Polygonal Space Research Articles

The cobordism class of the moduli space of polygons in R3

The cobordism class of the moduli space of polygons in R3

Discrete monodromy, pentagrams, and the method of condensation

This paper studies the pentagram map, a projectively natural iteration on the space of polygons. Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly n algebraically independent invariants for the map, when it is defined on the space of n-gons. These invariants strongly suggest that the pentagram map is a discrete completely integrable system. The paper also relates the pentagram map to Dodgson’s method of condensation for computing determinants, also known as the octahedral recurrence.

Topology of random linkages

Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters ‐ the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R 3 . We also prove results about higher moments of Betti numbers.

Polygons in Minkowski space and the Gelfand–Tsetlin method for pseudo-unitary groups

We study the symplectic geometry of the moduli spaces of polygons in the Minkowski 3-space. These spaces naturally carry completely integrable systems with periodic flows. We extend the Gelfand–Tsetlin method to pseudo-unitary groups and show that the action variables are given by the Minkowski lengths of non-intersecting diagonals.

Path integrals on manifolds by finite dimensional approximation

Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic polygons. We also show a uniform convergence result for the heat kernels. This yields a simple and natural proof for the Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of the heat operator.

Betti numbers of random manifolds

AbstractBetti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters – the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure. The main results of the paper apply to planar linkages as well as for linkages in R3. We also prove results about higher moments of Betti numbers. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Twelve-Sided Polygonal Voltage Space Vector Based Multilevel Inverter for an Induction Motor Drive With Common-Mode Voltage Elimination

In this paper, a 12-sided polygonal voltage space vector generation with common-mode voltage elimination (CME) is proposed for an induction motor drive. An open-end winding configuration is used for the motor. The proposed multilevel structure is achieved by cascading only the conventional two-level inverters with asymmetrical dc link voltages. By appropriately selecting the voltage vectors with identical common-mode voltage from the opposite ends, a pulsewidth modulation (PWM) strategy with zero common-mode voltage variation can be achieved for the entire modulation range with a common dc link requirement for both the inverter systems. Along with the CME, the proposed 12-sided polygonal space vector based multilevel inverter structure has increased modulation range with the absence of 5th, 7th, 17th, 19th, etc., harmonics up to 12-step operation. The bandwidth problems associated with conventional hexagonal voltage space vector structure current controllers due to the presence of 5th and 7th harmonics in the overmodulation region are absent in the present 12-sided structure. So a simple PWM voltage control with linear voltage control up to 12-step operation is possible from the present 12-sided scheme with less current control complexity.

A Multiuser Multiperspective Stereographic QTVR Browser Complemented by Java3D Visualizer and Emulator

To support multiperspective and stereographic image display systems intended for multiuser applications, we have developed two integrated multiuser multiperspective stereographic browsers, respectively featuring IBR-generated egocentric and CG exocentric perspectives. The first one described, “VR4U2C” (‘virtual reality for you to see’), uses Apple's QuickTime VR technology and the Java programming language together with the support of the QuickTime for Java library. This unique QTVR browser allows coordinated display of multiple views of a scene or object, limited only by the size and number of monitors or projectors assembled around or among users (for panoramas or turnoramas) in various viewing locations. The browser also provides a novel solution to limitations associated with display of QTVR imagery: its multinode feature provides interactive stereographic QTVR (dubbed SQTVR) to display dynamically selected pairs of images exhibiting binocular parallax, the stereoscopic depth percept enhanced by motion parallax from displacement of the viewpoint through space coupled with rotation of the view through a 360° horizontal panorama. This navigable approach to SQTVR allows proper occlusion/disocclusion as the virtual standpoint shifts, as well as natural looming of closer objects compared to more distant ones. We have integrated this stereographic panoramic browsing application in a client/server architecture with a sibling client, named “Just Look at Yourself!” which is built with Java3D and allows realtime visualization of the dollying and viewpoint adjustment as well as juxtaposition and combination of stereographic CG and IBR displays. “Just Look at Yourself!” visualizes and emulates VR4U2C, embedding avatars associated with cylinder pairs wrapped around the stereo standpoints texture-mapped with a set of panoramic scenes into a 3D CG model of the same space as that captured by the set of panoramas. The transparency of the 3D CG polygon space and the photorealistic stereographic 360° scenes, as well as the size of the stereo goggles through which the CG space is conceptually viewed and upon which the 360° scenes are texture-mapped, can be adjusted at runtime to understand the relationship of the spaces.

