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class="profile--tab_heading_container">Papers by Stephan Mertens</h3></div><div class="js-work-strip profile--work_container" data-work-id="110736397"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736397/A_physicists_approach_to_number_partitioning"><img alt="Research paper thumbnail of A physicist's approach to number partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461455/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736397/A_physicists_approach_to_number_partitioning">A physicist's approach to number partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 15, 2000</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The statistical physics approach to the number partioning problem, a classical NPhard problem, is...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The statistical physics approach to the number partioning problem, a classical NPhard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the "easy-to-solve" from the "hard-to-solve" phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size N to some extent corresponds to locating the minimum in an unsorted list of O(2 N) numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="33c4de5ea1d16434d29faca364603709" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461455,"asset_id":110736397,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461455/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736397"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736397"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736397; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736396"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736396/The_Easiest_Hard_Problem_Number_Partitioning"><img alt="Research paper thumbnail of The Easiest Hard Problem: Number Partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461454/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736396/The_Easiest_Hard_Problem_Number_Partitioning">The Easiest Hard Problem: Number Partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Oct 14, 2003</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It ha...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like k-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736395"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736395/Proof_of_the_local_REM_conjecture_for_number_partitioning_I_Constant_energy_scales"><img alt="Research paper thumbnail of Proof of the local REM conjecture for number partitioning. I: Constant energy scales" class="work-thumbnail" src="https://attachments.academia-assets.com/108461450/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736395/Proof_of_the_local_REM_conjecture_for_number_partitioning_I_Constant_energy_scales">Proof of the local REM conjecture for number partitioning. I: Constant energy scales</a></div><div class="wp-workCard_item"><span>Random Structures and Algorithms</span><span>, Dec 15, 2008</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The number partitioning problem is a classic problem of combinatorial optimization in which a set...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The number partitioning problem is a classic problem of combinatorial optimization in which a set of n numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the n numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies-corresponding to the costs of the partitions, and overlaps-corresponding to the correlations between partitions. Although the energy levels of this model are a priori highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0e047f9546767b1ba0e50d4bd0bce8d6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461450,"asset_id":110736395,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461450/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736395"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736395"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736395; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110736395]").text(description); $(".js-view-count[data-work-id=110736395]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 110736395; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110736395']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0e047f9546767b1ba0e50d4bd0bce8d6" } } $('.js-work-strip[data-work-id=110736395]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110736395,"title":"Proof of the local REM conjecture for number partitioning. 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We show that limn→∞ a...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Consider "lagged" Fibonacci sequences a(n) = a(n − 1) + a(⌊n/k⌋) for k > 1. We show that limn→∞ a(kn)/a(n) • ln n/n = k ln k and we demonstrate the slow numerical convergence to this limit and how to deal with this slow convergence. We also discuss the connection between two classical results of N.G. de Bruijn and K. Mahler on the asymptotics of a(n).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a336b0a859d0da655649d4bbcfb35075" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461445,"asset_id":110736394,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461445/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736394"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736394"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736394; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736393"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736393/Proof_of_the_local_REM_conjecture_for_number_partitioning"><img alt="Research paper thumbnail of Proof of the local REM conjecture for number partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461449/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736393/Proof_of_the_local_REM_conjecture_for_number_partitioning">Proof of the local REM conjecture for number partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jan 31, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The number partitioning problem is a classic problem of combinatorial optimization in which a set...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The number partitioning problem is a classic problem of combinatorial optimization in which a set of $n$ numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the $n$ numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies -- corresponding to the costs of the partitions, and overlaps -- corresponding to the correlations between partitions. Although the energy levels of this model are {\em a priori} highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5f0d757d175411894c932780ea06a37a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461449,"asset_id":110736393,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461449/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736393"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736393"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736393; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736392"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736392/Random_Number_Generators_A_Survival_Guide_for_Large_Scale_Simulations"><img alt="Research paper thumbnail of Random Number Generators: A Survival Guide for Large Scale Simulations" class="work-thumbnail" src="https://attachments.academia-assets.com/108461448/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736392/Random_Number_Generators_A_Survival_Guide_for_Large_Scale_Simulations">Random Number Generators: A Survival Guide for Large Scale Simulations</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, May 26, 2009</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Monte Carlo simulations are an important tool in statistical physics, complex systems science, an...