Complex Calculations Research Articles

The effect of extralinguistic variables on verb selection in Italian requests

Abstract Politeness relies on interlocutors’ frames, which are cognitive concepts that include a linguistic expression and extralinguistic variables. Politeness research has highlighted the importance of extralinguistic variables on speakers’ linguistic choices. Despite many studies that touch on these topics, questions about the comparative importance of contextual variables and the joint effect of them on speakers’ utterances remain unanswered. To examine these questions, a quantitative approach using a conditional inference tree was employed to investigate the influence of power, distance, and imposition on the use of verb forms in requests in Italian. Verb forms were selected as the dependent variable because they are essential for performing speech acts and they can be placed on a politeness continuum. Considering the importance and hierarchical relationship of the predictor variables of power, distance, and imposition, the results indicated that the three variables were predictors of verb form. While power was the main predictor, the effect of distance and imposition depended on whether the other variables were considered, showing a varying and complex effect of contextual variables. The findings enhance the understanding of Italian politeness, and represent the complex calculations that speakers make when selecting linguistic forms by considering interacting contextual variables.

Coastal Forum: The latest sea level rise projections of the Intergovernmental Panel on Climate Change

The Intergovernmental Panel on Climate Change (IPCC) is a multi-country organization of the United Nations formed in 1988 to assess science related to climate change (IPCC 2024). In March 2023, the IPCC published its Sixth Assessment Report (AR6), its sixth assessment of climate change since its first in 1990 (IPCC 2023). IPCC (2023) and its supporting reports total almost 8,000 pages (Boehm and Schumer 2023). Hundreds of scientists from around the world worked to develop the IPCC reports, and thousands of scientists peer-reviewed the reports(Boehm and Schumer 2023; Houston 2022).A chapter of AR6 deals with the physical basis for climate change and includes sea level rise projections (Fox-Kemper et al. 2021). These projections were supported by updated projections of the contributions to sea level rise from Antarctica (Oppenheimer 2019) and Greenland (The IMBIE Team 2020). IPCC (2023) presents the final projections. Updated IPCC projections are not expected until its seventh assessment in 2029-2030. I will report on the sea level projections in AR6, and on an online tool that provides sea level rise projections for the world as a whole and at individual tide gauge locations around the world. The IPCC bases its projections on climate response to five scenarios that cover the range of possible future anthropogenic drivers of climate change. “These scenarios combine socio-economic assumptions, levels of climate mitigation, land use, and air pollution controls for aerosols” (Climate Neutral Group 2021). IPCC then determines levels of greenhouse gases(largely carbon dioxide) and temperatures to 2050 and 2100 using complex calculations that depend on how quickly humans curb greenhouse gas emissions. It estimates sea level rise using projected temperatures. Scenarios based on IPCC (2023) are summarized below.

Simulated annealing algorithm for the optimal design of diffractive structures under fluctuation models

In this paper, to improve the optimization capability of the simulated annealing algorithm (SA) in the optimal design of diffractive structures, the random fluctuation problem in the algorithm is investigated, and a new fluctuation calculation model is proposed. The optimal design of phase-type diffraction gratings and beam-shaping devices verified the fast accuracy of the proposed model. The simulation experimental results show that the model’s introduction is particularly significant in substantially reducing computation time (a threefold increase in computation speed) and improving the algorithm’s optimization finding capability. Finally, the setting of the model parameters is discussed analytically to make the proposed model more applicable and robust. The study results provide guidance and assistance for higher speed and effectiveness in the optimal design using simulated annealing algorithms. It can be applied to any complex and large-scale optimization calculation problem.

Designing Accurate Moment Tensor Potentials for Phonon-Related Properties of Crystalline Polymers.

The phonon-related properties of crystalline polymers are highly relevant for various applications. Their simulation is, however, particularly challenging, as the systems that need to be modeled are often too extended to be treated by ab initio methods, while classical force fields are too inaccurate. Machine-learned potentials parametrized against material-specific ab initio data hold the promise of being extremely accurate and also highly efficient. Still, for their successful application, protocols for their parametrization need to be established to ensure an optimal performance, and the resulting potentials need to be thoroughly benchmarked. These tasks are tackled in the current manuscript, where we devise a protocol for parametrizing moment tensor potentials (MTPs) to describe the structural properties, phonon band structures, elastic constants, and forces in molecular dynamics simulations for three prototypical crystalline polymers: polyethylene (PE), polythiophene (PT), and poly-3-hexylthiophene (P3HT). For PE, the thermal conductivity and thermal expansion are also simulated and compared to experiments. A central element of the approach is to choose training data in view of the considered use case of the MTPs. This not only yields a massive speedup for complex calculations while essentially maintaining DFT accuracy, but also enables the reliable simulation of properties that, so far, have been entirely out of reach.

