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On Simultaneous Approximation of Algebraic Power Series over a Finite Field 论有限域上代数幂级数的同时逼近
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-07-30 DOI: 10.1134/s2070046624030063
Khalil Ayadi, Chiheb Ben Bechir, Samir Elkadri
{"title":"On Simultaneous Approximation of Algebraic Power Series over a Finite Field","authors":"Khalil Ayadi, Chiheb Ben Bechir, Samir Elkadri","doi":"10.1134/s2070046624030063","DOIUrl":"https://doi.org/10.1134/s2070046624030063","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In 1970, W. M. Schmidt [6] generalized Roth’s well-known theorem on rational approximation to a single algebraic irrational, to include simultaneous rational approximation for a given <span>(n)</span> algebraic irrationals. As no analogue of Roth’s theorem for algebraic irrational power series over a finite field exists, we will show that there is no analogue of Schmidt’s theorem for such <span>(n)</span> elements. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863713","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}
引用次数: 0
$$H_A$$ -Weakly Periodic $$p$$ -Adic Generalized Gibbs Measures for the $$p$$ -Adic Ising Model on the Cayley Tree of Order Two 二阶 Cayley 树上 $$p$$ -Adic Ising 模型的弱周期 $$H_A$$ -p$$ -Adic 广义吉布斯度量
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-07-30 DOI: 10.1134/s2070046624030038
Muzaffar Rahmatullaev, Zulxumor Abdukaxorova
{"title":"$$H_A$$ -Weakly Periodic $$p$$ -Adic Generalized Gibbs Measures for the $$p$$ -Adic Ising Model on the Cayley Tree of Order Two","authors":"Muzaffar Rahmatullaev, Zulxumor Abdukaxorova","doi":"10.1134/s2070046624030038","DOIUrl":"https://doi.org/10.1134/s2070046624030038","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In the present paper, we consider a <span>(p)</span>-adic Ising model on a Cayley tree. For this model, <span>(p)</span>-adic analogue of the notion of weakly periodic Gibbs measures is introduced. For some normal subgroup of the group representation of the Cayley tree, the existence of such Gibbs measures is proved. We also study fixed points and their behaviour of the mapping which coincides with weakly periodic quantities of the functional equation. Moreover, the boundedness of such kinds of measures is established, which yields the occurrence of a phase transition. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
$$p$$ -Adic Welch Bounds and $$p$$ -Adic Zauner Conjecture $$p$$ -阿迪克-韦尔奇边界和 $$p$$ -阿迪克-扎乌纳猜想
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-07-30 DOI: 10.1134/s207004662403004x
K. M. Krishna
{"title":"$$p$$ -Adic Welch Bounds and $$p$$ -Adic Zauner Conjecture","authors":"K. M. Krishna","doi":"10.1134/s207004662403004x","DOIUrl":"https://doi.org/10.1134/s207004662403004x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Let <span>(p)</span> be a prime. For <span>(din mathbb{N})</span>, let <span>(mathbb{Q}_p^d)</span> be the standard <span>(d)</span>-dimensional p-adic Hilbert space. Let <span>(m in mathbb{N})</span> and <span>(text{Sym}^m(mathbb{Q}_p^d))</span> be the <span>(p)</span>-adic Hilbert space of symmetric m-tensors. We prove the following result. Let <span>({tau_j}_{j=1}^n)</span> be a collection in <span>(mathbb{Q}_p^d)</span> satisfying (i) <span>(langle tau_j, tau_jrangle =1)</span> for all <span>(1leq j leq n)</span> and (ii) there exists <span>(b in mathbb{Q}_p)</span> satisfying <span>(sum_{j=1}^{n}langle x, tau_jrangle tau_j =bx)</span> for all <span>( x in mathbb{Q}^d_p.)</span> Then </p><span>$$begin{aligned} , max_{1leq j,k leq n, j neq k}{|n|, |langle tau_j, tau_krangle|^{2m} }geq frac{|n|^2}{left|{d+m-1 choose m}right| }. end{aligned}$$</span>(0.1)<p> We call Inequality (0.1) as the <span>(p)</span>-adic version of Welch bounds obtained by Welch [<i>IEEE Transactions on Information Theory, 1974</i>]. Inequality (0.1) differs from the non-Archimedean Welch bound obtained recently by M. Krishna as one can not derive one from another. We formulate <span>(p)</span>-adic Zauner conjecture. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863710","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}
引用次数: 0
Finite Adelic Wavelet Bases and a Pseudodifferential Equation 有限阿德利小波基与伪微分方程
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-07-30 DOI: 10.1134/s2070046624030051
V. A. Aguilar-Arteaga, S. M. Delfín-Prieto, S. Estala-Arias
