The famous theorem of Cobham says that, for multiplicatively independent integers k and l, any subset of N, which is both kand l-recognizable, is recognizable. Many of its proofs are based on so called Hansel’s lemma stating that such a kand l-recognizable set is syndetic. We consider these proofs and point out that some of them are inadequate.

In this note, we give an alternative proof of the following result. Let p, q ≥ 2 be two multiplicatively independent integers. If an infinite set of integers is both pand q-recognizable, then it is syndetic. Notice that this result is needed in the classical proof of the celebrated Cobham’s theorem. Therefore the aim of this paper is to complete [13] and [1] to obtain an accessible proof of Cob...

In this paper the piecewise level set method is combined with phase field method to solve the shape and topology optimization problem. First, the optimization problem is formed based on piecewise constant level set method then is updated using the energy term of phase field equations. The resulting diffusion equation which updates the level set function and optimization ...

occasion of his 60th birthday. Abstract For a piecewise aane function f : R n ! R m , the recession function is deened by f 1 (x) := lim !1 f(x) : In this paper, we study the zero set and error bound properties of f via f 1. We show, for example, that f has a zero when f 1 has a unique zero (at the origin) with a nonvanishing index. We also characterize the global error bound property of a piec...

This paper discusses an approach to combining user and system models within a common mathematical framework, to yield an integrated view of human-system interaction. Aspects of systems that affect usability can be described and reasoned about in terms of the conjoint behaviour of user and computer. The paper outlines the basic ideas behind the approach through a case study. The paper then revie...

let $a$ be a non-trivial abelian group and $a^{*}=asetminus {0}$. a graph $g$ is said to be $a$-magic graph if there exists a labeling$l:e(g)rightarrow a^{*}$ such that the induced vertex labeling$l^{+}:v(g)rightarrow a$, define by $$l^+(v)=sum_{uvin e(g)} l(uv)$$ is a constant map.the set of all constant integerssuch that $sum_{uin n(v)} l(uv)=c$, for each $vin n(v)$,where $n(v)$ denotes the s...

It is not difficult to see that every group homomorphism from Z to R extends to a homomorphism from R to R. We discuss other examples of discrete subgroups Γ of connected Lie groups G, such that the homomorphisms defined on Γ can (“virtually”) be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that Γ has this property if it is Zariski dens...

We have recently developed an approach to modelling interaction that encompasses not just the device, but also aspects of a cognitive model. This integrated framework, called syndetic modelling, has been used to reason about the interplay between cognitive and computational resources deployed within an interaction. Here, a new view on this integrated framework is described. We consider the inte...