Symmetric Neighborhood Research Articles

Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules

We study two-dimensional critical bootstrap percolation models. We establish that a class of these models including all isotropic threshold rules with a convex symmetric neighbourhood, undergoes a sharp metastability transition. This extends previous instances proved for several specific rules. The paper supersedes a draft by Alexander Holroyd and the first author from 2012. While it served a role in the subsequent development of bootstrap percolation universality, we have chosen to adopt a more contemporary viewpoint in its present form.

DRRN: Differential rectification & refinement network for ischemic infarct segmentation

AbstractAccurate segmentation of infarct tissue in ischemic stroke is essential to determine the extent of injury and assess the risk and choose optimal treatment for this life‐threatening disease. With the prior knowledge that asymmetric analysis of anatomical structures can provide discriminative information, plenty of symmetry‐based approaches have emerged to detect abnormalities in brain images. However, the inevitable non‐pathological noise has not been fully alleviated and weakened, leading to unsatisfactory results. A novel differential rectification and refinement network (DRRN) for the automatic segmentation of ischemic strokes is proposed. Specifically, a differential feature perception encoder (DFPE) is developed to fully exploit and propagate the bilateral quasi‐symmetry of healthy brains. In DFPE, an erasure‐rectification (ER) module is devised to rectify pseudo‐lesion features caused by non‐pathological noise through utilising discriminant features within the symmetric neighbourhood of the original image. And a differential‐attention (DA) mechanism is also integrated to fully perceive the differences in cross‐axial features and estimate the similarity of long‐range spatial context information. In addition, a crisscross differential feature reinforce module embedded with multiple boundary enhancement attention modules is designed to effectively integrate multi‐scale features and refine textual details and margins of the infarct area. Experimental results on the public ATLAS and Kaggle dataset demonstrate the effectiveness of DRRN over state‐of‐the‐arts.

Time Distribution from Zero up to Beginning: of the Final Stop of Semi-Markov Diffusion Process on Interval with Unattainable Boundaries

A one-dimensional process with continuous trajectories on a non-negative semiaxis is considered. The process has the Markov property with respect to the first exit time from any open interval (semi-Markov process). This process is called diffusion if the probability of its first exit from any symmetric neighborhood of its initial point across any boundary tends to 1/2 as the length of this neighborhood tends to zero. The time distribution from zero up to the beginning of the final interval of constancy is investigated. This distribution depends on semi-Markov transition generating functions of the process. Representation for Laplace transform of this distribution is obtained in an integral form. The integrand of this representation explains the meaning of quadratic terms of Taylor expansion of a semi-Markov transition generating function by powers of the diameter of the symmetric neighborhood of the initial point of the process. Namely, the trajectory of the process has no final interval of constancy if the coefficient of such a quadratic term is equal to zero.

A Novel Autoencoder Deep Architecture for Detecting the Outlier in Heterogeneous Data Sets

Neighborhood-based unsupervised approaches like LDOF, LOF and symmetric neighborhood (INFLO) have proven effective over decades. These techniques principally utilize the information of either the [Formula: see text]-nearest neighbor or the reverse [Formula: see text]-nearest neighbors to detect the outlierness of each object in a data set. However, these methodologies fail to detect genuine outliers in heterogeneous data sets located between two dense clusters, between dense and sparse clusters or the scattered data sets. In addition, LOF treats a normal point of a sparse cluster as an outlier if the sparse cluster is close to a dense cluster. This paper proposes a novel autoencoder deep learning architecture to overcome the limitations of the aforementioned techniques. In the proposed approach, we identify the potential outliers intelligently from a given data set and mark them to generate training samples for the autoencoder. These marked points are not included in the training samples. Finally, the trained autoencoder is used to compute the outlierness of each data point in the whole data set (training samples + marked points). Experimental results with synthetic and real-world data sets show that the proposed model outperforms these widely applied techniques along with state-of-the-art works.

