General Lattice Research Articles

New transference theorems on lattices possessing [formula omitted]-unique shortest vectors

In this paper, we first discuss lattices possessing nϵ-unique shortest vectors. We obtain three optimal transference theorems by establishing close relationships among successive minima, the covering radius and the minimal length of generating vectors. These results can be used to get finer reductions between GapSV Pγ′ and GapSIV Pγ for this class of lattices. Our work improves related results in the literature. In the second part of this paper, we prove a new transference theorem for general lattices where an optimal lower bound relating the successive minima of a lattice with its dual is given. As an application, we compare the respective advantages of current upper bounds on the smoothing parameters related to discrete Gaussian measures on lattices and give a more appropriate bound for lattices with duals possessing n-unique shortest vectors.

Evolution of non‐linear dynamics using Graphical Processing Units

AbstractA new Graphical Processing Unit (GPU) accelerated general lattice PDE solver and real‐time visualisation framework, called Gripper, is presented. Besides its scientific goals, the purpose of this project is to provide a publicly available, general purpose GPU enabled framework with visualisation capabilities. One of the application of this newly developed software, is aimed to provide valuable insight into non‐linear evolution of relativistic dynamical systems, such, as for example, neutron stars or black holes. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Magnetic responses of randomly depleted spin ladders

The magnetic responses of a spin-1/2 ladder doped with non-magnetic impurities are studied using various methods and including the regime where frustration induces incommensurability. Several improvements are made on the results of the seminal work of Sigrist and Furusaki [J. Phys. Soc. Jpn. 65, 2385 (1996)]. Deviations from the Brillouin magnetic curve due to interactions are also analyzed. First, the magnetic profile around a single impurity and effective interactions between impurities are analyzed within the bond-operator mean-field theory and compared to density-matrix renormalization group calculations. Then, the temperature behavior of the Curie constant is studied in details. At zero-temperature, we give doping-dependent corrections to the results of Sigrist and Furusaki on general bipartite lattice and compute exactly the distribution of ladder cluster due to chain breaking effects. Using exact diagonalization and quantum Monte-Carlo methods on the effective model, the temperature dependence of the Curie constant is compared to a random dimer model and a real-space renormalization group scenario. Next, the low-part of the magnetic curve corresponding to the contribution of impurities is computed using exact diagonalization. The random dimer model is shown to capture the bulk of the curve, accounting for the deviation from the Brillouin response. At zero-temperature, the effective model prediction agrees relatively well with density-matrix renormalization group calculations. Finite-temperature effects are displayed within the effective model and for large depleted ladder models using quantum Monte-Carlo simulations. In all, the effect of incommensurability does not display a strong qualitative effect on both the magnetic susceptibility and the magnetic curve. Consequences for experiments on the BiCu2PO6 compound and other spin-gapped materials are briefly discussed.

Cation-Dependent Magnetic Ordering and Room-Temperature Bistability in Azido-Bridged Perovskite-Type Compounds

A series of end-to-end azido-bridged perovskite-type compounds [(CH3)nNH4-n][Mn(N3)3] (n = 1-4) were synthesized and characterized. Structural phase transitions indicating the general lattice flexibility were observed and confirmed by the crystal structures of different phases. These materials show cation-dependent magnetic ordering at up to 92 K and magnetic bistability near room temperature.

Prime Filters and Ideals in Distributive Lattices

Summary. The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding [9] proven some results; we tried however to follow [21], [12], and [13]. All three were devoted to the Stone representation theorem [18] for Boolean or Heyting lattices. The main aim of the present article was to bridge this gap between general distributive lattices and Boolean algebras, having in mind that the more general approach will eventually replace the common proof of aforementioned articles.1 Because in Boolean algebras the notions of ultrafilters, prime filters and maximal filters coincide, we decided to construct some concrete examples of ultrafilters in nontrivial Boolean lattice. We proved also the Prime Ideal Theorem not as BPI (Boolean Prime Ideal), but in the more general setting. In the final section we present Nachbin theorems [15],[1] expressed both in terms of maximal and prime filters and as the unordered spectra of a lattice [11], [10]. This shows that if the notion of maximal and prime filters coincide in the lattice, it is Boolean.

Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for L 2 ( ? d ) ${\boldsymbol {L}{^{2}}(\mathbb {R}^d)}$ are obtained by applying time-frequency shifts from a lattice in ? ? ? d × ? ? $\boldsymbol {\Lambda } \vartriangleleft {\mathbb {R}^{d} \times \mathbb {\widehat {R}}}$ to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some g ? S 0 ( ? d ) $g \in {\boldsymbol {S}_{0}(\mathbb {R}^{d})}$ . There is always a canonical dual frame, generated by the dual Gabor atom g ~ ${\widetilde g}$ . The paper promotes a numerical approach for the efficient calculation of good approximations to the dual Gabor atom for general lattices, including the non-separable ones (different from a ? d × b ? d ${a\mathbb {Z}^{d}\,{\times }\,b\mathbb {Z}^{d}}$ ). The theoretical foundation for the approach is the well-known Wexler-Raz biorthogonality relation and the more recent theory of localized frames. The combination of these principles guarantees that the dual Gabor atom can be approximated by a linear combination of a few time-frequency shifted atoms from the adjoint lattice ? ° $\boldsymbol {\Lambda }\circ$ . The effectiveness of this approach is justified by a new theoretical argument and demonstrated by numerical examples.

A Randomized Sieving Algorithm for Approximate Integer Programming

The Integer Programming Problem (IP) for a polytope $P \subseteq \mathbb{R} ^{n}$ is to find an integer point in P or decide that P is integer free. We give a randomized algorithm for an approximate version of this problem, which correctly decides whether P contains an integer point or whether a (1+∈)-scaling of P about its center of gravity is integer free in 2 O(n)(1/∈ 2) n -time and 2 O(n)(1/∈) n -space with overwhelming probability. Our algorithm proceeds by reducing the approximate IP problem to an approximate Closest Vector Problem (CVP) under a "near-symmetric" norm. Our main technical contribution is an extension of the AKS randomized sieving technique, first developed by Ajtai et al. (Proceedings of 33rd Symposium on the Theory of Computing (STOC), pp. 601---610, 2001) for lattice problems under the l 2 norm, to the setting of asymmetric norms. We also present an application of our techniques to exact IP, where we give a nearly optimal algorithmic implementation of the Flatness Theorem, a central ingredient for many IP algorithms. Our results also extend to general convex bodies and lattices.

Generalized Courant-Snyder theory for charged-particle dynamics in general focusing lattices.

The Courant-Snyder (CS) theory for one degree of freedom is generalized to the case of coupled transverse dynamics in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D sympletic rotation. The envelope equation, the transfer matrix, and the CS invariant of the original CS theory all have their counterparts, with remarkably similar expressions, in the generalized theory.

Potts models with invisible states on general Bethe lattices

The number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order has attracted recent attention. In the q = 2 case, a Bragg–Williams (mean-field) approach necessitates four such invisible states while a 3-regular random graph formalism requires seventeen. In both of these cases, the changeover from second- to first-order behaviour induced by the invisible states is identified through the tricritical point of an equivalent Blume–Emery–Griffiths model. Here we investigate the generalized Potts model on a Bethe lattice with z neighbours. We show that, in the q = 2 case, invisible states are required to manifest the equivalent Blume–Emery–Griffiths tricriticality. When z = 3, the 3-regular random graph result is recovered, while z → ∞ delivers the Bragg–Williams (mean-field) result.

Resuspension of small particles from multilayer deposits in turbulent boundary layers

This paper describes a hybrid kinetic model for the resuspension of micron-size particles from multilayer deposits in a fully developed turbulent boundary layer. The rate of removal of particles from any given layer depends upon the rate of removal of particles from the layer above which acts as a source of uncovering and exposure of particles to the resuspending flow. The primary resuspension rate constant for an individual particle within a layer is based on the Rock'n'Roll (R'n'R) model using non-Gaussian statistics for the aerodynamic removal forces acting on the particles (Zhang et al., 2013). The coupled layer equations that describe multilayer resuspension of all the particles in each layer are based on the generic lattice model of Friess & Yadigaroglu (2001) which is extended here to include the influence of layer coverage and particle size distribution. The model is used to investigate a range of effects, including the influence of layer thickness on resuspension, the spread of inter-particle adhesive forces within layers, Gaussian and non-Gaussian pdfs for the removal forces and the associated timescales. Finally model predictions are compared with data from two resuspension experiments – STORM (Castelo et al., 1999) and BISE (Alloul-Marmor, 2002).

