Linear Map Research Articles

New PID parameter tuning based on improved dung beetle optimization algorithm

AbstractIn this paper, a proportional‐integral‐derivative (PID) controller parameter optimization method based on the improved dung beetle optimization (IDBO) algorithm is proposed, which improves the balance between the global exploration and local exploitation capabilities of the dung beetle optimization (DBO) and significantly enhances the convergence speed and optimization accuracy. Initially, the dung beetle population is initialized using piecewise linear chaotic map (PWLCM) chaotic mapping in order to increase its variety and the DBO algorithm's capacity for global exploration. Furthermore, adaptive weighting in the DBO algorithm is now balanced between the capabilities of local exploitation and global exploration with the addition of adaptive weights. After that, in order to improve the DBO algorithm's capacity for local exploitation, a triangle wandering strategy is included during the dung beetle reproductive phase. Finally, using both Lévy flying wandering and greedy strategy (GS) together make it easier to take advantage of opportunities in both local and global areas. The outcomes of the traditional benchmark function test demonstrate a significant improvement in both convergence speed and optimization accuracy when the particle swarm optimization (PSO), DBO, grey wolf optimization (GWO), and sparrow search algorithm (SSA) algorithms are compared. The performance index function incorporates an overshooting penalty term to prevent the overshooting phenomenon in the control system. Simulation experiments are carried out for the DC motor control system, and the time domain performance, frequency domain performance, and robustness performance of the closed‐loop control system with ZN‐PID, Lambda‐PID, PSO‐PID, and IDBO‐PID rectified PID controller parameters are comparatively analyzed, which verifies the validity and practicability of the IDBO algorithm.

FusPB-ESM2: Fusion model of ProtBERT and ESM-2 for cell-penetrating peptide prediction

Cell-penetrating peptides have attracted much attention for their ability to break through cell membrane barriers, which can improve drug bioavailability, reduce side effects, and promote the development of gene therapy. Traditional wet-lab prediction methods are time-consuming and costly, and computational methods provide a short-time and low-cost alternative. Still, the accuracy and reliability need to be further improved. To solve this problem, this study proposes a feature fusion-based prediction model, where the protein pre-trained language models ProtBERT and ESM-2 are used as feature extractors, and the extracted features from both are fused to obtain a more comprehensive and effective feature representation, which is then predicted by linear mapping. Validated by many experiments on public datasets, the method has an AUC value as high as 0.983 and shows high accuracy and reliability in cell-penetrating peptide prediction.

PRGNN: Modeling high-order proximity with relational graph neural network for knowledge graph completion

Relational Graph Neural Networks (RGNNs) are designed to extract structural information from relational graphs and have garnered attention in the domain of Knowledge Graph Completion (KGC). However, recent empirical investigations have indicated that some prominent RGNN-based methodologies have not significantly enhanced precision, prompting questions regarding the efficacy of RGNNs in KGC applications.In this paper, we introduce a novel RGNN-based KGC approach, the Proximity Relational Graph Neural Network (PRGNN), which excels at modeling high-order proximities among entities. PRGNN is founded on a notably straightforward yet effective RGNN framework that discards unnecessary components commonly incorporated in previous approaches, such as attention layers, and linear and non-linear mappings. We demonstrate that PRGNN empowers traditional KGC techniques to apprehend high-order proximities among entities more effectively. Through extensive experimentation on benchmark datasets, we establish that PRGNN consistently outperforms conventional KGC methods and achieves state-of-the-art results. Furthermore, we show that PRGNN necessitates considerably less training time (ranging from one-third to one-fifth) and fewer parameters (ranging from half to two-thirds), rendering it an exceptionally efficient approach. All data and code have been made available at 11https://github.com/zhudanhao/PRGNN..

Cryptographic algorithm for multi-path distribution of entangled states of orbital angular momentum based on Fibonacci values

Drawing inspiration from the Fibonacci sequence and its complementary Lucas sequence, this paper introduces an innovative encryption and decryption algorithm tailored for multi-path quantum key distribution. The algorithm capitalizes on the high-quality orbital angular momentum entangled states, harnessing the mathematical elegance of Fibonacci numbers to construct block diagonal matrices. These matrices serve as the foundation for the simultaneous execution of key distribution across multiple communication paths in a structured block distribution format. The encryption process is facilitated through a combination of linear mappings, employing specific transition matrices to manage the cryptographic flow. The security underpinning of this method is firmly rooted in the Heisenberg Uncertainty Principle, a fundamental tenet of quantum mechanics, which ensures the confidentiality and integrity of the quantum communication channel. This approach paves the way for a novel encryption paradigm, fortifying the security framework of quantum communication networks.

