搜索结果：Journal of computational and applied mathematics

Previous article Next article The NP-Completeness of Edge-ColoringIan HolyerIan Holyerhttps://doi.org/10.1137/0210055PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.[1] S. Fiorini and , Robin James Wilson, Edge-colourings of graphs, Pitman, London, 1977iii+154 58:27599 0421.05023 Google Scholar[2] Michael R. Garey and , David S. Johnson, Computers and intractability, W. H. Freeman and Co., San Francisco, Calif., 1979x+338 80g:68056 0411.68039 Google Scholar[3] Rufus Isaacs, Infinite families of nontrivial trivalent graphs which are not Tait colorable, Amer. Math. Monthly, 82 (1975), 221–239 52:2940 0311.05109 CrossrefISIGoogle Scholar[4] R. Isaacs, Loupekine's snarks: A bifamily of non-Tait-colorable graphs, unpublished Google ScholarKeywordscomputational complexityNP-complete problemschromatic indexedge-coloring Previous article Next article FiguresRelatedReferencesCited ByDetails Regular pattern-free coloringDiscrete Applied Mathematics, Vol. 321 | 1 Nov 2022 Cross Ref Chromatic index of dense quasirandom graphsJournal of Combinatorial Theory, Series B, Vol. 157 | 1 Nov 2022 Cross Ref Subcubic planar graphs of girth 7 are class IDiscrete Mathematics, Vol. 345, No. 10 | 1 Oct 2022 Cross Ref Independence number of edge‐chromatic critical graphsJournal of Graph Theory, Vol. 101, No. 2 | 20 March 2022 Cross Ref Complexity Dichotomy for List-5-Coloring with a Forbidden Induced SubgraphSepehr Hajebi, Yanjia Li, and Sophie SpirklSIAM Journal on Discrete Mathematics, Vol. 36, No. 3 | 30 August 2022AbstractPDF (592 KB)The Overfullness of Graphs with Small Minimum Degree and Large Maximum DegreeYan Cao, Guantao Chen, Guangming Jing, and Songling ShanSIAM Journal on Discrete Mathematics, Vol. 36, No. 3 | 13 September 2022AbstractPDF (395 KB)Vertex-Distinguishing Edge Colorings Of Some Complete Multipartite Graphs“Katchar” Collection of Scientific Articles International Scientific-Educational Center NAS RA | 22 September 2022 Cross Ref Orientation‐based edge‐colorings and linear arboricity of multigraphsJournal of Graph Theory, Vol. 11 | 20 September 2022 Cross Ref Edge coloring graphs with large minimum degreeJournal of Graph Theory, Vol. 244 | 18 September 2022 Cross Ref Switching 3-edge-colorings of cubic graphsDiscrete Mathematics, Vol. 345, No. 9 | 1 Sep 2022 Cross Ref Overfullness of critical class 2 graphs with a small core degreeJournal of Combinatorial Theory, Series B, Vol. 156 | 1 Sep 2022 Cross Ref Using the minimum maximum flow degree to approximate the flow coloring problemAnnals of Operations Research, Vol. 316, No. 2 | 30 June 2021 Cross Ref Colouring generalized claw-free graphs and graphs of large girth: Bounding the diameterTheoretical Computer Science, Vol. 931 | 1 Sep 2022 Cross Ref An improvement to the Hilton-Zhao vertex-splitting conjectureDiscrete Mathematics, Vol. 345, No. 8 | 1 Aug 2022 Cross Ref Parameterized Complexity of Maximum Edge Colorable SubgraphAlgorithmica, Vol. 35 | 14 July 2022 Cross Ref On 3-Coloring of ($$2P_4,C_5$$)-Free GraphsAlgorithmica, Vol. 84, No. 6 | 15 February 2022 Cross Ref On the Nash number and the diminishing Grundy number of a graphDiscrete Applied Mathematics, Vol. 314 | 1 Jun 2022 Cross Ref Colouring graphs of bounded diameter in the absence of small cyclesDiscrete Applied Mathematics, Vol. 314 | 1 Jun 2022 Cross Ref How many matchings cover the nodes of a graph?Mathematical Programming, Vol. 81 | 31 May 2022 Cross Ref Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective PlaneCombinatorica, Vol. 12 | 19 May 2022 Cross Ref Fast and Heavy Disjoint Weighted Matchings for Demand-Aware Datacenter TopologiesIEEE INFOCOM 2022 - IEEE Conference on Computer Communications | 2 May 2022 Cross Ref Dichotomizing k -vertex-critical H -free graphs for H of order fourDiscrete Applied Mathematics, Vol. 312 | 1 May 2022 Cross Ref Approximate separable multichoice optimization over monotone systemsDiscrete Optimization, Vol. 44 | 1 May 2022 Cross Ref Weighted 2-sections and hypergraph reconstructionTheoretical Computer Science, Vol. 915 | 1 May 2022 Cross Ref Rolling weight-matching methods for the inter-satellite link assignment in global navigation satellite systemsGPS Solutions, Vol. 26, No. 2 | 19 January 2022 Cross Ref Characterization of saturated graphs related to pairs of disjoint matchingsIllinois Journal of Mathematics, Vol. 66, No. 1 | 1 Apr 2022 Cross Ref On the Maximum Number of Edges in Chordal Graphs of Bounded Degree and Matching NumberAlgorithmica, Vol. 309 | 22 March 2022 Cross Ref Colouring simplicial complexes via the Lechuga–Murillo’s modelApplicable Algebra in Engineering, Communication and Computing, Vol. 33, No. 2 | 12 June 2020 Cross Ref The smallest pair of cospectral cubic graphs with different chromatic indexesDiscrete Applied Mathematics, Vol. 309 | 1 Mar 2022 Cross Ref GAN-Based Pareto Optimization for Self-Healing of Radio Access Network SlicesIEEE Transactions on Network and Service Management, Vol. 19, No. 1 | 1 Mar 2022 Cross Ref Bounds for the Rainbow Disconnection Numbers of GraphsActa Mathematica Sinica, English Series, Vol. 38, No. 2 | 15 February 2022 Cross Ref Coloring temporal graphsJournal of Computer and System Sciences, Vol. 123 | 1 Feb 2022 Cross Ref A proof of the Erdös-Faber-Lovász conjecture: Algorithmic aspects2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) | 1 Feb 2022 Cross Ref Multi-Sender Index Coding Over Linear NetworksIEEE Communications Letters, Vol. 26, No. 2 | 1 Feb 2022 Cross Ref Open Shop Scheduling with Simultaneity ConstraintsA Book of Open Shop Scheduling | 31 October 2021 Cross Ref The Complexity of L(p, q)-Edge-LabellingWALCOM: Algorithms and Computation | 16 March 2022 Cross Ref Optimization of the Functional Decomposition of Parallel and Distributed Computations in Graph Coloring With the Use of High-Performance ComputingIEEE Access, Vol. 10 | 1 Jan 2022 Cross Ref On coloring a class of claw-free and hole-twin-free graphsDiscrete Applied Mathematics, Vol. 9 | 1 Nov 2021 Cross Ref The complexity of counting edge colorings for simple graphsTheoretical Computer Science, Vol. 889 | 1 Oct 2021 Cross Ref Complexity and algorithms for injective edge-coloring in graphsInformation Processing Letters, Vol. 170 | 1 Sep 2021 Cross Ref Your rugby mates don't need to know your colleagues: Triadic closure with edge colorsJournal of Computer and System Sciences, Vol. 120 | 1 Sep 2021 Cross Ref Decomposition of university course timetablingAnnals of Operations Research, Vol. 302, No. 2 | 25 October 2019 Cross Ref Disjointness graphs of segments in the spaceCombinatorics, Probability and Computing, Vol. 30, No. 4 | 4 November 2020 Cross Ref On Graph Coloring Analysis Through Visualization2021 International Conference on Information and Digital Technologies (IDT) | 22 Jun 2021 Cross Ref Revising Johnson’s table for the 21st centuryDiscrete Applied Mathematics, Vol. 3 | 1 Jun 2021 Cross Ref On Generalisations of the AVD Conjecture to DigraphsGraphs and Combinatorics, Vol. 37, No. 2 | 20 January 2021 Cross Ref Fly-path: Traffic-based multi-hop routing approach for hybrid wireless data centersComputer Communications, Vol. 170 | 1 Mar 2021 Cross Ref On the average degree of edge chromatic critical graphsJournal of Combinatorial Theory, Series B, Vol. 147 | 1 Mar 2021 Cross Ref Algorithmic Aspects of the Maximum 2-edge-colorable Subgraph ProblemAdvanced Information Networking and Applications | 1 May 2021 Cross Ref Colouring Graphs of Bounded Diameter in the Absence of Small CyclesAlgorithms and Complexity | 4 May 2021 Cross Ref Applications and ExtensionsGuide to Graph Colouring | 8 October 2021 Cross Ref Colorability in Line GraphsLine Graphs and Line Digraphs | 23 July 2021 Cross Ref Bears with Hats and Independence PolynomialsGraph-Theoretic Concepts in Computer Science | 20 September 2021 Cross Ref On 3-Coloring of $$(2P_4,C_5)$$-Free GraphsGraph-Theoretic Concepts in Computer Science | 20 September 2021 Cross Ref Disjoint Stable Matchings in Linear TimeGraph-Theoretic Concepts in Computer Science | 20 September 2021 Cross Ref Allocating Indivisible Items with Minimum Dissatisfaction on Preference GraphsAlgorithmic Decision Theory | 27 October 2021 Cross Ref New Steiner 2-designs from old ones by paramodificationsDiscrete Applied Mathematics, Vol. 288 | 1 Jan 2021 Cross Ref Superposition of snarks revisitedEuropean Journal of Combinatorics, Vol. 91 | 1 Jan 2021 Cross Ref The chromatic index of split-interval graphsProcedia Computer Science, Vol. 195 | 1 Jan 2021 Cross Ref Coverability of Graphs by Parity Regular SubgraphsMathematics, Vol. 9, No. 2 | 18 January 2021 Cross Ref Better 3-coloring algorithms: Excluding a triangle and a seven vertex pathTheoretical Computer Science, Vol. 850 | 1 Jan 2021 Cross Ref Evolution through segmental duplications and losses: a Super-Reconciliation approachAlgorithms for Molecular Biology, Vol. 15, No. 1 | 26 May 2020 Cross Ref Restricted extension of sparse partial edge colorings of hypercubesDiscrete Mathematics, Vol. 343, No. 11 | 1 Nov 2020 Cross Ref Grundy coloring in some subclasses of bipartite graphs and their complementsInformation Processing Letters, Vol. 163 | 1 Nov 2020 Cross Ref High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and QualitySC20: International Conference for High Performance Computing, Networking, Storage and Analysis | 1 Nov 2020 Cross Ref Complete Complexity Dichotomy for $$\boldsymbol 7 $$-Edge Forbidden Subgraphs in the Edge Coloring ProblemJournal of Applied and Industrial Mathematics, Vol. 14, No. 4 | 29 January 2021 Cross Ref Graph fractal dimension and the structure of fractal networksJournal of Complex Networks, Vol. 8, No. 4 | 18 November 2020 Cross Ref 1‐Factorizations of pseudorandom graphsRandom Structures & Algorithms, Vol. 57, No. 2 | 21 May 2020 Cross Ref Connected greedy coloring of H -free graphsDiscrete Applied Mathematics, Vol. 284 | 1 Sep 2020 Cross Ref On determining when small embeddings of partial Steiner triple systems existJournal of Combinatorial Designs, Vol. 28, No. 8 | 17 March 2020 Cross Ref b-continuity and partial Grundy coloring of graphs with large girthDiscrete Mathematics, Vol. 343, No. 8 | 1 Aug 2020 Cross Ref Colouring (Pr + Ps)-Free GraphsAlgorithmica, Vol. 82, No. 7 | 25 January 2020 Cross Ref A $$\frac{5}{2}$$-approximation algorithm for coloring rooted subtrees of a degree 3 treeJournal of Combinatorial Optimization, Vol. 40, No. 1 | 9 April 2020 Cross Ref Complexity-separating graph classes for vertex, edge and total colouringDiscrete Applied Mathematics, Vol. 281 | 1 Jul 2020 Cross Ref Edge-colouring graphs with bounded local degree sumsDiscrete Applied Mathematics, Vol. 281 | 1 Jul 2020 Cross Ref On the complexity of cd-coloring of graphsDiscrete Applied Mathematics, Vol. 280 | 1 Jun 2020 Cross Ref Co-density and fractional edge cover packingJournal of Combinatorial Optimization, Vol. 39, No. 4 | 4 February 2020 Cross Ref On the Relation of Strong Triadic Closure and Cluster DeletionAlgorithmica, Vol. 82, No. 4 | 16 August 2019 Cross Ref Group connectivity: Z4 vs Z22Journal of Graph Theory, Vol. 93, No. 3 | 28 August 2019 Cross Ref The multi-league sports scheduling problem, or how to schedule thousands of matchesOperations Research Letters, Vol. 48, No. 2 | 1 Mar 2020 Cross Ref Mixed Graph Colorings: A Historical ReviewMathematics, Vol. 8, No. 3 | 9 March 2020 Cross Ref A QoS‐aware energy‐efficient memetic flower pollination routing protocol for underwater acoustic sensor networkConcurrency and Computation: Practice and Experience, Vol. 32, No. 4 | 13 February 2019 Cross Ref Distributed Edge Coloring and a Special Case of the Constructive Lovász Local LemmaACM Transactions on Algorithms, Vol. 16, No. 1 | 11 Jan 2020 Cross Ref Obstructions for Three-Coloring and List Three-Coloring $H$-Free GraphsMaria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, and Mingxian ZhongSIAM Journal on Discrete Mathematics, Vol. 34, No. 1 | 19 February 2020AbstractPDF (669 KB)Hardness and Ease of Curing the Sign Problem for Two-Local Qubit HamiltoniansJoel Klassen, Milad Marvian, Stephen Piddock, Marios Ioannou, Itay Hen, and Barbara M. TerhalSIAM Journal on Computing, Vol. 49, No. 6 | 17 December 2020AbstractPDF (701 KB)Parameterized Complexity of Maximum Edge Colorable SubgraphComputing and Combinatorics | 27 August 2020 Cross Ref The Complexity of the Partition Coloring ProblemTheory and Applications of Models of Computation | 9 October 2020 Cross Ref Fully Dynamic Graph Algorithms Inspired by Distributed ComputingACM Journal of Experimental Algorithmics, Vol. 24 | 17 Dec 2019 Cross Ref Colouring square-free graphs without long induced pathsJournal of Computer and System Sciences, Vol. 106 | 1 Dec 2019 Cross Ref Low-cost quantum circuits for classically intractable instances of the Hamiltonian dynamics simulation problemnpj Quantum Information, Vol. 5, No. 1 | 27 May 2019 Cross Ref Tight Bounds for Online Edge Coloring2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) | 1 Nov 2019 Cross Ref Optimization design for parallel coloring of a set of graphs in the High-Performance Computing2019 IEEE 15th International Scientific Conference on Informatics | 1 Nov 2019 Cross Ref Goldberg's conjecture is true for random multigraphsJournal of Combinatorial Theory, Series B, Vol. 138 | 1 Sep 2019 Cross Ref Puzzle—More Logic Puzzle Apps Solved by Mathematical ProgrammingINFORMS Transactions on Education, Vol. 20, No. 1 | 1 Sep 2019 Cross Ref Approximately Coloring Graphs Without Long Induced PathsAlgorithmica, Vol. 81, No. 8 | 26 April 2019 Cross Ref H -colouring Pt -free graphs in subexponential timeDiscrete Applied Mathematics, Vol. 267 | 1 Aug 2019 Cross Ref The Chromatic Index of Proper Circular-arc Graphs of Odd Maximum Degree which are ChordalElectronic Notes in Theoretical Computer Science, Vol. 346 | 1 Aug 2019 Cross Ref On Coloring a Class of Claw-free GraphsElectronic Notes in Theoretical Computer Science, Vol. 346 | 1 Aug 2019 Cross Ref Weighted and locally bounded list-colorings in split graphs, cographs, and partial k-treesTheoretical Computer Science, Vol. 782 | 1 Aug 2019 Cross Ref On strong proper connection number of cubic graphsDiscrete Applied Mathematics, Vol. 265 | 1 Jul 2019 Cross Ref Perfect 1-factorizationsMathematica Slovaca, Vol. 69, No. 3 | 21 May 2019 Cross Ref Classifying k-edge colouring for H-free graphsInformation Processing Letters, Vol. 146 | 1 Jun 2019 Cross Ref An upper bound for the choice number of star edge coloring of graphsApplied Mathematics and Computation, Vol. 348 | 1 May 2019 Cross Ref Certifying coloring algorithms for graphs without long induced pathsDiscrete Applied Mathematics, Vol. 261 | 1 May 2019 Cross Ref Independent Feedback Vertex Set for $$P_5$$ P 5 -Free GraphsAlgorithmica, Vol. 81, No. 4 | 26 June 2018 Cross Ref Detecting an induced subdivision of K4Journal of Graph Theory, Vol. 90, No. 2 | 25 June 2018 Cross Ref Graphs with small fall-spectrumDiscrete Applied Mathematics, Vol. 254 | 1 Feb 2019 Cross Ref Algorithmic problems in right-angled Artin groups: Complexity and applicationsJournal of Algebra, Vol. 519 | 1 Feb 2019 Cross Ref Densities, Matchings, and Fractional Edge-ColoringsXujin Chen, Wenan Zang, and Qiulan ZhaoSIAM Journal on Optimization, Vol. 29, No. 1 | 22 January 2019AbstractPDF (504 KB)The Hilton--Zhao Conjecture is True for Graphs with Maximum Degree 4Daniel W. Cranston and Landon RabernSIAM Journal on Discrete Mathematics, Vol. 33, No. 3 | 11 July 2019AbstractPDF (351 KB)Multi-channel Assignment and Link Scheduling for Prioritized Latency-Sensitive ApplicationsAlgorithms for Sensor Systems | 5 November 2019 Cross Ref Graph Edge Coloring: A SurveyGraphs and Combinatorics, Vol. 35, No. 1 | 4 January 2019 Cross Ref List-edge-colouring planar graphs with precoloured edgesEuropean Journal of Combinatorics, Vol. 75 | 1 Jan 2019 Cross Ref Space-efficient Euler partition and bipartite edge coloringTheoretical Computer Science, Vol. 754 | 1 Jan 2019 Cross Ref Complexity of edge coloring with minimum reload/changeover costsNetworks, Vol. 38 | 7 December 2018 Cross Ref Improved Process of Running Tasks in the High Performance Computing System2018 16th International Conference on Emerging eLearning Technologies and Applications (ICETA) | 1 Nov 2018 Cross Ref Minimum multiplicity edge coloring via orientationDiscrete Applied Mathematics, Vol. 247 | 1 Oct 2018 Cross Ref 1-Factorizations of Pseudorandom Graphs2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) | 1 Oct 2018 Cross Ref New algorithms for maximum disjoint paths based on tree-likenessMathematical Programming, Vol. 171, No. 1-2 | 14 November 2017 Cross Ref Star chromatic index of subcubic multigraphsJournal of Graph Theory, Vol. 88, No. 4 | 8 December 2017 Cross Ref Three-Coloring and List Three-Coloring of Graphs Without Induced Paths on Seven VerticesCombinatorica, Vol. 38, No. 4 | 31 May 2017 Cross Ref On the chromatic index of join graphs and triangle-free graphs with large maximum degreeDiscrete Applied Mathematics, Vol. 245 | 1 Aug 2018 Cross Ref Node-Based Service-Balanced Scheduling for Provably Guaranteed Throughput and Evacuation Time PerformanceIEEE Transactions on Mobile Computing, Vol. 17, No. 8 | 1 Aug 2018 Cross Ref A theory of linear typings as flows on 3-valent graphsProceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science | 9 July 2018 Cross Ref Chromatic index determined by fractional chromatic indexJournal of Combinatorial Theory, Series B, Vol. 131 | 1 Jul 2018 Cross Ref The Backbone Packet Radio Network coloring for Time Division Multiple Access link scheduling in Wireless Multihop NetworksNetworks, Vol. 71, No. 4 | 30 August 2017 Cross Ref Trees, Paths, Stars, Caterpillars and SpidersAlgorithmica, Vol. 80, No. 6 | 25 October 2016 Cross Ref On colouring (2P2,H)-free and (P5,H)-free graphsInformation Processing Letters, Vol. 134 | 1 Jun 2018 Cross Ref On f-colorings of nearly bipartite graphsInformation Processing Letters, Vol. 134 | 1 Jun 2018 Cross Ref On the algorithmic aspects of strong subcoloringJournal of Combinatorial Optimization, Vol. 35, No. 4 | 6 March 2018 Cross Ref Optimal Joint Routing and Scheduling in Millimeter-Wave Cellular NetworksIEEE INFOCOM 2018 - IEEE Conference on Computer Communications | 1 Apr 2018 Cross Ref Underwater Acoustic Sensor Node Scheduling using an Evolutionary Memetic AlgorithmJournal of Telecommunications and Information Technology, Vol. 1, No. 2018 | 28 March 2018 Cross Ref 3-Colorable Subclasses of $P_8$-Free GraphsMaria Chudnovsky and Juraj StachoSIAM Journal on Discrete Mathematics, Vol. 32, No. 2 | 15 May 2018AbstractPDF (668 KB)On the structure of (pan, even hole)-free graphsJournal of Graph Theory, Vol. 87, No. 1 | 29 June 2017 Cross Ref On the Relation of Strong Triadic Closure and Cluster DeletionGraph-Theoretic Concepts in Computer Science | 2 September 2018 Cross Ref Reconstructing the History of Syntenies Through Super-ReconciliationComparative Genomics | 8 September 2018 Cross Ref ApproximationsalgorithmenKombinatorische Optimierung | 25 July 2018 Cross Ref A coloring algorithm for 4K1 -free line graphsDiscrete Applied Mathematics, Vol. 234 | 1 Jan 2018 Cross Ref Complexity and Algorithms for a Perfect from Mixed Transactions on and Vol. 15, No. 1 | 1 Jan 2018 Cross Ref Optimization | 14 March 2018 Cross Ref Conjecture Large Maximum of Graph Theory, Vol. No. 4 | 2 March 2017 Cross Ref On the Complexity of Computing of Combinatorics, Vol. No. 4 | 7 August 2017 Cross Ref a Graph Using Mobile Vol. No. 3 | 18 October 2016 Cross Ref The of of graphs with large maximum Mathematics and Computation, Vol. | 1 Nov 2017 Cross Ref graphsJournal of Combinatorial Optimization, Vol. 34, No. 3 | 5 January 2017 Cross Ref On some colorings of graphsDiscrete Applied Mathematics, Vol. | 1 Oct 2017 Cross Ref algorithms in A IEEE 16th International Symposium on Network Computing and Applications | 1 Oct 2017 Cross Ref On the of the of and Combinatorics, Vol. 33, No. 5 | 12 August 2017 Cross Ref scheduling and for on Computing, Vol. 20, No. 3 | 29 November 2016 Cross Ref for paths and of Computer and System Sciences, Vol. | 1 Aug 2017 Cross Ref for the chromatic number of some Programming, Vol. No. 1-2 | 11 November 2016 Cross Ref Subcubic in Steiner triple systemsDiscrete Mathematics, Vol. No. 6 | 1 Jun 2017 Cross Ref and of graphsJournal of Combinatorial Theory, Series B, Vol. | 1 May 2017 Cross Ref A on the Complexity of Coloring Graphs with Forbidden of Graph Theory, Vol. 84, No. 4 | 2 March 2016 Cross Ref An to the via graph coloringDiscrete Mathematics, Vol. No. 4 | 1 Apr 2017 Cross Ref Algorithmic complexity of and the Computer Science, Vol. | 1 Apr 2017 Cross Ref With An Algorithmic Transactions on Networking, Vol. No. 2 | 1 Apr 2017 Cross Ref Graphs with Induced of Graph Theory, Vol. 84, No. 3 | 12 February 2016 Cross Ref Algorithms for and GraphsAlgorithmica, Vol. No. 3 | 4 January 2016 Cross Ref The complexity of for simple Journal of Combinatorics, Vol. | 1 Mar 2017 Cross Ref Euler Partition and Edge and Complexity | 14 April 2017 Cross Ref Minimum and Combinatorial Optimization | 24 May 2017 Cross Ref Approximately Coloring Graphs Without Long Induced Concepts in Computer Science | 2 November 2017 Cross Ref Complexity of coloring graphs without paths and cyclesDiscrete Applied Mathematics, Vol. | 1 Jan 2017 Cross Ref of Applied Mathematics, Vol. | 1 Jan 2017 Cross Ref and of through International Conference on Computing, Networking and Communications | 1 Jan 2017 Cross Ref graph a Letters, Vol. No. 8 | 26 December Cross Ref The complexity of problems on graphs of bounded and Computation, Vol. | 1 Dec 2016 Cross Ref The complexity of counting edge colorings and a for some in the Mathematical Sciences, Vol. No. 1 | 1 September 2016 Cross Ref The scheduling problem with Journal of Research, Vol. No. 3 | 1 Nov 2016 Cross Ref On the algorithmic complexity of Processing Letters, Vol. No. 11 | 1 Nov 2016 Cross Ref Chromatic and maximum Notes in Discrete Mathematics, Vol. | 1 Oct 2016 Cross Ref On a of the of for Mathematics, Vol. No. 9 | 1 Sep 2016 Cross Ref Research Letters, Vol. No. 4 | 1 Jul 2016 Cross Ref On the Complexity of Computing the of a of Graph Theory, Vol. 82, No. 1 | 8 June Cross Ref Optimal and of dense quasirandom graphsJournal of Combinatorial Theory, Series B, Vol. | 1 May 2016 Cross Ref by Edge Coloring for Wireless Sensor IEEE Conference | 1 May 2016 Cross Ref The edge chromatic number of graphsDiscrete Mathematics, Vol. No. 4 | 1 Apr 2016 Cross Ref scheduling for and INFOCOM 2016 - The Annual IEEE International Conference on Computer Communications | 1 Apr 2016 Cross Ref of graphsDiscrete Mathematics, Algorithms and Vol. No. | 26 February 2016 Cross Ref to cover a graphDiscrete Mathematics, Vol. No. 2 | 1 Feb 2016 Cross Ref of Theory | 20 October 2016 Cross Ref of -Free GraphsGraph-Theoretic Concepts in Computer Science | 28 September 2016 Cross Ref Algorithms and Tight for 3 of Small Diameter GraphsAlgorithmica, Vol. No. 1 | 21 October Cross Ref Improved complexity on k Pt -free Journal of Combinatorics, Vol. | 1 Jan 2016 Cross Ref complexity of Computer Science, Vol. | 1 Jan 2016 Cross Ref partition of of bipartite graph Mathematics and Vol. 26, No. 6 | 1 Jan 2016 Cross Ref Vol. 28, No. 1 | 1 Jan 2016 Cross Ref A for in Notes in Discrete Mathematics, Vol. | 1 Dec Cross Ref On the chromatic index of and join Notes in Discrete Mathematics, Vol. | 1 Dec Cross Ref Colorings with and Combinatorics, Vol. No. 6 | 10 January Cross Ref Mathematics, Vol. No. 11 | 1 Nov Cross Ref Optimal and of dense quasirandom Notes in Discrete Mathematics, Vol. | 1 Nov Cross Ref Joint in with and scheduling Vol. 91 | 1 Sep Cross Ref the and problems in of the of Sciences, Vol. No. 3 | 1 Sep Cross Ref On the maximum of by matchings in a cubic graphDiscrete Mathematics, Vol. No. 8 | 1 Aug Cross Ref Edge Colorings of the of and Combinatorics, Vol. No. 4 | 23 March Cross Ref The Chromatic Index of a Graph Maximum Degree and Combinatorics, Vol. No. 4 | 12 April Cross Ref index for networksJournal of Discrete Algorithms, Vol. | 1 Jul Cross Ref of of line graphsDiscrete Applied Mathematics, Vol. | 1 May Cross Ref How are Research Letters, Vol. No. 3 | 1 May Cross Ref coloring and in graphsDiscrete Applied Mathematics, Vol. | 1 Mar Cross Ref scheduling with Annual Conference on Information and Systems | 1 Mar Cross Ref The complexity of the Processing Letters, Vol. No. 2 | 1 Feb Cross Ref of Weighted Algorithms and Computation | 1 Jan Cross Ref Matching in Algorithms | 7 June Cross Ref Scheduling via and Edge and Wireless | 19 June Cross Ref a Perfect from Mixed in | 28 August Cross Ref a Graph Using Mobile and Computation | 27 November Cross Ref List Coloring in the Absence of a Linear Vol. 71, No. 1 | 6 April Cross Ref On of bipartite and Vol. No. 1 | 4 February Cross Ref The of for International Conference on with Applications in and | 1 Dec Cross Ref in Radio NetworksIEEE Transactions on Mobile Computing, Vol. No. 11 | 1 Nov Cross