{"title":"Finite Adelic Wavelet Bases and a Pseudodifferential Equation","authors":"V. A. Aguilar-Arteaga, S. M. Delfín-Prieto, S. Estala-Arias","doi":"10.1134/s2070046624030051","DOIUrl":"https://doi.org/10.1134/s2070046624030051","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this article we apply a polyadic approach to obtain very explicit description of a novel kind of wavelets on the ring of finite adèles, <span>(mathbb{A}_{f})</span>, which are also eigenfunctions of a Vladimirov-type pseudodifferential operator on <span>(L^2(mathbb{A}_{f}))</span>. As an accompaniment, we solve the Cauchy problem for a certain pseudodifferential equation. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863712","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}
引用次数: 0
Rough Hardy-Littlewood Operators on $$p$$ -Adic Function Spaces with Variable Exponents 具有可变指数的$$p$$自变函数空间上的粗糙哈代-利特尔伍德算子
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-07-30 DOI: 10.1134/s2070046624030026
K. H. Dung, P. T. K. Thuy
{"title":"Rough Hardy-Littlewood Operators on $$p$$ -Adic Function Spaces with Variable Exponents","authors":"K. H. Dung, P. T. K. Thuy","doi":"10.1134/s2070046624030026","DOIUrl":"https://doi.org/10.1134/s2070046624030026","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this paper, we establish some sufficient conditions for the boundedness of rough Hardy-Littlewood operators on the <span>(p)</span>-adic local central Morrey, <span>(p)</span>-adic Morrey-Herz, and <span>(p)</span>-adic local block spaces with variable exponents. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Compactly Supported Distributions on $$p$$ -Adic Lie Groups 关于$$p$$-自洽李群的紧凑支持分布
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-07-30 DOI: 10.1134/s2070046624030014
Dubravka Ban, Jeremiah Roberts
{"title":"Compactly Supported Distributions on $$p$$ -Adic Lie Groups","authors":"Dubravka Ban, Jeremiah Roberts","doi":"10.1134/s2070046624030014","DOIUrl":"https://doi.org/10.1134/s2070046624030014","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Let <span>(K)</span> be a finite extension of <span>(mathbb{Q}_p)</span> and let <span>(G)</span> be a <span>(p)</span>-adic Lie group. In this paper, we define the Iwasawa algebra <span>(K[[G]])</span> and prove that it is isomorphic to the convolution algebra of compactly supported distributions on <span>(G)</span>. This has important applications in the theory of admissible representations of <span>(G)</span> on <span>(p)</span>-adic Banach spaces. In particular, we prove the Frobenius reciprocity for continuous principal series representations. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863669","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}
引用次数: 0
Approximation by Vallée-Poussin Type Means of Vilenkin-Fourier Series 维伦金-傅里叶级数的 Vallée-Poussin 型近似手段
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-07-30 DOI: 10.1134/s2070046624030075
S. S. Volosivets
{"title":"Approximation by Vallée-Poussin Type Means of Vilenkin-Fourier Series","authors":"S. S. Volosivets","doi":"10.1134/s2070046624030075","DOIUrl":"https://doi.org/10.1134/s2070046624030075","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We estimate the degree of approximation by linear means of Vallée-Poussin type of Vilenkin-Fourier series in classical Lebesgue spaces and in a space of generalized continuous functions. These results generalize ones obtained by I. Blahota and G.Gat for means of Walsh-Fourier series. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863717","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}
引用次数: 0
On the Theory of Relativistic Brownian Motion 论相对论布朗运动
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-05-06 DOI: 10.1134/s207004662402002x
E. A. Kurianovich, A. I. Mikhailov, I. V. Volovich
{"title":"On the Theory of Relativistic Brownian Motion","authors":"E. A. Kurianovich, A. I. Mikhailov, I. V. Volovich","doi":"10.1134/s207004662402002x","DOIUrl":"https://doi.org/10.1134/s207004662402002x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The approach to the theory of a relativistic random process is considered by the path integral method as Brownian motion taking into account the boundedness of speed. An attempt was made to build a relativistic analogue of the Wiener measure as a weak limit of finite-difference approximations. A formula has been proposed for calculating the probability particle transition during relativistic Brownian motion. Calculations were carried out by three different methods with identical results. Along the way, exact and asymptotic formulas for the volume of some parts and sections of an N-1-dimensional unit cube were obtained. They can have independent value. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932400","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}
