Pub Date : 2023-10-17DOI: 10.1017/s0013091523000615
Jaume Llibre, Claudia Valls
Abstract For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{prime}=f_1(x,y)$ , $y^{prime}=f_2(x,y)$ , where f 1 has degree one, f 2 has degree $nge 1$ and has degree one in the variable y . As a consequence, we provide sufficient conditions, weaker than the Markus–Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form $x^{prime}=y$ , $y^{prime}=g_1(x) +y g_2(x)$ with g 1 and g 2 polynomials of degrees n and m , respectively.
{"title":"A Weakened Markus–Yamabe Condition for Planar Polynomial Differential Systems of Degree ","authors":"Jaume Llibre, Claudia Valls","doi":"10.1017/s0013091523000615","DOIUrl":"https://doi.org/10.1017/s0013091523000615","url":null,"abstract":"Abstract For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{prime}=f_1(x,y)$ , $y^{prime}=f_2(x,y)$ , where f 1 has degree one, f 2 has degree $nge 1$ and has degree one in the variable y . As a consequence, we provide sufficient conditions, weaker than the Markus–Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form $x^{prime}=y$ , $y^{prime}=g_1(x) +y g_2(x)$ with g 1 and g 2 polynomials of degrees n and m , respectively.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136033057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-05DOI: 10.1017/s0013091523000536
José RodrÍguez, Abraham Rueda Zoca
Abstract We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(Omega,Sigma)$ be a measurable space, let E be a Banach lattice and let $nu:Sigma to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(nu)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue–Bochner space $L_1(mu,Y)$ is weakly almost square for any Banach space Y and for any non-atomic finite measure µ . On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space, which fails the diameter two property. In this line, we prove that if X is any Banach space containing a complemented isomorphic copy of c 0 , then for every $0 lt varepsilon lt 1$ , there exists an equivalent norm $|cdot|$ on X satisfying the following: (i) every slice of the unit ball $B_{(X,|cdot|)}$ has diameter 2; (ii) $B_{(X,|cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter and (iii) $(X,|cdot|)$ is ( r , s )-SQ for all $0 lt r,s lt frac{1-varepsilon}{1+varepsilon}$ .
{"title":"On Weakly Almost Square Banach Spaces","authors":"José RodrÍguez, Abraham Rueda Zoca","doi":"10.1017/s0013091523000536","DOIUrl":"https://doi.org/10.1017/s0013091523000536","url":null,"abstract":"Abstract We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(Omega,Sigma)$ be a measurable space, let E be a Banach lattice and let $nu:Sigma to E^+$ be a non-atomic countably additive measure having relatively norm compact range. Then the space $L_1(nu)$ is weakly almost square. This result applies to some abstract Cesàro function spaces. Similar arguments show that the Lebesgue–Bochner space $L_1(mu,Y)$ is weakly almost square for any Banach space Y and for any non-atomic finite measure µ . On the other hand, we make some progress on the open question of whether there exists a locally almost square Banach space, which fails the diameter two property. In this line, we prove that if X is any Banach space containing a complemented isomorphic copy of c 0 , then for every $0 lt varepsilon lt 1$ , there exists an equivalent norm $|cdot|$ on X satisfying the following: (i) every slice of the unit ball $B_{(X,|cdot|)}$ has diameter 2; (ii) $B_{(X,|cdot|)}$ contains non-empty relatively weakly open subsets of arbitrarily small diameter and (iii) $(X,|cdot|)$ is ( r , s )-SQ for all $0 lt r,s lt frac{1-varepsilon}{1+varepsilon}$ .","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134976021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-18DOI: 10.1017/s0013091523000366
Brian Cook, Kevin Hughes, Eyvindur Palsson
We prove discrete restriction estimates for a broad class of hypersurfaces arising in seminal work of Birch. To do so, we use a variant of Bourgain’s arithmetic version of the Tomas–Stein method and Magyar’s decomposition of the Fourier transform of the indicator function of the integer points on a hypersurface.
{"title":"Discrete restriction estimates for forms in many variables","authors":"Brian Cook, Kevin Hughes, Eyvindur Palsson","doi":"10.1017/s0013091523000366","DOIUrl":"https://doi.org/10.1017/s0013091523000366","url":null,"abstract":"We prove discrete restriction estimates for a broad class of hypersurfaces arising in seminal work of Birch. To do so, we use a variant of Bourgain’s arithmetic version of the Tomas–Stein method and Magyar’s decomposition of the Fourier transform of the indicator function of the integer points on a hypersurface.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135153866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1017/s0013091523000482
A. Gauvan
We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $left{X_k : k geq 1right}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions begin{equation*}Omega_{text{rand},text{lin}} := left{ frac{pi X_k}{k}: k geq 1right}end{equation*} and begin{equation*}Omega_{text{rand},text{lac}} := left{frac{pi X_k}{2^k} : kgeq 1 right}.end{equation*} We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({mathbb{R}}^2)$ for any $1 lt p lt infty$ .
