Orthogonality Preserving Research Articles

An identity for (δ,ε)-approximately orthogonality preserving mappings

Suppose T is a Hilbert space operator. Given δ∈[0,1), we define εˆδ(T) to be the smallest ε for which T is (δ,ε)-approximately orthogonality preserving, and then obtain an exact formula for εˆδ(T) in terms of δ,‖T‖ and the minimum modulus m(T) of T. For two nonzero operators T,S, it follows from the formula that T is (εˆ(S),ε)-AOP if and only if S is (εˆ(T),ε)-AOP, where εˆ(T)=εˆ0(T). Finally, we show that an operator T is (δ,ε)-AOP if and only if there exists a “special” δ-AOP operator S such that TS is ε-AOP [Theorem 3.8].

Mappings preserving approximate orthogonality in Hilbert $C^*$-modules

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta , \varepsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta , \varepsilon )$-orthogonality preserving. In particular, if $\mathscr {E}$ is a full Hilbert $\mathscr {A}$-module with $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T, S\colon \mathscr {E}\to \mathscr {E}$ are two linear mappings satisfying $|\langle Sx, Sy\rangle | = \|S\|^2|\langle x, y\rangle |$ for all $x, y\in \mathscr {E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta , \varepsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T\colon \mathscr {E} \to \mathscr {F}$ is a nonzero $\mathscr {A}$-linear $(\delta , \varepsilon )$-orthogonality preserving mapping between $\mathscr {A}$-modules, then \[ \bigl \|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle \bigr \|\leq \frac {4(\varepsilon - \delta )}{(1 - \delta )(1 + \varepsilon )} \|Tx\|\|Ty\|\qquad (x, y\in \mathscr {E}). \] As a result, we present some characterizations of the orthogonality preserving mappings.

Nonlinear Model for Condition Monitoring and Fault Detection Based on Nonlocal Kernel Orthogonal Preserving Embedding

The dimension reduction methods have been proved powerful and practical to extract latent features in the signal for process monitoring. A linear dimension reduction method called nonlocal orthogonal preserving embedding (NLOPE) and its nonlinear form named nonlocal kernel orthogonal preserving embedding (NLKOPE) are proposed and applied for condition monitoring and fault detection. Different from kernel orthogonal neighborhood preserving embedding (KONPE) and kernel principal component analysis (KPCA), the NLOPE and NLKOPE models aim at preserving global and local data structures simultaneously by constructing a dual‐objective optimization function. In order to adjust the trade‐off between global and local data structures, a weighted parameter is introduced to balance the objective function. Compared with KONPE and KPCA, NLKOPE combines both the advantages of KONPE and KPCA, and NLKOPE is also more powerful in extracting potential useful features in nonlinear data set than NLOPE. For the purpose of condition monitoring and fault detection, monitoring statistics are constructed in feature space. Finally, three case studies on the gearbox and bearing test rig are carried out to demonstrate the effectiveness of the proposed nonlinear fault detection method.

Necessary condition for local distinguishability of maximally entangled states: Beyond orthogonality preservation

The (im)possibility of local distinguishability of orthogonal multipartite quantum states still remains an intriguing question. Beyond ${\mathbb{C}}^{3}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, the problem remains unsolved even for maximally entangled states (MESs). So far, the only known condition for the local distinguishability of states is the well-known orthogonality preservation (OP). Using an upper bound on the locally accessible information for bipartite states, we derive a very simple necessary condition for any set of pairwise orthogonal MESs in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ to be perfectly locally distinguishable. It is seen that particularly when the number of pairwise orthogonal MES states in ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ is equal to $d$, then this necessary condition, along with the OP condition, imposes more constraints (for said states to be perfectly locally distinguishable) than the OP condition does. When testing this condition for the local distinguishability of all sets of four generalized Bell states in ${\mathbb{C}}^{4}\ensuremath{\bigotimes}{\mathbb{C}}^{4}$, we find that it is not only necessary but also sufficient to determine their local distinguishability. This demonstrates that the aforementioned upper bound may play a significant role in the general scenario of local distinguishability of bipartite states.

