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We prove error estimates and convergence rates for the FBP approximation of target functions from Sobolev spaces $$\\mathrm H^\\alpha ({\\mathbb {R}}^2)$$ of fractional order $$\\alpha >0$$, where we bound the FBP approximation error, which is incurred by the application of a low-pass filter, with respect to the weaker norms of the rougher Sobolev spaces $$\\mathrm H^\\sigma ({\\mathbb {R}}^2)$$, for $${0 \\le \\sigma \\le \\alpha }$$. In particular, we generalize our previous results to non band-limited filter functions and show that the decay rate of the error saturates at fractional order depending on smoothness properties of the filter’s window function at the origin. The theoretical results are supported by numerical simulations.\n",{"EN":103},"Saturation rates of filtered back projection approximations",{"VOID":105},"[\"2971340172418410776\"]",{"VOID":107},"Beckmann, M., Iske, A.: Sobolev error estimates for filtered back projection reconstructions. In: 2017 International Conference on Sampling Theory and Applications (SampTA), pp. 251-255 (2017)\nBeckmann, M., Iske, A.: Error estimates and convergence rates for filtered back projection. Math. Comput. 88, 801–835 (2019)\nBeckmann, M., Iske, A.: Convergence rates for Hölder-windows in filtered back projection. In: 2019 International Conference on Sampling Theory and Applications (SampTA)\nEngl, H., Hanke, M., Neubauer, A.: Regularization of Ill-Posed Problems. Kluwer, Dordrecht (2000)\nFaridani, A., Ritman, E.: High-resolution computed tomography from efficient sampling. Inverse Probl. 16, 635–650 (2000)\nFeeman, T.: The Mathematics of Medical Imaging: A Beginner’s Guide. Springer, New York (2015)\nHelgason, S.: The Radon Transform. Birkhäuser, Boston (1999)\nMadych, W.: Summability and approximate reconstruction from Radon transform data. In: Grinberg, E., Quinto, T. (eds.) Integral Geometry and Tomography, pp. 189–219. American Mathematical Society, Providence (1990)\nMunshi, P., Rathore, R., Ram, K., Kalra, M.: Error estimates for tomographic inversion. Inverse Probl. 7, 399–408 (1991)\nMunshi, P.: Error analysis of tomographic filters I: theory. NDT & E Int. 25, 191–194 (1992)\nMunshi, P., Rathore, R., Ram, K.S., Kalra, M.: Error analysis of tomographic filters II: results. NDT & E Int. 26, 235–240 (1993)\nNatterer, F.: A Sobolev space analysis of picture reconstruction. SIAM J. Appl. Math. 39, 402–411 (1980)\nNatterer, F.: The Mathematics of Computerized Tomography. SIAM, Philadelphia (2001)\nNatterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM, Philadelphia (2001)\nPopov, D.: On convergence of a class of algorithms for the inversion of the numerical Radon transform. In: Gelfand, I., Gindikin, S. (eds.) Mathematical Problems of Tomography, pp. 7–65. American Mathematical Society, Providence (1990)\nQu, G.: Convergence of FBP algorithm for tomography. Acta Math. Appl. Sin. 32, 963–968 (2016)\nRadon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte Sächsische Akademie der Wissenschaften 69, 262–277 (1917)\nRieder, A., Faridani, A.: The semidiscrete filtered backprojection algorithm is optimal for tomographic inversion. SIAM J. Numer. Anal. 41, 869–892 (2003)\nRieder, A., Schuster, T.: The approximate inverse in action II: convergence and stability. Math. Comput. 72, 399–1415 (2003)\nRieder, A., Schneck, A.: Optimality of the fully discrete filtered backprojection algorithm for tomographic inversion. Numer. Math. 108, 151–175 (2007)\nShepp, L., Logan, B.: The Fourier reconstruction of a head section. IEEE Trans. Nucl. Sci. 21, 21–43 (1974)\nSmith, K., Salmon, D., Wagner, S.: Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Am. Math. Soc. 83, 1227–1270 (1977)\nSmith, K., Keinert, F.: Mathematical foundations of computed tomography. Appl. 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In particular we deal with the Odd and Even Lidstone-type and the Generalized Lidstone interpolatory problems with respect to a linear functional $$L_1$$ and, respectively, $$L_2$$. Estimations of the remainder for the related interpolation polynomials are given. Numerical examples are provided. Some possible applications of these interpolant polynomials to BVPs, expansions of analytical real functions and numerical quadrature are sketched.",{"EN":214},"Odd and Even Lidstone-type polynomial sequences. Part 2: applications",{"VOID":216},"[\"15964318494032667326\"]",{"VOID":218},"Agarwal, R.P., Pinelas, S., Wong, P.J.: Complementary Lidstone interpolation and boundary value problems. J. Inequal. Appl. 2009(1), 624–631 (2009)\nAgarwal, R.P., Wong, P.J.: Quasilinearization and approximate quasilinearization for Lidstone boundary value problems. Int. J. Comput. Math. 42(1–2), 99–116 (1992)\nAgarwal, R.P., Wong, P.J.: Explicit error bounds for the derivatives of piecewise-Lidstone interpolation. J. Comput. Appl. Math. 58(1), 67–81 (1995)\nAgarwal, R.P., Wong, P.J.: Error Inequalities in Polynomial Interpolation and Their Applications, vol. 262. Springer, Berlin (2012)\nBoas, R.: Representations for completely convex functions. Am. J. Math. 81(3), 709–714 (1959)\nBoas, R., Buck, R.: Polynomial Expansions of Analytic Functions. Springer, Berlin (1958)\nCostabile, F., Dell’Accio, F., Luceri, R.: Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values. J. Comput. Appl. Math. 176(1), 77–90 (2005)\nCostabile, F., Napoli, A.: A special class of polynomials related to non-classic general interpolatory problems. Integral Transforms Spec. Funct. 20(7), 539–550 (2009)\nCostabile, F., Napoli, A.: A class of collocation methods for numerical integration of initial value problems. Comput. Math. Appl. 62(8), 3221–3235 (2011)\nCostabile, F., Napoli, A.: A new spectral method for a class of linear boundary value problems. J. Comput. Appl. Math. 292, 329–341 (2016)\nCostabile, F.A.: Modern Umbral Calculus: An Elementary Introduction with Applications to Linear Interpolation and Operator Approximation Theory, vol. 72. Walter de Gruyter, Berlin (2019)\nCostabile, F.A., Gualtieri, M.I., Napoli, A.: Relationship between interpolation and differential equations: a class of collocation methods. In: Reyhanoglu, M. (ed.) Dynamical Systems—Analytical and Computational Techniques. InTech, pp. 169–189 (2017)\nCostabile, F.A., Gualtieri, M.I., Napoli, A.: Recurrence relations and determinant forms for general polynomial sequences. Application to Genocchi polynomials. Integral Transforms Spec. Funct. 