Farah recently proved that many Borel P-ideals. on satisfy the following requirement: any measurable homomorphism has a continuous lifting which is a homomorphism itself. Ideals having such a property were called Radon–Nikodym (RN) ideals. Answering some Farah's questions, it is proved that many non-P ideals, including, for instance, Fin ⊗ Fin, are Radon–Nikodym. To prove this result, another property of ideals called the Fubini property, is introduced, which implies RN and is stable under some important transformations of ideals.

Radon's theorem [8] asserts that, if X is a finite set of s points in Rd and s ≥ d + 2, then X admits a. Radon partition, that is, a partition {X1; X2} of X into disjoint subsets X1 and X2, such that

It is shown that the cross-section body of a convex body K ⊂ ℝ3, that is the symmetric body which has for radial function in the direction u the maximal volume of a section of K by an hyperplane orthogonal to u, is a convex body in ℝ3.

The analytic paracommutators in the periodic case have been studied. Their boundedness, compactness, the Schatten-von Neumann properties and the cut-off phenomena have been proved. These results have been applied to some kind of operators on the Bergman spaces that have cut-off at any p∈(0, ∞).

This paper is concerned with infinitesimal transverse displacements of homogeneous isotropic elastic plates. The method uses moments of the fundamental equations of orders 0, 1, 2, 3. Assuming a form for the shear stresses tα3, these equations enable one to determine the mean values of the transverse displacements instead of the weighted mean values associated with plate theories of all but the classical type. The relevant moments of the stresses and displacements are expressed in terms of three functions satisfying three differential equations of the fourth order, the solutions of which may be expressed in terms of six independent functions. Thus six boundary conditions may be satisfied. Equating two, three and four of the above functions to zero in turn gives plate theories involving four, three and two boundary conditions respectively. The method is illustrated by assuming that the shear stresses are quadratic functions of the distance from the mid-plane of the material.

Let K be a field. We denote by K[t] the integral domain of all polynomials in an indeterminate t with coefficients in K, and by K(t) the quotient field of K[t], i.e. the field of all formal rational functions of t over K. A valuation |f| of the elements f of K(t) can be defined by

for f ≠ 0, and |0| = 0, where e > 1. This valuation is multiplicative, and has the properties

The existence of inductive limits in the category of (topological) measure spaces is proved. Next, permanence properties of inductive limits are investigated. If (X, , ) is the inductive limit of the measure spaces (X, , ), we prove, for 1 p 221E;, that LP(X, , ) is embeddible into the projectilimit of Lp(X, , ) in the category Ban, for p < , respectively in the category C* in the case p = +. As an application, we exten existence theorems of strong liftings to inductive limits.

Our result complements an interesting result of Roy O. Davies [1]; we assume familiarity with his paper. We use the details of the construction that he uses to prove his Theorem II.

We are concerned with invertible transformations of the unit n-dimensional cube In, 2 ≤ n ≤ ∞, which preserve n-dimensional Lebesgue measure μ. Following Halmos [4], we denote the space of all such transformations by G = G(In), and the subset of G consisting of homeomorphisms by M = M(In). We ask to what extent, and in what sense, can we approximate an arbitrary transformation g in G by a homeomorphism h in M. New results are obtained in the course of presenting a new proof of the theorem of J. Oxtoby and H. E. White, Jr., stated below.

The following theorem is typical of a group of results which have been found to be of importance in the theory of Cesàro summability.

Theorem A. If 0 ≤ ρ ≤ κ (κ, ρ integer), p > −1,p+q −1 and

and ifsn = 0(np) (C, κ), thensn εn = 0(np+q) (C, ρ).

Suppose that we have a system of congruences ai (mod ni) 1 < n1 < … < ni < … < nk such that every integer is congruent to at least one ai (mod ni), then we say that it is a covering system of congruences. If ni | m, 1 ≤ i ≤ k, we say that m is a covering number. We shall use the symbol ℕ to denote the natural numbers together with zero, then m is a covering number if, for each q there is an aq such that

The concept of mixed invariant set is due to Bandt [1], Bedford [2], Dekking [3, 4], Marion [4] and Schulz [10]. An m-tuple B = (B1, …, Bm) of closed and bounded subsets Bi of a complete finitely compact (bounded and closed subsets are compact) metric space X is called a mixed invariant set with respect to contractions f1, …, fm and a transition matrix M = (mij), if, and only if,

for every i ∈ {1, …, m}. In the papers quoted an essential condition is that all mappings f1, …, fm be contractions. We will show that, under certain conditions, the construction of mixed invariant sets also works in cases where some of the mappings are isometries or even expanding mappings.

