Decomposition Of Space Research Articles

Shifted model spaces and their orthogonal decompositions

We generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift \(S^*\). In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator \(T\), with respect to another operator, follow from certain commutation relations between the two operators involved.

Least Squares Projection Onto the Behavior for SISO LTI Models

We consider the least squares projection onto the behavior for linear time-invariant (LTI) single-input single-output (SISO) models, in which the observed input-output data are modified in a least squares (LS) sense by subtracting so-called misfits, so that the modified data satisfy a given linear dynamic relation. We show that the LS-criterion of the projection problem induces an orthogonal decomposition of the ambient data space and we characterize this decomposition by means of banded block-Toeplitz matrices, the elements of which are the coefficients of the difference equation describing the SISO LTI dynamics. We thereby generalize earlier results in the literature on autonomous LTI models to the more complicated SISO case. Additionally, we illustrate that the novel characterization is equivalent (up to a change of model representation) to results derived using (isometric) state-space representations in the literature on behavioral systems theory.

Bifurcation and Spatiotemporal Patterns of SI Epidemic Model with Diffusion

This paper investigates a spatiotemporal SI epidemiological model under homogeneous Neumann boundary conditions. First, the long-time behavior of the solutions is described, a priori estimates of nonconstant positive solutions are given, and the nonexistence of nonconstant positive steady states is proved by the energy method. Second, the Turing instability of the positive constant steady-state is discussed, and the existence of nonconstant positive steady states is shown by using the degree theory. Moreover, applying the bifurcation theory, we establish the local and global structures of the steady-state bifurcation from simple eigenvalues, and describe some conditions for determining the direction of bifurcation, where the techniques of space decomposition and implicit function theorem are adopted to deal with the local structure of the steady-state bifurcation from double eigenvalues. Finally, some analysis results are supplemented by numerical simulations.

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

Simplicial graviton from selfdual Ashtekar variables

In perturbative gravity, it is straight-forward to characterize the two local degrees of freedom of the gravitational field in terms of a mode expansion of the linearized perturbation. In the non-perturbative regime, we are in a more difficult position. It is not at all obvious how to construct Dirac observables that can separate the gauge orbits. Standard procedures rely on asymptotic boundary conditions or formal Taylor expansions of relational observables. In this paper, we lay out a new non-perturbative lattice approach to tackle the problem in terms of Ashtekar’s self-dual formulation. Starting from a simplicial decomposition of space, we introduce a local kinematical phase space at the lattice sites. At each lattice site, we introduce a set of constraints that replace the generators of the hypersurface deformation algebra in the continuum. We show that the discretized constraints close under the Poisson bracket. The resulting reduced phase space describes two complex physical degrees of freedom representing the two radiative modes at the discretized level. The paper concludes with a discussion of the key open problems ahead and the implications for quantum gravity.

Bifurcation Solutions to the Templator Model in Chemical Self-Replication

In this paper, we are concerned with a diffusive templator model in chemical self-replication, which describes the process of an individual molecule duplicating itself. Firstly, the stability of non-negative constant equilibrium solution is introduced. Then the existence of Hopf bifurcation is proved. Particularly, the stability and the direction of Hopf bifurcation for the spatially homogeneous model are discussed. Furthermore, by space decomposition and implicit function theorem, it is shown that the system may undergo a steady-state bifurcation with a two-dimensional kernel. Finally, several numerical simulations are completed to demonstrate the theoretical results.

A NOTE ON THE POLAR DECOMPOSITION IN METRIC SPACES

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions.Supplementary InformationThe online version contains the Armenian language version of the article available at https://doi.org/10.1007/s10958-023-06674-w.

A generic impulsive controller for Hamiltonian time-independent systems

In a recent work Briozzo (2021) the author introduced an impulsive Floquet controller for periodic orbits in time-periodic Hamiltonian systems, which expresses the control law analytically in terms of the linearized dynamics about the reference orbit and makes full use of the center manifold enhancing performance by minimizing the energy of the actual orbit in the co-moving frame of the reference one. The purpose of the present work is to introduce an equivalent controller for Hamiltonian linear time-periodic systems obtained by linearization of a general time-independent Hamiltonian one about a periodic orbit, so incorporating the presence of a shear manifold associated to energy conservation and the ensuing non-isolation of periodic orbits. Like its preceding version, the controller defines a target subspace akin to the sliding surface of sliding mode controllers, onto which the system state vector is repeatedly sent by control impulses, and is generic in the sense that the target subspace and the control law are explicitly given for any possible decomposition of state space into a shear manifold and saddle, centre, or saddle-focus ones.

