Radial Graphs Research Articles

“Spydergrams” to depict the results of the Short Form-36 questionnaire: a work in progress

In a recent article, Strand et al. [1] have suggested radar graphs (also termed radial graphs) to plot the results of the Short Form-36 questionnaire (SF-36). They have coined the term “spydergram” which relates to the web-like appearance of the graph but is slightly confusing because the term “spidergram” has been used for connected text boxes such as in a mind map [2]. Spydergrams constitute one of the attempts to graphically depict more than two dimensions on a flat surface. Tufte [3], an authority on scientific graphs, has termed this as “escaping flatland.” To escape flatland, compromises are inevitable, and the challenge is to compromise without distorting the story in the data.

Radial Moore graphs of radius three

The maximum number of vertices in a graph of specified degree and diameter cannot exceed the Moore bound. Graphs achieving this bound are called Moore graphs. Because Moore graphs are so rare, researchers have considered various relaxations of the Moore graph constraints. Since the diameter of a Moore graph is equal to its radius, one can consider graphs in which the condition on the diameter is relaxed, by one, while the condition on the radius is maintained. Such graphs are called radial Moore graphs. It has previously been shown that radial Moore graphs exist for all degrees when the radius is two. In this paper, we extend this result to radius three. We also construct examples that settle the existence question for a few new cases, and summarize the state of knowledge on the problem.

Eye tracking for visualization evaluation: Reading values on linear versus radial graphs

An eye tracking methodology can help uncover subtle cognitive processing stages that are otherwise difficult to observe in visualization evaluation studies. Pros and cons of eye tracking methods are discussed here, including common analysis metrics. One example metric is the initial time at which all elements of a visualization that are required to complete a task have been viewed. An illustrative eye tracking study was conducted to compare how radial and linear graphs support value lookup tasks for both one and two data-dimensions. Linear and radial versions of bar, line, area, and scatter graphs were presented to 32 participants, who each completed a counterbalanced series of tasks. Tasks were completed more quickly on linear graphs than on those with a radial layout. Scanpath analysis revealed that a three-stage processing model was supported: (1) find desired data dimension, (2) find its datapoint, and (3) map the datapoint to its value. Mapping a datapoint to its value was slower on radial than linear graphs, probably because eyes need to follow a circular, as opposed to a linear path. Finding a datapoint within a dimension was harder using line and area graphs than bar and scatter graphs, possibly due to visual confusion of the line representing a data series. Although few errors were made, eye tracking was also used here to classify error strategies. As a result of these analyses, guidelines are proposed for the design of radial and linear graphs.

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.

Chasing the reflected wave back into the heart: a new hypothesis while the jury is still out

Background:Arterial stiffness directly influences cardiac function and is independently associated with cardiovascular risk. However, the influence of the aortic reflected pulse pressure wave on left ventricular function has not been well characterized. The aim of this study was to obtain detailed information on regional ventricular wall motion patterns corresponding to the propagation of the reflected aortic wave on ventricular segments.Methods:Left ventricular wall motion was investigated in a group of healthy volunteers (n = 14, age 23 ± 3 years), using cardiac magnetic resonance navigator-gated tissue phase mapping. The left ventricle was divided into 16 segments and regional wall motion was studied in high temporal detail.Results:Corresponding to the expected timing of the reflected aortic wave reaching the left ventricle, a characteristic “notch” of regional myocardial motion was seen in all radial, circumferential, and longitudinal velocity graphs. This notch was particularly prominent in septal segments adjacent to the left ventricular outflow tract on radial velocity graphs and in anterior and posterior left ventricular segments on circumferential velocity graphs. Similarly, longitudinal velocity graphs demonstrated a brief deceleration in the upward recoil motion of the entire ventricle at the beginning of diastole.Conclusion:These results provide new insights into the possible influence of the reflected aortic waves on ventricular segments. Although the association with the reflected wave appears to us to be unambiguous, it represents a novel research concept, and further studies enabling the actual recording of the pulse wave are required.

