Let X be a smooth projective curve of genus g ≥ 2 over ℂ. A pair (E, φ) over X consists of an algebraic vector bundle E over X and a section φ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Here we prove that the third cohomology groups of the moduli spaces of τ-stable pairs with fixed determinant and rank n ≥ 2 are polarised pure Hodge structures, and they are isomorphic to H1(X) with its natural polarisation (except in very few exceptional cases). This implies a Torelli theorem for such moduli spaces. We recover that the third cohomology group of the moduli space of stable bundles of rank n ≥ 2 and fixed determinant is a polarised pure Hodge structure, which is isomorphic to H1(X). We also prove Torelli theorems for the corresponding moduli spaces of pairs and bundles with non-fixed determinant.

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney class

in the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,

(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and

(b) stable, i.e. w(V) = 1 when F is a trivial representation.

The problem of the two bodies has been treated on the new mechanics by Dirac, Pauli, and Schrödinger, who have independently derived the Balmer terms. The present paper is an attempt at a more complete solution. In particular, formulae are derived for the line intensities of the hydrogen spectrum, for the photoelectric effect and its inverse, and for the continuous absorption spectrum in the ultraviolet and in the X-ray regions. Also the probabilities of transition, deflection and capture are computed for the collision of an electron and an ion. Numerical values are only obtained, however, for the simplest line intensities. It is hoped to treat the problem in greater detail.

Pregroups were defined by Stallings[7] who showed that the elements of the group they define have a normal form up to an equivalence called interleaving. Recently Rimlinger[5] has shown that subject to a discreteness and a boundedness condition any pregroup P defines a graph of groups. We show here that closer analysis of P makes the boundedness condition superfluous. In § 1 we give results of Stallings and Rimlinger and prove some key lemmas. In §2 we show that the discreteness condition gives an integer-valued length function in the sense of Lyndon [4]. It follows from the work of Chiswell [2] and Serre [6] that this defines a graph of groups. I would like to thank the referee for his careful reading and useful comments on this paper.

Let $(R,M)$ be a Noetherian local domain, $F$ its quotient field and $v$ a valuation of $F$ centred at $R$. In the hope of understanding the valuation $v$, a useful tool is the so-called graded algebra relative to $v$, $gr_{v}R$. Its study can be understood as an attempt to approach the problem of classifying valuations. On the other hand, $gr_{v}R$ is also an object of interest since it can be provided with a geometrical meaning [15].

Renewal processes in discrete time (or as they are commonly termed, recurrent events) are appropriately described by renewal sequences {un} which are generated by discrete distributions , according to the equation

Any two renewal sequences {u′n}, {u″n} define another renewal sequence {un} by means of their term-by-term product {un} = {u′nu″n}, for the joint occurrence of two independent recurrent events ℰ′ and ℰ″ is also a recurrent event. Considering a renewal process in continuous time for which we shall suppose a frequency function f(x) of the lifetime distribution exists, so that a renewal density exists, the analogous property would be that for two renewal density functions h1(x) and h2(x), the function h(x) = h1(x) h2(x) is a renewal density function. A little intuitive reflexion shows that while h(x) dx has a probability density interpretation, this is not in general true of h1(x) h2(x) dx. It is not surprising therefore to find in example 1 a case where the product of two renewal densities is not a renewal density. Example 2, on the other hand, shows that in some cases it is true, and taken together with example 1, there is suggested the problem of characterizing the class of renewal densities h(x) for which αh(x) is a renewal density for all finite positive α and not merely α in 0 < α ≤ A < ∞. In turn this characterization enables us to define a class of renewal densities for which h1(x) and imply that .

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the coloured Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.

The theory of distributions, systematized by L. Schwartz (3), has many applications in pure mathematics and physics. A number of different approaches to the theory are possible. Schwartz's own account requires for its comprehension a familiarity with the topology of general spaces. I give in this note another approach to the theory based on a process of integration of the Stieltjes type.

A simple and elegant method of treating one-sided Hilbert transforms

has recently been described by Wong (7). This method immediately gives the form of the expansion but some of the coefficients in the expansion are in an inconvenient form for either analytical or numerical calculation. In the present paper it is shown that these coefficients can be readily determined when the Mellin transform of f(t) is known. The case is treated as an example.

Let G be a group, and let φ (x1, …, xn) be any word in the variables x1, …, xn. We denote by φ(G) and φ*(G) respectively the corresponding verbal and marginal subgroups of G, which are defined as follows.

(i) φ(G)is the subgroup generated by all elements of the form φ(a1; …, an) with ai in G(for i = 1, 2, …, n);

(ii) an element b of G lies in φ*(G) if and only if

for all choices of a1, …, an in G and i= 1, 2, …,n.

