Indian Journal of Pure and Applied Mathematics最新文献

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 (C^*) 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.

In this paper, we prove the existence of weak solutions for a class of nonlinear weighted elliptic equations in (Omega ) with p(x) 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)).

In this paper, we generalize the idea of expansive mappings and demonstrate various fixed point results within the context of (C^*)-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.

In this paper we show that spectral measures of the Laplacian on (ell ^2({mathbb {Z}}^d)) 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.

For positive real numbers (r, p_0,) and (p_1< cdots < p_n,) let (K_r) be the Kraus matrix whose (i, j) entry is equal to

In this article, we give a supplemental result to Sano and Takeuchi (J. Spectr. Theory, 2022) about the Kraus matrices (K_r): 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).

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.

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 (S_3(p)) which is the number of all integers (a in {1,2,ldots ,p-1}) such that (a+a^{-1}) and (a-a^{-1}) are both cubic residues modulo a prime (p equiv 1 pmod {3}). In this paper, we disprove these conjectures and use the theory of cubic residuosity to determine the specific formula for (S_3(p)) when 2 is a cubic non-residue modulo 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.

We are interested in Cauchy’s problem formed by a multidimensional incompressible Euler’s system and large amplitude oscillating initial data (w(x,varphi (x)/varepsilon )in mathcal {C}^1(Omega _r^0,mathbb {R}^n)), with (varepsilon in ]0,1]) is a parameter and (Omega ^0_rsubset mathbb {R}^n) the ball of centre zero and radius r. We determine the necessary and sufficient conditions that guarantee a solution on a domain of (mathbb {R}^+times mathbb {R}^n) independent of (varepsilon ) for the Cauchy’s problem previously mentioned. These conditions are a system of nonlinear partial differential equations uniform in (varepsilon ) involving the couple ((varphi ,w)), we show the existence of this couple, and we discuss its propagation over time.

By the Majorana representation, for any (d > 1) there is a one-one correspondence between a quantum state of dimension d and (d-1) qubits represented as (d-1) points in the Bloch sphere. Using the theory of symmetry class of tensors, we present a simple scheme for constructing (d-1) points on the Bloch sphere and the corresponding (d-1) qubits representing a d-dimensional quantum state. Additionally, we demonstrate how the inner product of two d-dimensional quantum states can be expressed as a permanent of a matrix related to their ((d-1))-qubit state representations. Extension of the result to mixed states is also considered.