Discrete Set Of Points Research Articles

Decomposition matrices for the special case of data on the triangular lattice of G(2)

A method for the decomposition of data functions sampled on a finite fragment of triangular lattice is described for the cases of lattices of any density corresponding to the simple Lie group G ( 2 ) . Its main advantage is the fact that the decomposition matrix needs to be calculated only once for arbitrary sets of data sampled on the same set of discrete points. The decomposition matrix applies to lattice of any density that carries data.

Recovering an unknown source in a fractional diffusion problem

A standard inverse problem is to determine a source which is supported in an unknown domain D from external boundary measurements. Here we consider the case of a time-independent situation where the source is equal to unity in an unknown subdomain D of a larger given domain Ω and the boundary of D has the star-like shape, i.e.∂D={q(θ)(cos⁡θ,sin⁡θ)⊤:θ∈[0,2π]}. Overposed measurements consist of time traces of the solution or its flux values on a set of discrete points on the boundary ∂Ω. The case of a parabolic equation was considered in [6]. In our situation we extend this to cover the subdiffusion case based on an anomalous diffusion model and leading to a fractional order differential operator. We will show a uniqueness result and examine a reconstruction algorithm. One of the main motives for this work is to examine the dependence of the reconstructions on the parameter α, the exponent of the fractional operator which controls the degree of anomalous behaviour of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.

Similarity Comparison Based High-Speed Pilot Protection for Transmission Line

To solve the following problem: the action speed of protection based on a phasor is restricted by the length of the filtering algorithm time window and the vulnerability to false data, a new pilot protection principle for transmission lines based on waveform similarity is proposed in this paper. By comparing the discrete point sets of current sampling values, a waveform similarity-based algorithm called the Hausdorff distance distinguishes the difference between both side currents of a protected transmission line, realizing fast identification, and removal of the fault. Based on the characteristics of the algorithm, setting methods and protection criterion are discussed. Finally, an improved Hausdorff distance algorithm is proposed to cope with false data. The effectiveness and advantages of the proposed algorithm are validated with theoretical analysis and extensive simulation test results.

Selfishness of convex bodies and discrete point sets

Let F be a family of sets in Rd. A set M⊂Rd is called F-convex if for any pair of distinct points x,y∈M, there is a set F∈F such that x,y∈F and F⊂M.A family F of compact sets is called complete if F contains all compact F-convex sets. Generalizing the definition in Yuan and Zamfirescu (2016), a compact set K will be called selfish, if the family FK of all sets similar to K contains all compact FK-convex sets.In this paper, we investigate the selfishness of rectangles, isosceles triangles, regular n-gons, and some finite sets.

A stochastic dynamic model of train-track-bridge coupled system based on probability density evolution method

• A new stochastic dynamic model of train-track-bridge coupled system (TTBS) based on PDEM is developed. • The multi-coupling effect of excitations can be simultaneously input into the new model. • The new developed approach NTM- mp succeeded in getting the discrete point design of TTBS. • Based on PDEM, the stochastic probability evolutionary mechanism of TTBS are reproduced accurately. An innovative stochastic dynamic model of a 3D train-track-bridge coupled system (TTBS) with refined wheel/rail interaction is established for a high-speed railway based on the random theory of probability density evolution method (PDEM). The multi-coupling effect of excitations can be simultaneously input into the new model, e.g. random track irregularity, random vehicle loads, stochastic system parameters, et al. Moreover, a new approach, named “Number theoretic method of multi-target probability functions” (NTM- mp ), is developed to obtain the discrete point sets of multidimensional random parameters in hypercube space, aims to solve the point design of system uncertainty. The stochastic harmonic function (SHF) is applied to generate representative random track irregularity samples. The results of TTBS got by PDEM are verified with several typical case studies for its efficiency and reliability, which are the deterministic results in the representative publication, the Monte Carlo method (MCM) results, and the field testing results on the high-speed railway. At last, a typical case study of TTBS on a high-speed railway is presented for numerical analysis. Discussions and significant conclusions on the random dynamic responses are presented.

Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel

We characterize the long-time behaviour of solutions to Smoluchowski's coagulation equation with a diagonal kernel of homogeneity $\gamma < 1$. Due to the property of the diagonal kernel, the value of a solution depends only on a discrete set of points. As a consequence, the long-time behaviour of solutions is in general periodic, oscillating between different rescaled versions of a self-similar solution. Immediate consequences of our result are a characterization of the set of data for which the solution converges to self-similar form and a uniqueness result for self-similar profiles.

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green’s functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.

ON THE POSSIBILITY OF GEOMETRIC CHANGES OF LARGE DAMS DETERMINATION BY GEODETIC METHODS

&amp;lt;jats:p&amp;gt;Large dams are exposed to the influence of various forces which in time cause changes in their position and geometry. Knowledge about changes in position and geometry of large dams is of crucial importance for their stability and security determination. Dams, in process of deformation analysis, are approximated by set of discrete points and on the base of the changes in position and through time of those points the conclusions of changes of dam position and geometry are made. For large dams&amp;#039; security of highest importance is to detect changes of position and geometry where they are small in order to discover the negative trends and to take measures for prevent consequences of jeopardized dams&amp;#039; security in time. This paper investigates the possibilities of dams&amp;#039; geometry changes detection on the base of geodetic measurements.&amp;lt;/jats:p&amp;gt;

Spectral locus interpolation with splines in optical instruments

The article deals with the problem of representation of the spectral locus curve on the CIE xy chromaticity diagram. The issue is that spectral locus is specified at discrete set of points corresponding to given set of wavelengths of monochromatic colors. Thus, the problem appears which way to calculate chromaticity coordinates for any other wavelength. Three different methods how to interpolate the spectral locus were considered. The method we proposed to build an interpolant curve is based on using of Bezier curves. This approach allows to calculate chromaticity coordinates for any arbitrary wavelength directly on the CIE xy chromaticity diagram. Obtained analytical expressions for the spectral locus curve will increase the evaluation accuracy of the luminous radiation dominant wavelength value which defines the hue and the saturation value of the emitting color and other things.

Multicommodity flows and Benders decomposition for restricted continuous location problems

The restricted continuous facility location problem arises when there is a need to locate a number of facilities to serve a discrete set of demand points, and where the location of a facility can be anywhere on the plane except for in restricted regions. The problem finds applications in urban planning, disaster management, and healthcare logistics. The restricted regions can occur randomly or are known in advance. The paper describes a new model for the problem that is based on multicommodity flows with unknown destinations and defined on a discretization of the plane. The model and discretization are applied to both the deterministic and the stochastic continuous restricted location problem, where the latter is converted into a deterministic equivalent problem by minimizing the expected value of the objective function weighted by the probabilities of scenarios. The paper also describes a Benders decomposition algorithm to optimally solve the model. Extensive computational results are presented on both benchmark instances from the literature and new instances, on both the deterministic and stochastic variant of the problem. The results indicate that the proposed algorithm is superior to an off-the-shelf solver in terms of computational time. To the best of the authors’ knowledge, the exact algorithm described here is the first to address both the deterministic and the stochastic variants of continuous restricted location problems with any number of facilities.

Determination of coupled mechanical and thermal fields using 2D digital image correlation and infrared thermography: Numerical procedures and results

The objective of the work is to develop numerical method for determining coupled thermo-mechanical fields based on experimental data obtained from two cameras working in the visible and infrared mode. The sequence of images recorded by the first camera is used to determine the displacement field on the sample surface using the 2D digital image correlation (DIC) method. The resulting field from DIC analysis in a form of a set of discrete points with the corresponding in-plane displacement vector is used as the input for the next step of analysis, where the coupled temperature field is computed. This paper provides a detailed description of the numerical procedures, that allow, to obtain coupled thermal and mechanical fields together with the specification of experimental data needed for calculations. The presented approach was tested on an experimental data obtained during uniaxial tension of the multicrystalline aluminum. The developed numerical routine has been implemented in dedicated software, which can be used for the testing of materials on both a macro and micro scales.