Bipartite network embedding with Symmetric Neighborhood Convolution

Mining bipartite networks, a ubiquitous type of model particularly developed for networks with two different classes of nodes, is of great importance in various applications. Recently, the advance of network representation learning has provided more tractable, effective and robust methods for several graph mining tasks. However, the strength of representation learning has not been sufficiently explored for bipartite structure, since that most existing learning methods either separately treat the nodes in different classes by defective projections, or independently sample instances for learning by adopting relation-aware random walks. Both methods can lose information and they hardly leverage the symmetry property of bipartite structure directly. In this paper, a novel method called Symmetric Neighborhood Convolution is proposed to fulfill the representation learning task in bipartite networks. The method takes into consideration the topological symmetry property and utilizes one-dimensional convolution kernels to extract features from neighborhood for the nodes in each class. Three different objectives are designated to compare the performance of the proposed convolution-based mechanism. The method is evaluated on five real-world datasets by the tasks of link prediction and recommendation. Comparing to eight competitive baselines, experimental results demonstrate the effectiveness of our method.

Time distribution from zero up to beginning of the final stop of semi-Markov diffusion process on interval with unattainable boundaries

A one-dimensional process with continuous trajectories on non-negative semi-axis is considered. The process has the Markov property with respect to the first exit time from any open interval (semi-Markov process). This process is called to be diffusion if probability for its first exit point from any symmetric neighborhood of its initial point across any boundary tends to 1/2 as length of this neighborhood tends to zero. Time distribution from zero up to beginning of the final interval of constancy is investigated. This distribution depends on semi-Markov transition generating functions of the process. Representation for Laplace transform of this distribution is obtained in an integral form. The integrand of this representation explains sense of quadratic members of Tailor decomposition of a semi-Markov transition generating function by powers of diameter of symmetric neighborhood of the process initial point. Namely trajectory of the process has no any final interval of constancy if and only if coefficient of such a quadratic member is equal to zero.

Pointwise k-Pseudo Metric Space

In this paper, the concept of a k-(quasi) pseudo metric is generalized to the L-fuzzy case, called a pointwise k-(quasi) pseudo metric, which is considered to be a map d:J(LX)×J(LX)⟶[0,∞) satisfying some conditions. What is more, it is proved that the category of pointwise k-pseudo metric spaces is isomorphic to the category of symmetric pointwise k-remote neighborhood ball spaces. Besides, some L-topological structures induced by a pointwise k-quasi-pseudo metric are obtained, including an L-quasi neighborhood system, an L-topology, an L-closure operator, an L-interior operator, and a pointwise quasi-uniformity.

Competitive drivers of interspecific deviations of crown morphology from theoretical predictions measured with Terrestrial Laser Scanning

AbstractTree crown morphology is a key driver of forest dynamics, determining not only the competitiveness of an individual but also the competitive effect exerted on neighbouring trees. Multiple ecological theories, including metabolic scaling theory (MST), predict crown morphology from first principles, but typically lack consideration of competition. The accurate quantification of crown morphology to test theoretical predictions, and the canopy interactions that could alter them, has historically been limited by the simplicity and associated error of traditional crown measurements.In this study, we calculate high‐resolution two‐ and three‐dimensional crown metrics from Terrestrial Laser Scanning data for 1,441Pinus sylvestris,P.nigra,Quercus fagineaandQ. ilextrees from a water‐limited forest community in central Spain and test height‐crown metric scaling relationships. We demonstrate new TLS methods to define symmetric and asymmetric neighbourhood metrics based on tree height, crown size and neighbour projected crown area, and test the importance of neighbourhood genus diversity on crown morphology by separating competition from congeneric and heterogeneric neighbours.Competition negatively impacted all crown metrics except crown depth where onlyP.nigrashowed sensitivity. Asymmetric competition was the strongest driver of pine crown morphology, but oaks were more sensitive to symmetric competition, in line with shade tolerance expectations. Congeneric competition reducedQ.fagineacrown size and changed its shape, but we found no significant effects of heterogeneric neighbours. Most species and crown dimensions had height‐crown scaling exponents below those predicted by MST, which may be due to water limitation effects. Pines and oaks showed large differences in crown depth to height scaling, with the former shallower and the latter deeper, in contrast to theoretical predictions.Synthesis. Our study is the first to demonstrate the ability of TLS to characterise crown morphology from leaf‐wood separated clouds and competitive neighbour effects in a water‐limited forest community, and to use TLS metrics to test ecological crown scaling theory. Most crown metrics scaled below theoretical predictions. Pines were more sensitive to competition by larger neighbours and oaks to crowding from all neighbours, with competition from neighbours of the same genus having a consistent negative effect.