Linear mappings approximately preserving [formula omitted]-orthogonality

We define the concept of approximate ρ∗-orthogonality preserving mapping between normed linear spaces and give some characterizations for a linear mapping to be approximately ρ∗-orthogonality preserving. Moreover, we show that every approximately ρ∗-orthogonality preserving linear mapping is necessarily a scalar multiple of an almost isometry.

Improved Artificial Neural Network (ANN) With Aid of Artificial Bee Colony (ABC) For Medical Data Classification

The ultimate aim of the proposed method is to establish a model for classification of medical data. Various methods have been generated to health related data to detect upcoming health fitness usage including detecting person's spending and illness related issues for diseased persons. In order to achieve promising results in medical data classification, we have planned to utilise orthogonal local preserving projection and optimal classifier. Initially, the pre-processing will be applied for extracting useful information and to convert suitable sample from raw medical datasets. Here, orthogonal local preserving projection (OLPP) is used to reduce the feature dimension. Once the feature reduction is formed, the prediction will be done based on the optimal classifier. In the optimal classifier, artificial bee colony algorithm will be used with neural network. The effectiveness of our proposed is measured in terms of accuracy, sensitivity and specificity. Here, Switzerland dataset achieves the maximum accuracy value 95.935%.

Optimal fuzzy min-max neural network for medical data classification using group search optimiser algorithm

Several techniques were applied to healthcare datasets for the prediction of future healthcare utilisation such as predicting individual expenditures and disease risks for patients. In order to achieve promising results in medical data classification, we have planned to utilise orthogonal local preserving projection and optimal classifier. Initially, the pre-processing will be applied to extract useful data and to convert suitable samples from raw medical datasets. Feature dimension reduction method will be applied to reduce the features' space without losing the accuracy of prediction. Here, orthogonal local preserving projection (OLPP) will be used. Once the feature reduction is formed, the prediction will be carried out based on the optimal classifier. In the optimal classifier, group search optimiser algorithm will be used for fuzzy min-max neural network. Here, the experimentation is done by using various datasets from UCI machine learning repository.

Orthogonality Preservers of JB*-triple-valued Functions

Let $X, Y$ be locally compact Hausdorff spaces, and $V, W$ be JB*-triples such that $W$ is prime. Let $T \colon C_0(X, V) \to C_0(Y, W)$ be an orthogonality preserving linear operator with dense range. We show that $T$ can be written as a weighted composition operator.

Orthogonal forms and orthogonality preservers on real function algebras revisited

In [Garcés J, Peralta AM. Orthogonal forms and orthogonality preservers on real function algebras. Linear Multilinear Algebra. 2014;62:275–296.] we determine the precise form of a continuous orthogonal form on a commutative real C-algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital real C-algebras. Among the consequences, we show that every orthogonality preserving linear bijection between commutative unital real C-algebras is continuous. In this note we revisit these results and their proofs with the aim of filling a gap in the arguments, and to extend the original conclusions.

Orthogonality breaking through few-mode optical fiber.

Polarization sensing and imaging through optical fibers is a technological challenge motivated by promising applications for in vivo, in situ polarimetric endoscopy for biomedical diagnosis. Among the recent approaches proposed to solve this issue, the depolarization/dichroism sensing by polarization orthogonality breaking (DSOB) technique was shown to perform remotely through single-mode optical fibers for depolarization/diattenuation measurements. In this article, we investigate the applicability of such a technique in slightly multimode waveguides. Through theoretical modeling and numerical simulations, we evidence the conditions required for the polarization orthogonality to be preserved after propagation in a few-mode fiber, notably in terms of detection geometry of the spatial modes. Original experiments realized in few-mode fibers both in transmission and reflection configurations are also reported and validate the theoretical predictions. These results allow us to analyze the influence of the experimental parameters, such as detection geometry, sample tilt, or fiber length, on orthogonality preservation and on the measurement dynamics of the DSOB technique in slightly multimode waveguides.

