Triangle Comparison Research Articles

A Finite Topological Type Theorem for Open Manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume

Abstract In this paper, we present finite topological type theorems for open manifolds with non-negative Ricci curvature, under almost maximal local rewinding volume. Unlike previous related research, our theorems remove the constraints of sectional curvature or conjugate radius, which were crucial additional assumptions on metric regularity in prior results. Notably, our settings do not necessarily satisfy a triangle comparison of Toponogov type. In fact, the method we adopt also extends to many previous related studies.

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds.

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54 (2018), no. 3, 399-447) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts such as timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature boundedbelow.

The triangle comparison theorem for spheres

The triangle comparison theorem for spheres

Necessary and sufficient conditions for a triangle comparison theorem

We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.

Comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces

We give an analogue of triangle comparison for Kähler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov–Hausdorff limits. We discuss tangent cones and singular Kähler spaces.

Application of tetrad testing to the evaluation of blinding strategies for ancillary supplies used in controlled clinical trials.

Ensuring verum and placebo cannot be visually distinguished from each other is a critical aspect of blinded controlled clinical trials. Our objective was to propose a rational approach to the visual evaluation of placebo matching candidates. Verum and placebo samples were prepared in clear clinical ancillary supplies (intravenous bags, syringes and administration lines) covered at different levels using opaque sleeves. Triangle and tetrad tests, two sensory discriminative testing methods widely used in the food industry, were applied to assess visual differences between verum and placebo. Triangle and tetrad test results allowed defining the level of opaque coverage required to ensure blinding for three biological drug molecules of therapeutic interest. While the limited number of panelists did not allow a statistically sound comparison of triangle and tetrad test methodologies, tetrad test has a theoretical higher power than triangle test, meaning fewer panelists are needed to reach the same statistical conclusion. Tetrad test offers a rational approach to define a blinding strategy for ancillary supplies used in a controlled clinical trial.

Harmonic branched coverings and uniformization of CAT(k) spheres

Abstract Let S be a surface with a metric d satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into ( S , d ) {(S,d)} is a branched covering. As a consequence, if ( S , d ) {(S,d)} is homeomorphically equivalent to the 2-sphere 𝕊 2 {\mathbb{S}^{2}} , then it is conformally equivalent to 𝕊 2 {\mathbb{S}^{2}} .

Cohomogeneity one Alexandrov spaces in low dimensions

Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions 5, 6 and 7. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions at most 7.

A Comparison of Triangle, Diamond and Pentagon Fraud Models in Fraud Risk Assessment

A Comparison of Triangle, Diamond and Pentagon Fraud Models in Fraud Risk Assessment

Finiteness and realization theorems for Alexandrov spaces with bounded curvature

Every closed Alexandrov space with a lower and upper curvature bound (in the triangle comparison sense) is a space of bounded curvature (in the sense of Berestovskii and Nikolaev). These spaces are topological manifolds and admit a canonical smooth structure such that, in its charts, the metric is induced by a Riemannian metric tensor with low regularity. In this note, we show that there are finitely many diffeomorphism types of closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound, provided $$2\le n\le 6$$. We also show that an analogous diffeomorphism finiteness result for such spaces does actually hold for general dimension n if, in addition, the second homotopy group is finite. Moreover, we prove that all closed, simply connected n-dimensional Alexandrov spaces with bounded curvature and upper diameter bound can be realized as quotients of finitely many closed, simply connected smooth manifolds with finite second homotopy group by some free torus action. These results extend well-known finiteness and realization theorems in Riemannian geometry to the more general setting of Alexandrov geometry. Indeed, our results may be simply viewed as concrete instances of the more general phenomenon that virtually any result that holds for a given class of closed Riemannian manifolds with bounded curvature and diameter may be extended to the Lipschitz closure of this class.

Modified SVPWM Algorithm for Three Level VSI With Synchronized and Symmetrical Waveforms

To generate the required reference vector than triangle comparison based PWM techniques for three-level inverters the space vector based PWM (SVPWM) strategies contain broader choice of switching sequences. This space vector based PWM technique involves in various steps. These steps are computationally exhaustive. The SVPWM has been used in three phase inverter control system. The center-aligned PWM is the most effective way for the Microprocessor Control Unit implementation of the SVPWM, because it can easily generate the center aligned PWM of the multilevel inverters for generation of the signal of space vector pulse width modulation (SVPWM), this concept brings out the method. The inverter leg switching times are generated by this algorithm and middle vector switching times are centered in a sampled interval. The proposed algorithm does not require any sector identification. And it reduces the computational time as a result. The adjacent voltage space vectors are forming the small triangles it is called sectors. Multilevel converters can meet the increasing demand of power ratings and power quality associated with reduced harmonic distortion and lower electromagnetic interference. Furthermore to optimize switching waveforms, space vector pulse-width modulation algorithms offer great flexibility among them. Finally the results are verified through MATLAB/SIMULINK