A finite element method for stochastic diffusion equations using fluctuating hydrodynamics

We present a finite element approach for diffusion problems with thermal fluctuations based on a fluctuating hydrodynamics model. The governing equations are stochastic partial differential equations with a fluctuating forcing term. We propose a discrete formulation of the fluctuating forcing term that has the correct covariance matrix up to a standard discretization error. Furthermore, we derive a linear mapping to transform the finite element solution into an equivalent discrete solution that is free of the artificial correlations introduced by the spatial discretization. The method is validated by applying it to two diffusion problems: a second-order diffusion equation and a fourth-order diffusion equation. The theoretical (continuum) solution to the first case presents spatially decorrelated fluctuations, while the second case presents fluctuations correlated over a finite length. In both cases, the numerical solution presents a structure factor that approximates well the continuum one.

Identification of linear flat outputs using neural networks—Examples of two-degree-of-freedom underactuated mechanical systems

This paper proposes a neural networks-based approach of finding flat output of linearized underactuated mechanical systems (UMS). Given that differential flatness and controllability are equivalent for linear systems, the problem is equivalent to finding the Brunovsky canonical form of linearized UMSs. We use a two degree-of-freedom (2DOF) system to illustrate the theoretical development. The proposed method identifies the local flat output of nonlinear mechanical systems from the measurements only, without a detailed mathematical model. The identification allows us to combine the well-known active disturbance rejection control (ADRC) and differential flatness control. A time-domain direct identification (TDDI) algorithm and its variant based on the algebraic method are proposed for flat output identification (FOID). The neural networks for implementing the TDDI algorithm called FOID-NN are created to evaluate flat output candidates and to use the flat output to reconstruct the system states in terms of a linear mapping. The neural networks are trained with the loss functions defined by reconstruction errors. Two special layers, namely tracking differentiator (TD) and algebraic layer, are inserted in the FOID-NN to handle noisy signal differentiations. Simulations of a cart–pole system and a stable 2DOF nonlinear UMS are carried out to show the range of applications of the FOID-NN. The experimental data of an underactuated rotary crane are used to validate the identified flat output.

Ergodic quantum processes on finite von Neumann algebras

Let (M,τ) be a tracial von Neumann algebra with a separable predual and let (Ω,P) be a probability space. A bounded positive random linear operator on L1(M,τ) is a map γ:Ω×L1(M,τ)→L1(M,τ) so that τ(γω(x)a) is measurable for all x∈L1(M,τ) and a∈M, and x↦γω(x) is bounded, positive, and linear almost surely. Given an ergodic T∈Aut(Ω,P), we study quantum processes of the form γTnω∘γTn−1ω∘⋯∘γTmω for m,n∈Z. Using the Hennion metric introduced in [21], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when γω is the predual of a normal positive linear map on M. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [21].

Characterizations of derivations on incidence algebras by local actions

Let ( X , ≤ ) be a locally finite pre-ordered set and R be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra I ( X , R ) which is derivable at zero is a generalized derivation and every local derivation on I ( X , R ) is a derivation.

Generalizability of foot placement control strategies during unperturbed and perturbed gait.

Control of foot placement is an essential strategy for maintaining balance during walking. During unperturbed, steady-state walking, foot placement can be accurately described as a linear function of the body's centre of mass (CoM) state at midstance. However, it is uncertain if this mapping from CoM state to foot placement generalizes to larger perturbations that could potentially cause falls. Recovery from these perturbations may require reactive control strategies not observed during unperturbed walking. Here, we used unpredictable changes in treadmill belt speed to assess the generalizability of foot placement mappings identified during unperturbed walking. We found that foot placement mappings generalized poorly from unperturbed to perturbed walking and differed for forward perturbation versus backward perturbation. We also used the singular value decomposition of the mapping matrix to reveal that people were more sensitive to backward versus forward perturbations. Together, these results indicate that a single linear mapping cannot describe the foot placement control during both forward and backward losses of balance induced by treadmill belt speed perturbations. Better characterization of human balance control strategies could improve our understanding of why different neuromotor disorders result in heightened fall risk and inform the design of controllers for balance-assisting devices.

Bifurcation structures of a two-dimensional piecewise linear discontinuous map: analysis of a cobweb model with regime-switching expectations

We consider the bifurcations occurring in a two-dimensional piecewise-linear discontinuous map that describes the dynamics of a cobweb model in which firms rely on a regime-switching expectation rule. In three different partitions of the phase plane, separated by two discontinuity lines, the map is defined by linear functions with the same Jacobian matrix, having two real eigenvalues, one of which is negative and one equal to 0. This leads to asymptotic dynamics that can belong to two or three critical lines. We show that when the basic fixed point is attracting, it may coexist with at most three attracting cycles. We have determined their existence regions, in the two-dimensional parameter plane, bounded by border collision bifurcation curves. At parameter values for which the basic fixed point is repelling, chaotic attractors may exist - either one that is symmetric with respect to the basic fixed point, or, if not symmetric, the symmetric one also exists. The homoclinic bifurcations of repelling cycles leading to the merging of chaotic attractors are commented by using the first return map on a suitable line. Moreover, four different kinds of homoclinic bifurcations of a saddle 2-cycle, leading to divergence of the generic trajectory, are determined.

