Pub Date : 2014-04-04DOI: 10.2478/s11533-014-0407-0
D. Buhagiar, E. Chetcuti, H. Weber
We study the behaviour of ℵ-compactness, extent and Lindelöf number in lexicographic products of linearly ordered spaces. It is seen, in particular, that for the case that all spaces are bounded all these properties behave very well when taking lexicographic products. We also give characterizations of these notions for generalized ordered spaces.
{"title":"κ-compactness, extent and the Lindelöf number in LOTS","authors":"D. Buhagiar, E. Chetcuti, H. Weber","doi":"10.2478/s11533-014-0407-0","DOIUrl":"https://doi.org/10.2478/s11533-014-0407-0","url":null,"abstract":"We study the behaviour of ℵ-compactness, extent and Lindelöf number in lexicographic products of linearly ordered spaces. It is seen, in particular, that for the case that all spaces are bounded all these properties behave very well when taking lexicographic products. We also give characterizations of these notions for generalized ordered spaces.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"26 1","pages":"1249-1264"},"PeriodicalIF":0.0,"publicationDate":"2014-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72993023","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 : 2014-04-01DOI: 10.2478/s11533-013-0392-8
L. Kleprlík
Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to Lq(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W1X to W1X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.
{"title":"Composition operators on W1X are necessarily induced by quasiconformal mappings","authors":"L. Kleprlík","doi":"10.2478/s11533-013-0392-8","DOIUrl":"https://doi.org/10.2478/s11533-013-0392-8","url":null,"abstract":"Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to Lq(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W1X to W1X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"53 1","pages":"1229-1238"},"PeriodicalIF":0.0,"publicationDate":"2014-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85343596","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 : 2014-04-01DOI: 10.2478/s11533-013-0362-1
M. S. Riveros, M. Urciuolo
AbstractIn this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) cdots k_m left( {x - A_m y} right),$$$$k_i left( x right) = {{Omega _i left( x right)} mathord{left/ {vphantom {{Omega _i left( x right)} {left| x right|}}} right. kern-nulldelimiterspace} {left| x right|}}^{{n mathord{left/ {vphantom {n {q_i }}} right. kern-nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, Ai are certain invertible matrices, and n/q1 +…+n/qm = n−α, 0 ≤ α < n. We obtain the appropriate weighted Lp-Lq estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.
{"title":"Weighted inequalities for some integral operators with rough kernels","authors":"M. S. Riveros, M. Urciuolo","doi":"10.2478/s11533-013-0362-1","DOIUrl":"https://doi.org/10.2478/s11533-013-0362-1","url":null,"abstract":"AbstractIn this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) cdots k_m left( {x - A_m y} right),$$$$k_i left( x right) = {{Omega _i left( x right)} mathord{left/\u0000 {vphantom {{Omega _i left( x right)} {left| x right|}}} right.\u0000 kern-nulldelimiterspace} {left| x right|}}^{{n mathord{left/\u0000 {vphantom {n {q_i }}} right.\u0000 kern-nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, Ai are certain invertible matrices, and n/q1 +…+n/qm = n−α, 0 ≤ α < n. We obtain the appropriate weighted Lp-Lq estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"116 1","pages":"636-647"},"PeriodicalIF":0.0,"publicationDate":"2014-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74202514","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 : 2014-04-01DOI: 10.2478/s11533-013-0360-3
E. D’Aniello, T. H. Steele
Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ωf: 2ℕ → K(2ℕ) defined as ωf (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ωf is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ωf is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ωf and some forms of chaos are investigated.
{"title":"Chaotic behaviour of the map x ↦ ω(x, f)","authors":"E. D’Aniello, T. H. Steele","doi":"10.2478/s11533-013-0360-3","DOIUrl":"https://doi.org/10.2478/s11533-013-0360-3","url":null,"abstract":"Let K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ωf: 2ℕ → K(2ℕ) defined as ωf (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ωf is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ωf is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ωf and some forms of chaos are investigated.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"2 1","pages":"584-592"},"PeriodicalIF":0.0,"publicationDate":"2014-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77638873","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 : 2014-03-05DOI: 10.2478/s11533-013-0384-8
N. Raulf
In this paper we compute the trace formula for Hecke operators acting on automorphic forms on the hyperbolic 3-space for the group PSL2($mathcal{O}_K $) with $mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field K of class number HK > 1. Furthermore, as a corollary we obtain an asymptotic result for class numbers of binary quadratic forms.
