In this paper, we investigate the polynomial numerical index (n^{(k)}(l_p),) the symmetric multilinear numerical index (n_s^{(k)}(l_p),) and the multilinear numerical index (n_m^{(k)}(l_p)) of (l_p) spaces, for (1leq pleq infty.) First we prove that (n_{s}^{(k)}(l_1)=n_{m}^{(k)}(l_1)=1,) for every (kgeq 2.) We show that for (1 lt p lt infty,) (n_I^{(k)}(l_p^{j+1})leq n_I^{(k)}(l_p^j),) for every (jin mathbb{N}) and (n_I^{(k)}(l_p)=lim_{jto infty}n_I^{(k)}(l_p^j),) for every (I=s, m,) where (l_p^j=(mathbb{C}^j, |cdot|_p)) or ((mathbb{R}^j, |cdot|_p).) We also show the following inequality between ( n_s^{(k)}(l_p^j)) and (n^{(k)}(l_p^j)): let (1 lt p lt infty) and (kin mathbb{N}) be fixed. Then [ c(k: l_p^j)^{-1}~n^{(k)}(l_p^j)leq n_s^{(k)}(l_p^j)leq n^{(k)}(l_p^j), ] for every (jin mathbb{N}cup{infty},) where (l_p^{infty}:=l_p,) [ c(k: l_p)=infBig{M>0: |check{Q}|leq M|Q|,mbox{ for every}~Qin {mathcal P}(^k l_p)Big} ] and (check{Q}) denotes the symmetric (k)-linear form associated with (Q.) From this inequality, we deduce that if (l_{p}) is a complex space, then (lim_{jto infty} n_s^{(j)}(l_p)=lim_{jto infty} n_m^{(j)}(l_p)=0,) for every (1lt p lt infty.)
{"title":"Three kinds of numerical indices of (l_p)-spaces","authors":"Sung Guen Kim","doi":"10.3336/gm.57.1.04","DOIUrl":"https://doi.org/10.3336/gm.57.1.04","url":null,"abstract":"In this paper, we investigate the polynomial numerical index (n^{(k)}(l_p),) the symmetric multilinear numerical index\u0000(n_s^{(k)}(l_p),) and the multilinear numerical index (n_m^{(k)}(l_p)) of (l_p) spaces, for (1leq pleq infty.) First we prove that (n_{s}^{(k)}(l_1)=n_{m}^{(k)}(l_1)=1,) for every (kgeq 2.)\u0000We show that for (1 lt p lt infty,) (n_I^{(k)}(l_p^{j+1})leq n_I^{(k)}(l_p^j),) for every (jin mathbb{N}) and (n_I^{(k)}(l_p)=lim_{jto infty}n_I^{(k)}(l_p^j),) for every (I=s, m,) where (l_p^j=(mathbb{C}^j, |cdot|_p)) or ((mathbb{R}^j, |cdot|_p).)\u0000We also show the following inequality between ( n_s^{(k)}(l_p^j)) and (n^{(k)}(l_p^j)): let (1 lt p lt infty) and (kin mathbb{N}) be\u0000fixed. Then\u0000[\u0000\u0000\u0000\u0000c(k: l_p^j)^{-1}~n^{(k)}(l_p^j)leq n_s^{(k)}(l_p^j)leq n^{(k)}(l_p^j),\u0000\u0000\u0000]\u0000for every (jin mathbb{N}cup{infty},) where\u0000(l_p^{infty}:=l_p,)\u0000[\u0000\u0000\u0000\u0000c(k: l_p)=infBig{M>0: |check{Q}|leq M|Q|,mbox{ for every}~Qin {mathcal P}(^k l_p)Big}\u0000\u0000\u0000]\u0000and (check{Q}) denotes the symmetric (k)-linear form associated with (Q.) From this inequality, we deduce that if (l_{p}) is a complex space, then (lim_{jto infty} n_s^{(j)}(l_p)=lim_{jto infty} n_m^{(j)}(l_p)=0,) for every (1lt p lt infty.)","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85497616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to studying the first simultaneous hitting time of a given set by two discrete-time, inhomogeneous Markov chains with values in general phase space. Established conditions for the existence of the hitting time's exponential moment. Computable bounds for the exponential moment are obtained under the condition of stochastic dominance.