Lorentzian length spaces

We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way, we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions

We use Toponogov’s triangle comparison theorem from Riemannian geometry along with quantitative scale oriented variants of classical propagation of singularities arguments to obtain logarithmic improvements of the Kakeya–Nikodym norms introduced in [22] for manifolds of nonpositive sectional curvature. Using these and results from our paper [4] we are able to obtain log-improvements of $L^p (M)$ estimates for such manifolds when $2 \lt p \lt \frac{2(n+1)}{n-1}$. These in turn imply $(\log \lambda)^{\sigma_n} , \sigma_n \approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of the estimates of the second author and Zelditch [28], and using a result from Hezari and the second author [18], under this curvature assumption, we are able to improve the lower bounds for the size of nodal sets of Colding and Minicozzi [12] by a factor of $(\log \lambda)^{\mu}$ for any $\mu \lt \frac{2(n+1)^2}{n-1}$, if $n \geq 3$.

Comparison of triangle and tetrad discrimination methodology in an applied manner

Choosing a discrimination test can involve numerous factors, one of which is statistical power. The tetrad test has been shown to possess statistical advantages over the more traditional triangle method. However, these statistical advantages may not be present when the effect size decreases by more than 1/3. The frequency of large decreases in effect size, as well as other measures of test performance has not been characterized. In this study, over thirty products were tested using both triangles and tetrads in order to compare the two methods. The products tested ranged from canned vegetables and fresh fruits to deli meats and baked goods. After testing, inconsistencies were found within and across product categories. Only 16% of the tests were found to end with a different conclusion regarding a statistical difference (p < 0.05). In six of the experiments, the triangle test showed a significantly higher effect size than the tetrad. In eight of the experiments, the reduction in effect size for the tetrad led to no power advantage of the tetrad test. Participants also noted that the product being tested affected their impression of test difficulty in multiple experiments. This study creates a functional comparison of tetrad and triangle testing and quantifies the frequency in which the tetrad method effect size decreases by more than 1/3, leading to decreased statistical power.

Replacing the lower curvature bound in Toponogov's comparison theorem by a weaker hypothesis

Toponogov's triangle comparison theorem and its generalizations are important tools for studying the topology of Riemannian manifolds. In these theorems, one assumes that the curvature of a given manifold is bounded from below by the curvature of a model surface. The models are either of constant curvature, or, in the generalizations, rotationally symmetric about some point. One concludes that geodesic triangles in the manifold correspond to geodesic triangles in the model surface which have the same corresponding side lengths, but smaller corresponding angles. In addition, a certain rigidity holds: Whenever there is equality in one of the corresponding angles, the geodesic triangle in the surface embeds totally geodesically and isometrically in the manifold. In this paper, we discuss a condition relating the geometry of a Riemannian manifold to that of a model surface which is weaker than the usual curvature hypothesis in the generalized Toponogov theorems, but yet is strong enough to ensure that a geodesic triangle in the manifold has a corresponding triangle in the model with the same corresponding side lengths, but smaller corresponding angles. In contrast, it is interesting that rigidity fails in this setting.

Stability of stationary wave maps from a curved background to a sphere

We study time and space equivariant wave maps from $M\times\mathbb{R}\rightarrow S^2,$ where $M$ is diffeomorphic to a two dimensional sphere and admits an action of $SO(2)$ by isometries. We assume that metric on $M$ can be written as $dr^2+f^2(r)d\theta^2$ away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where $M$ is isometric to the standard sphere (considered by Shatah and Tahvildar-Zadeh [34]), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.

Advanced spatio-temporal filtering techniques for photogrammetric image sequence analysis in civil engineering material testing

The paper shows advanced spatial, temporal and spatio-temporal filtering techniques which may be used to reduce noise effects in photogrammetric image sequence analysis tasks and tools. As a practical example, the techniques are validated in a photogrammetric spatio-temporal crack detection and analysis tool applied in load tests in civil engineering material testing. The load test technique is based on monocular image sequences of a test object under varying load conditions. The first image of a sequence is defined as a reference image under zero load, wherein interest points are determined and connected in a triangular irregular network structure. For each epoch, these triangles are compared to the reference image triangles to search for deformations. The result of the feature point tracking and triangle comparison process is a spatio-temporally resolved strain value field, wherein cracks can be detected, located and measured via local discrepancies. The strains can be visualized as a color-coded map. In order to improve the measuring system and to reduce noise, the strain values of each triangle must be treated in a filtering process. The paper shows the results of various filter techniques in the spatial and in the temporal domain as well as spatio-temporal filtering techniques applied to these data. The best results were obtained by a bilateral filter in the spatial domain and by a spatio-temporal EOF (empirical orthogonal function) filtering technique.

