Integer Lattice Research Articles

A solution for the greedy approximation of a step function with a waveform dictionary

In this paper we consider a step function characterized by an arbitrary sequence of real-valued scalars and approximate it with a matching pursuit (MP) algorithm. We utilize a waveform dictionary with rectangular window functions as part of this algorithm. We show that the waveform dictionary is not necessary when all of the scalars are either non positive or non negative and the parameters of a wavelet dictionary on an integer lattice yields a closed-form solution for the initial optimization problem as part of the MP. Additionally, for any real-valued scalar sequence, we provide a solution with a related wavelet dictionary at each iteration of the algorithm. This allows for practical calculation of the approximating function, which we use to provide examples on simulated and real univariate time series data that display discontinuities in its underlying structure where the step function can be thought of as a sample from a signal of interest.

Shape of the asymptotic maximum sum-free sets in integer lattice grids

We determine the shape of all sum-free sets in {1,2,…,n}2 of size close to the maximum 35n2, solving a problem of Elsholtz and Rackham. We show that all such asymptotic maximum sum-free sets lie completely in the stripe 45n−o(n)≤x+y≤85n+o(n). We also determine for any positive integer p the maximum size of a subset A⊆{1,2,…,n}2 which forbids the triple (x,y,z) satisfying px+py=z.

On unique recovery of finite-valued integer signals and admissible lattices of sparse hypercubes

The paper considers the problem of unique recovery of sparse finite-valued integer signals using a single linear integer measurement. For l-sparse signals in {mathbb {Z}}^n, 2l<n, with absolute entries bounded by r, we construct an 1times n measurement matrix with maximum absolute entry Delta =O(r^{2l-1}). Here the implicit constant depends on l and n and the exponent 2l-1 is optimal. Additionally, we show that, in the above setting, a single measurement can be replaced by several measurements with absolute entries sub-linear in Delta. The proofs make use of results on admissible (n-1)-dimensional integer lattices for m-sparse n-cubes that are of independent interest.

Cantor spectrum for CMV matrices with almost periodic Verblunsky coefficients

We consider extended CMV matrices with analytic quasi-periodic Verblunsky coefficients with Diophantine frequency vector in the perturbatively small coupling constant regime and prove the analyticity of the tongue boundaries. As a consequence we establish that, generically, all gaps of the spectrum that are allowed by the Gap Labeling Theorem are open and hence the spectrum is a Cantor set. We also prove these results for a related class of almost periodic Verblunsky coefficients and present an application to suitable quantum walk models on the integer lattice.

A Drainage Network with Dependence and the Brownian Web

We study a system of coalescing random walks on the integer lattice \( {\mathbb {Z}}^{d} \) in which the walk is oriented in the d-th direction and follows certain specified rules. We first study the geometry of the paths and show that, almost surely, the paths form a graph consisting of just one tree for dimensions \( d=2,3 \) and infinitely many disjoint trees for dimensions \( d\ge 4 \). Also, there is no bi-infinite path in the graph almost surely for \( d\ge 2 \). Subsequently, we prove that for \( d=2 \) the diffusive scaling of this system converges in distribution to the Brownian web.

Integer polynomial recovery from outputs and its application to cryptanalysis of a protocol for secure sorting

AbstractWe investigate the problem of recovering integer inputs (up to an affine scaling) when given only the integer monotonic polynomial outputs. Givennninteger outputs of a degree-ddinteger monotonic polynomial whose coefficients and inputs are integers within known bounds andn≫dn\gg d, we give an algorithm to recover the polynomial and the integer inputs (up to an affine scaling). A heuristic expected time complexity analysis of our method shows that it is exponential in the size of the degree of the polynomial but polynomial in the size of the polynomial coefficients. We conduct experiments with real-world data as well as randomly chosen parameters and demonstrate the effectiveness of our algorithm over a wide range of parameters. Using only the polynomial evaluations at specific integer points, the apparent hardness of recovering the input data served as the basis of security of a recent protocol proposed by Kesarwani et al. for securekk-nearest neighbor computation on encrypted data that involved secure sorting. The protocol uses the outputs of randomly chosen monotonic integer polynomial to hide its inputs except to only reveal the ordering of input data. By using our integer polynomial recovery algorithm, we show that we can recover the polynomial and the inputs within a few seconds, thereby demonstrating an attack on the protocol of Kesarwani et al.

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

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 r and a preconditioning parameter γ, and the speed of sound in the collision model is naturally adapted to r 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 γ, 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.

Screening Tool to Identify Patients with Advanced Aortic Valve Stenosis.

(1) Background: The clinical burden of aortic stenosis (AS) remains high in Western countries. Yet, there are no screening algorithms for this condition. We developed a risk prediction model to guide targeted screening for patients with AS. (2) Methods: We performed a cross-sectional analysis of all echocardiographic studies performed between 2013 and 2018 at a tertiary academic care center. We included reports of unique patients aged from 40 to 95 years. A logistic regression model was fitted for the risk of moderate and severe AS, with readily available demographics and comorbidity variables. Model performance was assessed by the C-index, and its calibration was judged by a calibration plot. (3) Results: Among the 38,788 reports yielded by inclusion criteria, there were 4200 (10.8%) patients with ≥moderate AS. The multivariable model demonstrated multiple variables to be associated with AS, including age, male gender, Caucasian race, Body Mass Index ≥ 30, and cardiovascular comorbidities and medications. C-statistics of the model was 0.77 and was well calibrated according to the calibration plot. An integer point system was developed to calculate the predicted risk of ≥moderate AS, which ranged from 0.0002 to 0.7711. The lower 20% of risk was approximately 0.15 (corresponds to a score of 252), while the upper 20% of risk was about 0.60 (corresponds to a score of 332 points). (4) Conclusions: We developed a risk prediction model to predict patients’ risk of having ≥moderate AS based on demographic and clinical variables from a large population cohort. This tool may guide targeted screening for patients with advanced AS in the general population.

Background-Oriented Schlieren technique using Fast Checkerboard Demodulation for extending measurement range of pressure of underwater shock waves

We have successfully extended the measurement range by applying Fast Checkerboard Demodulation (FCD) to Background-Oriented Schlieren (BOS) technique, which is a contactless measurement technique. In the experiment, the apparent displacement field on the background image distorted by underwater shock waves were measured using FCD-BOS and contact measurement pressure (hydrophone). The results show that the measurement limitation of FCD is higher than the methods of previous studies which used Optical Flow (OF) and Digital Image Correlation (DIC) algorithms. Since the displacement is detected from the frequency change due to the distortion of the background image, FCD that combines fast Fourier transform (FFT) and a background image with a two-dimensional periodic pattern, can be applied to the large distortion which is difficult to be detected using OF and DIC. The numerical approach in our previous study(Shimazaki et al., 2022) showed that the measurement range of FCD-BOS is strongly influenced by the background pattern. Therefore, this study also performed numerical investigation on the use of synthetic background images with the frequency of the resultant wave of two sine waves of different wavelengths for FCD-BOS. The numerical results show that the measurement range using the synthetic background images with wavelengths of (60,60/12) μm is three times larger than the experimental measurement range using the lattice grid (width: 20 μm) background. In addition to the synthetic background of wavelengths of 60,60/12, the measurement accuracy, phase wrapping, overlap, and aliasing artifacts of those of (40,40/12) and (30,30/12) were investigated. Such results show the superiority of the background with the frequency of sine wave over that of square wave (lattice grid).

Post-Quantum Secure Identity-Based Encryption Scheme using Random Integer Lattices for IoT-enabled AI Applications

Identity-based encryption is an important cryptographic system that is employed to ensure confidentiality of a message in communication. This article presents a provably secure identity based encryption based on post quantum security assumption. The security of the proposed encryption is based on the hard problem, namely Learning with Errors on integer lattices. This construction is anonymous and produces pseudo random ciphers. Both public-key size and ciphertext-size have been reduced in the proposed encryption as compared to those for other relevant schemes without compromising the security. Next, we incorporate the constructed identity based encryption (IBE) for Internet of Things (IoT) applications, where the IoT smart devices send securely the sensing data to their nearby gateway nodes(s) with the help of IBE and the gateway node(s) secure aggregate the data from the smart devices by decrypting the messages using the proposed IBE decryption. Later, the gateway nodes will securely send the aggregated data to the cloud server(s) and the Big data analytics is performed on the authenticated data using the Artificial Intelligence (AI)/Machine Learning (ML) algorithms for accurate and better predictions.

Survival probability of random walks and Lévy flights with stochastic resetting

We perform a thorough analysis of the survival probability of symmetric random walks with stochastic resetting, defined as the probability for the walker not to cross the origin up to time n. For continuous symmetric distributions of step lengths with either finite (random walks) or infinite variance (Lévy flights), this probability can be expressed in terms of the survival probability of the walk without resetting, given by Sparre Andersen theory. It is therefore universal, i.e. independent of the step length distribution. We analyze this survival probability at depth, deriving both exact results at finite times and asymptotic late-time results. We also investigate the case where the step length distribution is symmetric but not continuous, focusing our attention onto arithmetic distributions generating random walks on the lattice of integers. We investigate in detail the example of the simple Polya walk and propose an algebraic approach for lattice walks with a larger range.

