Non-convex Curves Research Articles

Partial shading mitigation in PV arrays through dragonfly algorithm based dynamic reconfiguration

The susceptibility of the PV array towards partial shading has raised a major reliability concern for efficient power generation. The partial shading forces the arrays to generate lower power along with forming non-convex curves causing complicated operation of power tracking algorithms. Hence, to reduce the losses, various conventional configurations and reconfiguration techniques exist with vulnerabilities in terms of reliable power enhancement and complexity. In this paper, a highly reliable, less complex, and fast array reconfiguration based on the Dragonfly algorithm (DA) optimization with a higher power enhancement capability, lower computational time, and hasty convergence is proposed for unwanted shading losses reduction in arrays. The effectiveness of the proposed reconfiguration is evaluated against conventional configurations and three pre-existing reconfiguration techniques under various artificial shading cases via power generation, losses, and efficiencies using simulation and experimental analysis for 3 × 3 and 9 × 9 arrays. From the conducted analysis, it has been established that the DA reconfiguration has 22%, 10.10%, 15.36%, 5.85%, 2.95%, 2.55%, and 1.07% higher power generation than the conventional configurations, electrical reconfiguration, SD-PAR, Sudoku, GA, PSO, and EAR respectively with reduced switches counts.

A unified approach towards computing Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves using touching discs

This paper proposes a unified approach towards computing geometry structures viz. Voronoi diagram, medial axis, Delaunay graph and α-hull of planar closed curves. It initially presents an algorithm for computing the Voronoi diagram of a set of planar freeform closed curves without approximating the curves using points, lines or biarcs. The algorithm starts by computing the minimum antipodal discs (MADs) for all pairs of curves and these MADs are systematically processed to identify all branch points. The key feature of the algorithm is that it computes a branch point without computing any of the bisectors a priori. Local computations of Voronoi segments are then done using the identified pairs of the segments of curves. The theoretical foundation of the algorithm has been first laid for a set of convex curves and then extended to non-convex curves. It has also been shown that the developed algorithm for the Voronoi diagram can also be used to compute related structures such as medial axis, Delaunay graph and α-hull. They have also been addressed without computing Voronoi edges/segments. Results of the implementation have been provided along with a detailed discussion of the algorithm.

Density of bivariate homogeneous polynomials on non-convex curves

The density of bivariate homogeneous polynomials is studied in the space of continuous functions on the $L_{\alpha }$ sphere given by $K_{\alpha }:=\{(x,y)\in \mathbb R^2: |x|^{\alpha }+|y|^{\alpha }= 1\}, \; \alpha >0.$ The goal is to approximate functions $f\in C(K_{\alpha })$ by sums of the form $h_{2n}+h_{2n+1}$, where $h_{2n},h_{2n+1}$ are bivariate homogeneous polynomials of degree $2n$ and $2n+1$, respectively. It is known that whenever $\alpha \geq 1$, i.e., when $K_{\alpha }$ is convex, a Weierstrass-type approximation result holds, namely for every $f\in C(K_{\alpha })$ there are homogeneous polynomials $h_{2n},h_{2n+1}$ for which $f=\lim _{n\rightarrow \infty }(h_{2n}+h_{2n+1})$ uniformly on $K_{\alpha }$. In this note the problem is solved in the non-convex case $0<\alpha <1$. It is verified that $f(x,y)$ is a uniform limit on $K_{\alpha }$ of sums $h_{2n}+h_{2n+1}$ of homogeneous polynomials if and only if $f(\pm 1,0)=f(0,\pm 1)=0.$ The theorem is proven in an equivalent form: $g\in C(\mathbb R)$ is a uniform limit as $n\rightarrow \infty$ of weighted polynomials $(1+|t|^{\alpha })^{-n/\alpha }p_{n}(t)$ (degree $p_{n}\leq n$) if and only if $g(0)=g(\infty )=g(-\infty )=0.$

Robust Semi-Automatic Identification of Compliantly Coupled Two-Mass Systems

We present a method to identify the physical parameters of rotational systems with compliant load coupling in a robust manner. To ensure sufficient excitation, a pseudo random binary signal (PRBS) is automatically parameterized based on the noise level in the available feedback signal. To identify the plant parameters, we rely on a two step procedure: Firstly, the user is asked to mark distinct features in the non-parametric estimate of the frequency response. These features serve to calculate initial parameter guesses. Secondly, a non-convex curve fitting problem is solved to refine the parameter estimates. Experimental results verify that the method allows to robustly identify parameters even for small amounts of raw data resulting from short measurement times.