Homology of planar polygon spaces

In this paper, we study topology of the variety of closed planar n-gons with given side lengths $$l_1, \dots, l_n$$ . The moduli space $$M_\ell$$ where $$\ell =(l_1, \dots, l_n)$$ , encodes the shapes of all such n-gons. We describe the Betti numbers of the moduli spaces $$M_\ell$$ as functions of the length vector $$\ell=(l_1, \dots, l_n)$$ . We also find sharp upper bounds on the sum of Betti numbers of $$M_\ell$$ depending only on the number of links n. Our method is based on an observation of a remarkable interaction between Morse functions and involutions under the condition that the fixed points of the involution coincide with the critical points of the Morse function.

On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards

Following a recent paper by Baryshnikov and Zharnitsky, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has empty interior.

A Novel Split Phase Induction Motor Drive without Harmonic Filters and with Linear Voltage Control for the Full Modulation Range

A modulation scheme for a split phase (six phase) induction motor is presented in this paper. For a six phase induction motor drive harmonics of the order 6n ± 1 (n = 1, 3, 5 etc.,) will not contribute to the air gap flux and hence these harmonic currents are limited by the stator impedance only. So harmonic filters are needed for the suppression of these (6n ±1( n = 1, 3, 5 etc.,) high amplitude harmonic currents, when a six phase induction motor is used for variable speed application. In the proposed scheme, a modulation technique to eliminate the 6n ± 1 (n = 1, 3, 5 etc.,) harmonic currents, without the need for harmonic filters, from the stator phases of a six phase induction motor drive is explained. An openend winding drive configuration with appropriate DC-link voltages, is chosen in such away that a 12-sided polygonal voltage space phasor combinations are achieved for each 3-phase group, independently. A modulation scheme based on 12-sided polygonal voltage space phasors cancel the 6n ± 1 (n = 1, 3, 5 etc.,) harmonic voltages from all the motor phases and hence currents due to these harmonics can not flow in the motor phases of the six phase machine. For the same flux level of operation, nearly 11.5 % increase in modulation range is possible from this proposed drive (while still being in the linear PWM range), which is more than that of a six-step over modulation region in a conventional 3-phase induction motor drive. The proposed drive scheme is experimentally verified, for variable speed application, using a 1.5 kW six-phase induction motor drive.

The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer k ≥ 2 k\geq 2 , let S ( k ) {\mathcal S}(k) be the space of complete embedded singly periodic minimal surfaces in R 3 \mathbb {R}^3 , which in the quotient have genus zero and 2 k 2k Scherk-type ends. Surfaces in S ( k ) {\mathcal S}(k) can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that S ( 2 ) {\mathcal S}(2) consists of the 1 1 -parameter family of singly periodic Scherk minimal surfaces. We prove that for each k ≥ 3 k\geq 3 , there exists a natural one-to-one correspondence between S ( k ) {\mathcal S}(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M ∈ S ( k ) M\in {\mathcal S}(k) the polygon whose edges are the flux vectors at the ends of M M (a special polygon is a parallelogram with two sides of length 1 1 and two sides of length k − 1 k-1 ). As consequence, S ( k ) {\mathcal S}(k) reduces to the saddle towers constructed by Karcher.

Quantization of bending flows

We briefly review the Kapovich-Millson notion of Bending flows as an integrable system on the space of polygons in ${\bf R}^3$, its connection with a specific Gaudin XXX system, as well as the generalisation to $su(r), r>2$. Then we consider the quantisation problem of the set of Hamiltonians pertaining to the problem, quite naturally called Bending Hamiltonians, and prove that their commutativity is preserved at the quantum level.

Symplectic Capacities of Toric Manifolds and Related Results

AbstractIn this paper we give concrete estimations for the pseudo symplectic capacities of toric manifolds in combinatorial data. Some examples are given to show that our estimates can compute their pseudo symplectic capacities. As applications we also estimate the symplectic capacities of the polygon spaces. Other related results are impacts of symplectic blow-up on symplectic capacities, symplectic packings in symplectic toric manifolds, the Seshadri constant of an ample line bundle on toric manifolds, and symplectic capacities of symplectic manifolds withS1-action.

