Orthogonality Relations Research Articles

Partial eigenvalue assignment of descriptor fractional discrete-time linear systems by parametric state feedback

In this paper, we present a nonlinear parametric method to stabilize descriptor fractional discrete time linear system practically. Parametric methods with the free parameters can be adjusted to obtain better performance responses like minimum norm in state feedback. The aim is assigning desirable eigenvalues to obtain satisfactory responses by forward state feedback and forward and propositional state feedback in new systems with large matrices. However, finding the solution to nonlinear parametric equations makes some errors. In partial eigenvalue assignment, just a part of the open-loop spectrum of the standard linear systems is reassigned, while leaving the rest of the spectrum invariant. The size of matrices, state, and input vectors are decreased and the stability is kept. At the end, summary and conclusions are proposed and the convergence of state vectors in the descriptor fractional discrete-time system to zero is also shown by figures in a numerical example. Our method is also compared with another method with one of orthogonality relations in our article and example.

A ( p,q )-analogue of Qi-type formula for r -Dowling numbers

In this paper, \((p,q)\)-analogues of \(r\)-Whitney numbers of the first and second kinds are defined using horizontal generating functions. Several fundamental properties such as orthogonality and inverse relations, an explicit formula, and a kind of exponential generating function are obtained. Moreover, a \((p,q)\)-analogue of \(r\)-Whitney-Lah numbers is also defined in terms of a horizontal generating function, where necessary properties are obtained. These properties help develop a \((p,q)\)-analogue of the \(r\)-Dowling numbers, particularly, a \((p,q)\)-analogue of a Qi-type formula.

Free Vibrations of Cylindrical Shells by The Nodal Line Finite Difference Method-Part-I Theory.(Dept.C)

In this paper, the nodal line finite difference method (NLFDM) is extended to the analysis of a thin circular cylindrical shell, as shown in Fig. 1, undergoing free vibrations [1]. The material of the circular cylindrical shell is assumed to be linear elastic isotropic. The complexity of a theoretical analysis of a vibration problem depends largely on the number of degrees of freedom of the structural system in question. Since a thin circular cylindrical shell is a continuous system it has an infinite number of degrees of freedom. The NLFDM [10] transforms this continuous system into a system having a finite number of degrees of freedom. While carrying out the dynamic (or static) analysis by the NLFDM, the number of degrees of freedom depends on the number of the used nodal lines. The natural frequencies and modes are treated in more detail because they are basic to understanding the dynamic response under any kind of excitation. It will be shown that the number of natural frequencies and that of normal modes, for axial wave number different from zero, are each equal to three times the number of nodal lines used. The orthogonality relationships of the normal modes obtained by the nodal line finite difference method are also included. It should be remarked that these relationships are slightly different from those obtained by the finite element method; in fact the overall matrix of coefficients replaces the stiffness matrix, and any two different modes are orthogonal with resp ect to the unit matrix instead of the mass matrix.

Complete solutions of inverse quantum orthogonal equivalence classes

In the year 1939, the mathematician G.H. Hardy proved that the only functions f which satisfy the classical orthogonality relation ∫01f(λmt)f(λnt)dt=0,m≠n,are the Bessel functions Jν(t) under certain constraints, where ν>−1 is the order of the Bessel function, and λm, λn are the nontrivial zeros of the Bessel function. By example, herein we present a complete equivalence class of inverse q-periodic functions, satisfying the inverse q-orthogonality relation ∫01f(λmt)f(λnt)d1qt=0,m≠n.The inverse q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points q−ℓ, with the step size at the point q−ℓ being q, ∀ℓ∈N0, where N0≔N∪{0}, and |q|≠1. An equivalence class of entire functions f∈Lq−12(0,1) that are inverse q-orthogonal with respect to their own nontrivial zeros is proven by invoking the Fundamental Theorem of Algebra, as the inverse q-periodic functions constitute an equivalence class of nonzero inverse q-periodic constants on the complex plane.

UKF based high-precision data fusion approach of orthogonal IMU sensors

Accurate attitude angle measurement is very important for some automatic robots, but the noise uncertainty in the sensor can cause measurement error. In order to solve the problem that these pose sensors have low estimation accuracy, this paper proposes a data fusion approach using the IMU sensor including gyroscope, accelerometer and magnetometer. Through placing multi-sensors with spatial orthogonal relation, global optimization is achieved, and the UKF algorithm is adopted to fuse data. This fusion uses the gyroscope data as the state vector, and regards the accelerometer and magnetometer data as the observation vector to improve the accuracy of attitude angle measurement. The experimental result is performed to compare the proposed method with traditional method, and the proposed method can obtain better attitude measurement accuracy, which illustrates the effectiveness of the proposed method.

