We study the group structure of the Schönbucher–Wilmott equation with a free parameter, which models the pricing options. We find a five-dimensional group of equivalence transformations for this equation. By means of this group we find four-dimensional Lie algebras of the admitted operators of the equation in the cases of two cases of the free term and we find a three-dimensional Lie algebra for other nonequivalent specifications. For each algebra we find optimal systems of subalgebras and the corresponding invariant solutions or invariant submodels.
{"title":"Symmetries and exact solutions of a nonlinear pricing options equation","authors":"M. Dyshaev, V. Fedorov","doi":"10.13108/2017-9-1-29","DOIUrl":"https://doi.org/10.13108/2017-9-1-29","url":null,"abstract":"We study the group structure of the Schönbucher–Wilmott equation with a free parameter, which models the pricing options. We find a five-dimensional group of equivalence transformations for this equation. By means of this group we find four-dimensional Lie algebras of the admitted operators of the equation in the cases of two cases of the free term and we find a three-dimensional Lie algebra for other nonequivalent specifications. For each algebra we find optimal systems of subalgebras and the corresponding invariant solutions or invariant submodels.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"17 1","pages":"29-40"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84530071","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}
L. K. Zhapsarbaeva, B. Kanguzhin, M. N. Konyrkulzhaeva
. In the work we study differential operators on arbitrary geometric graphs without loops. We extend the known results for differential operators on an interval to the differential operators on the graphs. In order to define properly the maximal operator on a given graph, we need to choose a set of boundary vertices. The non-boundary vertices are called interior vertices. We stress that the maximal operator on a graph is determined not only by the given differential expressions on the edges, but also by the Kirchhoff conditions at the interior vertices of the graph. For the introduced maximal operator we prove an analogue of the Lagrange formula. We provide an algorithm for constructing adjoint boundary forms for an arbitrary set of boundary conditions. In the conclusion of the paper, we present a complete description of all self-adjoint restrictions of the maximal operator.
{"title":"Self-adjoint restrictions of maximal operator on graph","authors":"L. K. Zhapsarbaeva, B. Kanguzhin, M. N. Konyrkulzhaeva","doi":"10.13108/2017-9-4-35","DOIUrl":"https://doi.org/10.13108/2017-9-4-35","url":null,"abstract":". In the work we study differential operators on arbitrary geometric graphs without loops. We extend the known results for differential operators on an interval to the differential operators on the graphs. In order to define properly the maximal operator on a given graph, we need to choose a set of boundary vertices. The non-boundary vertices are called interior vertices. We stress that the maximal operator on a graph is determined not only by the given differential expressions on the edges, but also by the Kirchhoff conditions at the interior vertices of the graph. For the introduced maximal operator we prove an analogue of the Lagrange formula. We provide an algorithm for constructing adjoint boundary forms for an arbitrary set of boundary conditions. In the conclusion of the paper, we present a complete description of all self-adjoint restrictions of the maximal operator.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"60 1","pages":"35-43"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77894209","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}
E. Muhamadiev, A. Naimov, Akhmad Khasanovich Sattorov
We study the existence and uniqueness of a solution bounded in the entire space for a class of higher order linear partial differential equations. We prove the theorem on the necessary and sufficient condition for the existence and uniqueness of a bounded solution for a studied class of equations. This theorem is an analogue of the Bohl theorem known in the theory of ordinary differential equations. In a partial case the unique solvability conditions are expressed in terms of the coefficients of the equation and we provide the integral representation for the bounded solution.
{"title":"Analogue of Bohl theorem for a class of linear partial differential equations","authors":"E. Muhamadiev, A. Naimov, Akhmad Khasanovich Sattorov","doi":"10.13108/2017-9-1-75","DOIUrl":"https://doi.org/10.13108/2017-9-1-75","url":null,"abstract":"We study the existence and uniqueness of a solution bounded in the entire space for a class of higher order linear partial differential equations. We prove the theorem on the necessary and sufficient condition for the existence and uniqueness of a bounded solution for a studied class of equations. This theorem is an analogue of the Bohl theorem known in the theory of ordinary differential equations. In a partial case the unique solvability conditions are expressed in terms of the coefficients of the equation and we provide the integral representation for the bounded solution.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"5 1","pages":"75-88"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75359753","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}
We consider normal unbounded operators acting in a real Hilbert space. The standard approach to solving spectral problems related with such operators is to apply the complexification, which is a passage to a complex space. At that, usually, the final results are to be decomplexified, that is, the reverse passage is needed. However, the decomplexification often turns out to be nontrivial. The aim of the present paper is to extend the classical results of the spectral theory for the case of normal operators acting in a real Hilbert space. We provide two real versions of the spectral theorem for such operators. We construct the functional calculus generated by the real spectral decomposition of a normal operator. We provide examples of using the obtained functional calculus for representing the exponent of a normal operator.
