Element Spacing Research Articles

Geometry of sample spaces

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of Mn modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces.We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov–Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fréchet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

A Study of Mutual Coupling Suppression between Two Closely Spaced Planar Monopole Antenna Elements for 5G New Radio Massive MIMO System Applications

5G NR (new radio) introduces the concept of massive MIMO (multiple-input multiple-output) technology, in which a larger number of antenna arrays are installed on the transceiver. Due to the increased number of antenna elements allocated close to each other (approximately at a distance of half a wavelength), mutual coupling becomes a serious problem leading to performance degradation of the MIMO communication system. In this communication, two different configurations of closely spaced antenna array elements are studied. In order to reduce the mutual coupling, a combination of a metamaterial-based frequency-selective surface (FSS), a metallic strip, and a slot element in the ground plane is examined. It is found that the proposed technique significantly suppresses mutual coupling from −12 dB to −25 dB. Both designs are fabricated and experimentally validated. The simulation results are in good agreement with the measurements. The proposed mutual coupling reduction technique may be suitable for massive MIMO systems in fifth-generation (5G) new radio applications.

Monitoring and control of water-ecological space in the Dongting Lake region

The territorial spatial planning in the new era strengthens the control of different functional spaces and emphasizes integrated and coordinated development of each functional space. Therefore, it is important to monitor the “structure-function” characteristics of water-ecological space based on the context of territorial spatial planning and develop a management and control framework. Based on land use and social statistics, and with the help of ArcGIS analysis, this paper examined the structure and function of the water-ecological space in the Dongting Lake study for 2010, 2015, and 2020, generating a control framework and proposing key initiatives. The main results are as follows: 1) The overall scale of the water ecological space in the Dongting Lake study is over 7,300 km2. Water bodies had the largest share, followed by coastal terrestrial areas, while the land-water ecotone was the smallest. There was a small decrease in the water ecological space during the study period. Yueyang had the largest overall scale of water ecological space, while Linli had the smallest. 2) The comprehensive function of water ecological space in the Dongting Lake study was about 0.4000, increasing somewhat during the study period. Among the units, the comprehensive function value was highest in Yuanjiang City, while Jincheng City had the lowest. 3) From the structural elements of water ecological space, scientific planning of functional zoning of water ecological space should be carried out and combined with the delineation of “three zones and three lines.” Key initiatives such as multi-scale and multi-level planning and control, use control and access restrictions, determination of water ecological space ownership, pollution control and accountability, and comprehensive water ecological treatment and restoration should be promoted.

A consistent projection integration for Galerkin meshfree methods

An efficient and inherently consistent integration method, named projection integration(PI), is proposed for Galerkin meshfree methods with arbitrary order. In contrast to traditional numerical integration methods, PI defines a projection operator based on a weighted norm to systematically approximate meshfree shape functions in an appropriate and consistent function space. Then a substitute function space is established by utilizing projected shape functions as a basis, and Galerkin weak form is approximately calculated by substituting the projected shape functions for the original meshfree shape functions. Owing to the projection consistency of the projection operator, the projected shape functions can still meet consistent conditions. For convenient implementations, the triangular finite element space is selected to approximate meshfree shape functions and the weight function in the weighted norm is carefully chosen to simplify the projection operator in this paper. Arbitrary order integration constraint conditions can be easily met by utilizing the corresponding order triangular finite element shape functions with low order Gauss quadrature rules, which means the optimal convergence rate can be obtained. A series of numerical examples are presented to demonstrate the proposed method’s efficiency and superior convergence performance.

De Rham compatible Deep Neural Network FEM

On general regular simplicial partitions T of bounded polytopal domains Ω⊂Rd, d∈{2,3}, we construct exact neural network (NN) emulations of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical “Raviart–Thomas element”, and the “Nédélec edge element”. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions T of Ω are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension d≥2. Our “FE-Nets” are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra Ω⊂R3. They are thus an essential ingredient in the application of e.g., the methodology of “physics-informed NNs” or “deep Ritz methods” to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the “Crouzeix–Raviart” elements and Hybridized, Higher Order (HHO) methods.

