Arbitrary Number Of Points Research Articles

Proposed theoretical value for TSP constant

The traveling salesman problem (TSP) involves determining the shortest length (optimal tour) for a complete circuit through an arbitrary number of points. In the special case that the points to be visited are randomly distributed in the plane within a unit square, and travel costs correspond to the Euclidean distance between points, it is known that the length of the optimal tour divided by the square root of the number of points ([Formula: see text]) asymptotically approaches a constant ([Formula: see text] as [Formula: see text] becomes large. If individual points are randomly drawn from a uniform distribution, the optimal lengths for a sufficiently large number of TSP instances of the same size ([Formula: see text] comprise a normal distribution whose mean approaches [Formula: see text] as [Formula: see text] increases. Moreover, as [Formula: see text] approaches infinity, this distribution gradually narrows such that its standard deviation asymptotically approaches zero. Although the precise value of [Formula: see text] is unknown, published studies indicate its magnitude is approximately 0.712. In this paper, possibly for the first time, precise formulae are proposed for the parameters of the underlying normal distribution and [Formula: see text], based on intuition, mathematical reasoning, and empirical fits to both published and experimentally derived data.

Arbitrary-Scale Point Cloud Upsampling by Voxel-Based Network with Latent Geometric-Consistent Learning

Recently, arbitrary-scale point cloud upsampling mechanism became increasingly popular due to its efficiency and convenience for practical applications. To achieve this, most previous approaches formulate it as a problem of surface approximation and employ point-based networks to learn surface representations. However, learning surfaces from sparse point clouds is more challenging, and thus they often suffer from the low-fidelity geometry approximation. To address it, we propose an arbitrary-scale Point cloud Upsampling framework using Voxel-based Network (PU-VoxelNet). Thanks to the completeness and regularity inherited from the voxel representation, voxel-based networks are capable of providing predefined grid space to approximate 3D surface, and an arbitrary number of points can be reconstructed according to the predicted density distribution within each grid cell. However, we investigate the inaccurate grid sampling caused by imprecise density predictions. To address this issue, a density-guided grid resampling method is developed to generate high-fidelity points while effectively avoiding sampling outliers. Further, to improve the fine-grained details, we present an auxiliary training supervision to enforce the latent geometric consistency among local surface patches. Extensive experiments indicate the proposed approach outperforms the state-of-the-art approaches not only in terms of fixed upsampling rates but also for arbitrary-scale upsampling. The code is available at https://github.com/hikvision-research/3DVision

Analyzing the effect of shot noise in indirect Time-of-Flight cameras

Continuous wave indirect Time-of-Flight cameras obtain depth images by emitting a modulated continuous light wave and measuring the delay of the received signal. In this paper we generalize the estimation of the effect of the shot noise when obtaining the phase delay with an arbitrary number of points in the Discrete Fourier Transform, extending and generalizing the analysis done in previous works for the case of four points. For that particular case, we compare our analysis with the state of art. Moreover, we extend the error model using a second order approximation in the error propagation analysis, which provides more accurate estimations according to the Montecarlo simulation experiments. The analysis, based on both analytical and numerical methods, shows that the phase error is, in general, related to the exposure time and weakly to the number of points in the Discrete Fourier Transform. It also depends on the background illumination level, on the amplitude of the received signal, and, when using a three point DFT, on the distance to the objects.

Adjoint Based Aerodynamic Shape Optimisation Using Kinetic Meshfree Method

The gradient based optimisation algorithms combined with the finite volume or element based adjoint approaches have been very successful in aerodynamic shape optimization (ASO). The meshfree least squares kinetic upwind method (LSKUM), which works on a cloud of points and its connectivity, has the advantage in terms of flexibility of generating a cloud of points and having an arbitrary number of points in the connectivity. The LSKUM based meshfree solvers have been successfully applied to problems involving multi-body configurations and moving boundaries. In the present work, the LSKUM based primal and discrete adjoint meshfree solvers for steady Euler flows have been used to perform aerodynamic shape optimisation. It is well-known that the raw shape sensitivities of a defined objective function are highly oscillatory. In this research, we have used the Sobolev gradient smoothing. An approach is developed to find a suitable choice for the smoothing parameter in the Sobolev gradient algorithm. Numerical results are presented for the shape optimisation of the NACA0012 airfoil at transonic and supersonic flows with aerodynamic and geometric constraints. The performance of the discrete adjoint LSKUM solver in accurate computation of shape sensitivities is demonstrated. The advantages of using LSKUM in aerodynamic shape optimisation have been shown.