{"title":"Spectral decomposition of normal operator in real Hilbert space","authors":"M. N. Oreshina","doi":"10.13108/2017-9-4-85","DOIUrl":"https://doi.org/10.13108/2017-9-4-85","url":null,"abstract":"We consider normal unbounded operators acting in a real Hilbert space. The standard approach to solving spectral problems related with such operators is to apply the complexification, which is a passage to a complex space. At that, usually, the final results are to be decomplexified, that is, the reverse passage is needed. However, the decomplexification often turns out to be nontrivial. The aim of the present paper is to extend the classical results of the spectral theory for the case of normal operators acting in a real Hilbert space. We provide two real versions of the spectral theorem for such operators. We construct the functional calculus generated by the real spectral decomposition of a normal operator. We provide examples of using the obtained functional calculus for representing the exponent of a normal operator.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"101 1","pages":"85-96"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76034473","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}
{"title":"Sharp Hardy type inequalities with weights depending on Bessel function","authors":"R. Nasibullin","doi":"10.13108/2017-9-1-89","DOIUrl":"https://doi.org/10.13108/2017-9-1-89","url":null,"abstract":"","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"15 1","pages":"89-97"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74342359","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}
This expository paper concerns local properties of Levi-flat real analytic manifolds with singularities. Levi-flat manifolds arise naturally in Complex Geometry and Foliation Theory. In many cases (global) compact Levi-flat manifolds without singularities do not exist. These global obstructions make natural the study of Levi-flat objects with singularities because they always exist. The present expository paper deals with some recent results on local geometry of Levi-flat singularities. One of the main questions concerns an extension of the Levi foliation as a holomorphic foliation to a full neighborhood of singularity. It turns out that in general such extension does not exist. Nevertheless, the Levi foliation always extends as a holomorphic web (a foliation with branching) near a non-dicritical singularity. We also present an efficient criterion characterizing these singularities.
{"title":"Levi-flat world: a survey of local theory","authors":"Sukhov Alexandre","doi":"10.13108/2017-9-3-172","DOIUrl":"https://doi.org/10.13108/2017-9-3-172","url":null,"abstract":"This expository paper concerns local properties of Levi-flat real analytic manifolds with singularities. Levi-flat manifolds arise naturally in Complex Geometry and Foliation Theory. In many cases (global) compact Levi-flat manifolds without singularities do not exist. These global obstructions make natural the study of Levi-flat objects with singularities because they always exist. The present expository paper deals with some recent results on local geometry of Levi-flat singularities. One of the main questions concerns an extension of the Levi foliation as a holomorphic foliation to a full neighborhood of singularity. It turns out that in general such extension does not exist. Nevertheless, the Levi foliation always extends as a holomorphic web (a foliation with branching) near a non-dicritical singularity. We also present an efficient criterion characterizing these singularities.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"24 1","pages":"172-185"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79200597","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}
In a bounded domain Ω ⊂ Rn, we consider the following hyperbolic equation {︃ vtt = Δpv + λ|v|p−2v − |v|α−2v, x ∈ Ω, v ⃒⃒ ∂Ω = 0. We assume that 1 < α < p < +∞; this implies that the nonlinearity in the right hand side of the equation is of a non-Lipschitz type. As a rule, this type of nonlinearity prevent us from applying standard methods from the theory of nonlinear differential equations. An additional difficulty arises due to the presence of the p-Laplacian Δp(·) := div(|∇(·)|p−2∇(·)) in the equation. In the first result, the theorem on the existence of the so-called stationary ground state of the equation is proved. The proof of this result is based on the Nehari manifold method. In the main result of the paper we state that each stationary ground state is unstable globally in time. The proof is based on the development of an approach by Payne and Sattinger introduced for studying the stability of solutions to hyperbolic equations.
