Orthogonal Complement Research Articles

Intertwining Property for Compressions of Multiplication Operators

Following Beurling’s theorem the natural compressions of the multiplication operator in the classical $$L^2$$ space are compressions to model spaces and to their orthogonal complements. Here, two possibly different model spaces are considered, hence asymmetric truncated Toeplitz and asymmetric dual truncated Toeplitz operators are investigated. The main purpose of the paper is to characterize operators which intertwine compressions of the unilateral shift.

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Restriction to Subgroups

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1\subset\mathfrak{p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space $\mathcal{H}_\lambda(D)\subset\mathcal{O}(D)$ on $D$. Its restriction to the subgroup $\widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the $K_1$-decomposition of the space $\mathcal{P}(\mathfrak{p}^+_2)$ of polynomials on the orthogonal complement $\mathfrak{p}^+_2$ of $\mathfrak{p}^+_1$ in $\mathfrak{p}^+$. The object of this article is to compute explicitly the inner product $\big\langle f(x_2),{\rm e}^{(x|\overline{z})_{\mathfrak{p}^+}}\big\rangle_\lambda$ for $f(x_2)\in\mathcal{P}(\mathfrak{p}^+_2)$, $x=(x_1,x_2)$, $z\in\mathfrak{p}^+=\mathfrak{p}^+_1\oplus\mathfrak{p}^+_2$. For example, when $\mathfrak{p}^+$, $\mathfrak{p}^+_2$ are of tube type and $f(x_2)=\det(x_2)^k$, we compute this inner product explicitly by introducing a multivariate generalization of Gauss' hypergeometric polynomials ${}_2F_1$. Also, as an application, we construct explicitly $\widetilde{G}_1$-intertwining operators (symmetry breaking operators) $\mathcal{H}_\lambda(D)|_{\widetilde{G}_1}\to\mathcal{H}_\mu(D_1)$ from holomorphic discrete series representations of $\widetilde{G}$ to those of $\widetilde{G}_1$, which are unique up to constant multiple for sufficiently large $\lambda$.

Adaptive estimation in multivariate response regression with hidden variables

A prominent concern of scientific investigators is the presence of unobserved hidden variables in association analysis. Ignoring hidden variables often yields biased statistical results and misleading scientific conclusions. Motivated by this practical issue, this paper studies the multivariate response regression with hidden variables, Y=(Ψ∗)TX+(B∗)TZ+E, where Y∈Rm is the response vector, X∈Rp is the observable feature, Z∈RK represents the vector of unobserved hidden variables, possibly correlated with X, and E is an independent error. The number of hidden variables K is unknown and both m and p are allowed, but not required, to grow with the sample size n. Though Ψ∗ is shown to be nonidentifiable due to the presence of hidden variables, we propose to identify the projection of Ψ∗ onto the orthogonal complement of the row space of B∗, denoted by Θ∗. The quantity (Θ∗)TX measures the effect of X on Y that cannot be explained through the hidden variables, and thus Θ∗ is treated as the parameter of interest. Motivated by the identifiability proof, we propose a novel estimation algorithm for Θ∗, called HIVE, under homoscedastic errors. The first step of the algorithm estimates the best linear prediction of Y given X, in which the unknown coefficient matrix exhibits an additive decomposition of Ψ∗ and a dense matrix due to the correlation between X and Z. Under the sparsity assumption on Ψ∗, we propose to minimize a penalized least squares loss by regularizing Ψ∗ and the dense matrix via group-lasso and multivariate ridge, respectively. Nonasymptotic deviation bounds of the in-sample prediction error are established. Our second step estimates the row space of B∗ by leveraging the covariance structure of the residual vector from the first step. In the last step, we estimate Θ∗ via projecting Y onto the orthogonal complement of the estimated row space of B∗ to remove the effect of hidden variables. Nonasymptotic error bounds of our final estimator of Θ∗, which are valid for any m,p,K and n, are established. We further show that, under mild assumptions, the rate of our estimator matches the best possible rate with known B∗ and is adaptive to the unknown sparsity of Θ∗ induced by the sparsity of Ψ∗. The model identifiability, estimation algorithm and statistical guarantees are further extended to the setting with heteroscedastic errors. Thorough numerical simulations and two real data examples are provided to back up our theoretical results.

