The system $\dot{X}=TX+Q(X)$ (in $\mathbb{R}^{n}$), where $T$ is linear and $Q$ is quadratic, is considered via commutative algebras. The case of the linearized system having a centre manifold spanned on vectors $E_{1}$, $E_{2}$ (and $TE_{1}=\omega E_{2}$, $TE_{2}=-\omega E_{1}$) is studied. It is shown that for $\Span(E_{1},E_{2})$ being a subalgebra (of the algebra corresponding to the form $Q(X)$), the system is stable. Necessary and sufficient conditions are given for stability of the system in the case where $\mathrm{span}(E_{1},E_{2})$ is not a subalgebra.

AMS 2000 Mathematics subject classification: Primary 34A34; 34C15; 34C27; 34C35. Secondary 13M99

Let $G$ be a periodic residually finite group containing a nilpotent subgroup $A$ such that $C_G(A)$ is finite. We show that if $\langle A,A^g\rangle$ is finite for any $g\in G$, then $G$ is locally finite.

AMS 2000 Mathematics subject classification: Primary 20F50

We prove the following.

(1) The inequalities

$$ \biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!a}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!a}\leq 2\leq\biggl(2-\frac{1}{\varGamma(x)}\biggr)^{\!b}+\biggl(2-\frac{1}{\varGamma(1/x)}\biggr)^{\!b} $$

hold for all $x>0$ if and only if

$$ -1.204\,64\ldots=2+\frac{1}{\gamma}-\frac{1}{6}\biggl(\frac{\pi}{\gamma}\biggr)^{\!2}\leq a\leq0\leq b. $$

(2) For all real numbers $x\in(0,1]$ we have

$$ x^{\alpha}\leq\frac{1}{2}\biggl(\frac{1}{\varGamma(x)}+\frac{1}{\varGamma(1/x)}\biggr)\leq x^{\beta}, $$

with the best possible constants

$$ \alpha=1.321\,76\dots\link{and}\beta=0. $$

These theorems extend and complement a result of Gautschi (from 1974), who proved that for all $x>0$ the harmonic mean of $\varGamma(x)$ and $\varGamma(1/x)$ is greater than or equal to $1$.

AMS 2000 Mathematics subject classification: Primary 33B15; 26D15

This paper gives a number which is used to determine the component number of links from their associated planar graphs. In particular, we use this number to determine the component numbers of links whose associated planar graphs are fans, wheels and 2-sums of graphs.

AMS 2000 Mathematics subject classification: Primary 05C10

We study a system of ordinary differential equations linked by parameters and subject to boundary conditions depending on parameters. We assume certain definiteness conditions on the coefficient functions and on the boundary conditions that yield, in the corresponding abstract setting, a right-definite case. We give results on location of the eigenvalues and oscillation of the eigenfunctions.

AMS 2000 Mathematics subject classification: Primary 34B08; 34B24

Felix and Murillo introduced the group $\mathrm{Aut}_\varOmega(X)$ of self-maps $f$ of $X$, which satisfy $\sOm f=1_{\varOmega X}$, and proved that the group is nilpotent with the order of nilpotency bounded by the Lusternik–Schnirelmann category of $X$. In this paper we construct a spectral sequence converging to the group $\mathrm{Aut}_\varOmega(X)$ and derive several interesting consequences.

AMS 2000 Mathematics subject classification: Primary 55P42

A weak form of faithfulness, depending on Green’s equivalence $\mathcal{D}$, is introduced for a ring $R$ graded by a semigroup $S$. Suppose that $R$ satisfies this condition. It is shown that if $e$ and $f$ are $\mathcal{D}$-equivalent idempotents of $S$ and $R_e$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_f$ is semiprime (respectively, prime, semiprimitive, right primitive). In addition, it is shown that if $G$ and $H$ are maximal subgroups of $S$ lying in the same $\mathcal{D}$-class and $R_G$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_H$ is semiprime (respectively, prime, semiprimitive, right primitive).

AMS 2000 Mathematics subject classification: Primary 16W50. Secondary 20M25

Precise bounds are given for the quantity

$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$

where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.

