Amplituhedron-like geometries

We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.

Higher order intrinsic weak differentiability and Sobolev spaces between manifolds

We define the notion of higher-order colocally weakly differentiable maps from a manifold M to a manifold N. When M and N are endowed with Riemannian metrics, {p\geq 1} and {k\geq 2}, this allows us to define the intrinsic higher-order homogeneous Sobolev space {\dot{W}^{k,p}(M,N)}. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of N in a Euclidean space; if the manifolds M and N are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the density of smooth maps in the intrinsic space {\dot{W}^{k,p}(M,N)} is that {\pi_{\lfloor kp\rfloor}(N)\simeq\{0\}}. We investigate the chain rule for higher-order differentiability in this setting.

Active faults sources for the Pátzcuaro–Acambay fault system (Mexico): fractal analysis of slip rates and magnitudes <i>M</i>_w estimated from fault length

The Pátzcuaro–Acambay fault system (PAFS), located in the central part of the Trans-Mexican Volcanic Belt (TMVB), is delimited by an active transtensive deformation area associated with the oblique subduction zone between the Cocos and North American plates, with a convergence speed of 55 mm yr−1 at the latitude of the state of Michoacán, Mexico. Part of the oblique convergence is transferred to this fault system, where the slip rates range from 0.009 to 2.78 mm yr−1. This has caused historic earthquakes in Central Mexico, such as the Acambay quake (Ms=6.9) on 19 November 1912 with surface rupture, and another in Maravatío in 1979 with Ms=5.6. Also, paleoseismic analyses are showing Quaternary movements in some faults, with moderate to large magnitudes. Notably, this zone is seismically active, but lacks a dense local seismic network, and more importantly, its neotectonic movements have received very little attention. The present research encompasses three investigations carried out in the PAFS. First, the estimation of the maximum possible earthquake magnitudes, based on 316 fault lengths mapped on a 15 m digital elevation model, by means of three empirical relationships. In addition, the Hurst exponent Hw and its persistence, estimated for magnitudes Mw (spatial domain) and for 32 slip-rate data (time domain) by the wavelet variance analysis. Finally, the validity of the intrinsic definition of active fault proposed here. The average results for the estimation of the maximum and minimum magnitudes expected for this fault population are 5.5≤Mw≤7. Also, supported by the results of H at the spatial domain, this paper strongly suggests that the PAFS is classified in three different zones (western PAFS, central PAFS, and eastern PAFS) in terms of their roughness (Hw=0.7,Hw=0.5,Hw=0.8 respectively), showing different dynamics in seismotectonic activity and; the time domain, with a strong persistence Hw=0.949, suggests that the periodicities of slip rates are close in time (process with memory). The fractal capacity dimension (Db) is also estimated for the slip-rate series using the box-counting method. Inverse correlation between Db and low slip-rate concentration was observed. The resulting Db=1.86 is related to a lesser concentration of low slip-rates in the PAFS, suggesting that larger faults accommodate the strain more efficiently (length ≥3 km). Thus, in terms of fractal analysis, we can conclude that these 316 faults are seismically active, because they fulfill the intrinsic definition of active faults for the PAFS.

THE GEOMETRY OF BLUEPRINTS PART II: TITS–WEYL MODELS OF ALGEBRAIC GROUPS

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_{1}$, the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category$\text{Sch}_{{\mathcal{T}}}$ comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for ${\mathcal{G}}$ in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of ${\mathcal{G}}$ defines a model $G$ in $\text{Sch}_{{\mathcal{T}}}$ whose Weyl extension is the Weyl group $W$ of ${\mathcal{G}}$. We call such models Tits–Weyl models. The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.

Lagrangian averaging with geodesic mean

This paper revisits the derivation of the Lagrangian averaged Euler (LAE), or Euler- α equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume-preserving diffeomorphism group. Under the additional assumption that first-order fluctuations are statistically isotropic and transported by the mean flow as a vector field, averaging of the kinetic energy Lagrangian of an ideal fluid yields the LAE Lagrangian. The derivation presented here assumes a Euclidean spatial domain without boundaries.

An intrinsic definition of the Rees algebra of a module

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

One invariant of intrinsic shape

Based on the intrinsic definition of shape by functions continuous over a covering and corresponding homotopy we will define proximate fundamental group. We prove that proximate fundamental group is an invariant of pointed intrinsic shape of a space.

REVISITING Xor-IMPLICATIONS: CLASSES OF FUZZY (CO)IMPLICATIONS BASED ON f-Xor (f-XNor) CONNECTIVES

The main contribution of this paper is the introduction of an intrinsic definition of the connective “fuzzy exclusive or” E (f-Xor E), based only on the properties of boundary conditions, commutativity and partial isotonicity-antitonicity on the the end-points of the unit interval U = [0,1], in a way that the classical definition of the boolean Xor is preserved. We show three classes of the f-Xor E that can be also obtained from the composition of fuzzy connectives, namely, triangular norms, triangular conorms and fuzzy negations. A discussion about extra properties satisfied by the f-Xor E is presented. Additionally, the paper introduces a class of fuzzy equivalences that generalizes the Fodor and Roubens's fuzzy equivalence, and four classes of fuzzy implications induced by the f-Xor E, discussing their main properties. The relationships between those classes of fuzzy implications and automorphisms are explored. The action of automorphisms on f-Xor E is analyzed.