Pair Of Arcs Research Articles

Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts

Historically, a contrived trisected line was used to trisect any other line, using the principle of projection. This is in essence about relationship and its accomplishment is about working backwards. Loosely speaking, any angle comprises two connecting lines. Attempts at trisecting any angle, which is dividing it into three equal parts, failed. In this paper any angle is defined as a unique pair of arc and chord of sector of a circle irrespective of arc radius. Two theorems viz. Equal arcs have equal central angles and equal chords have equal central angles are combined to establish a unique relationship between a pair of arc-chord and its composite of three identical pairs of arc-chord, thereby revealing a CYCLIC TRAPEZIUM, where the base defines the angle, and each equal edge defines each of the equal trisected parts of this angle. For a range of angles between 0o and 360o, this relationship is expressed as Lorna Graph, which becomes the practical tool for trisection of any angle, using the working backwards approach. This approach is extended to division of any angle into any number of equal parts.

The pillowcase and perturbations of traceless representations of knot groups

We introduce explicit holonomy perturbations of the Chern-Simons functional on a 3-ball containing a pair of unknotted arcs. These perturbations give us a concrete local method for making the moduli spaces of flat singular SO(3) connections relevant to Kronheimer and Mrowka's singular instanton knot homology non-degenerate. The mechanism for this study is a (Lagrangian) intersection diagram which arises, through restriction of representations, from a tangle decomposition of a knot. When one of the tangles is trivial, our perturbations allow us to study isolated intersections of two Lagrangians to produce minimal generating sets for singular instanton knot homology. The (symplectic) manifold where this intersection occurs corresponds to the traceless character variety of the four-punctured 2-sphere, which we identify with the familiar pillowcase. We investigate the image in this pillowcase of the traceless representations of tangles obtained by removing a trivial tangle from 2-bridge knots and torus knots. Using this, we compute the singular instanton homology of a variety of torus knots. In many cases, our computations allow us to understand non-trivial differentials in the spectral sequence from Khovanov homology to singular instanton homology.

A fast and robust ellipse-detection method based on sorted merging.

A fast and robust ellipse-detection method based on sorted merging is proposed in this paper. This method first represents the edge bitmap approximately with a set of line segments and then gradually merges the line segments into elliptical arcs and ellipses. To achieve high accuracy, a sorted merging strategy is proposed: the merging degrees of line segments/elliptical arcs are estimated, and line segments/elliptical arcs are merged in descending order of the merging degrees, which significantly improves the merging accuracy. During the merging process, multiple properties of ellipses are utilized to filter line segment/elliptical arc pairs, making the method very efficient. In addition, an ellipse-fitting method is proposed that restricts the maximum ratio of the semimajor axis and the semiminor axis, further improving the merging accuracy. Experimental results indicate that the proposed method is robust to outliers, noise, and partial occlusion and is fast enough for real-time applications.

Geometric Constructions for Symmetric 3-Configurations

A geometric 3-configuration is a collection of points and straight lines, typically in the Euclidean plane, in which every point has 3 lines passing through it and every line has 3 points lying on it, and we say that such configuration is symmetric if there are non-trivial isometries of the plane that map the configuration to itself. Many symmetric 3-configurations may be easily constructed with computer algebra systems using algebraic techniques: e.g., constructing a number of symmetry classes of points and lines, by various means, and then determining the position of a final class of points or lines by solving some polynomial equation. In contrast, this paper presents a number of ruler-and-compass-type constructions for exactly constructing various types of symmetric 3-configurations, as long as the vertices of an initial regular m-gon are explicitly provided. In addition, it provides methods for constructing chirally symmetric 3-configurations given an underlying unlabelled reduced Levi graph, for extending these constructions to produce dihedrally symmetric 3-configurations, and for constructing 3-configurations corresponding to all 3-orbit and 4-orbit reduced Levi graphs that contain a pair of parallel arcs. Notably, most of the configurations described are movable: that is, they have at least one continuous parameter.

MINIMAL CURVATURE-CONSTRAINED PATHS IN THE PLANE WITH A CONSTRAINT ON ARCS WITH OPPOSITE ORIENTATIONS

The declines that provide vehicle access in an underground mine are typically designed as paths formed by concatenating line segments and circular arcs. In order to reduce wear on the ore trucks and the road surfaces and to enhance driver safety, such paths may be subject to a further constraint: each pair of consecutive arcs with opposite orientations must be separated by a straight line segment of at least a certain specified length. In order to reduce the construction and operational costs of the mine, it is desirable to minimize the lengths of such paths between any given pair of directed points. Some necessary and sufficient conditions are obtained for paths of this form to be locally or globally minimal with respect to length. In particular, it is shown that there is always a globally minimal path that contains at most four circular arcs.

