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 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.

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but O(N1/3+ε) of the natural numbers not exceeding N.

Let K be a convex body of dimension at least 3, and let p0 be a point. If every section of K through p0 is centrally symmetric, then Rogers proved in [6] that K is centrally symmetric, although p0 may not be the centre of K. If this is the case, then Aitchison, Petty and Rogers [1] and Larman [2] proved that K must be an ellipsoid. Suppose now that, for every direction, we can choose continuously a section of K that is centrally symmetric; if K is strictly convex, then Montejano [3] proved that K must be centrally symmetric. Consider now the following example. Let D be a (euclidean) ball centred at the origin from which two symmetric caps are deleted. Then D is centrally symmetric with respect to the origin, and has a lot of circular sections whose centre is not the origin. In fact, we can choose continuously, for every direction, a section of D which is centrally symmetric, in such a way that not all these sections pass through the origin. Nevertheless, no matter how we choose these sections, there are always necessarily many of them that do pass through the origin. For those sections, of course, we have not imposed any condition, which explains the fact that D is not a quadric elsewhere.

It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space $X$ cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case for an arbitrary Banach space: there exists a separable, reflexive space $X$ and an unbounded, densely defined operator acting in $X$ with a compact resolvent whose norm is constant in a neighbourhood of zero; moreover $X$ is isometric to a Hilbert space on a subspace of co-dimension $2$. There is also a bounded linear operator acting on the same space whose resolvent norm is constant in a neighbourhood of zero. It is shown that similar examples cannot exist in the co-dimension $1$ case.

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.

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 $.

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 $\mathbf{P}^{n}$.

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.

One construction used to produce a random number table is to take a smooth function F(x) taking values between 0 and 1, to evaluate it at N points spaced 1/M apart, and to ignore the first t decimal digits. With T = 10t this corresponds to taking the fractional part of

where T>M>N. The grounds for assuming this sequence to be random are that it is so difficult to prove anything about it.

Green [8] has shown that a constitutive relation of the form

arises as a special case of an incompressible anisotropic simple fluid, where S is the stress tensor or matrix,

and V is the velocity gradient matrix at time t, all measured in a fixed rectangular cartesian coordinate system. Also, if F is the displacement gradient measured with respect to some curvilinear reference system θi, then

where R is a proper orthogonal matrix, and M and K are positive definite symmetric matrices. In addition

and

The impaction on symmetrical obstacles placed in uniform streams of aerosols is investigated. The governing equations of motion are nonlinear differential equations involving a parameter called the Stokes number. The study differentiates between the critical value of the Stokes number on the centre-line, kcr, below which no particles reach the stagnation point in finite time, and the critical value of the Stokes number on the obstacle, Kcr, below which no particles may be deposited on the obstacle in finite time. Based on the properties of the centre-line fluid velocity of the potential and viscous flows past a variety of symmetrically shaped obstacles, upper and lower bounds of Kcr and Kcr are established. Furthermore, using a numerical procedure for solving nonlinear differential equations with unknown parameters the critical values of Kcr and Kcr are obtained.

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),

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.

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.

Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure of K. For an element α ∈ \K, a chain α = α0, α1,…,αr of elements of , such that αi is of minimum degree over K with the property that ῡ(αi−1 − αi) = sup{ῡ(αi−1 − β) | [K (β) : K] < [K (αi−1) : K]} and that αr ∈ K, is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of (cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.

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).

Cardinal functions of topologies have been extensively studied. Cardinal functions of measures have attracted less interest, perhaps because there are fewer straightforward results which are independent of special axioms. In this paper I consider the “additivity” and “cofinality” of a measure (Definition 1) and show that they can often be calculated in terms of certain fundamental cardinals (Corollary 11 and Theorem 16).

§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.