The Graph of Triangulations of a Point Configuration with d +4 Vertices Is 3-Connected

We study the graph of bistellar flips between triangulations of a vector configuration A with d+4 elements in rank d+1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension d with d+4 elements. Hence, every pair of triangulations can be joined by a finite sequence of bistellar flips and, in the acyclic case, every triangulation has at least three geometric bistellar neighbours. In corank 4, connectivity is not known and having at least four flips is false. In corank 2, the results are trivial since the graph is a cycle. Our methods are based on a dualization of the concept of triangulation of a point or vector configuration A to that of a virtual chamber of its Gale transform B , introduced by de Loera et al. in 1996. As an additional result we prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector configuration B can be realized as a cell in some pseudo-chamber complex of B in the same way that regular triangulations appear as cells in the usual chamber complex. All the results in this paper generalize to triangulations of corank 3 oriented matroids and virtual chambers of rank 3 oriented matroids, realizable or not. The details for this generalization are given in the Appendix. Received March 1, 1999, and in revised form September 7, 1999.     

An Intrinsic Geometric Formulation of the Equilibrium Equations in Continuum Mechanics

The theory of invariant continuum mechanics is based on the concept that forces and stresses are defined as elements of the cotangent bundle of the configuration manifold. While body and physical space are modeled as differentiable manifolds, the infinite dimensional configuration manifold is given by all configurations of the body in the physical space. In this paper a virtual work principle is postulated which leads together with an induced traction stress and Stokes' theorem directly to the local equilibrium equations and the traction boundary conditions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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  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.     

Virtual stability: Constructing a simulation model

After a discussion of the importance of stability and instability for complex systems theory, we define the concept of virtual stability as a state in which a system employs self‐monitoring and adaptive control to maintain itself in a configuration that would otherwise be unstable. The energy expended in this gains the system an increase in its flexibility of behavioral response to environmental changes. A model designed to illustrate virtual stability is presented, followed by a brief discussion of the evolutionary advantage this capacity provides. This leads to the suggestion that such advantage gives an argument both for the directionality of evolution and for the emergence of self‐consciousness. © 2008 Wiley Periodicals, Inc. Complexity, 2009     

An alternative perspective on the concept of stress in classical continuum mechanics

Within a variational formulation of continuum mechanics, as proposed for instance by Germain [1], the internal virtual work contribution of a continuum is postulated as a smooth density integrated over the deformed configuration of the body. In this smooth density the stress field appears as dual quantity to the gradient of the virtual displacement field. Since the mathematical definition of the volume integral naturally provides a macro-micro relation between infinitesimal volume elements and the continuous body, we propose in this paper an alternative definition of stress on the micro level of the infinitesimal volume elements. In particular, the stress is defined as the internal forces of the body that model the mutual force interaction between neighboring volume elements. The existence of the stress tensor on the macro level is then obtained from the summation of all virtual work contributions within the body, followed by a limit process in which the volume elements are sent to zero. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lagrangian description of the variational equations

A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all the virtual displacements joining any two integral curves. After discussing certain features of the formulation, we introduce the so-called inherited constants of the motion and relate them to the Noether constants of the extended system.     

一种基于新型插值单元的稳态传热边界元法

为了提高边界元法在求解稳态热问题时的计算精度,通过使用一种新型单元插值方法(称为扩展单元插值法),实现对稳态传热问题的求解。扩展单元是在传统不连续单元的边界配置虚拟节点,把原非连续单元变成高阶的连续单元,并将其作为新型的插值单元。利用虚拟节点和内部源节点构造出的插值函数,可以精确插值边界上的连续和不连续物理场,插值精度要比原始不连续单元高两阶。另外,边界积分方程只在传统的不连续单元的内部节点处建立,只包含内部源节点的自由度,而虚拟节点的自由度可通过与内部源节点之间的关系消除掉,因此最终系统方程的求解规模不会增加。这种新型的插值单元继承了传统连续和不连续单元的优点,克服了它们的缺点。数值结果表明,此种单元插值方法用于求解稳态传热问题时可获得较高的计算精度和收敛性。     

A Categorical Structure for the Virtual Braid Group

This article gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. This point of view makes many relationships between the virtual braid group and the pure virtual braid group apparent, and makes representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter Equation equally transparent. In this categorical framework, the virtual braid group has nothing to do with the plane and nothing to do with virtual crossings. It is a natural group associated with the structure of algebraic braiding.     

