Constant Approximation Research Articles

Fully polarized Compton scattering in plane waves and its polarization transfer

Fully polarized Compton scattering from a beam of spin-polarized electrons is investigated in plane-wave backgrounds in a broad intensity region from the perturbative to the nonperturbative regimes. In the perturbative regime, polarized linear Compton scattering is considered for investigating polarization transfer from a single laser photon to a scattered photon, and in the high-intensity region, the polarized locally monochromatic approximation and locally constant field approximation are established and are employed to study polarization transfer from an incoming electron to a scattered photon. The numerical results suggest an appreciable improvement of about 10% in the scattering probability in the intermediate-intensity region if the electron’s longitudinal spin is parallel to the laser rotation. The longitudinal spin of the incoming electron can be transferred to the scattered photon with an efficiency that increases with laser intensity and collisional energy. For collision between an optical laser with frequency ∼1 eV and a 10 GeV electron, this polarization transfer efficiency can increase from about 20% in the perturbative regime to about 50% in the nonperturbative regime for scattered photons with relatively high energy.

A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem

We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation.

Pandora’s Problem with Deadlines

Pandora’s problem is a fundamental model that studies optimal search under costly inspection. In the classic version, there are n boxes, each associated with a known cost and a known distribution over values. A strategy inspects the boxes sequentially and obtains a utility that equals the difference between the maximum value of an inspected box and the total inspection cost. Weitzman (1979) presented a surprisingly simple strategy that obtains the optimal expected utility. In this work we introduce a new variant of Pandora’s problem in which every box is also associated with a publicly known deadline, indicating the final round by which its value may be chosen. This model captures many real-life scenarios where alternatives admit deadlines, such as candidate interviews and college admissions. Our main result is an efficient threshold-based strategy that achieves a constant approximation relative to the performance of the optimal strategy for the deadlines setting.

Parameterized Approximation Algorithms for Sum of Radii Clustering and Variants

Clustering is one of the most fundamental tools in artificial intelligence, machine learning, and data mining. In this paper, we follow one of the recent mainstream topics of clustering, Sum of Radii (SoR), which naturally arises as a balance between the folklore k-center and k-median. SoR aims to determine a set of k balls, each centered at a point in a given dataset, such that their union covers the entire dataset while minimizing the sum of radii of the k balls. We propose a general technical framework to overcome the challenge posed by varying radii in SoR, which yields fixed-parameter tractable (fpt) algorithms with respect to k (i.e., whose running time is f(k) ploy(n) for some f). Our framework is versatile and obtains fpt approximation algorithms with constant approximation ratios for SoR as well as its variants in general metrics, such as Fair SoR and Matroid SoR, which significantly improve the previous results.

On Optimal Tradeoffs between EFX and Nash Welfare

A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. The goal of this work is to characterize the tradeoffs between two well-studied measures of fairness and efficiency --- envy freeness up to any item (EFX) for fairness, and Nash welfare for efficiency --- by saying, for given constants α and β, whether there exists an α-EFX allocation that guarantees a β-fraction of the maximum Nash welfare (β-MNW). For additive valuations, we show that for any α ∈ [0,1], there exists a partial allocation that is α-EFX and 1/(α+1)-MNW. This tradeoff turns out to be tight (for every α) as demonstrated by an impossibility result that we give. We also show that for α ∈ [0, φ-1 ≃ 0.618] these partial allocations can be turned into complete allocations where all items are assigned. Furthermore, for any α ∈ [0, 1/2], we show that the tight tradeoff of α-EFX and 1/(α+1)-MNW with complete allocations holds for the more general setting of subadditive valuations. Our results improve upon the current state of the art, for both additive and subadditive valuations, and match the best-known approximations of EFX under complete allocations, regardless of Nash welfare guarantees. Notably, our constructions for additive valuations also provide EF1 and constant approximations for maximin share guarantees.

