Pub Date : 1992-02-28DOI: 10.1070/SM1992V071N02ABEH001399
A. Ivanov
This paper is devoted to the development of methods of investigating the stability and global minimality of specific surfaces in Euclidean space and more generally in the Riemannian manifold. The author has obtained an effective sufficient condition for the stability of symmetric cones of any codimension in Euclidean space. By means of this sufficient condition he has proved the stability of several new series of cones of codimension two and higher. The author has constructed a new class of globally minimal surfaces in locally trivial vector bundles. The proof of the basic theorems is carried out by means of the construction of suitable calibration forms.
{"title":"CALIBRATION FORMS AND NEW EXAMPLES OF STABLE AND GLOBALLY MINIMAL SURFACES","authors":"A. Ivanov","doi":"10.1070/SM1992V071N02ABEH001399","DOIUrl":"https://doi.org/10.1070/SM1992V071N02ABEH001399","url":null,"abstract":"This paper is devoted to the development of methods of investigating the stability and global minimality of specific surfaces in Euclidean space and more generally in the Riemannian manifold. The author has obtained an effective sufficient condition for the stability of symmetric cones of any codimension in Euclidean space. By means of this sufficient condition he has proved the stability of several new series of cones of codimension two and higher. The author has constructed a new class of globally minimal surfaces in locally trivial vector bundles. The proof of the basic theorems is carried out by means of the construction of suitable calibration forms.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"4 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":"127051619","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/SM1992V073N02ABEH002551
V. E. Maiorov
Inequalities of the following form are proved: if is an arbitrary function and , then where depends only on . The exponent is a limiting exponent. With the inequalities as a basis, imbedding theorems are constructed for classes of solutions of nonlinear singular differential equations in the space of times differentiable functions.
{"title":"Multiplicative Inequalities for Derivatives, and a Priori Estimates of Smoothness of Solutions of Nonlinear Differential Equations","authors":"V. E. Maiorov","doi":"10.1070/SM1992V073N02ABEH002551","DOIUrl":"https://doi.org/10.1070/SM1992V073N02ABEH002551","url":null,"abstract":"Inequalities of the following form are proved: if is an arbitrary function and , then where depends only on . The exponent is a limiting exponent. With the inequalities as a basis, imbedding theorems are constructed for classes of solutions of nonlinear singular differential equations in the space of times differentiable functions.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"4 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":"127376213","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/SM1992V072N01ABEH001410
A. Grigor’yan
The behavior of the Green function G(x, y, t) of the Cauchy problem for the heat equation on a connected, noncompact, complete Riemannian manifold is investigated. For manifolds with boundary it is assumed that the Green function satisfies a Neumann condition on the boundary.
研究了热方程在连通非紧化完备黎曼流形上的柯西问题的格林函数G(x, y, t)的性质。对于有边界的流形，假定格林函数在边界上满足诺伊曼条件。
{"title":"THE HEAT EQUATION ON NONCOMPACT RIEMANNIAN MANIFOLDS","authors":"A. Grigor’yan","doi":"10.1070/SM1992V072N01ABEH001410","DOIUrl":"https://doi.org/10.1070/SM1992V072N01ABEH001410","url":null,"abstract":"The behavior of the Green function G(x, y, t) of the Cauchy problem for the heat equation on a connected, noncompact, complete Riemannian manifold is investigated. For manifolds with boundary it is assumed that the Green function satisfies a Neumann condition on the boundary.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"37 2 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":"114156231","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/SM1992V073N01ABEH002540
A. Gushchin, V. P. Mikhaĭlov
A test is established for the existence of a boundary value for the solution of the elliptic second order equation In this connection, it is proved that the solution has a property () similar to continuity with respect to all variables in , and that its boundary value is the limit in of the traces of the solution on surfaces in a large class (which are not necessarily "parallel" to the boundary).
