Pub Date : 1993-02-28DOI: 10.1070/SM1993V074N01ABEH003332
V. Goryainov
An infinitesimal description is obtained for fractional iterates of analytic functions on the unit disk under the condition that the functions and their iterates do not move fixed points on the unit circle at which they have finite angular derivatives.
{"title":"FRACTIONAL ITERATES OF FUNCTIONS ANALYTIC IN THE UNIT DISK, WITH GIVEN FIXED POINTS","authors":"V. Goryainov","doi":"10.1070/SM1993V074N01ABEH003332","DOIUrl":"https://doi.org/10.1070/SM1993V074N01ABEH003332","url":null,"abstract":"An infinitesimal description is obtained for fractional iterates of analytic functions on the unit disk under the condition that the functions and their iterates do not move fixed points on the unit circle at which they have finite angular derivatives.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123722189","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 : 1993-02-28DOI: 10.1070/SM1993V074N02ABEH003362
A. Ivanov, A. Tuzhilin
The famous Steiner problem in the Euclidean plane, which is that of investigating minimal nets spanning fixed finite subsets M of points in the plane, is solved when M is extremal, i.e. when M lies on the boundary of its convex hull, and the nets are nondegenerate, i.e. have no vertices of degree 2.
{"title":"THE STEINER PROBLEM IN THE PLANE OR IN PLANE MINIMAL NETS","authors":"A. Ivanov, A. Tuzhilin","doi":"10.1070/SM1993V074N02ABEH003362","DOIUrl":"https://doi.org/10.1070/SM1993V074N02ABEH003362","url":null,"abstract":"The famous Steiner problem in the Euclidean plane, which is that of investigating minimal nets spanning fixed finite subsets M of points in the plane, is solved when M is extremal, i.e. when M lies on the boundary of its convex hull, and the nets are nondegenerate, i.e. have no vertices of degree 2.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129452462","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 : 1993-02-28DOI: 10.1070/SM1993V074N01ABEH003341
V. Burichenko
A special commutative Moufang loop of order is described. With the help of this loop, a trilinear Dickson form is constructed whose automorphism group is a Chevalley group of type . Next, with the help of , a 27-dimensional representation is constructed for over , . This makes it possible to prove anew the embedding . A similar construction concerning the embedding is described.
{"title":"ON A SPECIAL LOOP, THE DICKSON FORM, AND THE LATTICE CONNECTED WITH $ O_7(3)$","authors":"V. Burichenko","doi":"10.1070/SM1993V074N01ABEH003341","DOIUrl":"https://doi.org/10.1070/SM1993V074N01ABEH003341","url":null,"abstract":"A special commutative Moufang loop of order is described. With the help of this loop, a trilinear Dickson form is constructed whose automorphism group is a Chevalley group of type . Next, with the help of , a 27-dimensional representation is constructed for over , . This makes it possible to prove anew the embedding . A similar construction concerning the embedding is described.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129733574","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 : 1993-02-28DOI: 10.1070/SM1993V074N01ABEH003340
V. S. Klimov, N. V. Senchakova
Conditions are presented under which the relative index of a critical set realizing a local minimum of a nonsmooth functional coincides with the Euler-Poincare characteristic of this set. An analogous result is obtained for the index of a functional increasing at .
{"title":"ON THE RELATIVE ROTATION OF MULTIVALUED POTENTIAL VECTOR FIELDS","authors":"V. S. Klimov, N. V. Senchakova","doi":"10.1070/SM1993V074N01ABEH003340","DOIUrl":"https://doi.org/10.1070/SM1993V074N01ABEH003340","url":null,"abstract":"Conditions are presented under which the relative index of a critical set realizing a local minimum of a nonsmooth functional coincides with the Euler-Poincare characteristic of this set. An analogous result is obtained for the index of a functional increasing at .","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132951274","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 : 1993-02-28DOI: 10.1070/SM1993V074N01ABEH003335
A. I. Subbotin
Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton-Jacobi equations to prove the equivalence of various types of definitions of generalized solutions.
{"title":"ON A PROPERTY OF THE SUBDIFFERENTIAL","authors":"A. I. Subbotin","doi":"10.1070/SM1993V074N01ABEH003335","DOIUrl":"https://doi.org/10.1070/SM1993V074N01ABEH003335","url":null,"abstract":"Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton-Jacobi equations to prove the equivalence of various types of definitions of generalized solutions.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114208090","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 : 1993-02-28DOI: 10.1070/SM1993V074N01ABEH003339
V. Medvedev
Let be continuous mappings of a compactum onto compacta , . The following theorem is known for : if any bounded function on can be represented in the form , where and are bounded functions on and , then any continuous can be represented in the same form with continuous and . An example is constructed showing that the analogous theorem is false for .
