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Relating singular values and discrepancy of weighted directed graphs

Steven Butler
<span title="">2006</span> <i title="ACM Press"> Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA &#39;06 </i> &nbsp;
We consider two such parameters for directed graphs: the singular values of the (normalized) adjacency matrix and discrepancy (a measurement of how randomly edges have been placed).  ...  We will also give a related result for discrepancy of walks when the indegree and out-degree at each vertex is equal.  ...  Acknowledgment: The author wishes to express his appreciation to Fan Chung for introducing him to this problem and for many stimulating discussions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1109557.1109680">doi:10.1145/1109557.1109680</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lldxcc7zsnaq5dz2kttnxhd34u">fatcat:lldxcc7zsnaq5dz2kttnxhd34u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809114501/http://orion.math.iastate.edu/butler/papers/06_01_discrepancy.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ee/87/ee877b7f2a89d56eddcc89d4d9ab8e4b9b5b7a9d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1109557.1109680"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Relating multiway discrepancy and singular values of nonnegative rectangular matrices

Marianna Bolla
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
The result naturally extends to the singular values of the normalized adjacency matrix of a weighted undirected or directed graph.  ...  The minimum k-way discrepancy md k (C) of a rectangular matrix C of nonnegative entries is the minimum of the maxima of the within-and between-cluster discrepancies that can be obtained by simultaneous  ...  Formerly (in 2012), in the framework of the same course, Max Del Giudice and Joan Wang applied the biclustering algorithm to directed graphs for finding migration patterns, based on sociological data.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2015.09.013">doi:10.1016/j.dam.2015.09.013</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zetc72oiprb3jhnsuq5wfjvfmu">fatcat:zetc72oiprb3jhnsuq5wfjvfmu</a> </span>
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Relating multiway discrepancy and singular values of graphs and contingency tables [article]

Marianna Bolla
<span title="2015-02-02">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The result naturally extends to the singular values of the normalized adjacency matrix of a weighted undirected or directed graph.  ...  The k-way discrepancy _k () of a rectangular array of nonnegative entries is the minimum of the maxima of the within- and between-cluster discrepancies that can be obtained by simultaneous k-clusterings  ...  US students James Drain, Cristina Mata, Matthew Willian, and in particular, Calvin Cheng whose computer processing of real-word data helped in formulating the main theorem.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1408.6443v3">arXiv:1408.6443v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2b3jqnmasrbh3pzdi2ol3voxzm">fatcat:2b3jqnmasrbh3pzdi2ol3voxzm</a> </span>
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SVD, discrepancy, and regular structure of contingency tables

Marianna Bolla
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
eigenvalues and the distances of the corresponding eigen-subspaces of the normalized modularity matrix of an edge-weighted graph (several clusters, symmetric case).  ...  table by the largest non-trivial singular value of the normalized table (one-cluster, rectangular case), and partly the result of Bolla for estimating the constant of volume-regularity by the structural  ...  case); moreover, he proves a two-sided relation between this singular value and the discrepancy.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2014.03.022">doi:10.1016/j.dam.2014.03.022</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q66gxo75cffd7egnnygb6x2q6e">fatcat:q66gxo75cffd7egnnygb6x2q6e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808162652/http://math.bme.hu/~marib/bsmeur/bollaDAM2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d7/74/d7745b889a96c19a274afef79db06b25fbabe7c5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2014.03.022"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Using discrepancy to control singular values for nonnegative matrices

Steve Butler
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
Our main result is to show that these are related in that discrepancy can be bounded by the second largest singular value and vice versa.  ...  These matrix results are then used to derive some (edge/alternating walks) discrepancy properties of edgeweighted directed graphs.  ...  Acknowledgment The author expresses his appreciation to Fan Chung for helpful discussions and guidance, and also to the anonymous referee whose comments helped improve the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.laa.2006.05.015">doi:10.1016/j.laa.2006.05.015</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/b7gw7avjmjfbbejnb6agveoed4">fatcat:b7gw7avjmjfbbejnb6agveoed4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190414214143/https://core.ac.uk/download/pdf/82316009.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b7/1b/b71bfe7c268904bb63936deea632b976ac3e4028.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.laa.2006.05.015"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Matrix and discrepancy view of generalized random and quasirandom graphs

