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The Traveling Salesman Problem
This book presents the latest findings on one of the most intensely investigated subjects in computational mathematics--the traveling salesman problem. It sounds simple enough: given a set of cities and the cost of travel between each pair of them, the problem challenges you to find the cheapest route by which to visit all the cities and return home to where you began. Though seemingly modest, this exercise has inspired studies by mathematicians, chemists, and physicists. Teachers use it in the classroom. It has practical applications in genetics, telecommunications, and neuroscience.
The authors of this book are the same pioneers who for nearly two decades have led the investigation into the traveling salesman problem. They have derived solutions to almost eighty-six thousand cities, yet a general solution to the problem has yet to be discovered. Here they describe the method and computer code they used to solve a broad range of large-scale problems, and along the way they demonstrate the interplay of applied mathematics with increasingly powerful computing platforms. They also give the fascinating history of the problem--how it developed, and why it continues to intrigue us.
eBook
Word problems in mathematics education: a survey
Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.
Journal Article
What part of the brain is involved in graphic design thinking in landscape architecture?
Graphic design thinking is a key skill for landscape architects, but little is known about the links between the design process and brain activity. Based on Goel’s frontal lobe lateralization hypothesis (FLLH), we used functional magnetic resonance imaging (fMRI) to scan the brain activity of 24 designers engaging in four design processes—viewing, copy drawing, preliminary ideas, and refinement—during graphic design thinking. The captured scans produced evidence of dramatic differences between brain activity when copying an existing graphic and when engaging in graphic design thinking. The results confirm that designs involving more graphic design thinking exhibit significantly more activity in the left prefrontal cortex. These findings illuminate the design process and suggest the possibility of developing specific activities or exercises to promote graphic design thinking in landscape architecture.
Journal Article
Predicting protein structures with a multiplayer online game
Many hands make light work
A natural polypeptide chain can fold into a native protein in microseconds, but predicting such stable three-dimensional structure from any given amino-acid sequence and first physical principles remains a formidable computational challenge. Aiming to recruit human visual and strategic powers to the task, Seth Cooper, David Baker and colleagues turned their 'Rosetta' structure-prediction algorithm into an online multiplayer game called Foldit, in which thousands of non-scientists competed and collaborated to produce a rich set of new algorithms and search strategies for protein structure refinement. The work shows that even computationally complex scientific problems can be effectively crowd-sourced using interactive multiplayer games.
Predicting the structure of a folded protein from first principles for any given amino-acid sequence remains a formidable computational challenge. To recruit human abilities to the task, these authors turned their Rosetta structure prediction algorithm into an online multiplayer game in which thousands of non-scientists competed and collaborated to produce new algorithms and search strategies for protein structure refinement. This shows that computationally complex problems can be effectively 'crowd-sourced' through interactive multiplayer games.
People exert large amounts of problem-solving effort playing computer games. Simple image- and text-recognition tasks have been successfully ‘crowd-sourced’ through games
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, but it is not clear if more complex scientific problems can be solved with human-directed computing. Protein structure prediction is one such problem: locating the biologically relevant native conformation of a protein is a formidable computational challenge given the very large size of the search space. Here we describe Foldit, a multiplayer online game that engages non-scientists in solving hard prediction problems. Foldit players interact with protein structures using direct manipulation tools and user-friendly versions of algorithms from the Rosetta structure prediction methodology
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, while they compete and collaborate to optimize the computed energy. We show that top-ranked Foldit players excel at solving challenging structure refinement problems in which substantial backbone rearrangements are necessary to achieve the burial of hydrophobic residues. Players working collaboratively develop a rich assortment of new strategies and algorithms; unlike computational approaches, they explore not only the conformational space but also the space of possible search strategies. The integration of human visual problem-solving and strategy development capabilities with traditional computational algorithms through interactive multiplayer games is a powerful new approach to solving computationally-limited scientific problems.
Journal Article
Algorithm discovery by protein folding game players
Foldit is a multiplayer online game in which players collaborate and compete to create accurate protein structure models. For specific hard problems, Foldit player solutions can in some cases outperform state-of-the-art computational methods. However, very little is known about how collaborative gameplay produces these results and whether Foldit player strategies can be formalized and structured so that they can be used by computers. To determine whether high performing player strategies could be collectively codified, we augmented the Foldit gameplay mechanics with tools for players to encode their folding strategies as \"recipes\" and to share their recipes with other players, who are able to further modify and redistribute them. Here we describe the rapid social evolution of player-developed folding algorithms that took place in the year following the introduction of these tools. Players developed over 5,400 different recipes, both by creating new algorithms and by modifying and recombining successful recipes developed by other players. The most successful recipes rapidly spread through the Foldit player population, and two of the recipes became particularly dominant. Examination of the algorithms encoded in these two recipes revealed a striking similarity to an unpublished algorithm developed by scientists over the same period. Benchmark calculations show that the new algorithm independently discovered by scientists and by Foldit players outperforms previously published methods. Thus, online scientific game frameworks have the potential not only to solve hard scientific problems, but also to discover and formalize effective new strategies and algorithms.
