Orthonormal Research Articles

Novel method of constructing generalized Hoberman sphere mechanisms based on deployment axes

This study proposes a method of constructing type II generalized angulated elements (GAEs II) Hoberman sphere mechanisms on the basis of deployment axes that intersect at one point. First, the constraint conditions for inserting n GAEs II into n deployment axes to form a loop are given. The angle constraint conditions of the deployment axes are obtained through a series of linear equations. Second, the connection conditions of two GAEs II loops that share a common deployable center are discussed. Third, a flowchart of constructing the generalized Hoberman sphere mechanism on the basis of deployment axes is provided. Finally, four generalized Hoberman sphere mechanisms based on a fully enclosed regular hexahedron, arithmetic sequence axes, orthonormal arithmetic sequence axes, and spiral-like axes are constructed in accordance with the given arrangement of deployment axes that satisfy the constraint conditions to verify the feasibility of the proposed method.

Learning the tangent space of dynamical instabilities from data.

For a large class of dynamical systems, the optimally time-dependent (OTD) modes, a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory, are known to depend "pointwise" on the state of the system on the attractor but not on the history of the trajectory. We leverage the power of neural networks to learn this "pointwise" mapping from the phase space to OTD space directly from data. The result of the learning process is a cartography of directions associated with strongest instabilities in the phase space. Implications for data-driven prediction and control of dynamical instabilities are discussed.

A Pseudo-Random Beamforming Technique for Improving Physical-Layer Security of MIMO Cellular Networks

In this paper, we propose a pseudo-random beamforming (PRBF) technique for improving physical-layer security (PLS) in multiple input multiple output (MIMO) downlink cellular networks consisting of a legitimate base station (BS), multiple legitimate mobile stations (MSs) and potential eavesdroppers. The legitimate BS can obtain available potential eavesdroppers’ channel state information (CSI), which is registered in an adjacent cell. In the proposed PRBF technique, the legitimate BS pseudo-randomly generates multiple candidates of the transmit beamforming (BF) matrix, in which each transmit BF matrix consists of multiple orthonormal BF vectors and shares BF information with legitimate MSs before data transmission. Each legitimate MS generates receive BF vectors to maximize the receive signal-to-interference-plus-noise (SINR) for all pseudo-randomly generated transmit beams and calculates the corresponding SINR. Then, each legitimate MS sends a single beam index and the corresponding SINR value of the BF vector that maximizes the received SINR for each BF matrix since a single spatial stream is sent to each legitimate MS. Based on the feedback information from legitimate MSs and the CSI from the legitimate BS to eavesdroppers, the legitimate BS selects the optimal transmit BF matrix and the legitimate MSs that maximizes secrecy sum-rate. We also propose a codebook-based opportunistic feedback (CO-FB) strategy to reduce feedback overhead at legitimate MSs. Based on extensive computer simulations, the proposed PRBF with the proposed CO-FB significantly outperforms the conventional random beamforming (RBF) with the conventional opportunistic feedback (O-FB) strategies in terms of secrecy sum-rate and required feedback bits.

On infinite Jacobi matrices with a trace class resolvent

Let {Pn(x)} be an orthonormal polynomial sequence and denote by {wn(x)} the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for {Pn(x)} is determinate and denote by J the corresponding Jacobi matrix operator on ℓ2. We show that if J is positive definite and J−1 belongs to the trace class then the series on the right-hand side of the defining equation F(z)≔1−z∑n=0∞wn(0)Pn(z)converges locally uniformly on ℂ and it holds true that F(z)=∏n=1∞(1−z∕λn) where {λn;n=1,2,3,…}=specJ. Furthermore, the Al-Salam–Carlitz II polynomials are treated as an example of orthogonal polynomials to which this theorem can be applied.

Karhunen–Loève expansion of a set indexed fractional Brownian motion

This study presents the Karhunen–Loève expansion of a set indexed fractional Brownian motion (sifBM) XH={XAH}A∈A, based on “characterization of set indexed fractional Brownian motion by flows”. The characterization was proven by Herbin and Merzbach (2006), and it says that a set indexed process is a set indexed fractional Brownian motion if and only if its projections on all the increasing paths are one-parameter time changed fractional Brownian motions. The Karhunen–Loève expansion of a sifBM is: XAH=[μ(A)]H∑n=1∞eBnNnJν(γn(μ(A))H+12), for all A∈A.Where {eBn} is an orthonormal sequence of set indexed centered Gaussian variables, Jν is a Bessel function, Nn,γn are constants and {Bn}∈A. In addition, several cases of an expansion of a sifBM on [0,1]d are presented.

