Self-adjoint Operators Research Articles

Squares of Symmetric Operators

Squares of Symmetric Operators

Positivity and Entanglement of Polynomial Gaussian Integral Operators

Abstract Positivity preservation is an important issue in the dynamics of open quantum systems: positivity violations always mark the border of validity of the model. We investigate the positivity of self-adjoint polynomial Gaussian integral operators $\widehat{\kappa }_{\operatorname{PG}}$; i.e. the multivariable kernel $\kappa _{\operatorname{PG}}$ is a product of a polynomial $P$ and a Gaussian kernel $\kappa _G$. These operators frequently appear in open quantum systems. We show that $\widehat{\kappa }_{\operatorname{PG}}$ can only be positive if the Gaussian part is positive, which yields a strong and quite easy test for positivity. This has an important corollary for the bipartite entanglement of the density operators $\widehat{\kappa }_{\operatorname{PG}}$: if the Gaussian density operator $\widehat{\kappa }_G$ fails the Peres–Horodecki criterion, then the corresponding polynomial Gaussian density operators $\widehat{\kappa }_{\operatorname{PG}}$ also fail the criterion for all $P$; hence they are all entangled. We prove that polynomial Gaussian operators with polynomials of odd degree cannot be positive semidefinite. We introduce a new preorder $\preceq$ on Gaussian kernels such that if $\kappa _{G_0}\preceq \kappa _{G_1}$ then $\widehat{\kappa }_{\operatorname{PG}_0}\ge 0$, implying that $\widehat{\kappa }_{\operatorname{PG}_1}\ge 0$ for all polynomials $P$. Therefore, deciding the positivity of a polynomial Gaussian operator determines the positivity of a lot of other polynomial Gaussian operators having the same polynomial factor, which might improve any given positivity test by carrying it out on a much larger set of operators. We will show an example that this really can make positivity tests much more sensitive and efficient. This preorder has implications for the entanglement problem, too.

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.

Endpoint eigenfunction bounds for the Hermite operator

We establish the optimal L^{p} , p= 2(d+3)/(d+1) , eigenfunction bound for the Hermite operator \mathcal{H}=-\Delta+|x|^{2} on \mathbb{R}^{d} . Let \Pi_{\lambda} denote the projection operator to the vector space spanned by the eigenfunctions of \mathcal{H} with eigenvalue \lambda . The optimal L^{2} – L^{p} bounds on \Pi_{\lambda} , {2\le p\le \infty} , have been known by the works of Karadzhov and Koch–Tataru except p= 2(d+3)/(d+1) . For d\ge 3 , we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere \sqrt{\lambda} \mathbb{S}^{d-1} .

Essays on the Spectral Properties of a Polynomially Perturbed Hermitian Operator

Essays on the Spectral Properties of a Polynomially Perturbed Hermitian Operator

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

EHPR: Learning evolutionary hierarchy perception representation based on quaternion for temporal knowledge graph completion

Research on temporal knowledge graphs garners attention due to the intricate connection between facts and dynamic temporal factors. However, existing research uses timestamp as auxiliary data for representation learning and directly integrate it into facts, resulting in the inability to capture the intrinsic connections between relations under time evolution. To handle these challenges, we propose the Evolutionary Hierarchy Perception Representation (EHPR), which first leverages the Hamilton product to perform rotational transformations on relation and entity over time, aiming to learn temporal relation and temporal entity with close interactions with time information. Later, EHPR is divided into two modules: (a) Rotating the head entity towards the tail entity using temporal relation through Hamilton product to model complex patterns with quaternion rotation capabilities. (b) Adopting an evolutionary hierarchical factor to capture the differences in modulus distribution between the temporal head entity and the temporal tail entity, aiming to manage the evolutionary hierarchical information between different temporal entities. In this way, EHPR not only utilizes the rich quaternion rotation capabilities to model various relation patterns but also further enables modeling of evolutionary hierarchical patterns through evolutionary hierarchy factors. Experiments show that EHPR achieves remarkable performance on six mature benchmarks compared to state-of-the-art models. Furthermore, we successfully transferred the core idea of EHPR into complex embeddings, showcasing the framework's adaptability. Compared to complex embedding models, EHPR also demonstrates stronger expressive abilities with the Hamilton operator, surpassing the performance of complex Hermitian operator.