Impact fault detection for marine current turbines blade via MEGK-means and PCA under variable marine conditions

The advancement of the marine current turbine (MCT) technology has the potential to aid China in achieving its carbon peak and carbon neutrality goals. The performance of MCTs is influenced by fluctuating water velocities and erratic turbulence. The blades of MCTs are susceptible to cracking due to prolonged exposure to seawater. Unpredictable changes in marine currents contribute to unstable working environments. This study presents multiple envelope geometrical K-means to categorize stator current readings and develop fault detection models. It first constructs an envelope geometric feature matrix by using the extracted modulus signal. Then, the matrix is used to select the initial center for clustering, and a multi-scale principal component analysis is performed under each working condition to decrease data dimensionality. Finally, T 2 and squared prediction error (SPE) serve as the projection of the sample vector on the principal element space and the remaining subspace, respectively, and can be utilized to monitor fault cases. The experimental findings demonstrate that the proposed method has excellent recognition capabilities and detection accuracy for the impact faults of MCTs under variable working conditions.

Nonconforming modified Quasi-Wilson finite element method for convection–diffusion-reaction equation

The nonconforming modified Quasi-Wilson finite element method (FEM) is employed to investigate the unconditional superconvergence behavior of the convection–diffusion-reaction equation for the linearized energy stable Backward-Euler (BE) and the second order Backward differentiation formula (BDF2) fully discrete schemes on arbitrary quadrilateral meshes. Then, the energy stabilities of the discrete solutions for the proposed schemes are demonstrated through the mathematical induction, which is crucial to proving the unique solvabilities with Brouwer fixed point theorem. Based on the above achievements and the two typical characteristics of the element: one is that its consistency error in the broken H1-norm can be estimated as order O(h2) when the exact solution belongs to H3(Ω), which is one order higher than its interpolation error, and the other is that the nonconforming part of the function in the finite element space is orthogonal to the bilinear polynomials on each element of the subdivision in the integration sense, the superclose estimates without any restriction between spatial partition parameter h and the time step τ are obtained. Furthermore, the superconvergence properties for the aforementioned schemes are derived by the interpolation post-processing technique. Finally, the theoretical analysis is justified by the provided numerical experiments.

KAJIAN MODEL KELEMBAGAAN PERANAN DUNIA USAHA DALAM PENATAAN RUANG

One of the manifestations of development is the arrangement of areas according to spatial rules to create safe, comfortable, productive and sustainable spaces. As stated in the mandate of Law no. 26 of 2017 concerning Spatial Planning. Space includes land space, ocean, air and space in the earth. Judging from its nature, land is the center of all human activities, land as a strategic element of space is a basic need that concerns the livelihoods of many people. Land and Spatial Planning Policy within the investment framework is directed at efforts to encourage investment by reducing various obstacles, in order to foster competitiveness, it is this challenge that is the background of this study to initiate the need for a forum for the institutional model of Business World Awareness - Spatial Planning. This is intended to be able to carry out communication, consultative, participatory, advocacy in the implementation, utilization, supervision, and evaluation of spatial planning, especially among the business community which often experience problems/conflicts, as an effort to realize development that is productive, just and sustainable, by applying the principles, rules and norms of spatial planning.

Invariant Synchrony and Anti-synchrony Subspaces of Weighted Networks

The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.

A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen–Cahn equation

Abstract In this work, we apply an a posteriori error analysis for the space-time, discontinuous in time, Galerkin scheme, which has been proposed in Antonopoulou (2020, Space-time discontinuous Galerkin methods for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise. IMA J. Num. Analysis, 40, 2076–2105) for the $\varepsilon $-dependent stochastic Allen–Cahn equation with mild noise $\dot{W}^\varepsilon $ tending to rough as $\varepsilon \rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of Funaki (1999, Singular limit for stochastic reaction–diffusion equation and generation of random interfaces. Acta Math. Sinica, 15, 407–438). We prove a posteriori error estimates for the $m$-dimensional problem, $m\leq 4$ for a general class of space-time finite element spaces. The a posteriori bound is growing only polynomially in $\varepsilon ^{-1}$ if the step length $h$ is bounded by a positive power of $\varepsilon $. This agrees with the restriction posed so far in the a priori error analysis of continuous finite element schemes for the $\varepsilon $-dependent deterministic Allen–Cahn or deterministic and stochastic Cahn–Hilliard equation. As an application, we examine tensorial elements where the discrete solution is approximated by polynomial functions of separated space and time variables; the a posteriori estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $\dot{W}^\varepsilon $ as defined in Weber (2010, On the short time asymptotic of the stochastic Allen–Cahn equation. Ann. Inst. Henri Poincare Probab. Stat., 46, 965–975) through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one- and two-dimensional problems in various cases of interest, and compare with the deterministic ones.

