Functions In Plane Research Articles

Joint rotational and stretching multifractal spectra of mappings with integrable distortion

We establish bounds for both the stretching and the rotational multifractal spectra of planar homeomorphic mappings with p -integrable distortion. Moreover, we show that these bounds are sharp by presenting extremal examples of homeomorphic mappings with p -integrable distortion that attain the optimal stretching and rotation simultaneously, which additionally leads to the sharp bound for the joint rotational and stretching multifractal spectra. Finally, we will use the stretching multifractal spectra to obtain the sharp area contraction for homeomorphic mappings with p -integrable distortion.

HPM-TDP: An efficient hierarchical PatchMatch depth estimation approach using tree dynamic programming

Accurate and efficient estimation of the dense depth information from a pair of stereo images is a key step for many applications such as digital surface model production, 3D reconstruction and visualization, autonomous driving, and robotic navigation. Although great progress has been achieved in stereo matching over the past decade, the matching difficulties in poor and repetitive texture regions remain an issue. Aiming at solving the shortcomings of the current methods, this paper proposes HPM-TDP, which is an efficient hierarchical PatchMatch depth estimation approach that integrates a coarse-to-fine image pyramid strategy with a continuous Markov random field (MRF)-based global energy optimization framework, and minimizes the energy function by combining a hierarchical PatchMatch (HPM) framework and local α-expansion based tree dynamic programming (TDP). Firstly, the coarse-to-fine image pyramid strategy is integrated with the PatchMatch filter algorithm to quickly generate the hierarchical disparity plane prior for initializing each pixel’s disparity plane of the energy function optimization. Secondly, a multi-resolution cost aggregation strategy is adopted to boost the robustness of the matching cost function in the poor and repetitive texture areas. Finally, the HPM framework and local α-expansion based TDP are adopted to solve the non-submodular energy optimization problem, resulting in a globally optimized disparity plane map. Three benchmark datasets—the Middlebury 3.0, KITTI 2015, and Vaihingen datasets—were used to test the performance of HPM-TDP. The comprehensive experimental results demonstrate that HPM-TDP obtains a good performance on all datasets in terms of the (“Out-Noc”, “Avg-Noc”, “Out-All”, “Avg-All”) of (15.45%, 4.16px, 24.26%, 12.14px) and (5.46%, 1.20px, 6.55%, 1.54px) for Middlebury 3.0 and KITTI 2015 training datasets, and the (“Out-All”, “Avg-All”) of (26.32%, 4.04px) for Vaihingen dataset, respectively.

Stimulation of subchondral bone cyst healing by placement of a transcondylar screw in the equine medial femoral condyle

To predict the bone formation stimulus of a transcondylar screw across an equine subchondral bone cyst (SBC) in an equine medial femoral condyle (MFC). Finite element modeling (FEM) of an equine MFC with a 2 cm3 SBC under several transcondylar screw conditions. The right stifle of a yearling thoroughbred without stifle disease that had been euthanized for reasons unrelated to this study and donated to the University. The FEM was derived from computed tomography of a yearling thoroughbred and analyzed in ABAQUS v6.14. The transcondylar screw was modeled as a 4.5-mm stainless steel cylinder. The region of interest was the centrodistal MFC, and bone stimulus was calculated. The stimulus threshold for bone formation (BFT) was >60 MPa and is presented as the percentage of total bone surface area (BFA) and frontal plane maps. Principal compressive stress vectors were also determined. Tested variables were daily cycles, load, and screw compression and position. At 750 cycles and 900-N load, <3% of the BFA exceeded the BFT. Increases in BFA > BFT occurred proportionally with load, screw compression, and daily cycles (steps). Compressive stress was oriented vertically on the SBC surface without a screw but aligned with the long axis of well-placed lag screws. Screw placement through the void also increased the number and magnitude of compressive vectors. This model predicted that a transcondylar lag screw across an MFC SBC increased surface BFA stimulation and reoriented the compression vector. Increasing screw compression, load, and steps per day increased the bone formation stimulus. This study provides evidence that supports the use of a lag screw thorough an MFC SBC to promote bone formation.

