Multidimensional Lattice Research Articles

Data Compression With Low Distortion and Finite Blocklength

This paper considers lossy source coding of $n$ -dimensional memoryless sources and shows an explicit approximation to the minimum source coding rate required to sustain the probability of exceeding distortion $d$ no greater than $\epsilon $ , which is simpler than known dispersion-based approximations. Our approach takes inspiration in the celebrated classical result stating that the Shannon lower bound to rate-distortion function becomes tight in the limit $d \to 0$ . We formulate an abstract version of the Shannon lower bound that recovers both the classical Shannon lower bound and the rate-distortion function itself as special cases. Likewise, we show that a nonasymptotic version of the abstract Shannon lower bound recovers all previously known nonasymptotic converses. A necessary and sufficient condition for the Shannon lower bound to be attained exactly is presented. It is demonstrated that whenever that condition is met, the rate-dispersion function is given simply by the varentropy of the source. Remarkably, all finite alphabet sources with balanced distortion measures satisfy that condition in the range of low distortions. Most continuous sources violate that condition. Still, we show that lattice quantizers closely approach the nonasymptotic Shannon lower bound, provided that the source density is smooth enough and the distortion is low. This implies that fine multidimensional lattice coverings are nearly optimal in the rate-distortion sense even at finite $n$ . The achievability proof technique is based on a new bound on the output entropy of lattice quantizers in terms of the differential entropy of the source, the lattice cell size, and a smoothness parameter of the source density. The technique avoids both the usual random coding argument and the simplifying assumption of the presence of a dither signal.

Multi-Dimensional Random Walks and Integrable Phase Models

We consider random multi-dimensional lattice walks bounded by a hyperplane, calling them walks over multi-dimensional simplicial lattices. We demonstrate that generating functions of these walks are dynamical correlation functions of a certain type of exactly solvable quantum phase models describing strongly correlated bosons on a chain. Walks over oriented lattices are related to the phase model with a non-Hermitian Hamiltonian, while walks over disoriented ones are related to the model with a Hermitian Hamiltonian. The calculation of the generating functions is based on the algebraic Bethe Ansatz approach to the solution of integrable models. The answers are expressed through symmetric functions. Continuous-time quantum walks bounded by a onedimensional lattice of finite length are also studied. Bibliography: 40 titles.

Materialized View Selection using Artificial Bee Colony Optimization

Data warehouse is an essential component of almost every modern enterprise information system. It stores huge amount of subject-oriented, time-stamped, non-volatile and integrated data. It is highly required of the system to respond to complex online analytical queries posed against its data warehouse in seconds for efficient decision making. Optimization of online analytical query processing (OLAP) could substantially minimize delays in query response time. Materialized view is an efficient and effective OLAP query optimization technique to minimize query response time. Selecting a set of such appropriate views for materialization is referred to as view selection, which is a nontrivial task. In this regard, an Artificial Bee Colony (ABC) based view selection algorithm (ABCVSA), which has been adapted by incorporating N-point and GBFS based N-point random insertion operations, to select Top-K views from a multidimensional lattice is proposed. Experimental results show that ABCVSA performs better than the most fundamental view selection algorithm HRUA. Thus, the views selected using ABCVSA on materialization would reduce the query response time of OLAP queries and thereby aid analysts in arriving at strategic business decisions in an effective manner.

Synthesis, properties, and crystal structures of π-extended double [6]helicenes: contorted multi-dimensional stacking lattice.

The synthesis and properties of a new π-extended double [6]helicene 2 and a dithia[6]helicene 3 are described. Compared to the previously reported parent double-helicene molecule 1, the introduction of n-butyl groups successfully improved the solubility, which allowed an experimental investigation into the electronic structure of 2 and 3 by photophysical measurements and cyclic voltammetry. The characteristic two-blade propeller structures of 2 and 3 were unambiguously determined by single-crystal X-ray diffraction analysis. The crystal packing structure of 2 exhibited a contorted two-dimensional stacking, whereby molecules of n-pentane were incorporated in the stacks. Despite the presence of n-butyl groups, 3 formed a unique three-dimensional stacking lattice in the crystal. Time-resolved microwave conductivity measurements revealed that the double helicenes (1-3) exhibited transient conductivities. An organic field-effect transistor fabricated using 3 was found to function as a p-type transistor.

