Polynomial Operators Research Articles

Complex Eigenvalue Analysis of a Linear Pencil: An Experimental Study

This paper is concerned with a finite-dimensional example of a linear pencil which leads to a class of non-self-adjoint matrices. We consider the linear pencil $H_c-\lambda L$, where $H_c$ is a tri-diagonal matrix with a constant parameter $c$ on the main diagonal and off-diagonal entries equal to one, and $L$ is a diagonal matrix whose elements decrease linearly from one to minus one. In general, the spectra of operator polynomials may contain non-real eigenvalues as well as real eigenvalues. Nevertheless, they exhibit certain patterns. Our aim in this research is to carry out a variety of numerical investigation on the eigenvalues so as to understand the eigenvalue behaviour of such pencils from different points of view. In accordance with our numerical findings, a series of conjectures are offered and various heuristics has been discussed.

On singular pencils with commuting coefficients

We investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the matrix polynomial with commuting coefficients is singular, i.e. its spectrum is the whole complex plane, if and only if ( 0 , 0 , … , 0 ) belongs to the Taylor spectrum of its coefficients. On the other hand, we prove that this equivalence is no longer true if we consider the operators on infinite dimensional Hilbert space as coefficients of polynomial. As a consequence, we could propose a new description of the (Taylor) spectrum of k-tuple of matrices and we could disprove the conjecture previously proposed in the literature. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients.

Energy preserving evolutions over Bosonic systems

The exponential convergence to invariant subspaces of quantum Markov semigroups plays a crucial role in quantum information theory. One such example is in bosonic error correction schemes, where dissipation is used to drive states back to the code-space – an invariant subspace protected against certain types of errors. In this paper, we investigate perturbations of quantum dynamical semigroups that operate on continuous variable (CV) systems and admit an invariant subspace. First, we prove a generation theorem for quantum Markov semigroups on CV systems under the physical assumptions that (i) the generator is in GKSL form with corresponding jump operators defined as polynomials of annihilation and creation operators; and (ii) the (possibly unbounded) generator increases all moments in a controlled manner. Additionally, we show that the level sets of operators with bounded first moments are admissible subspaces of the evolution, providing the foundations for a perturbative analysis. Our results also extend to time-dependent semigroups and multi-mode systems. We apply our general framework to two settings of interest in continuous variable quantum information processing. First, we provide a new scheme for deriving continuity bounds on the energy-constrained capacities of Markovian perturbations of quantum dynamical semigroups. Second, we provide quantitative perturbation bounds for the steady state of the quantum Ornstein-Uhlenbeck semigroup and the invariant subspace of the photon dissipation used in bosonic error correction.

Deciding finiteness of bosonic dynamics with tunable interactions

Abstract In this work we are motivated by factorization of bosonic quantum dynamics and we study the corresponding Lie algebras, which can potentially be infinite dimensional. To characterize such factorization, we identify conditions for these Lie algebras to be finite dimensional. We consider cases where each free Hamiltonian term is itself an element of the generated Lie algebra. In our approach, we develop new tools to systematically divide skew-hermitian bosonic operators into appropriate subspaces, and construct specific sequences of skew-hermitian operators that are used to gauge the dimensionality of the Lie algebras themselves. The significance of our result relies on conditions that constrain only the independently controlled generators in a particular Hamiltonian, thereby providing an effective algorithm for verifying the finiteness of the generated Lie algebra. In addition, our results are tightly connected to mathematical work where the polynomials of creation and annihilation operators are known as the Weyl algebra. Our work paves the way for better understanding factorization of bosonic dynamics relevant to quantum control and quantum technology.

A Subclass of Bi-univalent Functions Defined by aSymmetric q-Derivative Operator and Gegenbauer Polynomials

This paper introduces a novel subclass of bi-univalent analytic functions by utilizing a symmetric q-derivative operator in conjunction with Gegenbauer polynomials. Within this newly defined subclass, we derive bounds for the first two Maclaurin coefficients and address the Fekete-Szeg o problem. By varying the parameters in our results between 0 and 1, we obtain a range of new insights and rediscover some previously established results. This approach not only broadens the scope of bi-univalent function theory but also deepens the understanding of coefficient bounds and extremal problems within this context.

Some New Applications of the Quantum Calculus for New Families of Sigmoid Activation Bi-univalent Functions Connected to Horadam Polynomials

The study of q-calculus is becoming increasingly prominent in the field of geometric function theory, reflecting a growing interest in its applications. In this research work, we first develop a new type of modified Sigmoid-Salagean q-differential operator in the open unit disk D, utilizing the concepts of quantum calculus and the Sigmoid activation function. Using this newly defined q analogous differential operator and Horadam polynomials, we introduce new subclasses of bi-univalent functions in D. We determine upper bounds on initial coefficients, as well as the Fekete-Szeg ̈o problems, for functions belonging to these special families. Additionally, we discuss several interesting consequences related to the findings presented in this study.