Hybrid deconvolution method based on mode composition beamforming for separating sound sources with different motion modes

The presence of sound sources that rotate can cause interference with stationary acoustic imaging, especially if the intensity of the rotating sound source is strong. Vice versa, stationary sound sources can also interfere with rotational acoustic imaging. Current algorithms for separating and locating rotating and stationary sound sources have problems such as weak separation capabilities, complex calculations, and limited array layouts. To address these challenges more effectively, a frequency-domain hybrid deconvolution method based on modal composition beamforming is proposed to separate and localize rotating and stationary sound sources. This algorithm operates within the frequency domain and is not restricted by array layout or the tonal single-frequency nature of sound sources. First, the point spread function and cross-point spread function matrices of the array are derived based on the modal transfer function. Then, a system of linear equations is constructed to separate rotating and stationary sound sources. Finally, the sound source separation and localization problems are solved through Gaussian Seidel and sparse-constrained deconvolution, respectively. Compared with the virtual rotating array method combined with matrix completion, simulation and experimental results demonstrate that this method can accurately locate and separate rotating and stationary sound sources, even when there is a substantial difference in intensity between rotating and stationary sound sources.

Nonlinear adaptive robust control with continuous error reduction mapping

Abstract Achieving high stability and high-precision control by traditional control algorithms is difficult due to the strong nonlinearity, parameter uncertainty, and random disturbance characteristics of electro-hydraulic proportional servo systems. Herein, we developed a nonlinear adaptive control algorithm based on Lyapunov functions. By estimating the parameters and converging their errors, this algorithm solved the parameter adaptation problem. Additionally, based on convergence inference from the nonnegativity of the derivative of Lyapunov functions, the algorithm designed a disturbance mapping through the derived function. The random disturbance was reduced through the mapping, improving the stability of the system under random disturbance. Finally, we carried out continuous derivation and moved the mapping relationship out of the intermediate temporary controller. This reduced the complexity of the calculation during controller design, prevented performance degradation of the error convergence algorithm after complex calculation, and enhanced the robustness of the algorithm. The algorithm was verified through the simulation model established by Simulink and experience. Results showed that the nonlinear adaptive control algorithm demonstrates fast response, high accuracy, and strong anti-interference ability compared with the high-gain robust control algorithm. It realizes the adaptation of unknown parameters and reduces the control error introduced by random disturbance.

Development of higher order deterministic approaches of forward and inverse uncertainty quantification for nonlinear engineering systems

This paper introduces an innovative numerical scheme that may accurately quantify the parameter and response distributions with minimal computational costs, with specific applications to the numerical computation of multiphase flows under various fluid conditions. The probabilistic approach of uncertainty quantification, e.g., the Monte Carlo simulation, although known to be effective in estimating the parameter/response distribution for highly nonlinear problems, is usually not applicable to multi-physics and multi-scale integrated systems due to its high computational cost. The objective of this study is first, to reduce the computing demand for complex physical system calculations and, at the same time, to accurately predict actual probability distributions by developing a higher order deterministic approach to uncertainty quantification. To achieve this, instead of executing simulation codes multiple times to estimate the probability distribution, the first to fourth moments of the probability density function are derived directly by employing a second order Taylor expansion form of the responses along with Bayesian statistics. This characterizes the parameter and response distributions computed by inverse and forward uncertainty quantification based on their mean value, mode, covariance, skewness, and kurtosis. The parameter distribution is estimated by the numerically calculated higher moments of the parameters with a posteriori probability density function being derived by Bayesian inference. The parameter distribution is then mapped into the second order approximated response function to compute the characteristics of the response distribution through both analytical calculation and numerical approximation of the moments of the responses. The proposed approach is numerically demonstrated through simulations of two benchmark problems: MIT pressurizer and FEBA tests. The prediction of the responses is improved by inverse uncertainty quantification as numerical calculations with the a posteriori model parameters are in better agreement with the measured data over a wide range of transient periods. Additionally, the response distribution estimated by the proposed approach of forward uncertainty quantification accurately predicts the distribution computed by the Monte Carlo simulation.

