Inequalities between overpartition ranks for all moduli

In this paper we give a full description of the inequalities that can occur between overpartition ranks. If $\overline{N}(a,c,n)$ denotes the number of overpartitions of $n$ with rank congruent to $a$ modulo $c,$ we prove that for any $c\ge7$ and $0\le a\overline{N}(b,c,n)$ for $n$ large enough. That the sign of the rank differences $\overline{N}(a,c,n)-\overline{N}(b,c,n)$ depends on the residue class of $n$ modulo $c$ in the case of small moduli, such as $c=6,$ is known due to the work of Ji, Zhang and Zhao (2018) and Ciolan (2020). We show that the same behavior holds for $ c\in\{2,3, 4,5\}.

Cyclotomic numerical semigroups

Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. We study cyclotomic numerical semigroups; these are numerical semigroups $S$ such that $\mathrm P_S(x)$ has all its roots in the unit disc. We conjecture that $S$ is a cyclotomic numerical semigroup if and only if $S$ is a complete intersection numerical semigroup and present some evidence for it. Aside from the notion of cyclotomic numerical semigroup we introduce the notion of cyclotomic exponents and polynomially related numerical semigroups. We derive some properties and give some applications of these new concepts.Comment: 17 pages, accepted for publication in SIAM J. Discrete Mat

Cyclotomic exponent sequences of numerical semigroups

We study the cyclotomic exponent sequence of a numerical semigroup $S,$ and we compute its values at the gaps of $S,$ the elements of $S$ with unique representations in terms of minimal generators, and the Betti elements $b\in S$ for which the set $\{a \in \operatorname{Betti}(S) : a \le_{S}b\}$ is totally ordered with respect to $\le_S$ (we write $a \le_S b$ whenever $a - b \in S,$ with $a,b\in S$). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, García-Sánchez and Moree (2016) stating that $S$ is cyclotomic if and only if it is a complete intersection

Comparing Aspects Of The Process Quality In Six European Early Childhood Educational Settings

The European project ‘Early Change’ (http://earlychange.teithe.gr) attempts to evaluate the quality of early childhood education (ECE) environments of six European countries, Greece, Portugal, Finland, Denmark, Cyprus and Romania. The purpose of this paper is to compare the level of two dimensions of the process quality of these environments a) Space & Furnishings, and b) Personal Care Routines. Theorists, practitioners and researchers agree that in order to provide qualitative education to young children, one of the basic needs of all children must be met; that need is the protection of their health and their safety. A high quality early childhood education program must contain a safe and stimulating environment for the child (Lindsey, 1998). Such an environment includes indoor space, outdoor space, furniture, and room arrangement, and it is considered an integral part of a high quality early childhood program. 117 early educators from the six participating countries attended the training seminars about the evaluation of ECE quality using the Early Childhood Environmental Rating Scale-R (ECERS-R). The trained educators evaluated the 8 indicators of the subscale ‘space & furnishings’ and the six indicators of the subscale ‘personal care routines’ in approximately 600 early childhood classrooms from six European countries. The results of this study highlight the similarities and differences concerning the specific dimensions of the process quality of ECE environments in six European countries, and reflect the diversity of ECE environment across these countries. The findings of this study may provide a valuable insight to researchers and educational policy makers for an enhanced understanding of the cultural diversities and the strengthening of the common values and targets of the European Union

Building consensus about eHealth in Slovene primary health care: Delphi study

<p>Abstract</p> <p>Background</p> <p>Slovenia's national eHealth strategy aims to develop an efficient, flexible and modern health care informatics framework that would be comparable to the most successful EU countries. To achieve this goal, the gap between availability and usage of information and communication technology by primary care physicians needs to be reduced.</p> <p>As recent efforts show, consensus on information and communication technology purpose and usage in primary care needs to be established before any national information and communication technology solutions are developed.</p> <p>The aim of this study was to identify the most appropriate measures in implementation of Slovene national eHealth strategy and to suggest an appropriate model for success by using the three round Delphi study.</p> <p>Methods</p> <p>An e-mail based, three-round Delphi study was undertaken to achieve consensus from a selected sample of nationally recognized experts from the fields of primary health care and medical informatics. The aim of this study was to identify the most appropriate measures and key obstacles in implementation of eHealth in Slovene primary health care by using the Delphi study.</p> <p>Results</p> <p>High levels of consensus on the majority of suggested measures were achieved among all study participants, as well as between the subgroups of experts from primary health care and medical informatics. All aims of the three-round Delphi study on eHealth implementation in Slovenian primary health care were achieved.</p> <p>Conclusions</p> <p>The three round decision Delphi process has proven to be effective for developing outcomes, ranking key priorities in primary care eHealth development, and achieving consensus among the most influential experts in that field. This consensus is an important contribution to future national eHealth strategies in the field of primary health care.</p

Cyclotomic exponent sequences of numerical semigroups

We study the cyclotomic exponent sequence of a numerical semigroup $S,$ and we compute its values at the gaps of $S,$ the elements of $S$ with unique representations in terms of minimal generators, and the Betti elements $b\in S$ for which the set $\{a \in \operatorname{Betti}(S) : a \le_{S}b\}$ is totally ordered with respect to $\le_S$ (we write $a \le_S b$ whenever $a - b \in S,$ with $a,b\in S$). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, Garc\'ia-S\'anchez and Moree (2016) stating that $S$ is cyclotomic if and only if it is a complete intersection.Comment: 25 page