Hypercomplex numbers in some geometries of two sets. I

The most important problem in the theory of phenomenologically symmetric geometries of two sets is that of classification of these geometries. In this paper, complexifying the metric functions of some known phenomenologically symmetric geometries of two sets (PSGTS) with the use of associative hypercomplex numbers, we find metric functions of new geometries in question. For these geometries, we find equations of the groups of motions and establish phenomenological symmetry, i.e., find functional relations between metric functions for certain finite number of arbitrary points. In particular, for one-component metric functions of PSGTS’s of ranks (2, 2), (3, 2), (3, 3), we find (n + 1)-component metric functions of the same ranks. For these metric functions, we find finite equations of the groups of motions and equations that express their phenomenological symmetry.

Proving and Generalizing Desargues’ Two-Triangle Theorem in 3-Dimensional Projective Space

With the use of only the incidence axioms we prove and generalize Desargues’ two-triangle Theorem in three-dimensional projective space considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the two planes to coincide and three points on any of the planes to be collinear. We provide three generalizations and we define the notions of a generalized line and a triangle-connected plane set of points.

Existence and construction of shifted lattice rules with an arbitrary number of points and bounded weighted star discrepancy for general decreasing weights

We study the problem of constructing shifted rank-1 lattice rules for the approximation of high-dimensional integrals with a low weighted star discrepancy, for classes of functions having bounded weighted variation, where the weighted variation is defined as the weighted sum of Hardy–Krause variations over all lower-dimensional projections of the integrand. Under general conditions on the weights, we prove the existence of rank-1 lattice rules such that for any δ > 0 , the general weighted star discrepancy is O ( n − 1 + δ ) for any number of points n > 1 (not necessarily prime), any shift of the lattice, general (decreasing) weights, and uniformly in the dimension. We also show that these rules can be constructed by a component-by-component strategy. This implies in particular that a single infinite-dimensional generating vector can be used for integrals in any number of dimensions, and even for infinite-dimensional integrands when they have bounded weighted variation. These same lattices are also good with respect to the worst-case error in weighted Korobov spaces with the same types of general weights. Similar results were already available for various special cases, such as general weights and prime n , or arbitrary n and product weights, but not for the most general combination of n composite, general weights, arbitrary shift, and star discrepancy, considered here. Our results imply tractability or strong tractability of integration for classes of integrands with finite weighted variation when the weights satisfy the conditions we give. These classes are a strict superset of those covered by earlier sufficient tractability conditions.

On the in situ estimation of surface acoustic impedance in interiors of arbitrary shape by acoustical inverse methods

A method for in situ estimation of the acoustic impedance of surfaces in interiors is introduced in this paper. The key difference with traditional in situ measurement techniques is the use of an inverse acoustic boundary framework which allows us to overcome some geometry constraints from previous methods (such as the planar-surface requirement and placement of microphone arrays). Furthermore, estimation of the acoustic impedance of not only one but all the surfaces is possible provided that local reaction is the predominant effect, and the following parameters are known: geometry of the surfaces, sound pressure at a number of arbitrary points in the interior field and the strength of the sound source. The estimation of the acoustic impedance at each surface is achieved by the solution of an optimization problem formulated from the linear equations of the boundary element method (BEM) applied to the discretized interior boundaries of an interior space. Previous work on similar methods have reported examples with numerical simulations. The work in this paper goes further and numerical examples together with results obtained with experimental data are presented.

Spacetime foam in twistor string theory

We show how a Kahler spacetime foam in four dimensional conformal (super)gravity may be mapped to twistor spaces carrying the D1 brane charge of the B model topological string theory. The spacetime foam is obtained by blowing up an arbitrary number of points in $\C^2$ and can be interpreted as a sum over gravitational instantons. Some twistor spaces for blowups of $\C^2$ are known explicitly. In these cases we write down a meromorphic volume form and suggest a relation to a holomorphic superform on a corresponding super Calabi-Yau manifold.

