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A note on Diophantine approximation with four squares and one k-th power of primes 关于用四个正方形和一个 k 次幂素数进行 Diophantine 近似的说明
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-08-06 DOI: 10.1007/s13226-024-00672-7
Yuhui Liu
{"title":"A note on Diophantine approximation with four squares and one k-th power of primes","authors":"Yuhui Liu","doi":"10.1007/s13226-024-00672-7","DOIUrl":"https://doi.org/10.1007/s13226-024-00672-7","url":null,"abstract":"<p>Let <span>(lambda _1, lambda _2, lambda _3, lambda _4, mu )</span> be non-zero real numbers, not all negative, with <span>(lambda _1/lambda _2)</span> irrational. Suppose that <span>(k geqslant 3)</span> be an integer and <span>(eta )</span> be any given real number. In this paper, it is proved that for any real number <span>(sigma )</span> with <span>(0&lt;sigma &lt;frac{1}{vartheta (k)})</span>, the inequality </p><span>$$begin{aligned} |lambda _1 p_1^2 + lambda _2 p_2^2+ lambda _3 p_3^2+ lambda _4 p_4^2 + mu p_5^k + eta | &lt; left( max limits _{1leqslant j leqslant 5}p_jright) ^{-sigma } end{aligned}$$</span><p>has infinitely many solutions in prime variables <span>(p_1,cdots ,p_5)</span>, where <span>(vartheta (k) = frac{32}{5}lceil {big (frac{k}{2} + 1 - [frac{k}{2}]big )2^{[frac{k}{2}]-1}}rceil )</span> for <span>(3leqslant k leqslant 9)</span> and <span>(vartheta (k) = frac{32}{5}lceil {big (frac{k}{2} - frac{1}{2}[frac{k}{2}]big )big ([frac{k}{2}]+1big )}rceil )</span> for <span>(k geqslant 10)</span>. This result constitutes an improvement upon that of Q. W. Mu, M. H. Zhu and P. Li [13].</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141940950","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
Numerical radius bounds for certain operators 某些算子的数值半径边界
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-08-01 DOI: 10.1007/s13226-024-00663-8
Pintu Bhunia
{"title":"Numerical radius bounds for certain operators","authors":"Pintu Bhunia","doi":"10.1007/s13226-024-00663-8","DOIUrl":"https://doi.org/10.1007/s13226-024-00663-8","url":null,"abstract":"<p>We provide sharp bounds for the numerical radius of bounded linear operators defined on a complex Hilbert space. We also provide sharp bounds for the numerical radius of <span>(A^{alpha }XB^{1-alpha })</span>, <span>(A^{alpha }XB^{alpha })</span> and the Heinz means of operators, where <i>A</i>, <i>B</i>, <i>X</i> are bounded linear operators with <span>(A,Bge 0)</span> and <span>(0le alpha le 1.)</span> Further, we study the <i>A</i>-numerical radius inequalities for semi-Hilbertian space operators. We prove that <span>(w_A(T) le left( 1-frac{1}{2^{n-1}}right) ^{1/n} Vert TVert _A)</span> when <span>(AT^n=0)</span> for some least positive integer <i>n</i>. Some equalities for the <i>A</i>-numerical radius inequalities are also studied.\u0000</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863417","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
A note on Majorana representation of quantum states 关于量子态的马约拉纳表征的说明
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-27 DOI: 10.1007/s13226-024-00649-6
Chi-Kwong Li, Mikio Nakahara
{"title":"A note on Majorana representation of quantum states","authors":"Chi-Kwong Li, Mikio Nakahara","doi":"10.1007/s13226-024-00649-6","DOIUrl":"https://doi.org/10.1007/s13226-024-00649-6","url":null,"abstract":"<p>By the Majorana representation, for any <span>(d &gt; 1)</span> there is a one-one correspondence between a quantum state of dimension <i>d</i> and <span>(d-1)</span> qubits represented as <span>(d-1)</span> points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing <span>(d-1)</span> points on the Bloch sphere and the corresponding <span>(d-1)</span> qubits representing a <i>d</i>-dimensional quantum state. Additionally, we demonstrate how the inner product of two <i>d</i>-dimensional quantum states can be expressed as a permanent of a matrix related to their <span>((d-1))</span>-qubit state representations. Extension of the result to mixed states is also considered.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776886","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 odd-normal numbers 关于奇正态数
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-23 DOI: 10.1007/s13226-024-00642-z
Malabika Pramanik, Junqiang Zhang
