Integer Lattice Research Articles

Efficient Schemes for Optimizing Load Balancing and Communication Cost in Edge Computing Networks

Edge computing architectures promise increased quality of service with low communication delays by bringing cloud services closer to the end-users, at the distributed edge servers of the network edge. Hosting server capabilities at access nodes, thereby yielding edge service nodes, offers service proximity to users and provides QoS guarantees. However, the placement of edge servers should match the level of demand for computing resources and the location of user load. Thus, it is necessary to devise schemes that select the most appropriate access nodes to host computing services and associate every remaining access node with the most proper service node to ensure optimal service delivery. In this paper, we formulate this problem as an optimization problem with a bi-objective function that aims at both communication cost minimization and load balance optimization. We propose schemes that tackle this problem and compare their performance against previously proposed heuristics that have been also adapted to target both optimization goals. We study how these algorithms behave in lattice and random grid network topologies with uniform and non-uniform workloads. The results validate the efficiency of our proposed schemes in addition to the significantly lower execution times compared to the other heuristics.

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

Conformal symplectic and constraint-preserving model order reduction of constrained conformal Hamiltonian systems

Dissipative and constrained dynamical systems model various physical phenomena comprising damping and constraining forces. These forces give rise to specific geometric properties of the dynamical system's governing equations. In this paper, we reduce and solve high-dimensional parameterized constrained and damped nonlinear differential equations with structure-preserving algorithms. These algorithms preserve the constrained equation's geometric properties, such as conformal symplecticness and phase-space, through model order reduction and numerical integration. We develop a reduced model based on an orthosymplectic reduced basis and establish the well-posedness and conformal symplecticness of the reduced model. Moreover, we derive a priori error bounds for state space and constraints errors. Subsequently, we utilize two RATTLE-based integrators to simulate the full and reduced models while preserving their geometric properties. Numerical experiments on constrained damped oscillators and a constrained lattice grid justify theoretical results and demonstrate the advantages of the structure-preserving computational methods for constrained dynamical systems.

Fast-Track Score to Predict the Feasibility of Early Extubation Post Liver Transplant

Background Over the past several years, the liver transplant community has embraced the concept of fast-tracking patients to facilitate earlier postoperative recovery. Aim Derive and validate a novel “fast-track” risk score that captures the demographic and clinical characteristics of DDLT patients to predict the likelihood of early extubation after surgery. Design Adult patients who underwent non-fulminant DDLT between January 2014 and July 2019 were included. The cohort was divided in 2 groups: patients extubated within 4 hours of surgery vs extubated after 4 h. Logistic regression was performed to identify the independent predictors of early extubation. The area under the curve (AUC) was calculated to measure the ability of the risk score to predict early extubation. The score was validated by applying coefficients of the regression model to the validation cohort and calculating the AUC. Results A total of 290 DDLT patients were included, of which 175 (60%) were in the “delayed extubation” group and 115 (40%) were in the “fast-track” group. Patients with a MELD <29, transfused <4 units of pRBCs, and transfused <5 units of FFP during surgery were 2.30 times, 5.74 times, and 3.09 times more likely to be extubated early, respectively. A risk score with an integer point scale was derived and exhibited an AUC of .80. The proportion of patients who were extubated early increased from 2.78% at a score of 0 to 66.67% at a score of 4. Conclusions The proposed score provides a fast and easy method to help identify DDLT patients suitable for early extubation.

Quantum geodesic flows on graphs

We revisit the construction of quantum Riemannian geometries on graphs starting from a hermitian metric compatible connection, which always exists. We use this method to find quantum Levi-Civita connections on the n-leg star graph for n=2,3,4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2,3,4$$\\end{document} and find the same phenomenon as recently found for the An\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_n$$\\end{document} Dynkin graph that the metric length for each inbound arrow has to exceed the length in the other direction by a multiple, here n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{n}$$\\end{document}. We then study quantum geodesics on graphs and construct these on the 4-leg graph and on the integer lattice line Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document} with a general edge-symmetric metric.

