Praxisnahe Ausbildung in Technik und Betrieb an der Abteilung Chemie der Ingenieurschule beider Basel

Practice-oriented training in technical theory and operational procedures in the Chemisty Department of the Basle State Institute of Technology, Switzerland. The process technology center is presented

The Ring of Polyfunctions over Z/nZ\mathbb Z/n\mathbb Z

We study the ring of polyfunctions over a commutative ring $R$ with unit element, i.e., the ring of functions $f:R\to R$ which admit a polynomial representative $p\in R[x]$ in the sense that $f(x)= p(x)$ for all $x\in R$. This allows to define a ring invariant $s$ which associates to a commutative ring $R$ with unit element a value in $\mathbb N\cup\{\infty\}$. The function $s$ generalizes the number theoretic Smarandache function. For the ring $R=\mathbb Z/n\mathbb Z$ we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number $\Psi(n)$ of polyfunctions over $\mathbb Z/n\mathbb Z$. We also investigate algebraic properties of the ring of polyfunctions over $\mathbb Z/n\mathbb Z$. In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive a new formula for the size of the ring of polyfunctions in several variables over $\mathbb Z/n\mathbb Z$.Comment: 22 page

Training Hazardous-Materials Response Teams and Chemistry Students through Practical Experimentation

Large quantities of hazardous substances are required to meet the needs of today's industrial society. During the manufacture, transport, and use of these substances – whether they serve as raw materials, intermediate products or energy carriers – accidents and damage cannot be totally excluded despite all the efforts and technical knowledge that may go into their prevention. Incidents involving hazardous chemicals always represent a substantial risk for accident response teams, the general population, and the environment. In order to keep damage to a minimum, special attention must be given to training the specialists involved. It is possible to demonstrate the risks resulting from hazardous materials and the most suitable methods of effectively combating such risks by the use of thought-provoking practical experiments. This is shown with a number of examples

The ring of polyfunctions over Z/nZ

We study the ring of polyfunctions over Z/nZ. The ring of polyfunctions over a commutative ring R with unit element is the ring of functions f:R→R which admit a polynomial representative p∈R[x] in the sense that f(x)=p(x) for all x∈R. This allows to define a ring invariant s which associates to a commutative ring R with unit element a value in N∪{∞}. The function s generalizes the number theoretic Smarandache function. For the ring R=Z/nZ we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number Ψ(n) of polyfunctions over Z/nZ. We also investigate algebraic properties of the ring of polyfunctions over Z/nZ. In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive formulas for the size of the ring of polyfunctions in several variables over Z/nZ, and we compute the number of polyfunctions which are units of the ring.ISSN:0092-7872ISSN:1532-412

Polyfunctions over commutative rings

A function f:R→R, where R is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative p∈R[x]. Based on this notion, we introduce ring invariants which associate to R the numbers s(R) and s(R′;R), where R′ is the subring generated by 1. For the ring R=Z/nZ the invariant s(R) coincides with the number theoretic Smarandache or Kempner functions(n). If every function in a ring R is a polyfunction, then R is a finite field according to the Rédei–Szele theorem, and it holds that s(R)=|R|. However, the condition s(R)=|R| does not imply that every function f:R→R is a polyfunction. We classify all finite commutative rings R with unit element which satisfy s(R)=|R|. For infinite rings R, we obtain a bound on the cardinality of the subring R′ and for s(R′;R) in terms of s(R). In particular we show that |R′|≤s(R)!. We also give two new proofs for the Rédei–Szele theorem which are based on our results

The ring of polyfunctions over ℤ/nℤ

We study the ring of polyfunctions over Z/nZ. The ring of polyfunctions over a commutative ring R with unit element is the ring of functions f:R→R which admit a polynomial representative p∈R[x] in the sense that f(x)=p(x) for all x∈R. This allows to define a ring invariant s which associates to a commutative ring R with unit element a value in N∪{∞}. The function s generalizes the number theoretic Smarandache function. For the ring R=Z/nZ we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number Ψ(n) of polyfunctions over Z/nZ. We also investigate algebraic properties of the ring of polyfunctions over Z/nZ. In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive formulas for the size of the ring of polyfunctions in several variables over Z/nZ, and we compute the number of polyfunctions which are units of the ring