Star-shaped Polygon Research Articles

Ricci limit flows and weak solutions

In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner–Lott, Haslhofer–Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler’s precompactness theorem, as well as every singular Ricci flow of Kleiner–Lott, is a weak solution in the sense of Haslhofer–Naber. We also generalize all path-space estimates of Haslhofer–Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.

Multi-target detection and sizing of single nanoparticles using an optical star polygon microcavity.

We present a miniaturized single nanoparticle detector that utilizes an optical star polygon microcavity with a 3 µm-radius. The microcavity supports high-quality factor resonant modes, with light localized at the corners of the star-shaped polygon, where the air region is situated. When nanoparticles are positioned at the corners of the microcavity, the light-matter interactions are enhanced. Notably, increasing the number of particles has little effect on the quality factor of the cavity, making it ideal for the simultaneous detection of multiple targets. Our numerical simulations demonstrate the high precision detection of polystyrene nanoparticles with a radius of 3 nm using this method. Furthermore, the size and number of nanoparticles can be determined by utilizing the triangular corners of the cavity as rulers. These findings represent a significant advancement in miniaturized and multi-target simultaneous nanoparticle detection. The proposed detector is expected to have a wide range of applications in various fields, including biomedicine and environmental monitoring.

Computational design of nanoantennas with improved power enhancement capabilities via shape optimization

Computational design and analyses of nanoantennas obtained via surface shape optimization are presented. Starting with a kernel geometry, free deformations are applied on selected surfaces to reach optimal designs that can provide improved power enhancement capabilities at desired frequencies. An in-house implementation of genetic algorithms is efficiently combined with the multilevel fast multipole algorithm developed for accurate solutions of plasmonic problems to construct the effective optimization environment. The geometries obtained via optimization do not only represent optimal shapes within the allowed deformation limits but also reveal certain types of modifications on kernel geometries to improve their performances.

Contour Models for Physical Boundaries Enclosing Star-Shaped and Approximately Star-Shaped Polygons

Abstract Boundaries on spatial fields divide regions with particular features from surrounding background areas. Methods to identify boundary lines from interpolated spatial fields are well established. Less attention has been paid to how to model sequences of connected spatial points. Such models are needed for physical boundaries. For example, in the Arctic ocean, large contiguous areas are covered by sea ice, or frozen ocean water. We define the ice edge contour as the ordered sequences of spatial points that connect to form a line around set(s) of contiguous grid boxes with sea ice present. Polar scientists need to describe how this contiguous area behaves in present and historical data and under future climate change scenarios. We introduce the Gaussian Star-shaped Contour Model (GSCM) for modelling boundaries represented as connected sequences of spatial points such as the sea ice edge. GSCMs generate sequences of spatial points via generating sets of distances in various directions from a fixed starting point. The GSCM can be applied to contours that enclose regions that are star-shaped polygons or approximately star-shaped polygons. Metrics are introduced to assess the extent to which a polygon deviates from star-shapedness. Simulation studies illustrate the performance of the GSCM in different situations.

The New Ag-S Cluster [Ag50S13(StBu)20][CF3COO]4 with a Unique hcp Ag14 Kernel and Ag36 Keplerian-Shell-Based Structural Architecture and Its Photoresponsivity.

In metal nanoclusters (NCs), the kernel geometry and the nature of the surface protecting ligands are very crucial for their structural stability and properties. The synthesis and structural elucidation of Ag NCs is challenging because the zerovalent oxidation state of Ag is very reactive and prone to oxidization. Here, we report the NC [Ag50S13(StBu)20][CF3COO]4 with a hexagonal close-packed (hcp) cagelike Ag14 kernel. A truncated cubic shell and an octahedral shell encapsulate the hcp-layered kernel via an interstitial S2- anionic shell to form an Ag36 Keplerian outer shell of the NC. A theoretical study indicates the stability of this NC in its 4+ charge state and the charge distribution between the kernel and Keplerian shell. The unprecedented electronic structure facilitates its application toward sustainable photoresponse properties. The new insights into this novel Ag NC kernel and Keplerian shell structure may pave the way to understanding the unique structure and developing electronic structure-based applications.

