Pub Date : 1992-04-30DOI: 10.1070/IM1992V038N02ABEH002197
V. Bulitko
The traditional method for constructing criteria for completeness with respect to reducibility is by describing the property of (in general, weakened) productiveness satisfied by the complement of a set which is complete with respect to the given reducibility. Originally this property was tied to the reducibility of a creative set to the complete set. Such a method appeals directly to the universality of the creative set in the class of all recursively enumerable sets.However, for several reducibilities it is possible to determine the completeness of a recursively enumerable set from the fact that a certain set of degree below the degree of the creative sets is reducible to the given set. This second, "test" set is, of course, not recursively enumerable. In addition, in place of the property of effective nonrecursive enumerability which productive sets have, it is possible to substitute variants of the property of diagonal nonrecursiveness, although not for all reducibilities.In this paper we examine the connection between these two approaches--specifically, between different weakenings of the property of productiveness on the one hand, and diagonal nonrecursiveness on the other--and we present results obtained by these methods for Turing and truth-table reducibilities.
{"title":"ON WAYS OF CHARACTERIZING COMPLETE SETS","authors":"V. Bulitko","doi":"10.1070/IM1992V038N02ABEH002197","DOIUrl":"https://doi.org/10.1070/IM1992V038N02ABEH002197","url":null,"abstract":"The traditional method for constructing criteria for completeness with respect to reducibility is by describing the property of (in general, weakened) productiveness satisfied by the complement of a set which is complete with respect to the given reducibility. Originally this property was tied to the reducibility of a creative set to the complete set. Such a method appeals directly to the universality of the creative set in the class of all recursively enumerable sets.However, for several reducibilities it is possible to determine the completeness of a recursively enumerable set from the fact that a certain set of degree below the degree of the creative sets is reducible to the given set. This second, \"test\" set is, of course, not recursively enumerable. In addition, in place of the property of effective nonrecursive enumerability which productive sets have, it is possible to substitute variants of the property of diagonal nonrecursiveness, although not for all reducibilities.In this paper we examine the connection between these two approaches--specifically, between different weakenings of the property of productiveness on the one hand, and diagonal nonrecursiveness on the other--and we present results obtained by these methods for Turing and truth-table reducibilities.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134452421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002194
V. Klyachin, V. Miklyukov
Consider -solutions of the equations for maximal surfaces in Minkowski space The hypersurface is tubular if for every the level sets are compact. The girth function of a tubular hypersurface is given by . In this paper it is shown that the girth function of a maximal tubular surface satisfies the differential inequality . As a consequence of this assertion it is established that the union of the rays tangent to the hypersurface at an isolated singular point forms the light cone; a bound is obtained, in the neighborhood of an isolated singularity, to the spread of the maximal tube in the direction of the time axis in terms of its deviation from the light cone.
{"title":"MAXIMAL TUBULAR HYPERSURFACES IN MINKOWSKI SPACE","authors":"V. Klyachin, V. Miklyukov","doi":"10.1070/IM1992V038N01ABEH002194","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002194","url":null,"abstract":"Consider -solutions of the equations for maximal surfaces in Minkowski space The hypersurface is tubular if for every the level sets are compact. The girth function of a tubular hypersurface is given by . In this paper it is shown that the girth function of a maximal tubular surface satisfies the differential inequality . As a consequence of this assertion it is established that the union of the rays tangent to the hypersurface at an isolated singular point forms the light cone; a bound is obtained, in the neighborhood of an isolated singularity, to the spread of the maximal tube in the direction of the time axis in terms of its deviation from the light cone.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127158589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002223
V. A. Krasnov
For a vector bundle on a real algebraic variety , the author studies the connections between the characteristic classes It is proved that for -varieties the equality implies the congruence . Sufficient conditions are found also for the converse to hold; this requires the construction of new characteristic classes . The results are applied to study the topology of .