A note on fractal dimensions of graphs of certain continuous functions

This article investigates fractal dimensions variation of fractal functions under certain operations, which corroborates the conjecture raised in our previous work Yu and Liang (2024) [1]. It has been shown that when polynomial operations are applied to the independent variable of a function, the fractal dimension of the function’s graph remains unchanged. This finding holds significance for the construction of fractal functions.

Dynamical exploration of kink and lump interaction solutions for the integrable (3+1)-dimensional Ito equation

In this research, we have delved into the investigation of an integrable extension of the Ito equation in a (3+1)-dimensional space with the aim of discovering novel analytical solutions. Our approach involves the utilization of mathematical tools such as Hirota’s bilinear operator and Bell polynomials, to derive the bilinear form of the considered equation. Additionally, we have explored different test functions f in the corresponding bilinear equation, which leads to the emergence of various families of exact solutions accompanied by multiple free parameters. To enhance the understanding of physical implications, the graphical representations of bright solitons and periodic solutions, kink waveforms and interaction solutions, lumps and interaction solutions, and breather solutions are depicted.

Integrating fully homomorphic encryption to enhance the security of blockchain applications

Blockchain has been widely used for secure transactions among untrusted parties, but the current design of blockchain does not provide sufficient privacy and security for the data on the chain, limiting its application in sensitive information scenarios. To address this problem, we propose integrating fully homomorphic encryption (FHE) to enhance the security of blockchain applications, which can extend the application scope of blockchain and improve the privacy and security of blockchain by the features of FHE. Our scheme classifies FHE into those supporting polynomial and non-polynomial operations and introduces the concept of ciphertext computation conversion into Ethereum, enabling conversion between different ciphertext computation types. Moreover, we analyse the security and correctness to explain the feasibility and availability of the scheme. We carry out comparative experiments using different open-source libraries for fully homomorphic encryption and the time performance evaluation of the ciphertext computation conversion under different thread counts. The experiment results demonstrate the efficiency and usability of our scheme.

Silicon-Proven ASIC Design for the Polynomial Operations of Fully Homomorphic Encryption

Silicon-Proven ASIC Design for the Polynomial Operations of Fully Homomorphic Encryption

PENGGUNAAN ALAT PERAGA PAPAN POLINOMIAL UNTUK MENSTIMULASI PEMAHAMAN KONSEP MATEMATIS SISWA

To increase students' understanding of mathematical concepts, manipulative teaching aids are needed to support the continuity of the learning process. This research aims to describe the use of polynomial board mathematics teaching aids, as well as determine the level of effectiveness of the polynomial board created during the learning process. The type of research used is R&D (Research and Development) research. This research method is used for product development through needs research and then development is carried out to produce tested products. Research data uses a qualitative approach through direct peer teaching trials together with media expert lecturers in the Department of Mathematics, State University of Malang. Peer teaching carried out by polynomial boards has a high level of effectiveness based on active responses when learning is carried out using models as well as input and suggestions from media expert lecturers after peer teaching activities are carried out. This teaching aid can provide students with a concrete understanding of the concept of polynomial operations, both solving contextual problems and making it easier for teachers to provide explanations regarding the concept of polynomial operations. Researchers provide suggestions, especially that manipulative media in learning is very important with the aim of deepening understanding of concepts and needing to pay attention to aspects of the relevance of the material to the teaching aids used.

Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators

The Quantum Max Cut (QMC) problem has emerged as a test-problem for designing approximation algorithms for local Hamiltonian problems. In this paper we attack this problem using the algebraic structure of QMC, in particular the relationship between the quantum max cut Hamiltonian and the representation theory of the symmetric group.The first major contribution of this paper is an extension of non-commutative Sum of Squares (ncSoS) optimization techniques to give a new hierarchy of relaxations to Quantum Max Cut. The hierarchy we present is based on optimizations over polynomials in the qubit swap operators. This is in contrast to the "standard" quantum Lasserre Hierarchy, which is based on polynomials expressed in terms of the Pauli matrices. To prove correctness of this hierarchy, we exploit a finite presentation of the algebra generated by the qubit swap operators. This presentation allows for the use of computer algebraic techniques to manipulate and simplify polynomials written in terms of the swap operators, and may be of independent interest. Surprisingly, we find that level-2 of this new hierarchy is numerically exact (up to tolerance 10−7) on all QMC instances with uniform edge weights on graphs with at most 8 vertices.The second major contribution of this paper is a polynomial-time algorithm that computes (in exact arithmetic) the maximum eigenvalue of the QMC Hamiltonian for certain graphs, including graphs that can be "decomposed" as a signed combination of cliques. A special case of the latter are complete bipartite graphs with uniform edge-weights, for which exact solutions are known from the work of Lieb and Mattis \cite{lieb1962ordering}. Our methods, which use representation theory of the symmetric group, can be seen as a generalization of the Lieb-Mattis result.