Finding Largest Common Point Sets

Let [Formula: see text] and [Formula: see text] be two discrete point sets in [Formula: see text] of sizes [Formula: see text] and [Formula: see text], respectively, and let [Formula: see text] be a given input threshold. The largest common point set problem (LCP) seeks the largest subsets [Formula: see text] and [Formula: see text] such that [Formula: see text] and there exists a transformation [Formula: see text] that makes the bottleneck distance between [Formula: see text] and [Formula: see text] at most [Formula: see text]. We present two algorithms that solve a relaxed version of this problem under translations in [Formula: see text] and under rigid motions in the plane, and that takes an additional input parameter [Formula: see text]. Let [Formula: see text] be the largest subset size achievable for the given [Formula: see text]. Our algorithm finds subsets [Formula: see text] and [Formula: see text] of size [Formula: see text] and a transformation [Formula: see text] such that the bottleneck distance between [Formula: see text] and [Formula: see text] is at most [Formula: see text]. For rigid motions in the plane, the running time is [Formula: see text]. For translations in [Formula: see text], the running time is [Formula: see text], where [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text].

An Efficient Boundary Integral Scheme for the MBO Threshold Dynamics Method via the NUFFT

The MBO threshold dynamics method consists of two steps. The first step solves a pure initial value problem of the heat equation with the initial data being the indicator function of some bounded domain. In the second step, the new sharp interface is generated via thresholding either by some prescribed solution value or by volume preserving. We propose an efficient boundary integral scheme for simulating the threshold dynamics via the nonuniform fast Fourier transform (NUFFT). The first step is carried out by evaluating a boundary integral via the NUFFT, and the second step is performed applying a root-finding algorithm along the normal directions of a discrete set of points at the interface. Unlike most existing methods where volume discretization is needed for the whole computational domain, our scheme requires the discretization of physical space only in a small neighborhood of the interface and thus is meshfree. The algorithm is spectrally accurate in space for smooth interfaces and has \(O(N\log N)\) complexity, where N is the total number of discrete points near the interface when the time step \(\Delta t\) is not too small. The performance of the algorithm is illustrated via several numerical examples in both two and three dimensions.

On the existence of optimal meshes in every convex domain on the plane

In this paper we study the so called optimal polynomial meshes for domains in K⊂Rd,d≥2. These meshes are discrete point sets Yn of cardinality cnd which have the property that ‖p‖K≤A‖p‖Yn for every polynomial p of degree at most n with a constant A>1 independent of n. It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d=2.

Monte Carlo simulation of radiative transfer in a medium with varying refractive index specified at discrete points

• Monte Carlo simulation for radiative transfer in refractive media is presented. • Discrete ray tracing using scattered discrete refractive indices is developed. • Refractive-index values and gradients are retreived by a moving least square method. • Ray-tracing results for various irregularly-spaced refractive indices are examined. A Monte Carlo method with discrete ray tracing is developed to simulate radiative transfer in a medium with a spatially varying refractive-index distribution available merely at a set of arbitrary discrete points. We solve the ray equation by a Runge−Kutta Dormand−Prince method to carry out the numerical ray tracing. To retrieve the refractive-index values and gradients needed in the discrete ray tracing, we apply cubic spline interpolation for one-dimensional simulation and a moving least square (MLS) method for two-dimensional simulation. The influence of the basis vectors and the numbers of sampled data used by the MLS method on ray tracing based on the retrieved refractive-index values and gradients has been examined. The results of radiative equilibrium in a planar medium and radiative transfer in two-dimensional media with different geometries and conditions obtained by the present methods are compared with those obtained by solving the integral equations of radiative transfer and the discrete ordinates method. The comparisons show that the present methods generate accurate results for radiative transfer with various geometries, parameters and refractive-index distributions specified at discrete points.