Electron-microscopic and XPS study of the phase transitions in a Co70Mo30 alloy

Electron-microscopic analysis and X-ray photoelectron spectroscopy are used to study the structure of the Co70Mo30 alloy formed at various temperatures. It is found that, at temperatures below the solidus temperature, the structure-phase transformations in the alloy have a tendency toward phase separation, which results in the formation of a cellular structure in the form of two types of clusters. These clusters are strongly enriched in molybdenum or cobalt and exist in the general fcc lattice of the solvent (cobalt). The allotropic α-Co → ɛ-Co phase transition occurs in the alloy at 532°C, and the transition-induced deformation contrast is superimposed on the cellular structure.

Numerical Simulation of Filling Process in HPDC by Lattice Boltzmann Method

A general lattice Boltzmann Method (LBM) to simulate filling process of high pressure die casting (HPDC) is investigated. Boundary conditions are studied and the free surface model is established by combine the LBM with VOF method. The final model was substantiated by simulating filling process in HPDC in three dimensions. The simulated results from LBM and finite difference method (FDM) were compared with the experiments. The results show the former is in a better agreement with experiments. It demonstrates the efficiency and precision of this LBM model in describing flow pattern in filling process.

Localization of disordered bosons and magnets in random fields

We study localization properties of disordered bosons and spins in random fields at zero temperature. We focus on two representatives of different symmetry classes, hard-core bosons (XY magnets) and Ising magnets in random transverse fields, and contrast their physical properties. We describe localization properties using a locator expansion on general lattices. For 1d Ising chains, we find non-analytic behavior of the localization length as a function of energy at ω=0, ξ−1(ω)=ξ−1(0)+A|ω|α, with α vanishing at criticality. This contrasts with the much smoother behavior predicted for XY magnets. We use these results to approach the ordering transition on Bethe lattices of large connectivity K, which mimic the limit of high dimensionality. In both models, in the paramagnetic phase with uniform disorder, the localization length is found to have a local maximum at ω=0. For the Ising model, we find activated scaling at the phase transition, in agreement with infinite randomness studies. In the Ising model long range order is found to arise due to a delocalization and condensation initiated at ω=0, without a closing mobility gap. We find that Ising systems establish order on much sparser (fractal) subgraphs than XY models. Possible implications of these results for finite-dimensional systems are discussed.

Influence of Solid Area Distribution on the Drag of a Two-Dimensional Lattice Frame

AbstractMany wind design codes provide recommended values of the drag coefficient for generic lattice frame geometries based on the solidity ratio. The solidity ratio of a lattice structure often varies along its length or height, resulting in a piecewise treatment in design. By definition, the solidity ratio is only dependent on the amount of solid area to the total enclosed area of a frame, and therefore does not account for member spacing or the number of members within the framework; this has implications for some nonuniform sections used in lattice tower design. Drag coefficients for many geometric configurations were measured in a wind tunnel with the focus placed on configurations having an identical solidity ratio but different member spacing ratios. It was observed that the drag coefficient varied with the member spacing ratio as well as with the number of members in the frame. The results suggest that the solidity ratio alone may not be sufficient for describing the drag coefficient of a lattice...

Symmetric Multiwavelet Frames with General Lattice

The multiwavelet has more desired properties than any single wavelet, such as short support, symmetry, and smoothness. In this paper, symmetric multiwavelet frames with general lattice and with any symmetric points are constructed from multiwavelet frames given. The existing result is generalized to the case of general lattice.

Vaporization Processes and Thermodynamic Properties of Oxide Systems Studied by High Temperature Mass Spectrometry

Information on the vaporization processes and thermodynamic properties of the ternary silicate systems Na2O-B2O3-SiO2, Cs2O-B2O3-SiO2, Rb2O-B2O3-SiO2, MgO-B2O3-SiO2, CaO-B2O3-SiO2, SrO-B2O3-SiO2, BaO-B2O3-SiO2, MgO-Al2O3-SiO2, CaO-Al2O3-SiO2, CaO-TiO2-SiO2, BaO-TiO2-SiO2, PbO-B2O3-SiO2 and ZnO-B2O3-SiO2 together with previously obtained data is discussed. Various types of vapour species were found over the oxide systems studied such as the associated, dissociated and polymerized products of vaporization. The systematic trends in the vaporization of binary and multicomponent oxide systems are illustrated and discussed from the point of view of the acid-base concept. Results on determination of thermodynamic functions in oxide systems were considered taking into account the main requirements for the confirmation of their reliability. The general lattice theory of associated solutions was used for the calculation of thermodynamic properties of the ternary silicate melts studied. Some industrial applications of multicomponent oxide systems are also illustrated.