A semi-analytical model for the modal density of periodic mediums based on the symplectic method

In this paper, a semi-analytical approach is provided for the modal density of periodic mediums based on the symplectic method. For two-dimensional periodic mediums with a plate component and one-dimensional periodic mediums with a beam component and truss component, the symplectic method is introduced to describe the conditions of continuity and periodicity of the unit cell. And then by virtue of the adjoint symplectic orthogonal relations, an eigenproblem is first established for the dispersion relation of the periodic mediums. The group velocity is then obtained semi-analytically by differentiating the eigenproblem with respect to frequency. Since the expressions of the kinematic and the kinetic variables of the unit cell involved in derivation processes are expressed in terms of symplectic analytical waves, the modal density of periodic mediums can be obtained with high efficiency and with high accuracy. Numerical examples including two-dimensional periodic mediums with a plate component and one-dimensional periodic mediums with a beam component and truss component are provided. The comparison of the present results with the results obtained from the finite element model confirms the effectiveness of the proposed method.

An inhomogeneous problem for an elastic half-strip: An exact solution

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Gold remobilization in gossans of the Amani area, southwestern Tanzania

Oxide facies gold deposits are characterised by high Au-tenor free-milling gold nuggets and differ to their associated sulphide assemblages which have a lower Au-tenor. These changes point to an active gold geochemical cycle during the transitioning from sulphide- to oxide facies, which commonly also correlates with the formation of an iron oxide-rich gossan. Surficial gold occurrences in the Amani area of southwestern Tanzania are hosted by gossanous outcrops comprising weathered- to highly weathered quartz-carbonate veins and high Au-tenor gold, and thus represent an ideal natural laboratory in which to study supergene gold enrichment processes. The gossan mineralogy comprises goethite, hematite, malachite, azurite, siderite and quartz assemblages, with supergene gold particles up to 6 mm and with an average gold concentration of 95 wt%. The supergene gold particles have irregular and elongate morphologies, with elongate gold occurring exclusively along narrow ribs of neoformed goethite. The crystal habit of this gold shows a persistent orthogonal relationship between adjacent sub-micrometre-sized terraces, reflecting a possible crystallographic control on supergene gold precipitation. Novel results from 3-D X-ray computed tomography reveal that the micrometre- to millimetre scale distribution of gold follows a near-orthogonal pattern controlled by the textures and fluid flow pathways within the developing gossan. These elongate gold particles form interconnected 3-D skeletal gold networks which coalesce at discrete intersections to form larger nuggets (up to 6 mm in length) of high Au-tenor supergene gold. A detailed geochemical consideration of the mineralogy of the gossan, its precursor vein paragenesis and the surrounding host rocks suggest that the physicochemical conditions within the developing gossan favour supergene gold growth by dissolution-re-precipitation mechanisms involving an intermediate gold thiosulphate ligand complex.

Comparative analysis of Rayleigh and Love waves detected propagating in the Nobi and Kanto basins during the 2004-, 2007- Chuetsu and 2011 Tohoku earthquakes

Propagation of long-period ground motion in sedimentary basins has been a subject of great interest among seismologists and engineers. Intense long-period ground motions consist primarily of surface waves that get trapped or generated locally as seismic energy travels through sedimentary deposits. In the present work, we investigate the propagation of surface waves in the basins of Kanto and Nobi in Japan, during three relatively recent events: The Mw 6.6 2004 Niigata Chuetsu, the Mw 6.6 2007 Chuetsu-Oki and the Mw 9.0 2011 Tohoku earthquakes. We identify the surface waves using a signal processing technique that detects their polarization characteristics, in the time-frequency space, using orthogonality relations among phase vectors. Then, by applying the “Normalized Inner Product” (NIP), regions of a particular type of polarization are delineated and filters are applied to isolate the associated surface waves, along with their direction of propagation. With our investigation, we attempt to follow the ‘flow’ of seismic energy as it approaches just outside the basins, and then how it evolves once inside the basin. Our analysis shows that the long period (<0.1 Hz) surface wave energy approaching the Kanto basin during the 2011 Tohoku earthquake consists of Rayleigh waves, and that part of the seismic energy is converted to Love waves. In a higher frequency range (0.1–0.5 Hz), prograde Rayleigh and Love waves were detected in selected areas such as the Chiba sub-basin and the Tokyo lowlands. Regarding the Nobi basin, we find that whereas the Rayleigh waves in the frequency range (0.1–0.5 Hz) radiated during the 2011 Tohoku earthquake strongly interact with the basin, the Rayleigh waves radiated by the Chuetsu events appear to propagate through the basin unaffected. This difference in basin response is attributed to the different azimuthal direction of incidence of the surface wave energy.

