Pub Date : 2014-02-01DOI: 10.2478/s11533-013-0334-5
We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.
{"title":"Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations","authors":"","doi":"10.2478/s11533-013-0334-5","DOIUrl":"https://doi.org/10.2478/s11533-013-0334-5","url":null,"abstract":"We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"25 1","pages":"308-321"},"PeriodicalIF":0.0,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74245319","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 : 2014-02-01DOI: 10.2478/s11533-013-0332-7
A. Bonciocat
We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as well. For spaces that satisfy a rough curvature-dimension condition we prove a generalized Brunn-Minkowski inequality and a Bonnet-Myers type theorem.
{"title":"A rough curvature-dimension condition for metric measure spaces","authors":"A. Bonciocat","doi":"10.2478/s11533-013-0332-7","DOIUrl":"https://doi.org/10.2478/s11533-013-0332-7","url":null,"abstract":"We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as well. For spaces that satisfy a rough curvature-dimension condition we prove a generalized Brunn-Minkowski inequality and a Bonnet-Myers type theorem.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"28 1","pages":"362-380"},"PeriodicalIF":0.0,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82601113","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 : 2014-02-01DOI: 10.2478/s11533-013-0340-7
A. Trepalin
AbstractLet $$Bbbk$$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $$Bbbk$$. Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $$Bbbk$$ is algebraically closed. In this paper we prove that $${{mathbb{P}_Bbbk ^2 } mathord{left/ {vphantom {{mathbb{P}_Bbbk ^2 } G}} right. kern-nulldelimiterspace} G}$$ is rational for an arbitrary field $$Bbbk$$ of characteristic zero.
{"title":"Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero","authors":"A. Trepalin","doi":"10.2478/s11533-013-0340-7","DOIUrl":"https://doi.org/10.2478/s11533-013-0340-7","url":null,"abstract":"AbstractLet $$Bbbk$$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $$Bbbk$$. Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $$Bbbk$$ is algebraically closed. In this paper we prove that $${{mathbb{P}_Bbbk ^2 } mathord{left/\u0000 {vphantom {{mathbb{P}_Bbbk ^2 } G}} right.\u0000 kern-nulldelimiterspace} G}$$ is rational for an arbitrary field $$Bbbk$$ of characteristic zero.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"44 1","pages":"229-239"},"PeriodicalIF":0.0,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83226901","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 : 2014-02-01DOI: 10.2478/s11533-013-0344-3
Changlu Guo, P. Koskela
We establish the basic properties of the class of generalized simply connected John domains.
建立了一类广义单连通John域的基本性质。
{"title":"Generalized John disks","authors":"Changlu Guo, P. Koskela","doi":"10.2478/s11533-013-0344-3","DOIUrl":"https://doi.org/10.2478/s11533-013-0344-3","url":null,"abstract":"We establish the basic properties of the class of generalized simply connected John domains.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"10 1","pages":"349-361"},"PeriodicalIF":0.0,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75791887","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 : 2014-02-01DOI: 10.2478/s11533-013-0339-0
M. Bonanzinga, M. V. Cuzzupè, B. Pansera
Two variations of Arhangelskii’s inequality $$left| X right| leqslant 2^{chi (X) - L(X)}$$ for Hausdorff X [Arhangel’skii A.V., The power of bicompacta with first axiom of countability, Dokl. Akad. Nauk SSSR, 1969, 187, 967–970 (in Russian)] given in [Stavrova D.N., Separation pseudocharacter and the cardinality of topological spaces, Topology Proc., 2000, 25(Summer), 333–343] are extended to the classes with finite Urysohn number or finite Hausdorff number.
{"title":"On the cardinality of n-Urysohn and n-Hausdorff spaces","authors":"M. Bonanzinga, M. V. Cuzzupè, B. Pansera","doi":"10.2478/s11533-013-0339-0","DOIUrl":"https://doi.org/10.2478/s11533-013-0339-0","url":null,"abstract":"Two variations of Arhangelskii’s inequality $$left| X right| leqslant 2^{chi (X) - L(X)}$$ for Hausdorff X [Arhangel’skii A.V., The power of bicompacta with first axiom of countability, Dokl. Akad. Nauk SSSR, 1969, 187, 967–970 (in Russian)] given in [Stavrova D.N., Separation pseudocharacter and the cardinality of topological spaces, Topology Proc., 2000, 25(Summer), 333–343] are extended to the classes with finite Urysohn number or finite Hausdorff number.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"16 1","pages":"330-336"},"PeriodicalIF":0.0,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75239335","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 : 2014-02-01DOI: 10.2478/s11533-013-0343-4
M. Sakai
We show that (I) there is a Lindelöf space which is not weakly Menger, (II) there is a Menger space for which TWO does not have a winning strategy in the game Gfin(O,Do). These affirmatively answer questions posed in Babinkostova, Pansera and Scheepers [Babinkostova L., Pansera B.A., Scheepers M., Weak covering properties and infinite games, Topology Appl., 2012, 159(17), 3644–3657]. The result (I) automatically gives an affirmative answer of Wingers’ problem [Wingers L., Box products and Hurewicz spaces, Topology Appl., 1995, 64(1), 9–21], too. The selection principle S1(Do,Do) is also discussed in view of a special base. We show that every subspace of a hereditarily Lindelöf space with an ortho-base satisfies S1(Do,Do).
