A Survey of the Homotopy Properties of Inclusion of Certain Types of Configuration Spaces into the Cartesian Product

Let X be a topological space.In this survey the authors consider several types of configuration spaces,namely,the classical(usual)configuration spaces F_n(X)and D_n(X),the orbit configuration spaces F_n^G(X)and F_n^G(X)/S_nwith respect to a free action of a group G on X,and the graph configuration spaces F_n~Γ(X)and F_n~Γ(X)/H,whereΓis a graph and H is a suitable subgroup of the symmetric group S_n.The ordered configuration spaces F_n(X),F_n^G(X),F_n~Γ(X)are all subsets of the n-fold Cartesian product ∏_1~nX of X with itself,and satisfy F_n^G(X)?F_n(X)?F_n~Γ(X)?∏_1~nX.If A denotes one of these configuration spaces,the authors analyse the difference between A and ∏_1~nXfrom a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusionι:A-→∏_1~nX,the homotopy type of the homotopy fibre I_ιof the mapιvia certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I_ιand arising from the inclusionι.In this respect,if X is either a surface without boundary,in particular if X is the 2-sphere or the real projective plane,or a space whose universal covering is contractible,or an orbit space S^k/Gof the k-dimensional sphere by a free action of a Lie group G,the authors present recent results obtained by themselves for the first case,and in collaboration with Golasi′nski for the second and third cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In order to motivate various questions,for the remaining types of configuration spaces,a few of their basic properties are described and proved.A list of open questions and problems is given at the end of the paper.

Orbit configuration spaces of small covers and quasi-toric manifolds

In this article,we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology,such as small covers,quasi-toric manifolds and(real)moment-angle manifolds;especially for the cases of small covers and quasi-toric manifolds.These kinds of orbit configuration spaces have non-free group actions,and they are all noncompact,but still built via simple convex polytopes.We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope.As a by-product of our method,we also obtain a formula of the Euler characteristic for the classical configuration space,which generalizes the Félix-Thomas formula.In addition,we also study the homotopy type of such orbit configuration spaces.In particular,we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold,which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

The Cross-ratio Compactification of the Configuration Space of Ordered Points on

A natural compactification of the virtual configuration space of N points on the Riemann sphere C is constructed by using cross-ratios. We show that this compactification is homeomorphic to the Bers＇ compactification of the virtual moduli space of a punctured Riemann sphere of type N. In particular, the system of global and explicit coordinates of this standard compactification is given by cross-ratios.

Crossing-changeable braids from chromatic configuration spaces

Motivated by the work in Li et al.(2019),this paper deals with the theory of the braids from chromatic configuration spaces.These kinds of braids possess the property that some strings of each braid may intersect together and can also be untangled,so they are quite different from the ordinary braids in the sense of Artin(1925).This enriches and extends the theory of ordinary braids.

Evaluation and Optimization of Electrode Configuration of Multi-Channel Corona Discharge Plasma for Dye-Containing Wastewater Treatment

A discharge plasma reactor with a point-to-plane structure was widely studied experimentally in wastewater treatment.In order to improve the utilization efficiency of active species and the energy efficiency of this kind of discharge plasma reactor during wastewater treatment,the electrode configuration of the point-to-plane corona discharge reactor was studied by evaluating the effects of discharge spacing and adjacent point distance on discharge power and discharge energy density,and then dye-containing wastewater decoloration experiments were conducted on the basis of the optimum electrode configuration.The experimental results of the discharge characteristics showed that high discharge power and discharge energy density were achieved when the ratio of discharge spacing to adjacent point distance（d/s） was 0.5.Reactive Brilliant Blue（RBB） wastewater treatment experiments presented that the highest RBB decoloration efficiency was observed at d/s of 0.5,which was consistent with the result obtained in the discharge characteristics experiments.In addition,the biodegradability of RBB wastewater was enhanced greatly after discharge plasma treatment under the optimum electrode configuration.RBB degradation processes were analyzed by GC-MS and IC,and the possible mechanism for RBB decoloration was also discussed.

An alternative view of condensed-phase photoionization

This paper proposes an accurate valuable interpretation scheme to study the evolvement of the photoionization processes from the isolated to the condensed atoms by a unique ab initio method. The variations of the photoionization cross sections of the atomic sodium with the photoelectron energy and the boundary radius of the atomic configuration space are studied in this new scheme by the R-matrix method. The discrepancy in the photoionization spectra of the isolated and the condensed sodium has been explained quantitatively and understood successfully by this alternative view in detail for the first time.

