Symmetric Kernel Research Articles

A different approach to Eringen's nonlocal integral stress model with applications for beams

The nonlocal continuum theory, either integral or differential form, has been formulated, evolved and widely used in our era to explain size effect phenomena in micro– and nano– structures. In the case of Euler Bernoulli beam theory (EBBT), the nonlocal integral form produces energy consistent formulas, unlike the nonlocal differential form revealing inconsistencies and giving rise to paradoxes. In this work, our overall research objective mainly focuses on nonlocal integral elasticity analysis of beams by employing a normalized symmetric kernel. This kernel corresponds to a finite domain (and will be called modified kernel from now on), that satisfies all the properties of a probability density function as well as successfully handles the physical inconsistencies of classic types of kernel. Our concern is to investigate the static response of a beam with various types of loading and boundary conditions (BCs) by making use the modified kernel and the kernel corresponding to the two phase nonlocal integral (TPNI) model. Carrying out numerical methods to our problems, the deducing results appear more flexible behavior than those of classic-local and the common nonlocal differential, respectively. What is more, a comparison is made between the results of the aforementioned kernels to demonstrate the advantages of the modified kernel. Unlike the common nonlocal differential form, the nonlocal integral forms do not give rise to paradoxes. Moreover, it is critical that the TPNI model does not raise paradoxes as has been observed in other publications presented in the literature.

Symmetric function kernels and sweeping of measures

This is a potential theoretic study of balayage (sweeping) of a positive Radon measure ω on a locally compact (Hausdorff) space X onto a closed, or, more generally, a quasiclosed set A ⊂ X (that is, a set which can be approximated in outer capacity by closed sets). The setting is that of potentials with respect to a suitable symmetric function kernel G: X × X → [0,+∞]. We consider energy capacity, not as a set function, but as a functional, acting on positive numerical functions on X. The finiteness of the upper capacity of the function 1AGω is sufficient for the possibility of the sweeping in question (1A denoting the indicator function of A and Gω the G-potential of ω).

MO-FG-CAMPUS-TeP1-05: Rapid and Efficient 3D Dosimetry for End-To-End Patient-Specific QA of Rotational SBRT Deliveries Using a High-Resolution EPID

Purpose: EPID-based patient-specific quality assurance provides verification of the planning setup and delivery process that phantomless QA and log-file based virtual dosimetry methods cannot achieve. We present a method for EPID-based QA utilizing spatially-variant EPID response kernels that allows for direct calculation of the entrance fluence and 3D phantom dose. Methods: An EPID dosimetry system was utilized for 3D dose reconstruction in a cylindrical phantom for the purposes of end-to-end QA. Monte Carlo (MC) methods were used to generate pixel-specific point-spread functions (PSFs) characterizing the spatially non-uniform EPID portal response in the presence of phantom scatter. The spatially-variant PSFs were decomposed into spatially-invariant basis PSFs with the symmetric central-axis kernel as the primary basis kernel and off-axis representing orthogonal perturbations in pixel-space. This compact and accurate characterization enables the use of a modified Richardson-Lucy deconvolution algorithm to directly reconstruct entrance fluence from EPID images without iterative scatter subtraction. High-resolution phantom dose kernels were cogenerated in MC with the PSFs enabling direct recalculation of the resulting phantom dose by rapid forward convolution once the entrance fluence was calculated. A Delta4 QA phantom was used to validate the dose reconstructed in this approach. Results: The spatially-invariant representation of the EPID response accurately reproduced the entrance fluence with >99.5% fidelity with a simultaneous reduction of >60% in computational overhead. 3D dose for 106 voxels was reconstructed for the entire phantom geometry. A 3D global gamma analysis demonstrated a >95% pass rate at 3%/3mm. Conclusion: Our approach demonstrates the capabilities of an EPID-based end-to-end QA methodology that is more efficient than traditional EPID dosimetry methods. Displacing the point of measurement external to the QA phantom reduces the necessary complexity of the phantom itself while offering a method that is highly scalable and inherently generalizable to rotational and trajectory based deliveries. This research was partially supported by Varian.

Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence

In this paper, we show that smooth solutions to the Dirichlet problem for the parabolic equation $$v_t(x,t)=\sum_{i,j=1}^{N} a_{ij}(x)\frac{\partial^{2}v(x,t)}{\partial{x}_{i}\partial{x}_{j}} + \sum_{i =1}^{N} b_{i}(x)\frac{\partial{v}(x,t)}{\partial{x}_i} \qquad x \in \Omega,$$ with v(x, t) = g(x, t), \({x \in \partial \Omega,}\) can be approximated uniformly by solutions of nonlocal problems of the form $$u_{t}^{\varepsilon}(x,t)=\int_{\mathbb{R}^{n}} K_{\varepsilon}(x,y)(u^{\varepsilon}(y,t)-u^{\varepsilon}(x,t))dy, \,\,\quad x \in \Omega,$$ with \({u^{\varepsilon}(x,t)=g(x,t)}\), \({x \notin \Omega}\), as \({\varepsilon \to 0}\), for an appropriate rescaled kernel \({K_{\varepsilon}}\). In this way, we show that the usual local evolution problems with spatial dependence can be approximated by nonlocal ones. In the case of an equation in divergence form, we can obtain an approximation with symmetric kernels, that is, \({K_{\varepsilon}(x,y) = K_{\varepsilon}(y,x)}\).

Volcanic field elongation, vent distribution, and tectonic evolution of a continental rift: The Main Ethiopian Rift example

Magmatism and faulting operate and interact in continental rifts at a variety of scales. The East African Rift, an example of an active magmatic continental rift, provides the ideal location to study the interplay between these two mechanisms. Previous work has shown that the distribution of volcanic vents and the shape of volcanic fields are linked to their tectonic environment and their magmatic system. In order to distinguish the impact of each mechanism, we analyzed four distinct volcanic fields located in the Main Ethiopian Rift; three of them (Debre Zeyit, Wonji, and Kone) are located within the rift valley, and one (Akaki) lies on the western rift shoulder. The elongation and shape of each field were analyzed based on their vent distribution using three statistical methods: the principal component analysis, the vent-to-vent distance, and the two-dimensional, symmetric Gaussian kernel density estimation method. The results of these analyses show that the elongations of the fields increase from the western shoulder toward the rift axis and are inversely proportional to their angular dispersion. In addition, we observed that none of the analyzed fields have elongation that parallels either the trends of the youngest faults or the volcano alignments. Dike intrusions inferred from the alignment analysis of volcanic centers that are located within the rift axis tend to parallel the orientation of the recent active faults. This parallelism, however, decreases from the rift axis toward the rift shoulder, which has a lower strain rate and shows a larger number of preexisting fault orientations. Our results suggest that the shape of volcanic fields in the Main Ethiopian Rift is controlled mainly by large crustal structures and/or by the lithosphere-asthenosphere–boundary geometry. Diking is principally controlled by the location of the field within the rift zone; intrusions within the rift zone are controlled mostly by the state of stress and strain rate; while intrusions on the rift shoulder are controlled mostly by the presence of preexisting fractures. This study provides a combination of statistical analyses and geological observations to study and differentiate the mechanisms involved in the formation of volcanic fields on Earth and, potentially, other planets.

On single valued neutrosophic relations

Smarandache initiated neutrosophic sets (NSs) which can be used as a mathematical tool for dealing with indeterminate and inconsistent information. In order to apply NSs conveniently, single valued neutrosophic sets (SVNSs) were proposed by Wang et al. In this paper, we propose single valued neutrosophic relations (SVNRs) and study their properties. The notions of anti-reflexive kernel, symmetric kernel, reflexive closure, and symmetric closure of a SVNR are introduced, respectively. Their accurate calculate formulas and some properties are explored. Some examples are also given. Finally, single valued neutrosophic relation mappings and inverse single valued neutrosophic relation mappings are introduced, and some interesting properties are also obtained.

One-dimensional solutions of non-local Allen–Cahn-type equations with rough kernels

We are interested in the study of local and global minimizers for an energy functional of the type14∬R2N∖(RN∖Ω)2|u(x)−u(y)|2K(x−y)dxdy+∫ΩW(u(x))dx, where W is a smooth, even double-well potential and K is a non-negative symmetric kernel in a general class, which contains as a particular case the choice K(z)=|z|−N−2s, with s∈(0,1), related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space RN and obtain sharp estimates for some quantities associated to it. In particular, we deduce the existence of solutions of the non-local Allen–Cahn equationp.v.∫RN(u(x)−u(y))K(x−y)dy+W′(u(x))=0for any x∈RN, which possess one-dimensional symmetry.The results presented here were proved in [9,10,36] for the model case K(z)=|z|−N−2s. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.

