Self-adjoint Operators Research Articles

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

Remark on square roots of self-adjoint weighted composition operators on H2

In this paper, we study square roots of self-adjoint weighted composition operators on H2. More precisely, we focus on square roots Wf,φ of a self-adjoint operator Wg,ψ=Wf,φ2 when φ is a linear fractional selfmap of D. We also investigate several properties of such Wf,φ. Finally, we show that the square roots Wf,φ may be other, nonself-adjoint weighted composition operators and have nontrivial invariant subspaces.

Submanifolds with constant scalar curvature in space forms

Let [Formula: see text] be an [Formula: see text]-dimensional oriented compact submanifold with constant normalized scalar curvature [Formula: see text] in the space form [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the mean curvature and the Ricci curvature of [Formula: see text] respectively. By applying Cheng-Yau’s self-adjoint operator, we first prove that if [Formula: see text] is a hypersurface in a unit sphere, and [Formula: see text], then [Formula: see text] is totally umbilical. Furthermore, we investigate the submanifolds in [Formula: see text] with flat normal bundle satisfying [Formula: see text], and obtain a complete classification theorem.

Research on Asymmetrical Operation of Multilevel Converter-Type Solid-State Transformers Based on High-Frequency Link Interconnection

The large size of the sub-module (SM) capacitor is a typical problem in traditional modular multilevel converter-type solid-state transformers (MMC-SSTs). The MMC-SST based on high-frequency link interconnection is an effective solution for achieving lightweight capacitance. This structure can help to eliminate the symmetric SM fluctuating power, thereby reducing the SM capacitance. In a three-phase interconnected MMC-SST with low capacitance, potential risks may arise during transient processes, especially in cases of three-phase voltage asymmetry, such as large fluctuations in the SM voltage and unstable DC bus voltage. Aiming to solve this problem, this article re-analyzes the internal power characteristics of the MMC-SST under asymmetric operation and re-derives the SM capacitance constraint suitable for different degrees of three-phase voltage asymmetry. The new SM capacitance constraint enhances the asymmetric voltage ride-through capability of the MMC-SST. The new capacitance constraint is higher than that in symmetric operation, but it still has significant advantages in capacitance compared with the traditional MMC-SST.

First‐order asymptotic perturbation theory for extensions of symmetric operators

AbstractThis work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.

New Results on Differential Subordination and Superordination for Multivalent Functions Involving New Symmetric Operator

This article aims to significantly advance geometric function theory by providing a valuable contribution to analytic and multivalent functions. It focuses on differential subordination and superordination, which characterize the interactions between analytic functions. To achieve our goal, we employ a method that relies on the characteristics of differential subordination and superordination. As one of the latest advancements in this field, this technique is able to derive several results about differential subordination and superordination for multivalent functions defined by the new operator within the open unit disk . Additionally, by employing the technique, the differential sandwich outcome is achieved. Therefore, this work presents crucial exceptional instances that follow the results. The findings of this paper can be applied to a wide range of mathematical and engineering problems, including system identification, orthogonal polynomials, fluid dynamics, signal processing, antenna technology, and approximation theory. Furthermore, this work significantly advances the knowledge and understanding of the analytical functions of the unit and its interactive higher relations. The characteristics and consequences of differential subordination theory are symmetric to those of differential superordination theory. By combining them, sandwich-type theorems can be derived.

Determination of reduced density matrices in the doubly occupied configuration interaction space: A Hellmann-Feynman theorem approach.

In this work, the Hellmann-Feynman theorem is extended within the doubly occupied configuration interaction space to enable practical calculations of reduced density matrices and expected values. This approach is straightforward, employing finite energy differences, yet remains reliable and accurate even with approximate energies from successive approximation methods. The method's validity is rigorously tested against the Richardson-Gaudin-Kitaev and reduced Bardeen-Cooper-Schrieffer models using approximate excitation energies procured from the Hermitian operator method within the same space, effectively proving the approach's reliability with median error rates for reduced density matrix calculations around 0.1%. These results highlight the procedure's potential as a practical tool for computing reduced density matrices and expected values, particularly valuable as an ad hoc method in scenarios where only system energies are easily available.

Interval-valued picture fuzzy decision-making framework with partitioned maclaurin symmetric mean aggregation operators

Interval-valued picture fuzzy (IVPF) set is an extension of the picture fuzzy set theory used to represent uncertainty and vagueness in the processes of decision-making. This study focuses on exploring the interrelationships among multiple IVPFSs and criteria partitions. We investigate the IVPF partitioned Maclaurin symmetric mean operator and the weighted IVPF partitioned Maclaurin symmetric mean operator and discuss their respective properties. Subsequently, we identify certain special cases of these operators based on IVPF sets. Furthermore, we deploy a multi-criteria decision-making procedure utilizing the suggested IVPF partition operators. Through a numerical example, we demonstrate the practicality and validity of the presented approach. Finally, a thorough comparison with existing approaches is conducted to elucidate the superiority of the proposed method.

