§1. Let f(x) = x′Ax be a positive definite or semi–definite n-ary quadratic form with real symmetric matrix A. Then, f is Minkowski-reduced, if for all sets of integers m1, …, mn with gcd (mi, …, mn) = 1,

We show that a measure on ℝd is linearly rectifiable if, and only if, the lower l-density is positive and finite and agrees with the lower average l-density almost everywhere.

We consider a tiling of a square with convex tiles of areas a1,… aN and perimeters p1,…,pN and pose the following problems.

PROBLEM 1. For given values of a1, …, aN find the minimum of P = p1 + … + pN.

PROBLEM 2. For given values of p1,…,pN find the maximum of A = a1+…+aN.

We provide a unified and simplified proof that for any partition of (0, 1] into sets that are measurable or have the property of Baire, one cell will contain an infinite sequence together with all of its sums (finite or infinite) without repetition. In fact any set which is large around 0 in the sense of measure or category will contain such a sequence. We show that sets with 0 as a density point have very rich structure. Call a sequence and its resulting all-sums set structured provided for each We show further that structured all-sums sets with positive measure are not partition regular even if one allows shifted all-sums sets. That is, we produce a two cell measurable partition of (0, 1 ] such that neither set contains a translate of any structured all-sums set with positive measure.

The classification of conies in the real or complex projective plane under the action of the appropriate group is simple and well known. We consider here the more complicated question of classifying conies in the complex projective plane ℂℙ2 under the action of the real projective group PGL (3, ℝ).

Let be a plane bounded convex set whose width in the direction φ is . It has been shown by L. A. Santaló that

is invariant under unimodular affine transformations. Santaló [1] established a number of properties of this invariant and conjectured that if the area of is A() then and that equality characterizes triangles. In this note Santaló's conjecture is shown to be true.

New upper and lower bounds are given for Fh(g, N), the maximum size of a Bh[g] sequence contained in [1, N]. It is proved that and that

and that, for any ε > 0 and g > g(ε, h),

Let R be a ring, not necessarily commutative, with an identity element, and let A be a left R-module. We shall describe this situation by writing (RA). If

is an exact sequence of left. R-modules and R-homomorphisms in which each Pi (i ≥ 0) is R-projective, then the sequence

which we denote by P, is called an R-projective resolution of A. Suppose now that A is non-trivial; if Pi = 0 when i > n and if there are no R-projective resolutions of A containing fewer non-zero terms, then A is said to have left projective (or homological) dimension n, and we write 1.dim RA = n. If no finite resolutions of this type exist, we write l.dim RA = ∞. As a convention, we put l.dim R0 = −1. If M denotes a variable left R-module, then is called the left global dimension of the ring R and is denoted by l.gl. dim R. It is well known that l.dim RA < n if and only if for all left R-modules B and that l.gl.dim R < m if and only if regarded as a functor of left R-modules, takes only null values.

A metric is defined for a plane affine geometry with coordinates in GF(2n) and is applied to the geometry of a triangle.

§1. Let k be an algebraic number field of finite degree over the field Q of rational numbers and Ki be an extension of k of degree (Ki:k) = ni, i = 1,2. We choose a Hecke Grössencharakter Xi in Ki and consider the L-function

associated with Xi see [1, 2]. It is known to be a meromorphic function on the whole complex plane. We are interested here in the properties of the convolution of functions LK1, LK2 over k defined by

where

and a (and n) run over integral ideals of Ki (and k), whose norm NKi/k a is equal to n. The function (2) is sometimes called the “scalar product of the Hecke L-functions” «3—11». The object of the paper is the following theorem.

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_{K}^{m}$ satisfies the Hasse principle, weak approximation, and the Manin–Peyre conjecture if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner’s number field version of the Hardy–Littlewood circle method. As a by-product, we point out and correct an error in Skinner’s treatment of the singular integral.

It is shown that a separable Hilbert space can be covered by non-overlapping closed convex sets Ci with outer radii uniformly bounded from above and inner radii uniformly bounded from below. This answers a question originating from the work of Klee.