{"title":"Trace formulae and applications to class numbers","authors":"N. Raulf","doi":"10.2478/s11533-013-0384-8","DOIUrl":"https://doi.org/10.2478/s11533-013-0384-8","url":null,"abstract":"In this paper we compute the trace formula for Hecke operators acting on automorphic forms on the hyperbolic 3-space for the group PSL2($mathcal{O}_K $) with $mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field K of class number HK > 1. Furthermore, as a corollary we obtain an asymptotic result for class numbers of binary quadratic forms.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"34 1","pages":"824-847"},"PeriodicalIF":0.0,"publicationDate":"2014-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80951048","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 : 2014-03-01DOI: 10.2478/s11533-013-0350-5
S. B. Tabaldyev
Let A be a unital strict Banach algebra, and let K+ be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K+), the algebra of continuous functions on K+. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .
{"title":"An additivity formula for the strict global dimension of C(Ω)","authors":"S. B. Tabaldyev","doi":"10.2478/s11533-013-0350-5","DOIUrl":"https://doi.org/10.2478/s11533-013-0350-5","url":null,"abstract":"Let A be a unital strict Banach algebra, and let K+ be the one-point compactification of a discrete topological space K. Denote by the weak tensor product of the algebra A and C(K+), the algebra of continuous functions on K+. We prove that if K has sufficiently large cardinality (depending on A), then the strict global dimension is equal to .","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"7 1","pages":"470-475"},"PeriodicalIF":0.0,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80261279","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 : 2014-03-01DOI: 10.2478/s11533-013-0349-y
J. Rachunek, Z. Svoboda
Bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate multiplicative interior and additive closure operators (mi- and ac-operators) generalizing topological interior and closure operators on such algebras. We describe connections between mi- and ac-operators, and for residuated lattices with Glivenko property we give connections between operators on them and on the residuated lattices of their regular elements.
{"title":"Interior and closure operators on bounded residuated lattices","authors":"J. Rachunek, Z. Svoboda","doi":"10.2478/s11533-013-0349-y","DOIUrl":"https://doi.org/10.2478/s11533-013-0349-y","url":null,"abstract":"Bounded integral residuated lattices form a large class of algebras containing some classes of algebras behind many valued and fuzzy logics. In the paper we introduce and investigate multiplicative interior and additive closure operators (mi- and ac-operators) generalizing topological interior and closure operators on such algebras. We describe connections between mi- and ac-operators, and for residuated lattices with Glivenko property we give connections between operators on them and on the residuated lattices of their regular elements.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"32 1","pages":"534-544"},"PeriodicalIF":0.0,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82629148","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 : 2014-03-01DOI: 10.2478/s11533-013-0359-9
O. T. Alas, V. Tkachuk, R. Wilson
We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.
{"title":"Maximal pseudocompact spaces and the Preiss-Simon property","authors":"O. T. Alas, V. Tkachuk, R. Wilson","doi":"10.2478/s11533-013-0359-9","DOIUrl":"https://doi.org/10.2478/s11533-013-0359-9","url":null,"abstract":"We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"3 1","pages":"500-509"},"PeriodicalIF":0.0,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79097367","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 : 2014-03-01DOI: 10.2478/s11533-013-0357-y
L. Volkmann
Let k ≥ 2 be an integer. A function f: V(G) → {−1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v]f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G)f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number αsk(G) of G.In this work, we mainly present upper bounds on αsk (G), as for example αsk(G) ≤ n − 2⌈(Δ(G) + 2 − k)/2⌉, and we prove the Nordhaus-Gaddum type inequality $$alpha _S^k left( G right) + alpha _S^k left( {bar G} right) leqslant n + 2k - 3$$, where n is the order, Δ(G) the maximum degree and $$bar G$$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.
{"title":"Signed k-independence in graphs","authors":"L. Volkmann","doi":"10.2478/s11533-013-0357-y","DOIUrl":"https://doi.org/10.2478/s11533-013-0357-y","url":null,"abstract":"Let k ≥ 2 be an integer. A function f: V(G) → {−1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k − 1. That is, Σx∈N[v]f(x) ≤ k − 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σv∈V(G)f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number αsk(G) of G.In this work, we mainly present upper bounds on αsk (G), as for example αsk(G) ≤ n − 2⌈(Δ(G) + 2 − k)/2⌉, and we prove the Nordhaus-Gaddum type inequality $$alpha _S^k left( G right) + alpha _S^k left( {bar G} right) leqslant n + 2k - 3$$, where n is the order, Δ(G) the maximum degree and $$bar G$$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"72 1","pages":"517-528"},"PeriodicalIF":0.0,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76547695","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 : 2014-03-01DOI: 10.2478/s11533-013-0347-0
M. H. Jafari, A. Madadi
In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.
{"title":"Generalized matrix functions and determinants","authors":"M. H. Jafari, A. Madadi","doi":"10.2478/s11533-013-0347-0","DOIUrl":"https://doi.org/10.2478/s11533-013-0347-0","url":null,"abstract":"In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.","PeriodicalId":50988,"journal":{"name":"Central European Journal of Mathematics","volume":"102 1","pages":"464-469"},"PeriodicalIF":0.0,"publicationDate":"2014-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79377665","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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