Snowdrift‐Permitting Simulations of Seasonal Snowpack Processes Over Large Mountain Extents

AbstractThe melt of seasonal snowpack in mountain regions provides downstream river basins with a critical supply of freshwater. Snowdrift‐permitting models have been proposed as a way to accurately simulate snowpack heterogeneity that stems from differences in energy inputs, over winter redistribution, sublimation, melt, and variations in precipitation. However, these spatial scales can be computationally intractable for large extents. In this work, the multiscale Canadian Hydrological Model (CHM) was applied to simulate snowpacks at snowdrift‐permitting scales (≈50 m) across the Canadian Cordillera and adjacent regions (1.37 million km2) forced by downscaled atmospheric data. The use of a multiscale land surface representation resulted in a reduction of computational elements of 98% while preserving land‐surface heterogeneity. CHM includes complex terrain windflow and radiative transfer calculations, lapses temperature, humidity, and precipitation with elevation, redistributes snow by avalanching, wind transport and forest canopy interception and calculates the energetics of canopy and surface snowpacks. Model outputs were compared to a set of multiscale observations including snow‐covered area (SCA) from Sentinel and Landsat imagery, snow depth from uncrewed aerial system lidar, and point surface observations of depth and density. Including snow redistribution and sublimation processes improved the summer SCA r2 from 0.7 to 0.9. At larger scales, inclusion of snow redistribution processes delayed full snowpack ablation by an average of 33 days, demonstrating process emergence with scale. These simulations show how multiscale modeling can improve snowpack predictions to support prediction of water supply, droughts, and floods.

Globally optimal sequencing of optimal reactive dispatch control adjustments to minimize operational losses in transmission systems by graph shortest path, parallel computing, and dynamic programming

Minimizing operational losses in transmission systems through the Optimal Reactive Dispatch (ORD), a non-convex mixed-integer nonlinear programming problem, is crucial for operational cost reduction, resource optimization, and greenhouse gas emission mitigation. Besides all intricacies associated with solving ORDs, transmission system operators encounter the challenge of determining sequences in which ORD control adjustments must be implemented before significant changes occur in generators scheduled power output and system loading. Sequencing ORD control adjustments, in spite of not being novel, remains modestly scrutinized in the literature. This paper introduces a two-phase framework that tackles the globally optimal sequencing of n ORD control adjustments over n! potential paths by solving the ORD to minimize operational losses in transmission systems in the first phase, and optimally sequencing ORD control adjustments employing fast power flow calculations, graph shortest path, parallel computing, and dynamic programming in the second phase. We discuss the framework’s second phase asymptotic time complexity, which is exponential over factorial for brute-force approaches, and its capability to guarantee globally optimal paths toward minimal operational losses determined in the framework’s first phase. ORD control adjustments for transmission systems with up to 27 controllable variables are benchmarked against two mixed-integer nonlinear programming solvers: BARON, a global non-convex solver, and Knitro, a local solver (assuming convexity around local optima). Globally optimal sequences of ORD control adjustments over n! potential paths (more than 1028 for sequencing 27 control adjustments) and average algorithm runtimes validate the straightforward application and, more importantly, effectiveness of such a comprehensive framework.

Exploiting Cascaded Channel Signature for PHY-Layer Authentication in RIS-Enabled UAV Communication Systems

Reconfigurable Intelligent Surface (RIS)-assisted Unmanned Aerial Vehicle (UAV) communications face a critical security threat from impersonation attacks, where adversaries impersonate legitimate entities to infiltrate networks to obtain private data or unauthorized access. To combat such security threats, this paper proposes a novel physical layer (PHY-layer) authentication scheme for validating UAV identity in RIS-enabled UAV wireless networks. Considering that most existing works focus on traditional communication systems such as IoT and millimeter wave multiple-input multiple-output (MIMO) systems, there is currently no mature PHY-layer authentication scheme to serve RIS-UAV communication systems. To this end, our scheme leverages the unique characteristics of cascaded channels related to RIS to verify the legitimacy of UAV transmitting signals to the base station (BS). To be more precise, we first use the least squares estimate method and coordinate a descent-based algorithm to extract the cascaded channel feature. Next, we explore a quantizer to quantize the fluctuations of the channel gain that are related to the extracted channel feature. The 1-bit quantizer’s output findings are exploited to generate the authentication decision criteria, which are then tested using a binary hypothesis. The statistical signal processing technique is utilized to obtain the analytical formulations for detection and false alarm probabilities. We also conduct a computational complexity analysis of the proposed scheme. Finally, the numerical results validate the effectiveness of the proposed performance metric models and show that our detection performance can reach over 90% accuracy at a low signal-to-noise ratio (e.g., −8 dB), with a 10% improvement in detection accuracy compared with existing schemes.