{"title":"ON THE EXISTENCE OF BOUNDARY VALUES OF SOLUTIONS OF AN ELLIPTIC EQUATION","authors":"A. Gushchin, V. P. Mikhaĭlov","doi":"10.1070/SM1992V073N01ABEH002540","DOIUrl":"https://doi.org/10.1070/SM1992V073N01ABEH002540","url":null,"abstract":"A test is established for the existence of a boundary value for the solution of the elliptic second order equation In this connection, it is proved that the solution has a property () similar to continuity with respect to all variables in , and that its boundary value is the limit in of the traces of the solution on surfaces in a large class (which are not necessarily \"parallel\" to the boundary).","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"55 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":"127577075","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/SM1992V073N01ABEH002538
A. Premet
The ring of invariant polynomial functions on the general algebra of Cartan type is described explicitly. It is assumed that the ground field is algebraically closed and its characteristic is greater than 2. This result is used to prove that the variety of nilpotent elements in is an irreducible complete intersection and contains an open orbit whose complement consists of singular points. Moreover, a criterion for orbits in to be closed is obtained, and it is proved that the action of the commutator subgroup of the automorphism group in is stable.
{"title":"THE THEOREM ON RESTRICTION OF INVARIANTS, AND NILPOTENT ELEMENTS IN","authors":"A. Premet","doi":"10.1070/SM1992V073N01ABEH002538","DOIUrl":"https://doi.org/10.1070/SM1992V073N01ABEH002538","url":null,"abstract":"The ring of invariant polynomial functions on the general algebra of Cartan type is described explicitly. It is assumed that the ground field is algebraically closed and its characteristic is greater than 2. This result is used to prove that the variety of nilpotent elements in is an irreducible complete intersection and contains an open orbit whose complement consists of singular points. Moreover, a criterion for orbits in to be closed is obtained, and it is proved that the action of the commutator subgroup of the automorphism group in is stable.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"37 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":"133175214","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/SM1992V071N01ABEH001393
V. Palamodov
Properties of sheaves of graded Lie algebras associated with a flat mapping of complex spaces are established. In particular, for a minimal versal deformation the tangent algebra of a fiber defines a linearization of the algebra of liftable fields on the base, which in turn enables one to find the discriminant of the deformation and its modular subspace. A criterion is obtained for the nilpotency of the tangent algebra of the germ of a hypersurface with a unique singular point. It is proved that in the algebra of liftable fields on the base of a minimal versal deformation of such a germ there always exists a basis with symmetric coefficient matrix.
{"title":"TANGENT FIELDS ON DEFORMATIONS OF COMPLEX SPACES","authors":"V. Palamodov","doi":"10.1070/SM1992V071N01ABEH001393","DOIUrl":"https://doi.org/10.1070/SM1992V071N01ABEH001393","url":null,"abstract":"Properties of sheaves of graded Lie algebras associated with a flat mapping of complex spaces are established. In particular, for a minimal versal deformation the tangent algebra of a fiber defines a linearization of the algebra of liftable fields on the base, which in turn enables one to find the discriminant of the deformation and its modular subspace. A criterion is obtained for the nilpotency of the tangent algebra of the germ of a hypersurface with a unique singular point. It is proved that in the algebra of liftable fields on the base of a minimal versal deformation of such a germ there always exists a basis with symmetric coefficient matrix.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"87 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":"133188966","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/SM1992V071N01ABEH001391
E. Ferapontov
Weakly nonlinear semi-Hamiltonian systems of n differential equations of hydrodynamic type in Riemann invariants are considered, and the geometry of the (n + 2)-web formed by the characteristics and the level lines of the independent variables are studied. It is shown that the rank of this web on the general solution of the system is equal to n. This result is used to obtain formulas for the general integral of the systems under consideration, with the necessary arbitrariness in n functions of a single argument. Separate consideration is given to the cases n = 3 and n = 4, for which it is possible not only to integrate the corresponding systems, but also to give a complete classification of them to within so-called transformations via a solution (reciprocal transformations). It turns out that for n = 3 they can all be linearized (and are thus equivalent), while for n = 4 there exist exactly five mutually nonequivalent systems, and any other system can be reduced to one of them by a transformation via a solution. There is a discussion of the connection between weakly nonlinear semi-Hamiltonian systems and Dupin cyclides-hypersurfaces of Euclidean space whose principal curvatures are constant along the corresponding principal directions. Some unsolved problems are formulated at the end of the paper.