Tensional twist-folding of sheets into multilayered scrolled yarns.

Twisting sheets as a strategy to form functional yarns relies on millennia of human practice in making catguts and fabric wearables, but it still lacks overarching principles to guide their intricate architectures. We show that twisted hyperelastic sheets form multilayered self-scrolled yarns, through recursive folding and twist localization, that can be reconfigured and redeployed. We combine weakly nonlinear elasticity and origami to explain the observed ordered progression beyond the realm of perturbative models. Incorporating dominant stretching modes with folding kinematics, we explain the measured torque and energetics originating from geometric nonlinearities due to large displacements. Complementarily, we show that the resulting structures can be algorithmically generated using Schläfli symbols for star-shaped polygons. A geometric model is then introduced to explain the formation and structure of self-scrolled yarns. Our tensional twist-folding framework shows that origami can be harnessed to understand the transformation of stretchable sheets into self-assembled architectures with a simple twist.

Non-uniform ergodic properties of Hamiltonian flows with impacts

AbstractThe ergodic properties of two uncoupled oscillators, one horizontal and one vertical, residing in a class of non-rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly greater than $1$ . When at least one of the oscillators is unharmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is uniquely ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components coexist. We prove that for almost all partial energies in such segments the motion is uniquely ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

Flow dynamics of laser-induced breakdown at a fuel–oxidizer interface and its effect on ignition

Motivated by experimental observations, we analyze the ignition dynamics following a laser-induced breakdown near a distinct fuel–oxidizer interface using detailed simulations of the flow following a model energy deposition. Cases simulated with a detailed H2-O2 combustion model show the dependence of ignition time, and its success or failure, on the post-breakdown kernel geometry. For asymmetric kernels, ignition can occur by ejections of hot gas that propagate several times the kernel size. The extreme conditions induced near the laser focus lead to near-complete dissociation, allowing heat release by radical recombination to be a primary mechanism by which the ejected gas maintains ignition temperatures as it advects. For kernels close to ehe interface, a pronounced hydrodynamic interaction between the density gradient and expanding kernel results in ignition-suppressing flow that repels hot gas away so completely in some cases that ignition fails. Mechanisms for a wide range of parameters are studied with a four-species reduced combustion model, and we show that this ignition-adverse flow response, particularly salient in hydrogen–oxygen cases, is absent for heavier fuels. For disparate fuel and oxidizer molecular weights, the varied flow response due to the breakdown location is a primary determinant of outcome.

Attraction-convexity and normal visibility

Beacon attraction, or simply attraction, is a movement system whereby a point moves in a free space so as to always locally minimize its Euclidean distance to an activated beacon (also a point). This results in the point moving directly towards the beacon when it can, and otherwise sliding along the edge of an obstacle or being stuck (unable to move). When the point can reach the activated beacon by this method, we say that the beacon attracts the point. In this paper, we study attraction-convex polygons, which are those where every point in the polygon attracts every other point. We find that these polygons are a subclass of weakly externally visible polygons, which are those where every point on the boundary is visible from some point arbitrarily distant (or at infinity on the projective plane). We propose a new class of polygons called normally visible, and show that this is exactly the class of attraction-convex polygons. This alternative characterization of attraction-convex polygons leads to a simple linear-time attraction-convex polygon recognition algorithm. We also give a Helly-type characterization of inverse-attraction star-shaped polygons.

A locally segmented reconstruction method for parallel imaging.

