Motivated by the result that an ‘approximate’ evaluation of the Jones polynomial of a braid at a 5th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalizations. However, we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.

JumbleG is a Maker–Breaker game. Maker and Breaker take turns in choosing edges from the complete graph $K_n$. Maker's aim is to choose what we call an $\epsilon$-regular graph (that is, the minimum degree is at least $(\frac12-\epsilon) n$ and, for every pair of disjoint subsets $S,T\subset V$ of cardinalities at least $\epsilon n$, the number of edges $e(S,T)$ between $S$ and $T$ satisfies $\bigl|\frac{e(S,T)}{|S|\,|T|}-\frac12\bigr|\leq \epsilon$.) In this paper we show that Maker can create an $\epsilon$-regular graph, for $\epsilon\geq 2(\log n/n)^{1/3}$. We also consider a similar game, JumbleG2, where Maker's aim is to create a graph with minimum degree at least $\bigl(\frac12-\epsilon\bigr)n$ and maximum co-degree at most $\bigl(\frac14+\epsilon\bigr)n$, and show that Maker has a winning strategy for $\epsilon> 3 (\log n/n)^{1/2}$. Thus, in both games Maker can create a pseudo-random graph of density $\frac12$. This guarantees Maker's win in several other positional games, also discussed here.

A set of n triangles sharing a common edge is called a book with n pages and is denoted by $B_{n}$. It is known that the Ramsey number $r ( B_{n} ) $ satisfies $r ( B_{n} ) = ( 4+o ( 1 ) ) n.$ We show that every red–blue edge colouring of $K_{ \lfloor ( 4-\varepsilon ) n \rfloor }$ with no monochromatic $B_{n}$ exhibits quasi-random properties when $\varepsilon$ tends to 0. This implies that there is a constant $c>0$ such that for every red–blue edge colouring of $K_{r ( B_{n} ) }$ there is a monochromatic $B_{n}$ whose vertices span at least $ \lfloor cn^{2} \rfloor $ edges of the same colour as the book.

As an application we find the Ramsey number for a class of graphs.

Let ${\cal H}$ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= \{x: abx\in E({\cal H})\} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ${\cal H}$. This is equivalent to ${\cal H}$ not containing a copy of $\mathbb{F} :=\{ abx$, $aby$, $abz$, $xyz\}$.

In this paper we prove an analogue of the Andrásfai–Erdös–Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any $\mathbb{F}$-free 3-graph with minimum degree exceeding $(\frac{4}{9}-\frac{1}{125})\binom{n}{2}$ is bipartite, (for $n> n_0$), i.e., the vertices of ${\cal H}$ can be split into two parts so that every triple meets both parts.

This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that

\[\ex(n,\mathbb{F}_{3,2})=\max\limits_{\alpha}(n-\alpha)\left({\alpha\atop2}\right)\].

Here the right-hand side is $\frac{4}{9}\binom{n}{3}+O(n^2)$. Moreover $e({\cal H})={\rm ex}(n,\mathbb{F})$ is possible only if $V({\cal H})$ can be partitioned into two sets $A$ and $B$ so that each triple of ${\cal H}$ intersects $A$ in exactly two vertices and $B$ in one.

In 1978, Bollobás and Eldridge [5] made the following two conjectures.

1. (C1) There exists an absolute constant $c>0$ such that, if k is a positive integer and $G_1$ and $G_2$ are graphs of order n such that $\Delta(G_1),\Delta(G_2)\leq n-k$ and $e(G_1),e(G_2)\leq ck n$, then the graphs $G_1$ and $G_2$ pack.

2. (C2) For all $0<\alpha<1/2$ and $0<c<\sqrt{1/8}$, there exists an $n_0=n_0(\alpha,c)$ such that, if $G_1$ and $G_2$ are graphs of order $n>n_0$ satisfying $e(G_1)\leq \alpha n$ and $e(G_2)\leq c\sqrt{n^3/ \alpha}$, then the graphs $G_1$ and $G_2$ pack.

We extend a result by Füredi and Komlós and show that the first eigenvalue of a random graph is asymptotically normal, both for $G_{n,p}$ and $G_{n,m}$, provided $np\geq n^\delta$ or $m/n\geq n^\delta$ for some $\delta>0$. The asymptotic variance is of order $p$ for $G_{n,p}$, and $n^{-1}$ for $G_{n,m}$. This gives a (partial) solution to a problem raised by Krivelevich and Sudakov.