{"title":"Perron’s capacity of random sets","authors":"A. Gauvan","doi":"10.1017/s0013091523000482","DOIUrl":"https://doi.org/10.1017/s0013091523000482","url":null,"abstract":"\u0000 We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables \u0000 \u0000 \u0000 $left{X_k : k geq 1right}$\u0000 \u0000 uniformly distributed in \u0000 \u0000 \u0000 $(0,1)$\u0000 \u0000 and independent, we consider the following random sets of directions\u0000\u0000 \u0000 \u0000 begin{equation*}Omega_{text{rand},text{lin}} := left{ frac{pi X_k}{k}: k geq 1right}end{equation*}\u0000 \u0000 and\u0000\u0000 \u0000 \u0000 begin{equation*}Omega_{text{rand},text{lac}} := left{frac{pi X_k}{2^k} : kgeq 1 right}.end{equation*}\u0000 \u0000 \u0000 We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on \u0000 \u0000 \u0000 $L^p({mathbb{R}}^2)$\u0000 \u0000 for any \u0000 \u0000 \u0000 $1 lt p lt infty$\u0000 \u0000 .","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41978588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.1017/s0013091523000524
{"title":"PEM series 2 volume 66 issue 3 Cover and Back matter","authors":"","doi":"10.1017/s0013091523000524","DOIUrl":"https://doi.org/10.1017/s0013091523000524","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42257992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.1017/S0013091523000421
Jin-Hui Fang
Abstract For a nonempty set A of integers and an integer n, let $r_{A}(n)$ be the number of representations of n in the form $n=a+a'$, where $aleqslant a'$ and $a, a'in A$, and $d_{A}(n)$ be the number of representations of n in the form $n=a-a'$, where $a, a'in A$. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion of n, i.e., $n=sum_{iin S(n)} 2^i$, $S(-n)=-S(n)$ and $S(0)=emptyset$. For real number x, let $A(-x,x)$ be the number of elements $ain A$ with $-xleqslant aleqslant x$. The famous Erdős-Turán Conjecture states that if A is a set of positive integers such that $r_A(n)geqslant 1$ for all sufficiently large n, then $limsup_{nrightarrowinfty}r_A(n)=infty$. In 2004, Nešetřil and Serra initially introduced the notation of “bounded” property and confirmed the Erdős-Turán conjecture for a class of bounded bases. They also proved that, there exists a set A of integers satisfying $r_A(n)=1$ for all integers n and $|S(x)bigcup S(y)|leqslant 4|S(x+y)|$ for $x,yin A$. On the other hand, Nathanson proved that there exists a set A of integers such that $r_A(n)=1$ for all integers n and $2log x/log 5+c_1leqslant A(-x,x)leqslant 2log x/log 3+c_2$ for all $xgeqslant 1$, where $c_1,c_2$ are absolute constants. In this paper, following these results, we prove that, there exists a set A of integers such that: $r_A(n)=1$ for all integers n and $d_A(n)=1$ for all positive integers n, $|S(x)bigcup S(y)|leqslant 4|S(x+y)|$ for $x,yin A$ and $A(-x,x) gt (4/log 5)loglog x+c$ for all $xgeqslant 1$, where c is an absolute constant. Furthermore, we also construct a family of arbitrarily spare such sets A.
