Efficient Computation of Squarefree Separator Polynomials

Given a finite set of distinct points, a separator family is a set of polynomials, each one corresponding to a point of the given set, such that each of them takes value one at the corresponding point, whereas it vanishes at any other point of the set. Separator polynomials are fundamental building blocks for polynomial interpolation and they can be employed in several practical applications. Ceria and Mora recently developed a new algorithm for squarefree separator polynomials. The algorithm employs as a tool the point trie structure, first defined by Felszeghy-R\ue1th-R\uf3nyai in their Lex game algorithm, which gives a compact representation of the relations among the points\u2019 coordinates. In this paper, we propose a fast implementation in C of the aforementioned algorithm, based on an efficient storing and visiting of the point trie. We complete the implementation with tests on some sets of points, giving different configurations of the corresponding tries

The vanishing ideal of a finite set of points with multiplicity structures

Given a finite set of arbitrarily distributed points in affine space with arbitrary multiplicity structures, we present an algorithm to compute the reduced Groebner basis of the vanishing ideal under the lexicographic ordering. Our method discloses the essential geometric connection between the relative position of the points with multiplicity structures and the quotient basis of the vanishing ideal, so we will explicitly know the set of leading terms of elements of I. We split the problem into several smaller ones which can be solved by induction over variables and then use our new algorithm for intersection of ideals to compute the result of the original problem. The new algorithm for intersection of ideals is mainly based on the Extended Euclidean Algorithm.Comment: 12 pages,12 figures,ASCM 201

A note on linear Sperner families

In an earlier work we described Gr\"obner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors $\mathbf{v} \in \{0,1\}^n$ of the complete $d$ unifom set family over the ground set $[n]$. In particular, it turns out that the standard monomials of the above ideal are {\em ballot monomials}. We give here a partial extension of the latter fact. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation $a_1v_1+\cdots +a_nv_n=k$, where $0<a_q\leq a_2\leq \cdots \leq a_n$ and $k$ are integers. As an application, we confirm a conjecture of Frankl for linear Sperner systems.Comment: 12 page

Long-Lasting Deficient Dexamethasone Suppression of Hypothalamic–Pituitary–Adrenocortical Activation Following Peripheral CRF Challenge in Socially Defeated Rats

The present study focuses on the long-term changes in the regulation of the hypothalamic-pituitary-adrenocortical (HPA) axis following two short-lasting episodes of intensive stress in the rat stress model of social defeat and the possible similarities with HPA functioning in human affective disorders. Male Wistar rats experienced social defeats on 2 consecutive days by an aggressive male conspecific. The long-term effect of these defeats on resting and ovine corticotropin-releasing factor (oCRF; intravenous (i.v.) 0.5 µg/kg) induced levels of plasma ACTH and corticosterone (CORT) were measured 1 and 3 weeks later. In a second experiment the glucocorticoid feedback regulation of HPA function was tested in a combined dexamethasone (DEX)/CRF test (DEX; 25 µg/kg s.c., 90 min before oCRF injection, 0.5 µg/kg). The oCRF challenges were performed between 11.00 and 13.00 h (about three hours after start of the light phase). One week after defeat the ACTH response to CRF was significantly enhanced in defeated rats as compared to controls. Three weeks after defeat the ACTH response was back to control levels. The increased ACTH response 1 week after the stressor was not reflected in higher CORT levels. Neither were baseline ACTH and CORT levels affected by the prior stress exposure. DEX pretreatment inhibited pituitary adrenocortical activity, reflected both in reduced baseline and response values of ACTH and CORT. The ACTH response to CRF following DEX administration was significantly higher in defeated rats as compared to controls both at one and three weeks after defeat. A reduced DEX suppression of baseline secretion of ACTH appeared 3 weeks after defeat. The same tendency was apparent in response and baseline values of CORT. The differences in CORT between socially stressed and control treated rats, however, did not reach significance. The possible role of changes in glucocorticoid-(GR) and mineralocorticoid receptor (MR) binding in the altered regulation of HPA activity following defeat were studied in brain and pituitary of male Wistar rats 1 and 3 weeks after defeat. One week after defeat GR-binding decreased in hippocampus and hypothalamus. No changes were observed in GR-binding in the pituitary nor in MR-binding in any of the regions analysed. Three weeks after defeat GR-binding recovered in hippocampus and hypothalamus but at this time MR-binding in hippocampal tissue was seriously decreased. In a fourth experiment vasopressin (AVP) and CRF stores in the external zone of the median eminence (ZEME) were measured by quantitative immunocytochemistry one and three weeks after defeat and compared with controls. Social defeat failed to induce a change in the immunocytochemical stores of AVP or CRF. The present findings show that in rats short-lasting stressors like defeat induce long-lasting, temporal dynamic changes in the regulation of the HPA axis. Since these changes in time are reflected in GRs and MRs in different brain areas an altered corticosteroid receptor binding might play an important role in the affected HPA activity following defeat.