scholarly journals Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings

Author(s):  
David R. Wood
2014 ◽  
Vol 13 (08) ◽  
pp. 1450050 ◽  
Author(s):  
Irene Platoni

In a three-dimensional Galois space of odd order q, the known infinite families of complete caps have size far from the theoretical lower bounds. In this paper, we investigate some caps defined from elliptic curves. In particular, we show that for each q between 100 and 350 they can be extended to complete caps, which turn out to be the smallest complete caps known in the literature.


2020 ◽  
Author(s):  
Jinmei Zheng ◽  
Bin Sun ◽  
Ruolan Lin ◽  
Yongqi Teng ◽  
Xihai Zhao ◽  
...  

Abstract Background Atherosclerotic plaques are often present in regions with complicated flow patterns. Vascular morphology plays a role in hemodynamics. In this study, we investigate the relationship between the geometry of the vertebrobasilar artery system and the basilar artery (BA) plaque prevalence.Methods We enrolled 290 patients with posterior circulation ischemic stroke. We distinguished four configurations of the vertebrobasilar artery: Walking, Tuning Fork, Lambda, and No Confluence. The diameter of the vertebral artery (VA) and the number of bends in the intracranial VA segment was assessed using three-dimensional time-of-flight magnetic resonance angiography. We differentiated between multi-bending (≥ 3 bends) and oligo-bending (< 3 bends) VAs. High-resolution magnetic resonance imaging was used to evaluate BA plaques. Logistic regression models examined the relationship between the geometry type and BA plaque prevalence.Results After adjusting for sex, age, body mass index ≥ 28, hypertension, and diabetes mellitus, the Walking, Lambda, and No Confluence geometries were associated with the presence of BA plaque. Patients with multi-bending VAs in both the Walking (71.43%, P = 0.003) and Lambda group (40.43%, P = 0.018) had more plaques compared to patients with oligo-bending VAs in these groups. In the Lambda group, the diameter difference between the VAs was larger in patients with BA plaques than that in patients without BA plaques (1.4 mm vs. 0.9 mm, P < 0.001).Conclusions The Walking, Lambda, and No Confluence geometry, ≥ 3 bends in the VAs, and a large diameter difference between the VAs were associated with the presence of BA plaque.


2013 ◽  
Vol 11 (01) ◽  
pp. 1450006 ◽  
Author(s):  
JUAN ÁNGEL ALEDO ◽  
ALFONSO ROMERO ◽  
RAFAEL M. RUBIO

We provide upper and lower bounds for the volume of a compact spacelike hypersurface in an (n + 1)-dimensional Generalized Robertson–Walker (GRW) spacetime in terms of the volume of the fiber, the hyperbolic angle function and the warping function. Under several geometrical and physical assumptions, we characterize the spacelike slices as the only spacelike hypersurfaces where these bounds are attained. As a consequence of these results, we get an upper bound for the first eigenvalue of a compact spacelike surface in a three-dimensional GRW spacetime whose fiber is a topological sphere, which includes the case of the three-dimensional De Sitter spacetime, and show that the bound is attained if and only if M is a spacelike slice.


1993 ◽  
Vol 03 (02) ◽  
pp. 129-138
Author(s):  
STEVEN CHEUNG ◽  
FRANCIS C.M. LAU

We present time lower bounds for the permutation routing problem on three- and higher-dimensional n x…x n meshes with buses. We prove an (r–1)n/r lower bound for the general case of an r-dimensional bused mesh, r≥2, which is not as strong for low-dimensional as for higher-dimensional cases. We then use a different approach to construct a 0.705n lower bound for the three-dimensional case.


2008 ◽  
Vol 22 (23) ◽  
pp. 2163-2175 ◽  
Author(s):  
MIKLÓS HORVÁTH

We consider three-dimensional inverse scattering with fixed energy for which the spherically symmetrical potential is nonvanishing only in a ball. We give exact upper and lower bounds for the phase shifts. We provide a variational formula for the Weyl–Titchmarsh m-function of the one-dimensional Schrödinger operator defined on the half-line.


2010 ◽  
Vol 06 (04) ◽  
pp. 731-766 ◽  
Author(s):  
L. YA. VULAKH

Let H3 be the upper half-space model of the three-dimensional hyperbolic space. For certain cocompact Fuchsian subgroups Γ of an extended Bianchi group Bd, the extremality of the axis of hyperbolic F ∈ Γ in H3 with respect to Γ implies its extremality with respect to Bd. This reduction is used to obtain sharp lower bounds for the Hurwitz constants and lower bounds for the highest limit points in the Markov spectra of Bd for some d < 1000. In particular, such bounds are found for all non-Euclidean class one imaginary quadratic fields. The Hurwitz constants for the imaginary quadratic fields with discriminants -120 and -132 are given. The second minima are also indicated for these fields.


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