Recently Bergelson and Leibman proved an extension to van der Waerden's theorem involving polynomials. They also generalised the Hales–Jewett theorem in a similar way. The paper presents short and purely combinatorial proofs of those results.

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.

Suppose that G(x) is a form, or homogeneous polynomial, of odd degree d in s variables, with real coefficients. Schmidt [15] has shown that there exists a positive integer s0(d), which depends only on the degree d, so that if s [ges ] s0(d), then there is an x ∈ ℤs\{0} satisfying the inequality

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In other words, if there are enough variables, in terms of the degree only, then there is a nontrivial solution to (1). Let s0(d) be the minimum integer with the above property. In the course of proving this important result, Schmidt did not explicitly give upper bounds for s0(d). His methods do indicate how to do so, although not very efficiently. However, in fact much earlier, Pitman [13] provided explicit bounds in the case when G is a cubic. We consider a general cubic form F(x) with real coefficients, in s variables, and look at the inequality

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Specifically, Pitman showed that if

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then inequality (2) is non-trivially soluble in integers. We present the following improvement of this bound.

A formula of Poisson summation type is established for sums involving relative norms in an extension of number fields.

A certain property of some type-definable subgroups of superstable groups with finite U-rank is closely related to the Mordell–Lang conjecture. This property is discussed in the context of algebraic groups.

The paper studies the permutation representations of a finite general linear group, first on finite projective space and then on the set of vectors of its standard module. In both cases the submodule lattices of the permutation modules are determined. In the case of projective space, the result leads to the solution of certain incidence problems in finite projective geometry, generalizing the rank formula of Hamada. In the other case, the results yield as a corollary the submodule structure of certain symmetric powers of the standard module for the finite general linear group, from which one obtains the submodule structure of all symmetric powers of the standard module of the ambient algebraic group.

A finite non-trivial group G is called a Hurwitz group if it is an image of the infinite triangle group

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Thus G is a Hurwitz group if and only if it can be generated by an involution and an element of order 3 whose product has order 7. The history of Hurwitz groups dates back to 1879, when Klein [9] was studying the quartic

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of genus 3. The automorphism group of this curve has order 168 = 84(3−1), and it is isomorphic to the simple group PSL2(7), which is generated by the projective images of the matrices

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with product

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and so is a Hurwitz group. In 1893, Hurwitz [7] proved that the automorphism group of an algebraic curve of genus g (or, equivalently, of a compact Riemann surface of genus g) always has order at most 84(g−1), and that, moreover, a finite group of order 84(g−1) can act faithfully on a curve of genus g if and only if it is an image of Δ(2, 3, 7).

The problem of determining which finite simple groups are Hurwitz groups has received considerable attention. In [10], Macbeath classified the Hurwitz groups of type PSL2(q); there are infinitely many of them. In [1] Cohen proved that no group PSL3(q) is a Hurwitz group except PSL3(2), which is isomorphic to PSL2(7). Certain exceptional groups of Lie type, and some of the sporadic groups, are known to be Hurwitz groups. For discussions of the results on these groups we refer the reader to [3, 5, 11].

In this paper we define three elements of a certain generalised cohomology ring BP〈m, n〉* BVk. Here m, n and k are non-negative integers with k+m[les ]n+1, there is a fixed prime p not exhibited in the notation, and Vk is an elementary Abelian p-group of rank k. We show that these elements are equal; this is striking, because the three definitions are very different. The significance of our equation is not yet entirely clear, but it makes contact with other work in the literature in a number of fascinating ways.

Let G be a locally compact Abelian group and consider the group algebra L1(G) of all complex-valued absolutely integrable functions on G. Some invariants are found that are necessarily shared by groups with isomorphic group algebras, such as dimension, ranks or minimal divisible extensions in the case of torsion-free groups. This sheds some light on the question of finding out to what extent the algebra structure of L1(G) reflects the topological group structure of G. It is shown in particular that some classes of torsion- free locally compact Abelian groups are characterized by their group algebra.

The commonly known properties of continuous measures such as the symmetry of the A∞ condition, its equivalence to the reverse Hölder inequality, the left-openness of the Ap condition, etc., are no longer necessarily true when the underlying measure is allowed to have atoms. The measures that preserve these properties are called good measures. The class of good measures is investigated and various criteria for a measure to belong to this class are presented.

Let T be an ergodic and free ℤdrotation on the d-dimensional torus [ ]d given by

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where (m1, …, md) ∈ ℤd, (z1, …, zd) ∈ [ ]d and [αjk]j,k=1 …, d ∈ Md(ℝ). For a continuous circle cocycle ϕ[ratio ]ℤd × [ ]d → [ ](ϕm+n(z) = ϕm(Tnz)ϕn(z) for any m, n ∈ ℤd), the winding matrix W(ϕ) of a cocycle ϕ, which is a generalization of the topological degree, is defined. Spectral properties of extensions given by

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are studied. It is shown that if ϕ is smooth (for example ϕ is of class C1) and det W(ϕ) ≠ 0, then Tϕ is mixing on the orthocomplement of the eigenfunctions of T. For d = 2 it is shown that if ϕ is smooth (for example ϕ is of class C4), det W(ϕ) ≠ 0 and T is a ℤ2-rotation of finite type, then Tϕ has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(ϕ) = 1, then Tϕ has singular spectrum.

Certain results which have been proved or conjectured for the subclass consisting of those in L2(ℝ) are generalized to entire functions of exponential type bounded on the real axis.