Simultaneous approximation problems of p-adic numbers and p-adic knapsack cryptosystems - Alice in p-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l ∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.

Large scale quantum walks by means of optical fiber cavities

We demonstrate a platform for implementing quantum walks that overcomes many of the barriers associated with photonic implementations. We use coupled fiber-optic cavities to implement time-bin encoded walks in an integrated system. We show that this platform can achieve very low losses combined with high-fidelity operations, enabling an unprecedented large number of steps in a passive system, as required for scenarios with multiple walkers. Furthermore the platform is reconfigurable, enabling variation of the coin, and readily extends to multidimensional lattices. We demonstrate variation of the coin bias experimentally for three different values.

Discrete inverse problems for the Schrödinger operator on the multi-dimensional square lattice with partial Cauchy data

Motivated by the known results for an ongoing problem, in this paper we define the partial Cauchy data set for the discrete Schrödinger operator on a finite subset of the grid, and we prove a reconstruction theorem for the potential. Our methods are applicable in some general situations; we give a potential reconstruction theorem in the case of the hexagonal lattice as an example.

A compressible lattice Boltzmann finite volume model for high subsonic and transonic flows on regular lattices

A multi-dimensional double distribution function thermal lattice Boltzmann model has been developed to simulate fully compressible flows at moderate Mach number. The lattice Boltzmann equation is temporally and spatially discretizated by an asymptotic preserving finite volume scheme. The micro-velocities discretization is adopted on regular low-symmetry lattices (D1Q3, D2Q9, D3Q15, D3Q19, D3Q27). The third-order Hermite polynomial density distribution function on low-symmetry lattices is used to solve the flow field, while a second-order energy distribution is employed to compute the temperature field. The fully compressible Navier–Stokes equations are recovered by standard order Gauss–Hermite polynomial expansions of Maxwell distribution with cubic correction terms, which are added by an external force expressed in orthogonal polynomials form. The proposed model is validated considering several benchmark cases, namely the Sod shock tube, thermal Couette flow and two-dimensional Riemann problem. The numerical results are in very good agreement with both analytical solution and reference results.

Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails

We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of ‘symmetry’ in the spatial configuration of branching sources. The presented results are based on Green’s function representation of transition probabilities of an underlying random walk and cover not only the case of the finite variance of jumps but also a less studied case of infinite variance of jumps.

Multidimensional Watson lemma and its applications

We prove the multidimensional analog of the well-knownWatson lemma and then apply it to prove a local limit theorem for the transition probabilities of symmetric random walks on the multidimensional lattice with infinite variance of jumps.

Concentrations for the simple random walk in un- bounded nonnegative potentials

We consider the simple random walk in i.i.d.nonnegative potentials on the multidimensional cubic lattice. Our goal is to investigate the cost paid by the simple random walk for traveling from the origin to a remote location in a landscape of potentials. In particular, we obtain concentration inequalities for the travel cost in unbounded nonnegative potentials.

Multidimensional Circlet Lattice based Approach to Mine Spatially Co-Located Patterns

With immense extent of applications, one of the important tasks in data mining is the spatial pattern identification. Discovering spatial co-location patterns presents challenges as spatial objects are embedded in a continuous space. For co-location pattern mining, previous studies often emphasize on the association analysis of every spatial feature. As a result, interesting patterns involving events with different frequency cannot be captured. In this paper, we study the problem of efficiently mining co-location patterns with lattice structure called, Multi-dimensional Circlet Lattice-based Spatial Co-location Pattern Mining (MCL-SCPM), frequently located together in spatial proximity. MCL-SCPM initially identifies the candidate co-location using feature inclusive ratio which incorporates the spatial co-location patterns at minimum time interval with maximum coverage ratio. Next, MCL-SCPM discovers co-location patterns in a multi-dimensional spatial structure for different movements of an object by measuring the cohesion of a pattern. To extract maximal co-located patterns, MCL-SCPM method finally, uses Circlet Lattice-based structure to extract maximal co-located patterns. We conduct an extensive performance study to test and evaluate the effectiveness of MCL-SCPM method. Our experimental results show that MCL-SCPM method minimizes the execution time for mining co-location patterns without producing large computational overheads.