ON THE GEOMETRY OF LORENTZ SPACES AS A LIMIT SPACE

Abstract. In this paper, we prove that there is no branch point in theLorentz space (M,d) which is the limit space of a sequence {(M α ,d α )}ofcompact globally hyperbolic interpolating spacetimes with C α± -propertiesand curvature bounded below. Using this, we also obtain that every max-imal timelike geodesic in the limit space (M,d) can be expressed as thelimit curve of a sequence of maximal timelike geodesics in {(M α ,d α )}.Finally, we show that the limit space (M,d) satisfies a timelike trian-gle comparison property which is analogous to the case of Alexandrovcurvature bounds in length spaces. 1. IntroductionLorentzian Gromov-Hausdorff theory was first introduced by J. Noldus in[3] and the notion of Gromov-Hausdorff (GH) distance d GH ((M,g),(N,h))between two compact, globally hyperbolic (CGH),interpolating spacetimes(M,d) and (N,h) was defined in a similar way as in the Riemannian case.Specifically, we say (M,d) and (N,h) ǫ-close if and only if there exist map-pings ψ: M→ N, ξ: N→ Msuch that| d

TOPOLOGY OF COMPLETE FINSLER MANIFOLDS WITH RADIAL FLAG CURVATURE BOUNDED BELOW

We recently established a Toponogov type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution (arXiv:1205.3913). In this article, as its applications, we prove the finiteness of topological type and a diffeomorphism theorem to Euclidean spaces.

Fpga Controlled Cascaded Mli With Advanced Multilevel Strategies For Harmonic Reduction

This paper focused on the implementation of Cascaded H-bridge multilevel inverter fed induction motor by using the modulating technique of OVPWM with U-type carrier to the Five-level inverter. The PWM switching signals based on offset voltage injected in sine reference with U-type carrier been generated. The multilevel inverter fed induction motor drive with SPWM and proposed strategies are performed and results are analyzed. Xilinx FPGA is a programmable logic device developed by Xilinx, which is considered as an efficient hardware for rapid prototyping. In addition to Xilinx FPGA, MATLAB/Simulink software was used for simulation and verification of the proposed circuit before implementation. Hardware is implemented for a single phase cascaded five level inverter with the proposed modulating technique by using FPGA ,which drives a single phase induction motor of 0.5 H.P Simulation results and hardware results are compared. It is justified that the proposed modulation technique can be recommended to three phase five level Cascaded H-bridge inverter fed three phase induction motor for better performance with reduced harmonics at output. Index Terms- Multilevel concept, Cascade Multilevel inverter, FPGA, Multi level carrier signals, Pulse width modulation, Total Harmonic Distortion. I. INTRODUCTION Modulation in multilevel inverters has recently been widely investigated, but the overvoltage caused by multilevel modulation has not been discussed thoroughly. Several modulation methods, such as space vector pulse width modulation (SVPWM), space vector control (SVC), duty cycle modulation (DCM) and several sine- triangle comparison modulation variations have been presented, for example, in the literature given by Wei.S et.al.(19) and Naumanen et.al.(24). All the modulation methods, except for the SVC, are based on pulse width modulation.. The output voltage waveform of the SVC resembles staircase that approximates the shape of the reference wave. An example of a multilevel inverter using SVC is presented by Kouro.S et.al. (22). Power electronic switches, especially IGBTs have become faster in terms of turn-on and turn-off times, which has led to the high dv/dt of the edges of the PWM voltage. The short rise time compared with the propagation delay causes an oscillating overvoltage at the motor terminals. The propagation delay is caused by the length and insulation of the cable, and with the impedance a mismatch between the cable and the motor an overvoltage occurs. As dv/dt becomes higher, an overvoltage will occur at even shorter cable lengths. This phenomenon has been thoroughly reported by Persson et.al.(6) and Skibinski(9). The classification of PWM techniques or modulating techniques can be classified as shown in Fig.1. Wang et.al (15) in his investigation on sine triangle modulation technique stated that SPWM is the most commonly used modulating technique suffers from certain drawbacks like low fundamental output voltage. The Modified Reference Modulation techniques that offer improved performances as discussed by M.H.Rashid (20) and are as Trapezoidal ,Stair case ,Stepped , Harmonic injected, Space Vector PWM (SVPWM) and Offset voltage injected in reference(OVPWM) The above PWM techniques are applicable to three-phase inverters. However the last three techniques are commonly used for three-phase inverters.