Star points on smooth hypersurfaces

Star sampling (SS) is a random sampling procedure on a graph wherein each sample consists of a randomly selected vertex (the star center) and its (one-hop) neighbors (the star points). We consider the use of SS to find any member of a target set of vertices in a graph, where the figure of merit (cost) is either the expected number of samples (unit cost) or the expected number of star centers plus star points (linear cost) until a vertex in the target set is encountered, either as a star center or as a star point. We analyze these two performance measures on three related star sampling paradigms: SS with replacement (SSR), SS without center replacement (SSC), and SS without star replacement (SSS). Exact and approximate expressions are derived for the expected unit and linear costs of SSR, SSC, and SSS on Erdős-Rényi (ER) random graphs. The approximations are seen to be accurate. SSC/SSS are notably better than SSR under unit cost for low-density ER graphs, while SSS is notably better than SSR/SSC under linear cost for low- to moderate-density ER graphs. Simulations on twelve "real-world" graphs shows the cost approximations to be of variable quality: the SSR and SSC approximations are uniformly accurate, while the SSS approximation, derived for an ER graph, is of variable accuracy.

Star sensor is disturbed by strong straylight, which increases the gray level of the captured star map, and this leads to invalid detection of star points and affects the high-precision location of the centroid. To address this issue, we propose a star centroid localization method based on gradient-oriented multi-directional local contrast enhancement. First, the background gray level distribution patterns of star sensors under various actual straylight interference conditions are analyzed. Based on this analysis, a background imaging model for complex operational scenarios is established. Finally, simulations are conducted under complex conditions with straylight images to test the star point detection rate, false detection rate, centroid localization accuracy, and statistical significance testing. The results show that the proposed algorithm outperforms the TOP-HAT, MAX-BACKG (Max-Background Filtering), LCM (Local Contrast Measure), MPCM (Multiscale Patch-Based Contrast Measure), and CMLCM (Curvature-Based Multidirectional Local Contrast Method for Star Detection of Star Sensor) algorithms in terms of star point detection rate. Additionally, the RMSE centroid localization error is achieved with 0.1 pixels, demonstrating its ability to effectively locate star centroids under complex conditions and meet certain engineering application requirements.

An external magnetic field in configuration space coupled to quantum dynamics induces noncommutativity in its velocity momentum space. By phase space duality, an external vector potential in the conjugate momentum sector of the system induces noncommutativity in its configuration space. Such a rationale for noncommutativity is explored herein for an arbitrary configuration space of Euclidean geometry. Ordinary quantum mechanics with a commutative configuration space is revisited first. Through the introduction of an arbitrary positive definite ∗-product, a one-to-one correspondence between the Hilbert space of abstract quantum states and that of the enveloping algebra of the position quantum operators is identified. A parallel discussion is then presented when configuration space is noncommutative, and thoroughly analysed when the conjugate magnetic field is momentum independent and nondegenerate. Once again the space of quantum states may be identified with the enveloping algebra of the noncommutative position quantum operators. Furthermore, when the positive definite ∗-product is adapted to the conjugate magnetic field, the coordinate operators span a Fock algebra of which the coherent states are the analogues of the structureless points in a commutative configuration space. These results generalise and justify a posteriori within ordinary canonical quantisation the heuristic approach to quantum mechanics in the noncommutative Euclidean plane as constructed and developed by F. G. Scholtz and his collaborators.

The problem of the global Lagrangian description of a dynamical system which has a global Hamiltonian description is considered; the motion of a charged particle in the field of a static magnetic monopole is a prototype of such dynamical problems. The symplectic structure on the (cotangent bundle of the) manifold is associated with a closed but not necessarily exact two-form. In conventional language this means that the tensor field inverse to the symplectic metric is divergence free but not necessarily expressible globally as the curl of a vector field; and hence the usual passage to Hamilton's principle and the Lagrangian can be carried out in patches but not necessarily globally. Generalizing techniques developed elsewhere, we show that the fundamental expression for the action integral is like the surface integral of the closed but not necessarily exact antisymmetric tensor field (two-form) rather than like the line integral of a vector field (one-form). This naturally leads to the path-space formalism in which the action functional depends not only on a point in configuration space but also a path from a chosen fixed point terminating in the relevant point in configuration space. We then show how this is a universal unfolding and how the topological obstructions in the configuration-space description are circumvented in the path-space description. The path-space description contains redundancies and they can be reduced; locally the new manifold is obtained by associating an angle of rotation [a U(1) fiber] with each configuration-space point, but globally there would be a nontrivial fiber-bundle structure. These questions naturally carry over to the quantum theory of such systems and it is shown how quantization conditions arise. The structure of the quantum-theory formalism for such systems is analyzed. The considerations are applied to the monopole problem where the enlarged space in which the Lagrangian description is possible is explicitly identified as the manifold of the group SU(2).

