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In this paper, we will place clusters in type 픸̃ (equivalently triangulations of an annulus) into infinite families parametrized by winding numbers of certain arcs in the corresponding triangulation. We will count how many such families there are and provide an algebraic description of these families in terms of the corresponding cluster category. [ABSTRACT FROM AUTHOR]

The efficient and accurate tracking of a target in complex scenes has always been one of the challenges to tackle. At present, the most effective tracking algorithms are basically neural network models based on deep learning. Although such algorithms have high tracking accuracy, the huge number of parameters and computations in the network models makes it difficult for such algorithms to meet the real-time requirements under limited hardware conditions, such as embedded platforms with small size, low power consumption and limited computing power. Tracking algorithms based on a kernel correlation filter are well-known and widely applied because of their high performance and speed, but when the target is in a complex background, it still can not adapt to the target scale change and occlusion, which will lead to template drift. In this paper, a fast multi-scale kernel correlation filter tracker based on adaptive template updating is proposed for common rigid targets. We introduce a simple scale pyramid on the basis of Kernel Correlation Filtering (KCF), which can adapt to the change in target size while ensuring the speed of operation. We propose an adaptive template updater based on the Mean of Cumulative Maximum Response Values (MCMRV) to alleviate the problem of template drift effectively when occlusion occurs. Extensive experiments have demonstrated the effectiveness of our method on various datasets and significantly outperformed other state-of-the-art methods based on a kernel correlation filter. [ABSTRACT FROM AUTHOR]

The efficient and accurate tracking of a target in complex scenes has always been one of the challenges to tackle. At present, the most effective tracking algorithms are basically neural network models based on deep learning. Although such algorithms have high tracking accuracy, the huge number of parameters and computations in the network models makes it difficult for such algorithms to meet the real-time requirements under limited hardware conditions, such as embedded platforms with small size, low power consumption and limited computing power. Tracking algorithms based on a kernel correlation filter are well-known and widely applied because of their high performance and speed, but when the target is in a complex background, it still can not adapt to the target scale change and occlusion, which will lead to template drift. In this paper, a fast multi-scale kernel correlation filter tracker based on adaptive template updating is proposed for common rigid targets. We introduce a simple scale pyramid on the basis of Kernel Correlation Filtering (KCF), which can adapt to the change in target size while ensuring the speed of operation. We propose an adaptive template updater based on the Mean of Cumulative Maximum Response Values (MCMRV) to alleviate the problem of template drift effectively when occlusion occurs. Extensive experiments have demonstrated the effectiveness of our method on various datasets and significantly outperformed other state-of-the-art methods based on a kernel correlation filter.

A frieze in the modern sense is a map from the set of objects of a triangulated category C to some ring. A frieze X is characterised by the property that if τx→y→x is an Auslander-Reiten triangle in C, then X(τx)X(x)-X(y)=1. The canonical example of a frieze is the (original) Caldero-Chapoton map, which send objects of cluster categories to elements of cluster algebras. Holm and Jørgensen (Nagoya Math J 218:101-124, 2015; Bull Sci Math 140:112-131, 2016), the notion of generalised friezes is introduced. A generalised frieze X′ has the more general property that X′(τx)X′(x)-X′(y)∈{0,1}. The canonical example of a generalised frieze is the modified Caldero-Chapoton map, also introduced in Holm and Jørgensen (2015, 2016). Here, we develop and add to the results in Holm and Jørgensen (2016). We define Condition F for two maps α and β in the modified Caldero-Chapoton map, and in the case when C is 2-Calabi-Yau, we show that it is sufficient to replace a more technical “frieze-like” condition from Holm and Jørgensen (2016). We also prove a multiplication formula for the modified Caldero-Chapoton map, which significantly simplifies its computation in practice. [ABSTRACT FROM AUTHOR]

We present a system for the 6-DOF pose estimation of rigid objects which can be used for tool tracking in manual manufacturing applications like quality assurance or assembly assistance. To be practical and widely adaptable, the system must be low-cost, passive, impact resistant, retrofittable and applicable to arbitrary rigid objects. Therefore we attach circular markers of retroreflective self-adhesive film to relevant objects. A stereo camera system with IR illumination tracks the 3D positions of these markers and uses known marker constellations to estimate the object pose. Additionally a simple teach-in procedure which learns marker constellations of new objects is presented. [ABSTRACT FROM AUTHOR]

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]