Classical echoes of quantum boundary conditions

Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.

Some new results on the property (R) under compact perturbations

The property (R) for a bounded linear operator T on a Hilbert space means that the points λ in the approximate point spectrum of T for which T − λI is upper semi-Browder are exactly the isolated points of the spectrum of T which are eigenvalues of finite multiplicity. For an operator T, if T + K satisfies the property (R) for all compact operators K, then T is said to have the stability of the property (R). In this paper, we give new criteria for the fulfillment of the stability of the property (R) for an operator. Moreover, the property (R) for complex symmetric operators is discussed.

Secure and Efficient Authentication using Linkage for permissionless Bitcoin network

The cryptocurrency’s permissionless and large-scale broadcasting requirements prohibit the traditional authentication implementation on the blockchain’s underlying peer-to-peer (P2P) networking. Thus, blockchain networking implementations remain vulnerable to networking integrity threats such as spoofing or hijacking. We design Secure and Efficient Authentication using Linkage (SEAL) to build connection security for permissionless Bitcoin networking. SEAL uses the linkage between the packets for a symmetric operation, in contrast to the traditional authentication approach relying on identity-credential-based trust. To make it appropriate for cryptocurrency networking, SEAL utilizes the packet header, protects the end-to-end connection, and separates the online process and the offline process so that the real-time overhead is minimal for greater efficiency and practicality. We implement SEAL on a functioning Bitcoin node and demonstrate that SEAL operates efficiently with minimal overhead. Specifically, it reduces the hash rate by only 1.3% compared to an unsecured node. Additionally, we use a network simulator to emulate the Bitcoin Mainnet and analyze SEAL’s impact on block propagation delay. SEAL yields 2.04 times smaller delay and 1.25 times smaller delay in block propagation than HMAC and ChaCha20-Poly1305, respectively. The key advantage of SEAL is that it requires fewer hash computations and simpler mixing operations, resulting in significantly lower computational overhead compared to traditional authentication schemes based on message authentication codes (MACs).

Effect of parity–time symmetry operation on entanglement distribution in two parallel Heisenberg spin chains

Abstract The effect of the parity–time ( PT ) symmetry operation on the entanglement transfer in two parallel Heisenberg spin chains is investigated. We give two schemes, i.e. pre- and post- PT symmetric operations, to demonstrate the performance of the PT symmetric operation on the entanglement dynamics of two qubits. In our investigation, we find that the effect of pre- PT symmetric operation schemes on the protection of entanglement of two qubits in the case of double spin chains is superior to that of the single spin chain case. Conversely, the effect of the post- PT symmetric operation scheme on the protection of entanglement of two qubits in the single spin chain case is better than that in the double spin chain case.

New subclass of meromorphic harmonic functions defined by symmetric q-calculus and domain of Janowski functions

In this article, by applying the convolution principle and symmetric q-calculus, we develop a new generalized symmetric q-difference operator of convolution type, which is applicable in the domain E⁎={τ:τ∈C and 0<|τ|<∞}. Utilizing this operator, we construct, analyze, and evaluate two new sets of meromorphically harmonic functions in the Janowski domain. Furthermore, we investigate the convolution properties and necessary conditions for a function F to belong to the class MSHP,R(q,q−1), examining the sufficiency conditions for F to satisfy these properties. Moreover, we examine key geometric properties of the function F in the class MSH‾P,R(q,q−1), including the distortion bound, convex combinations, the extreme point theorem, and weighted mean estimates.

A stable difference scheme for the solution of a source identification problem for telegraph-parabolic equations

In the present paper, we construct a first order of accuracy difference scheme for the approximate solution of the inverse problem for telegraph-parabolic equations with an unknown spacewise dependent source term. The unique solvability of constructed difference scheme and the stability estimates for its solution were obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

Squares of Symmetric Operators

Squares of Symmetric Operators

A multi attribute decision making framework based on partitioned dual Maclaurin symmetric mean operators under Fermatean fuzzy environment