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Monte Carlo simulations are an important tool in statistical physics, complex systems science, and many other fields. An increasing number of these simulations is run on parallel systems ranging from multicore desktop computers to supercomputers with thousands of CPUs. This raises the issue of generating large amounts of random numbers in a parallel application. In this lecture we will learn just enough of the theory of pseudo random number generation to make wise decisions on how to choose and how to use random number generators when it comes to large scale, parallel simulations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2c518b98f9572f4ec6e79cb333e15241" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461448,"asset_id":110736392,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461448/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736392"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736392"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736392; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736391"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736391/A_complete_anytime_algorithm_for_balanced_number_partitioning"><img alt="Research paper thumbnail of A complete anytime algorithm for balanced number partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461444/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736391/A_complete_anytime_algorithm_for_balanced_number_partitioning">A complete anytime algorithm for balanced number partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Mar 11, 1999</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so th...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest differencing method (BLDM) and Korf's complete Karmarkar-Karp algorithm to get a new algorithm that optimally solves the balanced partitioning problem. For numbers with twelve significant digits or less, the algorithm can optimally solve balanced partioning problems of arbitrary size in practice. 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We demonstrate this effect for the Edwards-Anderson model, but we also discuss its universality.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="48a69bf8d5f1757ea7acdf17b1b7185f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461443,"asset_id":110736390,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461443/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736390"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736390"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736390; 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We derive an explicit formula for pn and compute exact values of pn for n ≤ 12.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2ea6a0a526866c819f2a2d5288bd81f0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461432,"asset_id":110736374,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461432/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736374"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736374"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736374; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736373"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736373/Pseudo_Random_Coins_Show_More_Heads_Than_Tails"><img alt="Research paper thumbnail of Pseudo Random Coins Show More Heads Than Tails" class="work-thumbnail" src="https://attachments.academia-assets.com/108461429/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736373/Pseudo_Random_Coins_Show_More_Heads_Than_Tails">Pseudo Random Coins Show More Heads Than Tails</a></div><div class="wp-workCard_item"><span>Journal of Statistical Physics</span><span>, Feb 1, 2004</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced by a pseudo random number generator. It can be shown analytically and by exact enumerations that popular random number generators are not capable of imitating a fair coin: pseudo random coins show more "heads" than "tails". This bias explains the empirically observed failure of some random number generators in random walk experiments. It can be traced down to the special role of the value zero in the algebra of finite fields.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="261f745958c075f4d714453f9ea8215f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461429,"asset_id":110736373,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461429/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736373"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736373"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736373; 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dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "261f745958c075f4d714453f9ea8215f" } } $('.js-work-strip[data-work-id=110736373]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110736373,"title":"Pseudo Random Coins Show More Heads Than Tails","internal_url":"https://www.academia.edu/110736373/Pseudo_Random_Coins_Show_More_Heads_Than_Tails","owner_id":160629230,"coauthors_can_edit":true,"owner":{"id":160629230,"first_name":"Stephan","middle_initials":null,"last_name":"Mertens","page_name":"StephanMertens","domain_name":"sfb-trr-62","created_at":"2020-06-09T09:47:12.388-07:00","display_name":"Stephan Mertens","url":"https://sfb-trr-62.academia.edu/StephanMertens"},"attachments":[{"id":108461429,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/108461429/thumbnails/1.jpg","file_name":"0307138.pdf","download_url":"https://www.academia.edu/attachments/108461429/download_file","bulk_download_file_name":"Pseudo_Random_Coins_Show_More_Heads_Than.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/108461429/0307138-libre.pdf?1701892696=\u0026response-content-disposition=attachment%3B+filename%3DPseudo_Random_Coins_Show_More_Heads_Than.pdf\u0026Expires=1739831985\u0026Signature=N~ORvo0lve4c19QS2p23UKUMzZ2iQCvSKKCQ2qVzW4HNYuycKGoiYSyKW6kRB9MYKoQ3Ya3NE-zX6R4upZ7nvf7SX4bcgMhF-k6Mst2PL8FtO1XY5~mLN1YqHjz61NEOlT7kpGmwK~1u36QvXV7gimoOvmnCAW2gTtVDAiW1n3M8TkQSnPn6WGRzroguRPiUFqzYrKl-ivOxyPiMlhigBQK7rxYq4WIXvNOp8ge4EfbvRhzz2uhftcKVjvc5k8UiRHi2tp3iB3lKIRGRwssWx2P58gAE~bWiP52yUXGZSxk5z9C5v1Kk0o~UL~fpOpHPLDlp4u9EApVmQRhV0v-30A__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":108461431,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/108461431/thumbnails/1.jpg","file_name":"0307138.pdf","download_url":"https://www.academia.edu/attachments/108461431/download_file","bulk_download_file_name":"Pseudo_Random_Coins_Show_More_Heads_Than.