Lattice Construction Research Articles

Low-Dimensional Shaping for High-Dimensional Lattice Codes

We propose two low-complexity lattice code constructions that have competitive coding and shaping gains. The first construction, named systematic Voronoi shaping , maps short blocks of integers to the dithered Voronoi integers, which are dithered integers that are uniformly distributed over the Voronoi region of a low-dimensional shaping lattice. Then, these dithered Voronoi integers are encoded using a high-dimensional lattice retaining the same shaping and coding gains of low- and high-dimensional lattices. A drawback to this construction is that there is no isomorphism between the underlying message and the lattice code, preventing its use in applications such as compute-and-forward. Therefore, we propose a second construction, called mixed nested lattice codes , in which a high-dimensional coding lattice is nested inside a concatenation of low-dimensional shaping lattices. This construction not only retains the same shaping/coding gains as first construction but also provides the desired algebraic structure. We numerically study these methods, for point-to-point channels as well as compute-and-forward using low-density lattice codes as coding lattices and $E_{8}$ and Barnes–Wall as shaping lattices. Numerical results indicate a shaping gain of up to 0.86 dB, compared with the state-of-the-art of 0.4 dB; furthermore, the proposed method has lower complexity than the state-of-the-art approaches.

Granular reducts of formal fuzzy contexts

Knowledge reduction is one of the key issues in knowledge discovery and data mining. During the construction of a concept lattice, it has been recognized that computational complexity is a major obstacle in deriving all the concept from a database. In order to improve the computational efficiency, it is necessary to preprocess the database and reduce its size as much as possible. Focusing on formal fuzzy contexts, we introduce in the paper the notions of granular consistent sets and granular reducts and propose granular reduct methods in the sense of reducing the attributes. With the proposed approaches, the attributes that are not essential to all the object concepts can be removed without loss of knowledge and, consequently, the computational complexity of constructing the concept lattice is reduced. Furthermore, the relationship between the granular reducts and the classification reducts in a formal fuzzy context is investigated.

Constructing lattice based on irreducible concepts

The construction of concept lattices is one of the key issues of formal concept analysis. Many methods and algorithms are proposed to build a lattice, among which, incremental algorithms are more appropriate in real-life applications that work with dynamic datasets. But they cost much time to locate generators before generating a real concept. The batch algorithms generate concepts quickly. However, they ignore the procedure of building lattice relationship. In this paper, we build the lattice from meet-irreducible attribute concepts by using generators directly, and make optimizations in key steps. We yield the relationship among concepts during the generating process that saves much time in contrast to other batch algorithms. In addition to proving the correctness of our algorithm, we evaluate its performance on some real datasets and compare it with an incremental algorithm called FastAddIntent. The results show that our algorithm mainly depends on the numbers of concepts and the numbers of attributes, which achieves good performance, especially to large formal contexts.

Doubled lattice Chern–Simons–Yang–Mills theories with discrete gauge group

We construct doubled lattice Chern–Simons–Yang–Mills theories with discrete gauge group G in the Hamiltonian formulation. Here, these theories are considered on a square spatial lattice and the fundamental degrees of freedom are defined on pairs of links from the direct lattice and its dual, respectively. This provides a natural lattice construction for topologically-massive gauge theories, which are invariant under parity and time-reversal symmetry. After defining the building blocks of the doubled theories, paying special attention to the realization of gauge transformations on quantum states, we examine the dynamics in the group space of a single cross, which is spanned by a single link and its dual. The dynamics is governed by the single-cross electric Hamiltonian and admits a simple quantum mechanical analogy to the problem of a charged particle moving on a discrete space affected by an abstract electromagnetic potential. Such a particle might accumulate a phase shift equivalent to an Aharonov–Bohm phase, which is manifested in the doubled theory in terms of a nontrivial ground-state degeneracy on a single cross. We discuss several examples of these doubled theories with different gauge groups including the cyclic group Z(k)⊂U(1), the symmetric group S3⊂O(2), the binary dihedral (or quaternion) group D̄2⊂SU(2), and the finite group Δ(27)⊂SU(3). In each case the spectrum of the single-cross electric Hamiltonian is determined exactly. We examine the nature of the low-lying excited states in the full Hilbert space, and emphasize the role of the center symmetry for the confinement of charges. Whether the investigated doubled models admit a non-Abelian topological state which allows for fault-tolerant quantum computation will be addressed in a future publication.

