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Truncated versions of three identities of Euler and Gauss 欧拉和高斯三个恒等式的截断形式
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-08-01 DOI: 10.1017/S0013091522000347
Olivia X. M. Yao
{"title":"Truncated versions of three identities of Euler and Gauss","authors":"Olivia X. M. Yao","doi":"10.1017/S0013091522000347","DOIUrl":"https://doi.org/10.1017/S0013091522000347","url":null,"abstract":"Abstract In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"775 - 798"},"PeriodicalIF":0.7,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48205889","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}
引用次数: 2
Singular elliptic equations with nonlinearities of subcritical and critical growth 具有非线性亚临界和临界增长的奇异椭圆方程
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-07-27 DOI: 10.1017/S0013091522000268
Matheus F. Stapenhorst
{"title":"Singular elliptic equations with nonlinearities of subcritical and critical growth","authors":"Matheus F. Stapenhorst","doi":"10.1017/S0013091522000268","DOIUrl":"https://doi.org/10.1017/S0013091522000268","url":null,"abstract":"Abstract In this paper, we consider the problem $-Delta u =-u^{-beta }chi _{{u>0}} + f(u)$ in $Omega$ with $u=0$ on $partial Omega$, where $0<beta <1$ and $Omega$ is a smooth bounded domain in $mathbb {R}^{N}$, $Ngeq 2$. We are able to solve this problem provided $f$ has subcritical growth and satisfy certain hypothesis. We also consider this problem with $f(s)=lambda s+s^{frac {N+2}{N-2}}$ and $Ngeq 3$. In this case, we are able to obtain a solution for large values of $lambda$. We replace the singular function $u^{-beta }$ by a function $g_epsilon (u)$ which pointwisely converges to $u^{-beta }$ as $epsilon rightarrow 0$. The corresponding energy functional to the perturbed equation $-Delta u + g_epsilon (u) = f(u)$ has a critical point $u_epsilon$ in $H_0^{1}(Omega )$, which converges to a non-trivial non-negative solution of the original problem as $epsilon rightarrow 0$.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"652 - 690"},"PeriodicalIF":0.7,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45433185","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}
引用次数: 0
Upper and lower limit oscillation conditions for first-order difference equations 一阶差分方程的上下限振荡条件
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-07-07 DOI: 10.1017/S0013091522000244
Ö. Öcalan
{"title":"Upper and lower limit oscillation conditions for first-order difference equations","authors":"Ö. Öcalan","doi":"10.1017/S0013091522000244","DOIUrl":"https://doi.org/10.1017/S0013091522000244","url":null,"abstract":"Abstract In this work, we consider the first-order difference equation with general argument [ varDelta x(n)+p(n)xleft( tau (n)right) =0,quad ngeq 0, ] where $(p(n))$ is a sequence of non-negative real numbers, $(tau (n))$ is a sequence of integers such that $tau (n)< n$ for$ nin mathbf {N},,$and$ lim _{nrightarrow infty }tau (n)=infty.$ Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"618 - 631"},"PeriodicalIF":0.7,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47676017","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}
引用次数: 0
On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field 完全非阿基米德值域中单变量WLUD∞函数和多变量WLUD∞函数的可解析性
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-07-07 DOI: 10.1017/S001309152200027X
K. Shamseddine
{"title":"On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field","authors":"K. Shamseddine","doi":"10.1017/S001309152200027X","DOIUrl":"https://doi.org/10.1017/S001309152200027X","url":null,"abstract":"Abstract Let $mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{infty }$ functions at a point or on an open subset of $mathcal {N}$. Then, we show under what conditions a WLUD$^{infty }$ function at a point $x_0in mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{infty }$ to functions from $mathcal {N}^{n}$ to $mathcal {N}$, for any $nin mathbb {N}$. Then, we formulate conditions under which a WLUD$^{infty }$ function at a point $boldsymbol {x_0} in mathcal {N}^{n}$ is analytic at that point.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"691 - 704"},"PeriodicalIF":0.7,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43827547","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}
引用次数: 0
Which states can be reached from a given state by unital completely positive maps? 从一个给定的状态通过完全正的单映射可以到达哪些状态?
