Discrete Set Of Points Research Articles

Fractal Dimension Through Numerical Integration of Fractal Interpolation Functions

The initial objective of the paper is to propose an explicit relationship between the fractal dimension and fractal numerical integration of curves approximated through fractal interpolation from a discrete set of data points. Once the proposed relation is established, it is shown to be accurate by considering the data points of certain functions. The conventionalbox-counting dimension method has several drawbacks including the proper positioning of the boxes and determining the size of the boxes. The proposed relation becomes significant for such situations in providing the accurate determination of fractal dimension. Secondly, the paper aims to apply the derived relationship in the evaluation of two integral transforms of fractal interpolation functions, namely, the Fourier transform and the Laplace transform. The two integral transforms have been provided with alternate expressions, primarily, using the fractal dimension of fractal interpolation functions. Finally, considering these newly derived expressions and the proposed relation between fractal dimension and fractal numerical integration, this paper provides another method for the evaluation of the two integral transforms. This newly introduced method calculates the two integral transforms, via the fractal numerical integration of fractal interpolation functions.

2D nonlinear Acoustic Wave equation in heterogeneous fluid

In this work, the non-linear equation of Acoustic Wave propagation in a heterogeneous fluid (CHACALTANA et al., 2015; PERES et. al., 2023) is solved in two dimensions by Finite Element Method (FEM). The Weighted Residual Method (Petrov-Galerkin) and linear and parabolic approximation basis functions are used to minimize the residue in each element. Ricker-type pressure source (CHACALTANA et al., 2015; PICOLLI et al., 2020) and the sinusoidal type are used to generate the P-wave. Reflective Neumann (natural) and non-reflective ABC (Absorbing Boundary Condition) boundary conditions based on the non-reflective boundaries of Reynolds (1978) were implemented. The numerical code is written in Fortran 95 language and the Octave graphical interface is used to analyze the results. The GMSH mesher (v. 4.8.4) is used to represent the continuous domain by a set of discrete points that group together form a non-uniform mesh of triangular elements. Numerical tests on square and circular geometries are performed to verify the implementations of the Neumann and ABC boundary conditions. Finally, a good agreement is found between the results of numerical simulations when compared with existing results in the literature.

Inverse problem of reconstructing source term for a class of non‐divergence parabolic equations

This paper explores an inverse problem pertaining to the determination of a source function in non‐divergence parabolic equations, where the solution is known at a discrete set of points. Being different from other ordinary inverse source problems, which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable but also depends on the time . On the basis of the optimal control framework, the existence of the optimal solution of the control function is proved. The necessary conditions to be satisfied by the optimal solution are given. The convergence of the optimal solution when the mesh parameters tend to zero is obtained. The conjugate gradient method is applied to the inverse problem and some numerical results are presented for various typical test examples.

Improving modular bootstrap bounds with integrality

We propose methods that efficiently impose integrality — i.e., the condition that the coefficients of characters in the partition function must be integers — into numerical modular bootstrap. We demonstrate the method with a number of examples where it can be used to strengthen modular bootstrap results. First, we show that, with a mild extra assumption, imposing integrality improves the bound on the maximal allowed gap in dimensions of operators in theories with a U(1)c symmetry at c = 3, and reduces it to the value saturated by the SU(4)1 WZW model point of c = 3 Narain lattices moduli space. Second, we show that our method can be used to eliminate all but a discrete set of points saturating the bound from previous Virasoro modular bootstrap results. Finally, when central charge is close to 1, we can slightly improve the upper bound on the scaling dimension gap.

USING THE CUBIC SPLINE INTERPOLATION METHOD TO APPROXIMATE SOME REAL DATA

In engineering and science problems, the data being considered are of-ten known in the form of sets of discrete points, not as a continuous function. In applications, the values of the discrete data at specific points may be required to estimate. To solve these problems, a common mathematical ap-proach is to use the interpolation method. In this paper, we use the cubic natural spline interpolation method to build approxi-mate functions for some real data in Vietnam such as data on rice out-put and rice area of cultivation, data on expenditure and income per capita. The data used in this paper are extracted from the General Statis-tics Office of Vietnam. We develop some Matlab programs to find the coefficients of splines and calculate the values of the interpolation functions at specific points. Using the obtained interpolation function, some missing values in the data on income and expenditure per capita in the Southeast region of Vietnam in 2019 and 2021 are estimated. Fur-ther-more, some unknown data on rice productivity and cultivation area are obtained in a similar manner. A good agreement between the calcu-lated values and actual values are found.

