A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0\leqslant p\leqslant n-1$. Using local parallel sets, Hinderer constructed examples of $\text{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\text{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\text{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger’s theorem.

Much recent attention has been given to geometric representation of elements of the stable homotopy groups of spheres, π*s A particular example concerns non-singular bilinear maps ℝm+1 × ℝn+1 → ℝm+n+1−p; on restriction and normalisation these become biskew maps Sm × Sn ℝ Sm+n-p;. Now the Hopf construction ℋ applied to any map f: Sm × Sn → Sm+n-p yields

Let Ω be a compact metric space of non-σ-finite Λh-measure, where Λh denotes the Hausdorff measure corresponding to some continuous increasing real function h(t), denned for t ≥ 0 and with h(t) > 0 and h(t) > 0 for t > 0. C. A. Rogers has drawn attention to two problems that have remained unsolved for some years, and which seem to present considerable difficulty:

I. Does Ω necessarily have subsets of finite positive Λh-measure?

II. Does Ω. necessarily contain a system ofdisjoint closed subsets each of non-σ-finite Λh-measure ?

The answers are affirmative when Ω is a subset of a Euclidean space (see [1], [2], [3], [4], [5]); recently D. G. Larman ([8], [9]) was able to extend each of these results to the case of the “finite-dimensional” compact metric spaces that he introduced in [7]. Both of his proofs depend on the detailed structure of such spaces; the aim of the present paper is to establish a general theorem, from which it follows in particular that Larman's second result is a consequence of his first, and that an affirmative answer to I would imply an affirmative answer to II. (Since it seems likely that I must be answered negatively in general, and II affirmatively, the present theorem may well be of only ephemeral significance; the nature of the proof reinforces this expectation.)

One may perhaps doubt whether in the geometry of numbers any particular family of lattices deserves such an attention as, for example, BCH codes receive in coding theory. However, only recently a quite interesting family has emerged. The general case of these lattices considered by Rosenbloom and Tsfasman [5, Section 2] parallels Goppa's construction of codes from algebraic curves. Here we shall take a closer look at the case of genus zero where some special features of Goppa's early codes will show up again: There is a lattice Λ (L, g) in n-dimensional euclidean space associated with a subset L of the field, and a polynomial g satisfying g(є) ≠ for all λ є L. For g = zd previously known sphere packings are recovered and generalized. A nonconstructive argument shows that for n → ∞ and some irreducible polynomials g Minkowski's lower packing bound is met (this being not achieved in [5] where q is fixed, but the genus grows; cf. also [4]).

In this paper, we prove that if $X$ is an infinite-dimensional real Hilbert space and $J: X\rightarrow \mathbb{R} $ is a sequentially weakly lower semicontinuous ${C}^{1} $ functional whose Gâteaux derivative is non-expansive, then there exists a closed ball $B$ in $X$ such that $(\mathrm{id} + {J}^{\prime } )(B)$ intersects every convex and dense subset of $X$.

In the middle of the last century, Kummer's studies on the famous Fermat conjecture led him to the question: when does a given prime p > 2 divide the class number of the p-th cyclotomic field? His conclusion was that this happens, if, and only if, p divides at least one of the Bernoulli numbers B2, B4,…, Bp_3. Such a prime is called irregular. Carlitz [1] has given the simplest proof of the fact that the number of irregular primes is infinite. However, it is not known whether there are infinitely many regular primes.

One of the substantial differences between real and complex analysis is the behaviour of pointwise sequential limits of functions. It is well known that, if f(z) is a bounded analytic function in D = {z∈ C: |z| < 1}, then there exists a sequence {pn(z): n = 1,2,…} of polynomials such that

(i) ‖Pn‖ ≤ ‖f‖ for all n = 1,2,3,…, and

(ii) for each z ∈ D, Pn(Z) → f(z) as n → ∞,

where we have used the notation

Let $K$ be an algebraic number field of degree $d\geqslant 3$, $\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$ the embeddings of $K$ into $\mathbb{C}$, $\unicode[STIX]{x1D6FC}$ a non-zero element in $K$, $a_{0}\in \mathbb{Z}$, $a_{0}>0$ and