If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.

If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.

AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

Given a graph G on n vertices with average degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that, if the edge connectivity k(G) satisfies k(G) [Gt] d/log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

We study a model motivated by the minesweeper game. In this model one starts with the percolation of mines on the lattice ℤd, and then tries to find an infinite path of mine-free sites. At every recovery of a free site, the player is given some information on the sites adjacent to the current site. We compare the parameter values for which there exists a strategy such that the process survives to the critical parameter of ordinary percolation. We then derive improved bounds for these values for the same process, when the player has some complexity restrictions in computing his moves. Finally, we discuss some monotonicity issues which arise naturally for this model.

This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [12, 11] and another studied by Tsilevich [30, 31] and Mayer-Wolf, Zeitouni and Zerner [21]. The invariance under this split-and-merge transformation of the interval partition generated by a suitable Poisson process yields a simple proof of the recent result of [21] that a Poisson–Dirichlet distribution is invariant for a closely related fragmentation–coagulation process. Uniqueness and convergence to the invariant measure are established for the split-and-merge transformation of interval partitions, but the corresponding problems for the fragmentation–coagulation process remain open.

We present improved lower and upper bounds for the time constant of first-passage percolation on the square lattice. For the case of lower bounds, a new method, using the idea of a transition matrix, has been used. Numerical results for the exponential and uniform distributions are presented. A simulation study is included, which results in new estimates and improved upper confidence limits for the time constants.

For two stochastically dependent random variables X and Y taking values in {0,…, m−1}, we study the distribution of the random residue U = XY mod m. Our main result is an upper bound for the distance Δm = supx∈[0,1] [mid ] P(U/m [les ] x)−x[mid ]. For independent and uniformly distributed X and Y, the exact distribution of U is derived and shown to be stochastically smaller than the uniform distribution on {0,…, m−1}. Moreover, in this case Δm is given explicitly.

Let q be an integer with q [ges ] 2. We give a new proof of a result of Erdös and Turán determining the proportion of elements of the finite symmetric group Sn having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group An. In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.

We apply these results to estimate the proportion of elements of order 2f in Sn, and of order 3f in An and Sn, where gcd(2, f) = 1, and gcd(3, f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in An or Sn, given as a black-box group with an order oracle, is discussed.

We consider k-uniform set systems over a universe of size n such that the size of each pairwise intersection of sets lies in one of s residue classes mod q, but k does not lie in any of these s classes. A celebrated theorem of Frankl and Wilson [8] states that any such set system has size at most (ns) when q is prime. In a remarkable recent paper, Grolmusz [9] constructed set systems of superpolynomial size Ω(exp(c log2n/log log n)) when q = 6. We give a new, simpler construction achieving a slightly improved bound. Our construction combines a technique of Frankl [6] of ‘applying polynomials to set systems’ with Grolmusz's idea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also extend Frankl's original argument to arbitrary prime-power moduli: for any ε > 0, we construct systems of size ns+g(s), where g(s) = Ω(s1−ε). Our work overlaps with a very recent technical report by Grolmusz [10].

Let G be a connected graph that is 2-cell embedded in a surface S, and let G* be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G*.

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

In this paper we will study solution pairs $(u,D)$ of the minimal surface equation defined over an unbounded domain $D$in $R^2$, with $u=0$ on $\partial D$. It is well known that there are severe limitations on the geometry of $D$; forexample $D$ cannot be contained in any proper wedge (angle less than $\pi$). Under the assumption of sublinear growthin a suitably strong sense, we show that if $u$ has order of growth $\alpha$ in the sense of complex variables, thenthe ‘asymptototic angle’ of $D$ must be at least $\pi/\alpha$. In particular, there are at most two such solution pairsdefined over disjoint domains. If $\alpha<1$ then $u$ cannot change sign and there is no other disjoint solution pair.This result is sharp as can be seen by a suitable piece of Enneper’s surface which has order$\alpha=\tfrac{2}{3}$ and asymptotic angle $\tfrac{3}{2}\pi$.

AMS 2000 Mathematics subject classification: Primary 35J60; 53A10