A morphological and morphometrical study on the sacculus rotundus and ileum of the Angora rabbit

The aim of this study was to reveal morphological and morphometrical properties of the sacculus rotundus (SR) and ileum in the Angora rabbit. For this purpose, a total of thirteen adult healthy Angorarabbits of both sexes were used. At the level of the junction of the ileum and SR, the ileum invaginated into the SR by protruding in 9 Angorarabbits, but joined directly the SR in 21. Numerous aggregate lymph follicles located just under the tunica serosa formed outstanding macroscopic polygonal areas on the external surface of the SR. In the inner wall of the SR, irregular projections were seen grossly. The saccorotundocecal orifice was found to be bordered laterally by two folds facing the cecum. These folds enclosed small polygonal spaces with mushroom shaped protrusions. The mean lengths and weights of SR and ileum in male were larger than those in female, and the related values also had statistical significance (P < 0.05). Compared with the ileum, the SR had short and thick villi, had a large amount of crypts and aggregated lymphoid follicles, and had a much thicker wall and much wider lumen. The crescent-like-hollows were detected between the lamina propria and the apical portions of the lymph follicles. The results from this study are thought to shed light on future studies on the digestive system and proper diagnosis of pathological disorders related to it in the Angora rabbit, and to contribute to the present morphological knowledge on the SR and ileum in this species.

Multilevel inverter with 12-sided polygonal voltage space vector locations for induction motor drive

Multilevel inverters are preferred over two-level inverters in high-voltage, high-power applications mainly because low-switching-frequency PWM schemes can be implemented with power devices of low-voltage ratings. Common multilevel inverter structures such as NPC, H-bridge and flying capacitor will produce a hexagonal voltage space vector structure. In the overmodulation range a multilevel inverter with a hexagonal voltage space vector structure will produce high amplitude 5th and 7th harmonic voltages in the output voltages. The maximum phase peak fundamental voltage is 0.637 Vdc in six-step mode operation (where Vdc is the radii of the hexagonal voltage space vector structure). A 12-sided polygonal voltage space vector structure will have a maximum phase peak fundamental voltage of 0.64 Vdc in the linear modulation range itself (where Vdc is the radii of the 12-sided polygonal voltage space vector structure), which is more than the output in six-step operation in the over-modulation range for a conventional multilevel structure. Also with a 12-sided polygonal space vector structure, an additional boost of the fundamental phase voltage is possible in the 12-step operation with the elimination of the 5th- and 7th- order harmonics (6n +/- 1, n = 1, 3, 5... etc). A multilevel inverter structure with 12-sided polygonal voltage space vector structure is proposed for an induction motor drive. The proposed inverter structure is realised by cascading three conventional two-level inverters. This makes the power bus structure very simple.

Conjugation spaces

There are classical examples of spaces X with an involution tau whose mod 2-comhomology ring resembles that of their fixed point set X^tau: there is a ring isomorphism kappa: H^2*(X) --> H^*(X^tau). Such examples include complex Grassmannians, toric manifolds, polygon spaces. In this paper, we show that the ring isomorphism kappa is part of an interesting structure in equivariant cohomology called an H^*-frame. An H^*-frame, if it exists, is natural and unique. A space with involution admitting an H^*-frame is called a conjugation space. Many examples of conjugation spaces are constructed, for instance by successive adjunctions of cells homeomorphic to a disk in C^k with the complex conjugation. A compact symplectic manifold, with an anti-symplectic involution compatible with a Hamiltonian action of a torus T, is a conjugation space, provided X^T is itself a conjugation space. This includes the co-adjoint orbits of any semi-simple compact Lie group, equipped with the Chevalley involution. We also study conjugate-equivariant complex vector bundles (`real bundles' in the sense of Atiyah) over a conjugation space and show that the isomorphism kappa maps the Chern classes onto the Stiefel-Whitney classes of the fixed bundle.

Spaces of Polygons in the Plane and Morse Theory

Spaces of Polygons in the Plane and Morse Theory

Spaces of Polygons in the Plane and Morse Theory

Spaces of Polygons in the Plane and Morse Theory

On the symplectic volume of the moduli space of spherical and Euclidean polygons

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the polygons with fixed perimeter in S3 or E3 the regular one is the most flexible and that among all the spherical polygons the regular one with side-length π/2 is the most flexible.