{"title":"CHARACTERISTIC CLASSES OF VECTOR BUNDLES ON A REAL ALGEBRAIC VARIETY","authors":"V. A. Krasnov","doi":"10.1070/IM1992V039N01ABEH002223","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002223","url":null,"abstract":"For a vector bundle on a real algebraic variety , the author studies the connections between the characteristic classes It is proved that for -varieties the equality implies the congruence . Sufficient conditions are found also for the converse to hold; this requires the construction of new characteristic classes . The results are applied to study the topology of .","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124323537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002225
A. Khrennikov
A mathematical apparatus is developed for non-Archimedean physics: a theory of generalized functions, a theory of integration, and a harmonic analysis. Both finite-dimensional and infinite-dimensional non-Archimedean spaces are considered. Gaussian and Feynman path integrals on non-Archimedean function spaces are introduced. Quantization of a non-Archimedean scalar bosonic field is carried out in the formalism of path integrals. Linear differential equations in spaces of test functions and spaces of generalized functions on infinite-dimensional non-Archimedean spaces are studied (in particular, the heat equation and the Schrodinger equation with a potential).
{"title":"GENERALIZED FUNCTIONS AND GAUSSIAN PATH INTEGRALS OVER NON-ARCHIMEDEAN FUNCTION SPACES","authors":"A. Khrennikov","doi":"10.1070/IM1992V039N01ABEH002225","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002225","url":null,"abstract":"A mathematical apparatus is developed for non-Archimedean physics: a theory of generalized functions, a theory of integration, and a harmonic analysis. Both finite-dimensional and infinite-dimensional non-Archimedean spaces are considered. Gaussian and Feynman path integrals on non-Archimedean function spaces are introduced. Quantization of a non-Archimedean scalar bosonic field is carried out in the formalism of path integrals. Linear differential equations in spaces of test functions and spaces of generalized functions on infinite-dimensional non-Archimedean spaces are studied (in particular, the heat equation and the Schrodinger equation with a potential).","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124425215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002185
I. Babenko, S. Bogatyi
This paper strengthens a theorem due to A. Dold on the algebraic properties of sequences of integers which are Lefschetz numbers of the iterates of a continuous map from a finite polyhedron to itself. The realizability of sequences satisfying Dold's condition at a single fixed point of a continuous map on R3 is proved. Indices of a fixed point (under iteration) are investigated in the case of a smooth mapping. A linear lower bound on the number of periodic points of a smooth map, which strengthens a result of Shub and Sullivan, is obtained.
{"title":"THE BEHAVIOR OF THE INDEX OF PERIODIC POINTS UNDER ITERATIONS OF A MAPPING","authors":"I. Babenko, S. Bogatyi","doi":"10.1070/IM1992V038N01ABEH002185","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002185","url":null,"abstract":"This paper strengthens a theorem due to A. Dold on the algebraic properties of sequences of integers which are Lefschetz numbers of the iterates of a continuous map from a finite polyhedron to itself. The realizability of sequences satisfying Dold's condition at a single fixed point of a continuous map on R3 is proved. Indices of a fixed point (under iteration) are investigated in the case of a smooth mapping. A linear lower bound on the number of periodic points of a smooth map, which strengthens a result of Shub and Sullivan, is obtained.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121228289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002187
A. Bolsinov
This paper presents a method for checking completeness of families of functions which are in involution with respect to compatible Poisson brackets. Several examples of compatible Poisson brackets on duals of Lie algebras are considered, as well as the associated involutive function families and Hamiltonian systems. The transitions of Liouville tori for some nonintegrable Hamiltonian systems, notably the equations of motion for a higher dimensional rigid body, are described.
{"title":"COMPATIBLE POISSON BRACKETS ON LIE ALGEBRAS AND COMPLETENESS OF FAMILIES OF FUNCTIONS IN INVOLUTION","authors":"A. Bolsinov","doi":"10.1070/IM1992V038N01ABEH002187","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002187","url":null,"abstract":"This paper presents a method for checking completeness of families of functions which are in involution with respect to compatible Poisson brackets. Several examples of compatible Poisson brackets on duals of Lie algebras are considered, as well as the associated involutive function families and Hamiltonian systems. The transitions of Liouville tori for some nonintegrable Hamiltonian systems, notably the equations of motion for a higher dimensional rigid body, are described.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123026943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002186
F. Bogomolov
The author proves that every Abelian subgroup of rank in the Galois group of the algebraic closure of a rational function field is contained in a ramification subgroup, and also that the unramified Brauer group equals the unramified Brauer group defined in [2], §3, where is the quotient group .