Experimental investigation on the uniqueness of a center of a body

The object of our investigation is a point that gives the maximum value of a potential with a strictly decreasing radially symmetric kernel. It defines a center of a body in Rm. When we choose the Riesz kernel or the Poisson kernel as the kernel, such centers are called a radial center or an illuminating center, respectively. The existence of a center is easily shown but the uniqueness does not always hold. Sufficient conditions of the uniqueness of a center have been studied by some researchers. The main results in this paper are some new sufficient conditions for the uniqueness of a center of a body.

Asymptotic behavior for a nonlocal diffusion equation in exterior domains: The critical two-dimensional case

We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, ∂tu=J⁎u−u, where J is a smooth, radially symmetric kernel with support Bd(0)⊂R2. The problem is set in an exterior two-dimensional domain which excludes a hole H, and with zero Dirichlet data on H. In the far field scale, ξ1≤|x|t−1/2≤ξ2 with ξ1,ξ2>0, the scaled function log⁡tu(x,t) behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by J. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic ‘logarithmic momentum' of the solution, limt→∞⁡∫R2u(x,t)log⁡|x|dx. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, |x|≤t1/2h(t) with limt→∞⁡h(t)=0, the scaled function t(log⁡t)2u(x,t)/log⁡|x| converges to a multiple of ϕ(x)/log⁡|x|, where ϕ is the unique stationary solution of the problem that behaves as log⁡|x| when |x|→∞. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, |x|≥t1/2g(t) with g(t)→∞, the solution is proved to be of order o((tlog⁡t)−1).

Simulation of the 3D viscoelastic free surface flow by a parallel corrected particle scheme**Project supported by the Natural Science Foundation of Jiangsu Province, China (Grant Nos. BK20130436 and BK20150436) and the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant No. 15KJB110025).

In this work, the behavior of the three-dimensional (3D) jet coiling based on the viscoelastic Oldroyd-B model is investigated by a corrected particle scheme, which is named the smoothed particle hydrodynamics with corrected symmetric kernel gradient and shifting particle technique (SPH_CS_SP) method. The accuracy and stability of SPH_CS_SP method is first tested by solving Poiseuille flow and Taylor–Green flow. Then the capacity for the SPH_CS_SP method to solve the viscoelastic fluid is verified by the polymer flow through a periodic array of cylinders. Moreover, the convergence of the SPH_CS_SP method is also investigated. Finally, the proposed method is further applied to the 3D viscoelastic jet coiling problem, and the influences of macroscopic parameters on the jet coiling are discussed. The numerical results show that the SPH_CS_SP method has higher accuracy and better stability than the traditional SPH method and other corrected SPH method, and can improve the tensile instability.

A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves: an Extension From Small to Finite Amplitude

The reduced 3-wave and 4-wave Hamiltonian equations for ocean surface waves were widely used for the simplified structure with symmetric polynomial kernels and for the conservation of energy, etc. However, according to the related assumption for approximation in derivation, the range of applicability was limited to weakly nonlinear waves of small amplitude. Here the following issue was further studied: for nonlinear waves of finite amplitude within a certain range, was it also possible to obtain reduced Hamiltonian equations with symmetric polynomial kernels in a sense of sufficient accuracy? Because of complicated strongly nonlinear coupling, few development in this significant respect had been made as yet. A new approach was proposed based on the Chebyshev polynomials to best approximate the primitive water wave equations in the exact sense of strongly nonlinear coupling and derive new reduced Hamiltonian equations with symmetric polynomial kernels. The new results exhibit an extension from a weakly nonlinear case in which the product of the wave number and the wave steepness is small to a nonlinear case in which this product goes up to about 1.035. Moreover, within this range, the approximation errors are lower than 5%, and in particular, the new results prove exact whenever the said product lies close to 0.9.

Unleashing GPU acceleration for symmetric band linear algebra kernels and model reduction

Linear algebra operations arise in a myriad of scientific and engineering applications and, therefore, their optimization is targeted by a significant number of high performance computing research ...

Asymptotic behavior for a nonlocal model of neural fields

In this paper we consider the non local evolution problem $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u =- u + K(f\circ u ) \ \ in \ \ \Omega ,\\ u = 0 \ \ in \ \ \mathbb {R}^N\backslash \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^N\), \(f: \mathbb {R}\rightarrow \mathbb {R}\) and K is an integral operator with a symmetric kernel. We prove existence and some regularity properties of the global attractor. We also characterize the global attractor, using the properties of a Lyapunov functional for this model.

A Hilbert-type operator with a symmetric homogeneous kernel of two parameters and its applications

We introduce a general homogeneous kernel whose degree is given by two parameters to establish the equivalent inequalities with the norm of a new Hilbert-type operator. As applications, we provide new extended Hilbert-type inequalities with the best possible constant factors.

