Exponential Vector Research Articles (Page 1)

Comparability of Metrics and Norms in terms of Basis of Exponential Vector Space

We construct three representations of the [Formula: see text] current algebra: in extended Fock space, with Gamma random measures, and with negative binomial (Pascal) point processes. For the second and third representations, the lowering and neutral operators are generators of measure-valued branching processes (Dawson–Watanabe superprocesses) and spatial birth–death processes. The vacuum is the constant function [Formula: see text] and iterated application of raising operators yields Laguerre and Meixner polynomials. In addition, we prove a Baker–Campbell–Hausdorff formula and give an explicit formula for the action of unitaries [Formula: see text] on exponential vectors. We explain how the representations fit in with a general scheme proposed by Araki and with representations of the [Formula: see text] current group with Vershik, Gelfand and Graev’s multiplicative measure.

Quasi-exact l-states of Klein–Gordon equation with position-dependent effective mass for exponential scalar and vector fields

Abstract In this study, an efficient calculation method of matrix exponential function and vector product is proposed to reduce the calculation error. This method is based on the flexible exponential integral scheme; using B-series theory and two-color tree theory, the numerical series expression is obtained, and the exponential integral of the matrix exponential function is solved. By constructing a specific numerical integration scheme and analyzing its convergence under the theoretical framework of analytic semigroups, the convergence analysis of numerical schemes is studied and perfected. The numerical results show that the proposed method of matrix exponential function and vector product has a small error, which shows obvious advantages in reducing the calculation error.

Linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries equations

This study emphasizes the dual exposition of the impact of convective condition and the significant effects of magnetic field, heat source and velocity slip on hybrid nanofluid flow over the exponentially shrinking sheet. The hybrid nanofluid is regarded as a contemporary variety of nanofluid; consequently, it is utilized to enhance the efficiency of heat transfer. By applying the Tiwari–Das model, the key objective of the current research is to investigate the impact of involving factors Biot number, Eckert number, volume fraction, magnetohydrodynamic (MHD), slip and suction on the temperature and velocity profiles. In addition, the Nusselt number and skin friction variations have been explored against the suction effect on the solid volume fraction of copper [Formula: see text] [1–3%] and the Biot number [Formula: see text] [1–3%]. The nonlinear partial differential equations are converted to a set of an ordinary differential equations by incorporating exponential similarity vectors. Eventually, ordinary differential equations are rectified utilizing MATLAB bvp4c solver. The results demonstrate the presence of dual exposition for suction with varying values of copper volume fraction and the Biot number. The heat transmission rate is observed to escalate in both cases as the strength of the Biot and Eckert numbers intensifies in the range of 1–3%. In summary, the results show that duality and non-unique solutions occur in the aiding flow situation when the suction [Formula: see text], while there is no flow and unique solutions of hybrid nanofluid feasible when [Formula: see text].

In this paper we have found a necessary and sufficient condition for equivalence of two norms on a linear space using the theory of exponential vector space. Exponential vector space (‘evs’ in short) is an ordered algebraic structure which can be considered as an algebraic ordered extension of vector space. This structure is axiomatised on the basis of the intrinsic properties of the hyperspace C(X) comprising all nonempty compact subsets of a Hausdorff topological vector space X. Exponential vector space is a conglomeration of a semigroup structure, a scalar multiplication and a compatible partial order. We have shown that the collection of all norms defined on a linear space, together with the constant function zero, forms a topological exponential vector space. Then using the concept of comparing function (a concept defined on a topological exponential vector space) we have proved the aforesaid necessary and sufficient condition; also we have proved using comparing function that in an infinite dimensional linear space there are uncountably many non-equivalent norms.

Abstract The human brain, the primary constituent of the nervous system, exhibits distinctive complexities that present considerable difficulties for healthcare practitioners, specifically in categorizing brain tumours. Magnetic resonance imaging is a widely favoured imaging modality for detecting brain tumours due to its extensive range of image characteristics and utilization of non‐ionizing radiation. The primary objective of the current investigation is to differentiate between three distinct classifications of brain tumours by introducing a novel methodology. The utilization of a combined feature extraction technique that integrates novel global grey level co‐occurrence matrix and local binary patterns is employed, thereby offering a comprehensive representation of the structural and textural information contained within the images. Principal component analysis is used to improve the model's efficiency for effective feature selection and dimensionality reduction. This study presents a novel framework incorporating four separate kernel functions, Minkowski–Gaussian, exponential support vector machine (SVM), histogram intersection SVM, and wavelet kernel, into a SVM classifier. The ensemble kernel employed in this study is specifically designed to classify glioma, meningioma, and pituitary tumours. Its implementation enhances the model's robustness and adaptability, surpassing the performance of conventional single‐kernel SVM approaches. This study substantially contributes to medical image classification by utilizing innovative kernel functions and advanced machine‐learning techniques. The findings demonstrate the potential for enhanced diagnostic accuracy in brain tumour cases. The presented approach shows promise in effectively addressing the intricate challenges associated with classifying brain tumours.

