Algebraic Interpolation Research Articles

A bound for the torsion on subvarieties of abelian varieties

We give a uniform bound on the degree of the maximal torsion cosets for subvarieties of an abelian variety. The proof combines algebraic interpolation and a theorem of Serre on homotheties in the Galois representation associated to the torsion subgroup of an abelian variety.

Estimating the Lebesgue constant for the Chebyshev distribution of nodes

In this paper an approach to estimation of the Lebesgue constant for the Lagrange interpolation process with nodes in the zeros of Chebyshev polynomials of the first kind is done. Two-sided estimation of this constant is carried out by using the logarithmic derivative of the Euler gamma function and of the Riemann zeta function. The choice of interpolation nodes is due to the fact that with a fixed number of Chebyshev nodes, the Lebesgue constant tends to its minimum value, thus reducing the error of algebraic interpolation and providing less sensitivity to rounding errors. The expressions for the upper and the lower bounds of this constant are represented as finite sums of an asymptotic alternating series. Based on the expressions obtained, these boundaries are calculated depending on the number of nodes of the interpolation process. The error of each of the boundaries’ value is estimated based on the first discarded term in the corresponding asymptotic series. The results of the calculations are presented in tables showing deviations of the Lebesgue constant from its lower and upper estimated bounds. Dependence of the values’ errors on the number of Chebyshev nodes is depicted in these tables as well. It is numerically shown that with an increase in the number of these nodes, the estimation boundaries rapidly get close to each other. The presented results can be used in the theory of interpolation to estimate the norm of the operator matching a function to its interpolation polynomial and to estimate a deviation of the constructed perturbed polynomial from the unperturbed one.

Interpolation and duality in algebras of multipliers on the ball

We study the multiplier algebras A(\mathcal{H}) obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces \mathcal{H} on the ball \mathbb{B}_d of \mathbb{C}^d . Our results apply, in particular, to the Drury–Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of A(\mathcal H) in terms of the complementary bands of Henkin and totally singular measures for \operatorname{Mult}(\mathcal{H}) . This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact \operatorname{Mult}(\mathcal{H}) -totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is \operatorname{Mult}(\mathcal{H}) -totally null.

Physics super-resolution of thermal environment maps in urban area by deep neural network with attention

Real-time micrometeorology predictions are important for heat-stroke risk reduction and safe drone logistics in urban streets. However, the extensive calculation cost hinders operational real-time predictions of micrometeorology. In order to solve this, it is proposed to use a super-resolution simulation system that can provide high-resolution (HR) predictions at the low cost for low-resolution (LR) simulations. The system utilizes the super-resolution that converts LR maps into HR ones by means of machine-learning technologies. The super-resolution technology has been advancing rapidly and its neural networks can now learn physics to some extent. It has, however, the room for improvement. We propose the Pixel Attention Super-Resolution Network (PA-SRN), which gives attention weights to features and space at the same time. The training and test data for the deep neural networks were generated from the building-resolving urban micrometeorology simulations with a multi-scale atmosphere-ocean coupled model named the Multi-Scale Simulator for the Geoenvironment (MSSG). The proposed PA-SRN model has improved learning accuracy compared to a conventional algebraic interpolation and other existing neural network models. In addition, the analysis of attention weight distributions has clarified how the proposed neural network learns the physics in the micrometeorology data.

A numerical investigation on the heat flow in the process of oil-water displacement in SDPSO

Spar Drilling Production Storage and Offloading (SDPSO) is a new type of deep ocean platform developed in recent years. The process of oil-water displacement is used for oil storage and offloading and the research on the accompanying heat flow has been significant. The wax precipitation and solidification at low temperature will particularly affect the flowing of oil in the displacement process. When the heat flow is concerned, numerical simulation requires large computation. It is necessary to develop an efficient numerical method for this calculation. As a kind of interface tracking method, the volume of fluid (VOF) method needs less computing resources compared with other multiphase numerical methods. As for thermal expressions, there are mainly two kinds of governing equations, i.e. temperature equation and enthalpy equation, and two kinds of interpolation scheme of heat conductivity, i.e. algebraic interpolation scheme and harmonic interpolation scheme. There is a need to find out which combination of governing equation and interpolation scheme of heat conductivity would result in a better precision for the heat flow. Therefore, four non-isothermal solvers, corresponding to the four combinations, are established. After comparison with an analytical solution, it is found that the temperature equation together with the harmonic interpolation scheme of heat conductivity results in better precision.