30, 1–16 (2018)\nCostabile, F.A., Gualtieri, M.I., Napoli, A., Altomare, M.: Odd and even Lidstone-type polynomial sequences. Part 1: basic topics. Adv. Differ. Equ. 2018(1), 299 (2018)\nCostabile, F.A., Longo, E., Luceri, R.: A new proof of uniform convergence of Bernoulli and Lidstone series for entire real functions of exponential type. Rendiconti dell’Istituto Lombardo Accademia di Scienze e Lettere, Classe di Scienze Matematiche Fisiche e Naturali 143, 63–70 (2009)\nCostabile, F.A., Napoli, A.: Collocation for high-order differential equations with Lidstone boundary conditions. J. Appl. Math. 2012, 1–20 (2012)\nCostabile, F.A., Napoli, A.: A class of Birkhoff–Lagrange-collocation methods for high order boundary value problems. Appl. Numer. Math. 116, 129–140 (2017)\nCostabile, F.A., Serpe, A.: An algebraic approach to Lidstone polynomials. Appl. Math. Lett. 20(4), 387–390 (2007)\nDavis, P., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York (1975)\nDavis, P.J.: Interpolation and Approximation. Courier Corporation, New York (1975)\nEngels, H.: Numerical Quadrature and Cubature. Academic Press, New York (1980)\nKrylov, V.I., Stroud, A.H.: Approximate calculation of integrals. ACM monograph series. Macmillan, New York, NY (1962)\nLidstone, G.J.: Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types. Proc. Edinb. Math. Soc. (Ser. 2) 2(1), 16–19 (1930)\nNapoli, A., Abd-Elhameed, W.: An innovative harmonic numbers operational matrix method for solving initial value problems. Calcolo 54(1), 57–76 (2017)\nPethe, S., Sharma, A.: Modified Abel expansion and a subclass of completely convex functions. SIAM J. Math. Anal. 3(3), 546–558 (1972)\nPortisky, H.: On certain polynomial and other approximations to analytic functions. Proc. Natl. Acad. Sci. 16(1), 83–85 (1930)\nRenka, R., Cline, A.: A triangle-based C\\(^1\\) interpolation method. Rocky Mt. J. Math. 14, 223–237 (1984)\nSchoenberg, I.: On certain two-point expansions of integral functions of exponential type. Bull. Am. Math. Soc. 42(4), 284–288 (1936)\nWhittaker, J.M.: On Lidstone’s series and two-point expansions of analytic functions. Proc. Lond. Math. Soc. 2(1), 451–469 (1934)\nWidder, D.: Completely convex functions and Lidstone series. Trans. Am. Math. Soc. 51(2), 387–398 (1942)\nWong, P., Agarwal, R.: Sharp error bounds for the derivatives of Lidstone-spline interpolation. Comput. Math. Appl. 28(9), 23–53 (1994)\nWong, P., Agarwal, R.: Sharp error bounds for the derivatives of Lidstone-spline interpolation II. Comput. Math. 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classical buckling eigenvalue problem for a square plate clamped along its boundary is here considered. By using the Rayleigh-Ritz method and the method of orthogonal invariants, we obtain upper and lower bounds for the first 60 eigenvalues. Numerical tables are given. The multiplicity of the first eigenvalues and the symmetries of the corresponding eigenfunctions are also studied.",{"EN":326},"Difetti ed eccessi degli autovalori del classico problema di «buckling»",{"VOID":328},"[]",{"EN":330},"",{"VOID":332},"10.1007\u002FBF02576188","2024-06-26T03:45:42.990+00:00",[335],"EN","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02576188",[338,353],{"id":339,"sortIndex":20,"researcher":19,"roles":340,"affiliations":341,"properties":350,"displayName":352,"givenName":19,"familyName":19},"ea35e318-fb16-4578-9d63-979eb4f614b1",[],[342],{"id":343,"sortIndex":20,"affiliation":344,"properties":19},"7cf1b905-39c5-49ca-858e-a0f482ef541a",{"id":343,"createTime":19,"updateTime":19,"relativeEntities":345,"slug":19,"properties":346,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":349,"statistic":19},[],{"title":347},{"EN":348},"Departimento di Meccanica, Facoltà di Ingegneria, Università di Brescia, Brescia, Italy",[],{"title":351},{"EN":352},"A. Aimi",{"id":354,"sortIndex":85,"researcher":19,"roles":355,"affiliations":356,"properties":363,"displayName":365,"givenName":19,"familyName":19},"1d4dc494-a04c-4a39-a909-c264aa12f033",[],[357],{"id":343,"sortIndex":20,"affiliation":358,"properties":19},{"id":343,"createTime":19,"updateTime":19,"relativeEntities":359,"slug":19,"properties":360,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":362,"statistic":19},[],{"title":361},{"EN":348},[],{"title":364},{"EN":365},"M. Diligenti",{"url":19,"publisher":367,"properties":19},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":368,"slug":10,"properties":369,"entityType":17,"verifyStatus":18,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":20,"subjectFields":373,"manageAffiliations":382,"indexDatabases":388,"url":79,"thumbnailPath":19,"statistic":403,"gsStatistic":19,"type":88,"analyzePriority":19},[],{"issn":370,"title":371,"eissn":372},{"VOID":13},{"EN":10},{"VOID":16},[374,378],{"id":23,"createTime":19,"updateTime":19,"relativeEntities":375,"label":376,"description":377,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":26},{},{"id":29,"createTime":19,"updateTime":19,"relativeEntities":379,"label":380,"description":381,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":32},{},[383],{"id":36,"createTime":19,"updateTime":19,"relativeEntities":384,"slug":19,"properties":385,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":387,"statistic":19},[],{"title":386},{"EN":40},[42],[389,396],{"id":45,"indexDatabase":390,"url":58,"indexYears":19,"academicFieldIds":395,"indexDatabaseRanking":19},{"id":47,"createTime":19,"updateTime":19,"relativeEntities":391,"label":392,"description":393,"key":54,"publicationTags":394,"standard":19},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":397,"url":73,"indexYears":74,"academicFieldIds":402,"indexDatabaseRanking":78},{"id":64,"createTime":19,"updateTime":19,"relativeEntities":398,"label":399,"description":400,"key":70,"publicationTags":401,"standard":19},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76,77],{"impactFactor":20,"impactFactorByYear":404,"i10Index":20,"i10IndexLast5Year":20,"totalPublication":82,"totalPublicationByYear":405,"totalCitation":20,"totalCitationByYear":406,"totalCitationPerPublication":20,"totalCitationPerPublicationByYear":407,"hindexLast5Year":20,"hindex":20},{},{"1966":84,"1968":85,"1970":85,"1976":85,"1977":85,"1980":85,"1986":85,"1991":85,"1992":85,"2005":85,"2007":85,"2013":85,"2017":85,"2019":85,"2023":85},{},{},"1992-09-01",1992,"ERROR_IN_GET_PLATFORM_ID","2026-07-21T00:49:17.100+00:00",[78,56],[414,416,418,420,422,424,426,428,430,432,434,436,438,440,442,444,446,448],{"id":19,"text":415,"url":19,"identifiers":19},"N. Aronszajn,Approximation methods for eigenvalues of completely continuous symmetric operators, Proc. Symp. Spectral Theory and Differential Problems, Stillwater, Oklahoma, (1951), 179–201.",{"id":19,"text":417,"url":19,"identifiers":19},"L. Bassotti,Su un problema di autovalori per l'elasticità piana, Riv. Mat. Univ. Parma (2)8 (1967), 259–289.",{"id":19,"text":419,"url":19,"identifiers":19},"C. De Boor,A practical guide to