We consider configuration spaces of nonidentical pointlike particles. The physically motivated assumption that any two particles cannot be located at the same point in space–time leads to nontrivial topological structure of the configuration space. For a quantum mechanical description of such a system, we classify complex vector bundles over the configuration space and obtain potentials of topological origin, similar to those that occur in the fiber bundle approach to Dirac’s magnetic monopole or in Yang–Mills theory.

Let $\Delta $ denote a nondegenerate k-simplex in $\mathbb {R}^k$ . The set $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices in $\mathbb {R}^k$ similar to $\Delta $ is diffeomorphic to $\operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$ , where the factor in $\operatorname {O}(k)$ is a matrix called the pose. Among $(k-1)$ -spheres smoothly embedded in $\mathbb {R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in \operatorname {O}(k)$ . Further, the intersection of $\operatorname {\mathrm {Sim}}(\Delta )$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $\operatorname {O}(k)$ via the pose map. This gives a high-dimensional generalisation of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces.

A new heuristic approach to the robot free-path problem by stepwise path improvement

Let $\Sigma \subset \mathbb{P}^4$ be an integral hypersurface of degree $s$ with a $(s-2)$-uple plane. We show that the degrees of smooth surfaces $S \subset \Sigma$ with $q(S)=0$ are bounded by a function of $s$. We also show that if $S \subset \mathbb{P}^4$ is a smooth surface with $q(S)=0$ and if $S$ lies on a quartic hypersurface $\Sigma$ such that $\dim(\Sing(\Sigma))=2$, then $\deg(S) \leq 40$.

Robotic manipulators designed for home assistance and new surgical procedures often have significant uncertainty in their actuation due to compliance requirements, cost constraints, and size limits. We introduce a new integrated motion planning and control algorithm for robotic manipulators that makes safety a priority by explicitly considering the probability of unwanted collisions. We first present a fast method for estimating the probability of collision of a motion plan for a robotic manipulator under the assumptions of Gaussian motion and sensing uncertainty. Our approach quickly computes distances to obstacles in the workspace and appropriately transforms this information into the configuration space using a Newton method to estimate the most relevant collision points in configuration space. We then present a sampling-based motion planner based on executing multiple independent rapidly exploring random trees that returns a plan that, under reasonable assumptions, asymptotically converges to a plan that minimizes the estimated collision probability. We demonstrate the speed and safety of our plans in simulation for (1) a 3-D manipulator with 6 DOF, and (2) a concentric tube robot, a tentacle-like robot designed for surgical applications.

Definitions Symmetry operations leave a set of objects in indistinguishable configurations which are said to be equivalent . A set of symmetry operators always contains at least one element, the identity operator E . When operating with E the final configuration is not only indistinguishable from the initial one, it is identical to it. A proper rotation , or simply rotation , is effected by the operator R ( φ n ), which means “carry out a rotation of configuration space with respect to fixed axes through an angle φ about an axis along some unit vector n .” The range of φ is − π φ ≤ π . Configuration space is the three-dimensional (3-D) space ℛ 3 of real vectors in which physical objects such as atoms, molecules, and crystals may be represented. Points in configuration space are described with respect to a system of three space-fixed right-handed orthonormal axes x , y , z , which are collinear with OX, OY, OZ (Figure 2.1(a)). (A right-handed system of axes means that a right-handed screw advancing from the origin along OX would rotate OY into OZ, or advancing along OY would rotate OZ into OX, or along OZ would rotate OX into OY.) The convention in which the axes x , y , z remain fixed, while the whole of configuration space is rotated with respect to fixed axes, is called the active representation .

A two-dimensional, motion planning approach that derives inspiration from the formation of shadows by opaque objects in the presence of a Gaussian light source is presented. It utilises the ‘potential field’-like varying intensity of a Gaussian light source coupled with the rectilinear property of light. The method borrows advantages of artificial potential field (APF) methods, such as single path generation and scope of feedback control. At the same time, with the selection of adequate parameters, this method is immune to the formation of critical points in the configuration space – a drawback associated with most APF-based methods. Simulations are carried out and results are presented to demonstrate the applicability of the approach.