Computation of program controls performed nonstop by gyrodynes

A new method for calculating space vehicle (SV) attitude controls ensuring their effective implementation by a system of collinear pairs of single-gimbal forced unrestrained gyros (gyrodynes) has been proposed. The novelty of the method consists in a virtual kinematic configuration of the gyro system, i.e., the precession of gyro units in the collinear pairs of gyrodynes is coupled in a nonmechanical manner. In addition, the angular momentum of the system as a state variable for describing the dynamics of the SV permanent rotation was used for the first time at the stage of computing controls performed nonstop by gyrodynes. In the general formulation, when the desired final state of the SV is arbitrary, the SV attitude control problem can be reduced to a sequence of permanent rotations. The performance of the method is demonstrated as applied to the calculation of program gyrodyne controls with a permanent reduction in the SV angular velocity around its center of mass with a nonzero SV angular momentum after its discharge.

     

Virtual Teams: What are their Characteristics, and Impact on Team Performance?

To date, efforts to understand virtual teaming have been largely anecdotal and atheoretical. Therefore, drawing from the extant research in the groups domain, we attempt to ground the definition of a virtual team in well-established group-level constructs, and design a simulation study to investigate the impact of different virtual team characteristics on team performance. Essentially, we argue that the virtual team is defined by three key characteristics—the virtual team context, the virtual team composition, and the virtual team structure. Using the VDT computational discrete event simulation model as our experimental platform, we simulated different virtual team models, and examined their impact on various team performance dimensions. We found that virtual team characteristics have different effects on different aspects of team performance. The virtual context team had a lower rework volume but higher coordination volume and longer project duration than the virtual composition team. Interestingly, we also found that the virtual structure team performed better than the software development team baseline model in all aspects of team performance. Based on these results, we proposed strategies to improve performance in different types of virtual team. Specifically, we propose (1) increasing the ease of communication and availability of routines in the virtual context team; (2) clarifying role expectations and fostering a team culture in the virtual composition team; and (3) implementing a lateral structure in the virtual team. Our results also suggest that firms should consider situational demands, specifically tolerance for errors and coordination volume, when considering the design of virtual teams.     

Virtual Braids, Virtual Temperley-Lieb Algebra and f-Polynomial

The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra,which is an approach similar to Jones's original construction.     

Job routing and operations scheduling: a network-based virtual cell formation approach

Virtual cellular manufacturing inherits the benefits of traditional cellular manufacturing and maintains the responsiveness to the changing market and routing flexibility of a job shop by integrating machine-grouping, shop layout design and intercellular flow handling. The primary goal of virtual cell formation is to minimize the throughput time of a given job. This paper proposes a method for virtual cell formation by adopting the double-sweep algorithm for the k-shortest path problem, and a heuristic is devised to schedule the virtual cells for the multiple job orders. Results generated from this method include not only the optimal candidates of the virtual cell with the shortest throughput time with sub-optimal alternative route(s) and throughput time(s) as the alternative candidates in case some resources are restricted or are not available. The procedure of virtual cell creation and scheduling is illustrated explicitly with examples. Since most of the scheduling problems are NP-hard and virtual cell scheduling is even more complex due to the bottleneck machines that are demanded by jobs at other cells. For multiplicity of possible virtual cell candidates, in addition to the precedence and resource constraints, heuristic solutions are found to be reasonable.     

Virtual manifolds and localization

In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects ＂virtual manifolds/orbifolds＇, on which we develop the integration theory. In particular, the virtual localization formula is obtained.     