{"title":"INTEGRATION OF WEAKLY NONLINEAR SEMI-HAMILTONIAN SYSTEMS OF HYDRODYNAMIC TYPE BY METHODS OF THE THEORY OF WEBS","authors":"E. Ferapontov","doi":"10.1070/SM1992V071N01ABEH001391","DOIUrl":"https://doi.org/10.1070/SM1992V071N01ABEH001391","url":null,"abstract":"Weakly nonlinear semi-Hamiltonian systems of n differential equations of hydrodynamic type in Riemann invariants are considered, and the geometry of the (n + 2)-web formed by the characteristics and the level lines of the independent variables are studied. It is shown that the rank of this web on the general solution of the system is equal to n. This result is used to obtain formulas for the general integral of the systems under consideration, with the necessary arbitrariness in n functions of a single argument. Separate consideration is given to the cases n = 3 and n = 4, for which it is possible not only to integrate the corresponding systems, but also to give a complete classification of them to within so-called transformations via a solution (reciprocal transformations). It turns out that for n = 3 they can all be linearized (and are thus equivalent), while for n = 4 there exist exactly five mutually nonequivalent systems, and any other system can be reduced to one of them by a transformation via a solution. There is a discussion of the connection between weakly nonlinear semi-Hamiltonian systems and Dupin cyclides-hypersurfaces of Euclidean space whose principal curvatures are constant along the corresponding principal directions. Some unsolved problems are formulated at the end of the paper.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"39 10 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":"133138238","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/SM1992V071N02ABEH002130
F. K. Mukminov
The first mixed problem with a homogeneous boundary condition is considered for a linear parabolic equation of second order. It is assumed that the unbounded domain satisfies the following condition: there exists a positive constant such that for any point of the boundary For a certain class of initial functions , which includes all bounded functions, the following condition is a necessary and sufficient condition for uniform stabilization of the solution to zero: The proof of the stabilization condition is based on an estimate of the Green function that takes account of its decay near the boundary.
{"title":"ON UNIFORM STABILIZATION OF SOLUTIONS OF THE FIRST MIXED PROBLEM FOR A PARABOLIC EQUATION","authors":"F. K. Mukminov","doi":"10.1070/SM1992V071N02ABEH002130","DOIUrl":"https://doi.org/10.1070/SM1992V071N02ABEH002130","url":null,"abstract":"The first mixed problem with a homogeneous boundary condition is considered for a linear parabolic equation of second order. It is assumed that the unbounded domain satisfies the following condition: there exists a positive constant such that for any point of the boundary For a certain class of initial functions , which includes all bounded functions, the following condition is a necessary and sufficient condition for uniform stabilization of the solution to zero: The proof of the stabilization condition is based on an estimate of the Green function that takes account of its decay near the boundary.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"96 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":"127290454","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/SM1992V071N01ABEH001390
E. Khukhro, V. M. Maksimov
It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.
{"title":"LIE RINGS AND LIE GROUPS ADMITTING AN ALMOST REGULAR AUTOMORPHISM OF PRIME ORDER","authors":"E. Khukhro, V. M. Maksimov","doi":"10.1070/SM1992V071N01ABEH001390","DOIUrl":"https://doi.org/10.1070/SM1992V071N01ABEH001390","url":null,"abstract":"It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"25 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":"115860449","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/SM1992V071N01ABEH001397
P. Gres
Theorems are proved that reduce the proof of the Brauer conjecture for finite groups G with a p-soluble centralizer of a p-element to the evaluation of the minimum of a suitable positive definite quadratic form, whose matrix is given in terms of the Cartan matrix of a p-block of a group of simpler structure than G.
{"title":"REDUCTION THEOREMS FOR THE BRAUER CONJECTURE ON THE NUMBER OF CHARACTERS IN A p-BLOCK","authors":"P. Gres","doi":"10.1070/SM1992V071N01ABEH001397","DOIUrl":"https://doi.org/10.1070/SM1992V071N01ABEH001397","url":null,"abstract":"Theorems are proved that reduce the proof of the Brauer conjecture for finite groups G with a p-soluble centralizer of a p-element to the evaluation of the minimum of a suitable positive definite quadratic form, whose matrix is given in terms of the Cartan matrix of a p-block of a group of simpler structure than G.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"2 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":"126560329","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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