A note on the squarefree density of polynomials

AbstractThe conjectured squarefree density of an integral polynomial in variables is an Euler product which can be considered as a product of local densities. We show that a necessary and sufficient condition for to be 0 when is a polynomial in variables over the integers, is that either there is a prime such that the values of at all integer points are divisible by or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density satisfies .

Equable parallelograms on the Eisenstein lattice

Abstract This paper studies equable parallelograms whose vertices lie on the Eisenstein lattice. Using Rosenberger’s Theorem on generalised Markov equations, we show that the set of these parallelograms forms naturally an infinite tree, all of whose vertices have degree 4, bar the root which has degree 3. This study naturally complements the authors’ previous study of equable parallelograms whose vertices lie on the integer lattice.

Cut‐Cell Microstructures for Two‐scale Structural Optimization

AbstractTwo‐scale topology optimization, combined with the design of microstructure families with a broad range of effective material parameters, is widely used in many fabrication applications to achieve a target deformation behavior for a variety of objects. The main idea of this approach is to optimize the distribution of material properties in the object partitioned into relatively coarse cells, and then replace each cell with microstructure geometry that mimics these material properties. In this paper, we focus on adapting this approach to complex shapes in situations when preserving the shape's surface is essential.Our approach extends any regular (i.e. defined on a regular lattice grid) microstructure family to complex shapes, by enriching it with tiles adapted to the geometry of the cut‐cell. We propose a fully automated and robust pipeline based on this approach, and we show that the performance of the regular microstructure family is only minimally affected by our extension while allowing its use on 2D and 3D shapes of high complexity.

Non-separable multidimensional multiresolution wavelets: A Douglas-Rachford approach

After re-casting the wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas–Rachford algorithm is employed in the search for multi-dimensional, non-separable, compactly supported, smooth, orthogonal, multiresolution wavelets in the case of translations along the integer lattice and isotropic dyadic dilations. An algorithm for the numerical construction of such wavelets is described. By applying the algorithm, new one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets.

Rearrangement inequalities on the lattice graph

AbstractThe Polya–Szegő inequality in states that, given a nonnegative function , its spherically symmetric decreasing rearrangement is ‘smoother’ in the sense of for all . We study analogues on the lattice grid graph . The spiral rearrangement is known to satisfy the Polya–Szegő inequality for , the Wang‐Wang rearrangement satisfies it for and no rearrangement can satisfy it for . We develop a robust approach to show that both these rearrangements satisfy the Polya–Szegő inequality up to a constant for all . In particular, the Wang‐Wang rearrangement satisfies for all . We also show the existence of (many) rearrangements on such that for all .

Sparse distribution of lattice points in annular regions

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ, where μ≥Clog⁡λ, such that intervals [λ,λ+μ] do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in R2 that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in R2. Specifically, we establish the existence of annuli {x∈R2:λ≤|x|2≤λ+κ} with arbitrarily large λ and κ≥Cλs for 0<s<14, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold s=14. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in R3.

Uniqueness and reconstruction of finite lattice sets from their line sums

If an unknown finite set C⊂Z2 is cut by lines parallel to given directions, then one may count the number of points of C that are intercepted by each line, that is, the projections of C in the given directions. The inverse problem consists in reconstructing the set C, interpreted as a binary image, from the knowledge of its projections.In general, this challenging combinatorial problem, also related to the tomographic reconstruction of an unknown homogeneous object by means of X-rays, is ill-posed, meaning that different binary images exist that match the available projections. Therefore, as a preliminary step, one can try to find conditions to be imposed on the considered directions in order to limit the number of allowed solutions.In this paper we address the above problems for sets C contained in a finite assigned lattice grid, and generalize some results known in literature.First, we describe special sets of lattice directions, called simple cycles, and focus on some of their properties. Then we prove that uniqueness of reconstruction for binary images is guaranteed if and only if the line sums are computed along suitable simple cycles having even cardinality.As a second item, we prove that the unique binary solution can be explicitly reconstructed from a real-valued solution having minimal Euclidean norm. This leads to an explicit reconstruction algorithm, tested on four different phantoms and compared with previous results, which points out a significant improvement of the corresponding performance.

