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Math., 251 (2005), 49–120.",{},{"id":434,"createTime":435,"updateTime":436,"relativeEntities":437,"slug":438,"properties":439,"entityType":108,"verifyStatus":109,"verifyTime":448,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":449,"fullTextUrl":22,"authors":450,"publicationType":161,"publisherRelationship":468,"citationCount":23,"citationInfo":515,"publishDate":518,"publishYear":516,"citationAnalyzeStatus":213,"lastCitationAnalyze":519,"indexDatabases":520,"openAccess":22,"references":521,"isForceReanalyzing":281},"20b02f78-4247-4890-800e-7d80e0047490","2023-12-11T12:18:47.971+00:00","2026-07-23T00:36:43.836+00:00",[],"Exponential-instability-in-the-inverse-scattering-problem-on-the-energy-interval",{"abstract":440,"title":442,"gsPaper":444,"doi":446},{"EN":441},"We consider the inverse scattering problem on the energy interval in three dimensions. We focus on stability and instability questions for this problem. In particular, we prove an exponential instability estimate which shows the optimality, up to the value of the exponent, of the logarithmic stability result obtained by P. Stefanov in 1990 with the use of some special norm for the scattering amplitude at fixed energy.",{"EN":443},"Exponential instability in the inverse scattering problem on the energy interval",{"VOID":445},"[\"16793291811774230781\"]",{"VOID":447},"10.1007\u002Fs10688-013-0025-9","2024-04-29T05:43:16.146+00:00","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs10688-013-0025-9",[451],{"id":452,"sortIndex":23,"researcher":22,"roles":453,"affiliations":454,"properties":463,"displayName":465,"givenName":22,"familyName":22},"7c67c1a9-eda7-433e-9f4b-a2fa7d114c1a",[117],[455],{"id":456,"sortIndex":23,"affiliation":457,"properties":22},"3119deb5-b479-492c-b274-20c50607cd8d",{"id":456,"createTime":22,"updateTime":22,"relativeEntities":458,"slug":22,"properties":459,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":462,"statistic":22},[],{"title":460},{"VI":461},"Centre de Mathématiques Appliquées, Ecole Polytechnique, Moscow Institute of Physics and Technology, Moscow, Russia",[],{"title":464,"gsAuthor":466},{"VI":465},"M. I. Isaev",{"VOID":467},"[\"V9D9Gf4jQQUC\"]",{"url":449,"publisher":469,"properties":510},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":470,"slug":10,"properties":471,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":475,"manageAffiliations":484,"indexDatabases":495,"url":22,"thumbnailPath":22,"statistic":22,"gsStatistic":22,"type":22,"analyzePriority":22},[],{"issn":472,"title":473,"eissn":474},{"VOID":15},{"EN":17},{"VOID":13},[476,480],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":477,"label":478,"description":479,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":481,"label":482,"description":483,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[485,490],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":486,"slug":22,"properties":487,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":489,"statistic":22},[],{"title":488},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":491,"slug":22,"properties":492,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":494,"statistic":22},[],{"title":493},{"EN":51},[],[496,503],{"id":55,"indexDatabase":497,"url":68,"indexYears":22,"academicFieldIds":502,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":498,"label":499,"description":500,"key":64,"publicationTags":501,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":504,"url":83,"indexYears":84,"academicFieldIds":509,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":505,"label":506,"description":507,"key":80,"publicationTags":508,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"pages":511,"volume":513},{"VOID":512},"187-194",{"VOID":514},"47",{"total":23,"publishYear":516,"statisticByYear":517},2013,{},"2013-09-11","2026-07-23T00:36:43.835+00:00",[88,66],[522,525,528,531,534,537,540,543,546,549,552,555,561,564,567,570,573],{"id":221,"text":523,"url":223,"identifiers":524},"G. Alessandrini, “Stable determination of conductivity by boundary measurements,” Appl. Anal., 27:1–3 (1988), 153–172.",{"doi":225},{"id":221,"text":526,"url":223,"identifiers":527},"N. V. Alekseenko, V. A. Burov, and O. D. Rumyantseva, “Solution of the three-dimensional acoustic inverse scattering problem: The modified Novikov algorithm,” Akust. Zh., 54:3 (2008), 469–482; English transl.: Acoust. Phys., 54:3 (2008), 407–419.",{"doi":225},{"id":221,"text":529,"url":223,"identifiers":530},"M. Di Cristo and L. Rondi, “Examples of exponential instability for inverse inclusion and scattering problems,” Inverse Problems, 19:3 (2003), 685–701.",{"doi":225},{"id":221,"text":532,"url":223,"identifiers":533},"I. M. Gelfand, “Some aspects of functional analysis and algebra,” in: Proc. Internat. Congress of Math., Amsterdam, 1954, vol. 1, North Holland, Amsterdam, 1957, 253–276.",{"doi":225},{"id":22,"text":535,"url":22,"identifiers":536},"L. D. Faddeev, “Uniqueness of the solution of the inverse scattering problem,” Vestn. Leningr. Univ., 11:7 (1956), 126–130.",{},{"id":221,"text":538,"url":223,"identifiers":539},"L. D. Faddeev, “Inverse problem of quantum scattering theory. II,” in: Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, vol. 3, VINITI, Moscow, 1974, 93–180; English transl.: J. Soviet Math., 5:3 (1976), 334–396.",{"doi":225},{"id":221,"text":541,"url":223,"identifiers":542},"G. M. Henkin and R. G. Novikov, “The \\(\\bar \\partial\\)-equation in the multidimensional inverse scattering problem,” Uspekhi Mat. Nauk, 42:3 (1987), 93–152; English transl.: Russian Math. Surveys, 42:3 (1987), 109–180.",{"doi":225},{"id":221,"text":544,"url":223,"identifiers":545},"M. I. Isaev, “Exponential instability in the Gel’fand inverse problem on the energy intervals,” J. Inverse Ill-Posed Probl., 19:3 (2011), 453–473.",{"doi":225},{"id":22,"text":547,"url":22,"identifiers":548},"A. N. Kolmogorov and V. M. Tikhomirov, “ɛ-entropy and ɛ-capacity in functional spaces,” Uspekhi Mat. Nauk, 14 (1959), 3–86; English transl.: Amer. Math. Soc. Transl., 17 (1961), 277–364.",{},{"id":221,"text":550,"url":223,"identifiers":551},"N. Mandache, “Exponential instability in an inverse problem for the Schrödinger equation,” Inverse Problems, 17:5 (2001), 1435–1444.",{"doi":225},{"id":22,"text":553,"url":22,"identifiers":554},"R. G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1989.",{},{"id":556,"text":557,"url":558,"identifiers":559},"3318393f-e5aa-4d8e-8843-0b971c7f41f3","R. G. Novikov, “Multidimensional inverse spectral problem for the equation −Δψ + (v(x) − Eu(x))Ψ = 0,” Funkts. Anal. Prilozhen., 22:4 (1988), 11–22; English transl.: Functional Anal. Appl., 22:4 (1988), 263–272","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF01077418",{"doi":560},"10.1007\u002FBF01077418",{"id":221,"text":562,"url":223,"identifiers":563},"R. G. Novikov, “The inverse scattering problem at fixed energy for the three-dimensional Schrödinger equation with an exponentially decreasing potential,” Comm. Math. Phys., 161:3 (1994), 569–595.",{"doi":225},{"id":221,"text":565,"url":223,"identifiers":566},"R. G. Novikov, “On determination of the Fourier transform of a potential from the scattering amplitude,” Inverse Problems, 17:5 (2001), 1243–1251.",{"doi":225},{"id":22,"text":568,"url":22,"identifiers":569},"R. G. Novikov, “The \\(\\bar \\partial\\)-approach to approximate inverse scattering at fixed energy in three dimensions,” IMRP Int. Math. Res. Pap., 6 (2005), 287–349.",{},{"id":221,"text":571,"url":223,"identifiers":572},"R. G. Novikov, “New global stability estimates for the Gelfand-Calderon inverse problem,” Inverse Problems, 27:1 (2011), 015001.",{"doi":225},{"id":221,"text":574,"url":223,"identifiers":575},"P. Stefanov, “Stability of the inverse problem in potential scattering at fixed energy,” Ann. Inst. Fourier (Grenoble), 40:4 (1990), 867–884.",{"doi":225},{"id":577,"createTime":578,"updateTime":579,"relativeEntities":580,"slug":581,"properties":582,"entityType":108,"verifyStatus":109,"verifyTime":593,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":594,"fullTextUrl":22,"authors":595,"publicationType":161,"publisherRelationship":645,"citationCount":23,"citationInfo":692,"publishDate":695,"publishYear":693,"citationAnalyzeStatus":696,"lastCitationAnalyze":697,"indexDatabases":698,"openAccess":22,"references":22,"isForceReanalyzing":281},"88527933-f16b-414a-811f-e32ef864fe4c","2023-12-28T20:09:27.751+00:00","2026-07-22T17:03:34.489+00:00",[],"Discrete-nonlinear-hyperbolic-equations-Classification-of-integrable-cases",{"abstract":583,"title":585,"gsPaper":587,"references":589,"doi":591},{"EN":584},"We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on ℤ2. The fields are associated with the vertices and an equation of the form Q(x\n                        1, x\n                        2, x\n                        3, x\n                        4) = 0 relates four vertices of one cell. The integrability of equations is understood as 3D-consistency, which means that it is possible to impose equations of the same type on all faces of a three-dimensional cube so that the resulting system will be consistent. This allows one to extend these equations also to the multidimensional lattices ℤ\n                  N\n                . We classify integrable equations with complex fields x and polynomials Q multiaffine in all variables. Our method is based on the analysis of singular solutions.",{"EN":586},"Discrete nonlinear hyperbolic equations. Classification of integrable cases",{"VOID":588},"[\"8862954288106837050\"]",{"VOID":590},"V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York etc., 1978.\nV. E. Adler, “Bäcklund transformation for the Krichever-Novikov equation,” Internat. Math. Res. Notices, 1 (1998), 1–4.\nV. E. Adler, A. I. Bobenko, and Yu. B. Suris, “Classification of integrable equations on quadgraphs. The consistency approach,” Comm. Math. Phys., 233:3 (2003), 513–543.\nV. E. Adler, A. I. Bobenko, and Yu. B. Suris, “Geometry of Yang-Baxter maps: pencils of conics and quadrirational mappings,” Comm. Anal. Geom., 12:5 (2004), 967–1007.\nV. E. Adler and Yu. B. Suris, “Q4: Integrable master equation related to an elliptic curve,” Internat. Math. Res. Notices, 47 (2004), 2523–2553.\nV. E. Adler and A. P. Veselov, “Cauchy problem for integrable discrete equations on quadgraphs,” Acta Appl. Math., 84:2 (2004), 237–262.\nV. Bazhanov, V. Mangazeev, and S. Sergeev, “Faddeev-Volkov solution of the Yang-Baxter equation and discrete conformal symmetry,” Nuclear Phys. B, 784:3 (2007), 234–258.\nV. Bazhanov and S. Sergeev, “Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups,” J. Phys. A., 39:13 (2006), 3295–3310.\nL. Bianchi, Vorlesungen über Differenzialgeometrie, Teubner, Leipzig, 1899.\nA. I. Bobenko and Yu. B. Suris, “Integrable systems on quad-graphs,” Internat. Math. Res. Notices, 11 (2002), 573–611.\nA. I. Bobenko and Yu. B. Suris, “Integrable non-commutative equations on quad-graphs. The consistency approach,” Lett. Math. Phys., 61:3 (2002), 241–254.\nA. I. Bobenko and Yu. B. Suris, Discrete Differential Geometry. Integrable Structure, Graduate Studies in Math., vol. 98, Amer. Math. Soc., Providence, RI, 2008.\nJ. Hietarinta, “A new two-dimensional lattice model that is ‘consistent around a cube’,” J. Phys. A, 37:6 (2004), L67–L73.\nJ. Hietarinta, “Searching for CAC-maps,” J. Nonlinear Math. Phys., 12,Suppl. 2 (2005), 223–230.\nR. M. Kashaev, I. G. Korepanov, and S. M. Sergeev, “The functional tetrahedron equation,” Teoret. Mat. Fiz., 117:3 (1998), 370–384; English transl.: Theoret. Math. Phys., 117:3 (1998), 1402–1413.\nI. G. Korepanov, Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics, http:\u002F\u002Farxiv.org\u002Fabs\u002Fsolv-int\u002F9506003.\nJ.-M. Maillet and F. W. Nijhoff, “Integrability for multidimensional lattice models,” Phys. Lett. B, 224:4 (1989), 389–396.\nA. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, “The symmetry approach to the classification of nonlinear equations. Complete lists of integrable systems,” Uspekhi Mat. Nauk, 42:4 (1987), 3–53; English transl.: Russ. Math. Surveys, 42:4 (1987), 1–63.\nF. W. Nijhoff, “Lax pair for the Adler (lattice Krichever-Novikov) system,” Phys. Lett. A, 297 (2002), 49–58.\nV. Papageorgiou, A. Tongas, and A. Veselov, “Yang-Baxter maps and symmetries of integrable equations on quad-graphs,” J. Math. Phys, 47:8 (2006), 083502.\nG. R. W. Quispel, F. W. Nijhoff, H. W. Capel, and J. van der Linden, “Linear integral equations and nonlinear difference-difference equations,” Phys. A, 125:2–3 (1984), 344–380.\nA. Ramani, N. Joshi, B. Grammaticos, and T. Tamizhmani, “Deconstructing an integrable lattice equation,” J. Phys. A, 39:8 (2006), L145–L149.\nYu. B. Suris and A. P. Veselov, “Lax pairs for Yang-Baxter maps,” J. Nonlinear Math. Phys., 10,Suppl. 2 (2003), 223–230.\nA. P. Veselov, “Integrable mappings,” Uspekhi Mat. Nauk, 46:5 (1991), 3–45; English transl.: Russian Math. Surveys, 46:5 (1991), 1–51.\nA. P. Veselov, “Yang-Baxter maps: dynamical point of view,” in: Combinatorial Aspect of Integrable Systems, MSJ Mem., vol. 17, Math. Soc. Japan, Tokyo, 2007, 145–167.\nC. Viallet, Algebraic Entropy for Lattice Equations, http:\u002F\u002Farxiv.org\u002Fabs\u002Fmath-ph\u002F0609043.\nA. Volkov, “Quantum lattice KdV equation,” Lett. Math. Phys., 39:4 (1997), 313–329.\nE. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 1927, reprinted in 1996.",{"VOID":592},"10.1007\u002Fs10688-009-0002-5","2024-06-23T02:39:14.624+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10688-009-0002-5",[596,613,630],{"id":597,"sortIndex":23,"researcher":22,"roles":598,"affiliations":599,"properties":608,"displayName":610,"givenName":22,"familyName":22},"5b28b7b4-5e4d-46cf-8cf6-f77595830571",[117],[600],{"id":601,"sortIndex":23,"affiliation":602,"properties":22},"f6a47ce8-15d2-4a5e-9763-b53b2ea68a2d",{"id":601,"createTime":22,"updateTime":22,"relativeEntities":603,"slug":22,"properties":604,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":607,"statistic":22},[],{"title":605},{"VI":606},"Landau Institute for Theoretical Physics, Moscow, Russia",[],{"title":609,"gsAuthor":611},{"VI":610},"V. E. Adler",{"VOID":612},"[\"763IQ5MAAAAJ\"]",{"id":614,"sortIndex":129,"researcher":22,"roles":615,"affiliations":616,"properties":625,"displayName":627,"givenName":22,"familyName":22},"1cef956e-dbc9-4f8c-b6bf-91617ee6e15d",[117],[617],{"id":618,"sortIndex":23,"affiliation":619,"properties":22},"ca20e686-2b8b-46f6-83bd-3e9d65a48a4d",{"id":618,"createTime":22,"updateTime":22,"relativeEntities":620,"slug":22,"properties":621,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":624,"statistic":22},[],{"title":622},{"VI":623},"Institut