A systematization for one-loop 4D Feynman integrals

We present a strategy for the systematization of manipulations and calculations involving divergent (or not) Feynman integrals, typical of the one-loop perturbative solutions of QFT, where the use of an explicit regularization is avoided. Two types of systematization are adopted. The divergent parts are put in terms of a small number of standard objects, and a set of structure functions for the finite parts is also defined. Some important properties of the finite structures, specially useful in the verification of relations among Green's functions, are identified. We show that, in fundamental (renormalizable) theories, all the finite parts of two-, three- and four-point functions can be written in terms of only three basic functions while the divergent parts require (only) five objects. The final results obtained within the proposed strategy can be easily converted into those corresponding to any specific regularization technique providing an unified point of view for the treatment of divergent Feynman integrals. Examples of physical amplitudes evaluation and their corresponding symmetry relations verification are presented as well as generalizations of our results for the treatment of Green's functions having an arbitrary number of points are considered.

FRETsg: Biomolecular structure model building from multiple FRET experiments

Fluorescence energy transfer (FRET) experiments of site-specifically labelled proteins allow one to determine distances between residues at the single molecule level, which provide information on the three-dimensional structural dynamics of the biomolecule. To systematically extract this information from the experimental data, we describe a program that generates an ensemble of configurations of residues in space that agree with the experimental distances between these positions. Furthermore, a fluctuation analysis allows to determine the structural accuracy from the experimental error. Program summary Title of program: FRETsg Catalogue identifier: ADTU Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTU Computer: SGI Octane, Pentium II/III, Athlon MP, DEC Alpha Operating system: Unix, Linux, Windows98/NT/XP Programming language used: ANSI C No. of bits in a word: 32 or 64 No. of processors used: 1 No. of bytes in distributed program, including test data, etc.: 11407 No. of lines in distributed program, including test data, etc.: 1647 Distribution format: gzipped tar file Nature of the physical problem: Given an arbitrary number of distance distributions between an arbitrary number of points in three-dimensional space, find all configurations (set of coordinates) that obey the given distances. Method of solution: Each distance is described by a harmonic potential. Starting from random initial configurations, their total energy is minimized by steepest descent. Fluctuations of positions are chosen to generate distance distribution widths that best fit the given values.

Parallel Quasirandom Walks on the Boundary

The Monte Carlo method called “random walks on boundary” has been successfully used for solving boundary-value problems. This method has significant advantages when compared with random walks on spheres, balls or on discrete grids when an exterior Dirichlet or Neumann problem is solved, or when we are interested in computing the solution to a problem at an arbitrary number of points using a single random walk. In this paper we will investigate ways: • to increase the convergence rate of this method by using quasirandom sequences instead of pseudorandom numbers for the construction of the boundary walks, • to find an efficient parallel implementation of this method on a cluster using MPI. In our parallel implementation we use disjoint contiguous blocks of quasirandom numbers extracted from a given quasirandom sequence for each processor. In this case, the increased convergence rate does not come at the cost of less trustworthy answers. We also present some numerical examples confirming both the increased rate of convergence and the good parallel efficiency of the method.

Real-time two-dimensional wavelet compression and its application to real-time modeling of ion mobility data

Real-time data analysis is important in many applications. However, many chemometric algorithms have difficulty processing data in real-time. A novel real-time two-dimensional wavelet compression (WC 2) has been developed to compress data as it is acquired from analytical instrumentation. The WC algorithm was enhanced so that data with an arbitrary number of points were compressed, and truncation or padding to a dyadic number was avoided. After compression, the noise level is reduced while useful chemical information is retained. A modified simple-to-use interactive self-modeling mixture analysis (SIMPLISMA) algorithm was applied to the wavelet-compressed data and the model was transformed back to the original representation while leaving the data compressed. The reduced size of the wavelet-compressed data furnished a faster implementation of SIMPLISMA that facilitates real-time acquisition. This real-time WC 2-SIMPLISMA algorithm was applied to the rapid identification of explosives by ion mobility spectrometry (IMS). SIMPLISMA resolved concentration profiles and component spectra were displayed simultaneously while the data was acquired from an ion mobility spectrometer with a LabVIEW virtual instrument (VI).

