Directed Lines Research Articles

Dual octonions and rigid body kinematics

In this paper, we first give the basic information about octonions and present the Euclidean rotation matrix formed by an octonion in seven‐dimensional Euclidean space. Next, we define and introduce the ‐module and dual vectors using dual numbers. Then, we provide the transformation that maps the points on the unit dual sphere one‐to‐one with the directed lines in . We also define a subset of the unit dual sphere, demonstrating that each element of this subset corresponds to two intersecting perpendicular directed lines in seven‐dimensional Euclidean space. Following that, we introduce dual octonions with their basic algebraic properties and examine rigid body (screw) motions in seven‐dimensional Euclidean space using dual octonions. Finally, we define an operator and express that this operator transforms two perpendicular intersecting directed lines in seven‐dimensional Euclidean space into two perpendicular intersecting directed lines.

Single-particle fabric tensors for assemblies of spherical particles

A novel method to calculate single-particle fabric tensors and related geometric characteristics (contact, free and particle area, particle volume and contact centroid) of assemblies of spherical particles has been developed. A standard contact detection stage is used to identify overlapping particles. The shape of each contact region is obtained as the intersection of a circle, obtained in the absence of any contact impingement, and a convex polygon, determined by half-plane intersection based on directed lines obtained from the intersection of contact planes. An underlying power diagram is used to subdivide volume and area between particles, but calculations are performed particle-wise, resulting in an efficient algorithm with a structure ideally suited for parallelisation. Numerical tests performed on loose and dense particle assemblies and comparison (of single-particle free area and particle volume) with a reference method (POWERSASA) indicate that the method is robust and highly accurate. The method will be useful not only for geometric analysis but also as a basis for subsequent developments of strain measures and computational procedures for highly deformed granular materials. Moreover, it constitutes a viable alternative for calculation of solvent accessible surface areas, considering its robustness, efficiency and ease of parallelisation.

Evaluation of color image interpolation based on incompressible Navier Stokes technique

Color image interpolation encompasses reconstructing parts of a video or an image based on information from the neighbor. Technique involves the restoration of noised photos and animation or image denoising. The Navier-Stokes (NS) technique has been widely investigated as an essential research by image restoration. These NS equations contribute spectacular results for producing an animation as they augment reality. They can boost real-time video games to be more sensible than ever. In this paper, we present the Incompressible NS approach (INS) for color image interpolating. The method per se is based on fluid flow concept to circulate directed lines from the peripheral into the area to be interpolated. The image intensity represents stream function in a computational flow of 2D fluid dynamics. The algorithm is implemented to carry on lines regarding gradient vectors at the edge of the interpolating region. It uses the improvement of powerful numerical analysis. It is also proven as an innovative idea for easing problems in image analytics as well as computer vision.

Topological relations between a directed line and a directed region

AbstractRepresenting the topological relations between directed spatial objects has gained increasing attention in recent years. Although topological relations between directed lines and other types of spatial objects, such as regions and bodies, have been widely investigated, few studies have focused on the topological relations between directed lines and directed regions. This research focuses on the representation and application of directed line–directed region (DLDR) topological relations, and may contribute to spatial querying and spatial analyses related to directed spatial objects or time‐varying objects. Compared with other topological relation models, a DLDR model that considers the starting and ending points of the directed line and the front and back faces of directed regions is proposed in this research to describe the topological relations between directed lines and directed regions. DLDR topological relations are presented, the completeness of the 111 DLDR topological relations is proved, and the topological relations based on the 9‐intersection model (9IM), 9+‐intersection model (9+‐IM), and DLDR model are compared. The formalism of the DLDR model and the corresponding geometric interpretations of the 111 DLDR topological relations are presented, seven propositions are stated to prove the completeness of the 111 DLDR topological relations, and the case study shows that more detailed topological relation information can be obtained based on the DLDR model.