Plaintext related image hybrid encryption scheme using algebraic interpolation and generalized chaotic map

In this paper, a plaintext related image hybrid encryption scheme is proposed based on Lagrange interpolation, generalized Henon map and nonlinear operations of matrices. The proposed scheme consists of three parts. In the first part, a generalized chaotic map is constructed on the basis of Henon map. Using the novel map, a chaotic sequence is built. And then, both the chaotic sequence and the plaintext pixels are used to implement the first nonlinear operation for generating the first cipher matrix associated with the plaintext. By performing an exclusive XOR operation between the original pixels matrix and the first cipher matrix, the diffusion encryption is carried out. In the second part, Lagrange interpolation is used to create the second cipher matrix related to the diffused image; the second nonlinear transformation is developed between the diffused image and the second cipher matrix; and sequence rearrangement is adopted to scramble the diffused image. In the third part, the third nonlinear transformation of matrices based on point operation and rounding operation is implemented on the scrambled image to complete the image encryption. Accordingly, the decryption process is executed by the inverse operations in the opposite order. The proposed algorithm has some distinctive features: a variety of nonlinear tools such as nonlinear polynomial interpolation, nonlinear chaotic map, and nonlinear operations were involved in the scheme. The cryptosystem is designed with the plaintext to enhance the algorithm security. Due to the combination of multiple nonlinear methods and random factors, the scheme is one time pad, which can withstand multiple types of attacks. The algorithm has a clear structure and a simple calculation, so it is easy to program. In addition, encryption simulation and performance analysis are carried out. The feasibility and effectiveness of the algorithm are verified by the simulated results. The security of the algorithm are proved by the objective indicators such as the running time, key space, statistical properties, key sensitivity, and differential analysis, etc.

Interpolation and amalgamation in modal cylindric algebras

Let α be an ordinal and L be a unimodal logic (like S4 or S5). A modal cylindric algebra of dimension α, an LCA α, is a cylindric algebra of dimension α, expanded with α-many L modalities. For a frame (U, R) of L, each k < α, one defines a diamond box operator on . This defines the semantics of the L modalities in set algebras, with the rest of the operations defined like in cylindric set algebras of dimension α. We study interpolation properties for the corresponding predicate logic having α-many variables. Our results are valid for any reflexive L whose frames contain the universal frames (U, U × U ). In particular, they hold for K5CA α, S4CA α (which is an algebraizable extension of topological predicate logic with semantics induced by Alexandrov topologies).

A versatile embedded boundary adaptive mesh method for compressible flow in complex geometry

We present an embedded ghost fluid method for numerical solutions of the compressible Navier Stokes (CNS) equations in arbitrary complex domains. A PDE multidimensional extrapolation approach is used to reconstruct the solution in the ghost fluid regions and imposing boundary conditions on the fluidsolid interface, coupled with a multi-dimensional algebraic interpolation for freshly cleared cells. The CNS equations are numerically solved by the second order multidimensional upwind method. Block-structured adaptive mesh refinement, implemented with the Chombo framework, is utilized to reduce the computational cost while keeping high resolution mesh around the embedded boundary and regions of high gradient solutions. The versatility of the method is demonstrated via several numerical examples, in both static and moving geometry, ranging from low Mach number nearly incompressible flows to supersonic flows. Our simulation results are extensively verified against other numerical results and validated against available experimental results where applicable. The significance and advantages of our implementation, which revolve around balancing between the solution accuracy and implementation difficulties, are briefly discussed as well.

Order theory and interpolation in operator algebras

We continue our study of operator algebras with and contractive approximate identities (cais). In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator algebra and the C*-algebra it generates. In much of this it is not necessary that the algebra have an approximate identity. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.

On the Lebesgue constant of subperiodic trigonometric interpolation

We solve a recent conjecture, proving that the Lebesgue constant of Chebyshev-like angular nodes for trigonometric interpolation on a subinterval [ − ω , ω ] of the full period [ − π , π ] is attained at ± ω , and its value is independent of ω and coincides with the Lebesgue constant of algebraic interpolation at the classical Chebyshev nodes in ( − 1 , 1 ) .