Optimal design of continuous time irrational filter with a set of fractional order gammatone components via norm relaxed sequential quadratic programming approach

This paper proposes a continuous time irrational filter structure via a set of the fractional order Gammatone components instead of via a set of integer order Gammatone components. The filter design problem is formulated as a nonsmooth and nonconvex infinite constrained optimization problem. The nonsmooth function is approximated by a smooth operator. The domain of the constraint functions is sampled into a set of finite discrete points so the infinite constrained optimization problem is approximated by a finite constrained optimization problem. To find a near globally optimal solution, the norm relaxed sequential quadratic programming approach is applied to find the locally optimal solutions of this nonconvex optimization problem. The current or the previous locally optimal solutions are kicked out by adding the random vectors to them. The locally optimal solutions with the lower objective functional values are retained and the locally optimal solutions with the higher objective functional values are discarded. By iterating the above procedures, a near globally optimal solution is found. The designed filter is applied to perform the denoising. It is found that the signal to noise ratio of the designed filter is higher than those of the filters designed by the conventional gradient descent approach and the genetic algorithm method, while the required computational power of our proposed method is lower than those of the conventional gradient descent approach and the genetic algorithm method. Also, the signal to noise ratio of the filter with the fractional order Gammatone components is higher than those of the filter with the integer order Gammatone components and the conventional rational infinite impulse response filters.

A Functional Estimate of Covariation

ABSTRACTThe analysis of functional data calls for a bivariate functional covariance function σ(s, t) that may be evaluated at any discrete set of points to define a variance-covariance matrix This article uses finite element methodology to construct a representation of a functional Choleski factor λ(w, s) to define σ(s, t) = ∫λ(w, s)λ(w, t) dw. An estimate of is especially important for applications and, where the eigenstructure of the covariance permits, this is readily available since the resulting is almost always positive definite. A simulation study compares the performance of estimates of and to those from the classic covariance matrix estimate and an estimate using glasso package in R. The method’s capability of constraining estimates of to be strongly band-structured resulted in superior estimates. The real data application is to the smoothing of the Fels female growth data where σ(s, t) estimates the residual covariance structure in the presence of sampling points varying from one case to another. Supplementary materials are available online.

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gerard Cohen influenced important parts of my research. The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalising the notion of separating systems in the context of metric convexity. The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry - sets of points forming only acute angles - who seem not to be aware of each other, while they are working on precisely the same problem! Of course, these three themes are closely intertwined.

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gerard Cohen influenced important parts of my research. The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalizing the notion of separating systems in the context of metric convexity. The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry --- sets of points forming only acute angles --- who seem not to be aware of each other, while they are working on precisely the same problem! Of course, these three themes are closely intertwined.

Comparison of the Stress Concentration Factors for GFRP Plate having Centered Circular Hole by Three Resource-Conserving Methods

Fiber reinforced plastic (FRP) composites have drawn increasing attentions worldwide for decades due to its outstanding properties. Stress concentration factor (SCF) as an essential parameter in materials science are critically considered in structure design and application, strength assessment and failure prediction. However, investigation of stress concentration in FRP composites has been rarely reported so far. In this study, three resource-conserving analyses (Isotropic analysis, Orthotropic analysis and Finite element analysis) were introduced to plot the <TEX>$K_T^A-d/W$</TEX> curve for E-glass/epoxy composite plate with the geometrical defect of circular hole placed centrally. The plates were loaded to uniaxial direction for simplification. Finite element analysis (FEA) was carried out via ACP (ANSYS composite prepost module). Based on the least squares method, a simple expression of fitting equation could be given based on the simulated results of a set of discrete points. Finally, all three achievable solutions were presented graphically for explicit comparison. In addition, the investigation into customized efficient SCFs has also been carried out for further reference.