Journal Article
Probabilistic estimation of model parameters through grid search approaches: applications to geomagnetic anomaly source estimations
Model parameters, extracted from observed data that inherently contain uncertainties, necessitate estimation as probability distributions. In geophysical problem-solving, especially when dealing with a few model parameters, the conventional approach employing a grid search is widely used to determine model parameters that explain observed data. However, the metrics of the results derived from the grid search approach are predominantly based on residuals between observed data and the model’s anticipated response, such as the root mean square misfit, which lacks representation as a probability distribution. This study introduces a straightforward technique to transform the distributions of root mean square misfits acquired via grid search into probability distributions, facilitating a statistical evaluation grounded in a Bayesian framework. The outcomes of this methodology are effectively visualized through marginal probability distributions. Employing this method, we investigated synthetic geomagnetic anomaly datasets to evaluate the location and magnitude of magnetic moments of the source. The synthetic tests showed that the method is applicable not only for well-posed problems, but also for ill-posed problems, which are challenging to evaluate solely using root mean square misfits. Subsequently, we applied this methodology to real geomagnetic anomaly data reflecting temporal magnetic fluctuations induced by volcanic activity within the Nishinoshima volcano. The method’s versatility allows its broad application across various geophysical problems, including identification of earthquake epicenters, analysis of gravity anomalies and surface geodetic deformation, and their concurrent analyses. Furthermore, this approach easily utilizes prior grid search outcomes to evaluate the probability of model parameters.
Graphical Abstract
Journal Article
Comparative value of a simulation by gaming and a traditional teaching method to improve clinical reasoning skills necessary to detect patient deterioration: a randomized study in nursing students
Background
Early detection and response to patient deterioration influence patient prognosis. Nursing education is therefore essential. The objective of this randomized controlled trial was to compare the respective educational value of simulation by gaming (SG) and a traditional teaching (TT) method to improve clinical reasoning (CR) skills necessary to detect patient deterioration.
Methods
In a prospective multicenter study, and after consent, 2nd year nursing students were randomized into two groups:
Simulation by gaming “SG”: the student played individually with a serious game consisting of 2 cases followed by a common debriefing with an instructor;
Traditional Teaching “TT”: the student worked on the same cases in text paper format followed by a traditional teaching course with a PowerPoint presentation by an instructor.
CR skill was measured by script concordance tests (80 SCTs, score 0–100) immediately after the session (primary outcome) and on month later. Other outcomes included students’ satisfaction, motivation and professional impact.
Results
One hundred forty-six students were randomized. Immediately after training, the SCTs scores were 59 ± 9 in SG group (
n
= 73) and 58 ± 8 in TT group (
n
= 73) (
p
= 0.43). One month later, the SCTs scores were 59 ± 10 in SG group (
n
= 65) and 58 ± 8 in TT group (
n
= 54) (
p
= 0.77). Global satisfaction and motivation were highly valued in both groups although significantly greater in the SG group (
p
< 0.05). The students declared that the training course would have a positive professional impact, with no difference between groups.
Conclusions
In this study assessing nursing student CR to detect patient deterioration, no significant educational difference (SCT), neither immediate nor 1 month later, was observed between training by SG and the TT course. However, satisfaction and motivation were found to be greater with the use of SG.
Trial registration
ClinicalTrials.gov;
NCT03428269
. Registered 30 january 2018.
Journal Article
Modeling human intuitions about liquid flow with particle-based simulation
Humans can easily describe, imagine, and, crucially, predict a wide variety of behaviors of liquids-splashing, squirting, gushing, sloshing, soaking, dripping, draining, trickling, pooling, and pouring-despite tremendous variability in their material and dynamical properties. Here we propose and test a computational model of how people perceive and predict these liquid dynamics, based on coarse approximate simulations of fluids as collections of interacting particles. Our model is analogous to a \"game engine in the head\", drawing on techniques for interactive simulations (as in video games) that optimize for efficiency and natural appearance rather than physical accuracy. In two behavioral experiments, we found that the model accurately captured people's predictions about how liquids flow among complex solid obstacles, and was significantly better than several alternatives based on simple heuristics and deep neural networks. Our model was also able to explain how people's predictions varied as a function of the liquids' properties (e.g., viscosity and stickiness). Together, the model and empirical results extend the recent proposal that human physical scene understanding for the dynamics of rigid, solid objects can be supported by approximate probabilistic simulation, to the more complex and unexplored domain of fluid dynamics.
Journal Article