively presentable ≤_r-ideal I and some recursive ≤_r-hard set B for I which is not contained in I, there is some recursive set C, where B and C are an exact pair for I, that is, I is equal to the intersection of the lower ≤_r-cones of B and C, where C is not in I. In particular, if the relation ≤_r is in addition transitive and there are least sets, then every recursive set which is not in the least degree is half of a minimal pair of recursive sets.     

The best quadrature based on given Hermite information for the Sobolev class KWr[a, b]

As usual, denote by KW ^r[a, b] the Sobolev class consisting of every function whose (r − 1)th derivative is absolutely continuous on the interval [a, b] and rth derivative is bounded by K a.e. in [a, b]. For a function f ∈ KW ^r [a, b], its values and derivatives up to r − 1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KW ^r [a, b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KW ^r [a, b] is also obtained.     

A note on a quadratic rational map with two Siegel disks

Suppose f（z） is a quadratic rational map with two Siegel disks. If the rotation numbers of the Siegel disks are both of bounded type, the Hausdorff dimension of the Julia set satisfies Dim （J（f））〈2.     

Every incomplete computably enumerable truth-table degree is branching

If r is a reducibility between sets of numbers, a natural question to ask about the structure ? _r of the r-degrees containing computably enumerable sets is whether every element not equal to the greatest one is branching (i.e., the meet of two elements strictly above it). For the commonly studied reducibilities, the answer to this question is known except for the case of truth-table (tt) reducibility. In this paper, we answer the question in the tt case by showing that every tt-incomplete computably enumerable truth-table degree a is branching in ? _tt . The fact that every Turing-incomplete computably enumerable truth-table degree is branching has been known for some time. This fact can be shown using a technique of Ambos-Spies and, as noticed by Nies, also follows from a relativization of a result of Degtev. We give a proof here using the Ambos-Spies technique because it has not yet appeared in the literature. The proof uses an infinite injury argument. Our main result is the proof when a is Turing-complete but tt-incomplete. Here we are able to exploit the Turing-completeness of a in a novel way to give a finite injury proof. Received: 22 January 1999 / Revised version: 12 July 1999 / Published online: 21 December 2000     

The Number of Permutations with Exactlyr132-Subsequences IsP-Recursive in the Size!

Proving a first nontrivial instance of a conjecture of Noonan and Zeilberger we show that the numberS_r(n) of permutations of lengthncontaining exactlyrsubsequences of type 132 is aP-recursive function ofn. We show that this remains true even if we impose some restrictions on the permutations. We also show the stronger statement that the ordinary generating functionG_r(x) ofS_r(n) is algebraic, in fact, it is rational in the variablesxand . We use this information to show that the degree of the polynomial recursion satisfied byS_r(n) isr.     

Integral Schur algebras forGL 2

The structure of Schur algebrasS(2,r) over the integral domainZ is intensively studied from the quasi-hereditary algebra point of view. We introduce certain new bases forS(2,r) and show that the Schur algebraS(2,r) modulo any ideal in the defining sequence is still such a Schur algebra of lower degree inr. A Wedderburn-Artin decomposition ofS _K (2,r) over a fieldK of characteristic 0 is described. Finally, we investigate the extension groups between two Weyl modules and classify the indecomposable Weyl-filtered modules for the Schur algebrasS _Zp(2,r) withr<p ² . Research supported by ARC Large Grant L20.24210     

Extremal graphs without a semi‐topological wheel

For 2≤r∈?, let S_r denote the class of graphs consisting of subdivisions of the wheel graph with r spokes in which the spoke edges are left undivided. Let ex(n, S_r) denote the maximum number of edges of a graph containing no S_r‐subgraph, and let Ex(n, S_r) denote the set of all n‐vertex graphs containing no S_r‐subgraph that are of size ex(n, S_r). In this paper, a conjecture is put forth stating that for r≥3 and n≥2r + 1, ex(n, S_r) = (r ? 1)n ? ?(r ? 1)(r ? 3/2)? and for r≥4, Ex(n, S_r) consists of a single graph which is the graph obtained from K_{r ? 1, n ? r + 1} by adding a maximum matching to the color class of cardinality r ? 1. A previous result of C. Thomassen [A minimal condition implying a special K₄‐subdivision, Archiv Math 25 (1974), 210–215] implies that this conjecture is true for r = 3. In this paper it is shown to hold for r = 4. © 2011 Wiley Periodicals, Inc. J Graph Theory 68:326‐339, 2011     

Propagation of round-off error in a model of quadratic rational rotations

We investigate the propagation of round-off error for a discrete map modeling a one-dimensional linear oscillator viewed stroboscopically in phase space, with uniform, non-dissipative round-off. The probability P(r,t) of a net displacement r during t time steps can be reduced, essentially, to a weighted sum over contributions from a small number of infinite scaling sequences of periodic orbits. We show that the successive members of each scaling sequence can be built up by application of a set of substitution rules. This implies recursion relations, not only for the geometry of the orbits, but also for P(r,t) and its moments, allowing these quantities to be calculated exactly as algebraic numbers. For asymptotically large t, the moments have power-law increase, modulated by log-periodic or (in one particularly interesting case) log-quasi-periodic oscillations.