Asymptotics of Arithmetic Functions of GCD and LCM of Random Integers in Hyperbolic Regions

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.

Lattice-based public-key encryption with equality test supporting flexible authorization in standard model

As cloud computing has developed rapidly, outsourcing data to cloud servers for remote storage has become an attractive trend. However, when cloud clients store their data in the cloud, the security and privacy of cloud data would be threatened due to accidental corruptions or purposive attacks caused by a semi-trusted cloud server. The widely used method of addressing the security and privacy of cloud data is to store encrypted data instead of plain data. As the resulting system is unusable, since the cloud can no longer search throughout the data, new cryptographic primitive such as public-key encryption with equality test (PKEET) has been introduced. PKEET has many interesting applications such as keyword search on encrypted data, encrypted data partitioning for efficient encrypted data management, personal health record systems, spam filtering in encrypted email systems and so forth. PKEET allows checking whether two ciphertexts encrypted under different public keys contain the same message or not. However, unrestricted access to equality tests can reveal information about the underlying data. This is not acceptable in respect of users' privacy. In 2015, Ma et al. introduce the notion of PKEET with flexible authorization (PKEET-FA) which strengthens privacy protection. Since 2015, there are several follow-up works on PKEET-FA. However, all are secure in the random oracle model and vulnerable to quantum attacks. In this paper, we provide two constructions of quantum-safe PKEET-FA secure in the standard model based on the hardness assumptions of integer lattices and ideal lattices. Finally, we implement the PKEET-FA scheme over ideal lattices.

Markov chains generated by convolutions of orthogonality measures

About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms of convolutions of orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, q-Hahn, q-Meixner and little q-Jacobi polynomials. By construction, the stationary probability distributions, the complete sets of eigenvalues and eigenvectors are provided by the polynomials and the orthogonality measures. An interesting property possessed by these stationary probability distributions, called ‘convolutional self-similarity,’ is demonstrated.

A fast and deterministic algorithm for Knapsack-constrained monotone DR-submodular maximization over an integer lattice

A fast and deterministic algorithm for Knapsack-constrained monotone DR-submodular maximization over an integer lattice

Some New Results in Quantitative Diophantine Approximation

Abstract In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the second, we examine inhomogeneous forms of arbitrary degree taking values at integer points.

Overrelaxation in a diffusive integer lattice gas

One of the most striking drawbacks of standard lattice gas methods over lattice Boltzmann methods is a much more limited range of transport parameters that can be achieved. It is common for lattice Boltzmann methods to use over-relaxation to achieve arbitrarily small transport parameters in the hydrodynamic equations. Here, we show that it is possible to implement over-relaxation for integer lattice gases. For simplicity, we focus here on lattice gases for the diffusion equation. We demonstrate that adding a flipping operation to lattice gases results in a multirelaxation time lattice Boltzmann scheme with over-relaxation in the Boltzmann limit.

Stationary determinantal processes: [formula omitted]-mixing property and correlation dimensions

We study the stationary determinantal processes on the integer lattice, induced by strictly positive and strictly contractive convolution kernels. A necessary and sufficient condition of ψ-mixing property for such a process is given. Some new results on the correlation dimension of the stationary determinantal measure on the symbolic space are obtained. As an application, the precise decreasing rate of the shortest distances between two orbits in the corresponding dynamical system is obtained.

Counting basis extensions in a lattice

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by T T , producing an asymptotic estimate as T → ∞ T \to \infty . This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the 2 2 -dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in R n \mathbb R^n . Our main result on counting basis extensions also generalizes to arbitrary lattices in R n \mathbb R^n . Finally, we establish some basic properties of sparse representations of integers by multilinear forms.

Integer and fractionalized vortex lattices and off-diagonal long-range order

We analyze the implication of off-diagonal long-range order (ODLRO) for inhomogeneous periodic field configurations and multi-component order parameters. For single component order parameters we show that the only static, periodic field configuration consistent with ODLRO is a vortex lattice with integer flux in units of the flux quantum in each unit cell. For a superconductor with g degenerate components, fractional vortices are allowed. Depending on the precise order-parameter manifold, they tend to occur in units of 1/g of the flux quantum. These results are well known to emerge from the Ginzburg-Landau or BCS theories of superconductivity. Our results imply that they are valid even if these theories no-longer apply. Integer and fractional vortex lattices are transparently seen to emerge as a consequence of the macroscopic coherence and single valuedness of the condensate.