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Estimates for the discrete fundamental solution of the discrete Laplace operator on a rectangular lattice

MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2024年第10期47卷 8002-8024页

作者： Legatiuk, Anastasiia Guerlebeck, Klaus Hommel, Angela Bauhaus Univ Weimar Chair Intelligent Tech Design Weimar Germany Bauhaus Univ Weimar Chair Appl Math Weimar Germany Westsachs Hsch Zwickau Fac Business & Econ Zwickau Germany

This paper presents numerical analysis of the discrete fundamental solution of the discrete Laplace operator on a rectangular lattice. Additionally, to provide estimates in interior and exterior domains, two different regularisations of the discrete fundamental solution are considered. Estimates for the absolute difference and l(p)-estimates are constructed for both regularisations. Thus, this work extends the classical results in the discrete potential theory to the case of a rectangular lattice and serves as a basis for future convergence analysis of the method of discrete potentials on rectangular lattices.

关键词： discrete fourier transform discrete fundamental solution discrete potential theory estimates laplace operator rectangular lattice

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Comparison of two loop photonic crystal: ring-resonator with hexagonal and rectangular lattice structure

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JOURNAL OF OPTICS-INDIA 2021年第2期50卷 223-236页

作者： Kaur, Harsimran Jit Goyal, Phalguni Chitkara Univ Inst Engn & Technol Rajpura Punjab India

Recent years have witnessed significant exploration in photonic crystal (PC)-supported optical devices owing their competence in incorporation of optoelectronic components and miniaturization. Various structures of these devices have been explored in the literature with the aim to improve the performance. This paper compares two loop ring-resonators (horizontal and vertical arrangements) with hexagonal and rectangular lattice structures. Paper reports the outperformance of two loop PC ring-resonator for wavelengths of 0.85 mu m, 1.3 mu m and 1.55 mu m, respectively, for hexagonal lattice-based two loop ring-resonator having vertical arrangement at width of 30 with Q-factor that are 22,500, 26,000 and 38,700 due to effective confinement of light and out-of-plane loss and propagation loss.

关键词： FDTD Hexagonal lattice Photonic crystals Ring-resonators rectangular lattice

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Magnetic properties of Cu[C6H2(COO)4][H3N-(CH2)2-NH3]⋅3H2O-A quasi-two-dimensional S=1/2 antiferromagnet on rectangular lattice

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JOURNAL OF PHYSICS AND CHEMISTRY OF SOLIDS 2023年第1期182卷

作者： Danylchenko, Petro Tarasenko, Robert Orendacova, Alzbeta Strecka, Jozef Cizmar, Erik Tkac, Vladimir Orendac, Martin Safarik Univ Inst Phys Fac Sci Pk Angelinum 9 Kosice 04154 Slovakia Univ Chem & Technol Dept Solid State Engn Techn 5 Prague 16628 Czech Republic

The analysis of specific heat, magnetic susceptibility and magnetization identified the studied compound Cu [C6H2(COO)4][H3N-(CH2)2-NH3]& sdot;3H2O as a quasi-two-dimensional S = 1/2 Heisenberg antiferromagnet on the rectangular lattice with the intrachain coupling J1/kB approximate to 7.39 K and the interaction ratio R approximate to 0.44. A phase transition to a magnetically ordered state was observed in zero magnetic field at TN = 1.28 K. The analysis of magnetic specific heat in non-zero magnetic fields revealed features characteristic of the field-induced Bere-zinskii-Kosterlitz-Thouless phase transition theoretically predicted for ideal two-dimensional magnets. The electron paramagnetic resonance studies of Cu[C6H2(COO)4][H3N-(CH2)2-NH3]& sdot;3H2O revealed the increase of gx and gy and decrease of gz below 25 K due to the presence of dipolar coupling and the exchange anisotropy. The upturn of linewidth appearing below 30 K can be ascribed to the development of intralayer magnetic correlations.

