Mahler's measure of a polynomial can be written as a logarithmic integral over the torus. We propose a definition when the underlying group is an elliptic curve. Having reviewed some of the classical results in the toral case, we take some first steps towards realising elliptic analogues. In particular, we focus on elliptic analogues of Kronecker's theorem and Lehmer's problem. We wish to stress the fundamental role played by Jensen's formula in both the toral and elliptic formulations of these results.

Projections are constructed in the rotation algebra that are orthogonal to their Fourier transform and which are fixed under the flip automorphism. Such projections are expected in a construction of an inductive limit structure for the irrational rotation algebra that is invariant under the Fourier transform (namely, as two circle algebras of the same dimension, which are swapped by the Fourier transform, plus a few points). The calculation is based on Rieffel's construction of the Schwartz space as an equivalence bimodule of rotation algebras as well as on the theory of theta functions.

Asymptotics on the diagonal of the Green function and, as a consequence, asymptotic distribution of the spectrum for the semibounded Sturm–Liouville operator are obtained.

Motivated by, but independent of, some recent work in quantum stochastic calculus, a theory of differential and integral calculus is developed which is intrinsic to the universal enveloping algebra of a Lie algebra whose Lie bracket is obtained by taking commutators in an associative algebra. The differential map satisfies a generalisation of Leibniz' formula called the Leibniz–Itô formula, which involves the associative multiplication. There is an analogue of the Taylor–Maclaurin expansion. Through passing to formal power series, a theory of product integrals is developed; such integrals are characterised by a group-like property with respect to the coproduct.

Let U be a domain, convex in $x$ and symmetric about the $y$-axis, which is contained in a centered and oriented rectangle $S$. It is proved that $H_t{(U^+)}/H_t{(U)}\,{\leq}\, H_t{(S^+)}/H_t{(S)}$ where $H_t$ stands for heat content, that is, the remaining heat in the domain at time $t$ if it initially has uniform temperature 1, with Dirichlet boundary conditions, where $A^+\,{=}\,A\,{\cap}\, \{(x,y)\,{:}\,x\,{>}\,0\}$. It is also shown that the analog of this inequality holds for some other Schrödinger operators.

Let M be an ω-categorical structure (that is, M is countable and Th(M) is ω-categorical). A nice enumeration of M is a total ordering [pr ] of M having order-type ω and satisfying the following. Whenever ai, i<ω, is a sequence of elements from M, there exist some i<j<ω and an automorphism σ of M such that σ(ai) = aj and whenever b[pr ]ai, then σ(b)[pr ]aj.

Such enumerations were introduced by Ahlbrandt and Ziegler in [1] where they showed that any Grassmannian of an infinite-dimensional projective space over a finite field (or of a disintegrated set) admits a nice enumeration; this combinatorial property played an essential role in their proof that almost strongly minimal totally categorical structures are quasi-finitely axiomatisable.

Recall that if M is ω-categorical and a is a k-tuple of distinct elements from M (with tp(a) non-algebraic), then the Grassmannian Gr(M; a) is defined as follows. The domain of Gr(M; a) is the set of realisations of tp(a) in Mk, modulo the equivalence relation xEy if x and y are equal as sets. This is a 0-definable subset of Meq, and now the relations on Gr(M; a) are by definition precisely those which are 0-definable in the structure Meq. (In particular, Gr(M; a) is also ω-categorical.)

Notice that it is by no means clear that if M admits a nice enumeration, then so do Grassmannians of M. However, there is a strengthening of the notion of nice enumeration for which this is the case.

A metric space X has the unique midset property if there is a topology-preserving metric d on X such that for every pair of distinct points x, y there is one and only one point p such that d(x, p) = d(y, p). The following are proved. (1) The discrete space with cardinality [nfr ] has the unique midset property if and only if [nfr ] ≠ 2, 4 and [nfr ] [les ] [cfr ], where [cfr ] is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than [cfr ], then the countable power DN of D has the unique midset property. In particular, the Cantor set and the space of irrational numbers have the unique midset property.

A finite discrete space with n points has the unique midset property if and only if there is an edge colouring ϕ of the complete graph Kn such that for every pair of distinct vertices x, y there is one and only one vertex p such that ϕ(xp) = ϕ(yp). Let ump(Kn) be the smallest number of colours necessary for such a colouring of Kn. The following are proved. (3) For each k [ges ] 0, ump(K2k+1) = k. (4) For each k [ges ] 3, k [les ] ump(K2k) [les ] 2k−1.