Previous article Next article On the Construction and Comparison of Difference SchemesGilbert StrangGilbert Stranghttps://doi.org/10.1137/0705041PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. Barkley Rosser, A Runge-Kutta for all seasons, SIAM Rev., 9 (1967), 417–452 10.1137/1009069 MR0219242 0243.65041 LinkISIGoogle Scholar[2] Robert D. Richtmyer and , K. W. Morton, Difference methods for initial-value problems, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967xiv+405 MR0220455 0155.47502 Google Scholar[3] Peter D. Lax and , Burton Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math., 17 (1964), 381–398 MR0170484 0233.65050 CrossrefISIGoogle Scholar[4] Samuel Z. Burstein, Numerical methods in multidimensional shocked flows, AIAA J., 2 (1964), 2111–2117 MR0180067 CrossrefGoogle Scholar[5] S. Z. Burstein, Finite-difference calculations for hydrodynamic flows containing discontinuities, J. Comp. Phys., 2 (1967), 198–222 0166.22103 Google Scholar[6] C. Berger, On the numerical range of powers of an operator, to appear Google Scholar[7] W. P. Crowley, Second-order numerical advection, J. Comp. Phys., 1 (1967), 471–484 10.1016/0021-9991(67)90053-8 0149.44403 CrossrefGoogle Scholar[8] Gilbert Strang, Accurate partial difference methods. I. Linear Cauchy problems, Arch. Rational Mech. Anal., 12 (1963), 392–402 MR0146970 0113.32303 CrossrefISIGoogle Scholar[9] A. R. Gourlay and , J. Ll. Morris, A multistep formulation of the optimized Lax-Wendroff method for nonlinear hyperbolic systems in two space variables, Math. Comp., 22 (1968), 715–719 MR0251931 0172.19904 CrossrefISIGoogle Scholar[10] Peter Lax and , Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), 217–237 MR0120774 0152.44802 CrossrefISIGoogle Scholar[11] R. D. Richtmyer, A survey of difference methods for nonsteady fluid dynamics, Tech. Note, 63-2, National Center for Atmospheric Research, Boulder, Colorado, 1962 Google Scholar[12] Gilbert Strang, Accurate partial difference methods. II. Non-linear problems, Numer. Math., 6 (1964), 37–46 10.1007/BF01386051 MR0166942 0143.38204 CrossrefGoogle Scholar[13] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons Inc., New York, 1962xi+407 MR0135729 0112.34901 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Semiexplicit symplectic integrators for non-separable Hamiltonian systems23 September 2022 | Mathematics of Computation, Vol. 92, No. 339 Cross Ref A combined volume penalization / selective frequency damping approach for immersed boundary methods applied to high-order schemesJournal of Computational Physics, Vol. 472 Cross Ref Numerical computation for rogue waves in the coupled nonlinear Schrödinger equations with the coherent coupling effect29 April 2022 | International Journal of Computer Mathematics, Vol. 99, No. 12 Cross Ref A second-order maximum bound principle preserving operator splitting method for the Allen–Cahn equation with applications in multi-phase systemsMathematics and Computers in Simulation, Vol. 202 Cross Ref Ultrafast laser matter interactions: modeling approaches, challenges, and prospects27 September 2022 | Modelling and Simulation in Materials Science and Engineering, Vol. 30, No. 8 Cross Ref An efficient IMEX-DG solver for the compressible Navier-Stokes equations for non-ideal gasesJournal of Computational Physics, Vol. 471 Cross Ref Implications of CO 2 mass transport dynamics for large-scale CCS in basalt formationsInternational Journal of Greenhouse Gas Control, Vol. 121 Cross Ref A charge-preserving compact splitting method for solving the coupled stochastic nonlinear Schrödinger equationsApplied Numerical Mathematics, Vol. 181 Cross Ref Local effects in vortex-flame interactions: Implications for turbulent premixed flame scaling and observablesCombustion and Flame, Vol. 245 Cross Ref Hamiltonian preserving nonlinear opticsPhysica D: Nonlinear Phenomena, Vol. 439 Cross Ref A fully coupled, fully implicit simulation method for unsteady flames using Jacobian approximation and clusteringCombustion and Flame, Vol. 245 Cross Ref Reactive Transport Modeling with a Coupled OpenFOAM®-PHREEQC Platform7 October 2022 | Transport in Porous Media, Vol. 145, No. 2 Cross Ref Multirate Exponential Rosenbrock MethodsVu Thai Luan, Rujeko Chinomona, and Daniel R. Reynolds13 October 2022 | SIAM Journal on Scientific Computing, Vol. 44, No. 5AbstractPDF (764 KB)Assessment of a new sub-grid model for magnetohydrodynamical turbulence. I. Magnetorotational instability10 October 2022 | Monthly Notices of the Royal Astronomical Society, Vol. 517, No. 3 Cross Ref Consistent submodel coupling in hybrid particle/finite volume algorithms for zone-adaptive modelling of turbulent reactive flows12 October 2022 | Combustion Theory and Modelling, Vol. 1 Cross Ref Real-time propagator eigenstates2 September 2022 | Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 55, No. 19 Cross Ref Operator splitting scheme based on barycentric Lagrange interpolation collocation method for the Allen-Cahn equation19 November 2021 | Journal of Applied Mathematics and Computing, Vol. 68, No. 5 Cross Ref Dynamics of Electron Pearcey– Gaussian Beams in a Constant Magnetic Field5 August 2022 | Annalen der Physik, Vol. 534, No. 10 Cross Ref Synergistic interactions of thermodiffusive instabilities and turbulence in lean hydrogen flamesCombustion and Flame, Vol. 244 Cross Ref General time-dependent configuration-interaction singles. II. Atomic case10 October 2022 | Physical Review A, Vol. 106, No. 4 Cross Ref A high order accurate numerical algorithm for the space-fractional Swift-Hohenberg equationComputers & Mathematics with Applications, Vol. 15 Cross Ref A self-adaptive theta scheme using discontinuity aware quadrature for solving conservation laws19 October 2021 | IMA Journal of Numerical Analysis, Vol. 42, No. 4 Cross Ref Isolating gasdynamic and chemical effects on the detonation cellular structure: A combined experimental and computational studyProceedings of the Combustion Institute, Vol. 9 Cross Ref Improved Uniform Error Bounds on Time-Splitting Methods for the Long-Time Dynamics of the Dirac Equation with Small Potentials Weizhu Bao , Yue Feng , and Jia Yin 27 September 2022 | Multiscale Modeling & Simulation, Vol. 20, No. 3AbstractPDF (16068 KB)Improving the Performance of Reactive Transport Simulations Using Artificial Neural Networks27 September 2022 | Transport in Porous Media, Vol. 513 Cross Ref Signatures of coherent vibronic exciton dynamics and conformational control in the two-dimensional electronic spectroscopy of conjugated polymers1 January 2022 | Faraday Discussions, Vol. 237 Cross Ref Robust estimation of charge carrier diffusivity using transient photoluminescence microscopyThe Journal of Chemical Physics, Vol. 157, No. 10 Cross Ref Electron Symmetric Pearcey Gaussian Vortex Beams5 August 2022 | Annalen der Physik, Vol. 534, No. 9 Cross Ref An enhanced semi-explicit particle finite element method for incompressible flows30 May 2022 | Computational Mechanics, Vol. 70, No. 3 Cross Ref A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo modelApplied Numerical Mathematics, Vol. 179 Cross Ref Far-field sonic boom prediction considering atmospheric turbulence effects: An improved approachChinese Journal of Aeronautics, Vol. 35, No. 9 Cross Ref An Improved Method for Coupling Hydrodynamics with Astrophysical Reaction Networks25 August 2022 | The Astrophysical Journal, Vol. 936, No. 1 Cross Ref High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems14 June 2022 | Calcolo, Vol. 59, No. 3 Cross Ref Effects of wedge-angle change on the evolution of oblique detonation wave structurePhysics of Fluids, Vol. 34, No. 9 Cross Ref HPC-enabling technologies for high-fidelity combustion simulationsProceedings of the Combustion Institute, Vol. 33 Cross Ref A Second-Order Accurate, Operator Splitting Scheme for Reaction-Diffusion Systems in an Energetic Variational FormulationChun Liu, Cheng Wang, and Yiwei Wang2 August 2022 | SIAM Journal on Scientific Computing, Vol. 44, No. 4AbstractPDF (1148 KB)Improved Uniform Error Bounds on Time-Splitting Methods for Long-Time Dynamics of the Nonlinear Klein--Gordon Equation with Weak NonlinearityWeizhu Bao, Yongyong Cai, and Yue Feng1 August 2022 | SIAM Journal on Numerical Analysis, Vol. 60, No. 4AbstractPDF (41582 KB)Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics11 August 2022 | BIT Numerical Mathematics, Vol. 48 Cross Ref An effective operator splitting scheme for two-dimensional conservative nonlocal Allen–Cahn equationApplied Mathematics Letters, Vol. 130 Cross Ref A new accelerating technique for low speed flow: pseudo high speed methodChinese Journal of Aeronautics, Vol. 35, No. 8 Cross Ref The hydrodynamics of LERNAJournal of Computational Physics, Vol. 463 Cross Ref An asymptotic Green's function method for time-dependent Schrödinger equations with application to Kohn-Sham equationsJournal of Computational Physics, Vol. 463 Cross Ref Effects of differential diffusion on hydrogen flame kernel development under engine conditionsProceedings of the Combustion Institute, Vol. 14 Cross Ref Instability seeding mechanisms due to internal defects in inertial confinement fusion targetsPhysics of Plasmas, Vol. 29, No. 8 Cross Ref Investigation of droplet grouping in monodisperse streams by direct numerical simulationsPhysics of Fluids, Vol. 34, No. 8 Cross Ref The development of a high-resolution Eulerian radiation-hydrodynamics simulation capability for laser-driven HohlraumsPhysics of Plasmas, Vol. 29, No. 8 Cross Ref Exponential Bhatnagar-Gross-Krook integrator for multiscale particle-based kinetic simulations4 August 2022 | Physical Review E, Vol. 106, No. 2 Cross Ref Casimir preserving spectrum of two-dimensional turbulence24 August 2022 | Physical Review Fluids, Vol. 7, No. 8 Cross Ref Learning PDE to Model Self-Organization of Matter9 August 2022 | Entropy, Vol. 24, No. 8 Cross Ref Numerical Estimation of Nozzle Throat Heat Flux in Oxygen-Methane Rocket Engines18 July 2022 | Journal of Propulsion and Power, Vol. 87 Cross Ref A sixth order entropy condition scheme for compressible flowComputers & Fluids, Vol. 243 Cross Ref Explore artificial neural networks for solving complex hydrocarbon chemistry in turbulent reactive flowsFundamental Research, Vol. 2, No. 4 Cross Ref High-order Scharfetter-Gummel-based schemes and applications to gas discharge modelingJournal of Computational Physics, Vol. 461 Cross Ref Higher order Haar wavelet method integrated with strang splitting for solving regularized long wave equationMathematics and Computers in Simulation, Vol. 197 Cross Ref Graphite Nozzle Erosion Trends in Paraffin/Oxygen Hybrid RocketsJournal of Propulsion and Power, Vol. 38, No. 4 Cross Ref A Massively Parallel Eulerian-Lagrangian Method for Advection-Dominated Transport in Viscous FluidsNils Kohl, Marcus Mohr, Sebastian Eibl, and Ulrich Rüde24 June 2022 | SIAM Journal on Scientific Computing, Vol. 44, No. 3AbstractPDF (52728 KB)Modeling of Wall Heat Flux in Oxygen-Methane Liquid Rocket Thrust Chambers20 June 2022 Cross Ref Verification and Validation of Immersed Boundary Solvers for Hypersonic Flows with Gas-Surface Interactions20 June 2022 Cross Ref Numerical simulation and entropy dissipative cure of the carbuncle instability for the shallow water circular hydraulic jump23 February 2022 | International Journal for Numerical Methods in Fluids, Vol. 94, No. 6 Cross Ref Second-order maximum principle preserving Strang’s splitting schemes for anisotropic fractional Allen-Cahn equations30 September 2021 | Numerical Algorithms, Vol. 90, No. 2 Cross Ref Intrinsic instabilities in premixed hydrogen flames: Parametric variation of pressure, equivalence ratio, and temperature. Part 1 - Dispersion relations in the linear regimeCombustion and Flame, Vol. 240 Cross Ref Stability and convergence of Strang splitting. Part I: Scalar Allen-Cahn equationJournal of Computational Physics, Vol. 458 Cross Ref A dynamic relaxation method with operator splitting and random-choice strategy for SPHJournal of Computational Physics, Vol. 458 Cross Ref Numerical prediction of loudness metrics for N-waves and shaped sonic booms in kinematic turbulenceThe Journal of the Acoustical Society of America, Vol. 151, No. 6 Cross Ref Magneto-thermal instability in galaxy clusters – I. Theory and two-dimensional simulations9 April 2022 | Monthly Notices of the Royal Astronomical Society, Vol. 513, No. 3 Cross Ref Computational Fluid Dynamics of Reacting Flows at Surfaces: Methodologies and Applications26 February 2022 | Chemie Ingenieur Technik, Vol. 94, No. 5 Cross Ref Qualitative properties of different numerical methods for the inhomogeneous geometric Brownian motionJournal of Computational and Applied Mathematics, Vol. 406 Cross Ref An improved finite integration method for nonlocal nonlinear Schrödinger equationsComputers & Mathematics with Applications, Vol. 113 Cross Ref Time-discretization of a plasma-neutral MHD model with a semi-implicit leapfrog algorithmComputer Physics Communications, Vol. 274 Cross Ref A high-order unified stochastic particle method based on the Bhatnagar-Gross-Krook model for multi-scale gas flowsComputer Physics Communications, Vol. 274 Cross Ref A new stabilised scheme for the Richards’ equation with evapotranspirationGroundwater for Sustainable Development, Vol. 17 Cross Ref Well-balanced discrete unified gas-kinetic scheme for two-phase systemsPhysics of Fluids, Vol. 34, No. 5 Cross Ref On the Physical Nature of “Anomalous” High Energy Electrons in Vacuum Diodes24 June 2022 | Russian Physics Journal, Vol. 64, No. 12 Cross Ref Stability and convergence of Strang splitting. Part II: Tensorial Allen-Cahn equationsJournal of Computational Physics, Vol. 454 Cross Ref A Model for Characteristic X-Ray Emission in Electron Probe Microanalysis Based on the (Filtered) Spherical Harmonic () Method for Electron Transport11 April 2022 | Microscopy and Microanalysis, Vol. 28, No. 2 Cross Ref Restoring the conservativity of characteristic-based segregated models: Application to the hybrid lattice Boltzmann methodPhysics of Fluids, Vol. 34, No. 4 Cross Ref An unconditionally stable, time-implicit algorithm for solving the one-dimensional Vlasov–Poisson system8 March 2022 | Journal of Plasma Physics, Vol. 88, No. 2 Cross Ref Atomistic simulation of ultra-short pulsed laser ablation of Al: an extension for non-thermalized electrons and ballistic transport29 December 2021 | Journal of Physics D: Applied Physics, Vol. 55, No. 13 Cross Ref Enhancing the Quantum Linear Systems Algorithm Using Richardson ExtrapolationACM Transactions on Quantum Computing, Vol. 3, No. 1 Cross Ref Reacting condensed phase explosives in direct contactJournal of Applied Physics, Vol. 131, No. 9 Cross Ref On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs11 February 2022 | Journal of Scientific Computing, Vol. 90, No. 3 Cross Ref A three-dimensional Laguerre one-way wave equation solverApplied Numerical Mathematics, Vol. 173 Cross Ref Higher Order Auxiliary Field Quantum Monte Carlo MethodsJournal of Physics: Conference Series, Vol. 2207, No. 1 Cross Ref Numerical method for modeling photosynthesis of algae on pulsing soft corals31 March 2022 | Physical Review Fluids, Vol. 7, No. 3 Cross Ref Numerical Method for Approximate Solution of Fisher's Equation1 March 2022 | Journal of the Institute of Science and Technology Cross Ref Algorithm for Solving a System of Coupled Nonlinear Schrödinger Equations by the Split-Step Method to Describe the Evolution of a High-Power Femtosecond Optical Pulse in an Optical Polarization Maintaining Fiber1 March 2022 | Fibers, Vol. 10, No. 3 Cross Ref CFD modeling of multiphase flows with detailed microkinetic description of the surface reactivityChemical Engineering Research and Design, Vol. 179 Cross Ref Nonlinear fourth order Taylor expansion of lattice Boltzmann schemesAsymptotic Analysis, Vol. 127, No. 4 Cross Ref Numerical scheme for solving a porous Saint-Venant type model for water flow on vegetated hillslopesApplied Numerical Mathematics, Vol. 172 Cross Ref A weak approximation method for irregular functionals of hypoelliptic diffusionsApplied Numerical Mathematics, Vol. 172 Cross Ref On pattern formation in reaction–diffusion systems containing self- and cross-diffusionCommunications in Nonlinear Science and Numerical Simulation, Vol. 105 Cross Ref Comparison of high-order Eulerian methods for electron hybrid modelJournal of Computational Physics, Vol. 451 Cross Ref Simulation of three-dimensional forced compressible isotropic turbulence by a redesigned discrete unified gas kinetic schemePhysics of Fluids, Vol. 34, No. 2 Cross Ref A coupled Lagrangian-Eulerian model for microplastics as vectors of contaminants applied to the Mediterranean Sea14 February 2022 | Environmental Research Letters, Vol. 17, No. 2 Cross Ref A fast convergence fourth-order Vlasov model for Hall thruster ionization oscillation analyses22 December 2021 | Plasma Science and Technology, Vol. 24, No. 2 Cross Ref Burgers equation and convexification method: Propagation of wave packets and narrowband noiseThe Journal of the Acoustical Society of America, Vol. 151, No. 2 Cross Ref A Robust Reacting Flow Solver with Computational Diagnostics Based on OpenFOAM and Cantera14 February 2022 | Aerospace, Vol. 9, No. 2 Cross Ref Development of a Discontinuous Galerkin Ionosphere‐Plasmasphere Model3 February 2022 | Journal of Geophysical Research: Space Physics, Vol. 127, No. 2 Cross Ref Accurate and efficient simulations of Hamiltonian mechanical systems with discontinuous potentialsJournal of Computational Physics, Vol. 450 Cross Ref Stellar evolution and tidal dissipation in REBOUNDx10 January 2022 | Monthly Notices of the Royal Astronomical Society, Vol. 510, No. 4 Cross Ref Mechanisms for severe combustion instabilities induced by low-temperature fuel injection of an H 2 /O 2 rocket-type combustor29 December 2022 Cross Ref New Trends on Multiphase Flow Simulations Using Slug Capturing Approach2 April 2022 Cross Ref Modelling and Numerical Simulation of Binary Droplet Collisions Under Extreme Conditions31 August 2022 Cross Ref Investigation of the Behaviour of Supercooled Droplets Concerning Evaporation, Sublimation and Freezing Under Different Boundary Conditions31 August 2022 Cross Ref Symmetric Splitting Methods for Wave Equations: First Approaches24 September 2021 Cross Ref Efficient Experimental and Numerical Methods for Solving Vertical Distribution of Sediments in Dam-Break Flows19 May 2022 Cross Ref Aerosol–climate modeling Cross Ref Flame edge dynamics in counterflow nonpremixed flames of CH4/He versus air at low strain rates: An experimental and numerical studyCombustion and Flame, Vol. 235 Cross Ref Numerical modelling of turbulent geophysical flows using a hyperbolic shear shallow water model: Application to powder snow avalanchesComputers & Fluids, Vol. 233 Cross Ref Stochastic population balance methods for detailed modelling of flame-made aerosol particlesJournal of Aerosol Science, Vol. 159 Cross Ref Immersed boundary method for high-order flux reconstruction based on volume penalizationJournal of Computational Physics, Vol. 448 Cross Ref A unified formulation of splitting-based implicit time integration schemesJournal of Computational Physics, Vol. 448 Cross Ref Face-based in of Computational Physics, Vol. Cross Ref A hybrid method for problems with boundary in of Computational Physics, Vol. Cross Ref A computational for of of in the Vol. Cross Ref A discontinuous Galerkin in for and Vol. 17 Cross Ref Modeling of CO 2 in Reactive A January 2022 | Research, Vol. No. 1 Cross Ref for numerical simulation of January 2022 | Institute No. Cross Ref to Schemes for and January 2022 | Vol. No. 1 Cross Ref of discrete and Systems - Vol. No. 10 Cross Ref February 2022 | Vol. 5 Cross Ref numerical schemes for Hamiltonian dynamics Cross Ref Numerical Simulations on Nonlinear Quantum with the April 2022 | The Journal of computational Vol. 8 Cross Ref April 2022 Cross Ref of nonlocal and Systems - Vol. No. Cross Ref Local boundary for nonlinear Schrödinger equation with wave operator on August 2021 | Methods in the Applied Vol. 44, No. Cross Ref methods for in October 2021 | International Journal for Numerical and Methods in Vol. No. Cross Ref modeling of by a of change and December 2021 | Journal of Vol. No. Cross Ref Operator splitting and in Mathematics and Computation, Vol. Cross Ref of flow and equivalence on oblique detonation Science and Technology, Vol. Cross Ref combustion of droplet under and Flame, Vol. Cross Ref A high order operator splitting method based on for the nonlocal equationJournal of Computational Physics, Vol. Cross Ref reconstruction and schemes for volume of fluid method in of Computational Physics, Vol. Cross Ref A solver based on method for the February 2021 | Scientific Vol. No. 1 Cross Ref of of Schrödinger June 2021 | Scientific Vol. No. 1 Cross Ref and formation droplet a September 2021 | Scientific Vol. No. 1 Cross Ref of and on October 2021 | Scientific Vol. No. 1 Cross Ref Wave in of in MHD December 2021 | The Astrophysical Journal, Vol. No. 2 Cross Ref splitting methods with Journal of Chemical Physics, Vol. No. Cross Ref for simulations with Modelling & Vol. Cross Ref and Schrödinger equation with of Computational Physics, Vol. Cross Ref semilinear and the of Analysis, Vol. No. 10 Cross Ref A Numerical of Coupling Schemes for Atmospheric November 2021 | Journal of in Modeling Vol. No. Cross Ref Boundary variation conservative method for compressible and incompressible of Fluids, Vol. No. Cross Ref of Schemes in the of Vortex December 2021 | Computational Mathematics and Physics, Vol. No. Cross Ref method for the of and equations in September 2021 | Journal of Vol. No. 6 Cross Ref Conservative Schemes for Nonlinear Schrödinger Equations in a with of October 2021 | Mathematics, Vol. 9, No. Cross Ref in reactive August 2021 | Vol. No. Cross Ref Hamiltonian based approach for time in May 2021 | Molecular Physics, Vol. No. Cross Ref for Conservative Schemes for the of Nonlinear Pulse with a October 2021 | Nonlinear in Vol. 24, No. 3 Cross Ref Numerical and the for August 2021 | Journal of the Royal Society, Vol. No. Cross Ref A fully coupled two-dimensional model in August 2021 | Environmental Fluid Mechanics, Vol. No. 5 Cross Ref The formation and development of oblique detonation wave with different chemical Science and Technology, Vol. Cross Ref A improved numerical method for the simulation of rogue waves of nonlinear Schrödinger in Nonlinear Science and Numerical Simulation, Vol. Cross Ref Heat by in turbulent boundary at Journal of Heat and Vol. Cross Ref A Lagrangian-Eulerian method for of Computational Physics, Vol. Cross Ref in linear and nonlinear January 2021 | Journal of Applied Mathematics, Vol. No. 5 Cross Ref High-order geometric integrators for Journal of Chemical Physics, Vol. No. 12 Cross Ref of Flow in Gas with by Electron Cross Ref chemistry for and using flow of Applied Physics, Vol. No. Cross Ref modelling of premixed May 2021 | Combustion Theory and Modelling, Vol. No. 6 Cross Ref A numerical of control August 2021 | Vol. No. 1 Cross Ref A Conservative Semi-Lagrangian Transport Model on the July 2021 | in Atmospheric Vol. 38, No. 9 Cross Ref A Second-Order Method for Schrödinger Equations with July 2021 | Journal of Scientific Computing, Vol. 88, No. 3 Cross Ref Strang technique for the August 2021 | Vol. No. 3 Cross Ref splitting method for space-fractional Schrödinger equationApplied Numerical Mathematics, Vol. Cross Ref Numerical simulation of in the of Journal of Multiphase Vol. Cross Ref approximation of and numerical October 2021 | in Applied Vol. No. 3 Cross Ref electron dynamics in high Accurate prediction of September 2021 | Physical Review Vol. No. 10 Cross Ref On the of by the hydrogen in July 2021 | Monthly Notices of the Royal Astronomical Society, Vol. No. 2 Cross Ref Reaction Effects on Nozzle Erosion in Paraffin/Oxygen Hybrid July 2021 Cross Ref modeling in July 2021 Cross Ref A fast and efficient numerical algorithm for equation with a nonlocal Mathematics Letters, Vol. Cross Ref An particle hydrodynamics method for modeling Methods in Applied and Engineering, Vol. Cross Ref A model for droplet Journal of Heat and Vol. Cross Ref Numerical for high frequency equation efficient time for Schrödinger equationJournal of Computational Physics, Vol. Cross Ref and for and simulation of Magnetic Vol. Cross Ref Local wave

