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Feynman–Kac perturbation of $$C^*$$  quantum stochastic flows C^*$$$量子随机流的费曼-卡克扰动
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-06 DOI: 10.1007/s13226-024-00648-7
Alexander C. R. Belton, Stephen J. Wills
{"title":"Feynman–Kac perturbation of $$C^*$$  quantum stochastic flows","authors":"Alexander C. R. Belton, Stephen J. Wills","doi":"10.1007/s13226-024-00648-7","DOIUrl":"https://doi.org/10.1007/s13226-024-00648-7","url":null,"abstract":"<p>The method of Feynman–Kac perturbation of quantum stochastic processes has a long pedigree, with the theory usually developed within the framework of processes on von Neumann algebras. In this work, the theory of operator spaces is exploited to enable a broadening of the scope to flows on <span>(C^*)</span> algebras. Although the hypotheses that need to be verified in this general setting may seem numerous, we provide auxiliary results that enable this to be simplified in many of the cases which arise in practice. A wide variety of examples is provided by way of illustration.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Nonlinear weighted elliptic problem with variable exponents and $$L^1$$ data 具有可变指数和 $$L^1$$ 数据的非线性加权椭圆问题
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-05 DOI: 10.1007/s13226-024-00627-y
Rabah Mecheter
{"title":"Nonlinear weighted elliptic problem with variable exponents and $$L^1$$ data","authors":"Rabah Mecheter","doi":"10.1007/s13226-024-00627-y","DOIUrl":"https://doi.org/10.1007/s13226-024-00627-y","url":null,"abstract":"<p>In this paper, we prove the existence of weak solutions for a class of nonlinear weighted elliptic equations in <span>(Omega )</span> with <i>p</i>(<i>x</i>) growth conditions and integrable data. The functional setting involves Lebesgue-Sobolev spaces with variable exponents. Our results are generalizations of the corresponding results in the constant exponent case in L. Boccardo et al (Boll. Unione Mat. Ital. 15, No. 4, 503-514 (2022)) and some results given in D. Arcoya et al ( Journal of Functional Analysis, 268(5), 1153-1166 (2015)).</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalized expansive mapping, equiexpansive mapping in $$C^{*}$$ -algebra valued metric space and some fixed points results 广义扩张映射、$$C^{*}$$-代数值度量空间中的等价扩张映射和一些定点结果
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-05 DOI: 10.1007/s13226-024-00664-7
Rishi Dhariwal, Deepak Kumar
{"title":"Generalized expansive mapping, equiexpansive mapping in $$C^{*}$$ -algebra valued metric space and some fixed points results","authors":"Rishi Dhariwal, Deepak Kumar","doi":"10.1007/s13226-024-00664-7","DOIUrl":"https://doi.org/10.1007/s13226-024-00664-7","url":null,"abstract":"<p>In this paper, we generalize the idea of expansive mappings and demonstrate various fixed point results within the context of <span>(C^*)</span>-algebra valued metric space. Furthermore, specific metric space results from the literature are generalized by our results. Some examples are presented here to illustrate the usability of obtained results. These illustrations show that generalized expansive mapping outperforms literature based expansive mappings in terms of benefits.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550929","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
Regularity of the integrated density of states in the continuous spectrum 连续谱中的积分态密度的规律性
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-05 DOI: 10.1007/s13226-024-00640-1
M. Krishna
{"title":"Regularity of the integrated density of states in the continuous spectrum","authors":"M. Krishna","doi":"10.1007/s13226-024-00640-1","DOIUrl":"https://doi.org/10.1007/s13226-024-00640-1","url":null,"abstract":"<p>In this paper we show that spectral measures of the Laplacian on <span>(ell ^2({mathbb {Z}}^d))</span> are smooth in some regions of its spectrum, a result that extends to parts of the absolutely continuous spectrum of some random perturbations of it. The spectral measures considered are associated with dense sets of vectors.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550930","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
Inertia of Kraus matrices II 克劳斯矩阵的惯性 II
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-05 DOI: 10.1007/s13226-024-00647-8
Takashi Sano
{"title":"Inertia of Kraus matrices II","authors":"Takashi Sano","doi":"10.1007/s13226-024-00647-8","DOIUrl":"https://doi.org/10.1007/s13226-024-00647-8","url":null,"abstract":"<p>For positive real numbers <span>(r, p_0,)</span> and <span>(p_1&lt; cdots &lt; p_n,)</span> let <span>(K_r)</span> be the Kraus matrix whose (<i>i</i>, <i>j</i>) entry is equal to </p><span>$$begin{aligned} frac{1}{p_i - p_j} Bigl ( frac{p_i^r - p_0^r}{p_i -p_0} - frac{p_j^r - p_0^r}{p_j -p_0} Bigr ). end{aligned}$$</span><p>In this article, we give a supplemental result to Sano and Takeuchi (J. Spectr. Theory, 2022) about the Kraus matrices <span>(K_r)</span>: the simplicity of non-zero eigenvalues. Our proof is accomplished by arguments similar to those for Loewner matrices given by Bhatia, Friedland and Jain (Indiana Univ. Math. J., 2016).</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550931","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
