Regular Subspaces Research Articles

On structure of regular Dirichlet subspaces for one-dimensional Brownian motion

The main purpose of this paper is to explore the structure of regular Dirichlet subspaces of one-dimensional Brownian motion. As stated in [Osaka J. Math. 42 (2005) 27–41], every such regular Dirichlet subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^{c}$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular Dirichlet subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular Dirichlet subspace on $F$ is still a regular Dirichlet subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover, we shall prove that the trace of Brownian motion on $F$ may be decomposed into two parts; one is the trace of the regular Dirichlet subspace on $F$, which has only the nonlocal part and the other comes from the orthogonal complement of the regular Dirichlet subspace, which has only the local part. Actually the orthogonal complement of regular Dirichlet subspace corresponds to a time-changed absorbing Brownian motion after a darning transform.

Weak convergence of regular Dirichlet subspaces

In this paper we shall prove the weak convergence of the associated diffusion processes of regular subspaces with monotone characteristic sets for a fixed Dirichlet form. More precisely, given a fixed 1-dimensional diffusion process and a sequence of its regular subspaces, if the characteristic sets of regular subspaces are decreasing or increasing, then their associated diffusion processes are weakly convergent to another diffusion process. This is an extended result of [14].

Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations

This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace (relation) below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspace, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.

The α-orthogonal complements of regular Dirichlet subspaces for one-dimensional Brownian motion

Roughly speaking, a regular subspace of a Dirichlet form is a subspace, which is also a regular Dirichlet form, on the same state space. In particular, the domain of regular subspace is a closed subspace of the Hilbert space induced by the domain and $\alpha$-inner product of original Dirichlet form. We shall investigate the orthogonal complement of regular subspace of 1-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the $\alpha$-harmonic equation under Neumann boundary condition.

Killing Transform on Regular Dirichlet Subspaces

In this paper, we shall consider the killing transform induced by a multiplicative functional on regular Dirichlet subspaces of a fixed Dirichlet form. Roughly speaking, a regular Dirichlet subspace is a closed subspace with Dirichlet and regular properties of fixed Dirichlet space. By using the killing transforms, our main results indicate that the big jump part of fixed Dirichlet form is not essential for discussing its regular Dirichlet subspaces. This fact is very similar to the status of killing measure when we consider the questions about regular Dirichlet subspaces in Li-and-Ying (2015).

Regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations

This paper is concerned with regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations. It is shown that the kth eigenvalue of any given self-adjoint subspace extension is exactly the limit of the kth eigenvalues of the induced regular self-adjoint subspace extensions in the case that each endpoint is either regular or in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, it is shown that isolated eigenvalues in every gap of the essential spectrum of any self-adjoint subspace extension are exactly the limits of eigenvalues of suitably chosen induced regular self-adjoint subspace extensions in the case that at least one endpoint is in the limit point case.

Regular subspaces and invariant subspaces of Boolean control networks

This study investigates some fundamental problems on regular subspaces and invariant subspaces of Boolean control networks. First, a new necessary and sufficient condition for regular subspaces is obtained, which exactly reveals the relationship between a regular subspace and its complementary subspaces. A new method to compute complementary subspaces is proposed. Second, invariant subspaces without the regularity presupposition are considered. Some necessary and sufficient conditions of invariant subspaces are given. Finally, two examples are given to illustrate the obtained theoretical results.

대용량 GPS 궤적 데이터를 위한 효율적인 클러스터링

Digital road map generation is primarily based on artificial satellite photographing or in-site manual survey work. Therefore, these map generation procedures require a lot of time and a large budget to create and update road maps. Consequently, people have tried to develop automated map generation systems using GPS trajectory data sets obtained by public vehicles. A fundamental problem in this road generation procedure involves the extraction of representative trajectory such as main roads. Extracting a representative trajectory requires the base data set of piecewise line segments(GPS-trajectories), which have close starting and ending points. So, geometrically similar trajectories are selected for clustering before extracting one representative trajectory from among them. This paper proposes a new divide- and-conquer approach by partitioning the whole map region into regular grid sub-spaces. We then try to find similar trajectories by sweeping. Also, we applied the distance measure to compute the similarity between a pair of trajectories. We conducted experiments using a set of real GPS data with more than 500 vehicle trajectories obtained from Gangnam-gu, Seoul. The experiment shows that our grid partitioning approach is fast and stable and can be used in real applications for vehicle trajectory clustering.