Probability of magnification in the HubbleFrontier Fields clusters

We present free-form gravitational lensing models derived with the WSLAP+ code for the six $\textit{Hubble Frontier Fields}$ clusters using the latest data available from the $\textit{Frontier Fields Lensing Models v.4}$ collaboration. We present magnifications maps in the lens plane and caustic maps in the source plane. From these maps, we derive the probability of magnification using two different, but related, methods. We confirm MACS 0717 as the cluster with the most complex structure, and having the largest lensing efficiency and Einstein radius. When comparing these results with the ones obtained by previous observations of these clusters, we obtain an increase in the lensing efficiency between 1.4 and 2.3. We also find a good correlation with a relatively small dispersion between the lensing efficiency and Einstein radius as a function of the source redshift ($z_s$). Finally, we estimate the lensing effects produced by the six $\textit{Hubble Frontier Fields}$ clusters on the luminosity function of galaxies at high redshift ($z=9$) for standard luminosity functions and an alternative luminosity function based on predictions from wave dark matter ($\psi$DM) models.

A note on F-planar mappings of manifolds with non-symmetric linear connection

Following the basic facts of [Formula: see text]-planar mappings introduced by Mikeš and Sinyukov and further developed by Hinterleitner and Mikeš, we consider the notions of an [Formula: see text]-planar curve and an [Formula: see text]-planar mapping, but in case of manifolds endowed with a non-symmetric linear connection and an affinor structure. Consequently, we investigate infinitesimal [Formula: see text]-planar and particularly geodesic transformations of manifolds with non-symmetric linear connection and find necessary and sufficient conditions for the existence of these transformations.

3D geochemical modeling for subsurface targets of Dashui Au deposit in Western Qinling (China)

Three dimensional (3D) geochemical modeling was used to identify deposit scale exploration targets combining the geological (110 boreholes and 6 tunnels with 709 assay borehole samples) dataset and 374 litho-geochemical samples in the Dashui Au deposit (China). The Dashui Au deposit, located in the Western Qinling orogenic belt, is a magmatic-hydrothermal type large Au deposit discovered in recent years. The methodologies are summarized as follows: (1) the 3D geological bodies (ore bodies, granitic stocks/dikes, fault fractures, and alteration zones) modeling using GOCAD software. (2) 3D deposit-scale geological modeling in mining zones combination the deposit-scale plane maps, cross sections, longitudinal sections, tunnel planes, borehole datasets, and the range of 3D model is 2.3 × 0.55 × 0.6 km3. (3) The ordinary kriging and discrete smooth interpolation (DSI) interpolation methods were used to analyze the distribution trend of Au grade features combined with borehole and the primary litho-geochemical halos dataset, and the Cu/Pb ratio model is analyzed and constructed by metallogenic model/rule using primary geochemical halos data with DSI interpolation method. (4) The concentration-volume (C–V) fractal method is used to analyze the anomaly classification of grade and Cu/Pb ratio in 3D space using GeoCube software. The results show as follows: (1) the high Au grade value zones are associated with ore-bearing intrusion and the NW- and NE-trending fault structures; (2) the high Au value blocks of Cu/Pb ratio can be used to delineate ore-bearing magmatic fluids channel and mineralization zones; (3) the integrated exploration criteria are Cu/Pb ratio, the NE- and NW-trending faults, and intrusions in the Dashui deposits, the potential targets at depth are located in the southwest and southeast zones under the mining levels of No. 2, No. 170, No. 20, and No. 35 Au orebodies, and the Au resource is >172 t (the depth is <1000 m) in the study area.

An efficient alternative to magnetic resonance (MR) composite relaxation (R2-R1ρ) mapping in human knee cartilage study at 3T

An efficient alternative to magnetic resonance (MR) composite relaxation (R2-R1ρ) mapping in human knee cartilage study at 3T

