Contact symmetry of time-dependent Schr\"odinger equation for a two-particle system: symmetry classification of two-body central potentials

Symmetry classification of two-body central potentials in a two-particle Schr\"{o}dinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem contact transformations are not essentially different from point transformations. We have also obtained the detailed algebraic structures of the corresponding Lie algebras and the functional bases of invariants for the transformation groups in all the four classes

Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.Comment: 13 pages, 12 figures, acknowledgments update

A class of solvable Lie algebras and their Casimir Invariants

A nilpotent Lie algebra n_{n,1} with an (n-1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n_{n,1} as their nilradical are obtained. Their dimension is at most n+2. The generalized Casimir invariants of n_{n,1} and of its solvable extensions are calculated. For n=4 these algebras figure in the Petrov classification of Einstein spaces. For larger values of n they can be used in a more general classification of Riemannian manifolds.Comment: 16 page

G Protein-Coupled Estrogen Receptor Immunoreactivity Fluctuates During the Estrous Cycle and Show Sex Differences in the Amygdala and Dorsal Hippocampus

G protein-coupled estrogen receptor (GPER) in the amygdala and the dorsal hippocampus mediates actions of estradiol on anxiety, social recognition and spatial memory. In addition, GPER participates in the estrogenic regulation of synaptic function in the amygdala and in the process of adult neurogenesis in the dentate gyrus. While the distribution of the canonical estrogen receptors α and β in the amygdala and dorsal hippocampus are well characterized, little is known about the regional distribution of GPER in these brain regions and whether this distribution is affected by sex or the stages of the estrous cycle. In this study we performed a morphometric analysis of GPER immunoreactivity in the posterodorsal medial, anteroventral medial, basolateral, basomedial and central subdivisions of the amygdala and in all the histological layers of CA1 and the dentate gyrus of the dorsal hippocampal formation. The number of GPER immunoreactive cells was estimated in these different structures. GPER immunoreactivity was detected in all the assessed subdivisions of the amygdaloid nucleus and dorsal hippocampal formation. The number of GPER immunoreactive cells was higher in males than in estrus females in the central (P = 0.001) and the posterodorsal medial amygdala (P < 0.05); higher in males than in diestrus females in the strata orients (P < 0.01) and radiatum-lacunosum-moleculare (P < 0.05) of CA1-CA3 and in the molecular layer of the dentate gyrus (P < 0.01); higher in diestrus females than in males in the basolateral amygdala (P < 0.05); higher in diestrus females than in estrus females in the central (P < 0.01), posterodorsal medial (P < 0.01) and basolateral amygdala (P < 0.01) and higher in estrus females than in diestrus females in the strata oriens (P < 0.05) and radiatum-lacunosum-moleculare (P < 0.05) of CA1-CA3 and in the molecular layer (P < 0.05) and the hilus of the dentate gyrus (P < 0.05). The findings suggest that estrogenic regulation of the amygdala and hippocampus through GPER may be different in males and in females and may fluctuate during the estrous cycle.This study was supported by Ministero dell'Istruzione, dell'Università e della Ricerca, Italy (MIUR project Dipartimenti di Eccellenza 2018–2022) to Department of Neuroscience Rita Levi Montalcini, Agencia Estatal de Investigación, Spain (BFU2017-82754-R, PSI2017-86396-P), Centro de Investigación Biomédica en Red Fragilidad y Envejecimiento Saludable (CIBERFES), Instituto de Salud Carlos III, Madrid and Fondos FEDER, GRUPOS UCM-BSCH 951579. MM fellowship was generously granted by Prof. G. C. Bergui

Investigating the potential of Sentinel-2 configuration to predict the quality of Mediterranean permanent grasslands in open woodlands

The assessment of pasture quality in permanent grasslands is essential for their conservation and management, as it can contribute to making real-time decisions for livestock management. In this study, we assessed the potential of Sentinel-2 configuration to predict forage quality in high diverse Mediterranean permanent grasslands of open woodlands. We evaluated the performance of Partial Least Squares Regression (PLS) models to predict crude protein (CP), neutral detergent fibre (NDF), acid detergent fibre (ADF) and enzyme digestibility of organic matter (EDOM) by using three different reflectance datasets: (i) laboratory measurements of reflectance of dry and ground pasture samples re-sampled to Sentinel-2 configuration (Spec-lab) (ii) field in-situ measurements of grasslands canopy reflectance resampled to Sentinel-2 configuration (Spec-field); (iii) and Bottom Of Atmosphere Sentinel-2 imagery. For the three reflectance datasets, the models to predict CP content showed moderate performance and predictive ability. Mean R2test = 0.68 were obtained using Spec-lab data, mean R2test decreased by 0.11 with Spec-field and by 0.18 when Sentinel-2 reflectance was used. Statistics for NDF showed worse predictions than those obtained for CP: predictions produced with Spec-lab showed mean R2test = 0.64 and mean RPDtest = 1.73. The mean values of R2test = 0.50 and RPDtest = 1.54 using Sentinel-2 BOA reflectance were marginally better than the values obtained with Spec-field (mean R2test = 0.48, mean RPDtest = 1.43). For ADF and EDOM, only predictions made with Spec-lab produced acceptable results. Bands from the red-edge region, especially band 5, and the SWIR regions showed the highest contribution to estimating CP and NDF. Bands 2, blue and 4, red also seem to be important. The implementation of field spectroscopy in combination with Sentinel-2 imagery proved to be feasible to produce forage quality maps and to develop larger datasets. This study contributes to increasing knowledge of the potential and applicability of Sentinel-2 to predict the quality of Mediterranean permanent grasslands in open woodlands

Bipartite Embeddings of Trees in the Plane

Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. The problem becomes more difficult if T is rooted and we want to root it at any particular point of P . The problem in this form was posed by Perles and partially solved by Pach and Torosick [5]. A complete solution was found by Ikebe et al. [4]. A related result by A. Tamura and Y. Tamura [6] is that, given a point set P = fp 1 ; : : : ; pn g and a sequence d = (d 1 ; : : : dn ) of positive integers with P d i = 2n \Gamma 2, there exists an embedding of some tree in P such that the degree of p i is equal to d i . Optimal algorithms for solving the above problems have been found by Bose, McAllister and Snoeyink [1]. In this paper we consider the following embedding problem. A point set P in the plane in general position (no three points collinear) is partitioned into two disjoint sets R and B (the red and the blue points), an..

Hamiltonian Alternating Paths on Bicolored Double-Chains

We find arbitrarily large finite sets S of points in general position in the plane with the following property. If the points of S are equitably 2-colored (i.e., the sizes of the two color classes differ by at most one), then there is a polygonal line consisting of straight-line segments with endpoints in S , which is Hamiltonian, non-crossing, and alternating (i.e., each point of S is visited exactly once, every two non-consecutive segments are disjoint, and every segment connects points of different colors). We show that the above property holds for so-called double-chains with each of the two chains containing at least one fifth of all the points. Our proof is constructive and can be turned into a linear-time algorithm. On the other hand, we show that the above property does not hold for double-chains in which one of the chains contains at most ≈ 1/29 of all the points

On structural and graph theoretic properties of higher order delaunay graphs

Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k < 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks