We study the minimal length of faithful nuclear representations of operators acting between finite-dimensional Banach spaces and the related minimal number of contact points of the John ellipsoid of maximal volume contained in the unit ball of a finite-dimensional Banach space. In both cases the classical upper estimates, which follow from the Caratheodory theorem, are shown to be exact. Related isometric characterizations of are proved.

It was shown by Davenport and Roth [7] that the values taken by

at integer points ( x1, …, x8) ∈ ℤ8 are dense on the real line, providing at least one of the ratios λi/λj, is irrational. Here and throughout, λi denote such nonzero real numbers. More precisely, Liu, Ng and Tsang [8] showed that for all the inequality

has infinitely many solutions in integers. Later Baker [1] obtained the same result in the enlarged range . In this note we improve this further, the progress being considerable.

The two examples of fluid motion in a container which are described in this paper can be easily demonstrated in any kitchen. The first motion was noticed by Professor C. A. Rogers while attempting to dissolve chlorine tablets in water to improve its drinkability. The water nearly filled a cylindrical jar and he had shaken it, with the axis of the jar horizontal, in such a way that the water had a considerable angular momentum about the axis. When the axis of the jar was suddenly moved into the vertical position, he noticed that the water was now rotating about the vertical, which prompted the question of the source of this vertical component of angular momentum. A simplified version of this motion is determined mathematically in §2, and the observations are found to be in general agreement with the theoretical prediction.

We examine the asymptotic form of a fundamental set of solutions of the third-order equation

as x → ∞, where the leading coefficient q is nowhere zero in some interval [a, ∞). The equation is self-adjoint in the case when p, r and iq are real. However, our analysis is not confined to this case, and we generally take the coefficients to be complex-valued.

In this paper, we investigate in various ways the representation of a large natural number as a sum of a fixed power of Piatetski-Shapiro numbers, thereby establishing a variant of the Hilbert–Waring problem with numbers from a sparse sequence.

Recently, an analogue over $\mathbb{F}_{q}[T]$ of Landau’s theorem on sums of two squares was considered by Bary-Soroker, Smilansky and Wolf. They counted the number of monic polynomials in $\mathbb{F}_{q}[T]$ of degree $n$ of the form $A^{2}+TB^{2}$, which we denote by $B(n,q)$. They studied $B(n,q)$ in two limits: fixed $n$ and large $q$; and fixed $q$ and large $n$. We generalize their result to the most general limit $q^{n}\rightarrow \infty$. More precisely, we prove

$$\begin{eqnarray}B(n,q)\sim K_{q}\cdot \binom{n-\frac{1}{2}}{n}\cdot q^{n},\quad q^{n}\rightarrow \infty ,\end{eqnarray}$$

$K_{q}=1+O(1/q)$

Let $A\subset \{1,\dots ,N\}$ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density ${\it\alpha}=|A|/{\it\pi}(N)$, where ${\it\pi}(N)$ denotes the number of primes in the set $\{1,\dots ,N\}$. By modifying Helfgott and De Roton’s work [Improving Roth’s theorem in the primes. Int. Math. Res. Not. IMRN2011 (4) (2011), 767–783], we improve their bound and show that

$$\begin{eqnarray}{\it\alpha}\ll \frac{(\log \log \log N)^{6}}{\log \log N}.\end{eqnarray}$$

We suggest an alternative mathematical model for the electron in dimension 1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose material points can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose a coframe and a density. We then add an extra (third) spatial dimension, extend our coframe and density into dimension 1+3, choose a conformally invariant Lagrangian proportional to axial torsion squared, roll up the extra dimension into a circle so as to incorporate mass and return to our original (1+2)-dimensional spacetime by separating out the extra coordinate. The main result of our paper is the theorem stating that our model is equivalent to the Dirac equation in dimension 1+2. In the process of analysing our model we also establish an abstract result, identifying a class of nonlinear second order partial differential equations which reduce to pairs of linear first order equations.

In Section 1 below I describe two measures of the complexity of a binary relation. J The theorem says that these two measures never disagree very much. Both measures of complexity arose in connection with Saharon Shelah's notion [5] of a stable firstt order theory; Shelah showed in effect that one measure is finite, if, and only if, the other is finite too. This follows trivially from the theorem below. I confess my main aim was not to get the extra information which the theorem provides, but to eliminate Shelah's use of uncountable cardinals, which seemed strangely heavy machinery for proving a purely finitary result. Section 2 below explains the modeltheoretic setting.