Određivanje lančanice nadzemnih strujnih vodova primenom geodetskih metoda merenja

The catenary of overhead power lines is described by cosine hyperbolic function. Cosine hyperbolic function describes the position of overhead power line which it should be taken in the space, and it is suitable in the process of overhead power lines design. The real position of overhead power line it is only possible to be determined by measuring certain values which could be used for determination of their coordinates. Overhead power line in that case is described by discrete set of points, while the parameters of catenary could be modelled on the base of that set of points by utilizing certain mathematical model. In this paper the position determination of power line DV405, which is located downstream of DJerdap 1 dam, is considered. The mathematical form of power line is based on the polynomial regression of second order. The results showed high concordance between values obtained by measurements and the utilized model. Some significant deviation between measured and modelled values might be explained by relatively large distance between the station point and the measured point of power line.

An atomic surface site interaction point description of non-covalent interactions.

Molecular electrostatic potential surfaces (MEPS) calculated using density functional theory have been used to develop a simplified description of the non-covalent interaction properties of organic molecules. The Atomic Interaction Point (AIP) model introduced here represents an evolution of the Surface Site Interaction Point (SSIP) model described previously, in which a molecule is represented by a discrete set of interaction points that define sites of interaction with other molecules. The interaction sites are described by interaction parameters that are equivalent to the experimentally determined H-bond donor and acceptor parameters α and β. By using high electron density MEPS that lie inside the van der Waals surface, it is possible to obtain accurate interaction parameters and locations for polar sites (s-holes, H-bond donors and acceptors), which are identified as local maxima and minima on the MEPS. For non-polar sites that represent π-systems and halogens, an approach based on molecular orbitals was used to assign the locations of the AIPs, and the interaction parameters were obtained using a lower electron density MEPS that lies close to the van der Waals surface. The AIP descriptions can be implemented directly in the Surface Site Interaction Point Model for Liquids at Equilibrium (SSIMPLE) to calculate solvation free energies, and the free energy of transfer of 1504 compounds from n-hexadecane to water was predicted with a root mean square error of 5 kJ mol-1. AIPs also provide a useful tool for mapping non-covalent interactions in intermolecular complexes, and examples are provided showing how X-ray crystal structures can be converted into AIP interaction maps that allow quantification of the free energy contributions of both polar and non-polar interactions to the stabilities of complexes in solution.

GRAPHICAL AND ANALYTICAL DEFINITION OF PARAMETRIC CURVES IN EXPLICIT FORM

The aim of the work is to establish the relationship between graphical and analytical methods of solving problems on the example of the definition of parametric curves in explicit form. The idea is to represent the system of parametric equations of a plane curve in the space of Otxy coordinate system as a system of two projections of the curve on a complex Monge drawing. Then the desired curve can be obtained explicitly as a discrete set of points on the profile plane of projections in the Otxy coordinate system. If we consider the system of curve projections as a single curve in three-dimensional space Otxy, then this curve can be flat and spatial. For planar curves located in a partial position plane, it is easy to make the transition from parametric setting of the curve to an explicit form due to the linear relationship between one of the variables (x or y) and the parameter t. If the curve in the Otxy coordinate system belongs to the general position plane, we can use the projection plane replacement method to transform it to a projecting position. This allows a linear relationship to be established between one of the variables and a parameter to obtain the equation of the curve in explicit form, followed by an inverse transformation of the planes of projections. Mathematically, replacing the plane of projections is equivalent to replacing the coordinate system and reduces it to a rotation around the origin of the coordinates. Formulas for transforming the coordinate system, taking into account the peculiarities of using Monge's complex drawing and for determining the angle of rotation of the coordinate system are given.