Regular subspaces of skew product diffusions

Abstract Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space E 1 × E 2 ${E_{1}\times E_{2}}$ and expressed as X t = ( X t 1 , X A t 2 ) , t ≥ 0 , $X_{t}=(X^{1}_{t},X^{2}_{A_{t}}),\quad t\geq 0,$ where X i ${X^{i}}$ is a symmetric diffusion on E i ${E_{i}}$ for i = 1 , 2 ${i=1,2}$ , and A is a positive continuous additive functional of X 1 ${X^{1}}$ . One of our main results indicates that any skew product type regular subspace of X, say Y t = ( Y t 1 , Y A ~ t 2 ) , t ≥ 0 , $Y_{t}=(Y^{1}_{t},{Y^{2}_{\tilde{A}_{t}}}),\quad t\geq 0,$ can be characterized as follows: the associated smooth measure of A ~ ${\tilde{A}}$ is equal to that of A, and Y i ${Y^{i}}$ corresponds to a regular subspace of X i ${X^{i}}$ for i = 1 , 2 ${i=1,2}$ . Furthermore, we shall make some discussions on rotationally invariant diffusions on ℝ d ∖ { } ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ , which are special skew product diffusions on ( 0 , ∞ ) × S d - 1 ${(0,\infty)\times S^{d-1}}$ . Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ℝ d ∖ { } ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ to a new regular Dirichlet form on ℝ d ${\mathbb{R}^{d}}$ . More precisely, fix a regular Dirichlet form ( ℰ , ℱ ) ${(\mathcal{E,F}\kern 0.569055pt)}$ on the state space ℝ d ${\mathbb{R}^{d}}$ . Its part Dirichlet form on ℝ d ∖ { } ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is denoted by ( ℰ 0 , ℱ ) 0 ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ . Let ( ℰ ~ 0 , ℱ ~ ) 0 ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ be a regular subspace of ( ℰ 0 , ℱ ) 0 ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ . We want to find a regular subspace ( ℰ ~ , ℱ ~ ) ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ of ( ℰ , ℱ ) ${(\mathcal{E,F}\kern 0.569055pt)}$ such that the part Dirichlet form of ( ℰ ~ , ℱ ~ ) ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ on ℝ d ∖ { } ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is exactly ( ℰ ~ 0 , ℱ ~ ) 0 ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ . If ( ℰ ~ , ℱ ~ ) ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ exists, we call it a regular extension of ( ℰ ~ 0 , ℱ ~ ) 0 ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ . We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of ( ℰ 0 , ℱ ) 0 ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ has a unique regular extension.

Regular synchrony lattices for product coupled cell networks.

There are several ways for constructing (bigger) networks from smaller networks. We consider here the cartesian and the Kronecker (tensor) product networks. Our main aim is to determine a relation between the lattices of synchrony subspaces for a product network and the component networks of the product. In this sense, we show how to obtain the lattice of regular synchrony subspaces for a product network from the lattices of synchrony subspaces for the component networks. Specifically, we prove that a tensor of subspaces is of synchrony for the product network if and only if the subspaces involved in the tensor are synchrony subspaces for the component networks of the product. We also show that, in general, there are (irregular) synchrony subspaces for the product network that are not described by the synchrony subspaces for the component networks, concluding that, in general, it is not possible to obtain the all synchrony lattice for the product network from the corresponding lattices for the component networks. We also make the following remark concerning the fact that the cartesian and Kronecker products, as graph operations, are quite different, implying that the associated coupled cell systems have distinct structures. Although, the kinds of dynamics expected to occur are difficult to compare, we establish an inclusion relation between the lattices of synchrony subspaces for the cartesian and Kronecker products.

Optimization Algorithm Analysis on Satellite Communication Signal Processing

Aiming at the difference between satellite’s signal frequency spectrum and satellite signals under specific channel condition, larger fluctuation and ineffective threshold searching method by adopting traditional energy, the paper improved singular point’s performance detection of traditional wavelet transform, selected suitable searching threshold by ultimately combining with both advantages and promoted searching performance of satellite signals under specific channel condition. Based on preprocessed order-1 subspace tracing algorithm of space sign function and infinite norm normalization, the paper solved the problem of degenerating performance of regular order-1 subspace tracing algorithm under noise environment of a stationary distribution effectively.