Marianna Bolla, Ahmed Elbanna
<span title="2016-01-01">2016</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ecjasvn2cjesrhoardoygsassy" style="color: black;">Special Matrices</a> </i> &nbsp;
We also investigate relations between spectral properties, multiway discrepancies, and degree distribution of generalized random graphs.  ...  The notions are extended to rectangular arrays of nonnegative entries and to directed graphs.  ...  Sós and Gergely Kiss for useful advises on the quasirandom graph construction. Ahmed Elbanna's research was partly done under the auspices of the MTA-BME Stochastic Research Group.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/spma-2016-0004">doi:10.1515/spma-2016-0004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oxnaigfua5adddzmyhu23m4the">fatcat:oxnaigfua5adddzmyhu23m4the</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201202183601/https://www.degruyter.com/downloadpdf/journals/spma/open-issue/article-10.1515-spma-2016-0004/article-10.1515-spma-2016-0004.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/12/30/123052f0620438c934bc430a5b2b36d7f02e1ae5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/spma-2016-0004"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> degruyter.com </button> </a>

Discrepancy inequalities for directed graphs

Fan Chung, Franklin Kenter
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
We show that various isoperimetric parameters, as measured by the stationary distribution of the random walks, including the Cheeger constant and discrepancy are related to the singular values of the normalized  ...  Finally, we apply our results to construct extremal families of directed graphs with large differences between the discrepancy of the underlying graph and the skew-discrepancy. 1 fan@math.ucsd.edu; Research  ...  In this paper, we expand upon this method of using three graph theoretic matrices and show their eigen-and singular values in order to bound the notions of discrepancy described above.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2014.03.023">doi:10.1016/j.dam.2014.03.023</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3gehkylgl5h6napu2kvzc455ym">fatcat:3gehkylgl5h6napu2kvzc455ym</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170812002512/http://www.caam.rice.edu/~fhk2/DirectedDiscrepancy.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/de/df/dedf8d775fce04c13c4687ed9ea0abba37aab9c5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2014.03.023"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

SVD, discrepancy, and regular structure of contingency tables [article]

Marianna Bolla
<span title="2013-01-22">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
eigenvalues and the distances of the corresponding eigen-subspaces of the normalized modularity matrix of an edge-weighted graph (several clusters, symmetric case).  ...  table by the second largest singular value of the normalized table (one-cluster, rectangular case), and partly a former result of the author for estimating the constant of volume-regularity by the structural  ...  Acknowledgements We are indebted to the Tesco Hungary for making their data available and Tamás Kói for computer processing the data.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1301.5259v1">arXiv:1301.5259v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eoxx2qogqbdqzb252i3aqkenzi">fatcat:eoxx2qogqbdqzb252i3aqkenzi</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1301.5259/1301.5259.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1301.5259v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Regularity based spectral clustering and mapping the Fiedler-carpet [article]

Marianna Bolla, Vilas Winstein, Tao You, Frank Seidl, Fatma Abdelkhalek
<span title="2022-01-05">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Near optimal clusters are obtained with singular value decomposition and with the weighted k-means algorithm.  ...  In case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, and in case of edge-weighted graphs, a normalized Laplacian based clustering.  ...  participation of US undergraduate stu-  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2112.10637v2">arXiv:2112.10637v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/72uh32lao5hnppuko4qehbjuam">fatcat:72uh32lao5hnppuko4qehbjuam</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20211226122735/https://arxiv.org/pdf/2112.10637v1.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/49/8d/498dc2dcbc9a945f20157b26aaf98ca66da35455.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2112.10637v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Open surfaces of small volume

Valery Alexeev
<span title="2019-05-01">2019</span> <i title="Foundation Compositio Mathematica"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vikxtlicyvamphqvvfgh6k7swm" style="color: black;">Algebraic Geometry</a> </i> &nbsp;
We construct a surface with log terminal singularities and ample canonical class that has K 2 X = 1/48983 and a log canonical surface pair (X, B) with a nonempty reduced divisor B and ample K X + B that  ...  Acknowledgements The first author would like to thank Giancarlo Urzúa and José Ignacio Yáñez for helpful discussions.  ...  The second author would like to thank Sönke Rollenske and Stephen Coughlan for helpful discussions about log canonical surfaces with small volumes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14231/ag-2019-015">doi:10.14231/ag-2019-015</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7dsgk2j6ijaorhh335dpofu6tu">fatcat:7dsgk2j6ijaorhh335dpofu6tu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200215141847/http://content.algebraicgeometry.nl/2019-3/2019-3-015.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/29/81/2981ecf7890086eaca557b945de25d19b1888170.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14231/ag-2019-015"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Open surfaces of small volume [article]