Explosion in a growth model with cooperative interaction on an infinite graph

In the process of adsorption, which is the process of accumulation of molecules on a surface, particles are allocated to vertices of a graph accordingly to a specific rule. Such phenomenon has great technological importance in the purification and separation of mixtures in petroleum industries, in effluent treatment, and in the manufacture of biomaterials such as prostheses and implants. The subject of this paper is to study the asymptotic behavior of a growth process, in our case it is the process of adsorption of particles in an infinite number of sites, using a probabilistic approach. For this we distinguish explosion/non-explosion of a continuos time growth process with cooperative interaction on $${\mathbb {Z}}_+$$ . We consider symmetric neighborhood and different types of rate functions and prove that explosion occurs for exponential rates, and for some cases with polynomial rates, but not for other cases. We also present some simulations to illustrate the types of explosions to check whether the explosion occurs horizontally or vertically.

Local Approximation by Parabolic Splines in the Mean for Large Averaging Intervals

Local parabolic splines on the axis $$\mathbb R$$ with equidistant nodes realizing the simplest local approximation scheme are considered. But, instead of the values of functions at the nodes, its formula approximates their mean values in symmetric neighborhoods of these nodes. For an arbitrary averaging step $$H$$ more than twice as large as the grid step $$h$$ of the spline, the approximation errors are calculated exactly in the uniform metric of these functions and their derivatives for the function class $$W_\infty^2$$ . For small averaging steps $$H\le 2h$$ , these quantities were calculated by E. V. Strelkova (Shevaldina) in 2007.

On Product of Smooth Neutrosophic Topological Spaces

In this paper, we develop the notion of the basis for a smooth neutrosophic topology in a more natural way. As a sequel, we define the notion of symmetric neutrosophic quasi-coincident neighborhood systems and prove some interesting results that fit with the classical ones, to establish the consistency of theory developed. Finally, we define and discuss the concept of product topology, in this context, using the definition of basis.

Local Peaks-Based Clustering Algorithm in Symmetric Neighborhood Graph

Density-based clustering methods have achieved many applications in data mining, whereas most of them still likely suffer poor performances on data sets with extremely uneven distributions, like the manifold or ring data. The paper proposes a novel method for clustering with local peaks in the symmetric neighborhood. Local peaks are points with maximum densities at the local level. During the searching of local peaks, all data, except those outliers, can be easily divided into a number of small clusters in accordance with the local peaks in each point's neighborhood. Especially, a graph-based scheme is adopted here to merge similar clusters based on their similarity in the symmetric neighborhood graph, followed by assigning each outlier to the closest cluster. A variety of artificial, real data sets and a real building data set have been tested for clustering by the proposed method and compared against other popular density-based methods and other algorithms.

SOME MULTIPLICATIVE NEIGHBORHOOD TOPOLOGICAL INDICES OF NANOCONES AND DENDRIMERS

A topological index is a numerical parameter mathematically derived from the graph structure. In this paper, we introduce the modified multiplicative first and second neighborhood indices, multiplicative F-neighborhood index, general multiplicative neighborhood index, multiplicative inverse sum indeg neighborhood index, multiplicative harmonic neighborhood index and multiplicative symmetric division neighborhood index, first and second multiplicative Gourava neighborhood indices of a graph. We compute these newly defined multiplicative neighborhood indices for nanocones and dendrimers.

Numerical investigation of the collapse of a static bubble at the free surface in the presence of neighbors

Formation of a liquid jet and drop as a consequence of the bursting of a bubble in symmetric and asymmetric neighborhoods is modeled numerically. We find that unbalanced interaction of neighbors forces the jet and drop to bend towards the absent bubbles.

Efficient Clustering Method Based on Density Peaks With Symmetric Neighborhood Relationship

Efficient Clustering Method Based on Density Peaks With Symmetric Neighborhood Relationship

Surface reconstruction from discrete indicator functions.