The Miyazawa polynomial for long virtual knots

We introduce a two-variable polynomial invariant of a long virtual knot, which dominates the Kauffman f-polynomial and the Miyazawa polynomial of the closure. Our invariant satisfies a product formula for the concatenation product of long virtual knots. It describes a formula of the Miyazawa polynomial of a ‘connected sum’ of two virtual knots. It also gives lower bounds for the real crossing number and the virtual crossing number of a long virtual knot.     

A counterexample to questions on the integrality property of virtual signature

Wall (1961), defined the virtual Euler Characteristic χ(Γ) of an arbitrary group Γ of finite homological type as , where Γ′ is any torsion free subgroup of finite index in Γ. Analogous to virtual Euler Characteristic, we define the Virtual signature of an oriented virtual Poincare Duality group, a rational number. We show that two of Ken Brown's results on questions regarding the integrality property of virtual Euler Characteristics when formulated in the Virtual signature case is false.     

Pointed spherical tilings and hyperbolic virtual polytopes

The paper presents a shortcut introduction to the theory of hyperbolic virtual polytopes from the point of view of combinatorial rigidity. (It is assumed that the reader is acquainted with the notions of Laman graph, 3D lifting, and pointed tiling.) From this point of view, a hyperbolic virtual polytope is a stressed pointed graph embedded in the sphere S ². The advantage of such a presentation is that it gives an alternative and most convincing proof of existence of hyperbolic virtual polytopes. Bibliography: 20 titles.     

Virtual knots and links

This paper is an introduction to the subject of virtual knot theory and presents a discussion of some new specific theorems about virtual knots. The new results are as follows: Using a 3-dimensional topology approach, we prove that if a connected sum of two virtual knots K ₁ and K ₂ is trivial, then so are both K ₁ and K ₂. We establish an algorithm for recognizing virtual links that is based on the Haken-Matveev technique. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 252, pp. 114–133.     

Recurrence time and range for a generalized random walk

The notions of recurrence time, range, and the limit of probabilities P_k of return to the origin arise in the study of random walks on groups. We examine these notions and develop relationships among them in an ergodic theory setting in which the usual requirement of independence of the increments of the random walk can be relaxed to simply an ergodic requirement. Thus we consider generalized random walks or GRWs. The ergodic theory setting is related to Mackey's virtual group theory in that the GRW determines a virtual group homomorphism (or cocycle). We relate the condition- that the homomorphism is trivial (the cocycle is a coboundary) to the Cesáro limit of P_k. The basic ideas of virtual group theory were established by Mackey and further developed by Ramsay. Our virtual group homomorphism result does not require familiarity with the technicalities of virtual group theory.     

The virtual level of the Schrödinger equation

The three-dimensional Schrödinger operator H is considered with a so-called virtual level at the start of the spectrum. The existence of a virtual level means, roughly speaking, that the corresponding Lippman-Schwinger equation has a nontriviai, zero-energy solution. The concept of a virtual level is discussed in detail, and the structure of the resolvent of the operator H is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 203–216, 1975.     

Self‐organized trail systems in groups of humans

We have developed an experimental platform for studying the trail systems that spontaneously emerge when people are motivated to take advantage of the trails left by others. In this virtual environment, the participants' task is to reach randomly selected destinations while minimizing travel costs. The travel cost of every patch in the environment is inversely related to the number of times the patch was visited by others. The resulting trail systems are a compromise between people going to their destinations and going where many people have previously traveled. We compare the results from our group experiments to the Active Walker model of pedestrian motion from biophysics. The Active Walker model accounted for deviations of trails from the beeline paths, the gradual merging of trails over time, and the influences of scale and configuration of destinations on trail systems, as well as correctly predicting the approximate spatial distribution of people's steps. Two deviations of the model from empirically obtained results were corrected by (1) incorporating a distance metric sensitive to canonical horizontal and vertical axes, and (2) increasing the influence of a trail's travel cost on an agent's route as the agent approaches its destination. © 2006 Wiley Periodicals, Inc. Complexity 11: 43–50, 2006