{"title":"On global instability of solutions to hyperbolic equations with non-Lipschitz nonlinearity","authors":"Y. Il'yasov, E. E. Kholodnov","doi":"10.13108/2017-9-4-44","DOIUrl":"https://doi.org/10.13108/2017-9-4-44","url":null,"abstract":"In a bounded domain Ω ⊂ Rn, we consider the following hyperbolic equation {︃ vtt = Δpv + λ|v|p−2v − |v|α−2v, x ∈ Ω, v ⃒⃒ ∂Ω = 0. We assume that 1 < α < p < +∞; this implies that the nonlinearity in the right hand side of the equation is of a non-Lipschitz type. As a rule, this type of nonlinearity prevent us from applying standard methods from the theory of nonlinear differential equations. An additional difficulty arises due to the presence of the p-Laplacian Δp(·) := div(|∇(·)|p−2∇(·)) in the equation. In the first result, the theorem on the existence of the so-called stationary ground state of the equation is proved. The proof of this result is based on the Nehari manifold method. In the main result of the paper we state that each stationary ground state is unstable globally in time. The proof is based on the development of an approach by Payne and Sattinger introduced for studying the stability of solutions to hyperbolic equations.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"115 1","pages":"44-53"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80835618","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}
We study a spectral problem in a bounded domain Ω ⊂ Rm depending on a bounded operator coefficient Q > 0 and a dissipation parameter α > 0. In the general case we establish sufficient conditions ensuring that the problem has a discrete spectrum consisting of countably many isolated eigenvalues of finite multiplicity accumulating at infinity. We also establish the conditions, under which the system of root elements contains an Abel-Lidskii basis in the space L2(Ω). In model oneand two-dimensional problems we establish the localization of the eigenvalues and find critical values of α.
{"title":"On spectral properties of one boundary value problem with a surface energy dissipation","authors":"O. A. Andronova, V. I. Voititskii","doi":"10.13108/2017-9-2-3","DOIUrl":"https://doi.org/10.13108/2017-9-2-3","url":null,"abstract":"We study a spectral problem in a bounded domain Ω ⊂ Rm depending on a bounded operator coefficient Q > 0 and a dissipation parameter α > 0. In the general case we establish sufficient conditions ensuring that the problem has a discrete spectrum consisting of countably many isolated eigenvalues of finite multiplicity accumulating at infinity. We also establish the conditions, under which the system of root elements contains an Abel-Lidskii basis in the space L2(Ω). In model oneand two-dimensional problems we establish the localization of the eigenvalues and find critical values of α.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"152 1","pages":"3-16"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90689878","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}
. In the paper we show that each analytic solution of a homogeneous convolution equation with the characteristic function of minimal exponential type is represented by a series of exponential polynomials in its domain. This series converges absolutely and uniformly on compact subsets in this domain. It is known that if the characteristic function is of minimal exponential type, the density of its zero set is equal to zero. This is why in the work we consider the sequences of exponents having zero density. We provide a simple description of the space of the coefficients for the aforementioned series. Moreover, we provide a complete description of all possible system of functions constructed by rather small groups, for which the representation by the series of exponential polynomials holds.
{"title":"Invariant subspaces with zero density spectrum","authors":"O. Krivosheeva","doi":"10.13108/2017-9-3-100","DOIUrl":"https://doi.org/10.13108/2017-9-3-100","url":null,"abstract":". In the paper we show that each analytic solution of a homogeneous convolution equation with the characteristic function of minimal exponential type is represented by a series of exponential polynomials in its domain. This series converges absolutely and uniformly on compact subsets in this domain. It is known that if the characteristic function is of minimal exponential type, the density of its zero set is equal to zero. This is why in the work we consider the sequences of exponents having zero density. We provide a simple description of the space of the coefficients for the aforementioned series. Moreover, we provide a complete description of all possible system of functions constructed by rather small groups, for which the representation by the series of exponential polynomials holds.","PeriodicalId":43644,"journal":{"name":"Ufa Mathematical Journal","volume":"13 1","pages":"100-108"},"PeriodicalIF":0.5,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89871669","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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