关键词： rectangular lattice Heisenberg antiferromagnet Heat capacity Susceptibility Magnetization

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Preconditioned central moment lattice Boltzmann method on a rectangular lattice grid for accelerated computations of inhomogeneous flows

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JOURNAL OF COMPUTATIONAL SCIENCE 2022年第0期63卷

作者： Yahia, Eman Premnath, Kannan N. Univ Colorado Coll Engn Design & Comp Dept Mech Engn 1200 Larimer St Denver CO 80217 USA

Convergence acceleration of flow simulations to their steady states at lower Mach numbers can be achieved via preconditioning the lattice Boltzmann (LB) schemes that alleviate the associated numerical stiffness, which have so far been constructed on square lattices. We present a new central moment LB method on rectangular lattice grids for efficient computations of inhomogeneous and anisotropic flows by solving the preconditioned Navier-Stokes (PNS) equations. Moment equilibria corrections are derived via a Chapman-Enskog analysis for eliminating the truncation errors due to grid-anisotropy arising from the use of the rectangular lattice and the non-Galilean invariant cubic velocity errors resulting from an aliasing effect on the standard D2Q9 lattice for consistently recovering the PNS equations. Such corrections depend on the diagonal components of the velocity gradients, which are locally obtained from the second order non-equilibrium moments and parameterized by an associated grid aspect ratio gamma and a preconditioning parameter gamma, and the speed of sound in the collision model is naturally adapted to gamma via a physically consistent strategy. We develop our approach by using a robust non-orthogonal moment basis and the central moment equilibria are based on a matching principle, leading to simpler expressions for the corrections for using the rectangular grids and for representing the viscosities as functions of the relaxation parameters, r and gamma, and its implementation is modular allowing a ready extension of the existing LB schemes based on the square lattice. Numerical simulations of inhomogeneous and anisotropic shear-driven bounded flows using the preconditioned rectangular central moment LB method demonstrate the accuracy and significant reductions in the numbers of steps to reach the steady states for various sets of characteristic parameters.

关键词： lattice Boltzmann method rectangular lattice Central moments Preconditioning Inhomogeneous and anisotropic flows Convergence acceleration

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Study on the characteristics of a photonic crystal sensor with rectangular lattice based on bound states in the continuum

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JOURNAL OF PHYSICS D-APPLIED PHYSICS 2022年第17期55卷 175106-175106页

作者： Wang, Zhan Xue, Qi Zhao, SaiLi Zhang, XiaoRong Liu, HuiMin Sun, XiaoHong Zhengzhou Univ Sch Informat Engn Henan Key Lab Laser & Optoelect Informat Technol Zhengzhou 450001 Peoples R China Zhengzhou Univ Inst Intelligent Sensing Zhengzhou 450001 Peoples R China

In this paper, a sensor model of a photonic crystal with a rectangular lattice is proposed and its sensing characteristics based on bound states in the continuum are studied using the finite element method. Using the different resonances of the rectangular lattice in two axes, the resonance line-width can be adjusted by the polarization direction of incident light. This structure can achieve a sensitivity of 148 nm RIU-1 and the figure of merit above 821 RIU-1, which shows a great potential in high-quality sensors, filter, laser line-width modulation as well as measurement of angle and angular velocity.

关键词： photonic crystal sensor rectangular lattice BIC figure of merit polarization

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Extraordinary Transmission Characteristics of Subwavelength Nanoholes with rectangular lattice

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PLASMONICS 2017年第3期12卷 655-661页

作者： Cetin, Arif E. Drsata, Martin Eksioglu, Yasa Petracek, Jiri MIT Koch Inst Integrat Canc Res 77 Massachusetts Ave Cambridge MA 02139 USA Brno Univ Technol Fac Mech Engn Inst Engn Phys Tech 2 Brno 61669 Czech Republic Istanbul Kemerburgaz Univ Dept Elect & Elect Engn Mahmutbey Dilmenler Cad 26 TR-34217 Istanbul Turkey Brno Univ Technol CEITEC Cent European Inst Technol Tech 10 Brno 61600 Czech Republic

We investigate the extraordinary optical transmission (EOT) properties of nanohole arrays with a rectangular lattice for label-free refractive index sensing applications. We show that the deviation within the periodicities along the two axes at the nanohole plane leads to more advantageous spectral quality of EOT signal compared to the conventional square lattice geometries. We introduce a way to further improve the sensitivity of the aperture system by carefully choosing the periodicities. We introduce nanohole arrays with a rectangular lattice supporting EOT signals with larger figure-of-merit values as well as enabling much stronger light transmission. We also model a nanohole system covered with a thin dielectric layer, mimicking biomolecules captured on the gold surface, in order to show its biosensing capability. We also show that certain deviation amounts between periodicities create spectral splitting within the EOT signal leading to larger spectral shifts in the presence of a thin dielectric film.