This paper introduces a procedure for the calculation of the vertex positions in Marching-Cubes-like surface reconstruction methods, when the surface to reconstruct is characterised by a discrete indicator function. Linear or higher order methods for the vertex interpolation problem require a smooth input function. Therefore, the interpolation methodology to convert a discontinuous indicator function into a triangulated surface is non-trivial. Analytical formulations for this specific vertex interpolation problem have been derived for the 2D case by Manson et al. [Eurographics (2011) 30, 2] and the straightforward application of their method to a 3D case gives satisfactory visual results. A rigorous extension to 3D, however, requires a least-squares problem to be solved for the discrete values of a symmetric neighbourhood. It thus relies on an extra layer of information, and comes at a significantly higher cost. This paper proposes a novel vertex interpolation method which yields second-order-accurate reconstructed surfaces in the general 3D case, without altering the locality of the method. The associated errors are analysed and comparisons are made with linear vertex interpolation and the analytical formulations of Manson et al. [Eurographics (2011) 30, 2].

Learnable despeckling framework for optical coherence tomography images.

Optical coherence tomography (OCT) is a prevalent, interferometric, high-resolution imaging method with broad biomedical applications. Nonetheless, OCT images suffer from an artifact called speckle, which degrades the image quality. Digital filters offer an opportunity for image improvement in clinical OCT devices, where hardware modification to enhance images is expensive. To reduce speckle, a wide variety of digital filters have been proposed; selecting the most appropriate filter for an OCT image/image set is a challenging decision, especially in dermatology applications of OCT where a different variety of tissues are imaged. To tackle this challenge, we propose an expandable learnable despeckling framework, we call LDF. LDF decides which speckle reduction algorithm is most effective on a given image by learning a figure of merit (FOM) as a single quantitative image assessment measure. LDF is learnable, which means when implemented on an OCT machine, each given image/image set is retrained and its performance is improved. Also, LDF is expandable, meaning that any despeckling algorithm can easily be added to it. The architecture of LDF includes two main parts: (i)an autoencoder neural network and (ii)filter classifier. The autoencoder learns the FOM based on several quality assessment measures obtained from the OCT image including signal-to-noise ratio, contrast-to-noise ratio, equivalent number of looks, edge preservation index, and mean structural similarity index. Subsequently, the filter classifier identifies the most efficient filter from the following categories: (a)sliding window filters including median, mean, and symmetric nearest neighborhood, (b)adaptive statistical-based filters including Wiener, homomorphic Lee, and Kuwahara, and (c)edge preserved patch or pixel correlation-based filters including nonlocal mean, total variation, and block matching three-dimensional filtering.

Serrin’s overdetermined problem on the sphere

We study Serrin’s overdetermined boundary value problem $$\begin{aligned} -\Delta _{S^N}\, u=1 \quad \text { in }\Omega ,\quad u=0, \; \partial _\eta u={\text {const}} \quad \text {on }\partial \Omega \end{aligned}$$ in subdomains $$\Omega $$ of the round unit sphere $$S^N \subset \mathbb {R}^{N+1}$$ , where $$\Delta _{S^N}$$ denotes the Laplace–Beltrami operator on $$S^N$$ . A subdomain $$\Omega $$ of $$S^N$$ is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $$S^N$$ , $$N \ge 2$$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in $$S^{N}$$ which are not bounded by geodesic spheres.

The uniformly normal spaces

A topological space X is uniformly normal if the family U of all symmetric neighborhoods of the diagonal Δ ⊂ X × X forms a uniformity on X. A neighborhood of the diagonal is any subset whose interior contains the diagonal. It is proved that the Σ-product of Lindelof p-spaces of countable tightness is uniformly normal.

Final Distribution of a Diffusion Process: Semi-Markov Approach

A one-dimensional diffusion process is considered. This process is proposed to have the homogeneous Markov property with respect to the first exit time from any open interval (semi-Markov property). A diffusion property of the process is defined as asymptotical equiprobable exit through each of two edges of any symmetric neighborhood interval of an initial point of a sample process trajectory while length of the neighborhood tends to zero. Such a process is proved to have a limit as $t\to\infty$ if probability of the process not leaving this neighborhood is decreased as square of its length. In particular this condition is satisfied for a diffusion Markov process with a break for which the nonexit condition is replaced by the break condition. A semi-Markov method is applied to a derivation of the formula of a conditional final distribution of the diffusion process with a limit at infinity.