Central to the Fourier development of automorphic forms on GL(n, ℝ) are the class 1 principal series Whittaker functions Wn,a(z), which were first studied systematically by Jacquet [13]. (See Section 2 below for the definition of Wn,a(z).)

Of particular interest are the Mellin transforms Mn,a(s) (s ∈ [Copf ]n−1) of Wn,a(z). (See equation (2.11) below.) For example, such transforms, and analogous Mellin transforms of products of Whittaker functions, arise as archimedean Euler factors for certain automorphic L-functions (see [5, 6, 8, 21, 22] for discussions and examples.) Moreover, Mn,a(s) has relevance to the problem of special values of Whittaker functions; cf. [7] in the case n = 3.

Friedberg and Goldfeld [11] have shown Mn,a(s), for general n, to have analytic continuation and to satisfy certain recurrence relations. However, explicit formulae for these transforms have, until the present work, been deduced only for n [les ] 4. In particular, both M2,a(s) (cf. [4, 25]) and M3,a(s) (cf. [7, 9, 19]) are expressible as (some powers of 2 and π times) ratios of gamma functions. On the other hand, M4,a(s) (cf. [22]) may be realized essentially as a hypergeometric series of type 7F6(1), or equivalently as a sum of two series of type 4F3(1). (See Section 2 below for a brief general discussion of hypergeometric series.)

Monotonicity with respect to the order v of the magnitude of general Bessel functions [Cscr ]v(x) = aJv(x)+bYv(x) at positive stationary points of associated functions is derived. In particular, the magnitude of [Cscr ]v at its positive stationary points is strictly decreasing in v for all positive v. It follows that supx[mid ]Jv(x)[mid ] strictly decreases from 1 to 0 as v increases from 0 to ∞. The magnitude of x1/2[Cscr ]v(x) at its positive stationary points is strictly increasing in v. It follows that supx[mid ]x1/2[Cscr ]v(x)[mid ] equals √2/π for 0 [les ] v [les ] 1/2 and strictly increases to ∞ as v increases from 1/2 to ∞.

It is shown that v1/3supx[mid ]Jv(x)[mid ] strictly increases from 0 to b = 0.674885… as v increases from 0 to ∞. Hence for all positive v and real x,

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where b is the best possible such constant. Furthermore, for all positive v and real x,

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where c = 0.7857468704… is the best possible such constant.

Additionally, errors in work by Abramowitz and Stegun and by Watson are pointed out.

The purpose of the paper is to introduce a new linearization method for local well-posedness of nonlinear evolution equations. This approach is based on an implicit function theorem of Nash–Moser type. The technique is illustrated by an application to a general class of fully nonlinear parabolic partial differential equations of arbitrary order on ℝn. The estimates required by the Nash–Moser technique are derived for the higher order Sobolev norms of the solutions of the linearized parabolic equation using semigroup theory and elliptic theory. In particular, a priori estimates, resolvent estimates and commutator estimates are involved. The general method based on a combination of Nash–Moser techniques with semigroup theory is applicable to other problems and has already been used to prove short-time solvability for some nonlinear Schrödinger type equation. This approach might be useful in other situations as well since it compensates for a loss of derivatives in the estimates of the solutions of the linearized equation.

It is proved that Thom spectra of generalized braid groups are the wedges of suspensions over the Eilenberg–MacLane spectrum for ℤ/2. The precise structure of the Thom spectra of the generalized braid groups of the types C and D is obtained. For the generalized braid groups of type C the natural pairing analogous to the pairing of the classical braids is studied. This pairing generates the multiplicative structure of the Thom spectrum such that the corresponding bordism theory has the coefficient ring isomorphic to the polynomial ring over ℤ/2 on one generator of dimension 1[ratio ]ℤ/2[s].

The ideas of Friedman and Qin are extended to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with pg = 0, q > 0 and anticanonical divisor −K effective, for any wall ζ with lζ = ¼(ζ2 − p1) being 0 or 1.

McMichael proved that the convolution with the (euclidean) arclength measure supported on the curve t [xrarr ] (t, t2, …, tn), 0 < t < 1, maps Lp(ℝn) boundedly into Lp′(ℝn) if and only if 2n(n+1)/(n2+n+2) [les ] p [les ] 2. In proving this, a uniform estimate on damping oscillatory integrals with polynomial phase was crucial. In this paper, a remarkably simple proof of the same estimate on oscillatory integrals is presented. In addition, it is shown that the convolution operator with the affine arclength measure on any polynomial curve in ℝn maps Lp(ℝn) boundedly into Lp′(ℝn) if p = 2n(n+1)/(n2+n+2).

Let ϕ be a hyperbolic map. Cocycle equations of the form f = uϕ·g·αu−1 are considered, with f, g, u taking values in a compact connected Lie group G, α being an automorphism of G and f, g being Hölder continuous. When the eigenvalues of the derivative of α have modulus 1, it is proved that any measurable solution u has a Hölder continuous version. This condition on α is optimal. When f, g are Ck then u may be taken to be Ck−1+ε for any ε ∈ (0, 1).

Consider a discrete-time regenerative phenomenon with associated renewal sequence un. General results for the supremum of um+1, …, um+n are developed for those renewal sequences {un} for which the first m + 1 elements match those of a fixed renewal sequence {vn}, that is, u0 = v0, …, um = vm. A series of associated lemmas are developed in the process.