Scaling of temperature-dependent thermal conductivities for one-dimensional nonlinear lattices

In general nonlinear lattices, the existence of renormalized phonons due to the nonlinear interactions has been independently discovered by many research groups. Regarding these renormalized phonons as the energy carriers responsible for the heat transport, the scaling laws of temperature-dependent thermal conductivities of one-dimensional nonlinear lattices can be derived from the phenomenological effective phonon approach. For the paradigmatic nonlinear φ(4) lattice, κ(T)[proportionality]T(-1.35), which was numerically obtained more than a decade ago, can be well explained by the current approach. Most importantly, this approach is able to predict the scaling laws of temperature-dependent thermal conductivities of generalized nonlinear Klein-Gordon lattices. These theoretical predictions are compared by numerical simulations, and perfect agreements have been found.

Periodic ordering of clusters in a one-dimensional lattice model.

A generic lattice model for systems containing particles interacting with short-range attraction long-range repulsion (SALR) potential that can be solved exactly in one dimension is introduced. We assume attraction J1 between the first neighbors and repulsion J2 between the third neighbors. The ground state of the model shows existence of two homogeneous phases (gas and liquid) for J2/J1 <1/3. In addition to the homogeneous phases, the third phase with periodically distributed clusters appears for J2/J1 > 1/3. Phase diagrams obtained in the self-consistent mean-field approximation for a range of values of J2/J1 show very rich behavior, including reentrant melting, and coexistence of two periodic phases (one with strong and the other one with weak order) terminated at a critical point. We present exact solutions for the equation of state as well as for the correlation function for characteristic values of J2/J1. Based on the exact results, for J2/J1 > 1/3 we predict pseudo-phase transitions to the ordered cluster phase indicated by a rapid change of density for a very narrow range of pressure, and by a very large correlation length for thermodynamic states where the periodic phase is stable in mean field. For 1/9 < J2/J1 < 1/3 the correlation function decays monotonically below certain temperature, whereas above this temperature exponentially damped oscillatory behavior is obtained. Thus, even though macroscopic phase separation is energetically favored and appears for weak repulsion at T = 0, local spatial inhomogeneities appear for finite T. Monte Carlo simulations in canonical ensemble show that specific heat has a maximum for low density ρ that we associate with formation of living clusters, and if the repulsion is strong, another maximum for ρ = 1/2.

Grobner Bases for Lattices and an Algebraic Decoding Algorithm

In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grdot{o}bner bases for binary linear codes were introduced by Borges-Quintana et al. . We extend their work to non-binary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Λ as the Grobner basis of its label code L. Since the associated label code for integer and non-integer lattices are group codes, the assigned Grobner bases can be obtained for both cases. Using this Grobner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E_7 and E_8) show that for low SNR the performance of these lattices is near to the lower bounds given in .

On the Characterization and Software Implementation of General Protein Lattice Models

models of proteins have been widely used as a practical means to computationally investigate general properties of the system. In lattice models any sterically feasible conformation is represented as a self-avoiding walk on a lattice, and residue types are limited in number. So far, only two- or three-dimensional lattices have been used. The inspection of the neighborhood of alpha carbons in the core of real proteins reveals that also lattices with higher coordination numbers, possibly in higher dimensional spaces, can be adopted. In this paper, a new general parametric lattice model for simplified protein conformations is proposed and investigated. It is shown how the supporting software can be consistently designed to let algorithms that operate on protein structures be implemented in a lattice-agnostic way. The necessary theoretical foundations are developed and organically presented, pinpointing the role of the concept of main directions in lattice-agnostic model handling. Subsequently, the model features across dimensions and lattice types are explored in tests performed on benchmark protein sequences, using a Python implementation. Simulations give insights on the use of square and triangular lattices in a range of dimensions. The trend of potential minimum for sequences of different lengths, varying the lattice dimension, is uncovered. Moreover, an extensive quantitative characterization of the usage of the so-called “move types” is reported for the first time. The proposed general framework for the development of lattice models is simple yet complete, and an object-oriented architecture can be proficiently employed for the supporting software, by designing ad-hoc classes. The proposed framework represents a new general viewpoint that potentially subsumes a number of solutions previously studied. The adoption of the described model pushes to look at protein structure issues from a more general and essential perspective, making computational investigations over simplified models more straightforward as well.