The human visual system uses a global closure mechanism

Research asserting that the visual system instantiates a global closure heuristic in contour integration has been challenged by an argument that behaviorally-detected closure enhancement could be accounted for by low-level local mechanisms driven by collinearity or “good continuation” interacting with proximity. The present study investigated this issue in three experiments. Exp. 1 compared the visibility of closed and open contours using circles and S-contours from low to moderately high angles of path curvature in a temporal alternative-forced choice task. Circles were more detectable than S-contours, an effect that increased with curvature. The closure enhancement observed can, however, be explained by the fact that circles contain more ‘contiguity’ than S-contours. Additional tests added discontinuities to otherwise closed paths to control for the effects of good continuation and closure independently. Exp. 2 compared the visibility of incomplete circles (C-contours) and S-contours derived from the full circles and S-contours in Exp. 1. Exp. 3a compared the visibility of arc pairs arranged in an enclosed position similar to “()” and a non-enclosed position similar to “)(”. Results consistently showed enhanced visibility of contour configurations enclosing a region even after controlling for differences in contiguity and changes of curvature direction. A control test (Exp. 3b) demonstrated that the gap in the contours of Exp. 3a was too large to be bridged by local-level collinearity/proximity alone. The combination of good continuation and proximity alone does not explain the closure effects observed across these tests, as demonstrated through the application of a Bayesian model of collinearity and proximity (Geisler et al., 2001) to the stimuli in Exps. 3a and 3b. These results argue for the presence of a global closure-driven contour enhancing mechanism in human vision.

Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than $${\frac{1}{2}n + 7\sqrt{n}/3}$$ contains an anti-directed hamiltonian cycle. In addition, we show that D contains anti-directed cycles of all possible (even) lengths when n is sufficiently large and has minimum in- and out-degree at least $${(1/2+ \epsilon)n}$$ for any $${\epsilon > 0}$$ .

The maximum flow problem with disjunctive constraints

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly $\mathcal{NP}$ -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly $\mathcal{NP}$ -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.

A sufficient condition for the existence of an anti-directed 2-factor in a directed graph

Let D be a directed graph with vertex set V, arc set A, and order n. The graph underlyingD is the graph obtained from D by replacing each arc (u,v)∈A by an undirected edge {u,v} and then replacing each double edge by a single edge. An anti-directed (hamiltonian) cycleH in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. An anti-directed 2-factor in D is a vertex-disjoint collection of anti-directed cycles in D that span V. It was proved in Busch et al. (submitted for publication) [3] that if the indegree and the outdegree of each vertex of D is greater than 916n then D contains an anti-directed Hamilton cycle. In this paper we prove that given a directed graph D, the problem of determining whether D has an anti-directed 2-factor is NP-complete, and we use a proof technique similar to the one used in Busch et al. (submitted for publication) [3] to prove that if the indegree and the outdegree of each vertex of D is greater than 2446n then D contains an anti-directed 2-factor.

A COMPACT BAND-NOTCHED UWB PLANAR MONOPOLE ANTENNA WITH PARASITIC ELEMENTS

In this paper, a planar microstrip UWB monopole antenna with a good band-rejection is presented. By using a pair of arc shaped parasitic elements around the patch, an excellent notched frequency band for rejecting the WLAN band (5{6GHz) can be obtained. The arc shaped strips are parametrically studied and the efiects of them on the radiation patterns and time domain behaviour of the UWB antenna are investigated.

