Criteria for 2 to be an e-th power residue of a prime

p ≡ 1 (mod e = ef+1,

have been obtained in various forms for e = 2, 3, 5. Euler proved the well known result that 2 is a quadratic residue of a prime p ≡ ± l (mod 8). Dickson [1] showed that 2 is a cubic residue of p ≡ 1 (mod 3) if and only if p = L2+27M2 is soluble in integers L, M.

A double-normal pair of a finite set $S$ of points that spans $\mathbb{R}^{d}$ is a pair of points $\{\mathbf{p},\mathbf{q}\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $\mathbf{p}$ and $\mathbf{q}$ perpendicular to $\mathbf{p}\mathbf{q}$. A double-normal pair $\{\mathbf{p},\mathbf{q}\}$ is strict if$S\setminus \{\mathbf{p},\mathbf{q}\}$ lies in the open strip. The problem of estimating the maximum number $N_{d}(n)$ of double-normal pairs in a set of $n$ points in $\mathbb{R}^{d}$, was initiated by Martini and Soltan [Discrete Math. 290 (2005), 221–228]. It was shown in a companion paper that in the plane, this maximum is $3\lfloor n/2\rfloor$, for every $n>2$. For $d\geqslant 3$, it follows from the Erdős–Stone theorem in extremal graph theory that $N_{d}(n)=\frac{1}{2}(1-1/k)n^{2}+o(n^{2})$ for a suitable positive integer $k=k(d)$. Here we prove that $k(3)=2$ and, in general, $\lceil d/2\rceil \leqslant k(d)\leqslant d-1$. Moreover, asymptotically we have $\lim _{n\rightarrow \infty }k(d)/d=1$. The same bounds hold for the maximum number of strict double-normal pairs.

For a non-empty polyhedral set $P\subset \mathbb{R}^{d}$, let ${\mathcal{F}}(P)$ denote the set of faces of $P$, and let $N(P,F)$ be the normal cone of $P$ at the non-empty face $F\in {\mathcal{F}}(P)$. We prove the identity

$$\begin{eqnarray}\mathop{\sum }_{F\in {\mathcal{F}}(P)}(-1)^{\operatorname{dim}F}\unicode[STIX]{x1D7D9}_{F-N(P,F)}=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }P\text{ is bounded},\\ 0\quad & \text{if }P\text{ is unbounded and line-free}.\end{array}\right.\end{eqnarray}$$

$0$

A small value estimate is a statement providing necessary conditions for the existence of certain sequences of non-zero polynomials with integer coefficients taking small values at points of an algebraic group. Such statements are desirable for applications to transcendental number theory to analyze the outcome of the construction of an auxiliary function. In this paper, we present a result of this type for the product $ \mathbb {G}_{\mathrm {a}}\times \mathbb {G}_{\mathrm {m}}$ whose underlying group of complex points is $\mathbb {C}\times \mathbb {C}^{*}$. It shows that if a certain sequence of non-zero polynomials in $ \mathbb {Z}[X_1,X_2]$ takes small values at a point $(\xi ,\eta )$ together with their first derivatives with respect to the invariant derivation $\partial /\partial X_1 + X_2 (\partial /\partial X_2)$, then both $\xi $ and $\eta $ are algebraic over $\mathbb {Q}$. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on the absolute values of their derivatives. It improves on a direct application of Philippon’s criterion for algebraic independence and compares favorably with constructions coming from Dirichlet’s box principle.

The paper is devoted to the study of Fefferman–Stein inequalities for stochastic integrals. If $X$ is a martingale, $Y$ is the stochastic integral, with respect to $X$, of some predictable process taking values in $[-1,1]$, then for any weight $W$ belonging to the class $A_{1}$ we have the estimates $\Vert Y_{\infty }\Vert _{L^{p}(W)}\leqslant 8pp^{\prime }[W]_{A_{1}}\Vert X_{\infty }\Vert _{L^{p}(W)},$$1<p<\infty ,$ and $\Vert Y_{\infty }\Vert _{L^{1,\infty }(W)}\leqslant c[W]_{A_{1}}(1+\log [W]_{A_{1}})\Vert X_{\infty }\Vert _{L^{1}(W)}.$ The proofs rest on the Bellman function method: the inequalities are deduced from the existence of certain special functions, enjoying appropriate majorization and concavity. As an application, related statements for Haar multipliers are indicated. The above estimates can be regarded as probabilistic counterparts of the recent results of Lerner, Ombrosi and Pérez concerning singular integral operators.

In this paper, we study how small a box contains at least two points from a modular hyperbola $xy\equiv c\;(\text{mod}\;p)$ . There are two such points in a square of side length $p^{1/4+\unicode[STIX]{x1D716}}$ . Furthermore, it turns out that either there are two such points in a square of side length $p^{1/6+\unicode[STIX]{x1D716}}$ or the least quadratic non-residue is less than $p^{1/(6\sqrt{e})+\unicode[STIX]{x1D716}}$ .

Let be the class of non-constant functions f(z), holomorphic in |z| < 1, which have asymptotic values at a dense set of points on |z| = 1. MacLane [4; p. 18] asked whether the sum and product of two functions in had to be either constant or in . Recently Barth and Ryan [2] have shown that this was not necessarily so; in this note we will demonstrate, in a totally elementary way, why not. Our principal result is:

Theorem 1. Any non-constant function R(z), holomorphic in |z| < 1, may be represented both as a sum and as a product of pairs of functions in.

