Discrete Square Research Articles

Functionalised Clathrochelate Complexes--New Building Blocks for Supramolecular Structures.

Tris(dioxime) iron(II) clathrochelate complexes functionalised with 3- and 4-pyridyl groups have been employed as building blocks in the preparation of supramolecular structures by coordination-driven self-assembly. These complexes possess a number of desirable characteristics, being straightforward to synthesise and offering ample opportunity for steric and functional modification. Clathrochelate-based 4,4'-bipyridyl metalloligands from 1.5 nm to 5.4 nm in length were prepared in up to two steps and their potential as building blocks for supramolecular architectures demonstrated through the preparation of a discrete molecular square and a three dimensional (3D) coordination polymer. Furthermore, the structure-directing capability of clathrochelate building blocks was illustrated through the synthesis of octahedral cage compounds, which are capable of encapsulating the large, hydrophobic BPh4- anion in aqueous solvent mixtures.

APPLICATION OF VECTOR SPHERICAL HARMONICS AND KERNEL REGRESSION TO THE COMPUTATIONS OF OMM PARAMETERS

The high quality of Hipparcos data in position, proper motion, and parallax has allowed for studies about stellar kinematics with the aim of achieving a better physical understanding of our galaxy, based on accurate calculus of the Ogorodnikov–Milne model (OMM) parameters. The use of discrete least squares is the most common adjustment method, but it may lead to errors mainly because of the inhomogeneous spatial distribution of the data. We present an example of the instability of this method using the case of a function given by a linear combination of Legendre polynomials. These polynomials are basic in the use of vector spherical harmonics, which have been used to compute the OMM parameters by several authors, such as Makarov & Murphy, Mignard & Klioner, and Vityazev & Tsvetkov. To overcome the former problem, we propose the use of a mixed method (see Marco et al.) that includes the extension of the functions of residuals to any point on the celestial sphere. The goal is to be able to work with continuous variables in the calculation of the coefficients of the vector spherical harmonic developments with stability and efficiency. We apply this mixed procedure to the study of the kinematics of the stars in our Galaxy, employing the Hipparcos velocity field data to obtain the OMM parameters. Previously, we tested the method by perturbing the Vectorial Spherical Harmonics model as well as the velocity vector field.

Analysis of least squares pseudo-spectral method for the interface problem of the Navier–Stokes equations

The aim of this paper is to propose and analyze the first order system least squares method for the incompressible Navier–Stokes equation with discontinuous viscosity and singular force along the interface as the earlier work of the first author on Stokes interface problem (Hessari, 2014). Interface conditions are derived, and the Navier–Stokes equation transformed into a first order system of equations by introducing velocity gradient as a new variable. The least squares functional is defined based on L2 norm applied to the first order system. Both discrete and continuous least squares functionals are put into the canonical form and the existence and uniqueness of branch of nonsingular solutions are shown. The spectral convergence of the proposed method is given. Numerical studies of the convergence are also provided.

Littlewood-Paley functionals on graphs

Let Γ be a graph equipped with a Markov operator P. We introduce discrete fractional Littlewood-Paley square functionals and prove their -boundedness under various geometric assumptions on Γ.

Analysis of discrete least squares on multivariate polynomial spaces with evaluations at low-discrepancy point sets

We analyze the stability and accuracy of discrete least squares on multivariate polynomial spaces to approximate a given function depending on a multivariate random variable uniformly distributed on a hypercube. The polynomial approximation is calculated starting from pointwise noise-free evaluations of the target function at low-discrepancy point sets. We prove that the discrete least-squares approximation, in a multivariate anisotropic tensor product polynomial space and with evaluations at low-discrepancy point sets, is stable and accurate under the condition that the number of evaluations is proportional to the square of the dimension of the polynomial space, up to logarithmic factors. This result is analogous to those obtained in Cohen et al. (2013), Migliorati et al. (2014), Migliorati (2013) and Chkifa et al. (in press) for discrete least squares with random point sets, however it holds with certainty instead of just with high probability. The result is further generalized to arbitrary polynomial spaces associated with downward closed multi-index sets, but with a more demanding (and probably nonoptimal) proportionality between the number of evaluation points and the dimension of the polynomial space.