{"title":"On odd-normal numbers","authors":"Malabika Pramanik, Junqiang Zhang","doi":"10.1007/s13226-024-00642-z","DOIUrl":"https://doi.org/10.1007/s13226-024-00642-z","url":null,"abstract":"<p>A real number <i>x</i> is considered normal in an integer base <span>(b geqslant 2)</span> if its digit expansion in this base is “equitable”, ensuring that for each <span>(k geqslant 1)</span>, every ordered sequence of <i>k</i> digits from <span>({0, 1, ldots , b-1})</span> occurs in the digit expansion of <i>x</i> with the same limiting frequency. Borel’s classical result [4] asserts that Lebesgue-almost every <span>(x in {mathbb {R}})</span> is normal in every base <span>(b geqslant 2)</span>. This paper serves as a case study of the measure-theoretic properties of Lebesgue-null sets containing numbers that are normal only in certain bases. We consider the set <span>({mathscr {N}}({mathscr {O}}, {mathscr {E}}))</span> of reals that are normal in odd bases but not in even ones. This set has full Hausdorff dimension [30] but zero Fourier dimension. The latter condition means that <span>({mathscr {N}}({mathscr {O}}, {mathscr {E}}))</span> cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that <span>({mathscr {N}}({mathscr {O}}, {mathscr {E}}))</span> supports a Rajchman measure <span>(mu )</span>, whose Fourier transform <span>({widehat{mu }}(xi ))</span> approaches 0 as <span>(|xi | rightarrow infty )</span> by definiton, albeit slower than any negative power of <span>(|xi |)</span>. Moreover, the decay rate of <span>({widehat{mu }})</span> is essentially optimal, subject to the constraints of its support. The methods draw inspiration from the number-theoretic results of Schmidt [38] and a construction of Lyons [24]. As a consequence, <span>(mathscr {N}({mathscr {O}}, {mathscr {E}}))</span> emerges as a set of multiplicity, in the sense of Fourier analysis. This addresses a question posed by Kahane and Salem [17] in the special case of <span>({mathscr {N}}({mathscr {O}}, {mathscr {E}}))</span>.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"110 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785622","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
Further results on alternating two-stage iterative method 两阶段交替迭代法的进一步结果
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-22 DOI: 10.1007/s13226-024-00669-2
Vaibhav Shekhar
{"title":"Further results on alternating two-stage iterative method","authors":"Vaibhav Shekhar","doi":"10.1007/s13226-024-00669-2","DOIUrl":"https://doi.org/10.1007/s13226-024-00669-2","url":null,"abstract":"<p>Matrix splitting is an efficient and readily used technique for study of solution of linear systems, iteratively. Migallón <i>et al.</i> [Adv. Eng. Softw. 41:13-21, 2010] proposed alternating two-stage methods in which the inner iterations are accomplished by an alternating method. However, the convergence theory of an alternating two-stage iteration scheme for various class of matrix splittings is a literature gap. In this article, we establish convergence theory of alternating two-stage iterative methods for nonsingular, consistent singular and inconsistent rectangular (or singular) linear systems for different class of matrix splittings. Finally, numerical computations are performed which illustrate that this method has some advantages over simple two-stage iterative method.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785624","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
Construction of permutation polynomials with specific cycle structure over finite fields 构建有限域上具有特定循环结构的置换多项式
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-22 DOI: 10.1007/s13226-024-00668-3
Anitha G, P. Vanchinathan
{"title":"Construction of permutation polynomials with specific cycle structure over finite fields","authors":"Anitha G, P. Vanchinathan","doi":"10.1007/s13226-024-00668-3","DOIUrl":"https://doi.org/10.1007/s13226-024-00668-3","url":null,"abstract":"<p>For a finite field of odd order <i>q</i>, and a divisor <i>n</i> of <span>(q-1)</span>, we construct families of permutation polynomials of <i>n</i> terms with one fixed-point (namely zero) and remaining elements being permuted as disjoint cycles of same length. Our polynomials will all be of same format: that is the degree, the terms are identical. For our polynomials their compositional inverses are also polynomials in the same format and are easy to write down. The special cases of <span>(n=2,3)</span> give very simple families of permutation binomials and trinomials. For example, in the field of 121 elements our methods provide 4080 permutation trinomials all decomposing into three disjoint cycles of length 40 along with a unique fixed point.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776888","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
Krasnosel’skii iterative process for approximating fixed points of generalized Bianchini mappings in Banach space and applications to variational inequality and split feasibility problems 近似巴拿赫空间广义边沁映射定点的克拉斯诺瑟尔迭代过程及其在变分不等式和分割可行性问题中的应用
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-18 DOI: 10.1007/s13226-024-00625-0
Ravindra K. Bisht