Model of directed lines for square ice with second-neighbor and third-neighbor interactions

The investigation of the properties of nanoconfined systems is one of the most rapidly developing scientific fields. Recently it has been established that water monolayer between two graphene sheets forms square ice. Because of the energetic disadvantage, in the structure of the square ice there are no longitudinally arranged molecules. The result is that the structure is formed by unidirectional straight-lines of hydrogen bonds only. A simple but accurate discrete model of square ice with second-neighbor and third-neighbor interactions is proposed. According to this model, the ground state includes all configurations which do not contain three neighboring unidirectional chains of hydrogen bonds. Each triplet increases the energy by the same value. This new model differs from an analogous model with long-range interactions where in the ground state all neighboring chains are antiparallel. The new model is suitable for the corresponding system of point electric (and magnetic) dipoles on the square lattice. It allows separately estimating the different contributions to the total binding energy and helps to understand the properties of infinite monolayers and finite nanostructures. Calculations of the binding energy for square ice and for point dipole system are performed using the packages TINKER and LAMMPS.

Topological Relations between Rivers in Updating Increments

Topological relations between rivers (TRR) are sophisticated for rivers are shown as various patterns in Geographical Information Sciences (GIS). In this paper, basic models which were simplified into directed lines (DL) and directed bandings (DB) were used to represent TRR. Based on the nine-intersection model (NIM), topological relations (TR) were represented into 3 types of DL-DL TR, 7 types of DL-DB TR, and 8 types of DB-DB TR. Otherwise, the four-intersection model (FIM) and common region-region topological relations were testified not suitable to represent TRR in updating increments (UI).

Prolog to, "Geometric Algebra for Electrical and Electronic Engineers."

When Irish mathematician William Hamilton first described the number system called quaternions in 1843, it was applied to the field of mechanics only. The quaternion, the quotient of two directed lines or vectors in 3-D space, was for Hamilton a fundamental structural component of the universe. Hamilton inspired Maxwell to write his famous electromagnetic equations in terms of quaternions; however, they proved to be unwieldy and ineffective for that purpose. This prompted others (namely, Heaviside, Gibbs, and Helmholtz) to develop a simpler system for vector analysis that would be more readily adopted by physicists and engineers.

Depinning of stiff directed lines in random media.

Driven elastic manifolds in random media exhibit a depinning transition to a state with nonvanishing velocity at a critical driving force. We study the depinning of stiff directed lines, which are governed by a bending rigidity rather than line tension. Their equation of motion is the (quenched) Herring-Mullins equation, which also describes surface growth governed by surface diffusion. Stiff directed lines are particularly interesting as there is a localization transition in the static problem at a finite temperature and the commonly exploited time ordering of states by means of Middleton's theorems [Phys. Rev. Lett. 68, 670 (1992)] is not applicable. We employ analytical arguments and numerical simulations to determine the critical exponents and compare our findings with previous works and functional renormalization group results, which we extend to the different line elasticity. We see evidence for two distinct correlation length exponents.

Organization of Strongly Interacting Directed Polymer Liquids in the Presence of Stringent Constraints

The impact of impenetrable obstacles on the energetics and equilibrium structure of strongly repulsive directed polymers is investigated. As a result of the strong interactions, regions of severe polymer depletion and excess are found in the vicinity of the obstacle, and the associated free-energy cost is found to scale quadratically with the average polymer density. The polymer-polymer interactions are accounted for via a sequence of transformations: from the 3D line liquid to a 2D fluid of Bose particles to a 2D composite fermion fluid and, finally, to a 2D one-component plasma. The results presented here are applicable to a range of systems consisting of noncrossing directed lines.

Stiff directed lines in random media

We investigate the localization of stiff directed lines with bending energy by a short-range random potential. We apply perturbative arguments, Flory scaling arguments, a variational replica calculation, and functional renormalization to show that a stiff directed line in 1+d dimensions undergoes a localization transition with increasing disorder for d>2/3. We demonstrate that this transition is accessible by numerical transfer matrix calculations in 1+1 dimensions and analyze the properties of the disorder-dominated phase in detail. On the basis of the two-replica problem, we propose a relation between the localization of stiff directed lines in 1+d dimensions and of directed lines under tension in 1+3d dimensions, which is strongly supported by identical free-energy distributions. This shows that pair interactions in the replicated Hamiltonian determine the nature of directed line localization transitions with consequences for the critical behavior of the Kardar-Parisi-Zhang equation. We support the proposed relation to directed lines via multifractal analysis, revealing an analogous Anderson transition-like scenario and a matching correlation length exponent. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder.

