What's in the Numbers? Child Welfare Statistics and the Children (NI) Order

This article draws attention to the importance of routinely collected administrative data as an important source for understanding the characteristics of the Northern Ireland child welfare system as it has developed since the Children (Northern Ireland) Order 1995 became its legislative base. The article argues that the availability of such data is a strength of the Northern Ireland child welfare system and urges local politicians, lobbyists, researchers, policy-makers, operational managers, practitioners and service user groups to make more use of them. The main sources of administrative data are identified. Illustration of how these can be used to understand and to ask questions about the system is provided by considering some of the trends since the Children Order was enacted. The “protection” principle of the Children Order provides the focus for the illustration. The statistical trends considered relate to child protection referrals, investigations and registrations and to children and young people looked after under a range of court orders available to ensure their protection and well-being.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).