$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$

$\unicode[STIX]{x1D710}$

$K$

$a\in \mathbb{Z}$

$F_{0}(X,Y)\in \mathbb{Z}[X,Y]$

$\unicode[STIX]{x1D710}^{a}$

$a\in \mathbb{Z}$

$\unicode[STIX]{x1D710}$

$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$

$m>0$

$(x,y,a)\in \mathbb{Z}^{3}$

$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$

$xy\not =0$

$\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$

$m$

$K$

$F_{0}$

$\unicode[STIX]{x1D710}$

$d$

The following problem was proposed by Professor N. J. Fine: to prove that there do not exist rational functions F1, F2, F3 of x1, x2, x3, x4, with real coefficients, such that

identically. In the present note I give a proof.†

Large sieve inequalities have been developed and applied to a host of arithmetical problems since their inception by Linnik in 1941. Such inequalities provide mean square estimates for a trigonometric polynomial over a set of well-spaced points. In particular, let x ∈ ℝ and let

The following theorem, due to Artin [1], is one of a number of statements often referred to as Chebotarev density theorem.

The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

In this paper we study the Duffin–Schaeffer conjecture, which claims that $\unicode[STIX]{x1D706}(\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n})=1$ if and only if $\sum _{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n})=\infty$ , where $\unicode[STIX]{x1D706}$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$ ,

$$\begin{eqnarray}{\mathcal{E}}_{n}={\mathcal{E}}_{n}(\unicode[STIX]{x1D713})=\mathop{\bigcup }_{\substack{ m=1 \\ (m,n)=1}}^{n}\bigg(\frac{m-\unicode[STIX]{x1D713}(n)}{n},\frac{m+\unicode[STIX]{x1D713}(n)}{n}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D713}$

$\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n}$

$\bigcup _{n=1}^{\infty }{\mathcal{E}}_{n}$

$C>0$

$$\begin{eqnarray}\unicode[STIX]{x1D706}\bigg(\mathop{\bigcup }_{n=1}^{\infty }{\mathcal{E}}_{n}\bigg)\geqslant C\min \bigg\{\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n}),1\bigg\}.\end{eqnarray}$$

$p$

Let $C\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ be a cubic form. Assume that $C$ splits into four forms. Then $C(x_{1},\ldots ,x_{n})=0$ has a non-trivial integer solution provided that $n\geqslant 10$.

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in ℝ n with smaller i-dimensional central sections necessarily have smaller volume. A generalization of this problem is studied, when the volumes are measured with weights satisfying certain conditions. The case of hyperplane sections (i = n − 1) has been studied by A. Zvavitch.

Let where a0≠0, m≥2, n = e1 + … + em and the ξi(1≤i≤m) are the distinct zeros of f in some algebraic closure of the p-adic field . Then K = (ξ1, ξ2, …, ξm) is a finite separable extension of and we denote by “ord” the unique extension of the (additive) p-adic valuation on to K, normalized with ord p = 1.

In this paper we discuss some dimension results for triangle sets of compact sets in $\mathbb{R}^{2}$. In particular we prove that for any compact set $F$ in $\mathbb{R}^{2}$, the triangle set $\unicode[STIX]{x1D6E5}(F)$ satisfies

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant {\textstyle \frac{3}{2}}\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>1$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant 1+\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>4/3$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant \min \{{\textstyle \frac{5}{2}}\dim _{\text{A}}F-1,3\}.\end{eqnarray}$$

$F$

$$\begin{eqnarray}\text{}\underline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\text{}\underline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F)\quad \text{and}\quad \overline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\overline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F).\end{eqnarray}$$

Let G1,… Gn be groups, let *Gi be their free product, and let ´ Gi be their direct product. A homomorphism may be defined by requiring it to be trivial on Gj and the identity on Gi for i ¹ j. Let [G1 … Gn] = Çker pj. P. J. Hilton [2] proves that [G1, …, Gn] is a free group, and, if HiÌGi, i = 1, …, n, that [H1,… Hn] is a free factor of [G1 … Gn]. He asks whether, if Hλi Ì Gi, λ = 1, …, k, i = 1, …, n, and Hλ = [Hλ1, …, Hλn] the group generated by the Hλ is a free factor of [G1, …, Gn].

Ramsey's theorem implies that every function f:0, 1 ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r ). A typical result: if f is bounded from below and is not convex on any infiniteset then there exists an interval on which the graph of f can be covered by the graphs of countably many strictly concave functions.

Let A = {ala2,…, an} be a finite set of (not necessarily distinct) positive integers and

be the corresponding set of multiples. My primary object here is to show that in fairly general circumstances there are significant irregularities in B(A), regarded as an ordered sequence.