Localization transition of stiff directed lines in random media

We investigate the localization of stiff directed lines with bending energy by a short-range random potential. Using perturbative arguments, Flory arguments, and a replica calculation, we show that a stiff directed line in 1+d dimensions undergoes a localization transition with increasing disorder for d>2/3. We demonstrate that this transition is accessible by numerical transfer matrix calculations in 1+1 dimensions and analyze the properties of the disorder-dominated phase. On the basis of the two-replica problem, we propose a relation between the localization of stiff directed lines in 1+d dimensions and of directed lines under tension in 1+3d dimensions, which is strongly supported by identical free energy distributions. This shows that pair interactions in the replicated Hamiltonian determine the nature of directed line localization transitions with consequences for the critical behavior of the Kardar-Parisi-Zhang (KPZ) equation. Furthermore, we quantify how the persistence length of the stiff directed line is reduced by disorder.

On Martin Disteli's spatial cycloidal gearing

The modeling of the tooth flanks of gears with skew axes still represents a challenge to geometers. This paper is a contribution along these lines, as pertaining to gears with skew axes and cycloidal teeth. To this end, the authors follow and extend the results reported by Martin Disteli at the turn of the 20th century concerning the general synthesis of gears with skew axes. The main goal is the rendering of the tooth profiles of cycloidal gears with skew axes. This is done by means of the dualization of the geometric relations proposed by Disteli for bevel gears with cycloidal teeth, which leads to ruled surfaces as conjugate tooth flanks such that at any instant the contact points are located on a straight line. All results are proven by means of a consistent use of dual vectors representing directed lines and rigid-body twists.

A Comparison of Spatial and Movement Patterns between Sympatric Predators: Bull Sharks (Carcharhinus leucas) and Atlantic Tarpon (Megalops atlanticus)

BackgroundPredators can impact ecosystems through trophic cascades such that differential patterns in habitat use can lead to spatiotemporal variation in top down forcing on community dynamics. Thus, improved understanding of predator movements is important for evaluating the potential ecosystem effects of their declines.Methodology/Principal FindingsWe satellite-tagged an apex predator (bull sharks, Carcharhinus leucas) and a sympatric mesopredator (Atlantic tarpon, Megalops atlanticus) in southern Florida waters to describe their habitat use, abundance and movement patterns. We asked four questions: (1) How do the seasonal abundance patterns of bull sharks and tarpon compare? (2) How do the movement patterns of bull sharks and tarpon compare, and what proportion of time do their respective primary ranges overlap? (3) Do tarpon movement patterns (e.g., straight versus convoluted paths) and/or their rates of movement (ROM) differ in areas of low versus high bull shark abundance? and (4) Can any general conclusions be reached concerning whether tarpon may mitigate risk of predation by sharks when they are in areas of high bull shark abundance?Conclusions/SignificanceDespite similarities in diet, bull sharks and tarpon showed little overlap in habitat use. Bull shark abundance was high year-round, but peaked in winter; while tarpon abundance and fishery catches were highest in late spring. However, presence of the largest sharks (>230 cm) coincided with peak tarpon abundance. When moving over deep open waters (areas of high shark abundance and high food availability) tarpon maintained relatively high ROM in directed lines until reaching shallow structurally-complex areas. At such locations, tarpon exhibited slow tortuous movements over relatively long time periods indicative of foraging. Tarpon periodically concentrated up rivers, where tracked bull sharks were absent. We propose that tarpon trade-off energetic costs of both food assimilation and osmoregulation to reduce predation risk by bull sharks.

A visualization method for geographic conceptual modelling

Because of the difficulty in sharing of modelling conceptions, the complexity of modelling methods, the difficulty of cooperative modelling for experts in different domains and so on, many problems have arisen during the geographic conceptual modelling process. A formalization and visualization method of geographic conceptual modelling is studied in this article for solving these problems. With respect to the traditional entity–relationship model, two issues are studied in this article: one is the establishment of a supporting system, in which the expression methods of geographic concept, the use of geographic conceptual icons as well as the designing methods are illustrated in detail. The other step is using the former to build geographic conceptual models as follows: the concept of geo-entities is represented by geographic conceptual icons and the concept of relationships between geo-entities is represented by directed lines; then geographic conceptual models can be built by dragging those elements into a geographic conceptual scenario in a guided way. Moreover, the geographic conceptual scenarios can be saved as templates for reuse and further modification. Experiments have shown that compared with traditional graphical conceptual modelling methods, apart from enhancing the intelligence of the modelling environment and promoting the visual ability during conceptual modelling process, this approach can also support conceptual model sharing and knowledge reuse.