Quality Evaluation of Park Green Space Based on Multi-Source Spatial Data in Shenyang

Park green spaces serve as essential urban public recreation spaces and represent significant components of urban construction land. Investigating the quality of park green spaces is a vital requirement for the establishment of a high-quality urban living environment and the enhancement of land use efficiency. This study developed a quality evaluation index system for park green spaces in Shenyang, considering the constituent elements of recreational spaces. By categorizing 168 park spaces into four size grades and employing the entropy-TOPSIS method with multiple data sources, the research assessed and analyzed the generally quality of park green spaces in Shenyang. Additionally, the study analyzed factors limiting park green spaces and introduced the Recreational Quality Index (RQI) to represent the distribution of park green space resources within the study area. The results demonstrated that 46 low-quality park green spaces require urgent improvement, and the riverside area and the vicinity of the second ring road possess high-quality park green space resources. Barriers of the same scale in park green spaces exhibited similar characteristics. Park green spaces in Huanggu District were allocated recreational resources optimally. The analysis determined that system construction and deep cultural heritage are crucial factors. This study can offer a scientific basis and recommendations for the development of park green spaces and the enhancement of efficient urban land use.

Influence of the Size of Measurement Area Determined by Smooth-Rough Crossover Scale and Mean Profile Element Spacing on Topography Parameters of Samples Produced with Additive Methods

Surface topography measurements are becoming more and more popular and complement the 2D analysis of surface texture. The selection of the measurement area is not yet included in the standards, and the size of this area affects the values of the determined parameters. The article presents the results of research on determining the measurement area based on the smooth-rough crossover scale (SCR) and mean profile element spacing (Rsm) parameters. The tests focused on measuring the surface topography of random and directional types of polymer parts produced by various additive manufacturing techniques. The measurements were conducted using the focus variation method. Surface topography parameters were determined for large evaluation areas determined based on the cut-off filter length Lc and for small areas defined based on the SCR and Rsm parameters. The values of parameters determined from large areas constituted the reference values to which the values determined from small areas were compared. In the case of random-type samples, it was shown that the values of the parameters calculated from smaller areas determined based on the SCR significantly differed from the reference values. For both types of samples, determination of the evaluation area based on the Rsm yielded good results. In most cases, the greatest differences between the values of parameters calculated for small and large areas were noted for the Ssk and Smr1 parameters. Based on the test results, it could be advantageous to replace the measurement of a larger area with the measurement of several smaller areas located at different places on the sample.

Continuous interior penalty stabilization for divergence-free finite element methods

Abstract In this paper, we propose, analyze and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac 12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

A Low-Profile Dual-Polarized Substrate Integrated Magneto-Electric Dipole MIMO Antenna

A magneto-electric dipole (MED) MIMO antenna with low-profile, dual-polarized, and substrate integrated properties is proposed. The antenna element is composed of four fan-shaped patches lying on the radiation plane and four vertical metal vias linking the patches and the ground. Compared with the common MED antennas, the proposed MED antenna element has a lower profile of 0.1λ. Besides, it is fed by microstrip lines via two orthogonal H-shaped slots, resulting in a broad impedance bandwidth of 40% (3-4.5 GHz) and a steady gain in the entire operational band. The ports isolation is higher than 27 dB, the cross-polarization discrimination (XPD) is higher than 37 dB, and the measured radiation efficiency is greater than 70%. Furthermore, four antenna elements are utilized to form a MIMO antenna, which can obtain better ports isolation characteristics under the condition that the element spacing is half wavelength. Finally, the MIMO antenna is simulated, fabricated and measured. The results verify the rationality of the design.

SASA: Super-Resolution and Ambiguity-Free Sparse Array Geometry Optimization With Aperture Size Constraints for MIMO Radar

To improve the performance of multiple-input-multiple-output (MIMO) radar, various sparse arrays have been employed. However, the angular resolution of existing non-uniform arrays optimized by either combinatorial algorithms or heuristic ones is limited by the Rayleigh criterion, which is strictly related to the aperture size. Based on the angular ambiguity function (AAF) analysis, two new models are established in this work for directly optimizing the sidelobe level (SLL) or the main lobe width (MLW) with the constraints of aperture size and element spacing. The aforementioned designs result in non-convex and nonlinear optimization problems, and solutions are derived via the alternating direction multiplier method (ADMM). Furthermore, considering a parametric trade-off between SLL and MLW, a hybrid algorithm is proposed to search for the SLL-MLW Pareto front boundary. Finally, simulations are provided to demonstrate the high angular resolution and ambiguity-free properties of the optimized sparse arrays.