Modeling of a wheeled humanoid robot and hybrid algorithm-based path planning of wheel base for the dynamic obstacles avoidance

PurposeMost of the conventional humanoid modeling approaches are not successful in coupling different branches of the tree-type humanoid robot. In this paper, a tree-type upper body humanoid robot with mobile base is modeled. The main purpose of this work is to model a non holonomic mobile platform and to develop a hybrid algorithm for avoiding dynamic obstacles. Decoupled Natural Orthogonal Complement methodology effectively combines different branches of the humanoid body during dynamic analysis. Collision avoidance also plays an important role along with modeling methods for successful operation of the upper body wheeled humanoid robot during real-time operations. The majority of path planning algorithms is facing problems in avoiding dynamic obstacles during real-time operations. Hence, a multi-fusion approach using a hybrid algorithm for avoiding dynamic obstacles in real time is introduced.Design/methodology/approachThe kinematic and dynamic modeling of a humanoid robot with mobile platform is done using screw theory approach and Newton–Euler formulations, respectively. Dynamic obstacle avoidance using a novel hybrid algorithm is carried out and implemented in real time. D star lite and a geometric-based hybrid algorithms are combined to generate the optimized path for avoiding the dynamic obstacles. A weighting factor is added to the D star lite variant to optimize the basic version of D star lite algorithm. Lazy probabilistic road map (PRM) technique is used for creating nodes in configuration space. The dynamic obstacle avoidance is experimentally validated to achieve the optimum path.FindingsThe path obtained using the hybrid algorithm for avoiding dynamic obstacles is optimum. Path length, computational time, number of expanded nodes are analysed for determining the optimality of the path. The weighting function introduced along with the D star lite algorithm decreases computational time by decreasing the number of expanding nodes during path generation. Lazy evaluation technique followed in Lazy PRM algorithm reduces computational time for generating nodes and local paths.Originality/valueModeling of a tree-type humanoid robot along with the mobile platform is combinedly developed for the determination of the kinematic and dynamic equations. This paper also aims to develop a novel hybrid algorithm for avoiding collision with dynamic obstacles with minimal computational effort in real-time operations.

Characterizations of modalities and lex modalities

A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of $\infty$-categories. Working in homotopy type theory, we give new characterizations of the following conditions on a reflective subuniverse $L$: (1) the associated subuniverse $L'$ of $L$-separated types is a modality; (2) $L$ is a modality; (3) $L$ is a lex modality; and (4) $L$ is a cotopological modality. In each case, we give several necessary and sufficient conditions. Our characterizations involve various families of maps associated to $L$, such as the $L$-\'etale maps, the $L$-equivalences, the $L$-local maps, the $L$-connected maps, the unit maps $\eta_X$, and their left and/or right orthogonal complements. More generally, our main theorem gives an overview of how all of these classes related to each other. We also give examples that show that all of the inclusions we describe between these classes of maps can be strict.

Projection and Rescaling Algorithm for Finding Maximum Support Solutions to Polyhedral Conic Systems

We propose a simple projection and rescaling algorithm that finds maximum support solutions to the pair of feasibility problems: [Formula: see text] where L is a linear subspace in [Formula: see text] and [Formula: see text] is its orthogonal complement. The algorithm complements a basic procedure that involves only projections onto L and [Formula: see text] with a periodic rescaling step. The number of rescaling steps and, thus, overall computational work performed by the algorithm are bounded above in terms of a condition measure of the above pair of problems. Our algorithm is a natural but significant extension of a previous projection and rescaling algorithm that finds a solution to the full support problem: [Formula: see text] when this problem is feasible. As a byproduct of our new developments, we obtain a sharper analysis of the projection and rescaling algorithm in the latter special case.