Isometries of Grassmann spaces

Botelho, Jamison, and Molnár have recently described the general form of surjective isometries of Grassmann spaces on complex Hilbert spaces under certain dimensionality assumptions. In this paper we provide a new approach to this problem which enables us first, to give a shorter proof and second, to remove dimensionality constraints completely. In one of the low dimensional cases, which was not covered by Botelho, Jamison, and Molnár, an exceptional possibility occurs. As a byproduct, we are able to handle the real case as well. Furthermore, in finite dimensions we remove the surjectivity assumption. A variety of tools is used in order to achieve our goal, such as topological, geometrical and linear algebra techniques. The famous two projections theorem for two finite rank projections will be re-proven using linear algebraic methods. A theorem of Györy and the second author on orthogonality preservers on Grassmann spaces will be strengthened as well. This latter result will be obtained by using Chow's fundamental theorem of geometry of Grassmannians.

High-Quality Mesh Deformation Using Quaternions for Orthogonality Preservation

Many mesh deformation techniques developed in the past have been widely used, but for large deformations, the mesh properties can be severely altered. In particular, the mesh lines orthogonality near the boundaries can be lost. In this paper, a method initially proposed by Samareh (“Application of Quaternions for Mesh Deformation,” NASA TM-2002-211646, April 2002) is developed to overcome this limitation and to provide a high-quality mesh-orthogonality preservation technique. Each node displacement is composed of a translation vector and a quaternion, which takes into account local mesh rotations. Algebraic unitary quaternion properties (Lie group algebra) are used to obtain a robust interpolation method. Two quaternion interpolation methods are compared, respectively: the spherical linear interpolation, and the Lie algebra linear interpolation. Furthermore, thanks to a new process (transformation division), the method becomes independent of the rotation center used. The accuracy and generality of the method is significantly improved by this new process, and different test cases (two-dimensional and three-dimensional) are presented to illustrate the benefits obtained. Global mesh quality is shown to be preserved, especially local mesh orthogonality near the moving surfaces. The application to an iced airfoil shows the capability of the method to propagate very severe deformations while providing deformed mesh that enables accurate Reynolds-averaged Navier–Stokes computational fluid dynamics calculations.

Orthogonal forms and orthogonality preservers on real function algebras

We initiate the study of orthogonal forms on a real C-algebra. Motivated by previous contributions, due to Ylinen, Jajte, Paszkiewicz and Goldstein, we prove that for every continuous orthogonal form on a commutative real C-algebra,, there exist functionals and in satisfying for every in. We describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between unital commutative real C-algebras. As a consequence, we show that every orthogonality preserving linear bijection between unital commutative real C-algebras is continuous.

Single SOA based simultaneous amplitude regeneration for WDM-PDM RZ-PSK signals

We propose and demonstrate all-optical amplitude regeneration for the wavelength division multiplexing and polarization division multiplexing (WDM-PDM) return-to-zero phase shift keying (RZ-PSK) signals using a single semiconductor optical amplifier (SOA) and subsequent filtering. The regeneration is based on the cross phase modulation (XPM) effect in the saturated SOA and the subsequent narrow filtering. The spectrum of the distorted signal can be broadened due to the phase modulation induced by the synchronous optical clock signal. A narrow band pass filter is utilized to extract part of the broadened spectrum and remove the amplitude noise, while preserving the phase information. The working principle for multi-channel and polarization orthogonality preserving is analyzed. 4-channel dual polarization signals can be simultaneously amplitude regenerated without introducing wavelength and polarization demultiplexing. An average power penalty improvement of 1.75dB can be achieved for the WDM-PDM signals.

Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

We study holomorphic maps between C*-algebrasAandB, whenf:BA(0,ϱ)→Bis a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ballU=BA(0,δ). If we assume thatfis orthogonality preserving and orthogonally additive onAsa∩Uandf(U)contains an invertible element inB, then there exist a sequence(hn)inB**and Jordan*-homomorphismsΘ,Θ~:M(A)→B**such thatf(x)=∑n=1∞hnΘ~(an)=∑n=1∞Θ(an)hnuniformly ina∈U. WhenBis abelian, the hypothesis ofBbeing unital andf(U)∩inv(B)≠∅can be relaxed to get the same statement.

Orthogonal Complete Discriminant Locality Preserving Projections for Face Recognition

In this paper, we propose a novel orthogonal complete discriminant locality preserving projections for facial feature extraction and recognition (OCDLPP). All training samples are projected into the range of a so-called locality preserving total scatterto reduce dimensionality without loss of discriminative information. The transformation matrix of OCDLPP is orthogonal and is found simultaneously using QR decomposition technique. Moreover, a feasible and effective procedure is proposed to alleviate the computational burden of high dimensional matrix for typical face image data. Experiments results on the ORL, Yale, FERET and PIE face databases show the effectiveness of the proposed OCDLPP.

LINEAR ORTHOGONALITY PRESERVERS OF HILBERT BUNDLES

AbstractA ℂ-linear map θ (not necessarily bounded) between two Hilbert C*-modules is said to be ‘orthogonality preserving’ if 〈θ(x),θ(y)〉=0 whenever 〈x,y〉=0. We prove that if θ is an orthogonality preserving map from a full Hilbert C0(Ω)-module E into another Hilbert C0(Ω) -module F that satisfies a weaker notion of C0 (Ω) -linearity (called ‘localness’), then θ is bounded and there exists ϕ∈Cb (Ω)+ such that 〈θ(x),θ(y)〉=ϕ⋅〈x,y〉 for all x,y∈E.

LINEAR ORTHOGONALITY PRESERVERS OF STANDARD OPERATOR ALGEBRAS

In 2003, Araujo and Jarosz showed that every bijective linear map $\theta: A \to B$ between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both $A,B$ are unital standard algebras on Hilbert spaces and $\theta$ preserves range or domain orthogonality. In particular, such maps are automatically bounded. In this paper, we will study linear orthogonality preservers in a unified way. We will show that every surjective linear map between standard operator algebras preserving range/domain orthogonality carries a standard form, and is thus automatically bounded.

LINEAR ORTHOGONALITY PRESERVERS

In 2003, Araujo and Jarosz showed that every bijective linear map θ : A → B between unital standard operator algebras preserving zero products in two ways is a scalar multiple of an inner automorphism. Later in 2007, Zhao and Hou showed that similar results hold if both A,B are unital standard algebras on Hilbert spaces and θ preserves range or domain orthogonality. In particular, such maps are automatically bounded. In this paper, we will study linear orthogonality preservers in a unified way. We will show that every surjective linear map between standard operator algebras preserving range/domain orthogonality carries a standard form, and is thus automatically bounded.

Approximately orthogonality preserving mappings on [formula omitted]-modules

We study orthogonality preserving and approximately orthogonality preserving mappings in the setting of inner product C ∗ -modules. In particular, if V and W are inner product C ∗ -modules over the C ∗ -algebra A , any scalar multiple of an A -linear isometry is an A -linear orthogonality preserving mapping. The converse does not hold in general, but it holds if A contains K ( H ) (the C ∗ -algebra of all compact operators on a Hilbert space H ). Furthermore, we give the estimate of ‖ 〈 T x , T y 〉 − ‖ T ‖ 2 〈 x , y 〉 ‖ for an A -linear approximately orthogonality preserving mapping T : V → W when V and W are inner product C ∗ -modules over a C ∗ -algebra containing K ( H ) . In the case A = K ( H ) and V, W are Hilbert, we also prove that an A -linear approximately orthogonality preserving mapping can be approximated by an A -linear orthogonality preserving mapping.