Orthogonal Lattice Research Articles

Spontaneous Formation of Two-Dimensional Micropatterns with Straight and/or Curving Dendrites through Crystal Growth of Ba(NO3)2 in Polymer Matrix

Microscale patterns consisting of two-dimensional (2D) dendrites with trunks and branches 1–3 μm wide were precisely controlled through crystal growth in a thin polymer matrix by the use of a dipping technique. A variety of 2D micropatterns, such as orthogonal lattices, bull’s horns, randomly curving weaves, and aligned dots, were homogeneously formed with an aqueous solution of Ba(NO3)2 and poly(vinyl alcohol) in a wide area ranging over several centimeters on flat and rounded substrates. The micrometric dendritic growth that produced these several patterns was tuned by changing the withdrawal rate and the polymer concentration. The crystallographic orientation of the micropatterns was characterized to discuss on the formation mechanism of the specific morphologies. The curving branches were found to be induced by gradual change in the growth direction through low-angle grain boundaries under a highly diffusion-limited condition. This simple, bottom-up patterning process is applicable for various crystalline materials, including inorganic and organic substances.

Quantization error reduction for MIMO detection based on orthogonal lattices

Quantization error reduction for MIMO detection based on orthogonal lattices

Devonian salt dissolution-collapse breccias flooring the Cretaceous Athabasca oil sands deposit and development of lower McMurray Formation sinkholes, northern Alberta Basin, Western Canada

The sub-Cretaceous paleotopography underlying giant Lower Cretaceous Athabasca oil sands, northern Alberta, has an orthogonal lattice pattern of troughs up to 50km long and 100m deep between pairs of cross-cutting lineaments. These structures are interpreted to have been inherited from a similar pattern of dissolution collapse-subsidence troughs in the underlying Middle Devonian salt beds. Removal of more than 100m of halite salt fragmented the overlying Upper Devonian strata into fault blocks and collapse breccias that subsided into the underlying dissolution troughs. The unusually low 1:2 to 1:3 thickness ratios of halite salts to the overlying strata resulted in the Upper Devonian strata collapse-subsidence into underlying salt dissolution troughs being more cataclysmic during the first phase of salt removal. The second phase of slower but complete salt removal between the earlier troughs resulted in a more gradual subsidence of the overlying strata. This obliterated the earlier pattern of giant cross-cutting dissolution troughs bounded by major lineaments. The collapse breccia fabrics underlying the earlier troughs differ from those from areas between the troughs. Collapse breccias underlying the large troughs often have crushed fabrics distributed in zones that rapidly pinched out between fault blocks. Breccias between troughs developed as giant mosaics of detached carbonate blocks that formed breccia pipe complexes.Multiple sinkholes up to 100m deep aligned along multi-km linear valley trends that dissected the sub-Cretaceous paleotopography. These sinkhole trends formed orthogonal patterns inherited from underlying lattice of NW–SE and NE–SW salt structured lineaments. These cross-cutting sinkhole trends have a smaller 5km scale reticulate pattern similar to the giant 50km scale pattern of collapse-subsidence troughs. Other sinkholes developed as lower McMurray strata sagged when underlying Devonian fault blocks and breccia pipes differentially subsided.

Symmetric Self-Hilbertian Wavelets via Orthogonal Lattice Optimization

Orthonormal Hilbert pairs of wavelets that are time-reverse (mirror image) versions of each are termed Symmetric Self-Hilbertian wavelets and are used in the dual-tree complex wavelet transform. Previous methods for constructing the corresponding scaling low-pass filter are based on optimizing the product filter. These methods are practical only when the number of free-parameters is small due to the high computational load otherwise. An alternative method that is based on the orthogonal lattice is presented here and is practical with any number of free-parameters. Higher analytic quality Hilbert pairs can be obtained when there are more free-parameters. An effective strategy for optimizing the lattice parameters to give high quality filters is presented here.