KINEMATICS AND ERROR ANALYSIS FOR MULTIBODY SYSTEM

In this paper, a new concept, i.e. Generalized Jacobian Matrix of MBS (abbreviation for multibody system) is presented which unifies three concepts of different realms, Partial (angular) velocity in MBS, first order effect coefficient in mechanism, Jacobian Matrix in robotics. This new concept not only enriches and developes kinematics of MBS and makes kinematics theory of MBS concise and being a formal way. but also becomes a new way to kinematic analysis of spatial mechanisms. It is also enlarged the paper that the usage of kinematics of MBS, the identical relationship between kinematics of MBS and error analysis of MBS is revealed.

Learning real-time search on c-space GVDs

In the context of robotics, configuration space （c- space） is widely used for non-circular robots to engage tasks such as path planning, collision check, and motion planning. In many real-time applications, it is important for a robot to give a quick response to the user＇s command. Therefore, a constant bound on planning time per action is severely im- posed. However, existing search algorithms used in c-space gain first move lags which vary with the size of the under- lying problem. Furthermore, applying real-time search algo- rithms on c-space maps often causes the robots being trapped within local minima. In order to solve the above mentioned problems, we extend the learning real-time search （LRTS） algorithm to search on a set of c-space generalized Voronoi diagrams （c-space GVDs）, helping the robots to incremen- tally plan a path, to efficiently avoid local minima, and to ex- ecute fast movement. In our work, an incremental algorithm is firstly proposed to build and represent the c-space maps in Boolean vectors. Then, the method of detecting grid-based GVDs from the c-space maps is further discussed. Based on the c-space GVDs, details of the LRTS and its implemen- tation considerations are studied. The resulting experiments and analysis show that, using LRTS to search on the c-space GVDs can 1） gain smaller and constant first move lags which is independent of the problem size; 2） gain maximal clear- ance from obstacles so that collision checks are much re- duced; 3） avoid local minima and thus prevent the robot from visually unrealistic scratching.

A Theory of Orbit Braids

In this paper,the authors systematically discuss orbit braids in M×I with regards to orbit configuration space FG(M,n),where M is a connected topological manifold of dimension at least 2 with an effective action of a finite group G.These orbit braids form a group,named orbit braid group,which enriches the theory of ordinary braids.The authors analyze the substantial relations among various braid groups associated to those configuration spaces FG(M,n),F(M/G,n)and F(M,n).They also consider the presentations of orbit braid groups in terms of orbit braids as generators by choosing M=C with typical actions of Zpand(Z_(2))^(2).

A Topological Implementation for Motion Planning of a Robotic Arm

An efficient path planning algorithm based on topologic method is presented in this paper.The colli- sion free path planning for three-joint robotic arm consists of three parts:partition of C-space,construc- tion of CN and search for a path in CN.We mainly solved the problems of partitioning the C-space and judging the connectivity between connected blocks,etc.For the motion planning of a robotic arm with a gripper,we developed the concepts of global planning and local planning,and discussed the basic fac- tors for constructing the planning system.In the paper,some evaluation and analysis of the complexity and reliability of the algorithm are given,together with some ideas to improve the efficiency and increase the reliability.At last,some experimental results are presented to show the efficiency and accuracy of the nigorithm.

Direct 3D Painting with a Metaball-Based Paintbrush

This paper presents a direct 3D painting algorithm for polygonalmodels in 3D object-space with a metaball-based paintbrush in virtual environment.The user is allowed to directly manipulate the parameters used to shade the surfaceof the 3D shape by applying the pigment to its surface with direct 3D manipulationthrough a 3D flying mouse.

Poincar and Weak Poincar Inequalities for the Mixed Poisson Measure

By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincare and weak Poincare inequalities are established, while another Poincare type inequality is disproved under some reasonable assumptions.

Electronic States of Some Semiconductor Clusters

Potential surfaces and equilibrium geometries of InAs 2, In 2As, InAs 2 + and In 2As + were studied using the complete active space multi configuration self consistent field (CASMCSCF) technique. Two electronic states, namely 2B 2 and 2B 1, were found to prevail as the ground states for the InAs 2 and In 2As trimers, respectively. The corresponding adiabatic ionization energies were computed and the leading configurations of the ground states were analyzed according to the wavefunctions.