Bayesian networks: a powerful tool for systems biology study

The wide application of omics research approaches caused a burst of biological data in the past decade, and also promoted the growth of systems biology, a research field that studies biological questions from a genome-wide point of view. One feature of systems biology study is to integrate and identify. Not only experiments are carried out at whole-genome scales, but also data from various resources, such as genomics, transcriptomics, proteomics, and metabolics data, need to be integrated to identify correlations among targeted entities. Therefore, plenty amounts of experimental data, robust statistical methods, and reliable network construction models are indispensable for systems biology study. Among the available network construction models, Bayesian network is considered as one of the most effective methods available so far for biological network predictions (Pe’er, 2005). Bayesian networks are constructed based on the Bayes’ theorem. The Bayes’ theorem (often called Bayes’ law or Bayes’ rule) is well-known in probability analysis. It is named for Rev. Thomas Bayes, a British mathematician. The basis of Bayes’ theorem is to show the relation between one conditional probability and its reverse. That is, given that event B happened, the occurrence probability of event A depends not only on the relationship between A and B, but also on the absolute probability of A independent of B as well as the absolute probability of B independent of A. Bayesian networks can be considered as a mechanism to automatically apply Bayes’ theorem to complex problems. A Bayesian network is a probabilistic graphic model that represents a set of random variables and their conditional independencies. In Bayesian networks, random variables are presented by nodes, and the conditional dependencies of variables are shown by directed lines. Therefore, Bayesian networks are directional, and are also named as “directed acyclic graphical model”. Bayesian network is very robust to identify relationships among variables, and to find out hidden dependencies among a subset of variables when the dependencies of other variables are given. Thus, it is ideal for relationship identification among large-scale datasets and has been widely applied in systems biology studies (Campos et al., 2004). As an outstanding molecular systems biologist, Jing-Dong Jackie Han at the Institute of Genetics & Developmental Biology, Chinese Academy of Sciences, Beijing, China, and her research group have accomplished many pioneer works using Bayesian networks to decipher protein interactomes in various species (Xia et al., 2006; Xue et al., 2007; Xia et al., 2008; Yu et al., 2008). To generalize the application of Bayesian networks, in a review of this issue’s Frontiers in Biology, Liu and Han (Liu and Han, 2010) introduced some basic concepts of probabilistic models and Bayesian networks, explained the structures of Bayesian networks and their building algorithms, and summarized the appropriate applications of Bayesian networks on biological data. The review provides readers a general picture about Bayesian networks along concise and clear written lines. It is undoubtful that with the increment of omics data of various types, the application of Bayesian networks in systems biology analysis will bring more striking discoveries.

Topological relations between directed lines and simple geometries

Directed lines are fundamental geometric elements to represent directed linear entities. The representations of their topological relations are so different from those of simple lines that they cannot be solved exactly with normal methods. In this paper, a new model based on point-set topology is defined to represent the topological relations between directed lines and simple geometries. Through the intersections between the start-points, end-points, and interiors of the directed lines and the interiors, boundaries, and exteriors of the simple geometries, this model identifies 5 cases of topological relations between directed lines and points, 39 cases of simple lines, and 26 cases of simple polygons. Another 4 cases of simple lines and one case of simple polygons are distinguished if considering the exteriors of the directed lines. All possible cases are furthermore grouped into an exclusive and complete set containing 11 named predicts. And the conceptual neighborhood graph is set up to illustrate their relationship and similarity. This model can provide a basis for natural language description and spatial query language to present the dynamic semantics of directed lines relative to the background features.

