Provable properties of asymptotic safety in f(R) approximation

We study an f(R) approximation to asymptotic safety, using a family of non-adaptive cutoffs, kept general to test for universality. Matching solutions on the four-dimensional sphere and hyperboloid, we prove properties of any such global fixed point solution and its eigenoperators. For this family of cutoffs, the scaling dimension at large n of the nth eigenoperator, is λn ∝ b n ln n. The coefficient b is non-universal, a consequence of the single-metric approximation. The large R limit is universal on the hyperboloid, but not on the sphere where cutoff dependence results from certain zero modes. For right-sign conformal mode cutoff, the fixed points form at most a discrete set. The eigenoperator spectrum is quantised. They are square integrable under the Sturm-Liouville weight. For wrong sign cutoff, the fixed points form a continuum, and so do the eigenoperators unless we impose square-integrability. If we do this, we get a discrete tower of operators, infinitely many of which are relevant. These are f(R) analogues of novel operators in the conformal sector which were used recently to furnish an alternative quantisation of gravity.

Partial associativity and rough approximate groups

Suppose that a binary operation $$\circ $$ ∘ on a finite set X is injective in each variable separately and also associative. It is easy to prove that $$(X,\circ )$$ ( X , ∘ ) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples $$(x,y,z)\in X^3$$ ( x , y , z ) ∈ X 3 satisfy the equation $$x\circ (y\circ z)=(x\circ y)\circ z$$ x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z . Other results in additive combinatorics would lead one to expect that there must be an underlying ‘group-like’ structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.

Forcing axioms and coronas of C∗-algebras

We prove rigidity results for large classes of corona algebras, assuming the Proper Forcing Axiom. In particular, we prove that a conjecture of Coskey and Farah holds for all separable [Formula: see text]-algebras with the metric approximation property and an increasing approximate identity of projections.

The introduction of the bow and arrow in the Argentine Andes (29–34º S): A preliminary metric approximation

The study size patterns in projectile points (n=39) from six sites in the Argentine Andes (29–34°S) associated with 17 radiocarbon dates with medians spanning 3080–470 cal BP. This is the region’s first attempt to metrically distinguish arrows and darts, which is based on shoulder or maximum width, following Shott. The northern part of the study area (29°S) includes the earliest arrow point, slightly after 3080 cal BP. This suggests a rapid spread of this technology from the central Andes 16–26°S, where early arrows are dated ~3500–3000 cal BP. However, at 32 and 34°S, arrows are not clearly present until 1280 cal BP. For 1280–400 cal BP (European contact), 96% of points were identified as arrows, suggesting the bow and arrow replaced spear-based weapon systems. A single late dart from 34°S may reflect a late use of this space by hunter-gatherers. The predominance of arrows beginning at 1280 cal BP is associated with broader changes such as demographic growth, reduced mobility, low-level food production, and herding economies, following similar trends in other regions.

Programs and Practices for Identifying and Nurturing High Intellectual Abilities in Spain

The recent educational legislation in Spain shows a great interest in enhancing the talents of all citizens. Different models of identification and intervention for students with high intellectual abilities (HIAs) coexist. The assessment model based on intelligence is still in force in the psychoeducational guidance field; however, from the research, other multidimensional and developmental models are prevalent, rethinking the nature of giftedness and talent, as well as identification and educational practices. These models consider HIA as potential in development, depending on the interrelation among neurobiological bases, personal, and environmental conditions. Efforts are being made to detect high-ability students. The most common intervention measures are the school enrichment of the curriculum, curricular adaptations, and acceleration. Several universities and some autonomous communities (i.e., school districts in the states) have organized extracurricular enrichment programs, some for longer than 10 years. The training of specialized teachers in high abilities has substantially increased, both in the Ministry of Education and autonomous local communities. Universities have also included some subjects in their programs related to this issue of gifted education with specific training designed in postgraduate courses. The research agendas of HIAs currently focus on studying metric approximation, identification and profiles, cognitive functioning and creativity, management of cognitive resources, socioemotional characteristics, gender, enrichment programs, and their effectiveness.