On a variant of Flory model

We consider a one-dimensional variant of a recently introduced settlement planning problem in which houses can be built on finite portions of the rectangular integer lattice subject to certain requirements on the amount of insolation they receive. In our model, each house occupies a unit square on a 1×n strip, with the restriction that at least one of the neighboring squares must be free. We are interested mostly in situations in which no further building is possible, i.e. in maximal configurations of houses in the strip. We reinterpret the problem as a problem of restricted packing of vertices in a path graph and then apply the transfer matrix method in order to compute the bivariate generating functions for the sequences enumerating all maximal configurations of a given length with respect to the number of houses. This allows us to determine the asymptotic behavior of the enumerating sequences and to compute some interesting statistics. Along the way, we establish close connections between our maximal configurations and several other types of combinatorial objects, including restricted permutations and walks on certain small oriented graphs. We then generalize our results in several directions by considering multi-story houses, by varying the insolation restrictions, and, finally, by considering strips of width 2 and 3. At the end we comment on several possible directions of future research.

Norm bounds on Eisenstein series

We study the sup-norm bound (both individually and on average) for Eisenstein series on certain arithmetic hyperbolic orbifolds producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta functions, and counting restricted values of indefinite quadratic forms at integer points.

On the number of lattice points in thin sectors

On the circle of radius R centred at the origin, consider a “thin” sector about the fixed line y=αx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y = \\alpha x$$\\end{document} with edges given by the lines y=(α±ϵ)x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y = (\\alpha \\pm \\epsilon ) x$$\\end{document}, where ϵ=ϵR→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon = \\epsilon _R \\rightarrow 0$$\\end{document} as R→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ R \\rightarrow \\infty $$\\end{document}. We establish an asymptotic count for Sα(ϵ,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\\alpha }(\\epsilon ,R)$$\\end{document}, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} and on the rationality/irrationality type of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. In particular, we demonstrate that if α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is Diophantine, then Sα(ϵ,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\\alpha }(\\epsilon ,R)$$\\end{document} is asymptotic to the area of the sector, so long as ϵRt→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon R^{t} \\rightarrow \\infty $$\\end{document} for some t<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t<2 $$\\end{document}.

On the asymmetric additive energy of polynomials

We prove a general result concerning the paucity of integer points on a certain family of 4-dimensional affine hypersurfaces. As a consequence, we deduce that integer-valued polynomials have small asymmetric additive energy.

Bayesian hidden mark interaction model for detecting spatially variable genes in imaging-based spatially resolved transcriptomics data.

Recent technology breakthroughs in spatially resolved transcriptomics (SRT) have enabled the comprehensive molecular characterization of cells whilst preserving their spatial and gene expression contexts. One of the fundamental questions in analyzing SRT data is the identification of spatially variable genes whose expressions display spatially correlated patterns. Existing approaches are built upon either the Gaussian process-based model, which relies on ad hoc kernels, or the energy-based Ising model, which requires gene expression to be measured on a lattice grid. To overcome these potential limitations, we developed a generalized energy-based framework to model gene expression measured from imaging-based SRT platforms, accommodating the irregular spatial distribution of measured cells. Our Bayesian model applies a zero-inflated negative binomial mixture model to dichotomize the raw count data, reducing noise. Additionally, we incorporate a geostatistical mark interaction model with a generalized energy function, where the interaction parameter is used to identify the spatial pattern. Auxiliary variable MCMC algorithms were employed to sample from the posterior distribution with an intractable normalizing constant. We demonstrated the strength of our method on both simulated and real data. Our simulation study showed that our method captured various spatial patterns with high accuracy; moreover, analysis of a seqFISH dataset and a STARmap dataset established that our proposed method is able to identify genes with novel and strong spatial patterns.