Time Series Forecasting Model Based on Discrete Grey LS-SVM

The advantages and disadvantages of discrete GM(1,1) model and least squares support vector machine are analyzed respectively, this article proposes a new time series forecasting model of discrete grey least squares support vector machine. The new model adopts structural risk minimization principle, at the same time develops the advantages of accumulation generation in the grey forecasting method, weakens the effect of stochastic-disturbing factors in original sequence, and avoids the theoretical defects existing in the grey forecasting model. The simulation results show that the forecasting model is effective and reliable, and consolidates the advantage of the discrete GM(1,1) model and least squares support vector machine. It offers a new way to improve the time series forecasting accuracy. Index Terms—Time series prediction, Least square support vector machines algorithm, Grey system, Small samples, Discrete GM(1,1) model, Discrete grey least squares support vector machine .

Gradient Recovery for the Crouzeix–Raviart Element

A gradient recovery method for the Crouzeix---Raviart element is proposed and analyzed. The proposed method is based on local discrete least square fittings. It is proven to preserve quadratic polynomials and be a bounded linear operator. Numerical examples indicate that it can produce a superconvergent gradient approximation for both elliptic equations and Stokes equations. In addition, it provides an asymptotically exact posteriori error estimators for the Crouzeix---Raviart element.

Solvable non-Hermitian discrete square well with closed-form physical inner product

A new Hermitizable quantum model is proposed in which the bound-state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two Chebyshev polynomials. As an -site lattice version of square well with complex Robin-type two-parametric boundary conditions the model is unitary with respect to the Hilbert space metric Θ which becomes equal to the most common Diracʼs metric in the conventional textbook Hermitian–Hamiltonian limit. This metric is constructed in closed form at all .

An adaptive node regeneration technique for the efficient solution of elasticity problems using MDLSM method

An efficient adaptive node regeneration method is proposed in this paper for solving elasticity problems using the mixed discrete least squares meshless (MDLSM) method. The method starts with a point-wise error estimation of the solution produced on an arbitrary initial configuration defined by the user using the MDLSM method. The point-wise error estimate is associated with the support domain of the nodal points and used to calculate the required nodal spacing at each support domain and subsequently generate new nodes at support domain level. A node-removing process is then used to remove some of the nodes created at the overlapping regions of the support domains. To improve the quality of the final configuration, a node-moving procedure based on interpolation of the errors on the original configuration is used to create the final nodal configuration. The proposed method is a single step refinement procedure and is capable of producing nodal configurations of desired accuracy for different problems. The proposed method is used to simulate three benchmark examples from the literature and the results are produced and compared with those of the conventional multi-stage node enrichment method. The results indicate the superior efficiency and effectiveness of the proposed method compared to the available methods.

High-temperature, high-pressure hydrothermal synthesis, characterization, and structural relationships of layered uranyl arsenates.

