Complete Problem Research Articles

Formulating the complete initial boundary value problem in numerical relativity to model black hole echoes

Abstract In an attempt to simulate black hole echoes (generated by potential quantum-gravitational structure) in numerical relativity, we recently described how to implement a reflecting boundary outside of the horizon of a black hole in spherical symmetry. Here, we generalize this approach to spacetimes with no symmetries and implement it numerically using the generalized harmonic formulation. We cast the evolution equations and the numerical implementation into a Summation By Parts (SBP) scheme, which seats our method closer to a class of provably numerically stable systems. We implement an embedded boundary numerical framework that allows for arbitrarily shaped domains on a rectangular grid and even boundaries that evolve and move across the grid. As a demonstration of this framework, we study the evolution of gravitational wave scattering off a boundary either inside, or just outside, the horizon of a black hole. This marks a big leap toward the goal of a generic framework to obtain gravitational waveforms for behaviors motivated by quantum gravity near the horizons of merging black holes.

Completion problems and sparsity for Kemeny’s constant

Abstract For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny’s constant. We prove that for any partially specified stochastic matrix for which the problem is well defined, there is a minimizing completion that is as sparse as possible. We also find the minimum value of Kemeny’s constant in two special cases: when the diagonal has been specified and when all specified entries lie in a common row.

Effect of Critical Thinking Skills on Misconceptions of Students in Learning Mathematics at the Elementary School Level

The aim of mathematics education in schools is to create logical reasoning and critical thinking ability in students. These skills development not only help students understand better but can also help remove the misconceptions they have about mathematical concepts. The purpose of this study was to investigate how critical thinking skills can help address misconceptions in mathematics learning at the elementary level. The research used an A-B-A design to determine the effects of an intervention, which is a perfect design for this purpose. 35 eighth-grade students who were enrolled in the current academic year were participants in this study. Three researcher-developed tools facilitated data collection. The first and initial tool was called the Test of Critical Thinking Skills (ToCTS), which measured students' critical thinking math skills. The second tool, the Test of Misconceptions (ToM), was developed to identify specifically how the students thought they were solving the problems. Finally, an interview protocol was used to learn about the students' reasoning regarding their answers on the tests. The mathematics curriculum of grades three to eight formed the basis of the development of these instruments, both close-ended and open-ended questions. Descriptive and inferential statistical methods were used in data analysis, and qualitative content from the interviews was applied to thematic analysis. A poor level of critical thinking skills in factoring equations and selecting proper solving methods for linear equations was shown by students. In addition, students had difficulty solving complete mathematical problems with linear equations. A significant gap was identified in their understanding and application of the BODMAS rule during algebraic operations. Based on these findings, the study recommended fostering critical thinking skills to improve students' abilities in factoring equations. Teachers should provide clear and comprehensive guidance to help students solve mathematical problems effectively. To address misconceptions, educators could introduce simplified problem-solving methods and reinforce students’ understanding of fundamental operations and rules. Specifically, it is essential to provide targeted instruction on the application of the BODMAS rule in algebraic contexts to enhance students' problem-solving capabilities.

A Majorization-Minimization Gauss-Newton Method for 1-Bit Matrix Completion

In 1-bit matrix completion, the aim is to estimate an underlying low-rank matrix from a partial set of binary observations. We propose a novel method for 1-bit matrix completion called Majorization-Minimization Gauss-Newton (MMGN). Our method is based on the majorization-minimization principle, which converts the original optimization problem into a sequence of standard low-rank matrix completion problems. We solve each of these sub-problems by a factorization approach that explicitly enforces the assumed low-rank structure and then apply a Gauss-Newton method. Using simulations and a real data example, we illustrate that in comparison to existing 1-bit matrix completion methods, MMGN outputs comparable if not more accurate estimates. In addition, it is often significantly faster, and less sensitive to the spikiness of the underlying matrix. In comparison with three standard generic optimization approaches that directly minimize the original objective, MMGN also exhibits a clear computational advantage, especially when the fraction of observed entries is small.