In (1), Burgoyne and Fong have shown that the Schur multiplier of the Mathieu group M12 is of order 2. It is shown in Theorem 2·4 that the Schur multiplier of Aut M12, the automorphism group of M12, is also of order 2. It is therefore possible to choose a complex 2-cocycle α of Aut M12, taking only the values 1 and − 1, such that the cohomology class of α is of order 2 and the cohomology class of the restriction of α to M12 is of order 2. The characters of the irreducible α-projective representations of Aut M12 are calculated in § 2.

A radial basis function approximation in n variables has the form

where ø:ℝn → ℝ denotes the n-variate, spherically symmetric function associated with a prescribed radial basis function ø+:ℝ+ → ℝ, i.e. ø = ø+(‖ · ‖), the norm being Euclidean. The are real coefficients (often, approximants s above are considered where only finitely many λjs are non-zero), and is a fixed set of points in ℝn (of course, only the xj with non-zero coefficient λj affect s). Thus s is a linear combination of translates of a radially symmetric function which can be of global support, the simplest choice being , where c is a positive parameter. The latter is referred to as the multiquadric function and is usefull in applications.

Let U denote the variety of pencils of binary cubics without base point defined over an algebraically closed field k whose characteristic is not equal to 2 or 3. (For full details of the terminology, see § 2.) There is a natural action of SL(2) on U; our main object is to show that U possesses a good quotient for this action in the sense of ((7), Definition 1·5) or ((4), p. 70), and to identify this quotient with the affine line over k. In fact, we prove

Theorem 1. There exists a morphism φ: such that (, φ) is a good quotient of U by SL(2). Moreover, all but one of the fibres of φ are orbits; the exceptional fibre consists of two orbits corresponding to pencils with one or two triple points.

In (1) the authors introduced the idea of a syntopogenous preordered space (SPS), thereby generalizing simultaneously the concepts of ‘uniform preordered space’ and ‘proximity preordered space’. In the case of a classical topology the SPS condition was seen to be equivalent to the condition, ‘convex plus T1-preordered’. In the present paper we show that, in the general case, the SPS condition may be characterized as the sum of two generalizations of the logically independent conditions of convexity and T1-preorderedness.

The main result of this paper gives an explicit computation of the $L_4$ norm of any cyclotomic polynomial of the form \[ f(x)=\Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1 p_2\cdots p_{r-1}}). \] Here $\Phi_{p}$ is the $p$th cyclotomic polynomial and the $p_i$ are primes that are not necessarily distinct.

A corollary of this is the following theorem.

THEOREM. If\[ f(x)=\Phi_{p_1}(\e_1 x)\Phi_{p_2}(\e_2 x^{p_1})\cdots \Phi_{p_r}(\e_r x^{p_1 p_2\cdots p_{r-1}}), \]where N = p1p2···prand εj = ±, then \[ \frac{\| f\|_4^4}{N^2} & \ge & \frac{\big\|\Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big) \big\|_4^4}{4^r}\\& = & \frac{\big(\frac12 +\frac{5}{34}\sqrt{17}\big)(1+\sqrt{17})^r - \big({-}\frac12 +\frac{5}{34}\sqrt{17}\big)(1-\sqrt{17})^r}{4^r}. \] In particular the minimum possible normalized L4 norm of any polynomial of the above form is attained by \[ \Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big). \]

We develop new criteria for the stability of a periodic solution of a given newtonian equation based on the $L^1$-norm of the coefficients of the third order approximation, by proving that the twist coefficient is different form zero.

We study strong triviality of 2-bridge knots via Dehn surgery techniques and prove that a 2-bridge knot is strongly $n$-trivial for $n\geq 1$ if and only if it is the trivial knot, the trefoil knot or the figure-eight knot.

We prove a robust version of Freiman's 3k – 4 theorem on the restricted sumset A+ΓB, which applies when the doubling constant is at most (3+$\sqrt{5}$)/2 in general and at most 3 in the special case when A = −B. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz–Sobolev inequality.

1. Let (S, S,μ) be a measure-space, and let f(x) and ø(x) be two μ-integrable functions defined in S. We shall assume that f(x) is a ‘difficult’ function, of which all that is known is how to calculate f(x), generally by a rather complicated process, for any given x∈S; and that ø(x) is an ‘easy’ function, whose value for any x, and whose integral over any subset T of S belonging to the class S of μ-measurable sets, are relatively simple to obtain. We shall define the sets

and assume that

It will be of interest to consider those functions ø(x) which may be regarded as approximations to f(x) in S.