Approximation of holomorphic Legendrian curves with jet-interpolation

We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in C2n+1 lies on an injectively immersed isotropic surface with a prescribed complex structure. If the set has no accumulation points, the surface may be taken properly embedded. We also prove a Carleman-type theorem for holomorphic Legendrian curves with interpolation. Namely, a Legendrian curve, defined on a certain type of unbounded closed set in a given open Riemann surface R, may be approximated in the C0-topology by an entire Legendrian curve with prescribed finite-order Taylor polynomials at a closed discrete set of points in R. Under suitable conditions, the approximating map may be made into a proper embedding.

Polyharmonic splines interpolation on scattered data in 2D and 3D with applications

Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines r2nlnr and r2n+1 “failed to produce acceptable results”.In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions’ values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).

Grid-based Atmospheric Retrievals for Reflected-light Spectra of Exoplanets Using PSGnest

Techniques to retrieve the atmospheric properties of exoplanets via direct observation of their reflected light have often been limited in scope owing to computational constraints imposed by the forward-model calculations. We have developed a new set of techniques that significantly decrease the time required to perform a retrieval while maintaining accurate results. We constructed a grid of 1.4 million precomputed geometric albedo spectra valued at discrete sets of parameter points. Spectra from this grid are used to produce models for a fast and efficient nested sampling routine called PSGnest. Beyond the upfront time to construct a spectral grid, the amount of time to complete a full retrieval using PSGnest is on the order of seconds to minutes using a personal computer. An extensive evaluation of the error induced from interpolating intermediate spectra from the grid indicates that this bias is insignificant compared to other retrieval error sources, with an average coefficient of determination between interpolated and true spectra of 0.998. We apply these new retrieval techniques to help constrain the optimal bandpass centers for retrieving various atmospheric and bulk parameters from a LuvEx-type mission observing several planetary archetypes. We show that spectral observations made using a 20% bandpass centered at 0.73 μm can be used alongside our new techniques to make detections of H2O and O2 without the need to increase observing time beyond what is necessary for a signal-to-noise ratio of 10. The methods introduced here will enable robust studies of the capabilities of future observatories to characterize exoplanets.

Quantum Dynamical Investigation of Dihydrogen-Hydride Exchange in a Transition-Metal Polyhydride Complex.

The ongoing shift toward clean, sustainable energy is a primary driving force behind hydrogen fuel research. Safe and effective storage of hydrogen is a major challenge (particularly for mobile applications) and requires a detailed understanding of the atomic level interactions of hydrogen with its host materials. The light mass of hydrogen, however, implies that quantum effects are important, so a quantum dynamical treatment is required to properly account for these effects in computational simulations. As one such example, we describe herein the hydrogen exchange dynamics between a hydride and a dihydrogen ligand in the [FeH(H2)(PH3)4]+ model complex. A global three-dimensional (3D) potential energy surface (PES) was constructed by fitting to and interpolating from a discrete set of grid points computed using density functional theory; exact quantum dynamical calculations were then carried out on the 3D PES using discrete variable representation basis sets. Energy levels and their quantum tunneling splittings were computed up to 3000 cm-1 above the ground state. Within that energy range, all three fundamentals have been identified using wave function plots, as well as the first three overtones of the exchange (reaction coordinate) motion and several of its combination bands. From the tunneling splittings, the Boltzmann-averaged tunneling rates were computed. The Arrhenius plot of the total exchange rate shows a clear transition around 150 K, below which the activation energy is essentially zero and above which it is less than half of the electronic structure barrier. This indicates that exchange rates are governed by quantum tunneling throughout the relevant temperature range with the low-temperature regime dominated by a single quantum (ground) state. This work is the first-ever fully quantum dynamical study to investigate the hydrogen exchange dynamics between hydride and dihydrogen ligands coordinated to a transition-metal complex.

