Equilateral Polygons Research Articles

Generating equilateral random polygons in confinement

One challenging problem in biology is to understand the mechanism of DNA packing in a confined volume such as a cell. It is known that confined circular DNA is often knotted and hence the topology of the extracted (and relaxed) circular DNA can be used as a probe of the DNA packing mechanism. However, in order to properly estimate the topological properties of the confined circular DNA structures using mathematical models, it is necessary to generate large ensembles of simulated closed chains (i.e. polygons) of equal edge lengths that are confined in a volume such as a sphere of certain fixed radius. Finding efficient algorithms that properly sample the space of such confined equilateral random polygons is a difficult problem. In this paper, we propose a method that generates confined equilateral random polygons based on their probability distribution. This method requires the creation of a large database initially. However, once the database has been created, a confined equilateral random polygon of length n can be generated in linear time in terms of n. The errors introduced by the method can be controlled and reduced by the refinement of the database. Furthermore, our numerical simulations indicate that these errors are unbiased and tend to cancel each other in a long polygon.

Microfluidics-based assay on the effects of microenvironmental geometry and aqueous flow on bacterial adhesion behaviors

A new microfluidic system with four different microchambers (a circle and three equilateral concave polygons) was designed and fabricated using poly(dimethylsiloxane) (PDMS) and the soft lithography method. Using this microfluidic device at six flow rates (5, 10, 20, 30, 40, and 50μL/h), the effects of microenvironmental geometry and aqueous flow on bacterial adhesion behaviors were investigated. Escherichia coli HB101 pGLO, which could produce a green fluorescent protein induced by l-arabinose, was utilized as the model bacteria. The results demonstrated that bacterial adhesion was significantly related to culture time, microenvironment geometry, and aqueous flow rates. Adhered bacterial density increased with the culture time. Initially, the adhesion occurred at the microchamber sides, and then the entire chamber was gradually covered with increased culture time. Adhesion densities in the side zones were larger than those in the center zones because of the lower shearing force in the side zone. Also, the adhesion densities in the complex chambers were larger than those in the simple chambers. At low flow rates, the orientation of adhered bacteria was random and disorderly. At high flow rates, bacterial orientation became close to the streamline and oriented toward the flow direction. All these results implied that bacterial adhesion tended to occur in complicated aqueous flow areas. The present study provided an on-chip flow system for physiological behavior of biological cells, as well as provided a strategic cue for the prevention of bacterial infection and biofilm formation.

On the mean and variance of the writhe of random polygons

We here address two problems concerning the writhe of random polygons. First, we study the behavior of the mean writhe as a function length. Second, we study the variance of the writhe. Suppose that we are dealing with a set of random polygons with the same length and knot type, which could be the model of some circular DNA with the same topological property. In general, a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons; if the mean writhe is non-zero then the knot is chiral. How accurate is this method? For example, if for a specific knot type K the mean writhe decreased to zero as the length of the polygons increased, then this method would be limited in the case of long polygons. Furthermore, we conjecture that the sign of the mean writhe is a topological invariant of chiral knots. This sign appears to be the same as that of an ‘ideal’ conformation of the knot. We provide numerical evidence to support these claims, and we propose a new nomenclature of knots based on the sign of their expected writhes. This nomenclature can be of particular interest to applied scientists. The second part of our study focuses on the variance of the writhe, a problem that has not received much attention in the past. In this case, we focused on the equilateral random polygons. We give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons (of length n) behaves as a linear function of the length of the equilateral random polygon.