MTML: An Efficient Multitrait Multilocus GWAS Method Based on the Cauchy Combination Test.

Genome-wide association study (GWAS) by measuring the joint effect of multiple loci on multiple traits, has recently attracted interest, due to the decreased costs of high-throughput genotyping and phenotyping technologies. Previous studies mainly focused on either multilocus models that identify associations with a single trait or multitrait models that scan a single marker at a time. Since these types of models cannot fully utilize the association information, the powers of the tests are usually low. To potentially address this problem, we present here a multitrait multilocus (MTML) modeling framework that implements in three steps: (1) simplify the complex calculation; (2) reduce the model dimension; (3) integrate the joint contribution of single markers to multiple traits by Cauchy combination. The performances of MTML are evaluated and compared with other three published methods by Monte Carlo simulations. Simulation results show that MTML is more powerful for quantitative trait nucleotide detection and robust for various numbers of traits. In the meanwhile, MTML can effectively control type I error rate at a reasonable level. Real data analysis of Arabidopsis thaliana shows that MTML identifies more pleiotropic genetic associations. Therefore, we conclude that MTML is an efficient GWAS method for joint analysis of multiple quantitative traits. The R package MTML, which facilitates the implementation of the proposed method, is publicly available on GitHub https://github.com/Guohongping/MTML.

Kolmogorov complexity and nondeterminism versus determinism for polynomial time computations

We call any consistent and sufficiently powerful formal theory that enables to algorithmically verify whether a text is a proof algorithmically verifiable mathematics (av-mathematics). We study the fundamental question whether nondeterminism is more powerful than determinism for polynomial time computations in the framework of av-mathematics.Our goal is to show strong indications that nondeterminism is more powerful than determinism for polynomial time computations. To do that, we do not consider decision problems only, but also compression algorithms. We show that at least one of the following three claims must be true:(i)▪(ii)non-determinism is more powerful than determinism for polynomial-time compression(iii)for each polynomial-time compression algorithm there exists another one of the same asymptotic time complexity that compresses infinitely many strings logarithmically strongerAnother surprising consequence of P = NP would be that time-bounded Kolmogorov complexity for any polynomial bound can be computed by deterministic algorithms in polynomial time.

Centrally-chosen versus user-selected swaps: How the selection of swapping stations impacts standby battery inventories

Swapping depleted batteries of electric vehicles promises much better driving-to-total-travel-time ratios than plug-in charging. Nonetheless, large-scale battery swapping systems have not successfully established yet. One obstacle, on top of the high infrastructure cost, is certainly the additional invest into extra standby batteries that await their swaps at stations. Existing research is focused on systems in which users decide individually where they want to swap batteries. This system requires significant standby battery inventories to protect against uncertain swapping demand. This paper evaluates another system where users must register their trips on a central platform, so that battery swaps can be coordinated based on central optimization results. To benchmark central optimization and user choice regarding their impact on standby battery inventories, we formulate the min-battery swapping problem: For a given set of vehicle trips, this optimization problem minimizes the number of standby batteries, distributes them in a given station network, and derives detailed swapping plans to feasibly power all trips. First, we present a thorough analysis of computational complexity. Then, we provide an efficient mixed-integer programming formulation that is adaptable to different swapping policies and (when fed into a default solver) solves instances with up to 200 trips to proven optimality in just a few seconds. Our computational study reveals that central optimization promises a significant reduction of standby battery inventories. This potential is shown to increase if swaps of not yet fully-charged batteries are allowed and swaps are pooled at a reduced number of swapping stations.