Research on Multi-Objective Process Parameter Optimization Method in Hard Turning Based on an Improved NSGA-II Algorithm

To address the issue of local optima encountered during the multi-objective optimization process with the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm, this paper introduces an enhanced version of the NSGA-II. This improved NSGA-II incorporates polynomial and simulated binary crossover operators into the genetic algorithm’s crossover phase to refine its performance. For evaluation purposes, the classic ZDT benchmark functions are employed. The findings reveal that the enhanced NSGA-II algorithm achieves higher convergence accuracy and surpasses the performance of the original NSGA-II algorithm. When applied to the machining of the high-hardness material 20MnCrTi, four algorithms were utilized: the improved NSGA-II, the conventional NSGA-II, NSGA-III, and MOEA/D. The experimental outcomes show that the improved NSGA-II algorithm delivers a more optimal combination of process parameters, effectively enhancing the workpiece’s surface roughness and material removal rate. This leads to a significant improvement in the machining quality of the workpiece surface, demonstrating the superiority of the improved algorithm in optimizing machining processes.

Reproducing property of bounded linear operators and kernel regularized least square regressions

We consider bounded linear operators from the view of functional reproducing property. For some bounded linear operators associated with orthogonal polynomials we define an inner product space associated with a kernel constructed with orthogonal polynomials, show that it is a functional reproducing kernel Hilbert space (FRKHS) associated with these bounded linear operators and give decay rate for the best FRKHS approximation with a [Formula: see text]-functional associated with the FRKHS. On this basis, we provide a learning rate for kernel regularized regression whose hypothesis space is the defined FRKHS. As applications, we define some concrete FRKHSs associated with polynomial operators such as the Bernstein–Durrmeyer operators, the de la Vallée Poussin operators on both the unit sphere [Formula: see text] and the unit ball [Formula: see text]. We show that these polynomial operators have reproducing property with respect to the corresponding concrete FRKHSs and show the learning rate for the kernel regularized regression. In short, we provide a way of constructing FRKHS operators with Fourier multipliers and show a learning framework from the view of operator approximation.

Adomian Decomposition Method With Inverse Differential Operator and Orthogonal Polynomials for Nonlinear Models

A proficient Adomian decomposition method is proposed amidst the presence of inverse differential operator and orthogonal polynomials for solving nonlinear differential models. The method is indeed a reformation of the standard Adomian method thereby improving the rapidity of the solution's convergence rate. A generalized recurrent scheme for a general nonlinear model was derived and further utilized to solve certain nonlinear test models. Lastly, numerical results are reported in comparative tables, demonstrating absolute error differences between the exact and approximate solutions with regards to various employed orthogonal polynomials.

Multi-Objective Optimization Technology for Building Energy-Saving Renovation Strategy Based on Genetic Algorithm

Building energy-saving design is significant for the industry to achieve carbon reduction and sustainable development. Firstly, a multi-objective model for energy consumption, cost, and carbon emissions is established based on the three-dimensional perspectives of society, nature, and economy. Then, a polynomial operator is used to improve the non dominated sorting genetic algorithm to calculate the optimal solution set. The low computational efficiency caused by direct coupling of algorithms in traditional optimization processes is expected to be addressed. According to the results, for the Square1 dataset and Iris dataset, the algorithm proposed in this study improved the reverse distance and convergence metrics by more than 70% compared to support vector machine-genetic algorithm and multi-objective clustering algorithm, with values closer to 0. The solution solved by this algorithm had lower building costs, energy consumption, and carbon emissions, with values of 345200 yuan, 2374 KWh/year, and 26 tons, respectively. This validates the effectiveness of the multi-objective model and solving algorithm established in the study, which helps to obtain the optimal energy-saving design scheme and provides reference for low-carbon optimization of building.

