Operator Norm Research Articles

A Comparative Analysis of the Peritextual Features of the English-Urdu Translated Versions of ‘Salome’: Utilizing the Poly-Systemic Approach

This study employed poly-systemic approach to Urdu-English translated version of the seminal text “Salome” where the researcher tried to find out the peri-textual feature in both the versions while comparing them. Furthermore. This study aimed to explore the cultural influence on language spoken within a specific area. The method used in this study was descriptive-qualitative in nature. It used two different versions of the same text. The textual analysis established, while taking into consideration the initial, preliminary and operational norms (taken from the peritexts), that majority of the translators and translation critics aligned with the target-oriented or acceptability paradigm and reflected as well as conformed to the secondary position of translation in literary (poly) systems of Pakistan. However, the study found that found that people usually share their culture with other while using language. Moreover, the two versions have specific terminologies representing same ideologies or concepts but they have different sentence constructions. The study was significant as it not only offers contextual information about the process of translation provided by the translators and critics but also because of being first of its kind in Pakistan. Keywords: Salome; compative translation analysis; poly-systemic approach; peri-textual feature

Upper bounds for the Euclidean spectral radius of operators via joint norms

Let r e ( S ) be the Euclidean spectral radius associated with a q-tuple S = ( S 1 , … , S q ) of bounded linear operators on a complex Hilbert space. The principal objective of our study is to establish various compelling upper bounds involving r e ( ⋅ ) . In particular, our findings demonstrate that, for all t ∈ [ 0 , 1 ] , we have r e ( S ) ≤ 1 2 max { ω ( S ) , ω ( Δ t ( S ) ) } + 1 2 ‖ S ‖ 1 2 ‖ S ‖ e 1 2 . Here, Δ t ( S ) represents the generalized spherical Aluthge transform of S , while the notations ω ( ⋅ ) , ‖ ⋅ ‖ , and ‖ ⋅ ‖ e pertain to the joint numerical radius, joint operator norm, and Euclidean operator norm, respectively, of operators in Hilbert spaces. Furthermore, we extend the notions of spherical and Duggal transforms and derive multiple upper bounds for r e ( S ) in relation to these transforms. Additionally, there are some applications that are derived as well.

ON THE NORM OF FINITE LENGTH ELEMENTARY OPERATOR IN TENSOR PRODUCT OF C*-ALGEBRAS

Many properties of Elementary operators, including spectrum, numerical ranges, compactness, rank, and norm have been studied in depth and some results have been obtained. However, little has been done in determining the norm of finite length elementary operator in tensor product of C*-algebras. The norm of basic elementary operator in tensor product of C*-algebras have been determined and results obtained. This paper determines the norm of finite length elementary operator in a tensor product of C*-algebras. More precisely, the bounds of the norm of finite length elementary operator in a tensor product of C*-algebras are investigated. The paper employs the techniques of tensor products and finite rank operators to express the norm of an elementary operator in terms of its coefficient operators.

Homogenization of elliptic and parabolic equations with periodic coefficients in a bounded domain under the Neumann condition

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a selfadjoint matrix elliptic second-order differential operator $B_{N,\varepsilon}$, $0<\varepsilon\leqslant1$, with the Neumann boundary condition. The principal part of this operator is given in a factorized form. The operator involves first-order and zero-order terms. The coefficients of $B_{N,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the generalized resolvent $(B_{N,\varepsilon}-\zeta Q_0( {\cdot} /\varepsilon))^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ with two-parametric error estimates (with respect to $\varepsilon$ and $\zeta$). The results are applied to study the behaviour of the solutions of the initial boundary value problem with the Neumann condition for the parabolic equation $Q_0(\mathbf{x}/\varepsilon) \partial_t \mathbf{u}_\varepsilon(\mathbf{x},t) = -(B_{N,\varepsilon} \mathbf{u}_\varepsilon)(\mathbf{x},t)$ in a cylinder $\mathcal{O} \times (0,T)$, where $0<T \le \infty$.