A q-analogue of generalized translated Whitney numbers: Cigler's approach

Using a certain combinatorial interpretation in terms of set partition, a \(q\)-analogue of generalized translated Whitney numbers of the second kind is defined in this paper. Some properties such as the recurrence relation, explicit formula, and certain symmetric formula are obtained. Moreover, a \(q\)-analogue of generalized translated Whitney numbers of the first kind is introduced to obtain a \(q\)-analogue of the orthogonality and inverse relations of the two kinds of generalized translated Whitney numbers.

Special affine wavelet transform and the corresponding Poisson summation formula

In the article, “Windowed special affine Fourier transform” in J. Pseudo-Differ. Oper. Appl. (2020), we introduced the notion of windowed special affine Fourier transform (WSAFT) as a ramification of the special affine Fourier transform. Keeping in view the fact that the WSAFT is not befitting for in the context of non-stationary signals, we continue our endeavor and introduce the notion of the special affine wavelet transform (SAWT) by combining the merits of the special affine Fourier and wavelet transforms. Besides studying the fundamental properties of the SAWT including orthogonality relation, inversion formula and range theorem, we also demonstrate that the SAWT admits the constant [Formula: see text]-property in the time–frequency domain. Moreover, we formulate an analog of the well-known Poisson summation formula for the proposed SAWT.

An orthogonality relation for (with an appendix by Bingrong Huang)

Abstract Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

Octonion Short-Time Fourier Transform for Time-Frequency Representation and Its Applications

The octonion Fourier transform (OFT) is a useful tool for signal processing and analysis. However, due to the lack of time localization information, it is not suitable for processing signals whose frequencies vary with time. In this paper, we utilize octonion algebra to propose a new method for timefrequency representation (TFR) called the octonion short-time Fourier transform (OSTFT). The originality of the method is based on the quaternion short-time Fourier transform (QSTFT). First, we generalize the QSTFT to the OSTFT by substituting the quaternion kernel function with the octonion kernel function in the definition of the QSTFT, and the physical significance of the OSTFT is also presented. Then, several essential properties of the OSTFT are derived, such as linearity, inversion formulas, timefrequency shifts and orthogonality relation. Based on the classic Fourier convolution operation, the convolution theorem for the OSTFT is derived. We apply the relationship between the OFT and OSTFT to establish Pitts inequality and Liebs inequality for the OSTFT. According to the logarithmic uncertainty principle of the OFT, the logarithmic uncertainty principle associated with the OSTFT is investigated. Finally, an application in which OSTFT can be used to study linear time varying (LTV) systems is proposed, and some potential applications of the OSTFT are also introduced.

Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as a more encompassing characterization of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree ℓD≥1, the three term recurrence relations are broken. The extra ℓD “lower degree polynomials,” which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree ℓD+N. The discrete orthogonality relations are shown to hold for another group of “new” orthogonal polynomials called Krein–Adler polynomials based on the Hermite, Laguerre, and Jacobi polynomials.

Q−Orthogonal dualities for asymmetric particle systems

We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP(q,θ), asymmetric exclusion process, with a repulsive interaction, allowing up to θ∈N particles in each site, and the ASIP(q,θ), θ∈R+, asymmetric inclusion process, that is its attractive counterpart. We extend to the asymmetric setting the investigation of orthogonal duality properties done in [8] for symmetric processes. The analysis leads to multivariate q−analogues of Krawtchouk polynomials and Meixner polynomials as orthogonal duality functions for the generalized asymmetric exclusion process and its asymmetric inclusion version, respectively. We also show how the q-Krawtchouk orthogonality relations can be used to compute exponential moments and correlations of ASEP(q,θ).