A locally segmented parallel imaging reconstruction method is proposed that efficiently utilizes sensitivity distribution of multichannel receiver coil. A method of locally segmenting a MR signal is introduced to maximize the differences in sensitivity between receiver channels. A 1D Fourier transformation of the undersampled k-space data is performed along the readout direction, which generates a hybrid 2D space. The hybrid space is partitioned into localized segments along the readout direction. In every localized segment, kernels representing relation between adjacent signals are estimated from autocalibration signals, and data at unsampled points are estimated using the kernels. Then, the images are reconstructed from full k-space data that consists of the sampled data and the estimated data at unsampled points. In a computer simulation and in vivo experiments, the locally segmented reconstruction method produced fewer residual artifacts compared to the conventional parallel imaging reconstruction methods with the same kernel geometry. The performance gain of the proposed method comes from maximizing encoding capability of receiver channels, thus resulting in the accurately estimated kernel weights that reflect the relation between adjacent signals. The proposed spatial segmentation method maximally utilizes differences in the sensitivity of receiver channels to reconstruct images with reduced artifacts.

Primitive Points in Rational Polygons

AbstractLet ${\mathcal{A}}$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t{\mathcal{A}}$ is asymptotically $\frac{6}{\unicode[STIX]{x1D70B}^{2}}\text{Area}(t{\mathcal{A}})$ as $t\rightarrow \infty$. We show that the error term is both $\unicode[STIX]{x1D6FA}_{\pm }(t\sqrt{\log \log t})$ and $O(t(\log t)^{2/3}(\log \log t)^{4/3})$. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler’s $\unicode[STIX]{x1D719}(n)$.

Relationship between the vertex-centered linearity-preserving scheme and the lowest-order virtual element method for diffusion problems on star-shaped polygons

As a generalization of the finite element method, the virtual element method (VEM) has made remarkable achievements in the algorithm and analysis of simulating various problems recently. Vertex-centered linearity-preserving scheme (VLPS, Wu et al., 2016) is a linear finite volume scheme which is currently only applied to the numerical simulation of diffusion problems. In this paper, we study the relationship between VLPS and the lowest-order VEM for diffusion problems on star-shaped polygonal meshes from an algebraic point of view. The global stiffness matrix of VLPS with a special stabilization term coincides with that of the lowest-order VEM while the load terms are generally different. Specifically, we find that the global stiffness matrices of the two methods can be split as the consistency parts and the stability parts, and the consistency parts are the same while the stability parts coincide under some assumptions. As a by-product, a new stability term is obtained for VLPS. Besides, a post-processing procedure is suggested for the lowest-order VEM to preserve the positivity of the cell-centered unknowns and maintain at the same time the local conservation on the primary meshes. The positivity, existence and uniqueness of the cell-centered unknowns are studied. Numerical experiments confirm the theoretical findings and demonstrate the efficiency of the post-processing procedure on various polygonal meshes.

Machine classification of spatiotemporal patterns: automated parameter search in a rebounding spiking network.

Various patterns of electrical activities, including travelling waves, have been observed in cortical experimental data from animal models as well as humans. By applying machine learning techniques, we investigate the spatiotemporal patterns, found in a spiking neuronal network with inhibition-induced firing (rebounding). Our cortical sheet model produces a wide variety of network activities including synchrony, target waves, and travelling wavelets. Pattern formation is controlled by modifying a Gaussian derivative coupling kernel through varying the level of inhibition, coupling strength, and kernel geometry. We have designed a computationally efficient machine classifier, based on statistical, textural, and temporal features, to identify the parameter regimes associated with different spatiotemporal patterns. Our results reveal that switching between synchrony and travelling waves can occur transiently and spontaneously without a stimulus, in a noise-dependent fashion, or in the presence of stimulus when the coupling strength and level of inhibition are at moderate values. They also demonstrate that when a target wave is formed, its wave speed is most sensitive to perturbations in the coupling strength between model neurons. This study provides an automated method to characterize activities produced by a novel spiking network that phenomenologically models large scale dynamics in the cortex.

Self-calibrated interpolation of non-Cartesian data with GRAPPA in parallel imaging.