Near-Optimal Resilient Aggregation Rules for Distributed Learning Using 1-Center and 1-Mean Clustering with Outliers

Byzantine machine learning has garnered considerable attention in light of the unpredictable faults that can occur in large-scale distributed learning systems. The key to secure resilience against Byzantine machines in distributed learning is resilient aggregation mechanisms. Although abundant resilient aggregation rules have been proposed, they are designed in ad-hoc manners, imposing extra barriers on comparing, analyzing, and improving the rules across performance criteria. This paper studies near-optimal aggregation rules using clustering in the presence of outliers. Our outlier-robust clustering approach utilizes geometric properties of the update vectors provided by workers. Our analysis show that constant approximations to the 1-center and 1-mean clustering problems with outliers provide near-optimal resilient aggregators for metric-based criteria, which have been proven to be crucial in the homogeneous and heterogeneous cases respectively. In addition, we discuss two contradicting types of attacks under which no single aggregation rule is guaranteed to improve upon the naive average. Based on the discussion, we propose a two-phase resilient aggregation framework. We run experiments for image classification using a non-convex loss function. The proposed algorithms outperform previously known aggregation rules by a large margin with both homogeneous and heterogeneous data distributions among non-faulty workers. Code and appendix are available at https://github.com/jerry907/AAAI24-RASHB.

Solving Satisfiability Modulo Counting for Symbolic and Statistical AI Integration with Provable Guarantees

Satisfiability Modulo Counting (SMC) encompasses problems that require both symbolic decision-making and statistical reasoning. Its general formulation captures many real-world problems at the intersection of symbolic and statistical AI. SMC searches for policy interventions to control probabilistic outcomes. Solving SMC is challenging because of its highly intractable nature (NP^PP-complete), incorporating statistical inference and symbolic reasoning. Previous research on SMC solving lacks provable guarantees and/or suffers from suboptimal empirical performance, especially when combinatorial constraints are present. We propose XOR-SMC, a polynomial algorithm with access to NP-oracles, to solve highly intractable SMC problems with constant approximation guarantees. XOR-SMC transforms the highly intractable SMC into satisfiability problems by replacing the model counting in SMC with SAT formulae subject to randomized XOR constraints. Experiments on solving important SMC problems in AI for social good demonstrate that XOR-SMC outperforms several baselines both in solution quality and running time.

A detailed kinetic submechanism for OH[formula omitted] chemiluminescence in hydrogen combustion revisited. Part 2

In this series of two papers, we present a novel physically sound kinetic submechanism aimed at modeling the formation and decay of chemiluminescent (electronically excited) OH∗ molecules in the ignition and combustion of hydrogen and hydrogen-based mixtures over a wide range of temperatures and pressures. The present Part describes the construction of this detailed reaction submechanism based on the groundwork and findings of the preceding paper. In so doing, we relied both on critical evaluation of known and at times conflicting experimental and theoretical data on the elementary reaction kinetics of OH∗ and on our quantum chemical calculations and rate constant estimates for the key processes. The kinetic model under development was validated against the representative dataset of the observed OH(A2Σ+→X2Π) chemiluminescent emission accompanying the high-temperature (post-shock) oxidation of various H2-containing mixtures. To improve its performance against the collected OH∗ emission data, the rate constants of some reactive and quenching processes, to which the OH∗ kinetics is the most sensitive and for which there is a particular uncertainty, were jointly varied within the limits imposed by theoretical considerations and/or the scatter in the available experimental evidence. Thus, by refining the rate constants of these elementary processes, we obtained a reasonable reaction submechanism for OH∗ that describes the known experimental data better than previous models.Novelty and significance statement Although ultraviolet OH∗ chemiluminescence as an optical signature of the combustion process and the underlying chemistry is known to provide unique capabilities for combustion diagnostics, the development of the detailed reaction mechanisms intended for quantitative interpretation of the OH∗ emission measurements has now almost stagnated. Indeed, existing kinetic models contain a discouragingly small number of elementary processes and, apparently, do not account for the full variety of possible reaction pathways of OH∗ formation and depletion. Therefore, in this serial work, we have intended to revise the current ideas about the kinetics of OH∗ production and consumption during ignition and combustion and focused first on the OH∗ kinetics in a reacting H2/O2/Ar/N2 mixture. The principal outcome of this Part of the series is a new physically based kinetic submechanism aimed at modeling the formation and decay of electronically excited OH∗ molecules in the ignition and combustion of hydrogen and hydrogen-containing mixtures. Its incorporation into any conventional kinetic model of hydrogen oxidation enables quantitative interpretation of the OH(A2Σ+−X2Π) chemiluminescent emission observed experimentally in flames and shock-heated reacting flows with an accuracy higher than that of previous analogous mechanisms. No less importantly, the rate constant approximations for a number of individual elementary processes with OH∗, recommended here based on the critical review of literature data and on our quantum chemical calculations, are also of independent interest for the kinetics of electronically excited OH∗ particles. Finally, the OH∗ reaction subset we suggested for hydrogen flames can form the basis of more complex OH∗ chemiluminescence modeling reaction mechanisms for other fuels and fuel compositions.