Trunk muscle activation and associated lumbar spine joint shear forces under different levels of external forward force applied to the trunk

High anterior intervertebral shear loads could cause low back injuries and therefore the neuromuscular system may actively counteract these forces. This study investigated whether, under constant moment loading relative to L3L4, an increased externally applied forward force on the trunk results in a shift in muscle activation towards the use of muscles with more backward directed lines of action, thereby reducing the increase in total joint shear force. Twelve participants isometrically resisted forward forces, applied at several locations on the trunk, while moments were held constant relative to L3L4. Surface EMG and lumbar curvature were measured, and an EMG–driven muscle model was used to calculate compression and shear forces at all lumbar intervertebral joints. Larger externally applied forward forces resulted in a flattening of the lumbar lordosis and a slightly more backward directed muscle force. Furthermore, the overall muscle activation increased. At the T12L1 to L3L4 joint, resulting joint shear forces remained small (less than 200 N) because the average muscle force pulled backward relative to those joints. However, at the L5S1 joint the average muscle force pulled the trunk forward so that the increase in muscle force with increasing externally applied forward force caused a further rise in shear force (by 102.1 N, SD = 104.0 N), resulting in a joint shear force of 1080.1 N (SD = 150.4 N) at 50 Nm moment loading. It is concluded that the response of the neuromuscular system to shear force challenges tends to increase rather than reduce the shear loading at the lumbar joint that is subjected to the highest shear forces.

Global Existence of Solutions to the Coupled Einstein and Maxwell-Higgs System in the Spherical Symmetry

We prove the global unique existence of classical solutions to the Einstein equations coupled with Maxwell-Higgs system for small initial data under the spherical symmetry. We also obtain the decay estimates of the solutions, and find that the corresponding space-time is time-like and null geodesically complete toward the future. For the proof we reduce the system to a single first order integrodifferential equation, and use the contraction mapping theorem in the appropriate function spaces. We also obtain the completeness of space-time along the future directed time-like lines exterior to a region which resembles the even horizon of the Reissner-Nordstrom black hole.

A new continuous-curvature line/path-tracking method for car-like vehicles

In this paper, we first investigate the problem of finding an algorithm for the movement of a car-like vehicle to track a given directed straight line. Car-like vehicles' motions possess 2 d.o.f., speed v and path curvature κ. We compute the 'derivative of path curvature', λ = dκ/ds, rather than the path curvature κ itself, to enforce continuous-curvature vehicle motions. Specifically, we compute λ as a linear function of the current path curvature, orientation and positional difference of the vehicle against the given line. We call this function λ the steering function. The uniform asymptotic stability of the feedback rule is proved through linearization and a Lyapunov function. Next, this basic line-tracking result is applied to the problem of path tracking where a path consists of directed straight lines. The main advantages and features of the results are summarized as follows. (i) Since the derivative dκ/ds of path curvature is obtained, the path curvature is always continuous while a vehicle converges to the line. (ii) By assuming the critical damping condition, the steering function contains one parameter σ that controls the smoothness (or equivalently, sharpness) of vehicle motions. (iii) This theory can be applied to a vehicle with any wheel architecture, i.e. this theory is vehicle independent. (iv) The theory can easily be extended to the more general problem of tracking a path consisting of any number of directed lines with a new principle of neutral switching. (v) The simplicity and machine independence of this theory makes implementation of this theory on vehicles easy. We present numerous simulation results of line/path-tracking motions. These results verify the effectiveness of this continuous-curvature motion control method. Some successful results obtained on the autonomous robot Yamabico at the Naval Postgraduate School are also included.

Superfluid bosons and flux liquids: disorder, thermal fluctuations, and finite-size effects

The influence of different types of disorder (both uncorrelated and correlated) on the superfluid properties of a weakly interacting or dilute Bose gas, as well as on the corresponding quantities for flux line liquids in high-temperature superconductors at low magnetic fields are reviewed, investigated and compared. We exploit the formal analogy between superfluid bosons and the statistical mechanics of directed lines, and explore the influence of the different “imaginary time” boundary conditions appropriate for a flux line liquid. For superfluids, we discuss the density and momentum correlations, the condensate fraction, and the normal-fluid density as function of temperature for two- and three-dimensional systems subject to a space- and time-dependent random potential as well as conventional point-, line-, and plane-like defects. In the case of vortex liquids subject to point disorder, twin boundaries, screw dislocations, and various configurations of columnar damage tracks, we calculate the corresponding quantities, namely, density and tilt correlations, the “boson” order parameter, and the tilt modulus. The finite-size corrections due to periodic vs. open “imaginary time” boundary conditions differ in interesting and important ways. Experimental implications for vortex lines are described briefly.