Abstract In information aggregation, the Maclaurin symmetric mean (MSM) operator has drawn a lot of interest to the researchers. And, partitioned dual MSM (PDMSM) has a precondition that all attributes are grouped into several partitions and the attributes in the same partition are relevant to other attributes in the same group, while the attributes located in different groups have no relation. The Fermatean fuzzy set (FFS), on the other hand, is a potent mathematical model that effectively manages uncertain data. The existing FFS-based multi attribute decision making (MADM) techniques fail to evaluate the partitions of the relative attributes, the interdependencies between various criteria, and the ability to mitigate the detrimental impacts of irrelevant criteria. Motivated by these issues, this paper proposes novel operators named FFPDMSM and weighted FFPDMSM to handle the scenarios where criteria are divided into distinct parts and there are interconnections among multiple criteria within the same part. The proposed operators deal not only with interrelationships between criteria but also with partitioned relationships among criteria. Some properties of the proposed operators are discussed in detail. Further, an MADM approach is developed based on the proposed operators in the FF environment. A realistic numerical illustration with sensitivity analysis is demonstrated to validate the proposed approach. Finally, the method is compared with different existing techniques to demonstrate the proposed method’s applicability and feasibility.

Investigation of Solid Polymer Electrolytes for NASICON-Type Solid-State Symmetric Sodium-Ion Battery.

The inevitable shift toward renewable energy and electrification necessitates earth-abundant sodium reserves for next-generation Na-based energy storage technologies. By coupling the benefits of solid electrolytes over traditional nonaqueous electrolytes due to their safety hazards, solid-state sodium-ion batteries hold huge prospects in the future. This work presents a comprehensively developed solid-state sodium-ion symmetric full cell operating at room temperature enabled through a poly(vinylidene fluoride-co-hexafluoropropylene) (PVDF-HFP)-based polymer electrolyte and modified NASICON-structured positive and negative electrodes. Among the investigated polymer electrolytes, PVDF-HFP-NaTFSI was found to outperform other counterparts by achieving a higher ionic conductivity and delivered an appreciable electrochemical stability window. By further delving into the properties of PVDF-HFP-NaTFSI, it was found to possess the least crystallinity, minimal porous structure, lowest melting point, and fusion enthalpy, indicating better ion transport than other investigated polymer electrolytes. The as-assembled solid-state battery revealed a storage capacity of 74 mAh g-1 at 0.1 C with a specific energy density of 130 Wh kgcathode_active_material-1 and demonstrated an impressive capacity retention of 84% of the initial capacity after 200 cycles. The structure and morphology retention of the cycled electrode and electrolyte through postmortem analysis bolster the electrochemo-mechanical stability of the developed solid cell. The findings reported here on polymer electrolytes persuade expedient solutions for developing ambient temperature solid-state sodium-ion batteries with promising electrochemical performance for commercialization in the near future.

Assessment of learning management systems based on Schweizer–Sklar picture fuzzy Maclaurin symmetric mean aggregation operators

Assessment of learning management systems based on Schweizer–Sklar picture fuzzy Maclaurin symmetric mean aggregation operators

Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

Non-defective degeneracy in non-Hermitian bipartite system

Starting from a Hermitian operator with two distinct eigenvalues, we construct a non-Hermitian bipartite system in Gaussian orthogonal ensemble according to random matrix theory, where we introduce the off-diagonal fluctuations through random eigenkets and realizing the bipartite configuration consisting of two D × D subsystems (with D being the Hilbert space dimension). As required by the global thermalization (chaos), one of the two subsystems is fully ranked, while the other is rank deficient. For the latter (rank-deficient) subsystem, there is a block with non-defective degeneracies that contains non-local symmetries, as well as the accumulation effect of the linear map in adjacent eigenvectors. The maximally mixed state formed by the eigenvectors of this special region does not exhibit thermal ensemble behavior (neither canonical or Gibbs), and displays similar characteristics to the corresponding reduced density, which can be verified through the Loschmidt echo and variance of the imaginary spectrum. This non-defective degeneracy region partially meets the Lemma in 10.1103/PhysRevLett.122.220603 and theorem in 10.1103/PhysRevLett.120.150603. The coexistence of strong entanglement and initial state fidelity in this region make it possible to achieve a maximally mixed density, which, however, does not correspond to a thermal canonical ensemble (with complete insensitivity to the environmental energy or temperature). Outside this region, the collection of eigenstates (reduced density) always exhibit restriction on the corresponding Hilbert space dimension (with, e.g., infinite number of bound states), and thus suppress the global thermalization. There are abundant physics for those densities in Hermitian and non-Hermitian bases, which we investigate separately in this work. For example, the disentangling effect originates from non-Hermitian skin effect where the coherence exists along the direction orthogonal to the entangled boundaries of the Loschmidt echo spectrum in the Hermitian basis, while it originates from the many-body localization with the coherence among echo boundaries in the non-Hermitian basis which is disorder-free.