THREE-DIMENSIONAL COMPOSITION MATRICES AND THREE-PARAMETER POINT POLYNOMIALS

It is informed that three-dimensional composite matrices (comp matrices) are intended for the formation, in compact records, of three-parameter point polynomials and that the basis for compiling three-dimensional composite matrices is always a real three-dimensional object, the elements of which have location-time characteristics. These objects are represented as a discrete set of base points with an unlimited finite number of coordinates. Such a set of base points is a geometric composition, which is a frame of points of this object. On the basis of the frame of points, frames of lines are created in three parametric directions, which are parameterized, and because of this, each base point is determined by three parameters and has a triple index. An example of a geometric body is given, on which the rules for indexing its base points are explained. Detailed explanations are provided for the creation of a point compomatrix for this geometric body, its conventional notation and indexing of the elements of this compomatrix. Explanations are also provided for the creation of three parametric compo-matrices corresponding to the point compo-matrix, their notation and indexing of the elements of these compo-matrices. It is emphasized that each element of the point compomatrix corresponds to an element of each of the three parametric compomatrixes. It is shown that the product of the elements of the point compomatrix by the corresponding elements of the parametric compomatrix gives the elements of all the edges, along the corresponding parametric direction, which are part of the geometric body. The formation of compomatrixes of a geometric figure for the surfaces included in the composition of a geometric body, and which can be presented as its basic states, is shown. It is emphasized that the formation of elements of compomatrixes of geometric shapes for both edges and surfaces is carried out by multiplying the elements-points and elements-characteristic functions only of those that have the same triple indices. An explanation is provided for the formation of the compomatrix of a geometric figure for a geometric body, the elements of which are also obtained by multiplying the elements-points and elements-characteristic functions, in three parametric directions, only those with the same triple indices. It is indicated that the sum of all elements of the compomatrix of a geometric figure is a three-parameter point polynomial, which continuously determines the current points both on the surface of the geometric body and inside it. It is proposed to submit three-dimensional comp matrices in the form of two-comp matrices. An explanation of the creation of two-comp matrices and the implementation of operations on them is given.

Finite temperature negativity Hamiltonians of the massless Dirac fermion

The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation of mixed-state entanglement. However, so far, it has only been studied for the mixed-state density matrices corresponding to subsystems of globally pure states. Here, we consider as a genuine example of a mixed state the one-dimensional massless Dirac fermions in a system at finite temperature and size. As subsystems, we consider an arbitrary set of disjoint intervals. The structure of the corresponding negativity Hamiltonian resembles the one for the entanglement Hamiltonian in the same geometry: in addition to a local term proportional to the stress-energy tensor, each point is non-locally coupled to an infinite but discrete set of other points. However, when the lengths of the transposed and non-transposed intervals coincide, the structure remarkably simplifies and we retrieve the mild non-locality of the ground state negativity Hamiltonian. We also conjecture an exact expression for the negativity Hamiltonian associated to the twisted partial transpose, which is a Hermitian fermionic matrix. We finally obtain the continuum limit of both the local and bi-local operators from exact numerical computations in free-fermionic chains.

Exact solution approaches for the discrete α-neighbor p-center problem.

The discrete -neighbor -center problem (d--CP) is an emerging variant of the classical -center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate facilities on these points in such a way that the maximum distance between each point where no facility is located and its -closest facility is minimized. The only existing algorithms in literature for solving the d--CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d--CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&Calgorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances.

Qualitative inverse problems: mapping data to the features of trajectories and parameter values of an ODE model

Our recent work on linear and affine dynamical systems has laid out a general framework for inferring the parameters of a differential equation model from a discrete set of data points collected from a system being modeled. It introduced a new class of inverse problems where qualitative information about the parameters and the associated dynamics of the system is determined for regions of the data space, rather than just for isolated experiments. Rigorous mathematical results have justified this approach and have identified common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka–Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter and dynamical features in the data space that we call the P n -diagram, which is particularly useful for visualization of the qualitative dependence of the system dynamics on data for low-dimensional (small n) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associated P 2-diagrams.