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/108461431/0307138-libre.pdf?1701892694=\u0026response-content-disposition=attachment%3B+filename%3DPseudo_Random_Coins_Show_More_Heads_Than.pdf\u0026Expires=1739831985\u0026Signature=UkKYX4yqsA0PXLCPKAEXuyD4E2X7wRnXttz6eRrDcdzCfz6UmjxkIb0KblxpXsgnIYGSwV2U2YjDbkbhZUVJVCPw3cCjXjVNdGLN6x0wUGhz9dVTD9zqF03qK7UmSvC0Bq1yiD9SJdIBq3xNyUYW4jFk-n1c5D8g1OthSxf~9T~57zk9LBH2KRU3XAlZHOaRWy3LsUgfLVEMa7OWWoNebdGyw86U6uznBS1TzbRxVYBsyzHpZRGPuO7-py~qIYToQqxHn687Bhmvtl5zqWfop3YX2l7FZW0kik6lD97dzrrt1U649vQp6s3zhV0csJ7ga4ER4ZvNhHue3kq0foktVQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> </div><div class="profile--tab_content_container js-tab-pane tab-pane" data-section-id="12060089" id="papers"><div class="js-work-strip profile--work_container" data-work-id="110736397"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736397/A_physicists_approach_to_number_partitioning"><img alt="Research paper thumbnail of A physicist's approach to number partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461455/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736397/A_physicists_approach_to_number_partitioning">A physicist's approach to number partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Sep 15, 2000</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The statistical physics approach to the number partioning problem, a classical NPhard problem, is...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The statistical physics approach to the number partioning problem, a classical NPhard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the "easy-to-solve" from the "hard-to-solve" phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size N to some extent corresponds to locating the minimum in an unsorted list of O(2 N) numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="33c4de5ea1d16434d29faca364603709" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461455,"asset_id":110736397,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461455/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736397"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736397"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736397; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736396"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736396/The_Easiest_Hard_Problem_Number_Partitioning"><img alt="Research paper thumbnail of The Easiest Hard Problem: Number Partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461454/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736396/The_Easiest_Hard_Problem_Number_Partitioning">The Easiest Hard Problem: Number Partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Oct 14, 2003</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It ha...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like k-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9a85a2f8b3a8d44e65c63447c7d948d6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461454,"asset_id":110736396,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461454/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736396"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736396"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736396; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736395"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736395/Proof_of_the_local_REM_conjecture_for_number_partitioning_I_Constant_energy_scales"><img alt="Research paper thumbnail of Proof of the local REM conjecture for number partitioning. I: Constant energy scales" class="work-thumbnail" src="https://attachments.academia-assets.com/108461450/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736395/Proof_of_the_local_REM_conjecture_for_number_partitioning_I_Constant_energy_scales">Proof of the local REM conjecture for number partitioning. I: Constant energy scales</a></div><div class="wp-workCard_item"><span>Random Structures and Algorithms</span><span>, Dec 15, 2008</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The number partitioning problem is a classic problem of combinatorial optimization in which a set...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The number partitioning problem is a classic problem of combinatorial optimization in which a set of n numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the n numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies-corresponding to the costs of the partitions, and overlaps-corresponding to the correlations between partitions. Although the energy levels of this model are a priori highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="0e047f9546767b1ba0e50d4bd0bce8d6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461450,"asset_id":110736395,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461450/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736395"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736395"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736395; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110736395]").text(description); $(".js-view-count[data-work-id=110736395]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 110736395; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110736395']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "0e047f9546767b1ba0e50d4bd0bce8d6" } } $('.js-work-strip[data-work-id=110736395]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110736395,"title":"Proof of the local REM conjecture for number partitioning. 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We show that limn→∞ a...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Consider "lagged" Fibonacci sequences a(n) = a(n − 1) + a(⌊n/k⌋) for k > 1. We show that limn→∞ a(kn)/a(n) • ln n/n = k ln k and we demonstrate the slow numerical convergence to this limit and how to deal with this slow convergence. We also discuss the connection between two classical results of N.G. de Bruijn and K. Mahler on the asymptotics of a(n).</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a336b0a859d0da655649d4bbcfb35075" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461445,"asset_id":110736394,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461445/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736394"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736394"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736394; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736393"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736393/Proof_of_the_local_REM_conjecture_for_number_partitioning"><img alt="Research paper thumbnail of Proof of the local REM conjecture for number partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461449/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736393/Proof_of_the_local_REM_conjecture_for_number_partitioning">Proof of the local REM conjecture for number partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Jan 31, 2005</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The number partitioning problem is a classic problem of combinatorial optimization in which a set...