Previous article Next article Fractional Brownian Motions, Fractional Noises and ApplicationsBenoit B. Mandelbrot and John W. Van NessBenoit B. Mandelbrot and John W. Van Nesshttps://doi.org/10.1137/1010093PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. Adelman, Long cycles—fact or artifact?, Amer. Economic Rev., 60 (1965), 444–463 Google Scholar[2] William Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statistics, 22 (1951), 427–432 MR0042626 0043.34201 CrossrefISIGoogle Scholar[3] I. M. Gel'fand and , N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]xiv+384 MR0435834 Google Scholar[4] C. W. J. Granger, The typical spectral shape of an economic variable, Econometrica, 34 (1966), 150–161 CrossrefISIGoogle Scholar[5] G. A. Hunt, Random Fourier transforms, Trans. Amer. Math. Soc., 71 (1951), 38–69 MR0051340 0043.30601 CrossrefISIGoogle Scholar[6] H. E. Hurst, , R. P. Black and , Y. M. Sinaika, Long Term Storage in Reservoirs. An Experimental Study, Constable, London, 1965 Google Scholar[7] A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 115–118 MR0003441 0022.36001 Google Scholar[8] John Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc., 104 (1962), 62–78 MR0138128 0286.60017 CrossrefGoogle Scholar[9] Paul Lévy, Random functions: General theory with special reference to Laplacian random functions, Univ. California Publ. Statist., 1 (1953), 331–390 MR0055607 0052.14402 Google Scholar[10] Michel Loève, Probability theory, 2nd ed. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-New York-London, 1960xvi+685 MR0123342 0095.12201 Google Scholar[11] Benoı⁁t Mandelbrot, Une classe processus stochastiques homothétiques à soi; application à la loi climatologique H. E. Hurst, C. R. Acad. Sci. Paris, 260 (1965), 3274–3277 MR0176521 0127.09501 Google Scholar[12] B. Mandelbrot, Self-similar error-clusters in communication systems and the concept of conditional stationarity, IEEE Trans. Comet. Tech., COM-13 (1965), 71–90 10.1109/TCOM.1965.1089090 CrossrefISIGoogle Scholar[13] B. Mandelbrot, Noises with an $l/f$ spectrum, a bridge between direct current and white noise, IEEE Trans. Information Theory, IT-13 (1967), 289–298 10.1109/TIT.1967.1053992 0148.40507 CrossrefISIGoogle Scholar[14] Benoit Mandelbrot, Sporadic random functions and conditional spectral analysis: Self-similar examples and limitsProc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif., 1965/66), Vol. III: Physical Sciences, Univ. California Press, Berkeley, Calif., 1967, 155–179 MR0224243 0189.18302 Google Scholar[15] B. Mandelbrot and , J. R. Wallis, Noah, Joseph and operational hydrology, Water Resources Research, to appear Google Scholar[16] B. Mandelbrot and , J. R. Wallis, Computer experiments with fractional Gaussian noise, Water Resources Research, to appear Google Scholar[17] B. Mandelbrot and , J. R. Wallis, Some long run properties of geophysical records, Water Resources Research, to appear. Google Scholar[18] G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyusyu Univ. A., 4 (1949), 45–106 MR0032127 0045.40602 CrossrefGoogle Scholar[19] M. Rosenblatt, Independence and dependenceProc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, 431–443 MR0133863 0105.11802 Google Scholar[20] G. I. Taylor, Statistical theory of turbulence, Proc. Roy. Sac. Ser. A, 151 (1935), 421–478 CrossrefGoogle Scholar[21] H. Weyl, Bemerkungen zum Begriff der Differential-Quotenten gebrochener Ordnung, Vierteljschr. Naturforsch. Ges. Zürich, 62 (1967), 296–302 Google Scholar[22] A. M. Yaglom, Correlation theory of processes with random stationary nth increments, Amer. Math. Soc. Transl. (2), 8 (1958), 87–141 MR0091554 0080.34903 CrossrefGoogle Scholar[23] A. M. Yaglom, L. M. LeCam and , J. Neyman, Stationary Gaussian processes satisfying the strong mixing condition and best predictable functionalsBernoulli-Bayes-Laplace Anniversary Volume, Springer-Verlag, New York, 1965, 241–252 0142.13904 CrossrefGoogle Scholar[24] A. M. Yaglom, Outline of some topics in linear extrapolation of stationary random processesProc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II:Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, Calif., 1967, 259–278 MR0215358 0189.18401 Google Scholar[25] Antoni Zygmund, Trigonometrical series, Chelsea Publishing Co., New York, 1952vi+329, 2nd ed. MR0076084 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Mean square stability of stochastic theta method for stochastic differential equations driven by fractional Brownian motionJournal of Computational and Applied Mathematics, Vol. 420 | 1 Mar 2023 Cross Ref ARMA–GARCH model with fractional generalized hyperbolic innovationsFinancial Innovation, Vol. 8, No. 1 | 15 May 2022 Cross Ref Fokker–Planck equation of the fractional Brownian motionInternational Journal of Non-Linear Mechanics, Vol. 147 | 1 Dec 2022 Cross Ref A generative model for fBm with deep ReLU neural networksJournal of Complexity, Vol. 73 | 1 Dec 2022 Cross Ref Emergence of non-Fickian transport in truncated pluri-Gaussian permeability fieldsGEM - International Journal on Geomathematics, Vol. 13, No. 1 | 1 October 2022 Cross Ref A comparison of maximum likelihood and absolute moments for the estimation of Hurst exponents in a stationary frameworkCommunications in Nonlinear Science and Numerical Simulation, Vol. 114 | 1 Nov 2022 Cross Ref Multiscaling and rough volatility: An empirical investigationInternational Review of Financial Analysis, Vol. 84 | 1 Nov 2022 Cross Ref Multivariate claim processes with rough intensities: Properties and estimationInsurance: Mathematics and Economics, Vol. 107 | 1 Nov 2022 Cross Ref Exploring the dynamic nonlinear relationship between crude oil price and implied volatility indices: A new perspective from MMV-MFDFAPhysica A: Statistical Mechanics and its Applications, Vol. 603 | 1 Oct 2022 Cross Ref On the model of random walk with multiple memory structurePhysica A: Statistical Mechanics and its Applications, Vol. 603 | 1 Oct 2022 Cross Ref Self-similarity and response of fractional differential equations under white noise inputProbabilistic Engineering Mechanics, Vol. 70 | 1 Oct 2022 Cross Ref The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motionStatistics & Probability Letters, Vol. 189 | 1 Oct 2022 Cross Ref Discrete scaling and criticality in a chain of adaptive excitable integratorsChaos, Solitons & Fractals, Vol. 163 | 1 Oct 2022 Cross Ref Novel Neuron-like Procedure of Weak Signal Detection against the Non-Stationary Noise Background with Application to Underwater SoundRemote Sensing, Vol. 14, No. 19 | 29 September 2022 Cross Ref Chaos Analysis Framework: How to Safely Identify and Quantify Time-Series DynamicsNonlinear Systems - Recent Developments and Advances [Working Title] | 22 September 2022 Cross Ref Asymptotic of the running maximum distribution of a Gaussian BridgeStochastic Analysis and Applications, Vol. 28 | 19 September 2022 Cross Ref Wavelet eigenvalue regression in high dimensionsStatistical Inference for Stochastic Processes, Vol. 1 | 18 September 2022 Cross Ref Data-Driven Modeling of Non-Markovian Noise in Semiconductor Lasers2022 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) | 12 Sep 2022 Cross Ref Continuous wavelet estimation for multivariate fractional Brownian motionPakistan Journal of Statistics and Operation Research | 10 September 2022 Cross Ref Persistence in daily returns of stocks with highest market capitalization in the Indian marketDigital Finance, Vol. 503 | 9 September 2022 Cross Ref Bayesian semiparametric long memory models for discretized event dataThe Annals of Applied Statistics, Vol. 16, No. 3 | 1 Sep 2022 Cross Ref Scale invariance in fNIRS as a measurement of cognitive loadCortex, Vol. 154 | 1 Sep 2022 Cross Ref Efficient calculation of fractal properties via the Higuchi methodNonlinear Dynamics, Vol. 109, No. 4 | 23 June 2022 Cross Ref N-Fold Compound Option Fuzzy Pricing Based on the Fractional Brownian MotionInternational Journal of Fuzzy Systems, Vol. 24, No. 6 | 9 May 2022 Cross Ref On the nature and propagation of errors in roughness parameters obtained from spectral analysis of atomic force microscopy topographic imagesJournal of Vacuum Science & Technology A, Vol. 40, No. 5 | 1 Sep 2022 Cross Ref Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processesChaos, Solitons & Fractals, Vol. 162 | 1 Sep 2022 Cross Ref Study on the Effect of Judgment Excitation Mode to Relieve Driving Fatigue Based on MF-DFABrain Sciences, Vol. 12, No. 9 | 6 September 2022 Cross Ref Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitionsChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, No. 9 | 1 Sep 2022 Cross Ref How critical is brain criticality?Trends in Neurosciences, Vol. 21 | 1 Sep 2022 Cross Ref Estimation of hurst exponent for sequential monitoring of clinical trials with covariate adaptive randomizationContemporary Clinical Trials, Vol. 120 | 1 Sep 2022 Cross Ref A GMM approach to estimate the roughness of stochastic volatilityJournal of Econometrics, Vol. 57 | 1 Sep 2022 Cross Ref Frequency–frequency correlations of single-trajectory spectral densities of Gaussian processesNew Journal of Physics, Vol. 24, No. 9 | 27 September 2022 Cross Ref Dispersion heterogeneous recurrence analysis and its use on fault detectionCommunications in Nonlinear Science and Numerical Simulation, Vol. 3 | 1 Sep 2022 Cross Ref UNIFORM CONTINUITY OF FRACTIONAL ORDER INTEGRAL OF FRACTAL INTERPOLATION FUNCTIONFractals, Vol. 30, No. 06 | 6 June 2022 Cross Ref Order pattern recurrence for the analysis of complex systemsPhysica A: Statistical Mechanics and its Applications, Vol. 65 | 1 Sep 2022 Cross Ref Multifractal detrended cross-correlation analysis of radioactivity borne radon, thoron and meteorological time seriesPhysica A: Statistical Mechanics and its Applications, Vol. 10 | 1 Sep 2022 Cross Ref Recovering mixtures of fast-diffusing states from short single-particle trajectorieseLife, Vol. 11 | 30 August 2022 Cross Ref Shadow Price Approximation for the Fractional Black Scholes ModelInternational Journal of Mathematics and Mathematical Sciences, Vol. 2022 | 29 Aug 2022 Cross Ref Roughness and intermittency within metropolitan regions - Application in three French conurbationsEnvironment and Planning B: Urban Analytics and City Science, Vol. 215 | 27 August 2022 Cross Ref Understanding the structure of cognitive noisePLOS Computational Biology, Vol. 18, No. 8 | 17 August 2022 Cross Ref Edgeworth expansion of the t-statistic of the whittle MLE for linear regression processes with long-memory disturbancesCommunications in Statistics - Theory and Methods, Vol. 3 | 16 August 2022 Cross Ref Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and modelsPhysical Chemistry Chemical Physics, Vol. 24, No. 31 | 1 January 2022 Cross Ref UV Slopes at All Redshifts Are Consistent with H = 1 Stochastic Star Formation HistoriesResearch Notes of the AAS, Vol. 6, No. 8 | 10 August 2022 Cross Ref A stochastic model to reproduce the star formation history of individual galaxies in hydrodynamic simulationsMonthly Notices of the Royal Astronomical Society, Vol. 515, No. 3 | 13 July 2022 Cross Ref Forecasting with fractional Brownian motion: a financial perspectiveQuantitative Finance, Vol. 22, No. 8 | 20 May 2022 Cross Ref Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferentialInternational Journal of Nonlinear Sciences and Numerical Simulation, Vol. 0, No. 0 | 3 August 2022 Cross Ref Pairs trading with fractional Ornstein–Uhlenbeck spread modelApplied Economics, Vol. 10 | 3 August 2022 Cross Ref Stochastic particle unbinding modulates growth dynamics and size of transcription factor condensates in living cellsProceedings of the National Academy of Sciences, Vol. 119, No. 31 | 26 July 2022 Cross Ref Option pricing based on a type of fuzzy processJournal of Ambient Intelligence and Humanized Computing, Vol. 13, No. 8 | 26 June 2021 Cross Ref Entropic Compressibility of Lévy ProcessesIEEE Transactions on Information Theory, Vol. 68, No. 8 | 1 Aug 2022 Cross Ref Comment on “A computational technique to classify several fractional Brownian motion processes” [Chaos Solitons and Fractals 150(2021) 111152]Chaos, Solitons & Fractals, Vol. 161 | 1 Aug 2022 Cross Ref Fractional Gaussian fields on the Sierpiński Gasket and related fractalsJournal d'Analyse Mathématique, Vol. 146, No. 2 | 11 July 2022 Cross Ref Multifractal Characterization and Modeling of Blood Pressure SignalsAlgorithms, Vol. 15, No. 8 | 26 July 2022 Cross Ref Robustness of Detrended Cross-correlation Analysis Method Under Outliers ObservationsFluctuation and Noise Letters, Vol. 21, No. 04 | 21 April 2022 Cross Ref Entropy fluctuation and correlation transfer in tunable discrete-time quantum walk with fractional Gaussian noisePhysical Review E, Vol. 106, No. 2 | 15 August 2022 Cross Ref Poincaré Plot Nonextensive Distribution Entropy: A New Method for Electroencephalography (EEG) Time SeriesSensors, Vol. 22, No. 16 | 21 August 2022 Cross Ref A Model Based on Fractional Brownian Motion for Temperature Fluctuation in the Campi Flegrei CalderaFractal and Fractional, Vol. 6, No. 8 | 30 July 2022 Cross Ref Acceleration scaling and stochastic dynamics of a fluid particle in turbulencePhysical Review Fluids, Vol. 7, No. 8 | 24 August 2022 Cross Ref Multifractional Brownian motion characterization based on Hurst exponent estimation and statistical learningChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 32, No. 8 | 1 Aug 2022 Cross Ref Stability analysis of the Iraqi GNSS stationsJournal of Applied Geodesy, Vol. 16, No. 3 | 31 March 2022 Cross Ref Multifractal test for nonlinearity of interactions across scales in time seriesBehavior Research Methods, Vol. 61 | 19 July 2022 Cross Ref Rolling Bearing Fault Diagnosis Based on MFDFA-SPS and ELMMathematical Problems in Engineering, Vol. 2022 | 18 Jul 2022 Cross Ref Encoding Protest Duration In An Agent-Based Model As Characteristic Phase Transitions2022 Annual Modeling and Simulation Conference (ANNSIM) | 18 Jul 2022 Cross Ref Decomposing the effect of anomalous diffusion enables direct calculation of the Hurst exponent and model classification for single random pathsJournal of Physics A: Mathematical and Theoretical, Vol. 55, No. 27 | 13 June 2022 Cross Ref Classification of stochastic processes by convolutional neural networksJournal of Physics A: Mathematical and Theoretical, Vol. 55, No. 27 | 24 June 2022 Cross Ref A generalized two‐factor square‐root framework for modeling occurrences of natural catastrophesJournal of Forecasting, Vol. 4 | 6 July 2022 Cross Ref The first exit time of fractional Brownian motion from the minimum and maximum parabolic domainsStatistics & Probability Letters, Vol. 186 | 1 Jul 2022 Cross Ref Fractional stochastic Loewner evolution and scaling curvesPhysica A: Statistical Mechanics and its Applications, Vol. 597 | 1 Jul 2022 Cross Ref Stochastic pursuit-evasion curves for foraging dynamicsPhysica A: Statistical Mechanics and its Applications, Vol. 597 | 1 Jul 2022 Cross Ref Power‐type derivatives for rough volatility with jumpsJournal of Futures Markets, Vol. 42, No. 7 | 13 May 2022 Cross Ref Simultaneous Identification of Volatility and Mean-Reverting Parameter for European Option under Fractional CKLS ModelFractal and Fractional, Vol. 6, No. 7 | 21 June 2022 Cross Ref Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentialsNew Journal of Physics, Vol. 24, No. 7 | 7 July 2022 Cross Ref Sublinear diffusion in the generalized triangle mapPhysical Review E, Vol. 106, No. 1 | 11 July 2022 Cross Ref Multivariate long memory structure in the cryptocurrency market: The impact of COVID-19International Review of Financial Analysis, Vol. 82 | 1 Jul 2022 Cross Ref Unravelling the origins of anomalous diffusion: From molecules to migrating storksPhysical Review Research, Vol. 4, No. 3 | 18 July 2022 Cross Ref Fractality of Borsa Istanbul during the COVID-19 PandemicMathematics, Vol. 10, No. 14 | 18 July 2022 Cross Ref Probing local chromatin dynamics by tracking telomeresBiophysical Journal, Vol. 121, No. 14 | 1 Jul 2022 Cross Ref Are the top six cryptocurrencies efficient? Evidence from time‐varying long memoryInternational Journal of Finance & Economics, Vol. 27, No. 3 | 7 December 2020 Cross Ref The anomalous process in singlet fission kinetic model with time-dependent coefficientPhysica B: Condensed Matter, Vol. 117 | 1 Jul 2022 Cross Ref A Unified Convergence Analysis for the Fractional Diffusion Equation Driven by Fractional Gaussian Noise with Hurst Index $H\in(0,1)$Daxin Nie and Weihua DengSIAM Journal on Numerical Analysis, Vol. 60, No. 3 | 28 June 2022AbstractPDF (498 KB)Elimination of Image Saturation Effects on Multifractal Statistics Using the 2D WTMM in Vol. 13 | 28 June 2022 Cross Ref the relationship between processes with fractional Brownian motion errors of Statistical and Simulation, Vol. 8 | 28 June 2022 Cross Ref under a fractional Brownian motion with Stochastic in and Vol. 2 | 27 June 2022 Cross Ref of rough stochastic volatility models on Journal, Vol. 19 | June 2022 Cross Ref the of anomalous diffusion with the of of Physics A: Mathematical and Theoretical, Vol. 55, No. 24 | 26 May 2022 Cross Ref impact on Evidence from a Hurst A: Statistical Mechanics and its Applications, Vol. | 1 2022 Cross Ref method to the of in fractal and Vol. 10, No. 2 | 18 May 2022 Cross Ref of Using and Statistical in Journal of Science and Transactions of Engineering, Vol. No. 3 | 26 September 2021 Cross Ref The of fractional Brownian Annals of Applied Vol. 32, No. 3 | 1 2022 Cross Ref behavior during the a comparison between and European based on Economic Vol. 12, No. 2 | 20 September 2021 Cross Ref On fractional Lévy motion: and in Applied Vol. No. 2 | 6 June 2022 Cross Ref Time and Persistence in An Using Fractional Journal of Research, Vol. 16, No. 3 | 7 April 2022 Cross Ref A for time fractional with Vol. No. | 2 2021 Cross Ref and Vol. No. 2 | 13 June 2022 Cross Ref Long short time the rough and in and Finance, Vol. No. 1 | 8 2021 Cross Ref properties for a fractional and Applied Analysis, Vol. No. 3 | 13 May 2022 Cross Ref Fractional Lévy motion with for and analysis of Vol. | 1 2022 Cross Ref random in Review E, Vol. No. 6 | 27 June 2022 Cross Ref of of fractional Brownian Review E, Vol. No. 6 | 30 June 2022 Cross Ref of for fractional impulsive differential equations driven by fractional Brownian and Stochastic Vol. 0, No. 0 | 31 May 2022 Cross Ref of fractional stochastic differential equation with and fractional Brownian Vol. | 30 May 2022 Cross Ref the of scales in and Planning B: Urban Analytics and City Science, Vol. 11 | May 2022 Cross Ref the of on of International Conference on and | 23 May 2022 Cross Ref the of in Multivariate by in the Wavelet 2022 - 2022 IEEE International Conference on and Signal | 23 May 2022 Cross Ref estimation for the model driven by the complex fractional Brownian Vol. No. 4 | 28 July 2021 Cross Ref Anomalous diffusion and memory in Computational Biology, Vol. 18, No. 5 | 18 May 2022 Cross Ref OF OF A | 13 May 2022 Cross Ref On the of Vol. 10, No. 9 | 22 April 2022 Cross Ref Convergence of for for with of Computing, Vol. No. 2 | 21 March 2022 Cross Ref Entropy of in a Review E, Vol. No. 5 | 2 May 2022 Cross Ref Analysis of during 26 August Vol. 24, No. 5 | 14 May 2022 Cross Ref stability of stochastic A and Dynamics, Vol. 22, No. | 18 April 2022 Cross Ref behavior in a model with A, Vol. | 1 May 2022 Cross Ref Asymptotic for driven by fractional Brownian motion in the of Mathematical Analysis and Applications, Vol. No. 1 | 1 May 2022 Cross Ref Generalized and rough Ornstein–Uhlenbeck Vol. | 30 April 2022 Cross Ref The the distribution of processes for the estimation of the Sciences Journal, Vol. No. 6 | 12 April 2022 Cross Ref and Application of Some Fractional Driven by Fractional Economics, Vol. 2 | April 2022 Cross Ref for processes and to Vol. No. 4 | 1 March 2022 Cross Ref Stochastic with to fractional processes in of Analysis, Vol. No. 8 | 1 2022 Cross Ref of and in Hurst exponents from spectral analysis of atomic force microscopy topographic Science, Vol. | 1 2022 Cross Ref in Distribution to Stochastic Driven by a Fractional Brownian Journal of Mathematics, Vol. No. 2 | 21 2022 Cross Ref a Under Random and Stochastic and Transactions on Science and Engineering, Vol. No. 2 | 1 2022 Cross Ref A new fractional based on nonlinear structure to systems with and of and Vol. No. | 30 January 2021 Cross Ref and The Transactions on and Systems, Vol. No. 2 | 1 2022 Cross Ref Analysis on with of and Fractional, Vol. 6, No. 4 | 14 April 2022 Cross Ref The of the Fractional and Fractional, Vol. 6, No. 4 | 15 April 2022 Cross Ref A and for complex Review E, Vol. No. 4 | 18 April 2022 Cross Ref of under in Review E, Vol. No. 4 | April 2022 Cross Ref theory of of an Review A, Vol. No. 4 | 26 April 2022 Cross Ref method for model in fractal Solitons & Fractals, Vol. | 1 2022 Cross Ref Time fractional equation with a based fractional Solitons & Fractals, Vol. | 1 2022 Cross Ref stochastic and of with complex in Applied Mechanics and Engineering, Vol. | 1 2022 Cross Ref OF Journal, Vol. No. 2 | 18 August 2022 Cross Ref The of of Fractional by in Vol. 2 | 24 March 2022 Cross Ref Bayesian of scaled fractional Brownian motionJournal of Physics A: Mathematical and | 24 March 2022 Cross Ref Anomalous diffusion: fractional Brownian motion fractional motionJournal of Physics A: Mathematical and Theoretical, Vol. 55, No. 11 | 23 2022 Cross Ref process with noise as a anomalous diffusion of Physics A: Mathematical and | 8 March 2022 Cross Ref by and Analysis and Vol. | 4 March 2022 Cross Ref Stochastic for the stochastic differential equations with & Probability Letters, Vol. | 1 Mar 2022 Cross Ref Multivariate range A: Statistical Mechanics and its Applications, Vol. | 1 Mar 2022 Cross Ref the memory for and time with Vol. No. 3 | 31 January 2022 Cross Ref for the Fractional Ornstein–Uhlenbeck of Vol. No. 1 | 22 October 2020 Cross Ref Properties of a Generalized Fractional Brownian of Vol. No. 1 | 9 January 2021 Cross Ref The for Time of Stochastic Vol. 10, No. 3 | 23 2022 Cross Ref A for Long in Higher Vol. 10, No. 5 | 24 2022 Cross Ref The of time Journal of Statistics, Vol. No. 1 | 14 August 2021 Cross Ref Estimation on of time generalized Dynamics, Vol. No. 4 | 12 January 2022 Cross Ref of the in the of fields by the of the Vol. | 1 Mar 2022 Cross Ref Statistical of nonlinear time and application to Review Research, Vol. 4, No. 1 | 17 March 2022 Cross Ref Price Modeling of under for Vol. 13, No. 3 | 18 March 2022 Cross Ref and in Advances of Physical Sciences, Vol. 06 | 27 April 2022 Cross Ref of of under fractional Brownian motionStatistics & Probability Letters, Vol. | 1 Mar 2022 Cross Ref Fractional and Statistical An of the Indian of Science, Vol. 29 | 24 2022 Cross Ref diffusion with stochastic of Physics A: Mathematical and Theoretical, Vol. 55, No. 7 | 28 January 2022 Cross Ref On the fractional stochastic for random Analysis and Applications, Vol. | 2 2022 Cross Ref of and Systems to