Exponential algebra is a new algebraic structure consisting of a semigroup structure, a scalar multiplication, an internal multiplication and a partial order [introduced in [4]]. This structure is based on the structure `exponential vector space' which is thoroughly developed by Priti Sharma et. al. in [11] [This structure was actually proposed by S. Ganguly et. al. in [1] with the name `quasi-vector space'] Exponential algebra can be considered as an algebraic ordered extension of the concept of algebra. In the present paper we have shown that the function space $ C^+(\mathbf X) $ of all non-negative continuous functions on a topological space $\mathbf X$ is a topological exponential algebra under the compact open topology. Also we have discussed the ideals and maximal ideals of $ C^+(\mathbf X) $. We find an ideal of $ C^+(\mathbf X)$ which is not a maximal ideal in general; actually maximality of that ideal depends on the topology of $\mathbf{X}$. The concept of ideals of exponential algebra was introduced by us in [4].

Computing high dimensional multiple integrals involving matrix exponentials

High-accuracy level underwater acoustical surveying plays an important role in ocean engineering applications, such as subaqueous tunnel construction, oil and gas exploration, and resources prospecting. This novel imaging method is eager to break through the existing theory to achieve a higher accuracy level of surveying. Multibeam Synthetic Aperture Sonar (MBSAS) is a kind of underwater acoustical imaging theory that can achieve 3D high-resolution detecting and overcome the disadvantages of traditional imaging methods, such as Multibeam Echo Sounder (MBES) and Synthetic Aperture Sonar (SAS). However, the resolution in the across-track direction inevitably decreases with increasing range, limited by the beamwidth of the transducer array of MBES. Furthermore, the sidelobe problem is also a significant interference of imaging sonar that introduces image noise and false peaks, which reduces the accuracy of the underwater images. Therefore, we proposed an accelerated deconvolved MBSAS beamforming method that introduces exponential acceleration and vector extrapolation to improve the convergence velocity of the classical Richardson-Lucy (R-L) iteration. The method proposed achieves a narrow beamwidth with a high sidelobe ratio in a few iterations. It can be applied to actual engineering applications, which breaks through the limitation of the actual transducer array scale. Simulations, tank, and field experiments also demonstrate the feasibility and advantages of the method proposed. 3D high-accuracy level underwater acoustical surveying can be achieved through this 2D MBES transducer array system, which can be widely promoted in the field of underwater acoustical remote sensing.

As one of the most important properties, energy preservation is a natural requirement for numerical integrators of Hamiltonian systems. Considering the limited second-order accuracy of the existing exponential average vector filed (or discrete gradient) method, this paper is devoted to developing high-order energy-preserving exponential integrators for general semilinear Hamiltonian systems. To this end, we first formulate and analyze the continuous-stage exponential Runge--Kutta (ERK) method. After deriving the order conditions, energy-preserving conditions, and symmetric conditions for the continuous-stage ERK method, we construct a novel class of fourth-order symmetric energy-preserving continuous-stage ERK methods with a free parameter based on the established conditions and present a convergence theorem for the fixed-point iteration associated with the continuous-stage ERK method. Furthermore, we extend the proposed continuous-stage ERK method to noncanonical Hamiltonian systems and discuss implementation issues in detail. Finally, numerical results from the FPU problem, the charged-particle dynamics, and the Klein--Gordon equation demonstrate the high efficiency, good energy-preserving behavior, and applicability of large time stepsizes of the proposed fourth-order energy-preserving continuous-stage ERK method.

We consider singular rank one perturbations of the semigroup of shifts on the half-axis changing the domain of its generator. Only one example using the projection on the exponential vector is considered.

Charged-particle dynamics in a strong constant magnetic field can yield a fast gyromotion with high frequency around the center. Considering the superior of exponential integrators for highly oscillatory problems and the benefit of energy preservation of numerical integrators in solving the charged-particle dynamics, this paper is devoted to developing a fourth-order energy-preserving exponential integrator for the charged-particle dynamics in a strong constant magnetic field. To this end, we first rewrite the problem in the form of a semilinear Poisson system, to which the exponential average vector field (EAVF) method can be applied with energy preservation. Then, by deriving the truncated modified differential equation of the EAVF method, we propose a fourth-order energy-preserving exponential integrator according to the modifying integrator theory. Finally, numerical results soundly support the good energy preservation and high efficiency of the proposed fourth-order integrator in solving the problem considered in this paper.

A fast null-space method for the unsteady Stokes equations

Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

We point out a misleading treatment in the literature regarding to bound-state solutions for the $s$-wave Klein-Gordon equation with exponential scalar and vector potentials. Following the appropriate procedure for an arbitrary mixing of scalar and vector couplings, we generalize earlier works and present the correct solution to bound states and additionally we address the issue of scattering states. Moreover, we present a new effect related to the polarization of the charge density in the presence of weak short-range exponential scalar and vector potentials.

The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature.

Global fixed-time attitude tracking control for the rigid spacecraft with actuator saturation and faults

On the existence problem of solutions to a class of fuzzy mixed exponential vector variational inequalities