Sets of orthogonal three-dimensional polarization states and their physical interpretation

The spectral and characteristic decompositions of the polarization matrix provide fruitful frameworks for the physical interpretation of three-dimensional (3D) partially polarized light fields. The decompositions are formulated in terms of the three pure eigenstates, which in turn are represented through their associated orthogonal complex 3D Jones vectors. This mathematical orthogonality does not correspond, in general, to orthogonality of the polarization-ellipse planes of the respective eigenstates. Consequently, due to such inherent mathematical complexity, the geometric and physical interpretation of these sets of orthogonal complex vectors, being essential for the best understanding of the structure and properties of partially polarized 3D light, has not been addressed thoroughly. In this work, the geometric and physical features of sets of three orthonormal 3D Jones vectors are identified and analyzed, allowing one to obtain meaningful interpretations of any given mixed (partially polarized) 3D polarization state in terms of either the spectral or the characteristic decompositions. Among other results, it is found that, given a pure polarization state, any plane in space contains the polarization ellipse of a pure state that is orthogonal to it, and the mathematical expressions for the azimuth and ellipticity of such an ellipse are calculated in terms of the angular parameters determining said plane and the ellipticity of the given state. Furthermore, the spin vectors of the three polarization eigenstates are arranged in a peculiar spatial manner, such that they lie in a common plane. Beyond polarization phenomena, the approach presented also has potential applications in areas where 3 \ifmmode\times\else\texttimes\fi{} 3 unitary matrices play a key role, like three-level quantum systems and gates for ternary quantum logic circuits.

Parseval Frames of Exponentially Localized Magnetic Wannier Functions

Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension $d \le 3$, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model $m$ occupied energy bands by a real-analytic and $\mathbb Z^{d}$-periodic family $\{P({\bf k})\}_{{\bf k} \in \mathbb R^{d}}$ of orthogonal projections of rank $m$. A moving orthonormal basis of $\mathrm{Ran} P({\bf k})$ consisting of real-analytic and $\mathbb Z^d$-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of $P$ vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of $m-1$ orthonormal, real-analytic, and periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of $m+1$ real-analytic and periodic Bloch vectors which generate $\mathrm{Ran} P({\bf k})$. Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by $2d$ discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schr\"{o}dinger operators as well.

Mutually disjoint, maximally commuting set of physical observables for optimum state determination

We consider the state determination problem using mutually unbiased bases (MUBs). For spin-1, spin-3/2 and spin-2 systems, analogous to Pauli operators of spin-1/2 system, which are experimentally implementable and correspond to the optimum measurement in characterizing the density matrix, we describe a procedure to construct an orthonormal set of operators from MUBs. The constructed operators are maximally commuting, can be physically realized, and correspond to physical observables. The method of construction is general enough to allow for extensions to higher-dimensional spin systems and arbitrary density matrices in finite dimensions for which MUBs are known to exist.

Pipes Construction Based on the Pipes' Centre Curve Shapes

Parts of pipeline need a long piece, a short segment, and various inflate-deflate models. They require as well the thickness and curvature of the pipes. The objective of this paper is to obtain some formulas for modeling the long pipe, the short tube, and various inflate-deflate pipe patches. Relating to the purposes, we use their cross-section, longitudinal section, and center curves of the pipe parts. The methods are, the using of the polar coordinates and of the real functions, to define the cross and longitudinal section of the pipe patches, respectively. Then, we calculate three orthonormal vectors that are determined by the tangent vectors of the pipe center curves and two unit vectors that are perpendicular to the tangent vectors. After that, we evaluate the formulas to model the long pipes and the short pipes, both inflate-deflate and thickness shapes. The results show that, using its center curves of the pipe, it is handy to design the long and short pipes, multiple thicknesses, various volume fluctuations of the pipes, and useful to model the inflate-deflate pipe parts.