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-07-05 DOI: 10.1017/S0013091522000256
B. Magajna
{"title":"Which states can be reached from a given state by unital completely positive maps?","authors":"B. Magajna","doi":"10.1017/S0013091522000256","DOIUrl":"https://doi.org/10.1017/S0013091522000256","url":null,"abstract":"Abstract For a state $omega$ on a C$^{*}$-algebra $A$, we characterize all states $rho$ in the weak* closure of the set of all states of the form $omega circ varphi$, where $varphi$ is a map on $A$ of the form $varphi (x)=sum nolimits _{i=1}^{n}a_i^{*}xa_i,$ $sum nolimits _{i=1}^{n}a_i^{*}a_i=1$ ($a_iin A$, $nin mathbb {N}$). These are precisely the states $rho$ that satisfy $|rho |J|leq |omega |J|$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $mathcal {R}$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $omega circ psi$, where $psi$ is a quantum channel on $mathcal {R}$ (that is, a map of the form $psi (x)=sum nolimits _ja_j^{*}xa_j$, where $a_jin mathcal {R}$ are such that the sum $sum nolimits _ja_j^{*}a_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C$^{*}$-algebra and for properly infinite von Neumann algebras the converse also holds.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"632 - 651"},"PeriodicalIF":0.7,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42749946","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}
引用次数: 0
On the Dales–Żelazko conjecture for Beurling algebras on discrete groups 关于离散群上Beurling代数的Dales–Żelazko猜想
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-06-28 DOI: 10.1017/S0013091523000354
Jared T. White
{"title":"On the Dales–Żelazko conjecture for Beurling algebras on discrete groups","authors":"Jared T. White","doi":"10.1017/S0013091523000354","DOIUrl":"https://doi.org/10.1017/S0013091523000354","url":null,"abstract":"Abstract Let G be a group that is either virtually soluble or virtually free, and let ω be a weight on G. We prove that if G is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $ell^1(G, omega)$, which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Żelazko holds for these Banach algebras. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance, we describe a Beurling algebra on an infinite group in which every closed left ideal of finite codimension is finitely generated and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Żelazko’s conjecture.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"613 - 624"},"PeriodicalIF":0.7,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43427150","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}
引用次数: 0
Corrigendum: Liftings of Jordan and Super Jordan Planes 勘误表:升降乔丹和超级乔丹飞机
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-06-14 DOI: 10.1017/s0013091522000207
N. Andruskiewitsch, I. Angiono, I. Heckenberger
{"title":"Corrigendum: Liftings of Jordan and Super Jordan Planes","authors":"N. Andruskiewitsch, I. Angiono, I. Heckenberger","doi":"10.1017/s0013091522000207","DOIUrl":"https://doi.org/10.1017/s0013091522000207","url":null,"abstract":"\u0000 We complete the classification of the pointed Hopf algebras with finite Gelfand-Kirillov dimension that are liftings of the Jordan plane over a nilpotent-by-finite group, correcting the statement in [N. Andruskiewitsch, I. Angiono and I. Heckenberger, Liftings of Jordan and super Jordan planes, Proc. Edinb. Math. Soc., II. Ser. 61(3) (2018), 661–672.].","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44681700","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}
引用次数: 1
On a planar Schrödinger–Poisson system involving a non-symmetric potential 关于包含非对称势的平面Schrödinger–Poisson系统
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-06-04 DOI: 10.1017/S0013091522000517
R. Molle, Andrea Sardilli
{"title":"On a planar Schrödinger–Poisson system involving a non-symmetric potential","authors":"R. Molle, Andrea Sardilli","doi":"10.1017/S0013091522000517","DOIUrl":"https://doi.org/10.1017/S0013091522000517","url":null,"abstract":"Abstract We prove the existence of a ground state positive solution of Schrödinger–Poisson systems in the plane of the form [ -Delta u + V(x)u + frac{gamma}{2pi} left(log|cdot| ast u^2 right)u = b |u|^{p-2}u quadtext{in} mathbb{R}^2, ]where $pge 4$, $gamma,b>0$ and the potential $V$ is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"1133 - 1146"},"PeriodicalIF":0.7,"publicationDate":"2022-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48155473","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}
引用次数: 0
The Reidemeister Spectrum of Finite Abelian Groups 有限阿贝尔群的Reidemeister谱
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-05-31 DOI: 10.1017/s0013091523000500
Pieter Senden
{"title":"The Reidemeister Spectrum of Finite Abelian Groups","authors":"Pieter Senden","doi":"10.1017/s0013091523000500","DOIUrl":"https://doi.org/10.1017/s0013091523000500","url":null,"abstract":"\u0000 For a finite abelian group A, the Reidemeister number of an endomorphism φ is the same as the number of fixed points of φ, and the Reidemeister spectrum of A is completely determined by the Reidemeister spectra of its Sylow p-subgroups. To compute the Reidemeister spectrum of a finite abelian p-group P, we introduce a new number associated to an automorphism ψ of P that captures the number of fixed points of ψ and its (additive) multiples, we provide upper and lower bounds for that number, and we prove that every power of p between those bounds occurs as such a number.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43411815","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}
引用次数: 0
On the derivative of a polynomial 关于多项式的导数
IF 0.7 3区 数学
Proceedings of the Edinburgh Mathematical Society Pub Date : 2022-05-24 DOI: 10.1017/s0013091521000687
Prasanna Kumar
{"title":"On the derivative of a polynomial","authors":"Prasanna Kumar","doi":"10.1017/s0013091521000687","DOIUrl":"https://doi.org/10.1017/s0013091521000687","url":null,"abstract":"In this paper, we prove the well-known Erdős–Lax inequality [4] in a sharpened form. As a consequence, another widely used inequality due to Ankeny and Rivlin [1] gets sharpened. These results may be useful in various applications that required the Erdős–Lax and the Ankeny–Rivlin inequalities.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44685720","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}
引用次数: 0
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