From extreme values of i.i.d. random fields to extreme eigenvalues of finite-volume Anderson Hamiltonian

The aim of this paper is to study asymptotic geometric properties almost surely or/and in probability of extreme order statistics of an i.i.d. random field (potential) indexed by sites of multidimensional lattice cube, the volume of which unboundedly increases. We discuss the following topics: (I) high level exceedances, in particular, clustering of exceedances; (II) decay rate of spacings in comparison with increasing rate of extreme order statistics; (III) minimum of spacings of successive order statistics; (IV) asymptotic behavior of values neighboring to extremes and so on. The conditions of the results are formulated in terms of regular variation (RV) of the cumulative hazard function and its inverse. A relationship between RV classes of the present paper as well as their links to the well-known RV classes (including domains of attraction of max-stable distributions) are discussed. The asymptotic behavior of functionals (I)--(IV) determines the asymptotic structure of the top eigenvalues and the corresponding eigenfunctions of the large-volume discrete Schr\" odinger operators with an i.i.d. potential (Anderson Hamiltonian). Thus, another aim of the present paper is to review and comment a recent progress on the extreme value theory for eigenvalues of random Schr\" odinger operators as well as to provide a clear and rigorous understanding of the relationship between the top eigenvalues and extreme values of i.i.d. random potentials. We also discuss their links to the long-time intermittent behavior of the parabolic problems associated with the Anderson Hamiltonian via spectral representation of solutions.

Lagrangian and Hamiltonian structures in an integrable hierarchy and space–time duality

We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schrödinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zero-curvature representation of the NLEE can be realized by two Hamiltonian formulations stemming from two distinct choices of the configuration space, yielding two inequivalent Poisson structures on the corresponding phase space and two distinct Hamiltonians. This is fundamentally different from the standard bi-Hamiltonian or generally multitime structure. The first formulation chooses purely space-dependent fields as configuration space; it yields the standard Poisson structure for NLS. The other one is new: it chooses purely time-dependent fields as configuration space and yields a different Poisson structure at each level of the hierarchy. The corresponding NLEE becomes a space evolution equation. We emphasize the role of the Lagrangian formulation as a unifying framework for deriving both Poisson structures, using ideas from covariant field theory. One of our main results is to show that the two matrices of the Lax pair satisfy the same form of ultralocal Poisson algebra (up to a sign) characterized by an r-matrix structure, whereas traditionally only one of them is involved in the classical r-matrix method. We construct explicit dual hierarchies of Hamiltonians, and Lax representations of the triggered dynamics, from the monodromy matrices of either Lax matrix. An appealing procedure to build a multi-dimensional lattice of Lax pair, through successive uses of the dual Poisson structures, is briefly introduced.

Multidimensional discrete compactons in nonlinear Schrödinger lattices with strong nonlinearity management

The existence of multidimensional lattice compactons in the discrete nonlinear Schr\"odinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, a new effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kind. These terms allow one to solve, at this averaged level, for exact discrete compacton solution configurations in the corresponding stationary equation. We focus on seven types of compacton solutions: single site and vortex solutions are found to be always stable in the parametric regimes we examined. Other solutions such as double site in- and out-of-phase, four site symmetric and anti-symmetric, and a five site compacton solution are found to have regions of stability and instability in two-dimensional parametric planes, involving variations of the strength of the coupling and of the nonlinearity. We also explore the time evolution of the solutions and compare the dynamics according to the averaged with those of the original dynamical equations without the averaging. Possible observation of compactons in the BEC loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is discussed.