{"title":"Exponential moments of simultaneous hitting time for non-atomic Markov chains","authors":"V. Golomoziy","doi":"10.3336/gm.57.1.09","DOIUrl":"https://doi.org/10.3336/gm.57.1.09","url":null,"abstract":"This paper is devoted to studying the first simultaneous hitting time of a given set by two discrete-time, inhomogeneous Markov chains with values in general phase space. Established conditions for the existence of the hitting time's exponential moment. Computable bounds for the exponential moment are obtained under the condition of stochastic dominance.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89365397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Finite W-algebras associated to truncated current Lie algebras are studied in this paper. We show that some properties of finite W-algebras in the semisimple case hold in the truncated current case. In particular, Kostant's theorem and Skryabin equivalence hold in our case. As an application, we give a classification of simple Whittaker modules for truncated current Lie algebras in the (sell_2) case.
{"title":"Finite W-algebras associated to truncated current Lie algebras","authors":"Xiao He","doi":"10.3336/gm.57.1.02","DOIUrl":"https://doi.org/10.3336/gm.57.1.02","url":null,"abstract":"Finite W-algebras associated to truncated current Lie algebras are studied in this paper. We show that some properties of finite W-algebras in the semisimple case hold in the truncated current case. In particular, Kostant's theorem and Skryabin equivalence hold in our case. As an application, we give a classification of simple Whittaker modules for truncated current Lie algebras in the (sell_2) case.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87105436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The contracting boundary of a proper geodesic metric space generalizes the Gromov boundary of a hyperbolic space. It consists of contracting geodesics up to bounded Hausdorff distances. Another generalization of the Gromov boundary is the (kappa)–Morse boundary with a sublinear function (kappa). The two generalizations model the Gromov boundary based on different characteristics of geodesics in Gromov hyperbolic spaces. It was suspected that the (kappa)–Morse boundary contains the contracting boundary. We will prove this conjecture: when (kappa =1) is the constant function, the 1-Morse boundary and the contracting boundary are equivalent as topological spaces.
{"title":"Equivalent topologies on the contracting boundary","authors":"Vivian He","doi":"10.3336/gm.58.1.06","DOIUrl":"https://doi.org/10.3336/gm.58.1.06","url":null,"abstract":"The contracting boundary of a proper geodesic metric space generalizes the Gromov boundary of a hyperbolic space. It consists of contracting geodesics up to bounded Hausdorff distances. Another generalization of the Gromov boundary is the (kappa)–Morse boundary with a sublinear function (kappa). The two generalizations model the Gromov boundary based on different characteristics of geodesics in Gromov hyperbolic spaces. It was suspected that the (kappa)–Morse boundary contains the contracting boundary. We will prove this conjecture: when (kappa =1) is the constant function, the 1-Morse boundary and the contracting boundary are equivalent as topological spaces.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84976005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Assaf Bar-Natan, Advay Goel, Brendan Halstead, P. Hamrick, Sumedh Shenoy, R. Verma
In this paper, we study the relationship between the mapping class group of an infinite-type surface and the simultaneous flip graph, a variant of the flip graph for infinite-type surfaces defined by Fossas and Parlier [6]. We show that the extended mapping class group is isomorphic to a proper subgroup of the automorphism group of the flip graph, unlike in the finite-type case. This shows that Ivanov's metaconjecture, which states that any “sufficiently rich" object associated to a finite-type surface has the extended mapping class group as its automorphism group, does not extend to simultaneous flip graphs of infinite-type surfaces.
{"title":"Big flip graphs and their automorphism groups","authors":"Assaf Bar-Natan, Advay Goel, Brendan Halstead, P. Hamrick, Sumedh Shenoy, R. Verma","doi":"10.3336/gm.58.1.09","DOIUrl":"https://doi.org/10.3336/gm.58.1.09","url":null,"abstract":"In this paper, we study the relationship between the mapping class\u0000 group of an infinite-type surface and the simultaneous flip graph,\u0000 a variant of the flip graph for infinite-type surfaces defined by\u0000 Fossas and Parlier [6]. We show that the extended\u0000 mapping class group is isomorphic to a proper subgroup of the\u0000 automorphism group of the flip graph, unlike in the finite-type\u0000 case. This shows that Ivanov's metaconjecture, which states that\u0000 any “sufficiently rich\" object associated to a finite-type surface\u0000 has the extended mapping class group as its automorphism group, does\u0000 not extend to simultaneous flip graphs of infinite-type surfaces.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75792637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation -17ex p^n - q^m = f. We prove that for any non-constant polynomial ( f ) there are only finitely many quadruples ( (n,m,deg p,deg q) ) consisting of integers ( n,m geq 2 ) and non-constant polynomials ( p,q ) such that Pillai's equation holds. Moreover, we will give some examples that there can still be infinitely many possibilities for the polynomials ( p,q ).
