. We study the notion of strong r -stability for the context of closed hypersurfaces Σ n ( n ≥ 3) with constant ( r + 1)-th mean curvature H r +1 immersed into the Euclidean sphere S n +1 , where r ∈ { 1 ,...,n − 2 } . In this setting, under a suitable restriction on the r -th mean curvature H r , we establish that there are no r -strongly stable closed hypersurfaces immersed in a certain region of S n +1 , a region that is determined by a totally umbilical sphere of S n +1 . We also provide a rigidity result for such hypersurfaces.
。我们研究了闭超曲面Σ n (n≥3)的强r -稳定性的概念，该闭超曲面具有常数(r +1)-平均曲率H r +1浸入欧几里得球S n +1，其中r∈{1，…，n−2}。在这种情况下，在对r -平均曲率H的适当限制下，我们建立了在S n +1的某一区域中不存在r -强稳定闭超曲面，该区域由S n +1的完全脐带球确定。我们还提供了这种超曲面的刚性结果。
{"title":"A half-space type property in the Euclidean sphere","authors":"M. Velásquez","doi":"10.5817/am2022-1-49","DOIUrl":"https://doi.org/10.5817/am2022-1-49","url":null,"abstract":". We study the notion of strong r -stability for the context of closed hypersurfaces Σ n ( n ≥ 3) with constant ( r + 1)-th mean curvature H r +1 immersed into the Euclidean sphere S n +1 , where r ∈ { 1 ,...,n − 2 } . In this setting, under a suitable restriction on the r -th mean curvature H r , we establish that there are no r -strongly stable closed hypersurfaces immersed in a certain region of S n +1 , a region that is determined by a totally umbilical sphere of S n +1 . We also provide a rigidity result for such hypersurfaces.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"29 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82228974","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}
. Let P ( z ) = P n ν =0 a ν z ν be a polynomial of degree at most n which does not vanish in the disk | z | < 1, then for 1 ≤ p < ∞ and R > 1, Boas and Rahman proved In this paper, we improve the above inequality for 0 ≤ p < ∞ by involving some of the coefficients of the polynomial P ( z ). Analogous result for the class of polynomials P ( z ) having no zero in | z | > 1 is also given.
。设P (z) = P n ν =0 a ν z ν是一个不消失于圆盘| z | < 1的最多n次的多项式，那么对于1≤P <∞和R > 1, Boas和Rahman证明了在0≤P <∞时，我们通过引入多项式P (z)的一些系数来改进上述不等式。给出了一类多项式P (z)在| z | > 1范围内无零的类似结果。
{"title":"$L_{p}$ inequalities for the growth of polynomials with restricted zeros","authors":"N. A. Rather, Suhail Gulzar, A. Bhat","doi":"10.5817/am2022-3-159","DOIUrl":"https://doi.org/10.5817/am2022-3-159","url":null,"abstract":". Let P ( z ) = P n ν =0 a ν z ν be a polynomial of degree at most n which does not vanish in the disk | z | < 1, then for 1 ≤ p < ∞ and R > 1, Boas and Rahman proved In this paper, we improve the above inequality for 0 ≤ p < ∞ by involving some of the coefficients of the polynomial P ( z ). Analogous result for the class of polynomials P ( z ) having no zero in | z | > 1 is also given.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"48 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73911132","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 paper is concerned with certain generalized subclasses of bi-univalent functions defined with subordination in the open unit disc E = { z : | z | < 1 } . The bounds for the initial coefficients for the functions in these classes are studied. The earlier known results follow as special cases.