Elucidation of kinetics for use with redundant manipulators

Robotics have numerous applications in industry and rely on manipulators, which are devices that can manipulate materials without the need for direct human contact. Manipulators are able to perform tasks thanks to so-called 'degrees of freedom'. Researchers are working to elucidate kinetics for use with redundant manipulators with potential benefits for industries across the globe in terms of time, energy and cost savings. Dr Kousuke Okabe, Department of Electrical Engineering, National Institute of Technology, which forms part of Wakayama College, Japan, is exploring redundant manipulators and their algorithms and wants to fill some of the existing knowledge gaps in this area. Okabe has found the orthogonal complement of the space spanned by the column vector of the inverse matrix of a generalised manipulator with a degree of redundancy of two or more and is working to find the generalised vector of this orthogonal complement in the future. Okabe and his team are also interested in exploring the effects of kinetics based on the state of movement of the hands. The researchers believe that their research results will be relevant to research on energy conservation in the field of high-speed manipulation, as well as the industrial world.

Robust dynamic output feedback stabilization for uncertain singular Markovian jump systems with time‐varying delays

AbstractThe article investigates robust dynamic output feedback (DOF) stabilization for uncertain time‐varying delays singular Markovian jump systems (SMJSs). First, by using improved Lyapunov–Krasovskii functional, sufficient conditions of robust admissibility (namely regularity, non‐impulsiveness, and asymptotic stability) for unforced nominal time‐varying delays SMJSs are given. Second, by constructing a DOF controller such that the resulting closed‐loop system satisfies robust admissibility, the robust DOF stabilization problem is solved in terms of linear matrix inequalities via orthogonal complement technique. Finally, the availability and practicality of the prime results are certified by two numerical examples and a practical example of the oil catalytic cracking process.

On angles between convex cones

There are two basic angles associated with a pair of linear subspaces: the Diximier angle and the Friedrichs angle. These classical notions, which date back to the first half of the 20th century, have been thoroughly studied due to their utility in the description of convergence rates for various projection-based algorithms for solving feasibility problems. The Dixmier angle of the pair of orthogonal complements is the same as the Dixmier angle of the original pair provided that the original pair gives rise to a direct (not necessarily orthogonal) sum of the underlying Hilbert space. Even more satisfyingly, the Friedrichs angles of the original pair and the pair of the orthogonal complements always coincide. These two beautiful results are due to Krein, Krasnoselskii, and Milman and to Solmon, respectively. In 1995, Deutsch provided a very nice survey with complete proofs and interesting historical comments. One key result in Deutsch's survey was an inequality for Dixmier angles provided by Hundal. In this paper, we present extensions of these results to the case when the linear subspaces are only required to be convex cones. It turns out that Hundal's result has a nice conical extension while the situation is more technical for the results by Krein et al. and by Solmon. Our analysis is based on Deutsch's survey and our recent work on angles between convex sets. Throughout, we also provide examples illustrating the sharpness of our results.

Geodesic and Conformally Reeb Vector Fields on Flat 3-Manifolds

A unit vector field on a Riemannian manifold $M$ is called geodesic if all of its integral curves are geodesics. We show, in the case of $M$ being a flat 3-manifold not equal to $\mathbb{E}^3$, that every such vector field is tangent to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a geodesic vector field $X$ on a closed flat 3-manifold is (up to rescaling) the Reeb vector field of a contact form if and only if there is a contact structure transverse to $X$ that is given as the orthogonal complement of some other geodesic vector field. An explicit description of the lifted contact structures (up to diffeomorphism) on the 3-torus is given in terms of the volume of $X$. Finally, similar results for non-closed flat 3-manifolds are discussed.