The following theorem is proved: if $(M^{4n+1},g)$ is a complete Riemannian manifold and $\Sigma\subset M$ is an oriented hypersurface partitioning $M$ and with non-zero signature, then the spectrum of the Hodge–deRham Laplacian is $[0,\infty[$ . This result is obtained by a new Callias-type index. This new formula links half-bound harmonic forms (that is, nearly $L^2$ but not in $L^2$ ) with the signature of $\Sigma$ .

We give a necessary and sufficient condition on an elliptic symbol σ of order m to ensure that the unique closed extension in Lp(ℝn) for 1 < p < ∞, of the pseudo-differential operator Tσ, initially defined on the Schwartz space, is a Fredholm operator from Lp(ℝn) into Lp(ℝn) with domain Hm, p, where Hm, p is the Lp Sobolev space of order m.

A real algebraic, plane, projective curve $A$ of degree $m$ is given by a homogeneous polynomial of degree $m$ in three variables, with real coefficients, defined up to multiplication by a non-zero scalar. If $F$ is such a polynomial defining $A$ , we denote by ${\bb C} A$ and ${\bb R} A$ , respectively, the sets of solutions of the equation $F = 0$ in ${\bb C} P^2$ , respectively ${\bb R} P^2$ . We suppose that the curve $A$ is non-singular, that is, $F$ has no critical points in ${\bb C}^3\backslash 0$ . Then ${\bb C} A$ is a Riemannian surface of genus $g = (m-1)(m-2)/2$ , and ${\bb R} A$ is a collection of $L \le g+1$ circles embedded in ${\bb R} P^2$ . If $L = g+1$ , we say that $A$ is an $M$ -curve. A circle embedded in ${\bb R} P^2$ is called oval, or pseudo-line, depending on whether it realizes the class 0 or 1 of $H_1({\bb R} P^2)$ . If $m$ is even, the $L$ components of ${\bb R} A$ are ovals; if $m$ is odd, ${\bb R} A$ contains exactly one pseudo-line, which will be denoted by ${\cal J}$ . Note that an oval separates ${\bb R} P^2$ into two pieces, a Möbius band and a disc. The latter is called the interior of the oval. An oval of ${\bb R} A$ is said to be empty if its interior contains no other oval of ${\bb R} A$ . Two ovals form an injective pair if one of them lies in the interior of the other one.

It is shown that the natural action of the absolute Galois group on the ideals defining principal congruence subgroups of certain nonarithmetic Fuchsian triangle groups is compatible with its action on the algebraic curves that these congruence groups uniformize.

Let $U^+$ be the plus part of the quantized enveloping algebra of a simple Lie algebra of type $A_n$ and let $\mathcal{B}^*$ be the dual canonical basis of $U^+$. Let $b$, $b'$ be in $\mathcal{B}^*$, and suppose that one of the two elements is a $q$-commuting product of quantum flag minors. It is shown that $b$ and $b'$ are multiplicative if and only if they $q$-commute.

Scharlemann and Thompson introduced an invariant $\rho(K,\gamma)\in \mathbb{Q}/2\mathbb{Z}$ for the pair of a knot $K$ in the 3-sphere and an unknotting tunnel $\gamma$ for $K$. The paper studies the relationship between the invariant $\rho(K,\gamma)$ of a (1,1)-knot and the distance of its (1,1)-splitting introduced by the author.

The paper continues the investigation of the Hecke structure of spaces of half-integral weight cusp forms ${S_{k + 1/2}}(4N, \chi)$, where $k$ and $N$ are positive integers with $N$ odd, and $\chi$ is an even quadratic Dirichlet character modulo $4N$. In the Hecke decomposition of these spaces, contributions are determined arising from newforms which are quadratic twists of newforms at lower levels. Combining this result with its counterpart in a previous paper regarding non-twists gives this paper's main result: necessary and sufficient conditions under which a given newform has equivalent cusp forms in ${S_{k + 1/2}}(4N, \chi)$. The result reformulates, in classical terms, the representation-theoretic conditions given by Flicker and Waldspurger. The conditions involve easily-verified information about the primes dividing the level of the newform, and about the behavior of the newform under certain quadratic twists and Atkin–Lehner involutions. The theorem is applied to give explicit examples of twisted newforms having no equivalent half-integral weight cusp forms in any space ${S_{k + 1/2}}(4N, \chi)$ as above.