Regular approximations of spectra of singular second-order symmetric linear difference equations

This paper is concerned with regular approximations of spectra of singular second-order symmetric linear difference equations. For any given self-adjoint subspace extension of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of regular self-adjoint subspace extensions generated by the same difference expression on smaller finite intervals.

Real-time intrusion detection in power system operations

Increasing power system security is crucial to the future of the electric power grid. In this paper, we define a new class of cyberattacks to power systems-malicious modification of network data stored in an accessible database. Grounded in multivariate statistical process control, our approach to intrusion detection ensures data integrity in power system operations. We develop an algorithm that monitors power flow results and detects anomalies in the input values that could have been modified by cyberattacks. Our algorithm uses principal component analysis to separate power flow variability into regular and irregular subspaces. Analysis of the information in the irregular subspace determines whether the power system data has been compromised. We verify the efficacy of the algorithm using both the IEEE 24-bus and 118-bus reliability test systems. The results show that the developed algorithm is a promising enhancement to data security procedures in a control center.

Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states

We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials constitute orthogonal families of vectors in ambient quaternionic L2-spaces. Using these polynomials, we then define regular and anti-regular subspaces of these L2-spaces, the associated reproducing kernels and the ensuing quaternionic coherent states.

Regular lattices of tensor products

It is proved that the collection of all regular invariant subspaces of tensor products of operators is a lattice. Further characterization of regular lattices are considered for injective operators. The approach is carried out keeping pace with the results on intrinsic invariant subspaces of direct sums, extending them to regular invariant subspaces of tensor products.

On nonlinear functional spaces based on triangular conorms

In this paper, we discuss the nonlinear functional spaces based on triangular conorms. Particularly, we discuss the properties of the upper-closures of the regular subspaces of the nonlinear functional space based on a continuous triangular conorm. Furthermore, we prove that with respect to a strict triangular conorm, a subset of the nonlinear functional space is an upper-complete normal subspace if and only if the family of all sets whose characteristic functionals are contained in the given subset of the nonlinear functional space is a sigma-algebra.

The POWDER protocol as infrastructure to serving and compressing semantic data

The POWDER protocol is a Semantic Web technology that takes advantage of natural groupings of URIs to annotate all the resources in a regular expression-delineated sub-space of the URI space. POWDER is a mechanism for accreditation, trustmarking and resource discovery, emphasising the publishing of attributed metadata by third parties and trusted authorities. Demonstrating its versatility, it has also been deployed in unforeseen use cases, such as repository compression. In this paper, we present the POWDER protocol, explain its position in the Semantic Web architecture, expose and discuss current implementations and use cases and future directions.

Dirichlet forms associated with linear diffusions

One-dimensional local Dirichlet spaces associated with linear diffusions are studied. The first result is to give a representation for any 1-dim local, irreducible and regular Dirichlet space. The second result is a necessary and sufficient condition for a Dirichlet space to be regular subspace of another Dirichlet space.

Grassmann spaces of regular subspaces

The underlying metric affine geometry, or metric projective geometry, can be recovered from Grassmann spaces associated with the family of regular subspaces of respective space. In other words, automorphisms of such Grassmann spaces are collineations witch preserve orthogonality of the respective underlying space. This generalizes results of Prazmowska et al. (Linear Algebra Appl 430:3066–3079, 2009) and Prazmowska and Żynel (Adv Geom, to appear).

State–Space Analysis of Boolean Networks

This paper provides a comprehensive framework for the state-space approach to Boolean networks. First, it surveys the authors' recent work on the topic: Using semitensor product of matrices and the matrix expression of logic, the logical dynamic equations of Boolean (control) networks can be converted into standard discrete-time dynamics. To use the state-space approach, the state space and its subspaces of a Boolean network have been carefully defined. The basis of a subspace has been constructed. Particularly, the regular subspace, Y-friendly subspace, and invariant subspace are precisely defined, and the verifying algorithms are presented. As an application, the indistinct rolling gear structure of a Boolean network is revealed.