Proper condensates

In this article, a novel characterization of Bose–Einstein condensates is proposed. Instead of relying on occupation numbers of a few dominant modes, which become macroscopic in the limit of infinite particle numbers, it focuses on the regular excitations whose numbers stay bounded in this limit. In this manner, subspaces of global, respectively, local regular wave functions are identified. Their orthogonal complements determine the wave functions of particles forming proper (infinite) condensates in the limit. In contrast to the concept of macroscopic occupation numbers, which does not sharply fix the wave functions of condensates in the limit states, the notion of proper condensates is unambiguously defined. It is outlined how this concept can be used in the analysis of condensates in models. The method is illustrated by the example of trapped non-interacting ground states and their multifarious thermodynamic limits, differing by the structure of condensates accompanying the Fock vacuum. The concept of proper condensates is also compared with the Onsager–Penrose criterion based on the analysis of eigenvalues of one-particle density matrices. It is shown that the concept of regular wave functions is also useful in that approach for the identification of wave functions forming proper condensates.

Projection-aided Adaptive Residual Generator for Disturbance-decoupled Process Monitoring

Adaptive residual generator is a powerful technique which delivers a simultaneous co-estimation of the state variables and unknown parameters with stability and convergence guarantee. This paper reports our recent study on a projection-aided adaptive residual generator design for disturbance-decoupled process monitoring. The core of the study is the implementation of a projection on the orthogonal complement of disturbance subspace, which results in a robust state and parameter estimation against unknown disturbance, furthermore, accurate process monitoring can be achieved.

Blind Estimation of Self-Synchronous Scrambler Using Orthogonal Complement Space in DSSS Systems

In a non-cooperative context, a receiver has to estimate the communication parameters without any prior knowledge of the transmitter, which is highly demanding. Estimating a self-synchronous scrambler is even more challenging because the scrambling sequence of the self-synchronous scrambler is affected by the input sequence. This paper proposes an improved algorithm for blind estimation of a self-synchronous scrambler using the orthogonal property without any bias condition of the received signals in direct sequence spread spectrum (DSSS) systems. We first examine the linear relation of a scrambling sequence using the repetitive property of the spreading code used in a DSSS system. Using the obtained linear relation and the basis of the orthogonal complement space, we then acquire the feedback polynomial candidates of the scrambler. Finally, by calculating the greatest common divisor polynomial of the feedback polynomial candidates, we estimate the correct feedback polynomial. Through computer simulations, we verify that the proposed method achieves superior estimation performance compared to the existing method. Furthermore, we show that the proposed method has practically acceptable computational complexity. For these reasons, it is expected that the proposed method can be applied to blind estimation of a self-synchronous scrambler in a practical non-cooperative system.

Improving MRAC Transient Response Using Generalized Dynamic Inversion and Nullspace Parametrization

We demonstrate how the transient response of Model Reference Adaptive Control (MRAC) is improved by utilizing Generalized Dynamic Inversion (GDI) and the corresponding nullspace parametrization in the powerful Greville formula. The proposed approach neither alters the reference model structure, nor it modifies it through feedback. Rather, the proposed approach utilizes the control redundancy in matching the reference model for the purpose of designing an MRAC control law that yields good transient closed loop adaptive control response. The adaptive GDI-MRAC control law is composed of two parts: a particular part that enforces the reference model dynamics, and an auxiliary part that is affine in a null control vector. The particular and the auxiliary parts of the adaptive control law are confined in the range space of the transposed control matrix and its orthogonal complement nullspace, respectively. The null control vector is projected onto the nullspace of the control matrix via a nullprojection matrix. Hence, the null control vector design does not interfere with the reference model following, yet it can be designed via state feedback to improve closed loop state transient response. An improved Lyapunov control function is used to design the adaptation gains such that asymptotic vanishing of the state error dynamics is guaranteed. Extensive simulations are conducted to demonstrate the efficacy of the proposed method.

Properties of dual Toeplitz operator on the orthogonal complement of the pluriharmonic Bergman space of the unit ball

In this paper, we characterize the hyponormal dual Toeplitz operators with special symbols on the orthogonal complement of the pluriharmonic Bergman space of the unit ball. Also we completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators.