Pairs of continuous linear bijective maps preserving fixed products of operators

Let X X be a complex Banach space, and denote by B ( X ) \mathcal {B}(X) the algebra of all bounded linear operators on X X . Let C , D ∈ B ( X ) C,D\in \mathcal {B} \left ( X\right ) be fixed operators. In this paper, we characterize linear, continuous and bijective maps φ \varphi and ψ \psi on B ( X ) \mathcal {B}\left ( X\right ) for which there exist invertible operators T 0 , W 0 ∈ B ( X ) T_0, W_0 \in \mathcal { B}(X) such that φ ( T 0 ) , ψ ( W 0 ) ∈ B ( X ) \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that φ ( A ) ψ ( B ) = D \varphi \left ( A\right ) \psi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C in B ( X ) \mathcal {B}(X) . As a corollary, we deduce the form of linear, bijective and continuous maps φ \varphi on B ( X ) \mathcal {B}(X) having the property that φ ( A ) φ ( B ) = D \varphi \left ( A\right ) \varphi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C .

Development of a linear decoupling cable-driven manipulator with independent driving joints,

A cable-driven manipulator demonstrates significant application in cramped environments, such as space maintenance and equipment monitoring, owing to its slender body and excellent flexibility. However, in traditional designs, the mapping between the operational space and the joint space is nonlinear and non-consistent, and the driving cables are also coupled. Consequently, the kinematics and dynamics become highly complex, posing challenges in enhancing efficiency and precision in trajectory planning and control. This paper introduces a novel linear decoupling cable-driven manipulator with independent driving joints. Two sets of nonlinear transmission mechanisms are designed and serially connected to form an equivalent linear transmission mechanism. This arrangement establishes a proportional relationship between the motor angle and joint angle, with the proportionality coefficient representing the equivalent transmission ratio. Moreover, a two-way wire-pulling mechanism is designed to achieve one-to-one driving between the motor and the joint. The nonlinear coupling problem between driving cables is solved by connecting the driving cable to the target joint through a constant-length cable sleeve. Based on the aforementioned design, the linear and consistent mapping between the operational space and the joint space is realized, significantly simplifying the kinematic model. Prototype experiments validate the manipulator's extensive range of motion and high motion accuracy.

Optimality of generalized Choi maps in M 3

A family of linear positive maps in the algebra of 3×3 complex matrices proposed recently by Bera et al 2024 Linear and Multilinear Algebra 1–16) is further analyzed. It provides a generalization of a seminal Choi nondecomposable extremal map in M 3. We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.

A study on Fibo-Pascal sequence spaces and associated matrix transformations and applications of Hausdorff measure of non-compactness

Abstract In this article, we introduce Fibo-Pascal sequence spaces P p F {P_{p}^{F}} , 0 < p < ∞ {0<p<\infty} , and P ∞ F {P_{\infty}^{F}} through the utilization of the Fibo-Pascal matrix P F {P^{F}} . We establish that both P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} are BK-spaces, enjoying a linear isomorphism with the classical spaces ℓ p {\ell_{p}} and ℓ ∞ {\ell_{\infty}} , respectively. Further contributing to the depth of our investigation, we proceed to derive the Schauder basis of the space P p F {P_{p}^{F}} , alongside an exhaustive computation of the α-, β-, and γ-duals for both spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . Additionally, we undertake the task of characterizing certain classes of matrix mappings pertaining to the spaces P p F {P_{p}^{F}} and P ∞ F {P_{\infty}^{F}} . The final section of this study is dedicated to the meticulous characterization of compact operators acting in the spaces P 1 F {P_{1}^{F}} , P p F {P_{p}^{F}} , and P ∞ F {P_{\infty}^{F}} by using Hausdorff measure of non-compactness.

A unified treatment of tractability for approximation problems defined on Hilbert spaces

A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions 1,2,… grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus on the situation where the space of available information consists of all linear functionals, and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.