The formula for the asymptotic mean involves a mysterious power series.

For a graph G, let f(G) denote the maximum number of edges in a cut of G. For an integer m and for a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as G ranges over all graphs on m edges that contain no copy of H. In this paper we study this function for various graphs H. In particular we show that for any graph H obtained by connecting a single vertex to all vertices of a fixed nontrivial forest, there is a $c(H) >0$ such that $f(m,H) \geq \frac{m}{2} + c(H) m^{4/5}$, and that this is tight up to the value of $c(H)$. We also prove that for any even cycle $C_{2k}$ there is a $c(k)>0$ such that $f(m,C_{2k}) \geq \frac{m}{2} + c(k) m^{(2k+1)/(2k+2)}$, and that this is tight, up to the value of $c(k)$, for $2k\in \{4,6,10\}$. The proofs combine combinatorial, probabilistic and spectral techniques.

We consider the two-dimensional autonomous systems of differential equations of the formwhere P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

We give a simple explicit description of the norm in the complex interpolation space (Cp[Lp(M)], Rp[Lp(M)])θ for any von Neumann algebra M and any 1 ≤ p ≤ ∞.

Maz'ja and Sinnamon proved a characterization of the boundedness of the Hardy operator from Lp(v) into Lq(w) in the case 0 < q < p, 1 < p < ∞. We present here a new simple proof of the sufficiency part of that result.

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

In this article we investigate the regularizing properties of generalized continua of Cosserat micropolar type in the elasto-plastic case. We propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects. It is shown that the new model is thermodynamically admissible and allows for a unique, global-in-time solution of the corresponding rate-independent initial–boundary-value problem. The method of choice is the Yosida approximation and a passage to the limit.

Maz'ja and Sinnamon proved a characterization of the boundedness of the Hardy operator from Lp(v) into Lq(w) in the case 0 < q < p, 1 < p < ∞. We present here a new simple proof of the sufficiency part of that result.

In this article we investigate the regularizing properties of generalized continua of Cosserat micropolar type in the elasto-plastic case. We propose an extension of classical infinitesimal elasto-plasticity to include consistently non-dissipative micropolar effects. It is shown that the new model is thermodynamically admissible and allows for a unique, global-in-time solution of the corresponding rate-independent initial–boundary-value problem. The method of choice is the Yosida approximation and a passage to the limit.

This note extends the results in ‘Optimal coercivity inequalities in W1,p(Ω)’ (G. Auchmuty, Proc. R. Soc. Edinb. A 135, 915–933.) describing the dependence of the optimal constant in the p-version of Friedrichs' inequality on the boundary integral term. In particular, it is shown that this constant is continuous, increasing, concave and increases to the optimal constant for the Dirichlet problem as s → ∞.

In this work we consider the Cauchy problem associated with dissipative perturbations of infinite-dimensional Hamiltonian systems. we describe abstract conditions under which the problem is locally and globally well posed. Moreover, we establish the existence of global attractor. Finally, we present several applications of the theory.

This paper describes the characterization of optimal constants for some coercivity inequalities in W1,p(Ω), 1 < p < ∞. A general result involving inequalities of p-homogeneous forms on a reflexive Banach space is first proved. The constants are shown to be the least eigenvalues of certain eigenproblems with equality holding for the corresponding eigenfunctions. This result is applied to three different classes of coercivity results on W1,p(Ω). The inequalities include very general versions of the Friedrichs and Poincaré inequalities. Scaling laws for the inequalities are also described.

In this paper, we consider a ring of neurons with self-feedback and delays. As a result of our approach based on global bifurcation theorems of delay differential equations coupled with representation theory of Lie groups, the coexistence of its asynchronous periodic solutions (i.e. mirror-reflecting waves, standing waves and discrete waves), bifurcated simultaneously from the trivial solution at some critical values of the delay, will be established for delay not only near to but also far away from the critical values. Therefore, we can obtain wave solutions of large amplitudes. In addition, we consider the coincidence of these periodic solutions.

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.