splines, (1978), Springer-Verlag, New York.",{"id":19,"text":421,"url":19,"identifiers":19},"L. De Vito, G. Fichera, A. Fusciardi, M. Schaerf,Sugli autovalori della piastra quadrata incastrata lungo il suo bordo, Rend. Acc. Naz. Lincei, Classe Sci. Fis. Mat. Nat.XL, (1966), 725–733.",{"id":19,"text":423,"url":19,"identifiers":19},"M. Diligenti,Sulla dipendenza degli autovalori di un problema di elasticità tridimensionale da un parametro, Calcolo,19, (1982), 209–229.",{"id":19,"text":425,"url":19,"identifiers":19},"G. Fichera,Linear elliptic differential systems and eigenvalue problems, Lecture Notes in Math.,8, Springer-Verlag, (1965), Berlin.",{"id":19,"text":427,"url":19,"identifiers":19},"G. Fichera,Numerical and quantitative Analysis, (1978), Pitman, London.",{"id":19,"text":429,"url":19,"identifiers":19},"G. Fichera,Abstract and numerical aspects of eigenvalue theory, Università di Alberta (U.S.A.), (1973), Edmonton.",{"id":19,"text":431,"url":19,"identifiers":19},"G. Fichera,Approximation and estimates for eigenvalues, Proc. Symp. Numerical solution of partial diff. equations, Univ. Maryland 1965, Acad. Press, (1966), New York.",{"id":19,"text":433,"url":19,"identifiers":19},"G. Fichera,Upper bounds for orthogonal invariants of some positive linear operators, Rend. Ist. Mat. Univ. Trieste,1 (1969), 1–8.",{"id":19,"text":435,"url":19,"identifiers":19},"G. H. Golub, C. F. Van Loan,Matrix Computations, (1983), North Oxford Academic, Oxford.",{"id":19,"text":437,"url":19,"identifiers":19},"S. G. Mikhlin,Variational methods in mathematical physics, (1964), Pergamon Press, New York.",{"id":19,"text":439,"url":19,"identifiers":19},"P. M. Prenter,Splines and Variational Methods, (1975), John Wiley & Sons, New York.",{"id":19,"text":441,"url":19,"identifiers":19},"A. Weinstein, W. Stenger,Methods of Intermediate Problems for Eigenvalues, (1972), Academic Press, New York.",{"id":19,"text":443,"url":19,"identifiers":19},"A. Weinstein,On a minimal problem in the theory of elasticity, J. London Math. Soc.10, (1935), 184–192.",{"id":19,"text":445,"url":19,"identifiers":19},"A. Weinstein,Sur la stabilitè de plaques encastrées, Compt. Rend.200, (1935), 107–109.",{"id":19,"text":447,"url":19,"identifiers":19},"A. Weinstein,Etudes des spectres des équations aux derivées partielles de la théorie des plaques élastiques, Memor. Sci. Math.88, Gauthier-Vilars (1937).",{"id":19,"text":449,"url":19,"identifiers":19},"J. H. Wilkinson,The Algebraic Eigenvalue Problem, (1965), Claredon Univ. Press, Oxford.",{"id":451,"createTime":452,"updateTime":453,"relativeEntities":454,"slug":455,"properties":456,"entityType":110,"verifyStatus":111,"verifyTime":467,"verifyNote":113,"languages":19,"translateLanguages":19,"viewCount":20,"primaryUrl":468,"fullTextUrl":19,"authors":469,"publicationType":149,"publisherRelationship":498,"citationCount":85,"citationInfo":545,"publishDate":548,"publishYear":546,"citationAnalyzeStatus":549,"lastCitationAnalyze":453,"indexDatabases":550,"openAccess":19,"references":19,"isForceReanalyzing":203},"09244208-f1c8-4ae3-80a2-ef06f7f6468a","2023-12-04T11:33:17.707+00:00","2026-07-20T09:01:46.273+00:00",[],"Un-metodo-per-la-determinazione-di-un-albero-con-lati-pre-assegnati-o-meno-in-un-grafo-non-orientato-descritto-mediante-la-matrice-di-adiacenza",{"abstract":457,"title":459,"gsPaper":461,"references":463,"doi":465},{"EN":458},"A computer oriented algorithm is given for the determination of a tree in a nondirected linear graph described by its adjacency matrix. The algorith allows two types of constraints for the tree to be found; it can be requested: 1. to involve some given edges. 2. not to involve some other given edges. The computation speed can be improved, when a centre of the graph is nown; an optimization of the procedure is given for this case. The storage requirement is pratically reduced to the only required by the adjacency matrix.",{"EN":460},"Un metodo per la determinazione di un albero con lati pre-assegnati o meno in un grafo non-orientato descritto mediante la matrice di adiacenza",{"VOID":462},"[\"11634630726590137329\"]",{"VOID":464},"Biondi, E., Lunelli L.:Procedimenti per la determinazione delle maglie in una rete a lati orientati. Automazione e Strumentazione, vol. 14, n. 8 (agosto 1966), pp. 345–351.\nGotlieb, G. C., Corneil, D. G.:Algorithms for finding a fundamental set of cycles for an undirected linear graph. Comm. A.C.M., vol. 10, n. 12 (dicembre 1967), pp. 780–783.\nBerge C.:Théorie des graphes et ses applications, Dunod ed., Paris 1963.\nRighi R.:I grafi in Problemi attuali di teoria dei controlli automatici-1° seminario, Bressanone 1963, Ed. C.N.R.",{"VOID":466},"10.1007\u002FBF02576058","2024-05-07T12:00:47.420+00:00","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF02576058",[470,485],{"id":471,"sortIndex":20,"researcher":19,"roles":472,"affiliations":473,"properties":482,"displayName":484,"givenName":19,"familyName":19},"4aaa40d3-c06a-40c3-9f8f-a1a890f75ac5",[119],[474],{"id":475,"sortIndex":20,"affiliation":476,"properties":19},"a827f6f9-61a2-45ba-b17d-ac8a9166a50e",{"id":475,"createTime":19,"updateTime":19,"relativeEntities":477,"slug":19,"properties":478,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":481,"statistic":19},[],{"title":479},{"VI":480},"Istituto di Elettrotecnic ed Elettronica del Politecnico di Milano, Milano, Italia",[],{"title":483},{"VI":484},"V. Amoia",{"id":486,"sortIndex":85,"researcher":19,"roles":487,"affiliations":488,"properties":495,"displayName":497,"givenName":19,"familyName":19},"515486d7-407b-4041-8125-908d52df9439",[119],[489],{"id":475,"sortIndex":20,"affiliation":490,"properties":19},{"id":475,"createTime":19,"updateTime":19,"relativeEntities":491,"slug":19,"properties":492,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":494,"statistic":19},[],{"title":493},{"VI":480},[],{"title":496},{"VI":497},"G. Cottafava",{"url":468,"publisher":499,"properties":540},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":500,"slug":10,"properties":501,"entityType":17,"verifyStatus":18,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":20,"subjectFields":505,"manageAffiliations":514,"indexDatabases":520,"url":79,"thumbnailPath":19,"statistic":535,"gsStatistic":19,"type":88,"analyzePriority":19},[],{"issn":502,"title":503,"eissn":504},{"VOID":13},{"EN":10},{"VOID":16},[506,510],{"id":23,"createTime":19,"updateTime":19,"relativeEntities":507,"label":508,"description":509,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":26},{},{"id":29,"createTime":19,"updateTime":19,"relativeEntities":511,"label":512,"description":513,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":32},{},[515],{"id":36,"createTime":19,"updateTime":19,"relativeEntities":516,"slug":19,"properties":517,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":519,"statistic":19},[],{"title":518},{"EN":40},[42],[521,528],{"id":45,"indexDatabase":522,"url":58,"indexYears":19,"academicFieldIds":527,"indexDatabaseRanking":19},{"id":47,"createTime":19,"updateTime":19,"relativeEntities":523,"label":524,"description":525,"key":54,"publicationTags":526,"standard":19},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":529,"url":73,"indexYears":74,"academicFieldIds":534,"indexDatabaseRanking":78},{"id":64,"createTime":19,"updateTime":19,"relativeEntities":530,"label":531,"description":532,"key":70,"publicationTags":533,"standard":19},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76,77],{"impactFactor":20,"impactFactorByYear":536,"i10Index":20,"i10IndexLast5Year":20,"totalPublication":82,"totalPublicationByYear":537,"totalCitation":20,"totalCitationByYear":538,"totalCitationPerPublication":20,"totalCitationPerPublicationByYear":539,"hindexLast5Year":20,"hindex":20},{},{"1966":84,"1968":85,"1970":85,"1976":85,"1977":85,"1980":85,"1986":85,"1991":85,"1992":85,"2005":85,"2007":85,"2013":85,"2017":85,"2019":85,"2023":85},{},{},{"pages":541,"volume":543},{"VOID":542},"109-120",{"VOID":544},"5",{"total":85,"publishYear":546,"statisticByYear":547},1968,{"1968":85},"1968-03-01","DONE_ANALYZE_CITATION",[78,56],{"id":552,"createTime":553,"updateTime":554,"relativeEntities":555,"slug":556,"properties":557,"entityType":110,"verifyStatus":111,"verifyTime":567,"verifyNote":113,"languages":19,"translateLanguages":19,"viewCount":20,"primaryUrl":568,"fullTextUrl":19,"authors":569,"publicationType":149,"publisherRelationship":585,"citationCount":19,"citationInfo":19,"publishDate":631,"publishYear":546,"citationAnalyzeStatus":410,"lastCitationAnalyze":632,"indexDatabases":633,"openAccess":19,"references":19,"isForceReanalyzing":203},"b281e2c4-13b5-40f9-8133-afdf342a3804","2023-12-07T03:22:33.448+00:00","2026-07-11T01:08:02.296+00:00",[],"Automatic-analysis-of-nervous-activity-in-a-steady-state-or-during-modulated-stimulation",{"abstract":558,"title":560,"gsPaper":562,"references":563,"doi":565},{"EN":559},"In the area of computer processed bio-electrical signals from implanted micro-electrodes, same computer programs are here presented for investigating statistical properties of the retinal ganglion cell activity in the cat, either in a steady state or during a sinusoidally modulated light. In the latter case, the celle activity is examined in relation to the stimulus phase. The results quoted here have been presented by our research group in previous papers (See Refs 1–5); the automatic analysis techniques discussed here have been found well suited for studying how information may be coded into nervous pulses.",{"EN":561},"Automatic analysis of nervous activity, in a steady state or during modulated stimulation",{"VOID":328},{"VOID":564},"G. B. Gerace, G. Gestri:Un sistema automatico per l’analisi dell’attività nervosa. Alta Frequenza32, 639–644 (1963).\nG. Gestri, L. Mafferi, D. Petracchi:Statistical independence of events in neighbouring retinal untis. Brain Research2, 397–398 (1966).\nG. Gestri, L. Maffei, D. Petracchi:Spatial and Temporal Organization in Retinal Units. Kibernetik,3, 196–202 (1966).\nG. Gestri L. Maffei, D. Petracchi:Amplitude modulation of the retinal ganglion cell impulses. Kibernetik4 (2) (1967).\nG. Gestri, L. Maffei, D. Petracchi:The transformation induced by light stimulus on the retinal discharge. International Symposium on Information Theory of the IEEE-San Remo-Settembre 1967.",{"VOID":566},"10.1007\u002FBF02576106","2024-06-23T21:28:51.958+00:00","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF02576106",[570],{"id":571,"sortIndex":20,"researcher":19,"roles":572,"affiliations":573,"properties":582,"displayName":584,"givenName":19,"familyName":19},"125bec01-1430-4a93-981f-1d504e1cc85d",[119],[574],{"id":575,"sortIndex":20,"affiliation":576,"properties":19},"d4e9c9d8-3620-474b-a6b7-703481f6d66e",{"id":575,"createTime":19,"updateTime":19,"relativeEntities":577,"slug":19,"properties":578,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":581,"statistic":19},[],{"title":579},{"VI":580},"Gruppo di Cibernetica del C. N. R. presso l’Università di Pisa, Pisa, Italia",[],{"title":583},{"VI":584},"D. Petracchi",{"url":568,"publisher":586,"properties":627},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":587,"slug":10,"properties":588,"entityType":17,"verifyStatus":18,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":20,"subjectFields":592,"manageAffiliations":601,"indexDatabases":607,"url":79,"thumbnailPath":19,"statistic":622,"gsStatistic":19,"type":88,"analyzePriority":19},[],{"issn":589,"title":590,"eissn":591},{"VOID":13},{"EN":10},{"VOID":16},[593,597],{"id":23,"createTime":19,"updateTime":19,"relativeEntities":594,"label":595,"description":596,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":26},{},{"id":29,"createTime":19,"updateTime":19,"relativeEntities":598,"label":599,"description":600,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":32},{},[602],{"id":36,"createTime":19,"updateTime":19,"relativeEntities":603,"slug":19,"properties":604,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":606,"statistic":19},[],{"title":605},{"EN":40},[42],[608,615],{"id":45,"indexDatabase":609,"url":58,"indexYears":19,"academicFieldIds":614,"indexDatabaseRanking":19},{"id":47,"createTime":19,"updateTime":19,"relativeEntities":610,"label":611,"description":612,"key":54,"publicationTags":613,"standard":19},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":616,"url":73,"indexYears":74,"academicFieldIds":621,"indexDatabaseRanking":78},{"id":64,"createTime":19,"updateTime":19,"relativeEntities":617,"label":618,"description":619,"key":70,"publicationTags":620,"standard":19},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76,77],{"impactFactor":20,"impactFactorByYear":623,"i10Index":20,"i10IndexLast5Year":20,"totalPublication":82,"totalPublicationByYear":624,"totalCitation":20,"totalCitationByYear":625,"totalCitationPerPublication":20,"totalCitationPerPublicationByYear":626,"hindexLast5Year":20,"hindex":20},{},{"1966":84,"1968":85,"1970":85,"1976":85,"1977":85,"1980":85,"1986":85,"1991":85,"1992":85,"2005":85,"2007":85,"2013":85,"2017":85,"2019":85,"2023":85},{},{},{"pages":628,"volume":630},{"VOID":629},"195-204",{"VOID":544},"1968-06-01","2026-07-11T01:08:02.295+00:00",[78,56],{"id":635,"createTime":636,"updateTime":637,"relativeEntities":638,"slug":639,"properties":640,"entityType":110,"verifyStatus":111,"verifyTime":650,"verifyNote":113,"languages":19,"translateLanguages":19,"viewCount":20,"primaryUrl":651,"fullTextUrl":19,"authors":652,"publicationType":149,"publisherRelationship":683,"citationCount":19,"citationInfo":19,"publishDate":730,"publishYear":731,"citationAnalyzeStatus":410,"lastCitationAnalyze":637,"indexDatabases":732,"openAccess":19,"references":19,"isForceReanalyzing":203},"f41d73a5-5d2f-41b2-8141-da862f2eb064","2023-12-29T14:21:30.788+00:00","2026-07-06T12:00:57.038+00:00",[],"Explicit-multi-frequency-symmetric-extended-RKN-integrators-for-solving-multi-frequency-and-multidimensional-oscillatory-reversible-systems",{"abstract":641,"title":643,"gsPaper":645,"references":646,"doi":648},{"EN":642},"This