Evaluation of the uncertainty of the length of energy distribution networks

A statistical evaluation of the uncertainty of the length of a utility network is frequently omitted in current geographic information system (GIS) applications, although this information is important for management purposes. Nevertheless, the spatial database storing geographic information about the network might have been set up through a complex process (ground surveying, digitising of analogue maps, integration of different data sources), resulting in a dataset with unclassified uncertainty. This paper proposes a practical but ‘rigorous’ approach to evaluate the influence of the uncertainty in the determination of the length of a network. Starting from the definition of length uncertainty in terms of simple geometric elements (arc, pair of arcs, open and closed path), some rules for its evaluation in the case of complex networks have been developed. The statistical concept applied for the evaluation of uncertainty is quite standard because it is based on the covariance propagation theory. On the other hand, the influence of correlations between the lengths of adjacent elements has been considered as well. The application of this method only requires the topological structure of the network that can be built in a GIS environment. Furthermore, an alternative ‘approximate’ method is proposed to find an approximation of the uncertainty of the network length with a simple formula, which requires neither any particular processing step nor the network topology. Both methods have been tested and validated on simulated and real datasets.

Vertex-Oriented Hamilton Cycles in Directed Graphs

Let $D$ be a directed graph of order $n$. An anti-directed Hamilton cycle $H$ in $D$ is a Hamilton cycle in the graph underlying $D$ such that no pair of consecutive arcs in $H$ form a directed path in $D$. We prove that if $D$ is a directed graph with even order $n$ and if the indegree and the outdegree of each vertex of $D$ is at least ${2\over 3}n$ then $D$ contains an anti-directed Hamilton cycle. This improves a bound of Grant. Let $V(D) = P \cup Q$ be a partition of $V(D)$. A $(P,Q)$ vertex-oriented Hamilton cycle in $D$ is a Hamilton cycle $H$ in the graph underlying $D$ such that for each $v \in P$, consecutive arcs of $H$ incident on $v$ do not form a directed path in $D$, and, for each $v \in Q$, consecutive arcs of $H$ incident on $v$ form a directed path in $D$. We give sufficient conditions for the existence of a $(P,Q)$ vertex-oriented Hamilton cycle in $D$ for the cases when $|P| \geq {2\over 3}n$ and when ${1\over 3}n \leq |P| \leq {2\over 3}n$. This sharpens a bound given by Badheka et al.

Motivic integrals and functional equations

A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to multiplication by a constant and there is a simple recursive algorithm to find its coefficients. The method is universal enough and gives, for example, equations for the integral corresponding to the intersection number over the space of pairs of arcs and over the space of unordered tuples of arcs.

Strong oriented chromatic number of planar graphs without cycles of specific lengths

Abstract A strong oriented k-coloring of an oriented graph G is a homomorphism ϕ from G to H having k vertices labelled by the k elements of an abelian additive group M, such that for any pairs of arcs u v → and z t → of G, we have ϕ ( v ) − ϕ ( u ) ≠ − ( ϕ ( t ) − ϕ ( z ) ) . The strong oriented chromatic number χ s ( G ) is the smallest k such that G admits a strong oriented k-coloring. In this paper, we consider the following problem: Let i ⩾ 4 be an integer. Let G be an oriented planar graph without cycles of lengths 4 to i. What is the strong oriented chromatic number of G?

A note on Duchet's conjecture

A kernel in a digraph is a set of vertices that is both an absorbing and independent set. In [4] P. Duchet conjectured that a digraph such that every odd circuit has two short chords is kernel perfect digraph (i.e. every induced subdigraph has a kernel) and he proved the conjecture for digraphs where every 3 -circuit is reversal. We give another simple proof of this result by using reorientation and constructive methods. We also show that the conjecture is valid for digraphs D where the underlying undirected spanning subgraph deduced from the spanning subdigraph of D with only pairs of symmetric arcs is a comparability graph.

A Strong-Lens Survey in AEGIS: The Influence of Large-Scale Structure

We report on a visual search for galaxy-scale strong gravitational lenses over 650 arcmin2 of HST ACS (V606 and I814) imaging in the DEEP2 Extended Groth Strip. This field has Keck DEIMOS spectroscopy of ~14,000 galaxies (~75% complete to RAB < 24.1). We identify three strong galaxy-galaxy lenses: HST J141735+522646 is a previously known four-image lens (the Cross; zl = 0.8106, zs = 3.40); HST J141820+523611 (the Dewdrop; zl = 0.5798, zs = 0.9818) features two pairs of arcs; and HST J141833+524352 (the Anchor; zl = 0.4625, no zs) has one pair of arcs. Based on a normalized local density (1 + δ3), lenses are found to be in both under- and overdense local environments. All three lenses are fit well by singular isothermal ellipsoid models including external shear, with χ ~ 1. The model shears are ~10%. Approximating all line-of-sight galaxies as singular isothermal sphere halos truncated at 200 h-1 kpc, with masses estimated through the Faber-Jackson relation, infers shears of ~2%, much smaller than those required by the models. Therefore, the corresponding convergence estimates must also be suspect. If more realistic treatment of galaxies (and the large-scale structure that they are embedded in) were to match the inferred shears to the model shears, then the true convergence could be measured and the mass-sheet degeneracy broken.