A construction of lattices in splittable solvable Lie groups

In this paper, we consider a unified constructions of lattices in splittable solvable Lie groups.

Nucleic Acid Nanostructures for Chemical and Biological Sensing.

The nanoscale features of DNA have made it a useful molecule for bottom-up construction of nanomaterials, for example, two- and three-dimensional lattices, nanomachines, and nanodevices. One of the emerging applications of such DNA-based nanostructures is in chemical and biological sensing, where they have proven to be cost-effective, sensitive and have shown promise as point-of-care diagnostic tools. DNA is an ideal molecule for sensing not only because of its specificity but also because it is robust and can function under a broad range of biologically relevant temperatures and conditions. DNA nanostructure-based sensors provide biocompatibility and highly specific detection based on the molecular recognition properties of DNA. They can be used for the detection of single nucleotide polymorphism and to sense pH both in solution and in cells. They have also been used to detect clinically relevant tumor biomarkers. In this review, recent advances in DNA-based biosensors for pH, nucleic acids, tumor biomarkers and cancer cell detection are introduced. Some challenges that lie ahead for such biosensors to effectively compete with established technologies are also discussed.

Congruence Lattices of a Finite Isoform Lattices

This result is a stronger version of the result obtained in the previous chapter. To prove this result, we introduce a new lattice construction which is described in section 1.2. The congruence structure of this new construct is described section 1.3. In section 1.4, we present the proof of the main theorem. We start with the definition of isoform lattices.

Construction of an Artificial Ferrimagnetic Lattice by Lithium Ion Insertion into a Neutral Donor/Acceptor Metal-Organic Framework.

Construction of a molecular system in which the magnetic lattice exhibits long-range order is one of the fundamental goals in materials science. In this study, we demonstrate the artificial construction of a ferrimagnetic lattice by doping electrons into acceptor sites of a neutral donor/acceptor metal-organic framework (D/A-MOF). This doping was achieved by the insertion of Li-ions into the D/A-MOF, which was used as the cathode of a Li-ion battery cell. The neutral D/A-MOF is a layered system composed of a carboxylate-bridged paddlewheel-type diruthenium(II,II) complex as the donor and a TCNQ derivative as the acceptor. The ground state of the neutral form was a magnetically disordered paramagnetic state. Upon discharge of the cell, spontaneous magnetization was induced; the transition temperature was variable. The stability of the magnetically ordered lattice depended on the equilibrium electric potential of the D/A-MOF cathode, which reflected the electron-filling level.

Construction of an Artificial Ferrimagnetic Lattice by Lithium Ion Insertion into a Neutral Donor/Acceptor Metal–Organic Framework

AbstractConstruction of a molecular system in which the magnetic lattice exhibits long‐range order is one of the fundamental goals in materials science. In this study, we demonstrate the artificial construction of a ferrimagnetic lattice by doping electrons into acceptor sites of a neutral donor/acceptor metal–organic framework (D/A‐MOF). This doping was achieved by the insertion of Li‐ions into the D/A‐MOF, which was used as the cathode of a Li‐ion battery cell. The neutral D/A‐MOF is a layered system composed of a carboxylate‐bridged paddlewheel‐type diruthenium(II,II) complex as the donor and a TCNQ derivative as the acceptor. The ground state of the neutral form was a magnetically disordered paramagnetic state. Upon discharge of the cell, spontaneous magnetization was induced; the transition temperature was variable. The stability of the magnetically ordered lattice depended on the equilibrium electric potential of the D/A‐MOF cathode, which reflected the electron‐filling level.