Singularities of base polynomials and Gau–Wu numbers

In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n×n matrix A, which counts the maximal number of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range W(A). We refer to k(A) as the Gau–Wu number of the matrix A. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify k(A). This continues the work of Wang and Wu [14] classifying the Gau–Wu numbers for 3×3 matrices. Our focus on singularities is inspired by Chien and Nakazato [3], who classified W(A) for 4×4 unitarily irreducible A with irreducible base curve according to singularities of that curve. When A is a unitarily irreducible n×n matrix, we give necessary conditions for k(A)=2, characterize k(A)=n, and apply these results to the case of unitarily irreducible 4×4 matrices. However, we show that knowledge of the singularities is not sufficient to determine k(A) by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different k(A). In addition, we extend Chien and Nakazato's classification to consider unitarily irreducible A with reducible base curve and show that we can find corresponding matrices with identical base curve but different k(A). Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky [5], generalizing their result in the process.

MLACE-MLFMA Combined With Reduced Basis Method for Efficient Wideband Electromagnetic Scattering From Metallic Targets

In this paper, a fast interpolation algorithm [multilevel accelerated Cartesian expansion (MLACE)-multilevel fast multipole algorithm (MLFMA)-reduced basis method (RBM)] is proposed for the first time by leveraging the benefits of the MLACE, the classical MLFMA, and the RBM; furthermore, a novel nonblind sampling method (NBSM) is proposed to approximate an orthogonal subspace for the MLACE-MLFMA-RBM. By integrating the empirical interpolation method (EIM) and the error-based compensation method (EBCM) into the NBSM, we first efficiently establish a robust and precision-controllable orthogonal subspace. Next, with the accurate orthogonal subspace, the MLACE-MLFMA-RBM can simplify the iterative solution process of the MLACE-MLFMA or the MLFMA into N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> matrix-vector multiplications (MVMs) and the solution of a reduced N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> -dimensional linear equation (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> is the number of orthonormal vectors). Note that only the MVMs between the far-region impedance matrix and the orthogonal vector bases are calculated. In comparison with the classical MLACE-MLFMA and the MLACE-MLFMA-RBM based on different sampling methods for wideband electromagnetic (EM) responses, our proposed algorithms require less memory and CPU time and integrate higher flexibility. The numerical results are provided to demonstrate the accuracy, efficiency, and flexibility.

Multipole expansion for magnetic structures: A generation scheme for a symmetry-adapted orthonormal basis set in the crystallographic point group

We propose a systematic method to generate a complete orthonormal basis set of multipole expansion for magnetic structures in arbitrary crystal structure. The key idea is the introduction of a virtual atomic cluster of a target crystal, on which we can clearly define the magnetic configurations corresponding to symmetry-adapted multipole moments. The magnetic configurations are then mapped onto the crystal so as to preserve the magnetic point group of the multipole moments, leading to the magnetic structures classified according to the irreducible representations of crystallographic point group. We apply the present scheme to pyrhochlore and hexagonal ABO3 crystal structures, and demonstrate that the multipole expansion is useful to investigate the macroscopic responses of antiferromagnets.

Some results on the Laplacian Spread Conjecture

For a graph G of order n, let λ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that λ2(G)+λ2(G‾)≥1, where G‾ is the complement of G. For any x∈Rn, let ∇x∈R(n2) be the vector whose {i,j}-th entry is |xi−xj|. In this paper, we show the aforementioned conjecture is equivalent to prove that every two orthonormal vectors f,g∈Rn with zero mean satisfy‖∇f−∇g‖2≥2. In this article, it is shown that for the validity of the conjecture it suffices to prove that the conjecture holds for all permutation graphs.

Deep Learning-Based Pilot Design for Multi-User Distributed Massive MIMO Systems

This letter proposes a deep learning based pilot design scheme to minimize the sum mean square error (MSE) of channel estimation for multi-user distributed massive multiple-input multiple-output (MIMO) systems. The pilot signal of each user is expressed as a weighted superposition of orthonormal pilot sequence basis, where the power assigned to each pilot sequence is the corresponding weight. A multi-layer fully connected deep neural network (DNN) is designed to optimize the power allocated to each pilot sequence to minimize the sum MSE, which takes the channel large-scale fading coefficients as input and outputs the pilot power allocation vector. The loss function of the DNN is defined as the sum MSE, and we leverage the unsupervised learning strategy to train the DNN. Simulation results show that the proposed scheme achieves better sum MSE performance than other methods with low complexity.