Analytical solution for a Mode III crack in a 3D beam lattice

The brittle fracture behavior of an open cell foam is considered. The foam is modeled by an infinite lattice composed of elastic straight-line beam elements (struts) having uniform cross-sections and rigidly connected at the nodal points. The beams are parallel to the three mutually orthogonal lattice vectors thus forming a microstructure with rectangular parallelepiped cells.A semi-infinite Mode III crack is embedded in the lattice and, for the considered antiplane deformation, each node has three degrees of freedom, namely, the displacement parallel to the crack front and two rotations about the axes perpendicular to this direction. The analysis method hinges on the discrete Fourier transform, which allows to formulate the crack problem by means of the Wiener–Hopf equation. Its solution yields closed-form analytical expressions for the forces and the displacements at any cross-section, and, in particular, at the crack plane. An eigensolution for the traction-free crack faces and K-field remote loading is derived from the solution for the loaded crack using a limiting procedure. An analytical expression for the fracture toughness is derived from the eigensolution by comparing the remote stress field and the stresses in the near-tip struts. The obtained expression is found to be consistent with the known analytical and experimental results for Mode I deformation. It appears, that the dependence of the fracture toughness upon shape anisotropy ratio of the lattice material is non-monotonic. The optimal value of this parameter, which provides the maximum crack arresting ability is determined.

Hardware oriented, quasi-optimal detectors for iterative and non-iterative MIMO receivers

Abstract In this article we study hardware-oriented versions of the recently appeared Layered ORthogonal lattice Detector (LORD) and Turbo LORD (T-LORD). LORD and T-LORD are attractive Multiple-Input Multiple-Output (MIMO) detection algorithms, that aim to approach the optimal Maximum-Likelihood (ML) and Maximum-A-Posteriori (MAP) performance, respectively, yet allowing a complexity quadratic in the number of transmitting antennas rather than exponential. LORD and T-LORD are also well suited to a hardware (e.g., ASIC or FPGA) implementation because of their regularity, parallelism, deterministic latency, and complexity. Nevertheless, their complexity is still high in case of high cardinality constellations, such as the 64-QAM foreseen by the 802.11n standard. We show that, when only global latency constraints exist, e.g., a fixed time to detect the whole OFDM symbol, the LORD and T-LORD complexity can be remarkably reduced, still approaching the ML and MAP performance, respectively. Notwithstanding the suboptimal low-complexity and hardware-oriented implementation, LORD and T-LORD approach the EXtrinsic Information Transfer of the ML and MAP detectors, respectively. To focus on a specific setting, we consider the indoor MIMO wireless LAN 802.11n standard, taking into account errors in channel estimation and a frequency selective, spatially correlated channel model.

Optochemical self-organisation of white light in a photopolymerisable gel: a single-step route to intersecting and interleaving 3-D optical and waveguide lattices

We report a minute-long, single step, room temperature and spontaneous route to complex 3-D optical lattices and their corresponding waveguide architectures. The method exploits the instability and spontaneous filamentation of broad light beams propagating through a photopolymerisable medium and in this way, combines the ease and elegance of self-organisation with the precision and directionality that characterise beam-directed lithography. We found that when launched into a photopolymer, an orthogonal pair of white incandescent beams spontaneously divided into light filaments that self-organised into a hexagonal lattice. When made to intersect, the orthogonal hexagonal lattices generated a 3-D optical lattice with BCC symmetry and when made to interleave, yielded a lattice with the woodpile arrangement. These optical lattices permanently inscribed microstructures composed of thousands of discrete cylindrical, multimode and polychromatic waveguides. The structural lattices spanned a volume of ≅1000 mm3, at least an order of magnitude greater than volumes that are typically constructed through lithography or assembled through self-organisation processes. The densely packed and multidirectional waveguide lattices could serve as fundamentally new and remarkably inexpensive devices with highly efficient, angle-independent light collection and guiding capability. As a result, they hold significant potential to improve the light-harnessing capabilities of photovoltaic devices and separately, to function as nonlinear photonic crystals.

