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 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.

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.

This paper treats the L2-discrepancy of digital (0, 1)-sequences over ℤ2, and gives conditions on the generator matrix of such a sequence which guarantee minimal possible order of L2-discrepancy of the generated sequence. The existence is proved for the first time of digital (0; 1)-sequences over ℤ2 with L2-discrepancy of order . This order is best possible by a result of K. Roth. The existence proof is constructive.

In this paper, an improvement of a large sieve type inequality in high dimensions is presented, and its implications on a related problem are discussed.

This paper concerns the spectrum of the r-dimensional Sturm–Liouville equation y″ + (λI − Q(x))y = 0 with the Dirichlet boundary conditions, where Q is an r × r symmetric matrix. It is proved that, under certain conditions on Q, this problem can only have a finite number of eigenvalues with multiplity r. Further discussion is given for the multiplicities of eigenvalues when Q is an r × r Jacobian matrix.

Let K be a convex body and let p0∈int K. Suppose that in every direction we can choose continuously a section of K which is a translated copy of the corresponding parallel section of K through p0. Our main result essentially claims that if all these pairs of sections are different almost everywhere, then K is an ellipsoid.

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.

Let k be a positive integer and g(k) be the least integer n > k + 1 such that all prime factors of are >k. We prove that

where c is an absolute positive constant. We also establish a new theorem on the distribution of fractional parts of a smooth function.

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

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

$0$

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs mainly rely on a versatile Cassels–Gallagher type theorem and the cross fibering principle of Beresnevich, Haynes and Velani.

We relate Catalan numbers and Catalan determinants to random matrix integrals and to moments of spin representations of odd orthogonal groups.

We study solvability of convolution equations for functions with discrete support in $\mathbf{R}^{n}$, a special case being functions with support in the integer points. The more general case is of interest for several grids in Euclidean space, like the body-centred and face-centred tessellations of 3-space, as well as for the non-periodic grids that appear in the study of quasicrystals. The theorem of existence of fundamental solutions by de Boor et al is generalized to general discrete supports, using only elementary methods. We also study the asymptotic growth of sequences and arrays using the Fenchel transformation.

We introduce a precise framework for transferring strategies from simpler to more complex games, and use it to construct strategies in certain finite and infinite combinations of games. In particular, we give a finitary characterization of finite hypergraphs X such that the first player can win the positional game on infinitely many copies of X. This resolves a conjecture of Leader.

We study ultrasymmetric spaces in the case in which the fundamental function belongs to a limiting class of quasiconcave functions. In the process, we study limiting cases of $J$ interpolation spaces and establish new $J$ – $K$ identities as well as a reiteration theorem for these limiting interpolation methods.

Arvanitakis [A simultaneous selection theorem. Preprint] recently established a theorem which is a common generalization of Michael’s convex selection theorem [Continuous selections I. Ann. of Math. (2) 63 (1956), 361–382] and Dugundji’s extension theorem [An extension of Tietze’s theorem, Pacific J. Math.1 (1951), 353–367]. In this note we provide a short proof of a more general version of Arvanitakis’s result.

In the above paper [1] there is an error in the statements of Proposition 1 and Theorem 1 which we shall now correct. The principal results, Theorems 2 and 3, are unaffected.

This paper is motivated by Davenport’s problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidean space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate ${\mathcal{C}}^{1}$ submanifold of $\mathbb{R}^{n}$ has full dimension. In the second approach, we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

For a positive integer $n$ let

$$\begin{eqnarray}\mathfrak{P}_{n}=\mathop{\prod }_{\substack{ p \\ s_{p}(n)\geqslant p}}p,\end{eqnarray}$$

$p$

$s_{p}(n)$

$p$

$n$

$n$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n}$

$\sqrt{n}$

$n^{20/37}$

$\sqrt{n}$

$\unicode[STIX]{x1D705}\sqrt{n}/\text{log}\,n$

$\unicode[STIX]{x1D705}$

$\unicode[STIX]{x1D705}=2$

$\mathfrak{P}_{n}$

$\mathfrak{P}_{n+1}$

$\mathfrak{P}_{n}$

$n$

$\mathfrak{P}_{n}>\mathfrak{P}_{n+1}$

We study the meromorphic continuation and the spectral expansion of the opposite-sign Kloosterman sum zeta function

$$\begin{eqnarray}\displaystyle (2{\it\pi}\sqrt{mn})^{2s-1}\mathop{\sum }_{\ell =1}^{\infty }\frac{S(m,-n,\ell )}{\ell ^{2s}} & & \displaystyle \nonumber\end{eqnarray}$$

$m,n$

$s\in \mathbb{C}$