{"title":"Pillai's conjecture for polynomials","authors":"Sebastian Heintze","doi":"10.3336/gm.58.1.05","DOIUrl":"https://doi.org/10.3336/gm.58.1.05","url":null,"abstract":"In this paper we study the polynomial version of Pillai's conjecture on the exponential Diophantine equation\u0000 \u0000 -17ex p^n - q^m = f.\u0000\u0000 We prove that for any non-constant polynomial ( f ) there are only finitely many quadruples ( (n,m,deg p,deg q) ) consisting of integers ( n,m geq 2 ) and non-constant polynomials ( p,q ) such that Pillai's equation holds.\u0000 Moreover, we will give some examples that there can still be infinitely many possibilities for the polynomials ( p,q ).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73306295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hyungryul Baik, Juhun Baik, Changsub Kim, Philippe Tranchida
We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface (S) of genus (g) with (n) punctures, we show that the minimal entropy of a pseudo-Anosov map is bounded from above by (dfrac{(k+1)log(k+3)}{|chi(S)|}) up to a constant multiple when the rank of the first homology of the mapping torus is (k+1) and (k, g, n) satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.
{"title":"Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology of mapping tori","authors":"Hyungryul Baik, Juhun Baik, Changsub Kim, Philippe Tranchida","doi":"10.3336/gm.57.2.09","DOIUrl":"https://doi.org/10.3336/gm.57.2.09","url":null,"abstract":"We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface (S) of genus (g) with (n) punctures, we show that the minimal entropy of a pseudo-Anosov map is bounded from above by (dfrac{(k+1)log(k+3)}{|chi(S)|}) up to a constant multiple when the rank of the first homology of the mapping torus is (k+1) and (k, g, n) satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80884767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We review established and recent results on the homotopy nilpotence of spaces. In particular, the homotopy nilpotency of the loop spaces (Omega(G/K)) of homogenous spaces (G/K) for a compact Lie group (G) and its closed homotopy nilpotent subgroup (K lt G) is discussed.
{"title":"On homotopy nilpotency","authors":"M. Golasiński","doi":"10.3336/gm.56.2.10","DOIUrl":"https://doi.org/10.3336/gm.56.2.10","url":null,"abstract":"We review established and recent results on the homotopy nilpotence of spaces. In particular, the homotopy nilpotency of the loop spaces (Omega(G/K)) of homogenous spaces (G/K) for a compact Lie group (G) and its closed homotopy nilpotent subgroup (K lt G) is discussed.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72538159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we discuss the continuity of generalized Riesz potentials ( I_{rho}f) of functions in Morrey spaces (L^{Phi,nu(cdot)}(G)) of double phase functionals with variable exponents.
{"title":"Continuity of generalized Riesz potentials for double phase functionals with variable exponents","authors":"T. Ohno, T. Shimomura","doi":"10.3336/gm.56.2.07","DOIUrl":"https://doi.org/10.3336/gm.56.2.07","url":null,"abstract":"In this note, we discuss the continuity of generalized Riesz potentials ( I_{rho}f) of functions in Morrey spaces (L^{Phi,nu(cdot)}(G)) of double phase functionals with variable exponents.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77683534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove that the Ramanujan-Nagell type Diophantine equation (Dx^2+k^n=B) has at most three nonnegative integer solutions ((x, n)) for (k) a prime and (B, D) positive integers.
{"title":"On the Ramanujan-Nagell type Diophantine equation (Dx^2+k^n=B)","authors":"Zhong-Can Zhang, A. Togbé","doi":"10.3336/gm.56.2.04","DOIUrl":"https://doi.org/10.3336/gm.56.2.04","url":null,"abstract":"In this paper, we prove that the Ramanujan-Nagell type Diophantine equation (Dx^2+k^n=B) has at most three nonnegative integer solutions ((x, n)) for (k) a prime and (B, D) positive integers.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88532954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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