Effect of Column Deformation for Steel Frames with Semi-rigid Connection

As semi-rigid behaviour will give a more economical design, it should be applied in the structural steel design. The properties of semi-rigid connection may affect the column design and is the main focus in this study. Complex calculation with semi-rigid consideration in design may cause unfavourable time consuming. Therefore, a simplified column design is needed to eliminate complicated procedures while accommodating the semi-rigid condition. In this analysis, the validated finite element procedure has been used for the analysis of a semi-rigid column in a non-sway frame. Parametric study was performed within variables of types of connection, loading patterns, locations of investigated column and column base fixity. The influences of column slenderness and beam flexibilities were also included and the implications towards the column design were then identified.

Investigation of Criteria for Comparing of Natural and LED Radiation Spectral Distribution

The difference in the spectral composition of artificial and natural lighting can negatively affect health, as well as lead to a distorted perception of the color of surrounding objects. At the same time, a certain correction of the spectral composition of visible radiation in medical institutions and workplaces has a positive effect on human health, while can be carried lighting control out taking into account the data of personal sensor devices that determine the human condition. The purpose of the research was to select criteria for comparing natural and LED optical and visible radiation by spectral composition and by the visibility of color differences in natural and LED lighting. The effectiveness of the application of known and developed criteria for assessing the difference in the spectral composition of optical and visible radiation from natural and LED sources was investigated, as well as for the visibility of color differences in natural and LED lighting. To minimize the values of criteria are proposed additive and subtractive methods for calculating LED parameters. Their comparison allowed us to conclude that a more complex calculation algorithm, but higher performance for an additive technique than for a subtractive one with the same minimization results.It was found that to simulate the spectral composition of natural radiation using LEDs, it is most effective to use the criteria "standard deviations of the relative differences between the optical and visible spectral components of natural and LED radiation". A comparison of the criteria for the visibility of color differences in natural and LED lighting showed approximately the same effectiveness of using the criteria "small color differences" and "standard deviation by photoreceptors" at the present stage and the prospects for applying the second criterion, provided that its acceptable values are established.

Data Processing of Bridge IoT Monitoring Based on Task Scheduling of Cloud Service Listening Signal

For the long-term continuous monitoring of bridge-related indicators, it is necessary to arrange relatively perfect acquisition equipment on the bridge, which can feedback various information parameters of the bridge. However, there are many parameters to feedback the bridge information, which leads to the complex and overstaffed structure of the monitoring system. Furthermore, the huge amount of data collected and the complex calculation process also increase the difficulty of the operation of the monitoring system. In this regard, we should choose more scientific and reasonable indicators, lightweight data structure, stable data transmission, and analysis programs to improve the accuracy of continuous monitoring. To establish a stable and efficient bridge monitoring system, we use the distance coefficient-effective independent algorithm to optimize. Then, we calculate the relevant information of the strain environment with the help of a neural network model, strengthen the training of deep learning through the YOLOv5s model, and improve the task scheduling strategy of attention concentration. Through that, we solve the problem of embedded systems with relatively low computing power. Different weights are assigned to each fused feature map, and the nodes at the highest level and the lowest level are deleted so that a concise and efficient lightweight network model is constructed. Multiple iterations are performed to achieve deeper feature fusion. Therefore, the complexity of the model is effectively reduced, and the monitoring performance can be effectively improved. Finally, through the experimental analysis, it is proved that compared with the traditional fusion model, the number of parameters of the improved fusion network structure in bridge health monitoring is reduced by 7.37%. The detection speed is increased by 18.2%. The amount of computation is reduced by 42.92%, and the average detection accuracy is required to reach 95.33%. It is verified that the proposed method can effectively improve the accuracy and risk control ability of the detection data by learning from the samples with small labels. It also has great practical significance and market value for the design and optimization of the bridge health monitoring system, which is suitable for the monitoring data of large-scale construction projects.

Tensile fracture of blunt V-notches in rectangular plates additively manufactured from Acrylonitrile Butadiene Styrene with different raster angles

In this research, the tensile fracture behavior of rectangular specimens made of Acrylonitrile Butadiene Styrene (ABS) containing double-edge blunt V-notches is evaluated. The specimens are printed using the fused deposition modeling (FDM) technique with five different raster angles and three different notch tip radii. To simplify the complex calculations related to the anisotropic behavior of these specimens, the Virtual Isotropic Material Concept (VIMC) is used in conjunction with three well-known linear elastic notch fracture mechanics (LENFM) criteria, i.e., the mean stress (MS), point stress (PS), and averaged strain energy density (ASED) criteria as well as the extended finite element method (XFEM). It is shown that all these fracture models can predict the load-bearing capacity of the 3D-printed notched specimens by acceptable accuracies, where the best predictions belong to the MS criterion with the critical distance obtained directly from the finite element method, and the least accurate predictions belong to the XFEM method.

