Two-dimensional Euclidean Space Research Articles

The use of fractals to quantify the morphology of cluster microforms

The emphasis of this paper is on the fractal-analysis of irregular objects like particle clusters that fill the two-dimensional Euclidean space R 2. A common fractal descriptor of such objects is D L (1 < D L < 2), which defines the fractal dimension of the object outline and is determined by the divider method. In this case, an object may possibly exhibit only two statistically self-similar scales, namely ‘structural’ and ‘textural’. The structural fractal dimension describes the form structure (or the degree of convolution) of the object perimeter and the textural describes the crenellations (or edge effects) found on the object perimeter. Recent studies have shown that the D L does not uniquely define the morphology of complex objects like clusters as it does not include the morphologic attributes, associated with the objects enclosed projected area. A key contribution of this work is the recognition that the ‘total areal’ fractal dimension of the object, D AT (1 < D AT < 2) offers a unique description of cluster morphology as it accounts for the collective contributions from the crenellations found on the object perimeter and its form structure, as well as contributions from the object core structure or equivalently object projected area. The D AT , which is theoretically interrelated to D L , satisfies the key self-similarity principle of fractals and is employed here to differentiate among four cluster morphologic groups observed in field and laboratory studies, namely, in-line clusters with elongation ratio, ER < 5 and ER > 5, pseudo-triangle, and -rhomboid. The analysis for this study is performed by applying the box-counting method for analyzing cluster planview images from four groups of cluster morphologies derived from synthetic, laboratory, and field data sets. The D AT is determined via Richardson plots by estimating the slope of the best-fit line over the entire range of box sizes. Doing so, provides a single aggregate dimension that includes the embedded effects from the objects projected area and perimeter. The results show that the average D AT ¯ and its standard deviation can effectively discriminate the four cluster morphologic groups. The D AT ¯ provides a clear demarcation between the nonlinear morphologies and the in-line morphology with ER > 5. The analysis also shows that the in-line clusters with ER < 5 change between pseudo-linear and pseudo-nonlinear morphologies. Moreover, it is found that most morphologies in the field are statistically well represented by the corresponding laboratory and synthetic morphologies. Finally, the analysis of the laboratory cluster data reveals the utility of D AT ¯ for portraying the correspondence between cluster morphology and physical processes responsible for cluster formation and morphologic evolution, such as flow and sediment supply. Future research should build on the knowledge gained on this topic and further examine the links between the fractal dimension of different cluster morphologies and the physical processes that produce the characteristic scale captured by D AT ¯ .

Mathematical Model for Evaluating Roundness Errors by Minimum Circumscribed Circle Method

An unconstrained optimization model is established for assessing roundness errors by the minimum circumscribed circle method based on radial deviation measurement. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the two-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function in order to get the wanted roundness errors by the minimun circumscribed circle assessment. One example is given to verify the theoretical results presented.

Mathematical Model for Evaluating Roundness Errors by Maximum Inscribed Circle Method

An unconstrained optimization model is established for assessing roundness errors by the maximum inscribed circle method based on radial deviation measurement. The properties of the objective function in the optimization model are thoroughly researched. On the basis of the modern theory on convex functions, it is strictly proved that the objective function is a continuous and non-differentiable and convex function defined on the two-dimensional Euclidean space. The minimal value of the objective function is unique and any of its minimal point must be its global minimal point. Any existing optimization algorithm, so long as it is convergent, can be used to solve the objective function in order to get the wanted roundness errors by the maximum inscribed circle assessment. One example is given to verify the theoretical results presented.