Effect of atomic ordering on the magnetic anisotropy of single crystal Ni80Fe20

We investigate the effect of atomic ordering on the magnetic anisotropy of Ni80Fe20 at.% (Py). To this end, Py films were grown epitaxially on MgO(001) using dc magnetron sputtering (dcMS) and high power impulse magnetron sputtering (HiPIMS). Aside from twin boundaries observed in the latter case, both methods present high quality single crystals with cube-on-cube epitaxial relationship as verified by the polar mapping of important crystal planes. However, X-ray diffraction results indicate higher order for the dcMS deposited film towards L12 Ni3Fe superlattice. This difference can be understood by the very high deposition rate of HiPIMS during each pulse which suppresses adatom mobility and ordering. We show that the dcMS deposited film presents biaxial anisotropy while HiPIMS deposition gives well defined uniaxial anisotropy. Thus, higher order achieved in the dcMS deposition behaves as predicted by magnetocrystalline anisotropy i.e. easy axis along the [111] direction that forced in the plane along the [110] direction due to shape anisotropy. The uniaxial behaviour in HiPIMS deposited film then can be explained by pair ordering or more recent localized composition non-uniformity theories. Further, we studied magnetoresistance of the films along the [100] directions using an extended van der Pauw method. We find that the electrical resistivities of the dcMS deposited film are lower than in their HiPIMS counterparts verifying the higher order in the dcMS case.

Incremental 3D Cuboid Modeling with Drift Compensation.

This paper presents a framework of incremental 3D cuboid modeling by using the mapping results of an RGB-D camera based simultaneous localization and mapping (SLAM) system. This framework is useful in accurately creating cuboid CAD models from a point cloud in an online manner. While performing the RGB-D SLAM, planes are incrementally reconstructed from a point cloud in each frame to create a plane map. Then, cuboids are detected in the plane map by analyzing the positional relationships between the planes, such as orthogonality, convexity, and proximity. Finally, the position, pose, and size of a cuboid are determined by computing the intersection of three perpendicular planes. To suppress the false detection of the cuboids, the cuboid shapes are incrementally updated with sequential measurements to check the uncertainty of the cuboids. In addition, the drift error of the SLAM is compensated by the registration of the cuboids. As an application of our framework, an augmented reality-based interactive cuboid modeling system was developed. In the evaluation at cluttered environments, the precision and recall of the cuboid detection were investigated, compared with a batch-based cuboid detection method, so that the advantages of our proposed method were clarified.

Extension to 3-Colorable Triangulations

In order to attack some problems in computational geometry, Hoffmann and Kriegel [SIAM J. Discrete Math., 9 (1996), pp. 210--224] considered the problem of whether a plane map can be extended to a ...

Бифуркация рождения замкнутой инвариантной кривой в однопараметрическом семействе квадратичных отображений плоскости

Бифуркация рождения замкнутой инвариантной кривой в однопараметрическом семействе квадратичных отображений плоскости

Landau-Bloch theorems for bounded biharmonic mappings

We determine coefficient bounds for bounded planar biharmonic mappings and generalize the Landau-Bloch univalency theorems for such bounded biharmonic functions. The univalence radii presented here improve many related results published to date, including the most recent one [Complex Var. Elliptic Equ. 58(12) (2013), 1667-1676] and are sharp in some given cases.

Application of an extended van der Pauw method to anisotropic magnetoresistance measurements of ferromagnetic films

We demonstrate anisotropic resistivity measurements using the extended van der Pauw (vdP) method in ferromagnetic Ni80Fe20 (Py) films. We apply it to measure anisotropic magnetoresistance (AMR) and compare the results of the vdP method with the more conventional Hall-bar method along the hard and easy axis of the film and show that the vdP method gives more reliable AMR result. For instance the AMR result along the hard and easy axis of the film are in close agreement. Further, we applied the vdP method to study AMR in a series of Py films with thicknesses ranging between 10–250 nm. The films were grown by sputtering deposition at an angle with respect to the substrate normal and with an in situ magnetic field, both conditions assisting in the definition of in-plane uniaxial anisotropy. The microstructure of Py films was characterized using x-ray reflectivity, diffraction and polar mapping of (1 1 1) planes. We detected no off-normal texture and negligible surface roughness, which indicates that self-shadowing is not dominating in our growth. Yet the films have well defined uniaxial anisotropy. Abrupt changes in the average resistivity versus film thickness were observed, which cannot be explained by the models accounting for the thickness and grain size but strongly correlate with the changes in (1 1 1) texture in the films. We compared our results with the literature and show that independent of growth method, substrate and deposition temperature, the AMR value presents a saturation behavior with thickness at about 100 nm.