Previous article Next article Computational Aspects of Three-Term Recurrence RelationsWalter GautschiWalter Gautschihttps://doi.org/10.1137/1009002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] M. Abramowitz, Review 58, Math. Tables Aids Comput, 10 (1956), 176– Google Scholar[2] Renata Babušková, Über numerische Stabilität einiger Rekursionsformeln, Apl. Mat., 9 (1964), 186–193 MR0187365 0123.11104 Google Scholar[3] T. A. Bancroft, Some recurrence formulae in the incomplete beta function ratio, Ann. Math. Statistics, 20 (1949), 451–455 MR0030653 0041.03402 CrossrefISIGoogle Scholar[4] G. Blanch, Numerical evaluation of continued fractions, SIAM Rev., 6 (1964), 383–421 10.1137/1006092 MR0171374 0133.38904 LinkISIGoogle Scholar[5] British Association For The Advancement Of Science, Bessel functions. Part II. Functions of positive integer order, British Association for the Advancement of Science, Mathematical Tables, vol. X, University Press, Cambridge, 1952xl+255 MR0050973 0049.09409 Google Scholar[6] Herbert Buchholz, Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen, Ergebnisse der angewandten Mathematik. Bd. 2, Springer-Verlag, Berlin, 1953xvi+234 MR0054783 0050.07402 CrossrefGoogle Scholar[7] C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc., 53 (1957), 134–149 MR0082196 0077.32503 CrossrefGoogle Scholar[8] Lothar Collatz, The numerical treatment of differential equations. 3d ed, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin, 1960xv+568 pp. (1 plate) MR0109436 0086.32601 CrossrefGoogle Scholar[9] F. J. Corbató, On the computation of auxiliary functions for two-center integrals by means of a high-speed computer, J. Chem. Phys., 24 (1956), 452–453 CrossrefGoogle Scholar[10] Fernando J. Corbató and , Jack L. Uretsky, Generation of spherical Bessel functions in digital computers, J. Assoc. Comput. Mach., 6 (1959), 366–375 MR0105792 0087.28104 CrossrefISIGoogle Scholar[11] A. Erdélyi, Asymptotic expansions, Dover Publications Inc., New York, 1956vi+108 MR0078494 0070.29002 Google Scholar[12] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xxvi+302 MR0058756 0051.30303 Google Scholar[13] Arthur Erdélyi, , Wilhelm Magnus, , Fritz Oberhettinger and , Francesco G. Tricomi, Higher transcendental functions. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xvii+396 MR0058756 0052.29502 Google Scholar[14] M. A. Efgrafov, A new proof of Perron's theorem, Izvestiya Akad. Nauk SSSR. Ser. Mat., 17 (1953), 77–82 MR0058109 Google Scholar[15] L. Fox, A short table for Bessel functions of integer orders and large arguments, Royal Society Shorter Mathematical Tables, No. 3, Cambridge, at the University Press, 1954, 28– MR0065245 0058.11402 Google Scholar[16] Walter Gautschi, Recursive computation of the repeated integrals of the error function, Math. Comp., 15 (1961), 227–232 MR0136074 0099.11401 Google Scholar[17] W. Gautschi, Algorithm 222—Incomplete beta function ratios, Comm. ACM, 7 (1964), 143–144, Certification of Algorithm 222, Ibid., p. 244 10.1145/363958.363967 CrossrefISIGoogle Scholar[18] W. Gautschi, Algorithm 236—Bessel functions of the first kind, Comm. ACM, 7 (1964), 479–480, Certification of Algorithm 236, Ibid., 8 (1965), pp. 105–106 10.1145/355586.355587 CrossrefISIGoogle Scholar[19] W. Gautschi, Algorithm 259—Legendre functions for arguments larger than one, Comm. ACM, 8 (1965), 488–492 10.1145/365474.355591 CrossrefISIGoogle Scholar[20] W. Gautschi, Algorithm 292—Regular Coulomb wave functions, Comm. ACM, 9 (1966), 793–795 10.1145/365876.365884 CrossrefISIGoogle Scholar[21] A. O. Gel'Fond, Calculus of Finite Differences, Gosud. Izdat. Fiz.-Mat. Lit., Moscow, 1959, (German and French translations of the first edition are available.) Google Scholar[22] M. Goldstein and , R. M. Thaler, Recurrence techniques for the calculation of Bessel functions, Math. Tables Aids Comput., 13 (1959), 102–108 MR0105794 0092.33002 CrossrefGoogle Scholar[23] Marion C. Gray, Bessel functions of integral order and complex argument, Comm. ACM, 4 (1961), 169– 10.1145/355578.366318 MR0151652 CrossrefISIGoogle Scholar[24] Philip Hartman, Ordinary differential equations, John Wiley & Sons Inc., New York, 1964xiv+612 MR0171038 0125.32102 Google Scholar[25] D. R. Hartree, Some properties and applications of the repeated integrals of the error function, Mem. Proc. Manchester Lit. Philos. Soc., 80 (1936), 85–102 Google Scholar[26] Sin Hitotumatu, Note on the computation of Bessel functions through recurrence formula, J. Math. Soc. Japan, 15 (1963), 353–359 MR0156008 0131.34401 CrossrefGoogle Scholar[27] C. W. Jones, A short table for the Bessel functions In+1/2(x),2πKn+1/2(x), Cambridge, at the University Press, 1952, 20–, Royal Society Shorter Mathematical Tables, No. 1 MR0051573 0047.37203 Google Scholar[28] W. Kahan, Note on bounds for generating Bessel functions by recurrence, un published Google Scholar[29] P. Kreuser, Masters Thesis, Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen, Dissertation, University of Tübingen, Leipzig, 1914 Google Scholar[30] N. A. Logan, Survey of some early studies of the scattering of plane waves by a sphere, Proc. IEEE, 53 (1965), 773–785 CrossrefISIGoogle Scholar[31] A. V. Luk'Yanov, , I. B. Teplov and , M. K. Akimov, Tablicy volnovih kulonovskih funkciinodoť, Izdat. Akad. Nauk SSSR., Moscow, 1961, (English translation by D. E. Brown, Pergamon Press, New York, 1965) Google Scholar[32] Saburo Makinouchi, Note on the recurrence techniques for the calculation of Bessel functions $J\sb{v}(x)$, Tech. Rep. Osaka Univ., 16 (1965), 185–201 MR0198658 Google Scholar[33] Saburo Makinouchi, Note on the recurrence techniques for the calculation of Bessel functions $I\sb{v}(x)$, Tech. Rep. Osaka Univ., 16 (1965), 203–216 MR0198659 Google Scholar[34] Herbert Meschkowski, Differenzengleichungen, Studia Mathematica, Bd. XIV, Vandenhoeck & Ruprecht, Göttingen, 1959, 243– MR0109264 0084.29202 Google Scholar[35] L. M. Milne-Thomson, The Calculus of Finite Differences, Macmillan, London, 1933 0008.01801 Google Scholar[36] Google Scholar[37] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer, Berlin, 1924 CrossrefGoogle Scholar[38] F. W. J. Olver, Error analysis of Miller's recurrence algorithm, Math. Comp., 18 (1964), 65–74 MR0169406 0115.34502 ISIGoogle Scholar[39] F. W. J. Olver, Bessel functions of integer order, Handbook of Mathematical Functions, NBS Appl. Math. Ser. 55, 1964, Chap. 9 Google Scholar[40] Morio Onoe, Tables of modified quotients of Bessel functions of the first kind for real and imaginary arguments, Columbia University Press, New York, 1958vii+338 MR0095590 0085.12301 Google Scholar[41] O. Perron, Über einen Satz des Herrn Poincaré, J. Reine Angew. Math., 136 (1909), 17–37 CrossrefGoogle Scholar[42] O. Perron, Über lineare Differenzengleichungen, Acta Math., 34 (1911), 109–137 CrossrefGoogle Scholar[43A] Oskar Perron, Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954vi+194 MR0064172 0056.05901 Google Scholar[43B] Oskar Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957vi+316 MR0085349 0077.06602 Google Scholar[44] S. Pincherle, Sur la génération de systèmes récurrents au moyen d'une équation linéaire différentielle, Acta Math., 16 (1892), 341–363 CrossrefGoogle Scholar[45] S. Pincherle, Delle funzioni ipergeometriche e di varie questioni ad esse attinenti, Giorn. Mat. Battaglini, 32 (1894), 209–291, esp. Ch. III. 15. (Also in: Opere Scelte, vol. 1, pp. 273–357.) Google Scholar[46] H. Poincare, Sur les Equations Lineaires aux Differentielles Ordinaires et aux Differences Finies, Amer. J. Math., 7 (1885), 203–258, (Also in: Oeuvres Henri Poincaré, vol. 1, pp. 226–289.) MR1505385 CrossrefGoogle Scholar[47] J. B. Randels and , R. F. Reeves, Note on empirical bounds for generating Bessel functions, Comm. ACM, 1 (1958), 3–5 10.1145/368819.368846 0087.28103 CrossrefISIGoogle Scholar[48] Lord (J. W. Strutt) Rayleigh, The incidence of light upon a transparent sphere of dimensions comparable with the wave-length, Proc. Roy. Soc. London Ser. A, 84 (1910), 25–46, (Also in: Scientific Papers by John William Strutt, Baron Rayleigh, vol. 5, pp. 547–568.) CrossrefGoogle Scholar[49] Louis Robin, Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II, Gauthier-Villars, Paris, 1958viii+384 MR0101928 0088.05102 Google Scholar[50] A. Rotenberg, The calculation of toroidal harmonics, Math. Comput., 14 (1960), 274–276 MR0115264 0096.32205 CrossrefGoogle Scholar[51] F. W. Schäfke, Lösungstypen von Differenzengleichungen und Summengleichungen in normierten abelschen Gruppen, Math. Z., 88 (1965), 61–104 MR0176247 0134.06401 CrossrefISIGoogle Scholar[52] Hisayoshi Shintani, Note on Miller's recurrence algorithm, J. Sci. Hiroshima Univ. Ser. A-I Math., 29 (1965), 121–133 MR0201048 0135.38703 CrossrefGoogle Scholar[53] Irene A. Stegun and , Milton Abramowitz, Generation of Coulomb wave functions by means of recurrence relations, Phys. Rev. (2), 98 (1955), 1851–1852 10.1103/PhysRev.98.1851 MR0070259 0065.43202 CrossrefISIGoogle Scholar[54] Irene A. Stegun and , Milton Abramowitz, Generation of Bessel functions on high speed computers, Math. Tables Aids Comput, 11 (1957), 255–257 MR0093939 0084.12101 CrossrefGoogle Scholar[55] Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23. Revised ed, American Mathematical Society, Providence, R.I., 1959ix+421 MR0106295 0089.27501 Google Scholar[56] D. Teichroew, Use of continued fractions in high speed computing, Math. Tables and Other Aids to Computation, 6 (1952), 127–133 MR0049650 0047.12002 CrossrefGoogle Scholar[57] E. Thieleker, Generating functions of confluent hypergeometric functions, unpublished Google Scholar[58] A. Tubis, Tables of nonrelativistic Coulomb wave functions, LA-2150, Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1958 Google Scholar[59] H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948xiii+433 MR0025596 0035.03601 Google Scholar[60] H. Werner and , R. Collinge, Chebyshev approximations for the gamma function, Math. Comput., 15 (1961), 195–197 0099.11901 CrossrefGoogle Scholar[61] D. J. Wheeler, Personal communication, 1960 Google Scholar[62] Ion Zamfirescu, An extension of Gauss' method for the calculation of improper integrals, Acad. R. P. Romine Stud. Cerc. Mat., 14 (1963), 615–631 MR0184434 0235.65022 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Quasinormal modes and shadow of noncommutative black holeScientific Reports, Vol. 12, No. 1 | 20 May 2022 Cross Ref Hidden Spectral Symmetries and Mode Stability of Subextremal Kerr(-de Sitter) Black HolesCommunications in Mathematical Physics, Vol. 394, No. 2 | 5 July 2022 Cross Ref Glimpses of violation of strong cosmic censorship in rotating black holesPhysical Review D, Vol. 106, No. 4 | 26 August 2022 Cross Ref Computing Gaussian quadrature rules with high relative accuracyNumerical Algorithms, Vol. 18 | 2 June 2022 Cross Ref Nonlinear effects in the black hole ringdown: Absorption-induced mode excitationPhysical Review D, Vol. 105, No. 6 | 28 March 2022 Cross Ref Computation of solutions to linear difference and differential equations with a prescribed asymptotic behaviorAdvances in Difference Equations, Vol. 2021, No. 1 | 19 March 2021 Cross Ref Eigenvalue problems for a class of infinite complex symmetric tridiagonal matrices with related three-term recurrence relationJournal of Computational and Applied Mathematics, Vol. 194 | 1 Dec 2021 Cross Ref Orthogonal polynomials relative to a generalized Marchenko–Pastur probability measureNumerical Algorithms, Vol. 88, No. 3 | 18 March 2021 Cross Ref Quasinormal modes and shadow of a Schwarzschild black hole with GUPAnnals of Physics, Vol. 434 | 1 Nov 2021 Cross Ref Teukolsky master equation and Painlevé transcendents: Numerics and extremal limitPhysical Review D, Vol. 104, No. 8 | 13 October 2021 Cross Ref Resolving Singularities and Monodromy Reduction of Fuchsian ConnectionsAnnales Henri Poincaré, Vol. 22, No. 9 | 27 April 2021 Cross Ref Efficient numerical method for evaluating normal and anomalous time-domain equilibrium Green's functions in inhomogeneous systemsPhysical Review B, Vol. 104, No. 12 | 2 September 2021 Cross Ref Computation of Gabor coefficients for objects with polygonal cross section2021 International Conference on Electromagnetics in Advanced Applications (ICEAA) | 9 Aug 2021 Cross Ref VPA: Computer program for the computation of the phase shift in atom–atom potential scattering using the Variable Phase ApproachComputer Physics Communications, Vol. 263 | 1 Jun 2021 Cross Ref The aging effect in evolving scientific citation networksScientometrics, Vol. 126, No. 5 | 12 March 2021 Cross Ref Solutions of Heun's general equation and elliptic Darboux equationMathematical Methods in the Applied Sciences, Vol. 33 | 29 March 2021 Cross Ref Clenshaw–Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integralsJournal of Computational and Applied Mathematics, Vol. 385 | 1 Mar 2021 Cross Ref Alternative quantisation condition for wavepacket dynamics in a hyperbolic double wellJournal of Physics A: Mathematical and Theoretical, Vol. 54, No. 3 | 30 December 2020 Cross Ref Solving Poisson’s equation for birth–death chains: Structure, instability, and accurate approximationPerformance Evaluation, Vol. 145 | 1 Jan 2021 Cross Ref Additional methods for the stable calculation of isotropic Gaussian filter coefficients: The case of a truncated filter kernelComputers & Geosciences, Vol. 145 | 1 Dec 2020 Cross Ref Evolutionary dynamics and eigenspectrum of confluent Heun equationJournal of Physics A: Mathematical and Theoretical, Vol. 53, No. 39 | 28 August 2020 Cross Ref Analytic solutions of the Teukolsky equation for massless perturbations of any spin in de Sitter backgroundJournal of Mathematical Physics, Vol. 61, No. 10 | 1 Oct 2020 Cross Ref Encyclopedia of Special Functions: The Askey-Bateman Project14 September 2020 Cross Ref Excited hairy black holes: Dynamical construction and level transitionsPhysical Review D, Vol. 102, No. 4 | 6 August 2020 Cross Ref The Mott transition as a topological phase transitionPhysical Review B, Vol. 102, No. 8 | 14 August 2020 Cross Ref Matrix Calculus-Based Approach to Orthogonal Polynomial SequencesMediterranean Journal of Mathematics, Vol. 17, No. 4 | 25 June 2020 Cross Ref Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetimeClassical and Quantum Gravity, Vol. 37, No. 11 | 19 May 2020 Cross Ref Noncommutative scalar field in the nonextremal Reissner-Nordström background: Quasinormal mode spectrumPhysical Review D, Vol. 101, No. 11 | 15 June 2020 Cross Ref Fast algorithms using orthogonal polynomialsActa Numerica, Vol. 29 | 30 November 2020 Cross Ref Quadrature rules from a RII type recurrence relation and associated quadrature rules on the unit circleNumerical Algorithms, Vol. 83, No. 3 | 3 May 2019 Cross Ref Complementary Romanovski–Routh Polynomials, Orthogonal Polynomials on the Unit Circle, and Extended Coulomb Wave FunctionsResults in Mathematics, Vol. 75, No. 1 | 13 February 2020 Cross Ref Classification of Laguerre–Hahn orthogonal polynomials of class oneMathematische Nachrichten, Vol. 293, No. 2 | 27 December 2019 Cross Ref Computing Hitting Probabilities of Markov Chains: Structural Results with regard to the Solution Space of the Corresponding System of EquationsJournal of Applied Mathematics, Vol. 2020 | 13 Jan 2020 Cross Ref The Mathematical Landscape up to the Mid-Twentieth CenturyExtrapolation and Rational Approximation | 1 December 2020 Cross Ref Constraint polynomial approach: an alternative to the functional Bethe Ansatz method?The European Physical Journal Plus, Vol. 135, No. 1 | 15 January 2020 Cross Ref The spectra of gravitational atomsJournal of Cosmology and Astroparticle Physics, Vol. 2019, No. 12 | 2 December 2019 Cross Ref Perturbations of an extremal Kerr spacetime: Analytic framework and late-time tailsPhysical Review D, Vol. 100, No. 12 | 10 December 2019 Cross Ref Generalized continued fractions: a unified definition and a Pringsheim-type convergence criterionAdvances in Difference Equations, Vol. 2019, No. 1 | 23 September 2019 Cross Ref Stable recurrent calculation of isotropic Gaussian filter coefficientsComputers & Geosciences, Vol. 133 | 1 Dec 2019 Cross Ref Contiguous relations, Laplace’s methods, and continued fractions for $${}_3F_2(1)$$ 3 F 2 ( 1 )The Ramanujan Journal, Vol. 49, No. 1 | 4 September 2018 Cross Ref Miscellaneous Topics on Symplectic SystemsSymplectic Difference Systems: Oscillation and Spectral Theory | 6 September 2019 Cross Ref Multilayered neural architectures evolution for computing sequences of orthogonal polynomialsAnnals of Mathematics and Artificial Intelligence, Vol. 84, No. | 18 September 2018 Cross Ref evolution in the general of & of Physics, Vol. No. 12 | 3 December 2018 Cross Ref Black from a Review Vol. No. 6 | 7 August 2018 Cross Ref equations for Generalized polynomials of type of Difference Equations and Vol. No. 6 | 25 February 2018 Cross Ref Polynomial approximation for of associated Legendre et Vol. 53, No. 2 | 28 April 2018 Cross Ref of dynamics in and Space Science, Vol. No. 5 | 19 April 2018 Cross Ref Recursive Computation of and of and Bessel Functions and of Scientific Vol. 75, No. 1 | 5 August Cross Ref Computational Mathematics A of the of | 23 May 2018 Cross Ref linear and Vol. | 1 Jan 2018 Cross Ref Stable Chebyshev for the numerical solution of differential Vol. No. 1 | 19 July Cross Ref with Legendre of Computational and Applied Mathematics, Vol. | 1 Oct Cross Ref On the and in Mathematical Physics, Vol. No. | 4 May Cross Ref A method for computing integrals of oscillatory of Computational and Applied Mathematics, Vol. | 1 May Cross Ref of and of Theory and Vol. No. 4 | 5 April Cross Ref and toroidal and of Review B, Vol. No. 16 | 17 April Cross Ref The of the of Physics A: Mathematical and Theoretical, Vol. No. 11 | 14 February Cross Ref Numerical methods for the computation of the confluent and hypergeometric Algorithms, Vol. No. 3 | 27 August Cross Ref of the using Vol. | 1 Jan Cross Ref Numerical in Computing of Markov Mathematics, Vol. No. | 1 Jan Cross Ref imaginary modes of the Kerr and Quantum Gravity, Vol. No. 24 | 18 November Cross Ref A for a in Difference Equations, Vol. No. 1 | March Cross Ref of the Kerr with imaginary Review D, Vol. No. 10 | 30 November Cross Ref and applications of some solutions of the confluent Heun Mathematics and Computation, Vol. | 1 Cross Ref Some of Stability Perturbations of Continued with and a Variable of of Mathematical Sciences, Vol. No. 1 | 5 April Cross Ref Quasinormal modes of extremal black holesPhysical Review D, Vol. No. 6 | 25 March Cross Ref of integrals by and of Mathematical Vol. 54, No. 2 | 8 December Cross Ref Spectral of and Vol. 58, No. 1 | 4 February Solutions of in Quantum Mathematics | 19 September Cross Ref and in Matrix | 26 July Cross Ref A for in a of Computational Physics, Vol. | 1 Jan Cross Ref Numerical of of Vol. | 9 February Cross Ref at and the of Physics, Vol. No. | 1 Jan Cross Ref Efficient and accurate of for the in Numerical Mathematics, Vol. | 1 Cross Ref Computational of of Computational Vol. 22, No. 9 | 1 Cross Ref The Kerr and Quantum Gravity, Vol. No. 12 | 1 June Cross Ref for solutions of the confluent Heun Mathematics and Computation, Vol. | 1 Cross Ref of the integrals with the Journal of Physics, Vol. No. 12 | 28 Mar Cross Ref waves from a in a rotating black hole to the of and Physics, Vol. No. 3 | 10 March Cross Ref Spectral functions and evolution from the Chebyshev Review B, Vol. No. 11 | March Cross Ref solutions for and Vol. | 1 Mar Cross Ref Special functions and of and Vol. | 1 Jan Cross Ref Numerical computation of the for Jan Cross Ref Continued of the Journal of and Vol. No. | 1 Jan Cross Ref Quantum with by the of polynomial of Physics A: Mathematical and Theoretical, Vol. No. | 20 November Cross Ref solution of of Physics, Vol. | 1 Dec Cross Ref Polynomials and the Solution to the in an of Computational and Vol. No. | December Cross Ref Quasinormal modes of a Reissner-Nordström black Review D, Vol. No. 12 | 3 December Cross Ref perturbations of the Kerr Review D, Vol. No. 12 | 5 December Cross Ref of the calculation of integrals I. Coulomb and Review Vol. No. 6 | 29 December Cross Ref of the calculation of integrals II. of the Review Vol. No. 6 | 29 December Cross Ref for functions in the of Computer Vol. | 1 Cross Ref symmetric A new of Physics, Vol. | 1 Aug Cross Ref Journal of Vol. No. | 9 May Cross Ref and to on with and Vol. No. 1 | 19 December Cross Ref Asymptotic of Functions of of methods with applications to | 1 Jan Cross Ref A of the of Physics, Vol. No. 1 | 1 Jan Cross Ref On Fast and Stable of and Quadrature and Applied Vol. | 1 Jan Cross Ref Solving the and the in of Physics A: Mathematical and Theoretical, Vol. No. | 24 October Cross Ref On and of the of Physics, Vol. | 1 Nov Cross Ref to of Physics A: Mathematical and Theoretical, Vol. No. | 11 October Cross Ref Stability of rotating black in 5 and Quantum Gravity, Vol. No. 12 | 16 May Cross Ref The function for matrices with and Vol. No. 11 | 1 Jun Cross Ref Continued fractions and the of Physics A: Mathematical and Theoretical, Vol. No. 17 | 9 April Cross Ref Computation of integrals with oscillatory and singular using Chebyshev of Computational and Applied Mathematics, Vol. | 1 Cross Ref A generalized for the Quantum der Vol. No. 3 | 14 February Cross Ref A Pringsheim-type convergence for continued fractions in of Approximation Vol. | 1 Cross Ref On the of a class of Vol. 100, No. 6 | 7 January Cross Ref rotating black and Analytic for of Physics, Vol. No. 8 | August Cross Ref evaluation of the function for an of Vol. No. 7 | 1 Cross Ref Mathematical and Mathematics and Computation, Vol. No. 20 | 1 Jun Cross Ref An Efficient Algorithm for the Computation of the on and Vol. 60, No. 5 | 1 May Cross Ref of the Coulomb Computation of auxiliary Journal of Physics, Vol. No. 10 | 14 Mar Cross Ref Numerical calculation of and Physics Communications, Vol. No. 3 | 1 Mar Cross Ref for on Vol. No. 3 | 1 Mar Cross Ref of the Wave | 1 Jan Cross Ref Quantum with in of Physics, Vol. No. 5 | 24 September Cross Ref computation of hypergeometric on Mathematical Vol. No. 2 | 1 Dec Cross Ref Algorithm for the Computation of the Generalized on and Vol. No. 10 | 1 Oct Cross Ref la recurrence for and hypergeometric Journal, Vol. 55, No. 1 | 19 April Cross Ref Orthogonal polynomial for the and Vol. No. 3 | 1 Aug Cross Ref dimensions in Review D, Vol. 84, No. 2 | 18 July Cross Ref Functions | 1 March Cross Ref Methods for Computing Special in Computational and Applied