Even-odd effect in higher-order holographic production of electron vortex beams with nontrivial radial structures

Structured electron beams carrying orbital angular momentum are currently of considerable interest, both from a fundamental point of view and for application in electron microscopy and spectroscopy. Until recently, most studies have focused on the azimuthal structure of electron vortex beams with well-defined orbital angular momentum. To unambiguously define real electron-beam states and realise them in the laboratory, the radial structure must also be specified. Here we use a specific set of orthonormal modes of electron (vortex) beams to describe both the radial and azimuthal structures of arbitrary electron wavefronts. The specific beam states are based on truncated Bessel beams localised within the lens aperture plane of an electron microscope. We show that their Fourier transform set of beams can be realised at the focal planes of the probe-forming lens using a binary computer generated electron hologram. Using astigmatic transformation optics, we demonstrate that the azimuthal indices of the diffracted beams scale with the order of the diffraction through phase amplification. However, their radial indices remain the same as those of the encoding beams for all the odd diffraction orders or are reduced to the zeroth order for the even-order diffracted beams. This simple even-odd rule can also be explained in terms of the phase amplification of the radial profiles. We envisage that the orthonormal cylindrical basis set of states could lead to new possibilities in phase contrast electron microscopy and spectroscopy using structured electron beams.

Teaching about Schimidt's orthogonalization

Teaching about Schimidt's orthogonalization

Orthonormal sequences and time frequency localization related to the Riemann-Liouville operator

Orthonormal sequences and time frequency localization related to the Riemann-Liouville operator

Characteristic Vectors in p/q-Channel Orthonormal Wavelet

Wavelet analysis is a newly rapidly developing subject in the late twentieth century. As a time-frequency analysis tool, wavelet analysis has many advantages over other time-frequency tools, such as in signal processing, image processing, speech processing, pattern recognition, quantum physics and other fields. Multiresolution analysis (MRA for short) is an important method for studying wavelet orthonormal wavelet bases with rational dilation 2. However, p/q-band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing and attracted more and more interest in recent years. But there are relatively less results for the case of p/q-band. This paper studies the orthonormal wavelet bases with rational dilation factor p/q based on multiresolution analysis by a polyphase decomposition technique. First, we gave the concept of Multiresolution analysis with rational dilation p/q and deduced an identity of the masks matrix. Also, a perfect reconstruction condition in terms of masks was presented. Further, we gave the refinement and wavelet matrices respectively and derived the characteristic roots and the corresponding orthonormal characteristic vectors of the wavelet matrix, and then a method with characteristic vectors was reduced to achieve the orthonormal wavelet bases with rational dilation factor p/q. In the end, an example is offered to verify this theory.

A Reinforced Blind Color Image Watermarking Scheme Based on Schur Decomposition

In this paper, the deficiency of Schur decomposition (SD) for image watermarking has been comprehensively investigated. A remedial scheme is developed not only to fix the existing problems but also to reinforce the performance in robustness and imperceptibility. This scheme starts with partitioning the host image into non-overlapping blocks of 4 × 4 pixels and then applying SD to each block individually. Level shifting serves as a controlling gauge for embedding strength. The use of intentional perturbation prevents nil orthonormal vectors derived from zero eigenvalues. The dominant vector is identified by analyzing the entire Schur matrix instead of just diagonal elements. To achieve effective watermark embedding and extraction, the orthonormality of the acquired unitary matrix ought to be preserved while the resulting distortion can be compensated via the systematic modification of the Schur matrix. Finally, a recursive regulation guarantees the retrieval of watermark bits. Experiment results indicate that the proposed scheme is free of errors in the absence of attacks and can withstand a variety of image processing attacks as well. Moreover, the inclusion of distortion compensation contributes an improvement of 2.47 dB in terms of peak signal-tonoise ratio. In comparison with previous SD-based schemes, the proposed one exhibits superior robustness and imperceptibility while operating at the same payload capacity.

Behavior of a Scale Factor for Wiener Integrals of an Unbounded Function

The purpose of this paper is to investigate the behavior of a scale factor for Wiener integrals about the unbounded function , where {a1,a2,...,an} is an orthonormal set of elements in L2[0,T] on the Wiener space C0[0,T].