Multiple positive bound state solutions for a critical Choquard equation

In this paper we consider the problem(P lambda()) {-Delta u + V-lambda(x)u = (I-mu * vertical bar u vertical bar(2 mu*))vertical bar u vertical bar(2 mu*-2)u in R-N,u > 0 in R-N,where V-lambda = lambda + V-0 with lambda >= 0, V-0 is an element of L-N/2(R-N), I-mu = 1/vertical bar x vertical bar(mu) is the Riesz potential with 0 < mu < min{N, 4} and 2(mu)* = 2N-mu/N-2 with N >= 3. Under some smallness assumption on V-0 and lambda we prove the existence of two positive solutions of (P-lambda). In order to prove the main results, we used variational methods combined with degree theory

A compositional turnover zone of biogeographical magnitude within lowland Amazonia

Aim To assess the relative roles of geologically defined terrain types ( environmental heterogeneity) and a major river ( physical dispersal barrier) as predictors of ecological structuring and biogeographical differentiation within Amazonian forests.Location Western Brazilian Amazonia, where the Jurua river and its terraces cross a 1000-km-long boundary between two geological formations ( the Solimoes and Ica Formations).Methods We sampled a 500-km stretch of the Jurua with 71 transects ( 5 m by 500 m) that spanned both the river and the geological boundary. All transects were inventoried for pteridophytes ( ferns and lycophytes) and Melastomataceae, and a subset of 39 transects also for palms and Zingiberales. Three surface soil samples were collected from each transect. The data were analysed using ordinations, regression trees, indicator species analyses and Mantel tests.Results All plant groups showed congruent species turnover between geologically defined terrain types, but little evidence of isolation by the river or geographical distance. Soil cation concentration differed between the Solimoes Formation and other terrain types and emerged as the main explanatory factor for species turnover. A large proportion of the plant species were significant indicators for specific parts of the soil cation concentration gradient, and these edaphic associations were congruent with those found in other parts of Amazonia. Pteridophytes had a larger proportion of species in the cation-rich soils than the other plant groups did, and palms had a higher proportion of generalists.Main conclusions The geological boundary between the Solimoes and Ica formations is confirmed as significant floristic turnover zone. As it runs in a north-south orientation for more than 1000 km, the edaphic differences associated with this boundary have wide-ranging implications for speciation and biogeographical patterns in Amazonia

Using digital soil maps to infer edaphic affinities of plant species in Amazonia: Problems and prospects

Amazonia combines semi-continental size with difficult access, so both current ranges of species and their ability to cope with environmental change have to be inferred from sparse field data. Although efficient techniques for modeling species distributions on the basis of a small number of species occurrences exist, their success depends on the availability of relevant environmental data layers. Soil data are important in this context, because soil properties have been found to determine plant occurrence patterns in Amazonian lowlands at all spatial scales. Here we evaluate the potential for this purpose of three digital soil maps that are freely available online: SOTERLAC, HWSD, and SoilGrids. We first tested how well they reflect local soil cation concentration as documented with 1,500 widely distributed soil samples. We found that measured soil cation concentration differed by up to two orders of magnitude between sites mapped into the same soil class. The best map-based predictor of local soil cation concentration was obtained with a regression model combining soil classes from HWSD with cation exchange capacity (CEC) from SoilGrids. Next, we evaluated to what degree the known edaphic affinities of thirteen plant species (as documented with field data from 1,200 of the soil sample sites) can be inferred from the soil maps. The species segregated clearly along the soil cation concentration gradient in the field, but only partially along the model-estimated cation concentration gradient, and hardly at all along the mapped CEC gradient. The main problems reducing the predictive ability of the soil maps were insufficient spatial resolution and/or georeferencing errors combined with thematic inaccuracy and absence of the most relevant edaphic variables. Addressing these problems would provide better models of the edaphic environment for ecological studies in Amazonia

Multiplicity of Positive Solutions for an Obstacle Problem in R

In this paper we establish the existence of two positive solutions for the obstacle problem \displaystyle \int_{\Re}\left[u'(v-u)'+(1+\lambda V(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka where $f$ is a continuous function verifying some technical conditions and \Ka is the convex set given by \Ka =\left\{v\in H^{1}(\Re); v \geq \varphi \right\}, with $\varphi \in H^{1}(\Re)$ having nontrivial positive part with compact support in $\Re$. \vspace{0.2cm} \noindent \emph{2000 Mathematics Subject Classification} : 34B18, 35A15, 46E39. \noindent \emph{Key words}: Obstacle problem, Variational methods, Positive solutions.Comment: To appear in Progress in Nonlinear Differential Equations and their Application

How manipulating task constraints in small-sided and conditioned games shapes emergence of individual and collective tactical behaviours in football: A systematic review

Background: Small-Sided and Conditioned Games are characterised by modifications of field dimensions, number of players, rules of the game, manipulations used to shape the key task constraints that performers need to satisfy in practice. Evidence has already demonstrated the importance of designing practice to enhance understanding of tactical behaviours in football, but there is a lack of information about how coaches can manipulate task constraints to support tactical learning. Objective: To investigate which task constraints have been most often manipulated in studies of SSCGs; and what impact each manipulation had on emerging tactical behaviours, technical–tactical actions, and positional relationships between players. Methods: PubMed, Web of Science, Scielo, and Academic Google databases were searched for relevant reports without time limits. The criteria adopted for inclusion were: a) studies performed with football players; b) studies that included SSCGs as an evaluation method; c) studies that investigated tactical behaviours in SSCGs; and d), articles in English and Portuguese. Results: The electronic database search included 24 articles in the review. Of these, five manipulated field dimensions, six manipulated number of players involved, five manipulated field dimensions and number of players, five used different scoring targets, two altered the number of players and scoring target, and one manipulated the number of players, field dimension, and scoring target. Conclusion: Among the task constraints analyzed in this systematic review, manipulation of number of players and playing field dimensions concomitantly occurred most frequentl

How manipulating task constraints in small-sided and conditioned games shapes emergence of individual and collective tactical behaviours in football: A systematic review

Background: Small-Sided and Conditioned Games are characterised by modifications of field dimensions, number of players, rules of the game, manipulations used to shape the key task constraints that performers need to satisfy in practice. Evidence has already demonstrated the importance of designing practice to enhance understanding of tactical behaviours in football, but there is a lack of information about how coaches can manipulate task constraints to support tactical learning. Objective: To investigate which task constraints have been most often manipulated in studies of SSCGs; and what impact each manipulation had on emerging tactical behaviours, technical–tactical actions, and positional relationships between players. Methods: PubMed, Web of Science, Scielo, and Academic Google databases were searched for relevant reports without time limits. The criteria adopted for inclusion were: a) studies performed with football players; b) studies that included SSCGs as an evaluation method; c) studies that investigated tactical behaviours in SSCGs; and d), articles in English and Portuguese. Results: The electronic database search included 24 articles in the review. Of these, five manipulated field dimensions, six manipulated number of players involved, five manipulated field dimensions and number of players, five used different scoring targets, two altered the number of players and scoring target, and one manipulated the number of players, field dimension, and scoring target. Conclusion: Among the task constraints analyzed in this systematic review, manipulation of number of players and playing field dimensions concomitantly occurred most frequentl