Previous article Next article The Numerical Solution of Parabolic and Elliptic Differential EquationsD. W. Peaceman and H. H. Rachford, Jr.D. W. Peaceman and H. H. Rachford, Jr.https://doi.org/10.1137/0103003PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. H., Bruce, , D. W., Peaceman, , H. H., Rachford and , J. D. Rice, Calculation of unsteady-state gas flow through porous media, Trans. Amer. Inst. Mining and Met. Engrs., 198 (1953), 79– ISIGoogle Scholar[2] H. S. Carslaw and , J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947viii+386 MR0022294 Google Scholar[3] Stanley P. Frankel, Convergence rates of iterative treatments of partial differential equations, Math. Tables and Other Aids to Computation, 4 (1950), 65–75 MR0046149 CrossrefGoogle Scholar[4] George G. O'Brien, , Morton A. Hyman and , Sidney Kaplan, A study of the numerical solution of partial differential equations, J. Math. Physics, 29 (1951), 223–251 MR0040805 0042.13204 CrossrefISIGoogle Scholar[5] Jim Douglas, Jr., On the numerical integration of $\partial\sp 2u/\partial x\sp 2+\partial\sp 2u/\partial y\sp 2=\partial u/\partial t$ by implicit methods, J. Soc. Indust. Appl. Math., 3 (1955), 42–65 10.1137/0103004 MR0071875 0067.35802 LinkISIGoogle Scholar[6] Jim Douglas, Jr. and , T. M. Gallie, Jr., Variable time steps in the solution of the heat flow equation by a difference equation, Proc. Amer. Math. Soc., 6 (1955), 787–793 MR0078754 0066.10502 CrossrefGoogle Scholar[7] J. Douglas, Jr. and , D. W. Peaceman, Numerical solution of two-dimensional heat flow problems, to be presented at the May, 1955 meeting of The American Institute of Chemical Engineers at Houston, Texas Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A General Alternating-Direction Implicit Framework with Gaussian Process Regression Parameter Prediction for Large Sparse Linear SystemsKai Jiang, Xuehong Su, and Juan ZhangSIAM Journal on Scientific Computing, Vol. 44, No. 4 | 7 July 2022AbstractPDF (1383 KB)Convergence Analysis of the Nonoverlapping Robin--Robin Method for Nonlinear Elliptic EquationsEmil Engström and Eskil HansenSIAM Journal on Numerical Analysis, Vol. 60, No. 2 | 21 March 2022AbstractPDF (565 KB)Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDEM. RakhubaSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 4 March 2021AbstractPDF (935 KB)A Parallel Cyclic Reduction Algorithm for Pentadiagonal Systems with Application to a Convection-Dominated Heston PDEAbhijit Ghosh and Chittaranjan MishraSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 29 April 2021AbstractPDF (643 KB)Operator Splitting for a Homogeneous Embedding of the Linear Complementarity ProblemBrendan O'DonoghueSIAM Journal on Optimization, Vol. 31, No. 3 | 4 August 2021AbstractPDF (695 KB)Adaptive Douglas--Rachford Splitting Algorithm from a Yosida Approximation StandpointZihan Liu and Kannan RamchandranSIAM Journal on Optimization, Vol. 31, No. 3 | 4 August 2021AbstractPDF (721 KB)Splitting with Near-Circulant Linear Systems: Applications to Total Variation CT and PETErnest K. Ryu, Seyoon Ko, and Joong-Ho WonSIAM Journal on Scientific Computing, Vol. 42, No. 1 | 4 February 2020AbstractPDF (1366 KB)Operator Splitting Performance Estimation: Tight Contraction Factors and Optimal Parameter SelectionErnest K. Ryu, Adrien B. Taylor, Carolina Bergeling, and Pontus GiselssonSIAM Journal on Optimization, Vol. 30, No. 3 | 13 August 2020AbstractPDF (569 KB)AMF-type W-methods for Parabolic Problems with Mixed DerivativesS. González-Pinto, E. Hairer, D. Hernández-Abreu, and S. Pérez-RodríguezSIAM Journal on Scientific Computing, Vol. 40, No. 5 | 13 September 2018AbstractPDF (702 KB)On MultiScale ADI Methods for Parabolic PDEs with a Discontinuous CoefficientZhilin Li, Xiaohong Chen, and Zhengru ZhangMultiscale Modeling & Simulation, Vol. 16, No. 4 | 18 October 2018AbstractPDF (7287 KB)On the Quasi-unconditional Stability of BDF-ADI Solvers for the Compressible Navier--Stokes Equations and Related Linear ProblemsOscar P. Bruno and Max CubillosSIAM Journal on Numerical Analysis, Vol. 55, No. 2 | 18 April 2017AbstractPDF (1278 KB)A Selective Linearization Method For Multiblock Convex OptimizationYu Du, Xiaodong Lin, and Andrzej RuszczyńskiSIAM Journal on Optimization, Vol. 27, No. 2 | 15 June 2017AbstractPDF (259 KB)Convergence Analysis of Douglas--Rachford Splitting Method for “Strongly + Weakly” Convex ProgrammingKe Guo, Deren Han, and Xiaoming YuanSIAM Journal on Numerical Analysis, Vol. 55, No. 4 | 6 July 2017AbstractPDF (1088 KB)Algebraic Multigrid Preconditioners for Multiphase Flow in Porous MediaQuan M. Bui, Howard C. Elman, and David J. MoultonSIAM Journal on Scientific Computing, Vol. 39, No. 5 | 26 October 2017AbstractPDF (511 KB)Fiber Orientation Distribution Estimation Using a Peaceman--Rachford Splitting MethodSIAM Journal on Imaging Sciences, Vol. 9, No. 2 | 3 May 2016AbstractPDF (1273 KB)Convergence Study on the Symmetric Version of ADMM with Larger Step SizesSIAM Journal on Imaging Sciences, Vol. 9, No. 3 | 22 September 2016AbstractPDF (836 KB)Isogeometric Preconditioners Based on Fast Solvers for the Sylvester EquationSIAM Journal on Scientific Computing, Vol. 38, No. 6 | 15 November 2016AbstractPDF (692 KB)Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive ConvolutionSIAM Journal on Numerical Analysis, Vol. 54, No. 3 | 2 June 2016AbstractPDF (1298 KB)Computational Methods for Linear Matrix EquationsSIAM Review, Vol. 58, No. 3 | 4 August 2016AbstractPDF (851 KB)A Proximal Strictly Contractive Peaceman--Rachford Splitting Method for Convex Programming with Applications to ImagingSIAM Journal on Imaging Sciences, Vol. 8, No. 2 | 24 June 2015AbstractPDF (1523 KB)On the Numerical Behavior of Matrix Splitting Iteration Methods for Solving Linear SystemsSIAM Journal on Numerical Analysis, Vol. 53, No. 4 | 14 July 2015AbstractPDF (364 KB)A Generalized Proximal Point Algorithm and Its Convergence RateSIAM Journal on Optimization, Vol. 24, No. 4 | 14 October 2014AbstractPDF (1948 KB)Fourth Order Accurate Scheme for the Space Fractional Diffusion EquationsSIAM Journal on Numerical Analysis, Vol. 52, No. 3 | 12 June 2014AbstractPDF (453 KB)A Strictly Contractive Peaceman--Rachford Splitting Method for Convex ProgrammingSIAM Journal on Optimization, Vol. 24, No. 3 | 17 July 2014AbstractPDF (930 KB)Balanced Splitting and Rebalanced SplittingSIAM Journal on Numerical Analysis, Vol. 51, No. 6 | 19 November 2013AbstractPDF (587 KB)A Convergence Analysis of the Peaceman--Rachford Scheme for Semilinear Evolution EquationsSIAM Journal on Numerical Analysis, Vol. 51, No. 4 | 2 July 2013AbstractPDF (226 KB)An $h$-Adaptive Operator Splitting Method for Two-Phase Flow in 3D Heterogeneous Porous MediaSIAM Journal on Scientific Computing, Vol. 35, No. 1 | 29 January 2013AbstractPDF (1869 KB)Domain Decomposition Approaches for Mesh Generation via the Equidistribution PrincipleSIAM Journal on Numerical Analysis, Vol. 50, No. 4 | 23 August 2012AbstractPDF (429 KB)Fast Multiple-Splitting Algorithms for Convex OptimizationSIAM Journal on Optimization, Vol. 22, No. 2 | 24 May 2012AbstractPDF (1115 KB)A Variational Approach for Sharpening High Dimensional ImagesSIAM Journal on Imaging Sciences, Vol. 5, No. 1 | 24 January 2012AbstractPDF (4775 KB)An LQP-Based Decomposition Method for Solving a Class of Variational InequalitiesSIAM Journal on Optimization, Vol. 21, No. 4 | 22 November 2011AbstractPDF (176 KB)An Error Analysis for Rational Galerkin Projection Applied to the Sylvester EquationSIAM Journal on Numerical Analysis, Vol. 49, No. 6 | 22 November 2011AbstractPDF (279 KB)Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov EquationSIAM Journal on Numerical Analysis, Vol. 49, No. 5 | 20 September 2011AbstractPDF (334 KB)Energy-Conserved Splitting Finite-Difference Time-Domain Methods for Maxwell's Equations in Three DimensionsSIAM Journal on Numerical Analysis, Vol. 48, No. 4 | 31 August 2010AbstractPDF (552 KB)Inverse Iteration for Purely Imaginary Eigenvalues with Application to the Detection of Hopf Bifurcations in Large-Scale ProblemsSIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 4 | 7 May 2010AbstractPDF (281 KB)From Functional Analysis to Iterative MethodsSIAM Review, Vol. 52, No. 2 | 6 May 2010AbstractPDF (279 KB)Nested Iterative Algorithms for Convex Constrained Image Recovery ProblemsSIAM Journal on Imaging Sciences, Vol. 2, No. 2 | 4 June 2009AbstractPDF (854 KB)General Projective Splitting Methods for Sums of Maximal Monotone OperatorsSIAM Journal on Control and Optimization, Vol. 48, No. 2 | 25 February 2009AbstractPDF (332 KB)Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and ConvergenceSIAM Journal on Optimization, Vol. 19, No. 3 | 31 October 2008AbstractPDF (335 KB)Efficient Preconditioning of Sequences of Nonsymmetric Linear SystemsSIAM Journal on Scientific Computing, Vol. 29, No. 5 | 28 September 2007AbstractPDF (242 KB)Robin–Robin Domain Decomposition Methods for the Stokes–Darcy CouplingSIAM Journal on Numerical Analysis, Vol. 45, No. 3 | 22 May 2007AbstractPDF (251 KB)An Alternating-Direction Implicit Orthogonal Spline Collocation Scheme for Nonlinear Parabolic Problems on Rectangular PolygonsSIAM Journal on Scientific Computing, Vol. 28, No. 3 | 4 August 2006AbstractPDF (241 KB)An Accelerated Splitting-up Method for Parabolic EquationsSIAM Journal on Mathematical Analysis, Vol. 37, No. 4 | 1 August 2006AbstractPDF (267 KB)An ADI-Like Preconditioner for Boltzmann TransportSIAM Journal on Scientific Computing, Vol. 26, No. 3 | 25 July 2006AbstractPDF (206 KB)Solving Degenerate Reaction-Diffusion Equations via Variable Step Peaceman--Rachford SplittingSIAM Journal on Scientific Computing, Vol. 25, No. 4 | 16 May 2012AbstractPDF (8559 KB)Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 24, No. 3 | 31 July 2006AbstractPDF (257 KB)Stabilized Explicit-Implicit Domain Decomposition Methods for the Numerical Solution of Parabolic EquationsSIAM Journal on Scientific Computing, Vol. 24, No. 1 | 25 July 2006AbstractPDF (216 KB)Difference Graphs of Block ADI MethodSIAM Journal on Numerical Analysis, Vol. 38, No. 3 | 26 July 2006AbstractPDF (162 KB)A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov EquationsSIAM Journal on Scientific Computing, Vol. 21, No. 4 | 25 July 2006AbstractPDF (377 KB)Domain Decomposition Operator Splittings for the Solution of Parabolic EquationsSIAM Journal on Scientific Computing, Vol. 19, No. 3 | 25 July 2006AbstractPDF (406 KB)Discrete-time Orthogonal Spline Collocation Methods for Schrödinger Equations in Two Space VariablesSIAM Journal on Numerical Analysis, Vol. 35, No. 2 | 25 July 2006AbstractPDF (458 KB)An ADI Method for Hysteretic Reaction-Diffusion SystemsSIAM Journal on Numerical Analysis, Vol. 34, No. 3 | 25 July 2006AbstractPDF (459 KB)Application of ADI Iterative Methods to the Restoration of Noisy ImagesSIAM Journal on Matrix Analysis and Applications, Vol. 17, No. 1 | 17 February 2012AbstractPDF (2458 KB)Coordination in Coarse-Grained DecompositionSIAM Journal on Optimization, Vol. 4, No. 4 | 13 July 2006AbstractPDF (1888 KB)Alternating Direction Preconditioning for Nonsymmetric Systems of Linear EquationsSIAM Journal on Scientific Computing, Vol. 15, No. 2 | 13 July 2006AbstractPDF (1688 KB)Alternating Direction Implicit Iteration for Systems with Complex SpectraSIAM Journal on Numerical Analysis, Vol. 28, No. 3 | 14 July 2006AbstractPDF (1238 KB)Vectorization of the Odd–Even Hopscotch Scheme and the Alternating Direction Implicit Scheme for the Two-Dimensional Burgers EquationsSIAM Journal on Scientific and Statistical Computing, Vol. 11, No. 2 | 13 July 2006AbstractPDF (1591 KB)The Solution of Two-Point Boundary Value Problems by the Alternating Group Explicit (AGE) MethodSIAM Journal on Scientific and Statistical Computing, Vol. 9, No. 3 | 13 July 2006AbstractPDF (639 KB)Tensor Product Generalized ADI Methods for Separable Elliptic ProblemsSIAM Journal on Numerical Analysis, Vol. 24, No. 1 | 14 July 2006AbstractPDF (1728 KB)Iterated Splitting Method of High Order for Time-Dependent Partial Differential EquationsSIAM Journal on Numerical Analysis, Vol. 21, No. 4 | 17 July 2006AbstractPDF (1855 KB)Multistep Splitting Methods of High Order for Initial Value ProblemsSIAM Journal on Numerical Analysis, Vol. 17, No. 3 | 17 July 2006AbstractPDF (1937 KB)Alternating Direction Implicit Methods for Parabolic Equations with a Mixed DerivativeSIAM Journal on Scientific and Statistical Computing, Vol. 1, No. 1 | 16 May 2012AbstractPDF (2857 KB)Splitting Algorithms for the Sum of Two Nonlinear OperatorsSIAM Journal on Numerical Analysis, Vol. 16, No. 6 | 17 July 2006AbstractPDF (1402 KB)Dynamic ADI Methods for Elliptic EquationsSIAM Journal on Numerical Analysis, Vol. 16, No. 5 | 17 July 2006AbstractPDF (2283 KB)The Alternating Phase Truncation Method for Numerical Solution of a Stefan ProblemSIAM Journal on Numerical Analysis, Vol. 16, No. 4 | 17 July 2006AbstractPDF (2605 KB)The Extrapolation of First Order Methods for Parabolic Partial Differential Equations. ISIAM Journal on Numerical Analysis, Vol. 15, No. 6 | 14 July 2006AbstractPDF (1229 KB)Numerical Solution of a Diffusion Consumption Problem with a Free BoundarySIAM Journal on Numerical Analysis, Vol. 12, No. 4 | 14 July 2006AbstractPDF (2304 KB)A Survey of Modern Numerical AnalysisSIAM Review, Vol. 15, No. 2 | 2 August 2006AbstractPDF (2464 KB)Iterative Solution of Implicit Approximations of Multidimensional Partial Differential EquationsSIAM Journal on Numerical Analysis, Vol. 5, No. 3 | 14 July 2006AbstractPDF (2718 KB)Rounding Errors in Alternating Direction Methods for Parabolic ProblemsSIAM Journal on Numerical Analysis, Vol. 5, No. 2 | 3 August 2006AbstractPDF (1267 KB)A New Computational Procedure for A.D.I. MethodsSIAM Journal on Numerical Analysis, Vol. 4, No. 2 | 14 July 2006AbstractPDF (593 KB)Multistage Alternating Direction MethodsSIAM Journal on Numerical Analysis, Vol. 3, No. 4 | 14 July 2006AbstractPDF (829 KB)Two-Level Difference Schemes for Hyperbolic SystemsSIAM Journal on Numerical Analysis, Vol. 3, No. 3 | 14 July 2006AbstractPDF (882 KB)A New Alternating Direction Method for Parabolic Equations in Three Space VariablesJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 4 | 13 July 2006AbstractPDF (641 KB)Nonsymmetric Difference EquationsJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 3 | 13 July 2006AbstractPDF (501 KB)Alternating Direction Schemes for the Heat Equation in a General DomainJournal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 2, No. 3 | 14 July 2006AbstractPDF (1121 KB)The Solution of Elliptic Difference Equations by Semi-Explicit Iterative TechniquesJournal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 2, No. 1 | 3 August 2006AbstractPDF (1718 KB)An Alternating Direction Method for Operator EquationsJournal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 4 | 13 July 2006AbstractPDF (461 KB)The Numerical Solution of the Dirichlet Problem for Laplace’s Equation by Linear ProgrammingJournal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 1 | 13 July 2006AbstractPDF (374 KB)Another Alternating-Direction-Implicit MethodJournal of the Society for Industrial and Applied Mathematics, Vol. 11, No. 4 | 13 July 2006AbstractPDF (290 KB)Numerical Studies of Transition From Laminar to Turbulent Flow Over a Flat PlateJournal of the Society for Industrial and Applied Mathematics, Vol. 10, No. 4 | 13 July 2006AbstractPDF (2277 KB)On Incomplete Iteration for Implicit Parabolic Difference EquationsJournal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 3 | 10 July 2006AbstractPDF (466 KB)An Alternating-Direction-Implicit Iteration TechniqueJournal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 2 | 10 July 2006AbstractPDF (1508 KB)Simultaneous, Successive and Alternating Direction Iteration SchemesJournal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 1 | 10 July 2006AbstractPDF (1569 KB)A Method of Block IterationJournal of the Society for Industrial and Applied Mathematics, Vol. 4, No. 4 | 10 July 2006AbstractPDF (667 KB)On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit MethodsJim Douglas, Jr.Journal of the Society for Industrial and Applied Mathematics, Vol. 3, No. 1 | 10 July 2006AbstractPDF (1365 KB) Volume 3, Issue 1| 1955Journal of the Society for Industrial and Applied Mathematics1-65 History Submitted:18 October 1954Published online:10 July 2006 InformationCopyright © 1955 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0103003Article page range:pp. 28-41ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