Two-dimensional small leak detection of pipeline based on time sequence coding

In this study, a two-dimensional pipeline small leak detection based on time series imaging is introduced. First, the sampling points and data of pipeline signals are converted into polar diameters and polar angles in polar coordinate system to form vectors, and then the inner product between vectors is mapped to 2-dimensional pixels through the use of Gram angular field (GAF). The resulting image preserves both the temporal and spatial features of the original signal. Wavelet threshold (WT) denoising and adaptive segmentation constant approximation (APCA) are used to denoise and fit the original signal in order to ensure the noise immunity and computational efficiency of the imaging. The image generated after WT-APCA-GAF processing is input into Swin Transformer for training, and the final identification accuracy of small leakage can reach 100%, effectively avoiding the problem of missing and false positive in detecting small leaks.

Which Link Matters? Maintaining Connectivity of Uncertain Networks Under Adversarial Attack

This paper studies connectivity maintenance in uncertain networks under adversarial attack, where a defender conceals crucial links to prevent the largest connected component from being decomposed by an attacker. In contrast with its static counterpart, connectivity maintenance in uncertain networks involves additional probing on links to determine their existence. Therefore, by modeling an uncertain network as a random graph with each link associated with an existence probability and a probing cost, our goal is to design a defensive strategy for link selection that maximizes the expected size of the largest remaining connected component with the minimum expected probing cost, and moreover, the strategy should be independent of the attacking patterns. To this end, we first unravel the computational complexity of the problem by proving its NP-hardness, and then propose optimal defensive strategies based on dynamic programming and multi-objective optimization. Due to the prohibitive computational cost of optimality, two approximate defensive strategies are further designed to pursue decent performance with quasilinear complexity, in which the first one is a heuristic approach that quantifies the link vulnerability through an analogy from the degree centrality of a vertex in static networks to the connectivity weight of a link in uncertain networks, and the second one is an adaptive greedy policy incorporating the minimax rule from game theory, which minimizes the possible loss suffered by the defender in a worst-case scenario and has a constant approximation ratio. Extensive experiments on both synthetic and real-world network datasets under diverse attacking patterns demonstrate the superiority of the proposed strategies over baselines.

Two-stage BP maximization under p-matroid constraint

The BP problem maximizes the sum of a suBmodular function and a suPermodular function(BP) subject to some constraints, where both functions are nonnegative and monotonic. This problem has been widely studied under the single-stage setting. In this paper, we consider a variant of the BP maximization problem. The problem is a two-stage BP maximization problem subject to a p-matroid constraint, for which we propose an approximation algorithm with constant approximation ratio parameterized by the curvatures of the two functions involved.

A bubble VEM-fully discrete polytopal scheme for mixed-dimensional poromechanics with frictional contact at matrix–fracture interfaces

This article addresses the discretisation of fractured/faulted poromechanical models using 3D polyhedral meshes in order to cope with the geometrical complexity of faulted geological models. A new polytopal scheme is proposed for contact-mechanics, based on a mixed formulation combining a fully discrete space and suitable reconstruction operators for the displacement field together with a face-wise constant approximation of the Lagrange multiplier, accounting for the surface tractions along the fracture/fault network. To ensure the inf–sup stability of the mixed formulation, a bubble-like degree of freedom is included in the discrete space of displacements and used in the reconstruction operators. This fully discrete scheme for the displacement is equivalent to a low-order Virtual Element scheme, with a bubble enrichment of the virtual space. This P1-bubble VEM–P0 mixed discretisation is combined with an Hybrid Finite Volume scheme for the Darcy flow. The proposed approach is adapted to complex geometry with a network of planar faults/fractures which can include corners, tips and intersections; it leads to efficient semi-smooth Newton solvers for the contact-mechanics, and preserves the dissipative properties of the fully coupled model. The scheme is numerically investigated in terms of convergence and robustness on several 2D and 3D test cases, using either analytical or numerical reference solutions and considering both the stand alone static contact-mechanics model and the fully coupled poromechanical model.

