New lower bounds are given for the size of a point set in a Desarguesian projective plane over a finite field that contains at least a prescribed number s of points on every line. These bounds are best possible when q is square and s is small compared with q. In this case the smallest set is shown to be the union of disjoint Baer subplanes. The results are based on new results on the structure of certain lacunary polynomials, which can be regarded as a generalization of Rédei's results in the case when the derivative of the polynomial vanishes.

The paper shows that any graph G with the maximum degree Δ(G) [ges ] 8, which is embeddable in a surface Σ of Euler characteristic χ(Σ) [ges ] 0, is totally (Δ(G)+2)-colorable. In general, it is shown that any graph G which is embeddable in a surface Σ and satisfies the maximum degree Δ(G) [ges ] (20/9) (3−χ(Σ))+1 is totally (Δ(G)+2)-colorable.

A proper vertex-colouring of a graph is acyclic if there are no 2-coloured cycles. It is known that every planar graph is acyclically 5-colourable, and that there are planar graphs with acyclic chromatic number χa = 5 and girth g = 4. It is proved here that a planar graph satisfies χa [les ] 4 if g [ges ] 5 and χa [les ] 3 if g [ges ] 7.

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 paper describes the cell structure of the affine Temperley–Lieb algebra with respect to a monomial basis. A diagram calculus is constructed for this algebra.

Let T be a d×d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus X = ℝd/ℤd. Choose for each natural number n a ball B(n) in X and suppose that B(n+1) has smaller radius than B(n) for all n. Now let W be the set of points x ∈ X such that Tn(x) ∈ B(n) for infinitely many n ∈ ℕ. The Hausdorff dimension of W is studied by analogy with the Jarník–Besicovitch theorem on the dimension of the set of well-approximable real numbers. The dimension depends on the quantity

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A complete description is given only when the matrix is diagonalizable over ℚ. In other cases a result is obtained for sufficiently large τ. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of τ which is piecewise of the form (Aτ+B)/(Cτ+D). The numbers A, B, C and D which arise in this way are typically sums of logarithms of the absolute values of eigenvalues of T.

It is known from Vaughan and Wooley's work on Waring's problem that every sufficiently large natural number is the sum of at most 17 fifth powers [13]. It is also known that at least six fifth powers are required to be able to express every sufficiently large natural number as a sum of fifth powers (see, for instance, [5, Theorem 394]). The techniques of [13] allow one to show that almost all natural numbers are the sum of nine fifth powers. A problem of related interest is to obtain an upper bound for the number of representations of a number as a sum of a fixed number of powers. Let R(n) denote the number of representations of the natural number n as a sum of four fifth powers. In this paper, we establish a non-trivial upper bound for R(n), which is expressed in the following theorem.

To a reduced plane curve singularity with r branches there is associated its semigroup, a subsemigroup of ℤr[ges ]0. For an irreducible curve singularity, the description of its semigroup in terms of its set of generators is well known. For a reducible curve there exists a combinatorial description of the semigroup. The semigroup of a plane curve singularity with several branches is described in terms of the minimal set of generators constructed in a natural geometric way.

Thompson's famous theorems on singular values–diagonal elements of the orbit of an n×n matrix A under the action (1) U(n) [otimes ] U(n) where A is complex, (2) SO(n) [otimes ] SO(n), where A is real, (3) O(n) [otimes ] O(n) where A is real are fully examined. Coupled with Kostant's result, the real semi-simple Lie algebra [sfr ][ofr ]n, n yields (2) and hence (3) and the sufficient part (the hard part) of (1). In other words, the curious subtracted term(s) are well explained. Although the diagonal elements corresponding to (1) do not form a convex set in [Copf ]n, the projection of the diagonal elements into ℝn (or iℝn) is convex and the characterization of the projection is related to weak majorization. An elementary proof is given for this hidden convexity result. Equivalent statements in terms of the Hadamard product are also given. The real simple Lie algebra [sfr ][ufr ]n, n shows that such a convexity result fits into the framework of Kostant's result. Convexity properties and torus relations are studied. Thompson's results on the convex hull of matrices (complex or real) with prescribed singular values, as well as Hermitian matrices (real symmetric matrices) with prescribed eigenvalues, are generalized in the context of Lie theory. Also considered are the real simple Lie algebras [sfr ][ofr ]p, q and [sfr ][ofr ]p, q, p < q, which yield the rectangular cases. It is proved that the real part and the imaginary part of the diagonal elements of complex symmetric matrices with prescribed singular values are identical to a convex set in ℝn and the characterization is related to weak majorization. The convex hull of complex symmetric matrices and the convex hull of complex skew symmetric matrices with prescribed singular values are given. Some questions are asked.

In the 1960s, Richard J. Thompson introduced a triple of groups F ⊆ T ⊆ G which, among them, supplied the first examples of infinite, finitely presented, simple groups [14] (see [6] for published details), a technique for constructing an elementary example of a finitely presented group with an unsolvable word problem [12], the universal obstruction to a problem in homotopy theory [8], and the first examples of torsion free groups of type FP∞ and not of type FP [5]. In abstract measure theory, it has been suggested by Geoghegan (see [3] or [9, Question 13]) that F might be a counterexample to the conjecture that any finitely presented group with no non-cyclic free subgroup is amenable (admits a bounded, non-trivial, finitely additive measure on all subsets that is invariant under left multiplication). Recently, F has arisen in the theory of groups of diagrams over semigroup presentations [10], and as the object of questions in the algebra of string rewriting systems [7]. For more extensive bibliographies and more results on Thompson's groups and their generalizations see [1, 4, 6].