All one-term tight descriptions of 3-paths in normal plane maps without [formula omitted

A description of paths on three vertices (3-paths) in normal plane maps (NPMs) is tight if no parameter of this description can be improved and no term dropped.We know that there are precisely seven tight descriptions of 3-paths in triangle-free NPMs: {(5,3,6),(4,3,7)}, {(3,5,3),(3,4,4)}, {(5,3,6),(3,4,3)}, {(3,5,3),(4,3,4)}, (5,3,7), (3,5,4), and (5,4,6).The problem of describing all tight descriptions of 3-paths in arbitrary NPMs is widely open. It is only known that there are precisely three tight descriptions consisting of one triplet of restrictions on the vertex degrees of 3-paths: (5,∞,6), (5,10,∞), or (10,5,∞).If NPMs are allowed to have K4−e as subgraphs, then it can happen that every 3-path goes through a vertex of unbounded degree: for example, we delete every other horizontal edge from a double 2n-pyramid. This fact explains the appearance of ∞ in the above description.The purpose of this paper is to prove that there are precisely four tight one-term descriptions of 3-paths in normal plane maps without K4−e: (10,5,20), (10,6,15), (5,10,15), and (5,15,6). In particular, the same description holds for 3-polytopes.

The Landau–Bloch type theorems for planar harmonic mappings with bounded dilation

In this paper, we first establish the sharp version of Landau–Bloch type theorem for holomorphic functions with bounded derivative by applying Schwarz–Pick lemma. Next, we derive the sharp version of Landau–Bloch type theorem for planar harmonic mappings with bounded dilation by applying geometric method. Furthermore, all extremal functions are also provided under suitable restriction.

On a problem of Bharanedhar and Ponnusamy involving planar harmonic mappings

In this paper, we give a negative answer to a problem presented by Bharanedhar and Ponnusamy concerning univalency of a class of harmonic mappings. More precisely, we show that for all values of the involved parameter, this class contains a non-univalent function. Moreover, several results on a new subclass of close-to-convex harmonic mappings, motivated by the work of Ponnusamy and Sairam Kaliraj, are obtained.

Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with Lp data and applications to nonlinear elasticity

We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function f∈Lp with p>1. More precisely, for any 1<q<(p+1)/2 we construct W1,q-bi-Sobolev maps with identity boundary conditions; for f∈L∞, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.

On periodic solutions of a second-order, time-delayed, discontinuous dynamical system

This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.

A Pure Mathematical Proof of the Four Color Problem

As stated originally the four-color problem asked whether it is always possible to color the regions of a plane map with four colors such that regions which share a common boundary (and not just a point) receive different colors. In the long and arduous history of attacks to prove the four colortheorem many attempts came close, but what finally succeeded in the Apple-Haken proof of 1976 and also in the recent proof of Robertson, Sanders, Seymour and Thomas 1997 was a combination of some old ideas and the calculating powers of modern-day computers. Thirty years after the original proof, the situation is still basically the same,no pure mathematical proof is in sight.Now I give in my paper such a pure mathematical proof.

Seismic sedimentology study of braided river — A case study in He8 and Shan1 members of Permian, Sulige western region, Ordos Basin

The He8 and Shan1 members deposited by braided river are major pay zones of the Sulige western region, Ordos Basin. In these two members, the P-wave impedance ranges of gas-bearing reservoir and the surrounding lithologies are highly overlapped. The gas-bearing reservoir cannot be feasibly recognized by the conventional poststack seismic data. In addition, the seismic reflection of the Shan1 member showing mid-weak amplitude distributes discontinuously within the whole seismic survey, leading to difficulties in tracking seismic events and delineating the plane map of reservoir distribution. To solve the problems above, a seismic sedimentology study was conducted in the study area. First, the seismic reflection characterization of braided river was studied according to its sedimentary model. Also, the amplitude variation with offset (AVO) seismic data were used to identify gas-bearing reservoir because the Poisson’s ratios of gas-bearing reservoir and the surrounding rocks are distinctly different. The AVO attribute volume is therefore converted into the lithologic volume, which is useful for the subsequent lithologic interpretation work. The isochronous surface of the Shan1 member seismic reflection can be achieved with the aid of a stratal slicing technique. The practice indicates the feasibility of lithologic converting of the AVO attribute volume. The gas-bearing reservoir of the He8 and Shan1 members can be precisely predicted, and the evolutional history and plane-distribution character of the gas-bearing reservoir are clearly delineated by the stratal slices with isochronous significance.