Maxwell’s True Current

Maxwell defined a ‘true’ or ‘total’ current in a way not widely used today. He said that “… true electric current … is not the same thing as the current of conduction but that the time-variation of the electric displacement must be taken into account in estimating the total movement of electricity”. We show that the true or total current is a universal property of electrodynamics independent of the properties of matter. We use mathematics without the approximation of a dielectric constant. The resulting Maxwell current law is a generalization of the Kirchhoff law of current used in circuit analysis, that also includes the displacement current. The generalization is not a long-time low-frequency approximation in contrast to the traditional presentation of Kirchhoff’s law.

Efficient first-principles electronic transport approach to complex band structure materials: the case of n-type Mg3Sb2

We present an efficient method for accurately computing electronic scattering rates and transport properties in materials with complex band structures. Using ab initio simulations, we calculate a limited number of electron–phonon matrix elements, and extract scattering rates for acoustic and optical processes based on deformation potential theory. Polar optical phonon scattering rates are determined using the Fröhlich model, and ionized impurity scattering rates are derived from the Brooks-Herring theory. Subsequently, electronic transport coefficients are computed within the Boltzmann transport theory. We exemplify our approach with n-type Mg3Sb2, a promising thermoelectric material with a challenging large unit cell and low symmetry. Notably, our method attains competitive accuracy, requiring less than 10% of the computational cost compared to state-of-the-art ab initio methods, dropping to 1% for simpler materials. Additionally, our approach provides explicit information on individual scattering processes, offering an alternative that combines efficiency, robustness, and flexibility beyond the commonly employed constant relaxation time approximation with the accuracy of fully first-principles calculations.

A computational study of nematic core structure and disclination interactions in elastically anisotropic nematics.

A singular potential method in the Q tensor order parameter representation of a nematic liquid crystal is used to study the equilibrium configuration of a disclination dipole. Unlike the well studied isotropic limit (the so called one constant approximation), we focus on the case of anisotropic Frank elasticity (bend/splay elastic constant contrast). Prior research has established that the singular potential method provides an accurate description of the tensor order parameter profile in the vicinity of a disclination core of a highly anisotropic lyotropic chromonic liquid crystal. This research is extended here to two interacting disclinations forming a dipole configuration. The director angle is shown to decay in the far field inversely with distance to the dipole as is the case in the isotropic limit, but with a different angular dependence. Therefore elastic constant anisotropy modifies the elastic screening between disclinations, with implications for the study of ensembles of defects as seen, for example, in active matter in the extended system limit.

Construction of Piano Performing Arts Talent Cultivation and Creative Thinking Based on the Constant Differential Error Approximation Method

Abstract Piano talent cultivation mostly focuses on repetitive training only, and lacks a clear cultivation program for theoretical knowledge and skill points. In this paper, we adopt the curriculum knowledge graph to extract the knowledge point subgraph of piano learning and provide an initialized knowledge graph structure for the learning path under the cultivation scheme. A multi-objective optimization model of the path containing constraints such as the difficulty and mastery of knowledge points is set, and the relevant parameters of the model are defined and calculated to establish a multi-optimization objective function. A solution method that uses constant differential error approximation is proposed as a means of approaching the Pareto optimal solution. To develop the LSPIA algorithm with reciprocal weights, the adjustment vectors are modified using a least squares asymptotic iterative approximation with minimum squares values. Following the discovery of the most suitable route, a teaching experiment was carried out. The results show that the posttest P-value of the total creative thinking score of the experimental and control classes is 0.022 less than 0.05, which indicates that there is a significant difference between the creative thinking of the students in the two classes in terms of affective characteristics. The experimental group achieved a mean score of 122.69 on the post-test, which was a significant improvement from the mean score on the pre-test. An innovative approach to cultivating piano playing talents involves optimizing learning paths.