Riemannian geometry of resonant optical responses

The geometry of quantum states is well-established as a basis for understanding the response of electronic systems to static electromagnetic fields, as exemplified by the theory of the quantum and anomalous Hall effects. However, it has been challenging to relate quantum geometry to resonant optical responses. The main obstacle is that optical transitions involve a pair of states, while existing geometrical properties are defined for a single state. As a result, a concrete geometric understanding of optical responses has so far been limited to two-level systems, where the Hilbert space is completely determined by a single state and its orthogonal complement. Here, we construct a general theory of Riemannian geometry for resonant optical processes by identifying transition dipole moment matrix elements as tangent vectors. This theory applies to arbitrarily high-order responses, suggesting that optical responses can generally be thought of as manifestations of the Riemannian geometry of quantum states. We use our theory to show that third-order photovoltaic Hall effects are related to the Riemann curvature tensor and demonstrate an experimentally accessible regime where they dominate the response.

Sensitivity Operator Framework for Analyzing Heterogeneous Air Quality Monitoring Systems

Air quality monitoring systems differ in composition and accuracy of observations and their temporal and spatial coverage. A monitoring system’s performance can be assessed by evaluating the accuracy of the emission sources identified by its data. In the considered inverse modeling approach, a source identification problem is transformed to a quasi-linear operator equation with the sensitivity operator. The sensitivity operator is composed of the sensitivity functions evaluated on the adjoint ensemble members. The members correspond to the measurement data element aggregates. Such ensemble construction allows working in a unified way with heterogeneous measurement data in a single-operator equation. The quasi-linear structure of the resulting operator equation allows both solving and predicting solutions of the inverse problem. Numerical experiments for the Baikal region scenario were carried out to compare different types of inverse problem solution accuracy estimates. In the considered scenario, the projection to the orthogonal complement of the sensitivity operator’s kernel allowed predicting the source identification results with the best accuracy compared to the other estimate types. Our contribution is the development and testing of a sensitivity-operator-based set of tools for analyzing heterogeneous air quality monitoring systems. We propose them for assessing and optimizing observational systems and experiments.

Cauchy’s surface area formula in the Heisenberg groups

We show an analogue of Cauchy's surface area formula for the Heisenberg groups $\mathbb{H}\_n$ for $n\geq 1$, which states that the p-area of any compact hypersurface $\Sigma$ in $\mathbb{H}\_n$ with its p-normal vector defined almost everywhere on $\Sigma$ is the average of its projected p-areas onto the orthogonal complements of all p-normal vectors of the Pansu spheres (up to a constant). The formula provides a geometric interpretation of the p-areas defined by Cheng–Hwang–Malchiodi–Yang in $\mathbb{H}\_1$ and Cheng–Hwang– Yang in $\mathbb{H}\_n$ for $n\geq 2$. We also characterize the projected areas for rotationally symmetric domains in $\mathbb{H}\_n$; namely, for any rotationally symmetric domain with boundary in $\mathbb{H}\_n$, its projected p-area onto the orthogonal complement of any normal vector of the Pansu spheres is a constant, independent of the choice of the projected directions.

A two-level preconditioned Helmholtz-Jacobi-Davidson method for the Maxwell eigenvalue problem

In this paper, based on a domain decomposition method, we propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as γ = c ( H ) ( 1 − C δ 2 H 2 ) \gamma =c(H)(1-C\frac {\delta ^{2}}{H^{2}}) , where C C is a constant independent of the mesh size h h and the diameter of subdomains H H , δ \delta is the overlapping size among the subdomains, and c ( H ) c(H) decreasing monotonically to 1 1 as H → 0 H\to 0 , which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory are given.

Sums of Dual Toeplitz Products on the Orthogonal Complements of the Hardy–Sobolev Spaces

Sums of Dual Toeplitz Products on the Orthogonal Complements of the Hardy–Sobolev Spaces