To develop a non-Cartesian k-space reconstruction method using self-calibrated region-specific interpolation kernels for highly accelerated acquisitions. In conventional non-Cartesian GRAPPA with through-time GRAPPA (TT-GRAPPA), the use of region-specific interpolation kernels has demonstrated improved reconstruction quality in dynamic imaging for highly accelerated acquisitions. However, TT-GRAPPA requires the acquisition of a large number of separate calibration scans. To reduce the overall imaging time, we propose Self-calibrated Interpolation of Non-Cartesian data with GRAPPA (SING) to self-calibrate region-specific interpolation kernels from dynamic undersampled measurements. The SING method synthesizes calibration data to adapt to the distinct shape of each region-specific interpolation kernel geometry, and uses a novel local k-space regularization through an extension of TT-GRAPPA. This calibration approach is used to reconstruct non-Cartesian images at high acceleration rates while mitigating noise amplification. The reconstruction quality of SING is compared with conjugate-gradient SENSE and TT-GRAPPA in numerical phantoms and in vivo cine data sets. In both numerical phantom and in vivo cine data sets, SING offers visually and quantitatively similar reconstruction quality to TT-GRAPPA, and provides improved reconstruction quality over conjugate-gradient SENSE. Furthermore, temporal fidelity in SING and TT-GRAPPA is similar for the same acceleration rates. G-factor evaluation over the heart shows that SING and TT-GRAPPA provide similar noise amplification at moderate and high rates. The proposed SING reconstruction enables significant improvement of acquisition efficiency for calibration data, while matching the reconstruction performance of TT-GRAPPA.

An Image-Based Multimedia Database and Efficient Detection though Features

Accurate feature detection during Image retrieval is important, data retrieves through image retrieval methods like CBIR and CBIR higher dimension data also need storage and access through different methods, content-based Image retrieval uses query like query by feature and query by example. More focus has made on accurate feature detection because need accurate feature retrieval. In simple words objectives are, to develop methods with sequence to classify features with normalization for efficient image retrieval from bulk dataset and also to improve method for local and global feature retrieval with automatic feature detection along accuracy. After study of different detection-based system, a methodology has been proposed which improves retrieval based on feature detection and feature detection had been improve with combination DWT+PCA+KSVM (polygon kernel +RBF kernel + Linear Kernel).

Torsion of elastic solids with sparse voids parallel to the twist axis

With the purpose of investigating a linear elastic solid containing a dilute distribution of cylindrical and prismatic holes parallel to the torsion axis, the full-field solution for an infinite elastic plane containing a single void and subject to torsion is derived. The obtained solution is exploited to derive the analytic expressions for the stress concentration factor related to the presence of an elliptical hole, the stress intensity factor for hypocycloidal holes and star-shaped cracks and the notch stress intensity factor for star-shaped polygons. Special sets of the void location are obtained for which peculiar mechanical behaviours are displayed, such as stress annihilation at some points along the boundary of elliptical voids and stress singularity removal at the cusps or points of hypocycloidal or isotoxal star-shaped polygonal voids. By means of finite-element simulations, it is finally shown that the presented closed-form expressions for the stress intensification provide reliable predictions, even for finite-domain realizations; in particular, the infinite-plane solution remains highly accurate when the the smooth and non-smooth external boundaries are larger than twice and five times the void dimension, respectively. Under these geometrical conditions, the derived analytical expressions represent a valid ‘guide tool’ in mechanical design.

The effects of focusing power on TEA CO2 laser-induced gas breakdown and the consequent pulse shaping effects

Transversely Excited Atmospheric (TEA) CO2 laser pulses were used in order to generate an optical breakdown in a variety of mono- and polyatomic molecules using different focusing powers. The dependence of the spark kernel geometry and the transmitted pulse shapes on the focusing power as well as the pressure, molecular weight, and ionization energy of the gases was investigated in detail. Partial removal of the transmitted pulse tail in the 0.05–2.6 μs range together with shortened spikes in the 10–60 ns range has been observed by applying a 2.5 cm focal length lens for all the gases. At higher focal lengths, this effect is only incompletely observed for He gas. Spatial-temporal analyses of the laser beams and the relevant plasma plumes indicate that this behavior is due to the drop in the plasma density below the critical level, before the laser pulse tail is completed.