paper studies explicit multi-frequency symmetric extended Runge–Kutta–Nyström (ERKN) integrators tailored to numerically computing the multi-frequency and multidimensional oscillatory reversible second-order differential equations \n                  \n                    \n                  \n                  $$q''(t)+Mq(t)=f\\big (q(t)\\big )$$\n                  \n                    \n                  \n                . We establish the symmetry conditions in a simplified way for multi-frequency ERKN integrators. Five explicit multi-frequency symmetric ERKN integrators are derived based on the simplified symmetry conditions. The arbitrary high-order explicit multi-frequency symmetric ERKN integrators can be achieved by the application of the symmetric composition. The stability and phase properties of the new integrators are discussed. Five numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new explicit multi-frequency symmetric integrators when applied to the multi-frequency and multidimensional oscillatory reversible second-order differential equations.",{"EN":644},"Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems",{"VOID":328},{"VOID":647},"Chen, Z., You, X., Shi, W., Liu, Z.: Symmetric and symplectic ERNK methods for oscillatory Hamiltonian systems. Comput. Phys. Commun. 183, 86–98 (2012)\nCohen, D., Hairer, E., Lubich, C.: Numerical energy conservation for multi-frequency oscillatory differential equations. BIT 45, 287–305 (2005)\nFang, Y., Wu, X.: A new pair of explicit ARKN methods for the numerical integration of general perturbed oscillators. Appl. Numer. Math. 57, 166–175 (2007)\nFranco, J.M.: Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)\nFranco, J.M.: A 5(3) pair of explicit ARKN methods for the numerical integration of perturbed oscillators. J. Comput. Appl. Math. 161, 283–293 (2003)\nGarcía, A., Martín, P., González, A.B.: New methods for oscillatory problems based on classical codes. Appl. Numer. Math. 42, 141–157 (2002)\nGarcía-Archilla, B., Sanz-Serna, J.M., Skeel, R.D.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 20, 930–963 (1999)\nGonzález, A.B., Martín, P., Farto, J.M.: A new family of Runge–Kutta type methods for the numerical integration of perturbed oscillators. Numer. Math. 82, 635–646 (1999)\nHairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)\nHairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin, Heidelberg (2006)\nHairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993)\nHochbruck, M., Lubich, C.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)\nJimánez, S., Vázquez, L.: Analysis of four numerical schemes for a nonlinear Klein–Gordon equation. Appl. Math. Comput. 35, 61–93 (1990)\nSanz-Serna, J.M., Calvo, M.P.: Numerical Hmiltonian Problem. In: Applied Mathematics and Matematical Computation, vol. 7. Chapman & Hall, London (1994)\nStavroyiannis, S., Simos, T.E.: Optimization as a function of the phase-lag order of two-step P-stable method for linear periodic IVPs. App. Numer. Math. 59, 2467–2474 (2009)\nTocino, A., Vigo-Aguiar, J.: Symplectic conditions for exponential fitting Runge–Kutta–Nyström methods. Math. Comput. Modell. 42, 873–876 (2005)\nVan der Houwen, P.J., Sommeijer, B.P.: Explicit Runge–Kutta(–Nyström) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. Anal. 24, 595–617 (1987)\nVan de Vyver, H.: Stability and phase-lag analysis of explicit Runge-Kutta methods with variable coefficients for oscillatory problems. Comput. Phys. Comm. 173, 115–130 (2005)\nVigo-Aguiar, J., Simos, T.E., Ferrándiz, J.M.: Controlling the error growth in long-term numerical integration of perturbed oscillations in one or more frequencies. Proc. Roy. Soc. Lond. Ser. A 460, 561–567 (2004)\nWang, B., Liu, K., Wu, X.: A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems. J. Comput. Phys. 243, 210–223 (2013)\nWang, B., Wu, X.: A new high precision energy-preserving integrator for system of oscillatory second-order differential equations. Phys. Lett. A 376, 1185–1190 (2012)\nWang, B., Wu, X., Xia, J.: Error bounds for explicit ERKN integrators for systems of multi-frequency oscillatory second-order differential equations. Appl. Numer. Math. 74, 17–34 (2013)\nWang, B., Wu, X., Zhao, H.: Novel improved multidimensional Strömer–Verlet formulas with applications to four aspects in scientific computation. Math. Comput. Modell. 57, 857–872 (2013)\nWu, X.: A note on stability of multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems. Appl. Math. Modell. 36, 6331–6337 (2012)\nWu, X., Wang, B.: Multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems. Comput. Phys. Commun. 181, 1955–1962 (2010)\nWu, X., Wang, B., Shi, W.: Efficient energy-preserving integrators for oscillatory Hamiltonian systems. J. Comput. Phys. 235, 587–605 (2013)\nWu, X., Wang, B., Xia, J.: Explicit symplectic multidimensional exponential fitting modified Runge–Kutta–Nyström methods. BIT 52, 773–795 (2012)\nWu, X., You, X., Shi, W., Wang, B.: ERKN integrators for systems of oscillatory second-order differential equations. Comput. Phys. Commun. 181, 1873–1887 (2010)\nWu, X., You, X., Wang, B.: Structure-Preserving Algorithms for Oscillatory Differential Equations. Springer, Berlin, Heidelberg (2013)",{"VOID":649},"10.1007\u002Fs10092-014-0114-z","2024-06-26T21:36:43.857+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10092-014-0114-z",[653,668],{"id":654,"sortIndex":20,"researcher":19,"roles":655,"affiliations":656,"properties":665,"displayName":667,"givenName":19,"familyName":19},"411217c0-9316-4934-859d-77b1fbd46c13",[119],[657],{"id":658,"sortIndex":20,"affiliation":659,"properties":19},"acf7f48d-1010-4bf7-8002-c7bb90562e3b",{"id":658,"createTime":19,"updateTime":19,"relativeEntities":660,"slug":19,"properties":661,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":664,"statistic":19},[],{"title":662},{"VI":663},"School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, People’s Republic of China",[],{"title":666},{"VI":667},"Bin