Copying quantum states 复制量子态
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-05 DOI: 10.1007/s13226-024-00634-z
Hans Maassen, Burkhard Kümmerer
{"title":"Copying quantum states","authors":"Hans Maassen, Burkhard Kümmerer","doi":"10.1007/s13226-024-00634-z","DOIUrl":"https://doi.org/10.1007/s13226-024-00634-z","url":null,"abstract":"<p>The no-broadcasting theorem in quantum information says that a set of states on a quantum system admits a common broadcasting (copying) operation if and only if their density matrices belong to a commuting family. We discuss and prove this theorem, as well as the closely related “no-cloning theorem” in the context of quantum probability theory, i.e. in the category of (finite dimensional) C*-algebras with unital completely positive maps.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"137 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550927","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 two conjectures related to cubic residues 与立方残差有关的两个猜想
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-04 DOI: 10.1007/s13226-024-00626-z
Xiaopeng Zhao, Zhenfu Cao
{"title":"On two conjectures related to cubic residues","authors":"Xiaopeng Zhao, Zhenfu Cao","doi":"10.1007/s13226-024-00626-z","DOIUrl":"https://doi.org/10.1007/s13226-024-00626-z","url":null,"abstract":"<p>In a recent paper by Yuan and Zhang (Indian J. Pure Appl. Math. 54(3):806–815, 2023), the authors put forward two conjectures regarding <span>(S_3(p))</span> which is the number of all integers <span>(a in {1,2,ldots ,p-1})</span> such that <span>(a+a^{-1})</span> and <span>(a-a^{-1})</span> are both cubic residues modulo a prime <span>(p equiv 1 pmod {3})</span>. In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for <span>(S_3(p))</span> when 2 is a cubic non-residue modulo <i>p</i>.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550933","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
Positivity and matrix inequalities: some reminiscences of K. R. Parthasarathy 正性与矩阵不等式：对 K. R. Parthasarathy 的一些回忆
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-04 DOI: 10.1007/s13226-024-00639-8
Rajendra Bhatia
{"title":"Positivity and matrix inequalities: some reminiscences of K. R. Parthasarathy","authors":"Rajendra Bhatia","doi":"10.1007/s13226-024-00639-8","DOIUrl":"https://doi.org/10.1007/s13226-024-00639-8","url":null,"abstract":"<p>One of K. R. Parthasarathy’s major and abiding interests was the theory of positive definite functions and especially its relation to classical and quantum probability. Here we recall his contribution in providing a seminal idea that stimulated a lot of work on matrix inequalities.\u0000</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550926","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
Compatibility conditions allowing mono phasic oscillating solutions for the multidimensional incompressible Euler system 多维不可压缩欧拉系统单相振荡解的兼容条件
Indian Journal of Pure and Applied Mathematics Pub Date : 2024-07-03 DOI: 10.1007/s13226-024-00630-3
Mekki Houbad
{"title":"Compatibility conditions allowing mono phasic oscillating solutions for the multidimensional incompressible Euler system","authors":"Mekki Houbad","doi":"10.1007/s13226-024-00630-3","DOIUrl":"https://doi.org/10.1007/s13226-024-00630-3","url":null,"abstract":"<p>We are interested in Cauchy’s problem formed by a multidimensional incompressible Euler’s system and large amplitude oscillating initial data <span>(w(x,varphi (x)/varepsilon )in mathcal {C}^1(Omega _r^0,mathbb {R}^n))</span>, with <span>(varepsilon in ]0,1])</span> is a parameter and <span>(Omega ^0_rsubset mathbb {R}^n)</span> the ball of centre zero and radius <i>r</i>. We determine the necessary and sufficient conditions that guarantee a solution on a domain of <span>(mathbb {R}^+times mathbb {R}^n)</span> independent of <span>(varepsilon )</span> for the Cauchy’s problem previously mentioned. These conditions are a system of nonlinear partial differential equations uniform in <span>(varepsilon )</span> involving the couple <span>((varphi ,w))</span>, we show the existence of this couple, and we discuss its propagation over time.</p>","PeriodicalId":501427,"journal":{"name":"Indian Journal of Pure and Applied Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523070","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 : 2024-07-03 DOI: 10.1007/s13226-024-00602-7
Chi-Kwong Li, Mikio Nakahara
{"title":"A note on Majorana representation of quantum states","authors":"Chi-Kwong Li, Mikio Nakahara","doi":"10.1007/s13226-024-00602-7","DOIUrl":"https://doi.org/10.1007/s13226-024-00602-7","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":"2012 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523072","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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