Cn-MOVE AND ITS DUPLICATED MOVE OF LINKS

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, Cn-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by Cn-moves. We define a local move, βn-move, which is obtained from a Cn-move by duplicating a single pair of arcs with same end points. Then we immediately have that a Cn+1-move is realized by a βn-move and that a βn-move is realized by twice Cn-moves. In this note we study the relation between Cn-move and βn-move, and in particular, give answers to the following questions: (1) Is a βn-move realized by a finite sequence of Cn+1-moves? (2) Is Cn-move realized by a finite sequence of βn-moves?

Constraints on dark energy models from galaxy clusters with multiple arcs

We make an exploratory study of how well dark energy models can be constrained using lensed arcs at different redshifts behind cluster lenses. Arcs trace the critical curves of clusters, and the growth of critical curves with source redshift is sensitive to the dark-energy equation of state. Using analytical models and numerically simulated clusters, we explore the key factors involved in using cluster arcs as a probe of dark energy. We quantify the sensitivity to lens mass, concentration and ellipticity with analytical models that include the effects of dark energy on halo structure. We show with simple examples how degeneracies between mass models and cosmography may be broken using arcs at multiple redshifts or additional constraints on the lens density profile. However we conclude that the requirements on the data are so stringent that it is very unlikely that robust constraints can be obtained from individual clusters. We argue that surveys of clusters, analyzed in conjunction with numerical simulations, are a more promising prospect for arc-cosmography. We use such numerically simulated clusters to estimate how large a sample of clusters/arcs could provide interesting constraints on dark energy models. We focus on the scatter produced by differences in the mass distribution of individual clusters. We find from our sample of simulated clusters that at least 1000 pairs of arcs are needed to obtain constraints if the mass distribution of individual clusters is taken to be undetermined. We discuss several unsolved problems that need study to fully develop this method for precision studies with future surveys.

Disjoint paths in arborescences

An arborescence in a digraph is a tree directed away from its root. A classical theorem of Edmonds characterizes which digraphs have λ arc-disjoint arborescences rooted at r. A similar theorem of Menger guarantees that λ strongly arc disjoint rv -paths exist for every vertex v , where “strongly” means that no two paths contain a pair of symmetric arcs. We prove that if a directed graph D contains two arc-disjoint spanning arborescences rooted at r, then D contains two such arborences with the property that for every node v the paths from r to v in the two arborences satisfy Menger's theorem.

Pseudo-Line Arrangements: Duality, Algorithms, and Applications

A finite collection of x-monotone unbounded Jordan curves in the plane is called a family of pseudo-lines if every pair of curves intersect in at most one point, and the two curves cross each other there. Let L be such a collection of n pseudo-lines, and let P be a set of m points in $\reals^2$. Extending a result of Goodman [Discrete Math., 32 (1980), pp. 27--35], we define a duality transform that maps L to a set L* of points in $\reals^2$ and P to a set P* of (x-monotone) pseudo-lines in $\reals^2$, so that the incidence and the above-below relations between the points and the pseudo-lines are preserved. We present an efficient algorithm for computing the dual arrangement {\eus A}$(P^*)$ under an appropriate model of computation. We also present a dynamic data structure for reporting, in $O(m^\eps + k)$ time, all k points of P that lie below a query arc, which is either a circular arc or a portion of the graph of a polynomial of fixed degree. This result is needed for computing the dual arrangement for certain classes of pseudo-lines arising in several applications, but is also interesting in its own right. We present a few applications of our dual arrangement algorithm, such as computing incidences between points and pseudo-lines and computing a subset of faces in a pseudo-line arrangement. Next, we present an efficient algorithm for cutting a set of circles into arcs so that every pair of arcs intersect in at most one point, i.e., the resulting arcs constitute a collection of pseudo-segments. By combining this algorithm with our algorithm for computing the dual arrangement of pseudo-lines, we obtain efficient algorithms for several problems involving arrangements of circles or circular arcs, such as reporting or counting incidences between points and circles and computing a set of marked faces in arrangements of circles.