Valery Alexeev, Wenfei Liu
<span title="2017-01-31">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We construct a surface with log terminal singularities and ample canonical class that has K_X^2=1/48 983 and a log canonical pair (X,B) with a nonempty reduced divisor B and ample K_X+B that has (K_X+B  ...  The blue small numbers underneath are the weights, and the red fractional numbers on top are b i , the negatives of discrepancies.  ...  The winning weights are (1, 2, 3, 5). Both pairs achieve the minimal possible value of 1 = 1 − 1 2 − 1 3 − 1 7 = 1 42 . On the other hand, one has K X + B ≡ 1 11 B and δ 1 = 1 11 by Lemma 3.4.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1612.09116v2">arXiv:1612.09116v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rj662mwa6rfebjgygczepbmixi">fatcat:rj662mwa6rfebjgygczepbmixi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200902164750/https://arxiv.org/pdf/1612.09116v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/57/fb/57fb050b9e0be6a46374fba315fa5c8b16a6081e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1612.09116v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Graph Fourier Transform: A Stable Approximation [article]

João Domingos, José M. F. Moura
<span title="2020-05-04">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also confirm our conclusions by applying SGFA on very sparse and very dense directed Erdős-Rényi graphs.  ...  The (right) eigenvectors of the shift A (graph spectral components) diagonalize A and lead to a graph Fourier basis F that provides a graph spectral representation of the graph signal.  ...  This relation for the graph in figure 2a is symmetric, hence all edges are undirected and have the same weight, with possible exceptions of boundary nodes that may have directed edges and/or different  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.05042v2">arXiv:2001.05042v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ojy6kawzvncvbiagxuyigmbg54">fatcat:ojy6kawzvncvbiagxuyigmbg54</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200826220849/https://arxiv.org/pdf/2001.05042v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4e/aa/4eaac3c7811a8a3313cc489b934a51591d52eea1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.05042v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Low-density series expansions for directed percolation: III. Some two-dimensional lattices

Iwan Jensen
<span title="2004-06-22">2004</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zwsatfxasfac5lr2ajk3hqfpaq" style="color: black;">Journal of Physics A: Mathematical and General</a> </i> &nbsp;
In addition we also study the non-physical singularities of the series.  ...  We use very efficient algorithms to calculate low-density series for bond and site percolation on the directed triangular, honeycomb, kagomé, and (4.8^2) lattices.  ...  Acknowledgments The calculations presented in this paper were in part performed on the facilities of the Australian Partnership for Advanced Computing (APAC) and the Victorian Partnership for Advanced  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/0305-4470/37/27/003">doi:10.1088/0305-4470/37/27/003</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/so6dvuifcrghpffvasofux4a5y">fatcat:so6dvuifcrghpffvasofux4a5y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20091009212444/http://www.ms.unimelb.edu.au/%7Eiwan/Publications/2004/JPA_37_6899.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b6/6c/b66cf980db0f53ae65239db9e561a5f9088f44b6.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/0305-4470/37/27/003"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> iop.org </button> </a>

Quasirandom Cayley graphs

David Conlon, Yufei Zhao
<span title="2017-03-08">2017</span> <i title="Discrete Analysis"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/apu7pie435bspdnsjakb3rps6i" style="color: black;">Discrete Analysis</a> </i> &nbsp;
We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, R\"odl, and Schacht, who treated the abelian  ...  As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.  ...  Z. would like to thank the Simons Foundation and the organisers of the Simons Symposium on Analysis of Boolean Functions for facilitating these conversations.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.19086/da.1294">doi:10.19086/da.1294</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gxwb4uizxfaxbgvzsg6wqi24q4">fatcat:gxwb4uizxfaxbgvzsg6wqi24q4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180721195536/http://discreteanalysisjournal.com/api/v1/articles/1294-quasirandom-cayley-graphs.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e9/2d/e92dc274faf5f717001aa00b47b42102ab79bdc5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.19086/da.1294"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones

Patrick Popescu-Pampu
<span title="">2011</span> <i title="Duke University Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qiuuwifskbf4foqk2k7hj4ilce" style="color: black;">Duke mathematical journal</a> </i> &nbsp;
Laufer proved in 1977 that, given a numerically Gorenstein topological type of singularity, every analytical realization of it is Gorenstein if and only if one has either a Kleinian or a minimally elliptic  ...  of the starting singularity.  ...  I am extremely grateful to him for his kind encouragements and for his remarks on a previous version of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/00127094-1433412">doi:10.1215/00127094-1433412</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iqwvuq2x6fel7hwjjphzgvemkm">fatcat:iqwvuq2x6fel7hwjjphzgvemkm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20160705141313/http://math.univ-lille1.fr:80/~popescu/2011-Duke.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3c/48/3c48e2bc1647e0b2a53ef69553083fedc2c8debf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1215/00127094-1433412"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>
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