The Toda flow on Hessenberg elements of real, split simple Lie algebras

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits.

I. Annensky’s poem «Bow and strings» in the framework of an intermedial analysis

In the article it examines the features of the poetics of one of the unusual authors of the Silver Age, I. Annensky. The aim of the research is to analyze the aesthetic issue of the interaction of music and words within the framework of intermediality and traditions formed in Russian literature. Extensive material of Russian literature of the XIX century is involved. The analysis and comparison of the intermediate image of the string from the poem by I. Annensky «Bow and Strings» with a similar element in the works of F.I. Tyutchev, A.A. Fet, L.N. Tolstoy, I.S. Turgenev, A.P. Chekhov, V.V. Mayakovsky is carried out. The study is based on methods of intermediate analysis. The specificity of the «string» image is revealed and the difference in its semantics in I. Annensky's poem in relation to its predecessors is emphasized. The influence of the poet on subsequent generations of writers is shown. A musical instrument is endowed in the poetics of the author with greater sensitivity and sincerity, which gives them special power and significance. The image is contrasted with the world of people and becomes a sign of true exalted being. And Annensky continues the established intermedial tradition of presenting the image of the «string» within the framework of combining music and words, but complicates it, complements it with complex aesthetic and ethical content. In previous studies, these points were not reflected, which shows the novelty of this work. Associative and intertextual connections can be defined as the basis of intermedial interactions in the works of I. Annensky. The author relies on the same elements of time and space as his predecessors, but significantly transforms them already within the framework of his artistic concept of being.

A modified weak Galerkin method for (curl)-elliptic problem

In this paper, we design and analyze a modified weak Galerkin (MWG) finite element method for H ( curl ) -elliptic problem. We first introduce a new discrete weak curl operator and the MWG finite element space. The modified weak Galerkin method does not require the penalty parameter by comparing with traditional DG methods. We prove an optimal error estimate in energy norm. At last, we provide the numerical results to confirm these theoretical results.

A Joint Array Parameters Design Method Based on FDA-MIMO Radar

Frequency diverse array (FDA)-multiple-input and multiple-output (MIMO) radar is characterized by a range-angle-dependent beampattern owing to extra frequency offset across the transmit array elements. In this respect, the ideal beamforming performance can be obtained by designing the transmit and receive array parameters. On this basis, this paper aims to design the ideal beampattern for FDA-MIMO radar by introducing a joint array parameters optimization design method including receive filter (RF) and transmit frequency offset-receive element spacing (TFO-RS). The cyclic iteration is introduced, and this joint optimization method is divided into three sub-problems, including RF, TFO, and RS design. As for the TFO and RS design subproblems, function scaling is applied to transform the objection function to the quadratic form. Furthermore, the quadratic programming (QP) optimization problem with linear constraints can be constructed. The simulation results show that the proposed optimization design method can improve the ability of FDA-MIMO radar to detect multiple targets at the same time compared with the given conventional TFO and RS schemes.

Intelligently Wireless Batteryless RF-Powered Reconfigurable Surface: Theory, Implementation & Limitations

This work exploits ultra-low cost, commodity, radio frequency identification (RFID) tags as elements of a reconfigurable intelligent surface (RIS). Such batteryless tags are powered and controlled by a software-defined radio reader, with properly modified software, so that a source-destination link is assisted, operating at a different band. Signal model includes small-scale and large-scale fading, direct link, as well as specific parameters relevant to reflection (i.e., backscatter) radio, such as antenna structural mode and reflection efficiency, typically overlooked in the literature. An algorithm is offered that computes the optimal RIS configuration with complexity of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ) in number of elements <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> , instead of intractable exponential complexity of exhaustive search, while accommodating any number <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> ≥ 2 of loads. With the proposed algorithm, it is shown that performance gains reach a plateau for constant element spacing and increasing number of elements, suggesting that the weak, passive nature of backscattered links limits the performance gains, even with perfect channel estimation. Channel estimation with linear minimum mean squared error (LMMSE) estimator is shown to be effective, provided that there are sufficient number of pilot symbols. A concrete way is offered to design and prototype a wireless, batteryless, RF-powered, reconfigurable surface and a proof-of-concept is experimentally demonstrated.