On Vector Perturbation Precoding for the MIMO Gaussian Broadcast Channel

Precoding schemes in the framework of vector perturbation (VP) for the multiple-input multiple-output (MIMO) Gaussian broadcast channel (GBC) are investigated. The VP scheme, originally a “one-shot” technique, is generalized to encompass processing over multiple time instances. Using lattice-based extended alphabets (“perturbations”), and considering the infinite time-span extension limit, a lower bound on the achievable sum-rate using the generalized VP scheme is analytically obtained. The lower bound is shown to asymptotically achieve the optimum sum-rate in the high signal-to-noise ratio (SNR) regime (both in terms of degrees-of-freedom and power offset), for any number of users and transmit antennas. For the two-user cases, it is shown that the lower bound coincides with the sum-capacity for low SNR. The above lower bound is constructively obtained by means of an efficient practically oriented suboptimal transmit energy minimization algorithm, which exhibits a polynomial complexity in the number of users. The proposed precoding scheme demonstrates that the “shaping gain” is achievable for VP schemes, when employing “good” multidimensional lattices. It is also shown that the suboptimum algorithm has its merits, even when processing over multiple time instances is not employed. For the $2\times 2$ MIMO GBC, the VP scheme is generalized further, and an inner bound for the entire achievable rate region is obtained, by which an interesting correspondence is identified with the ultimate capacity region, as obtained by “dirty paper coding”.

Long time dynamics of a multidimensional nonlinear lattice with memory

This work is devoted to study the nature of vibrations arising in a multidimensional nonlinear periodic lattice structure with memory. We prove the existence of a global attractor. In the homogeneous case under a restriction on the nonlinear term we obtain decay rates of the total energy. These rates could be exponential, polynomial or several other intermediate types.

The structure of the positive discrete spectrum of the evolution operator arising in branching random walks

A branching random walk (BRW) with continuous time and a finite number of branching sources located at points of a multidimensional lattice is considered. The definition of weakly supercritical BRWs, whose discrete spectrum contains a unique positive eigenvalue, is introduced. Conditions for a supercritical BRW to be weakly supercritical are determined.

The injectivity of the canonical signal module for multidimensional linear systems of difference equations with variable coefficients

We consider discrete behaviors with varying coefficients. Our results are new also for one-dimensional systems over the time-axis of natural numbers and for varying coefficients in a field, we derive the results, however, in much greater generality: Instead of the natural numbers we use an arbitrary submonoid N of an abelian group, for instance the standard multidimensional lattice of r-dimensional vectors of natural numbers or integers. We replace the base field by any commutative self-injective ring F, for instance a direct product of fields or a quasi-Frobenius ring or a finite factor ring of the integers. The F-module W of functions from N to F is the canonical discrete signal module and is a left module over the natural associated noncommutative ring A of difference operators with variable coefficients. Our main result states that this module is injective and therefore satisfies the fundamental principle: An inhomogeneous system of linear difference equations with variable coefficients has a solution if and only if the right side satisfies the canonical compatibility conditions. We also show that for the typical cases of partial difference equations and in contrast to the case of constant coefficients the A-module W is not a cogenerator. We also generalize the standard one-dimensional theory for periodic coefficients to the multidimensional situation by invoking Morita equivalence.

Materialized view selection using HBMO

Strategic business decision making has become far more complex, challenging and consequential in today’s modern and highly competitive economy. So, managers have been using decision support systems to assist them in making accurate, efficient and effective decisions. These systems takes hours and days to process massive data sets in order to find relevant information for answering analytical queries. As a result the query response times are high. This response time can be reduced substantially by selecting and materializing pre-computed views that can provide answers to analytical queries. In this paper, an attempt has been made to select optimal sets of views, which would significantly reduce response time of analytical queries. In this regard, honey bee mating optimization based view selection algorithm (HBMOVSA) is proposed that selects Top-K views, from amongst all possible views, in a multidimensional lattice. Experimental results show that HBMOVSA is able to select comparatively better quality of views when compared with those selected by the most fundamental view selection algorithm HRUA.