In the kernel clustering problem we are given a large n × n positive-semidefinite matrix A = (aij) with and a small k × k positive-semidefinite matrix B = (bij). The goal is to find a partition S1, …, Sk of {1, … n} which maximizes the quantity

We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song et al. In some cases we manage to compute the sharp approximation threshold for this problem assuming the unique games conjecture (UGC). In particular, when B is the 3 × 3 identity matrix the UGC hardness threshold of this problem is exactly 16π/27. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when B is the k × k identity matrix is (8π/9)(1 – 1/k) for every k ≥ 3.

§1. Introduction, By a remarkable result of Erdos and Selfridge [3] in 1975. the diophantine equation

with integers k≥2 and m≥2, has only the trivial solutions. x = −j(j = i, …, m), y = 0. This put an end to the old question whether the product of consecutive positive integers could ever be a perfect power; for a brief account of its history see [7].

A local $\mathit{Tb}$ theorem is an $L^{2}$ boundedness criterion by which the question of the global behavior of an operator is reduced to its local behavior, acting on a family of test functions $b_{Q}$ indexed by the dyadic cubes. We present two versions of such results, in particular, treating square function operators whose kernels do not satisfy the standard Littlewood–Paley pointwise estimates. As an application of one version of the local $\mathit{Tb}$ theorem, we show how the solvability of the Kato problem (which was implicitly based on local $\mathit{Tb}$ theory) may be deduced from this general criterion.

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

On a convex surface S ⊂ Rd, two points x, y are conjugate if there are at least two shortest paths, called segments, from x to y. This paper is about the set of points conjugate to some fixed point xєS.

An original linear algebraic approach to the basic notion of Freiman's isomorphism is developed and used in conjunction with a combinatorial argument to answer two questions, posed by Freiman about 35 years ago.

First, the order of growth is established of t(n), the number of classes isomorphic n-element sets of integers: t(n) = n(2 + σ(1))n. Second, it is proved linear Roth sets (sets of integers free of arithmetic progressions and having Freiman rank 1) exist and, moreover, the number of classes of such cardinality n is amazingly large; in fact, it is “the same as above”: .

§1. Let E = E(A) be the set of real numbers x ε (0, 1) whose regular continued fraction expansion

contains only partial denominators ai from a given set A of positive integers. For finite A the (Hausdorff-) dimensional numbers dim E have been studied by I. J. Good ‘2’ and T. W. Cusick ‘1’. C. A. Rogers ‘8’ introduced a natural probability measure on E. He showed that excluding sets with measure between 0 and 1 (in the strict sense) from E does not reduce the dimensionality more than excluding sets of measure zero, and that the minimal (“essential”) dimensions ess dim E arising in this way is smaller than dim E, at least for A = {1,…, r} when r = 2 or r is sufficiently large.

Our main object in this note is to establish (Theorem 1) a necessary and sufficient condition to be satisfied by a sequence {εn} so that a series Σ an εnmay be summable | A |whenever the series Σanis summable (C, — 1). We suppose that an and εn are complex numbers. The condition is unchanged if the an are restricted to be real, but our proof is adapted to the case where they may be complex. Theorem 1 has been quoted by Bosanquet and Chow [12] in order to fill a gap in the theory of summability factors. We also obtain some related results, which are discussed in the Appendix.

Let K/k be a normal extension of algebraic number fields whose Galois group G is a Frobenius group. Then K/k is said to be a Frobenius extension. Most of the structure of the unit group and of the ideal class group of K is determined by that; of the subfields fixed by the Frobenius kernel N and by a complement F. Here this is investigated when G is a maximal or metacyclic Frobenius group. In particular, the results apply firstly to the normal closure of where a ∊ k and p is a rational prime, and, secondly, when G is a dihedral group of order 2n for an odd integer n. A. Scholz, taking n = p = 3, was the first to consider this problem.

The following theorem is proved in [2[.

Let K be a finitely generated field over its prime field. Then for almost all e-tuples σ = (σ1, …, σe) of elements of the abstract Galois group G(K) of K and for all elliptic curves E defined over the following results hold.

Let X be a completely regular space, and Cb(X) the space of all bounded continuous real valued functions on X equipped with the metric associated to the uniform norm. For f∈Cb(X) and γ∈¡ we use the following standard notations: inf(f) = infx∈Xf(x) and {f<γ} = {x∈X:f(x)<γ}