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The number partitioning problem is a classic problem of combinatorial optimization in which a set of $n$ numbers is partitioned into two subsets such that the sum of the numbers in one subset is as close as possible to the sum of the numbers in the other set. When the $n$ numbers are i.i.d. variables drawn from some distribution, the partitioning problem turns out to be equivalent to a mean-field antiferromagnetic Ising spin glass. In the spin glass representation, it is natural to define energies -- corresponding to the costs of the partitions, and overlaps -- corresponding to the correlations between partitions. Although the energy levels of this model are {\em a priori} highly correlated, a surprising recent conjecture asserts that the energy spectrum of number partitioning is locally that of a random energy model (REM): the spacings between nearby energy levels are uncorrelated. In other words, the properly scaled energies converge to a Poisson process. The conjecture also asserts that the corresponding spin configurations are uncorrelated, indicating vanishing overlaps in the spin glass representation. In this paper, we prove these two claims, collectively known as the local REM conjecture.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="5f0d757d175411894c932780ea06a37a" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461449,"asset_id":110736393,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461449/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736393"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736393"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736393; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736392"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736392/Random_Number_Generators_A_Survival_Guide_for_Large_Scale_Simulations"><img alt="Research paper thumbnail of Random Number Generators: A Survival Guide for Large Scale Simulations" class="work-thumbnail" src="https://attachments.academia-assets.com/108461448/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736392/Random_Number_Generators_A_Survival_Guide_for_Large_Scale_Simulations">Random Number Generators: A Survival Guide for Large Scale Simulations</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, May 26, 2009</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Monte Carlo simulations are an important tool in statistical physics, complex systems science, an...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Monte Carlo simulations are an important tool in statistical physics, complex systems science, and many other fields. An increasing number of these simulations is run on parallel systems ranging from multicore desktop computers to supercomputers with thousands of CPUs. This raises the issue of generating large amounts of random numbers in a parallel application. In this lecture we will learn just enough of the theory of pseudo random number generation to make wise decisions on how to choose and how to use random number generators when it comes to large scale, parallel simulations.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2c518b98f9572f4ec6e79cb333e15241" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461448,"asset_id":110736392,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461448/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736392"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736392"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736392; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736391"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736391/A_complete_anytime_algorithm_for_balanced_number_partitioning"><img alt="Research paper thumbnail of A complete anytime algorithm for balanced number partitioning" class="work-thumbnail" src="https://attachments.academia-assets.com/108461444/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736391/A_complete_anytime_algorithm_for_balanced_number_partitioning">A complete anytime algorithm for balanced number partitioning</a></div><div class="wp-workCard_item"><span>arXiv (Cornell University)</span><span>, Mar 11, 1999</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so th...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest differencing method (BLDM) and Korf's complete Karmarkar-Karp algorithm to get a new algorithm that optimally solves the balanced partitioning problem. For numbers with twelve significant digits or less, the algorithm can optimally solve balanced partioning problems of arbitrary size in practice. 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We demonstrate this effect for the Edwards-Anderson model, but we also discuss its universality.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="48a69bf8d5f1757ea7acdf17b1b7185f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461443,"asset_id":110736390,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461443/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736390"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736390"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736390; 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We derive an explicit formula for pn and compute exact values of pn for n ≤ 12.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="2ea6a0a526866c819f2a2d5288bd81f0" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461432,"asset_id":110736374,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461432/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736374"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736374"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736374; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110736373"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/110736373/Pseudo_Random_Coins_Show_More_Heads_Than_Tails"><img alt="Research paper thumbnail of Pseudo Random Coins Show More Heads Than Tails" class="work-thumbnail" src="https://attachments.academia-assets.com/108461429/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110736373/Pseudo_Random_Coins_Show_More_Heads_Than_Tails">Pseudo Random Coins Show More Heads Than Tails</a></div><div class="wp-workCard_item"><span>Journal of Statistical Physics</span><span>, Feb 1, 2004</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">Tossing a coin is the most elementary Monte Carlo experiment. In a computer the coin is replaced by a pseudo random number generator. It can be shown analytically and by exact enumerations that popular random number generators are not capable of imitating a fair coin: pseudo random coins show more "heads" than "tails". This bias explains the empirically observed failure of some random number generators in random walk experiments. It can be traced down to the special role of the value zero in the algebra of finite fields.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="261f745958c075f4d714453f9ea8215f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108461429,"asset_id":110736373,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108461429/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="110736373"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110736373"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110736373; 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