Inspector Gadget

We introduce InspectorGadget, an Open-Source Python-based software for assessing and comparing the complexity of masking gadgets. By providing a limited set of characteristics of a hardware platform, our tool allows to estimate the cost of a masking gadget in terms of cycle count equivalent and memory footprint. InspectorGadget is highly flexible. It enables the user to define her own estimation functions, as well as to expand the set of gadgets and predefined microcontrollers. As a case-study, we produce a fair comparison of several masked versions of Kyber compression function from the literature, together with novel alternatives automatically generated by our tool. Our results confirm that an interesting middle ground exists between theoretical performance measures (asymptotic complexity or operations count) and real implementations benchmarks (clock cycle accurate evaluations). InspectorGadget offers both simplicity and genericity while capturing the main performance-related parameters of a hardware platform.

Copy Number Profiling Implicates Thin High-Grade Squamous Intraepithelial Lesions as a True Precursor of Cervical Human Papillomavirus-Induced Squamous Cell Cancer

Full-thickness high-grade squamous intraepithelial lesions (HSIL) are precursors of invasive cervical squamous cell carcinoma (SCC). The World Health Organization and Lower Anogenital Squamous Terminology Standardization Project for human papilloma virus (HPV)–associated lesions divide full-thickness HSIL of the cervix into thin HSIL with thickness of 1 to 9 cell layers and the typical full-thickness HSIL of >10 cell layers. Although HPV oncogene transcripts and p16ink4a overexpression, as markers of transforming HPV infection, are detectable in thin HSIL, the biological significance of thin HSIL in cervical carcinogenesis remains poorly understood. To further characterize thin HSIL, we performed a comparative study of chromosomal copy number variations (CNV), an analysis of dysregulated genes present in the segments with CNV, and a generalized genetic complexity calculation for 31 thin HSIL, 31 thick HSIL, 24 microinvasive SCC (pT1a SCC), and 22 highly invasive SCC samples. Thin HSIL share various CNV and specific dysregulated gene pathways with thick HSIL and invasive SCC. Thin HSIL exhibited an average CNV of 11.6% compared with 14.1% for thick HSIL, 15.5% for pT1a SCC, and 26.6% for highly invasive SCC. The CNV included gains at 1q and 3q (40% and 43%, respectively), partial loss of 3p, and loss of chromosomes 11 (18%), 16 (50%), 20 (35%), and 22 (40%). Pathways affected solely in thin HSIL were those enhancing immune evasion and primarily involved the (interleukin) IL6, IL21, and IL23 genes. ILs are transiently upregulated in response to infection and play a crucial role in mounting antitumor T-cell activity. Deregulation reflects an attempt by the HPV to evade the initial immune response of the host. The primary pathways shared by thick HSIL and invasive SCC were interactions between lymphoid and nonlymphoid cells, NOTCH2 signaling, tight junction interactions (primarily of the claudin family), and FGR2 alternative splicing. Our results show that thin HSIL carry similar genetic changes as thick HSIL and SCC, indicating that thin HSIL are true precursor lesions that can progress to thick HSIL and SCC.

No existence of a linear algorithm for the one-dimensional Fourier phase retrieval

Fourier phase retrieval, which aims to reconstruct a signal from its Fourier magnitude, is of fundamental importance in fields of engineering and science. In this paper, we provide a theoretical understanding of algorithms for the one-dimensional Fourier phase retrieval problem. Specifically, we demonstrate that if an algorithm exists which can reconstruct an arbitrary signal x∈CN in Poly(N)log⁡(1/ϵ) time to reach ϵ-precision from its magnitude of discrete Fourier transform and its initial value x(0), then P=NP. This partially elucidates the phenomenon that, despite the fact that almost all signals are uniquely determined by their Fourier magnitude and the absolute value of their initial value |x(0)|, no algorithm with theoretical guarantees has been proposed in the last few decades. Our proofs employ the result in computational complexity theory that the Product Partition problem is NP-complete in the strong sense.