English

&nbsp; Although traditional census can present unbiased information about different land uses, it is spatial independent and do not present particular information about spatial distribution of studied characteristic. In this study, we used geostatistic and Geographical Information System (GIS) to estimate some different land uses allometric characteristics in Isfahan Province (Iran). Thus, samples information was surveyed considering their geographic position in the studied area. After optimizing variogram parameters, empirical variogram was prepared to investigate spatial structure of different land uses allometric characteristics. Our results confirme that spatial structure for the quantitative characteristics of different land uses has a moderate degree of spatial correlation, except for type variable that has no spatial structure. Nugget effect for variogram obtained from the quantitative characteristics of different land uses was equal to 35 to 64%. We used ordinary Kriging for preparing Kriging map and Kriging standard deviation of different land uses. Also, we used geostatistic and GIS to compare geostatistical and algebraic interpolation methods and nine different interpolation methods (Kriging, local polynomial methods, inverse distance weighted, radial basis functions, global polynomial, moving average weighted, natural neighbor, nearest neighbor and triangulation with Linear Interpolation) were investigated. Spatial distribution of different land uses quantitative characteristics were validated with ordinary Kriging and algebraic methods. Our results confirm that ordinary Kriging has more accuracy than other methods for spatial prediction of different land uses quantitative characteristics. &nbsp; Key words:&nbsp;Geostatistic, interpolation method, land use allometric characteristics, Kriging

A new boundary interpolation technique for parameterisation of planar surfaces with four arbitrary boundary curves

This article is concerned with construction of planar parametric surfaces enclosed by four arbitrary boundary curves. The available conventional methods for this purpose are algebraic interpolation method (like the Coons method) and partial differential equation method. These methods usually yield some problems. This article proposes a new method namely boundary interpolation method where the geometric properties of the boundary curves have been taken into account. A technique of simultaneous displacement of the interpolated curves from the opposite boundaries has been adopted. The geometric properties considered for displacement include weighted combination of geometric and linear displacement vectors of the two opposite generating curves. The algorithm has two adjustable parameters that control the characteristics of interpolation of one set of curves from their parents. Case studies show that this algorithm gives reasonably smooth boundary-fitted planar parametric surface compared with the other two conventional methods.

Construction of 2-D parametric surfaces bounded with four irregular curves

Conventional methods of boundary-conformed 2D surface generation including algebraic interpolation method (like the linear Coons method) usually yield some problems. This paper proposes a new method namely boundary transformation method. This new method applies a mechanism of simultaneous displacement of the interpolated curves from the opposite boundaries to generate the intermediate curves. The geometric properties considered for displacements include weighted combination of geometric and linear displacement vectors of the two opposite generating curves. The algorithm has one adjustable parameter that controls the characteristics of transformation of one set of curves from their parents. Case studies show that this algorithm gives reasonably smooth transformation of the boundaries and manipulation of the surface constructed is quite user friendly. The new algorithm is capable enough to resolve the 2D boundary-conformed parameterisation problems.

Towards toric absolute factorization

This article presents an algorithmic approach to study and compute the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry.

The Beth property and interpolation in lattice-based algebras and logics

We deal with logics based on lattices with an additional unary operation. Interrelations of different versions of interpolation, the Beth property, and amalgamation, as they bear on modal logics and varieties of modal algebras, superintuitionistic logics and varieties of Heyting algebras, positive logics and varieties of implicative lattices, have been studied in many works. Sometimes these relations can and sometimes cannot be extended to the logics without implication considered in the paper.

Interpolation in semigroupoid algebras

A seminal result of Agler characterizes the so-called Schur-Agler class of functions on the polydisk in terms of a unitary colligation transfer function representation. We generalize this to the unit ball of the algebra of multipliers for a family of test functions over a broad class of semigroupoids. There is then an associated interpolation theorem. Besides leading to solutions of the familiar Nevanlinna-Pick and Carathéodory-Fejér interpolation problems and their multivariable commutative and noncommutative generalizations, this approach also covers more exotic problems.