A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures

Development of advanced synthetic materials that can mimic the mechanical properties of non-mineralized soft biological materials has important implications in a wide range of technologies. Hierarchical lattice materials constructed with horseshoe microstructures belong to this class of bio-inspired synthetic materials, where the mechanical responses can be tailored to match the nonlinear J-shaped stress–strain curves of human skins. The underlying relations between the J-shaped stress–strain curves and their microstructure geometry are essential in designing such systems for targeted applications. Here, a theoretical model of this type of hierarchical lattice material is developed by combining a finite deformation constitutive relation of the building block (i.e., horseshoe microstructure), with the analyses of equilibrium and deformation compatibility in the periodical lattices. The nonlinear J-shaped stress–strain curves and Poisson ratios predicted by this model agree very well with results of finite element analyses (FEA) and experiment. Based on this model, analytic solutions were obtained for some key mechanical quantities, e.g., elastic modulus, Poisson ratio, peak modulus, and critical strain around which the tangent modulus increases rapidly. A negative Poisson effect is revealed in the hierarchical lattice with triangular topology, as opposed to a positive Poisson effect in hierarchical lattices with Kagome and honeycomb topologies. The lattice topology is also found to have a strong influence on the stress–strain curve. For the three isotropic lattice topologies (triangular, Kagome and honeycomb), the hierarchical triangular lattice material renders the sharpest transition in the stress–strain curve and relative high stretchability, given the same porosity and arc angle of horseshoe microstructure. Furthermore, a demonstrative example illustrates the utility of the developed model in the rapid optimization of hierarchical lattice materials for reproducing the desired stress–strain curves of human skins. This study provides theoretical guidelines for future designs of soft bio-mimetic materials with hierarchical lattice constructions.

The Structure of Finite Distributive Lattices

This paper is devoted to the structure that describes the construction of finite distributive lattices. From the viewpoint of application, we consider algorithms of construction and enumeration of distributive lattices and partially ordered sets for finite distributive lattices: A formula for finding the maximum anti-chain with respect to nonintersection is given, it is shown that elements of the lattice can be split into pairs according to comparison, the point of the maximum number of elements in the lattices is considered, and the structure of lattice congruence is described.

Hydrogen-Bonded Anionic Host Lattices Constructed with Isocyanurate and Thiourea/Urea

A series of seven inclusion complexes containing the isocyanurate ion, thiourea/urea, and selected quaternary ammonium ions as templates, namely, (CH3)4N+C3H2N3O3–· 3(NH2)2CS·H2O (1), (CH3)4N+C3H2N3O3–·(NH2)2CS (2), (C2H5)4N+C3H2N3O3–·(NH2)2CS (3), (n-C3H7)4N+C3H2N3O3–·(NH2)2CS·H2O (4), 3[(n-C4H9)4N+C3H2N3O3–]·5(NH2)2CS·3H2O (5), (C2H5)4N+C3H2N3O3–·(NH2)2CO (6) and (n-C3H7)4N+C3H2N3O3–·2(NH2)2CO·H2O (7), have been prepared and characterized by X-ray crystallography. Complex (1) features a hydrogen-bonded molecular cage in which two (CH3)4N+ cations can be accommodated. Complexes (2) and (7) are channel-type inclusion compounds. In (2), the thiourea sulfur atom does not participate in host lattice construction, while (7) contains several intersecting channel systems that are formed by the cross-linkage of two sets of isocyanurate–urea–water ribbons. Complexes (3)–(6) exhibit layer-type architectures. The host layers of complexes (3) and (6) are constructed by two distinct rosette motifs produced by isocyan...

Optimal design of lattice metal constructions of overhead power transmission lines

The article presents a new concept of design of power lines which allowed us to develop new unified power transmission supports, efficient and reliable ones. For the first time, a method of optimum design of overhead power lines has been developed which considered the power line as a united structural system, the relief, the atmospheric and climatic conditions, the environment pollution, the efficiency requirements, the technological and operational expenditures. The article describes the changes introduced into the requirements for the dimensions and parameters of grid metal works with the minimum mass. The optimum design, the computer-aided assembly process and the use of advanced technologies guarantee the reduction of the construction period and the increase in the maintenance period of power lines.