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott–Halberstam conjecture, we show that all large natural numbers $n$ with $8\nmid n$, $n\not\equiv 2~(\text{mod} ~3)$ and $n\not\equiv 14~(\text{mod} ~16)$ are the sum of two squares and three biquadrates.

We prove that the exponent of distribution of $\unicode[STIX]{x1D70F}_{3}$ in arithmetic progressions can be as large as $\frac{1}{2}+\frac{1}{34}$, provided that the moduli is squarefree and has only sufficiently small prime factors. The tools involve arithmetic exponent pairs for algebraic trace functions, as well as a double $q$-analogue of the van der Corput method for smooth bilinear forms.

We prove bounds for the truncated directional Hilbert transform in $L^{p}(\mathbb{R}^{2})$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.

We consider a class of singular Schrödinger operators H that act in L2(0,∞), each of which is constructed from a positive function ϕ on (0,∞). Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of ϕ. For a large class of H that have discrete spectrum, we prove that the eigenvalue asymptotics of H does not depend on rapid oscillations of ϕ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

We examine correlations of the Möbius function over $\mathbb{F}_{q}[t]$ with linear or quadratic phases, that is, averages of the form 1

$$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f<n}\unicode[STIX]{x1D707}(f)\unicode[STIX]{x1D712}(Q(f))\end{eqnarray}$$

$\unicode[STIX]{x1D712}$

$\mathbb{F}_{q}$

$Q\in \mathbb{F}_{q}[x_{0},\ldots ,x_{n-1}]$

$x_{0},\ldots ,x_{n-1}$

$f=\sum _{i<n}x_{i}t^{i}$

$\unicode[STIX]{x1D707}$

$O_{\unicode[STIX]{x1D716}}(q^{(-1/4+\unicode[STIX]{x1D716})n})$

$\unicode[STIX]{x1D716}>0$

$Q$

$O(q^{-n^{c}})$

$c>0$

$Q$

$O(q^{-c^{\prime }n})$

$c^{\prime }>0$

$Q(f)$

$f^{2}$

This paper extends Hlawka’s theorem (from the point of view of Siegel and Weil) on SL(n,ℝ)/SL(n,ℤ) to Sp(n,ℝ)/Sp(n,ℤ). Namely, if Vn=vol(Sp(n,ℝ)/Sp(n,ℤ), where the measure is the Sp(n,ℝ)-invariant measure on Sp(n,ℝ)/Sp(n,ℤ), then Vn can be expressed in terms of the Riemann zeta function by As a consequence, let D be a domain of a sufficiently regular set in ℝ2n. Then:

1. (i) if vol(D)>Vn, then some lattice in ℝ2n contains a non-zero point of D;

2. (ii) if vol(D)<Vn, then some lattice in ℝ2n contains only the zero point of D;

3. (iii) if D is star-shaped about the origin and vol(D)<ζ(2n)Vn, then some lattice in ℝ2n contains only the zero point of D.

We study the metric entropy of the metric space ${\mathcal{B}}_{n}$ of all $n$-dimensional Banach spaces (the so-called Banach–Mazur compactum) equipped with the Banach–Mazur (multiplicative) “distance” $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({\mathcal{B}}_{n},d,1+{\it\varepsilon})$ is the smallest number of “balls” of “radius” $1+{\it\varepsilon}$ that cover ${\mathcal{B}}_{n}$, then for any ${\it\varepsilon}>0$ we have

$$\begin{eqnarray}0<\liminf _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})\leqslant \limsup _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})<\infty .\end{eqnarray}$$

$n$

$d_{N}$

$N\times N$

$N$

$N$

$N=1$

$n$

$N\times N$

One of the oldest questions concerning primitive roots is that of the actual order df magnitude of the least positive primitive root m(p) to a large prime modulus p. During the last sixty years the interest of several mathematicians has been attracted by this problem. The history of the major results is too well known to need elaboration. The best known bound for m(p) is due to Wang [8] and Burgess [1] who independently obtained