For more than three decades, the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) has provided a framework to quantify health loss due to diseases, injuries, and associated risk factors. This paper presents GBD 2023 findings on disease and injury burden and risk-attributable health loss, offering a global audit of the state of world health to inform public health priorities. This work captures the evolving landscape of health metrics across age groups, sexes, and locations, while reflecting on the remaining post-COVID-19 challenges to achieving our collective global health ambitions. The GBD 2023 combined analysis estimated years lived with disability (YLDs), years of life lost (YLLs), and disability-adjusted life-years (DALYs) for 375 diseases and injuries, and risk-attributable burden associated with 88 modifiable risk factors. Of the more than 310 000 total data sources used for all GBD 2023 (about 30% of which were new to this estimation round), more than 120 000 sources were used for estimation of disease and injury burden and 59 000 for risk factor estimation, and included vital registration systems, surveys, disease registries, and published scientific literature. Data were analysed using previously established modelling approaches, such as disease modelling meta-regression version 2.1 (DisMod-MR 2.1) and comparative risk assessment methods. Diseases and injuries were categorised into four levels on the basis of the established GBD cause hierarchy, as were risk factors using the GBD risk hierarchy. Estimates stratified by age, sex, location, and year from 1990 to 2023 were focused on disease-specific time trends over the 2010-23 period and presented as counts (to three significant figures) and age-standardised rates per 100 000 person-years (to one decimal place). For each measure, 95% uncertainty intervals [UIs] were calculated with the 2·5th and 97·5th percentile ordered values from a 250-draw distribution. Total numbers of global DALYs grew 6·1% (95% UI 4·0-8·1), from 2·64 billion (2·46-2·86) in 2010 to 2·80 billion (2·57-3·08) in 2023, but age-standardised DALY rates, which account for population growth and ageing, decreased by 12·6% (11·0-14·1), revealing large long-term health improvements. Non-communicable diseases (NCDs) contributed 1·45 billion (1·31-1·61) global DALYs in 2010, increasing to 1·80 billion (1·63-2·03) in 2023, alongside a concurrent 4·1% (1·9-6·3) reduction in age-standardised rates. Based on DALY counts, the leading level 3 NCDs in 2023 were ischaemic heart disease (193 million [176-209] DALYs), stroke (157 million [141-172]), and diabetes (90·2 million [75·2-107]), with the largest increases in age-standardised rates since 2010 occurring for anxiety disorders (62·8% [34·0-107·5]), depressive disorders (26·3% [11·6-42·9]), and diabetes (14·9% [7·5-25·6]). Remarkable health gains were made for communicable, maternal, neonatal, and nutritional (CMNN) diseases, with DALYs falling from 874 million (837-917) in 2010 to 681 million (642-736) in 2023, and a 25·8% (22·6-28·7) reduction in age-standardised DALY rates. During the COVID-19 pandemic, DALYs due to CMNN diseases rose but returned to pre-pandemic levels by 2023. From 2010 to 2023, decreases in age-standardised rates for CMNN diseases were led by rate decreases of 49·1% (32·7-61·0) for diarrhoeal diseases, 42·9% (38·0-48·0) for HIV/AIDS, and 42·2% (23·6-56·6) for tuberculosis. Neonatal disorders and lower respiratory infections remained the leading level 3 CMNN causes globally in 2023, although both showed notable rate decreases from 2010, declining by 16·5% (10·6-22·0) and 24·8% (7·4-36·7), respectively. Injury-related age-standardised DALY rates decreased by 15·6% (10·7-19·8) over the same period. Differences in burden due to NCDs, CMNN diseases, and injuries persisted across age, sex, time, and location. Based on our risk analysis, nearly 50% (1·27 billion [1·18-1·38]) of the roughly 2·80 billion total global DALYs in 2023 were attributable to the 88 risk factors analysed in GBD. Globally, the five level 3 risk factors contributing the highest proportion of risk-attributable DALYs were high systolic blood pressure (SBP), particulate matter pollution, high fasting plasma glucose (FPG), smoking, and low birthweight and short gestation-with high SBP accounting for 8·4% (6·9-10·0) of total DALYs. Of the three overarching level 1 GBD risk factor categories-behavioural, metabolic, and environmental and occupational-risk-attributable DALYs rose between 2010 and 2023 only for metabolic risks, increasing by 30·7% (24·8-37·3); however, age-standardised DALY rates attributable to metabolic risks decreased by 6·7% (2·0-11·0) over the same period. For all but three of the 25 leading level 3 risk factors, age-standardised rates dropped between 2010 and 2023-eg, declining by 54·4% (38·7-65·3) for unsafe sanitation, 50·5% (33·3-63·1) for unsafe water source, and 45·2% (25·6-72·0) for no access to handwashing facility, and by 44·9% (37·3-53·5) for child growth failure. The three leading level 3 risk factors for which age-standardised attributable DALY rates rose were high BMI (10·5% [0·1 to 20·9]), drug use (8·4% [2·6 to 15·3]), and high FPG (6·2% [-2·7 to 15·6]; non-significant). Our findings underscore the complex and dynamic nature of global health challenges. Since 2010, there have been large decreases in burden due to CMNN diseases and many environmental and behavioural risk factors, juxtaposed with sizeable increases in DALYs attributable to metabolic risk factors and NCDs in growing and ageing populations. This long-observed consequence of the global epidemiological transition was only temporarily interrupted by the COVID-19 pandemic. The substantially decreasing CMNN disease burden, despite the 2008 global financial crisis and pandemic-related disruptions, is one of the greatest collective public health successes known. However, these achievements are at risk of being reversed due to major cuts to development assistance for health globally, the effects of which will hit low-income countries with high burden the hardest. Without sustained investment in evidence-based interventions and policies, progress could stall or reverse, leading to widespread human costs and geopolitical instability. Moreover, the rising NCD burden necessitates intensified efforts to mitigate exposure to leading risk factors-eg, air pollution, smoking, and metabolic risks, such as high SBP, BMI, and FPG-including policies that promote food security, healthier diets, physical activity, and equitable and expanded access to potential treatments, such as GLP-1 receptor agonists. Decisive, coordinated action is needed to address long-standing yet growing health challenges, including depressive and anxiety disorders. Yet this can be only part of the solution. Our response to the NCD syndemic-the complex interaction of multiple health risks, social determinants, and systemic challenges-will define the future landscape of global health. To ensure human wellbeing, economic stability, and social equity, global action to sustain and advance health gains must prioritise reducing disparities by addressing socioeconomic and demographic determinants, ensuring equitable health-care access, tackling malnutrition, strengthening health systems, and improving vaccination coverage. We live in times of great opportunity. Gates Foundation and Bloomberg Philanthropies.