{"title":"On the density of bounded bases","authors":"Jin-Hui Fang","doi":"10.1017/S0013091523000421","DOIUrl":"https://doi.org/10.1017/S0013091523000421","url":null,"abstract":"Abstract For a nonempty set A of integers and an integer n, let $r_{A}(n)$ be the number of representations of n in the form $n=a+a'$, where $aleqslant a'$ and $a, a'in A$, and $d_{A}(n)$ be the number of representations of n in the form $n=a-a'$, where $a, a'in A$. The binary support of a positive integer n is defined as the subset S(n) of nonnegative integers consisting of the exponents in the binary expansion of n, i.e., $n=sum_{iin S(n)} 2^i$, $S(-n)=-S(n)$ and $S(0)=emptyset$. For real number x, let $A(-x,x)$ be the number of elements $ain A$ with $-xleqslant aleqslant x$. The famous Erdős-Turán Conjecture states that if A is a set of positive integers such that $r_A(n)geqslant 1$ for all sufficiently large n, then $limsup_{nrightarrowinfty}r_A(n)=infty$. In 2004, Nešetřil and Serra initially introduced the notation of “bounded” property and confirmed the Erdős-Turán conjecture for a class of bounded bases. They also proved that, there exists a set A of integers satisfying $r_A(n)=1$ for all integers n and $|S(x)bigcup S(y)|leqslant 4|S(x+y)|$ for $x,yin A$. On the other hand, Nathanson proved that there exists a set A of integers such that $r_A(n)=1$ for all integers n and $2log x/log 5+c_1leqslant A(-x,x)leqslant 2log x/log 3+c_2$ for all $xgeqslant 1$, where $c_1,c_2$ are absolute constants. In this paper, following these results, we prove that, there exists a set A of integers such that: $r_A(n)=1$ for all integers n and $d_A(n)=1$ for all positive integers n, $|S(x)bigcup S(y)|leqslant 4|S(x+y)|$ for $x,yin A$ and $A(-x,x) gt (4/log 5)loglog x+c$ for all $xgeqslant 1$, where c is an absolute constant. Furthermore, we also construct a family of arbitrarily spare such sets A.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42559236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.1017/S0013091523000457
Olivia X. M. Yao
Abstract In recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.
{"title":"New mock theta functions and formulas for basic hypergeometric series","authors":"Olivia X. M. Yao","doi":"10.1017/S0013091523000457","DOIUrl":"https://doi.org/10.1017/S0013091523000457","url":null,"abstract":"Abstract In recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56897244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.1017/S0013091523000469
Jingcheng Liu, Min-Wei Tang, Shan Wu
Abstract Given a Borel probability measure µ on $mathbb{R}^n$ and a real matrix $Rin M_n(mathbb{R})$. We call R a spectral eigenmatrix of the measure µ if there exists a countable set $Lambdasubset mathbb{R}^n$ such that the sets $E_Lambda=big{{rm e}^{2pi i langlelambda,xrangle}:lambdain Lambdabig}$ and $E_{RLambda}=big{{rm e}^{2pi i langle Rlambda,xrangle}:lambdain Lambdabig}$ are both orthonormal bases for the Hilbert space $L^2(mu)$. In this paper, we study the structure of spectral eigenmatrix of the planar self-affine measure $mu_{M,D}$ generated by an expanding integer matrix $Min M_2(2mathbb{Z})$ and the four-elements digit set $D = {(0,0)^t,(1,0)^t,(0,1)^t,(-1,-1)^t}$. Some sufficient and/or necessary conditions for R to be a spectral eigenmatrix of $mu_{M,D}$ are given.
{"title":"The spectral eigenmatrix problems of planar self-affine measures with four digits","authors":"Jingcheng Liu, Min-Wei Tang, Shan Wu","doi":"10.1017/S0013091523000469","DOIUrl":"https://doi.org/10.1017/S0013091523000469","url":null,"abstract":"Abstract Given a Borel probability measure µ on $mathbb{R}^n$ and a real matrix $Rin M_n(mathbb{R})$. We call R a spectral eigenmatrix of the measure µ if there exists a countable set $Lambdasubset mathbb{R}^n$ such that the sets $E_Lambda=big{{rm e}^{2pi i langlelambda,xrangle}:lambdain Lambdabig}$ and $E_{RLambda}=big{{rm e}^{2pi i langle Rlambda,xrangle}:lambdain Lambdabig}$ are both orthonormal bases for the Hilbert space $L^2(mu)$. In this paper, we study the structure of spectral eigenmatrix of the planar self-affine measure $mu_{M,D}$ generated by an expanding integer matrix $Min M_2(2mathbb{Z})$ and the four-elements digit set $D = {(0,0)^t,(1,0)^t,(0,1)^t,(-1,-1)^t}$. Some sufficient and/or necessary conditions for R to be a spectral eigenmatrix of $mu_{M,D}$ are given.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43000130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.1017/s0013091523000470
Allan P. Donsig, Adam H. Fuller, David R. Pitts
An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the ‘Save PDF’ action button.
{"title":"Corrigendum: Von Neumann Algebras and Extensions of Inverse Semigroups","authors":"Allan P. Donsig, Adam H. Fuller, David R. Pitts","doi":"10.1017/s0013091523000470","DOIUrl":"https://doi.org/10.1017/s0013091523000470","url":null,"abstract":"An abstract is not available for this content. As you have access to this content, full HTML content is provided on this page. A PDF of this content is also available in through the ‘Save PDF’ action button.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136222850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-01DOI: 10.1017/s0013091523000512
{"title":"PEM series 2 volume 66 issue 3 Cover and Front matter","authors":"","doi":"10.1017/s0013091523000512","DOIUrl":"https://doi.org/10.1017/s0013091523000512","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43749073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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