Algorithm for computing positive α-hull for a set of planar closed curves

In this paper, the computation of positive α-hull for a set of planar closed C1-continuous curves has been addressed without sampling the curves into point-sets or polylines. Positive α-hull, so far, has been computed only for a set of points, using the farthest Delaunay triangulation, a dual of farthest Voronoi diagram. However, Delaunay triangulation does not exist for a set of curved boundaries and the computation of Voronoi diagram for such a set is still a topic of active research. The key insight behind our algorithm is to merge adjacent pairs of curves on the convex hull into a set of triplets. Along with a directed-cyclic graph and a R-List (list of radii), α-neighbours are derived. Using the constraint equations, α-discs are then computed. The algorithm is first provided for convex non-intersecting closed curves, but later explained how it can be generalized for non-convex curves. We show that the algorithm has time complexity of O(n2) time where n is the number of curves, which leads to a practical implementation with a reasonable running time in seconds for a few dozen curves. By directly operating on the curves, our method is both robust and accurate thus avoiding the problems that arise on polyline/point-set approximations of the curve networks.

On some simple examples of non-parabolic curve flows in the plane

We discuss several examples of non-parabolic curve flows in the plane. In these flows, the speed functions do not involve the curvature at all. Although elementary in nature, there are some interesting properties. In particular, certain non-parabolic flows can be employed to evolve a convex closed curve to become circular or to evolve a non-convex curve to become convex eventually, like what we have seen in the classical curve shortening flow (parabolic flow) by Gage and Hamilton (J Differ Geom 23:69–96, 1986), Grayson (J Differ Geom 26:285–314, 1987).

Drawing Area-Proportional Euler Diagrams Representing Up To Three Sets

Area-proportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact area-proportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn-2 and Venn-3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler-3 diagrams in an area-proportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where nonconvex curves are necessary, our method draws an appropriate diagram using nonconvex polygons. Thus, we are now always able to automatically visualize data for up to three sets.

The maximal Hilbert transform along nonconvex curves

The Hilbert transform along curves is defined by the principal value integral. The pointwise existence of the principal value Hilbert transform can be educed from the appropriate estimates for the corresponding maximal Hilbert transform. By using the estimates of Fourier transforms and Littlewood-Paley theory, we obtain L p -boundedness for the maximal Hilbert transform associated to curves (t,P(γ(t))), where 1<p<∞, P is a real polynomial and γ is convex on [0,∞). Then, we can conclude that the Hilbert transform along curves (t,P(γ(t))) exists in pointwise sense.MSC:42B20, 42B25.

Closed-Loop Measurements of Iso-Response Stimuli Reveal Dynamic Nonlinear Stimulus Integration in the Retina

Neurons often integrate information from multiple parallel signaling streams. How a neuron combines these inputs largely determines its computational role in signal processing. Experimental assessment of neuronal signal integration, however, is often confounded by cell-intrinsic nonlinear processes that arise after signal integration has taken place. To overcome this problem and determine how ganglion cells in the salamander retina integrate visual contrast over space, we used automated online analysis of recorded spike trains and closed-loop control of the visual stimuli to identify different stimulus patterns that give the same neuronal response. These iso-response stimuli revealed a threshold-quadratic transformation as a fundamental nonlinearity within the receptive field center. Moreover, for a subset of ganglion cells, the method revealed an additional dynamic nonlinearity that renders these cells particularly sensitive to spatially homogeneous stimuli. This function is shown to arise from a local inhibition-mediated dynamic gain control mechanism.

The DNA Inequality in non-convex regions

Abstract A simple plane closed curve Γ satisfies the DNA Inequality if the average curvature of any closed curve contained inside Γ exceeds the average curvature of Γ. In 1997 Lagarias and Richardson proved that all convex curves satisfy the DNA Inequality and asked whether this is true for some non-convex curve. They conjectured that the DNA Inequality holds for certain L-shaped curves. In this paper, we disprove this conjecture for all L-shapes and construct a large class of non-convex curves for which the DNA Inequality holds. We also give a polynomial-time procedure for determining whether any specific curve in a much larger class satisfies the DNA Inequality.

$L^p$-boundedness for the Hilbert transform and maximal operator along a class of nonconvex curves

Some sufficient conditions on a real polynomial P and a convex function γ are given in order for the Hilbert transform and maximal operator along (t, P(γ(t))) to be bounded on L p , for all p in (1, ∞), with bounds independent of the coefficients of P. The same conclusion is shown to hold for the corresponding hypersurface in R d+1 (d > 2) under weaker hypotheses on γ.