Classification of high-dimensional imbalanced biomedical data based on spectral clustering SMOTE and marine predators algorithm

The research of biomedical data is crucial for disease diagnosis, health management, and medicine development. However, biomedical data are usually characterized by high dimensionality and class imbalance, which increase computational cost and affect the classification performance of minority class, making accurate classification difficult. In this paper, we propose a biomedical data classification method based on feature selection and data resampling. First, use the minimal-redundancy maximal-relevance (mRMR) method to select biomedical data features, reduce the feature dimension, reduce the computational cost, and improve the generalization ability; then, a new SMOTE oversampling method (Spectral-SMOTE) is proposed, which solves the noise sensitivity problem of SMOTE by an improved spectral clustering method; finally, the marine predators algorithm is improved using piecewise linear chaotic maps and random opposition-based learning strategy to improve the algorithm’s optimization seeking ability and convergence speed, and the key parameters of the spectral-SMOTE are optimized using the improved marine predators algorithm, which effectively improves the performance of the over-sampling approach. In this paper, five real biomedical datasets are selected to test and evaluate the proposed method using four classifiers, and three evaluation metrics are used to compare with seven data resampling methods. The experimental results show that the method effectively improves the classification performance of biomedical data. Statistical test results also show that the proposed PRMPA-Spectral-SMOTE method outperforms other data resampling methods.

Robust-to-Dynamics Optimization

A robust-to-dynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function [Formula: see text] and a feasible set [Formula: see text]) and (ii) a dynamical system (a map [Formula: see text]). Its goal is to minimize f over the set [Formula: see text] of initial conditions that forever remain in [Formula: see text] under g. The focus of this paper is on the case where the mathematical program is a linear program and where the dynamical system is either a known linear map or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to [Formula: see text]. Our inner approximations are optimized with respect to the objective function f, and their semidefinite characterization—which has a semidefinite constraint of fixed size—is obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations to [Formula: see text] from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case. Funding: O. Günlük was partially supported by the Office of Naval Research [Grant N00014-21-1-2575]. This work was partially funded by the Alfred P. Sloan Foundation, the Air Force Office of Scientific Research, Defense Advanced Research Projects Agency [Young Faculty Award], the National Science Foundation [Faculty Early Career Development Program Award], and Google [Faculty Award].

General spatial photonic Ising machine based on the interaction matrix eigendecomposition method

The spatial photonic Ising machine has achieved remarkable advancements in solving combinatorial optimization problems. However, it still remains a huge challenge to flexibly map an arbitrary problem to the Ising model. In this paper, we propose a general spatial photonic Ising machine based on the interaction matrix eigendecomposition method. The arbitrary interaction matrix can be configured in the two-dimensional Fourier transformation based spatial photonic Ising model by using values generated by matrix eigendecomposition. The error in the structural representation of the Hamiltonian decreases substantially with the growing number of eigenvalues utilized to form the Ising machine. In combination with the optimization algorithm, as low as ∼65% of the eigenvalues are required by intensity modulation to guarantee the best probability of optimal solution for a 20-vertex graph Max-cut problem, and this percentage decreases to below ∼20% for near-zero probability. The 4-spin experiments and error analysis demonstrate the Hamiltonian linear mapping and ergodic optimization. Our work provides a viable approach for spatial photonic Ising machines to solve arbitrary combinatorial optimization problems with the help of the multi-dimensional optical property.

Biderivations of the mirror Heisenberg–Virasoro algebra

The mirror Heisenberg–Virasoro algebra [Formula: see text] is the semi-direct product of the Virasoro algebra and the twisted Heisenberg algebra. In this paper, all biderivations of the mirror Heisenberg–Virasoro algebra are determined by using the gradation of biderivations. As an application, we show that any linear commuting map [Formula: see text] on the mirror Heisenberg–Virasoro algebra is the form [Formula: see text], where [Formula: see text] is an element from the basic field and [Formula: see text] is a linear map having its range in the center of [Formula: see text].

Linear Hashing with $\ell_\infty$ guarantees and two-sided Kakeya bounds

We show that a randomly chosen linear map over a finite field gives a good hash function in the $\ell_\infty$ sense. More concretely, consider a set $S \subset \mathbb{F}_q^n$ and a randomly chosen linear map $L : \mathbb{F}_q^n \to \mathbb{F}_q^t$ with $q^t$ taken to be sufficiently smaller than $ |S|$. Let $U_S$ denote a random variable distributed uniformly on $S$. Our main theorem shows that, with high probability over the choice of $L$, the random variable $L(U_S)$ is close to uniform in the $\ell_\infty$ norm. In other words, {\em every} element in the range $\mathbb{F}_q^t$ has about the same number of elements in $S$ mapped to it. This complements the widely-used Leftover Hash Lemma (LHL) which proves the analog statement under the statistical, or $\ell_1$, distance (for a richer class of functions) as well as prior work on the expected largest 'bucket size' in linear hash functions [ADMPT99]. By known bounds from the load balancing literature [RS98], our results are tight and show that linear functions hash as well as trully random function up to a constant factor in the entropy loss. Our proof leverages a connection between linear hashing and the finite field Kakeya problem and extends some of the tools developed in this area, in particular the polynomial method.