General Synthesis Method for Dispersively Coupled Resonator Filters With Cascaded Topologies

This article presents a new synthesis technique for circuits that may include dispersively coupled resonators and admitting an overall cascaded topology. A decomposition technique of the Darlington type is first introduced to split the original response S of the filter, taken as its scattering matrix, into m subresponses S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ,⋯, S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> corresponding to each subblock of the cascaded circuital structure. Each individual subresponse Sk is then synthesized separately. In this article, the state space equations governing the model of dispersively coupled resonators are detailed. An extension to the case of dispersive coupling of the shortest path rule, which determines the maximum number of finite TZs realizable by a given topology, is then introduced. Congruent transformations that extend the concept of rotations or similarity transformations while preserving the filter response are exploited to reduce the individual synthesis problems to the determination of a basis of vectors verifying certain orthogonality relations. A direct synthesis technique for dispersive building blocks, such as duplets, triplets, and quadruplets, is then given in the form of an orthogonalization procedure used for the computation of the desired basis. This approach is then combined with the aforementioned decomposition technique to produce a versatile algorithm able to synthesize hybrid circuits made of cascaded subblocks of different orders and types that implement each a subset of the overall TZs by means of coupling topologies containing a mixture of dispersive and nondispersive couplings. The first synthesis example is detailed where two dispersive duplets are combined with a classical quadruplet to realize a symmetric 6-4 response. A hardware implementation of the synthesized circuit is presented in combline technology. The second example proposes a slightly more involved coupling topology able to realize 10-8 asymmetric responses by means of four cascaded basic dispersive blocks.

Near-Field to Far-Field RCS Prediction on Arbitrary Scanning Surfaces Based on Spherical Wave Expansion.

Near-field to far-field transformation (NFFFT) is a frequently-used method in antenna and radar cross section (RCS) measurements for various applications. For weapon systems, most measurements are captured in the near-field area in an anechoic chamber, considering the security requirements for the design process and high spatial costs of far-field measurements. As the theoretical RCS value is the power ratio of the scattered wave to the incident wave in the far-field region, a scattered wave measured in the near-field region needs to be converted into field values in the far-field region. Therefore, this paper proposes a near-field to far-field transformation algorithm based on spherical wave expansion for application in near-field RCS measurement systems. If the distance and angular coordinates of each measurement point are known, the spherical wave functions in an orthogonal relationship can be calculated. If each weight is assumed to be unknown, a system of linear equations as numerous as the number of samples measured in the near electric field can be generated. In this system of linear equations, each weight value can be calculated using the iterative least squares QR-factorization method. Based on this theory, the validity of the proposed NFFFT is verified for several scatterer types, frequencies and measurement distances.

Rich p-type-doping phenomena in boron-substituted silicene systems.

The essential properties of monolayer silicene greatly enriched by boron substitutions are thoroughly explored through first-principles calculations. Delicate analyses are conducted on the highly non-uniform Moire superlattices, atom-dominated band structures, charge density distributions and atom- and orbital-decomposed van Hove singularities. The hybridized 2pz–3pz and [2s, 2px, 2py]–[3s, 3px, 3py] bondings, with orthogonal relations, are obtained from the developed theoretical framework. The red-shifted Fermi level and the modified Dirac cones/π bands/σ bands are clearly identified under various concentrations and configurations of boron-guest atoms. Our results demonstrate that the charge transfer leads to the non-uniform chemical environment that creates diverse electronic properties.

ON ORTHOGONALIZATION OF BOUBAKER POLYNOMIALS

In this work, we explore some unknown properties of the Boubaker polynomials. The orthogonalization of the Boubaker polynomials has not been discussed in the literature. Since most of the application areas of such polynomial sequences demand orthogonal polynomials, the orthogonality of the Boubaker polynomials will help extend its theareas of application. We investigate orthogonality of classical Boubaker polynomials using Sturm-Liouville form and then apply the Gram-Schmidt orthogonalization process to develop modified Boubaker polynomials which are also orthogonal. Some classical properties, like orthogonality and orthonormality relation and zeros, of the modified Boubaker polynomials, have been proved. The contributions from this study have an impact on the further application of modified Boubaker polynomials to not only the cases where classical polynomials could be used but also in cases where the classical ones could not be used due to orthogonality issue.

Stable visualization of connected components in dynamic graphs

One of the primary goals of many systems for the visual analysis of dynamically changing networks is to maintain the stability of the drawing throughout the sequence of graph changes. We investigate the scenario where the changes are determined by a stream of events, each being either an edge addition or an edge removal. The visualization must be updated immediately after each new event is received. Our main goal is to provide the user with an intuitive visualization that highlights the different connected components of the graph while preserving the user’s mental map after each event. The drawing stability is measured in terms of changes in the orthogonal relationships between vertices of two consecutive drawings. We describe two different visualization models, one for the 1-dimensional space and the other for the 2-dimensional space. In both models the connected components are drawn inside rectangular regions. To validate our approach, we report the results of an experimental analysis that compares the drawing stability of the online algorithm with that of an offline algorithm that knows in advance the whole sequence of events. We also present a case study of our online algorithm on a collaboration network.