Star-Shaped and L-Shaped Orthogonal Drawings

An orthogonal drawing of a plane graph $G$ is a planar drawing of $G$, denoted by $D(G)$, such that each vertex of $G$ is drawn as a point on the plane, and each edge of $G$ is drawn as a sequence of horizontal and vertical line segments with no crossings. An orthogonal polygon $P$ is called orthogonally convex if the intersection of any horizontal or vertical line $L$ and $P$ is either a single line segment or empty. An orthogonal drawing $D(G)$ is called orthogonally convex if all of its internal faces are orthogonally convex polygons. An orthogonal polygon $P$ is called a star-shaped polygon if there is a point $p\in P$ such that the entire $P$ is visible from $p$. An orthogonal drawing $D(G)$ is called a star-shaped orthogonal drawing (SSOD) if all of its internal faces are star-shaped polygons. Every SSOD is an orthogonally convex drawing, but the reverse is not true. SSOD is visually more appealing than orthogonally convex drawings. Recently, Chang et al. gave a necessary and sufficient condition for a plane graph to have an orthogonally convex drawing. In this paper, we show that if $G$ satisfies the same condition given by Chang et al., it not only has an orthogonally convex drawing, but also a SSOD, which can be constructed in linear time. An orthogonal drawing $D(G)$ is called an $L$-shaped drawing if each face of $D(G)$ is an $L$-shaped polygon. In this paper we also show that an $L$-shaped orthogonal drawing can be constructed in $O(n)$ time. The same algorithmic technique is used for solving both problems. It is based on regular edge labeling and is quite different from the methods used in previous results.

Limiting Forms of Iterated Circular Convolutions of Planar Polygons

We consider a complex representation of an arbitrary planar polygon $$\mathcal {P}$$ centered at the origin. Let $$\mathcal {P}^{(1)}$$ be the normalized polygon obtained from $$\mathcal {P}$$ by connecting the midpoints of its sides and normalizing the complex vector of vertex coordinates. We say that $$\mathcal {P}^{(1)}$$ is a normalized average of $$\mathcal {P}$$ . We identify this averaging process with a special case of a circular convolution. We show that if the convolution is repeated many times, then for a large class of polygons the vertices of the limiting polygon lie either on an ellipse or on a star-shaped polygon. We derive a complete and compact analytical description of the limiting elliptical envelope using discrete Fourier transforms and circular convolutions. One of the key insights of this approach is the realization that the repeated circular convolution removes all higher harmonic pairs leaving only the principal harmonic pair from the discrete Fourier transform of the original polygon to dominate the Fourier transform of the repeatedly convolved polygon thereby controlling the limiting behavior.

Computer-aided identification of the water diffusion coefficient for maize kernels dried in a thin layer

Abstract Uncertainties in mathematical modelling of water transport in cereal grain kernels during drying and storage are mainly due to implementing unreliable values of the water diffusion coefficient and simplifying the geometry of kernels. In the present study an attempt was made to reduce the uncertainties by developing a method for computer-aided identification of the water diffusion coefficient and more accurate 3D geometry modelling for individual kernels using original inverse finite element algorithms. The approach was exemplified by identifying the water diffusion coefficient for maize kernels subjected to drying. On the basis of the developed method, values of the water diffusion coefficient were estimated, 3D geometry of a maize kernel was represented by isoparametric finite elements, and the moisture content inside maize kernels dried in a thin layer was predicted. Validation of the results against experimental data showed significantly lower error values than in the case of results obtained for the water diffusion coefficient values available in the literature.