Complete Visibility Algorithms of Luminous Robots With Two‐Color Lights on Grid

ABSTRACTAn autonomous mobile robot system is a distributed system consisting of multiple mobile computational entities, called robots, which autonomously and repeatedly perform three fundamental operations: look, compute, and move. Various challenges in such systems, including gathering, pattern formation, and flocking, have been extensively studied to explore the relationship between the robots' capabilities and the feasibility of solving these problems (i.e., solvability). In this study, we focus on the complete visibility problem, which aims to relocate all robots on an infinite grid plane so that every robot is visible to every other robot (i.e., complete visibility). We assume that each robot is a luminous robot (i.e., has a light with a constant number of colors) and opaque (non‐transparent). This paper primarily examines the number of light colors required for each robot to solve the complete visibility problem. Specifically, we investigate the question: “how many colors can we reduce while still achieving complete visibility?” (even under certain stronger assumptions). As an answer to the above question, we show the existence of a deterministic algorithm to achieve complete visibility (i.e., every robot can observe all the other robots) using only two colors of light, if the robots agree on the directions and orientations of both axes. The proposed algorithm correctly works even if robots operate asynchronously and have no knowledge of the total number of robots. Moreover, its spatial complexity (i.e., the area of the smallest enclosed rectangle that includes all robots in the final configuration) ensures the optimal one, , where is the number of robots.

Unifying lower bounds for algebraic machines, semantically

We present a new abstract method for proving lower bounds in computational complexity based on the notion of topological and measurable entropy for dynamical systems. It is shown to generalise several previous lower bounds results from the literature in algebraic complexity, thus providing a unifying framework for “topological” proofs of lower bounds. We further use this method to prove that maxflow, a ▪ complete problem, is not computable in polylogarithmic time on parallel random access machines (prams) working with real numbers. This improves on a result of Mulmuley since the class of machines considered extends the class “prams without bit operations”, making more precise the relationship between Mulmuley's result and similar lower bounds on real prams.

Mobility Prediction via Rule-enhanced Knowledge Graph

With the rapid development of location acquisition technologies, massive mobile trajectories have been collected and made available to us, which support a fantastic way of understanding and modeling individuals’ mobility. However, existing data-driven methods either fail to capture the long-range dependency or suffer from a high computational cost. To overcome these issues, we propose a knowledge-driven framework for mobility prediction, which leverages knowledge graphs (KG) to formulate the mobility prediction task into the KG completion problem through integrating the structured “knowledge” from the mobility data. However, most related mobility prediction works only focus on the structured information encoded in existing triples, which ignores the rich semantic information of relation paths composed of multiple relation triples. In this article, we apply a dedicated module to extract the supplementary semantic structure of paths in KG, which contributes to the interpretability and accuracy of our model. Specifically, the extracted rules are applied to capture the dependencies between relational facts. Moreover, by incorporating user information in the entity-relation space with the corresponding hyperplane, our method could capture diverse user mobility patterns and model the personal characteristics of users to improve the accuracy of mobility prediction. Extensive evaluations illustrate that our proposed model beats state-of-the-art mobility prediction algorithms, which verifies the superiority of utilizing logical rules and user hyperplanes. Our implementation code is available at https://github.com/tsinghua-fib-lab/RulekG-MobiPre.git

The effectiveness of perspective video modelling training on one‐stage word problem‐solving skills in children with autism spectrum disorder

AbstractSingle‐subject research can provide adequate justification when developing successful evidence‐based educational practices for children with autism spectrum disorder (ASD), which can help such students thrive academically. This research investigates whether point‐of‐view video modelling effectively improved the word problem‐solving addition performances of primary‐school‐aged students with ASD. The effectiveness of the intervention on each participant's ability to solve single‐digit addition by a single digit, effectively view the video on a tablet and generalise a learned skill was investigated by utilising a design that included several participant baselines. All participants saw an improvement in their ability to solve simple addition problems due to using point‐of‐view video modelling. Between the baseline and intervention phases, each participant's rate of digits correctly entered per minute and total number of steps completed significantly increased. A generalisation phase was performed at home. After receiving the intervention, people with ASD could independently complete word problems involving addition with a single digit. The findings suggest that this technology could practically support the education of the growing number of children and young people with ASD by mollifying the particular learning obstacles their impairment brings.

Self representation based methods for tensor completion problem

Tensor, the higher-order data array, naturally arises in many fields, such as information sciences, seismic data reconstruction, physics, video inpainting and so on. In this paper, we intend to provide a new model to recover a tensor, based on self-representation, for the all-mode unfoldings of the desired tensor, regardless of the tensor rank. The suggested idea generalizes self-representation to tensor and recovers an incomplete tensor by reconstructing one fiber by others in such a way that they all belong to the same subspace. We design least-square and low-rank self-representation algorithms based on the Linearized Alternating Direction Method utilizing this concept. We show that the proposed algorithms converge to the rank-minimization of the incomplete tensor.