Molecular dynamics simulation of the interaction of ethanol-water mixture with a Pt surface

An analysis of the molecular dynamics of ethanol solvated by water molecules in the absence and presence of a Pt surface has been performed using DL_POLY_2.19 code. The structure and diffusion properties of an ethanol–water system have been studied at various temperatures from 250 to 600 K. We have measured the self-diffusion coefficients of the 50:50% ethanol–water solution; in the absence of a Pt surface our results show an excellent agreement–within an error of 7.4% – with the experimental data. An increase in the self-diffusion coefficients with the inclusion of a Pt surface has been observed. The estimation of the diffusion coefficients of both water and ethanol in the presence of a Pt surface shows that they obey the Arrhenius equation; the calculated activation energies of diffusion of ethanol and water are 2.47 and 2.98 Kcal/mole, respectively. The radial distribution function graphs and density profiles have been built; their correlations with the self-diffusion coefficients of both ethanol and water molecules are also illustrated.

Global k-Level Crossing Reduction

Directed graphs are commonly drawn by a four phase framework introduced by Sugiyama et al. in 1981. The vertices are placed on parallel horizontal levels. The edge routing between consecutive levels is computed by solving one-sided 2-level crossing minimization problems, which are repeated in up and down sweeps over all levels. Crossing minimization problems are generally NP-hard. We introduce a global crossing reduction, which at any particular time considers all crossings between all levels. Our approach is based on the sifting technique. It yields an improvement of 5 ‐ 10% in the number of crossings over the level-by-level one-sided 2-level crossing reduction heuristics. In addition, it avoids type 2 conflicts which are crossings between edges whose endpoints are dummy vertices. This helps straightening long edges spanning many levels. Finally, the global crossing reduction approach can directly be extended to cyclic, radial, and clustered level graphs achieving similar improvements. The running time is quadratic in the size of the input graph, whereas the common level-by-level approaches are faster but operate on larger graphs with many dummy vertices for long edges.

On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs

The subject of the paper is Schrödinger operators on tree graphs which are radial, having the branching number \documentclass[12pt]{minimal}\begin{document}$b_n$\end{document}bn at all the vertices at the distance \documentclass[12pt]{minimal}\begin{document}$t_n$\end{document}tn from the root. We consider a family of coupling conditions at the vertices characterized by \documentclass[12pt]{minimal}\begin{document}$(b_n-1)^2+4$\end{document}(bn−1)2+4 real parameters. We prove that if the graph is sparse so that there is a subsequence of \documentclass[12pt]{minimal}\begin{document}$\lbrace t_{n+1}-t_n\rbrace$\end{document}{tn+1−tn} growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrödinger operator can be purely absolutely continuous.

Further results on radial graphs

Further results on radial graphs

Anderson localization for radial tree-like random quantum graphs

We prove that certain random models associated with radial, tree-like, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.

CONVENIENT POSITION-VALUE AND GRAVITY MODEL

The aim of this paper is to analyze the distribution of convenient position-values defined on a connected undirected graph. Such values are defined only by the connectiveity of the graph, but meet equilbrium condition on the graph. First, we prove the convenient position-value are distributed according to gravity model using a negative exponential distribution. Then, we exhibit the echo effect and characterize the distribution of convenient position-value on a monocentric city. Finally, we compare radial, external ring and radial-arc graphs routings in terms of convenient position-values.

Some existence results about radial graphs of constant mean curvature with boundary in parallel planes

In this work, for given H, we investigate the existence of radial cmc H annulus spanning two given non necessarily convex Jordan curves in parallel planes of \({\mathbb{R}^{3}}\). We established some existence results under hypotheses relating the geometry of the curves and the distance between the planes.

Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature

We prove the existence and uniqueness in Rn,1 of entire spacelike hypersurfaces contained in the future of the origin O and asymptotic to the light-cone, with scalar curvature prescribed at their generic point M as a negative function of the unit vector pointing in the direction of OM→, divided by the square of the norm of OM→ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from O (the hyperbolic space); radial upper and lower solutions are constructed which, relying on a previous result in the Cartesian setting, imply their existence.