für Mathematik, Technische Universität Berlin, Berlin, Germany",[],{"title":626,"gsAuthor":628},{"VI":627},"A. I. Bobenko",{"VOID":629},"[\"BqbscJEAAAAJ\"]",{"id":631,"sortIndex":390,"researcher":22,"roles":632,"affiliations":633,"properties":642,"displayName":644,"givenName":22,"familyName":22},"52203dd1-c65d-41f3-a9ae-c78ad43cded9",[117],[634],{"id":635,"sortIndex":23,"affiliation":636,"properties":22},"9e251118-bb5e-491a-b813-d8c98d0da47e",{"id":635,"createTime":22,"updateTime":22,"relativeEntities":637,"slug":22,"properties":638,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":641,"statistic":22},[],{"title":639},{"VI":640},"Zentrum Mathematik, Technische Universität München, München, Germany",[],{"title":643},{"VI":644},"Yu. B. Suris",{"url":594,"publisher":646,"properties":687},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":647,"slug":10,"properties":648,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":652,"manageAffiliations":661,"indexDatabases":672,"url":22,"thumbnailPath":22,"statistic":22,"gsStatistic":22,"type":22,"analyzePriority":22},[],{"issn":649,"title":650,"eissn":651},{"VOID":15},{"EN":17},{"VOID":13},[653,657],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":654,"label":655,"description":656,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":658,"label":659,"description":660,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[662,667],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":663,"slug":22,"properties":664,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":666,"statistic":22},[],{"title":665},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":668,"slug":22,"properties":669,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":671,"statistic":22},[],{"title":670},{"EN":51},[],[673,680],{"id":55,"indexDatabase":674,"url":68,"indexYears":22,"academicFieldIds":679,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":675,"label":676,"description":677,"key":64,"publicationTags":678,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":681,"url":83,"indexYears":84,"academicFieldIds":686,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":682,"label":683,"description":684,"key":80,"publicationTags":685,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"pages":688,"volume":690},{"VOID":689},"3-17",{"VOID":691},"43",{"total":23,"publishYear":693,"statisticByYear":694},2009,{},"2009-03-18","ERROR_IN_ANALYZE_CITATION","2026-07-22T17:03:34.488+00:00",[88,66],{"id":700,"createTime":701,"updateTime":702,"relativeEntities":703,"slug":704,"properties":705,"entityType":108,"verifyStatus":109,"verifyTime":716,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":717,"fullTextUrl":22,"authors":718,"publicationType":161,"publisherRelationship":736,"citationCount":23,"citationInfo":783,"publishDate":786,"publishYear":784,"citationAnalyzeStatus":213,"lastCitationAnalyze":702,"indexDatabases":787,"openAccess":22,"references":22,"isForceReanalyzing":281},"eff1b060-5a94-49ac-8123-c58a7d7fcf24","2024-01-10T21:26:42.741+00:00","2026-07-19T02:18:35.719+00:00",[],"Asymptotic-Relations-for-the-Distributional-Stockwell-and-Wavelet-Transforms",{"abstract":706,"title":708,"gsPaper":710,"references":712,"doi":714},{"EN":707}," Abelian- and Tauberian-type results characterizing the quasiasymptotic behavior of distributions in \n                  \n                    \n                  \n                  $$\\mathcal{S}_{0}'(\\mathbb{R})$$\n                 in terms of their Stockwell transforms are obtained. An Abelian-type result relating the quasiasymptotic boundedness of Lizorkin distributions to the asymptotic behavior of their Stockwell transforms is given. Several asymptotic results for the distributional wavelet transform are also presented. ",{"EN":709},"Asymptotic Relations for the Distributional Stockwell and Wavelet Transforms",{"VOID":711},"[\"17494278849790253272\"]",{"VOID":713},"R. G. Stockwell, L. Mansinha, and R. P. Lowe, “Localization of the complex spectrum: the S transform”, IEEE Trans. Signal Process., 44 (1996), 998–1001.\nJ. Du, M. W. Wong, and H. Zhu, “Continuous and discrete inversion formulas for the Stockwell transform”, Integral Transforms Spec. Func., 18:8 (2007), 537–543.\nL. Riba, Multi-Dimensional Stockwell Transforms and Applications (PhD Thesis), Universitá degli Studii di Torino, Torino, 2014.\nQ. Guo, S. Molahajloo, and M. W. Wong, “Modified Stockwell transforms and time-frequency analysis”, New Developments in Pseudo-Differential Operators, Operator Theory: Advances and Applications, 189, Basel, Birkhäuser, 2009, 275–285.\nK. H.-V. Saneva, S. Atanasova, and J. V. Buralieva, “Tauberian theorems for the Stockwell transform of Lizorkin distributions”, Appl. Anal., 99:4 (2020), 596–610.\nV. Catană, “Abelian and Tauberian results for the one-dimensional modified Stockwell transforms”, Appl. Anal., 96:6 (2017), 1047–1057.\nR. Estrada and R. P. Kanwal, A Distributional Approach to Asymptotics. Theory and Applications, Birkhäuser, Boston, 2002.\nS. Pilipović, B. Stankovic, and J. Vindas, Asymptotic behavior of generalized functions, World Scientific Publishing Co., Hackensack, NJ, 2012.\nV. S. Vladimirov, Yu. N. Drozinov, and B. I. Zavialov, Tauberian Theorems for Generalized Functions, Kluwer Academic, Dordrecht, 1988.\nJ. Vindas, S. Pilipović, and D. Rakić, “Tauberian theorems for the wavelet transform”, J. Fourier Anal. Appl., 17:1 (2011), 65–95.\nK. Saneva, R. Aceska, and S. Kostadinova, “Some Abelian and Tauberian results for the short-time Fourier transform”, Novi Sad J. Math., 43:2 (2013), 81–89.\nJ. V. Buralieva, K. Saneva, and S. Atanasova, “Directional short-time Fourier transform and quasiasymptotics of distributions”, Funkts. Anal. i Prilozhen., 53:1 (2019), 6–15; English transl.: Funct. Anal. Appl., 53:1 (2019), 3–10.\nS. Pilipović and J. Vindas, “Multidimensional Tauberian theorems for vector-valued distributions”, Publ. Inst. Math. (Beograd) (N.S.), 95 (2017), 1–28.\nF. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York–London, 1967.\nL. Schwartz, “Thèorie des distributions à valeurs vectorielles. I”, Ann. Inst. Fourier Grenoble, 7 (1957), 1–141.\nM. Holschneider, Wavelets. An Analysis Tool, The Clarendon Press, Oxford University Press, New York, 1995.\nK. Gröchenig, Foundations of Time-Frequency Analysis, App. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2001.",{"VOID":715},"10.1134\u002FS0016266323010033","2024-05-28T04:38:05.840+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1134\u002FS0016266323010033",[719],{"id":720,"sortIndex":23,"researcher":22,"roles":721,"affiliations":722,"properties":731,"displayName":733,"givenName":22,"familyName":22},"9e2482ca-a532-4724-8538-33cbe8b26492",[117],[723],{"id":724,"sortIndex":23,"affiliation":725,"properties":22},"7f642a44-2020-4792-8b21-142f37b6b599",{"id":724,"createTime":22,"updateTime":22,"relativeEntities":726,"slug":22,"properties":727,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":730,"statistic":22},[],{"title":728},{"VI":729},"University Goce Delcev, Faculty of Computer Science, Shtip, Macedonia",[],{"title":732,"gsAuthor":734},{"VI":733},"J. V. Buralieva",{"VOID":735},"[\"9Vm9nkAAAAAJ\"]",{"url":717,"publisher":737,"properties":778},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":738,"slug":10,"properties":739,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":743,"manageAffiliations":752,"indexDatabases":763,"url":22,"thumbnailPath":22,"statistic":22,"gsStatistic":22,"type":22,"analyzePriority":22},[],{"issn":740,"title":741,"eissn":742},{"VOID":15},{"EN":17},{"VOID":13},[744,748],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":745,"label":746,"description":747,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":749,"label":750,"description":751,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[753,758],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":754,"slug":22,"properties":755,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":757,"statistic":22},[],{"title":756},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":759,"slug":22,"properties":760,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":762,"statistic":22},[],{"title":761},{"EN":51},[],[764,771],{"id":55,"indexDatabase":765,"url":68,"indexYears":22,"academicFieldIds":770,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":766,"label":767,"description":768,"key":64,"publicationTags":769,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":772,"url":83,"indexYears":84,"academicFieldIds":777,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":773,"label":774,"description":775,"key":80,"publicationTags":776,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"pages":779,"volume":781},{"VOID":780},"29-39",{"VOID":782},"57",{"total":23,"publishYear":784,"statisticByYear":785},2023,{},"2023-09-05",[88,66],{"id":789,"createTime":790,"updateTime":791,"relativeEntities":792,"slug":793,"properties":794,"entityType":108,"verifyStatus":109,"verifyTime":803,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":804,"fullTextUrl":22,"authors":805,"publicationType":161,"publisherRelationship":821,"citationCount":23,"citationInfo":868,"publishDate":871,"publishYear":869,"citationAnalyzeStatus":21,"lastCitationAnalyze":872,"indexDatabases":873,"openAccess":22,"references":874,"isForceReanalyzing":281},"73017937-4b15-4b36-8076-cb7d4a0585d9","2023-12-05T03:12:03.465+00:00","2026-07-12T04:55:27.222+00:00",[],"Strongly-elliptic-second-order-systems-with-boundary-conditions-on-a-nonclosed-Lipschitz-surface",{"abstract":795,"title":797,"gsPaper":799,"doi":801},{"EN":796},"We consider boundary value problems and transmission problems for strongly elliptic second-order systems with boundary conditions on a compact nonclosed Lipschitz surface S with Lipschitz boundary. The main goal is to find conditions for the unique solvability of these problems in the spaces H\n                        \n                  s\n                , the simplest L\n                        2-spaces of the Sobolev type, with the use of potential type operators on S. We also discuss, first, the regularity of solutions in somewhat more general Bessel potential spaces and Besov spaces and, second, the spectral properties of problems with spectral parameter in the transmission conditions on S, including the asymptotics of the eigenvalues.",{"EN":798},"Strongly elliptic second-order systems with boundary conditions on a nonclosed Lipschitz surface",{"VOID":800},"[\"12445937772238922323\"]",{"VOID":802},"10.1007\u002Fs10688-011-0001-1","2024-05-03T11:15:02.782+00:00","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs10688-011-0001-1",[806],{"id":807,"sortIndex":23,"researcher":22,"roles":808,"affiliations":809,"properties":818,"displayName":820,"givenName":22,"familyName":22},"087a8092-bd9d-4992-b866-58f263e2872a",[117],[810],{"id":811,"sortIndex":23,"affiliation":812,"properties":22},"c3397635-6090-4ed5-8abb-222c63887b39",{"id":811,"createTime":22,"updateTime":22,"relativeEntities":813,"slug":22,"properties":814,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":817,"statistic":22},[],{"title":815},{"VI":816},"Moscow Institute of Electronics and Mathematics, Moscow, Russia",[],{"title":819},{"VI":820},"M. S. Agranovich",{"url":804,"publisher":822,"properties":863},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":823,"slug":10,"properties":824,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":828,"manageAffiliations":837,"indexDatabases":848,"url":22,"thumbnailPath":22,"statistic":22,"gsStatistic":22,"type":22,"analyzePriority":22},[],{"issn":825,"title":826,"eissn":827},{"VOID":15},{"EN":17},{"VOID":13},[829,833],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":830,"label":831,"description":832,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":834,"label":835,"description":836,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[838,843],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":839,"slug":22,"properties":840,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":842,"statistic":22},[],{"title":841},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":844,"slug":22,"properties":845,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":847,"statistic":22},[],{"title":846},{"EN":51},[],[849,856],{"id":55,"indexDatabase":850,"url":68,"indexYears":22,"academicFieldIds":855,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":851,"label":852,"description":853,"key":64,"publicationTags":854,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":857,"url":83,"indexYears":84,"academicFieldIds":862,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":858,"label":859,"description":860,"key":80,"publicationTags":861,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"pages":864,"volume":866},{"VOID":865},"1-12",{"VOID":867},"45",{"total":23,"publishYear":869,"statisticByYear":870},2011,{},"2011-03-20","2026-07-12T04:55:27.221+00:00",[88,66],[875,881,884,887,890,893,896,899,902,905,908,911,914,917,920,923,926,929,932,935,938,941,945,948,951,954,960,963,966,969,972],{"id":876,"text":877,"url":878,"identifiers":879},"da870a1c-86b9-4ec9-acbc-d41c7a8c74dd","M. S. Agranovich, “Regularity of variational solutions to linear boundary value problems in Lipschitz domains,” Funkts. Anal. Prilozhen., 40:4 (2006), 83–103; English transl.: Functional Anal. Appl., 40:4 (2006), 313–329.","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10688-006-0048-6",{"doi":880},"10.1007\u002Fs10688-006-0048-6",{"id":221,"text":882,"url":223,"identifiers":883},"M. S. Agranovich, “Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces H σp and B σ p ,” Funkts. Anal. Prilozhen., 42:4 (2008), 2–23; English transl.: Functional Anal. Appl., 42:4 (2008), 249–267.",{"doi":225},{"id":221,"text":885,"url":223,"identifiers":886},"M. S. Agranovich, “Potential type operators and transmission problems for strongly elliptic second-order systems in Lipschitz domains,” Funkts. Anal. Prilozhen., 43:3 (2009), 3–25; English transl.: Functional Anal. Appl., 43:3 (2009), 165–183.",{"doi":225},{"id":22,"text":888,"url":22,"identifiers":889},"M. S. Agranovich, “Spectral boundary value problems in Lipschitz domains,” Sovremennaya Matematika. Fundamental’nye Napravleniya, 39 (to appear).",{},{"id":221,"text":891,"url":223,"identifiers":892},"M. S. Agranovich, “Mixed problems in Lipschitz domains for strongly elliptic second-order systems,” Funkts. Anal. Prilozhen., 45 (to appear).",{"doi":225},{"id":221,"text":894,"url":223,"identifiers":895},"M. S. Agranovich, “Strongly elliptic second order systems with spectral parameter in transmission conditions