The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3$ are also considered.

It is proved that the 3-part of the class number of a quadratic field $\mathbb{Q}(\sqrt{D})$ is $O(|D|^{55/112 + \ep})$ in general and $O(|D|^{5/12+\ep})$ if $|D|$ has a divisor of size $|D|^{5/6}$. These bounds follow as results of nontrivial estimates for the number of solutions to the congruence $x^a \,{\con}\, y^b$ modulo $q$ in the ranges $x \,{\leqslant}\,X$ and $y\,{\leqslant}\, Y$, where $a,b$ are nonzero integers and $q$ is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over $\Q$ with conductor $N$ is $O(N^{55/112 + \ep})$ in general and $O(N^{5/12 + \ep})$ if $N$ has a divisor of size $N^{5/6}$. These results are the first improvements to the trivial bound $O(|D|^{1/2 + \ep})$ and the resulting bound $O(N^{1/2 + \ep})$ for the 3-part and the number of elliptic curves, respectively.

Let g [ratio ] M → N be a quasiconformal harmonic diffeomorphism between noncompact Riemann surfaces M and N. In this paper we study the relation between the map g and the complex structures given on M and N. In the case when M and N are of finite analytic type we derive a precise estimate which relates the map g and the Teichmüller distance between complex structures given on M and N. As a corollary we derive a result that every two quasiconformally related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. In addition, we study the question of whether every quasisymmetric selfmap of the unit circle has a quasiconformal harmonic extension to the unit disk. We give a partial answer to this problem. We show the existence of the harmonic quasiconformal extensions for a large class of quasisymmetric maps. In particular it is proved that all symmetric selfmaps of the unit circle have a unique quasiconformal harmonic extension to the unit disk.

It is shown that for every non-negative integer $n$, there is a real $n$-dimensional family of minimal Lagrangian tori in $\mathbb{CP}^2$, and hence of special Lagrangian cones in $\mathbb{C}^3$ whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curve) data.

It is shown that if a plane of PG(3,q), q even, meets an ovoid in a conic, then the ovoid must be an elliptic quadric. This is proved by using the generalized quadrangles T2([Cscr ]) ([Cscr ] a conic), W(q) and the isomorphism between them to show that every secant plane section of the ovoid must be a conic. The result then follows from a well-known theorem of Barlotti.

It is shown that if S is an aperiodic random walk on the integers, S* is the Markov chain that arises when S is killed when it leaves the non-negative integers, and H+ is the renewal process of weak increasing ladder heights in S, then there is a 1[ratio ]1 correspondence between functions which are non-negative and superregular for S* and H+. This allows all the regular functions for S* to be described, and thus a result due to Spitzer to be completed for the recurrent case. This result is then applied to give a ratio limit theorem for Px(τ* =n)/P0{τ*=n}, where τ* is the lifetime of S*, in the case when S drifts to −∞, and the right-hand tail of its step distribution is ‘locally sub-exponential’.

Let Δ be a finite quiver, that is, a finite directed graph, k be a commutative ring, and kΔ be the semigroup ring of paths in Δ. In this paper we compute the Hochschild homology of the following classes of algebras:

(1) kΔ/[mfr ]n, where [mfr ] is the arrow ideal;

(2) kΔ/I where I is an ideal generated by quadratic monomials.

In Section 2 we establish the notation, and recall a projective resolution of kΔ0, the degree 0 part in kΔ, over a monomial algebra which is due to Anick and Green. This resolution is then used in Section 3 as the building block in the construction of a resolution of a truncated or quadratic monomial algebra A, over its enveloping algebra (in the Hochschild sense), Ae. The fact that a monomial algebra possesses a fine grading then enables us to compute the homology.

The results obtained extend results by Liu and Zhang [3] and Geller, Reid and Weibel [2].

Jordan superalgebras defined by brackets on associative commutative superalgebras are studied. It is proved that any such superalgebra is imbedded into a superalgebra defined by Poisson brackets. In particular, all Jordan superalgebras of brackets are i-special. The speciality of these superalgebras is also examined, and it is proved, in particular, that the Cheng–Kac superalgebra is special.

In this paper we consider an aperiodic integer-valued random walk $S$ and a process $S^{\ast }$ that is a harmonic transform of $S$ killed when it first enters the negative half; informally, $S^{\ast }$ is ‘$S$ conditioned to stay non-negative’. If $S$ is in the domain of attraction of the standard normal law, without centring, a suitably normed and linearly interpolated version of $S$ converges weakly to standard Brownian motion, and our main result is that under the same assumptions a corresponding statement holds for $S^{\ast }$, the limit of course being the three-dimensional Bessel process. As this process can be thought of as Brownian motion conditioned to stay non-negative, in essence our result shows that the interchange of the two limit operations is valid. We also establish some related results, including a local limit theorem for $S^{\ast }$, and a bivariate renewal theorem for the ladder time and height process, which may be of independent interest.

Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an additional row and column. The extra row and column in the magic square correspond to the sextonions. This is a six-dimensional subalgebra of the split octonions which contains the split quaternions.

Let us consider the boundary value problem

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where Ω ⊂ ℝN is a bounded domain with smooth boundary (for example, such that certain Sobolev imbedding theorems hold). Let

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Then, if ϕ(s) = [mid ]s[mid ]p−1s, p > 1, problem (1) is fairly well understood and a great variety of existence results are available. These results are usually obtained using variational methods, monotone operator methods or fixed point and degree theory arguments in the Sobolev space W1,0p(Ω). If, on the other hand, we assume that ϕ is an odd nondecreasing function such that

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and

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then a Sobolev space setting for the problem is not appropriate and very general results are rather sparse. The first general existence results using the theory of monotone operators in Orlicz–Sobolev spaces were obtained in [5] and in [9, 10]. Other recent work that puts the problem into this framework is contained in [2] and [8].

It is in the spirit of these latter papers that we pursue the study of problem (1) and we assume that F[ratio ]Ω×ℝ→ℝ is a Carathéodory function that satisfies certain growth conditions to be specified later.

We note here that the problems to be studied, when formulated as operator equations, lead to the use of the topological degree for multivalued maps (cf. [4, 16]).

We shall see that a natural way of formulating the boundary value problem will be a variational inequality formulation of the problem in a suitable Orlicz–Sobolev space. In order to do this we shall have need of some facts about Orlicz–Sobolev spaces which we shall give now.

In 1998, the second author of this paper raised the problem of classifying the irreducible characters of Sn of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work provides a complete solution. In this article we first classify all the irreducible characters of Sn of prime power degree (Theorem 2.4), and then we deduce the corresponding classification for the alternating groups (Theorem 5.1), thus providing the answer for one of the two remaining families in Zalesskii's problem. This classification has another application in group theory. With it, we are able to answer, for alternating groups, a question of Huppert: which simple groups G have the property that there is a prime p for which G has an irreducible character of p-power degree > 1 and all of the irreducible characters of G have degrees that are relatively prime to p or are powers of p?

The case of the double covers of the symmetric and alternating groups will be dealt with in a forthcoming paper; in particular, this completes the answer to Zalesskii's problem.

The paper is organized as follows. In Section 2, some results on hook lengths in partitions are proved. These results lead to an algorithm which allows us to show that every irreducible representation of Sn with prime power degree is labelled by a partition having a large hook. In Section 3, we obtain a new result concerning the prime factors of consecutive integers (Theorem 3.4). In Section 4 we prove Theorem 2.4, the main result. To do so, we combine the algorithm above with Theorem 3.4 and work of Rasala on minimal degrees. This implies Theorem 2.4 for large n. To complete the proof, we check that the algorithm terminates appropriately for small n (that is, those n [les ] 9.25 · 108) with the aid of a computer. In the last section we derive the classification of irreducible characters of An of prime power degree, and we solve Huppert's question for alternating groups.

The existence of $2\pi$-periodic solutions of the second-order differential equation \[ x''+f(x)x'+ax^+-bx^-+g(x)=p(t), \qquad n\in \mathbb{N},\] where $a, b$ satisfy $1/\sqrt{a}+1/\sqrt{b}=2/n$ and $p(t)=p(t+2\pi)$, $t\in \mathbb{R}$, is examined. Assume that limits $\lim_{x\to\pm\infty}F(x)=F(\pm\infty)$ ($F(x)=\int_0^xf(u) du$) and $\lim_{x\to\pm\infty}g(x)=g(\pm\infty)$ exist and are finite. It is proved that the equation has at least one $2\pi$-periodic solution provided that the zeros of the function $\Sigma_1$ are simple and the zeros of the functions $\Sigma_1, \Sigma_2$ are different and the signs of $\Sigma_2$ at the zeros of $\Sigma_1$ in $[0,2\pi/n)$ do not change or change more than two times, where $\Sigma_1$ and $\Sigma_2$ are defined as follows: \[\Sigma_1(\theta)=\frac{n}{\pi}\left[\frac{g(+\infty)}{a}-\frac{g(-\infty)}{b} \right]-\frac {1}{2\pi}\int_0^{2\pi}p(t)\varphi(t+\theta)\,dt,\qquad \theta\in [0, 2\pi/n],\]\[\Sigma_2(\theta)=\frac{n}{\pi}[F(+\infty)-F(-\infty)]-\frac{1}{2\pi} \int_0^{2\pi}p(t)\varphi'(t+\theta)\,dt, \qquad\theta\in [0, 2\pi/n].\] Moreover, it is also proved that the given equation has at least one $2\pi$-periodic solution provided that the following conditions hold: \begin{eqnarray*} -\infty&<&\liminf_{x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}\le \limsup_ {x\to\pm\infty}\frac{F(x)}{|x|^{p-2}x}<+\infty,\\ 0&<&\liminf_{x\to\pm\infty}\frac{g(x)}{|x|^{q-2}x}\leq \limsup_{x\to\pm\infty} \frac{g(x)}{|x|^{q-2}x}<+\infty, \end{eqnarray*} with $1\leq p<q<2$.