Norm of composition operator on mixed norm spaces

Norm of composition operator on mixed norm spaces

Estimates for the norm of the spherical maximal operator on finite graphs

Estimates for the norm of the spherical maximal operator on finite graphs

The Teaching of Reading to English Majors in China (1950-2020): Historical Development, Current Predicament and Potential Solutions

This paper examines the historical development, current predicament, and potential solutions pertaining to the teaching of reading to English majors in China, with a specific focus on the course commonly known as "Intensive Reading" or "Comprehensive English". It investigates its evolution over a span of more than seventy years. It reveals that the educational objective for English majors has increasingly emphasized the development of language proficiency and humanistic literacy. The positioning of the course, however, has largely remained as a comprehensive language-skills training course, lacking clear objectives and effective operational norms. To better align with current educational goals, this paper proposes the course positioning as an enlightening and inspiring course that promotes language proficiency, enhances humanistic literacy, and establishes connections with disciplinary knowledge. It also sets the objectives as providing perspectives from literature, linguistics and related disciplines, and guidance on learning methods, aiming to guide students in exploring and studying texts to achieve a profound understanding in both language and ideas. Based on an analysis of the challenges associated with the traditional practice, content-based instruction and production-oriented approaches, this paper proposes potential solutions that involve integrating autonomous learning with in-class instruction and incorporating disciplinary knowledge into language learning.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

Norm and numerical radius inequalities for operator matrices

Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices with positive real and imaginary parts will be discussed, and sharper bounds will be shown for such classes.

On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

Cauchy problems are considered for families of, generally speaking, non-Volterra functional differential equations of the second order. For each family considered, in terms of the parameters of this family, necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family are obtained. Such necessary and sufficient conditions are obtained for the following four kinds of families: integral restrictions are imposed on positive and negative functional operators, namely, operator norms are specified; pointwise restrictions are imposed on positive and negative functional operators in the form of values of operators’ actions on the unit function; an integral constraint is imposed on a positive functional operator, a pointwise constraint is imposed on a negative functional operator; a pointwise constraint is imposed on a positive functional operator, an integral constraint is imposed on a negative functional operator. In all cases, effective conditions for the solvability of the Cauchy problem for all equations of the family are obtained, expressed through some inequalities regarding the parameters of the families. The set of parameters of families of equations for which Cauchy problems are uniquely solvable can be easily calculated approximately with any accuracy. The resulting solvability conditions improve the solvability conditions following from the Banach contraction principle. An example of the Cauchy problem for an equation with a coefficient changing sign is given. Taking into account various restrictions for the positive and negative parts of functional operators allows us to significantly improve the known solvability conditions.

Operator norm-based determination of failure probability of nonlinear oscillators with fractional derivative elements subject to imprecise stationary Gaussian loads

An approximate analytical technique is developed for bounding the first-passage probability of lightly damped nonlinear and hysteretic oscillators endowed with fractional derivative elements and subjected to imprecise stationary Gaussian loads. In particular, the statistical linearization and stochastic averaging methodologies are integrated with an operator norm-based approach to formulate a numerically efficient proxy for the first-passage probability. This proxy is employed to determine the realizations of the interval-valued parameters of the excitation model that yield the extrema of the failure probability function. Ultimately, each failure probability bound is determined in a fully decoupled manner by solving a standard optimization problem followed by a single evaluation of the first-passage probability. The proposed approximate technique can be construed as an extension of a recently developed operator norm scheme to account for oscillators with fractional derivative elements. In addition, it can readily treat a wide range of nonlinear and hysteretic behaviors. To illustrate the applicability and effectiveness of the proposed technique, a hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements subject to imprecise stationary Gaussian loads are considered as numerical examples.

The multiple-set split feasibility problem and its application in intensity-modulated radiation therapy

In this paper, we apply a parallel iterative method for solving the split common fixed point problem in Hilbert spaces, without prior knowledge of the transfer operator norm, to the multiple-set split feasibility problem, which plays an important role in medical image reconstruction and signal processing. By formulating a model of the intensity-modulated radiation therapy treatment in a mathematical framework based on the multiple-set split feasibility problem, we also offer an effective approach for solving the intensity-modulated radiation therapy treatment planning.

Dynamic stepsize iteration process for solving split common fixed point problems with applications

In this paper, we study the split common fixed point problem for two nonlinear mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an algorithm which uses dynamic stepsize, it allows to be easily implemented without prior information about operator norm. We further apply our result to solve the split variational inclusion problem, equilibrium problem and convexly constrained linear inverse problem. Moreover, we provide numerical examples to verify efficiency of our algorithm.