Barannikov and Kontsevich [Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Not. IMRN1998(4) (1998), 201–215], constructed a DGBV (differential Gerstenhaber–Batalin–Vilkovisky) algebra $\mathbf{t}$ for a compact smooth Calabi–Yau complex manifold $M$ of dimension $m$, which gives rise to the $B$-side formal Frobenius manifold structure in the homological mirror symmetry conjecture. The cohomology of the DGBV algebra $\mathbf{t}$ is isomorphic to the total singular cohomology $H^{\bullet }(M)=\bigoplus _{k=0}^{2m}H^{k}(M,\mathbb{C})$ of $M$. If $M=X_{G}(\mathbb{C})$, where $X_{G}$ is the hypersurface defined by a homogeneous polynomial $G(\text{}\underline{x})$ in the projective space $\mathbb{P}^{n}$, then we give a purely algorithmic construction of a DGBV algebra ${\mathcal{A}}_{U}$, which computes the primitive part $\bigoplus _{k=0}^{m}\mathbf{PH}^{k}$ of the middle-dimensional cohomology $\bigoplus _{k=0}^{m}H^{k}(M,\mathbb{C})$, using the de Rham cohomology of the hypersurface complement $U_{G}:=\mathbb{P}^{n}\setminus X_{G}$ and the residue isomorphism from $H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$ to $\mathbf{PH}^{k}$. We observe that the DGBV algebra ${\mathcal{A}}_{U}$ still makes sense even for a singular projective Calabi–Yau hypersurface, i.e. ${\mathcal{A}}_{U}$ computes $\bigoplus _{k=0}^{m}H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$ even for a singular $X_{G}$. Moreover, we give a precise relationship between ${\mathcal{A}}_{U}$ and $\mathbf{t}$ when $X_{G}$ is smooth in $\mathbf{P}^{n}$.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

In this paper we shall consider some axisymmetrical problems involving the determination of a harmonic function which satisfies prescribed conditions on two given spherical surfaces. The latter may be exterior to one another or one inside the other. The classical problems which come under this heading are the electrostatic two sphere condenser and the motion of two spheres along the line of centres in an unbounded inviscid fluid. The early investigators of these problems used the well known method of images (Kelvin [1], Hicks [2]). In the electrostatic case (Dirichlet boundary conditions) the images are sets of point monopoles, and in the hydrodynamic case (Neumann boundary conditions) sets of dipole sources. Later Neumann [3] and Jeffery [4] gave solutions using a bipolar coordinate system.

When K is a convex body in d-dimensional euclidean space E and 0 ≤ s ≤ d, the s-skeleton of K, denoted skel(s)K, consists of those points of K which are not centres of (s + l)-dimensional balls contained in K. The s-skeleton is thus the union of the extreme faces of K having dimension at most s. The s-skeleton is a -set [see 6] and it is therefore measurable with respect to the s-dimensional Hausdorff measure ℋ(s) [see 7]; here we normalize ℋ(s) so that it assigns unit measure to the s-dimensional unit cube.

It is shown that every Banach space with a norming Markushevich basis has a strong Markushevich basis. It is also shown that the converse is false.

By a well-known theorem of Szemerdi 8 any set of integers that has positive density contains arithmetic progressions of arbitrary length. One might expect that there are conditions of similar generality, under which an integer set contains arbitrarily long strings of consecutive integers, i.e., arithmetic progressions with 1 as common difference. Results of this kind would be of great importance because of potential applications to arithmetically interesting sets such as the set n: (n) = 1, where (n) is the Liouville function, or the sets

where P(n) denotes the greatest prime factor of n and 0< < 1. One naturally expects that such sets contain arbitrarily long strings of consecutive integers, but no results in this direction are known, and the problem seems to be a very difficult one, perhaps comparable in depth to the prime k-tuple conjecture.

In a recent article [3] L. Mirsky proved a theorem which gives necessary and sufficient conditions for the existence of a finite integral matrix whose elements, row sums, and column sums all lie within prescribed bounds. Mirsky suggested to me the problem of extending his theorem to infinite matrices, and it is the solution of this problem that is presented in this note. To allow for extra generality, instead of prescribing upper and lower bounds for the row and column sums we shall prescribe upper and lower bounds for the row and column deficiencies (a term to be explained later). The theorem when upper and lower bounds for the row and column sums are prescribed is then a special case of the deficiency theorem. The solution of the problem depends on a construction of Mirsky [3] and a theorem of mine [1] concerning the existence of a partial transversal of a family of sets satisfying certain properties. As will be seen, we shall take a rather broad view of the notion of a matrix.

When a weak rotlet and a circular cylinder rotate together in a viscous fluid at low Reynolds number R, the Stokes' flow solution indicates a uniform stream as the radial distance r tends to infinity. It is shown here, when R is distinctly non-zero, that the flow is modified to form a spiral motion in the domain where R In r = O(l), but is not damped until the more distant domain R2 In r = O(l).