Hardware oriented, quasi-optimal detectors for iterative and non-iterative MIMO receivers

In this article we study hardware-oriented versions of the recently appeared Layered ORthogonal lattice Detector (LORD) and Turbo LORD (T-LORD). LORD and T-LORD are attractive Multiple-Input Multiple-Output (MIMO) detection algorithms, that aim to approach the optimal Maximum-Likelihood (ML) and Maximum-A-Posteriori (MAP) performance, respectively, yet allowing a complexity quadratic in the number of transmitting antennas rather than exponential. LORD and T-LORD are also well suited to a hardware (e.g., ASIC or FPGA) implementation because of their regularity, parallelism, deterministic latency, and complexity. Nevertheless, their complexity is still high in case of high cardinality constellations, such as the 64-QAM foreseen by the 802.11n standard. We show that, when only global latency constraints exist, e.g., a fixed time to detect the whole OFDM symbol, the LORD and T-LORD complexity can be remarkably reduced, still approaching the ML and MAP performance, respectively. Notwithstanding the suboptimal low-complexity and hardware-oriented implementation, LORD and T-LORD approach the EXtrinsic Information Transfer of the ML and MAP detectors, respectively. To focus on a specific setting, we consider the indoor MIMO wireless LAN 802.11n standard, taking into account errors in channel estimation and a frequency selective, spatially correlated channel model.

Theoretical description of two ultracold atoms in finite three-dimensional optical lattices using realistic interatomic interaction potentials

A theoretical approach is described for an exact numerical treatment of a pair of ultracold atoms interacting via a central potential that are trapped in a finite three-dimensional optical lattice. The coupling of center-of-mass and relative-motion coordinates is treated using an exact diagonalization (configuration-interaction) approach. The orthorhombic symmetry of an optical lattice with three different but orthogonal lattice vectors is explicitly considered as is the Fermionic or Bosonic symmetry in the case of indistinguishable particles.

Molybdenum oxide crystals encapsulated inside carbon nanotubes by heat treatment in air

A simple method for encapsulating molybdenum oxide crystals inside single‐walled carbon nanotubes is introduced including a report on analyses of the crystal structure using a transmission electron microscope. Because the crystals were vulnerable under electron beam irradiation, a system that minimizes the electron dose before image acquisition was used to avoid lattice destruction. High‐resolution image and electron diffraction analysis showed that the encapsulated crystals had orthogonal lattice spacing of 0.391 and 0.363 nm. The result did not match any known molybdenum oxide crystals. Therefore, molybdenum oxide crystals that are different from those of bulk state seemed to be generated because of encapsulation in the strictly limited space. The encapsulation process and its driving force are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.

A polyominoes-permutations injection and tree-like convex polyominoes

Plane polyominoes are edge-connected sets of cells on the orthogonal lattice Z 2 , considered identical if their cell sets are equal up to an integral translation. We introduce a novel injection from the set of polyominoes with n cells to the set of permutations of [ n ] , and classify the families of convex polyominoes and tree-like convex polyominoes as classes of permutations that avoid some sets of forbidden patterns. By analyzing the structure of the respective permutations of the family of tree-like convex polyominoes, we are able to find the generating function of the sequence that enumerates this family, conclude that this sequence satisfies the linear recurrence a n = 6 a n − 1 − 14 a n − 2 + 16 a n − 3 − 9 a n − 4 + 2 a n − 5 , and compute the closed-form formula a n = 2 n + 2 − ( n 3 − n 2 + 10 n + 4 ) / 2 .