Timely and comprehensive analyses of causes of death stratified by age, sex, and location are essential for shaping effective health policies aimed at reducing global mortality. The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2023 provides cause-specific mortality estimates measured in counts, rates, and years of life lost (YLLs). GBD 2023 aimed to enhance our understanding of the relationship between age and cause of death by quantifying the probability of dying before age 70 years (70q0) and the mean age at death by cause and sex. This study enables comparisons of the impact of causes of death over time, offering a deeper understanding of how these causes affect global populations. GBD 2023 produced estimates for 292 causes of death disaggregated by age-sex-location-year in 204 countries and territories and 660 subnational locations for each year from 1990 until 2023. We used a modelling tool developed for GBD, the Cause of Death Ensemble model (CODEm), to estimate cause-specific death rates for most causes. We computed YLLs as the product of the number of deaths for each cause-age-sex-location-year and the standard life expectancy at each age. Probability of death was calculated as the chance of dying from a given cause in a specific age period, for a specific population. Mean age at death was calculated by first assigning the midpoint age of each age group for every death, followed by computing the mean of all midpoint ages across all deaths attributed to a given cause. We used GBD death estimates to calculate the observed mean age at death and to model the expected mean age across causes, sexes, years, and locations. The expected mean age reflects the expected mean age at death for individuals within a population, based on global mortality rates and the population's age structure. Comparatively, the observed mean age represents the actual mean age at death, influenced by all factors unique to a location-specific population, including its age structure. As part of the modelling process, uncertainty intervals (UIs) were generated using the 2·5th and 97·5th percentiles from a 250-draw distribution for each metric. Findings are reported as counts and age-standardised rates. Methodological improvements for cause-of-death estimates in GBD 2023 include a correction for the misclassification of deaths due to COVID-19, updates to the method used to estimate COVID-19, and updates to the CODEm modelling framework. This analysis used 55 761 data sources, including vital registration and verbal autopsy data as well as data from surveys, censuses, surveillance systems, and cancer registries, among others. For GBD 2023, there were 312 new country-years of vital registration cause-of-death data, 3 country-years of surveillance data, 51 country-years of verbal autopsy data, and 144 country-years of other data types that were added to those used in previous GBD rounds. The initial years of the COVID-19 pandemic caused shifts in long-standing rankings of the leading causes of global deaths: it ranked as the number one age-standardised cause of death at Level 3 of the GBD cause classification hierarchy in 2021. By 2023, COVID-19 dropped to the 20th place among the leading global causes, returning the rankings of the leading two causes to those typical across the time series (ie, ischaemic heart disease and stroke). While ischaemic heart disease and stroke persist as leading causes of death, there has been progress in reducing their age-standardised mortality rates globally. Four other leading causes have also shown large declines in global age-standardised mortality rates across the study period: diarrhoeal diseases, tuberculosis, stomach cancer, and measles. Other causes of death showed disparate patterns between sexes, notably for deaths from conflict and terrorism in some locations. A large reduction in age-standardised rates of YLLs occurred for neonatal disorders. Despite this, neonatal disorders remained the leading cause of global YLLs over the period studied, except in 2021, when COVID-19 was temporarily the leading cause. Compared to 1990, there has been a considerable reduction in total YLLs in many vaccine-preventable diseases, most notably diphtheria, pertussis, tetanus, and measles. In addition, this study quantified the mean age at death for all-cause mortality and cause-specific mortality and found noticeable variation by sex and location. The global all-cause mean age at death increased from 46·8 years (95% UI 46·6-47·0) in 1990 to 63·4 years (63·1-63·7) in 2023. For males, mean age increased from 45·4 years (45·1-45·7) to 61·2 years (60·7-61·6), and for females it increased from 48·5 years (48·1-48·8) to 65·9 years (65·5-66·3), from 1990 to 2023. The highest all-cause mean age at death in 2023 was found in the high-income super-region, where the mean age for females reached 80·9 years (80·9-81·0) and for males 74·8 years (74·8-74·9). By comparison, the lowest all-cause mean age at death occurred in sub-Saharan Africa, where it was 38·0 years (37·5-38·4) for females and 35·6 years (35·2-35·9) for males in 2023. Lastly, our study found that all-cause 70q0 decreased across each GBD super-region and region from 2000 to 2023, although with large variability between them. For females, we found that 70q0 notably increased from drug use disorders and conflict and terrorism. Leading causes that increased 70q0 for males also included drug use disorders, as well as diabetes. In sub-Saharan Africa, there was an increase in 70q0 for many non-communicable diseases (NCDs). Additionally, the mean age at death from NCDs was lower than the expected mean age at death for this super-region. By comparison, there was an increase in 70q0 for drug use disorders in the high-income super-region, which also had an observed mean age at death lower than the expected value. We examined global mortality patterns over the past three decades, highlighting-with enhanced estimation methods-the impacts of major events such as the COVID-19 pandemic, in addition to broader trends such as increasing NCDs in low-income regions that reflect ongoing shifts in the global epidemiological transition. This study also delves into premature mortality patterns, exploring the interplay between age and causes of death and deepening our understanding of where targeted resources could be applied to further reduce preventable sources of mortality. We provide essential insights into global and regional health disparities, identifying locations in need of targeted interventions to address both communicable and non-communicable diseases. There is an ever-present need for strengthened health-care systems that are resilient to future pandemics and the shifting burden of disease, particularly among ageing populations in regions with high mortality rates. Robust estimates of causes of death are increasingly essential to inform health priorities and guide efforts toward achieving global health equity. The need for global collaboration to reduce preventable mortality is more important than ever, as shifting burdens of disease are affecting all nations, albeit at different paces and scales. Gates Foundation.

Cancer is a leading cause of death globally. Accurate cancer burden information is crucial for policy planning, but many countries do not have up-to-date cancer surveillance data. To inform global cancer-control efforts, we used the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2023 framework to generate and analyse estimates of cancer burden for 47 cancer types or groupings by age, sex, and 204 countries and territories from 1990 to 2023, cancer burden attributable to selected risk factors from 1990 to 2023, and forecasted cancer burden up to 2050. Cancer estimation in GBD 2023 used data from population-based cancer registration systems, vital registration systems, and verbal autopsies. Cancer mortality was estimated using ensemble models, with incidence informed by mortality estimates and mortality-to-incidence ratios (MIRs). Prevalence estimates were generated from modelled survival estimates, then multiplied by disability weights to estimate years lived with disability (YLDs). Years of life lost (YLLs) were estimated by multiplying age-specific cancer deaths by the GBD standard life expectancy at the age of death. Disability-adjusted life-years (DALYs) were calculated as the sum of YLLs and YLDs. We used the GBD 2023 comparative risk assessment framework to estimate cancer burden attributable to 44 behavioural, environmental and occupational, and metabolic risk factors. To forecast cancer burden from 2024 to 2050, we used the GBD 2023 forecasting framework, which included forecasts of relevant risk factor exposures and used Socio-demographic Index as a covariate for forecasting the proportion of each cancer not affected by these risk factors. Progress towards the UN Sustainable Development Goal (SDG) target 3.4 aim to reduce non-communicable disease mortality by a third between 2015 and 2030 was estimated for cancer. In 2023, excluding non-melanoma skin cancers, there were 18·5 million (95% uncertainty interval 16·4 to 20·7) incident cases of cancer and 10·4 million (9·65 to 10·9) deaths, contributing to 271 million (255 to 285) DALYs globally. Of these, 57·9% (56·1 to 59·8) of incident cases and 65·8% (64·3 to 67·6) of cancer deaths occurred in low-income to upper-middle-income countries based on World Bank income group classifications. Cancer was the second leading cause of deaths globally in 2023 after cardiovascular diseases. There were 4·33 million (3·85 to 4·78) risk-attributable cancer deaths globally in 2023, comprising 41·7% (37·8 to 45·4) of all cancer deaths. Risk-attributable cancer deaths increased by 72·3% (57·1 to 86·8) from 1990 to 2023, whereas overall global cancer deaths increased by 74·3% (62·2 to 86·2) over the same period. The reference forecasts (the most likely future) estimate that in 2050 there will be 30·5 million (22·9 to 38·9) cases and 18·6 million (15·6 to 21·5) deaths from cancer globally, 60·7% (41·9 to 80·6) and 74·5% (50·1 to 104·2) increases from 2024, respectively. These forecasted increases in deaths are greater in low-income and middle-income countries (90·6% [61·0 to 127·0]) compared with high-income countries (42·8% [28·3 to 58·6]). Most of these increases are likely due to demographic changes, as age-standardised death rates are forecast to change by -5·6% (-12·8 to 4·6) between 2024 and 2050 globally. Between 2015 and 2030, the probability of dying due to cancer between the ages of 30 years and 70 years was forecasted to have a relative decrease of 6·5% (3·2 to 10·3). Cancer is a major contributor to global disease burden, with increasing numbers of cases and deaths forecasted up to 2050 and a disproportionate growth in burden in countries with scarce resources. The decline in age-standardised mortality rates from cancer is encouraging but insufficient to meet the SDG target set for 2030. Effectively and sustainably addressing cancer burden globally will require comprehensive national and international efforts that consider health systems and context in the development and implementation of cancer-control strategies across the continuum of prevention, diagnosis, and treatment. Gates Foundation, St Jude Children's Research Hospital, and St Baldrick's Foundation.