Digital Twin-Assisted, SFC-Enabled Service Provisioning in Mobile Edge Computing

Mobile Edge Computing (MEC) has been identified as a desirable computing paradigm that provides efficient and effective services for various applications, while meeting stringent service delay requirements. Orthogonal to the MEC computing paradigm, Network Function Virtualization (NFV) technology is another enabling technology that provides the network resource management with great flexibility and scalability, where the instances of Virtual Network Functions (VNFs) are deployed in edge servers as Service Function Chains (SFCs) for SFC-enabled services. Although reliable service provisioning in MEC environments is fundamentally important, the deployed VNF instances usually are not reliable, which can be affected by their software implementation, their execution duration, the workload among edge servers, and so on. Empowered by digital twin techniques, the states of VNF instances can be maintained by their digital twins in a real-time manner and their reliability can be accurately predicted through their digital twins. In this paper, we study digital twin-assisted, SFC-enabled reliable service provisioning in MEC networks by exploiting the dynamics of VNF instance reliability. We concentrate on two novel optimization problems of reliable service provisioning: the service cost minimization problem, and the dynamic service admission maximization problem. We first show their NP-hardness. We then formulate an Integer Linear Program (ILP) solution, and devise an approximation algorithm with a constant approximation ratio for the service cost minimization problem. We thirdly provide an ILP solution to the offline version of the dynamic service admission maximization problem. Built upon this offline ILP solution, we also develop an online algorithm with a provable competitive ratio for the problem, by adopting the primal-dual dynamic updating technique. We finally evaluate the performance of the proposed algorithms via simulations. Simulation results demonstrate that the proposed algorithms outperform their comparison benchmarks, and improve the performance of their comparison counterparts by no less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10.2 \%$</tex-math></inline-formula> .

Theoretical Investigations of the Structural, Dynamical, Electronic, Magnetic, and Thermoelectric Properties of CoMRhSi (M = Cr, Mn) Quaternary Heusler Alloys

The structural, dynamical, electrical, magnetic, and thermoelectric properties of CoMRhSi (M = Cr, Mn) quaternary Heusler alloys (QHAs) were investigated using density functional theory (DFT). The Y-type-II crystal structure was found to be the most stable configuration for these QHAs. Both CoCrRhSi and CoMnRhSi alloys possess a half-metallic behavior with a 100% spin-polarization as the majority spin channel is metallic. On the other hand, the minority spin channel is semiconducting with narrow indirect band gaps of 0.54 eV and 0.57 eV, respectively, along the Γ−X high symmetry line. In addition, both CoCrRhSi and CoMnRhSi alloys possess a ferromagnetic structure with total magnetic moments of 4 μB, and 5 μB, respectively, which are prominent for spintronics applications. The thermoelectric properties of the subject QHAs were calculated by using Boltzmann transport theory within the constant relaxation time approximation. The lattice thermal conductivities were also evaluated by Slack’s equation. The predicted values of the figure-of-merit (ZT) for CoCrRhSi and CoMnRhSi were found to be 0.84 and 2.04 at 800 K, respectively, making them ideal candidates for thermoelectric applications.

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

Structural, electronic, optical, and thermoelectric properties of CsPbI3-yBry (y = 0.5, 1.5, 2.5) compounds: First-principle study

BackgroundAmong photovoltaic materials, halide-based perovskites, such as CsPbI3, have been the subject of many research papers due to their suitable and direct bandgap, very good optical properties, and remarkable thermoelectric properties. Enhancing optoelectronic and thermoelectric properties can be achieved by combining halide atoms intelligently. The purpose of this article is to examine the physical properties of Br substitution at different concentrations in CsPbI3 perovskite. MethodsThe structural, electrical, and optical properties of CsPbI3-yBry (y = 0.5, 1.5, 2.5) compounds have been investigated by using the QUANTUM-ESPRESSO/PWSCF computational package which is based on the density functional theory and the pseudo-potential technique. The thermoelectric properties of CsPbI3-yBry perovskites were calculated by solving the Boltzmann transport equations within the constant relaxation time approximation as employed in the BoltzTraP package. Significant findingsThe structural investigation reveals that when the I atom is substituted by Br, the lattice constant decreases, increasing the bulk modulus and stiffness of compounds. The calculated negative formation energy confirmed the thermodynamic stability of CsPbI3-yBry compounds, and the stability improved with an increase in Br concentration. The electronic investigation indicates the semiconductor nature of compounds with a direct band gap, with a slight increase in electronic band gap as Br concentration increases. The noticeable optical absorption coefficient in the visible range of the electromagnetic spectrum, along with high and tunable optical conductivity, confirms the capability of the compounds for use in optoelectronic devices. The positive value of the Seebeck coefficient revealed the p-type semiconducting nature of CsPbI3-yBry perovskites. Lattice thermal conductivity calculation reveals that this quantity is decreased because of the increase in phonon-phonon interaction due to rising temperature. Thermoelectric investigation indicates that a precise mixture of halide I and Br atoms significantly enhances the electrical and thermal conductivity of perovskites. Also, the substitution of halide I atoms by Br can improve the figure of merit (ZT). The maximum value of ZT at room temperature is equal to 0.99, justifying the brilliant thermoelectric properties of the studied compounds.