Let $\Omega X$ be the space of Moore loops on a finite, $q$-connected, $n$-dimensional CW complex $X$, and let $R\subset\Q$ be a subring containing 1/2. Let $\rho\left(R\right)$ be the least non-invertible prime in $R$. For a graded $R$-module $M$ of finite type, let $FM = M / {\rm Torsion}\,M$. We show that the inclusion $P \subset FH_{*}\left(\Omega X;R\right)$ of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras $$UP \overset{\cong}{\longrightarrow} FH_{*}\left(\Omega X;R\right)},$$ provided that $\rho\left(R\right) \geqslant n/q$. Furthermore, the Hurewicz homomorphism induces an embedding of $F(\pi_{*}\left(\Omega X\right)\otimes R)$ in $P$, with $P/F(\pi_{*}\left(\Omega X\right)\otimes R)$ torsion. As a corollary, if $X$ is elliptic, then $FH_{*}\left(\Omega X;R\right)$ is a finitely generated $R$-algebra.

Let $A > 0$ be an integer. The equation $x^5 - y^5 = Az^5$ was first studied by Dirichlet and Lebesgue. Lebesgue conjectured in 1843 that if $A$ has no prime divisors of the form $10k+1$, the equation has no solutions except the visible ones. Partial results were obtained by Lebesgue and by Terjanian in 1987. The purpose of the paper is to prove Lebesgue's conjecture. The main tool used is the method known as the elliptic Chabauty method.

The classification of finite groups acting freely on a homology 3-sphere is still incomplete. However it is known that these groups have periodic cohomology and groups with periodic cohomology are classified by the Suzuki–Zassenhaus theorem. The paper considers, more generally, finite groups which may act on a homology 3-sphere possibly with fixed points and the natural generalization of the Suzuki–Zassenhaus theorem to this case is found. According to the Suzuki–Zassenhaus theorem a group with periodic cohomology is either solvable or has a unique composition factor which is not cyclic, isomorphic to PSL(2, q) for q an odd prime. The main point of the work is that this fact is still true in the more general situation.

The paper proves the existence and multiplicity of periodic solutions of the equation

formula here

where g[ratio ]R→R is of class C1, h[ratio ]R→R is continuous and 2π-periodic, and s is a parameter.

It is shown that the splitting modulo a prime p of a given monic, integral, irreducible cubic with non-square discriminant is equivalent to p being represented by forms in a certain subgroup of index 3 in the form class group of discriminant equal to the discriminant of the field defined by the cubic.

This paper deals with a formula satisfied by ‘r-reduced’ Schur functions. Schur functions originally appear as irreducible characters of general linear group over the complex number field. In this paper they are considered as weighted homogeneous polynomials with respect to the power sum symmetric functions. More precisely, for a Young diagram λ of size n, the Schur function indexed by λ reads

formula here

where χλ(v) is the character value of the irreducible representation Sλ of the group algebra ℚ[Sfr ]n, evaluated at the conjugacy class of the cycle type v = (1v12v2 … nvn). Setting tjr = 0 for j = 1, 2, … in Sλ(t), we have the r-reduced Schur function S(r)λ(t). The set of all r-reduced Schur functions spans the polynomial ring P(r) = ℚ[tj; j[nequiv ]0 (mod r)]. We show that a good choice of basis elements leads to an explicit description of all other r-reduced Schur functions involving the Littlewood–Richardson coefficients.

The formula has not only a purely combinatorial meaning, but also nice implications in two different fields. One is about the basic representation of the affine Lie algebra A(1)r−1. We show that the basis in the main theorem gives in turn a weight basis of the basic A(1)r−1-module realised in P(r). The other implication is about modular representations of the symmetric group. Our explicit formula implies that the determination of the decomposition matrices reduces to that for the basic set we give in this paper.

The paper is organised as follows. In Section 1 we introduce generalised Maya diagrams and associated r-reduced Schur functions. In Section 2 we discuss combinatorics of Young diagrams. Section 3 is devoted to the main theorem. In Section 4 we describe weight vectors of the basic A(1)r−1-module. In Section 5 the formula is translated into that in the modular representation theory.