{"title":"Some weak covering properties and infinite games","authors":"M. Sakai","doi":"10.2478/s11533-013-0343-4","DOIUrl":"https://doi.org/10.2478/s11533-013-0343-4","url":null,"abstract":"We show that (I) there is a Lindelöf space which is not weakly Menger, (II) there is a Menger space for which TWO does not have a winning strategy in the game Gfin(O,Do). These affirmatively answer questions posed in Babinkostova, Pansera and Scheepers [Babinkostova L., Pansera B.A., Scheepers M., Weak covering properties and infinite games, Topology Appl., 2012, 159(17), 3644–3657]. The result (I) automatically gives an affirmative answer of Wingers’ problem [Wingers L., Box products and Hurewicz spaces, Topology Appl., 1995, 64(1), 9–21], too. The selection principle S1(Do,Do) is also discussed in view of a special base. We show that every subspace of a hereditarily Lindelöf space with an ortho-base satisfies S1(Do,Do).","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"4 1","pages":"322-329"},"PeriodicalIF":0.0,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85593712","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 : 2014-01-17DOI: 10.2478/s11533-013-0371-0
M. F. Barrozo, U. Molter
We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {Fi: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps Fi are of the form Fi(x) = rix + bi on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supiri is strictly smaller than 1.Further, if ρ = {ρk}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.
{"title":"Countable contraction mappings in metric spaces: invariant sets and measure","authors":"M. F. Barrozo, U. Molter","doi":"10.2478/s11533-013-0371-0","DOIUrl":"https://doi.org/10.2478/s11533-013-0371-0","url":null,"abstract":"We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {Fi: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps Fi are of the form Fi(x) = rix + bi on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supiri is strictly smaller than 1.Further, if ρ = {ρk}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"109 1","pages":"593-602"},"PeriodicalIF":0.0,"publicationDate":"2014-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90424908","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 : 2014-01-17DOI: 10.2478/s11533-013-0368-8
Y. Povstenko
The central symmetric time-fractional heat conduction equation with Caputo derivative of order 0 < α ≤ 2 is considered in a ball under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of values of temperature and values of its normal derivative at the boundary, and the physical condition with the prescribed linear combination of values of temperature and values of the heat flux at the boundary, which is a consequence of Newton’s law of convective heat exchange between a body and the environment. The integral transform technique is used. Numerical results are illustrated graphically.
{"title":"Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition","authors":"Y. Povstenko","doi":"10.2478/s11533-013-0368-8","DOIUrl":"https://doi.org/10.2478/s11533-013-0368-8","url":null,"abstract":"The central symmetric time-fractional heat conduction equation with Caputo derivative of order 0 < α ≤ 2 is considered in a ball under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of values of temperature and values of its normal derivative at the boundary, and the physical condition with the prescribed linear combination of values of temperature and values of the heat flux at the boundary, which is a consequence of Newton’s law of convective heat exchange between a body and the environment. The integral transform technique is used. Numerical results are illustrated graphically.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"20 1","pages":"611-622"},"PeriodicalIF":0.0,"publicationDate":"2014-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87505364","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 : 2014-01-14DOI: 10.2478/s11533-013-0369-7
Hiroki Sano, Tamotsu Izumida, Ken-Ichi Mitani, T. Ohwada, K. Saito
In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. Cn that satisfy 0 ≤ Cn ≤ Σj=1n ‖xj‖ − ‖Σj=1nxj‖, x1,...,xn ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.
{"title":"Characterization of intermediate values of the triangle inequality II","authors":"Hiroki Sano, Tamotsu Izumida, Ken-Ichi Mitani, T. Ohwada, K. Saito","doi":"10.2478/s11533-013-0369-7","DOIUrl":"https://doi.org/10.2478/s11533-013-0369-7","url":null,"abstract":"In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. Cn that satisfy 0 ≤ Cn ≤ Σj=1n ‖xj‖ − ‖Σj=1nxj‖, x1,...,xn ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"1 1","pages":"778-786"},"PeriodicalIF":0.0,"publicationDate":"2014-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90727985","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 : 2014-01-10DOI: 10.2478/s11533-013-0366-x
A. Deitmar
For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.
{"title":"Fourier expansion along geodesics on Riemann surfaces","authors":"A. Deitmar","doi":"10.2478/s11533-013-0366-x","DOIUrl":"https://doi.org/10.2478/s11533-013-0366-x","url":null,"abstract":"For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"85 1","pages":"559-573"},"PeriodicalIF":0.0,"publicationDate":"2014-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79823030","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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