{"title":"Krasnosel’skii iterative process for approximating fixed points of generalized Bianchini mappings in Banach space and applications to variational inequality and split feasibility problems","authors":"Ravindra K. Bisht","doi":"10.1007/s13226-024-00625-0","DOIUrl":"https://doi.org/10.1007/s13226-024-00625-0","url":null,"abstract":"<p>In this paper, we establish existence, uniqueness, and convergence results for approximating fixed points using a Krasnosel’skii iterative process for generalized Bianchini mappings in Banach spaces. Additionally, we demonstrate the practical applications of our main fixed point theorems by solving variational inequality problems, split feasibility problems, and certain linear systems of equations.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141741317","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
Algebras of entire functions and representations of the twisted Heisenberg group 扭曲海森堡群的全函数代数和表征
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-14 DOI: 10.1007/s13226-024-00636-x
Sundaram Thangavelu
{"title":"Algebras of entire functions and representations of the twisted Heisenberg group","authors":"Sundaram Thangavelu","doi":"10.1007/s13226-024-00636-x","DOIUrl":"https://doi.org/10.1007/s13226-024-00636-x","url":null,"abstract":"<p>On the twisted Fock spaces <span>( mathcal {F}^lambda ({mathbb {C}}^{2n}) )</span> we consider a family of unitary operators <span>(rho _lambda (a,b) )</span> indexed by <span>( (a,b) in {mathbb {C}}^n times {mathbb {C}}^n.)</span> The composition formula for <span>( rho _lambda (a,b) circ rho _lambda (a^prime ,b^prime ) )</span> leads us to a group <span>( mathbb {H}^n_lambda ({mathbb {C}}) )</span> which contains two copies of the Heisenberg group <span>( mathbb {H}^n.)</span> The operators <span>( rho _lambda (a,b) )</span> lift to <span>( mathbb {H}_lambda ^n({mathbb {C}}) )</span> providing an irreducible unitary representation. However, its restriction to <span>( mathbb {H}^n_lambda (mathbb {R}) )</span> is not irreducible.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611237","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 distribution of Fourier coefficients of symmetric square L-functions over arithmetic progressions 算术级数上对称平方 L 函数傅里叶系数的分布
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-12 DOI: 10.1007/s13226-024-00628-x
Dan Wang
{"title":"The distribution of Fourier coefficients of symmetric square L-functions over arithmetic progressions","authors":"Dan Wang","doi":"10.1007/s13226-024-00628-x","DOIUrl":"https://doi.org/10.1007/s13226-024-00628-x","url":null,"abstract":"<p>Let <span>(L(s, mathrm{sym^2}f))</span> be the corresponding symmetric square <i>L</i>-function associated to <i>f</i>(<i>z</i>), where <i>f</i>(<i>z</i>) is a primitive holomorphic cusp form of even integral weight <i>k</i> for the full modular group. Suppose that <span>(lambda _{mathrm{sym^2}f} (n))</span> is the <i>n</i>th normalized Fourier coefficient of <span>(L(s, {mathrm{sym^2}f}))</span>. In this paper, we use the function equation and the large sieve inequality to study the asymptotic behaviour of the sums </p><span>$$begin{aligned} sum _{begin{array}{c} nleqslant x nequiv a(textrm{mod} q) end{array}}lambda ^{j}_{mathrm{sym^2}f}(n), 2leqslant jleqslant 4. end{aligned}$$</span>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611239","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
A note on outer quantum automorphisms of finite dimensional von Neumann algebras 关于有限维冯-诺依曼代数的外量子自动形态的说明
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-11 DOI: 10.1007/s13226-024-00637-w
Debashish Goswami
{"title":"A note on outer quantum automorphisms of finite dimensional von Neumann algebras","authors":"Debashish Goswami","doi":"10.1007/s13226-024-00637-w","DOIUrl":"https://doi.org/10.1007/s13226-024-00637-w","url":null,"abstract":"<p>This is part of an ongoing project of formulating notion(s) of quantum group of outer automorphisms of a <span>(C^*)</span> or von Neumann algebra. Motivated by the fact that the group of outer automorphism of a <span>(II_1)</span> factor can be viewed as a subgroup of the group of group-like or invertible objects in the category of Hilbert bimodules of finite ranks, we explore a natural class of objects in the bimodule category of a finite dimensional (i.e. direct sum of matrix algebras) von Neumann algebra <span>(mathcal{A})</span> which may come from the (co-action) of a discrete quantum group. In particular, we prove that any discrete quantum group giving an outer quantum symmetry on <span>(mathcal{A})</span> in a sense defined by us must be a finite dimensional quantum group. We relate the analysis of such quantum groups or the corresponding fusion rings with certain combinatorial objects involving matrices with nonnegative integer entries and do some explicit computations in a few simple examples.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611238","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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