A General Decomposition Theorem for the k-Server Problem

The first general decomposition theorem for the k-server problem is presented. Whereas all previous theorems are for the case of a finite metric with k+1 points, the theorem given here allows an arbitrary number of points in the underlying metric space. This theorem implies O(polylog(k))-competitive randomized algorithms for certain metric spaces consisting of a polylogarithmic number of widely separated subspaces and takes a first step toward a general O(polylog(k))-competitive algorithm. The only other cases for which polylogarithmic competitive randomized algorithms are known are the uniform metric space and the weighted cache metric space with two weights.

Use of window addressable memories for high speed geometrical analysis

Many real-time problems require assessment of the location of a moving platform with respect to a fixed or time varying set of objects. Examples of such problems are robots performing precision movement in a dense object space or low flying aircraft over enemy territory. The method described in this paper permits quick assessment of relative locations to two- or three-dimensional rectangular shaped objects with respect to an arbitrary number of points and subsequently to a real or fictive line of sight. The relative location assessment can be performed at a rate of up to 5 ns/data point/object and the line intersect evaluation at 5---500 ns/line/object dependent upon the geometry. Both the relative assessment algorithm and the intersect algorithm have been implemented in a small hardware unit called the TIGER, Three-dimensional Intersect & Geometrical Evaluation in Real-time, that can be incorporated in unmanned as well as manned systems in space, under water, and avionics. All quoted performance data is based upon analysis in three dimensional Cartesian space unless otherwise stated.

On Generating Test Problems for Nonlinear Programming Algorithms

We present techniques for the generation of test problems which provide a way of constructing nonlinear programming problems from a wide variety of given functions. These techniques permit one to specify an arbitrary number of points, at each one of which the problem should satisfy some “interesting”conditions (i.e. be optimal or stationary or degenerate), and to determine the characteristics of functions and derivatives at these points (i.e. choose predetermined values for functions, gradients, Hessians and Lagrange multipliers). Moreover, there is a limited capacity to create test problems with special structure.We give a sample of results obtained by using these techniques to generate test problems for two algorithms to solve problems having nonlinear-least-squares structure with nonlinear constraints.

Baryzentrische Formeln zur Trigonometrischen Interpolation (I)

Barycentric formulas for the interpolation of a periodic functionf by a trigonometric polynomial have been given by Salzer [11] in the case of an odd number of arbitrary (interpolating) points and by Henrici [7] in the special case of equidistant points. We present here formulas for the interpolation with an even number of arbitrary points as well as simpler versions for an even or an odd functionf.

Solution of the inverse boundary-value problem of heat conduction in an overdefined formulation

An interaction scheme is considered for the solution of a nonlinear inverse heat-conduction problem with the results of measuring the temperature at an arbitrary number of points within the body taken into account.

A New Algorithm for Discrete Cosine Transform of Arbitrary Number of Points

An alternate algorithm to compute the discrete cosine transform (DCT) of sequences of arbitrary number of points is proposed. The algorithm consists of partitioning the DCT kernel into submatrices which by proper row and column shuffling and negations can be made equivalent to the group tables (or parts of them) of appropriate Abelian groups. The computations pertaining to the submatrices can be carried out using multidimensional cyclic convolutions. Algorithms are also developed to perform the computations associated with the submatrices that are parts of larger group tables. The new algorithms are more versatile and generally better in terms of the computational complexity in comparison with the existing algorithms.

Vibration of a rectangular plate supported at an arbitrary number of points

A method is described for establishing the natural frequencies of a rectangular plate supported at points. The number and the location of these points may be completely arbitrary. The method is based on some extensions to the intermediate problem technique of Aronszjan and Weinstein [1–4] through the use of finite sets of constraints, which has been developed recently [5–8]. The method is called the modal constraint method. The merits of this method lie in the fact that the eigenvalues and eigenfunctions of a completely free vibrating rectangular plate are used as the reference structure. The modifications associated with the point supports are taken into account by Lagrangian generalized forces of constraint acting on the reference structure. The method has been verified with many known solutions. Furthermore the convergence is very fast with any desirable accuracy to exact known natural frequencies.