From approximate to exact integer programming

AbstractApproximate integer programming is the following: For a given convex body $$K \subseteq {\mathbb {R}}^n$$ K ⊆ R n , either determine whether $$K \cap {\mathbb {Z}}^n$$ K ∩ Z n is empty, or find an integer point in the convex body $$2\cdot (K - c) +c$$ 2 · ( K - c ) + c which is K, scaled by 2 from its center of gravity c. Approximate integer programming can be solved in time $$2^{O(n)}$$ 2 O ( n ) while the fastest known methods for exact integer programming run in time $$2^{O(n)} \cdot n^n$$ 2 O ( n ) · n n . So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point $$x^* \in (K \cap {\mathbb {Z}}^n)$$ x ∗ ∈ ( K ∩ Z n ) can be found in time $$2^{O(n)}$$ 2 O ( n ) , provided that the remainders of each component $$x_i^* \mod \ell $$ x i ∗ mod ℓ for some arbitrarily fixed $$\ell \ge 5(n+1)$$ ℓ ≥ 5 ( n + 1 ) of $$x^*$$ x ∗ are given. The algorithm is based on a cutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a $$2^{O(n)}n^n$$ 2 O ( n ) n n algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Carathéodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equation-standard form $$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$ A x = b , 0 ≤ x ≤ u , x ∈ Z n . Such a problem can be reduced to the solution of $$\prod _i O(\log u_i +1)$$ ∏ i O ( log u i + 1 ) approximate integer programming problems. This implies, for example that knapsack or subset-sum problems with polynomial variable range$$0 \le x_i \le p(n)$$ 0 ≤ x i ≤ p ( n ) can be solved in time $$(\log n)^{O(n)}$$ ( log n ) O ( n ) . For these problems, the best running time so far was $$n^n \cdot 2^{O(n)}$$ n n · 2 O ( n ) .

Differentially private submodular maximization with a cardinality constraint over the integer lattice

Differentially private submodular maximization with a cardinality constraint over the integer lattice

A clinical score to predict recovery in end-stage kidney disease due to acute kidney injury.

Acute kidney injury (AKI) is a major contributor to end-stage kidney disease (ESKD). About one-third of patients with ESKD due to AKI recover kidney function. However, the inability to accurately predict recovery leads to improper triage of clinical monitoring and impacts the quality of care in ESKD. Using data from the United States Renal Data System from 2005 to 2014 (n=22922), we developed a clinical score to predict kidney recovery within 90 days and within 12 months after dialysis initiation in patients with ESKD due to AKI. Multivariable logistic regressions were used to examine the effect of various covariates on the primary outcome of kidney recovery to develop the scoring system. The resulting logistic parameter estimates were transformed into integer point totals by doubling and rounding the estimates. Internal validation was performed. Twenty-four percent and 34% of patients with ESKD due to AKI recovered kidney function within 90 days and 12 months, respectively. Factors contributing to points in the two scoring systems were similar but not identical, and included age, race/ethnicity, body mass index, congestive heart failure, cancer, amputation, functional status, hemoglobin and prior nephrology care. Three score categories of increasing recovery were formed: low score (0-6), medium score (7-9) and high score (10-12), which exhibited 90-day recovery rates of 12%, 26% and 57%. For the 12-month scores, the low, medium and high groups consisted of scores 0-5, 6-8 and 9-11, with 12-month recovery rates of 16%, 33% and 62%, respectively. The internal validation assessment showed no overfitting of the models. A clinical score derived from information available at incident dialysis predicts renal recovery at 90 days and 12 months in patients with presumed ESKD due to AKI. The score can help triage appropriate monitoring to facilitate recovery and begin planning long-term dialysis care for others.