The Generation of Random Equilateral Polygons

Freely jointed random equilateral polygons serve as a common model for polymer rings, reflecting their statistical properties under theta conditions. To generate equilateral polygons, researchers employ many procedures that have been proved, or at least are believed, to be random with respect to the natural measure on the space of polygonal knots. As a result, the random selection of equilateral polygons, as well as the statistical robustness of this selection, is of particular interest. In this research, we study the key features of four popular methods: the Polygonal Folding, the Crankshaft Rotation, the Hedgehog, and the Triangle Methods. In particular, we compare the implementation and efficacy of these procedures, especially in regards to the population distribution of polygons in the space of polygonal knots, the distribution of edge vectors, the local curvature, and the local torsion. In addition, we give a rigorous proof that the Crankshaft Rotation Method is ergodic.

Close target reconnaissance with guaranteed collision avoidance

AbstractThis manuscript considers the problem of close target reconnaissance by a group of autonomous agents. The overall close target reconnaissance (CTR) involves subtasks of avoiding inter‐agent collisions, reaching a close vicinity of a specific target position, and forming an equilateral polygon formation around the target. The agents performing the task fly at a constant speed to mimic the velocity behavior of small fixed‐wing unmanned aerial vehicles (UAV). A decentralized control scheme is developed for this overall task and the finite‐time convergence of the system under the proposed control law is established. Furthermore, it is guaranteed that no collision occurs among the agent. The relevant analysis and simulation test results are provided. Copyright © 2010 John Wiley & Sons, Ltd.

Determination of the stressed state near edge cracks in a plate containing a hole of complex shape

We propose an approach to the determination of the stressed state of plates with edge cracks near holes of complex shapes based on the method of integral equations. We find the stress intensity factors for the cracks located near holes in the form of equilateral polygons and for systems of cracks of various lengths located near circular holes.

The linking number and the writhe of uniform random walks and polygons in confined spaces

Random walks and polygons are used to model polymers. In this paper we consider the extension of the writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n2). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form . Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length.

General solutions to spin transportation of electrons through equilateral polygon quantum rings with Rashba spin-orbit interaction

Based on the previous work (Li P, Deng W J 2009 Acta Phys. Sin. 58 02713), the quantum transportation of electron through arbitrary equilateral polygons quantum rings with Rashba spin-orbit interaction is studied. With the typical method of quantum network and the Landauer-Büttiker formalism, we analytically solve the scattering problem of electron through equilateral polygonal quantum ring, and obtain the relevant formula for spin transportation conductance. The characteristics of the conductance varying with the wave-vector of electron, the strength of spin-orbit interaction, the number of polygon edges, and the ways of leads connecting to quantum rings are discussed. In the limit of infinite number of edges of polygon, we prove that the formula is consistent with the results obtained directly from the circular model of quantum rings.

The growth of the mean average crossing number of equilateral polygons in confinement

The physical and biological properties of collapsed long polymer chains as well as of highly condensed biopolymers (such as DNA in all organisms) are known to be determined, at least in part, by their topological and geometrical properties. With this purpose of characterizing the topological properties of such condensed systems equilateral random polygons restricted to confined volumes are often used. However, very few analytical results are known. In this paper, we investigate the effect of volume confinement on the mean average crossing number (ACN) of equilateral random polygons. The mean ACN of knots and links under confinement provides a simple alternative measurement for the topological complexity of knots and links in the statistical sense. For an equilateral random polygon of n segments without any volume confinement constrain, it is known that its mean ACN ⟨ACN⟩ is of the order . Here we model the confining volume as a simple sphere of radius R. We provide an analytical argument which shows that ⟨ACN⟩ of an equilateral random polygon of n segments under extreme confinement (meaning R ≪ n) grows as O(n2). We propose to model the growth of ⟨ACN⟩ as a(R)n2 + b(R)nln(n) under a less-extreme confinement condition, where a(R) and b(R) are functions of R with R being the radius of the confining sphere. Computer simulations performed show a fairly good fit using this model.