Modulus fault attacks against RSA–CRT signatures

RSA–CRT fault attacks have been an active research area since their discovery by Boneh, DeMillo and Lipton in 1997. We present alternative key-recovery attacks on RSA–CRT signatures: instead of targeting one of the sub-exponentiations in RSA–CRT, we inject faults into the public modulus before CRT interpolation, which makes a number of countermeasures against Boneh et al.’s attack ineffective. Our attacks are based on orthogonal lattice techniques and are very efficient in practice: depending on the fault model, between 5 and 45 faults suffice to recover the RSA factorization within a few seconds. Our simplest attack requires that the adversary knows the faulty moduli, but more sophisticated variants work even if the moduli are unknown, under reasonable fault models. All our attacks have been fully validated experimentally with fault-injection laser techniques.

Space of symmetry matrices with elements 0, ±1 and complete geometric description; its properties and application

A fixed set, that is the set of all lattice metrics corresponding to the arithmetic holohedry of a primitive lattice, is a natural tool for keeping track of the symmetry changes that may occur in a deformable lattice [Ericksen (1979). Arch. Rat. Mech. Anal. 72, 1-13; Michel (1995). Symmetry and Structural Properties of Condensed Matter, edited by T. Lulek, W. Florek & S. Walcerz. Singapore: Academic Press; Pitteri & Zanzotto (1996). Acta Cryst. A52, 830-838; and references quoted therein]. For practical applications it is desirable to limit the infinite number of arithmetic holohedries, and simplify their classification and construction of the fixed sets. A space of 480 matrices with cyclic consecutive powers, determinant 1, elements from {0, ±1} and geometric description were analyzed and offered as the framework for dealing with the symmetry of reduced lattices. This matrix space covers all arithmetic holohedries of primitive lattice descriptions related to the three shortest lattice translations in direct or reciprocal spaces, and corresponds to the unique list of 39 fixed points with integer coordinates in six-dimensional space of lattice metrics. Matrices are presented by the introduced dual symbol, which sheds some light on the lattice and its symmetry-related properties, without further digging into matrices. By the orthogonal lattice distortion the lattice group-subgroup relations are easily predicted. It was proven and exemplified that new symbols enable classification of lattice groups on an absolute basis, without metric considerations. In contrast to long established but sophisticated methods for assessing the metric symmetry of a lattice, simple filtering of the symmetry operations from the predefined set is proposed. It is concluded that the space of symmetry matrices with elements from {0, ±1} is the natural environment of lattice symmetries related to the reduced cells and that complete geometric characterization of matrices in the arithmetic holohedry provides a useful tool for solving practical lattice-related problems, especially in the context of lattice deformation.

Adaptive synthesis of a wavelet transform using fast neural network

Adaptive synthesis of a wavelet transform using fast neural network This paper introduces a new method for an adaptive synthesis of a wavelet transform using a fast neural network with a topology based on the lattice structure. The lattice structure and the orthogonal lattice structure are presented and their properties are discussed. A novel method for unsupervised training of the neural network is introduced. The proposed approach is tested by synthesizing new wavelets with an expected energy distribution between low- and high-pass filters. Energy compaction of the proposed method and Daubechies wavelets is compared. Tests are performed using sound and image signals.

Low Complexity, Quasi-Optimal MIMO Detectors for Iterative Receivers

We propose a novel family of Soft-Input Soft-Output detectors for iterative, point-to-point, MIMO receivers. Compared to the optimal Maximum A Posteriori receiver, low complexity is achieved restricting the detector search to small subsets of the entire QAM hyper-symbol constellation, through simple criteria. These criteria are applied to an improved version of the non-iterative Layered ORthogonal lattice Detector. We show that, notwithstanding the suboptimal low-complexity implementation, this detector approaches the EXtrinsic Information Transfer of the MAP detector. Therefore, when included in an iterative receiver it delivers the same performance. Furthermore, the deterministic complexity and highly parallelizable structure of the proposed detector are well suited for HDL and ASIC implementation. To focus on a specific setting, we consider the indoor MIMO wireless LAN 802.11n standard, taking into account errors in Channel Estimation and a frequency selective, spatially correlated channel model.