． 研究开单位圆盘E = {z: | z | < 1}上有隶属定义的双单价函数的某些广义子类。研究了这类函数的初始系数的界。前面已知的结果是作为特殊情况出现的。
{"title":"Initial coefficients for generalized subclasses of bi-univalent functions defined with subordination","authors":"G. Singh, G. Singh, Gurmeet Singh","doi":"10.5817/am2022-2-105","DOIUrl":"https://doi.org/10.5817/am2022-2-105","url":null,"abstract":". This paper is concerned with certain generalized subclasses of bi-univalent functions defined with subordination in the open unit disc E = { z : | z | < 1 } . The bounds for the initial coefficients for the functions in these classes are studied. The earlier known results follow as special cases.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"57 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72522678","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 give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among finite graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counterpart of this fact for all bipartite graphs in the class of all graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the ordered sets of join-irreducible congruences of slim semimodular lattices can be described by finitely many axioms in the class of finite structures. Since a 2007 result of G. Grätzer and E. Knapp, slim semimodular lattices have constituted the most intensively studied part of lattice theory and they have already led to results even in group theory and geometry. In addition to the non-axiomatizability results mentioned above, we present a new property, called Decomposable Cyclic Elements Property, of the congruence lattices of slim semimodular lattices.
{"title":"Cyclic congruences of slim semimodular lattices and non-finite axiomatizability of some finite structures","authors":"G. Czédli","doi":"10.5817/am2022-1-15","DOIUrl":"https://doi.org/10.5817/am2022-1-15","url":null,"abstract":". We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among finite graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counterpart of this fact for all bipartite graphs in the class of all graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the ordered sets of join-irreducible congruences of slim semimodular lattices can be described by finitely many axioms in the class of finite structures. Since a 2007 result of G. Grätzer and E. Knapp, slim semimodular lattices have constituted the most intensively studied part of lattice theory and they have already led to results even in group theory and geometry. In addition to the non-axiomatizability results mentioned above, we present a new property, called Decomposable Cyclic Elements Property, of the congruence lattices of slim semimodular lattices.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"122 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87740124","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 this article, we obtain sufficient conditions so that every solution of neutral delay differential equation oscillates or tends to zero as t → ∞ , where, n ≥ 1 is any positive integer, p i , r i ∈ C ( n ) ([0 , ∞ ) , R ) and p i are bounded for each i = 1 , 2 ,...,k . Further, f ∈ C ([0 , ∞ ) , R ), g , h , v , u ∈ C ([0 , ∞ ) , [0 , ∞ )), G and H ∈ C ( R , R ). The functional delays r i ( t ) ≤ t , g ( t ) ≤ t and h ( t ) ≤ t and all of them approach ∞ as t → ∞ . The results hold when u ≡ 0 and f ( t ) ≡ 0. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature.
． 本文给出了中立型时滞微分方程在t→∞时的所有解振荡或趋近于零的充分条件，其中，n≥1是任意正整数，p i, r i∈C (n)([0，∞)，r)， p i对每一个i = 1,2，…, k。此外,f∈C([0,∞),R), g, h, v, u C∈([0,∞),[0,∞)),g和h∈C (R, R)。函数时滞r i (t)≤t, g (t)≤t, h (t)≤t，均在t→∞时趋于∞。当u≡0且f (t)≡0时，结果成立。本文扩展、概括和改进了最近的一些结果，并进一步回答了文献中一些未解决的问题。
{"title":"Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator","authors":"R. Rath, K. C. Panda, S. Rath","doi":"10.5817/am2022-2-65","DOIUrl":"https://doi.org/10.5817/am2022-2-65","url":null,"abstract":". In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation oscillates or tends to zero as t → ∞ , where, n ≥ 1 is any positive integer, p i , r i ∈ C ( n ) ([0 , ∞ ) , R ) and p i are bounded for each i = 1 , 2 ,...,k . Further, f ∈ C ([0 , ∞ ) , R ), g , h , v , u ∈ C ([0 , ∞ ) , [0 , ∞ )), G and H ∈ C ( R , R ). The functional delays r i ( t ) ≤ t , g ( t ) ≤ t and h ( t ) ≤ t and all of them approach ∞ as t → ∞ . The results hold when u ≡ 0 and f ( t ) ≡ 0. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"17 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84764671","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}
. Let { U n } = { U n ( P,Q ) } and { V n } = { V n ( P,Q ) } be the Lucas sequences of the first and second kind respectively at the parameters P ≥ 1 and Q ∈ {− 1 , 1 } . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation x 2 − 3 xy + y 2 + x = 0 , where ( x,y ) = ( U i ,U j ) or ( V i ,V j ) with i , j ≥ 1. Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.