In this paper we study the second-order parabolic equation

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in a domain [0,T]×ℝd ⊂ ℝd+1, where a = (aij)di,j=1 is matrix of bounded measurable coefficients, b = (bj)dj=1, and bˆ = (bˆj)dj=1 are measurable (in general, singular) vector fields, V is a measurable potential, T is a fixed positive number, and ∂tu = ∂u/∂t, and we employ the notation

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We introduce a new class of coefficients in the lower-order terms for which we prove the existence and the uniqueness of the weak fundamental solution, and for this we derive Gaussian upper and lower bounds.

The coefficients in asymptotics of regularized traces and associated trace (spectral) distributions for Schrödinger operators with short and long range potentials are computed. A kernel expansion for the Schrödinger semigroup is derived, and a connection with non-commutative Taylor formulas is established.

We are interested in Weil representations of Sp(2n, R), where R is the ring Z/plZ, p is an odd prime and l is a positive integer, or, more generally, R = [Oscr ]/[pfr ]l, where [Oscr ] is the ring of integers of a local field, [pfr ] is the maximal ideal of [Oscr ] and [Oscr ]/[pfr ] has odd characteristic. One reason for this interest is that a continuous finite-dimensional complex representation of Sp(2n, [Oscr ]) has to factor through a representation of Sp(2n, [Oscr ]/[pfr ]l) for some l.

It is proved that a connected infinite graph has a normal spanning tree (the infinite analogue of a depth-first search tree) if and only if it has no minor obtained canonically from either an (ℵ0, ℵ1)-regular bipartite graph or an order-theoretic Aronszajn tree. This disproves Halin's conjecture that only the first of these obstructions was needed to characterize the graphs with normal spanning trees. As a corollary Halin's further conjecture is deduced, that a connected graph has a normal spanning tree if and only if all its minors have countable colouring number.

The precise classification of the (ℵ0, ℵ1)-regular bipartite graphs remains an open problem. One such class turns out to contain obvious infinite minor-antichains, so as an unexpected corollary Thomas's result that the infinite graphs are not well-quasi-ordered as minors is reobtained.

We consider the classical universal covering $\exp: {{\mathbb C}}\To {{\mathbb C}}^*$ of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion-free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of ${{\mathbb C}}$. This can also be seen as a model-theoretic statement on the categoricity of a corresponding $L_{\omega_1,\omega}$-sentence. The proof is a combination of arithmetic and model-theoretic methods.

Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic when it is not a multiple of the identity. The operator of differentiation is hypercyclic on the space A−∞, but it need not be hypercyclic on radial weighted algebras of entire functions.

Arithmetic Fuchsian groups are the most interesting and most important Fuchsian groups owing to their significance for number theory and owing to their geometric properties. However, for a fixed signature there exist only finitely many non- conjugate arithmetic Fuchsian groups; it is therefore desirable to extend this class of Fuchsian groups. This is the motivation of our definition of semi-arithmetic Fuchsian groups. Such a group may be defined as follows (for the precise formulation see Section 2). Let Γ be a cofinite Fuchsian group and let Γ2 be the subgroup generated by the squares of the elements of Γ. Then Γ is semi-arithmetic if Γ is contained in an arithmetic group Δ acting on a product Hr of upper halfplanes. Equivalently, Γ is semi-arithmetic if all traces of elements of Γ2 are algebraic integers of a totally real field. Well-known examples of semi-arithmetic Fuchsian groups are the triangle groups (and their subgroups of finite index) which are almost all non-arithmetic with the exception of 85 triangle groups listed by Takeuchi [16].

While it is still an open question as to what extent the non-arithmetic Fuchsian triangle groups share the geometric properties of arithmetic groups, it is a fact that their automorphic forms share certain arithmetic properties with modular forms for arithmetic groups. This has been clarified by Cohen and Wolfart [5] who proved that every Fuchsian triangle group Γ admits a modular embedding, meaning that there exists an arithmetic group Δ acting on Hr, a natural group inclusion

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and a compatible holomorphic embedding

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that is with

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for all γ∈Γ and all z∈H.