Construction method of concept lattice based on improved variable precision rough set

This paper mainly focuses on how to construct concept lattice effectively and efficiently based on improved variable precision rough set. On the basis of preprocessing formal concept, one algorithm that can determine the value range of variable precision parameter β according to the approximate classification quality is proposed. An improved β-upper and lower distribution attribute reduction algorithm is also proposed based on the improved variable precision rough set, the algorithm can be used for attribute reduction on the original data of the concept lattice, and to eliminate the redundant knowledge or noises of the formal context. For the reduced formal context, the paper combines the concept construction algorithm with an improved rule acquisition algorithm seamlessly, and proposes a novel approach of concept lattice construction based on improved variable precision rough set. Finally, a concept lattice generation prototype system is developed, this paper also performs comprehensive experiments, and the effectiveness of the improved algorithm is proved through the experimental results.

An algorithm to construct arithmetic Fuchsian groups derived from quaternion algebras and the corresponding hyperbolic lattices

The aim of this paper is to propose an algorithm to construct arithmetic Fuchsian groups derived from quaternion algebras and quaternion orders which will lead to the construction of hyperbolic lattices. To achieve this goal a necessary condition for obtaining arithmetic Fuchsian groups Γp from a tessellation {p,q} whose regular hyperbolic polygon Pp generates an oriented surface with genus g≥2 is established. This necessary condition is called Fermat condition due to its identification with the Fermat primes. It is also shown an isomorphism between arithmetic Fuchsian groups derived from different edge-pairings sets of the regular fundamental region associated with the tessellation {4g,4g} for g=2n,3.2n,5.2n,and3.5.2n, and the tessellation {4g+2,2g+1} for g=2. One set uses the normal form whereas the other one uses diametrically opposite edge-pairings. All these transformations are hyperbolic and so result in an oriented compact Riemann surface.

Holographic lattices and metal–insulator transition

This paper is an extension of the talk given at the conference on Gravitation and Cosmology/The Fourth Galileo-Xu Guangqi Meeting. We intend to present a short review on recent progress on the construction of holographic lattices and its application to metal–insulator transition (MIT), which is a fundamentally important phenomenon in condensed matter physics. We will firstly implement the Peierls phase transition by constructing holographic charge density waves which are induced by the spontaneous breaking of translational symmetry. Then we turn to the holographic realization of metal–insulator transition as a quantum critical phenomenon with many strongly correlated electrons involved. The holographic entanglement entropy as a diagnostic for such quantum phase transitions will be briefly mentioned.

Construction of lattices of solvable Lie groups from a viewpoint of matrices

In this paper, we reconstruct lattices of famous solvable Lie groups from a viewpoint of matrices. As an application, we construct examples of solvable Lie groups which admit lattices.

An Approach Using Concept Lattice Structure for Data Mining and Information Retrieval

Since the 1980s, the concept lattice was studied and applied to the problems of text mining, frequent itemset, classification, etc. The formal concept analysis - FCA is one of the main techniques applied in the concept lattice. FCA is a mathematical theory which is applied to the data mining by setting a table with rows describing objects and columns describing attributes, with relationships between them, and then sets up the concept lattice structure. In the area of information retrieval, FCA considers the correlation of objects-attributes the same as those of documents-terms. In the process of setting up the lattice, FCA defines each node in the lattice as a concept. The algorithm for the construction of concept lattice will install a couple on each node, including a set of documents with common terms, and a set of terms which co-occurs in documents. In a larger scale, each concept in the lattice could be recognized as a couple of questions - answers. In the lattice, the action of browsing up or down of nodes will allow approaching more general concepts or more detail concepts, respectively.

On a projection-corrected component-by-component construction

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm.

A copula-based approach for generating lattices

Discrete approximations such as binomial and trinomial lattices have been developed to model the intertemporal dynamics of variables in a way that also allows contingent decisions to be included at the appropriate increments in time. In this paper we present an approach for developing these types of models based on copulas. In addition to ease of implementation, a primary benefit of this approach is its generality, and we show that various binomial and trinomial approximation methods for valuing contingent claim securities in the literature are special cases of this approach, each based on a choice of a particular set of probability and/or branching parameters. Because this approach encompasses these and other cases as feasible solutions, we also show how it can be used to optimize the construction of lattices so that discretization error is minimized, and we demonstrate its application for an option pricing example.