on a nonclosed surface,” in: Operator Theory: Advances and Applications, vol. 164, Birkhäuser, Basel, 2006, 1–21.",{"doi":225},{"id":221,"text":897,"url":223,"identifiers":898},"M. S. Agranovich and B. A. Amosov, “Estimates of s-numbers and spectral asymptotics for integral operators of potential type on nonsmooth surfaces,” Funkts. Anal. Prilozhen., 30:2 (1996), 1–18; English transl.: Functional Anal. Appl., 30:2 (1996), 75–89.",{"doi":225},{"id":221,"text":900,"url":223,"identifiers":901},"M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley-VCH, Berlin, 1999. (Revised English edition of [31].)",{"doi":225},{"id":22,"text":903,"url":22,"identifiers":904},"J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.",{},{"id":22,"text":906,"url":22,"identifiers":907},"M. Sh. Birman and M. Z. Solomyak, “Spectral asymptotics of nonsmooth elliptic operators, I,” Trudy Moskov. Mat. Obshch., 27 (1972), 3–52; English transl.: Trans. Moscow Math. Soc., 27 (1972), 1–52 (1975).",{},{"id":221,"text":909,"url":223,"identifiers":910},"M. Costabel and M. Dauge, “On representation formulas and radiation conditions,” Math. Methods Appl. Sci., 20:2 (1997), 133–150.",{"doi":225},{"id":221,"text":912,"url":223,"identifiers":913},"M. Costabel and E. Stephan, “An improved boundary element Galerkin method for threedimensional crack problems,” Integral Equations Operator Theory, 10:4 (1987), 467–504.",{"doi":225},{"id":221,"text":915,"url":223,"identifiers":916},"B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, “Boundary value problems for systems of elastostatics in Lipschitz domains,” Duke Math. J., 57:3 (1988), 795–818.",{"doi":225},{"id":221,"text":918,"url":223,"identifiers":919},"R. Duduchava and D. Natroshvili, “Mixed crack type problem in anisotropic elasticity,” Math. Nachr., 191 (1998), 83–107.",{"doi":225},{"id":22,"text":921,"url":22,"identifiers":922},"R. Duduchava, D. Natroshvili, and E. Shargorodsky, “Boundary value problems of the mathematical theory of cracks,” Trudy Inst. Prikl. Mat. Tbiliss. Gos. Univ., 39 (1990), 68–84.",{},{"id":221,"text":924,"url":223,"identifiers":925},"R. Duduchava and W. L. Wendland, “The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems,” Integral Equations Operator Theory, 23:3 (1995), 294–335.",{"doi":225},{"id":221,"text":927,"url":223,"identifiers":928},"G. I. Èskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, Amer. Math. Soc., Providence, RI, 1981.",{"doi":225},{"id":221,"text":930,"url":223,"identifiers":931},"P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.",{"doi":225},{"id":221,"text":933,"url":223,"identifiers":934},"G. C. Hsiao, E. P. Stephan, and W. L. Wendland, “An integral equation formulation for a boundary value problem of elasticity in the domain exterior to an arc,” in: Lecture Notes in Math., vol. 1121, Springer-Verlag, Berlin, 1983, 153–165.",{"doi":225},{"id":22,"text":936,"url":22,"identifiers":937},"G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer-Verlag, Berlin, 2008.",{},{"id":22,"text":939,"url":22,"identifiers":940},"W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge Univ. Press, Cambridge, 2000.",{},{"id":22,"text":942,"url":943,"identifiers":944},"S. E. Mikhailov, Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients, http:\u002F\u002Farxiv.org\u002Fabs\u002F0906.3875.","http:\u002F\u002Farxiv.org\u002Fabs\u002F0906.3875",{},{"id":221,"text":946,"url":223,"identifiers":947},"M. Mitrea and M. Taylor, “Boundary layer methods for Lipschitz domains in Riemannian manifolds,” J. Funct. Anal., 163:2 (1999), 181–251.",{"doi":225},{"id":22,"text":949,"url":22,"identifiers":950},"J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris; Academia, É diteurs, Prague, 1967.",{},{"id":22,"text":952,"url":22,"identifiers":953},"O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North Holland, Amsterdam, 1992.",{},{"id":955,"text":956,"url":957,"identifiers":958},"3fd6f6be-33d6-4458-ba7f-4325d9f58650","G. Rozenblum and G. Tashchiyan, “Eigenvalue asymptotics for potential type operators on Lipschitz surfaces,” Russian J. Math. Phys., 13:3 (2006), 326–339.","https:\u002F\u002Flink.springer.com\u002F10.1134\u002FS1061920806030083",{"doi":959},"10.1134\u002FS1061920806030083",{"id":22,"text":961,"url":22,"identifiers":962},"I. Ya. Shneiberg, “Spectral properties of linear operators in interpolation families of Banach spaces,” Mat. Issled., 9:2 (1974), 214–227.",{},{"id":22,"text":964,"url":22,"identifiers":965},"E. Stephan, Boundary integral equations for mixed boundary value problems, screen and transmission problems in ℝ3, Habilitationsschrift, Darmstadt, THD-preprint 848, 1984.",{},{"id":221,"text":967,"url":223,"identifiers":968},"E. Stephan, “A boundary integral equation method for three-dimensional crack problems in elasticity,” Math. Methods Appl. Sci., 8:4 (1986), 609–623.",{"doi":225},{"id":221,"text":970,"url":223,"identifiers":971},"E. Stephan, “Boundary integral equations for screen problems in ℝ3,” Integral Equations Operator Theory, 10:2 (1987), 236–257.",{"doi":225},{"id":22,"text":973,"url":22,"identifiers":974},"N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, Generalized Method of Eigenoscillations in Diffraction Theory, with supplement by M. S. Agranovich “Spectral properties of diffraction problems”, 289–416 [in Russian], Nauka, Moscow, 1977.",{},{"id":976,"createTime":977,"updateTime":978,"relativeEntities":979,"slug":980,"properties":981,"entityType":108,"verifyStatus":109,"verifyTime":990,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":991,"fullTextUrl":22,"authors":992,"publicationType":161,"publisherRelationship":1010,"citationCount":1057,"citationInfo":1058,"publishDate":1062,"publishYear":1059,"citationAnalyzeStatus":21,"lastCitationAnalyze":1063,"indexDatabases":1064,"openAccess":22,"references":1065,"isForceReanalyzing":281},"bb299aff-7a78-4a26-8015-bc90c461c936","2024-01-13T22:37:37.370+00:00","2026-07-11T20:51:24.960+00:00",[],"Resolution-of-Corank-1-Singularities-of-a-Generic-Front",{"abstract":982,"title":984,"gsPaper":986,"doi":988},{"EN":983},"We construct a resolution of singularities for wave fronts having only stable singularities of corank 1. It is based on a transformation that takes a given front to a new front with singularities of the same type in a space of smaller dimension. This transformation is defined by the class Aµ of Legendre singularities. The front and the ambient space obtained by the Aµ-transformation inherit topological information on the closure of the manifold of singularities Aµ of the original front. The resolution of every (reducible) singularity of a front is determined by a suitable iteration of Aµ-transformations. As a corollary, we obtain new conditions for the coexistence of singularities of generic fronts.",{"EN":985},"Resolution of Corank 1 Singularities of a Generic Front",{"VOID":987},"[\"1036831328300074187\"]",{"VOID":989},"10.1023\u002FA:1024456907021","2024-04-28T05:08:06.963+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1023\u002FA:1024456907021",[993],{"id":994,"sortIndex":23,"researcher":22,"roles":995,"affiliations":996,"properties":1005,"displayName":1007,"givenName":22,"familyName":22},"7912cbad-6c4a-46b2-b31a-4376f4ac8f8d",[117],[997],{"id":998,"sortIndex":23,"affiliation":999,"properties":22},"9593a735-c141-4047-91fa-eae0ceee3005",{"id":998,"createTime":22,"updateTime":22,"relativeEntities":1000,"slug":22,"properties":1001,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1004,"statistic":22},[],{"title":1002},{"VI":1003},"Russian