A Note on Finite Dimensional Odd Contact Lie Superalgebra in Prime Characteristic

Over a field of characteristic p>3, let KO(n,n+1;t̲) denote the odd contact Lie superalgebra. In this paper, the super-biderivations of odd Contact Lie superalgebra KO(n,n+1;t̲) are studied. Let TKO be a torus of KO(n,n+1;t̲), which is an abelian subalgebra of KO(n,n+1;t̲). By applying the weight space decomposition approach of KO(n,n+1;t̲) with respect to TKO, we show that all skew-symmetric super-biderivations of KO(n,n+1;t̲) are inner super-biderivations.

Current Harmonics and Unbalance Suppression of Dual Three-Phase PMSM Based on Adaptive Linear Neuron Controller

A strategy of current harmonics and unbalance suppression based on the vector space decomposition control with adaptive linear neuron (ALN) controller for the dual three-phase permanent magnet synchronous machine (DTP-PMSM) drive system is proposed in this paper. According to the conventional vector space decomposition theory, the compensation voltages from the ALN controller in the dq-frame and dqz-frame are employed to suppress the 5th and 7th current harmonics and unbalances. Compared with the VSD control using only the PI controller, the current harmonics and unbalance are sufficiently suppressed in the proposed strategy. In addition, unlike the resonant controller discretized by conventional Tustin transformation, the ALN controller has no resonant pole shift to the desired resonant frequency. Therefore, the harmonic suppression effect of the proposed strategy exhibits better performance than the VSD control with the resonant controller, especially in the high-speed region. Finally, the feasibility and effectiveness of the proposed strategy are validated by experimental results under both steady and transient states.

The First Cohomology of o s p ( 1,2 ) with Coefficients in Baby Verma Modules and Simple Modules

Over a field of characteristic [Formula: see text], the first cohomology of the orthogonal symplectic Lie superalgebra [Formula: see text] with coefficients in baby Verma modules and simple modules is determined by use of the weight space decompositions of these modules relative to a Cartan subalgebra of [Formula: see text]. As a byproduct, the first cohomology of [Formula: see text] with coefficients in the restricted enveloping algebra (under the adjoint action) is not trivial.

Dilational symmetries of decomposition and coorbit spaces

We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a certain metric in frequency domain. We formulate versions of this phenomenon both for the decomposition and coorbit space settings.We then apply the general results to a particular class of dilation groups, the so-called shearlet dilation groups. We present a general, algebraic characterization of matrices that are coorbit compatible with a given shearlet dilation group. We explicitly determine the groups of compatible dilations, for a variety of concrete examples.

Decoupled and compensated space vector pulse width modulation algorithm for dual‐channel PMSM with unbalanced power supplies

SummaryThis paper investigates an improved space vector pulse width modulation (SVPWM) and channel‐decoupled method for dual‐channel permanent magnet synchronous machine (PMSM) powered by unbalanced battery packs. Recently, individual power supplied voltage source inverter (VSI) is introduced considering insufficient power rating and unsafe driving in electrical vehicles (EVs). However, the issues of strong coupling and unbalanced power supplies exist between channels. Firstly, a modular topology is constructed to improve the reliability and security of the system. Secondly, the channel‐decoupled method is introduced based on multi‐dq rather than the conventional vector space decomposition (VSD) to eliminate coupling among multi‐channel. Thereafter, the variable distribution of voltage vectors (VVs) is corrected by the proposed SVPWM, where the switch can be operated appropriately. Finally, the comparative experiments are carried out focusing on decoupling and unbalanced power supplies, and the results present superior performance in terms of low torque fluctuation and balanced channel currents.

Thermal states of two identical two-level atoms interacting with the single-mode fields of the Arik–Coon and Biedenharn–Macfarlane q-oscillators

We consider the Jaynes–Cummings models (JCMs) of two identical two-level atoms interacting with a single mode of the Arik–Coon and Biedenharn–Macfarlane [Formula: see text]-boson fields in a cavity. It is shown that there is a constant of motion, so-called the total number of excitation, which leads to decomposition of the total Hilbert space of [Formula: see text]-deformed system into three invariant subspaces. So, we have the block diagonal structure with three blocks as the Hamiltonian matrices. The exact energy spectrum and eigenstates of the [Formula: see text]-deformed models are obtained by diagonalizing each block. Assuming that the cavity fields and atoms are in thermal equilibrium, the partition function and thermal density matrix of the models are calculated. Finally, the influence of deformation parameters of the Arik–Coon and Biedenharn–Macfarlane [Formula: see text]-oscillator light fields at low-temperatures on the entropy and internal energy of the Jaynes–Cummings models as well as on the atom–atom entanglement via the concurrence measure is analyzed.