Breast cancer is a leading cause of mortality and morbidity among females worldwide. As part of the Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2023, we provided an updated comprehensive assessment of the epidemiological trends, disease burden, and risk factors associated with breast cancer globally, regionally, and nationally from 1990 to 2023. Breast cancer incidence, mortality, prevalence, years lived with disability (YLDs), years of life lost (YLLs), and disability-adjusted life-years (DALYs) were estimated by age and sex for 204 countries and territories from 1990 to 2023. Mortality estimates were generated using GBD Cause of Death Ensemble models, leveraging data from population-based cancer registration systems, vital registration systems, and verbal autopsies. Mortality-to-incidence ratios were calculated to derive both mortality and incidence estimates. Prevalence was calculated by combining incidence and modelled survival estimates. YLLs were established by multiplying age-specific deaths with the GBD standard life expectancy at the age of death. YLDs were estimated by applying disability weights to prevalence estimates. The sum of YLLs and YLDs equalled the number of DALYs. Breast cancer burden attributable to seven risk factors was examined through the comparative risk assessment framework. The GBD forecasting framework was used to forecast breast cancer incidence and mortality from 2024 to 2050. Age-standardised rates were calculated for each metric using the GBD 2023 world standard population. In 2023, there were an estimated 2·30 million (95% uncertainty interval [UI] 2·01 to 2·61) breast cancer incident cases, 764 000 deaths (672 000 to 854 000), and 24·1 million (21·3 to 27·5) DALYs among females globally. In the World Bank low-income group, where a low age-standardised incidence rate (ASIR) was estimated (44·2 per 100 000 person-years [31·2 to 58·4]), the age-standardised mortality rate (ASMR) was the highest (24·1 per 100 000 [16·8 to 31·9]). The highest ASIR was in the high-income group (75·7 per 100 000 [67·1 to 84·0]), and the lowest ASMR was in the upper-middle-income group (11·2 per 100 000 [10·2 to 12·3]). Between 1990 and 2023, the ASIR in the low-income group increased by 147·2% (38·1 to 271·7), compared with a 1·2% (-11·5 to 17·2) change in the high-income group. The ASMR decreased in the high-income group, changing by -29·9% (-33·6 to -25·9), but increased by 99·3% (12·5 to 202·9) in the low-income group. The increase in age-standardised DALY rates followed that of ASMRs. Risk factors such as dietary risks, tobacco use, and high fasting plasma glucose contributed to 28·3% (16·6 to 38·9) of breast cancer DALYs in 2023. The risk factors with a decrease in attributable DALYs between 1990 and 2023 were high alcohol use and tobacco. By 2050, the global incident cases of breast cancer among females were forecast to reach 3·56 million (2·29 to 4·83), with 1·37 million (0·841 to 2·02) deaths. The stable incidence and declining mortality rates of female breast cancer in high-income nations reflect success in screening, diagnosis, and treatment. In contrast, the concurrent rise in incidence and mortality in other regions signals health system deficits. Without effective interventions, many countries will fall short of the WHO Global Breast Cancer Initiative's ambitious target of achieving an annual reduction of 2·5% in age-standardised mortality rates by 2040. The mounting breast cancer burden, disproportionately affecting some of the world's most vulnerable populations, will further exacerbate health inequalities across the globe without decisive immediate action. Gates Foundation, St Jude Children's Research Hospital.

Smoking is the leading behavioural risk factor for mortality globally, accounting for more than 175 million deaths and nearly 4·30 billion years of life lost (YLLs) from 1990 to 2021. The pace of decline in smoking prevalence has slowed in recent years for many countries, and although strategies have recently been proposed to achieve tobacco-free generations, none have been implemented to date. Assessing what could happen if current trends in smoking prevalence persist, and what could happen if additional smoking prevalence reductions occur, is important for communicating the effect of potential smoking policies. In this analysis, we use the Institute for Health Metrics and Evaluation's Future Health Scenarios platform to forecast the effects of three smoking prevalence scenarios on all-cause and cause-specific YLLs and life expectancy at birth until 2050. YLLs were computed for each scenario using the Global Burden of Disease Study 2021 reference life table and forecasts of cause-specific mortality under each scenario. The reference scenario forecasts what could occur if past smoking prevalence and other risk factor trends continue, the Tobacco Smoking Elimination as of 2023 (Elimination-2023) scenario quantifies the maximum potential future health benefits from assuming zero percent smoking prevalence from 2023 onwards, whereas the Tobacco Smoking Elimination by 2050 (Elimination-2050) scenario provides estimates for countries considering policies to steadily reduce smoking prevalence to 5%. Together, these scenarios underscore the magnitude of health benefits that could be reached by 2050 if countries take decisive action to eliminate smoking. The 95% uncertainty interval (UI) of estimates is based on the 2·5th and 97·5th percentile of draws that were carried through the multistage computational framework. Global age-standardised smoking prevalence was estimated to be 28·5% (95% UI 27·9-29·1) among males and 5·96% (5·76-6·21) among females in 2022. In the reference scenario, smoking prevalence declined by 25·9% (25·2-26·6) among males, and 30·0% (26·1-32·1) among females from 2022 to 2050. Under this scenario, we forecast a cumulative 29·3 billion (95% UI 26·8-32·4) overall YLLs among males and 22·2 billion (20·1-24·6) YLLs among females over this period. Life expectancy at birth under this scenario would increase from 73·6 years (95% UI 72·8-74·4) in 2022 to 78·3 years (75·9-80·3) in 2050. Under our Elimination-2023 scenario, we forecast 2·04 billion (95% UI 1·90-2·21) fewer cumulative YLLs by 2050 compared with the reference scenario, and life expectancy at birth would increase to 77·6 years (95% UI 75·1-79·6) among males and 81·0 years (78·5-83·1) among females. Under our Elimination-2050 scenario, we forecast 735 million (675-808) and 141 million (131-154) cumulative YLLs would be avoided among males and females, respectively. Life expectancy in 2050 would increase to 77·1 years (95% UI 74·6-79·0) among males and 80·8 years (78·3-82·9) among females. Existing tobacco policies must be maintained if smoking prevalence is to continue to decline as forecast by the reference scenario. In addition, substantial smoking-attributable burden can be avoided by accelerating the pace of smoking elimination. Implementation of new tobacco control policies are crucial in avoiding additional smoking-attributable burden in the coming decades and to ensure that the gains won over the past three decades are not lost. Bloomberg Philanthropies and the Bill & Melinda Gates Foundation.

Identification of 3D cephalometric landmarks that serve as proxy to the shape of human skull is the fundamental step in cephalometric analysis. Since manual landmarking from 3D computed tomography (CT) images is a cumbersome task even for the trained experts, automatic 3D landmark detection system is in a great need. Recently, automatic landmarking of 2D cephalograms using deep learning (DL) has achieved great success, but 3D landmarking for more than 80 landmarks has not yet reached a satisfactory level, because of the factors hindering machine learning such as the high dimensionality of the input data and limited amount of training data due to the ethical restrictions on the use of medical data. This paper presents a semi-supervised DL method for 3D landmarking that takes advantage of anonymized landmark dataset with paired CT data being removed. The proposed method first detects a small number of easy-to-find reference landmarks, then uses them to provide a rough estimation of the all landmarks by utilizing the low dimensional representation learned by variational autoencoder (VAE). The anonymized landmark dataset is used for training the VAE. Finally, coarse-to-fine detection is applied to the small bounding box provided by rough estimation, using separate strategies suitable for the mandible and the cranium. For mandibular landmarks, patch-based 3D CNN is applied to the segmented image of the mandible (separated from the maxilla), in order to capture 3D morphological features of mandible associated with the landmarks. We detect 6 landmarks around the condyle all at once rather than one by one, because they are closely related to each other. For cranial landmarks, we again use the VAE-based latent representation for more accurate annotation. In our experiment, the proposed method achieved a mean detection error of 2.88 mm for 90 landmarks using only 15 paired training data.

Recent studies have demonstrated that HER2 and MET receptor tyrosine kinases are co-overexpressed in a subset esophageal adenocarcinoma (EAC). We therefore studied the usefulness of combining HER2 and MET targeting by small-molecule inhibitors lapatinib and foretinib, respectively, both in in-vitro and in-vivo models of experimental EAC. We characterized MET and HER2 activation in a panel of human EAC cell lines, and the differential susceptibility of these EAC cell lines to single agent or combination of foretinib and lapatinib. We then explored the antitumor efficacy with survival advantage following foretinib and lapatinib monotherapy and in combination in murine subcutaneous xenograft and peritoneal metastatic survival models of human EAC. The OE33 EAC cell line with strong expression of phosphorylated both MET and HER2, demonstrated reduced sensitivity to foretinib and lapatinib when used as a single agent. The co-administration of foretinib and lapatinib effectively inhibited both MET and HER2 phosphorylation, enhanced inhibition of cell proliferation and xenograft tumor growth by inducing apoptosis, and significantly enhanced mouse overall survival, overcoming single agent resistance. In the OE19 EAC cell line with mainly HER2 phosphorylation, and the ESO51 EAC cell line with mainly MET phosphorylation, profound cell growth inhibition with induction of apoptosis was observed in response to single agent with lack of enhanced growth inhibition when the two agents were combined. These data suggest that combination therapy with foretinib and lapatinib should be tested as a treatment option for HER2 positive patients with MET-overexpressing EAC, and could be a novel treatment strategy for specific EAC patients.

Telemedicine use increased during the COVID-19 pandemic and has remained a regular component of health care delivery. However, the financial implications of this change for health systems' reimbursement and utilization remain unclear. To compare 30-day episode charges and subsequent visits after telemedicine and in-person index visits. The target trial emulation conducted in this comparative effectiveness research included ambulatory in-person and telemedicine visit data from an academic health system comprising 5 hospitals in Pennsylvania from January 1 to April 30, 2024. Analyses focused on 10 high-volume clinical conditions commonly managed through telemedicine. Telemedicine visits vs in-person visits. Outcomes included episode charges (the billed amount submitted for reimbursement to insurers and patients, excluding physician professional and facility fees for the index encounter) in an episode window from 7 days before to 30 days after the index visit and the number of subsequent visits within the episode window. Linear regression and Poisson regression with propensity score matching were conducted to adjust for demographic, clinical, socioeconomic, and contextual factors. A total of 163 308 visits (108 383 [66.4%] among females; mean [SD] patient age, 49.2 [19.1] years) were included in this study. After propensity score matching, the mean 30-day episode charge was $96.60 (95% CI, $92.24-$100.96) for telemedicine encounters and $509.21 (95% CI, $500.65-$517.77) for in-person encounters (mean difference, $412.62; 95% CI, $403.01-$422.22). Additionally, telemedicine visits were associated with fewer follow-up visits per 30-day episode than were in-person visits (mean [SD], 3.44 [5.38] vs 4.44 [7.41] visits; comparative reduction, 23% [95% CI, 20%-26%]). For mental and behavioral disorders, 3 categories-depressive disorders (-$69.47; 95% CI, -$100.90 to -$38.04), anxiety and fear-related disorders ($38.06; 95% CI, $23.14 to $52.99), and neurodevelopmental disorders (-$28.88; 95% CI, -$54.72 to -$3.04)-exhibited comparable episode charges for telemedicine vs in-person encounters. In this comparative effectiveness research using target trial emulation of outpatient telemedicine and in-person visits, telemedicine visits overall were associated with lower charges and fewer subsequent visits within the 30-day episode than were in-person visits. For mental and behavioral conditions, charges were comparable. These findings suggest that telemedicine may serve as a lower-charge alternative to in-person care without increasing the need for subsequent visits.

Within two years, the world has experienced a pandemic phenomenon that changed almost everything in the macro and micro-environment; the economy, the community's social life, education, and many other fields. Governments started to collaborate with health institutions and the WHO to control the pandemic spread, followed by many regulations such as wearing masks, maintaining social distance, and home office work. While the virus has a high transmission rate and shows many mutated forms, another discussion appeared in the community: the fear of getting infected and the side effects of the produced vaccines. The community started to face uncertain information spread through some networks keeping the discussions of side effects on-trend. However, this pollution spread confused the community more and activated multi fears related to the virus and the vaccines. This paper establishes a mathematical model of COVID-19, including the community's fear of getting infected and the possible side effects of the vaccines. These fears appeared from uncertain information spread through some social sources. Our primary target is to show the psychological effect on the community during the pandemic stage. The theoretical study contains the existence and uniqueness of the IVP and, after that, the local stability analysis of both equilibrium points, the disease-free and the positive equilibrium point. Finally, we show the global asymptotic stability holds under specific conditions using a suitable Lyapunov function. In the end, we conclude our theoretical findings with some simulations.

The increased adoption of the internet, social media, wearable devices, e-health services, and other technology-driven services in medicine and healthcare has led to the rapid generation of various types of digital data, providing a valuable data source beyond the confines of traditional clinical trials, epidemiological studies, and lab-based experiments. We provide a brief overview on the type and sources of real-world data and the common models and approaches to utilize and analyze real-world data. We discuss the challenges and opportunities of using real-world data for evidence-based decision making This review does not aim to be comprehensive or cover all aspects of the intriguing topic on RWD (from both the research and practical perspectives) but serves as a primer and provides useful sources for readers who interested in this topic. Real-world hold great potential for generating real-world evidence for designing and conducting confirmatory trials and answering questions that may not be addressed otherwise. The voluminosity and complexity of real-world data also call for development of more appropriate, sophisticated, and innovative data processing and analysis techniques while maintaining scientific rigor in research findings, and attentions to data ethics to harness the power of real-world data.

The same dataset can be analysed in different justifiable ways to answer the same research question, potentially challenging the robustness of empirical science1-3. In this crowd initiative, we investigated the degree to which research findings in the social and behavioural sciences are contingent on analysts' choices. We examined a stratified random sample of 100 studies published between 2009 and 2018, in which, for one claim per study, at least five reanalysts independently reanalysed the original data. The statistical appropriateness of the reanalyses was assessed in peer evaluations, and the robustness indicators were inspected along a range of research characteristics and study designs. We found that 34% of the independent reanalyses yielded the same result (within a tolerance region of ±0.05 Cohen's d) as the original report; with a four times broader tolerance region, this indicator increased to 57%. Of the reanalyses conducted, 74% reached the same conclusion as the original investigation, 24% yielded no effects or inconclusive results and 2% reported the opposite effect. This exploratory study indicates that the common single-path analyses in social and behavioural research should not be simply assumed to be robust to alternative analyses4. Therefore, we recommend the development and use of practices to explore and communicate this neglected source of uncertainty.

A defining challenge of the 21st century is meeting the nutritional demands of the growing human population, under a scenario of limited land and water resources and under the specter of climate change. The Vavilov seed bank contains numerous landraces collected nearly a hundred years ago, and thus may contain 'genetic gems' with the potential to enhance modern breeding efforts. Here, we analyze 407 landraces, sampled from major historic centers of chickpea cultivation and secondary diversification. Genome-Wide Association Studies (GWAS) conducted on both phenotypic traits and bioclimatic variables at landraces sampling sites as extended phenotypes resulted in 84 GWAS hits associated to various regions. The novel haploblock-based test identified haploblocks enriched for single nucleotide polymorphisms (SNPs) associated with phenotypes and bioclimatic variables. Subsequent bi-clustering of traits sharing enriched haploblocks underscored both non-random distribution of SNPs among several haploblocks and their association with multiple traits. We hypothesize that these clusters of pleiotropic SNPs represent co-adapted genetic complexes to a range of environmental conditions that chickpea experienced during domestication and subsequent geographic radiation. Linking genetic variation to phenotypic data and a wealth of historic information preserved in historic seed banks are the keys for genome-based and environment-informed breeding intensification.

The last few decades have verified the vital roles of microRNAs in the development of human diseases and witnessed the increasing interest in the prediction of potential disease-miRNA associations. Owning to the open access of many miRNA-related databases, up until recently, kinds of feasible in silico models have been proposed. In this work, we developed a computational model of Maximal Entropy Random Walk on heterogenous network for MiRNA-disease Association prediction (MERWMDA). MERWMDA integrated known disease-miRNA association, pair-wise functional relation of miRNAs and pair-wise semantic relation of diseases into a heterogenous network comprised of disease and miRNA nodes full of information. As a kind of widely-applied biased walk process with more randomness, MERW was then implemented on the heterogenous network to reveal potential disease-miRNA associations. Cross validation was further performed to evaluate the performance of MERWMDA. As a result, MERWMDA obtained AUCs of 0.8966 and 0.8491 respectively in the aspect of global and local leave-one-out cross validation. What' more, three different case study strategies on four human complex diseases were conducted to comprehensively assess the quality of the model. Specifically, one kind of case study on Esophageal cancer and Prostate cancer were conducted based on HMDD v2.0 database. 94% and 88% out of the top 50 ranked miRNAs were confirmed by recent literature, respectively. To simulate new disease without known related miRNAs, Lung cancer (confirmed ratio 94%) associated miRNAs were removed for case study. Lymphoma (verified ratio 88%) was adopted to assess the prediction robustness of MERWMDA based on HMDD v1.0 database. We anticipated that MERWMDA could offer valuable candidates for in vitro biomedical experiments in future.

Racial/ethnic differences are associated with the symptoms and conditions of post-acute sequelae SARS-CoV-2 infection (PASC) in adults. These differences may exist among children and warrant further exploration. We conducted a retrospective cohort study with difference-in-differences analyzes to assess these differences in children and adolescents under the age of 21. The study utilized data from the RECOVER Initiative in the United States, which aims to learn about the long-term effects of COVID-19. The cohort included 225,723 patients with SARS-CoV-2 infection or COVID-19 diagnosis between March 2020 and October 2022. The study compared minority racial/ethnic groups to Non-Hispanic White (NHW) individuals, stratified by severity during the acute phase of COVID-19. Within the severe group, Asian American/Pacific Islanders (AAPI) had a higher prevalence of fever/chills and respiratory signs and symptoms, Hispanic patients showed greater hair loss prevalence in severe COVID-19 cases, while Non-Hispanic Black (NHB) patients had fewer skin symptoms in comparison to NHW patients. Within the non-severe group, AAPI patients had increased POTS/dysautonomia and respiratory symptoms, and NHB patients showed more cognitive symptoms than NHW patients. In conclusion, racial/ethnic differences related to COVID-19 exist among PASC symptoms and conditions in pediatrics, and these differences are associated with the severity of illness during acute COVID-19.