State University of Oil and Gas (Gubkin), Russia",[],{"title":1006,"gsAuthor":1008},{"VI":1007},"V. D. Sedykh",{"VOID":1009},"[\"ZBHmc78AAAAJ\"]",{"url":991,"publisher":1011,"properties":1052},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":1012,"slug":10,"properties":1013,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":1017,"manageAffiliations":1026,"indexDatabases":1037,"url":22,"thumbnailPath":22,"statistic":22,"gsStatistic":22,"type":22,"analyzePriority":22},[],{"issn":1014,"title":1015,"eissn":1016},{"VOID":15},{"EN":17},{"VOID":13},[1018,1022],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":1019,"label":1020,"description":1021,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":1023,"label":1024,"description":1025,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[1027,1032],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":1028,"slug":22,"properties":1029,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1031,"statistic":22},[],{"title":1030},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":1033,"slug":22,"properties":1034,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1036,"statistic":22},[],{"title":1035},{"EN":51},[],[1038,1045],{"id":55,"indexDatabase":1039,"url":68,"indexYears":22,"academicFieldIds":1044,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":1040,"label":1041,"description":1042,"key":64,"publicationTags":1043,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":1046,"url":83,"indexYears":84,"academicFieldIds":1051,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":1047,"label":1048,"description":1049,"key":80,"publicationTags":1050,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"pages":1053,"volume":1055},{"VOID":1054},"123-133",{"VOID":1056},"37",12,{"total":1057,"publishYear":1059,"statisticByYear":1060},2003,{"2003":129,"2004":1061,"2005":390,"2006":129,"2007":390,"2012":1061},3,"2003-04-01","2026-07-11T20:51:24.959+00:00",[88,66],[1066,1069,1072,1075,1078,1081,1084,1087,1090,1093,1096,1099,1102,1108],{"id":22,"text":1067,"url":22,"identifiers":1068},"V. I. Arnold, \"Critical points of functions on a manifold with boundary, the simple Lie groups B k, C k, F 4, and singularities of evolutes,\" Usp. Mat. Nauk, 33, No. 5, 91–105 (1978).",{},{"id":22,"text":1070,"url":22,"identifiers":1071},"V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Maps, Vol. 1 [in Russian], Moscow, Nauka, 1982.",{},{"id":221,"text":1073,"url":223,"identifiers":1074},"V. A. Vasiliev, Lagrange and Legendre characteristic classes [in Russian], Moscow, MCCME, 2000.",{"doi":225},{"id":22,"text":1076,"url":22,"identifiers":1077},"V. M. Zakalyukin, Singularities of Lagrange and Legendre mappings, PhD thesis, Moscow State University, Moscow, 1977.",{},{"id":221,"text":1079,"url":223,"identifiers":1080},"M. E. Kazarian, \"Characteristic classes of Lagrange and Legendre singularities,\" Usp. Mat. Nauk, 50, No. 4, 45–70 (1995).",{"doi":225},{"id":22,"text":1082,"url":22,"identifiers":1083},"J. Mather, \"Stratifications and mappings,\" In: Dynamical Systems (Peixoto ed.), N.Y.: Academic Press, 1973, 195–232.",{},{"id":22,"text":1085,"url":22,"identifiers":1086},"V. D. Sedykh, \"The strict convexity of a convex generic manifold,\" Trudy Mat. Inst. Steklov., 209, 200–219 (1995); English transl. Proc. Steklov Inst. Math., 209, 174–190 (1995).",{},{"id":22,"text":1088,"url":22,"identifiers":1089},"V. D. Sedykh, \"Relations between Euler numbers of manifolds of corank 1 singularities of a generic front,\" Dokl. Ross. Akad. Nauk, 383, No. 6, 735–739 (2002); English transl. Russian Acad. Sci. Dokl. Math., 65, No. 2, 276–279 (2002).",{},{"id":221,"text":1091,"url":223,"identifiers":1092},"S. J. Colley, \"Enumerating stationary multiple-points,\" Adv. in Math., 66, No. 2, 149–170 (1987).",{"doi":225},{"id":221,"text":1094,"url":223,"identifiers":1095},"V. Goryunov, \"Semi-simplicial resolutions and homology of images and discriminants of mappings,\" Proc. London Math. Soc., 70, No. 3, 363–385 (1995).",{"doi":225},{"id":221,"text":1097,"url":223,"identifiers":1098},"S. Izumiya and W. L. Marar, \"The Euler characteristic of a generic wave front in a 3-manifold,\" Proc. Amer. Math. Soc., 118, No. 4, 1347–1350 (1993).",{"doi":225},{"id":221,"text":1100,"url":223,"identifiers":1101},"S. L. Kleiman, Multiple-point formulas I: Iteration, Acta Math., 147, No. 1-2, 13–49 (1981).",{"doi":225},{"id":1103,"text":1104,"url":1105,"identifiers":1106},"be5f5845-5cf5-45f5-a3d7-f6c582ef29cf","S. L. Kleiman, J. Lipman, and B. Ulrich, \"The multiple-point schemes of a finite curvilinear map of codimension one,\" Ark. Mat., 34, No. 2, 285–326 (1996).","https:\u002F\u002Fprojecteuclid.org\u002Fjournals\u002Farkiv-for-matematik\u002Fvolume-34\u002Fissue-2\u002FThe-multiple-point-schemes-of-a-finite-curvilinear-map-of\u002F10.1007\u002FBF02559549.full",{"doi":1107},"10.1007\u002FBF02559549",{"id":221,"text":1109,"url":223,"identifiers":1110},"W. L. Marar and D. Mond, \"Multiple-point schemes for corank 1 maps,\" J. London Math. Soc. (2), 39, No. 3, 553–567 (1989).",{"doi":225},{"id":1112,"createTime":1113,"updateTime":1114,"relativeEntities":1115,"slug":1116,"properties":1117,"entityType":108,"verifyStatus":109,"verifyTime":1128,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":1129,"fullTextUrl":22,"authors":1130,"publicationType":161,"publisherRelationship":1207,"citationCount":23,"citationInfo":1254,"publishDate":1257,"publishYear":1255,"citationAnalyzeStatus":21,"lastCitationAnalyze":1258,"indexDatabases":1259,"openAccess":22,"references":22,"isForceReanalyzing":281},"e9759fa2-29b1-41fa-80dd-26ee243bbd98","2024-01-15T13:30:59.019+00:00","2026-07-11T17:33:39.407+00:00",[],"Two-Dimensional-Periodic-Schr%C3%B6dinger-Operators-Integrable-at-an-Energy-Eigenlevel",{"abstract":1118,"title":1120,"gsPaper":1122,"references":1124,"doi":1126},{"EN":1119},"The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrödinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrödinger equation at the zero energy level is a smooth M-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrödinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov-Veselov construction.",{"EN":1121},"Two-Dimensional Periodic Schrödinger Operators Integrable at an Energy Eigenlevel",{"VOID":1123},"[\"10448423317515405322\"]",{"VOID":1125},"B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “The Schrödinger equation in a magnetic field and Riemann surfaces,” Dokl. Akad. Nauk SSSR, 229 (1976), 15–18; English transl.: Soviet Math. Dokl., 17 (1977), 947–951.\nB. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Non-linear Equations of Korteweg-de Vries Type, Finite-Zone Linear Operators, and Abelian Varieties,” Uspekhi Mat. Nauk, 31:1(187) (1976), 55–136; English transi.: Russian Math. Surveys, 31:1 (1976), 59–146.\nI. M. Krichever, “Potentials with zero coefficient of reflection on a background of finite-zone potentials,” Punkts. Anal. Prilozhen., 9:2 (1975), 77–78; English transi.: Functional Anal. Appl., 9:2 (1975), 161–163.\nI. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Funkts. Anal. Prilozhen., 11:1 (1977), 15–31; English transi.: Functional Anal. Appl., 11:1 (1977), 12–26.