Uniform Equicontinuity for a Family of Zero Order Operators Approaching the Fractional Laplacian

In this paper we consider a smooth bounded domain Ω ⊂ ℝ N and a parametric family of radially symmetric kernels K ε: ℝ N → ℝ+ such that, for each ε ∈ (0, 1), its L 1 − norm is finite but it blows up as ε → 0. Our aim is to establish an ε independent modulus of continuity in Ω, for the solution u ε of the homogeneous Dirichlet problem where and the operator ℐε has the form and it approaches the fractional Laplacian as ε → 0. The modulus of continuity is obtained combining the comparison principle with the translation invariance of ℐε, constructing suitable barriers that allow to manage the discontinuities that the solution u ε may have on ∂Ω. Extensions of this result to fully non-linear elliptic and parabolic operators are also discussed.

Improving partial mutual information-based input variable selection by consideration of boundary issues associated with bandwidth estimation

Input variable selection (IVS) is vital in the development of data-driven models. Among different IVS methods, partial mutual information (PMI) has shown significant promise, although its performance has been found to deteriorate for non-Gaussian and non-linear data. In this paper, the effectiveness of different approaches to improving PMI performance is investigated, focussing on boundary issues associated with bandwidth estimation. Boundary issues, associated with kernel-based density and residual computations within PMI, arise from the extension of symmetrical kernels beyond the feasible bounds of potential inputs, and result in an underestimation of kernel-based marginal and joint probability distribution functions in the PMI. In total, the effectiveness of 16 different approaches is tested on synthetically generated data and the results are used to develop preliminary guidelines for PMI IVS. By using the proposed guidelines, the correct inputs can be identified in 100% of trials, even if the data are highly non-linear or non-Gaussian.

Accurate Location and Manipulation of Nanoscaled Objects Buried under Spin-Coated Films.

Detection and precise localization of nanoscale structures buried beneath spin-coated films are highly valuable additions to nanofabrication technology. In principle, the topography of the final film contains information about the location of the buried features. However, it is generally believed that the relation is masked by flow effects, which lead to an upstream shift of the dry film's topography and render precise localization impossible. Here we demonstrate, theoretically and experimentally, that the flow-shift paradigm does not apply at the submicrometer scale. Specifically, we show that the resist topography is accurately obtained from a convolution operation with a symmetric Gaussian kernel whose parameters solely depend on the resist characteristics. We exploit this finding for a 3 nm precise overlay fabrication of metal contacts to an InAs nanowire with a diameter of 27 nm using thermal scanning probe lithography.

Spherically symmetric multivariate beta family kernels

The generalisation of univariate beta kernels to the multivariate spherically symmetric case is considered. By integrating the powers of quadratic forms over the unit ball, we exhibit closed form expressions, based on ratios of beta functions, for analysing these kernels.

Family of the generalised gamma kernels: a generator of asymmetric kernels for nonnegative data

Unlike symmetric kernels, so far exploring asymptotics on asymmetric kernels has relied on diversified approaches. This paper proposes a family of the generalised gamma (GG) kernels that is built on the probability density function of the GG distribution [Stacy, E.W. (1962), ‘A Generalization of the Gamma Distribution’, Annals of Mathematical Statistics, 33, 1187–1192] and a few common conditions. The family can generate asymmetric kernels that share appealing properties with the modified gamma kernel [Chen, S.X. (2000), ‘Probability Density Function Estimation Using Gamma Kernels’, Annals of the Institute of Statistical Mathematics, 52, 471–480]. Asymptotics on the kernels generated from the family can be delivered by manipulating the conditions directly, as with symmetric kernels.

An eigenvalue relation in spheroidal wave functions related to potential theory and its applications to contact problems

Eigenvalue and related integral relations for an integral operator with a symmetrical kernel, defined in a circular region, in the form of the ratio of an exponential function of the distance between two points to its distance, are established by methods of generalized potential theory, related to the Helmholtz equation, in an orthogonal system of coordinates of an oblate spheroid, where one of the coordinate surfaces is degenerate in a plane doubly covered circular disc. These relations extend corresponding relations for the integral operator with a symmetric kernel in the form of the Weber–Sonin integral and contain spheroidal wave functions. Using the integral relations obtained, an exact solution of the integral equation of the contact problem of the indentation of a punch, circular in plan, into a linearly deformed base of an elastic half-space with a kernel identical with a kernel, which varies exponentially with distance, is constructed. Other applications of the eigenvalue and related integral relations obtained are also indicated.