{"title":"ABELIAN SUBGROUPS OF GALOIS GROUPS","authors":"F. Bogomolov","doi":"10.1070/IM1992V038N01ABEH002186","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002186","url":null,"abstract":"The author proves that every Abelian subgroup of rank in the Galois group of the algebraic closure of a rational function field is contained in a ramification subgroup, and also that the unramified Brauer group equals the unramified Brauer group defined in [2], §3, where is the quotient group .","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125334855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002231
G. Tomanov
It is proved that if the Platonov-Margulis conjecture on the standard structure of normal subgroups holds for the division algebras of index r, then it also holds for the division algebras of index n=2mr, for any m. Thus the conjecture is proved for the division algebras of index n=2m, for any m, and its proof in the general case is reduced to the case of division algebras of odd index.
{"title":"ON THE REDUCED NORM 1 GROUP OF A DIVISION ALGEBRA OVER A GLOBAL FIELD","authors":"G. Tomanov","doi":"10.1070/IM1992V039N01ABEH002231","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002231","url":null,"abstract":"It is proved that if the Platonov-Margulis conjecture on the standard structure of normal subgroups holds for the division algebras of index r, then it also holds for the division algebras of index n=2mr, for any m. Thus the conjecture is proved for the division algebras of index n=2m, for any m, and its proof in the general case is reduced to the case of division algebras of odd index.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117202740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V038N01ABEH002190
S. Levendorskii
An algebra is constructed containing the operators of boundary value problems for pseudodifferential operators which are formally hypoelliptic in the sense of Hormander and elliptic at points of the boundary; parametrices for these boundary value problems are also constructed. The Fredholm property is demonstrated in a scale of weighted spaces, based on Lp, 1
{"title":"AN ALGEBRA OF BOUNDARY VALUE PROBLEMS FOR A CLASS OF PSEUDODIFFERENTIAL OPERATORS OF VARIABLE ORDER","authors":"S. Levendorskii","doi":"10.1070/IM1992V038N01ABEH002190","DOIUrl":"https://doi.org/10.1070/IM1992V038N01ABEH002190","url":null,"abstract":"An algebra is constructed containing the operators of boundary value problems for pseudodifferential operators which are formally hypoelliptic in the sense of Hormander and elliptic at points of the boundary; parametrices for these boundary value problems are also constructed. The Fredholm property is demonstrated in a scale of weighted spaces, based on Lp, 1","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125980550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1992-02-28DOI: 10.1070/IM1992V039N01ABEH002226
I. I. Golichev
For a problem with control of the right-hand side of the state equation and with observation at the terminal time instant, the representation of the solution in the form of a function of an operator is obtained. A criterion for solvability of the time-optimal control problem is found. An iterative process for solving the problem, whose rate of convergence does not depend on the regularization parameter, is constructed.
{"title":"THE METHOD OF FUNCTION OF AN OPERATOR, AND ITERATIVE PROCESSES IN SOME OPTIMAL CONTROL PROBLEMS","authors":"I. I. Golichev","doi":"10.1070/IM1992V039N01ABEH002226","DOIUrl":"https://doi.org/10.1070/IM1992V039N01ABEH002226","url":null,"abstract":"For a problem with control of the right-hand side of the state equation and with observation at the terminal time instant, the representation of the solution in the form of a function of an operator is obtained. A criterion for solvability of the time-optimal control problem is found. An iterative process for solving the problem, whose rate of convergence does not depend on the regularization parameter, is constructed.","PeriodicalId":159459,"journal":{"name":"Mathematics of The Ussr-izvestiya","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1992-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133960553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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