Rearrangements and radial graphs of constant mean curvature in hyperbolic space

We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results are proved via rearrangement techniques.

Interior regularity of conical capillary surfaces

We consider capillary problems that arise physically when the equilibrium surface of a fluid with fixed volume is situated in a cone. By using a variational approach in the space of functions of bounded variation on the sphere Sn, we obtain regularity results for a certain class of relative minima of the energy functional provided the volume is large enough. This special class of relative minima can be described by radial graphs.

Existence and non-existence for a mean curvature equation in hyperbolic space

There exists a well-known criterion for the solvability of theDirichlet Problem for the constant mean curvature equation in bounded smooth domains in Euclidean space. This classicalresult was established by Serrin in 1969. Focusing the DirichletProblem for radial vertical graphs P.-A. Nitsche has establishedan existence and non-existence results on account of a criterionbased on the notion of a hyperbolic cylinder. In this work wecarry out a similar but distinct result in hyperbolic spaceconsidering a different Dirichlet Problem based on another systemof coordinates. We consider a non standard cylindergenerated by horocycles cutting orthogonally a geodesic plane$\mathcal P$ along the boundary of a domain $\Omega\subset \mathcal P.$ We provethat a non strict inequality between the mean curvature $\mathcal H'_{\mathcal C}(y)$of this cylinder along $\partial \Omega$ and the prescribed meancurvature $\mathcal H(y),$ i.e $\mathcal H'_{\mathcal C}(y)\geq |\mathcal H(y)|, \forally\in\partial\Omega$ yields existence of our Dirichlet Problem. Thuswe obtain existence of surfaces whose graphs have prescribed meancurvature $\mathcal H(x)$ in hyperbolic space taking a smooth prescribedboundary data $\varphi.$ This result is sharp because if ourcondition fails at a point $y$ a non-existence result can beinferred.

Some a priori bounds for solutions of the constant Gauss curvature equation

In this work, we give a priori height and gradient estimates for solutions of the prescribed constant Gauss curvature equation in Euclidean space. We shall consider convex radial graphs with positive constant mean curvature. The estimates are established by considering in such a graph, the Riemannian metric given by the second fundamental form of the immersion.

Uniqueness without Growth Condition for the Mean Curvature Equation with Radial Initial Data

We prove the uniqueness of a solution to the mean curvature equation for graphs for any radial continuous initial data . The existence of a smooth solution to this problem comes from the work of Ecker, K., Huisken, G. ([1991]). Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105(3):547–569. We obtain in addition that the solution is radial. We point out that existence and uniqueness hold without any growth restriction on the solution or the initial data, a situation which is rather different to the related stationary problem: in this case, we show there is a limiting growth above which uniqueness does not hold anymore. An application of the uniqueness result to the evolution by mean curvature of entire radial graphs is given.

A note on radial graphs with constant mean curvature

Let Ω be a smooth domain on the unit sphere 𝕊n whose closure is contained in an open hemisphere and denote by ℋ the mean curvature of ∂Ω as a submanifold of Ω with respect to the inward unit normal. It is proved that for each real number H that satisfies inf ℋ > − H ≥ 0, there exists a unique radial graph on Ω bounded by ∂Ω with constant mean curvature H. The orientation on the graph is based on the normal that points on the opposite side as the radius vector.

Radial Graphs of Constant Mean Curvature and Doubly Connected Minimal Surfaces with Prescribed Boundary

In this paper we investigate existence and uniqueness of radial graphs ofconstant mean curvature (cmc) with prescribed boundary. Our main resultestablishes the existence of a minimal radial anullus spanning two givenconvex curves in parallel planes of R3; we also obtain a variant ofa well-known result of James Serrin about the existence of radial cmc graphsover convex domains in the sphere.