This paper deals with real-time implementation of visual-motor control of a 7 degree of freedom (DOF) robot manipulator using self-organized map (SOM) based learning approach. The robot manipulator considered here is a 7 DOF PowerCube manipulator from Amtec Robotics. The primary objective is to reach a target point in the task space using only a single step movement from any arbitrary initial configuration of the robot manipulator. A new clustering algorithm using Kohonen SOM lattice has been proposed that maintains the fidelity of training data. Two different approaches have been proposed to find an inverse kinematic solution without using any orientation feedback. In the first approach, the inverse Jacobian matrices are learnt from the training data using function decomposition. It is shown that function decomposition leads to significant improvement in accuracy of inverse kinematic solution. In the second approach, a concept called sub-clustering in configuration space is suggested to provide multiple solutions for the inverse kinematic problem. Redundancy is resolved at position level using several criteria. A redundant manipulator is dexterous owing to the availability of multiple configurations for a given end-effector position. However, existing visual motor coordination schemes provide only one inverse kinematic solution for every target position even when the manipulator is kinematically redundant. Thus, the second approach provides a learning architecture that can capture redundancy from the training data. The training data are generated using explicit kinematic model of the combined robot manipulator and camera configuration. The training is carried out off-line and the trained network is used on-line to compute the joint angle vector to reach a target position in a single step only. The accuracy attained is better than the current state of art.

Reducing the energy to carry out computational tasks is key to almost any computing application. We focus in this paper on iterative applications that have explicit computational deadlines per iteration. Our objective is to meet the computational deadlines while minimizing energy. We leverage the vast configuration space offered by heterogeneous multicore platforms which typically expose three dimensions for energy saving configurability: Voltage/frequency levels, thread count and core type (e.g. ARM big/LITTLE). We note that when choosing the most energy-efficient configuration that meets the computational deadline, an iteration will typically finish before the deadline and execution-time slack will build up across iterations. Our proposed slack management policy - SaC (Slack as a Currency) - proactively explores the configuration space to select configurations that can save substantial amounts of energy. To avoid the overheads of an exhaustive search of the configuration space, our proposal also comprises a low-overhead, on-line method by which one can assess each point in the configuration space by linearly interpolating between the endpoints in each configuration-space dimension. Overall, we show that our proposed slack management policy and linear-interpolation configuration assessment method can yield 62% energy savings on top of race-to-idle without missing any deadlines.

For a point p in a smooth real hypersurface M⊂ℂ n , where the Levi form has the nontrivial kernel K p 10 ≠0, we introduce an invariant cubic tensorτp3:ℂTp×Kp10×Kp10¯→ℂ⊗(Tp/Hp),which together with Ebenfelt’s 3rd order tensor, constitutes the full set of the 3rd order invariants of M at p.Next, in addition, assume M⊂ℂ n to be (weakly) pseudoconvex. Then τ p 3 must identically vanish. In this case we further define an invariant quartic tensorτp4:ℂTp×ℂTp×Kp10×Kp10¯→ℂ⊗(Tp/Hp),and for every q=0,⋯,n-1, an invariant submodule sheaf 𝒮 10 (q) of (1,0) vector fields in terms of the Levi form, and an invariant ideal sheaf ℐ(q) of complex functions generated by certain components and derivatives of the Levi form, such that the set of points of Levi rank q is locally contained in real smooth submanifolds defined by real parts of the functions in ℐ(q), whose tangent spaces have explicit algebraic description in terms of the quartic tensor τ 4 .Most recently, the constructions of τ 3 and τ 4 inspired Raich and Harrington [35] to discover new related invariants in the non-pseudoconvex case and connect them with the closed range property for the ∂ ¯ operator.Finally, we relate the introduced invariants with D’Angelo’s finite type, Catlin’s multitype and Catlin’s boundary systems.

The investigations repor ted at this sympos ium are in the process of publication elsewhere [1, 2]; since the la t te r papers are r a the r detailed, we will content ourselves here with a ra ther brief survey of the work, with only a few addit ional results. The t e rm (~ Monte Carlo ,) has come into use to designate numerical methods in which specifically stochastic elements are introduced, in contras t to the whole body of classical numerical techniques which consist of numerical evaluations of complete ly de te rminan t algebraic expressions. The par t icular Monte Carlo me thod used here was devised by METROPOHS et al. [3]; its essential fea ture is t ha t i t produces a Markov chain [4] in which the individual 5 Ia rkov s tates are points in the usual configuration space of s tat is t ical mechanics, for a sys tem of N molecules confined a t a t e m p e r a t u r e T in ~ volume Y. The t ransi t ion probabil i t ies character izing the Markov chain are de termined in such a way tha t the va lue of any funct ion of configuration state, averaged over all s tates developed in the Markov cha.in, approaches the pe t i t canonical ensemble average value of the same function. I t should be noticed t ha t the present me thod does no t consist in an evaluat ion of the configurational phase in tegral b y means of a random, selection of points in configuration space. A me thod of the l a t t e r sort, which m a y be compared with a sequence of independent throws of a die, is indeed