Affine-Distance symmetry sets

The affine distance symmetry set (ADSS) of a plane curve is an affinely invariant analogue of the euclidean symmetry set (SS) [7], [6]. We list all transitions on the ADSS for generic 1-parameter families of plane curves. We show that for generic convex curves the possible transitions coincide with those for the SS but for generic non-convex curves, further transitions occur which are generic in 1-parameter families of bifurcation sets, but are impossible in the euclidean case. For a non-convex curve there are also additional local forms and transitions which do not fit into the generic structure of bifurcation sets at all. We give computational and experimental details of these.

Modeling the Strategic Effects of Risk and Perceptions in Linkage Politics

We develop a new linkage model for multiple issue politics and negotiations, one that extends previous single and multidimensional work by Bueno de Mesquita, Stokman, Morgan, and Morrow by explicitly capturing risk in a spatial multidimensional analysis. In order to address the critiques of previous linkage models as cooperatively based, we allow risk to vary simultaneously across issues, which yields non-convex indifference curves. Consequently, our model incorporates a new perceptive and conflictual component in multiple issue contexts. This approach allows the possibility of misjudgment and misperceptions in behavior. We show that linkage solutions in a political context may or may not be taken advantage of by the participants, nor do they need be Pareto optimal. Besides more accurately modeling the negotiation process, these valuable insights yield strategies for unilateral maximization of linkages not captured by previous linkage approaches.

Approximation methods for non-convex curves

Based on the triangle sandwich and block sandwich methods, two approximation methods for non-convex curves are considered in this paper. Numerical algorithms corresponding to these methods are developed and quadratic convergence properties of two methods in the original space are established. Application is given to the approximation of an efficient frontier of a bi-criteria optimization problem.

Even Fitting Closed Curves: 2-D Algorithm and Assembly Applications

In certain assemblies there exists a requirement for the evenness of the gap between a “peg” and a “hole.” In these cases, when given a “peg” and a “hole” that must be fitted the task is to find the positioning of the “peg” such that the gap will be as even as possible. In this paper the even fitting requirement is formulated for two planar closed curves. The optimal fitting demand is found to be equivalent to the maximization of the minimal distance between the curves. Both convex and nonconvex curves with “tolerances” are considered and optimized. It is proven that for certain shape characteristics, fitting for evenness can be further improved by minimization of the Haussdorff distance between the curves. We demonstrate the algorithm by fitting an automobile door into a frame. The best door positioning is found in such a way that the variation gap is minimized according to the selected metrics.

On the affine heat equation for non-convex curves

In this paper, we extend to the non-convex case theaffine invariant geometric heat equationstudied by Sapiro and Tannenbaum for convex plane curves. We prove that a smooth embedded plane curve will converge to a point when evolving according to this flow. This result extends the analogy between the affine heat equation and the well-knownEuclidean geometric heat equation.

The Hilbert transform and maximal function along nonconvex curves in the plane

In this paper we study the Hilbert transform and maximal function related to a curve in R2.

On the performance of a discriminant function

In two-class discriminant problems, objects are allocated to one of the two classes by means of threshold rules based on discriminant functions. In this paper we propose to examine the quality of a discriminant functiong in terms of its performance curve. This curve is the plot of the two misclassification probabilities as the thresholdt assumes various real values. The role of such performance curves in evaluating and ordering discriminant functions and solving discriminant problems is presented. In particular, it is shown that: (i) the convexity of such a curve is a sufficient condition for optimal use of the information contained in the data reduced byg, and (ii)g with non-convex performance curve should be corrected by an explicitly obtained transformation.

Optimal destabilizing speculation or why the optimum tariff may oscillate

This paper uses general equilibrium tools to explore the issue of destabilizing speculation in the context of market power. We show how an individual or a nation may optimally use price destabilization to raise its welfare and show that the nonspeculator does not necessarily suffer. The cases considered are 1) inflection points in the foreign offer curve, 2) an upward sloping foreign excess demand curve which crosses the price axis more than once, 3) a nonconvex foreign social trade indifference curve, and 4) reversed transfers.

Proof that the Two-Dimensional Shape of Minimum Surface Free Energy is Convex

Published proofs that the Wulff construction yields the shape of minimum surface free-energy for a crystal of fixed volume (or area in the two-dimensional case) are all based on the tacit but unproven assumption that the equilibrium shape is convex. In the present work, this assumption is justified for the two-dimensional case by establishing from a special sequence of constructions that corresponding to every nonconvex curve of line (surface) energy E, there is a convex curve enclosing the same area but having a line energy less than E. Therefore the two-dimensional equilibrium form must be convex and the existing proofs of the Wulff construction in two dimensions are vindicated; unfortunately the method of proof cannot be extended to three dimensions.