Some results on caps and codes related to orthogonal Grassmannians — a preview

In this note we offer a short summary of some recent results, to be contained in a forthcoming paper [4], on projective caps and linear error correcting codes arising from the Grassmann embedding εgr k of an orthogonal Grassmannian ∆k . More precisely, we consider the codes arising from the projective system determined by εgr k (∆k ) and determine some of their parameters. We also investigate special sets of points of ∆k which are met by any line of ∆k in at most 2 points proving that their image under the Grassmann embedding is a projective cap

Geometries arising from trilinear forms on low-dimensional vector spaces

Let Gk(V) be the k-Grassmannian of a vector space V with dimV=n. Given a hyperplane H of Gk(V), we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating k-linear forms, J. Algebraic Combin. doi: 10.1007/s10801-016-0730-6] a point-line subgeometry of PG(V) called the geometry of poles of H. In the present paper, exploiting the classification of alternating trilinear forms in low dimension, we characterize the possible geometries of poles arising for k=3 and n≤7 and propose some new constructions. We also extend a result of [J.Draisma, R. Shaw, Singular lines of trilinear forms, Linear Algebra Appl. doi: 10.1016/j.laa.2010.03.040] regarding the existence of line spreads of PG(5,K) arising from hyperplanes of G3(V)

Grassmannians of codes

Consider the point line-geometry ${\mathcal P}_t(n,k)$ having as points all the $[n,k]$-linear codes having minimum dual distance at least $t+1$ and where two points $X$ and $Y$ are collinear whenever $X\cap Y$ is a $[n,k-1]$-linear code having minimum dual distance at least $t+1$. We are interested in the collinearity graph $\Lambda_t(n,k)$ of ${\mathcal P}_t(n,k).$ The graph $\Lambda_t(n,k)$ is a subgraph of the Grassmann graph and also a subgraph of the graph $\Delta_t(n,k)$ of the linear codes having minimum dual distance at least $t+1$ introduced in~[M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251--263, doi:10.1016/j.ffa.2016.02.004, arXiv:1506.00215]. We shall study the structure of $\Lambda_t(n,k)$ in relation to that of $\Delta_t(n,k)$ and we will characterize the set of its isolated vertices. We will then focus on $\Lambda_1(n,k)$ and $\Lambda_2(n,k)$ providing necessary and sufficient conditions for them to be connected.Comment: 20 pages/minor corrections/updated bibliograph

On the generation of some Lie-type geometries

Let $X_n(K)$ be a building of Coxeter type $X_n=A_n$ or $X_n=D_n$ or defined over a given division ring $K$ (a field when $X_n=D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $\Gamma(K)=Gr_J(X_n(K))$ be the $J$-grassmannian of $X_n(K)$ . We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$ . As a consequence, the generating rank of $\Gamma(K)$ is infinite when $K$ is not finitely generated. In particular, if $K$ is the algebraic closure of a finite field of prime order then the generating rank of $Gr_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1)^2-1$ or $(n+1)^2$

Neuroimmunoendocrinology of the cervical autonomic nervous system

This article reviews several peripheral neuroendocrine relationships in the cervical region, namely the effect of the sympathetic innervation on adenohypophysial, thyroid and parathyroid hormone release, and the effect of parasympathetic innervation in thyroid and parathyroid glands. The possible pathways through which the central nervous system modulates the circadian organization of the immune response are also reviewed and the relative importance of circadian control of immune reactivity through local sympathetic and parasympathetic nerves and of neuroendocrine signals, like melatonin, is also discussed. Altogether the present article argues in favor of the concept that nerves arriving to the endocrine and lymphoid tissue constitute alternate pathways through which the brain controls immunoendocrine phenomena.Biomedical Reviews 1998; 9: 47-59

On the generation of some Lie-type geometries

Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 of K. As a consequence, the generating rank of Γ(K) is infinite when K is not finitely generated. In particular, if K is the algebraic closure of a finite field of prime order then the generating rank of Gr1,n(An(K)) is infinite, although its embedding rank is either (n+1)2−1 or (n+1)2

Lyapunov Pairs in Semilinear Differential Problems with State-Dependent Impulses

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Clinical Aspects of Melatonin Intervention in Alzheimer’s Disease Progression

Melatonin secretion decreases in Alzheimer´s disease (AD) and this decrease has been postulated as responsible for the circadian disorganization, decrease in sleep efficiency and impaired cognitive function seen in those patients. Half of severely ill AD patients develop chronobiological day-night rhythm disturbances like an agitated behavior during the evening hours (so-called “sundowning”). Melatonin replacement has been shown effective to treat sundowning and other sleep wake disorders in AD patients. The antioxidant, mitochondrial and antiamyloidogenic effects of melatonin indicate its potentiality to interfere with the onset of the disease. This is of particularly importance in mild cognitive impairment (MCI), an etiologically heterogeneous syndrome that precedes dementia. The aim of this manuscript was to assess published evidence of the efficacy of melatonin to treat AD and MCI patients. PubMed was searched using Entrez for articles including clinical trials and published up to 15 January 2010. Search terms were “Alzheimer” and “melatonin”. Full publications were obtained and references were checked for additional material where appropriate. Only clinical studies with empirical treatment data were reviewed. The analysis of published evidence made it possible to postulate melatonin as a useful ad-on therapeutic tool in MCI. In the case of AD, larger randomized controlled trials are necessary to yield evidence of effectiveness (i.e. clinical and subjective relevance) before melatonin´s use can be advocated

Experimental allergic encephalomyelitis in pituitary-grafted Lewis rats

Treatment of susceptible rats with dopaminergic agonists that reduce prolactin release decreases both severity and duration of clinical signs of experimental allergic encephalomyelitis (EAE). To assess to what extent the presence of an ectopic pituitary (that produces an increase in plasma prolactin levels mainly derived from the ectopic gland) affects EAE, 39 male Lewis rats were submitted to pituitary grafting from littermate donors. Another group of 38 rats was sham-operated by implanting a muscle fragment similar in size to the pituitary graft. All rats received subcutaneous (s.c.) injections of complete Freund's adjuvant (CFA) plus spinal cord homogenate (SCH) and were monitored daily for clinical signs of EAE. Animals were killed by decapitation on days 1, 4, 7, 11 or 15 after immunization and plasma was collected for prolactin RIA. In a second experiment, 48 rats were immunized by s.c. injection of a mixture of SCH and CFA, and then received daily s.c. injections of bromocriptine (1 mg/kg) or saline. Groups of 8 animals were killed on days 8, 11 or 15 after immunization and plasma prolactin was measured. Only sham-operated rats exhibited clinical signs of the disease when assessed on day 15 after immunization. A progressive decrease in plasma prolactin levels was observed in pituitary-grafted rats, attaining a minimum 15 days after immunization, whereas plasma prolactin levels were increased during the course of the disease in sham-operated rats. Plasma prolactin levels were higher in pituitary-grafted rats than in sham-operated rats 1 day after immunization, but lower on days 7, 11 and 15 after immunogen injection. Further supporting a correlation of suppressed prolactin levels with absence of clinical signs of EAE, rats that were administered the dopaminergic agonist bromocriptine showed very low plasma prolactin levels and did not exhibit any clinical sign of EAE. These results indicate that low circulating prolactin levels coincide with absence of clinical signs of EAE in Lewis rats