Denote by Bn the n-dimensional unit ball centred at o. It is known that in every lattice packing of Bn there is a cylindrical hole of infinite length whenever n≥3. As a counterpart, this note mainly proves the following result: for any fixed ε with ε>0, there exist a periodic point set P(n, ε) and a constant c(n, ε) such that Bn + P(n, ε) is a packing in Rn, and the length of the longest segment contained in Rn\{int(εBn) + P(n, ε)} is bounded by c(n, ε) from above. Generalizations and applications are presented.

A tree is a connected graph that has no cycles. If x is any endnode of a tree, then the limb determined by x is the unique path that joins x with the nearest node other than x that does not have degree two in the tree; let l(x) denote the length of this path. (For definitions and results not given here see [2] or [3].) Different endnodes determine different limbs with one exception; when the tree is a path then both endnodes determine the same limb, namely, the tree itself. Our object here is to investigate the distribution of the length of limbs of trees Tn chosen at random from the set of nn-2 trees with n labelled nodes; in particular, it will follow from our results that the length of the longest limb in most trees Tn is approximately log n when n is large.

Let q be a prime number and let a = (a1, …, as) be an s-tuple of distinct integers modulo q. For any x coprime with q, let be such that . For fixed s and q→∞ an asymptotic formula is given for the number of residue classes x modulo q for which

The more general case, when q is not necessarily prime and x is restricted to lie in a given subinterval of [1, q], is also treated.

It is well-known that the σ -algebra of Borel subsets of a metric space coincides with the smallest family of sets which contains the open sets and is closed under countable intersections and countable disjoint unions «3, Th.3, p. 348». A deeper and less known result of Sierpiński is that for separable metric spaces the family of open sets may be replaced by the family of closed sets in the above result «16, p. 272–275» (and «17, p. 51» for the real line). This paper gives an in depth analysis of these and related generation processes. Several abstract formulations, generalizations and limiting examples are given.

A quasi-simplex of ℝd is the closure of a Choquet simplex of ℝd. We characterize these quasi-simplices and we use this characterization to describe the line-free Choquet simplices of ℝd.

Let D be an atomic integral domain (i.e., a domain in which each nonzero nonunit of D can be written as a product of irreducible elements) and k any positive integer. D is known as a half factorial domain (HFD) if for any irreducible elements α1, …, αn, β1, …, βm of D the equality α1… αn = β1… βm implies that n = m. In [5] the present authors define D to be a k-half factorial domain (k-HFD) if the equality above along with the fact that n or m ≤ k implies that n = m. In this paper we consider the k-HFD property in Dedekind domains with small class group and prove the following Theorem: if D is a Dedekind domain with class group of order less than 16 then D is k-HFD for some integer k > 1, if, and only if, D is HFD.

The cohomology of the symmetric groups with coefficients in a field has been studied by several authors, see [6] and [7] for example, but hardly anything has been published with the integers as coefficients. Given the connection between the infinite symmetric group and the classifying space BG for stable spherical fibrations, the computation of is an interesting problem, and the purpose of this paper is to solve the first non-trivial case, n = 4. (The symmetric groups on 2 and 3 letters have cohomology of period four, which is generated by c1 and c2 of the permutation matrix representation, [8].)

Dimension prints were developed in 1988 to distinguish between different fractal sets in Euclidean spaces having the same Hausdorff dimension but with very different geometric characteristics. In this paper we compute the dimension prints of some fractal sets, including generalized Cantor sets on the unit circle S1 in ℝ2 and the graphs of generalized Lebesgue functions, also in ℝ2. In this second case we show that the dimension print for the graphs of the Lebesgue functions can approach the maximal dimension print of a set of dimension 1. We study the dimension prints of Cartesian products of linear Borel sets and obtain the exact dimension print when each linear set has positive measure in its dimension and the dimension of the Cartesian product is the sum of the dimensions of the factors.

Novikov [7] or [5, Th. 2, p. 510] proves that: if{An} is a sequence of analytic sets in a complete separable metric space and

then there is a sequence {Bn} of Borel sets with

In this paper is derived Lagrange's expansion with remainder for a weak function whose argument satisfies an implicit relation. A necessary and sufficient condition is given for the associated infinite series expansion.

Let P be a d-dimensional convex polytope (briefly, a d-polytope) in d-dimensional euclidean space Ed. Associated with P is a vector f(P), known as the f-vector of P, defined by

where fj(P) is the number of j-faces of P for 0 ≤ j ≤ d − 1 and the superscript T denotes transposition. (We regard f(P) as a column vector, and identify it with the vector

where (e1, …, ed) is some fixed basis of Ed.) Let d be the set of all d-polytopes in Ed, and d be any subset of d. Using tghe notation of [1; §8.1], we donate by f(d) the set of vectors {f(P): P ε d}, and write aff f(d) for the (unique) affine subspace of lowest dimension in Ed which contains all the vectors of f(d). Then it is well-known that

the equation of the hyperplane aff f(d) being that given by the Euler relation between the numbers fj(P) [1; Theorem 8.1.1].