\nI. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications,” Uspekhi Mat. Nauk, 44:2(266) (1989), 121–184; English transi.: Russian Math. Surveys, 44:2 (1989), 145–225.\nI. M. Krichever, “Spectral theory of finite-zone nonstationary Schrödinger operators. A non-stationary Peierls model,” Funkts. Anal. Prilozhen., 20:3 (1986), 42–54; English transi.: Functional Anal. Appl., 20:3 (1986), 203–214.\nS. M. Natanzon, “Nonsingular finite-zone two-dimensional Schrödinger operators and prymians of real curves,” Funkts. Anal. Prilozhen., 22:1 (1988), 79–80; English transi.: Functional Anal. Appl., 22:1 (1988), 68–70.\nA. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations,” Dokl. Akad. Nauk SSSR, 279:1 (1984), 20–24.\nA. P. Veselov and S. P. Novikov, “Finite-zone, two-dimensional Schrödinger operators. Potential operators,” Dokl. Akad. Nauk SSSR, 279:1 (1984), 784–788; English transi.: Soviet Math. Dokl., 30 (1984), 588–591.\nI. A. Taimanov, “Two-dimensional Dirac operator and the theory of surfaces,” Uspekhi Mat. Nauk, 61:1(367) (2006), 85–164; English transi.: Russian Math. Surveys, 61:1 (2006), 79–159.",{"VOID":1127},"10.1007\u002Fs10688-019-0246-7","2024-05-14T05:05:39.084+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10688-019-0246-7",[1131,1155,1184],{"id":1132,"sortIndex":23,"researcher":22,"roles":1133,"affiliations":1134,"properties":1152,"displayName":1154,"givenName":22,"familyName":22},"17ec61d0-91a7-4dae-8808-ed386f156ea0",[117],[1135,1143],{"id":1136,"sortIndex":23,"affiliation":1137,"properties":22},"34368da3-8f22-4fe2-b028-3823b67b3f21",{"id":1136,"createTime":22,"updateTime":22,"relativeEntities":1138,"slug":22,"properties":1139,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1142,"statistic":22},[],{"title":1140},{"VI":1141},"Skolkovo Institute for Science and Technology, Moscow, Russia",[],{"id":1144,"sortIndex":129,"affiliation":1145,"properties":1151},"15d36eca-7f97-48eb-b5ce-7ab06ca8d82d",{"id":1144,"createTime":22,"updateTime":22,"relativeEntities":1146,"slug":22,"properties":1147,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1150,"statistic":22},[],{"title":1148},{"VI":1149},"National Research University, Higher School of Economics, Moscow, Russia",[],{},{"title":1153},{"VI":1154},"A. V. Ilina",{"id":1156,"sortIndex":129,"researcher":22,"roles":1157,"affiliations":1158,"properties":1179,"displayName":1181,"givenName":22,"familyName":22},"061b20f9-104c-42e3-a005-601ebd5fa1dc",[117],[1159,1165,1171],{"id":1136,"sortIndex":23,"affiliation":1160,"properties":22},{"id":1136,"createTime":22,"updateTime":22,"relativeEntities":1161,"slug":22,"properties":1162,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1164,"statistic":22},[],{"title":1163},{"VI":1141},[],{"id":1144,"sortIndex":129,"affiliation":1166,"properties":22},{"id":1144,"createTime":22,"updateTime":22,"relativeEntities":1167,"slug":22,"properties":1168,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1170,"statistic":22},[],{"title":1169},{"VI":1149},[],{"id":1172,"sortIndex":390,"affiliation":1173,"properties":22},"1550be97-f6da-42c8-8971-b4c1ffa60b37",{"id":1172,"createTime":22,"updateTime":22,"relativeEntities":1174,"slug":22,"properties":1175,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1178,"statistic":22},[],{"title":1176},{"EN":1177},"Columbia University,  New York,  USA",[],{"title":1180,"gsAuthor":1182},{"VI":1181},"I. M. Krichever",{"VOID":1183},"[\"uLThap0AAAAJ\"]",{"id":1185,"sortIndex":390,"researcher":22,"roles":1186,"affiliations":1187,"properties":1202,"displayName":1204,"givenName":22,"familyName":22},"0f282b97-9a9e-46bd-a6b3-bcfac4d47fd0",[117],[1188,1194],{"id":1136,"sortIndex":23,"affiliation":1189,"properties":22},{"id":1136,"createTime":22,"updateTime":22,"relativeEntities":1190,"slug":22,"properties":1191,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1193,"statistic":22},[],{"title":1192},{"VI":1141},[],{"id":1195,"sortIndex":129,"affiliation":1196,"properties":22},"edc00e94-ee95-4444-8bbd-103cc6cf5832",{"id":1195,"createTime":22,"updateTime":22,"relativeEntities":1197,"slug":22,"properties":1198,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1201,"statistic":22},[],{"title":1199},{"VI":1200},"Simons Center for Geometry and Physics, Stony Brook, USA",[],{"title":1203,"gsAuthor":1205},{"VI":1204},"N. A. Nekrasov",{"VOID":1206},"[\"bKV59LwAAAAJ\"]",{"url":1129,"publisher":1208,"properties":1249},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":1209,"slug":10,"properties":1210,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":1214,"manageAffiliations":1223,"indexDatabases":1234,"url":22,"thumbnailPath":22,"statistic":22,"gsStatistic":22,"type":22,"analyzePriority":22},[],{"issn":1211,"title":1212,"eissn":1213},{"VOID":15},{"EN":17},{"VOID":13},[1215,1219],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":1216,"label":1217,"description":1218,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":1220,"label":1221,"description":1222,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[1224,1229],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":1225,"slug":22,"properties":1226,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1228,"statistic":22},[],{"title":1227},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":1230,"slug":22,"properties":1231,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1233,"statistic":22},[],{"title":1232},{"EN":51},[],[1235,1242],{"id":55,"indexDatabase":1236,"url":68,"indexYears":22,"academicFieldIds":1241,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":1237,"label":1238,"description":1239,"key":64,"publicationTags":1240,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":1243,"url":83,"indexYears":84,"academicFieldIds":1248,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":1244,"label":1245,"description":1246,"key":80,"publicationTags":1247,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"pages":1250,"volume":1252},{"VOID":1251},"23-36",{"VOID":1253},"53",{"total":23,"publishYear":1255,"statisticByYear":1256},2019,{},"2019-06-01","2026-07-11T17:33:39.406+00:00",[88,66],{"id":1261,"createTime":1262,"updateTime":1263,"relativeEntities":1264,"slug":1265,"properties":1266,"entityType":108,"verifyStatus":109,"verifyTime":1277,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":1278,"fullTextUrl":22,"authors":1279,"publicationType":161,"publisherRelationship":1295,"citationCount":22,"citationInfo":22,"publishDate":1342,"publishYear":388,"citationAnalyzeStatus":21,"lastCitationAnalyze":1343,"indexDatabases":1344,"openAccess":22,"references":22,"isForceReanalyzing":281},"e772dadd-cb72-4a01-8144-5b22f3f64bf9","2024-01-14T06:14:09.095+00:00","2026-07-11T05:53:01.999+00:00",[],"On-local-solutions-of-degenerate-differential-inclusions",{"abstract":1267,"title":1269,"gsPaper":1271,"references":1273,"doi":1275},{"EN":1268},"We study the existence and properties of local solution sets for differential inclusions of the form (Ax)′ ∈ F(t, x), where A is a closed linear surjective operator with nontrivial null space and F is a compact set-valued mapping.",{"EN":1270},"On local solutions of degenerate differential inclusions",{"VOID":1272},"[\"14479978873857143390\"]",{"VOID":1274},"Yu. G. Borisovich, B. D. Gel’man, A. D. Myshkis, and V. V. Obukhovskii, Introduction to the Theory of Multivalued Mappings and Differential Inclusions [in Russian], KomKniga (URSS), Moscow, 2005.