Noise Radiation from a partially opened and elevated tunnel-tube

Road or railway tunnels are considered in noise prediction programs generally by neglecting any emission from the tunnel structure between the openings and replacing each tunnel mouth by additional sources simulating the sound radiated from inside to outside through these openings. The sound pressure level inside the tunnel can be derived from the analog model of a room formed by a slice cut out of the extended tunnel with two perfect reflecting planes perpendicular to the tunnel axis. The paper presents a methodology for the case of an elevated tunnel tube with an upper ventilation gap running over the entire length of the tunnel to calculate the sound pressure levels in the surrounding area more approximately using engineering models like ISO 9613-2 or others and more detailed using a particle method developed to account for complete 3-dimensional propagation problems.

An augmented swarm optimization algorithm for k-clustering minimum biclique completion problems

An augmented swarm optimization algorithm for k-clustering minimum biclique completion problems

CP decomposition-based algorithms for completion problem of motion capture data

Motion capture (MoCap) technology is an essential tool for recording and analyzing movements of objects or humans. However, MoCap systems frequently encounter the challenge of missing data, stemming from mismatched markers, occlusion, or equipment limitations. Recovery of these missing data is imperative to maintain the reliability and integrity of MoCap recordings. This paper introduces a novel application of the tensor framework for MoCap data completion. We propose three completion algorithms based on the canonical polyadic (CP) decomposition of tensors. The first algorithm utilizes CP decomposition to capture the low-rank structure of the tensor. However, relying only on low-rank assumptions may be insufficient to deal with complex motion data. Thus, we propose two modified CP decompositions that incorporate additional information, SmoothCP and SparseCP decompositions. SmoothCP integrates piecewise smoothness prior, while SparseCP incorporates sparsity prior, each aiming to improve the accuracy and robustness of MoCap data recovery. To compare and evaluate the merit of the proposed algorithms over other tensor completion methods in terms of several evaluation metrics, we conduct numerical experiments with different MoCap sequences from the CMU motion capture dataset.

Sinc method in spectrum completion and inverse Sturm–Liouville problems

Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

Robust Matrix Completion with Heavy-Tailed Noise

This article studies noisy low-rank matrix completion in the presence of heavy-tailed and possibly asymmetric noise, where we aim to estimate an underlying low-rank matrix given a set of highly incomplete noisy entries. Though the matrix completion problem has attracted much attention in the past decade, there is still lack of theoretical understanding when the observations are contaminated by heavy-tailed noises. Prior theory falls short of explaining the empirical results and is unable to capture the optimal dependence of the estimation error on the noise level. In this article, we adopt an adaptive Huber loss to accommodate heavy-tailed noise, which is robust against large and possibly asymmetric errors when the parameter in the Huber loss function is carefully designed to balance the Huberization biases and robustness to outliers. Then, we propose an efficient nonconvex algorithm via a balanced low-rank Burer-Monteiro matrix factorization and gradient descent with robust spectral initialization. We prove that under merely a bounded second-moment condition on the error distributions, rather than the sub-Gaussian assumption, the Euclidean errors of the iterates generated by the proposed algorithm decrease geometrically fast until achieving a minimax-optimal statistical estimation error, which has the same order as that in the sub-Gaussian case. The key technique behind this significant advancement is a powerful leave-one-out analysis framework. The theoretical results are corroborated by our numerical studies. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Mourre Theory and Asymptotic Observables in Local Relativistic Quantum Field Theory

We prove the convergence of Araki–Haag detectors in any Haag–Kastler quantum field theory with an upper and lower mass gap. We cover the case of a single Araki–Haag detector on states of bounded energy, which are selected from the absolutely continuous part of the energy-momentum spectrum sufficiently close to the lower boundary of the multi-particle spectrum. These states essentially encompass those states in the multi-particle spectrum lying below the three-particle threshold. In our proof, we draw on insights from proofs of asymptotic completeness in quantum mechanics. Notably, we apply Mourre’s conjugate operator method for the first time within the framework of Haag–Kastler quantum field theory. Furthermore, we discuss applications of our findings for the problem of asymptotic completeness in local relativistic quantum field theory.