Wang",{"id":669,"sortIndex":85,"researcher":19,"roles":670,"affiliations":671,"properties":680,"displayName":682,"givenName":19,"familyName":19},"c859b99a-6228-457b-8d41-2f2fd7ca19d6",[119],[672],{"id":673,"sortIndex":20,"affiliation":674,"properties":19},"cb106515-d338-4f91-9132-866e3d1a9c14",{"id":673,"createTime":19,"updateTime":19,"relativeEntities":675,"slug":19,"properties":676,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":679,"statistic":19},[],{"title":677},{"VI":678},"Department of Mathematics, Nanjing University, Nanjing, People’s Republic of China",[],{"title":681},{"VI":682},"Xinyuan Wu",{"url":651,"publisher":684,"properties":725},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":685,"slug":10,"properties":686,"entityType":17,"verifyStatus":18,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":20,"subjectFields":690,"manageAffiliations":699,"indexDatabases":705,"url":79,"thumbnailPath":19,"statistic":720,"gsStatistic":19,"type":88,"analyzePriority":19},[],{"issn":687,"title":688,"eissn":689},{"VOID":13},{"EN":10},{"VOID":16},[691,695],{"id":23,"createTime":19,"updateTime":19,"relativeEntities":692,"label":693,"description":694,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":26},{},{"id":29,"createTime":19,"updateTime":19,"relativeEntities":696,"label":697,"description":698,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":32},{},[700],{"id":36,"createTime":19,"updateTime":19,"relativeEntities":701,"slug":19,"properties":702,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":704,"statistic":19},[],{"title":703},{"EN":40},[42],[706,713],{"id":45,"indexDatabase":707,"url":58,"indexYears":19,"academicFieldIds":712,"indexDatabaseRanking":19},{"id":47,"createTime":19,"updateTime":19,"relativeEntities":708,"label":709,"description":710,"key":54,"publicationTags":711,"standard":19},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":714,"url":73,"indexYears":74,"academicFieldIds":719,"indexDatabaseRanking":78},{"id":64,"createTime":19,"updateTime":19,"relativeEntities":715,"label":716,"description":717,"key":70,"publicationTags":718,"standard":19},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76,77],{"impactFactor":20,"impactFactorByYear":721,"i10Index":20,"i10IndexLast5Year":20,"totalPublication":82,"totalPublicationByYear":722,"totalCitation":20,"totalCitationByYear":723,"totalCitationPerPublication":20,"totalCitationPerPublicationByYear":724,"hindexLast5Year":20,"hindex":20},{},{"1966":84,"1968":85,"1970":85,"1976":85,"1977":85,"1980":85,"1986":85,"1991":85,"1992":85,"2005":85,"2007":85,"2013":85,"2017":85,"2019":85,"2023":85},{},{},{"pages":726,"volume":728},{"VOID":727},"207-231",{"VOID":729},"52","2014-04-23",2014,[78,56],{"id":734,"createTime":735,"updateTime":736,"relativeEntities":737,"slug":738,"properties":739,"entityType":110,"verifyStatus":111,"verifyTime":750,"verifyNote":113,"languages":19,"translateLanguages":19,"viewCount":20,"primaryUrl":751,"fullTextUrl":19,"authors":752,"publicationType":149,"publisherRelationship":768,"citationCount":19,"citationInfo":19,"publishDate":631,"publishYear":546,"citationAnalyzeStatus":18,"lastCitationAnalyze":736,"indexDatabases":814,"openAccess":19,"references":19,"isForceReanalyzing":203},"ab72342d-e80a-459b-80bc-30727d23d396","2024-02-15T01:47:23.800+00:00","2026-07-06T07:45:16.384+00:00",[],"Pianificazione-e-controllo-della-produzione-mediante-I-generatori-indipendenti-siemens",{"abstract":740,"title":742,"gsPaper":744,"references":746,"doi":748},{"EN":741},"Planning and control problems are discussed. Siemens preprocessor PAN and SINETIK are oriented to CPM, PERT, MPM and other network methods. Preprocessor PLEIADES provides manufacturing organizations with a large kind of techniques.",{"EN":743},"Pianificazione e controllo della produzione mediante I generatori indipendenti siemens",{"VOID":745},"[\"11506035037496275060\"]",{"VOID":747},"De Ambrogio, W.:La tecnica PERTCOM per la pianificazione dei lavori relativi a più progetti contemporanei con livellamento dei carichi di risorse. Automazione e strumentazione, febraio 1965, 50–57.\nDe Ambrogio, W.:Tecniche reticolari di programmazione. In F. Brambilla, “Trattato di statistica”, vol.2, UTET.\nRicci, G.:Nucleo di una rete PERT e problema dei ritardi. Calcolo, vol. 5, fasc. 1, Gennaio-Marzo 1968.\nRicci, G.:Un nuovo formalismo per lo studio di certe proprietà dei grafi: La Pert-espresione. Istituto Lombardo (Rend SC.), A 100, 17 Marzo 1966, 439–454.\nSiemens Elettra:CPM europeo manuale informativo. Agosto 1968, Siemens Elettra S.p.A., V.C.E., Milano.\nSiemens Elettra:DIMTYP manuale informativo. Giugno 1968, Siemens Elettra S.p.A., V.C.E., Milano.\nSiemens Elettra:MPM manuale informativo. Luglio 1968, Siemens Elettra S.p.A., V.C.E., Milano.\nSiemens Elettra:SINETIK CPM-PERT manuale informativo. Agosto 1968, Siemens Elettra S.p.A., V.C.E., Milano.\nWiest, J. D.:The Scheduling of Large Projects with Limited Resources. O.N.R. Research Mem. No. 113, Carnegie Inst. of Techn., Pittsburgh, Penns. (1963).",{"VOID":749},"10.1007\u002FBF02576614","2024-05-13T11:31:10.152+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02576614",[753],{"id":754,"sortIndex":20,"researcher":19,"roles":755,"affiliations":756,"properties":765,"displayName":767,"givenName":19,"familyName":19},"a3a19eb6-e9b4-4718-945a-eab465347579",[119],[757],{"id":758,"sortIndex":20,"affiliation":759,"properties":19},"f83d121f-597a-490c-be74-f3be31ec8e19",{"id":758,"createTime":19,"updateTime":19,"relativeEntities":760,"slug":19,"properties":761,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":764,"statistic":19},[],{"title":762},{"VI":763},"Siemens Elettra, Milano",[],{"title":766},{"VI":767},"G. Ricci",{"url":751,"publisher":769,"properties":810},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":770,"slug":10,"properties":771,"entityType":17,"verifyStatus":18,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":20,"subjectFields":775,"manageAffiliations":784,"indexDatabases":790,"url":79,"thumbnailPath":19,"statistic":805,"gsStatistic":19,"type":88,"analyzePriority":19},[],{"issn":772,"title":773,"eissn":774},{"VOID":13},{"EN":10},{"VOID":16},[776,780],{"id":23,"createTime":19,"updateTime":19,"relativeEntities":777,"label":778,"description":779,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":26},{},{"id":29,"createTime":19,"updateTime":19,"relativeEntities":781,"label":782,"description":783,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":32},{},[785],{"id":36,"createTime":19,"updateTime":19,"relativeEntities":786,"slug":19,"properties":787,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":789,"statistic":19},[],{"title":788},{"EN":40},[42],[791,798],{"id":45,"indexDatabase":792,"url":58,"indexYears":19,"academicFieldIds":797,"indexDatabaseRanking":19},{"id":47,"createTime":19,"updateTime":19,"relativeEntities":793,"label":794,"description":795,"key":54,"publicationTags":796,"standard":19},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":799,"url":73,"indexYears":74,"academicFieldIds":804,"indexDatabaseRanking":78},{"id":64,"createTime":19,"updateTime":19,"relativeEntities":800,"label":801,"description":802,"key":70,"publicationTags":803,"standard":19},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76,77],{"impactFactor":20,"impactFactorByYear":806,"i10Index":20,"i10IndexLast5Year":20,"totalPublication":82,"totalPublicationByYear":807,"totalCitation":20,"totalCitationByYear":808,"totalCitationPerPublication":20,"totalCitationPerPublicationByYear":809,"hindexLast5Year":20,"hindex":20},{},{"1966":84,"1968":85,"1970":85,"1976":85,"1977":85,"1980":85,"1986":85,"1991":85,"1992":85,"2005":85,"2007":85,"2013":85,"2017":85,"2019":85,"2023":85},{},{},{"pages":811,"volume":813},{"VOID":812},"1011-1023",{"VOID":544},[78,56],{"id":816,"createTime":817,"updateTime":818,"relativeEntities":819,"slug":820,"properties":821,"entityType":110,"verifyStatus":111,"verifyTime":832,"verifyNote":113,"languages":19,"translateLanguages":19,"viewCount":20,"primaryUrl":833,"fullTextUrl":19,"authors":834,"publicationType":149,"publisherRelationship":867,"citationCount":19,"citationInfo":19,"publishDate":914,"publishYear":915,"citationAnalyzeStatus":18,"lastCitationAnalyze":818,"indexDatabases":916,"openAccess":19,"references":19,"isForceReanalyzing":203},"282f63e4-f749-4c6f-8114-a06aba8e4955","2023-12-28T20:52:59.328+00:00","2026-05-20T03:56:30.487+00:00",[],"Legendre-spectral-collocation-method-for-Volterra-integral-equations-with-non-vanishing-delay",{"abstract":822,"title":824,"gsPaper":826,"references":828,"doi":830},{"EN":823},"The main purpose of this paper is to propose the Legendre spectral-collocation method to solve the Volterra integral equations of the second kind with non-vanishing delay. We divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay. In each subinterval, where the solution is smooth enough, we can apply Legendre spectral-collocation method to approximate the solution. The provided convergence analysis shows that the numerical errors decay exponentially. Numerical examples are presented to confirm this theoretical predict.",{"EN":825},"Legendre spectral-collocation method for Volterra integral equations with non-vanishing delay",{"VOID":827},"[\"9076306507165564562\"]",{"VOID":829},"Ali, I.: Convergence analysis of spectral methods for integro-differential equations with vanishing proportional delays. J. Comput. Math. 28, 962–973 (2010)\nAli, I., Brunner, H., Tang, T.: A spectral method for pantograph-type delay differential equations and its convergence analysis. J. Comput. Math. 27, 254–265 (2009)\nAli, I., Brunner, H., Tang, T.: Spectral methods for pantograph-type differential and integral equations with multiple delays. Front. Math. China 4, 49–61 (2009)\nBellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations, Oxford Scientifc publications, Oxford (2003)\nBrunner, H.: Collocation methods for Volterra integral and related functional differential equations. Cambridge University Press, Lundon (2004)\nBrunner, H.: The numerical analysis of functional integral and integro-differential equations of Volterra type. Acta Numerica. 13, 55–145 (2004)\nBrunner, H.: Recent advances in the numerical analysis of Volterra functional differential equations with variable delays. J. Comput. Appl. Math. 228, 524–537 (2009)\nBrunner, H., Hu, Q., Lin, Q.: Geometric meshes in collocation method for Volterra integral equations with proportional delays. IMA J. Numer. Anal. 21, 783–798 (2001)\nBrunner, H., Liang, H.: Stability of collocation methods for delay differential equations with vanishing delays. BIT Numer. Math. 50, 693–711 (2010)\nBrunner, H., Xie, H., Zhang, R.: Analysis of collocation solutions for a class of functional equations with vanishing delays. IMA J. Numer. Anal. 31, 698–718 (2011)\nCanuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral method fundamentals in single domains, Spring-Verlag, New York (2006)\nChen, Y., Li, X., Tang, T.: A note on Jacobi spectral-collocation methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Math. 31, 47–56 (2013)\nChen, Y., Tang, T.: Spectral methods for weakly singular Volterra integral equations with smooth solutions. J. Comput. Appl. Math. 233, 938–950 (2009)\nChen, Y., Tang, T.: Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel. Math. Comput. 79, 147–167 (2010)\nJiang, Y.: On spectral methods for Volterra-type integro-differential equations. J. Comput. Appl. Math. 230, 333–340 (2009)\nLi, X., Tang, T.: Convergence analysis of the Javobi collocation methods foe Abel-Volterra integral equations of the second kind. Front. Math. China 7, 69–84 (2012)\nMessina, E., Russo, E., Vecchio, A.: A stable numerical method for Volterra integral equations with discontinuous kernel. J. Math. Anal. Appl. 337, 1383–1393 (2008)\nNevai, P.: Mean convergence of Lagrange interpolation, III. Trans. Amer. Math. Soc. 282, 669–698 (1984)\nShen, J., Tang, T.: Spectral and high-order methods with applications. Science Press, Beijing (2006)\nTang, T., Xu, X.: Accuracy enhancement using spectral postprocessing for differential equations and integral equations. Commun. Comput. Phys. 5, 779–792 (2009)\nTang, T., Xu, X., Cheng, J.: On Spectral methods for Volterra integral equation and the convergence analysis. J. Comput. Math. 26, 825–837 (2008)\nVermiglio, R.: On the stability of Runge-Kutta methods for delay integral equations. Numer. Math. 61, 561–577 (1992)\nWan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4, 181–193 (2009)\nWei, Y., Chen, Y.: Convergence analysis of the Legendre spectral collocation methods for second order Volterra integro-differential equations. Numer. Math. Theory Method Appl. 4, 419–438 (2011)\nWei, Y., Chen, Y.: Convergence analysis of the spectral methods for weakly singular Volterra integro-differential equations with smooth solutions. Adv. Appl. Math. Mech. 4, 1–20 (2012)\nXie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53, 414–434 (2012)\nYang, K., Zhang, R.: Analysis of continuous collocation solutions for a kind of Volterra functional integral equations with proportional delay. J. Comput. Appl. Math. 236, 743–752 (2011)",{"VOID":831},"10.1007\u002Fs10092-013-0083-7","2024-06-24T02:59:03.811+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10092-013-0083-7",[835,850],{"id":836,"sortIndex":20,"researcher":19,"roles":837,"affiliations":838,"properties":847,"displayName":849,"givenName":19,"familyName":19},"b8e72855-b950-4792-a5ec-09cd322ab900",[119],[839],{"id":840,"sortIndex":20,"affiliation":841,"properties":19},"4ede7c10-8641-4e57-911d-3ef36d45bbf7",{"id":840,"createTime":19,"updateTime":19,"relativeEntities":842,"slug":19,"properties":843,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":846,"statistic":19},[],{"title":844},{"VI":845},"School