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There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G3,3{G}_{3,3} generated by the linear fractional transformations a=1−1∕za=1-1/z and b=z+2b=z+2. This group is the unique subgroup of the modular group PSL(2,Z){\rm{PSL}}(2,{\mathbb{Z}}) with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number p∕qp/q, to obtain an element in G3,3{G}_{3,3} with cusp point p∕qp/q. As a result, we say that the action of G3,3{G}_{3,3} on rational numbers is transitive.

Most parallel manipulators have multiple solutions to the direct kinematic problem. The ability to perform assembly changing motions has received the attention of a few researchers. Cusp points play an important role in the kinematic behavior. This study investigates the cusp points and assembly changing motions in a two degrees of freedom planar parallel manipulator. The direct kinematic problem of the manipulator yields a quartic polynomial equation. Each root in the equation determines the assembly configuration, and four solutions are obtained for a given set of actuated joint coordinates. By regarding the discriminant of the repeated roots of the quartic equation as an implicit function of two actuated joint coordinates, the direct kinematic singularity loci in the joint space are determined by the implicit function. Cusp points are then obtained by the intersection of a quadratic curve and a cubic curve. Two assembly changing motions by encircling different cusp points are highlighted, for each pair of solutions with the same sign of the determinants of the direct Jacobian matrices. [ABSTRACT FROM AUTHOR]

In order to accomplish machine visualization, object detection algorithms strive to recognise all target items in the image and derive the categories and object location information. Numerous ways to solve this problem have been presented where most of them are inspired by computer vision and deep learning methodologies. Existing techniques, on the other hand, consistently fail to recognise small, dense objects and even fail to detect objects with random geometric modifications. Object skeletons can help in object representation and detection. Skeleton is a natural object’s inherent visual description that contains comprehensive shape semantics. They supplement the object outline by providing additional information, such as how the object scale in thickness that varies between object components. However, extracting object skeletons from natural photos is difficult because the extractor must be able to capture both local and non-local visual contexts in order to calculate the scale of each skeleton pixel. To solve this issue, a Cusp Pixel Labelled Model with Precise Tuned Outline using Machine Learning (CPLM-PTOML) is proposed in this paper that accurately detects the cusp points of the object in the image by extracting the skeleton of the object to recognize the exact object resided in the image. A multi scale associated outputs to each stage of the image by monitoring the relationship between the receptive field sizes of the different layers in the image and the skeleton scales they can capture. The model is trained via multi-task learning, with one job being skeleton localization, which determines whether or not a pixel is a skeleton pixel, and the other being skeleton scale prediction, which predicts the scale of each skeleton pixel. At various phases, supervision is imposed by directing the scale-associated side outputs toward the ground-truth skeletons at the appropriate scales. The proposed model is compared with the traditional model in terms Data Training Samples, Object Feature Extraction Time Levels, Feature Extraction Accuracy, Cusp Pixel Labelling Time Levels, Cusp Pixel Identification Accuracy, Cusp Point Linking Accuracy, Image Considered and Cusp Point Recognition levels and the results exhibit that the proposed model exhibits better outcome.

We study the exceptional loci of birational (bimeromorphic) contractions of a hyperkähler manifold M. Such a contraction locus is the union of all minimal rational curves in a collection of cohomology classes which are orthogonal to a wall of the Kähler cone. Cohomology classes which can possibly be orthogonal to a wall of the Kähler cone of some deformation of M are called MBM classes. We prove that all MBM classes of type (1,1) can be represented by rational curves, called MBM curves. Any MBM curve can be contracted on an appropriate birational model of M, unless b 2 (M) ⩽ 5 . When b 2 (M) > 5 , this property can be used as an alternative definition of an MBM class and an MBM curve. Using the results of Bakker and Lehn, we prove that the stratified diffeomorphism type of a contraction locus remains stable under all deformations for which these classes remains of type (1,1), unless the contracted variety has b 2 ⩽ 4 . Moreover, these diffeomorphisms preserve the MBM curves, and induce biholomorphic maps on the contraction fibers, if they are normal. [ABSTRACT FROM AUTHOR]

Let M be a compact hyperkähler manifold with maximal holonomy (IHS). The group $$H^2(M, {\mathbb {R}})$$ is equipped with a quadratic form of signature $$(3, b_2-3)$$ , called Bogomolov-Beauville-Fujiki form. This form restricted to the rational Hodge lattice $$H^{1,1}(M,{\mathbb {Q}})$$ has signature (1, k). This gives a hyperbolic Riemannian metric on the projectivization H of the positive cone in $$H^{1,1}(M,{\mathbb {Q}})$$ . Torelli theorem implies that the Hodge monodromy group $$\varGamma $$ acts on H with finite covolume, giving a hyperbolic orbifold $$X=H/\varGamma $$ . We show that there are finitely many geodesic hypersurfaces, which cut X into finitely many polyhedral pieces in such a way that each of these pieces is isometric to a quotient $$P(M')/{\text {Aut}}(M')$$ , where $$P(M')$$ is the projectivization of the ample cone of a birational model $$M'$$ of M, and $${\text {Aut}}(M')$$ the group of its holomorphic automorphisms. This is used to prove the existence of nef isotropic line bundles on a hyperkähler birational model of a simple hyperkähler manifold of Picard number at least 5 and also illustrates the fact that an IHS manifold has only finitely many birational models up to isomorphism (cf. Markman and Yoshioka in Int. Math. Res. Not. 2015(24), 13563-13574, ). [ABSTRACT FROM AUTHOR]

Abstract: When moving from one arbitrary location at another, a parallel manipulator may change its assembly-mode without crossing a singularity. Because the non-singular change of assembly-mode cannot be simply detected, the actual assembly-mode during motion is difficult to track. This paper proposes a global explanatory approach to help better understand non-singular assembly-mode changing motions for 3-R P R planar parallel manipulators. The approach consists in fixing one of the actuated joints and analyzing the configuration-space as a surface in a three-dimensional space. Such a global description makes it possible to display all possible non-singular assembly-mode changing trajectories. [Copyright &y& Elsevier]

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• Design of mechanism for Otologic Surgery. • Kinematic structure of the proposed mechanism. • Workspace and Joint space analysis of the proposed 2-DOF Spherical mechanism. • Influence of the design parameter on the workspace properties. • Workspace without non-singular assembly modes changing trajectories. During Otologic surgery, and more broadly during microsurgery, the surgeon encounters several difficulties due to the confined spaces and micro-manipulations. The purpose of the paper is to design a robot with a prescribed regular workspace shape to handle an endoscope to assist the Otologic surgery. A spherical parallel mechanism with two degrees of freedom is analysed in its design parameter space. This mechanism is composed of three legs (2USP-U) to connect the base to a moving platform connected to a double parallelogram to create a remote center of motion (RCM). Its kinematic properties, i.e. the singularity locus and the number of direct kinematic solutions, are investigated. For some design parameters, non-singular assembly modes changing trajectories may exist and have to be investigated inside the prescribed regular workspace shape. Two sets of design parameters are presented with their advantages and disadvantages. [ABSTRACT FROM AUTHOR]