引用次数: 0
The Collatz Conjecture & Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory 科拉茨猜想与非阿基米德谱理论 - 第一部分 - 算术动态系统与非阿基米德值分布理论
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-05-06 DOI: 10.1134/s2070046624020055
Maxwell C. Siegel
{"title":"The Collatz Conjecture & Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory","authors":"Maxwell C. Siegel","doi":"10.1134/s2070046624020055","DOIUrl":"https://doi.org/10.1134/s2070046624020055","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Let <span>(q)</span> be an odd prime, and let <span>(T_{q}:mathbb{Z}rightarrowmathbb{Z})</span> be the Shortened <span>(qx+1)</span> map, defined by <span>(T_{q}left(nright)=n/2)</span> if <span>(n)</span> is even and <span>(T_{q}left(nright)=left(qn+1right)/2)</span> if <span>(n)</span> is odd. The study of the dynamics of these maps is infamous for its difficulty, with the characterization of the dynamics of <span>(T_{3})</span> being an alternative formulation of the famous <b>Collatz Conjecture</b>. This series of papers presents a new paradigm for studying such arithmetic dynamical systems by way of a neglected area of ultrametric analysis which we have termed <span>(left(p,qright))</span><b>-adic analysis</b>, the study of functions from the <span>(p)</span>-adics to the <span>(q)</span>-adics, where <span>(p)</span> and <span>(q)</span> are distinct primes. In this, the first paper, working with the <span>(T_{q})</span> maps as a toy model for the more general theory, for each odd prime <span>(q)</span>, we construct a function <span>(chi_{q}:mathbb{Z}_{2}rightarrowmathbb{Z}_{q})</span> (the <b>Numen </b>of <span>(T_{q})</span>) and prove the <b>Correspondence Principle</b> (CP): <span>(xinmathbb{Z}backslashleft{ 0right} )</span> is a periodic point of <span>(T_{q})</span> if and only there is a <span>(mathfrak{z}inmathbb{Z}_{2}backslashleft{ 0,1,2,ldotsright} )</span> so that <span>(chi_{q}left(mathfrak{z}right)=x)</span>. Additionally, if <span>(mathfrak{z}inmathbb{Z}_{2}backslashmathbb{Q})</span> makes <span>(chi_{q}left(mathfrak{z}right)inmathbb{Z})</span>, then the iterates of <span>(chi_{q}left(mathfrak{z}right))</span> under <span>(T_{q})</span> tend to <span>(+infty)</span> or <span>(-infty)</span>. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932394","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}
引用次数: 0
Group Structure of the $$p$$ -Adic Ball and Dynamical System of Isometry on a Sphere 球面上 $$p$$ 阿迪克球的群结构和等值动态系统
IF 0.5
P-Adic Numbers Ultrametric Analysis and Applications Pub Date : 2024-05-06 DOI: 10.1134/s2070046624020031
I. A. Sattarov
{"title":"Group Structure of the $$p$$ -Adic Ball and Dynamical System of Isometry on a Sphere","authors":"I. A. Sattarov","doi":"10.1134/s2070046624020031","DOIUrl":"https://doi.org/10.1134/s2070046624020031","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this paper, the group structure of the <span>(p)</span>-adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations <span>(oplus)</span> and <span>(odot)</span> on a ball and sphere, respectively, and prove that these sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in <span>(mathbb Z_p)</span> is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere and show that the trajectory of any initial point that is not a fixed point is not convergent. We study ergodicity of this <span>(p)</span>-adic dynamical system with respect to normalized Haar measure reduced on the sphere. For <span>(pgeq 3)</span> we prove that the dynamical systems are not ergodic. But for <span>(p=2)</span> under some conditions the dynamical system may be ergodic. </p>","PeriodicalId":44654,"journal":{"name":"P-Adic Numbers Ultrametric Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932334","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}
引用次数: 0
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