Loop space decompositions of highly symmetric spaces with applications to polyhedral products

We generalise the fold map for the wedge sum and use this to give a loop space decomposition of topological spaces with a high degree of symmetry. This is applied to polyhedral products to give a loop space decomposition of polyhedral products associated to families of graphs.

Measurement of Angle Between Subspaces in Direct Sum Decomposition

The problems of Eigen structure assignment has been studied. As an immediate part of that, the problem addressed is the development of measures of “skewness” between subspaces which are in the direct sum decomposition of the state space and a concept of angle between a set of subspaces.

Efficient Mann turbulence generation for offshore wind farms with applications in fatigue load surrogate modelling

Surrogate models are vital for offshore wind farm optimisation, or digital twin for rapid execution. Reliable surrogates are difficult to make, especially for large wind farms and for fatigue loads. One limitation is the high resolution of turbulence needed for fatigue calculations and the vast size of a wind farm. We present code optimisations for implementing the Mann model for farm-sized flows, which are infeasible otherwise. Additionally, we present a case study of a full load surrogate for the Lillgrund offshore wind farm using large turbulence box generation in aeroelastic wind farm simulations using HAWC2Farm. Various mappings between the wind field in front of a turbine and the corresponding turbine structural fatigue loads are presented. The best-performing surrogate model, using a proper orthogonal decomposition of the input space with an artificial neural network, is able to significantly reduce the error in fatigue load estimates by as much as a factor of 5 compared to conventional methods. The presented recommendations are crucial to generating reliable wind farm load surrogates for wind farm optimisation purposes.

Local wellposedness of the relativistic Vlasov–Poisson equation in Besov space

We obtain the local existence and uniqueness of solution to the relativistic Vlasov–Poisson system in Besov space, extending the non-relativistic result to the relativistic case. Weaker regularity is required for the initial datum by optimizing the commutator estimates related to the field. Commutator estimates associated with relativistic velocity are proved by dyadic decomposition of the frequency space and the decreasing properties of Bessel potential.

Decomposition of Displacement Field into the Irrotational and Solenoidal Component Using Fast Fourier Transform

Purpose: A new code based on Helmholtz decomposition is presented to separate longitudinal (pressure) and transverse (shear) components of a mixed wave field. This algorithm will help isolate shear or pressure components of an elastic wave to further concentrate on each specific wave and its physical characteristics, particularly in medical imaging instrument development and image processing techniques.
 Materials and Methods: Using the combination of Fourier transform and Helmholtz decomposition, first, the mathematical basis of the work is prepared. After reaching a usable formula, this basis is embedded in the Code written in MATLAB program. Then, various test data containing shear and pressure waves were created and fed to the Code to evaluate its ability to decompose the displacements into the shear and pressure waves.
 Results: This new algorithm successfully isolated the transverse and longitudinal wavefront of the mixed wavefield. The Code demonstrated 100% accuracy for separating the shear wave and more than 99% for the pressure wave. Moreover, the background noise was kept under 0.03% in every step.
 Conclusion: The results show that using Helmholtz decomposition in Fourier space on 3D data can help decompose a displacement field into its irrotational and solenoidal components with high accuracy. A weak dependency on wave thickness and contrast was observed, but the algorithm's accuracy never fell below 99%.

A two-space-decomposition-based evolutionary algorithm for large-scale multiobjective optimization

Large-scale multiobjective optimization problems (LSMOPs) pose a great challenge to maintaining the diversity of solutions. However, existing large-scale multiobjective optimization algorithms (MOEAs) prefer to directly use environmental selection methods designed for small-scale optimization problems. These methods are not effective in solving complex LSMOPs. To address this issue, this paper proposes a two-space (decision space and objective space) decomposition (TSD)-based diversity maintenance mechanism. Its main idea is to explicitly decompose the decision space and objective space into a number of subspaces, each of which may contain some Pareto-optimal solutions. Searching for Pareto-optimal solutions in these subspaces may help maintain the diversity of solutions. To this end, a diversity design subspace (DDS) is constructed to decompose the decision space. Then, a large-scale MOEA (MOEA/TSD) is designed by using the proposed TSD-based diversity maintenance mechanism. Experimental studies validate the effectiveness of the proposed TSD mechanism. Compared with nine state-of-the-art large-scale MOEAs on 112 benchmark LSMOPs, our proposed algorithm offers considerable advantages in overall optimization performance. The source code of MOEA/TSD is available at https://github.com/yizhizhede/MOEATSD.