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It is an important aspect of cluster theory that cluster categories are “categorifications” of cluster algebras. This is expressed formally by the (original) Caldero–Chapoton map X which sends certain objects of cluster categories to elements of cluster algebras. Let τ c → b → c be an Auslander–Reiten triangle. The map X has the salient property that X ( τ c ) X ( c ) − X ( b ) = 1 . This is part of the definition of a so-called frieze, see [1] . The construction of X depends on a cluster tilting object. In a previous paper [14] , we introduced a modified Caldero–Chapoton map ρ depending on a rigid object; these are more general than cluster tilting objects. The map ρ sends objects of sufficiently nice triangulated categories to integers and has the key property that ρ ( τ c ) ρ ( c ) − ρ ( b ) is 0 or 1. This is part of the definition of what we call a generalised frieze. Here we develop the theory further by constructing a modified Caldero–Chapoton map, still depending on a rigid object, which sends objects of sufficiently nice triangulated categories to elements of a commutative ring A . We derive conditions under which the map is a generalised frieze, and show how the conditions can be satisfied if A is a Laurent polynomial ring over the integers. The new map is a proper generalisation of the maps X and ρ . [ABSTRACT FROM AUTHOR]

Let 풞 be a triangulated category which has Auslander-Reiten triangles, and ℛ a functorially finite rigid subcategory of 풞. It is well known that there exist Auslander-Reiten sequences in mod ℛ. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod ℛ and the Auslander-Reiten functors, triangles in 풞, respectively. Furthermore, if 풯 is a cluster-tilting subcategory of 풞 and mod 풯 is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod 풯 corresponding to the ones in 풞. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Frobenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d=2, and then by Dugas in the general case, using different methods. [ABSTRACT FROM AUTHOR]

Shape fabrics of elliptical objects in rocks are usually assumed to develop by passive behavior of inclusions with respect to the surrounding material leading to shape-based strain analysis methods belonging to the R f / ϕ family. A probability density function is derived for the orientational characteristics of populations of rigid ellipses deforming in a pure shear 2D deformation with both no-slip and slip boundary conditions. Using maximum likelihood a numerical method is developed for estimating finite strain in natural populations deforming for both mechanisms. Application to a natural example indicates the importance of the slip mechanism in explaining clast shape fabrics in deformed sediments. [ABSTRACT FROM AUTHOR]

We consider a Krull-Schmidt, Hom-finite, 2-Calabi-Yau triangulated category with a basic rigid object T, and show a bijection between the set of isomorphism classes of basic rigid objects in the finite presented category pr T of T and the set of isomorphism classes of basic τ-rigid pairs in the module category of the endomorphism algebra End( T). As a consequence, basic maximal objects in pr T are one-to-one correspondence to basic support τ-tilting modules over End( T). This is a generalization of correspondences established by Adachi-Iyama-Reiten. [ABSTRACT FROM AUTHOR]

Surface deformations without tearing or stretching, preserving the intrinsic properties, are called isometries. This paper presents a new definition of Gaussian curvature moments (GCMs) by the integral of n power of Gaussian curvature. Then a series of moment invariants, called Gaussian curvature moment invariants (GCMIs), are derived via GCMs. These moment invariants share many good properties under rigid transformations and isometric non-rigid transformations, and Gaussian-Bonnet theorem is a special case of GCMI. GCMIs are invariant under isometry and scaling transformations. We construct an invariant vector as a descriptor for a surface via GCMIs, and a modified χ2 distance is defined as a measure of similarity. Finally, experiments show that it is a reliable descriptor for isometric non-rigid shape. [ABSTRACT FROM AUTHOR]

Abstract: Penetration of rigid object into semi-infinite compressible solid is investigated in the present study. First a detailed numerical analysis of the penetration process is performed via smooth particle hydrodynamics. Based on the numerical results, estimate for the resistive force that the target exerts on the penetrating object is obtained. It is shown that in the computation of the resistive force the quasi-static coefficient can accurately be obtained through the spherical cavity expansion theory. However for a given target material the coefficient associated with the hydro-dynamic term significantly depends on the impact velocity unlike commonly assumed constant values. Penetration equations for rigid object with arbitrary nose geometries are derived. Developed analytical model is then verified through some experimental and analytical results reported in the literature. [Copyright &y& Elsevier]

Abstract: Shape fabrics of elliptical objects in rocks are usually assumed to develop by passive behaviour of inclusions together with the surrounding material leading to shape-based strain analysis methods belonging to the R f/ϕ family. By deriving the probability density function for populations of rigid ellipses deforming in a general 2D deformation, a method is developed which can be used to estimate both finite strain and the kinematic vorticity number. Statistical parameters are theoretically derived and their behaviours under various kinematic conditions are investigated. The maximum likelihood method from statistics is used to produce a numerical method for estimating deformation parameters from natural populations. A simulation study demonstrates that finite strain can be estimated well for both low and high applied finite strains, whereas the kinematic vorticity number is well estimated only in the case of high finite strains (R s >40), and that large sample numbers (≈1000) are required. [Copyright &y& Elsevier]

The paper presents two-dimensional numerical simulations of inclusion trail patterns of spherical and non-spherical synkinematic porphyroblasts rotating and growing over a pre-existing, passive foliation during a single progressive deformation taking into account the effects of heterogeneous strain field around the rigid porphyroblast. The analysis is based on the velocity functions around a rigid object hosted in a Newtonian viscous medium. Using Visual Basic software, the trail patterns have been simulated in a PC assuming that the porphyroblast grows by increments of constant radius, over a pre-existing foliation, which simultaneously gets distorted due to heterogeneous strain field around the porphyroblast and included within the porphyroblast. The numerical simulations show that the trail patterns in spherical porphyroblasts are controlled by two factors: (1) the initial orientation of the foliation marker relative to the shear direction (θ) and (2) the ratio of pure shear and simple shear rates in the bulk deformation (Sr). Complex trail structures form when θ<0° and Sr&les;0.75. The ratio of growth rates along axial directions of non-spherical porphyroblasts is an additional factor that controls the trail patterns. [Copyright &y& Elsevier]

We present a general search strategy called SANDROS for motion planning, and its applications to motion planning for three types of robots: (1) manipulator; (2) rigid object; and (3) multiple rigid objects. SANDROS is a dynamic-graph search algorithm, and can be described as a hierarchical, nonuniform-multiresolution, best-first search to find a heuristically short motion in the configuration space. The SANDROS planner is resolution complete, and its computation time is commensurate with the problem difficulty measured empirically by the solution-path complexity. For many realistic problems involving a manipulator or a rigid object with six degrees of freedom, its computation times are under 1 minute for easy problems involving wide free space, and several minutes for relatively hard problems. [ABSTRACT FROM PUBLISHER]