Five new uranyl arsenates, Na14[(UO2)5(AsO4)8]·2H2O (1), K6[(UO2)5O5(AsO4)2] (2a), K4[(UO2)3O2(AsO4)2] (2b), Rb4[(UO2)3O2(AsO4)2] (3), and Cs6[(UO2)5O2(AsO4)4] (4), were synthesized by high-temperature, high-pressure hydrothermal reactions at about 560 °C and 1440 bar and were characterized by single-crystal X-ray diffraction, thermogravimetric analysis, and photoluminescence spectroscopy. Crystal data for compound 1: triclinic, P1, a = 7.0005(3) Å, b = 12.1324(4) Å, c = 13.7428(5) Å, α = 64.175(2)°, β = 89.092(2)°, γ = 85.548(2)°, V = 1047.26(7) Å(3), Z = 1, R1 = 0.0185; compound 2a: monoclinic, P2₁/c, a = 6.8615(3) Å, b = 24.702(1) Å, c = 7.1269(3) Å, β = 98.749(2)°, V = 1193.89(9) Å(3), Z = 2, R1 = 0.0225; compound 2b: monoclinic, P2₁/c, a = 6.7852(3) Å, b = 17.3640(8) Å, c = 7.1151(3) Å, β = 98.801(3)°, V = 828.42(6) Å(3), Z = 2, R1 = 0.0269; compound 3: monoclinic, P2₁/m, a = 6.9783(3) Å, b = 17.4513(8) Å, c = 7.0867(3) Å, β = 90.808(3)°, V = 862.94(7) Å(3), Z = 2, R1 = 0.0269; compound 4: triclinic, P1, a = 7.7628(3) Å, b = 9.3324(4) Å, c = 11.9336(4) Å, α = 75.611(2)°, β = 73.136(2)°, γ = 86.329(2)°, V = 801.37(5) Å(3), Z = 1, R1 = 0.0336. The five compounds have layer structures consisting of uranyl square, pentagonal, and hexagonal bipyramids as well as AsO4 tetrahedra. Compound 1 contains chains of discrete uranyl square and pentagonal bipyramids, 2a contains three-polyhedron-wide ribbons of edge- and corner-sharing uranyl square and pentagonal bipyramids, 2b and 3 contain dimers of edge-shairing pentagonal bipyramids that share edges with hexagonal bipyramids to form chains, and 4 contains one-polyhedron-wide zigzag chains of edge-sharing uranyl polyhedra. The double sheet structure of 1 is new, but the chain topology has been observed in an organically templated uranyl sulfate. Compound 2b is a new geometrical isomer of the phosphuranylite group. The sheet anion topologies of 2a and 4 can be obtained by splitting the β-U3O8-type sheet into complex chains and connecting the chains by arsenates.

First order system least squares method for the interface problem of the Stokes equations

The first order system least squares method for the Stokes equation with discontinuous viscosity and singular force along the interface is proposed and analyzed. First, interface conditions are derived. By introducing a physical meaningful variable such as the velocity gradient, the Stokes equation transformed into a first order system of equations. Then the continuous and discrete norm least squares functionals using Legendre and Chebyshev weights for the first order system are defined. We showed that continuous and discrete homogeneous least squares functionals are equivalent to appropriate product norms. The spectral convergence of the proposed method is given. A numerical example is provided to support the method and its analysis.

The Maximum Principle of Pontryagin in Control of Twolegged Robot Based on Human Walking System

In the paper a hypothesis about state equations of human gait is presented. Instantaneous normalized power developed by human muscles at particular joints of a leg is a control vector in state equations of the human walking system. The maximum principle of Pontryagin in analysis of dynamic human knee joint was presented. The discrete Hamilton function of a knee joint is similar to a discrete square function of normalized power developed by muscles at the knee joint. The results satisfy optimal conditions and could be applied in control of exoskeleton and DAR type robot.

Initial Calibration and Online Error Compensation of a Resolver System

This paper presents a method for initial calibration and online compensation of systematic errors in a resolver system. Initial calibration at standstill condition is carried out for a few positions by applying Discrete Fourier Transform and Least Squares method. The error model is constructed based on complex Fourier expression of position angle signal obtained from non-ideal resolver signals due to amplitude imbalance, quadrature error and offset. Error parameters are determined online using the integral control loops which are initialized with the calibration results. Besides, single channel amplitude tracking is incorporated in the error compensation scheme. This makes the compensation adaptive to changes in error parameters. And thus accuracy of the resolver system is reasonably improved and the system becomes robust. The effectiveness of the proposed method is validated by simulation and experiments.

Computable Diagonalizations and Turing’s Cardinality Paradox

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers.

On the Application of Iterative Methods of Nondifferentiable Optimization to Some Problems of Approximation Theory

We consider the data fitting problem, that is, the problem of approximating a function of several variables, given by tabulated data, and the corresponding problem for inconsistent (overdetermined) systems of linear algebraic equations. Such problems, connected with measurement of physical quantities, arise, for example, in physics, engineering, and so forth. A traditional approach for solving these two problems is the discrete least squares data fitting method, which is based on discretel2-norm. In this paper, an alternative approach is proposed: with each of these problems, we associate a nondifferentiable (nonsmooth) unconstrained minimization problem with an objective function, based on discretel1- and/orl∞-norm, respectively; that is, these two norms are used as proximity criteria. In other words, the problems under consideration are solved by minimizing the residual using these two norms. Respective subgradients are calculated, and a subgradient method is used for solving these two problems. The emphasis is on implementation of the proposed approach. Some computational results, obtained by an appropriate iterative method, are given at the end of the paper. These results are compared with the results, obtained by the iterative gradient method for the corresponding “differentiable” discrete least squares problems, that is, approximation problems based on discretel2-norm.