Let

where φ denotes Euler's function. In this memoir we study the set w of sigmaphi numbers, that is, those composite natural numbers n which satisfy

The smallest such number is 65, and they appear to be moderately frequent. There are 290 sigma-phi numbers not exceeding 105 and 1,231 not exceeding 106. By comparison, we observe that the number of primes in these ranges is 9,592 and 78,498, respectively. Since the primes also satisfy the relationship (1.2) a sigma-phi number can be thought of as a kind of pseudo-prime. The motivation for studying sigma-phi numbers is that they should have similar properties to Carmichael numbers but be easier to study. A Carmichael number is a number n such that the least common multiple of the φ(pk) with pk||n divides n-1, i.e., by Korseldt's criterion, a number for which p-1||n-1 whenever p|n. The number of Carmichael numbers not exceeding 105 and 106 is 16 and 43, respectively. It seems that the counting functions for Carmichael and sigma-phi numbers have somewhat similar growth rates. The counts above are skewed by the fact that there are many sigma-phi numbers with exactly two prime factors but there are no Carmichael numbers of this kind.

Introduction. Throughout the paper K(x) is a simple transcendental extension of a field K; v is a valuation of K and w is an extension of v to K(x). Also koÍk and GoÍG denote respectively the residue fields and the value groups of the valuations v and w. A well-known theorem conjectured by Nagata asserts that either k is an algebraic extension of feo or k is a simple transcendental extension of a finite extension of ko (cf [4] or [6] or [1, Corollary 2.3]). We prove here an analogous result for the value groups viz. either G/ Go is a torsion group or there exists a subgroup G1 of G containing Go with [G1: Go] > ∞ such that G is the direct sum of G1 and an infinite cyclic group. Incidentally we obtain a description of the valuation w as well as of its residue field in the second case. Thus a characterization of all those extensions w of v to K(x), for which w(K(x)\{0})/Go is not a torsion group, is given. Corresponding to such a valuation w, we define three numbers N, S and T which satisfy the inequality N ≥= ST. This is analogous to the fundamental inequality established by Ohm (cf. [5, 1.2]) for residually transcendental extensions of v to K(x). We also investigate the conditions under which N = ST

We characterize straightness of digital curves in the integer plane by means of difference operators. Earlier definitions of digital rectilinear segments have used, respectively, Rosenfeld’s chord property, word combinatorics, Reveillès’ double Diophantine inequalities, and the author’s refined hyperplanes. We prove that all these definitions are equivalent. We also characterize convexity of integer-valued functions on the integers with the help of difference operators.

We study the minimal gap statistic for fractional parts of sequences of the form ${\mathcal{A}}^{\unicode[STIX]{x1D6FC}}=\{\unicode[STIX]{x1D6FC}a(n)\}$, where ${\mathcal{A}}=\{a(n)\}$ is a sequence of distinct integers. Assuming that the additive energy of the sequence is close to its minimal possible value, we show that for almost all $\unicode[STIX]{x1D6FC}$, the minimal gap $\unicode[STIX]{x1D6FF}_{\min }^{\unicode[STIX]{x1D6FC}}(N)=\min \{\unicode[STIX]{x1D6FC}a(m)-\unicode[STIX]{x1D6FC}a(n)\hspace{0.2em}{\rm mod}\hspace{0.2em}1:1\leqslant m\neq n\leqslant N\}$ is close to that of a random sequence.

Let $k\geq 2$ and $\Pi (n)= { \mathop{\prod }\nolimits}_{i= 1}^{k} ({a}_{i} n+ {b}_{i} )$ for some integers ${a}_{i} , {b}_{i} $ ($1\leq i\leq k$). Suppose that $\Pi (n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that the number of prime factors $\Omega (\Pi (n))$ of $\Pi (n)$ is at most ${r}_{k} $, for some integer ${r}_{k} $ depending only on $k$. We use a new kind of weighted sieve to improve the possible values of ${r}_{k} $ when $k\geq 4$.

For the superelliptic curves of the form

$$\begin{eqnarray}(x+1)\cdots (x+i-1)(x+i+1)\cdots (x+k)=y^{\ell }\end{eqnarray}$$

$y\neq 0$

$k\geqslant 3$

$\ell \geqslant 2,$

$i\in [2,k]\setminus \unicode[STIX]{x1D6FA}$

$\ell <\text{e}^{3^{k}}.$

$\unicode[STIX]{x1D6FA}$

$[p_{\unicode[STIX]{x1D703}},(k-p_{\unicode[STIX]{x1D703}}))$

$p_{\unicode[STIX]{x1D703}}$

$k/2$

$i=1$

In typical linear programming problems, we are concerned with finding non-negative integers {x1,…, xn} that maximize a linear form c1x1 + … + cnxn, subject to a number of linear inequalities, for The maximum is necessarily attained at one of the vertices of the convex hull of integer points defined by the inequalities, so we have an interest in estimating the number M of these vertices. We give two results; one improving an upper bound result for M of Hayes and Larman concerning the Knapsack polytope, the other an example showing that, in 3-dimensions, it is possible to choose the coefficients aij to obtain a lower bound for M.

In this paper, we consider an extended model of a coupled nonlinear reaction-diffusion equations with Neumann-Neumann boundary conditions. We obtain upper linear growth bound for one of the components. We also find the corresponding bound for the case of Dirichlet-Dirichlet boundary conditions.

If ω, ω′ are two points in the extended complex plane then the distance between their stereographic projections on the Riemann sphere is

with an obvious interpretation if ω or ω′ is ∞. is called the distance between ω and ω′ in the spherical or in the chordal metric.