Algebraic multigrid and algebraic multilevel methods: a theoretical comparison

AbstractWe consider algebraic methods of the two‐level type for the iterative solution of large sparse linear systems. We assume that a fine/coarse partitioning and an algebraic interpolation have been defined in one way or another, and review different schemes that may be built with these ingredients. This includes algebraic multigrid (AMG) schemes, two‐level approximate block factorizations, and several methods that exploit generalized hierarchical bases. We develop their theoretical analysis in a unified way, gathering some known results, rewriting some other and stating some new. This includes lower bounds, that is, we do not only investigate sufficient conditions of convergence, but also look at necessary conditions. Copyright © 2005 John Wiley &amp; Sons, Ltd.

CFD STUDIO: AN EDUCATIONAL SOFTWARE FOR CFD ANALYSIS

The main goal of this paper is to demonstrate the general characteristics of the educational user-friendly CFD Studio package for CFD teaching. The package was designed for teaching 2D fluid mechanics and heat transfer process, including conduction, coupled conduction/convection, natural and forced convection, external and internal flows, among other phenomena. The finite volume methodology and its related topics can also be taught using the software. Therefore, general aspects of the three main modules, pre-processor, solver and post-processor are discussed aiming to show the generality of the tool. These modules are integrated in the application by a so-called “numerical problem project” which guide the student through the steps to obtain the solution. To approximate the partial differential equations the finite volume approach is employed using a fully-implicit formulation with the interpolation schemes CDS, UDS and WUDS. Mesh editing and nonorthogonal boundary-fitted mesh generation, using algebraic interpolation and elliptic equations, are important features of the package. Coupled heat transfer problems are handled using the “solid-block” formulation and the pressure-velocity coupling uses the SIMPLE and SIMPLEC methods with non-staggered grids. To demonstrate the capabilities two fluid flow and heat transfer “problem projects” are presented.

CFD STUDIO: AN EDUCATIONAL SOFTWARE FOR CFD ANALYSIS

The main goal of this paper is to demonstrate the general characteristics of the educational&#x0D;user-friendly CFD Studio package for CFD teaching. The package was designed for&#x0D;teaching 2D fluid mechanics and heat transfer process, including conduction, coupled&#x0D;conduction/convection, natural and forced convection, external and internal flows, among&#x0D;other phenomena. The finite volume methodology and its related topics can also be taught&#x0D;using the software. Therefore, general aspects of the three main modules, pre-processor,&#x0D;solver and post-processor are discussed aiming to show the generality of the tool. These&#x0D;modules are integrated in the application by a so-called “numerical problem project”&#x0D;which guide the student through the steps to obtain the solution. To approximate the&#x0D;partial differential equations the finite volume approach is employed using a fully-implicit&#x0D;formulation with the interpolation schemes CDS, UDS and WUDS. Mesh editing and nonorthogonal&#x0D;boundary-fitted mesh generation, using algebraic interpolation and elliptic&#x0D;equations, are important features of the package. Coupled heat transfer problems are&#x0D;handled using the “solid-block” formulation and the pressure-velocity coupling uses the&#x0D;SIMPLE and SIMPLEC methods with non-staggered grids. To demonstrate the&#x0D;capabilities two fluid flow and heat transfer “problem projects” are presented.

Boundary-conformed toolpath generation for trimmed free-form surfaces

In this paper, we adopt a 2D reparameterization procedure to regenerate boundary-conformed toolpaths. Three methods for the 2D reparameterization of trimmed boundaries in parametric space are examined and compared. They are the Coons method, the Laplace method, and a newly developed boundary-blending method. These three methods represent three different approaches to 2D surface parameterization, namely, the algebraic interpolation approach, the partial differential equation approach, and a geometric offsetting approach, respectively. Complete algorithms for surface reparameterization and toolpath generation are developed and implemented. The results showed that the Coons method is relatively simple yet might cause anomalies when the complexities of the boundary are high. The Laplace method is robust but takes relatively more computational time and also has the problem of uneven distribution of iso-parametrics. For the newly developed boundary-blending method, both the computational efficiency and parameterization robustness are quite good, in addition, it alleviates the uneven distribution problem appeared in the Laplace results.