Error Estimate and Adaptive Refinement in Mixed Discrete Least Squares Meshless Method

The node moving and multistage node enrichment adaptive refinement procedures are extended in mixed discrete least squares meshless (MDLSM) method for efficient analysis of elasticity problems. In the formulation of MDLSM method, mixed formulation is accepted to avoid second-order differentiation of shape functions and to obtain displacements and stresses simultaneously. In the refinement procedures, a robust error estimator based on the value of the least square residuals functional of the governing differential equations and its boundaries at nodal points is used which is inherently available from the MDLSM formulation and can efficiently identify the zones with higher numerical errors. The results are compared with the refinement procedures in the irreducible formulation of discrete least squares meshless (DLSM) method and show the accuracy and efficiency of the proposed procedures. Also, the comparison of the error norms and convergence rate show the fidelity of the proposed adaptive refinement procedures in the MDLSM method.

How to maximize the contact perimeter and contact area of shapes composed of cells in the discrete domain

The contact perimeter of a shape composed of pixels is the number of joints between neighboring pixels of the shape, and the contact area of a shape composed of voxels (three dimensional cells) is the number of planar joints between neighboring voxels. In this paper, we present the maximal contact perimeter function for shapes composed of pixels, and the maximal contact area function for voxel-based shapes. Also, we see how each one of these functions induce the definition of the corresponding standard function that extends the concept of contact perimeter (area) to non discrete squares (cubes), and which is used in the definition of the normalized discrete compactness.

Transform domain Variable Step-size Griffiths Least Mean Square adaptive algorithm and its applications

This paper proposes discrete cosine Transform domain Variable Step-size Griffiths Least Mean Square (TVGLMS) algorithms with and without desired signal decomposition, which are robust to noise. For the TVGLMS algorithms, the robust gradient and robust Variable Step-size (VSS) are achieved by using the Griffiths gradient. The transform domain due to its input orthogonalization, improves the convergence rate without any requirement of explicit noise variance estimate. The TVGLMS algorithms are applied to system identification and linear/Decision Feedback Equalizers (DFE). For system identification with white observation noise, the Smoothed Ensemble Average Square Error (SEASE) index reduces by 20dB and 12dB for with and without the desired signal decomposition respectively, over the existing transform domain VSS. For colored observation noise the SEASE reduction is by 14dB and 7dB for the two cases in order. An improvement in the bit error for linear equalizer, 50% and for DFE, 70% is achieved by TVGLMS.

A Wigner distribution function for finite oscillator systems

We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called oscillator model in dimension 2j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.

Isolation of thorium benzoate polytypes with discrete ThO8 square antiprismatic units involved in chain-like assemblies

Two thorium benzoate polytypes (I- and II-Th[O2C-C6H5]4) have been identified. The I-type has been hydrothermally (H2O; 210°C for 20h) or solvothermally (THF; 130°C for 36h) synthesized as a pure phase, from the association of thorium salt (chloride or nitrate) with the monotopic ligand, benzoic acid. The second is obtained from a mixture of thorium chloride and THF in the presence of small amount of water (H2O/Th=15) after heating at 130°C for 36h. For the latter batch, the resulting white powder is composed of a mixture of I and II polytypes. Single-crystal X-ray diffraction analyses revealed that the thorium benzoates consist of the similar assembly of infinite chains of square antiprismatic units (ThO8) linked to each other through four bidentate carboxylates on each square face of the ThO8 moieties. They only differ by the β angle of the monoclinic crystallographic cell, which induces shift of the chain to each other in the I and II forms. Thermogravimetric analysis of I-Th[O2C-C6H5]4 indicated a two step weight loss from 280 up to 465°C. It is assigned to the decomposition of the organic part together with the possible formation of thorium carbonate. The final residue after 465°C is thorianite, ThO2.