Reconstruction of integrated hydraulic turbine characteristics curve based on classification weight neural network

To address the shortcomings in the drawing and processing of comprehensive characteristic curves of hydraulic turbines and the problems in the operation and management of small hydropower stations, this paper proposes a method for reconstructing the comprehensive characteristic curves based on classification weight neural network. Firstly, this study discretizes the comprehensive characteristic curve of the target turbine factory package to obtain multiple sets of discrete data points, and compares them with the actual operating data of small hydropower plants, and then pre-processes the operating data by the type and operating characteristics of the unit. Next, this paper uses XGBoost to classify the two types of data to obtain the leaf node optimal scores and uses the scores as the prediction weights of the deep neural network to grid the discrete points of the hydraulic turbine for prediction, and finally obtains the specific integrated characteristic graph of the unit. The method based on classification weight neural network can substantially improve the accuracy and reliability of composite characteristic curves while avoiding multivariate nonlinear mathematical problems, which provides a new method and idea for the reconstruction and expansion of the hydraulic turbine characteristic curve, and also provides a theoretical basis for the safe and economic operation of hydropower units.

Hydrodynamics for the ABC model with slow/fast boundary

In this article, we consider the ABC model in contact with slow/fast reservoirs. In this model, there is one particle per site, which can be of type α∈{A,B,C} and particles exchange positions in the discrete set of points {1,…,N−1} with a rate which is weakly asymmetric and depends on the type of particles involved in the exchange mechanism. At the boundary points x=1,N−1 particles can be injected or removed, and the rate at which this happens depends on the type of particles involved. We prove the hydrodynamic limit, in the diffusive time scale, given by a system of non-linear coupled equations with several boundary conditions, that depend on the strength of the reservoir’s action.

Digital Library Services in University Libraries in Nigeria: A Literature Study

Digitization of information resources for making digital library is the process of converting information into a digital (i.e. computer-readable) format. The result is the representation of an object, image, sound, document, or signal (usually an analog signal) obtained by generating a series of numbers that describe a discrete set of points or samples. The result is called digital representation or, more specifically, a digital image, for the object, and digital form, for the signal. On the one hand it can be argued that dramatic changes have been made at academic libraries in response to the new digital information environment. Most academic libraries, including University Libraries in Nigeria, offers a wide range of digital services and resources. But on the other hand, at a closer examination, one will find that these services and resources are mainly organized according to traditional library principles. In this case a guiding principle for the digital library need to be set up to create systematic and well organized general services and e-resources, for example subject specific gateway services. Thus, the digital part of the library functioned as an extension of the traditional library, an enterprise on the side with relatively high priority in this digital age, especially for of e-journals

An immersed boundary-lattice Boltzmann flux solver for simulation of flows around structures with large deformation

In this paper, we present an immersed boundary-lattice Boltzmann flux solver (IB-LBFS) to simulate the interactions of viscous flow with deformable elastic structures, namely, two-dimensional (2D) and three-dimensional (3D) capsules formed by elastic membranes. The IB-LBFS is based on a finite-volume formulation and makes use of hydrodynamic conservation equations with fluxes computed by a kinetic approach; thus, it is more flexible and efficient than the standard immersed boundary-lattice Boltzmann methods. The membrane of the 2D capsule is represented by a set of discrete Lagrangian points, with in-plane and bending forces acting on the membrane obtained by a finite difference method. In contrast, the membrane of a 3D capsule is discretized into flat triangular elements with membrane forces calculated by an energy-based finite-element method. The IB-LBFS is first validated by studying the deformation of a circular capsule in a linear Newtonian and a power-law shear flow. Next, the deformation dynamics of a spherical, an oblate spheroidal, and a biconcave capsule in a simple shear flow are simulated. For an initially spherical capsule, the tank-treading motion of its membrane is reproduced at the steady state; while for oblate spheroidal and biconcave capsules, the swinging and tumbling motions are observed. Furthermore, under certain parameter settings, the transient mode from tumbling to swinging motions is also found, showing a rich and complex dynamic behavior of non-spherical capsules. These results indicate that the IB-LBFS can be employed in future studies concerning the dynamics of a capsule suspension in more realistic flows.