of Mathematics and Computational Science, Xiangtan University, Xiangtan, China",[],{"title":848},{"VI":849},"Zhendong Gu",{"id":851,"sortIndex":85,"researcher":19,"roles":852,"affiliations":853,"properties":862,"displayName":864,"givenName":19,"familyName":19},"483e7560-5a5b-4380-85f6-8143f373ed92",[119],[854],{"id":855,"sortIndex":20,"affiliation":856,"properties":19},"c00efdf3-2685-4ffe-802c-a28613593feb",{"id":855,"createTime":19,"updateTime":19,"relativeEntities":857,"slug":19,"properties":858,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":861,"statistic":19},[],{"title":859},{"VI":860},"School of Mathematics Science, South China Normal University, Guangzhou, China",[],{"title":863,"gsAuthor":865},{"VI":864},"Yanping Chen",{"VOID":866},"[\"luXcZ1YAAAAJ\"]",{"url":833,"publisher":868,"properties":909},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":869,"slug":10,"properties":870,"entityType":17,"verifyStatus":18,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":20,"subjectFields":874,"manageAffiliations":883,"indexDatabases":889,"url":79,"thumbnailPath":19,"statistic":904,"gsStatistic":19,"type":88,"analyzePriority":19},[],{"issn":871,"title":872,"eissn":873},{"VOID":13},{"EN":10},{"VOID":16},[875,879],{"id":23,"createTime":19,"updateTime":19,"relativeEntities":876,"label":877,"description":878,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":26},{},{"id":29,"createTime":19,"updateTime":19,"relativeEntities":880,"label":881,"description":882,"parentId":19,"standard":19,"scholarHubFieldId":19},[],{"EN":32},{},[884],{"id":36,"createTime":19,"updateTime":19,"relativeEntities":885,"slug":19,"properties":886,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":888,"statistic":19},[],{"title":887},{"EN":40},[42],[890,897],{"id":45,"indexDatabase":891,"url":58,"indexYears":19,"academicFieldIds":896,"indexDatabaseRanking":19},{"id":47,"createTime":19,"updateTime":19,"relativeEntities":892,"label":893,"description":894,"key":54,"publicationTags":895,"standard":19},[],{"EN":50,"VI":50},{"EN":52,"VI":53},[56,57],[60],{"id":62,"indexDatabase":898,"url":73,"indexYears":74,"academicFieldIds":903,"indexDatabaseRanking":78},{"id":64,"createTime":19,"updateTime":19,"relativeEntities":899,"label":900,"description":901,"key":70,"publicationTags":902,"standard":19},[],{"EN":67,"VI":67},{"EN":67,"VI":69},[72],[76,77],{"impactFactor":20,"impactFactorByYear":905,"i10Index":20,"i10IndexLast5Year":20,"totalPublication":82,"totalPublicationByYear":906,"totalCitation":20,"totalCitationByYear":907,"totalCitationPerPublication":20,"totalCitationPerPublicationByYear":908,"hindexLast5Year":20,"hindex":20},{},{"1966":84,"1968":85,"1970":85,"1976":85,"1977":85,"1980":85,"1986":85,"1991":85,"1992":85,"2005":85,"2007":85,"2013":85,"2017":85,"2019":85,"2023":85},{},{},{"pages":910,"volume":912},{"VOID":911},"151-174",{"VOID":913},"51","2013-04-17",2013,[78,56],{"id":918,"createTime":919,"updateTime":920,"relativeEntities":921,"slug":922,"properties":923,"entityType":110,"verifyStatus":111,"verifyTime":934,"verifyNote":113,"languages":19,"translateLanguages":19,"viewCount":20,"primaryUrl":935,"fullTextUrl":19,"authors":936,"publicationType":149,"publisherRelationship":982,"citationCount":19,"citationInfo":19,"publishDate":1029,"publishYear":1030,"citationAnalyzeStatus":18,"lastCitationAnalyze":1031,"indexDatabases":1032,"openAccess":19,"references":19,"isForceReanalyzing":203},"53e70851-8398-45d2-8115-19841371bc0c","2023-12-02T22:28:59.907+00:00","2026-05-19T09:21:37.156+00:00",[],"Sulla-valutazione-dell-errore-nel-calcolo-numerico-degli-integrali-trigonometrici",{"abstract":924,"title":926,"gsPaper":928,"references":930,"doi":932},{"EN":925},"La tecnica di Davis è applicata per determinare una maggiorazione dell'errore di un metodo numerico per il calcolo degli integrali trigonometrici. Tale maggiorazione permette un confronto tra note formule che possono ottenersi come casi particolari del metodo considerato.",{"EN":927},"Sulla valutazione dell'errore nel calcolo numerico degli integrali trigonometrici",{"VOID":929},"[\"5214633011760682279\"]",{"VOID":931},"Murli A.,Il calcolo numerico degli integrali trigonometrici, Calcolo, Suppl. no 15 Atti del II Congresso Nazionale A.I.C.A. (1968), 922–933.\nMurli A.,Alcune formule per il calcolo numerico della trasformata di Fourier, Calcolo4 (1967), 647–676.\nFilon L. N. G.,On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburg49 (1928), 38–47.\nVan de Vooren A. I., Van Liñde H. J.,Numerical Calculations of Integrals with strongly oscillating Integrand, Math. Com.20 (1966), 232–245.\nClendenin W. N. A Method for Numerical Calculations of Fourier Integrals, Num. Math.8 (1966) 422–436.\nTuck E. O.,A simple Filon-Trapezoidal rule, Math. Comp.21 (1967), 239–241.\nDavis P. J.,Interapolation and approximation, (1963), Blaisdell, New York.\nDavis P. J.,Errors of Numerical Approximation for Analytic Functions, J. Radional Mech. Anal.,2 (1953), 303–313.\nDavis P. J. Errors of Numerical Approximation for Analytic Function, Survey of Numerical Analysis (1962), J. Todd (Ed.), Mc Graw-Hill, New York.\nChawla M. M.,On Davis's method for the estimation of errors of Gauss-Chebyshev quadrature, SIAM J. Num. Anal.6 (1969), 108–117.\nChawla M. M.,Hilbert Spaces for estimating errors of quadrature for analytic functions, BIT10 (1970), 145–155.\nKambo N. S.,Error bounds for a Chebyshev quadrature scheme, BIT13 (1973), 30–37.\nLanczos C.,Applied Analysis, (1964), Prentice Hall, Inc. Englewood Cliffs, N. I.\nGrosh,Computation of Bessel Functions of Integral Order., (1962), Stevens Institute of Technology Davison Laboratory-Castle Point Station-Hoboken, New Jersey.",{"VOID":933},"10.1007\u002FBF02576634","2024-06-24T01:49:43.757+00:00","http:\u002F\u002Flink.springer.com\u002F10.1007\u002FBF02576634",[937,954,969],{"id":938,"sortIndex":20,"researcher":19,"roles":939,"affiliations":940,"properties":949,"displayName":951,"givenName":19,"familyName":19},"c049153b-f4e2-4f9d-a798-33be5fec1c59",[119],[941],{"id":942,"sortIndex":20,"affiliation":943,"properties":19},"80276b19-7481-4299-abc7-dfc58cc1ce5f",{"id":942,"createTime":19,"updateTime":19,"relativeEntities":944,"slug":19,"properties":945,"entityType":19,"verifyStatus":19,"verifyTime":19,"verifyNote":19,"languages":19,"translateLanguages":19,"viewCount":19,"url":19,"parentIds":948,"statistic":19},[],{"title":946},{"VI":947},"Facoltà di Agraria, Università degli Studi di Napoli, Napoli, Italia",[],{"title":950,"gsAuthor":952},{"VI":951},"P. 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