Universal structure and universal equations (PDE) for unitary ensembles

Random matrix ensembles with unitary invariance of measure (UE) are described in a unified way using a combination of Tracy–Widom (TW) and Adler–Shiota–van Moerbeke approaches to the derivation of partial differential equations (PDEs) for spectral gap probabilities. First, general three-term recurrence relations for UE restricted to subsets of real line, or, in other words, for functions in the resolvent kernel, are obtained. Using them, simple universal relations between all TW dependent variables and one-dimensional Toda lattice τ-functions are found. A universal system of PDE for UE is derived from previous relations, which leads also to a single independent PDE for spectral gap probability of various UE. Thus, orthogonal function bases and Toda lattice are seen at the core of correspondence of different approaches. Moreover, Toda-AKNS system provides a common structure of PDE for unitary ensembles. Interestingly, this structure can be seen in two very different forms: one arises from orthogonal function-Toda lattice considerations, while the other comes from Schlesinger equations for isomonodromic deformations and their relation to TW equations. The simple example of Gaussian matrices most neatly exposes this structure.

Improving energy compaction of a wavelet transform using genetic algorithm and fast neural network

Improving energy compaction of a wavelet transform using genetic algorithm and fast neural networkIn this paper a new method for adaptive synthesis of a smooth orthogonal wavelet, using fast neural network and genetic algorithm, is introduced. Orthogonal lattice structure is presented. A new method of supervised training of fast neural network is introduced to synthesize a wavelet with desired energy distribution between output signals from low-pass and high-pass filters on subsequent levels of a Discrete Wavelet Transform. Genetic algorithm is proposed as a global optimization method for defined objective function, while neural network is used as a local optimization method to further improve the result. Proposed approach is tested by synthesizing wavelets with expected energy distribution between low- and high-pass filters. Energy compaction of proposed method and Daubechies wavelets is compared. Tests are performed using image signals.

Organization of the lamin scaffold in the internal nuclear matrix of normal and transformed hepatocytes

Nuclear lamins are among the more abundant proteins making up the internal nuclear matrix, but very little is known about their structure in the nucleoplasm. Using immunoelectron microscopy, we demonstrate the organization of lamins in the nuclear matrix isolated from rat hepatocytes for the first time. Lamin epitopes are arrayed both in locally ordered clusters and in quasi-regular rows. Fourier filtering of the images demonstrates that the epitopes are placed at the nodes and halfway between the nodes of square or rhombic lattices that are about 50 nm on each side, as well as along rows at regular ∼25-nm intervals. In addition, we have compared this structure with that of the internal nuclear matrix isolated from persistent hepatocyte nodules. In transformed hepatocytes, the islands of lamin lattice are lost, and only a partial regularity in the rows of gold particles remains. We suggest that orthogonal lattice assembly might be an intrinsic property of lamin molecules, and that the disassembly may be triggered by simple molecular events such as phosphorylation.

Chebyshev type lattice path weight polynomials by a constant term method

We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary ‘decorated’ weights as well as an arbitrary ‘background’ weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence, we also present an efficient method for finding explicit closed-form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed-form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as the models of steric stabilization and sensitized flocculation.

Textures on Rank‐1 Lattices

AbstractStoring textures on orthogonal tensor product lattices is predominant in computer graphics, although it is known that their sampling efficiency is not optimal. In two dimensions, the hexagonal lattice provides the maximum sampling efficiency. However, handling these lattices is difficult, because they are not able to tile an arbitrary rectangular region and have an irrational basis. By storing textures on rank‐1 lattices, we resolve both problems: Rank‐1 lattices can closely approximate hexagonal lattices, while all coordinates of the lattice points remain integer. At identical memory footprint texture quality is improved as compared to traditional orthogonal tensor product lattices due to the higher sampling efficiency. We introduce the basic theory of rank‐1 lattice textures and present an algorithmic framework which easily can be integrated into existing off‐line and real‐time rendering systems.