A Family of Semi-norms Between the Numerical Radius and the Operator Norm

A Family of Semi-norms Between the Numerical Radius and the Operator Norm

On renormalization and the local gap method for proving frustration-free local spin chains are gapped

Key properties of a physical system depend on whether it is gapped, i.e., whether its spectral gap has a positive lower bound that is independent of system size. Here, we provide a prescription for renormalizing a spin chain Hamiltonian in such a way that the renormalized Hamiltonian is gapped if and only if the original Hamiltonian is gapped. Then, we articulate a set of conditions that guarantees the renormalized Hamiltonian is gapped. These conditions are built on a certain strong notion of decaying correlations involving an operator norm of non-commuting terms in the renormalized Hamiltonian. We apply the method to show that two interesting models, with forms motivated by quantum circuits, are gapped. We also confirm the generality of the method by successfully applying it to a somewhat different case, the well-known Affleck-Kennedy-Lieb-Tasaki (AKLT) model.

Hermitians in matrix algebras with operator norm and associated Lie algebras

This is a continuation of Crabb et al. (2021) [3] and Duncan and McGregor (2022) [4] in which we began an investigation of the real space H of Hermitian matrices in Mn(C) with respect to norms on Cn. Here, we introduce a family of Lie algebras derived from the classical real Lie algebras, with the aim of determining whether or not they are Lie isomorphic to ideals of some H.

Mann-Type Inertial Projection and Contraction Method for Solving Split Pseudomonotone Variational Inequality Problem with Multiple Output Sets

In this paper, we study the concept of split variational inequality problem with multiple output sets when the cost operators are pseudomonotone and non-Lipschitz. We introduce a new Mann-type inertial projection and contraction method with self-adaptive step sizes for approximating the solution of the problem in the framework of Hilbert spaces. Under some mild conditions on the control parameters and without prior knowledge of the operator norms, we prove a strong convergence theorem for the proposed algorithm. We point out that while the cost operators are non-Lipschitz, our proposed method does not require any linesearch method but uses a more efficient self-adaptive step size technique that generates a non-monotonic sequence of step sizes. Finally, we apply our result to study certain classes of optimization problems and we present several numerical experiments to illustrate the applicability of the proposed method. Several of the existing results in the literature could be viewed as special cases of our result in this study.

On Bilevel Monotone Inclusion and Variational Inequality Problems

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted.

Statistical performance of quantile tensor regression with convex regularization

In this paper, we consider high-dimensional quantile tensor regression using a general convex decomposable regularizer and analyze the statistical performances of the estimator. The rates are stated in terms of the intrinsic dimension of the estimation problem, which is, roughly speaking, the dimension of the smallest subspace that contains the true coefficient. Previously, convex regularized tensor regression has been studied with a least squares loss, Gaussian tensorial predictors and Gaussian errors, with rates that depend on the Gaussian width of a convex set. Our results extend the previous work to nonsmooth quantile loss. To deal with the non-Gaussian setting, we use the concept of Rademacher complexity with appropriate concentration inequalities instead of the Gaussian width. For the multi-linear nuclear norm penalty, our Orlicz norm bound for the operator norm of a random matrix may be of independent interest. We validate the theoretical guarantees in numerical experiments. We also demonstrate advantage of quantile regression over mean regression, and compare the performance of convex regularization method and nonconvex decomposition method in solving quantile tensor regression problem in simulation studies.

Self-adaptive CQ-type algorithms for the split feasibility problem involving two bounded linear operators in Hilbert spaces

In this article, we consider and investigate a split convex feasibility problem involving two bounded linear operators in Hilbert spaces. We introduce a self-adaptive CQ-type algorithm by selecting the stepsize which is independent of the operator norms and establish a strong convergence result of the proposed algorithm under some mild control conditions. Moreover, we propose a self-adaptive relaxed CQ-type algo- rithm for solving the problem constrained by sub-level sets of convex functions. A numerical example and an application in compressed sensing are also given to illustrate the convergence behaviour of our proposed algorithms. Our results in this paper improve and generalize some existing results in the literature