{"title":"On the Representation of Functions as a Sum of Several Compositions","authors":"V. Medvedev","doi":"10.1070/SM1993V074N01ABEH003339","DOIUrl":"https://doi.org/10.1070/SM1993V074N01ABEH003339","url":null,"abstract":"Let be continuous mappings of a compactum onto compacta , . The following theorem is known for : if any bounded function on can be represented in the form , where and are bounded functions on and , then any continuous can be represented in the same form with continuous and . An example is constructed showing that the analogous theorem is false for .","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121125323","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 : 1993-02-28DOI: 10.1070/SM1993V074N01ABEH003345
L. A. Muravei, A. V. Filinovskii
Well-posed solvability is proved in an appropriate energy space of a boundary value problem with a nonlocal boundary condition for a one-dimensional parabolic equation; two-sided uniform estimates of the solution are obtained, which replace the maximum principle. The existence of an optimal control of the diffusion coefficient in the problem of minimizing the quality functional is established in the class of functions of bounded variation.
{"title":"ON A PROBLEM WITH NONLOCAL BOUNDARY CONDITION FOR A PARABOLIC EQUATION","authors":"L. A. Muravei, A. V. Filinovskii","doi":"10.1070/SM1993V074N01ABEH003345","DOIUrl":"https://doi.org/10.1070/SM1993V074N01ABEH003345","url":null,"abstract":"Well-posed solvability is proved in an appropriate energy space of a boundary value problem with a nonlocal boundary condition for a one-dimensional parabolic equation; two-sided uniform estimates of the solution are obtained, which replace the maximum principle. The existence of an optimal control of the diffusion coefficient in the problem of minimizing the quality functional is established in the class of functions of bounded variation.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"130 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128487047","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 : 1993-02-28DOI: 10.1070/SM1993V074N01ABEH003344
V. Rodin
This article is devoted to series in the Walsh-Paley system. In particular, for an integrable function a condition is obtained for p-strong summability of the Fourier-Walsh series a.e. on [0,1] along with a uniform analogue for a continuous, function on [0,1].
{"title":"THE SPACE BMO AND STRONG MEANS OF FOURIER-WALSH SERIES","authors":"V. Rodin","doi":"10.1070/SM1993V074N01ABEH003344","DOIUrl":"https://doi.org/10.1070/SM1993V074N01ABEH003344","url":null,"abstract":"This article is devoted to series in the Walsh-Paley system. In particular, for an integrable function a condition is obtained for p-strong summability of the Fourier-Walsh series a.e. on [0,1] along with a uniform analogue for a continuous, function on [0,1].","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116542908","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 : 1993-02-28DOI: 10.1070/SM1993V074N02ABEH003352
G. Magaril-Il'yaev
The concept of mean dimension is introduced for a broad class of subspaces of , and analogues of the Kolmogorov widths, Bernstein widths, Gel'fand widths, and linear widths are defined. The precise values of these quantities are computed for Sobolev classes of functions on in compatible metrics, and the corresponding extremal spaces and operators are described. A closely related problem of optimal recovery of functions in Sobolev classes is also studied.
{"title":"MEAN DIMENSION, WIDTHS, AND OPTIMAL RECOVERY OF SOBOLEV CLASSES OF FUNCTIONS ON THE LINE","authors":"G. Magaril-Il'yaev","doi":"10.1070/SM1993V074N02ABEH003352","DOIUrl":"https://doi.org/10.1070/SM1993V074N02ABEH003352","url":null,"abstract":"The concept of mean dimension is introduced for a broad class of subspaces of , and analogues of the Kolmogorov widths, Bernstein widths, Gel'fand widths, and linear widths are defined. The precise values of these quantities are computed for Sobolev classes of functions on in compatible metrics, and the corresponding extremal spaces and operators are described. A closely related problem of optimal recovery of functions in Sobolev classes is also studied.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130282076","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 : 1993-02-28DOI: 10.1070/SM1993V074N02ABEH003347
R. Varga, A. Ruttan, A. D. Karpenter
With denoting the error of best uniform rational approximation from to on , we determine the numbers , where each of these numbers was calculated with a precision of at least 200 significant digits. With these numbers, the Richardson extrapolation method was applied to the products , and it appears, to at least 10 significant digits, that which gives rise to an interesting new conjecture in the theory of rational approximation.
{"title":"NUMERICAL RESULTS ON BEST UNIFORM RATIONAL APPROXIMATION OF $ vert xvert$ ON $ lbrack-1,,+1rbrack$","authors":"R. Varga, A. Ruttan, A. D. Karpenter","doi":"10.1070/SM1993V074N02ABEH003347","DOIUrl":"https://doi.org/10.1070/SM1993V074N02ABEH003347","url":null,"abstract":"With denoting the error of best uniform rational approximation from to on , we determine the numbers , where each of these numbers was calculated with a precision of at least 200 significant digits. With these numbers, the Richardson extrapolation method was applied to the products , and it appears, to at least 10 significant digits, that which gives rise to an interesting new conjecture in the theory of rational approximation.","PeriodicalId":208776,"journal":{"name":"Mathematics of The Ussr-sbornik","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1993-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123006663","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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