关键词： Plasmonics Nanoholes Extraordinary optical transmission rectangular lattice Label-free biosensing

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On slim rectangular lattices

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ACTA SCIENTIARUM MATHEMATICARUM 2023年第1-2期89卷 13-22页

作者： Gratzer, George Univ Manitoba Winnipeg MB Canada

Let L be a slim, planar, semimodular lattice (slim means that it does not contain an M-3-sublattice). We call the interval I = [o, i] of L rectangular, if there are complementary a, b is an element of I such that a is to the left of b. We claim that a rectangular interval of a slim rectangular lattice is also a slim rectangular lattice. We will present some applications, including a recent result of G. Czedli. In a paper with E. Knapp about a dozen years ago, we introduced natural diagrams for slim rectangular lattices. Five years later, G. Czedli introduced e1 -diagrams. We prove that they are the same.

关键词： planar lattice slim lattice rectangular lattice rectangular interval natural diagram e(1)-diagram

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Quantum lattice transport along an infinitely extended perturbation

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2025年第14期58卷 145203-145203页

作者： Baradaran, Marzieh Exner, Pavel Khrabustovskyi, Andrii Univ Hradec Kralove Fac Sci Dept Phys Rokitanskeho 62 Hradec Kralove 50003 Czech Republic CzechTechn Univ Doppler Inst Math Phys & Appl Math Brehova 7 Prague 11519 Czech Republic Acad Sci Dept Theoret Phys NPI Hlavni 130 Rez 25068 Czech Republic

We consider a periodic quantum graph in the form of a rectangular lattice with the delta-coupling of strength gamma in the vertices perturbed by changing the latter at an infinite straight array of vertices to a gamma not equal gamma. We analyze the band spectrum of the system and show that it remains preserved as a set provided gamma > gamma > 0 while for all the other combinations additional band appear in some or all gaps of the unperturbed system. We also prove that for a randomly chosen positive energy, the probability of existence of a state exponentially localized in the vicinity of the perturbation equals 1/2.

关键词： quantum graphs rectangular lattice periodic structure non-local perturbation spectrum

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FAST AND ROBUST METHOD FOR SCREENED POISSON lattice GREEN'S FUNCTION USING ASYMPTOTIC EXPANSION AND FAST FOURIER TRANSFORM

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2025年第2期47卷 A1198-A1224页

作者： Hou, Wei Colonius, Tim CALTECH Div Engn & Appl Sci Pasadena CA 91125 USA

We study the lattice Green's function (LGF) of the screened Poisson equation on a two-dimensional rectangular lattice. This LGF arises in numerical analysis, random walks, solidstate physics, and other fields. Its defining characteristic is the screening term, which defines different regimes. When its coefficient is large, we can accurately approximate the LGF with an exponentially converging asymptotic expansion, and its convergence rate monotonically increases with the coefficient of the screening term. To tabulate the LGF when the coefficient is not large, we derive a one-dimensional integral representation of the LGF. We show that the trapezoidal rule can approximate this integral with exponential convergence, and we propose an efficient algorithm for its evaluation via the fast Fourier transform. We discuss applications including computing the LGF of the three-dimensional Poisson equation with one periodic direction and the return probability of a two-dimensional random walk with killing.

关键词： lattice Green's function screened Poisson equation fast Fourier transform rectangular lattice numerical methods random walk

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Length-preserving Extensions of a Semimodular lattice by Lowering a Join-irreducible Element

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ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS 2023年第2期40卷 403-421页

作者： Czedli, Gabor Univ Szeged Bolyai Inst Szeged Hungary

We extend the bijective correspondence between finite semimodular lattices and Faigle geometries to an analogous correspondence between semimodular lattices of finite lengths and a larger class of geometries. As the main application, we prove that if e is a joinirreducible element of a semimodular lattice L of finite length and h < e in L such that e does not cover h, then e can be "lowered" to a covering of h by taking a length-preserving semimodular extension K of L but not changing the rest of join-irreducible elements. With the help of our "lowering construction", we prove a general theorem on length-preserving semimodular extensions of semimodular lattices. This theorem implies earlier results proved by Grazer and Kiss (Order 2 351-365, 1986), Wild (Discrete Math. 112, 207-244, 1993), and Czedli and Schmidt (Adv. Math. 225, 2455-2463, 2010) on extensions to geometric lattices, and an unpublished result of E. T. Schmidt. Our approach offers shorter proofs of these results than the original ones.

关键词： Semimodular lattice Faigle geometry Geometric lattice Poset of join-irreducible elements Closure operator Cover-preserving extension rectangular lattice Slim semimodular lattice Planar semimodular lattice rectangular lattice

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