Intersection numbers of polygon spaces

We study the intersection ring of the space M ( α 1 , … , α m ) \mathcal {M}(\alpha _1,\ldots ,\alpha _m) of polygons in R 3 \mathbb {R}^3 . We find homology cycles dual to generators of this ring and prove a recursion relation in m m (the number of edges) for their intersection numbers. This result is an analog of the recursion relation appearing in the work of Witten and Kontsevich on moduli spaces of punctured curves and in the work of Weitsman on moduli spaces of flat connections on two-manifolds of genus g g with m m marked points. Based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples. Among others, we study the special case of equilateral polygon spaces (where all α i \alpha _i ’s are the same) and compare our results with the expressions for these particular spaces that have been determined by Kamiyama and Tezuka. Finally, we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of Yoshida, as well as with equivalent expressions for polygon spaces obtained by Takakura and Konno through different techniques.

A fast ergodic algorithm for generating ensembles of equilateral random polygons

Knotted structures are commonly found in circular DNA and along the backbone of certain proteins. In order to properly estimate properties of these three-dimensional structures it is often necessary to generate large ensembles of simulated closed chains (i.e. polygons) of equal edge lengths (such polygons are called equilateral random polygons). However finding efficient algorithms that properly sample the space of equilateral random polygons is a difficult problem. Currently there are no proven algorithms that generate equilateral random polygons with its theoretical distribution. In this paper we propose a method that generates equilateral random polygons in a ‘step-wise uniform’ way. We prove that this method is ergodic in the sense that any given equilateral random polygon can be generated by this method and we show that the time needed to generate an equilateral random polygon of length n is linear in terms of n. These two properties make this algorithm a big improvement over the existing generating methods. Detailed numerical comparisons of our algorithm with other widely used algorithms are provided.

Exact solutions to the transportation of electrons through equilateral polygonal quantum rings with Rashba spin-orbit interaction

The quantum transportation of electron through equilateral polygonal quantum rings with Rashba spin-orbit interaction is studied. By using the typical method of quantum network and the Landauer-Büttiker formalism, we solve analytically the scattering problem of electron through any equilateral polygonal quantum ring, and obtain the relevant formula for spin transportation conductance. The characters of conductance varying with wave-vector of electron and the strength of spin-orbit interaction are investigated, and the series of zero conductance points originating from spin-orbit interaction is determined. In the limit of infinite number of borders of equilateral polygon, we prove that the formula is consistent with the results obtained directly from the circular model of quantum rings.

Trajectory Planning of Satellite Formation Flying using Nonlinear Programming and Collocation

Recently, satellite formation flying has been a topic of significant research interest in aerospace society because it provides potential benefits compared to a large spacecraft. Some techniques have been proposed to design optimal formation trajectories minimizing fuel consumption in the process of formation configuration or reconfiguration. In this study, a method is introduced to build fuel-optimal trajectories minimizing a cost function that combines the total fuel consumption of all satellites and assignment of fuel consumption rate for each satellite. This approach is based on collocation and nonlinear programming to solve constraints for collision avoidance and the final configuration. New constraints of nonlinear equality or inequality are derived for final configuration, and nonlinear inequality constraints are established for collision avoidance. The final configuration constraints are that three or more satellites should form a projected circular orbit and make an equilateral polygon in the horizontal plane. Example scenarios, including these constraints and the cost function, are simulated by the method to generate optimal trajectories for the formation configuration and reconfiguration of multiple satellites.

AN INVARIANT SHAPE REPRESENTATION: INTERIOR ANGLE CHAIN

This paper proposes a novel shape representation scheme — Interior Angle Chain (IAC) — which is invariant to translation, rotation and scaling. The proposed method first approximates the contour of a planar shape with an equilateral polygon and then makes a representation using the polygon's interior angle chain. The difference between two shapes is measured by the distance between their IAC's. An algorithm to obtain equilateral polygon approximation and its associated IAC is proposed in this paper. The proposed shape representation scheme has been tested on two benchmarks and applied to lake recognition in SAR (Synthetic Aperture Radar) images. The results show that IAC is an effective shape representation scheme.

Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture

The conventional view holds that girih (geometric star-and-polygon, or strapwork) patterns in medieval Islamic architecture were conceived by their designers as a network of zigzagging lines, where the lines were drafted directly with a straightedge and a compass. We show that by 1200 C.E. a conceptual breakthrough occurred in which girih patterns were reconceived as tessellations of a special set of equilateral polygons ("girih tiles") decorated with lines. These tiles enabled the creation of increasingly complex periodic girih patterns, and by the 15th century, the tessellation approach was combined with self-similar transformations to construct nearly perfect quasi-crystalline Penrose patterns, five centuries before their discovery in the West.

90.47 A cyclic property of equilateral polygons

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Few-electron quantum rings in a magnetic field: Ground-state properties

The ground-state properties of few-electron quantum rings (QRs) subjected to an external magnetic field $B$ are studied. The evolution of the angular momentum ${L}_{0}$ and the spin ${S}_{0}$ of the ground state in accord with the radius $R$ of the QRs and $B$ has been calculated, the results are summarized and plotted in $({L}_{0},{S}_{0})$ phase diagrams. These diagrams indicate that ${L}_{0}$ and ${S}_{0}$ may jump when $B$ and/or $R$ vary. The spatial distribution of the electrons has also been investigated. It was found that the ground state of the QRs possesses an equilateral polygon (EP) configuration. Based on an analysis of the wave function, we show that each EP is accessible only to a specific group of states having specific ${L}_{0}$ and ${S}_{0}$.

Entropic Sampling of Free and Ring Polymer Chains

AbstractSummary: We extend Monte Carlo (MC) methods developed in our previous paper (J. Phys. A, Math. Gen. 2004, 37, 1573) and based on entropic sampling within Wang‐Landau (WL) algorithm to the simulation of lattice and continuous models of ring polymers. For a continuous freely joined ring chain (an equilateral polygon) with hard sphere monomer units, the excess entropy of rings relative to the corresponding reference system, a phantom ring chain, is obtained. The excess specific entropy is calculated for a set of various diameters of monomer units d and chain lengths N. Its limiting values for $N\to \infty$ ($N^{-1}\to 0$) are estimated for each d and coincidence with those for corresponding free chains is demonstrated. We also develop a WL approach to calculate thermal properties of free and ring continuous chains with an Lennard‐Jones attraction between nonbonded beads being added to hard core repulsion at fixed d. The obtained energy distributions provide calculation of canonical properties such as conformational energy, heat capacity, entropy, and mean square radius of inertia. Thermal results for free and ring chains are being finally compared. Analogous calculations are performed for lattice‐free chains and rings. magnified image

The average inter-crossing number of equilateral random walks and polygons

In this paper, we study the average inter-crossing number between two random walks and two random polygons in the three-dimensional space. The random walks and polygons in this paper are the so-called equilateral random walks and polygons in which each segment of the walk or polygon is of unit length. We show that the mean average inter-crossing number ICN between two equilateral random walks of the same length n is approximately linear in terms of n and we were able to determine the prefactor of the linear term, which is . In the case of two random polygons of length n, the mean average inter-crossing number ICN is also linear, but the prefactor of the linear term is different from that of the random walks. These approximations apply when the starting points of the random walks and polygons are of a distance ρ apart and ρ is small compared to n. We propose a fitting model that would capture the theoretical asymptotic behaviour of the mean average ICN for large values of ρ. Our simulation result shows that the model in fact works very well for the entire range of ρ. We also study the mean ICN between two equilateral random walks and polygons of different lengths. An interesting result is that even if one random walk (polygon) has a fixed length, the mean average ICN between the two random walks (polygons) would still approach infinity if the length of the other random walk (polygon) approached infinity. The data provided by our simulations match our theoretical predictions very well.

An isoperimetric problem for point interactions

We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all with the same coupling constant, placed at vertices of an equilateral polygon $\PP_N$. It is shown that the ground state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.