Multi-matrix correlators and localization

We study generating functions of 14\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{4} $$\\end{document}-BPS states in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills at finite N by attempting to generalize the Harish-Chandra-Itzykson-Zuber integral to multiple commuting matrices. This allows us to compute the overlaps of two or more generating functions; such calculations arise in the computation of two-point correlators in the free-field limit. We discuss the four-matrix HCIZ integral in the U(2) context and lay out a prescription for finding a more general formula for N > 2. We then discuss its connections with the restricted Schur polynomial operator basis. Our results generalize readily to arbitrary numbers of matrices, opening up the opportunity to study more generic BPS operators.

MemFHE: End-to-end Computing with Fully Homomorphic Encryption in Memory

The increasing amount of data and the growing complexity of problems have resulted in an ever-growing reliance on cloud computing. However, many applications, most notably in healthcare, finance, or defense, demand security and privacy, which today’s solutions cannot fully address. Fully homomorphic encryption (FHE) elevates the bar of today’s solutions by adding confidentiality of data during processing. It allows computation on fully encrypted data without the need for decryption, thus fully preserving privacy. To enable processing encrypted data at usable levels of classic security, e.g., 128-bit, the encryption procedure introduces noticeable data size expansion—the ciphertext is much bigger than the native aggregate of native data types. In this article, we present MemFHE, which is the first accelerator of both client and server for the latest Ring-GSW (Gentry et al. [ 17 ])-based homomorphic encryption schemes using Processing in Memory (PIM). PIM alleviates the data movement issues with large FHE encrypted data while providing in situ execution and extensive parallelism needed for FHE’s polynomial operations. While the client-PIM can homomorphically encrypt and decrypt data, the server-PIM can process homomorphically encrypted data without decryption. MemFHE’s server-PIM is pipelined and is designed to provide flexible bootstrapping, allowing two encryption techniques and various FHE security levels based on the application requirements. We evaluate MemFHE for various security levels and compare it with state-of-the-art CPU implementations for Ring-GSW-based FHE. MemFHE is up to 20 k × (265×) faster than CPU (GPU) for FHE arithmetic operations and provides on average 2,007× higher throughput than [ 36 ] while implementing neural networks with FHE.

Nonintrusive Operator Inference for Chaotic Systems

This work explores the physics-driven machine learning technique Operator Inference (OpInf) for predicting the state of chaotic dynamical systems. OpInf provides a non-intrusive approach to infer approximations of polynomial operators in reduced space without having access to the full order operators appearing in discretized models. Datasets for the physics systems are generated using conventional numerical solvers and then projected to a low-dimensional space via Principal Component Analysis (PCA). In latent space, a least-squares problem is set to fit a quadratic polynomial operator, which is subsequently employed in a time-integration scheme in order to produce extrapolations in the same space. Once solved, the inverse PCA operation is applied to reconstruct the extrapolations in the original space. The quality of the OpInf predictions is assessed via the Normalized Root Mean Squared Error (NRMSE) metric from which the Valid Prediction Time (VPT) is computed. Numerical experiments considering the chaotic systems Lorenz 96 and the Kuramoto-Sivashinsky equation show promising forecasting capabilities of the OpInf reduced order models with VPT ranges that outperform state-of-the-art machine learning (ML) methods such as backpropagation and reservoir computing recurrent neural networks [1], as well as Markov neural operators [2].

The SUSY partners of the QES sextic potential revisited

In this paper, the SUSY partner Hamiltonians of the quasi-exactly solvable (QES) sextic potential Vqes(x)=νx6+2νμx4+μ2−(4N+3)νx2 , N∈Z+ , are revisited from a Lie algebraic perspective. It is demonstrated that, in the variable z = x 2, the underlying sl2(R) hidden algebra of V qes(x) is inherited by its SUSY partner potential V 1(x) only for N = 0. At fixed N > 0, the algebraic polynomial operator h(x, ∂ x ; N) that governs the N exact eigenpolynomial solutions of V 1 is derived explicitly. These odd-parity solutions appear in the form of zero modes. The potential V 1 can be represented as the sum of a polynomial and rational parts. In particular, it is shown that the polynomial component is given by V qes with a different non-integer (cohomology) parameter N1=N−32 . A confluent second-order SUSY transformation is also implemented for a modified QES sextic potential possessing the energy reflection symmetry. By taking N as a continuous real constant and using the Lagrange-mesh method, highly accurate values (∼20 s. d.) of the energy E n = E n (N) in the interval N ∈ [ − 1, 3] are calculated for the three lowest states n = 0, 1, 2 of the system. The critical value N c above which tunneling effects (instanton-like terms) can occur is obtained as well. At N = 0, the non-algebraic sector of the spectrum of V qes is described by means of compact physically relevant trial functions. These solutions allow us to determine the effects in accuracy when the first-order SUSY approach is applied on the level of approximate eigenfunctions.