A persistent peculiarity of Thompson's groups is their ability to pop up in diverse areas of mathematics. This suggests that there might be something very natural about Thompson's groups. We support this idea by showing (Theorem 1.1 below) that PLo(I), the group of piecewise linear (finitely many changes of slope), orientation-preserving, self-homeomorphisms of the unit interval, is riddled with copies of F: a very weak criterion implies that a subgroup of PLo(I) must contain an isomorphic copy of F.

By considering branched coverings of piecewise Euclidean cubical complexes, the paper provides an example of a torsion free hyperbolic group containing a finitely presented subgroup which is not hyperbolic.

It is shown that the division ring of quotients of a skew polynomial ring of automorphism type, if infinite-dimensional over its centre k and satisfying suitable hypotheses, contains the group algebra of a free group of large rank (usually at least [mid ]k[mid ]). The result applies, in particular, to the skew polynomial rings constructed from rational function fields, and affirmatively settles the conjectures of Makar–Limanov and Lichtman in this case.

A minimax principle is derived for the eigenvalues in the spectral gap of a possibly non-semibounded self-adjoint operator. It allows the nth eigenvalue of the Dirac operator with Coulomb potential from below to be bound by the nth eigenvalue of a semibounded Hamiltonian which is of interest in the context of stability of matter. As a second application it is shown that the Dirac operator with suitable non-positive potential has at least as many discrete eigenvalues as the Schrödinger operator with the same potential.

The paper proves that a set which contains spheres centered at all points of a set of Hausdorff dimension greater than 1 must have positive Lebesgue measure. It also proves the corresponding result for circles, provided that the set of centers has Hausdorff dimension greater than 3/2.

The subject is the 2-point boundary value problem for a second order ordinary differential equation

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the simplest and oldest of all boundary value problems, which arose in Euler's work on calculus of variations in the eighteenth century. Accordingly, there is a vast literature, mostly in the context of applied mathematics, where one frequently uses the methods of functional analysis.

Here we adopt a different point of view: we look for the number of solutions of problem (1), if any, and how this number varies with the endpoints (t1, x1) and (t2, x2). The concept of focal decomposition, introduced in [15] and developed by Peixoto and Thom in [19], expresses precisely this point of view (see §2). It was further developed by Kupka and Peixoto in [10] in the context of geodesics. From there, one is led naturally to relationships with the arithmetic of positive definite quadratic forms, a line that is considered in [16]. In both [10] and [16] attention is drawn to the close formal relationship between focal decomposition and the Brillouin zones of solid state physics. In [17] this line is pursued further and it is pointed out that the focal decomposition associated to (1) appears naturally as a prerequisite for the semiclassical quantization of this equation via the Feynman path integral method.

The main goal of the present paper is to prove Theorem 2, stated at the end of §4. There we give a functional ∫t2t1L (t, x, x˙) dt and consider its Euler equation. To this second order differential equation we associate the corresponding focal decomposition. Theorem 2 gives a criterion on the Lagrangian L from which we get a rough geometric description of this focal decomposition. It turns out to be somewhat similar to the one associated to the pendulum equation x¨+sin x = 0, worked out by Peixoto and Thom in [19, pp. 631, 197]. In the case L = (x˙2/2)−V(x) this type of focal decomposition is generic in some topological sense.

A new proof is given of a theorem of Menšov which states that a measurable function can be changed on a set of arbitrarily small measure to obtain a function with uniformly convergent Fourier series. Other results of Menšov on the representation of functions by trigonometric series are obtained by using a related result.

Let 2 [les ] m [les ] n. The paper gives necessary and sufficient conditions on the parameters s1, s2, …, sm, p1, p2, …, pm such that the Jacobian determinant extends to a bounded operator from [Hscr ]s1p1 × [Hscr ]s2p2 × … × [Hscr ]smpm into [Sscr ]′. Here all spaces are defined on ℝn or on domains Ω⊂ℝn. In almost all cases the regularity of the Jacobian determinant is calculated exactly.

Let T = {T(t)}t[ges ]0 be a C0-semigroup on a Banach space X. The following results are proved.

(i) If X is separable, there exist separable Hilbert spaces X0 and X1, continuous dense embeddings j0[ratio ]X0 → X and j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that j0 ∘ T0(t) = T(t) ∘ j0 and T1(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0.

(ii) If T is [odot ]-reflexive, there exist reflexive Banach spaces X0 and X1 , continuous dense embeddings j[ratio ]D(A2) → X0, j0[ratio ]X0 → X, j1[ratio ]X → X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that T0(t) ∘ j = j ∘ T(t), j0 ∘ T0(t) = T(t) ∘ j0 and T(t) ∘ j1 = j1 ∘ T(t) for all t [ges ] 0, and such that σ(A0) = σ(A) = σ(A1), where Ak is the generator of Tk, k = 0, [emptyv ], 1.

The structure of a collineation group G preserving a unital in a projective plane of order m2 with m ≡ 1(4) is investigated and several results are obtained about G, when G is a 2-group. These results are then used to investigate the structure of G/O(G) in the general case and, in particular, that of F*(G/O(G)).

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theorems for the volume of extrinsic balls in minimal submanifolds of space forms. These results were extended by S. Markvorsen for minimal submanifolds of a riemannian manifold with just an upper bound on the sectional curvature. In the paper an isoperimetric inequality for extrinsic balls in minimal submanifolds of a riemannian manifold N with sectional curvatures bounded from above by a non-positive constant is found. As a corollary of this result an alternative proof is obtained of the comparison for the volume of extrinsic balls stated by the preceding authors, but now the equality case is characterized when the upper bound for the sectional curvatures of the ambient manifold is strictly negative. Finally, when the sectional curvatures of N are bounded from above for any constant (positive or negative), it is proved that the ∞-isoperimetric quotient of the extrinsic balls is bounded from below by the mean curvature of the geodesic spheres in the m-dimensional real space forms.