。令{U n} = {U n (P,Q)}和{V n} = {V n (P,Q)}分别为参数P≥1和Q∈{−1,1}时的第一类和第二类卢卡斯序列。本文给出了一种描述所谓Bartz-Marlewski方程x 2−3 xy + y 2 + x = 0，其中(x,y) = (U i,U j)或(V i,V j)且i, j≥1的解法。然后，将该方法应用于具有一定参数值的方程的完全解。
{"title":"Bartz-Marlewski equation with generalized Lucas components","authors":"H. Hashim","doi":"10.5817/am2022-3-189","DOIUrl":"https://doi.org/10.5817/am2022-3-189","url":null,"abstract":". Let { U n } = { U n ( P,Q ) } and { V n } = { V n ( P,Q ) } be the Lucas sequences of the first and second kind respectively at the parameters P ≥ 1 and Q ∈ {− 1 , 1 } . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation x 2 − 3 xy + y 2 + x = 0 , where ( x,y ) = ( U i ,U j ) or ( V i ,V j ) with i , j ≥ 1. Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"7 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85304368","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 this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$-almost periodic functions in ${mathbb R}^{n},$ slowly oscillating functions in ${mathbb R}^{n},$ and further analyze the recently introduced class of quasi-asymptotically $c$-almost periodic functions in ${mathbb R}^{n}.$ We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations and the ordinary differential equations.
{"title":"Remotely $c$-almost periodic type functions in ${mathbb{R}}^{n}$","authors":"M. Kostić, Vipin Kumar","doi":"10.5817/am2022-2-85","DOIUrl":"https://doi.org/10.5817/am2022-2-85","url":null,"abstract":"In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$-almost periodic functions in ${mathbb R}^{n},$ slowly oscillating functions in ${mathbb R}^{n},$ and further analyze the recently introduced class of quasi-asymptotically $c$-almost periodic functions in ${mathbb R}^{n}.$ We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations and the ordinary differential equations.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"78 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88142681","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}
Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. One particular family of examples have ends that collapse asymptotically to ${mathbb R}^2$.
{"title":"Four-dimensional Einstein metrics from biconformal deformations","authors":"P. Baird, J. Ventura","doi":"10.5817/am2021-5-255","DOIUrl":"https://doi.org/10.5817/am2021-5-255","url":null,"abstract":"Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. One particular family of examples have ends that collapse asymptotically to ${mathbb R}^2$.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"99 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81070161","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}
Abstract. In this paper, we analyze multi-dimensional quasi-asymptotically c-almost periodic functions and their Stepanov generalizations as well as multidimensional Weyl c-almost periodic type functions. We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically calmost periodic functions and reconsider the notion of semi-c-periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent. We provide certain applications of our results to the abstract Volterra integro-differential equations in Banach spaces.
{"title":"Generalized $c$-almost periodic type functions in ${mathbb{R}}^{n}$","authors":"M. Kosti'c","doi":"10.5817/am2021-4-221","DOIUrl":"https://doi.org/10.5817/am2021-4-221","url":null,"abstract":"Abstract. In this paper, we analyze multi-dimensional quasi-asymptotically c-almost periodic functions and their Stepanov generalizations as well as multidimensional Weyl c-almost periodic type functions. We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically calmost periodic functions and reconsider the notion of semi-c-periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent. We provide certain applications of our results to the abstract Volterra integro-differential equations in Banach spaces.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"20 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75278990","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 this paper we study properties of regular solutions of quaternionic Riccati equations. The obtained results we use for study of the asymptotic behavior of solutions of two first-order linear quaternionic ordinary differential equations.
{"title":"Properties of solutions of quaternionic Riccati equations","authors":"G. Grigorian","doi":"10.5817/am2022-2-115","DOIUrl":"https://doi.org/10.5817/am2022-2-115","url":null,"abstract":". In this paper we study properties of regular solutions of quaternionic Riccati equations. The obtained results we use for study of the asymptotic behavior of solutions of two first-order linear quaternionic ordinary differential equations.","PeriodicalId":45191,"journal":{"name":"Archivum Mathematicum","volume":"68 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75652552","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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