\nJ. Saint Raymond, C. R. Acad. Sci. Paris, 298 (1984), 71–74.\nB. D. Gel’man, Funkts. Anal. Prilozhen., 42:3 (2008), 78–81; English transl.: Functional Anal. Appl., 42:3 (2008), 227–229.",{"VOID":1276},"10.1007\u002Fs10688-012-0009-1","2024-06-26T16:11:43.857+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10688-012-0009-1",[1280],{"id":1281,"sortIndex":23,"researcher":22,"roles":1282,"affiliations":1283,"properties":1292,"displayName":1294,"givenName":22,"familyName":22},"693fd9bc-b81e-4102-8832-c9fe84790f8c",[117],[1284],{"id":1285,"sortIndex":23,"affiliation":1286,"properties":22},"17c64528-4cbb-48ec-9615-f180403887e9",{"id":1285,"createTime":22,"updateTime":22,"relativeEntities":1287,"slug":22,"properties":1288,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1291,"statistic":22},[],{"title":1289},{"VI":1290},"Voronezh State University, Voronezh, Russia",[],{"title":1293},{"VI":1294},"B. D. Gel’man",{"url":1278,"publisher":1296,"properties":1337},{"id":6,"createTime":7,"updateTime":8,"relativeEntities":1297,"slug":10,"properties":1298,"entityType":20,"verifyStatus":21,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":23,"subjectFields":1302,"manageAffiliations":1311,"indexDatabases":1322,"url":22,"thumbnailPath":22,"statistic":22,"gsStatistic":22,"type":22,"analyzePriority":22},[],{"issn":1299,"title":1300,"eissn":1301},{"VOID":15},{"EN":17},{"VOID":13},[1303,1307],{"id":26,"createTime":22,"updateTime":22,"relativeEntities":1304,"label":1305,"description":1306,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":29},{},{"id":32,"createTime":22,"updateTime":22,"relativeEntities":1308,"label":1309,"description":1310,"parentId":22,"standard":22,"scholarHubFieldId":22},[],{"EN":35},{},[1312,1317],{"id":39,"createTime":22,"updateTime":22,"relativeEntities":1313,"slug":22,"properties":1314,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1316,"statistic":22},[],{"title":1315},{"EN":43},[45],{"id":47,"createTime":22,"updateTime":22,"relativeEntities":1318,"slug":22,"properties":1319,"entityType":22,"verifyStatus":22,"verifyTime":22,"verifyNote":22,"languages":22,"translateLanguages":22,"viewCount":22,"url":22,"parentIds":1321,"statistic":22},[],{"title":1320},{"EN":51},[],[1323,1330],{"id":55,"indexDatabase":1324,"url":68,"indexYears":22,"academicFieldIds":1329,"indexDatabaseRanking":22},{"id":57,"createTime":22,"updateTime":22,"relativeEntities":1325,"label":1326,"description":1327,"key":64,"publicationTags":1328,"standard":22},[],{"EN":60,"VI":60},{"EN":62,"VI":63},[66,67],[70],{"id":72,"indexDatabase":1331,"url":83,"indexYears":84,"academicFieldIds":1336,"indexDatabaseRanking":88},{"id":74,"createTime":22,"updateTime":22,"relativeEntities":1332,"label":1333,"description":1334,"key":80,"publicationTags":1335,"standard":22},[],{"EN":77,"VI":77},{"EN":77,"VI":79},[82],[86,87],{"pages":1338,"volume":1340},{"VOID":1339},"66-68",{"VOID":1341},"46","2012-03-16","2026-07-11T05:53:01.998+00:00",[88,66],{"id":1346,"createTime":1347,"updateTime":1348,"relativeEntities":1349,"slug":1350,"properties":1351,"entityType":108,"verifyStatus":109,"verifyTime":1362,"verifyNote":111,"languages":22,"translateLanguages":22,"viewCount":23,"primaryUrl":1363,"fullTextUrl":1364,"authors":1365,"publicationType":161,"publisherRelationship":1410,"citationCount":22,"citationInfo":22,"publishDate":1459,"publishYear":1460,"citationAnalyzeStatus":1461,"lastCitationAnalyze":1462,"indexDatabases":1463,"openAccess":22,"references":22,"isForceReanalyzing":281},"7d5d8dd6-0179-481c-813d-a514c24a7c78","2024-01-09T14:55:53.408+00:00","2026-06-27T09:20:32.920+00:00",[],"On-Simple-mathbb-Z-3-Invariant-Function-Germs",{"abstract":1352,"title":1354,"gsPaper":1356,"references":1358,"doi":1360},{"EN":1353},"V. I. Arnold classified simple (i.e., having no moduli for classification) singularities (function germs) and also simple boundary singularities, that is, function germs invariant with respect to the action $$\\sigma(x_1; y_1,\\dots, y_n)=(-x_1; y_1,\\dots, y_n)$$ of the group $${\\mathbb Z}_2$$ . In particular, he showed that a function germ (a germ of a boundary singularity) is simple if and only if the intersection form (respectively, the restriction of the intersection form to the subspace of anti-invariant cycles) of a germ in $$3+4s$$ variables stably equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding space is finite. In a previous paper the authors obtained analogues of the latter statements for function germs invariant with respect to an arbitrary action of the group $${\\mathbb Z}_2$$ and also for corner singularities. This paper presents an analogue of the simplicity criterion in terms of the intersection form for functions invariant with respect to a number of actions (representations) of the group $${\\mathbb Z}_3$$ .",{"EN":1355},"On Simple $${\\mathbb Z}_3$$ -Invariant Function Germs",{"VOID":1357},"[]",{"VOID":1359},"citation_journal_title=Funkts. Anal. Prilozhen.; citation_title=Normal forms for functions near degenerate critical points, the Weyl groups \n                    \n                    \n                  , \n                    \n                    \n                  , \n                    \n                    \n                   and Lagrangian singularities; citation_author=V. I. Arnold; citation_volume=6; citation_issue=4; citation_publication_date=1972; citation_pages=3-25; citation_id=CR1\ncitation_journal_title=Uspekhi Mat. Nauk; citation_title=Critical points of functions on a manifold with boundary, the simple Lie groups \n                    \n                    \n                  , \n                    \n                    \n                  , \n                    \n                    \n                   and singularities of evolutes; citation_author=V. I. Arnold; citation_volume=33; citation_issue=5(203); citation_publication_date=1978; citation_pages=91-105; citation_id=CR2\ncitation_title=Singularities of differentiable maps.; citation_publication_date=1985; citation_id=CR3; citation_author=V. I. Arnold; citation_author=S. M. Gusein-Zade; citation_author=A. N. Varchenko; citation_publisher=Birkhauser\ncitation_journal_title=Phys. Lett. B; citation_title=The mirror map for invertible LG models; citation_author=M. Kreuzer; citation_volume=328; citation_issue=3–4; citation_publication_date=1994; citation_pages=312-318; citation_doi=10.1016\u002F0370-2693(94)91485-0; citation_id=CR4\ncitation_journal_title=Mat. Zametki; citation_title=On simple \n                    \n                    \n                  -invariant and corner function germs; citation_author=S. M. Gusein-Zade, A.-M. Ya. Rauch; citation_volume=107; citation_issue=6; citation_publication_date=2020; citation_pages=855-864; citation_doi=10.4213\u002Fmzm12512; citation_id=CR5\ncitation_journal_title=Math. Z.; citation_title=Einige Bemerkungen zur Entfaltung symmetrischer Funktionen; citation_author=P. Slodowy; citation_volume=158; citation_issue=2; citation_publication_date=1978; citation_pages=157-170; citation_doi=10.1007\u002FBF01320865; citation_id=CR6\ncitation_journal_title=Compositio Math.; citation_title=Intersection form for quasi-homogeneous singularities; citation_author=J. 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