We describe a toroidal polyhedral map which can be geometrically realized in R3 but not via a Schlegel diagram of a convex 4-polytope. Moreover, this map is not isomorphic to a subcomplex of the boundary complex of any convex polytope.

In recent work [2] we have investigated Borel isomorphisms at the first level, i.e. mappings that together with their inverses map Fσ-sets to Fσ-sets. We are grateful to Dr. F. Topsøe for writing to point out errors in the proofs of our Theorems 2.1, 2.2 and 2.3. The errors escaped our attention because we only wrote out the first step of a complicated inductive argument and carelessly and falsely claimed that the general step of the induction was similar to the first. To be more specific, in the proof of Theorem 2.1, although we do have

we do not, in general, have

Fortunately the proofs can be corrected.

Abstract. Let g(n) denote the largest number of positive integers not exceeding n such that the l.c.m. of any two of them does not exceed n. In this paper we obtain an improvement over a previously known upper estimate for g(n).

Based on the method of analytic continuation of Buchwald and Davies [1[, [2], the first boundary value problem of an elliptic plate is solved. The results are in agreement with the more complicated solution of Muskhelishvili [3]. The solution of the second boundary value problem is also obtained.

Hitherto, little use has been made of the well-known representation

of the Riemann zeta-function, ζ(s), within the critical strip l = {s:s = σ + it, 0 < σ < 1}. Certain variants of (1) have been used to deduce the functional equation of ζ(s), while a simple consequence of (1) itself is that ζ(s) does not vanish on the positive real axis.

For an ideal A of a commutative ring R with identity and a unitary R-module E the notion of an E-sequence of length d in A can be extended as follows. For d = 0 the E-sequence is empty, and for d = 1 it is a subset {ai|i ∈ I} = α ⊆ A such that . For d > 1 we may define, inductively, an E-sequence of length d in A as a sequence

of subsets of A such that a, is an E-sequence of length 1 in A and α2,…, αd is a -sequence of length d − 1 in A. Thus in the standard notion of E-sequence the sets αj, are singletons, and, in effect, the extended notion due to Hochster [1] and Northcott [3] the sets a; are finite. Many of the standard results concerning E-sequences when E is Noetherian extend to the above generalization when the Noetherian condition is dropped. For example it follows from the results of the present note that every maximal E-sequence has the same length (which may be infinite) and every E-sequence can be extended to a maximal E-sequence. This maximal length is inf which we call the homological grade of E in A and denote by hgrR (A; E). So 0 ≤ hgrR (A; E) ≤ ∞, hgrR (A; E) = 0, if, and only if, 0:EA≠0 and hgrg (A; E) = ∞, if, and only if, for all nєℤ.

Let Vo be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane [3] gave a method of constructing all real valuations of K(x) which are extensions of Vo. In this paper, we determine explicitly all rank 2 valuations of K(x) which extend Vo. One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski [5] in the case when K is an algebraically closed field of characteristic zero.

Motivated by questions of computational complexity, Rabin [7] introduced the notion of a complete proof of a system of inequalities. His work and the related paper of Spira [8] should interest geometers as well as computer scientists, for both papers involve convexity in an essential way. Spira's results concern the possibility of covering the intersection of a convex set C and a convex polyhedron Q with a finite collection P of polyhedra subject to certain conditions, while in Rabin's work the members of P may be more general than polyhedra. Both papers are interesting and treat important questions, but only Rabin's paper is correct in all respects. The present note contains counterexamples to some of Spira's results and establishes a correct version of one of them.

Let w = f(z) be regular and schlicht for |z|, and f (0) = 0.

Suppose that f maps the unit disc {z : |z| < 1} onto a domain D starlike with respect to w = 0. Let C(r, θ) be the image in D of the ray joining z = 0 to z = reiθ, and let

be its length. Sheil–Small [1] proved that l(r, θ) < (1 + log 4) | f (reiθ)|, and conjectured the following result, which it is my aim to prove in this paper.

Let Q(x) = Q(x1, x2,…, xn) be a quadratic form with integer coefficients. Schinzel, Schickewei and Schmidt [9, Theorem 1] have shown that for any modulus m there exists a nonzero such that

and ║x║≤m(1/2)+(1/2(n-1)), where ║x║ = max |xi|. When m is a prime Heath-Brown [8] has obtained a nonzero solution of (1) with ║x║≤m1/2 log m. Yuan [10] has extended Heath-Brown's work to all finite fields. We have proved related results in [5] and [6]. In this paper we extend Heath-Brown's work to moduli which are a product of two primes. Throughout the paper we shall assume that n is even and n>2. For any odd prime p let

where det Q is the determinant of the integer matrix representing Q and is the Legendre symbol.

Let be the sum of the m-th powers of the zeros of the Bessel polynomial yn(x). It is known that for m = 0, 1, 2, …,

where cm(v) is the Hawkins polynomial. In this paper we find rational functions wm(v) such that for m = 0, 1, 2, …

Let (X, ℱ, μ) be a topological measure space with X a completely regular Hausdorff space and ℱ the σ-algebra of all μ-measurable sets, containing all the Baire sets of X. Consider the following two conditions on (X, ℱ, μ).

Let →(X) be a function of the n variables (X) = (X1, …, Xn) defined for all real (X). A fundamental problem in the theory of Diophantine approximation is to prove the existence of real numbers (X) ≡ (x) (mod 1), where (x) = (x1, …, xn) is any given set of real numbers, for which