Direction-based surrounder queries for mobile recommendations

Location-based recommendation services recommend objects to the user based on the user’s preferences. In general, the nearest objects are good choices considering their spatial proximity to the user. However, not only the distance of an object to the user but also their directional relationship are important. Motivated by these, we propose a new spatial query, namely a direction-based surrounder (DBS) query, which retrieves the nearest objects around the user from different directions. We define the DBS query not only in a two-dimensional Euclidean space \({\mathbb{E}}\) but also in a road network \({\mathbb{R}}\) . In the Euclidean space \({\mathbb{E}}\) , we consider two objects a and b are directional close w.r.t. a query point q iff the included angle \({\angle aqb}\) is bounded by a threshold specified by the user at the query time. In a road network \({\mathbb{R}}\) , we consider two objects a and b are directional close iff their shortest paths to q overlap. We say object a dominates object b iff they are directional close and meanwhile a is closer to q than b. All the objects that are not dominated by others based on the above dominance relationship constitute direction-based surrounders (DBSs). In this paper, we formalize the DBS query, study it in both the snapshot and continuous settings, and conduct extensive experiments with both real and synthetic datasets to evaluate our proposed algorithms. The experimental results demonstrate that the proposed algorithms can answer DBS queries efficiently.

Scaling behaviors of the voltage distribution in dielectric breakdown networks

We study the distribution of voltage drops across bonds in dielectric breakdown networks and its qth moments in the two-dimensional Euclidean space. Performing numerical simulations, we find that the distribution is composed of three different power-law regimes which are distinguished by two crossover voltages V(1) and V(2). The scaling behaviors of these crossover voltages with respect to the system size govern those of the qth moments of the voltage distribution. This feature differs from the multifractal behavior of the qth moment in random resistor networks. We discuss the implications of these scaling behaviors in relation to the application of the dielectric breakdown network to memory devices.

Perturbations of generic Kasner spacetimes and their stability

This article investigates the stability of a generic Kasner spacetime to linear perturbations, both at late and early times. It demonstrates that the perturbation of the Weyl tensor diverges at late time in all cases but in the particular one in which the Kasner spacetime is the product of a two-dimensional Milne spacetime and a two-dimensional Euclidean space. At early times, the perturbation of the Weyl tensor also diverges unless one imposes a condition on the perturbations so as to avoid the most divergent modes to be excited.

On the mass spectrum of noncommutative Schwinger model in Euclidean ℝ2 space

The mass spectrum of noncommutative QED in two-dimensional Euclidean ℝ2 space is derived first in a perturbative approach at one-loop level and then in a nonperturbative approach using the equivalent bosonized noncommutative effective action. It turns out that the mass spectrum of noncommutative QED in two dimensions reduces to a single non-interacting meson with mass \(M_{\gamma}=\frac{g}{\sqrt{\pi}}\), as in commutative Schwinger model.

Nonlinear dynamical symmetries of Smorodinsky—Winternitz and Fokas—Lagerstorm systems

General solutions of the Smorodinsky—Winternitz system and the Fokas—Lagerstorm system, which are superintegrable in two-dimensional Euclidean space, are obtained using the algebraic method (structure function). Their dynamical symmetries, which are governed by deformed angular momentum algebras, are revealed.

Global existence and decay estimates of solutions to a parabolic-elliptic system of drift-diffusion type in $\mathbb R^2$

We consider a parabolic-elliptic system of drift-diffusion type in the entire two-dimensional Euclidean space, modeling chemotaxis and self-attracting particles. Under the assumption that the total mass of nonnegative initial data is less than $8\pi$, we give global existence and decay estimates of nonnegative solutions to the Cauchy problem for this system.

A plane graph representation of triconnected graphs

Given a graph G=(V,E), a set S={s1,s2,…,sk} of k vertices of V, and k natural numbers n1,n2,…,nk such that ∑i=1kni=|V|, the k-partition problem is to find a partition V1,V2,…,Vk of the vertex set V such that |Vi|=ni, si∈Vi, and Vi induces a connected subgraph of G for each i=1,2,…,k. For the tripartition problem on a triconnected graph, a naive algorithm can be designed based on a directional embedding of G in the two-dimensional Euclidean space. However, for graphs of large number of vertices, the implementing of this algorithm requires high precision real arithmetic to distinguish two close vertices in the plane. In this paper, we propose an algorithm for dealing with the tripartition problem by introducing a new data structure called the region graph, which represents a kind of combinatorial embedding of the given graph in the plane. The algorithm constructs a desired tripartition combinatorially in the sense that it does not require any geometrical computation with actual coordinates in the Euclidean space.