A new minimal element theorem and new generalizations of Ekeland’s variational principle in complete lattice optimization problem

A new minimal element theorem and new generalizations of Ekeland’s variational principle in complete lattice optimization problem

Neural shape completion for personalized Maxillofacial surgery

In this paper, we investigate the effectiveness of shape completion neural networks as clinical aids in maxillofacial surgery planning. We present a pipeline to apply shape completion networks to automatically reconstruct complete eumorphic 3D meshes starting from a partial input mesh, easily obtained from CT data routinely acquired for surgery planning. Most of the existing works introduced solutions to aid the design of implants for cranioplasty, i.e. all the defects are located in the neurocranium. In this work, we focus on reconstructing defects localized on both neurocranium and splanchnocranium. To this end, we introduce a new dataset, specifically designed for this task, derived from publicly available CT scans and subjected to a comprehensive pre-processing procedure. All the scans in the dataset have been manually cleaned and aligned to a common reference system. In addition, we devised a pre-processing stage to automatically extract point clouds from the scans and enrich them with virtual defects. We experimentally compare several state-of-the-art point cloud completion networks and identify the two most promising models. Finally, expert surgeons evaluated the best-performing network on a clinical case. Our results show how casting the creation of personalized implants as a problem of shape completion is a promising approach for automatizing this complex task.

The Roles of Rule Type and Word Term in the Deductive Reasoning of Adults with and without Dyslexia.

Despite its importance to everyday functioning, reasoning is underexplored in developmental dyslexia. The current study investigated verbal deductive reasoning on the Wason selection task, not previously used in dyslexia research despite its well-established pedigree. Reasoning rule was manipulated, with the conditional rules varying in the logical values presented. The word frequency and imageability of the word terms was also manipulated. Twenty-six adults with dyslexia and 31 adults without dyslexia completed Wason selection task problems. No group difference in reasoning accuracy or completion time was found. However, the participants were most accurate when reasoning with the rule type "If p, then not q" and least accurate with the rule type "If p then q". More trials were also answered correctly when the word terms were highly imageable but of average word frequency. These findings are in line with the general reasoning literature. Dyslexia status did not interact with either rule type or word term type. The study expands upon previous research by testing verbal deductive reasoning in dyslexia, highlighting the role of imageability in facilitating reasoning performance for all, regardless of the presence or absence of dyslexia. Implications for the design of educational materials are considered.

Noise radiation from a partially opened and elevated tunnel-tube

Road or railway tunnels are considered in noise prediction programs generally by neglecting any emission from the tunnel structure between the openings and replacing each tunnel mouth by additional sources simulating the sound radiated from inside to outside through these openings. The sound pressure level inside the tunnel can be derived from the analog model of a room formed by a slice cut out of the extended tunnel with two perfect reflecting planes perpendicular to the tunnel axis. The paper presents a methodology for the case of an elevated tunnel tube with an upper ventilation gap running over the entire length of the tunnel to calculate the sound pressure levels in the surrounding area more approximately using engineering models like ISO 9613-2 or others and more detailed using a particle method developed to account for complete 3-dimensional propagation problems.

Task Offloading and Execution in Edge-Cloud Collaborative Network Using Genetic Algorithm

Mobile Edge Computing (MEC) system is now one of the most emerging sectors in wireless communication. It provides computation capability near the edge users and reduces the dependency on the Internet Cloud (IC). The edge users offload the task to the MEC server due to the lack of computing resources for executing resource-hungry applications. MEC servers can serve faster as they are quite close to the edge users but have limited computation resources compared to IC. On the other hand, IC has abundant computation resources but offloading to IC will add extra latency to complete tasks execution. In this scenario, an MEC has to decide intelligently which tasks should be executed by itself and which should be sent to IC. In this paper, we addressed this issue of task assignment and execution either on MEC or on IC and formulated an optimization problem to reduce the task processing latency. The problem is Integer Linear and NP-Complete, thus having higher time complexity. In this regard, we proposed a Genetic algorithm-based Meta-Heuristic method to solve the task completion problem considering the delay constraint of the user. Simulation results of the proposed method indicated improvement in terms of successful task execution, task turnaround time, and better Quality of experience (QoE) compared to the state-of-the-art works. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 9(1), 2022 P 42-51