Order isomorphisms on unbounded self-adjoint operators

We provide a comprehensive description of all order isomorphisms between several types of unbounded self-adjoint operator sets. Namely, sets of all positive operators, sets of all positive boundedly invertible operators, and those of all self-adjoint operators.

The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time-dependent dissipation term of the form u″+Au+b(t)u′=g(t) in a real Hilbert space H is considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure.

Investigation of the dimension of the spectral projection of a self-adjoint second order quasidifferential operator

Investigation of the dimension of the spectral projection of a self-adjoint second order quasidifferential operator

Enhancing coherence by local parity-time symmetric operation under the non-Markovian environment

Abstract The dynamic behavior of the relative entropy of coherence for a two-level system is systematically investigated within non-Markovian environment. We derive the exact expressions of relative entropy quantifying coherence for an exactly solving model consisting of single qubit interacting with independent structured reservoir. By employing local non-Hermitian operation, i.e. parity-time ( PT ) symmetric operation, we show that the vanished quantum coherence can be effectively restored under the non-Markovian regimes. Furthermore, the freezing phenomenon of the coherence can be detected by using the optimal PT symmetric operation strength within the non-Markovian environments.

Oracle inequalities and upper bounds for kernel conditional U-statistics estimators on manifolds and more general metric spaces associated with operators

ABSTRACT U-statistics represent a fundamental class of statistics that emerge from modelling quantities of interest defined by multi-subject responses. These statistics generalize the empirical mean of a random variable X to summations encompassing all distinct k-tuples of observations drawn from X . A significant advancement was made by Stute [ConditionalU-statistics. Ann. Probab. 19 (1991), pp. 812–825], who introduced conditional U-statistics as a generalization of the Nadaraya–Watson estimates for regression functions. Stute demonstrated their robust pointwise consistency towards the conditional function r ( k ) ( φ , t ~ ) = E ( φ ( Y 1 , … , Y k ) ∣ ( X 1 , … , X k ) = t ~ ) , for t ~ ∈ R pk , where φ is a measurable function. In this investigation, we develop oracle inequalities and upper bounds for kernel-based estimators of conditional U-statistics of general order, applicable across a wide range of metric spaces associated with operators. Our analysis specifically targets doubling measure metric spaces, incorporating a non-negative self-adjoint operator characterized by Gaussian regularity in its heat kernel. Remarkably, our study achieves an optimal convergence rate in certain cases. To derive these results, we explore the regression function within a general framework, introducing several novel insights. These findings are established under sufficiently broad conditions on the underlying distributions. The theoretical results serve as essential tools for advancing the field of general-valued data, with potential applications including the examination of conditional distribution functions, relative-error prediction, the Kendall rank correlation coefficient, and discrimination problems – areas of significant independent interest.

Probability Bracket Notation for Probability Modeling

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman–Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

Trace formula for differential operators with frozen argument

We derive the formula for the regularized trace for rank-one perturbations of self-adjoint operators and then justify its applicability to a wide class of differential operators with frozen argument.

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (lq(·)(Lp(·))). Moreover, it was developed using classical Lebesgue space (Lp) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not.

Intercepting Dendrite Growth With a Heterogeneous Solid Electrolyte for Long-Life All-Solid-State Lithium Metal Batteries.

The application of lithium metal anode in all-solid-state batteries has the potential to achieve both high energy density and safety performance. However, the presence of serious dendrite issues hinders this potential. Here, the ion transport pathways and orientation of dendrite growth are regulated by utilizing the differences of ionic conductivity in heterogeneous electrolytes. The in situ formed Li-Ge alloy phases from the spontaneous reaction between Li10GeP2S12 and the attracted dendrites greatly enhance the ability to resist dendrite growth. As an outcome, the heterogeneous electrolyte achieves a high critical current density of 2.1mA cm-2 and long-term stable symmetrical battery operation (0.3mA cm-2 for 17000h and 1.0mA cm-2 for 2000h). Besides, due to the superior interfacial stability and low interface impedance between the heterogeneous electrolyte and lithium anode, the Li||LiNi0.8Co0.1Mn0.1O2 full battery exhibits great cycling stability (80.5% after 500 cycles at 1.0mA cm-2) and rate performance (125.4 mAh g at 2.0mA cm-2). This work provides a unique strategy of interface regulation via heterogeneous electrolytes design, offering insights into the development of state-of the-art all-solid-state batteries.