Improved algorithms for latency minimization in wireless networks

In the interference scheduling problem, one is given a set of n communication requests described by source–destination pairs of nodes from a metric space. The nodes correspond to devices in a wireless network. Each pair must be assigned a power level and a color such that the pairs in each color class can communicate simultaneously at the specified power levels. The feasibility of simultaneous communication within a color class is defined in terms of the Signal to Interference plus Noise Ratio (SINR) that compares the strength of a signal at a receiver to the sum of the strengths of other signals. The objective is to minimize the number of colors as this corresponds to the time needed to schedule all requests. We introduce an instance-based measure of interference, denoted by I , that enables us to improve on previous results for the interference scheduling problem. We prove the upper and lower bounds in terms of I on the number of steps needed for scheduling a set of requests. For general power assignments, we prove a lower bound of Ω ( I / ( log Δ log n ) ) steps, where Δ denotes the aspect ratio of the metric. When restricting to the two-dimensional Euclidean space (as in the previous work) the bound improves to Ω ( I / log Δ ) . Alternatively, when restricting to linear power assignments, the lower bound improves even to Ω ( I ) . The lower bounds are complemented by an efficient algorithm computing a schedule for linear power assignments using only O ( I log n ) steps. A more sophisticated algorithm computes a schedule using even only O ( I + log 2 n ) steps. For dense instances in the two-dimensional Euclidean space, this gives a constant factor approximation for scheduling under linear power assignments, which shows that the price for using linear (and, hence, energy-efficient) power assignments is bounded by a factor of O ( log Δ ) . In addition, we extend these results for single-hop scheduling to multi-hop scheduling and combined scheduling and routing problems, where our analysis generalizes the previous results towards general metrics and improves on the previous approximation factors.

Lie algebra contractions on two-dimensional hyperboloid

The Inonu-Wigner contraction from the SO(2, 1) group to the Euclidean E(2) and E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the four corresponding two-dimensional homogeneous spaces: two-dimensional hyperboloids and two-dimensional Euclidean and pseudo-Euclidean spaces. We show how the nine systems of coordinates on the two-dimensional hyperboloids contracted to the four systems of coordinates on E2 and eight on E1,1. The text was submitted by the authors in English.

Approximation theorems by positive linear operators in weighted spaces

In this study, we obtain some Korovkin type approximation theorems by positive linear operators on the weighted space of all real valued functions defined on the real two-dimensional Euclidean space \({\mathbb{R}^2}\). This paper is mainly consisted of two parts: a Korovkin type approximation theorem via the concept of A-statistical convergence and a Korovkin type approximation theorem via \({\mathcal {A}}\)-summability.

Essential self-adjointness of a discrete magnetic Schrödinger operator

We prove essential self-adjointness for a semibounded from below discrete magnetic Schrodinger operator in a space that represents a combinatorial model of the two-dimensional Euclidean space. The Dezin discretization scheme is used for constructing a discrete model.

The remapping of space in motor learning and human–machine interfaces

Studies of motor adaptation to patterns of deterministic forces have revealed the ability of the motor control system to form and use predictive representations of the environment. One of the most fundamental elements of our environment is space itself. This article focuses on the notion of Euclidean space as it applies to common sensory motor experiences. Starting from the assumption that we interact with the world through a system of neural signals, we observe that these signals are not inherently endowed with metric properties of the ordinary Euclidean space. The ability of the nervous system to represent these properties depends on adaptive mechanisms that reconstruct the Euclidean metric from signals that are not Euclidean. Gaining access to these mechanisms will reveal the process by which the nervous system handles novel sophisticated coordinate transformation tasks, thus highlighting possible avenues to create functional human–machine interfaces that can make that task much easier. A set of experiments is presented that demonstrate the ability of the sensory-motor system to reorganize coordination in novel geometrical environments. In these environments multiple degrees of freedom of body motions are used to control the coordinates of a point in a two-dimensional Euclidean space. We discuss how practice leads to the acquisition of the metric properties of the controlled space. Methods of machine learning based on the reduction of reaching errors are tested as a means to facilitate learning by adaptively changing he map from body motions to controlled device. We discuss the relevance of the results to the development of adaptive human–machine interfaces and optimal control.

Superintegrability with second- and third-order integrals of motion

Superintegrable systems in two-dimensional Euclidean space E 2 with scalar potentials and integrals of motion that are second- or third-order polynomials in the momenta are discussed. We review old results, obtained in collaboration with Ya.A. Smorodinsky more than 40 years ago and present new ones that are their continuation.

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials

We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painlevé transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions.

Uniform distributions in a class of convex polyhedrons with applications to drug combination studies

Motivated by experimental designs for drug combination studies, in this paper, we propose a novel approach for generating a uniform distribution on an arbitrary tetragon in two-dimensional Euclidean space R 2 . The key idea is to construct a one-to-one transformation between an arbitrary tetragon and the unit square [ 0 , 1 ] 2 . This transformation then provides a stochastic representation (SR) for the random vector uniformly distributed on the tetragon. An algorithm is proposed for generating a uniform distribution in an arbitrary triangular prism in R 3 . In addition, we develop methods for generating uniform distributions in a class of convex polyhedrons in n -dimensional Euclidean space R n . In particular, SRs for uniform distributions in regions with order restrictions are presented. We apply the proposed method to the experimental design for a drug combination study.

Fitting a geometric graph to a protein–protein interaction network

Finding a good network null model for protein-protein interaction (PPI) networks is a fundamental issue. Such a model would provide insights into the interplay between network structure and biological function as well as into evolution. Also, network (graph) models are used to guide biological experiments and discover new biological features. It has been proposed that geometric random graphs are a good model for PPI networks. In a geometric random graph, nodes correspond to uniformly randomly distributed points in a metric space and edges (links) exist between pairs of nodes for which the corresponding points in the metric space are close enough according to some distance norm. Computational experiments have revealed close matches between key topological properties of PPI networks and geometric random graph models. In this work, we push the comparison further by exploiting the fact that the geometric property can be tested for directly. To this end, we develop an algorithm that takes PPI interaction data and embeds proteins into a low-dimensional Euclidean space, under the premise that connectivity information corresponds to Euclidean proximity, as in geometric-random graphs. We judge the sensitivity and specificity of the fit by computing the area under the Receiver Operator Characteristic (ROC) curve. The network embedding algorithm is based on multi-dimensional scaling, with the square root of the path length in a network playing the role of the Euclidean distance in the Euclidean space. The algorithm exploits sparsity for computational efficiency, and requires only a few sparse matrix multiplications, giving a complexity of O(N(2)) where N is the number of proteins. The algorithm has been verified in the sense that it successfully rediscovers the geometric structure in artificially constructed geometric networks, even when noise is added by re-wiring some links. Applying the algorithm to 19 publicly available PPI networks of various organisms indicated that: (a) geometric effects are present and (b) two-dimensional Euclidean space is generally as effective as higher dimensional Euclidean space for explaining the connectivity. Testing on a high-confidence yeast data set produced a very strong indication of geometric structure (area under the ROC curve of 0.89), with this network being essentially indistinguishable from a noisy geometric network. Overall, the results add support to the hypothesis that PPI networks have a geometric structure. MATLAB code implementing the algorithm is available upon request.

Continuous Clustering of Moving Objects

This paper considers the problem of efficiently maintaining a clustering of a dynamic set of data points that move continuously in two-dimensional Euclidean space. This problem has received little attention and introduces new challenges to clustering. The paper proposes a new scheme that is capable of incrementally clustering moving objects. This proposal employs a notion of object dissimilarity that considers object movement across a period of time, and it employs clustering features that can be maintained efficiently in incremental fashion. In the proposed scheme, a quality measure for incremental clusters is used for identifying clusters that are not compact enough after certain insertions and deletions. An extensive experimental study shows that the new scheme performs significantly faster than traditional ones that frequently rebuild clusters. The study also shows that the new scheme is effective in preserving the quality of moving-object clusters.